<%BANNER%>

Cooperative Communication in Wireless Networks

Permanent Link: http://ufdc.ufl.edu/UFE0041615/00001

Material Information

Title: Cooperative Communication in Wireless Networks Flow-optimized Designs and Information-theoretic Characterizations
Physical Description: 1 online resource (126 p.)
Language: english
Creator: Chatterjee, Debdeep
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: achievable, bargaining, cognitive, communications, cooperative, discrete, flow, game, interference, memoryless, minimax, multiple, nbs, optimization, rates, relay, wireless
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: COOPERATIVE COMMUNICATION IN WIRELESS NETWORKS: FLOW-OPTIMIZED DESIGNS AND INFORMATION-THEORETIC CHARACTERIZATIONS The challenges involved in the design of efficient communication systems for the wireless medium can be attributed to the fact that the wireless medium possesses certain unique characteristics, the most important ones being the broadcast nature of the wireless medium, the susceptibility to interference effects, and the effects of path loss and fading on wireless link quality. Cooperation between different transceivers can potentially aid further development of next-generation wireless communication systems that demand high data rates and an excellent quality of service (QoS). This is possible by exploiting the broadcast nature of the wireless medium, and the diversity advantages that a multi-user system offers. We first consider a general single-source-single-destination wireless relay network and propose an information flow-optimized cooperative transmission design that achieves the optimal diversity-multiplexing tradeoff. Next, we apply game-theoretic techniques to the problems of resource allocation and characterization of cooperative behavior in a two-user fading multiple-access channel (MAC), with uncertainty about the channel state information at the transmitters (CSIT). In the third part of the dissertation, a more active form the above cooperative behavior is studied via a two-user fading cooperative multiple-access channel (CMAC), where each user, along with transmitting its own information to the destination, helps the other by forwarding the latter's information. We propose efficient cooperative transmission strategies based on a flow-theoretic approach, and evaluate their performances using numerical simulations. Finally, we consider communication through a two-user interference channel with unidirectional cooperation (ICUC), wherein one source uses its knowledge of the message of the other to reduce the interference to its own transmission, and simultaneously, help the other user-pair via cooperative relaying. We consider a very realistic scenario in which the cooperating source is subjected to a causality constraint. We derive a new achievable rate region for the discrete memoryless version of this form of ICUC, and demonstrate the contributions of the various coding strategies involved via numerical simulations for Gaussian channels. We also study the same channel with the cooperating source being subject to the half-duplex constraint as well. A discrete memoryless channel model incorporating the half-duplex constraint is presented, and a new achievable rate region, that enlarges the largest known rate region for the Gaussian version of this channel, is derived for this channel. The achievable rate region for the proposed coding scheme, specialized for Gaussian channels, is numerically evaluated and the strict inclusion of the previously known largest rate region is demonstrated.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Debdeep Chatterjee.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Wong, Tan F.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041615:00001

Permanent Link: http://ufdc.ufl.edu/UFE0041615/00001

Material Information

Title: Cooperative Communication in Wireless Networks Flow-optimized Designs and Information-theoretic Characterizations
Physical Description: 1 online resource (126 p.)
Language: english
Creator: Chatterjee, Debdeep
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: achievable, bargaining, cognitive, communications, cooperative, discrete, flow, game, interference, memoryless, minimax, multiple, nbs, optimization, rates, relay, wireless
Electrical and Computer Engineering -- Dissertations, Academic -- UF
Genre: Electrical and Computer Engineering thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: COOPERATIVE COMMUNICATION IN WIRELESS NETWORKS: FLOW-OPTIMIZED DESIGNS AND INFORMATION-THEORETIC CHARACTERIZATIONS The challenges involved in the design of efficient communication systems for the wireless medium can be attributed to the fact that the wireless medium possesses certain unique characteristics, the most important ones being the broadcast nature of the wireless medium, the susceptibility to interference effects, and the effects of path loss and fading on wireless link quality. Cooperation between different transceivers can potentially aid further development of next-generation wireless communication systems that demand high data rates and an excellent quality of service (QoS). This is possible by exploiting the broadcast nature of the wireless medium, and the diversity advantages that a multi-user system offers. We first consider a general single-source-single-destination wireless relay network and propose an information flow-optimized cooperative transmission design that achieves the optimal diversity-multiplexing tradeoff. Next, we apply game-theoretic techniques to the problems of resource allocation and characterization of cooperative behavior in a two-user fading multiple-access channel (MAC), with uncertainty about the channel state information at the transmitters (CSIT). In the third part of the dissertation, a more active form the above cooperative behavior is studied via a two-user fading cooperative multiple-access channel (CMAC), where each user, along with transmitting its own information to the destination, helps the other by forwarding the latter's information. We propose efficient cooperative transmission strategies based on a flow-theoretic approach, and evaluate their performances using numerical simulations. Finally, we consider communication through a two-user interference channel with unidirectional cooperation (ICUC), wherein one source uses its knowledge of the message of the other to reduce the interference to its own transmission, and simultaneously, help the other user-pair via cooperative relaying. We consider a very realistic scenario in which the cooperating source is subjected to a causality constraint. We derive a new achievable rate region for the discrete memoryless version of this form of ICUC, and demonstrate the contributions of the various coding strategies involved via numerical simulations for Gaussian channels. We also study the same channel with the cooperating source being subject to the half-duplex constraint as well. A discrete memoryless channel model incorporating the half-duplex constraint is presented, and a new achievable rate region, that enlarges the largest known rate region for the Gaussian version of this channel, is derived for this channel. The achievable rate region for the proposed coding scheme, specialized for Gaussian channels, is numerically evaluated and the strict inclusion of the previously known largest rate region is demonstrated.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Debdeep Chatterjee.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: Wong, Tan F.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041615:00001


This item has the following downloads:


Full Text





COOPERATIVE COMMUNICATION IN WIRELESS NETWORKS:
FLOW-OPTIMIZED DESIGNS AND INFORMATION-THEORETIC
CHARACTERIZATIONS


















By

DEBDEEP CHATTERJEE


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2010


































2010 Debdeep C'!I ili. ijee


































To the reader









ACKNOWLEDGMENTS

I would like to thank my advisor Prof. Tan Wong for his guidance, support, patience,

and the freedom I enjoi, .1 in choosing my research direction. I consider myself very

fortunate to have been able to pursue research under the guidance of someone who

encouraged me to try to define my own research problems, and at the same time, was

patient enough when a particular idea would fail to bear fruit as expected.

I would also like to thank Prof. John Shea for his general guidance and -~i-.-. -if..

regarding pursuing research, and more importantly, those on presenting one's research. I

thank Prof. Michael Fang and Prof. William Hager for their time and interest in my work.

I would like to take this opportunity to thank Dr. Ozgur Oyman of Intel Research for the

stimulating discussions that we had during my stay in Santa Clara, and for his valuable

comments and -.-.- -1 i. .'n regarding some of the later parts of this work.

My stay in the WING lab almost never had a dull moment, and credit for that is

due to my lab-mates, especially Surendra, Ryan, and Byong, who completely changed

the atmosphere of the lab ever since the summer of 2006 when I was practically the

only person present in the lab. There are way too many people I am indebted to for

all the help and support I have received in the past few years. Sridhar, Selvi, Manu,

Savya, Vaibhav, and Mallick are just a few people who have endured me over these years,

provided me with encouragement and hope (sometimes blatantly false, but they mostly

worked) when things have not worked out, and most importantly, been great friends.

Finally, I thank my parents, for no achievement, however big or small, may ever be

realized without their love and support.









TABLE OF CONTENTS
page

ACKNOW LEDGMENTS ................................. 4

LIST OF TABLES . . 7

LIST OF FIGURES .. .. .. .. ... .. .. .. .. ... .. .. .. .. ... .. 8

A B ST R A C T . . 10

CHAPTER

1 INTRODUCTION .................................. 12

1.1 Cooperative Transmission in Wireless Relay Networks ........... 13
1.2 Cooperative Behavior in a Fading Multiple-Access C!i Ii, I ......... 14
1.3 Active User Cooperation in a Fading Ci\ AC ................ 15
1.4 Achievable Rates in the ICUC with Causality Constraints ... 15
1.5 Organization of the Dissertation . 17

2 COOPERATIVE TRANSMISSION IN A WIRELESS RELAY NETWORK
BASED ON FLOW MANAGEMENT . 19

2.1 Introduction . . 19
2.2 General Design Using A Flow-theoretic Approach ... 22
2.3 Generalized-link Selection and Its Optimality . 26
2.3.1 The Three-node Relay Network . 26
2.3.2 Generalized-link Selection . 32
2.3.3 Diversity-multiplexing tradeoff . 34
2.4 Numerical Examples . 37
2.5 Sum m ary . . 40

3 RESOURCE ALLOCATION AND COOPERATIVE BEHAVIOR IN FADING
MULTIPLE-ACCESS CHANNELS UNDER UNCERTAINTY ... 48

3.1 Introduction . . 48
3.2 System Model . . 51
3.3 The Bargaining Problem for the Two-User Fading MAC ... 52
3.3.1 The Disagreement Point . ... 53
3.3.2 The Nash Bargaining Solution (NBS) . 54
3.4 The Modified Bargaining Problem . 58
3.5 Sum m ary . . 62

4 ACTIVE USER COOPERATION IN FADING MULTIPLE-ACCESS CHAN-
N E L S . . 64

4.1 Introduction . . 64









4.2 Flow-theoretic Transmission Protocols for the Cooperative Multiple-Access
C i i . .. 67
4.2.1 Orthogonal Relaying Protocol for the C'\! AC (OR-C \!AC) 68
4.2.2 Flow-optimized Protocol for the C\ 1. AC (FO-C \! AC) ... 70
4.3 Numerical Results . . 72
4.4 Sum m ary . . 74

5 INTERFERENCE CHANNELS WITH UNIDIRECTIONAL COOPERATION
AND CAUSALITY CONSTRAINTS . 79

5.1 Introduction . . 79
5.2 The CI, ,ii,, I M odel . 82
5.3 Achievable Rates for the ICUC-C . ... 83
5.4 The Gaussian ICUC-C ...... . 91
5.5 Discrete Memoryless CI, .i,,, I Model for the ICUC-HDC .. 94
5.6 An Achievable Rate Region for the ICUC-HDC ... 95
5.7 The Gaussian ICUC-HDC . 103
5.7.1 Inclusion Of Causal Achievable Region of [6] 105
5.7.2 Numerical Results . 107
5.8 Sum m ary . . 109

6 CONCLUSIONS AND FUTURE WORK . 116

6.1 Conclusions . . 116
6.2 Future Directions . . 118

REFERENCES . .. .. .. ... .. 120

BIOGRAPHICAL SKETCH . . 126









LIST OF TABLES


Table page

5-1 Description of Random Variables in Theorem 5.1 .... 84

5-2 Description of Random Variables in Theorem 5.2 .... 96









LIST OF FIGURES


Figure page

2-1 Basic graphs Gi and G2 for the three-node relay network with t1 + = 1. 42

2-2 FO protocol for the four-node relay network with tj + + t6 = 1. The flow
optimization is performed over all flows xi, ... x14, and all time slot lengths
t1, t6 . ... 4 2

2-3 Transmission strategy to obtain a lower bound on the outage probability for
the four-node relay network. Here t1 + + t4 = 1, and the optimization is
over xa, x14, and ti, t4, with the application of the max-flow-min-
cut theorem for the intermediate slots. . .... 43

2-4 Four-node relay network with uniform average power gains: Outage probabili-
ties for required rate R l= bit/s/Hz. . 44

2-5 Four-node relay network with uniform average power gains: Outage probabili-
ties for required rate R = 6bits/s/Hz. . 44

2-6 Five-node relay network with uniform average power gains: Outage probabili-
ties for required rate R l= bits/s/Hz. . 45

2-7 Five-node relay network with uniform average power gains: Outage probabili-
ties for required rate R = 6bits/s/Hz. . 45

2-8 Four-node relay network with non-uniform average power gains. Case A: E[Zs-,]
2.0, E[Zs72] 2.0, E[Zs] = 1.0, E[Z7IZ2] 1.0, E[ZD] 1.5, E[Z2D] -
1.0. Outage probabilities for required rate R = Ibit/s/Hz. 46

2-9 Four-node relay network with non-uniform average power gains. Case A: E[Zs-,]
2.0, E[Zsg7] 2.0, E[ZsD] = 1.0, E[ZR1Z] 1.0, E[ZRD] 1.5, E[Zgz] -
1.0. Outage probabilities for required rate R = 6bit/s/Hz. 46

2-10 Four-node relay network with non-uniform average power gains. Case B: E[Zs,] =
1.5, E[Zsg2] 0.75, E[Zs] = 1.0, E[ZR_1] 3.5, E[Zz] 0.2, E[Z2D] -
3.0. Outage probabilities for required rate R = Ibit/s/Hz. 47

2-11 Four-node relay network with non-uniform average power gains. Case B: E[Zs,] =
1.5, E[Zsg] 0.75, E[Zsv] = 1.0, E[ZR_1] 3.5, E[ZD] 0.2, E[Z2D] -
3.0. Outage probabilities for required rate R = 6bit/s/Hz. 47

3-1 Average rates with varying A . 63

4-1 Flow-theoretic transmission protocols for the C' \ AC: (a) OR-C'\!.AC, (b) FO-
C AC . .... 75

4-2 Achievable rate regions .,i-viiii 1i lic situation. . 76









4-3 Achievable rate regions .,i-ii,,, I i c situation with conventional MA slot (with-
out common information) . 76

4-4 Achievable rate regions symmetric situation. . 77

4-5 Outage performance .,-vmmetric situation. . 77

4-6 Outage performance symmetric situation. . 78

5-1 The discrete memoryless ICUC with causality constraint. 110

5-2 The Gaussian ICUC-C . . 110

5-3 Achievable Rates for the Gaussian ICUC-C: Weak interference for both cross-
links . . 111

5-4 Achievable Rates for the Gaussian ICUC-C: Strong interference from Sp to Dc
and weak interference from Sc to Dp. . 112

5-5 Achievable Rates for the Gaussian ICUC-C: Weak interference from Sp to DC
and strong interference from So to Dp. . 113

5-6 The discrete memoryless ICUC-HDC. . 114

5-7 Achievable Rates for the Gaussian ICUC-HDC: Weak interference for both cross-
links . . 114

5-8 Achievable Rates for the Gaussian ICUC-HDC: Strong interference from Sp to
Dc and weak interference from So to Dp. . 115









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

COOPERATIVE COMMUNICATION IN WIRELESS NETWORKS:
FLOW-OPTIMIZED DESIGNS AND INFORMATION-THEORETIC
CHARACTERIZATIONS

By

Debdeep C!I ,I1. ijee

August 2010

C!i ,r: Tan F. Wong
Major: Electrical and Computer Engineering

The challenges involved in the design of efficient communication systems for the

wireless medium can be attributed to the fact that the wireless medium possesses certain

unique characteristics, the most important ones being the broadcast nature of the wireless

medium, the susceptibility to interference effects, and the effects of path loss and fading on

wireless link quality. Cooperation between different transceivers can potentially aid further

development of next-generation wireless communication systems that demand high data

rates and an excellent quality of service (QoS). This is possible by exploiting the broadcast

nature of the wireless medium, and the diversity advantages that a multi-user system

offers.

We first consider a general single-source-single-destination wireless relay network

and propose an information flow-optimized cooperative transmission design that achieves

the optimal diversity-multiplexing tradeoff. Next, we apply game-theoretic techniques

to the problems of resource allocation and characterization of cooperative behavior in a

two-user fading multiple-access channel (\!AC), with uncertainty about the channel state

information at the transmitters (CSIT).

In the third part of the dissertation, a more active form the above cooperative

behavior is studied via a two-user fading cooperative multiple-access channel (C \! AC),

where each user, along with transmitting its own information to the destination, helps the









other by forwarding the latter's information. We propose efficient cooperative transmission

strategies based on a flow-theoretic approach, and evaluate their performances using

numerical simulations.

Finally, we consider communication through a two-user interference channel with

unidirectional cooperation (ICUC), wherein one source uses its knowledge of the message

of the other to reduce the interference to its own transmission, and simultaneously, help

the other user-pair via cooperative relaying. We consider a very realistic scenario in which

the cooperating source is subjected to a causality constraint. We derive a new achievable

rate region for the discrete memoryless version of this form of ICUC, and demonstrate

the contributions of the various coding strategies involved via numerical simulations

for Gaussian channels. We also study the same channel with the cooperating source

being subject to the half-duplex constraint as well. A discrete memoryless channel model

incorporating the half-duplex constraint is presented, and a new achievable rate region,

that enlarges the largest known rate region for the Gaussian version of this channel, is

derived for this channel. The achievable rate region for the proposed coding scheme,

specialized for Gaussian channels, is numerically evaluated and the strict inclusion of the

previously known largest rate region is demonstrated.









CHAPTER 1
INTRODUCTION

Over the last decade, wireless communication systems have gained popularity at a

very fast pace, and have currently become an integral part of our daily lives. With the

growing demands for higher data rates and better quality of service (QoS) guarantees

to support next-generation wireless systems, the demand for the design and develop-

ment of faster and more reliable multiuser communication systems, as compared to

the existing solutions, is more than ever before. It is well known that multiple-input-

multiple-output (M\! I \O) systems offer considerable performance improvements over

single-input-single-output (SISO) systems by efficient exploitation of the diversity benefits

of fading channels [1]. However, various practical considerations related to the cost, form

factor limitations and hardware implementation restrict the advantages of MIMO systems,

especially transmit diversity benefits, to base stations and access points of wireless net-

works. In this regard, cooperative communication offers a good alternative in providing

similar benefits by the sharing of antenna resources of multiple single-antenna nodes to

form virtual MIMO systems. Further, cooperative communication techniques can be em-

p1 ..l d for MIMO systems as well to bolster the overall system efficiency and reliability. In

general, cooperative communication comprises of special signal transmission and reception

schemes that effectively exploit the broadcast and diversity advantages of the wireless

medium, with due consideration to the detrimental effects of interference.

The aspect of cooperation in wireless systems can be broadly characterized into two

different manifestations of cooperative behavior:

Relay Cooperation: Dedicated relay nodes are available to help the source node(s)

transmit information to the destination node(s). For such a system, the source

and destination node(s) would constitute the class of users, while the relay nodes

would be part of the system's infrastructure. The three-node relay channel [2] is the

simplest example of such a system.









User Cooperation: In this case, the users, who themselves wish to transmit or receive

information, cooperate with each other by using some of their own resources to relay

other users' information. The cooperative multiple-access channel (C'\ AC) [3, 4],

the relay broadcast channel (RBC) [5], the interference channel with unidirectional

cooperation (ICUC), also known as the cognitive radio channel [6, 7], fall under this

class of cooperative communication systems.

Both these classes of cooperative communication techniques possess unique benefits

and limitations [9] in terms of different performance metrics, implementation cost and

complexity. Moreover, hybrid systems, that make use of both classes of cooperative

communication, can also be envisioned. For instance, cooperation within multiple cells of

an infrastructure network is expected to improve the efficiency and reliability of the overall

system [8]. The base stations (BSs) over multiple cells cooperate by jointly processing

the transmitted/received signals using high capacity backhaul connections, while the

mobile stations (\!SM) cooperate through relaying and "coiIl i- 's, between themselves.

Thus, depending on the application requirements and network configuration, different

cooperative communication systems demand different design approaches, and manifest

different forms of cooperative behavior amongst the participating nodes.

1.1 Cooperative Transmission in Wireless Relay Networks

A wireless relay network is a multiuser wireless communication network wherein

a wireless link exists between each pair of nodes. In general, it may be conceived as

a part of a larger network that itself may not be fully connected. It can also be seen

as a special form of a more general relay network. Although a considerable amount

of work on relay networks has been reported in the literature ( [10, 11, 12, 13, 14],

etc. ), the capacity of even the simplest of all relay networks the general three-node

relay channel, is still unknown. In the first part of the dissertation, we are concerned

with the problem of transmission of information from a single source node to the single

destination node in a wireless relay network, with the help of certain dedicated relay









nodes. Thus, the relay nodes, along with the source node, cooperate to form a virtual

multiple-input-single-output (\! ISO) system in order to achieve transmit diversity benefits.

In this work, we subject all the nodes in the relay network to the half-duplex constraint.

Previously proposed solutions to this problem include different path selection methods

and distributed space-time coding methods. Unfortunately, these methods fail to achieve

the optimal diversity-multiplexing tradeoff for the wireless relay network, and fail to be

efficient in the high data rate requirement regime. Under the assumption that the channel

state information (CSI) is available at all nodes, we develop a flow-optimized protocol, and

a suboptimal, but much simpler, generalized-link selection protocol, that are shown to be

optimal in terms of the diversity-multiplexing tradeoff, and that provide large coding gains

over direct transmission, even for high data rate requirements.

1.2 Cooperative Behavior in a Fading Multiple-Access Channel

In the previous section, we introduced a problem involving a single-source-single-

destination scenario. Next, we consider the simplest multi-source network: the multiple-

access channel (\!AC), wherein two users wish to transmit information to a single desti-

nation node. As the multiple transmitters share the same communication medium, unless

there exists a certain amount of cooperation or understanding between the users, they

could cause mutual interference to each other. Thus, even though it does not belong to the

class of cooperative communication systems that typically involve cooperation in the form

of active forwarding of information, there exists a certain level of cooperation between the

users as would be made explicit in C'!i pter 3.

Towards this, we consider the resource allocation problem for the two-user fading

MAC. We consider the case in which there exists some uncertainty in the channel state

information at the transmitters (CSIT). Under assumptions of perfect CSIT, the transmis-

sion strategies 1- -.-- -1. 1 by the solution to the bargaining problem yield optimal operation

points. On the other hand, owing to inaccuracies in the available CSIT, the conventional









bargaining problem solution may not be optimal, thereby resulting in considerable per-

formance degradation. To address this robustness issue, we propose a scheme in which

the conditions of the bargaining problem are relaxed to reduce the dependence of the

system performance on the solution to the bargaining problem. In the process, we develop

a game-theoretic framework to characterize the level of cooperation involved from an

individualistic viewpoint.

1.3 Active User Cooperation in a Fading CMAC

Next, we consider the problem of active user cooperation in a two-user fading C\! AC.

This alludes to a system in which each user, apart from transmitting its own information,

may cooperate with the other by actively forwarding the other's information to the des-

tination. Therefore, the C \! AC can also be considered as a basic example of cooperative

communication in a multi-source-single-destination system. Clearly, this involves a higher

level of cooperation as compared to the form of cooperation in a conventional MAC. We

use a flow-theoretic approach, and propose a flow-optimized solution and a much simpler

(but suboptimal) solution that decomposes the C'\! AC into orthogonal relay channels. The

performances of the proposed protocols are evaluated in terms of the achievable average

rate regions and outage probabilities, and the improvements over conventional MAC sys-

tems and a previously proposed method based on decode-and-forward (DF) relaying are

demonstrated through simulation results.

1.4 Achievable Rates in the ICUC with Causality Constraints

Finally, we study a manifestation of cooperation in a multi-source-multi-destination

system. The simplest multi-source-multi-destination system is the two-user interference

channel, wherein a pair of sources wish to transmit information to their respective

destinations resulting in interference caused to each other. The capacity region for the

interference channel, except for the special cases of strong interference, is still unknown,

and until recently, the original Han-Kc.1 i, -1i scheme [15] has been known to provide

the largest achievable rate region, without a clear idea of tightness of this region with









regard to the known outer bounds. Very recently, in [16], the authors have shown that

a Han-Kc., i, -,i-i type of coding scheme yields a rate region that is within 1 bit of the

tightest outer bound for the Gaussian interference channel.

Given this backdrop, we direct our focus to the interference channel with an .i-Cm-

metric cooperative relationship between the sources. Such a network is known as the

interference channel with unidirectional cooperation (ICUC)1 and is the simplest form

of general overlay cognitive networks [18]. The ICUC consists of a pair of sources that

demonstrate different behaviors. On the one hand, the primary source is solely interested

in transmitting information to its respective destination and does not actively cooperate

with the other user-pair. On the other hand, the secondary or cognitive source uses its

knowledge about the primary message to reduce the interference caused to its own link

by the primary transmission, and at the same time, aid the primary user-pair by relaying

the primary message to the primary destination, thereby reducing the effect of interference

caused by its own transmission to the primary link. Most of the work reported in the lit-

erature on this channel consider the scenario in which the cognitive source has non-causal

knowledge about the primary message. In this work, we impose a practical restriction

that the cognitive source may only obtain the message of the primary source in a causal

manner, i.e. the "cognitive" ability of the secondary source needs to be acquired. A new

achievable rate region for the discrete memoryless version of the ICUC with causality con-

straint (ICUC-C) is derived using block Markov superposition coding (SPC), conditional

rate-splitting, conditional Gel'fand-Pinkser (GP) inning, and cooperative relaying. This

rate region is evaluated for Gaussian channels, and numerical results are presented to

demonstrate the contributions of the various coding strategies used in the proposed coding

scheme towards enlarging the achievable rate region.



1 This network is also known in the literature as the cognitive radio channel [6] or inter-
ference channel with degraded message sets (IC-DMS) [17]









We then impose yet another very practical constraint on the cognitive source. It is

assumed that the cognitive source may only operate in a half-duplex fashion. A discrete

memoryless channel model for the ICUC with half-duplex and causality constraint (ICUC-

HDC) is presented, and a new achievable rate region is derived for this channel. The

random coding scheme used to obtain the rate region involves block Markov superposition

coding (SPC), conditional rate-splitting, cooperative relaying, conditional Gel'fand-Pinkser

(GP) inning, and a randomization of the listen-transmit schedule for the cognitive source.

We also prove that this rate region contains the largest achievable rate region for the

Gaussian ICUC-HDC previously reported in the literature, and further demonstrate this

using numerical results for the case of Gaussian channels.

1.5 Organization of the Dissertation

The rest of the dissertation is organized as follows. In C'!i pter 2, we present the

proposed protocols for efficient information transmission in a wireless relay network. We

first present the general design of the flow-optimized transmission protocol, and then,

the generalized-link selection protocol. Next, the optimality of the proposed protocol

in terms of the diversity-multiplexing tradeoff is established, and finally, we present the

performance evaluation of the proposed protocols for the finite signal-to-noise ratio (SNR)

regime through numerical results. In C'!i pter 3, we present the modified bargaining

problem formulation to model the resource allocation problem in a two-user fading MAC

under uncertainty. Solutions to these problems for the two choices of utility functions

are also presented, and numerical examples are shown to highlight various aspects of the

problem and the proposed solutions. We consider the problem of active user cooperation

in a two-user fading MAC in Ch! Ipter 4, and present the flow-optimized and orthogonal

relaying protocols for the C \! AC. Simulation results for different scenarios are then

presented for the performance evaluation of the proposed protocols. In C!h Ipter 5 we study

the problem of communicating through an ICUC with causality constraints, and present

new achievable rate regions for these networks first, for the scenario in which the









cognitive source operates in full-duplex manner, and then, for the situation wherein this

requirement is removed and instead a half-duplex constraint is imposed on the cognitive

source. Finally, in Ch'! pter 6, we conclude the dissertation, and discuss the possible

directions for future work.









CHAPTER 2
COOPERATIVE TRANSMISSION IN A WIRELESS RELAY NETWORK BASED ON
FLOW MANAGEMENT

2.1 Introduction

A wireless relay network is one in which a set of relay nodes assist a source node

transmit information to a destination node. Practically the wireless nodes can only sup-

port half-duplex communication [11], i.e., no nodes can receive and transmit information

simultaneously on the same frequency band. Different cooperative transmission schemes

for systems with half-duplex nodes have been proposed in the literature. Fundamentally,

these schemes consist of two basic steps. First, the source transmits to the destination,

and the relay listens and "< .i ni. [12] the transmission from the source at the same

time. Next, the relays send processed source information to the destination while the

source may still transmit to the destination directly. Variants of these techniques have

been proposed and have been shown to yield good performance under different circum-

stances [20, 19, 11].

Assuming channel state information (CSI) at the nodes, an opportunistic decode-and-

forward (DF) protocol for half-duplex relay channels is proposed in [21]. The maximum

delay-limited rate for this protocol is obtained by minimizing the average power over all

feasible resource allocation functions such that the required rate is achieved. In [22], the

authors present routing algorithms to optimize the rate from a source to a destination,

based on the DF technique that uses regular block Markov encoding and windowed

decoding [23, 24], for the Gaussian full-duplex multiple-relay channel. The achievable

rate of [23] for the Gaussian physically degraded full-duplex multi-relay channel has been

established as the capacity of this channel in [25]. In [26], it is shown that the cut-set

bound on the capacity of the Gaussian single source-multiple re! -v-,i-:.l' destination

mesh network can be achieved using the compress-and-forward (CF) method, as the relay

powers go to infinity.









Cooperative diversity methods based on network path selection have been proposed

in [27, 28]. These selection methods include: (i) the max-min selection method [27],

wherein the relay node with the maximum of the minimum of the source-relay and

relay-destination channel gains is selected; (ii) the harmonic mean selection method [27],

wherein the relay node with the highest harmonic mean of the source-relay and relay-

destination channel gains is selected; and (iii) the selection scheme of [28], in which the

relay that can correctly decode the information from the source and has the best relay-

destination channel is selected. These methods achieve a DMT of d(r) = (N 1)(1 2r)

for an N node relay network and multiplexing gain 0 < r < 0.5. This is close to what the

distributed space-time coding protocol [29] achieves, when N is large. Unfortunately, these

network path selection protocols perform poorly in high-rate scenarios (r > 0.5).

We have proposed a cooperative diversity design based on a flow optimization

approach for a three-node network in [30]. In this design, the source node broadcasts two

distinct flows to the destination and the relay node respectively during the relay's listen

period. Then the relay forwards this information using the DF approach while the source

may also send another flow of information to the destination during the relay's transmit

period. This scheme is shown to achieve the optimal diversity order for the three-node

relay channel and yield performance very close to optimal full-duplex relaying in both low-

and high-rate situations.

Here, we apply this cooperative transmission design to a general relay network,

wherein wireless links are present between each pair of nodes in the network. It is assumed

that the channels suffer from slow fading and hence, facilitate efficient estimation and dis-

semination of channel state information. The problem considered here is the transmission

of information from a source node to a corresponding destination node while the other

nodes act as re1hi-, to help in the transmission. As in [30], assuming CSI is available at all

nodes we use broadcasting (BC), multiple access (!\ A) and time sharing (TS) techniques









to formulate a flow theoretic convex optimization problem based on the channel condi-

tions. Instead of considering a total power constraint for all the transmitting nodes as

in [30], we subject each node to a maximum transmit power constraint. This yields a more

reasonable system model for a general wireless relay network, especially when the number

of nodes in the relay network is large. The resulting relaying protocol will be referred to

as the flow-optimized (FO) protocol. To obtain a more practical cooperative design we

develop the generalized-link selection (GLS) protocol, in which we select the best relay

node out of the available ones to form an equivalent three-node relay network to transmit

the information from the source to the destination. The benefit of this, over other network

path selection strategies, becomes evident when the rate requirement is high. It is shown

that the simple GLS protocol is optimal in terms of the DMT [31] and yields acceptable

performance even when the rate requirement is high.

Recently, in [32], the authors have shown that compress-and-forward (CF) relaying

achieves the optimal DMT for the three-node, half-duplex network, and that DF relaying

can achieve the optimal DMT of the four-node full-duplex network. In this work, we

show that the optimal DMT can be achieved for a general N-node (N > 3) half-duplex

network using the FO or GLS protocols. Here, it should be clarified that we consider

that the wireless links between each node-pair experience independent Rayleigh f ,ii:.

and this corresponds to the definition of non-' /.,;/ networked networks in [32]. The

performances of the FO and GLS protocols are evaluated numerically in terms of their

outage probabilities for four- and five-node relay networks for uniform and non-uniform

average power gains. The numerical results motivate the use of the GLS protocol for

situations where computation complexity is an issue and show a remarkable improvement

over the max-min selection method of [27]. The proposed designs, based on BC and

MA alone, are sub-optimal in general. For a fair appraisal of the proposed protocols, we

compare the proposed protocols to an upper bound on the maximum rate, derived using

the max-flow-min-cut theorem [33, Thm. 14.10.1].









The rest of the chapter is organized as follows. In the next section, we present

the design of the proposed flow-optimized protocol for a general N-node wireless relay

network. In Section 2.3, we use the flow-optimized solution to the three-node relay

network and use it to develop the GLS protocol and establish its optimality in terms of

the diversity-multiplexing tradeoff. Numerical examples comparing the performances of

the FO and GLS protocols to that corresponding to the work in [27] are presented in

Section 2.4. Lastly, the main contributions in this chapter are summarized in Section 2.5.

2.2 General Design Using A Flow-theoretic Approach

We propose a general design for the transmission of messages from a source to a

particular destination through a relay network using the idea of network flows with

the optimal application of BC, MA and TS techniques. This cooperative transmission

scheme is developed for a relay network of N nodes, with a wireless link between each

pair of nodes. We consider an N-node wireless relay network with a link joining each

pair of nodes. Each such wireless link is described by a bandpass Gaussian channel with

bandwidth W and one-sided noise spectral density No. We denote the power gain of the

link from node i to node j as Zi. The link power gains are assumed to be independent

and identically distributed (i.i.d.) exponential random variables with unit mean. This

corresponds to the case of independent Rayleigh fading channels with unit average power

gains. Moreover, we assume that each node has a maximum power limit of P and can

only support half-duplex transmission. Note that this model can be easily generalized to

the case where channels may have non-uniform average power gains (for which numerical

examples are presented in Section 2.4), and where different nodes may have different

maximum power constraints. More specifically, the latter case can be converted into

the uniform maximum power constraints case by absorbing the non-uniformity in the

transmit powers into the average power gains of the corresponding links. In the sequel, we

characterize the system in terms of the transmit signal-to-noise ratio (SNR), S = P

at the input of the links. Time is divided into unit intervals, and BC and MA are applied









with a TS strategy that is optimized to maximize the spectral efficiency (which we call

"rate" hereafter for convenience). To avoid interference between concurrent transmissions,

a time interval is divided into slots:

* During the first slot, the source may BC to all the other nodes in the network,
subject to its power constraint P.

During the subsequent slots, a relay may BC to all other nodes (except the source
node), or it may receive flows from all other nodes (except from the destination)
through MA.

During the very last slot, the source and the r-el ,- may send information flows to
the destination using MA, with the flows in the MA capacity region corresponding to
a maximum transmit power of P for each node.

Note that the forwarding of information by the rel ,i- is based on the DF approach. For

practicality consideration, it is assumed that the phases of the simultaneously transmitted

signals from different nodes are not synchronized. In general, for the above transmission

protocol, there would be a maximum of 2(N 2) + 2 = 2N 2 time slots of lengths

t1, t2, t2N-2 respectively.
Next, we describe the optimization problem using a graph-theoretic formulation.

Define a graph G (V, E), where V is the set of nodes, E is the set of all links joining

the nodes in the graph, and associate the vector r to represent the flow rates associated

with each link in E. Thus, the number of elements in r equals the cardinality of E.

For convenience, we write G = (V, E, r). Now denote the source by S, the destination

by D, and the relay nodes by R7,... ,7RN-2. The slotting of a unit time interval, as

described above, yields simpler graphs for each time slot, that we call basic '.i-'l, A

basic graph is either one in which a particular node may BC to several nodes, or in

which several nodes transmit via MA to a particular node. Thus for a basic graph,

we need to include only the links between the nodes that may participate during the

concerned time slot. For example, assume that the relay Ri broadcasts to all nodes

other than the source, during the i-th time slot. The basic graph is given by Gi =

(V, E, r) where V = {S, R1, RN-2, D}, E { = {RiR2, I 2,7 rN -








U T
S2 ... RIN -2 X )- where XAB is the flow from node A to node B during the

i-th time slot.

In general, the proposed design involves TS between the basic graphs to yield the

following equivalent graph G corresponding to a unit interval (see [34] for a similar idea):

G = V, [JE, tir) =tG, + t2G2 + ... 2N-2G2N-2. (2-1)

where the number of elements in each vector ri is extended to | U Ei by inserting zeros

appropriately. The second equality in (2-1) implies that G can be viewed as a linear
combination of the basic graphs Gis, with the equivalent set of edges given by the union

of the sets Ei, and the equivalent flow rate vector given by the linear combination of the

individual flow rate vectors ri. Further, this results in G being fully connected.
To maximize the data rate from the source to the destination through the relay

network, we need to consider each cut that partitions V into sets V8 and Vd with S E VS
and D) E Vd resulting cut sets are such that one set contains the source node (S) and the

other, the destination node (D). Clearly, there can be 2N-2 such possible cuts for the N-

node relay network. Let these cuts and the corresponding cut sets be denoted by Ck, Vk,
and Vkd, respectively, for k = 1, 2,... 2N-2. Further, for the graph G, for any two nodes

A c Vk and B c Vkd, there exists a cut edge AB that crosses the cut. Denote the total
flow through cut edge AB in a unit time interval by XAB Z= 2Ni- AB. Now recall from
network flow theory [35] that the maximum flow rate from the source to the destination is

specified by the minimal cut of the equivalent graph (2-1). Consequently, we arrive at the
following convex flow optimization problem that can be solved using standard optimization

techniques:


max min 5 XAB, S XAB, s XAB (2-2)
AEVf,BEVd AEV2,BEV2d AeV2_ ,BEVd /

over all flow allocations xB and all time slot lengths ti, subject to:









* the no'.-.' i ll.:.: ,; constraints: xAB, ti > 0 for all cut edges AB and i
1,2,... ,2N- 2,

the total-time constraint: t1 + + t2N-2 1,

the power (,-'"". /,'/) constraints:

for a BC slot the flow rates should lie in the capacity region of the BC channel
with the transmitting node having a power constraint of P,
for an MA slot the flow rates should lie in the capacity region of the MA
channel with a maximum power constraint P for each transmitting node,

the flow constraints: considering steady state operation, the total information flow
out of a relay should equal the flow into the relay in each unit time interval.

Note that the dependence of the objective function on the channel gains and the time

slot lengths is implicitly expressed through the capacity constraints. Denote the cut

separating S from all the other nodes and the cut separating ED from all nodes as Cs and

C-, respectively. Then we observe that the cost function in (2-2) above can be further

simplified to max mmin {x(Cs), x(C-)}, where

N-2 N-2
x(Cs) = XSD + X and x(CD) = xs, + xzi (2-3)
j= 1 i 1

are the total flows across the above-mentioned cuts Cs and CD, respectively. To see this,

consider the cut C with V8 = {S, 7RI, R }, and V = {- +l, RN-2, )} for some

I c {1, 2,... N 2}. The total flow across this cut is given by

N-2 I N-2
x(C) XS-D + x + XK-D + Yx -- (2-4)
j =1+1 i= 1 j=1+1

Now, consider node i for i c {1, 2,... 1}. According to the flow constraint for node i,

N-2 I N-2 I
X-R7iZ + Y X, ,+ > Xg^Za = XSm+ Xi,7 + > XiRzjz (25)
j=1+1 k 1,k i j 1+1 k 1,k i

Summing (2-5) over all i c {1, 2, .. l} we get

I 1 N-2 ) I N-2 ( -
X-i= D + f1 Xi= xSK +fxii7. (26)
J 1+1 j1+1









Since E3 '- 1 x > 0, combining (2-3), (2-4) and (2-6) gives x(C) > x(Cs).

Similarly, we have x(C) > x(CD). Thus the cost function in (2-2) reduces to the above-

mentioned form.

2.3 Generalized-link Selection and Its Optimality

In this section, we present the GLS protocol and establish the optimality of the FO

and GLS protocols in terms of the DMT. This is accomplished in three steps. First, we

apply the FO protocol to the three-node relay network. Next, we propose the GLS pro-

tocol based on a selection strategy that is sub-optimal to the FO protocol of Section 2.2.

Finally, the optimality of the GLS protocol, and thereby, that of the FO protocol, is

established.

2.3.1 The Three-node Relay Network

The three-node relay network consists of a source (S), a relay (R), and a destination

(D). We specialize the general design described in the previous section to this three-node

relay network. A unit time interval is divided into two time slots of lengths t1 and t2 with

t1 + t2 = 1, and information is divided into 3 flows of data xl, x2, and x3. During the first
time slot, S sends (via BC) two flows of rates xl%/tl = xl/t and x /ti = x2/t1 to

and R, respectively, resulting in the basic graph Gi as in Fig. 2-1. During the second time

slot, R and S send (via MA) two flows of rates x2D/t2 X4/t2 and x-)D/t2 X3/t2 to D,

respectively, resulting in the basic graph G2 as in Fig. 2-1.

Combining the two basic graphs yields the equivalent graph as G = tiG1 + t2G2. Note

that the information flow of rate x4/t2 sent by R during the MA time slot is from the flow

of rate x2/tl it received during the BC time slot. This gives rise to the flow constraint for

R, i.e, x4 x2. Thus, we have the flow constraint x4 x2. The rate for this network is

specified by the min-cut which is clearly min{(xi + x2 + x3), (Xi + x4 + x3)}. Hence, the

flow optimization problem is given by:

max min {(xi + X32 + X3), (X1 + X4 + X3)} (2-7)









over flow allocations xi, x2, x3, x4, and time slot lengths ti, t2, subject to:

* no, -t,' ii/,;, ,i;/ constraints: x1, X2, X3, x4 > 0, t1, t2 > 0,

* total-time constraint: t1 + t2 1,

* power constraints: SBC < S, x1 < tlC(ZSDS), X2 < tlC(Zs-RS) for the BC slot,
x3 < t2C(Zs-DS), X4 <_ t2C(ZzS), x3 + X4 <_ t2C(ZSDS + ZTDS) for the MA slot,

flow constraint: x2 4,

where C(x) = log(1 + x), and SBC, the minimum SNR required for the source to broadcast

at rates xl/ti and x2/tl to the destination and the relay, respectively, in the first time slot

with 0 < t1 < 1, is given by (see [30, Lemma 3.1] for proof)


S Z xl(elt 1) + Z e 1 1,ti 1) for ZsK > ZsD,
SBC \
{ ZS 2/2 1) + ZSD t (eli -L 1) for Zsg < ZsD.

For t1 = 0, SBC 0. Note that for the BC slot, the last two constraints are redundant

when t1 > 0, and complement the first constraint when t1 = 0.

As mentioned in Section 2.1, the above optimization problem formulation is different

from that in [30] wherein the sum of the source and relay powers, required to achieve

a certain data rate, is minimized. More specifically, when considering individual power

constraints for each node, we cannot use part 2 of [30, Lemma 3.1] to describe the power

constraints for the MA slot. This is because doing so would restrict the flows x2 and x3

such that the sum of powers expended at S and 7R is minimized. On the other hand, in

the present problem, the power constraints only dictate that the flow-rates should lie in

the MA capacity region specified by the maximum power available at each transmitting

node, for the particular fading state. With this modification in the constraint for the MA

slot, the solution approach to the above problem needs to be markedly different from that

in [30].

To solve the optimization problem in (2-7) we consider two cases with regard to the

link gains: (a) Zsv > ZS-R, and (b) Zsv < ZS-R. For both cases, we solve the optimization









problem in two stages: first, we fix ti, t2 > 0 such that tl + t2 = 1 and find the optimal
flows x1, X2, X3 in terms of t1, t2, and then, find the optimal values for t1, t2 to maximize
the objective function.

ZSD > ZS-R. For this case, the source-destination link is at least as good as the
source-relay link. To obtain an analytical solution to the optimization problem and better
insight into the nature of the solution to the flow optimization problem, we modify the
representation of the BC slot power constraint from that in (2-7) to the one that is
more conventionally used to describe the capacity region of the Gaussian BC channel, as
presented in (2-8). Using the flow constraint in (2-7), we first solve (2-8) for fixed ti, t2.

max(xi + X2 + X3) over x1, X2, X3, a, subject to (2-8)


x1, X2, X3 > 0, 0 < a < 1,

x ( + Zs -aS )
X2 < t2C(ZZDS), X3 < min t2C(ZsS), t2C(ZSDS + ZDS) x2} -

Here, a C [0, 1] is the fraction of total power spent at the source to transmit x, directly
to the destination during the BC slot, and a = 1 a. Although, this modification of the
BC slot power constraint apparently makes the optimization problem non-convex owing to
the non-convexity in a, as we shall see in the sequel, this issue can be handled easily by
utilizing the monotonicity of the logarithm function.
Denote the optimal solution by (x*, x*, x*, a*) and the corresponding maximum rate
by X(tl,t2). It is clear that x = tlC(ZsDa*S). Suppose that x* < tiC (l+Zs~i s Since

t1C ZsRas ) is a decreasing function of a, we can increase a from a* to ao such that
x tiC(ZsDvaS) > x and x tiC ( zs' ). Thus the objective function becomes
(xo + x< + x*) > (x* + x' + x*) at a This contradicts the optimality of (x, x, x ,*a*).

As a consequence, we have x* = t1C lZsa*s) <- t2C(ZKZS). This implies that a* >









max z -- 1 o and ao < 1. In essence, this means that the opti-
ZSa S (l + Z1 S)t2/t 1a .

mal x1 and X2 should lie on the boundary of the degraded BC capacity region. With this,
it is obvious then that x3 = min t2C(ZsDS), t2C(ZsDS + ZvRDS) tC ( zs) }s.
Therefore the optimization problem of (2-8) can be re-written as:

max (xi + x2 + 3) (2-9)

subject to max{0, a} < a < 1, xi = tiC(ZsvaS), x2 tC (y- as
1 + ZsnRaS)

x3 = min t2C(ZSDS), t2C(ZsVS + ZRS) tic ZSRas .
1 + Zs-RaaS)
We observe that X3 t2C(ZSDS) above if and only if

1 1 + ZasS A
SS 1+ S t2/t
SZ1 + +ZsvS
Comparing this to the expression for ao gives ao < a, < 1.
Next we consider two possible sub-cases:

i I+ n ZR-DS < 1 Z (7S):
In this case, we have a > 0 and t2 < C ZRS) A 2max. The maximum
a C V)+C(ZsRS)
rate can be expressed as X(t,t2) = max{Xi(ti,t2), X2(t1,t2)} where

X(t,, t2) = max tC(ZsVaS) + t2C(ZsDS + ZRS)
max{, a = tIC(ZsDvaS) + t2C(ZsDS + ZvRDS)
(1 + ZsVajS)(1 + ZSRS)
< t1 log + t2C(Zs-S)
1 + ZSRalSa
< C(ZsS) (2- 11)

X2(t1,t2) = max t,C(ZsvaS) + tiC ( z s t2Z S)
aal<< \1 + Zs-RaS
C(ZSDS) (2-12)

where the first inequality in (2-11) holds since (2-10) is not satisfied, and the second
inequality in (2-11) holds since a1 < 1 and that the first term in the previous step
is monotonically increasing in aa when ZSD > ZSR. This way, the last observation








helps avoid the non-convexity issue mentioned before. The value of X2(t1, 2)
in (2-12) is obtained using similar arguments.
ii. c(zs) > tIC(ZsHS):

In this case, we have aa < 0 and 2 > t2max. Thus the maximum rate is given by

X(tl,t2) max tlC(Zs-DaS)+tC 1 zsa S S}+t2C(ZSDS) = C(ZSDS). (2-13)
o
Hence, (2-11)-(2-13) imply that the solution to (2-8), for any ti, t2 pair, occurs at
a = 1, and the solution to the original problem of (2-7), when ZSD > ZS-, is given by
max{0_t,,t2 : tl+t2 1} X(tl, t2) = C(ZSDS) with any t1, t2 pair such that t1, t2 > 0 and
t + t2 1.
ZSD < ZS-R. In this case, the source-relay link is better than the source-destination
link. Again, we first fix time slot lengths t, and t2 and solve for the optimal values of
xa, X2, a3, and a, and then maximize the objective function of (2-7) over all feasible time
slot lengths. Following similar arguments as in Case a), the optimization problem of (2-7)
can be re-written as:

max (xa + x2 + X3) (2-14)

subject to 0 < a < min{1, a}, x = t1C ZSv-aS X2 tiC(ZsgZaS),
b1 + ZSvaS )
x3 min {t2C(ZsDS), t2C(ZsDS + ZDS) tiC(ZsgaS)}

where a 1 S [(1 + ZKRS)ts/tL 1] is an upper bound on a such that tiC(Zsg-aS) <
t2C(ZRDS). Note that a > 0. The optimization problem in (2-14) is non-convex in a,
but this technicality can be overcome by using the approach as in the previous case. As
before, X3 t2C(ZsvS) if and only if a < 1 + ai'.S ijtl A Also,
0 < a' < a. Again, we consider two possible sub-cases:

i. t2 -S







In this case we have a' < 1 and t2 < C(ZSRS) A t2max. The maximum
S-C ( V) +C(Zs S) )
rate X(ti,t2) can be expressed as X(ti, t2) max{Xi(ti,t2), X2(t1,t2)} where

XI(tl,t2) = max t1c 1ZSaS) + t1C(ZS aS) + t2C(ZsDS)
o
1- (1 ZsDab'S + t1C(ZsRalS) + t2C(ZSDS) (2-15)
1 + ZAS-DaS
X2(ti,t2) max tiC ZDaS + 2C(Z-DS + ZRDS)
a
tiC 1 ZsDabS + t2C(ZSDS + ZRDS), (2-16)
1 + ZsvaS

and both the maxima in (2-15) and (2-16) are attained at a a'. Substituting the
expression for a' in (2-15) and (2-16), we obtain XI(t1, t2) X21(t,t2) and


X(t ,t2) tilog 10 +ZS-DS ] + t2C(ZS-S + ZRvS). (2-17)
1 + z"S ( + z- s t 1
SIctC Zs SZS


In this case we have a' > 1 and t2 > t2max (with t2max as in sub-case i). Thus the
maximum rate is given by

X(t,,t2) = max ti Z S 1C(ZsaS) + t2C(ZSDS)
o = tlC(ZSR) + t2C(ZsDS) (2-18)

where the maximum occurs at a = 1, as the function to be maximized is monoton-
ically increasing in a when ZSR > ZSD. Again, the apparent non-convexity of the
optimization problem (2-14) in a is avoided by considering the sum of Xi and x2
together, and utilizing the last observation regarding the monotonicity property of
the objective function in (2-18).
Finally, we optimize the above solution to (2-14) over all possible time slot lengths to
obtain the solution to the original problem in (2-7) when ZSD < ZSR. Corresponding to
Case i. above, when t2 t2max, we note that

max X(t,,t2) > X(1 t2,t2) t,=t2max
{O c(Zs-S)c(ZsDS + ZRDS)
C(ZsRS) + C(ZRDS + ZSDS) C(ZsDS)









On the other hand, corresponding to Case ii., when t2 > t2max, from (2-18), we have

max X(t,t2) < -(1 t2 a)C(ZsS) +t2maxC(Z-DS)
{0 C(ZsKS)C(ZSDS + ZKDS) (220)
C(ZszS) + C(ZKDS + ZSDS) C(ZSDS)

where the inequality in (2-20) is obtained from (2-18) by using Zs- > ZsD and t2 > t2max.
Hence, from (2-19) and (2-20), we conclude that when ZS-R > ZsD, the maximum

achievable rate is given by X(S) maxo I+ZS[{ I ) l/
t2C(ZSDS + ZvDS).
Therefore, the maximum achievable rate of information transmission from the source
S to the destination D for different cases can be summarized as under:
The maximum information rate from the source S to the destination D for different
cases is summarized below:
a) ZSD > ZSR: The maximum rate is X(S) = C(ZSDS) with any ti, t2 pair such that

t1, t2 > 0 and tj + t2 = 1. This corresponds to directly transmitting all data through the
link from the source to the destination, without utilizing the relay.

b) ZsD < ZS-R: The maximum rate is X(S) maxo 1+ zsR| (I )t 11
t2C(ZsDS + ZKvS) with t2max C(ZsKS)/ [C(ZSKS) + C(ZKDS + ZsDS) -C(ZSDS)
and t\ = 1 t2.
Thus for a given power limit (i.e. a given S) at the nodes, relaying is advantageous
only when ZSD < ZSR. Further, the optimal solution aliv- allocates a non-zero flow to
the direct link. Also, the relay-destination link gain ZKD does not influence the strategy of
transmission (i.e. whether to use only the direct link or both the relay and direct links),
but only the amount of information through the relay link.
2.3.2 Generalized-link Selection

For the general N-node relay network, the flow optimization solution can be compu-
tationally demanding even for moderate values of N. Unfortunately, the existing simple









routing schemes based on different network path selection methods fail to provide ac-

ceptable outage performance in high-rate situations. The GLS protocol described below

provides a simple sub-optimal design to address this complexity issue. In essence, the GLS

protocol identifies the best relay path out of the possible N 2 relay paths and considers

only the chosen relay along with the source and destination to form a three-node relay net-

work, which we call a generalized-link from the source to the destination, for information

transmission. In other words, the aim is to choose the best relay such that the equivalent

three-node relay network obtained (containing the source, destination and the chosen

relay) gives the maximum rate over all possible equivalent three-node networks containing

the source and destination. More precisely, we need to consider the following possibilities:

* ZsD > Zs-Ri for all i c I {1, 2, .. N 2}: From the results of the optimization
problem (2-7), it is clear that the maximum rate would be C(ZSDS) with direct
transmission of all data from the source to the destination without using any relay.

There exists a k C I such that Zs-, > ZSD: Let the set of all such node indices
be K and for all i E I \ K, ZsD > ZS-,. For this case, choose the node 7R' as the
relay such that k' arg rn ::K =Xk (S), where Xk(S) is the maximum rate for the
three-node relay network with the source S, the relay )Rk and destination D.

In terms of the worst-case computational complexities for the FO and GLS protocols,

it can be seen that, for an N-node relay network with N > 3, the FO protocol involves

a max-min optimization over 2(N2 2N + 2) variables (all possible flows and time slot

lengths), subject to N-1 non-linear and 2(N2-N+1) linear constraints, whereas the GLS

protocol involves a maximum of N 2 maximizations of a non-linear concave function over

two variables, subject to two linear constraints, followed by finding the maximum of N 2

real numbers with a worst-case complexity of O(N 2). Moreover, for N > 3, for the FO

protocol, the BC slots potentially involve (N 1)- and (N 2)-level superposition coding

(SPC) or dirty paper coding (DPC) implementations for S and the rel-,,-, respectively,

while the MA slots at the rel'-iv, and D may involve a maximum of (N 3) and (N 2)

interference cancelation (IC) operations respectively. On the other hand, the GLS protocol









involves a maximum of 2-level SPC/DPC and one IC operation for the BC and MA slots

respectively, for any N > 3.

2.3.3 Diversity-multiplexing tradeoff

In this section, we show that both the FO and GLS protocols achieve the optimal

DMT. As in [31], the multiplexing gain is defined as r = lims,_ R(S) where S is the SNR
log S
and R(S) is the rate at an SNR level of S. Following [31], we parameterize the system, in

terms of the SNR S and the multiplexing gain, 0 < r < 1, with the rate increasing with

the SNR as R r log(S). With the parameterization (r, S), the diversity order achieved

by the transmission scheme is given by

1(r) lira log P, (r, S) (2 t)
S-oo log S

where P (r, S) is the average probability of error when the SNR is S and multiplexing

gain r. With the above definitions, we evaluate the performance of the proposed protocols

in terms of their diversity-multiplexing tradeoffs. The following theorem establishes the

optimality of the the GLS protocol (and hence the FO protocol) in terms of the diversity-

multiplexing tradeoff:

Theorem 2.1. The GLS protocol, and hence the FO protocol, achieve the optimal

diver.-,,iinllJ'..1, i.:,i tradeoff d(r) = (N 1)(1 r) for all 0 < r < 1, for the N-node

half-duplex wireless i/.,;/ network.

Proof. Here, we sketch the proof of the theorem. Part 3) of Theorem 4.2 of [30] can be

generalized for the N-node relay network to prove that, as the block length goes to infinity

(during any particular time interval), the average error probability for the FO protocol

is upper bounded by its corresponding outage probability. Here, the outage probability

denotes the probability that the data rate R cannot be supported by the system when

the SNR is S, i.e., Pou0t(r, S) = Pr[X(S) < rlog S] where X(S) denotes the maximum

rate possible for the given channel gain realizations when the SNR is S. Thus, from the









definition of diversity order (2-21), we have


log Pout (r, 5')
d(r) > lim -ilog P(2-22)
s-o log S

Moreover, the above result from [30] can be directly used to prove the same for the GLS

protocol. Using this fact, we derive a lower bound to the diversity-multiplexing tradeoff

that can be achieved by the GLS protocol. The sets I and K, used in the sequel, are the

sets of indices as described in Section 2.3.2. The outage probability for the GLS protocol

is given by

Pt (r, S) Pr (maxXk(S) < r logS (2-23)

where Xk(S) is the maximum rate achievable by the three-node relay network formed by

the source S, the relay Rk and the destination D. We have the following possibilities:

* Case A: IKI = 0, i.e. the cardinality of the set K is zero. This corresponds to the
case when ZSD > Zs-R for all k E I.

Case B: KI =i with i E I.

Note that for Case B there are (N -2) possibilities for the set K with cardinality i. Since

the link gains are assumed to be i.i.d., and the outage probability depends on the distribu-

tion of the maximum of Xk(S) over all k e I (or effectively, over all k e K when IK > 0),

only the cardinality of K is significant. Let the (N -2) possible constructions of the set

K be represented by a ;, i:, i c" set Ko with cardinality i. Without loss of generality, we

describe Ko as the set K corresponding to the case when the indices of the relay nodes are

ordered according to their source-relay link gains, i.e. ZsgRz > ZS.R2 > > ZS-Rg 2. Thus,

Case B now implies a solitary choice for set K, viz. Ko {1, 2,... i}. Therefore, from

(2-23), we have


gt(r,S) = Pr(C(Zs-DS) < rlog S IKo 0) Pr(Ko =0)
N-2
+ Pr maxXk(S) < r logS IKKo = i Pr(|Koo i). (2-24)
i 1i









We observe that using the right-most expression of (2-19) instead of Xk(S), for each
k c Ko, in (2-24) gives an upper bound on P, rt(r, S). This is utilized in obtain-

ing a lower bound on the diversity order of the GLS protocol. Let {S }- 1 be an

increasing unbounded sequence of SNRs with S1 > 1. Define the sequence of ran-
dom variables {3[} _, {Bk 1 and {Ak 1 with = c(zs -c(zss)
n-i n-i nn- C(ZS-DSn+ZR,-DS.)
Bk = c(zsRkS ), and A 1 = =S 1 respectively. Note that for all k e Ko,
logSI k log XII
3I' -+ 0 a.s. This implies that (B, Ak) -+ 0 a.s. Define A' = maxkeKo A and

B' = m--::, Ko B ( B1). Then using the above, it can be seen that (B, A') 0

a.s. Further, lim Pr (B' < rl IKol i) exists, and therefore the above implies that

lim Pr (A' < r| I Ko i) lim Pr (B, < r IKIKo i). Using this in (2-22) and (2-24),
n-*oo n-*oo
the diversity order for the GLS protocol, dgr(r), satisfies

dgr) > lira log Pgrt (r, S)
-s-oo log S

> lim log Pr ZsD < S- =I 0 Pr(|Ko| 0)
no log S L Sn.
N-2 S s
+ Pr ZS, < Kol i Pr(|Ko| i) (2-25)

-S1 S -1
= lim log Pr max{Zs ZS ,*** ,Z ZS R2 }< Ko- Pr(Ko 0)
n-oo log S, SS
N-2 (2 2)
+ Pr max{IZ s, Z ZSRN-} < |Ko| = i Pr(|Ko| = i) (2-26)
i= 1 n / -
S- log (Pr (max{Zs, zs,, zsRN } < s 1)


s-oo log S


where (2-26) is obtained from (2-25) by noting that max{ZsD, ZsR, 1 ZsR,_,} = ZSD

when Kol = 0, and max{Zs-, ZsRI, 1 ZR_ 2} 1= m-'i --Ko ZsR, = ZsR, when IKol > 0,

the first equality in (2-27) is due to the link gains being i.i.d., and the second equality in

(2-27) is obtained by using L'Hospital's rule.









Next given an N-node relay network, consider the multiple access cut that separates

the destination from all the other nodes. Clearly, the total flow across this cut gives an

upper bound on the maximum rate achievable in the N-node relay network. Consequently,

a lower bound on the outage probability PJ)t1(r, S) can be obtained using the maximum

sum-rate across this cut:


Put(r, S) > Pr [C((ZsD + Z, + + Z+Z 2D_)S) < r logS]

= Pr [(ZSv + Z,vD + ---+ Z _,2D) (N 1, (S- )/S)(228)
P(N- 1)

where 7(a, x) -= f' 1et dt is the lower incomplete gamma function and F(a) =

fo ta- letdt is the complete gamma function. The result in part 1) of Theorem 4.2 of

[30] can be extended to show that the diversity order of any transmission scheme over the

wireless relay network must satisfy
log (N-,(s"-)/s)
d(r) < lim log (Plut (r, S)) < lim log (N-2)
d~r) < hm -- ^ -- -< hm ----- -- '
s--oo log S s--oo log S

im6 xN-2[- (1 -- )- (N- 1)(1 -r). (2-29)
s- J0f ttN-2e tdt

Finally, from (2-27) and (2-29), we see that the GLS protocol, and hence the FO protocol,

achieve the optimal diversity-multiplexing tradeoff of (N 1)(1 r) for all 0 < r < 1. E

2.4 Numerical Examples

To demonstrate the performance of the proposed protocols, we consider the four- and

five-node relay networks, wherein information is to be sent from a source to a destination

with the help of two and three relay nodes, respectively. Using outage probability as the

performance metric, we compare the FO and GLS protocols against the max-min selection

method of [27], as it provides the best performance amongst previously proposed path

selection methods, and an outage probability lower bound derived using the max-flow-min-

cut theorem of [33, Thm. 14.10.1].









For the four-node relay network, there can be 6 possible time slots in the FO protocol

as shown in Fig. 2-2: three BC slots for the source and the two relays to transmit infor-

mation, and three MA slots for the two rel-iv- and the destination to receive information

respectively. To derive an upper bound on the achievable rate (and thereby a lower bound

on the outage probability), we use max-flow-min-cut type bounds for half-duplex commu-

nication. There are four possible time slots as shown in Fig. 2-3, with the first BC slot and

the last MA slot at the destination same as in the FO protocol, but now, the source and

a relay may transmit simultaneously to the other relay and the destination during each of

the intermediate slots over interference channels. We use the max-flow-min-cut theorem to

upper bound the maximum information flow in these two time slots.

For the five-node relay network, there can be 8 possible time slots in the FO protocol

- four BC slots for the source and the three rel-iv to transmit information, and four MA

slots for the three relays and the destination to receive information respectively. Similar

to the four-node relay network, for the max-flow-min-cut bound, there are 8 possible

time slots with the first BC slot and the last MA slot at the destination being the same

as for the FO protocol, and multi-source-multi-destination transmissions during the six

intermediate slots. In general, the following may occur during the six intermediate time

slots: the source and a relay may transmit simultaneously to the other r el1 and the

destination during the second, third and fourth slots, and the source and two rel-,v- may

transmit simultaneously to the remaining relay and the destination during the fifth, sixth

and seventh slots. As in the case of four-node relay network, we use the max-flow-min-cut

theorem to upper bound the maximum flow of information during the intermediate time

slots.

With the above division of time slots, the formalization of the problem is done as in

the previous sections, and we use the optimization routine of [36] to obtain the maximum

achievable rates and upper bounds for different values of required rates. In Figs. 2-4

and 2-5, we plot the outage probabilities of the various schemes with the required rate









R at lbit/s/Hz and 6bits/s/Hz respectively, for the four-node relay network. Figs. 2-6

and 2-7 present the same for the five-node relay network. When compared to the FO

protocol, the GLS protocol suffers a loss of around 1.0dB, and around 1.5dB (when R

is either lbit/s/Hz or 6bits/s/Hz), at an outage probability of 10-4, for the four- and

five-node relay networks respectively. On the other hand, the performance degradation for

the max-min selection method of [27], as compared to the FO protocol or even the GLS

protocol, is more than 12dB at an outage probability of 7.0 x 10-2, when R = 6bits/s/Hz

for the four-node relay network, and an exactly similar situation can be observed for

the five-node relay network. Moreover, for the four-node relay network, the FO protocol

is within 2.14dB (when R l= bit/s/Hz) to within 7.05dB (when R = 6bits/s/Hz) of

the lower bound on the outage probability when the outage probability is 10-4. For the

five-node relay network, the corresponding differences are approximately 3dB and 9.6dB

respectively. Thus, we see that as the number of nodes in the relay network increases, the

GLS protocol becomes more suboptimal, whereas the gap between the outage performance

of the FO protocol and the lower bound widens. With regard to the latter observation,

it should be kept in mind that the lower bound obtained using the max-flow-min-cut

theorem is, in general, not a tight bound.

The performances of the different protocols for the four-node relay network with

non-uniform average power gains are presented in Figs. 2-8 and 2-9, and Figs. 2-10

and 2-11 for cases A and B respectively, with the average power gains as stated in the

figures. Case A represents the situation when both the source-relay links are, on average,

better than the direct link, and one relay-destination link (the link between Ri and

D)is, on average, better than the other, resulting in relay Ri being a better candidate to

forward the information than the other relay. On the other hand, case B represents the

situation when no one relay has very good source-relay and relay-destination links. In

this case, one source-relay link is, on average, better than the direct link, which, in turn,

is better than the other source-relay link. The reverse is true for the relay-destination









links, and the inter-relay channel is, on average, very good. This situation promotes

inter-relay interactions for the FO protocol, and thereby increases the difference between

the performances of the FO and GLS protocols. The differences between the outage

performances of the FO and GLS protocols, at an outage probability of 10-4, are 1.2dB or

1.0dB, and 2.0dB or 1.3dB (when R l= bits/s/Hz or R = 6bits/s/Hz), for cases A and

B respectively. This reduction in the suboptimality of the GLS protocol with increase in

the required data rate can be explained by noting that, when the required rate is high,

the coding gain offered by a protocol heavily relies on the efficient use of the direct link,

and since the usage of the direct link is similar for both the FO and the GLS protocols,

the performance gap narrows as the required data rate increases. On the other hand, at

the same outage probability, the difference between the outage performance of the FO

protocol and the lower bound increases from 1.5dB to 7dB, and from 1.9dB to 6.0dB as

the required rate increases from lbit/s/Hz to 6bits/s/Hz, for cases A and B respectively.

Overall, these results demonstrate trends similar to the uniform average power gain case,

and confirm the generality of the proposed protocols.

2.5 Summary

In this chapter, we proposed a cooperative transmission design for a general multi-

node half-duplex wireless relay network where channel information is available at the

nodes. The proposed design is based on optimizing information flows, using the basic

components of BC and MA, to maximize the transmission rate from the source to the

destination, subject to maximum power constraints at individual nodes. Motivated by

the need for simpler network path selection schemes that perform well even in high-

rate scenarios, we developed the generalized-link selection protocol that combines relay

selection, and flow optimization for a three-node relay network. The proposed protocols

were shown to achieve the optimal diversity-multiplexing tradeoff for a general relay

network. Simulation results for the four- and five-node relay networks for uniform and

non-uniform average power gains demonstrate that the performance of the much simpler









GLS protocol is slightly worse than that of the FO protocol. This -i-:.-. -1 that the GLS

protocol can be used in systems with low-complexity requirements. We also note that the

proposed FO and GLS protocols can be used in wireless networks with topologies more

complicated that the wireless relay network considered here. For example, application of

similar ideas to a parallel relay network in which there is no direct connection between the

source and the destination is considered in [37].










R


S 3 D
X3


t1 (graph G1)


t2 (graph G2)


Figure 2-1. Basic graphs G1 and G2 for the three-node relay network with ti + t2 1.











R1 f R1 [


D {S


:x


D > S x7


D (S


Figure 2-2. FO protocol for the four-node relay network with tI + + t6 = 1. The flow
optimization is performed over all flows x1, X x14, and all time slot lengths
tl, t6.






































S
x 12


t3 t4

Figure 2-3. Transmission strategy to obtain a lower bound on the outage probability for
the four-node relay network. Here t1 + + t4 = 1, and the optimization is
over xi, x14, and ti, t4, with the application of the
max-flow-min-cut theorem for the intermediate slots.














100




10 -



10-1
-Q
o 2
10-4
0D


-3
0





10-4
0


Spectral efficiency, R=1bits/s/Hz


5 10
SNR (dB)


Figure 2-4. Four-node relay network with uniform average power gains: Outage
probabilities for required rate R l= bit/s/Hz.


Spectral efficiency, R=6bits/s/Hz
100



1011











10-3 Max-min selection routing *
-- Generalized-link selection
Flow-optimized protocol
S- -. Low er bound. ....... ............... .
..............


10 15 20 25
SNR (dB)


30 35


Figure 2-5. Four-node relay network with uniform average power gains: Outage
probabilities for required rate R = 6bits/s/Hz.


--- Max-min selection routing
- Generalized-link selection
-- Flow-optimized protocol
- -Lower bound












Spectral efficiency, R=1bits/s/Hz


10 -. Lower bound
. . .



10 .
o 2


10-3
10-4 .........



0 2 4 6 8 10
SNR (dB)


Figure 2-6. Five-node relay network with uniform average power
probabilities for required rate R lbits/s/Hz.
0 1 0 :: ::: ::: ; ::: ::: ::: ..........S ::: S :: :::
.. .. ...............M l





10-4----------------- X





probabilities for required rate R tbits/s/Hz.


Spectral efficiency, R=6bits/s/Hz


o
.Z

10


0
10


10 L
10


15 20 25 30 35
SNR (dB)


Figure 2-7. Five-node relay network with uniform average power
probabilities for required rate R = 6bits/s/Hz.


gains: Outage


-i- Max-min selection routing
-- Generalized-link selection
Flow-optimized protocol


12


gains: Outage











Non-uniform average power gains: Case A
Spectral efficiency, R=1bits/s/Hz


0 2 4 6 8 10 12 14
SNR (dB)


Figure 2-8. Four-node relay
E[Zsg] 2.0,
1.5, E[ZR2v]







10-1




_1-2
10 -----


10-1
10







1 I


network with non-uniform average power gains. Case A:
E[Zsg2] 2.0, E[Zsv] = 1.0, E[ZgZ ] 1.0, E[ZD] 1
1.0. Outage probabilities for required rate R l= bit/s/Hz.


Non-uniform average power gains: Case A


Max-min selection routing :
Generalized-link selection :
Flow-optimized protocol
Lower bound
15 20 25
SNR (dB)


30 35


Figure 2-9. Four-node relay network with non-uniform average power gains. Case A:
E[Zs,] 2.0, E[Zsg2] 2.0, E[Zsv] = 1.0, E[ZR1Z] 1.0, E[ZRD] -
1.5, E[Z2D] = 1.0. Outage probabilities for required rate R = 6bit/s/Hz.


100




10-1


-Q
o 10-

05
0





10-2
10-4













Non-uniform average power gains: Case B
Spectral efficiency, R=1bits/s/Hz


0 2 4 6 8 10 12 14 16 18
SNR (dB)


Figure 2-10. Four-node i

E[Zsi] l
0.2, E[ZK2I






10









10 -




10 1


10 L-
10


, network with non-uniform average power gains. Case B:

E[ZsZ2] 0.75, E[Zsv] = 1.0, E[ZRI2] 3.5, E[ZRD]
3.0. Outage probabilities for required rate R l= bit/s/Hz.


Non-uniform average power gains: Case B
Spectral efficiency, R=6bits/s/Hz
i .


15 20 25
SNR (dB)


30 35 40


Figure 2-11. Four-node relay network with non-uniform average power gains. Case B:

E[Zsg] 1.5, E[ZS2] 0.75, E[ZSD] = 1.0, E[ZgIg] 3.5, E[ZgzD]
0.2, E[ZR2D] = 3.0. Outage probabilities for required rate R = 6bit/s/Hz.


100





10



-Q
o
10
(D


0







10


-i- Max-min selection routing :\::
- Generalized-link selection:: ::
-- Flow-optimized protocol :::::
- Lower bound


.. ..... .... .... .... .... ...
. .
. .
....... ..................
11 .
. .. . .
. :\ . .
.............
.............
.............
.............
7 ..............
........................ \ .


. ...........................
. ........................
. . .
. . .
. . .
. . .
. .
.................. .....
.7 ....................
..........................
........................









CHAPTER 3
RESOURCE ALLOCATION AND COOPERATIVE BEHAVIOR IN FADING
MULTIPLE-ACCESS CHANNELS UNDER UNCERTAINTY

3.1 Introduction

The resource allocation problem for multi-user wireless systems has generated

considerable interest in the research community and has been considered from different

perspectives with regard to efficiency and fairness issues. The fading MAC is one of the

basic amongst such systems and different solutions have been proposed till date to the

above-mentioned problem. The throughput capacity region of the fading MAC, which

can be achieved by using dynamic power and rate allocation schemes to maximize the

average rate, has been characterized in [38]. Using ideas similar to those for long-term

power control in [39], the outage capacity region for this channel has been derived in [40],

where the users have average power constraints. The authors, in [40], obtain both the

common outage capacity region, when all the users have a common outage probability

constraint, and also the individual outage capacity region, when users may have different

outage requirements. For the latter case, the capacity region is characterized by a channel

usage reward vector, which determines the actual operating point for the system.

A game-theoretic approach towards solving the resource allocation problem with

the average rate utility and users subject to average power constraints, is considered

in [41]. They propose a Stackelberg formulation, where the receiver is the game leader

and the transmitters p1 iv a water-filling game, where the order of decoding of the users'

information, which implies a prioritizing of the users, may be decided by the receiver using

an auctioning process as in [42]. A low-complexity dynamic rate allocation policy, that

maximizes a general concave utility function of the rates in the throughput capacity region

for fixed transmission powers, is presented in [43]. In [44], the optimal power control

scheme for maximizing the sum-capacity in the multiple input multiple output (\!I\ 10)

fading MAC is characterized.









In the above works, the availability of perfect channel state information (CSI) at

the transmitters and the receiver is assumed. The optimal medium access and resource

allocation scheme, that maximizes the total throughput for the fading MAC with partial

CSI, is proposed in [45]. The partial CSI is either in the form of a threshold-based 1-bit

quantized feedback, or the best-user feedback. The outage rate regions with constraints

on individual user outage probabilities and no CSI at the transmitters, are derived in [46],

and they are used to obtain the best target rate vector to maximize the sum throughput

at the receiver.

An information-theoretic analysis of the achievable rate for the single-user fading

channel and the feasible rate region for the fading MAC, with imperfect CSI at both

transmitters) and the receiver, is presented in [47]. This is done by considering the

difference in the mutual information between the input(s) and the output with perfect

and imperfect CSI, by assuming optimal decoding at the receiver based on the knowledge

regarding the distribution of the fading process. In [48], the authors have quantified the

notion of "imperfect" CSI by showing that if the side information is such that the second

moment of the error is negligible compared to the reciprocal of the signal-to-noise ratio

(SNR), then it can be considered to be "perfect", whereas otherwise, the achievable rates

may be reduced as a result of the errors in the fading state estimation, and the use of

Gaussian codebooks and nearest neighbor decoding may not yield good performance.

This is established by considering the generalized mutual information (GMI), which gives

the highest rate for which the average error probability, averaged over the ensemble of

Gaussian codebooks, converges to zero. It is proved in [48] that in the absence of CSI,

the performance is poor for both low- and high-SNR operations. On the other hand,

with partial CSI available, for low-SNR regimes, the use of Gaussian codebooks and

maximum likelihood (\ I) decoding performs close (within a constant factor) to the

channel capacity, but not so in the high-SNR regime. The effect of imperfect CSI on the

finite-SNR diversity-multiplexing tradeoff for the quasi-static fading MAC is analyzed









in [49]. The common outage probability of the MAC is considered, and the bounds on the

fading MAC feasible rate region from [47] are used to demonstrate the effect of imperfect

CSI on the finite-SNR diversity-multiplexing tradeoff.

In this work, we model the resource allocation problem for the two-user fading MAC

using a two-person bargaining problem [50], wherein the extent of cooperative behavior

is determined by the outcome of the bargaining problem. In this work, we consider the

situation when the utility derived by each user is the average rate (over all fading states).

When the CSIT is perfect, the solution to the bargaining problem, specified by the NBS,

yields the optimal transmission strategy pair for the two users with regard to fairness

and efficiency considerations. Here, a transmission strl it; for each user corresponds to a

choice of transmission rate and power for a particular fading state.

We consider the situation wherein the receiver has access to perfect CSI, but there

exists a certain uncertainty regarding the CSIT that may stem from quantization (as,

in reality, the feedback channels are likely to have limited capacities) and/or prediction

errors. If the available CSIT is inaccurate, the transmission strategy pair -,- -1i ,1 by

the NBS may deviate from the true optimum, and thus, lead to considerable performance

degradation in terms of the true utilities. To overcome this lack of robustness we propose

a scheme in which the conventional two-person bargaining problem is relaxed to acknowl-

edge the fact that the NBS may not give the optimal strategy pair. According to this

modified bargaining problem formulation, each user independently decides its transmission

strategy via a maximin optimization from its respective set of possible strategies. For a

particular user, such a set is a range of transmission strategies about a nominal str iI

The nominal strategy is obtaining using the NBS to the original bargaining problem

and the available CSIT. This is in contrast to conventional bargaining problem formula-

tions, wherein, once the p1l -i, rs reach an agreement, they are bound to execute the exact

strategy pair 1- .-.: -1. I by the NBS.









In the following section, we present the system model. This is followed by the

description of the bargaining problem formulation in Section 3.3. The modified bargaining

problem is proposed in Section 3.4 along with numerical results. Finally, we conclude the

chapter in Section 3.5.

3.2 System Model

In this section, we give a brief description of the system model for the fading MAC

and introduce the formulation of the resource allocation problem to characterize the

cooperative behavior between two users (Users 1 and 2) who wish to transmit information

to a single receiver. Note that the form of cooperative behavior considered in the present

work is not the same as usually interpreted in the literature, wherein one user may

actually forward the information of another to the destination, and such a situation is

studied in C'! Ipter 4. As we shall see in the sequel, for the conventional two-user fading

MAC considered here, the level of cooperative behavior is manifested by how much a user

"backs off" from its maximum possible transmission rate for a certain transmission power

choice.

Consider a discrete-time two-user fading MAC with unit bandwidth, in the presence

of unit-variance Gaussian noise, with the fading state described by the power gain vector

Z = (ZI Z2). The power gains are assumed to be independent and identically distributed

(i.i.d.) exponential random variables with unit mean.

Let Ti be the maximum transmission power available to the ith user. We assume

that perfect CSI is available at the receiver, whereas the CSI at the transmitters may not

be accurate. The transmission strategies are determined in terms of the joint conditional

probability density functions (PDFs)1 of the transmission rate and power, where the

conditioning is on the fading state.



1 In this work, we use the term PDF to refer to both continuous and discrete probability
functions.









We model the resource allocation problem (i.e. the optimal choice of the transmit

powers and rates) as a two-user bargaining problem. This specifies the operating point

of the system. Note that a bargaining problem formulation is an appropriate choice to

model this problem as it does not presume any inherent cooperation between the two

users. Instead, the users negotiate to reach an agreement after evaluating (selfishly and

rationally) the potential benefits from cooperation over the event of them not arriving at

any mutual agreement. Moreover, it is well known that the NBS can be interpreted as a

generalized form of a proportional fairness solution, and coincides with the latter when the

p .,ioffs to the two pl -,v rs in the event of disagreement equal zero. Thus, the NBS provides

a fair and efficient (i.e. it is not possible to improve one user's performance without

degrading the performance of the other) solution to the resource allocation problem.

Unfortunately, owing to the dependence of the operating point on the fading state, if the

CSIT is not accurate, the operation point obtained with the erroneous CSIT may not

be optimal. In order to obtain a more robust solution to the resource allocation problem

under uncertainty, we propose a relaxed bargaining problem formulation in this work.

3.3 The Bargaining Problem for the Two-User Fading MAC

In this section, we solve the two-user bargaining problem to obtain the optimal

strategy for the two users using the available CSIT. Thus, for the two utility function

choices, we need to solve the two-person bargaining problem, defined as (T, Td). Here, T

is the set of feasible utilities, i.e. the achievable average rates for the two users, and

Td (Td, T2,d) represents the I. ireement point, i.e. the utility each user will derive

if they do not cooperate. Thus, the two users negotiate to reach an agreement regarding

the optimal transmission rates and powers, given the fading state. Moreover, it is assumed

that the users can agree to jointly randomized strategies regarding the their transmission

rates and powers. The disagreement points for this bargaining problem, followed by the

NBS, are derived next.









3.3.1 The Disagreement Point

Here, we consider the case when there is no agreement between the two users with

regard to their transmission strategies. Let Ri and Pi be the transmission rate and

power, respectively, of User i. Also, given a particular fading state and the transmis-

sion strategies for the two users, the achievable transmission rate for User i is given

by R'(P(Z), R(Z), Z). That is, if, for the fading state Z, the actions for the two

users are specified by the transmission powers and rates P(Z) = (Pi(Z), P2(Z)) and

R(Z) = (Ri(Z), R2(Z)) respectively, then the p ,ioff received by User i is given by

R'(P, R, Z), where the dependence of P and R on Z is not made explicit for brevity.

Particularly, these utilities correspond to the rates for the two users at which reliable

transmission can be supported with an arbitrarily small probability of error. Due to

the symmetric nature of the problem, it is sufficient to consider any one user's opera-

tion, iw User 1. In the absence of any agreement between the two users, the achievable

transmission rate for User 1, R', is given by (3-1), where C(x) = log(1 + x).

RI if R, < min{C(ZiPi), max{C(ZiPi + Z2P2)-

R (P, R, Z) < ( PR2, (3-t)

0 otherwise.

Note that for the scenario wherein the users fail to reach an agreement, there is no

restriction on the choice of the transmission strategies, as against the scenario wherein the

users reach a mutual agreement whereby each user's choice of the transmission strategy

is restricted in some specific way. Since each user is unaware of the strategy of the other,

we derive the optimal strategy of each user as the solution to a maximin problem, wherein

each user's aim is to maximize its own worst-case usage probability. Note that the users

being able to i,.../. p .J ;./i. decide from the set of all randomized strategies implies that,

in general, the users may use mixed strategies for the maximin games in this model. Thus,









for User 1, the optimal transmission strategy is


f plz = argmax min E[R'], (3-2)
fRllPlZ fR2P2 \Z

where E[.] denotes the expectation operator and the expectation is over all fading states.
Also, fRpplz is the joint conditional PDF of User i's transmission rate and power such that

Pr(Pi > Ti) = 0 for i = 1, 2, with the latter probabilities computed with respect to the

PDFs fpxz and fp2z, respectively.
Since R' in (3-1) is a monotonically non-increasing function of R2, a solution of the

minimization problem in (3-2) is given by


fpZ ,P2) 6 (r2 C (Z2T2)) 6 (P2 T2). (3-3)

Using (3-3) in (3-2) gives filpz argmax E[R[], such that Pr(Pi > TI) 0. In the
fRIPI|Z
above, R' is

Ri if R1 < C 1+2T

0 otherwise.

Clearly, it can be seen that the solution to the above problem is given by


fAlPlz(rlPl) = 6 r- C Z621T2 1i T1).

Therefore, when there is no agreement between the two users, the maximin optimal

achievable average rates are T1,d E C (ZT1 T2,d = E )221 for
Users 1 and 2 respectively. Since the operation point (for every fading state Z) is in the

interior of the MAC capacity region, a small perturbation in Z may not cause a significant
degradation in the actual achievable utilities.

3.3.2 The Nash Bargaining Solution (NBS)

We use the NBS to obtain the optimal utility allocation for this bargaining problem
using the available CSIT. The optimal allocation of the usage probabilities, using the NBS,









is given by [50]


Y* argmax( Yl,d)(Y2 Y2,d), (34)
Yey
where Y denotes the corresponding utility of interest, and Y is the feasible set of utility

allocations. The NBS yields the optimal allocation of utilities (Y*) between the two

pl! i., rs in the bargaining game. Thus, for the present problem, the NBS may be used to

obtain the optimal strategy pair fP Z (i.e. the optimal joint conditional PDF for the

powers P(Z) (PI(Z), P2(Z)) and rates R(Z) (RI(Z), R2(Z))), where the uniqueness

of fpRIz is up to the corresponding achievable average rate pair T*, that is, we identify

any two agreements which yield the same utility allocation. As a result, the possible

non-uniqueness of the transmission strategies does not affect the optimality of the resource

allocation solution.

Before proceeding with the derivation of the NBS we verify that the necessary

conditions [50] of a two-person bargaining problem are satisfied for the average rate utility

function. That the set of feasible utilities, 7, is compact can be proved by using the fact

that the capacity region of the MAC, for a particular fading state, is closed and bounded

in R2. Also, the disagreement point, defined by Td, is in T. The definition of the utility

function as an average of the p .,ioffs received in each fading state, and the fact that the

MAC capacity region is monotonically increasing in the transmission powers imply that

the users have continuous preference relations on the set of all conditional PDFs for the

nominal transmission rates and powers. Moreover, these preference relations satisfy the

von Neumann-Morgenstern (VNM) axioms of independence, continuity and of being

complete and transitive. Further, that the set {T cET : T > Td} is nonempty can be

concluded from the observation from the previous subsection that the disagreement point

correspond to the system operating strictly inside the capacity region of the two-user MAC

for any fading state. The convexity of T can be seen as follows: if T' = (TI, T2) can be

achieved by the joint conditional PDF fpRIz and T" by /p/ z, then, for any a c [0, 1], the








set of average rates defined by aT' + aT" can be achieved by time sharing between the
two agreements.
Hence, we may obtain an optimal conditional PDF for the transmission rates and
powers using equation (3-4). Thus, we have

T* argmax (E[Ri(Z)] T1,d) (E[R2(Z)] T2,d) (3-5)
{T: R(Z)E.MAAC(P(Z),Z)}

where P(Z) must satisfy the maximum power constraint for each user. It can be easily
seen that

T* (E C(ZITI) + C (zjj ] E C(Z2) + C Z2 )2 ] (36)

and this can be achieved with the following rate and power allocation:

2 czr) + rcz+c Q Z2T )] w. p.1,
S 1 C(Z ) + C w.p. 1, (37)
A 2 2 G + Z2T2

P* = Ti, w. p. 1, for i 1,2.

That is, for this case, employing the NBS for each fading state achieves the optimal
solution to the bargaining problem of (3-5). This solution is similar in flavor to the
one in [51], with the difference being in the nature of utility functions considered. More
specifically, the utility we consider here is an average metric, while the bargaining model
in [51] considers the utilities resulting from a single instance of the game for a particular
state of nature (the fading state in this work). One important property of the NBS-
- -...- -l,1 solution above that is of significance to the development of the modified
bargaining problem formulation in Section 3.4 is that the optimal choice of transmission
powers is deterministic and independent of the fading state.











R1 if R, < min{C(ZiT), max{C(ZiTi +Z2) R2,

R'(R, R*, Z) C ( )}},and R, S w. p. (1 ),

0 otherwise.

Here, we emphasize that the above choice of transmission rates as in (3-7) is not

the only possible solution to the Nash bargaining problem. Unless other constraints are

imposed, any choice of jointly randomized transmission rates satisfying (3-6) may be

selected as the NBS--,1-.- -i. -l optimal strategy, with the choice of the transmission powers

as in (3-7). Specifically, the above choice has been made to facilitate an easy exposition

of the modified bargaining problem formulation in Section 3.4 and also to yield a simple

solution for practical implementation with less overhead. With regard to the latter point,

note that for the solution of (3-7) each user only requires the knowledge of the channel

power gains Z (obtained from the CSIT) and can maximize the Nash product of (3-5)

independently, whereas for a choice of jointly randomized rates satisfying (3-6) some

form of communication between the two users needs to be established to enable the joint

randomization of the transmission rates.

Note that the NBS, as in (3-6), -,ti-.--i- that no user transmits at its maximum

possible transmission rate (i.e. C(ZiTi) for i = 1 or 2) with probability 1, and this

"backing off" of each user from its maximum possible rate may be interpreted as the

manifestation of its cooperative behavior, motivated by a rational and individualistic

evaluation of the benefits of cooperation as against any presumed altruism on its part.

Moreover, for any choice of jointly randomized transmission rates satisfying (3-6), the

transmission rate pair would aliv--, correspond to a point on the boundary of the MAC

capacity region for every fading state, thereby making the solution very sensitive to the

CSIT. In the following section, we propose a modified bargaining problem to handle this

robustness issue.









3.4 The Modified Bargaining Problem 2

The sensitivity of the operating point of the system to the uncertainty in CSIT

may be reduced by decreasing its dependence on the NBS--,-.;.; -I. I1 strategy pair as

described in this section. According to the modified bargaining problem, the NBS-

~-,-.; -1. .1 strategy pair (henceforth, the nominal strl' i'11 pair) is obtained using the

available CSIT as in Subsection 3.3.2, but instead of the two users being constrained

to implement these strategies, they get the flexibility of h.i. /*,p i'.1 ,.1;:i choosing their

transmission strategies from a certain set of strategies about this nominal strategy pair.

Let Ri(P*, R*, Z) and Pi(P*,R*, Z) (for i = 1, 2) be the actual transmission rates

and powers respectively, given the nominal transmission strategies and available CSIT.

Next, we utilize the fact that the proposed solution to (3-7) -,-i.; -1 using the maximum

available powers for all fading states, and hence, we set Pi(P*, R*, Z) = Ti w. p. 1.

Consequently, in what follows, we shall not represent the dependence of the transmission

rates on P* explicitly. Also, we define the transmission strategies for the two users by

the conditional PDF of only the transmission rates, with the conditioning on the fading

state and the nominal transmission rates R*. The sets of allowable transmission strategies

specify certain limited deviations from their nominal values to account for the uncertainty

regarding the CSIT. Define the sets of allowed transmission rates for the two users as

S, = {R : R, e [cR* AR, R* + AR] } for i = 1, 2. Then the choice of the actual

transmission strategy of User i (i = 1, 2) is subject to the following constraint:


RL(R*, Z) e Si w. p. (1 e), (3-8)



2 As would become clear from the subsequent discussion, we do not modify the bargain-
ing problem as such. Only the implications of reaching an agreement are modified. Thus,
the bargaining problem still stands valid as described in Section 3.3.









for some c > 0 that can be arbitrarily small. Here AR (a pre-determined non-negative

constant) denotes the maximum deviation (w. p. (1 e)) from the respective nominal

values3 This allows us to robustly handle the inaccuracy in the CSIT by reducing the

sensitivity of the operating point and hence, that of the true utilities, with respect to the

CSIT. Note that the probabilities in (3-8) are computed with respect to the joint PDF

fRiPRZ for User i.
With this flexibility in choosing the transmission rates, each user can no more be

certain of the other's exact transmission rate, i.e. for User 1, R2(R*, Z) is unknown, and

vice-versa. Hence, a natural option for each user would be select the transmission rate

using a maximin criterion to handle the uncertainty regarding the exact transmission

rate of the other. Moreover, for a particular fading state, the actual p ,,off received by

User 1, under the constraints defined by (3-8), can be interpreted in the same spirit as

R'(P, R, Z) in (3-1), and is given by (3-8). According to the maximin criterion, the

optimal choice of transmission rate for User 1 is given as:


fiR*,Z argmax min E [R'(R,R*,Z)] (3-10)
fRI|R*,Z fR21R*,Z

Here, we emphasize that this maximin optimization is carried out using the available

CSIT.

Using the values of R*(Z) from (3-7), define

A*/(Z) A C(ZiTi) R = C(Z2T2) R
1
2 [C(ZiTi) + C(Z2'2) C(ZITI + Z22)] .
2

We partition the entire space of power gains R2 into two sets Z = {Z : AR < A(Z)} and

its complement Z'. With this partitioning, we can derive the maximin-optimal strategies



3 Although we assume equal uncertainty values for the two users, this can easily be gen-
eralized to the case wherein they are different.








for the two users conditioned on the fading state belonging to one of these sets. Next, we
make the following important observation:

Z e Z' == R* + AR > C(ZT) for i = 1, 2, (3- 11)

( Z1TZ2 )

with a relation analogous to (3-12) being true for User 2 as well. For p(Z) > 0, let
co = min {1, c/p(Z)}. For any choice of fRpIR,z, the choice of User 2's transmission rate
that minimizes the average rate of User 1 in (3-10) can be derived as

2 (r2 C(Z2T2))+( o) (r2 (R2 +AR)) if Z e Z,
ftRt,z(r2) (r2C(3 13)
6 (2 + AR)) otherwise.

Substituting the above solution in (3-10), the maximin-optimal distribution for the
transmission rate can be shown to be the following mixed strategy:

\6 r( C ( {Z2 ( )) if Z e Z

.6 r( C+ (l ) 6 (ri (R A)) otherwise,

where
eo if C ( )>(- co) (R AR),

0 otherwise.
The maximin-optimal transmission rates for User 2 can be derived analogously.
Intuitively, when the channel power gains belong to the set Z', AR is comparatively
"1 i5, and the restriction imposed through (3-8) loses its effectiveness as the window
of strategies about the nominal strategy pair becomes "too wide". Hence, both users
select transmission rates corresponding to the disagreement point as in Section 3.3.1. On
the other hand, when Z e Z, the maximin-optimal strategy for each user is to respond
to the (worst case) mixed strategy of the other user (as in (3-13)) while satisfying
constraint (3-8). Hence, when the error in the CSIT (in terms of the power gains) is not









high, suitably selecting a -i ill" value of AR can ensure that both users back off from the

NBS-s,-'-. -1I 1 transmission rate pair of R* to (R* AR, R* AR) with high probability

(i.e. both /3* and eo can be made small enough with a choice of small AR and c in (3-8),

thereby resulting in p(Z) a 1). Note that, in this case, the modified bargaining problem

(that incorporates the maximin criterion) leads to both users backing off in a similar

fashion although each user independently chooses its respective strategy. Thus, it can

be seen that the modified bargaining problem formulation provides a general framework

for resource allocation from an individualistic perspective and a characterization of the

optimal strategy pairs in terms of AR.

As mentioned at the end of Subsection 3.3.2, the nominal transmission rate pair

selected in (3-7) is not the only possible choice, and other jointly randomized transmission

rates may also be selected. However, the choice of transmission rates as in (3-7), can

be shown to incur no loss of generality. For any choice of jointly randomized nominal

transmission rates, the maximin-optimal transmission rates can be derived in terms of

their conditional PDFs (with the conditioning on the nominal strategies and fading state)

in the same way as above to yield the same maximin-optimal values for the objective

functions (i.e. E[R'(R, R*, Z)] of (3-10) and its counterpart for User 2) as for the case

considered here.

In Fig. 3-1, we present the numerical results for the system described in 3.3.2 with

two different models for the error in the CSIT. For this example, the maximum available

powers at each user are Ti 100mW and T2 = 10mW respectively. We set e = 0.02

(cf. (3-8)) and consider two simple models for the error in the available CSIT: (i) a

5'. error in CSIT with the true channel power gains less than what the CSIT -,i-.-, -i-

and (ii) ,' error in the CSIT with x randomly chosen from a uniform distribution over

[-10, 10]. The true utilities are calculated in the following way. For a particular fading

state and choice of transmission strategy pair, if the transmission rates lie within the MAC









capacity region defined by the transmission powers and the true power gains, then these

rates are considered achievable. Else, both the users suffer outages.

For both error models, as AR increases from Obps/Hz to 2.5bps/Hz, the true utilities

increase from the minimum utility points (when the error in the CSIT is unaccounted for),

reach their respective maxima, that are very close to the achievable rates for the perfect

CSIT scenario, and then decrease to eventually converge at the disagreement points.

Hence, it can be concluded that with a proper choice of AR, the proposed solution can

provide the necessary robustness against inaccuracies in the available CSIT. Moreover, as

expected, it can be observed that the disagreement point is not affected significantly by

the error in CSIT as it corresponds to an interior point of the MAC capacity region for

any fading state.

3.5 Summary

Optimal resource allocation and cooperative behavior in a two-user fading MAC

with uncertainty regarding the CSIT are considered from a game-theoretic perspective.

The resource allocation problem is modeled as a two-user bargaining problem with the

average rate utility function. Owing to possible inaccuracies in the CSIT, the solution to

the bargaining problem (for instance, the NBS), that depends on the CSIT, may not be

optimal, and may cause system outages. To address this lack of robustness a modification

to the bargaining problem formulation is proposed. Numerical results demonstrate that

the proposed solution can be used to provide significant robustness without an explicit

modeling of the error in the CSIT.










Average rates for the two-user fading MAC subject to errors in CSIT

--- Perfect CSIT
U Disagreement point (Perfect CSIT)
O 5% error in CSIT (A=0Obps/Hz)
+ Disagreement point (5% error in CSIT) : :
...... 5% error in CSIT (AR=0 to 2.5bps/Hz)
CSIT error: Uniformly distd. '
on [-10%, 10%] (AR=Obps/Hz)
CSIT error: Uniformly distd.
- on [-10%, 10%] (AR=0 to 2.5bps/Hz) :


0.8-


0.6-


0.4-


0.2


I I I I I I I I I


S 0.5


1 1.5 2 2.5
T1 (bps/Hz)


3 3.5 4 4.5 5


Figure 3-1. Average rates with varying AR.


'









CHAPTER 4
ACTIVE USER COOPERATION IN FADING MULTIPLE-ACCESS CHANNELS

4.1 Introduction

The growing emphasis on multi-user wireless communication systems and the ever-

increasing demand for high data rates have heightened the importance of research in

the area of user cooperation. Although wireless systems bring forth design challenges

owing to multi-user interference and fading concerns, it also provides potential benefits,

like the broadcast nature of the wireless medium and diversity advantages in multi-user

systems. The MAC is one of the fundamental kinds of multi-user communication systems.

Conventionally, in a MAC, the users transmit directly to the destination, and the capacity

region of such a system is well known. Recently, it has been demonstrated [3, 4] that

the rate region of the fading MAC can be increased by providing spatial diversity that is

achieved by user cooperation in the form of forwarding each other's information to the

destination, giving rise to the C \!AC.

Although the capacity region for the C \! AC is yet to be determined, there have

been a number of different cooperative transmission strategies proposed in the literature.

Two broad classes of works reported in this regard can be distinguished based on the

transceiver capabilities of the wireless nodes, i.e. whether the nodes can support full-

duplex communication or not.

In [3], the authors provide a system-level description of the C '\!AC wherein the nodes

are capable of full-duplex communication. They present an achievable rate region based

on block Markov encoding and backward decoding, and show the potential increase in the

rate region as compared to the conventional MAC. It is assumed that the phase of the

fading is known to the transmitters and this is exploited to perform coherent combining

at the destination node, and obtain beamforming gain. In [4], the CDMA implementation

aspects of the scheme in [3] are considered, wherein the authors propose the use of

different spreading codes to obtain different channels for simultaneous transmission









and reception, without the use of complicated echo cancelation techniques. The power

allocation problem for the C \! AC with full-duplex nodes and full CSI available at all

nodes has been addressed in [55] and [56]. In [55], the authors consider average power

constraints and characterize the optimal power allocation policies that maximize the set

of ergodic rates achievable by block Markov encoding and backward decoding technique

as in [3], by a dimensionality reduction approach, i.e. by noting that some of the power

allocations are zero for every fading state. A more direct approach to solve the similar

problem of optimal power allocation, with an almost closed-form solution, is presented

in [56]. It has been established in [57] that windowed decoding is sufficient to achieve the

same sum-rate as backward decoding for the block Markov superposition encoding scheme

for the C \!AC.

The optimal power and resource allocation problem for the C\ I\AC with nodes

capable of half-duplex communication is considered in [58], where it is assumed that

full CSI is available at all the nodes, and the transmitters cooperate by relaying each

other's information over orthogonal frequency bands or time slots. The solution to the

problem is presented as a two-step convex optimization problem formulation: first, for a

particular bandwidth (or time) sharing parameter value, the optimal power allocation is

characterized by a convex optimization problem, and then, the optimal resource sharing

(time or bandwidth) parameter is obtained as a solution to the quasi-concave problem of

maximizing the rate of one user, given a target rate for the other. All the works mentioned

above use a DF approach for the relaying of information to the destination.

In [19], the authors present a cooperative transmission scheme, based on the non-

orthogonal amplify-and-forward (NAF) technique, that is proved to achieve the optimal

diversity-multiplexing tradeoff of N(1 r) for the N-user half-duplex C \! AC, with

symmetric data rate requirement and CSI available only at the receiving node of any link.

According to the proposed strategy, time is divided into cooperation frames of length N

cooperation symbols, and each user transmits only once during a cooperation frame. Every









user is allotted unique transmission and reception symbol intervals, using a particular

scheduling policy, and it transmits a linear combination of its own symbol and the signal

observed during its most recent reception symbol interval, thereby creating an artificial

inter-symbol interference (ISI) channel. A set of L cooperation frames are combined to

form a super-frame, and the assignment of the reception symbol intervals is scheduled

for each super-frame, with the lengths of super-frames and codes chosen such that a

coherence-interval consists of N 1 consecutive super-frames, and that all codewords span

the entire coherence interval.

Similar to the above NAF strategy, a cooperative transmission scheme for the two-

user C`\ IAC, based on superposition coding, has been proposed in [59]. This scheme uses

a time division approach in which a user simultaneously transmits its own information

and the other user's information by using the superposition coding (SPC) technique.

This scheme is demonstrated to achieve a gain of about 1.5 2 dB over traditional DF

approaches for relaying, and at the same level of system complexity of the latter. An

extension of this idea to the general N-user C \! AC is presented in [60], wherein the

authors prove the optimality of the proposed scheme in achieving the optimal diversity-

multiplexing tradeoff for the symmetric rate requirement scenario.

In this work, we propose flow-theoretic cooperative transmission protocols for the

two-user C\ AC. First, we present an orthogonal relaying protocol for the C I\!AC (OR-

C \! AC), wherein each user acts as a dedicated relay for the other in a time-division

fashion. The flow-optimized relaying approach of Chi Ipter 2, modified to incorporate

coherent combining at the destination is used for the constituent relay channels. This

relaying protocol has been shown to achieve the optimal diversity order and provide better

coding gains for the relay channel as compared to traditional DF relaying methods, by

efficiently utilizing the CSI available at all nodes. Next, we propose the flow-optimized

protocol for the C`\ !AC (FO-C \ IAC) that decomposes the C \! AC into two broadcast









channels (BC) and a multiple access ( \!A) channel with common information. The bound-

aries of the achievable rate regions are characterized by means of convex optimization

formulations. The improvement provided, in terms of the achievable rate region, by OR-

C \! AC and FO-C \! AC, over conventional MAC capacity and the DF strategy of [58]

without power control, increases as the amount of disparity between the channels from

the two sources to the destination increases. The outage performances of the proposed

protocols indicate that although the much simpler OR-C \! AC is suboptimal in terms of

the achievable rate region, it provides outage performance that is within 1 dB of that of

FO-C \ IAC. Moreover, both the proposed protocols achieve a diversity of order two for the

required rate region of interest.

The rest of the chapter is organized as follows. In Section 4.2, the flow-theoretic pro-

tocols of OR-C\ !.AC and FO-C \! AC are presented, and the boundaries of the achievable

rate regions are characterized by convex optimization formulations. This is followed by nu-

merical results in presenting the achievable average rate regions and outage performances

for different scenarios in Section 4.3. Finally, the primary contributions in this chapter are

summarized in Section 4.4.

4.2 Flow-theoretic Transmission Protocols for the Cooperative
Multiple-Access Channel

Consider a two-user C'\! AC where the two sources (S1 and S2) may actively coop-

erate to transmit information to a common destination (D). We use the phrase active

cooperation to distinguish between the cooperation involved in transmission strategies in

which one source may forward the other's information, and that in the conventional MAC,

wherein a user transmits at a rate lower than the maximum single-user rate possible for

the particular channel state and power expended. The quantification of this type of coop-

erative behavior was studied in C! Ipter 3. Thus, as a higher level of cooperative behavior,

the users may relay the information of each other by utilizing the broadcast advantage of









the wireless medium, giving rise to the C \! AC model. We consider a discrete-time two-

user fading MAC with unit bandwidth, in the presence of unit-variance Gaussian noise,

with the fading state described by the power gain vector Z (= Zss ZSiD ZS2D). Note

that, owing to the reciprocity of channels, we assume Zss = Zsas,. The power gains for

the wireless links are modeled as independent exponential random variables. We consider

two types of fading models in this work. First, in Section 4.3, we consider the situation

in which the channels are ergodic within a transmission block, for which we evaluate the

transmission protocols using average rates as the performance metric. Then, we present

the outage performance of the proposed protocols for the model in which the fading is not

fast enough and hence, the channels may not be ergodic during a transmission block. This

system model can be easily generalized to the case where the bandwidth W $ 1.

Let Pi be the maximum transmission power available to the source Si. In this work,

we consider short term power constraints only, and this precludes any potential advantage

of power allocation. We assume that full CSI is available at all nodes of the system.

Moreover, as a practical consideration, we assume that the nodes are not capable of

transmitting and receiving information simultaneously over the same frequency, i.e. they

are subjected to a half-duplex constraint. In the following subsections, we present two

protocols based on flow-theoretic designs to develop cooperative transmission schemes for

information transmission from sources S1 and S2 to destination D.

4.2.1 Orthogonal Relaying Protocol for the CMAC (OR-CMAC)

In this subsection, we present a simple cooperative transmission protocol based on

the conventional relaying approach. Time is divided into unit intervals. Owing to the

half-duplex limitation of the sources, the two sources cannot relay each other's information

at the same time. To address this, we divide each unit interval into time slots of lengths

T1 and T2. During time slot T1, source S2 solely assists source S1, by acting as a dedicated

relay to S1, to transmit the latter's information to the destination D. The reverse happens

during time slot T2. This is depicted in Figure 4-1.









Thus, we effectively have two relay channels over two orthogonal time slots. We use

the flow-optimized transmission scheme of C'!i pter 2 for the three-node relay channel.

According to the protocol presented therein, each time slot T1 (resp. T2) is further divided

into two sub-slots of lengths t1 and t2 (resp. t' and t'). Consider the time slot TI. During

the first sub-slot, S1 sends two independent flows of information xz and x2 to D and S2

respectively using a broadcast channel (BC), and in the second sub-slot, S2 forwards X2

to D and at the same time S1 sends out another information flow x3 to D, and X2 and

X3 are received at D via multiple-access ('\ A). To improve the achievable rates even

further, we modify the second sub-slot as follows. Since the flow X2, that S2 forwards to D

originated at SI, the latter is aware of it, and hence, we modify the second sub-slot from a

conventional MAC to a MAC with common information [61], where x2 forms the common

information between S1 and S2, X3 is the independent information from SI, and S2 does

not have any independent information to transmit. For this relaying scheme, maximizing

the overall transmission rate from Si to D can be formulated as the following optimization

problem:


max (xi + 2 + 3) overt 1, t2, Xl, x2, x3, P1, To (4 1)

subject to

non-negativity constraints: xI,x2, 3 > 0; tI,t2 > 0;

total-time constraint: t1 + t2 Ti;

capacity (power) constraints: PBc < PI;

x3 < t2C(ZsDPi); 2 + x3
0 < P I < P; To < ZS,D(PI Pi) + ZS2DP2 + 2ZS D(P P)ZS2DP2;


where C(x) = log(1 +x) and PBC, the minimum power required for the source to broadcast

at rates xz/tl and x2/ti to the destination and the relay, respectively, in the first sub-slot









with 0 < t1 < 1, is given by (see [30, Lemma 3.1] for proof)


Bc 1 (eXlti 1) + 1 e'1 Li(e12-ti 1) for Zs s2 > ZSID,
(2 ZL I 1) + Z 621 i(C61i-i 1) for Zs, s2 < ZSID.

For t1 = 0, PBC = 0. P1 is the power that Si allocates for the direct transmission of x3 to

D, and To denotes the power corresponding to the common information flow (x2), received

at D.

It can be checked that the above maximization problem belongs to the class of

convex optimization problems, and standard numerical techniques can be used to obtain a

solution. Let the solution of (4-1) be denoted by X(Z, TI). Then it can be easily seen that

X(Z, Ti) = TIX(Z, 1). For the second time slot, an exactly similar optimization problem

can be formulated as (4-1) with appropriate changes in the indices.

Therefore, for a particular fading state Z, the regular points on the boundary of the

achievable rate region can be obtained by maximizing a convex combination of the rates

of the two sources during the respective time slots, X(Z, TI) and Y(Z, T2). That is, by

solving the following optimization problem, for some 0 < p < 1,


max pX(Z, Ti) + (1 p)Y(Z, T2) (4-2)

subject to T1, T2 > 0, and T + T2 1.


The extreme points of the boundary region, i.e. maximizing only one source's rate are

given by T1 1, T2 = 0, etc. Unfortunately, this naive scheme of decomposing the C \! AC

into two orthogonal relay channels does not entail the best utilization of resources, and

as we shall see in the following subsection, this can be improved upon by a more efficient

flow-optimized transmission protocol.

4.2.2 Flow-optimized Protocol for the CMAC (FO-CMAC)

Instead of dividing the C \! AC into two separate relay channels, we divide a unit

interval into three time slots T1, T2, T3 with no orthogonalization of the relaying actions.









Now, the first two time slots are BC slots and the last one is an MA slot with common

information, as shown in Figure 4-1. During the first time slot, S1 transmits two indepen-

dent flows: xz + Y21 and x2 to D and S2 respectively using BC. Similarly, S2 transmits two

independent flows: yi + X21 and Y2 to D and $1 respectively using BC. Finally, Si and

S2 send two flows X3 + Y22 and y3 + a22 to D using MA with common information. Here

Y21 and Y22 are two parts of the information flow Y2 that Si received from S2 during the
previous unit interval. Similarly, x21 and X22 constitute the amount of information that S2

reh-,,-, for SI. Thus, for this scheme, the flow constraints imply that x2 x 21 + x22, and

Y2 21 + Y22. Also, for the last time slot, x22 and Y22 are known to both the sources, and
hence, they form the common information to be transmitted to D.

Hence, the total transmission rates from sources Si and S2 are given by X =

xi + X21 + X22 + X3 and Y = yi + y21 + Y22 + Y3 respectively. For this scheme, the boundary

of the achievable rate region can be characterized as follows: the regular points on the

boundary can be obtained by maximizing a convex combination of the rates X and Y, and

the extreme points correspond to the C \! AC degenerating into relay channels with one

source solely acting as a relay for the other. For compactness, let the information flows

corresponding to the two sources be represented by the vectors x = (Xa a21 22 a3) and

y = (yi Y21 Y22 Y3). The maximization problem that needs to be solved to obtain the
regular points can be formally stated as given below:


max IpX + (1 p)Y for p (0, 1) over TI, T2, T3, x, y, P, P2, To, (4-3)

subject to

non-negativity constraints: x, y > 0; TI, T2, T3 > 0;

total-time constraint: Ti + T2 + T3 1;

capacity (power) constraints: P1c < Pi; Pc < P2;

x3 < T3C(ZSDP); 3 < T3C(ZS2DP2);

X22 + X3 + Y22 + Y3








0 < Pi < Pi; 0
To < ZS,D(PI Pi) + ZSDP2 P2) + 2ZD(P Pl)ZSD(P2 P2).

As in the previous subsection, PBc and PFc are the minimum powers required by S1

and S2 respectively for the two BC slots. PFc is defined as in (4-4) and PFc is defined

similarly. Also, Pi and P2 are the powers allocated by S1 and S2 to transmit x3 and

Y3 respectively, and To denotes the received power at D corresponding to the common

information x22 + Y122-

1 (e(X+Y21)/T 1) + e(xl+w)/T(e/ 1) for Zss > ZSD, (44)
FPC S[D ZS1 S2 1 S
SS (X/TI 1) + eX2I ((1+Y21)1 1) for Zs1s2 < ZSID-.
For T1i 0, Pc 0.

Once again, it can be checked that the above maximization is a convex optimization

problem that can be solved using standard numerical optimization methods. Thus, FO-

C \! AC addresses the half-duplex limitation of the nodes by dividing the C \! AC into two

BC and one MAC with common information, and provides a more efficient utilization of

system resources as compared to protocols using two separate relay channels.

4.3 Numerical Results

In this section, we present some numerical results to demonstrate the performance

of the proposed protocols and compare them to the conventional MAC and the DF-based

strategy proposed in [58]. Figures 4-2 through 4-4 show the achievable average rate regions

for the various schemes for different scenarios. We consider different means of the fading

gains as stated in the figures. Considerable variations in the statistics of the fading gains

for the channels ZSID and ZS2D can occur in practical situations owing to different path

loss and shadowing effects and different amounts of scattering for the two direct links.

Thus, the .,i- ii ii. Ii ic situation corresponds to the case when the direct link from one

source to the destination is much worse than the other.









In Figure 4-2, the channel from S2 to D is much worse compared to that from Si

to D, and the large increase in the achievable rate region for the FO-C'\! AC rate region

is evident. Moreover, we see that the much simpler OR-C \ \ AC performs close to FO-

C \ \ AC for this scenario. On the other hand, the performance of the strategy of [58] is

much poorer than the above two. Recall that, in this work, we consider only short term

power constraint, and unlike [58], we do not consider the power constraint over the entire

unit interval. This eliminates any potential gains from optimizing the power allocations

for the different transmissions. So, for the present system, the cooperative transmission

strategy of [58] is clearly suboptimal. Figure 4-3 is presented to highlight the increase in

the achievable rate region resulting from the use of MA slots with common information (as

in Figure 4-2) instead of the conventional MA slots. As can be seen from Figure 4-3, with

the use of conventional MA slots, the maximum sum-rate for the FO-C'\ 1.AC coincides

with that of the conventional MAC without active cooperation between the transmitters.

The improvement in Figure 4-2 can be interpreted as the effect of the beamforming gain

as in a two-transmitter one-receiver MISO system. Figure 4-4 presents the symmetric

situation, when the two direct links are statistically identical. We see that in this case, the

increase in the rate region is not as pronounced as in the previous situation. Moreover,

the achievable rate region for the strategy of [58] lies strictly inside that for the baseline

system of MAC without active cooperation between the transmitters.

As mentioned in Section 4.2, for the situation in which the fading is not fast enough

so that the ergodic properties of the channels are observed, the outage performance of

the transmission strategies are a more reasonable performance metric as against average

rates. The outage performances of the proposed protocols are evaluated when the data

rate requirement is symmetric at K = 1 bit/s/Hz for both users. The outage event is

defined similar to the definition in [62], i.e. it is the union of the events that either one

or both of the users suffer an outage. The outage performances of the proposed protocols

are compared to that of the conventional MAC, the strategy of [58], and a lower bound on









the outage probability that is obtained by considering the case when the sources have a

perfect noiseless channel between them. Figure 4-5 presents the outage performances for

the .,i-,ii io. Ii ic situation as in the average rates case, and Figure 4-6 presents the same

for the symmetric situation. We see that the OR-C \!.AC suffers a loss of only 0.7 dB and

1.0 dB compared to FO-CM\iAC, at an outage probability of 10-3, for the .,i-i, i,. 1 lic

and symmetric situations respectively. The performance of FO-C \! AC is worse than the

lower bound by about 2.2 dB and 2.5 dB at the same outage level, for the .,i-iii,,. I lic

and symmetric situations respectively. On the other hand, the performance of the DF

strategy of [58] is significantly poorer, and even worse than the conventional MAC for the

symmetric situation. Another important observation from the outage performance plots is

that the slopes of the curves for FO-C \! AC and OR-C \! AC are identical to that for the

lower bound curve. Since the latter is identical to the 2 x 1 MISO point-to-point system, it

gives a diversity order of two. Hence, the above observation establishes the fact that both

the proposed protocols achieve the optimal diversity order of two for the two-user C'\!.AC

for the required rate region of interest.

4.4 Summary

In this chapter, we proposed flow-theoretic cooperative transmission protocols for the

two-user fading C'\ .AC, where the nodes are only capable of half-duplex communication

and have access to full CSI. We propose two such protocols, viz. a flow-optimized protocol

for the C'\! AC (FO-C \ IAC), and a suboptimal but simpler orthogonal relaying protocol

for the C'\ !AC (OR-C \ IAC). Both the proposed protocols are evaluated in terms of

achievable rate regions and outage performances. Numerical results show that FO-C'\!.AC

yields the largest achievable rate region amongst the different protocols considered here.

Although OR-C'\ !AC is clearly suboptimal in terms of the achievable rate region, its

outage performance is close (to within 1dB for the scenarios considered) to FO-C'\! AC,

and both the proposed protocols achieve the optimal diversity order of two for the

required rate region of interest.













S i s


x3+Y22


S2 y3+x22



T3


(b)

Figure 4-1. Flow-theoretic transmission protocols for the C \! AC: (a) OR-C\ A!.C, (b)
FO-C \! AC.


x+y21


S,

x2

S2


yi+x21


T2










E[Zs s ]=3.0, E[ZS D]=1.5, E[ZS D]=0.2
1 2 1 2


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8


Figure 4-2. Achievable rate regions


.i- ,iiiii., I I i situation.


E[Zs s ]=3.0, E[ZS D]=1.5, E[ZS D]=0.2 with conventional MA slot
1 2 1 2


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
R1


Figure 4-3. Achievable rate regions .-iiiii,. I i,: situation with conventional MA slot
(without common information).













E[Zs s ]=3.0, E[ZS D]=1.0, E[ZS D]=1.0
1 2 1 2

-. ---MAC
--- FO-CMAC
'' OR-CMAC
S- Reference [6]











.




I I I I I i


0.1 0.2 0.3 0.4
R


0.5 0.6 0.7 0.8


Figure 4-4. Achievable rate regions symmetric situation.


E[Zs s ]=3.0, E[ZS D]=1.5, E[ZS D]=0.2, K=1bit/s/Hz
1 2 1 2


0 5 10 15 20
SNR (dB)


O 0.4


100







. 10-1

-Q
g
0


-2


Figure 4-5. Outage performance .,-vmmetric situation.


25 30 35 40












































' 10-1
-Q
.-Q
2
0
CD

-2
0 10


E[Zs s ]=3.0, E[ZS D]=1.0, E[ZS D]=1.0, K=lbit/s/Hz
1 2 1 2


0 5 10 15 20 25 30 35 40
SNR (dB)


Figure 4-6. Outage performance symmetric situation.









CHAPTER 5
INTERFERENCE CHANNELS WITH UNIDIRECTIONAL COOPERATION AND
CAUSALITY CONSTRAINTS

5.1 Introduction

As mentioned in Ch ipter 1, the interference channel with unidirectional coopera-

tion (ICUC) is essentially the simplest realization of an overlay cognitive radio network.

Cognitive radios have a considerable potential in facilitating an efficient use of the li-

censed spectrum that is currently under-utilized [18]. The overlay paradigm for cognitive

radios not only provides an efficient technique for cognitive radio deployments but also

yields newer insights towards the understanding of interference channels and cooperative

behavior in multi-terminal networks, through the different manifestations of cognition, co-

operation and competition levels amongst different users or user groups [63]. In the overlay

form, the simplest of which being the ICUC (also known as cognitive radio channel [6, 7]),

the cognitive (or cooperating) radio utilizes the same spectrum as the primary user-pair

for its own data transmission. Whereas this would generally cause interference to the

primary link, the cognitive source may exploit its knowledge about the primary message

to improve its own transmission rates by preceding its information against the known

interference from the primary transmission and simultaneously alleviate the detrimental

effects of the interference to the primary destination owing to the secondary transmission

by cooperative relaying of the primary message.

Of late there has been a considerable body of work reported in the literature that

have helped improve the achievable rate region for the ICUC proposed in [6]. In [64], the

authors present inner bounds to the capacity region for joint and sequential decoding, and

encoding strategies that include rate-splitting for both primary and secondary messages,

conditional Gel'fand-Pinkser (GP) inning and cooperative relaying. They also present

a general outer bound that is very similar to an outer bound for the broadcast channel,

and a much simpler outer bound for the case wherein the primary destination experiences

strong interference. A slightly different coding scheme has been proposed in [17], wherein









there is no rate-splitting for the primary message and the two parts of the secondary

message are binned independently against the interference from primary transmission.

Very recently, another coding scheme for the discrete memoryless (non-causal) ICUC

has been proposed in [65]. According to this scheme, conditional rate-splitting is applied

to both primary and secondary messages, and the cognitive source uses GP inning and

a two-way inning strategy as in [67], conditioned on the codeword for the common part

of the primary message, to transmit the common part of the secondary message, and

the private parts of the primary and secondary messages respectively. The authors also

consider the case in which the primary message may be available at the cognitive source

in a causal fashion, but the channel model is modified to that for a Z interference channel

(ZIC), wherein the primary destination does not experience any interference due to the

secondary transmission. Another unified coding scheme, very similar to the one in [65],

that yields a rate region that includes the region of [65], has been proposed in [66].

Cognitive radio networks that are more involved than the two-user ICUC have also

been studied in some recent works. These networks combine the ICUC with other multi-

terminal networks like the multiple-access channels [68], broadcast channels [69], or relay

channels [70].

Cognitive multiple access networks, in which a set of cognitive users is provided

with a function of the messages of the set of non-cognitive users and each set has its

corresponding receiver, have been studied in terms of their achievable rate regions in [68].

Achievable rate regions for one-sided interference channels with a cognitive relay, that has

non-causal message- or signal-level information from both sources and a link to only the

destination experiencing interference, have been obtained in [70]. In [69], an achievable

rate region for the case of two interfering broadcast channels, with one source having

non-causal knowledge about the message of the other, has been derived.

Most of the above works consider the non-causal form of ICUC wherein it is assumed

that the primary message is available non-causally at the cognitive source. In reality,









some resources (in time or frequency) need to be expended by the system for the cognitive

source to acquire the primary message. The scenario in which the cognitive sources need

to obtain the information causally from the primary source explicitly models this overhead

and has been considered for the case of the two-user ICUC in [6] and [71]. In [6], the

authors consider half-duplex operation of the secondary source, and propose a two-phase

protocol. The first phase is used to transmit the primary message to the cognitive source

and a part of the primary message to the primary destination via a broadcast channel,

and once the cognitive source successfully decodes the primary message, the operation

proceeds as for the non-causal case. On the other hand, in [71], a full-duplex operation

of the secondary source is assumed, and block Markov SPC along with sliding-window

decoding, and rate-splitting for the two messages are used to obtain an achievable rate

region.

In this work, we consider the two-user ICUC with causality constraint (ICUC-C).

It is assumed that the cognitive source is capable of perfect echo cancellation, thereby

making full-duplex operation of the cognitive source feasible. We present a generalized

coding scheme for the discrete memoryless ICUC-C. The proposed achievable rate region

is based on block Markov SPC with backward decoding [72] for the primary message

transmission, conditional rate-splitting for the primary and secondary messages to

facilitate partial decoding at the non-pairing destinations, GP inning at the cognitive

source, and cooperative relaying of the primary message by the cognitive source. This

rate region is then evaluated for the case of Gaussian channels and numerical results for

different values of network parameters are presented. The results are used to shed light on

the performance of the coding strategies involved in the proposed scheme under different

scenarios.

Next, we relax the assumption that the cognitive source is capable of full-duplex

operation, and instead subject the cognitive source to the half-duplex constraint, i.e. it

cannot receive and transmit information simultaneously over the same band. Towards









this, we present a discrete memoryless channel model for the ICUC with half-duplex and

causality constraints (ICUC-HDC), and propose a generalized coding scheme for this

channel. Similar to the full-duplex case, we employ block Markov SPC with backward

coding, conditional rate-splitting, GP inning, and cooperative relaying by the cognitive

source. However, for the half-duplex case, the cognitive source employs a randomized

listen-transmit schedule [73] to encode and transmit information (via signaling). It is also

proved that the new rate region contains the previously known rate region of [6] for the

ICUC-HDC.

In the following section, we present the discrete memoryless channel model for the

ICUC-C. This is followed by Section 5.3, where we present the random coding scheme and

the corresponding achievable rate region for the ICUC-C. Section 5.4 details the Gaussian

ICUC-C along with numerical examples and a discussion on the role of different coding

techniques under different network scenarios. Following this, the discrete memoryless

ICUC-HDC is introduced in Section 5.5, and in Section 5.6, the proposed random coding

scheme and the new achievable rate region are presented. The Gaussian ICUC-HDC is

presented in Section 5.7, followed by analytical and numerical comparisons between the

new achievable rate region and the one in [6]. Finally, a summary of the contributions in

this chapter is presented in Section 5.8.

5.2 The Channel Model

Consider the communication scenario as in Fig. 5-1, wherein the primary source node

Sp intends to transmit information to its destination node Dp. Apart from the primary

source-destination pair, the wireless network consists of a secondary (or cognitive) source-

destination pair, Sc and Dc, with Sc having its own information to transmit to Dc. The

primary message is causally available at Sc, and the latter may use this knowledge to

assist Sp in the transmission of the primary message to Dp, and also transmit its own

information to Dc.









In n channel uses, the primary source, Sp, has a message wp E {1, 2,... 2"RP}

to transmit to Dp, while the secondary source Sc has a message wc E {1, 2,... 2nRc}

to transmit to its intended destination Dc. Let Xp, Xc, and Vc, Yp, Yc be the in-

put and output alphabets respectively. A rate pair (Rp, Rc) is achievable if there

exist an encoding function for Sp, X' fp(wp), fp : {1,2,-.. ,2nRP } XJ,

and a sequence of encoding functions for Sc, X5 = fj(wc, V ,-) with Xci =

fci(wcV, V1), fci {1, 2, ... 2R} x VS --1 Xc, and corresponding decoding functions
Wp gp(pY), gp yP {1, 2,... 2' } and c = gc (Yf'), gc : Y {1, 2,... 2nRc}

such that the average probability of error Pe = max{P(,) P)} -> 0, where
P2) = 1t 1 Pr [gM (Y) /t I (Wp, Wc) was sent] for M = P, C.
S 2n(Rp+Rc) tw

5.3 Achievable Rates for the ICUC-C

In this section, we present a new achievable rate region for the discrete memoryless

ICUC-C. We start with an overview of the coding scheme. In block b e {1, B}, Sp

splits the message Wp,b into two parts Wpco,b and Wppr,b. It uses superposition coding to

encode these two parts along with the message for the previous block (wpco,b-1, WPpr,b-1).

The latter acts as the resolution 'f. f., in.il. n for Dp and Dc that use backward decoding

to decode the primary message entirely or partially. In contrast to the rate-splitting tech-

nique in [71], wherein the two message parts are encoded independently and superposed,

Sp performs conditional rate-splitting [65] coupled with block Markov SPC. Sc decodes

the primary message for the current block and simultaneously performs a set of encod-

ing steps. In block b, Sc splits wc,b into two parts Wcco,b and WCpr,b, and conditioned on

the codeword (Tpco) for the resolution information for the common part of the primary

message [65], it uses conditional GP inning [64] to encode wcco,b and wcpr,b as Ucco and

Ucpr respectively, against the resolution information for the private part of the primary

message (Tppr) that causes interference at DC but is known at Sc. Finally, it transmits a

combination of the above codewords, along with the resolution information for the primary

message, with the latter part manifesting the cooperative relaying action by Sc. Unlike










the coding scheme in [71], Sc does not use block Markov SPC to encode its own message,

thereby giving rise to a simpler characterization of the rate region.

Dp waits until the last block B and uses backward decoding to jointly decode both

common and private parts of the primary message and the common part of the secondary

message. Similarly, Dc waits until the transmission for block B and then uses backward

decoding to jointly decode the common part of the primary message and both common

and private parts of the secondary message. Note that Dc performs backward decoding

only to decode the common part of the primary message to take advantage of the block

Markov SPC structure used to encode it. Further, the use of backward decoding yields a

much simpler rate region characterization compared to the one in [71]. Table 5-1 lists the

random variables involved in the code construction along with their significance.

Table 5-1. Description of Random Variables in Theorem 5.1
Random Variable Definition
Tpco Resolution information for common part of primary
message (known to Sc)
Tppr Resolution information for private part of primary
message (known to Sc)
Xpco New information for common part of primary message
Xp Transmitted codeword by Sp, generated by super-
posing new information for private part of primary
message on Tpco, Tppr, and Xpco (please refer to Code-
book Generation in the proof of Theorem 5.1
Ucco Common part of secondary message (generated by
conditional Gel'fand-Pinsker inning against Tpp,)
Ucpr Private part of secondary message (generated by con-
ditional Gel'fand-Pinkser inning against Tpp,)
Xc Transmitted codeword by Sc


Theorem 5.1. For the ICUC-C, the rate tuple (Rp, Rc), where Rp = Rpco + Rppr,

Rc Rcco + Rcpr, with non-negative reals Rpco Rppr, Rco pr i flying


Rpp, < min {I (Xp; Vc ITpco Tppr Xpo) ,

I(Tppr, Xp; Yp, Ucco|Tpo, Xpo)} (5-la)









Rpeo + Rpp, < I(Xp; Vc Tpo, Tp,) (5- b)

RpP, + Rco < I (Tppr, Xp, Ucco; YP TpCo, Xpco) (5 -c)

RPCO + Rppr + RCCO < I (Tpco, Tppr, Xp, Ucco; Yp) (5- d)

Rcpr < I(Ucpr; Yc, Ucco Tpco, Xpco) I(Ucpr; Tppr, UCco Tpco) (5 -e)

RcCO + Rcpr < I(Ucco, Ucpr; Yc Tpco, XpCo)

I (Ucco, Ucpr; Tppr TPco) (5-if)

RPCO + RCCO + Rcpr < I (TpCo, Xpco, Ucco, Ucpr; YC)

-I(Ucco, Ucpr; Tppr TPco) (5-1g)

is achievable for some joint distribution that factors as

P(tpco)p(tppr tpco)p(Xpco tpco)P (XpCtpco, tppr, Xpco)

x p(Ucco Pco)P UCpr tPco, UCco)p(xc \tpCo, tppr, Ucco, Ucpr)

xp(vcIxp,xc)P(ypxp,xc)p('/. p,XC), (5-2)

and for which the right-hand sides of (5-la) to (5- g) are non-negative.

Proof. Let A"(X, Y) denote set of jointly c-typical sequences according to the distribution
of random variables X, Y as induced by the same distribution used to generate the
codebooks (see [33]). For convenience, the dependence on the random variables will not be
stated explicitly, and should be clear from the context.
Codebook generation: Split the primary and cognitive users' rates as Rp = Rpco + Rppr
and Rc = RCco + Rcpr respectively. Thus, in block b c {1, ... B}, the primary
message can be represented as wp,b (wpco,b, wppr,b), and the secondary message as

wc,b (wCco,b, wcpr,b), where co and pr stand for the common and private part of
a message respectively. Fix a distribution p(tpotpp, xpp co Cxp UCo,cpr, C) as in
Theorem 5.1.








* Generate 2nRP-o i.i.d. codewords tcoPc(w o), wpo e {1 ... 2nRP~}, according to
i^ p(t Pcoi).
For each codeword tGco(Wpco), generate 2nRpP" conditionally i.i.d. codewords
tPr"(', "'1~.,), "',, E {1, C 2TRPP-}, according to Hi 1P(tPpri tpcoi(WP'o)).

For each codeword tcoP(wco), generate 2nRP,0 conditionally i.i.d. codewords
xco(,,' wpco), wpco e {1, ... 2nRP}, according to J p IP(xpcoi tpcoi(w'po))
For each codeword tuple (tPco(WP'co), pr(,', 'p,, ), x pco(po, wPco)), generate
2T'Rp i.i.d. codewords ,' (w'p, wpco, wpp., wppr), wppr e {, 2RPP }, according
to =H lp (Xpi tcoi(wPc), tpPri (WP, co Ppr) (coi w/p co))-

For each codeword t0co(w co), generate 2n(Rcc+R'cco) i.i.d. codewords ,, ,(w'co, wcco, bcco),
wcco e {1,... 2nRco} and bcco {1, ... 2TRC'o}, according to 7iJ lp(uccoi tpcoi(w'co)).

For each codeword pair (t'co(w co) ". (w'co, Cco, bcco)), generate 2T(Rc+pr+ pr)
i.i.d. codewords ,,, (wpo, wcco, bcco, WCpr, bcpr), wcpr {1,- -...- 2Rcp} and
br {1, ... 2nRc }, according to Y[ ip(UCpri tpcoi(w'pco) uccoi(,, wco, bcco)).
Generate x(, -', ,tr'p, wcco, bcco, ~ pr, bcpr) where xc is a deterministic function of
tPco, tPpr, UCco, UCpr.
Encoding:
At Sp: In block b {2, ... B 1}, Sp transmits i, (Wpco,b-1, WPco,b, WPpr,b-1, Wppr,b).
In the first block, Sp transmits xT"(1, Wpco, 1, wppr,1), while in block B, it transmits
x"'(wpco,B- 1, Wppr,B-, 1). Note that the actual rate for the primary message is BBRp,
but it converges to Rp as the number of blocks B goes to infinity.
At Sc: In block b E {1,.-- B}, to transmit WCco,b, Sc searches for bin index bcco,b
such that
(tco(wpo), u" o(wpo, WCco,b, bCco,b, pr (Pco, WPpr)) c AX, (5-3)

where Wpco and wpp, are Sc's estimates of WPco,b-1 and Wppr,b-1 respectively from the
previous block. Once bCco,b is determined, it searches for a bin index bcpr,b in order to
transmit WCpr,b such that

(tco((wpo)), aUo(wpco, WCco,b, bCco,b) ,~ r b( o, bCco,b, bCco,b, WCpr,b, bCpr,b),











It sets bcco,b = 1 or bcpr,b = 1 if the respective bin index is not found. It can be shown

using arguments similar to those in [64] that the probabilities of the events of Sc not able

to find a unique bc0o,b or bcpr,b satisfying (5-3) and (5-4) can be made arbitrarily small if

the following hold true:


Rfco > I(Ucco; TPpr TPco) + Co,

Rpr > I (Ucpr; Tpr Tpco, Ucco) + Co,

where eo > 0 may be arbitrarily small.

Sc transmits (, (wpco,b-1, WPpr,b- 1 WCco,b, bcco,b, WCpr,b, bcpr,b )*
Decoding:

At Sc: Assume that decoding till block b 1 has been successful. Sc declares that

(WPco,b, Wppr,b) (wpco, Wpr) was transmitted in block b if there exists a unique pair

(wPco, ppr)such that

(tpco WPco,b-1, tpr t WPco,b-1, WPpr,b-1) Xco (wPcob-1, WPco)

X'(WPco,b-1, WPco, WPpr,b-1, Ppr)I tC,b) C *A

Else, an error is declared.

At Dp: The primary destination Dp waits until block B, and then performs backward

decoding. We consider the decoding process using the output in block b C {B 1, 2}.

The decoding for the first and last blocks can be seen as special cases of the above. Thus,

for block b C {B 1,... ,2}, assuming that the decoding for the pair (wPco,b, Wppr,b) has

been successful, Dp searches for a unique pair (wpco, wpr) and some tuple (wcco, bcco) (for

(wcco,b, bcco,b)) such that

(tpco(wpPco), t pr wpco, wpr), Xpco (Pco, Wpco,b) XPo (w o, Wpcob, Wppr, WPpr,b)

uco Pco w cco, cco, y), b) A


Or (Pco, W Ppr )) c A (5-4)








At Dc: The cognitive destination Dc also waits until block B, and then performs
backward decoding to jointly decode the messages intended for it and the common part of
the primary message. For block b {B 1, ... 2}, Dc is assumed to have successfully
decoded WPco,b from block b + 1. With this knowledge, it searches for a unique tuple
(wpco, wco, bco, cpr, pr, bcpr) such that

(tco (Pco), xc ( p 0Pco, WPco,b), U0co wpco, wCco, Cco),
U'pr WPco, co, bcco, Oi. bcpr ) ,b) *YA.

Error Analysis:
Throughout the analysis, we assume, without loss of generality, that all transmitted
messages at the primary and cognitive sources, in any block b {1, ... B}, were ones.
Encoding errors at Sc: An encoding error occurs at Sc under two circumstances. An
error occurs if, in block b, Sc cannot find a bin index bcco,b such that (5-3) is not satisfied
for Wpo = 1, wppr 1, and Wcco,b = 1. This event occurs with probability
L ]
Pr U ( i PTo(1), Uro(1, bCcob), Tpr(1, t))
bCco,b=1

(t Pr [(Trco(t), Uco(t, t, bCco,b),r (tt)) pA
< ( -1 c)2-n[I(Ucco;TPpr TP)+co)]0

where e > 0 can be arbitrarily small, and the last two inequalities are due to the joint
Asymptotic Equipartition Property (AEP) [33] and the fact that (1 x)" < e-'
respectively. Clearly, the above probability can be made arbitrarily small if

Rcco > I (Ucco; TPpr Tpco) + 6o. (5-5)









Another possibility of an encoding error at Sc occurs if, in block b, it cannot find a bin
index bcpr,b such that (5-4) is not satisfied with wpco = 1, wpp, = 1, Wcco,b = 1, bcco,b
and WCpr,b = 1. Proceeding as for the first kind of error event, it can be shown that the
probability of this event can be made arbitrarily small if

RCpr > I(Ucpr; Tppr TpCo, Uco) + CO. (5-6)

Decoding errors at Sc: For the block Markov SPC encoded transmission from Sp,
the cognitive source Sc uses its knowledge about the information in the previous block
to jointly decode wp for the current block. For the superposition encoded common and
private parts of the primary message, it can be shown that the probability of error for this
decoding step can be made arbitrarily low as long as the following hold:

Rppr < I(Xp; Vc | Tpco, Tppr, Xpco),

RpCo + RPpr < I(Xp; Vc ITpco, Tppr).

Decoding errors at Dp: For block b c {B 1,... 2}, let Eijk be the event


(t"Pco()t"'P( (i, wP'CO Pco,b) (i, WPco,b ,Wppr,b (i, k, beco0Y b) A > I

wherein, it is assumed that the previous decoding step, starting with decoding for block B,
has been successful, i.e. (wpco,b, Pp,b) (wpco,b, wppr,b). Note that we need not consider
the probability of the events of Dp decoding wp,b-1 correctly, but not (wcco,b, bcco,b). Then
the probability of error at Dp is

Pe,Dp = Pr [E 1 U(iU)#(11,)Eikj

< Pr [EcJ] + 2nRP Pr [E121] + 2n(Rpr+Rcco+Rcco) Pr [E122

+2n(Rppr+Rppr+Rcco+R'ccO Pr [ E2221 (5-7)








The first term in (5-7) goes to zero with n due to joint AEP. The probabilities of the last
three error events can be shown to be upper bounded as listed in (5-8a) (5-8c).

Pr[E1211 < 2-n[I(Tpprxp;YpUccoTPco,'Xco)+6c] (5-8a)

Pr[E122 < 2- n[I(Tppr,Xp,Ucco;YPITPco,XPco)+I(Ucco;TppTpco)+8c] (5-8b)

Pr[E2221 < 2-n[I(TPco,TPpr,Xp,Ucco;YP)+I(Ucco;TPpr Tpco)+7c] (5 8c)

Thus, the above si..-- -.- that, for n large enough, (wpco, wpr) (Pco,b- 1, WPpr,b-1)
with arbitrarily small probability of error if Rppr is less than the second term inside the
minimum operator in equation (5-la), and (5- lc)-(5- d) are satisfied.
Decoding errors at DC: For block b c {B 1,... 2}, let E(Dc) denote the event

(tco (k), x co(k, 1), u" c, (k, i, bcco), u' (k, i, bcco, j, bcpr), yb) e .

Then, the total probability of error at DC can be upper bounded as

Pe,Dc < Pr [ (Dc)c + 2n(Rcpr+Rcpr) Pr[E2(c)1 + 2'(Rcco+R'cco+Rcpr pr) Pr[E22(Dc)

+ 2n(Rpco+RCco+R'cco+Rcpr+Cpr) Pr [E12c 1 (5-9)

Note that, owing to the coding structure, if k / 1, then all the transmitted codewords
would be independent of the received codewords. As a result, DC needs to correctly
decode wpco even though it may not be interested in this part of the primary message.
Again, by joint AEP, the first term of (5-9) goes to zero with n. The probabilities of the
other three events can be upper bounded as

Pr([E Dc) < 2-n(I(Ucpr;Yc Tpco,Xpco,Ucco)+6c) (5 10)

Pr[E (Dc) < 2-n(I(Ucco,Ucpr;Yc Tpco,Xpco)+6c) (5- 11)

Pr[E(Do)I < 2 -n((TpcoxpcoUcco Ucpr;Yc)+5e) (5-12)









Therefore, for n large enough, (wpco, wcco, bcco, bcpr) = wPco,b-1, WCco,b, bcco,b, WCpr,b, bcpr,b)
with arbitrarily low probability of error if (5-le)-(5- g) are satisfied.

Thus, the constraints on the rates as given in (5-la) (5- g) ensure that the average

probability of error at the two destinations can be driven to zero and thus, they describe

an achievable rate region for the ICUC-C. E

Remark 5.1. The achievable rates region described in Theorem 5.1 can be expressed

explicitly in terms of Rp and Rc using Fourier-Motzkin elimination. Denote the right

sides of (5-la)-(5- g) as I,, 12, .. I7. Then the achievable rate region of Theorem 5.1 can

be written as

Rp < min{J2,4}, (5-13a)

Rc < min{I6, 7}, (5-13b)

Rp + Rc < min{I4 + 5, 1i + I7}, (5-13c)

Rp + 2Rc < I3 + 1 + I7. (5-13d)

Remark 5.2. The achievable rate region described in Theorem 5.1 is convex and hence, no

time-sharing is required to enlarge the rate region. This can be proved using the Markov

chain structure of the code as was used in [74, Lemma 5], with the random variable Tpco in

Theorem 5.1 p1 i'ing a role similar to that of U in [74].

5.4 The Gaussian ICUC-C

We apply the result of Theorem 5.1 to the Gaussian ICUC-C. For Gaussian chan-

nels, with the direct links' channel gains normalized to unity (cf. [7], etc.), we have the

following input-output relationships as shown in Figure 5-2:

Vc = gpcXp + Zs, (5-14)

Yp = Xp + hcpXc + Zp, (5-15)

Yc hpcXp + Xc + ZD, (5-16)









where Zs0, Zp, ZDc ~ AV(O, 1) are i.i.d. random variables denoting the additive noise

at Sc, Dp, and Dc respectively. gpc, hpc, and hcp are positive reals that denote the

channel gains for the links from Sp to Sc, Sp to Dc, and from Sc to Dp respectively.

Also, the primary and cognitive sources are subjected to their respective power constraints

Pp and Pc:

X-ll |ll2 Pp, -XllX|ll2 Pc. (5-17)

Let ap, /3p, 7p, 6p, ac, /tc, 7c be real numbers in the interval [0, 1] such that ap + /3p +

7p + 6p < 1. Also, let I 1 TI for I E {ac, c,37c}. We evaluate the rate region of

Theorem 5.1 for the case of Gaussian channels with the following transmitted signals in

block be {1,... ,B}:


Xp(wp,b- 1, Wp,b) = Tpco + Xpo + Tpr + Xppr (5-18)

acycPc acycPc'
Xc(wp,b- 1, wc,b) = Xco + Xcpr + VTpco \P Tp +TP+p (5-19)
aprp 7p p

where Tp,, ~ N(O, apPp), Xpo ~ N(O, 0,pPp), Tpr ~ AN(0, 7pPp), Xpp ~ AN(0, pPp),

Xcco -~ (0, acicPc), and Xcpr -~ (0, acicPc) are i.i.d. random variables. Xcco and

Xcpr are transmitted to communicate the dirty paper coded messages wcco,b and "', I ,

respectively, with Ucco and Ucpr being the corresponding auxiliary random variables as

in [64].

For the coding scheme in Theorem 5.1, it is necessary that global channel state infor-

mation (CSI) is available at all nodes of the network. Also, the probability distributions of

the different codewords for the two encoders would be required to facilitate efficient rate

selection. Moreover, the different coding strategies used in obtaining the achievable rate

region of Theorem 5.1 involve certain assumptions regarding the knowledge of the code-

book at each node. In particular, cooperative relaying of the primary message by Sc, as

well as DPC at Sc, requires the knowledge of the primary codebook at Sc. On the other

hand, rate-splitting of the primary message requires Dc to know the primary codebook.









Lastly, rate-splitting for the secondary message requires Dp to know the secondary code-

book. Whereas the CSI and rate selection information may be provided through low-rate

control channels between the nodes, providing codebook knowledge at the non-pairing

nodes may be more difficult to achieve in practice. This can be seen to be especially true

for the last case if we consider the fundamental philosophy of cognitive radio networks

that they should be deploy. ,1 such that the already existing primary user-pair(s) should be

as oblivious as possible to the existence of the secondary user-pair(s).

Figs. 5-3 to 5-5 present some numerical results for the different coding strategies

for various choices of available powers and channel gains as given in the figures. In these

figures, "Coding I" refers to rate-splitting for both messages without use of DPC at

Sc (this coding scheme is the same as that proposed in [71]), "Coding II" refers to the

strategy of using DPC at So but without any rate-splitting, "Coding III" stands for the

scheme of using rate-splitting for only the primary message along with DPC at Sc, and

finally, "Coding IV" refers to the coding strategy described in the beginning of this section

(i.e. Theorem 5.1 specialized for Gaussian channels as outlined at the beginning of this

section), and involves rate-splitting for both messages as well as DPC at Sc. Note that

cooperative relaying is used in all of the above coding schemes.

Fig. 5-3 demonstrates that the coding scheme of [71] is strictly sub-optimal when both

destinations experience weak interference whereas using DPC and cooperative ing result in

a larger rate region. In light of the above discussion on codebook knowledge requirements,

it turns out that for scenarios as this, using DPC and cooperative relaying without any

rate-splitting is the best and most feasible coding scheme. The gain from rate-splitting

for the primary message when Dc experiences strong interference and Dp experiences

weak interference is evident from Fig. 5-4. In this case, as the interference at Dp is weak,

rate-splitting for the secondary message does not appear to provide any benefit towards

enlarging the rate region. Finally, Fig. 5-5 illustrates the gains from rate-splitting for

the secondary message coupled with DPC at So for the case of weak interference at Dc









and strong interference at Dp. Moreover, the increase in the difference in the regions

for "Coding I" and "Coding IV", as Pp increases from 1.5 to 6, shows the benefit of

DPC at Sc when the effective interference at Dc increases with Pp. From these results,

it appears that using cooperative relaying with DPC and rate-splitting of the primary

message may be practically more suitable strategies (in terms of codebook knowledge

requirements at the primary user-pair), except when the primary destination experiences

strong interference.

5.5 Discrete Memoryless Channel Model for the ICUC-HDC

Till the previous section we have been assuming that the cognitive source can operate

in full-duplex mode by performing perfect echo cancelation. Here, we remove the full-

duplex assumption, and introduce the discrete memoryless channel model for the ICUC

with half-duplex and causality constraints (ICUC-HDC). The ICUC-HDC is depicted

in Fig. 5-6, wherein the primary source node Sp intends to transmit information to its

destination node Dp. A cognitive (or secondary) source-destination pair, Soc and Dc,

wishes to communicate as well, with Soc having its own information to transmit to Dc.

As in the case of the ICUC-C, the primary message is only causally available at Sc. To

incorporate the half-duplex constraint for the discrete memoryless channel model, we

consider a second input at Sc, S, to indicate the state of Sc listening or transmitting.

With this, the channel transition probability is determined by the state of the

cognitive source as follows:



p(yp,yc,vc xpxcif s 1 (5-20)
p(yp, | p, xc)6e(vc) ifs t,

where e denotes an erasure at Sc, and 6e(vc) 1 if vc = e and 0 otherwise. To

incorporate the fact that So cannot transmit when in the listening state, we restrict the

joint probability distribution of the inputs as p(xp, xc, s) = p(xp s = 1)66(xc)p(s =

1) + p(xp,xc s t= )p(s = t), where Q is the "null" symbol.









In n channel uses, the primary source, Sp, has message wp c {1, 2,... 2RP } to

transmit to Dp, while the secondary source Sc has message wc E {1, 2,... 2nRc}

to transmit to Dc. Let Xp, Xc, S, and Vc, Yp, Yc be the input and output alphabets

respectively. Further, let S = {1, t}. A rate pair (Rp, Rc) is achievable if there exist

an encoding function for Sp, XT = fp(wp), fp {1, 2,... 2R} -} XJ, and a

sequence of encoding functions for Sc, (X, S"f) = f(wc, V ) with (Xci, S) =

fciwc, VS 1), fci {1, 2,... 2Rc} x V1' -- Xc x S, and corresponding decoding
functions wp gp(Y), gp : -> {1,2,--- ,2R 2 } and c = gc(Yc), gc : -Y

{1, 2, .. 2nRc } such that the average probability of error P( max{P(-, P)} }- 0,
where P() = Pr [g(Y1) /M '(wp, wc) was sent] for M = P, C.
(wp wc)
5.6 An Achievable Rate Region for the ICUC-HDC

First, we present a brief description of the coding scheme. In block b e {1,... B}, Sp

splits the message wp,b as wp,b ( wpi,b, wp2,b) where wi,b ( wpico,b, 'PF.,p,,b) for i = 1, 2.

Here, for any block, wpl is the message part that Sc decodes and uses for its cognitive and

cooperative actions, whereas wp2 is the message part that Sp directly transmits to Dp

when Sc is in transmit mode. As before, the subscripts co and pr indicate the common

and private message parts respectively. While the common message parts are decoded by

both destinations, the private message parts are decoded only by the intended destination.

WPIco,b is further divided into two parts Ws,b, that is forwarded by Sc in the next block
using the help of its random listen-transmit schedule [73], and We,b, that is transmitted

explicitly using a standard codebook.

Conditional rate-splitting [65] and superposition coding are used for the above

message splitting step. For block b c {1, B}, Sp transmits WpI,b during the Sc-

listen states, and it superposes wp2,b onto WpI,b-1 (using block Markov SPC) during the

Sc-transmit states, with wpi,b-1 acting as the resolution information for Dp and Dc to

decode wpl entirely or partially. In block b, Sc decodes Wpl,b from the received symbols

during the listen-states. In block b, Sc splits wC,b into two parts w co,b and wCpt,b, and










conditioned on the codeword pair (S, Tpico) for the resolution information for the common

part of wpi,b-1, it uses conditional GP inning [64] to encode wcco,b and wcpr,b as Ucco and

Ucpr respectively, against the resolution information for the private part of wpI,b-1 (TPpr).

It transmits a combination of the above codewords, along with the resolution information,

during the Sc-transmit states.

Both Dp and DC wait until the transmission in block B, and then use backward

decoding [72] to jointly decode both common and private parts of its intended message

and the common message parts) from the interfering transmission. Note that Dc

performs backward decoding only to decode WPlco,b-1 in order to take advantage of

the block Markov SPC structure used to encode it. Table 5-2 lists the random variables

involved in the code construction along with their significance.

Table 5-2. Description of Random Variables in Theorem 5.2
Random Variable Definition
S Listen-transmit state for Sc
Tpico Resolution information for common part of primary
message wpi (known to Sc)
Tplpr Resolution information for private part of primary
message wpi (known to Sc)
XPIco New information for common part of primary message
wpi (decoded by Sc)
Xp1pr New information for private part of primary message
wpi (decoded by Sc)
Xp2co Common part of primary message Wp2 (not decoded
by Sc)
Xp2pr Private part of primary message Wp2 (not decoded by
Sc)
Ucco Common part of secondary message (generated by
conditional Gel'fand-Pinsker inning)
Ucpr Private part of secondary message (generated by con-
ditional Gel'fand-Pinkser inning)
Xp Transmitted codeword by Sp
Xc Transmitted codeword by Sc


Let a = Pr[S = 1], and a = 1 a. Owing to the half-duplex constraint to the channel

model, we restrict the distributions for the codewords used in the codebook construction








as follows:


pAtPlco 1 P) 6(tpilco), (5-21a)

PtpPlpr tplco, S 1) 6(Plpr), (5-21b)

P(x"P2co tPlco, s 1) = (XP2co), (5-21c)

p(XP2pr Xp2co, tPpr, tplco, S 1) 6(XP2pr), (5-21d)

P(uccotplco, s 1) 60(ucco), (5-21e)

p(UCpr lCcotpico, S 1) O 60(ucpr), (5-21f)

p(xpico tplco, s t) 0 (XPlico), (5-21g)

p(XP1pr Xpico, tp1pr, tp1co, S t) 0 6("xplpr). (5-21h)

Theorem 5.2. For the discrete memornl' ICUC-HDC, all rate tuples (Rp, Rc), where
Rp Rp1 + RP2 RPlco + RPpr + RP2co + RP2pr, RPIco Rs + R Re C RCco + Rcr,
with non-negative reals Rs, Re, RPpr, RP2co, P2pr, Cco, Rpr f.'

Rpipr < al (Xpipr; Vc XpIco, S 1) (5-22a)
Rpi < aI(Xpipr; Vc IS 1) (5-22b)

RP2pr < aI (Xp2pr; Yp, Ucco Xp2co, TpIpr, TpIco, S t) (5-22c)
RP2 < aI (Xp2pr; YP, UccoI TPpr, Tlco, S t ) (5-22d)

P2pr + RCco aI (XP2pr, UCco; Yp Xp2co, Tp1pr, Tplco, S = t) (5-22e)
RP2 + RCco < a I (XP2pr, Ucco; YP Tppr, T lco, S = t) (5-22f)

RPlpr + RP2pr < OaI (Xpipr; Yp|XpIco, S = 1)
+aI (Tplpr, Xp2pr; Yp, UCcoXp2co, Tpco, S =t) (5-22g)

RPlpr + RP2 < aI (Xplpr; Yp XpIco, S = 1)
+aI (Tppr, Xp2pr; Yp, Ucco TPlco, S = t) (5-22h)

RPIpr + RP2pr + RCco < aI (Xplpr; Yp XpIco, S = 1)









+aI (Tp1pr, Xp2pr, Ucco; Yp Xp2co, TPlco, S


Rplpr + Rp2 + RCco < aI (Xpipr; Yp IXPco, S 1)


+aI (Tplpr, Xp2pr, Ucco; Yp Tplco, S


t) (5-22j)


+aI (Tpico, TPlpr, Xp2pr, Ucco; Y IS


t) (5-22k)


+aI (Tpico, TP.pr, Xp2pr, Ucco; Y IS


t) (5-221)


Rc < a [I (Ucco, Ucpr; Yc XP2co, T


RP2co + Rcpr <- a [I (Xp2co, Ucpr';


I (Ucpr; TPlpr, Ucco Tplco, S

Plco, S t)

I (Ucco, Ucpr; Tplpr Tplco, S

KC, Ucco Tplco, S = t)

I (Ucpr; TPlpr, Ucco Tplco, S


t)] (5-22m)


(5-22n)



(5-22o)


Rp2co + Rc < a [I (Xp2co, Ucco, Ucpr; Yc TPlco,


Re + Rp2co + Rc < al (X

+a [I(T


RPco + RP2co + RC < I (

+a [I (T


S t) I (Ucco, Ucpr; Tp pr Tplco, S

Cico; Y IS = 1)

Pico, Xp2co, Ucco, Ycpr; YC\S = t)

-I (Ucco, Ucpr; Tplpr Tplco, S

S; Yc) +aI (Xpico; YCIS =1)

Pico, Xp2co, Ucco Ucpr; YC S = t)

-I (Ucco, Ucpr; Tplpr Tplco, S


t)] (5-22p)


(5-22q)





(5-22r)


are achievable for some joint distribution that factors as

p(s)p(tplcol s)p(tplpr iPlco, 8p )p(Xplco Plcop 8)P( Ppr X Plco, tplpr, tPlco, 8)


Re + Rplpr + Rp2 + RCco < aI (Xplpr; Yp S 1)


Rp + Rcco < I (S; Yp) + al (Xp,; Yp S =1)


Rcpr < a [I (Ucpr; Yc, Ucco Xp2co, TPlco, S t)


- t) (5-22i)








xp(xP2co tpl, S)p (xp2pr XpI2c, tPlpr, tpico s) P X XP2pr, XP2co, Xplpr,

Xplco, tplpr, tpico, S) p (UCco tPlc ) p ucpr UCco, tpico, s)

Xp (xc Ucpr, UCco, tppr ,tplco, S) p (vc Xp, Xc, s)

xp(yp xp,xc,s) p(,/. I p,xc,s),

and -.,/.-. (5-21a)-(5-21h), and for which the right-hand sides of (5-22a)-(5-22r) are
non-negative.

Proof. Let A"(X, Y) denote set of jointly c-typical sequences according to the distribution
of random variables X, Y as induced by the same distribution used to generate the
codebooks. As in the proof for Theorem 5.1, the dependence on the random variables will
not be stated explicitly, and should be clear from the context. To avoid repetition, the
error analysis for the random coding scheme is not presented here, and can be derived in a
manner similar to the analysis in the proof of Theorem 5.1.
Codebook generation: Split the primary and cognitive users' rates as Rp = R, +
Re + RPipr + RP2co + RP2pr, and Rc = Rcco + Rcpr respectively. Fix a distribution
p (s, tpco, tpIpr, xPlco, PIpr, xp2co, XP2pr, xp, UCco UCpr, XC) as in Theorem 5.2.

* Generate 2"R, i.i.d. codewords s"(w') c S", w' E {I1,... 2"R,}, according to
Hi 1 p(Si).
For each codeword s"(w'), generate 2nR- conditionally i.i.d. codewords tlco (w, w'),
w e {1,... 2nR,}, according to H? 1p(tpicoi si).
For each codeword pair (sn(w'), tIpco (w' w')), generate 2nRlpip conditionally
i.i.d. codewords tp,(w's, w'p, ), Ww'p, {1, 1 E I 2nR" i'}, according to
HiP (tpipri Si, tplcoi).
For each codeword pair (s"(w), tlco(w'> we)), generate 2nRplc- conditionally
i.i.d. codewords x7co (w8, we, wpico), wplco {I1,'" 2nR c-}, according to
1? PXPlicoi sI, tpicoi)
For each codeword tuple (s"(w,), tGIco(w, We), x"P(co(W, We, wpco), tlp,(w We, pr)),
generate 2nRPip" conditionally i.i.d. codewords x'lpr (w', We, Wplco, Wpl, wppr
wipr e { 1, -- 2nRP-r }, according to YH Ip(xpIprisi, tpicoi, picoi, tPpri).








* For each codeword pair (s"(w'), tpo(,w'> w)), generate 2nRP2, conditionally
i.i.d. codewords x2c (w'w, ,wp2co), wpco C {1, 2 E 2n'P2o}, according to
I(X P2coi Si, tp Ioi).
For each codeword tuple (s~(w),t w,x2~co(w',, w e ,P),t ,(W we, wP1r)),
generate 2nRP2p, conditionally i.i.d. codewords x"2pr (s, We, p2co, W pr, I'i )
,-,, _, {1,... ,2"P2p-}, according to I p(xp2pri Si,tPlcoi, 2coitplpri).

For each codeword pair (s"(w'),tpIco(w> ,w2)), generate 2n(Rcco+Rcco) i.i.d. codewords
', ,(w', wcco, bcco), wcco C {1,... 2nRcco} and bcco c {1,... 2ncco}, according
to HT1 P' UCcoi i, tpilcoi)

For each codeword tuple (s"(w>), t Ico(wl, w'), uT'co (w', wcco, bcco)), gen-
erate 2n(Rc p+R cp) i.i.d. codewords u w'r (w, w, wcco, bcco, wcpr, bcpr), wcpr c
{ 1,... 2Rcp,} and bcpr e {1,... 2nRcp~}, according to ]iJ1 P(UCpri Si, tplcoi, uccoi).

Generate ,' (w', wse, W'lpr, wPlco, wp pr, WP2co, ', _- ) where xp is a deterministic
function of s, tPl tpIpr, XPlco, XpIpr, XP2co, Xp2pr.

Generate x'(w', w w'p1p, wcco, bcco, wcpr, bcpr) where xc is a deterministic function
of s, tPlco, tplpr, UCco, UCpr such that Xc if s 1.
Encoding:
At Sp: Sp transmits x" (ws,b-1, We,b-1, Plpr,b- wPlco,b, wPlpr,b, WP2co,b, b' -; ,b) in
block b {2, ... B 1}. In the first block, there is no resolution information to transmit,
and Sp transmits xT (1,, 1, co1, Wppr,i, Wp2co,1, t't -; ,1), while in block B, it transmits
x"7 (ws,B-1, We,B_-1, Plpr,B-l 1,1, lWP2co,B, "'/ ,B). Note that the actual rate for the
primary message is B(R + Re + Rppr) + Rp2co + 2pr, but it converges to Rp as the
number of blocks B goes to infinity.
At Sc: In block b E {1,.-- B}, to transmit WCco,b, Sc searches for bin index bCco,b
such that

(s"(ws), tlco(t'. We), ". ,( "'. '. WCco,b, bCco,b),t1pr(,. P W pr)) C A, (5-23)

where i'. we and wplpr are Sc's estimates of Ws,b-1, We,b-1 and WP1pr,b-1 respectively from
the previous block. Once bCco,b is determined, it searches for a bin index bcpr,b in order to









transmit Wcpr,b such that


((Ws), 'Plco'. We), uco(^'. I' WCco,b, bcco,b) ,upr '. I WCco,b bco,b, WCpr,b bpr,b) ,

P1pr (- Wpipr)) A,. (5-24)

It sets bCco,b = 1 or bcpr,b 1 if the respective bin index is not found. It can be shown
using arguments similar to those in [64] that the probabilities of the events of Sc not able

to find a unique bcco,b or bcpr,b satisfying (5-23) and (5-24) can be made arbitrarily small

if the following hold true:

RCcco > aI(Uco'; TP1pr Tplco, S t) + Co,

R!Cpr > aI(Ucpr; TPlpr Uco, TPlco, S = t) + Co,

where eo > 0 may be arbitrarily small.
Sc transmits x? (ws,b-1, We,b-1, WPIpr,b-1, WCco,b, bcco,b, WCpr,b, bCpr,b).

Decoding:
At Sc: Assume that decoding till block b 1 has been successful. Then, in

block b, Sc knows Wplco,b-1 (Ws,b-1, We,b-1) and Wplpr,b-1. It declares that the pair

(wplco,b, wp1pr,b) = wpico, wpipr) was transmitted in block b if there exists a unique pair
(wpico, wpipr) such that

(S' (Ws,b-1), lpco (Ws,b-1, We,b-1), tlpr (Ws,b-1, We,b-1,

Wplpr,b-1) "lco (Wes,b-l, We,b-l, Wplco) "lpr (Ws,b-l,

We,b- 1, Wplco, Wplpr,b- lp, ~1pr) b) A

Else, an error is declared. It can be shown that the probability of error for this decoding

step can be made arbitrarily low if (5-22a) and (5-22b) are satisfied.
At Dp: The primary destination Dp waits until block B, and then performs backward

decoding. We consider the decoding process using the output in block b C {B 1, ... 2}.

The decoding for the first and last blocks can be seen as special cases of the above. Thus,









for block b C {B 1,... 2}, assuming that the decoding for the pair (wpIco,b, wP1pr,b) has
been successful from block b+1, Dp searches for a unique tuple (,,. i,, wpipt, "/. "' .- )
and some tuple (wcco, bcco) such that


(S"(w ), tjit'.O. We),tp,('. i', wpip,) xp(r) c i

WPlco,b) Ppr (," 'I. WPlco,b, Wppr, WP1pr,b) xco ("'.
p2co) ,x 2pr ( '. It' P2co, Wplpr, i' ) co (''.

It'. Cco, co ) c A

The error analysis for this decoding step shows that, for n large enough,

(t'. i'. Wplpr, wP2co, "' ) (= Ws,b- 1, We,b-1, Wplpr,b-1, Wp2co,b, "' .- ,b)

with arbitrarily small probability of error if (5-22c)-(5-221) are satisfied.
At DC: The cognitive destination Dc also waits until block B, and then performs
backward decoding to jointly decode the messages intended for it and the common part of
the primary message. For block b e {B 1, ... 2}, Dc is assumed to have successfully
decoded Wplco,b from block b + 1. With this knowledge, it searches for a unique tuple

(" wcco, bcco, cpr, bcpr) and some it', such that

s( ws), th '. We), x 1p ( '. ', w' wpico,b ), 2co co Cco, b co),

'*. wcco, Cco, Cpr,bcpr), 1y c Ab .

Again, using the properties of joint typicality, it can be established that, for n large
enough, (n. co, bcc, ccpr, bcpr) (Ws,b- ,We,b- ,WCco,b, b, WCpr,b, bCpr,b) with an
arbitrarily low probability of error if (5-22m)-(5-22r) are satisfied.
Thus, the constraints on the rates as given in (5-22a) (5-22r) ensure that the
average probability of error at the two destinations can be driven to zero and thus, they
describe an achievable rate region for the ICUC-HDC. E









Remark 5.3. According to the above coding scheme, a part of the primary message (wp2)

is not decoded by Sc. This is different from the non-causal case. As Sc cannot receive

while it transmits, Sp may improve its rates by transmitting "fresh" information directly

to the destination during Sc-transmit states, thereby increasing the achievable rate region.

Compared to the situation wherein Sc is capable of full-duplex operation, transmitting a

part of the message directly to the destination provides potential gains even when the Sp

to So channel is much better than the direct link to Dp.

Remark 5.4. Note that the maximum increase in the achievable rates that may be realized

by using a random listen-transmit schedule for So is ibit [73].

Remark 5.5. The achievable rate region described in Theorem 5.2 is also convex (cf.

Remark 5.2) and consequently, time-sharing is not required to enlarge the rate region.

Here, the random variable S in Theorem 5.2 p1 i the role similar to that of U in [74,

Lemma 5].

Remark 5.6. For the Gaussian channel model with a fixed listen-transmit schedule, the

coding scheme of Theorem 5.2 yields the same rate region as with a time-division strategy

with the use of Gaussian parallel channels [75], instead of a block Markov structure, for

the decoding of wpp = (wpico, wpipr) at Dp and wpico at Dc. According to this strategy,

Sp transmits wpi during the first time-slot while So is in listening mode. In the second

time-slot, both Sp and So encode and transmit wpi as a (non-causal) ICUC, and Sp

also superposes wp2 on top of wpi (the latter acting as the resolution information for the

destinations). Both destinations decode only at the end of the second time-slot and exploit

the parallel Gaussian channel structure to decode wpi entirely or partially.

5.7 The Gaussian ICUC-HDC

As in Section 5.4, for the Gaussian ICUC-HDC, the direct links for each user-pair is

normalized to unity, gpc is the channel gain for the Sp -- Sc link, hpc is that for the

Sp -- Dc link, hcp is that for the Sc -- Dp link, and Sc, Dp, and Dc are assumed to

experience i.i.d. additive white Gaussian noise (AWGN) of unit-variance. As mentioned in









Remark 5.4, the increase in the achievable rate region using a randomized listen-transmit

schedule over that with a fixed schedule is upper bounded by one bit. Moreover, it should

be noted that, for a randomized listen-transmit schedule, the optimal distribution for

the random variable Xc may not be Gaussian [73]. For the above model, we consider a

fixed listen-transmit schedule and have the following input-output relationships for the

ICUC-HDC:


Vc, gpcXp,i + Zsc, (5-25)

YP,I = Xpl + Zp, (5-26)

Yc, = hpcXp,+l ZD, (5-27)

YP, t Xp,t + hcpXc + Zp, (5-28)

Yc,t = hpcXp,t + Xc + ZD, (5-29)


where Zsc, Zp, ZDc A/'(0, 1) are i.i.d. random variables corresponding to the additive

noise at Sc, Dp, and Dc respectively. In the above, Xp,i and Xp,t are the transmitted

signals during the listen and transmit states respectively. Similar notation is used to

describe the received signals at the concerned nodes as well. Finally, it is assumed that

the primary and cognitive sources are subject to the power constraints of (5-17) for each

state within any communication block. This may be interpreted as the scenario in which

both sources are constrained by their respective (listen/transmit) mode power constraints,

instead of an average power constraint over a block. It may be noted that average power

constraints, of a similar flavor as in [73], may be included along with the mode power

constraints in our system model, but is avoided here for the ease of presentation and

exposition of the main aspects of the coding scheme. To summarize, the power constraints

in this section may be expressed as:

S|X 112 < p, Xp,t 112 < Pp, IX | 112 < Pc, (5 30)
17 1Ut 1t









where, for any choice of positive nt and wn, ni + nt = n is the total number of channel uses

in a communication block.

Next, we provide an analytical proof for the inclusion of the achievable rate region for

the Gaussian ICUC-HDC presented in [6]. We show that an outer bound (not necessarily

achievable) to the rate region presented in [6] is contained in a subspace of the achievable

rate region of Theorem 5.2. We describe the specialization of the coding scheme of

Theorem 5.2 to the Gaussian case in further detail in Subsection 5.7.2, along with some

numerical examples.

5.7.1 Inclusion Of Causal Achievable Region of [6]

In [6], an achievable rate region for the Gaussian ICUC-HDC was presented. The

authors proposed four protocols and the overall achievable rate region (Ro) is given by the

convex hull of the four rate regions [6, Theorem 5]. In this section, we show that the rate

region of Theorem 5.2, R, contains Ro.

For the (non-causal) ICUC, the containment of the region of [6, Corollary 2], RDMT,

in the region RD of [76, Theorem 1] is clear. It is shown in [66] that RRTD [66, Theorem

1] contains RD. More specifically, [66] shows that RD C- RSt C RRTD C- RRTD, where

RSIt is obtained from RD by removing certain rate constraints, and R5-TD is obtained

from RRTD by restricting the input distribution to match that for RD. The coding scheme

of Theorem 5.2 may be specialized to yield a rate region for the ICUC. Towards this, we

set S = t w.p. 1, Xp2co = XP2pr = 4, and assume that a genie provides Sc with wp.

This gives us an achievable rate region RNC for the ICUC. Moreover, by restricting the

input distribution to independent rate-splitting and independent inning of the secondary

messages (as in [6, 76]) instead of conditional rate-splitting and conditional inning at Sc,

it can be shown using an appropriate mapping of the codebook random variables (omitted

due to lack of space), that the resulting region RWc is identical to RYRTD, and hence,

RDMT C_ RD C RNC'"









Next, we show that the rate regions obtained via each of the protocols proposed in [6]
are contained in R. Note that for all these protocols, wp = wpi, wp = (wpo, wppr), with
rates Rp = Rpco + Rppr, etc. To compare the two rate regions, we start with an alternate
description for the achievable rate region for Protocol 1 ([6, Lemma 3]). According to
Protocol 1, for any choice of a, the rate pair (Rp, Rc) is achievable if

Rp < log(l +gpcT1Pp) +log 1 + IPP (5-31)
2 1 +TIpP
(Rp,RC) E RDMT, RC = aRC, RPco = RPco, (5-32)

Rppr < log 1+ + aR (5-33)

where r cE [0, 1] is the power fraction allocated for transmitting a part (same as a in [6]) of
Wppr.
Consider the region corresponding to the fixed listen-transmit schedule and using par-
allel Gaussian channels as in Remark 5.6. For the first time-slot, set the input distribution
at Sp as p(xpicols l= )p(xpiprjxpico, s = 1). For the equivalent ICUC (during the second
time-slot), set Xp2co = XP2pr -= and restrict the input distribution to correspond to
independent rate-splitting and inning as in [66, (26)] to match the distribution corre-
sponding to RD. Let the overall rate region thereby obtained be R7. Clearly, R' C KR.
Using the result for parallel Gaussian channels, it can be shown that for any choice of a,
the rate pair (Rp, Rc) is achievable if

Rp < a log (1 + gpcPp) (Rp, R) E R (5-34)
2
Rc aRk Rco mmin -{log (l+ PP ,

2 1 + hpcTp~ p

Rppr < a log (1 + qrpPp) + aRp, (5-36)

where rIp E [0, 1] is the power fraction allocated for transmitting wpco in the first time-
slot. Note that, given an TI value, rIp may be chosen such that 1 < rIp < 1. Then,









comparing (5-31)-(5-33) to (5-34)-(5-36) establishes that the region corresponding to

Protocol 1 is contained in R7V.

The inclusion of the rate region corresponding to Protocol 2 can be easily proved by

considering the same coding structure and input distribution as used to obtain R", with

one further restriction the input distribution at Sp for the first time-slot is given by

p(xplcos = 1)p(xpip, s = 1). This yields an achievable rate region R7' (C 7), that has

exactly the same bounds as that for Protocol 2, except that the achievable rate region for

the NC-CRC (during the second time-slot) is 7'NC 7-DMT, thereby proving the above

inclusion.

The rate region for Protocol 3 can be obtained by setting S = t w.p. 1, Xppco =

XPIpr = TPIco = TPpr = in Theorem 5.2. Finally, the rate pair corresponding to

Protocol 4 may be obtained by using a fixed listen-transmit schedule, and by setting

Tpico = XIco = Xp2co XP2pr Ucco Ucpr = As the four rate regions of [6]
are contained in 7?, the convex hull of these regions (7Ro) is also contained in 7R (cf.

Remark 5.5).

5.7.2 Numerical Results

For a fixed listen-transmit schedule and considering parallel Gaussian channels (as in

Remark 5.6), the transmitted signals in any communication block, corresponding to the

coding scheme of Theorem 5.2, can be expressed as:


Xp,i(wp,) = XPI + XP+p,r (5-37)

Xp,t(wpl, Wp2) Tp1co + XP2o + TP1pr + Xppr, (5-38)

Xc(Wpi, Wc) = Xcco ~ XCpr ++ p TpCo + a TPc1pr (5-39)
apypPp apppPp

where Xpico -~ V(0, ]pPp), Xp'p, A~ (O, ]pPp), TpIco ~ /V(0 apypPp), Xp2co

f(O,ap7ypPp), T'p,- ~ Af(0,apPpPp), X'2p ~ A(0,apP3pPp), Xcco (O0,acf3cPc),
and Xcp, ~ A/(0, accPc) are i.i.d. random variables. In the above, Tp, ap, /3, 7p, ac,









/c, 7c are real numbers in the interval [0, 1]. Sc uses DPC to encode Xcco and Xcpr as

Ucco = Xcco + AcoTp1pr, (5-40)

Ucpr = Xcpr + prTPlpr, (5-41)


with Aco and Apr being non-negative real numbers that denote the correlation between the

known interference Tmpr and the auxiliary random variables Ucco and Ucpr respectively,

conditioned on Tpico. Note that according to the notation of Theorem 5.2, Xp,i = Xppr,

and Xp, = Xp2pr.

In the following, we present some numerical examples to compare the achievable rate

region corresponding to the transmission scheme proposed in [6] to that for Theorem 5.2,

with a fixed listen-transmit schedule and specialized for Gaussian channels. In these

examples, the link between the two sources is assumed to be better than the direct link,

and we compare the Han-Kc. li, i -,i rate region for the interference channel (without

any active cooperation between the user-pairs), the rate region of [6], and that for the

proposed coding scheme in this work. In Fig. 5-7, we consider the scenario when both

interfering links are weaker than the direct links, while in Fig. 5-8, the interfering link

from Sp to Dc is strong and that from Sc to Dp is weak.

Comparing the two figures shows that the improvement in the quality of the inter-

fering link from Sp to Dc may significantly increase the overall rate region for the two

user-pairs. Also, the manner in which the rate region of [6] is enlarged in both examples

~-,p--. -1 that the efficiency of the overall cooperative relaying scheme is the primary

contributor to the enlargement of the rate region. The advantage of the coding scheme

adapted from Theorem 5.2, and described in equations (5-37) through (5-41), lies in the

effective utilization of the direct link for the primary user-pair via the transmission of the

codewords corresponding to the message parts wP2 ('t i'I -_, ). Thus, not having the

entire primary message being decoded and transmitted through the cognitive source may









considerably increase the achievable rate region, especially in the direction of the primary

users' rate, Rp (cf. Remark 5.3).

5.8 Summary

In this chapter, a new achievable rate region for the discrete memoryless interference

channel with unidirectional cooperation (ICUC), wherein the primary message may only

be causally available at the cognitive source, is derived. The coding scheme, specialized

for Gaussian channels is also presented and is used to numerically evaluate different

coding strategies that are used as building blocks for the proposed coding scheme.

These results also demonstrate that the proposed coding scheme significantly enlarges

the previously known rate region for various network scenarios. A discrete memoryless

channel model for the ICUC-HDC was also presented in this chapter. A random coding

scheme, employing block Markov SPC, conditional rate-splitting of primary and secondary

messages, conditional inning, and a randomized listen-transmit schedule for the cognitive

source, was used to derive a new achievable rate region for this channel. For Gaussian

channels, the containment of the previously known rate region [6] in the new rate region

was analytically proved, and numerical examples were presented to supplement the

analytical comparison.






















Sp


Figure 5-1. The discrete memoryless ICUC with causality constraint.


Figure 5-2. The Gaussian ICUC-C.




















Weak Interference: c=1 0, h c=.55, hcp=0.55


....... Coding
. Coding
- Coding
- Coding
....... Coding
. Coding
- Coding
- Coding


I, Pp=1.5, Pc=6
II, Pp=1.5, PC=6
III, Pp=1.5, Pc=6
IV, Pp=1.5, Pc=6
I, Pp=6, Pc=1.5
II, Pp=6, Pc=1.5
III, Pp=6, Pc=1.5
IV, Pp=6, Pc=1.5


Single User Rate (Rp),
Pp=6, Pc=1.5


0.5
Single User Rate (Rp),
Pp=1.5, Pc=6


I


II I I I I


0 0.2


0.8 1
Rp (bits)


1.2 1.4 1.6 1.8


Figure 5-3. Achievable Rates for the Gaussian ICUC-C: Weak interference for both
cross-links.


I
I

fI ,




















Strong PC, Weak CP Interference: c=10, h c=1.5, hcp=0.55


I I


I II


!' Single User RatE
4; PP-1.5, Pc=f





I


..,Coding 1, PP=1.5, P =6
.....Coding I, Pp=1.5, PC=6
- Coding III, Pp=1.5, PC=6 I Ok
- Coding IV, Pp=1.5, Pc=6
....... Coding I, Pp=6, P =6
..... Coding II, Pp=6, P =6 :
- Coding III, Pp=6, PC=6
- Coding IV, Pp=6, Pc=6


0 0.2 0.4 0.6 0.8


Figure 5-4.


1
Rp (bits)


Single User Rate (Rp
Pp=6, Pc=6

/


1.2 1.4 1.6 1.8


Achievable Rates for the Gaussian ICUC-C: Strong interference from Sp to
Dc and weak interference from Sc to Dp.



















Weak PC, Strong CP Interference: c=10, h c=0.55, hcp=1.5


0.5 1 1.5 2 2.5
Rp (bits)


Figure 5-5. Achievable Rates for the Gaussian ICUC-C: Weak interference from Sp to Dc
and strong interference from Sc to Dp.













Sc


Sp


Figure 5-6. The discrete memoryless ICUC-HDC.


Achievable Rate Regions:


1pc=0.55, hcp=0.55, Pp=6, Pc=6
....... Interference Channel: HK regii
- Reference [6]
- This work


0.8 1
Rp (bits)


Figure 5-7. Achievable
cross-links.


Rates for the Gaussian ICUC-HDC: Weak interference for both




















Achievable Rate Regions: cgc=10, h C=1.5, hcP=0.55, Pp=6, Pc=6


' I Interference Channel: HK region
- Reference [6]
- This work












A \

\\ \


0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
Rp (bits)


Figure 5-8.


Achievable Rates for the Gaussian ICUC-HDC:
Dc and weak interference from Sc to Dp.


Strong interference from Sp to


0.5








0
0









CHAPTER 6
CONCLUSIONS AND FUTURE WORK

6.1 Conclusions

In this work, we have studied different levels of cooperation manifested in a variety

of multi-user communication system from flow-theoretic and information-theoretic

perspectives. For the single-source-single-destination wireless cluster with dedicated rel -i,

we proposed cooperative transmission protocols using a flow-theoretic approach. This

included the FO protocol and the suboptimal, but much simpler, GLS protocol. Both the

protocols are shown to achieve the optimal diversity-multiplexing tradeoff, and the GLS

protocol is shown to be a very good candidate for use in systems with low complexity

requirements. Simulation results for different cluster sizes, and uniform and non-uniform

average power gains indicate that the proposed protocols provide large coding gains by

efficiently utilizing the CSI available at all nodes, and that they perform much better than

other path selection methods previously proposed in the literature, especially in high data

rate requirement situations.

We next considered the two-user fading MAC as an example of a multi-source system.

We proposed a game-theoretic formulation involving bargaining and maximin games

to model the resource allocation problem and develop a characterization of cooperative

behavior for this system under uncertainty regarding the accuracy of the CSIT. To

improve the robustness of the system, we proposed that the conventional bargaining

problem be relaxed so that the users, instead of being bound to execute the strategy pair

- .-.- -. ,.1 by the solution to the conventional bargaining problem, may independently

choose their transmission strategy from their respective set of strategies defined by

the maximum deviation parameters about the nominal strategy pair. This reduces the

dependence of the system performance on the solution to the bargaining problem with the

(possibly inaccurate) available CSIT. From the development of this formulation, it can

be seen that even in the conventional two-user MAC, there exists a certain level of user









cooperation. Using numerical examples, we demonstrated the effects of uncertainty on the

achievable average rates and the improvement in the system robustness provided by the

proposed design.

The C \!.AC, that involves a higher level of cooperation between the users in the form

of active forwarding of each other's information, was studied in C!i Ipter 4. Again, using

the flow-theoretic approach, we developed two cooperative transmission protocols, based

on DF relaying, for cooperative transmission in the C'\! AC. We proposed the OR-C \! AC

that decomposes the C'\ !AC into two orthogonal relay channels, and the FO-C'\! AC that

decomposes the C'\! AC into two broadcast channels and one MAC. Moreover, with the

assumption of the availability of phase synchronization, we proposed the modification of

the MA slots of C!i Ipter 2 to MA with common information for further performance gain.

Simulation results for different scenarios indicate the potential performance improvements

over previously proposed transmission strategies, in terms of average rates and outage

probabilities.

Finally, we addressed the problem of communicating through the ICUC with a

causality constraint. This helps us avoid the somewhat unrealistic assumption of the

cognitive source having non-causal knowledge of the primary message that is considered

in most related works in the literature. We derived a new achievable rate region for the

discrete memoryless version of this channel, with an assumption of full-duplex operation

at the cognitive source. We specialized the coding scheme for Gaussian channels and

used it to numerically evaluate different coding strategies that are used as building

blocks for the proposed coding scheme. These results also demonstrate that the proposed

coding scheme significantly enlarges the previously known rate region for various network

scenarios. Following this, we removed the assumption of full-duplex capability at the

cognitive source, and presented a discrete memoryless channel model for the ICUC-HDC.

We developed a random coding scheme, employing block Markov SPC, conditional rate-

splitting of primary and secondary messages, conditional inning, and a randomized









listen-transmit schedule for the cognitive source, to derive a new achievable rate region for

this channel. For Gaussian channels, we proved the containment of the previously known

rate region [6] in the new rate region for the Gaussian ICUC-HDC, and demonstrated this

with numerical simulation results.

6.2 Future Directions

The flow-theoretic approach introduced in C'!i pter 2 could also be applied to multi-

user systems that involve more complex user cooperation than the ones considered in this

dissertation. For instance, the flow-theoretic approach would be suitable for the problem

of information transmission in a cooperative relay broadcast channel (RBC) [5], wherein

one source node broadcasts information to two receiver nodes, who now actively cooperate

(fully or partially) with each other in decoding their respective messages. Although

achievable rate regions employing block Markov coding along with decode-and-forward and

estimate-and-forward [2, Theorem 6] techniques have been proposed in the literature, the

appeal of the flow-theoretic approach lies in its simplicity. The flow-theoretic approach

essentially breaks down the original channel into much simpler channels for which the

capacity regions are known and practical coding schemes that perform close to the

random coding scheme have been extensively investigated in the literature. For the RBC,

a similar time-slotting approach of C'! Ipter 4 would now involve one BC and two MA

with common information time-slots, and it would be interesting to investigate as to how

the performance of the flow-optimized solution compares to the more complicated block

Markov methods in different channel conditions and SNR regimes.

In C'!i pter 5, we presented new achievable rate regions for the ICUC-C and ICUC-

HDC for both discrete memoryless and Gaussian channels. Although we have shown

the inclusion of previously proposed rate regions both analytically as well as through

numerical simulations, we still do not know how close we are to the capacity regions

for these channels. In this regard, new outer bounds for these channels that are tighter

than the MIMO broadcast channel capacity region [77] would be necessary. For Gaussian









channels, one way to approach this problem would be to consider the deterministic version

for the ICUC-C (without any randomness in the channels) and model the relationship

between the deterministic and their Gaussian counterparts as used in [16].

Yet another research direction that may be pursued in regard to the ICUC-C is

the study of the role of transmitter side information at both the primary and cognitive

sources. In Section 5.4, we demonstrated the interp-lv, between the different extent

of codebook knowledge at the different nodes and the effect of the different coding

building blocks like DPC, rate-splitting, and cooperative relaying for various channel and

transmit power conditions. This may be considered as a special case of the study of the

relationship between a general abstraction of side information at the nodes, the set of

coding techniques that may be feasible, and the resulting achievable rate regions. One

possible way to model this may be to consider the channels having different states with

different levels of information about these states available at the two sources. Related

to this, one may also ask the question whether a cognitive source with only signal-level

cognition, instead of message-level cognition can help in enlarging the Han-Ko-,-vi -hi

region for the traditional two-user interference channel. In other words, the problem would

be to determine the coding strategies that may be used by the cognitive source when

the level of "cognition" regarding the primary message is at the signal-level instead of

the message-level, and if, the resulting achievable rates could improve upon the usual

interference avoidance or interference control methods like the interweave or underlay

modes [78] of cognitive radio operation.









REFERENCES


[1] V. Tarokh, N. Seshadri, and A. R. Calderbank, "Space-time codes for high data rate
wireless communication: performance criterion and code construction," IEEE Trans.
If.'In, Th(.-,;' vol. 44, no. 2, pp. 744-765, Feb. 1998.

[2] T. Cover and A. El Gamal, "Capacity theorems for the relay channel," IEEE Trans.
I,.'f ,, Th.-.,,l vol. 25, no. 5, pp. 572-584, MA ,i- 1979.

[3] A. Sendonaris, E. Erkip, and B. Aazhang, "User cooperation diversity. Part I.
System description," IEEE Trans. Commun., vol. 51, no. 11, pp. 1927-1938, Nov.
2003.

[4] A. Sendonaris, E. Erkip, and B. Aazhang, "User cooperation diversity. Part II.
Implementation aspects and performance analysis," IEEE Trans. Commun., vol. 51,
no. 11, pp. 1939-1948, Nov. 2003.

[5] Y. Liang and V. V. Veeravalli, "Cooperative relay broadcast channels," IEEE Trans.
I,.'f i, Th(.-,,l vol. 53, no. 3, pp. 900-928, Mar. 2007.

[6] N. Devrc., P. Mitran, and V. Tarokh, "Achievable rates in cognitive radio chan-
nels," IEEE Trans. I,.[., I, Th(.-,.; vol. 52, no. 5, pp. 1813-1827, May 2006.

[7] A. Jovicic and P. Viswanath, "Cognitive radio: An information theoretic perspec-
tive," submitted to IEEE Trans. In. f', n, Th. ./ July 2006.

[8] S. Shamai (Shitz), 0. Somekh, 0. Simeone, A. Sanderovich, B. M. Zaidel, and H.
V. Poor, "Cooperative multi-cell networks: Impact of limited-capacity backhaul and
inter-users links," in Proc. Joint Workshop Coding and Commun. (JWCC 2007),
Durnstein, Austria, 2007.

[9] L. Sankaranarayan, G. Kramer, and N. B. Mand ,'ii-, "Cooperation vs. hierarchy:
an information-theoretic comparison," in Proc. IEEE International Symp. Inform.
Ti,.., j (ISIT 2005), Adelaide, Australia, Sep. 2005.

[10] P. Gupta and P. Kumar, "The capacity of wireless networks," IEEE Trans. Inform.
Ti, ..- ; vol. 46, no. 2, pp. 388-404, Feb. 2000.

[11] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, "Cooperative diversity in wireless
networks: Efficient protocols and outage behavior," IEEE Trans. I. f. 'ii Th(.,/
vol. 50, no. 12, pp. 3062-3080, Dec. 2004.

[12] A. Host-Madsen and J. Zhi :. "Capacity bounds and power allocation for wireless
relay channels," IEEE Trans. Ii[. ,in, Th.-.,.; vol. 51, no. 6, pp. 2020-2040, Jun.
2005.

[13] G. Kramer, M. Gastpar, and P. Gupta, "Cooperative strategies and capacity
theorems for relay networks," IEEE Trans. Inform. The..,, vol. 51, no. 9, pp.
3037-3063, Sep. 2005.









[14] M. Gastpar and M. Vetterli, "On the capacity of large Gaussian relay networks,"
IEEE Trans. In.[..,in The.-'i. vol. 51, no. 3, pp. 765-779, Mar. 2005.

[15] T. S. Han and K. Kob- v-l-hi "A new achievable rate region for the interference
channel," IEEE Trans. Inform. The.',;/ vol. IT-27, no. 1, pp. 49-60, Jan. 1981.

[16] R. H. Etkin, D. N. C. Tse, and H. Wang, "Gaussian interference channel capacity to
within one bit," IEEE Trans. Inf[..n The.' ,I vol. 54, no. 12, pp. 5534-5562, Dec.
2008.

[17] J. Jiang and Y. Xin, "On the achievable rate regions for interference channels with
degraded message sets," IEEE Trans. Inform. Tht'.,i vol. 54, no. 10, pp. 4707-4712,
Oct. 2008.

[18] A. Goldsmith, S. A. Jafar, I. Maric, and S. Srinivasa, "Breaking spectrum gridlock
with cognitive radios: An information theoretic perspective," Proc. IEEE, vol. 97,
issue 5, pp. 894-914, May 2009.

[19] K. Azarian, H. El Gamal, and P. Schniter, "On the achievable diversity-multiplexing
tradeoff in half-duplex cooperative channels," IEEE Trans. In.[.. tin Th-.;,I vol. 51,
no. 12, pp. 4152-4172, Dec. 2005.

[20] A. S. Avestimehr and D. N. C. Tse, "Outage capacity of the fading relay channel in
low-SNR regime," IEEE Trans. ITJ[..',n Th(..,'; vol. 53, no. 4, pp. 1401 1415, Apr.
2007.

[21] D. Gunduz and E. Erkip, "Opportunistic cooperation by dynamic resource alloca-
tion," IEEE Trans. Wireless Commun., vol. 6, no. 4, pp. 1446-1454, Apr. 2007.

[22] L. Ong and M. Motani, "Optimal routing for the Gaussian multiple-relay channel
with decode-and-forward," in Proc. IEEE Commun. Soc. Conf. Sensor, Mesh, and
Ad Hoc Commun. and Netw. (SECON 2007), San Diego, U.S.A., Jun. 2007.

[23] L.-L. Xie and P. R. Kumar, "A network information theory for wireless communica-
tion: Scaling laws and optimal Operation," IEEE Trans. In f. II,, Thu(..;, vol. 50, no.
5, pp. 748-767, May 2004.

[24] G. Kramer, M. Gastpar, and P. Gupta, "Capacity theorems for wireless relay
channels," IEEE Trans. Inr.f[..Im Tht.';,. vol. 51, no. 9, pp. 3037-3063, Sep. 2005.

[25] A. Reznik, S. R. Kulkarni, and S. Verdu, "Degraded Gaussian multirelay channel:
Capacity and optimal power Allocation," IEEE Trans. Inform. Th'(.i,. vol. 50, no.
12, pp. 3037-3046, Dec. 2004.

[26] L. Ong and M. Motani, "The capacity of the single source multiple relay single
destination mesh network," in Proc. IEEE International Symp. Inform. Theory
(ISIT .'/K,), Seattle, U.S.A., Jul. 2006.









[27] A. Bletsas, A. Khisti, D. P. Reed, and A. Lippman, "A simple cooperative diversity
method based on network path selection," IEEE J. Select. Areas Commun., vol. 24,
No. 3, pp. 659-672, Mar. 2006.

[28] E. Beres and R. S. Adve, "On selection cooperation in distributed networks," in
Proc. Conf. Inr.[.., i Sciences and Syst. (C'ISS .:W'), pp. 1056-1061, Mar. 2006.

[29] J. N. Laneman and G. W. Wornell, "Distributed space-time-coded protocols for
exploiting cooperative diversity in wireless networks," IEEE Trans. Inform. Th(.,
vol. 49, no. 10, pp. 2415-2425, Oct. 2003.

[30] T. F. Wong, T. M. Lok, and J. M. Shea, "Flow-optimized Cooperative Transmission
for the Relay C'!i ,i,,. I" submitted to IEEE Trans. Inform. Th(.. -; Dec. 2006.
[Online]. Available: http: //arxiv. org/PS_cache/cs/pdf/0701/0701019v3. pdf

[31] L. Zheng and D. N. C. Tse, "Diversity and multiplexing: A fundamental tradeoff
in multiple antenna channels," IEEE Trans. Inform. Th(.., vol. 49, no. 5, pp.
1073-1096, May 2003.

[32] M. Yuksel and E. Erkip, \!ili ipl! .-antenna cooperative wireless systems: A
diversity-multiplexing tradeoff perspective," IEEE Trans. Inr. ..t The.., ,/ vol.
53, no. 10, pp. 3371-3393, Oct. 2007.

[33] T. M. Cover and J. A. Thomas, Elements of information Ji,, .,;, Wiley, 1991.

[34] Y. Wu, P. A. Chou, and S.-Y. Kung, \Iiiiiiiiiiiii-energy multicast in mobile ad hoc
networks using network coding," IEEE Trans. Commun., vol. 53, no. 11, Nov. 2005.

[35] R. K. Alli T. L. Magnanti, and J. B. Orlin, Network flows: Th(..,; l1J..,':thms,
and applications, Prentice Hall, 1993.

[36] C. T. Lawrence, J. L. Zhou, and A. L. Tits, "User's guide for CFSQP version 2.5:
A C code for solving (large scale) constrained nonlinear (minimax) optimization,
generating iterates satisfying all inequality constraints," Technical Report TR-94-
16rl, University of Maryland, College Park, 1997.

[37] W. P. Tam, T. M. Lok, and T. F. Wong, "Flow optimization in parallel relay
networks with cooperative relaying," IEEE Trans. Wireless Commun., vol. 8, no. 1,
pp. 278-287, 2009.

[38] D. Tse and S. Hanly, \!!li i-access fading channels. Part I. Polymatroid structure,
optimal resource allocation and throughput capacities," IEEE Trans. I.i i[.
Ti .., -,; vol. 44, No. 7, pp. 2796 2815, Nov. 1998.

[39] G. Caire, G. Taricco, and E. Biglieri, "Optimum power control over fading chan-
nels," IEEE Trans. It. [..,, The..,,l vol. 45, no. 5, pp. 1468-1489, Jul. 1999.









[40] L. Li, N. Jindal, and A. Goldsmith, "Outage capacities and optimal power allocation
for fading multiple-access channels," IEEE Trans. hr.[..,,, Ti ..' .; vol. 51, no. 4, pp.
1326 1347, Apr. 2005.

[41] L. Lai and H. E. Gamal, "The water-filling game in fading multiple access channels,"
submitted to IEEE Trans. Inform. The.. ,. Nov. 2005. [Online]. Available:http:
//arxiv.org/abs/cs/0512013.

[42] J. Sun, L. Zi, i.- and E. Modiano, \\ i. I. -- channel allocation using an auction
algorithm," in Proc. Allerton Conf. Commun., Control and Con,,il','!, pp. 1114 -
1123, Oct. 2003.

[43] A. ParandehGheibi, A. Eryilmaz, A. Ozdaglar, and M. M6dard, "Dynamic rate allo-
cation in fading multiple access channels," in Proc. I f., 'i,, Theory and Applications
(ITA) Workshop, 2008.

[44] W. Yu, W. Rhee, J. Cioffi, "Optimal power control in multiple access fading
channels with multiple antennas," in Proc. IEEE International Conf. Commun.,
(ICC 2001), vol.2, pp.575 579, Jun. 2001.

[45] M. Mecking, Resource allocation for fading multiple-access channels with partial
channel state information," in Proc. IEEE International Conf. Commun., (ICC
2002), vol.3, pp.1419 1423, 2002.

[46] R. N i,-iiii, 'i, "Individual outage rate regions for fading multiple access channels,"
in Proc. IEEE International Symp. Inform. Theory (ISIT 2007), pp. 1571-1575, Jun.
2007.

[47] M. M6dard, "The effect upon channel capacity in wireless communications of perfect
and imperfect knowledge of the channel," IEEE Trans. Inform. Th(.. ,i vol. 46, no.
3, pp. 933 946, AT ,i- 2000.

[48] A. Lapidoth, and S. Shamai (Shitz), : ,l!iii channels: How perfect need "perfect
side information" be?," IEEE Trans. I, f[..' Th(.,, vol. 48, no. 5, pp. 1118 1134,
M, ,l- 2002.

[49] R. N o i-ii, ili,, "Effect of channel estimation errors on diversity-multiplexing trade-
off in multiple access channels," Proc. IEEE Global Commun. Conf., (GLOBECOM
.'ii,), pp. 1-5, Nov. 2006.

[50] M. J. Osborne and A. Rubinstein, "A course in game theory," MIT Press, 1994.

[51] W. C. Riddell, "Bargaining under uncertainty," Amer. Econ. Rev., vol. 71, no. 4, pp.
579 590, Sep. 1981.

[52] W. Bossert, E. Nosal, and V. Sadanand, "Bargaining under uncertainty and the
monotone path solutions," Games and Econ. Behavior, vol. 14, no. 2, pp. 173 189,
1996.









[53] W. Thomson, and R. B. Myerson, "Monotonicity and independence axioms,"
International J. Game Tht ;/ vol. 9, no. 1, pp. 37-49, Mar. 1980.

[54] E. Kalai, "Proportional solutions to bargaining situations: Interpersonal utility
comparisons," Econometrica, vol. 45, no. 7, pp. 1623 1630, 1977.

[55] 0. Kaya and S. Ulukus, "Power control for fading cooperative multiple access
channels," IEEE Trans. Commun., vol. 6, no. 8, pp. 2915-2923, Aug. 2007.

[56] W. Mesbah and T. N. Davidson, "Optimal power allocation for full-duplex coopera-
tive multiple access," in Proc. IEEE International Conf. Acoust., Speech, and S.:,j ,,i
Process. (ICASSP .'i/i"), Toulouse, France, vol. IV, pp. 689-692, MA ,i- 2006.

[57] 0. Kaya, "Window and backwards decoding achieve the same sum rate for the fad-
ing cooperative Gaussian multiple access channel," in Proc. IEEE Global Commun.
Conf., (GLOBECOM .'i,'), San Francisco, CA, Nov. 2006.

[58] W. Mesbah and T. N. Davidson, "Optimal power and resource allocation for half-
duplex cooperative multiple access," in Proc. IEEE International Conf. Commun.
(ICC .'i00,), vol. 10, Istanbul, Turkey, Jun. 2006.

[59] E. G. Larsson and B. R. Vojcic, "Cooperative transmit diversity based on superposi-
tion coding," IEEE Commun. Lett., vol. 9, no. 9, pp. 778-780, Sep. 2005.

[60] Z. Ding, T. R I i ,i i, i,! and C. C. F. Cowan, "On the diversity-multiplexing tradeoff
for wireless cooperative multiple access systems," IEEE Trans. S.:j,,i] Process., vol.
55, no. 9, pp. 4627 4638, Sep. 2007.

[61] N. Liu, and S. Ulukus, "Capacity Region and Optimum Power control strategies
for fading Gaussian multiple access channels with common data," IEEE Trans.
Commun., vol. 54, no. 10, pp. 1815 1826, Oct. 2006.

[62] D. N. C. Tse, P. Viswanath, and L. ZI, iw- "Diversity-multiplexing tradeoff in
multiple-access channels," IEEE Trans. If.'i,i, Th(-.,,l vol. 50, no. 9, pp. 1859-
1874, Sep. 2004.

[63] N. Devrc.,- P. Mitran, and V. Tarokh, "Cognitive decomposition of wireless
networks," in Proc. 1st International Symp. Cognitive Radio Oriented Wireless
Netw. and Commun.(CROWNCOM .'i"i), pp. 1-5, 2006.

[64] I. Maric, A. Goldsmith, G. Kramer, and S. Shamai (Shitz), "On the capacity of
interference channels with one cooperating transmitter," Euro Trans. Telecommuni-
cations, vol. 19, no. 4, pp. 405-420, 2008.

[65] Y. Cao and B. C'!., i I,-i I. i. -. channels with one cognitive transmitter,"
submitted to IEEE Trans. Inform. Th(.. ,-; Oct. 2009.









[66] S. Rini, D. Tuninetti, and N. Devrc.-,- "State of the cognitive interference chan-
nel: a new unified inner bound, and capacity to within 1.87 bits," Arxiv preprint
arXiv:0910.,:1..:' ,-, 2009.

[67] K. Marton, "A coding theorem for the discrete memoryless broadcast channel,"
IEEE Trans. Ir.f.'.,t, Th.. -;, vol. 25, no. 3, pp. 306-311, May 1979.

[68] N. Devrov., P. Mitran, and V. Tarokh, "Cognitive multiple access networks," in
Proc. International Symp. Inform. Theory (ISIT 2005), pp. 57-61, 2005.

[69] H. C'!i ,1 i,, .i, and M. N. -Kenari, "Achievable rates for two interfering broadcast
channels with a cognitive transmitter," in Proc. International Symp. Inform. Theory
(ISIT :',~), pp. 1358-1362, July 2008.

[70] 0. Sahin and E. Erkip, "Cognitive relaying with one-sided interference," in Proc.
Asilomar Conf. S/.'.1- Syst. and Comput., 2008.

[71] S. H. S. v. i, lhbi Y. Xin, and Y. Lian, "An achievable rate region for the causal
cognitive radio," in Proc. Allerton Conf. Commun. Control, and Computing, pp.
783-790, Sep. 2007.

[72] C.-M. Zeng, F. Kuhlmann, and A. Buzo, "Achievability proof of some multiuser
channel coding theorems using backward decoding," IEEE Trans. Inform. The.,
vol. 35, no. 6, pp. 1160 1165, Nov. 1989.

[73] G. Kramer, "Models and theory for relay channels with receive constraints," in Proc.
Allerton Conf. Commun., Control, and Computing, pp. 783-790, Sep. 2007.

[74] I. Csiszar and J. K6rner, "Broadcast channels with confidential messages," IEEE
Trans. Infr.',i Tht.-,,.l vol. IT-24, no. 3, pp. 339-348, Al ,i- 1978.

[75] A. Host-Madsen and J. Zhi ii "Capacity bounds and power allocation for wireless
relay channels," IEEE Trans. I.f[..'in, The..,.; vol. 51, no. 6, pp. 2020-2040, Jun.
2005.

[76] N. Devrc.-, "Information theoretic limits of cognition and cooperation in wireless
networks," Ph.D. dissertation, Harvard University, 2007.

[77] H. Weingarten, Y. Steinberg, and S. Shamai (Shitz), "The capacity region of the
Gaussian MIMO broadcast channel," IEEE Trans. Inr.[..,t, The..,.i vol. 52, no. 9,
pp. 3936-3964, Sep. 2006.

[78] E. Hossain, L. B. Le, N. Devrov,- and M. Vu, "Cognitive radio: From theory to
practical network (iiiii.. 1,iI,: in Advances in Wireless Communications, V. Tarokh
and I. Blake, Eds., Springer, 2009, pp. 251-289.









BIOGRAPHICAL SKETCH

Debdeep C!i i.1, ijee received the B.Tech. degree in electrical engineering in 2004

from the Indian Institute of Technology (IIT), Klio '-pur, India; and the M.S. and Ph.D.

degrees in electrical and computer engineering in 2006 and 2010 respectively from the

University of Florida, Gainesville. From August 2008 until July 2009, and from May

2010 until August 2010, he was with the Wireless Standards and Technologies team, Intel

Corporation, Santa Clara, California. His research interests include cooperative communi-

cations, multi-user information theory, game theory, and design of next-generation wireless

systems.





PAGE 1

COOPERATIVECOMMUNICATIONINWIRELESSNETWORKS: FLOW-OPTIMIZEDDESIGNSANDINFORMATION-THEORETIC CHARACTERIZATIONS By DEBDEEPCHATTERJEE ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2010 1

PAGE 2

c 2010DebdeepChatterjee 2

PAGE 3

Tothereader 3

PAGE 4

ACKNOWLEDGMENTS IwouldliketothankmyadvisorProf.TanWongforhisguidance,support,patience, andthefreedomIenjoyedinchoosingmyresearchdirection.Iconsidermyselfvery fortunatetohavebeenabletopursueresearchundertheguidanceofsomeonewho encouragedmetotrytodenemyownresearchproblems,andatthesametime,was patientenoughwhenaparticularideawouldfailtobearfruitasexpected. IwouldalsoliketothankProf.JohnSheaforhisgeneralguidanceandsuggestions regardingpursuingresearch,andmoreimportantly,thoseonpresentingone'sresearch.I thankProf.MichaelFangandProf.WilliamHagerfortheirtimeandinterestinmywork. IwouldliketotakethisopportunitytothankDr.OzgurOymanofIntelResearchforthe stimulatingdiscussionsthatwehadduringmystayinSantaClara,andforhisvaluable commentsandsuggestionsregardingsomeofthelaterpartsofthiswork. MystayintheWINGlabalmostneverhadadullmoment,andcreditforthatis duetomylab-mates,especiallySurendra,Ryan,andByong,whocompletelychanged theatmosphereofthelabeversincethesummerof2006whenIwaspracticallythe onlypersonpresentinthelab.TherearewaytoomanypeopleIamindebtedtofor allthehelpandsupportIhavereceivedinthepastfewyears.Sridhar,Selvi,Manu, Savya,Vaibhav,andMallickarejustafewpeoplewhohaveenduredmeovertheseyears, providedmewithencouragementandhopesometimesblatantlyfalse,buttheymostly workedwhenthingshavenotworkedout,andmostimportantly,beengreatfriends. Finally,Ithankmyparents,fornoachievement,howeverbigorsmall,mayeverbe realizedwithouttheirloveandsupport. 4

PAGE 5

TABLEOFCONTENTS page ACKNOWLEDGMENTS.................................4 LISTOFTABLES.....................................7 LISTOFFIGURES....................................8 ABSTRACT........................................10 CHAPTER 1INTRODUCTION..................................12 1.1CooperativeTransmissioninWirelessRelayNetworks............13 1.2CooperativeBehaviorinaFadingMultiple-AccessChannel.........14 1.3ActiveUserCooperationinaFadingCMAC.................15 1.4AchievableRatesintheICUCwithCausalityConstraints.........15 1.5OrganizationoftheDissertation........................17 2COOPERATIVETRANSMISSIONINAWIRELESSRELAYNETWORK BASEDONFLOWMANAGEMENT.......................19 2.1Introduction...................................19 2.2GeneralDesignUsingAFlow-theoreticApproach..............22 2.3Generalized-linkSelectionandItsOptimality................26 2.3.1TheThree-nodeRelayNetwork....................26 2.3.2Generalized-linkSelection........................32 2.3.3Diversity-multiplexingtradeo.....................34 2.4NumericalExamples..............................37 2.5Summary....................................40 3RESOURCEALLOCATIONANDCOOPERATIVEBEHAVIORINFADING MULTIPLE-ACCESSCHANNELSUNDERUNCERTAINTY..........48 3.1Introduction...................................48 3.2SystemModel..................................51 3.3TheBargainingProblemfortheTwo-UserFadingMAC..........52 3.3.1TheDisagreementPoint........................53 3.3.2TheNashBargainingSolutionNBS.................54 3.4TheModiedBargainingProblem.......................58 3.5Summary....................................62 4ACTIVEUSERCOOPERATIONINFADINGMULTIPLE-ACCESSCHANNELS.........................................64 4.1Introduction...................................64 5

PAGE 6

4.2Flow-theoreticTransmissionProtocolsfortheCooperativeMultiple-Access Channel.....................................67 4.2.1OrthogonalRelayingProtocolfortheCMACOR-CMAC.....68 4.2.2Flow-optimizedProtocolfortheCMACFO-CMAC........70 4.3NumericalResults................................72 4.4Summary....................................74 5INTERFERENCECHANNELSWITHUNIDIRECTIONALCOOPERATION ANDCAUSALITYCONSTRAINTS........................79 5.1Introduction...................................79 5.2TheChannelModel...............................82 5.3AchievableRatesfortheICUC-C.......................83 5.4TheGaussianICUC-C.............................91 5.5DiscreteMemorylessChannelModelfortheICUC-HDC..........94 5.6AnAchievableRateRegionfortheICUC-HDC...............95 5.7TheGaussianICUC-HDC...........................103 5.7.1InclusionOfCausalAchievableRegionof[6].............105 5.7.2NumericalResults............................107 5.8Summary....................................109 6CONCLUSIONSANDFUTUREWORK......................116 6.1Conclusions...................................116 6.2FutureDirections................................118 REFERENCES.......................................120 BIOGRAPHICALSKETCH................................126 6

PAGE 7

LISTOFTABLES Table page 5-1DescriptionofRandomVariablesinTheorem5.1.................84 5-2DescriptionofRandomVariablesinTheorem5.2.................96 7

PAGE 8

LISTOFFIGURES Figure page 2-1Basicgraphs G 1 and G 2 forthethree-noderelaynetworkwith t 1 + t 2 =1....42 2-2FOprotocolforthefour-noderelaynetworkwith t 1 + + t 6 =1.Theow optimizationisperformedoverallows x 1 ; ;x 14 ,andalltimeslotlengths t 1 ; ;t 6 .......................................42 2-3Transmissionstrategytoobtainalowerboundontheoutageprobabilityfor thefour-noderelaynetwork.Here t 1 + + t 4 =1,andtheoptimizationis over x 1 ; ;x 14 ,and t 1 ; ;t 4 ,withtheapplicationofthemax-ow-mincuttheoremfortheintermediateslots........................43 2-4Four-noderelaynetworkwithuniformaveragepowergains:Outageprobabilitiesforrequiredrate R =1bit/s/Hz.........................44 2-5Four-noderelaynetworkwithuniformaveragepowergains:Outageprobabilitiesforrequiredrate R =6bits/s/Hz.........................44 2-6Five-noderelaynetworkwithuniformaveragepowergains:Outageprobabilitiesforrequiredrate R =1bits/s/Hz.........................45 2-7Five-noderelaynetworkwithuniformaveragepowergains:Outageprobabilitiesforrequiredrate R =6bits/s/Hz.........................45 2-8Four-noderelaynetworkwithnon-uniformaveragepowergains.CaseA: E [ Z SR 1 ]= 2 : 0 ; E [ Z SR 2 ]=2 : 0 ; E [ Z SD ]=1 : 0 ; E [ Z R 1 R 2 ]=1 : 0 ; E [ Z R 1 D ]=1 : 5 ; E [ Z R 2 D ]= 1 : 0.Outageprobabilitiesforrequiredrate R =1bit/s/Hz.............46 2-9Four-noderelaynetworkwithnon-uniformaveragepowergains.CaseA: E [ Z SR 1 ]= 2 : 0 ; E [ Z SR 2 ]=2 : 0 ; E [ Z SD ]=1 : 0 ; E [ Z R 1 R 2 ]=1 : 0 ; E [ Z R 1 D ]=1 : 5 ; E [ Z R 2 D ]= 1 : 0.Outageprobabilitiesforrequiredrate R =6bit/s/Hz.............46 2-10Four-noderelaynetworkwithnon-uniformaveragepowergains.CaseB: E [ Z SR 1 ]= 1 : 5 ; E [ Z SR 2 ]=0 : 75 ; E [ Z SD ]=1 : 0 ; E [ Z R 1 R 2 ]=3 : 5 ; E [ Z R 1 D ]=0 : 2 ; E [ Z R 2 D ]= 3 : 0.Outageprobabilitiesforrequiredrate R =1bit/s/Hz.............47 2-11Four-noderelaynetworkwithnon-uniformaveragepowergains.CaseB: E [ Z SR 1 ]= 1 : 5 ; E [ Z SR 2 ]=0 : 75 ; E [ Z SD ]=1 : 0 ; E [ Z R 1 R 2 ]=3 : 5 ; E [ Z R 1 D ]=0 : 2 ; E [ Z R 2 D ]= 3 : 0.Outageprobabilitiesforrequiredrate R =6bit/s/Hz.............47 3-1Averagerateswithvarying R ............................63 4-1Flow-theoretictransmissionprotocolsfortheCMAC:aOR-CMAC,bFOCMAC.........................................75 4-2Achievablerateregions-asymmetricsituation...................76 8

PAGE 9

4-3Achievablerateregions-asymmetricsituationwithconventionalMAslotwithoutcommoninformation...............................76 4-4Achievablerateregions-symmetricsituation....................77 4-5Outageperformance-asymmetricsituation.....................77 4-6Outageperformance-symmetricsituation......................78 5-1ThediscretememorylessICUCwithcausalityconstraint..............110 5-2TheGaussianICUC-C.................................110 5-3AchievableRatesfortheGaussianICUC-C:Weakinterferenceforbothcrosslinks...........................................111 5-4AchievableRatesfortheGaussianICUC-C:Stronginterferencefrom S P to D C andweakinterferencefrom S C to D P ........................112 5-5AchievableRatesfortheGaussianICUC-C:Weakinterferencefrom S P to D C andstronginterferencefrom S C to D P ........................113 5-6ThediscretememorylessICUC-HDC.........................114 5-7AchievableRatesfortheGaussianICUC-HDC:Weakinterferenceforbothcrosslinks...........................................114 5-8AchievableRatesfortheGaussianICUC-HDC:Stronginterferencefrom S P to D C andweakinterferencefrom S C to D P ......................115 9

PAGE 10

AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy COOPERATIVECOMMUNICATIONINWIRELESSNETWORKS: FLOW-OPTIMIZEDDESIGNSANDINFORMATION-THEORETIC CHARACTERIZATIONS By DebdeepChatterjee August2010 Chair:TanF.Wong Major:ElectricalandComputerEngineering Thechallengesinvolvedinthedesignofecientcommunicationsystemsforthe wirelessmediumcanbeattributedtothefactthatthewirelessmediumpossessescertain uniquecharacteristics,themostimportantonesbeingthebroadcastnatureofthewireless medium,thesusceptibilitytointerferenceeects,andtheeectsofpathlossandfadingon wirelesslinkquality.Cooperationbetweendierenttransceiverscanpotentiallyaidfurther developmentofnext-generationwirelesscommunicationsystemsthatdemandhighdata ratesandanexcellentqualityofserviceQoS.Thisispossiblebyexploitingthebroadcast natureofthewirelessmedium,andthediversityadvantagesthatamulti-usersystem oers. Werstconsiderageneralsingle-source-single-destinationwirelessrelaynetwork andproposeaninformationow-optimizedcooperativetransmissiondesignthatachieves theoptimaldiversity-multiplexingtradeo.Next,weapplygame-theoretictechniques totheproblemsofresourceallocationandcharacterizationofcooperativebehaviorina two-userfadingmultiple-accesschannelMAC,withuncertaintyaboutthechannelstate informationatthetransmittersCSIT. Inthethirdpartofthedissertation,amoreactiveformtheabovecooperative behaviorisstudiedviaatwo-userfadingcooperativemultiple-accesschannelCMAC, whereeachuser,alongwithtransmittingitsowninformationtothedestination,helpsthe 10

PAGE 11

otherbyforwardingthelatter'sinformation.Weproposeecientcooperativetransmission strategiesbasedonaow-theoreticapproach,andevaluatetheirperformancesusing numericalsimulations. Finally,weconsidercommunicationthroughatwo-userinterferencechannelwith unidirectionalcooperationICUC,whereinonesourceusesitsknowledgeofthemessage oftheothertoreducetheinterferencetoitsowntransmission,andsimultaneously,help theotheruser-pairviacooperativerelaying.Weconsideraveryrealisticscenarioinwhich thecooperatingsourceissubjectedtoacausalityconstraint.Wederiveanewachievable rateregionforthediscretememorylessversionofthisformofICUC,anddemonstrate thecontributionsofthevariouscodingstrategiesinvolvedvianumericalsimulations forGaussianchannels.Wealsostudythesamechannelwiththecooperatingsource beingsubjecttothehalf-duplexconstraintaswell.Adiscretememorylesschannelmodel incorporatingthehalf-duplexconstraintispresented,andanewachievablerateregion, thatenlargesthelargestknownrateregionfortheGaussianversionofthischannel,is derivedforthischannel.Theachievablerateregionfortheproposedcodingscheme, specializedforGaussianchannels,isnumericallyevaluatedandthestrictinclusionofthe previouslyknownlargestrateregionisdemonstrated. 11

PAGE 12

CHAPTER1 INTRODUCTION Overthelastdecade,wirelesscommunicationsystemshavegainedpopularityata veryfastpace,andhavecurrentlybecomeanintegralpartofourdailylives.Withthe growingdemandsforhigherdataratesandbetterqualityofserviceQoSguarantees tosupportnext-generationwirelesssystems,thedemandforthedesignanddevelopmentoffasterandmorereliablemultiusercommunicationsystems,ascomparedto theexistingsolutions,ismorethaneverbefore.Itiswellknownthatmultiple-inputmultiple-outputMIMOsystemsoerconsiderableperformanceimprovementsover single-input-single-outputSISOsystemsbyecientexploitationofthediversitybenets offadingchannels[1].However,variouspracticalconsiderationsrelatedtothecost,form factorlimitationsandhardwareimplementationrestricttheadvantagesofMIMOsystems, especiallytransmitdiversitybenets,tobasestationsandaccesspointsofwirelessnetworks.Inthisregard, cooperativecommunication oersagoodalternativeinproviding similarbenetsbythesharingofantennaresourcesofmultiplesingle-antennanodesto formvirtualMIMOsystems.Further,cooperativecommunicationtechniquescanbeemployedforMIMOsystemsaswelltobolstertheoverallsystemeciencyandreliability.In general,cooperativecommunicationcomprisesofspecialsignaltransmissionandreception schemesthateectivelyexploitthebroadcastanddiversityadvantagesofthewireless medium,withdueconsiderationtothedetrimentaleectsofinterference. Theaspectofcooperationinwirelesssystemscanbebroadlycharacterizedintotwo dierentmanifestationsofcooperativebehavior: RelayCooperation: Dedicatedrelaynodesareavailabletohelpthesourcenodes transmitinformationtothedestinationnodes.Forsuchasystem,thesource anddestinationnodeswouldconstitutetheclassofusers,whiletherelaynodes wouldbepartofthesystem'sinfrastructure.Thethree-noderelaychannel[2]isthe simplestexampleofsuchasystem. 12

PAGE 13

UserCooperation: Inthiscase,theusers,whothemselveswishtotransmitorreceive information,cooperatewitheachotherbyusingsomeoftheirownresourcestorelay otherusers'information.Thecooperativemultiple-accesschannelCMAC[3,4], therelaybroadcastchannelRBC[5],theinterferencechannelwithunidirectional cooperationICUC,alsoknownasthecognitiveradiochannel[6,7],fallunderthis classofcooperativecommunicationsystems. Boththeseclassesofcooperativecommunicationtechniquespossessuniquebenets andlimitations[9]intermsofdierentperformancemetrics,implementationcostand complexity.Moreover,hybridsystems,thatmakeuseofbothclassesofcooperative communication,canalsobeenvisioned.Forinstance,cooperationwithinmultiplecellsof aninfrastructurenetworkisexpectedtoimprovetheeciencyandreliabilityoftheoverall system[8].ThebasestationsBSsovermultiplecellscooperatebyjointlyprocessing thetransmitted/receivedsignalsusinghighcapacitybackhaulconnections,whilethe mobilestationsMSscooperatethroughrelayingandconferencing"betweenthemselves. Thus,dependingontheapplicationrequirementsandnetworkconguration,dierent cooperativecommunicationsystemsdemanddierentdesignapproaches,andmanifest dierentformsofcooperativebehavioramongsttheparticipatingnodes. 1.1CooperativeTransmissioninWirelessRelayNetworks Awirelessrelaynetworkisamultiuserwirelesscommunicationnetworkwherein awirelesslinkexistsbetweeneachpairofnodes.Ingeneral,itmaybeconceivedas apartofalargernetworkthatitselfmaynotbefullyconnected.Itcanalsobeseen asaspecialformofamoregeneralrelaynetwork.Althoughaconsiderableamount ofworkonrelaynetworkshasbeenreportedintheliterature[10,11,12,13,14], etc.,thecapacityofeventhesimplestofallrelaynetworks-thegeneralthree-node relaychannel,isstillunknown.Intherstpartofthedissertation,weareconcerned withtheproblemoftransmissionofinformationfromasinglesourcenodetothesingle destinationnodeinawirelessrelaynetwork,withthehelpofcertaindedicatedrelay 13

PAGE 14

nodes.Thus,therelaynodes,alongwiththesourcenode,cooperatetoformavirtual multiple-input-single-outputMISOsysteminordertoachievetransmitdiversitybenets. Inthiswork,wesubjectallthenodesintherelaynetworktothehalf-duplexconstraint. Previouslyproposedsolutionstothisproblemincludedierentpathselectionmethods anddistributedspace-timecodingmethods.Unfortunately,thesemethodsfailtoachieve theoptimaldiversity-multiplexingtradeoforthewirelessrelaynetwork,andfailtobe ecientinthehighdataraterequirementregime.Undertheassumptionthatthechannel stateinformationCSIisavailableatallnodes,wedevelopaow-optimizedprotocol,and asuboptimal,butmuchsimpler,generalized-linkselectionprotocol,thatareshowntobe optimalintermsofthediversity-multiplexingtradeo,andthatprovidelargecodinggains overdirecttransmission,evenforhighdataraterequirements. 1.2CooperativeBehaviorinaFadingMultiple-AccessChannel Intheprevioussection,weintroducedaprobleminvolvingasingle-source-singledestinationscenario.Next,weconsiderthesimplestmulti-sourcenetwork:themultipleaccesschannelMAC,whereintwouserswishtotransmitinformationtoasingledestinationnode.Asthemultipletransmitterssharethesamecommunicationmedium,unless thereexistsacertainamountofcooperationorunderstandingbetweentheusers,they couldcausemutualinterferencetoeachother.Thus,eventhoughitdoesnotbelongtothe classofcooperativecommunicationsystemsthattypicallyinvolvecooperationintheform ofactiveforwardingofinformation,thereexistsacertainlevelofcooperationbetweenthe usersaswouldbemadeexplicitinChapter3. Towardsthis,weconsidertheresourceallocationproblemforthetwo-userfading MAC.Weconsiderthecaseinwhichthereexistssomeuncertaintyinthechannelstate informationatthetransmittersCSIT.UnderassumptionsofperfectCSIT,thetransmissionstrategiessuggestedbythesolutiontothebargainingproblemyieldoptimaloperation points.Ontheotherhand,owingtoinaccuraciesintheavailableCSIT,theconventional 14

PAGE 15

bargainingproblemsolutionmaynotbeoptimal,therebyresultinginconsiderableperformancedegradation.Toaddressthisrobustnessissue,weproposeaschemeinwhich theconditionsofthebargainingproblemarerelaxedtoreducethedependenceofthe systemperformanceonthesolutiontothebargainingproblem.Intheprocess,wedevelop agame-theoreticframeworktocharacterizethelevelofcooperationinvolvedfroman individualisticviewpoint. 1.3ActiveUserCooperationinaFadingCMAC Next,weconsidertheproblemof active usercooperationinatwo-userfadingCMAC. Thisalludestoasysteminwhicheachuser,apartfromtransmittingitsowninformation, maycooperatewiththeotherbyactivelyforwardingtheother'sinformationtothedestination.Therefore,theCMACcanalsobeconsideredasabasicexampleofcooperative communicationinamulti-source-single-destinationsystem.Clearly,thisinvolvesahigher levelofcooperationascomparedtotheformofcooperationinaconventionalMAC.We useaow-theoreticapproach,andproposeaow-optimizedsolutionandamuchsimpler butsuboptimalsolutionthatdecomposestheCMACintoorthogonalrelaychannels.The performancesoftheproposedprotocolsareevaluatedintermsoftheachievableaverage rateregionsandoutageprobabilities,andtheimprovementsoverconventionalMACsystemsandapreviouslyproposedmethodbasedondecode-and-forwardDFrelayingare demonstratedthroughsimulationresults. 1.4AchievableRatesintheICUCwithCausalityConstraints Finally,westudyamanifestationofcooperationinamulti-source-multi-destination system.Thesimplestmulti-source-multi-destinationsystemisthetwo-userinterference channel,whereinapairofsourceswishtotransmitinformationtotheirrespective destinationsresultingininterferencecausedtoeachother.Thecapacityregionforthe interferencechannel,exceptforthespecialcasesofstronginterference,isstillunknown, anduntilrecently,theoriginalHan-Kobayashischeme[15]hasbeenknowntoprovide thelargestachievablerateregion,withoutaclearideaoftightnessofthisregionwith 15

PAGE 16

regardtotheknownouterbounds.Veryrecently,in[16],theauthorshaveshownthat aHan-Kobayashitypeofcodingschemeyieldsarateregionthatiswithin1bitofthe tightestouterboundfortheGaussianinterferencechannel. Giventhisbackdrop,wedirectourfocustotheinterferencechannelwithanasymmetriccooperativerelationshipbetweenthesources.Suchanetworkisknownasthe interferencechannelwithunidirectionalcooperationICUC 1 ,andisthesimplestform ofgeneraloverlaycognitivenetworks[18].TheICUCconsistsofapairofsourcesthat demonstratedierentbehaviors.Ontheonehand,theprimarysourceissolelyinterested intransmittinginformationtoitsrespectivedestinationanddoesnotactivelycooperate withtheotheruser-pair.Ontheotherhand,thesecondaryor cognitive sourceusesits knowledgeabouttheprimarymessagetoreducetheinterferencecausedtoitsownlink bytheprimarytransmission,andatthesametime,aidtheprimaryuser-pairbyrelaying theprimarymessagetotheprimarydestination,therebyreducingtheeectofinterference causedbyitsowntransmissiontotheprimarylink.Mostoftheworkreportedintheliteratureonthischannelconsiderthescenarioinwhichthecognitivesourcehasnon-causal knowledgeabouttheprimarymessage.Inthiswork,weimposeapracticalrestriction thatthecognitivesourcemayonlyobtainthemessageoftheprimarysourceinacausal manner,i.e.thecognitive"abilityofthesecondarysourceneedstobe acquired .Anew achievablerateregionforthediscretememorylessversionoftheICUCwithcausalityconstraintICUC-CisderivedusingblockMarkovsuperpositioncodingSPC,conditional rate-splitting,conditionalGel'fand-PinkserGPbinning,andcooperativerelaying.This rateregionisevaluatedforGaussianchannels,andnumericalresultsarepresentedto demonstratethecontributionsofthevariouscodingstrategiesusedintheproposedcoding schemetowardsenlargingtheachievablerateregion. 1 Thisnetworkisalsoknownintheliteratureasthecognitiveradiochannel[6]orinterferencechannelwithdegradedmessagesetsIC-DMS[17] 16

PAGE 17

Wethenimposeyetanotherverypracticalconstraintonthecognitivesource.Itis assumedthatthecognitivesourcemayonlyoperateinahalf-duplexfashion.Adiscrete memorylesschannelmodelfortheICUCwithhalf-duplexandcausalityconstraintICUCHDCispresented,andanewachievablerateregionisderivedforthischannel.The randomcodingschemeusedtoobtaintherateregioninvolvesblockMarkovsuperposition codingSPC,conditionalrate-splitting,cooperativerelaying,conditionalGel'fand-Pinkser GPbinning,andarandomizationofthelisten-transmitscheduleforthecognitivesource. Wealsoprovethatthisrateregioncontainsthelargestachievablerateregionforthe GaussianICUC-HDCpreviouslyreportedintheliterature,andfurtherdemonstratethis usingnumericalresultsforthecaseofGaussianchannels. 1.5OrganizationoftheDissertation Therestofthedissertationisorganizedasfollows.InChapter2,wepresentthe proposedprotocolsforecientinformationtransmissioninawirelessrelaynetwork.We rstpresentthegeneraldesignoftheow-optimizedtransmissionprotocol,andthen, thegeneralized-linkselectionprotocol.Next,theoptimalityoftheproposedprotocol intermsofthediversity-multiplexingtradeoisestablished,andnally,wepresentthe performanceevaluationoftheproposedprotocolsforthenitesignal-to-noiseratioSNR regimethroughnumericalresults.InChapter3,wepresentthemodiedbargaining problemformulationtomodeltheresourceallocationprobleminatwo-userfadingMAC underuncertainty.Solutionstotheseproblemsforthetwochoicesofutilityfunctions arealsopresented,andnumericalexamplesareshowntohighlightvariousaspectsofthe problemandtheproposedsolutions.Weconsidertheproblemofactiveusercooperation inatwo-userfadingMACinChapter4,andpresenttheow-optimizedandorthogonal relayingprotocolsfortheCMAC.Simulationresultsfordierentscenariosarethen presentedfortheperformanceevaluationoftheproposedprotocols.InChapter5westudy theproblemofcommunicatingthroughanICUCwithcausalityconstraints,andpresent newachievablerateregionsforthesenetworks|rst,forthescenarioinwhichthe 17

PAGE 18

cognitivesourceoperatesinfull-duplexmanner,andthen,forthesituationwhereinthis requirementisremovedandinsteadahalf-duplexconstraintisimposedonthecognitive source.Finally,inChapter6,weconcludethedissertation,anddiscussthepossible directionsforfuturework. 18

PAGE 19

CHAPTER2 COOPERATIVETRANSMISSIONINAWIRELESSRELAYNETWORKBASEDON FLOWMANAGEMENT 2.1Introduction Awirelessrelaynetworkisoneinwhichasetofrelaynodesassistasourcenode transmitinformationtoadestinationnode.Practicallythewirelessnodescanonlysupporthalf-duplexcommunication[11],i.e.,nonodescanreceiveandtransmitinformation simultaneouslyonthesamefrequencyband.Dierentcooperativetransmissionschemes forsystemswithhalf-duplexnodeshavebeenproposedintheliterature.Fundamentally, theseschemesconsistoftwobasicsteps.First,thesourcetransmitstothedestination, andtherelaylistensandcaptures"[12]thetransmissionfromthesourceatthesame time.Next,therelayssendprocessedsourceinformationtothedestinationwhilethe sourcemaystilltransmittothedestinationdirectly.Variantsofthesetechniqueshave beenproposedandhavebeenshowntoyieldgoodperformanceunderdierentcircumstances[20,19,11]. AssumingchannelstateinformationCSIatthenodes,anopportunisticdecode-andforwardDFprotocolforhalf-duplexrelaychannelsisproposedin[21].Themaximum delay-limitedrateforthisprotocolisobtainedbyminimizingtheaveragepoweroverall feasibleresourceallocationfunctionssuchthattherequiredrateisachieved.In[22],the authorspresentroutingalgorithmstooptimizetheratefromasourcetoadestination, basedontheDFtechniquethatusesregularblockMarkovencodingandwindowed decoding[23,24],fortheGaussianfull-duplexmultiple-relaychannel.Theachievable rateof[23]fortheGaussianphysicallydegradedfull-duplexmulti-relaychannelhasbeen establishedasthecapacityofthischannelin[25].In[26],itisshownthatthecut-set boundonthecapacityoftheGaussiansinglesource-multiplerelay-singledestination meshnetworkcanbeachievedusingthecompress-and-forwardCFmethod,astherelay powersgotoinnity. 19

PAGE 20

Cooperativediversitymethodsbasedonnetworkpathselectionhavebeenproposed in[27,28].Theseselectionmethodsinclude:ithemax-minselectionmethod[27], whereintherelaynodewiththemaximumoftheminimumofthesource-relayand relay-destinationchannelgainsisselected;iitheharmonicmeanselectionmethod[27], whereintherelaynodewiththehighestharmonicmeanofthesource-relayandrelaydestinationchannelgainsisselected;andiiitheselectionschemeof[28],inwhichthe relaythatcancorrectlydecodetheinformationfromthesourceandhasthebestrelaydestinationchannelisselected.ThesemethodsachieveaDMTof d r = N )]TJ/F15 11.9552 Tf 12.188 0 Td [(1 )]TJ/F15 11.9552 Tf 12.187 0 Td [(2 r foran N noderelaynetworkandmultiplexinggain0 0 : 5. Wehaveproposedacooperativediversitydesignbasedonaowoptimization approachforathree-nodenetworkin[30].Inthisdesign,thesourcenodebroadcaststwo distinctowstothedestinationandtherelaynoderespectivelyduringtherelay'slisten period.ThentherelayforwardsthisinformationusingtheDFapproachwhilethesource mayalsosendanotherowofinformationtothedestinationduringtherelay'stransmit period.Thisschemeisshowntoachievetheoptimaldiversityorderforthethree-node relaychannelandyieldperformanceveryclosetooptimalfull-duplexrelayinginbothlowandhigh-ratesituations. Here,weapplythiscooperativetransmissiondesigntoageneralrelaynetwork, whereinwirelesslinksarepresentbetweeneachpairofnodesinthenetwork.Itisassumed thatthechannelssuerfromslowfadingandhence,facilitateecientestimationanddisseminationofchannelstateinformation.Theproblemconsideredhereisthetransmission ofinformationfromasourcenodetoacorrespondingdestinationnodewhiletheother nodesactasrelaystohelpinthetransmission.Asin[30],assumingCSIisavailableatall nodesweusebroadcastingBC,multipleaccessMAandtimesharingTStechniques 20

PAGE 21

toformulateaowtheoreticconvexoptimizationproblembasedonthechannelconditions.Insteadofconsideringatotalpowerconstraintforallthetransmittingnodesas in[30],wesubjecteachnodetoamaximumtransmitpowerconstraint.Thisyieldsamore reasonablesystemmodelforageneralwirelessrelaynetwork,especiallywhenthenumber ofnodesintherelaynetworkislarge.Theresultingrelayingprotocolwillbereferredto astheow-optimizedFOprotocol.Toobtainamorepracticalcooperativedesignwe developthegeneralized-linkselectionGLSprotocol,inwhichweselectthebestrelay nodeoutoftheavailableonestoformanequivalentthree-noderelaynetworktotransmit theinformationfromthesourcetothedestination.Thebenetofthis,overothernetwork pathselectionstrategies,becomesevidentwhentheraterequirementishigh.Itisshown thatthesimpleGLSprotocolisoptimalintermsoftheDMT[31]andyieldsacceptable performanceevenwhentheraterequirementishigh. Recently,in[32],theauthorshaveshownthatcompress-and-forwardCFrelaying achievestheoptimalDMTforthethree-node,half-duplexnetwork,andthatDFrelaying canachievetheoptimalDMTofthefour-nodefull-duplexnetwork.Inthiswork,we showthattheoptimalDMTcanbeachievedforageneral N -node N 3half-duplex networkusingtheFOorGLSprotocols.Here,itshouldbeclariedthatweconsider thatthewirelesslinksbetweeneachnode-pairexperienceindependentRayleighfading, andthiscorrespondstothedenitionof non-relaynetworked networksin[32].The performancesoftheFOandGLSprotocolsareevaluatednumericallyintermsoftheir outageprobabilitiesforfour-andve-noderelaynetworksforuniformandnon-uniform averagepowergains.ThenumericalresultsmotivatetheuseoftheGLSprotocolfor situationswherecomputationcomplexityisanissueandshowaremarkableimprovement overthemax-minselectionmethodof[27].Theproposeddesigns,basedonBCand MAalone,aresub-optimalingeneral.Forafairappraisaloftheproposedprotocols,we comparetheproposedprotocolstoanupperboundonthemaximumrate,derivedusing themax-ow-min-cuttheorem[33,Thm.14.10.1]. 21

PAGE 22

Therestofthechapterisorganizedasfollows.Inthenextsection,wepresent thedesignoftheproposedow-optimizedprotocolforageneral N -nodewirelessrelay network.InSection2.3,weusetheow-optimizedsolutiontothethree-noderelay networkanduseittodeveloptheGLSprotocolandestablishitsoptimalityintermsof thediversity-multiplexingtradeo.Numericalexamplescomparingtheperformancesof theFOandGLSprotocolstothatcorrespondingtotheworkin[27]arepresentedin Section2.4.Lastly,themaincontributionsinthischapteraresummarizedinSection2.5. 2.2GeneralDesignUsingAFlow-theoreticApproach Weproposeageneraldesignforthetransmissionofmessagesfromasourcetoa particulardestinationthrougharelaynetworkusingtheideaofnetworkowswith theoptimalapplicationofBC,MAandTStechniques.Thiscooperativetransmission schemeisdevelopedforarelaynetworkof N nodes,withawirelesslinkbetweeneach pairofnodes.Weconsideran N -nodewirelessrelaynetworkwithalinkjoiningeach pairofnodes.EachsuchwirelesslinkisdescribedbyabandpassGaussianchannelwith bandwidth W andone-sidednoisespectraldensity N 0 .Wedenotethepowergainofthe linkfromnode i tonode j as Z ij .Thelinkpowergainsareassumedtobeindependent andidenticallydistributedi.i.d.exponentialrandomvariableswithunitmean.This correspondstothecaseofindependentRayleighfadingchannelswithunitaveragepower gains.Moreover,weassumethateachnodehasamaximumpowerlimitof P andcan onlysupporthalf-duplextransmission.Notethatthismodelcanbeeasilygeneralizedto thecasewherechannelsmayhavenon-uniformaveragepowergainsforwhichnumerical examplesarepresentedinSection2.4,andwheredierentnodesmayhavedierent maximumpowerconstraints.Morespecically,thelattercasecanbeconvertedinto theuniformmaximumpowerconstraintscasebyabsorbingthenon-uniformityinthe transmitpowersintotheaveragepowergainsofthecorrespondinglinks.Inthesequel,we characterizethesystemintermsofthetransmitsignal-to-noiseratioSNR, S = P N 0 W attheinputofthelinks.Timeisdividedintounitintervals,andBCandMAareapplied 22

PAGE 23

withaTSstrategythatisoptimizedtomaximizethespectraleciencywhichwecall rate"hereafterforconvenience.Toavoidinterferencebetweenconcurrenttransmissions, atimeintervalisdividedintoslots: Duringtherstslot,thesourcemayBCtoalltheothernodesinthenetwork, subjecttoitspowerconstraint P Duringthesubsequentslots,arelaymayBCtoallothernodesexceptthesource node,oritmayreceiveowsfromallothernodesexceptfromthedestination throughMA. Duringtheverylastslot,thesourceandtherelaysmaysendinformationowsto thedestinationusingMA,withtheowsintheMAcapacityregioncorrespondingto amaximumtransmitpowerof P foreachnode. NotethattheforwardingofinformationbytherelaysisbasedontheDFapproach.For practicalityconsideration,itisassumedthatthephasesofthesimultaneouslytransmitted signalsfromdierentnodesarenotsynchronized.Ingeneral,fortheabovetransmission protocol,therewouldbeamaximumof2 N )]TJ/F15 11.9552 Tf 12.713 0 Td [(2+2=2 N )]TJ/F15 11.9552 Tf 12.713 0 Td [(2timeslotsoflengths t 1 ;t 2 ; ;t 2 N )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 respectively. Next,wedescribetheoptimizationproblemusingagraph-theoreticformulation. Deneagraph G = V;E ,where V isthesetofnodes, E isthesetofalllinksjoining thenodesinthegraph,andassociatethevectorr torepresenttheowratesassociated witheachlinkin E .Thus,thenumberofelementsinr equalsthecardinalityof E Forconvenience,wewrite G = V;E; r .Nowdenotethesourceby S ,thedestination by D ,andtherelaynodesby R 1 ;:::; R N )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 .Theslottingofaunittimeinterval,as describedabove,yieldssimplergraphsforeachtimeslot,thatwecall basicgraphs .A basicgraphiseitheroneinwhichaparticularnodemayBCtoseveralnodes,orin whichseveralnodestransmitviaMAtoaparticularnode.Thusforabasicgraph, weneedtoincludeonlythelinksbetweenthenodesthatmayparticipateduringthe concernedtimeslot.Forexample,assumethattherelay R 1 broadcaststoallnodes otherthanthesource,duringthe i -thtimeslot.Thebasicgraphisgivenby G i = V;E i ; r i where V = fS ; R 1 ; ; R N )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 ; Dg ;E i = fR 1 R 2 ; ; R 1 R N )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 ; R 1 Dg ; r i = 23

PAGE 24

x i R 1 R 2 t i x i R 1 R N )]TJ/F18 5.9776 Tf 5.757 0 Td [(2 t i x i R 1 D t i T ,where x i AB istheowfromnode A tonode B duringthe i -thtimeslot. Ingeneral,theproposeddesigninvolvesTSbetweenthebasicgraphstoyieldthe followingequivalentgraph G correspondingtoaunitintervalsee[34]forasimilaridea: G = V; [ i E i ; X i t i r i = t 1 G 1 + t 2 G 2 + ::: + t 2 N )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 G 2 N )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 : {1 wherethenumberofelementsineachvectorr i isextendedto j S i E i j byinsertingzeros appropriately.Thesecondequalityin2{1impliesthat G canbeviewedasalinear combinationofthebasicgraphs G i s,withtheequivalentsetofedgesgivenbytheunion ofthesets E i ,andtheequivalentowratevectorgivenbythelinearcombinationofthe individualowratevectorsr i .Further,thisresultsin G beingfullyconnected. Tomaximizethedataratefromthesourcetothedestinationthroughtherelay network,weneedtoconsidereachcutthatpartitions V intosets V s and V d with S2 V s and D2 V d resultingcutsetsaresuchthatonesetcontainsthesourcenode S andthe other,thedestinationnode D .Clearly,therecanbe2 N )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 suchpossiblecutsforthe N noderelaynetwork.Letthesecutsandthecorrespondingcutsetsbedenotedby C k V s k and V d k ,respectively,for k =1 ; 2 ; ; 2 N )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 .Further,forthegraph G ,foranytwonodes A 2 V s k and B 2 V d k ,thereexistsa cutedge AB thatcrossesthecut.Denotethetotal owthroughcutedge AB inaunittimeintervalby x AB = P 2 N )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 i =1 x i AB .Nowrecallfrom networkowtheory[35]thatthemaximumowratefromthesourcetothedestinationis speciedbytheminimalcutoftheequivalentgraph2{1.Consequently,wearriveatthe followingconvexowoptimizationproblemthatcanbesolvedusingstandardoptimization techniques: maxmin 0 B @ X A 2 V s 1 ;B 2 V d 1 x AB ; X A 2 V s 2 ;B 2 V d 2 x AB ; ; X A 2 V s 2 N )]TJ/F18 5.9776 Tf 5.756 0 Td [(2 ;B 2 V d 2 N )]TJ/F18 5.9776 Tf 5.756 0 Td [(2 x AB 1 C A {2 overallowallocations x i AB andalltimeslotlengths t i ,subjectto: 24

PAGE 25

the non-negativityconstraints : x i AB t i 0forallcutedges AB and i = 1 ; 2 ; ; 2 N )]TJ/F15 11.9552 Tf 11.955 0 Td [(2, the total-timeconstraint : t 1 + ::: + t 2 N )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 =1, the powercapacityconstraints : {foraBCslottheowratesshouldlieinthecapacityregionoftheBCchannel withthetransmittingnodehavingapowerconstraintof P {foranMAslottheowratesshouldlieinthecapacityregionoftheMA channelwithamaximumpowerconstraint P foreachtransmittingnode, the owconstraints :consideringsteadystateoperation,thetotalinformationow outofarelayshouldequaltheowintotherelayineachunittimeinterval. Notethatthedependenceoftheobjectivefunctiononthechannelgainsandthetime slotlengthsisimplicitlyexpressedthroughthecapacityconstraints.Denotethecut separating S fromalltheothernodesandthecutseparating D fromallnodesas C S and C D ,respectively.Thenweobservethatthecostfunctionin2{2abovecanbefurther simpliedtomaxmin f x C S ;x C D g ,where x C S = x SD + N )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 X j =1 x SR j and x C D = x SD + N )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 X i =1 x R i D {3 arethetotalowsacrosstheabove-mentionedcuts C S and C D ,respectively.Toseethis, considerthecut C with V s = fS ; R 1 ; ; R l g ,and V d = fR l +1 ; ; R N )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 ; Dg forsome l 2f 1 ; 2 ; ;N )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 g .Thetotalowacrossthiscutisgivenby x C = x SD + N )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 X j = l +1 x SR j + l X i =1 x R i D + N )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 X j = l +1 x R i R j : {4 Now,considernode i for i 2f 1 ; 2 ; ;l g .Accordingtotheowconstraintfornode i x R i D + N )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 X j = l +1 x R i R j + l X k =1 ;k 6 = i x R i R k = x SR i + N )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 X j = l +1 x R j R i + l X k =1 ;k 6 = i x R k R i : {5 Summing2{5overall i 2f 1 ; 2 ; ;l g weget l X i =1 x R i D + N )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 X j = l +1 x R i R j = l X i =1 x SR i + N )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 X j = l +1 x R j R i : {6 25

PAGE 26

Since P l i =1 P N )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 j = l +1 x R j R i 0,combining2{3,2{4and2{6gives x C x C S Similarly,wehave x C x C D .Thusthecostfunctionin2{2reducestotheabovementionedform. 2.3Generalized-linkSelectionandItsOptimality Inthissection,wepresenttheGLSprotocolandestablishtheoptimalityoftheFO andGLSprotocolsintermsoftheDMT.Thisisaccomplishedinthreesteps.First,we applytheFOprotocoltothethree-noderelaynetwork.Next,weproposetheGLSprotocolbasedonaselectionstrategythatissub-optimaltotheFOprotocolofSection2.2. Finally,theoptimalityoftheGLSprotocol,andthereby,thatoftheFOprotocol,is established. 2.3.1TheThree-nodeRelayNetwork Thethree-noderelaynetworkconsistsofasource S ,arelay R ,andadestination D .Wespecializethegeneraldesigndescribedintheprevioussectiontothisthree-node relaynetwork.Aunittimeintervalisdividedintotwotimeslotsoflengths t 1 and t 2 with t 1 + t 2 =1,andinformationisdividedinto3owsofdata x 1 x 2 ,and x 3 .Duringtherst timeslot, S sendsviaBCtwoowsofrates x 1 SD =t 1 = x 1 =t 1 and x 1 SR =t 1 = x 2 =t 1 to D and R ,respectively,resultinginthebasicgraph G 1 asinFig.2-1.Duringthesecondtime slot, R and S sendviaMAtwoowsofrates x 2 RD =t 2 = x 4 =t 2 and x 2 SD =t 2 = x 3 =t 2 to D respectively,resultinginthebasicgraph G 2 asinFig.2-1. Combiningthetwobasicgraphsyieldstheequivalentgraphas G = t 1 G 1 + t 2 G 2 .Note thattheinformationowofrate x 4 =t 2 sentby R duringtheMAtimeslotisfromtheow ofrate x 2 =t 1 itreceivedduringtheBCtimeslot.Thisgivesrisetotheowconstraintfor R ,i.e, x 4 = x 2 .Thus,wehavetheowconstraint x 4 = x 2 .Therateforthisnetworkis speciedbythemin-cutwhichisclearlymin f x 1 + x 2 + x 3 ; x 1 + x 4 + x 3 g .Hence,the owoptimizationproblemisgivenby: maxmin f x 1 + x 2 + x 3 ; x 1 + x 4 + x 3 g {7 26

PAGE 27

overowallocations x 1 ;x 2 ;x 3 ;x 4 ,andtimeslotlengths t 1 ;t 2 ,subjectto: non-negativityconstraints: x 1 ;x 2 ;x 3 ;x 4 0 ;t 1 ;t 2 0, total-timeconstraint: t 1 + t 2 =1, powerconstraints: S BC S;x 1 t 1 C Z SD S ;x 2 t 1 C Z SR S fortheBCslot, x 3 t 2 C Z SD S ;x 4 t 2 C Z RD S ;x 3 + x 4 t 2 C Z SD S + Z RD S fortheMAslot, owconstraint: x 2 = x 4 where C x =log+ x ,and S BC ,theminimumSNRrequiredforthesourcetobroadcast atrates x 1 =t 1 and x 2 =t 1 tothedestinationandtherelay,respectively,inthersttimeslot with0 < > : 1 Z SD e x 1 =t 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1+ 1 Z SR e x 1 =t 1 e x 2 =t 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1for Z SR >Z SD ; 1 Z SR e x 2 =t 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1+ 1 Z SD e x 2 =t 1 e x 1 =t 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1for Z SR Z SD : For t 1 =0, S BC =0.NotethatfortheBCslot,thelasttwoconstraintsareredundant when t 1 > 0,andcomplementtherstconstraintwhen t 1 =0. AsmentionedinSection2.1,theaboveoptimizationproblemformulationisdierent fromthatin[30]whereinthesumofthesourceandrelaypowers,requiredtoachieve acertaindatarate,isminimized.Morespecically,whenconsideringindividualpower constraintsforeachnode,wecannotusepart2of[30,Lemma3.1]todescribethepower constraintsfortheMAslot.Thisisbecausedoingsowouldrestricttheows x 2 and x 3 suchthatthesumofpowersexpendedat S and R isminimized.Ontheotherhand,in thepresentproblem,thepowerconstraintsonlydictatethattheow-ratesshouldliein theMAcapacityregionspeciedbythemaximumpoweravailableateachtransmitting node,fortheparticularfadingstate.WiththismodicationintheconstraintfortheMA slot,thesolutionapproachtotheaboveproblemneedstobemarkedlydierentfromthat in[30]. Tosolvetheoptimizationproblemin2{7weconsidertwocaseswithregardtothe linkgains:a Z SD Z SR ,andb Z SD
PAGE 28

problemintwostages:rst,wex t 1 ;t 2 0suchthat t 1 + t 2 =1andndtheoptimal ows x 1 ;x 2 ;x 3 intermsof t 1 ;t 2 ,andthen,ndtheoptimalvaluesfor t 1 ;t 2 tomaximize theobjectivefunction. Z SD Z SR .Forthiscase,thesource-destinationlinkisatleastasgoodasthe source-relaylink.Toobtainananalyticalsolutiontotheoptimizationproblemandbetter insightintothenatureofthesolutiontotheowoptimizationproblem,wemodifythe representationoftheBCslotpowerconstraintfromthatin2{7totheonethatis moreconventionallyusedtodescribethecapacityregionoftheGaussianBCchannel,as presentedin2{8.Usingtheowconstraintin2{7,werstsolve2{8forxed t 1 ;t 2 max x 1 + x 2 + x 3 over x 1 ;x 2 ;x 3 ;; subjectto{8 x 1 ;x 2 ;x 3 0 ; 0 1 ; x 1 t 1 C Z SD S ;x 2 t 1 C Z SR S 1+ Z SR S ; x 2 t 2 C Z RD S ;x 3 min f t 2 C Z SD S ;t 2 C Z SD S + Z RD S )]TJ/F21 11.9552 Tf 11.956 0 Td [(x 2 g : Here, 2 [0 ; 1]isthefractionoftotalpowerspentatthesourcetotransmit x 1 directly tothedestinationduringtheBCslot,and =1 )]TJ/F21 11.9552 Tf 12.252 0 Td [( .Although,thismodicationofthe BCslotpowerconstraintapparentlymakestheoptimizationproblemnon-convexowingto thenon-convexityin ,asweshallseeinthesequel,thisissuecanbehandledeasilyby utilizingthemonotonicityofthelogarithmfunction. Denotetheoptimalsolutionby x 1 ;x 2 ;x 3 ; andthecorrespondingmaximumrate by X t 1 ;t 2 .Itisclearthat x 1 = t 1 C Z SD S .Supposethat x 2 x 1 and x 2 = t 1 C Z SR 0 S 1+ Z SR 0 S .Thustheobjectivefunctionbecomes x 0 1 + x 2 + x 3 > x 1 + x 2 + x 3 at 0 .Thiscontradictstheoptimalityof x 1 ;x 2 ;x 3 ; Asaconsequence,wehave x 2 = t 1 C Z SR S 1+ Z SR S t 2 C Z RD S .Thisimpliesthat 28

PAGE 29

max 1 Z SR S 1+ Z SR S + Z RD S t 2 =t 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 | {z } a 0 ; 0 ; and 0 a 1 : Inessence,thismeansthattheoptimal x 1 and x 2 shouldlieontheboundaryofthedegradedBCcapacityregion.Withthis, itisobviousthenthat x 3 =min n t 2 C Z SD S ;t 2 C Z SD S + Z RD S )]TJ/F21 11.9552 Tf 11.955 0 Td [(t 1 C Z SR S 1+ Z SR S o Thereforetheoptimizationproblemof2{8canbere-writtenas: max x 1 + x 2 + x 3 {9 subjecttomax f 0 ; 0 a g 1 ;x 1 = t 1 C Z SD S ;x 2 = t 1 C Z SR S 1+ Z SR S ; x 3 =min t 2 C Z SD S ;t 2 C Z SD S + Z RD S )]TJ/F21 11.9552 Tf 11.955 0 Td [(t 1 C Z SR S 1+ Z SR S : Weobservethat x 3 = t 2 C Z SD S aboveifandonlyif 1 Z SR S 2 6 4 1+ Z SR S 1+ Z RD S 1+ Z SD S t 2 =t 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 3 7 5 1 a : {10 Comparingthistotheexpressionfor 0 a gives 0 a 1 a 1. Nextweconsidertwopossiblesub-cases: i. t 2 C Z RD S 1+ Z SD S t 1 C Z SR S : Inthiscase,wehave 1 a 0and t 2 C Z SR S C Z RD S 1+ Z SD S + C Z SR S t 2max .Themaximum ratecanbeexpressedas X t 1 ;t 2 =max f X 1 t 1 ;t 2 ;X 2 t 1 ;t 2 g where X 1 t 1 ;t 2 =max max f 0 ; 0 a g 1 a t 1 C Z SD S + t 2 C Z SD S + Z RD S = t 1 C Z SD 1 a S + t 2 C Z SD S + Z RD S t 1 log + Z SD 1 a S + Z SR S 1+ Z SR 1 a S + t 2 C Z SD S C Z SD S {11 X 2 t 1 ;t 2 =max 1 a 1 t 1 C Z SD S + t 1 C Z SR S 1+ Z SR S + t 2 C Z SD S = C Z SD S {12 wheretherstinequalityin2{11holdssince2{10isnotsatised,andthesecond inequalityin2{11holdssince 1 a 1andthattherstterminthepreviousstep ismonotonicallyincreasingin 1 a when Z SD Z SR .Thisway,thelastobservation 29

PAGE 30

helpsavoidthenon-convexityissuementionedbefore.Thevalueof X 2 t 1 ;t 2 in2{12isobtainedusingsimilararguments. ii. t 2 C Z RD S 1+ Z SD S >t 1 C Z SR S : Inthiscase,wehave 1 a < 0and t 2 >t 2max .Thusthemaximumrateisgivenby X t 1 ;t 2 =max 0 1 t 1 C Z SD S + t 1 C Z SR S 1+ Z SR S + t 2 C Z SD S = C Z SD S : {13 Hence,2{11{2{13implythatthesolutionto2{8,forany t 1 ;t 2 pair,occursat =1,andthesolutiontotheoriginalproblemof2{7,when Z SD Z SR ,isgivenby max f 0 t 1 ;t 2 : t 1 + t 2 =1 g X t 1 ;t 2 = C Z SD S withany t 1 ;t 2 pairsuchthat t 1 ;t 2 0and t 1 + t 2 =1. Z SD
PAGE 31

Inthiscasewehave 1 b 1and t 2 C Z SR S C Z RD S 1+ Z SD S + C Z SR S t 2max .Themaximum rate X t 1 ;t 2 canbeexpressedas X t 1 ;t 2 =max f X 1 t 1 ;t 2 ;X 2 t 1 ;t 2 g where X 1 t 1 ;t 2 =max 0 1 b t 1 C Z SD S 1+ Z SD S + t 1 C Z SR S + t 2 C Z SD S = t 1 C Z SD 1 b S 1+ Z SD 1 b S + t 1 C Z SR 1 b S + t 2 C Z SD S {15 X 2 t 1 ;t 2 =max 1 b min f 1 ; 0 b g t 1 C Z SD S 1+ Z SD S + t 2 C Z SD S + Z RD S = t 1 C Z SD 1 b S 1+ Z SD 1 b S + t 2 C Z SD S + Z RD S ; {16 andboththemaximain2{15and2{16areattainedat = 1 b .Substitutingthe expressionfor 1 b in2{15and2{16,weobtain X 1 t 1 ;t 2 = X 2 t 1 ;t 2 and X t 1 ;t 2 = t 1 log 0 B B @ 1+ Z SD S 1+ Z SD Z SR 1+ Z RD S 1+ Z SD S t 2 =t 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 1 C C A + t 2 C Z SD S + Z RD S : {17 ii. t 2 C Z RD S 1+ Z SD S >t 1 C Z SR S : Inthiscasewehave 1 b > 1and t 2 >t 2max with t 2max asinsub-casei.Thusthe maximumrateisgivenby X t 1 ;t 2 =max 0 1 t 1 C Z SD S 1+ Z SD S + t 1 C Z SR S + t 2 C Z SD S = t 1 C Z SR S + t 2 C Z SD S {18 wherethemaximumoccursat =1,asthefunctiontobemaximizedismonotonicallyincreasingin when Z SR >Z SD .Again,theapparentnon-convexityofthe optimizationproblem2{14in isavoidedbyconsideringthesumof x 1 and x 2 together,andutilizingthelastobservationregardingthemonotonicitypropertyof theobjectivefunctionin2{18. Finally,weoptimizetheabovesolutionto2{14overallpossibletimeslotlengthsto obtainthesolutiontotheoriginalproblemin2{7when Z SD
PAGE 32

Ontheotherhand,correspondingtoCaseii.,when t 2 >t 2max ,from2{18,wehave max f 0 t 1 ;t 2 : t 1 + t 2 =1 g X t 1 ;t 2 < )]TJ/F21 11.9552 Tf 11.955 0 Td [(t 2max C Z SR S + t 2max C Z SD S = C Z SR S C Z SD S + Z RD S C Z SR S + C Z RD S + Z SD S )]TJ/F21 11.9552 Tf 11.955 0 Td [(C Z SD S {20 wheretheinequalityin2{20isobtainedfrom2{18byusing Z SR >Z SD and t 2 >t 2max Hence,from2{19and2{20,weconcludethatwhen Z SR >Z SD ,themaximum achievablerateisgivenby X S =max 0 t 2 t 2max t 1 log 1+ Z SD S 1+ Z SD Z SR 1+ Z RD S 1+ Z SD S t 2 =t 1 )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 + t 2 C Z SD S + Z RD S Therefore,themaximumachievablerateofinformationtransmissionfromthesource S tothedestination D fordierentcasescanbesummarizedasunder: Themaximuminformationratefromthesource S tothedestination D fordierent casesissummarizedbelow: a Z SD Z SR :Themaximumrateis X S = C Z SD S withany t 1 ;t 2 pairsuchthat t 1 ;t 2 0and t 1 + t 2 =1.Thiscorrespondstodirectlytransmittingalldatathroughthe linkfromthesourcetothedestination,withoututilizingtherelay. b Z SD
PAGE 33

routingschemesbasedondierentnetworkpathselectionmethodsfailtoprovideacceptableoutageperformanceinhigh-ratesituations.TheGLSprotocoldescribedbelow providesasimplesub-optimaldesigntoaddressthiscomplexityissue.Inessence,theGLS protocolidentiesthe best relaypathoutofthepossible N )]TJ/F15 11.9552 Tf 12.226 0 Td [(2relaypathsandconsiders onlythechosenrelayalongwiththesourceanddestinationtoformathree-noderelaynetwork,whichwecalla generalized-link fromthesourcetothedestination,forinformation transmission.Inotherwords,theaimistochoosethebestrelaysuchthattheequivalent three-noderelaynetworkobtainedcontainingthesource,destinationandthechosen relaygivesthemaximumrateoverallpossibleequivalentthree-nodenetworkscontaining thesourceanddestination.Moreprecisely,weneedtoconsiderthefollowingpossibilities: Z SD Z SR i forall i 2 I = f 1 ; 2 ; ;N )]TJ/F15 11.9552 Tf 12.259 0 Td [(2 g :Fromtheresultsoftheoptimization problem2{7,itisclearthatthemaximumratewouldbe C Z SD S withdirect transmissionofalldatafromthesourcetothedestinationwithoutusinganyrelay. Thereexistsa k 2 I suchthat Z SR k >Z SD :Letthesetofallsuchnodeindices be K andforall i 2 I n K Z SD Z SR i .Forthiscase,choosethenode R 0 k asthe relaysuchthat k 0 =argmax k 2 K X k S ,where X k S isthemaximumrateforthe three-noderelaynetworkwiththesource S ,therelay R k anddestination D Intermsoftheworst-casecomputationalcomplexitiesfortheFOandGLSprotocols, itcanbeseenthat,foran N -noderelaynetworkwith N> 3,theFOprotocolinvolves amax-minoptimizationover2 N 2 )]TJ/F15 11.9552 Tf 12.47 0 Td [(2 N +2variablesallpossibleowsandtimeslot lengths,subjectto N )]TJ/F15 11.9552 Tf 10.153 0 Td [(1non-linearand2 N 2 )]TJ/F21 11.9552 Tf 10.154 0 Td [(N +1linearconstraints,whereastheGLS protocolinvolvesamaximumof N )]TJ/F15 11.9552 Tf 11.51 0 Td [(2maximizationsofanon-linearconcavefunctionover twovariables,subjecttotwolinearconstraints,followedbyndingthemaximumof N )]TJ/F15 11.9552 Tf 12.006 0 Td [(2 realnumberswithaworst-casecomplexityof O N )]TJ/F15 11.9552 Tf 12.148 0 Td [(2.Moreover,for N> 3,fortheFO protocol,theBCslotspotentiallyinvolve N )]TJ/F15 11.9552 Tf 12.102 0 Td [(1-and N )]TJ/F15 11.9552 Tf 12.102 0 Td [(2-levelsuperpositioncoding SPCordirtypapercodingDPCimplementationsfor S andtherelaysrespectively, whiletheMAslotsattherelaysand D mayinvolveamaximumof N )]TJ/F15 11.9552 Tf 12.304 0 Td [(3and N )]TJ/F15 11.9552 Tf 12.304 0 Td [(2 interferencecancelationICoperationsrespectively.Ontheotherhand,theGLSprotocol 33

PAGE 34

involvesamaximumof2-levelSPC/DPCandoneICoperationfortheBCandMAslots respectively,forany N> 3. 2.3.3Diversity-multiplexingtradeo Inthissection,weshowthatboththeFOandGLSprotocolsachievetheoptimal DMT.Asin[31],themultiplexinggainisdenedas r =lim S !1 R S log S where S istheSNR and R S istherateatanSNRlevelof S .Following[31],weparameterizethesystem,in termsoftheSNR S andthemultiplexinggain,0
PAGE 35

denitionofdiversityorder2{21,wehave d r lim S !1 )]TJ/F15 11.9552 Tf 11.291 0 Td [(log P out r;S log S : {22 Moreover,theaboveresultfrom[30]canbedirectlyusedtoprovethesamefortheGLS protocol.Usingthisfact,wederivealowerboundtothediversity-multiplexingtradeo thatcanbeachievedbytheGLSprotocol.Thesets I and K ,usedinthesequel,arethe setsofindicesasdescribedinSection2.3.2.TheoutageprobabilityfortheGLSprotocol isgivenby P gr out r;S =Pr max k 2 I X k S 0, onlythecardinalityof K issignicant.Letthe )]TJ/F22 7.9701 Tf 5.48 -4.379 Td [(N )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 i possibleconstructionsoftheset K berepresentedbyageneric"set K 0 withcardinality i .Withoutlossofgenerality,we describe K 0 astheset K correspondingtothecasewhentheindicesoftherelaynodesare orderedaccordingtotheirsource-relaylinkgains,i.e. Z SR 1 Z SR 2 Z SR N )]TJ/F18 5.9776 Tf 5.756 0 Td [(2 .Thus, CaseBnowimpliesasolitarychoiceforset K ,viz. K 0 = f 1 ; 2 ; ;i g .Therefore,from 2{23,wehave P gr out r;S =Pr C Z SD S
PAGE 36

Weobservethatusingtheright-mostexpressionof2{19insteadof X k S ,foreach k 2 K 0 ,in2{24givesanupperboundon P gr out r;S .ThisisutilizedinobtainingalowerboundonthediversityorderoftheGLSprotocol.Let f S n g 1 n =1 bean increasingunboundedsequenceofSNRswith S 1 > 1.Denethesequenceofrandomvariables f M k n g 1 n =1 f B k n g 1 n =1 and f A k n g 1 n =1 with M k n = C Z SR k S n )]TJ/F22 7.9701 Tf 6.587 0 Td [(C Z SD S n C Z SD S n + Z R k D S n B k n = C Z SR k S n log S n ,and A k n = X k S n log S n = B k n 1+ M k n ,respectively.Notethatforall k 2 K 0 M k n 0a.s.Thisimpliesthat B k n )]TJ/F21 11.9552 Tf 13.013 0 Td [(A k n 0a.s.Dene A 0 n =max k 2 K 0 A k n and B 0 n =max k 2 K 0 B k n = B 1 n .Thenusingtheabove,itcanbeseenthat B 0 n )]TJ/F21 11.9552 Tf 12.641 0 Td [(A 0 n 0 a.s.Further,lim n !1 Pr B 0 n 0, therstequalityin2{27isduetothelinkgainsbeingi.i.d.,andthesecondequalityin 2{27isobtainedbyusingL'Hospital'srule. 36

PAGE 37

Nextgivenan N -noderelaynetwork,considerthemultipleaccesscutthatseparates thedestinationfromalltheothernodes.Clearly,thetotalowacrossthiscutgivesan upperboundonthemaximumrateachievableinthe N -noderelaynetwork.Consequently, alowerboundontheoutageprobability P l out r;S canbeobtainedusingthemaximum sum-rateacrossthiscut: P l out r;S Pr C Z SD + Z R 1 D + + Z R N )]TJ/F18 5.9776 Tf 5.756 0 Td [(2 D S
PAGE 38

Forthefour-noderelaynetwork,therecanbe6possibletimeslotsintheFOprotocol asshowninFig.2-2:threeBCslotsforthesourceandthetworelaystotransmitinformation,andthreeMAslotsforthetworelaysandthedestinationtoreceiveinformation respectively.Toderiveanupperboundontheachievablerateandtherebyalowerbound ontheoutageprobability,weusemax-ow-min-cuttypeboundsforhalf-duplexcommunication.TherearefourpossibletimeslotsasshowninFig.2-3,withtherstBCslotand thelastMAslotatthedestinationsameasintheFOprotocol,butnow,thesourceand arelaymaytransmitsimultaneouslytotheotherrelayandthedestinationduringeachof theintermediateslotsoverinterferencechannels.Weusethemax-ow-min-cuttheoremto upperboundthemaximuminformationowinthesetwotimeslots. Fortheve-noderelaynetwork,therecanbe8possibletimeslotsintheFOprotocol -fourBCslotsforthesourceandthethreerelaystotransmitinformation,andfourMA slotsforthethreerelaysandthedestinationtoreceiveinformationrespectively.Similar tothefour-noderelaynetwork,forthemax-ow-min-cutbound,thereare8possible timeslotswiththerstBCslotandthelastMAslotatthedestinationbeingthesame asfortheFOprotocol,andmulti-source-multi-destinationtransmissionsduringthesix intermediateslots.Ingeneral,thefollowingmayoccurduringthesixintermediatetime slots:thesourceandarelaymaytransmitsimultaneouslytotheotherrelaysandthe destinationduringthesecond,thirdandfourthslots,andthesourceandtworelaysmay transmitsimultaneouslytotheremainingrelayandthedestinationduringthefth,sixth andseventhslots.Asinthecaseoffour-noderelaynetwork,weusethemax-ow-min-cut theoremtoupperboundthemaximumowofinformationduringtheintermediatetime slots. Withtheabovedivisionoftimeslots,theformalizationoftheproblemisdoneasin theprevioussections,andweusetheoptimizationroutineof[36]toobtainthemaximum achievableratesandupperboundsfordierentvaluesofrequiredrates.InFigs.2-4 and2-5,weplottheoutageprobabilitiesofthevariousschemeswiththerequiredrate 38

PAGE 39

R at1bit/s/Hzand6bits/s/Hzrespectively,forthefour-noderelaynetwork.Figs.2-6 and2-7presentthesamefortheve-noderelaynetwork.WhencomparedtotheFO protocol,theGLSprotocolsuersalossofaround1 : 0dB,andaround1 : 5dBwhen R iseither1bit/s/Hzor6bits/s/Hz,atanoutageprobabilityof10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(4 ,forthefour-and ve-noderelaynetworksrespectively.Ontheotherhand,theperformancedegradationfor themax-minselectionmethodof[27],ascomparedtotheFOprotocoloreventheGLS protocol,ismorethan12dBatanoutageprobabilityof7 : 0 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 ,when R =6bits/s/Hz forthefour-noderelaynetwork,andanexactlysimilarsituationcanbeobservedfor theve-noderelaynetwork.Moreover,forthefour-noderelaynetwork,theFOprotocol iswithin2 : 14dBwhen R =1bit/s/Hztowithin7 : 05dBwhen R =6bits/s/Hzof thelowerboundontheoutageprobabilitywhentheoutageprobabilityis10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(4 .Forthe ve-noderelaynetwork,thecorrespondingdierencesareapproximately3dBand9 : 6dB respectively.Thus,weseethatasthenumberofnodesintherelaynetworkincreases,the GLSprotocolbecomesmoresuboptimal,whereasthegapbetweentheoutageperformance oftheFOprotocolandthelowerboundwidens.Withregardtothelatterobservation, itshouldbekeptinmindthatthelowerboundobtainedusingthemax-ow-min-cut theoremis,ingeneral,notatightbound. Theperformancesofthedierentprotocolsforthefour-noderelaynetworkwith non-uniformaveragepowergainsarepresentedinFigs.2-8and2-9,andFigs.2-10 and2-11forcasesAandBrespectively,withtheaveragepowergainsasstatedinthe gures.CaseArepresentsthesituationwhenboththesource-relaylinksare,onaverage, betterthanthedirectlink,andonerelay-destinationlinkthelinkbetween R 1 and D is,onaverage,betterthantheother,resultinginrelay R 1 beingabettercandidateto forwardtheinformationthantheotherrelay.Ontheotherhand,caseBrepresentsthe situationwhennoonerelayhasverygoodsource-relayandrelay-destinationlinks.In thiscase,onesource-relaylinkis,onaverage,betterthanthedirectlink,which,inturn, isbetterthantheothersource-relaylink.Thereverseistruefortherelay-destination 39

PAGE 40

links,andtheinter-relaychannelis,onaverage,verygood.Thissituationpromotes inter-relayinteractionsfortheFOprotocol,andtherebyincreasesthedierencebetween theperformancesoftheFOandGLSprotocols.Thedierencesbetweentheoutage performancesoftheFOandGLSprotocols,atanoutageprobabilityof10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(4 ,are1 : 2dBor 1 : 0dB,and2 : 0dBor1 : 3dBwhen R =1bits/s/Hzor R =6bits/s/Hz,forcasesAand Brespectively.ThisreductioninthesuboptimalityoftheGLSprotocolwithincreasein therequireddataratecanbeexplainedbynotingthat,whentherequiredrateishigh, thecodinggainoeredbyaprotocolheavilyreliesontheecientuseofthedirectlink, andsincetheusageofthedirectlinkissimilarforboththeFOandtheGLSprotocols, theperformancegapnarrowsastherequireddatarateincreases.Ontheotherhand,at thesameoutageprobability,thedierencebetweentheoutageperformanceoftheFO protocolandthelowerboundincreasesfrom1 : 5dBto7dB,andfrom1 : 9dBto6 : 0dBas therequiredrateincreasesfrom1bit/s/Hzto6bits/s/Hz,forcasesAandBrespectively. Overall,theseresultsdemonstratetrendssimilartotheuniformaveragepowergaincase, andconrmthegeneralityoftheproposedprotocols. 2.5Summary Inthischapter,weproposedacooperativetransmissiondesignforageneralmultinodehalf-duplexwirelessrelaynetworkwherechannelinformationisavailableatthe nodes.Theproposeddesignisbasedonoptimizinginformationows,usingthebasic componentsofBCandMA,tomaximizethetransmissionratefromthesourcetothe destination,subjecttomaximumpowerconstraintsatindividualnodes.Motivatedby theneedforsimplernetworkpathselectionschemesthatperformwelleveninhighratescenarios,wedevelopedthegeneralized-linkselectionprotocolthatcombinesrelay selection,andowoptimizationforathree-noderelaynetwork.Theproposedprotocols wereshowntoachievetheoptimaldiversity-multiplexingtradeoforageneralrelay network.Simulationresultsforthefour-andve-noderelaynetworksforuniformand non-uniformaveragepowergainsdemonstratethattheperformanceofthemuchsimpler 40

PAGE 41

GLSprotocolisslightlyworsethanthatoftheFOprotocol.ThissuggeststhattheGLS protocolcanbeusedinsystemswithlow-complexityrequirements.Wealsonotethatthe proposedFOandGLSprotocolscanbeusedinwirelessnetworkswithtopologiesmore complicatedthatthewirelessrelaynetworkconsideredhere.Forexample,applicationof similarideastoaparallelrelaynetworkinwhichthereisnodirectconnectionbetweenthe sourceandthedestinationisconsideredin[37]. 41

PAGE 42

Figure2-1.Basicgraphs G 1 and G 2 forthethree-noderelaynetworkwith t 1 + t 2 =1. Figure2-2.FOprotocolforthefour-noderelaynetworkwith t 1 + + t 6 =1.Theow optimizationisperformedoverallows x 1 ; ;x 14 ,andalltimeslotlengths t 1 ; ;t 6 42

PAGE 43

Figure2-3.Transmissionstrategytoobtainalowerboundontheoutageprobabilityfor thefour-noderelaynetwork.Here t 1 + + t 4 =1,andtheoptimizationis over x 1 ; ;x 14 ,and t 1 ; ;t 4 ,withtheapplicationofthe max-ow-min-cuttheoremfortheintermediateslots. 43

PAGE 44

Figure2-4.Four-noderelaynetworkwithuniformaveragepowergains:Outage probabilitiesforrequiredrate R =1bit/s/Hz. Figure2-5.Four-noderelaynetworkwithuniformaveragepowergains:Outage probabilitiesforrequiredrate R =6bits/s/Hz. 44

PAGE 45

Figure2-6.Five-noderelaynetworkwithuniformaveragepowergains:Outage probabilitiesforrequiredrate R =1bits/s/Hz. Figure2-7.Five-noderelaynetworkwithuniformaveragepowergains:Outage probabilitiesforrequiredrate R =6bits/s/Hz. 45

PAGE 46

Figure2-8.Four-noderelaynetworkwithnon-uniformaveragepowergains.CaseA: E [ Z SR 1 ]=2 : 0 ; E [ Z SR 2 ]=2 : 0 ; E [ Z SD ]=1 : 0 ; E [ Z R 1 R 2 ]=1 : 0 ; E [ Z R 1 D ]= 1 : 5 ; E [ Z R 2 D ]=1 : 0.Outageprobabilitiesforrequiredrate R =1bit/s/Hz. Figure2-9.Four-noderelaynetworkwithnon-uniformaveragepowergains.CaseA: E [ Z SR 1 ]=2 : 0 ; E [ Z SR 2 ]=2 : 0 ; E [ Z SD ]=1 : 0 ; E [ Z R 1 R 2 ]=1 : 0 ; E [ Z R 1 D ]= 1 : 5 ; E [ Z R 2 D ]=1 : 0.Outageprobabilitiesforrequiredrate R =6bit/s/Hz. 46

PAGE 47

Figure2-10.Four-noderelaynetworkwithnon-uniformaveragepowergains.CaseB: E [ Z SR 1 ]=1 : 5 ; E [ Z SR 2 ]=0 : 75 ; E [ Z SD ]=1 : 0 ; E [ Z R 1 R 2 ]=3 : 5 ; E [ Z R 1 D ]= 0 : 2 ; E [ Z R 2 D ]=3 : 0.Outageprobabilitiesforrequiredrate R =1bit/s/Hz. Figure2-11.Four-noderelaynetworkwithnon-uniformaveragepowergains.CaseB: E [ Z SR 1 ]=1 : 5 ; E [ Z SR 2 ]=0 : 75 ; E [ Z SD ]=1 : 0 ; E [ Z R 1 R 2 ]=3 : 5 ; E [ Z R 1 D ]= 0 : 2 ; E [ Z R 2 D ]=3 : 0.Outageprobabilitiesforrequiredrate R =6bit/s/Hz. 47

PAGE 48

CHAPTER3 RESOURCEALLOCATIONANDCOOPERATIVEBEHAVIORINFADING MULTIPLE-ACCESSCHANNELSUNDERUNCERTAINTY 3.1Introduction Theresourceallocationproblemformulti-userwirelesssystemshasgenerated considerableinterestintheresearchcommunityandhasbeenconsideredfromdierent perspectiveswithregardtoeciencyandfairnessissues.ThefadingMACisoneofthe basicamongstsuchsystemsanddierentsolutionshavebeenproposedtilldatetothe above-mentionedproblem.ThethroughputcapacityregionofthefadingMAC,which canbeachievedbyusingdynamicpowerandrateallocationschemestomaximizethe averagerate,hasbeencharacterizedin[38].Usingideassimilartothoseforlong-term powercontrolin[39],theoutagecapacityregionforthischannelhasbeenderivedin[40], wheretheusershaveaveragepowerconstraints.Theauthors,in[40],obtainboththe commonoutagecapacityregion,whenalltheusershaveacommonoutageprobability constraint,andalsotheindividualoutagecapacityregion,whenusersmayhavedierent outagerequirements.Forthelattercase,thecapacityregionischaracterizedbyachannel usagerewardvector,whichdeterminestheactualoperatingpointforthesystem. Agame-theoreticapproachtowardssolvingtheresourceallocationproblemwith theaveragerateutilityanduserssubjecttoaveragepowerconstraints,isconsidered in[41].TheyproposeaStackelbergformulation,wherethereceiveristhegameleader andthetransmittersplayawater-llinggame,wheretheorderofdecodingoftheusers' information,whichimpliesaprioritizingoftheusers,maybedecidedbythereceiverusing anauctioningprocessasin[42].Alow-complexitydynamicrateallocationpolicy,that maximizesageneralconcaveutilityfunctionoftheratesinthethroughputcapacityregion forxedtransmissionpowers,ispresentedin[43].In[44],theoptimalpowercontrol schemeformaximizingthesum-capacityinthemultipleinputmultipleoutputMIMO fadingMACischaracterized. 48

PAGE 49

Intheaboveworks,theavailabilityofperfectchannelstateinformationCSIat thetransmittersandthereceiverisassumed.Theoptimalmediumaccessandresource allocationscheme,thatmaximizesthetotalthroughputforthefadingMACwithpartial CSI,isproposedin[45].ThepartialCSIiseitherintheformofathreshold-based1-bit quantizedfeedback,orthebest-userfeedback.Theoutagerateregionswithconstraints onindividualuseroutageprobabilitiesandnoCSIatthetransmitters,arederivedin[46], andtheyareusedtoobtainthebesttargetratevectortomaximizethesumthroughput atthereceiver. Aninformation-theoreticanalysisoftheachievablerateforthesingle-userfading channelandthefeasiblerateregionforthefadingMAC,withimperfectCSIatboth transmittersandthereceiver,ispresentedin[47].Thisisdonebyconsideringthe dierenceinthemutualinformationbetweentheinputsandtheoutputwithperfect andimperfectCSI,byassumingoptimaldecodingatthereceiverbasedontheknowledge regardingthedistributionofthefadingprocess.In[48],theauthorshavequantiedthe notionofimperfect"CSIbyshowingthatifthesideinformationissuchthatthesecond momentoftheerrorisnegligiblecomparedtothereciprocalofthesignal-to-noiseratio SNR,thenitcanbeconsideredtobeperfect",whereasotherwise,theachievablerates maybereducedasaresultoftheerrorsinthefadingstateestimation,andtheuseof Gaussiancodebooksandnearestneighbordecodingmaynotyieldgoodperformance. ThisisestablishedbyconsideringthegeneralizedmutualinformationGMI,whichgives thehighestrateforwhichtheaverageerrorprobability,averagedovertheensembleof Gaussiancodebooks,convergestozero.Itisprovedin[48]thatintheabsenceofCSI, theperformanceispoorforbothlow-andhigh-SNRoperations.Ontheotherhand, withpartialCSIavailable,forlow-SNRregimes,theuseofGaussiancodebooksand maximumlikelihoodMLdecodingperformsclosewithinaconstantfactortothe channelcapacity,butnotsointhehigh-SNRregime.TheeectofimperfectCSIonthe nite-SNRdiversity-multiplexingtradeoforthequasi-staticfadingMACisanalyzed 49

PAGE 50

in[49].ThecommonoutageprobabilityoftheMACisconsidered,andtheboundsonthe fadingMACfeasiblerateregionfrom[47]areusedtodemonstratetheeectofimperfect CSIonthenite-SNRdiversity-multiplexingtradeo. Inthiswork,wemodeltheresourceallocationproblemforthetwo-userfadingMAC usingatwo-personbargainingproblem[50],whereintheextentofcooperativebehavior isdeterminedbytheoutcomeofthebargainingproblem.Inthiswork,weconsiderthe situationwhentheutilityderivedbyeachuseristhe averagerate overallfadingstates. WhentheCSITisperfect,thesolutiontothebargainingproblem,speciedbytheNBS, yieldstheoptimaltransmissionstrategypairforthetwouserswithregardtofairness andeciencyconsiderations.Here,a transmissionstrategy foreachusercorrespondstoa choiceoftransmissionrateandpowerforaparticularfadingstate. WeconsiderthesituationwhereinthereceiverhasaccesstoperfectCSI,butthere existsacertainuncertaintyregardingtheCSITthatmaystemfromquantizationas, inreality,thefeedbackchannelsarelikelytohavelimitedcapacitiesand/orprediction errors.IftheavailableCSITisinaccurate,thetransmissionstrategypairsuggestedby theNBSmaydeviatefromthetrueoptimum,andthus,leadtoconsiderableperformance degradationintermsofthe trueutilities .Toovercomethislackofrobustnesswepropose aschemeinwhichtheconventionaltwo-personbargainingproblemisrelaxedtoacknowledgethefactthattheNBSmaynotgivetheoptimalstrategypair.Accordingtothis modiedbargainingproblemformulation,eachuserindependentlydecidesitstransmission strategyviaamaximinoptimizationfromitsrespectivesetofpossiblestrategies.Fora particularuser,suchasetisarangeoftransmissionstrategiesabouta nominalstrategy ThenominalstrategyisobtainingusingtheNBStotheoriginalbargainingproblem andtheavailableCSIT.Thisisincontrasttoconventionalbargainingproblemformulations,wherein,oncetheplayersreachanagreement,theyareboundtoexecutetheexact strategypairsuggestedbytheNBS. 50

PAGE 51

Inthefollowingsection,wepresentthesystemmodel.Thisisfollowedbythe descriptionofthebargainingproblemformulationinSection3.3.Themodiedbargaining problemisproposedinSection3.4alongwithnumericalresults.Finally,weconcludethe chapterinSection3.5. 3.2SystemModel Inthissection,wegiveabriefdescriptionofthesystemmodelforthefadingMAC andintroducetheformulationoftheresourceallocationproblemtocharacterizethe cooperativebehaviorbetweentwousersUsers1and2whowishtotransmitinformation toasinglereceiver.Notethattheformofcooperativebehaviorconsideredinthepresent workisnotthesameasusuallyinterpretedintheliterature,whereinoneusermay actuallyforwardtheinformationofanothertothedestination,andsuchasituationis studiedinChapter4.Asweshallseeinthesequel,fortheconventionaltwo-userfading MACconsideredhere,thelevelofcooperativebehaviorismanifestedbyhowmuchauser backso"fromitsmaximumpossibletransmissionrateforacertaintransmissionpower choice. Consideradiscrete-timetwo-userfadingMACwithunitbandwidth,inthepresence ofunit-varianceGaussiannoise,withthefadingstatedescribedbythepowergainvector Z = Z 1 Z 2 .Thepowergainsareassumedtobeindependentandidenticallydistributed i.i.d.exponentialrandomvariableswithunitmean. Let i bethemaximumtransmissionpoweravailabletothe i thuser.Weassume thatperfectCSIisavailableatthereceiver,whereastheCSIatthetransmittersmaynot beaccurate.Thetransmissionstrategiesaredeterminedintermsofthejointconditional probabilitydensityfunctionsPDFs 1 ofthetransmissionrateandpower,wherethe conditioningisonthefadingstate. 1 Inthiswork,weusethetermPDFtorefertobothcontinuousanddiscreteprobability functions. 51

PAGE 52

Wemodeltheresourceallocationproblemi.e.theoptimalchoiceofthetransmit powersandratesasatwo-userbargainingproblem.Thisspeciestheoperatingpoint ofthesystem.Notethatabargainingproblemformulationisanappropriatechoiceto modelthisproblemasitdoesnotpresumeanyinherentcooperationbetweenthetwo users.Instead,theusersnegotiatetoreachanagreementafterevaluatingselshlyand rationallythepotentialbenetsfromcooperationovertheeventofthemnotarrivingat anymutualagreement.Moreover,itiswellknownthattheNBScanbeinterpretedasa generalizedformofaproportionalfairnesssolution,andcoincideswiththelatterwhenthe payostothetwoplayersintheeventofdisagreementequalzero.Thus,theNBSprovides afairand ecient i.e.itisnotpossibletoimproveoneuser'sperformancewithout degradingtheperformanceoftheothersolutiontotheresourceallocationproblem. Unfortunately,owingtothedependenceoftheoperatingpointonthefadingstate,ifthe CSITisnotaccurate,theoperationpointobtainedwiththeerroneousCSITmaynot beoptimal.Inordertoobtainamorerobustsolutiontotheresourceallocationproblem underuncertainty,weproposearelaxedbargainingproblemformulationinthiswork. 3.3TheBargainingProblemfortheTwo-UserFadingMAC Inthissection,wesolvethetwo-userbargainingproblemtoobtaintheoptimal strategyforthetwousersusingtheavailableCSIT.Thus,forthetwoutilityfunction choices,weneedtosolvethetwo-personbargainingproblem,denedas T ; T d .Here, T isthesetoffeasibleutilities,i.e.theachievableaverageratesforthetwousers,and T d = T 1 ; d ;T 2 ; d representsthe disagreementpoint ,i.e.theutilityeachuserwillderive iftheydonotcooperate.Thus,thetwousersnegotiatetoreachanagreementregarding theoptimaltransmissionratesandpowers,giventhefadingstate.Moreover,itisassumed thattheuserscanagreetojointlyrandomizedstrategiesregardingthetheirtransmission ratesandpowers.Thedisagreementpointsforthisbargainingproblem,followedbythe NBS,arederivednext. 52

PAGE 53

3.3.1TheDisagreementPoint Here,weconsiderthecasewhenthereisnoagreementbetweenthetwouserswith regardtotheirtransmissionstrategies.Let R i and P i bethetransmissionrateand power,respectively,ofUser i .Also,givenaparticularfadingstateandthetransmissionstrategiesforthetwousers,theachievabletransmissionrateforUser i isgiven by R r i P Z ; R Z ; Z .Thatis,if,forthefadingstate Z ,theactionsforthetwo usersarespeciedbythetransmissionpowersandrates P Z = P 1 Z ;P 2 Z and R Z = R 1 Z ;R 2 Z respectively,thenthepayoreceivedbyUser i isgivenby R r i P ; R ; Z ,wherethedependenceof P and R on Z isnotmadeexplicitforbrevity. Particularly,theseutilitiescorrespondtotheratesforthetwousersatwhichreliable transmissioncanbesupportedwithanarbitrarilysmallprobabilityoferror.Dueto thesymmetricnatureoftheproblem,itissucienttoconsideranyoneuser'soperation,sayUser1.Intheabsenceofanyagreementbetweenthetwousers,theachievable transmissionrateforUser1, R r 1 ,isgivenby3{1,where C x =log+ x R r 1 P ; R ; Z = 8 > > > > > > < > > > > > > : R 1 if R 1 min f C Z 1 P 1 ; max f C Z 1 P 1 + Z 2 P 2 )]TJ/F21 11.9552 Tf -90.726 -28.69 Td [(R 2 ;C Z 1 P 1 1+ Z 2 P 2 oo ; 0otherwise : {1 Notethatforthescenariowhereintheusersfailtoreachanagreement,thereisno restrictiononthechoiceofthetransmissionstrategies,asagainstthescenariowhereinthe usersreachamutualagreementwherebyeachuser'schoiceofthetransmissionstrategy isrestrictedinsomespecicway.Sinceeachuserisunawareofthestrategyoftheother, wederivetheoptimalstrategyofeachuserasthesolutiontoamaximinproblem,wherein eachuser'saimistomaximizeitsownworst-caseusageprobability.Notethattheusers beingableto independently decidefromthesetofallrandomizedstrategiesimpliesthat, ingeneral,theusersmayusemixedstrategiesforthemaximingamesinthismodel.Thus, 53

PAGE 54

forUser1,theoptimaltransmissionstrategyis f R 1 P 1 j Z =argmax f R 1 P 1 j Z min f R 2 P 2 j Z E [ R r 1 ] ; {2 where E [ ]denotestheexpectationoperatorandtheexpectationisoverallfadingstates. Also, f R i P i j Z isthejointconditionalPDFofUser i 'stransmissionrateandpowersuchthat Pr P i > i =0for i =1 ; 2,withthelatterprobabilitiescomputedwithrespecttothe PDFs f P 1 Z and f P 2 Z ,respectively. Since R r 1 in3{1isamonotonicallynon-increasingfunctionof R 2 ,asolutionofthe minimizationproblemin3{2isgivenby f y R 2 P 2 j Z r 2 ;p 2 = r 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(C Z 2 2 p 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 2 : {3 Using3{3in3{2gives f R 1 P 1 j Z =argmax f R 1 P 1 j Z E [ ^ R r 1 ],suchthatPr P 1 > 1 =0.Inthe above, ^ R r 1 is ^ R r 1 = 8 > > < > > : R 1 if R 1 C Z 1 P 1 1+ Z 2 2 ; 0otherwise : Clearly,itcanbeseenthatthesolutiontotheaboveproblemisgivenby f R 1 P 1 j Z r 1 ;p 1 = r 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(C Z 1 1 1+ Z 2 2 p 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 1 : Therefore,whenthereisnoagreementbetweenthetwousers,themaximinoptimal achievableaverageratesare T 1 ; d = E h C Z 1 1 1+ Z 2 2 i and T 2 ; d = E h C Z 2 2 1+ Z 1 1 i for Users1and2respectively.Sincetheoperationpointforeveryfadingstate Z isinthe interioroftheMACcapacityregion,asmallperturbationin Z maynotcauseasignicant degradationintheactualachievableutilities. 3.3.2TheNashBargainingSolutionNBS WeusetheNBStoobtaintheoptimalutilityallocationforthisbargainingproblem usingtheavailableCSIT.Theoptimalallocationoftheusageprobabilities,usingtheNBS, 54

PAGE 55

isgivenby[50] Y =argmax Y 2Y Y 1 )]TJ/F21 11.9552 Tf 11.956 0 Td [(Y 1 ; d Y 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(Y 2 ; d ; {4 where Y denotesthecorrespondingutilityofinterest,and Y isthefeasiblesetofutility allocations.TheNBSyieldstheoptimalallocationofutilities Y betweenthetwo playersinthebargaininggame.Thus,forthepresentproblem,theNBSmaybeusedto obtaintheoptimalstrategypair f P R j Z i.e.theoptimaljointconditionalPDFforthe powers P Z = )]TJ/F15 11.9552 Tf 8.114 -6.662 Td [( P 1 Z ; P 2 Z andrates R Z = )]TJ/F15 11.9552 Tf 8.033 -6.662 Td [( R 1 Z ; R 2 Z ,wheretheuniqueness of f P R j Z isuptothecorrespondingachievableaverageratepair T ,thatis,weidentify anytwoagreementswhichyieldthesameutilityallocation.Asaresult,thepossible non-uniquenessofthetransmissionstrategiesdoesnotaecttheoptimalityoftheresource allocationsolution. BeforeproceedingwiththederivationoftheNBSweverifythatthenecessary conditions[50]ofatwo-personbargainingproblemaresatisedfortheaveragerateutility function.Thatthesetoffeasibleutilities, T ,iscompactcanbeprovedbyusingthefact thatthecapacityregionoftheMAC,foraparticularfadingstate,isclosedandbounded in R 2 .Also,thedisagreementpoint,denedby T d ,isin T .Thedenitionoftheutility functionasanaverageofthepayosreceivedineachfadingstate,andthefactthatthe MACcapacityregionismonotonicallyincreasinginthetransmissionpowersimplythat theusershavecontinuouspreferencerelationsonthesetofallconditionalPDFsforthe nominaltransmissionratesandpowers.Moreover,thesepreferencerelationssatisfythe vonNeumann-MorgensternVNMaxiomsofindependence,continuityandofbeing completeandtransitive.Further,thattheset f T 2T : T T d g isnonemptycanbe concludedfromtheobservationfromtheprevioussubsectionthatthedisagreementpoint correspondtothesystemoperatingstrictlyinsidethecapacityregionofthetwo-userMAC foranyfadingstate.Theconvexityof T canbeseenasfollows:if T 0 = T 0 1 ;T 0 2 canbe achievedbythejointconditionalPDF f 0 P R j Z ,and T 00 by f 00 P R j Z ,then,forany 2 [0 ; 1],the 55

PAGE 56

setofaverageratesdenedby T 0 + T 00 canbeachievedbytimesharingbetweenthe twoagreements. Hence,wemayobtainanoptimalconditionalPDFforthetransmissionratesand powersusingequation3{4.Thus,wehave T =argmax f T : R Z 2MAC P Z ; Z g )]TJ/F32 11.9552 Tf 5.48 -9.684 Td [(E [ R 1 Z ] )]TJ/F21 11.9552 Tf 11.955 0 Td [(T 1 ; d )]TJ/F32 11.9552 Tf 12.952 -9.684 Td [(E [ R 2 Z ] )]TJ/F21 11.9552 Tf 11.955 0 Td [(T 2 ; d ; {5 where P Z mustsatisfythemaximumpowerconstraintforeachuser.Itcanbeeasily seenthat T = 1 2 E C Z 1 1 + C Z 1 1 1+ Z 2 2 ; 1 2 E C Z 2 2 + C Z 2 2 1+ Z 1 1 ; {6 andthiscanbeachievedwiththefollowingrateandpowerallocation: R 1 = 1 2 C Z 1 1 + C Z 1 1 1+ Z 2 2 w.p.1 ; R 2 = 1 2 C Z 2 2 + C Z 2 2 1+ Z 1 1 w.p.1 ; {7 P i = i ; w.p.1 ; for i =1 ; 2 : Thatis,forthiscase,employingtheNBSforeachfadingstateachievestheoptimal solutiontothebargainingproblemof3{5.Thissolutionissimilarinavortothe onein[51],withthedierencebeinginthenatureofutilityfunctionsconsidered.More specically,theutilityweconsiderhereisanaveragemetric,whilethebargainingmodel in[51]considerstheutilitiesresultingfromasingleinstanceofthegameforaparticular stateofnature thefadingstateinthiswork.OneimportantpropertyoftheNBSsuggestedsolutionabovethatisofsignicancetothedevelopmentofthemodied bargainingproblemformulationinSection3.4isthattheoptimalchoiceoftransmission powersisdeterministicandindependentofthefadingstate. 56

PAGE 57

R r 1 R ; R ; Z = 8 > > > > > > < > > > > > > : R 1 if R 1 min f C Z 1 1 ; max f C Z 1 1 + Z 2 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(R 2 ; C Z 1 1 1+ Z 2 2 oo ; and R 1 2S 1 w.p. )]TJ/F21 11.9552 Tf 11.955 0 Td [( ; 0otherwise : Here,weemphasizethattheabovechoiceoftransmission rates asin3{7isnot theonlypossiblesolutiontotheNashbargainingproblem.Unlessotherconstraintsare imposed,anychoiceofjointlyrandomizedtransmissionratessatisfying3{6maybe selectedastheNBS-suggestedoptimalstrategy,withthechoiceofthetransmissionpowers asin3{7.Specically,theabovechoicehasbeenmadetofacilitateaneasyexposition ofthemodiedbargainingproblemformulationinSection3.4andalsotoyieldasimple solutionforpracticalimplementationwithlessoverhead.Withregardtothelatterpoint, notethatforthesolutionof3{7eachuseronlyrequirestheknowledgeofthechannel powergains Z obtainedfromtheCSITandcanmaximizetheNashproductof3{5 independently,whereasforachoiceofjointlyrandomizedratessatisfying3{6some formofcommunicationbetweenthetwousersneedstobeestablishedtoenablethejoint randomizationofthetransmissionrates. NotethattheNBS,asin3{6,suggeststhatnousertransmitsatitsmaximum possibletransmissionratei.e. C Z i i for i =1or2withprobability1,andthis backingo"ofeachuserfromitsmaximumpossibleratemaybeinterpretedasthe manifestationofitscooperativebehavior,motivatedbyarationalandindividualistic evaluationofthebenetsofcooperationasagainstanypresumedaltruismonitspart. Moreover,foranychoiceofjointlyrandomizedtransmissionratessatisfying3{6,the transmissionratepairwouldalwayscorrespondtoapointontheboundaryoftheMAC capacityregionforeveryfadingstate,therebymakingthesolutionverysensitivetothe CSIT.Inthefollowingsection,weproposeamodiedbargainingproblemtohandlethis robustnessissue. 57

PAGE 58

3.4TheModiedBargainingProblem 2 ThesensitivityoftheoperatingpointofthesystemtotheuncertaintyinCSIT maybereducedbydecreasingitsdependenceontheNBS-suggestedstrategypairas describedinthissection.Accordingtothemodiedbargainingproblem,theNBSsuggestedstrategypairhenceforth,the nominalstrategypair isobtainedusingthe availableCSITasinSubsection3.3.2,butinsteadofthetwousersbeingconstrained toimplementthesestrategies,theygettheexibilityof independently choosingtheir transmissionstrategiesfromacertainsetofstrategiesaboutthisnominalstrategypair. Let R i P ; R ; Z and P i P ; R ; Z for i =1 ; 2betheactualtransmissionrates andpowersrespectively,giventhenominaltransmissionstrategiesandavailableCSIT. Next,weutilizethefactthattheproposedsolutionto3{7suggestsusingthemaximum availablepowersforallfadingstates,andhence,weset P i P ; R ; Z = i w.p.1. Consequently,inwhatfollows,weshallnotrepresentthedependenceofthetransmission rateson P explicitly.Also,wedenethetransmissionstrategiesforthetwousersby theconditionalPDFofonlythetransmissionrates,withtheconditioningonthefading stateandthenominaltransmissionrates R .Thesetsofallowabletransmissionstrategies specifycertainlimiteddeviationsfromtheirnominalvaluestoaccountfortheuncertainty regardingtheCSIT.Denethesetsofallowedtransmissionratesforthetwousersas S i = R i : R i 2 R i )]TJ/F15 11.9552 Tf 11.955 0 Td [( R ; R i + R for i =1 ; 2.Thenthechoiceoftheactual transmissionstrategyofUser i i =1 ; 2issubjecttothefollowingconstraint: R i R ; Z 2S i w.p.1 )]TJ/F21 11.9552 Tf 11.955 0 Td [( ; {8 2 Aswouldbecomeclearfromthesubsequentdiscussion,wedonotmodifythebargainingproblemassuch.Onlytheimplicationsofreachinganagreementaremodied.Thus, thebargainingproblemstillstandsvalidasdescribedinSection3.3. 58

PAGE 59

forsome > 0thatcanbearbitrarilysmall.Here R apre-determinednon-negative constantdenotesthemaximumdeviationw.p. )]TJ/F21 11.9552 Tf 11.955 0 Td [( fromtherespectivenominal values 3 .ThisallowsustorobustlyhandletheinaccuracyintheCSITbyreducingthe sensitivityoftheoperatingpointandhence,thatofthetrueutilities,withrespecttothe CSIT.Notethattheprobabilitiesin3{8arecomputedwithrespecttothejointPDF f R i P RZ forUser i Withthisexibilityinchoosingthetransmissionrates,eachusercannomorebe certainoftheother'sexacttransmissionrate,i.e.forUser1, R 2 R ; Z isunknown,and vice-versa.Hence,anaturaloptionforeachuserwouldbeselectthetransmissionrate usingamaximincriteriontohandletheuncertaintyregardingtheexacttransmission rateoftheother.Moreover,foraparticularfadingstate,theactualpayoreceivedby User1,undertheconstraintsdenedby3{8,canbeinterpretedinthesamespiritas R r 1 P ; R ; Z in3{1,andisgivenby3{8.Accordingtothemaximincriterion,the optimalchoiceoftransmissionrateforUser1isgivenas: f R 1 j R ; Z =argmax f R 1 j R ; Z min f R 2 j R ; Z E R r 1 R ; R ; Z : {10 Here,weemphasizethatthismaximinoptimizationiscarriedoutusingtheavailable CSIT. Usingthevaluesof R Z from3{7,dene R Z C Z 1 1 )]TJ/F15 11.9552 Tf 14.508 3.022 Td [( R 1 = C Z 2 2 )]TJ/F15 11.9552 Tf 14.509 3.022 Td [( R 2 = 1 2 [ C Z 1 1 + C Z 2 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [(C Z 1 1 + Z 2 2 ] : Wepartitiontheentirespaceofpowergains R 2 + intotwosets Z = f Z : R < R Z g and itscomplement Z c .Withthispartitioning,wecanderivethemaximin-optimalstrategies 3 Althoughweassumeequaluncertaintyvaluesforthetwousers,thiscaneasilybegeneralizedtothecasewhereintheyaredierent. 59

PAGE 60

forthetwousersconditionedonthefadingstatebelongingtooneofthesesets.Next,we makethefollowingimportantobservation: Z 2Z c = R i + R C Z i i for i =1 ; 2 ; {11 and Z 2Z = R 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [( R >C Z 1 1 1+ Z 2 2 ; {12 witharelationanalogousto3{12beingtrueforUser2aswell.For Z > 0,let 0 =min f 1 ;= Z g .Foranychoiceof f R 1 j R ; Z ,thechoiceofUser2'stransmissionrate thatminimizestheaveragerateofUser1in3{10canbederivedas f y R 2 j R ; Z r 2 = 8 > > < > > : 0 r 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(C Z 2 2 + )]TJ/F21 11.9552 Tf 11.955 0 Td [( 0 )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [(r 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( R 2 + R if Z 2Z ; )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(r 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( R 2 + R otherwise : {13 Substitutingtheabovesolutionin3{10,themaximin-optimaldistributionforthe transmissionratecanbeshowntobethefollowingmixedstrategy: f R 1 j R ; Z r 1 = 8 > > < > > : r 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(C Z 1 1 1+ Z 2 2 if Z 2Z c ; 1 r 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(C Z 1 1 1+ Z 2 2 + )]TJ/F21 11.9552 Tf 11.955 0 Td [( 1 )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(r 1 )]TJ/F15 11.9552 Tf 11.956 0 Td [( R 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [( R otherwise ; where 1 = 8 > > < > > : 0 if C Z 1 1 1+ Z 2 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 0 )]TJ/F15 11.9552 Tf 8.033 -6.662 Td [( R 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [( R ; 0otherwise : Themaximin-optimaltransmissionratesforUser2canbederivedanalogously. Intuitively,whenthechannelpowergainsbelongtotheset Z c R iscomparatively large",andtherestrictionimposedthrough3{8losesitseectivenessasthewindow ofstrategiesaboutthenominalstrategypairbecomestoowide".Hence,bothusers selecttransmissionratescorrespondingtothedisagreementpointasinSection3.3.1.On theotherhand,when Z 2Z ,themaximin-optimalstrategyforeachuseristorespond totheworstcasemixedstrategyoftheotheruserasin3{13whilesatisfying constraint3{8.Hence,whentheerrorintheCSITintermsofthepowergainsisnot 60

PAGE 61

high,suitablyselectingasmall"valueof R canensurethatbothusersbackofromthe NBS-suggestedtransmissionratepairof R to R 1 )]TJ/F15 11.9552 Tf 12.073 0 Td [( R ; R 2 )]TJ/F15 11.9552 Tf 12.074 0 Td [( R withhighprobability i.e.both i and 0 canbemadesmallenoughwithachoiceofsmall R and in3{8, therebyresultingin Z 1.Notethat,inthiscase,themodiedbargainingproblem thatincorporatesthemaximincriterionleadstobothusersbackingoinasimilar fashionalthougheachuserindependentlychoosesitsrespectivestrategy.Thus,itcan beseenthatthemodiedbargainingproblemformulationprovidesageneralframework forresourceallocationfromanindividualisticperspectiveandacharacterizationofthe optimalstrategypairsintermsof R AsmentionedattheendofSubsection3.3.2,thenominaltransmissionratepair selectedin3{7isnottheonlypossiblechoice,andotherjointlyrandomizedtransmission ratesmayalsobeselected.However,thechoiceoftransmissionratesasin3{7,can beshowntoincurnolossofgenerality.Foranychoiceofjointlyrandomizednominal transmissionrates,themaximin-optimaltransmissionratescanbederivedintermsof theirconditionalPDFswiththeconditioningonthenominalstrategiesandfadingstate inthesamewayasabovetoyieldthesamemaximin-optimalvaluesfortheobjective functionsi.e. E [ R r 1 R ; R ; Z ]of3{10anditscounterpartforUser2asforthecase consideredhere. InFig.3-1,wepresentthenumericalresultsforthesystemdescribedin3.3.2with twodierentmodelsfortheerrorintheCSIT.Forthisexample,themaximumavailable powersateachuserare 1 =100mWand 2 =10mWrespectively.Weset =0 : 02 cf.3{8andconsidertwosimplemodelsfortheerrorintheavailableCSIT:ia 5%errorinCSITwiththetruechannelpowergainslessthanwhattheCSITsuggests, andii x %errorintheCSITwith x randomlychosenfromauniformdistributionover [ )]TJ/F15 11.9552 Tf 9.298 0 Td [(10 ; 10].The trueutilities arecalculatedinthefollowingway.Foraparticularfading stateandchoiceoftransmissionstrategypair,ifthetransmissionratesliewithintheMAC 61

PAGE 62

capacityregiondenedbythetransmissionpowersandthe true powergains,thenthese ratesareconsideredachievable.Else,boththeuserssueroutages. Forbotherrormodels,as R increasesfrom0bps/Hzto2 : 5bps/Hz,thetrueutilities increasefromtheminimumutilitypointswhentheerrorintheCSITisunaccountedfor, reachtheirrespectivemaxima,thatareveryclosetotheachievableratesfortheperfect CSITscenario,andthendecreasetoeventuallyconvergeatthedisagreementpoints. Hence,itcanbeconcludedthatwithaproperchoiceof R ,theproposedsolutioncan providethenecessaryrobustnessagainstinaccuraciesintheavailableCSIT.Moreover,as expected,itcanbeobservedthatthedisagreementpointisnotaectedsignicantlyby theerrorinCSITasitcorrespondstoaninteriorpointoftheMACcapacityregionfor anyfadingstate. 3.5Summary Optimalresourceallocationandcooperativebehaviorinatwo-userfadingMAC withuncertaintyregardingtheCSITareconsideredfromagame-theoreticperspective. Theresourceallocationproblemismodeledasatwo-userbargainingproblemwiththe averagerateutilityfunction.OwingtopossibleinaccuraciesintheCSIT,thesolutionto thebargainingproblemforinstance,theNBS,thatdependsontheCSIT,maynotbe optimal,andmaycausesystemoutages.Toaddressthislackofrobustnessamodication tothebargainingproblemformulationisproposed.Numericalresultsdemonstratethat theproposedsolutioncanbeusedtoprovidesignicantrobustnesswithoutanexplicit modelingoftheerrorintheCSIT. 62

PAGE 63

Figure3-1.Averagerateswithvarying R 63

PAGE 64

CHAPTER4 ACTIVEUSERCOOPERATIONINFADINGMULTIPLE-ACCESSCHANNELS 4.1Introduction Thegrowingemphasisonmulti-userwirelesscommunicationsystemsandtheeverincreasingdemandforhighdatarateshaveheightenedtheimportanceofresearchin theareaofusercooperation.Althoughwirelesssystemsbringforthdesignchallenges owingtomulti-userinterferenceandfadingconcerns,italsoprovidespotentialbenets, likethebroadcastnatureofthewirelessmediumanddiversityadvantagesinmulti-user systems.TheMACisoneofthefundamentalkindsofmulti-usercommunicationsystems. Conventionally,inaMAC,theuserstransmitdirectlytothedestination,andthecapacity regionofsuchasystemiswellknown.Recently,ithasbeendemonstrated[3,4]that therateregionofthefadingMACcanbeincreasedbyprovidingspatialdiversitythatis achievedbyusercooperationintheformofforwardingeachother'sinformationtothe destination,givingrisetotheCMAC. AlthoughthecapacityregionfortheCMACisyettobedetermined,therehave beenanumberofdierentcooperativetransmissionstrategiesproposedintheliterature. Twobroadclassesofworksreportedinthisregardcanbedistinguishedbasedonthe transceivercapabilitiesofthewirelessnodes,i.e.whetherthenodescansupportfullduplexcommunicationornot. In[3],theauthorsprovideasystem-leveldescriptionoftheCMACwhereinthenodes arecapableoffull-duplexcommunication.Theypresentanachievablerateregionbased onblockMarkovencodingandbackwarddecoding,andshowthepotentialincreaseinthe rateregionascomparedtotheconventionalMAC.Itisassumedthatthephaseofthe fadingisknowntothetransmittersandthisisexploitedtoperformcoherentcombining atthedestinationnode,andobtainbeamforminggain.In[4],theCDMAimplementation aspectsoftheschemein[3]areconsidered,whereintheauthorsproposetheuseof dierentspreadingcodestoobtaindierentchannelsforsimultaneoustransmission 64

PAGE 65

andreception,withouttheuseofcomplicatedechocancelationtechniques.Thepower allocationproblemfortheCMACwithfull-duplexnodesandfullCSIavailableatall nodeshasbeenaddressedin[55]and[56].In[55],theauthorsconsideraveragepower constraintsandcharacterizetheoptimalpowerallocationpoliciesthatmaximizetheset ofergodicratesachievablebyblockMarkovencodingandbackwarddecodingtechnique asin[3],byadimensionalityreductionapproach,i.e.bynotingthatsomeofthepower allocationsarezeroforeveryfadingstate.Amoredirectapproachtosolvethesimilar problemofoptimalpowerallocation,withanalmostclosed-formsolution,ispresented in[56].Ithasbeenestablishedin[57]thatwindoweddecodingissucienttoachievethe samesum-rateasbackwarddecodingfortheblockMarkovsuperpositionencodingscheme fortheCMAC. TheoptimalpowerandresourceallocationproblemfortheCMACwithnodes capableofhalf-duplexcommunicationisconsideredin[58],whereitisassumedthat fullCSIisavailableatallthenodes,andthetransmitterscooperatebyrelayingeach other'sinformationoverorthogonalfrequencybandsortimeslots.Thesolutiontothe problemispresentedasatwo-stepconvexoptimizationproblemformulation:rst,fora particularbandwidthortimesharingparametervalue,theoptimalpowerallocationis characterizedbyaconvexoptimizationproblem,andthen,theoptimalresourcesharing timeorbandwidthparameterisobtainedasasolutiontothequasi-concaveproblemof maximizingtherateofoneuser,givenatargetratefortheother.Alltheworksmentioned aboveuseaDFapproachfortherelayingofinformationtothedestination. In[19],theauthorspresentacooperativetransmissionscheme,basedonthenonorthogonalamplify-and-forwardNAFtechnique,thatisprovedtoachievetheoptimal diversity-multiplexingtradeoof N )]TJ/F21 11.9552 Tf 13.085 0 Td [(r forthe N -userhalf-duplexCMAC,with symmetricdataraterequirementandCSIavailableonlyatthereceivingnodeofanylink. Accordingtotheproposedstrategy,timeisdividedintocooperationframesoflength N cooperationsymbols,andeachusertransmitsonlyonceduringacooperationframe.Every 65

PAGE 66

userisallotteduniquetransmissionandreceptionsymbolintervals,usingaparticular schedulingpolicy,andittransmitsalinearcombinationofitsownsymbolandthesignal observedduringitsmostrecentreceptionsymbolinterval,therebycreatinganarticial inter-symbolinterferenceISIchannel.Asetof L cooperationframesarecombinedto formasuper-frame,andtheassignmentofthereceptionsymbolintervalsisscheduled foreachsuper-frame,withthelengthsofsuper-framesandcodeschosensuchthata coherence-intervalconsistsof N )]TJ/F15 11.9552 Tf 12.016 0 Td [(1consecutivesuper-frames,andthatallcodewordsspan theentirecoherenceinterval. SimilartotheaboveNAFstrategy,acooperativetransmissionschemeforthetwouserCMAC,basedonsuperpositioncoding,hasbeenproposedin[59].Thisschemeuses atimedivisionapproachinwhichausersimultaneouslytransmitsitsowninformation andtheotheruser'sinformationbyusingthesuperpositioncodingSPCtechnique. Thisschemeisdemonstratedtoachieveagainofabout1 : 5 )]TJ/F15 11.9552 Tf 12.531 0 Td [(2dBovertraditionalDF approachesforrelaying,andatthesamelevelofsystemcomplexityofthelatter.An extensionofthisideatothegeneral N -userCMACispresentedin[60],whereinthe authorsprovetheoptimalityoftheproposedschemeinachievingtheoptimaldiversitymultiplexingtradeoforthesymmetricraterequirementscenario. Inthiswork,weproposeow-theoreticcooperativetransmissionprotocolsforthe two-userCMAC.First,wepresentanorthogonalrelayingprotocolfortheCMACORCMAC,whereineachuseractsasadedicatedrelayfortheotherinatime-division fashion.Theow-optimizedrelayingapproachofChapter2,modiedtoincorporate coherentcombiningatthedestinationisusedfortheconstituentrelaychannels.This relayingprotocolhasbeenshowntoachievetheoptimaldiversityorderandprovidebetter codinggainsfortherelaychannelascomparedtotraditionalDFrelayingmethods,by ecientlyutilizingtheCSIavailableatallnodes.Next,weproposetheow-optimized protocolfortheCMACFO-CMACthatdecomposestheCMACintotwobroadcast 66

PAGE 67

channelsBCandamultipleaccessMAchannelwithcommoninformation.Theboundariesoftheachievablerateregionsarecharacterizedbymeansofconvexoptimization formulations.Theimprovementprovided,intermsoftheachievablerateregion,byORCMACandFO-CMAC,overconventionalMACcapacityandtheDFstrategyof[58] withoutpowercontrol,increasesastheamountofdisparitybetweenthechannelsfrom thetwosourcestothedestinationincreases.Theoutageperformancesoftheproposed protocolsindicatethatalthoughthemuchsimplerOR-CMACissuboptimalintermsof theachievablerateregion,itprovidesoutageperformancethatiswithin1dBofthatof FO-CMAC.Moreover,boththeproposedprotocolsachieveadiversityofordertwoforthe requiredrateregionofinterest. Therestofthechapterisorganizedasfollows.InSection4.2,theow-theoreticprotocolsofOR-CMACandFO-CMACarepresented,andtheboundariesoftheachievable rateregionsarecharacterizedbyconvexoptimizationformulations.Thisisfollowedbynumericalresultsinpresentingtheachievableaveragerateregionsandoutageperformances fordierentscenariosinSection4.3.Finally,theprimarycontributionsinthischapterare summarizedinSection4.4. 4.2Flow-theoreticTransmissionProtocolsfortheCooperative Multiple-AccessChannel Consideratwo-userCMACwherethetwosources S 1 and S 2 mayactivelycooperatetotransmitinformationtoacommondestination D .Weusethephrase active cooperation todistinguishbetweenthecooperationinvolvedintransmissionstrategiesin whichonesourcemayforwardtheother'sinformation,andthatintheconventionalMAC, whereinausertransmitsataratelowerthanthemaximumsingle-userratepossiblefor theparticularchannelstateandpowerexpended.ThequanticationofthistypeofcooperativebehaviorwasstudiedinChapter3.Thus,asahigherlevelofcooperativebehavior, theusersmayrelaytheinformationofeachotherbyutilizingthebroadcastadvantageof 67

PAGE 68

thewirelessmedium,givingrisetotheCMACmodel.Weconsideradiscrete-timetwouserfadingMACwithunitbandwidth,inthepresenceofunit-varianceGaussiannoise, withthefadingstatedescribedbythepowergainvector Z = Z S 1 S 2 Z S 1 D Z S 2 D .Note that,owingtothereciprocityofchannels,weassume Z S 1 S 2 = Z S 2 S 1 .Thepowergainsfor thewirelesslinksaremodeledasindependentexponentialrandomvariables.Weconsider twotypesoffadingmodelsinthiswork.First,inSection4.3,weconsiderthesituation inwhichthechannelsareergodicwithinatransmissionblock,forwhichweevaluatethe transmissionprotocolsusingaverageratesastheperformancemetric.Then,wepresent theoutageperformanceoftheproposedprotocolsforthemodelinwhichthefadingisnot fastenoughandhence,thechannelsmaynotbeergodicduringatransmissionblock.This systemmodelcanbeeasilygeneralizedtothecasewherethebandwidth W 6 =1. Let P i bethemaximumtransmissionpoweravailabletothesource S i .Inthiswork, weconsidershorttermpowerconstraintsonly,andthisprecludesanypotentialadvantage ofpowerallocation.WeassumethatfullCSIisavailableatallnodesofthesystem. Moreover,asapracticalconsideration,weassumethatthenodesarenotcapableof transmittingandreceivinginformationsimultaneouslyoverthesamefrequency,i.e.they aresubjectedtoahalf-duplexconstraint.Inthefollowingsubsections,wepresenttwo protocolsbasedonow-theoreticdesignstodevelopcooperativetransmissionschemesfor informationtransmissionfromsources S 1 and S 2 todestination D 4.2.1OrthogonalRelayingProtocolfortheCMACOR-CMAC Inthissubsection,wepresentasimplecooperativetransmissionprotocolbasedon theconventionalrelayingapproach.Timeisdividedintounitintervals.Owingtothe half-duplexlimitationofthesources,thetwosourcescannotrelayeachother'sinformation atthesametime.Toaddressthis,wedivideeachunitintervalintotimeslotsoflengths T 1 and T 2 .Duringtimeslot T 1 ,source S 2 solelyassistssource S 1 ,byactingasadedicated relayto S 1 ,totransmitthelatter'sinformationtothedestination D .Thereversehappens duringtimeslot T 2 .ThisisdepictedinFigure4-1. 68

PAGE 69

Thus,weeectivelyhavetworelaychannelsovertwoorthogonaltimeslots.Weuse theow-optimizedtransmissionschemeofChapter2forthethree-noderelaychannel. Accordingtotheprotocolpresentedtherein,eachtimeslot T 1 resp. T 2 isfurtherdivided intotwosub-slotsoflengths t 1 and t 2 resp. t 0 1 and t 0 2 .Considerthetimeslot T 1 .During therstsub-slot, S 1 sendstwoindependentowsofinformation x 1 and x 2 to D and S 2 respectivelyusingabroadcastchannelBC,andinthesecondsub-slot, S 2 forwards x 2 to D andatthesametime S 1 sendsoutanotherinformationow x 3 to D ,and x 2 and x 3 arereceivedat D viamultiple-accessMA.Toimprovetheachievablerateseven further,wemodifythesecondsub-slotasfollows.Sincetheow x 2 ,that S 2 forwardsto D originatedat S 1 ,thelatterisawareofit,andhence,wemodifythesecondsub-slotfroma conventionalMACtoaMACwithcommoninformation[61],where x 2 formsthecommon informationbetween S 1 and S 2 x 3 istheindependentinformationfrom S 1 ,and S 2 does nothaveanyindependentinformationtotransmit.Forthisrelayingscheme,maximizing theoveralltransmissionratefrom S 1 to D canbeformulatedasthefollowingoptimization problem: max x 1 + x 2 + x 3 over t 1 ;t 2 ;x 1 ;x 2 ;x 3 ;P 1 ; 0 {1 subjectto non-negativityconstraints: x 1 ;x 2 ;x 3 0; t 1 ;t 2 0; total-timeconstraint: t 1 + t 2 = T 1 ; capacitypowerconstraints: P BC P 1 ; x 3 t 2 C Z S 1 D P 1 ; x 2 + x 3 t 2 C Z S 1 D P 1 + 0 ; 0 P 1 P 1 ; 0 Z S 1 D P 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(P 1 + Z S 2 D P 2 +2 q Z S 1 D P 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(P 1 Z S 2 D P 2 ; where C x =log+ x and P BC ,theminimumpowerrequiredforthesourcetobroadcast atrates x 1 =t 1 and x 2 =t 1 tothedestinationandtherelay,respectively,intherstsub-slot 69

PAGE 70

with0 < > : 1 Z S 1 D e x 1 =t 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1+ 1 Z S 1 S 2 e x 1 =t 1 e x 2 =t 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1for Z S 1 S 2 >Z S 1 D ; 1 Z S 1 S 2 e x 2 =t 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1+ 1 Z S 1 D e x 2 =t 1 e x 1 =t 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1for Z S 1 S 2 Z S 1 D : For t 1 =0, P BC =0. P 1 isthepowerthat S 1 allocatesforthedirecttransmissionof x 3 to D ,and 0 denotesthepowercorrespondingtothecommoninformationow x 2 ,received at D Itcanbecheckedthattheabovemaximizationproblembelongstotheclassof convexoptimizationproblems,andstandardnumericaltechniquescanbeusedtoobtaina solution.Letthesolutionof4{1bedenotedby X Z ;T 1 .Thenitcanbeeasilyseenthat X Z ;T 1 = T 1 X Z ; 1.Forthesecondtimeslot,anexactlysimilaroptimizationproblem canbeformulatedas4{1withappropriatechangesintheindices. Therefore,foraparticularfadingstate Z ,theregularpointsontheboundaryofthe achievablerateregioncanbeobtainedbymaximizingaconvexcombinationoftherates ofthetwosourcesduringtherespectivetimeslots, X Z ;T 1 and Y Z ;T 2 .Thatis,by solvingthefollowingoptimizationproblem,forsome0 << 1, max X Z ;T 1 + )]TJ/F21 11.9552 Tf 11.955 0 Td [( Y Z ;T 2 {2 subjectto T 1 ;T 2 > 0 ; and T 1 + T 2 =1 : Theextremepointsoftheboundaryregion,i.e.maximizingonlyonesource'srateare givenby T 1 =1 ;T 2 =0,etc.Unfortunately,thisnaiveschemeofdecomposingtheCMAC intotwoorthogonalrelaychannelsdoesnotentailthebestutilizationofresources,and asweshallseeinthefollowingsubsection,thiscanbeimproveduponbyamoreecient ow-optimizedtransmissionprotocol. 4.2.2Flow-optimizedProtocolfortheCMACFO-CMAC InsteadofdividingtheCMACintotwoseparaterelaychannels,wedivideaunit intervalintothreetimeslots T 1 ;T 2 ;T 3 withnoorthogonalizationoftherelayingactions. 70

PAGE 71

Now,thersttwotimeslotsareBCslotsandthelastoneisanMAslotwithcommon information,asshowninFigure4-1.Duringthersttimeslot, S 1 transmitstwoindependentows: x 1 + y 21 and x 2 to D and S 2 respectivelyusingBC.Similarly, S 2 transmitstwo independentows: y 1 + x 21 and y 2 to D and S 1 respectivelyusingBC.Finally, S 1 and S 2 sendtwoows x 3 + y 22 and y 3 + x 22 to D usingMAwithcommoninformation.Here y 21 and y 22 aretwopartsoftheinformationow y 2 that S 1 receivedfrom S 2 duringthe previousunitinterval.Similarly, x 21 and x 22 constitutetheamountofinformationthat S 2 relaysfor S 1 .Thus,forthisscheme,theowconstraintsimplythat x 2 = x 21 + x 22 ,and y 2 = y 21 + y 22 .Also,forthelasttimeslot, x 22 and y 22 areknowntoboththesources,and hence,theyformthecommoninformationtobetransmittedto D Hence,thetotaltransmissionratesfromsources S 1 and S 2 aregivenby X = x 1 + x 21 + x 22 + x 3 and Y = y 1 + y 21 + y 22 + y 3 respectively.Forthisscheme,theboundary oftheachievablerateregioncanbecharacterizedasfollows:theregularpointsonthe boundarycanbeobtainedbymaximizingaconvexcombinationoftherates X and Y ,and theextremepointscorrespondtotheCMACdegeneratingintorelaychannelswithone sourcesolelyactingasarelayfortheother.Forcompactness,lettheinformationows correspondingtothetwosourcesberepresentedbythevectors x = x 1 x 21 x 22 x 3 and y = y 1 y 21 y 22 y 3 .Themaximizationproblemthatneedstobesolvedtoobtainthe regularpointscanbeformallystatedasgivenbelow: max X + )]TJ/F21 11.9552 Tf 11.956 0 Td [( Y for 2 ; 1over T 1 ;T 2 ;T 3 ; x ; y ;P 1 ;P 2 ; 0 ; {3 subjectto non-negativityconstraints: x ; y 0 ; T 1 ;T 2 ;T 3 0; total-timeconstraint: T 1 + T 2 + T 3 =1; capacitypowerconstraints: P 1 BC P 1 ; P 2 BC P 2 ; x 3 T 3 C Z S 1 D P 1 ; y 3 T 3 C Z S 2 D P 2 ; x 22 + x 3 + y 22 + y 3 T 3 C Z S 1 D P 1 + Z S 2 D P 2 + 0 ; 71

PAGE 72

0 P 1 P 1 ;0 P 2 P 2 ; 0 Z S 1 D P 1 )]TJ/F21 11.9552 Tf 11.956 0 Td [(P 1 + Z S 2 D P 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(P 2 +2 q Z S 1 D P 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(P 1 Z S 2 D P 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(P 2 : Asintheprevioussubsection, P 1 BC and P 2 BC aretheminimumpowersrequiredby S 1 and S 2 respectivelyforthetwoBCslots. P 1 BC isdenedasin4{4and P 2 BC isdened similarly.Also, P 1 and P 2 arethepowersallocatedby S 1 and S 2 totransmit x 3 and y 3 respectively,and 0 denotesthereceivedpowerat D correspondingtothecommon information x 22 + y 22 P 1 BC = 8 > < > : 1 Z S 1 D )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(e x 1 + y 21 =T 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 + 1 Z S 1 S 2 e x 1 + y 21 =T 1 )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(e x 2 =T 1 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 for Z S 1 S 2 >Z S 1 D ; 1 Z S 1 S 2 )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(e x 2 =T 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 + 1 Z S 1 D e x 2 =T 1 )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(e x 1 + y 21 =T 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 for Z S 1 S 2 Z S 1 D : {4 For T 1 =0 ;P 1 BC =0 : Onceagain,itcanbecheckedthattheabovemaximizationisaconvexoptimization problemthatcanbesolvedusingstandardnumericaloptimizationmethods.Thus,FOCMACaddressesthehalf-duplexlimitationofthenodesbydividingtheCMACintotwo BCandoneMACwithcommoninformation,andprovidesamoreecientutilizationof systemresourcesascomparedtoprotocolsusingtwoseparaterelaychannels. 4.3NumericalResults Inthissection,wepresentsomenumericalresultstodemonstratetheperformance oftheproposedprotocolsandcomparethemtotheconventionalMACandtheDF-based strategyproposedin[58].Figures4-2through4-4showtheachievableaveragerateregions forthevariousschemesfordierentscenarios.Weconsiderdierentmeansofthefading gainsasstatedinthegures.Considerablevariationsinthestatisticsofthefadinggains forthechannels Z S 1 D and Z S 2 D canoccurinpracticalsituationsowingtodierentpath lossandshadowingeectsanddierentamountsofscatteringforthetwodirectlinks. Thus,theasymmetricsituationcorrespondstothecasewhenthedirectlinkfromone sourcetothedestinationismuchworsethantheother. 72

PAGE 73

InFigure4-2,thechannelfrom S 2 to D ismuchworsecomparedtothatfrom S 1 to D ,andthelargeincreaseintheachievablerateregionfortheFO-CMACrateregion isevident.Moreover,weseethatthemuchsimplerOR-CMACperformsclosetoFOCMACforthisscenario.Ontheotherhand,theperformanceofthestrategyof[58]is muchpoorerthantheabovetwo.Recallthat,inthiswork,weconsideronlyshortterm powerconstraint,andunlike[58],wedonotconsiderthepowerconstraintovertheentire unitinterval.Thiseliminatesanypotentialgainsfromoptimizingthepowerallocations forthedierenttransmissions.So,forthepresentsystem,thecooperativetransmission strategyof[58]isclearlysuboptimal.Figure4-3ispresentedtohighlighttheincreasein theachievablerateregionresultingfromtheuseofMAslotswithcommoninformationas inFigure4-2insteadoftheconventionalMAslots.AscanbeseenfromFigure4-3,with theuseofconventionalMAslots,themaximumsum-ratefortheFO-CMACcoincides withthatoftheconventionalMACwithoutactivecooperationbetweenthetransmitters. TheimprovementinFigure4-2canbeinterpretedastheeectofthebeamforminggain asinatwo-transmitterone-receiverMISOsystem.Figure4-4presentsthesymmetric situation,whenthetwodirectlinksarestatisticallyidentical.Weseethatinthiscase,the increaseintherateregionisnotaspronouncedasintheprevioussituation.Moreover, theachievablerateregionforthestrategyof[58]liesstrictlyinsidethatforthebaseline systemofMACwithoutactivecooperationbetweenthetransmitters. AsmentionedinSection4.2,forthesituationinwhichthefadingisnotfastenough sothattheergodicpropertiesofthechannelsareobserved,theoutageperformanceof thetransmissionstrategiesareamorereasonableperformancemetricasagainstaverage rates.Theoutageperformancesoftheproposedprotocolsareevaluatedwhenthedata raterequirementissymmetricat K =1bit/s/Hzforbothusers.Theoutageeventis denedsimilartothedenitionin[62],i.e.itistheunionoftheeventsthateitherone orbothoftheuserssueranoutage.Theoutageperformancesoftheproposedprotocols arecomparedtothatoftheconventionalMAC,thestrategyof[58],andalowerboundon 73

PAGE 74

theoutageprobabilitythatisobtainedbyconsideringthecasewhenthesourceshavea perfectnoiselesschannelbetweenthem.Figure4-5presentstheoutageperformancesfor theasymmetricsituationasintheaverageratescase,andFigure4-6presentsthesame forthesymmetricsituation.WeseethattheOR-CMACsuersalossofonly0 : 7dBand 1 : 0dBcomparedtoFO-CMAC,atanoutageprobabilityof10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 ,fortheasymmetric andsymmetricsituationsrespectively.TheperformanceofFO-CMACisworsethanthe lowerboundbyabout2 : 2dBand2 : 5dBatthesameoutagelevel,fortheasymmetric andsymmetricsituationsrespectively.Ontheotherhand,theperformanceoftheDF strategyof[58]issignicantlypoorer,andevenworsethantheconventionalMACforthe symmetricsituation.Anotherimportantobservationfromtheoutageperformanceplotsis thattheslopesofthecurvesforFO-CMACandOR-CMACareidenticaltothatforthe lowerboundcurve.Sincethelatterisidenticaltothe2 1MISOpoint-to-pointsystem,it givesadiversityorderoftwo.Hence,theaboveobservationestablishesthefactthatboth theproposedprotocolsachievetheoptimaldiversityorderoftwoforthetwo-userCMAC fortherequiredrateregionofinterest. 4.4Summary Inthischapter,weproposedow-theoreticcooperativetransmissionprotocolsforthe two-userfadingCMAC,wherethenodesareonlycapableofhalf-duplexcommunication andhaveaccesstofullCSI.Weproposetwosuchprotocols,viz.aow-optimizedprotocol fortheCMACFO-CMAC,andasuboptimalbutsimplerorthogonalrelayingprotocol fortheCMACOR-CMAC.Boththeproposedprotocolsareevaluatedintermsof achievablerateregionsandoutageperformances.NumericalresultsshowthatFO-CMAC yieldsthelargestachievablerateregionamongstthedierentprotocolsconsideredhere. AlthoughOR-CMACisclearlysuboptimalintermsoftheachievablerateregion,its outageperformanceisclosetowithin1dBforthescenariosconsideredtoFO-CMAC, andboththeproposedprotocolsachievetheoptimaldiversityorderoftwoforthe requiredrateregionofinterest. 74

PAGE 75

Figure4-1.Flow-theoretictransmissionprotocolsfortheCMAC:aOR-CMAC,b FO-CMAC. 75

PAGE 76

Figure4-2.Achievablerateregions-asymmetricsituation. Figure4-3.Achievablerateregions-asymmetricsituationwithconventionalMAslot withoutcommoninformation. 76

PAGE 77

Figure4-4.Achievablerateregions-symmetricsituation. Figure4-5.Outageperformance-asymmetricsituation. 77

PAGE 78

Figure4-6.Outageperformance-symmetricsituation. 78

PAGE 79

CHAPTER5 INTERFERENCECHANNELSWITHUNIDIRECTIONALCOOPERATIONAND CAUSALITYCONSTRAINTS 5.1Introduction AsmentionedinChapter1,theinterferencechannelwithunidirectionalcooperationICUCisessentiallythesimplestrealizationofanoverlaycognitiveradionetwork. Cognitiveradioshaveaconsiderablepotentialinfacilitatinganecientuseofthelicensedspectrumthatiscurrentlyunder-utilized[18].Theoverlayparadigmforcognitive radiosnotonlyprovidesanecienttechniqueforcognitiveradiodeploymentsbutalso yieldsnewerinsightstowardstheunderstandingofinterferencechannelsandcooperative behaviorinmulti-terminalnetworks,throughthedierentmanifestationsofcognition,cooperationandcompetitionlevelsamongstdierentusersorusergroups[63].Intheoverlay form,thesimplestofwhichbeingtheICUCalsoknownascognitiveradiochannel[6,7], thecognitiveorcooperatingradioutilizesthesamespectrumastheprimaryuser-pair foritsowndatatransmission.Whereasthiswouldgenerallycauseinterferencetothe primarylink,thecognitivesourcemayexploititsknowledgeabouttheprimarymessage toimproveitsowntransmissionratesbyprecodingitsinformationagainsttheknown interferencefromtheprimarytransmissionandsimultaneouslyalleviatethedetrimental eectsoftheinterferencetotheprimarydestinationowingtothesecondarytransmission bycooperativerelayingoftheprimarymessage. Oflatetherehasbeenaconsiderablebodyofworkreportedintheliteraturethat havehelpedimprovetheachievablerateregionfortheICUCproposedin[6].In[64],the authorspresentinnerboundstothecapacityregionforjointandsequentialdecoding,and encodingstrategiesthatincluderate-splittingforbothprimaryandsecondarymessages, conditionalGel'fand-PinkserGPbinningandcooperativerelaying.Theyalsopresent ageneralouterboundthatisverysimilartoanouterboundforthebroadcastchannel, andamuchsimplerouterboundforthecasewhereintheprimarydestinationexperiences stronginterference.Aslightlydierentcodingschemehasbeenproposedin[17],wherein 79

PAGE 80

thereisnorate-splittingfortheprimarymessageandthetwopartsofthesecondary messagearebinnedindependentlyagainsttheinterferencefromprimarytransmission. Veryrecently,anothercodingschemeforthediscretememorylessnon-causalICUC hasbeenproposedin[65].Accordingtothisscheme,conditionalrate-splittingisapplied tobothprimaryandsecondarymessages,andthecognitivesourceusesGPbinningand atwo-waybinningstrategyasin[67],conditionedonthecodewordforthecommonpart oftheprimarymessage,totransmitthecommonpartofthesecondarymessage,and theprivatepartsoftheprimaryandsecondarymessagesrespectively.Theauthorsalso considerthecaseinwhichtheprimarymessagemaybeavailableatthecognitivesource inacausalfashion,butthechannelmodelismodiedtothatforaZinterferencechannel ZIC,whereintheprimarydestinationdoesnotexperienceanyinterferenceduetothe secondarytransmission.Anotheruniedcodingscheme,verysimilartotheonein[65], thatyieldsarateregionthatincludestheregionof[65],hasbeenproposedin[66]. Cognitiveradionetworksthataremoreinvolvedthanthetwo-userICUChavealso beenstudiedinsomerecentworks.ThesenetworkscombinetheICUCwithothermultiterminalnetworkslikethemultiple-accesschannels[68],broadcastchannels[69],orrelay channels[70]. Cognitivemultipleaccessnetworks,inwhichasetofcognitiveusersisprovided withafunctionofthemessagesofthesetofnon-cognitiveusersandeachsethasits correspondingreceiver,havebeenstudiedintermsoftheirachievablerateregionsin[68]. Achievablerateregionsforone-sidedinterferencechannelswithacognitiverelay,thathas non-causalmessage-orsignal-levelinformationfrombothsourcesandalinktoonlythe destinationexperiencinginterference,havebeenobtainedin[70].In[69],anachievable rateregionforthecaseoftwointerferingbroadcastchannels,withonesourcehaving non-causalknowledgeaboutthemessageoftheother,hasbeenderived. Mostoftheaboveworksconsiderthenon-causalformofICUCwhereinitisassumed thattheprimarymessageisavailablenon-causallyatthecognitivesource.Inreality, 80

PAGE 81

someresourcesintimeorfrequencyneedtobeexpendedbythesystemforthecognitive sourcetoacquiretheprimarymessage.Thescenarioinwhichthecognitivesourcesneed toobtaintheinformationcausallyfromtheprimarysourceexplicitlymodelsthisoverhead andhasbeenconsideredforthecaseofthetwo-userICUCin[6]and[71].In[6],the authorsconsiderhalf-duplexoperationofthesecondarysource,andproposeatwo-phase protocol.Therstphaseisusedtotransmittheprimarymessagetothecognitivesource andapartoftheprimarymessagetotheprimarydestinationviaabroadcastchannel, andoncethecognitivesourcesuccessfullydecodestheprimarymessage,theoperation proceedsasforthenon-causalcase.Ontheotherhand,in[71],afull-duplexoperation ofthesecondarysourceisassumed,andblockMarkovSPCalongwithsliding-window decoding,andrate-splittingforthetwomessagesareusedtoobtainanachievablerate region. Inthiswork,weconsiderthetwo-userICUCwithcausalityconstraintICUC-C. Itisassumedthatthecognitivesourceiscapableofperfectechocancellation,thereby makingfull-duplexoperationofthecognitivesourcefeasible.Wepresentageneralized codingschemeforthediscretememorylessICUC-C.Theproposedachievablerateregion isbasedonblockMarkovSPCwithbackwarddecoding[72]fortheprimarymessage transmission,conditionalrate-splittingfortheprimaryandsecondarymessagesto facilitatepartialdecodingatthenon-pairingdestinations,GPbinningatthecognitive source,andcooperativerelayingoftheprimarymessagebythecognitivesource.This rateregionisthenevaluatedforthecaseofGaussianchannelsandnumericalresultsfor dierentvaluesofnetworkparametersarepresented.Theresultsareusedtoshedlighton theperformanceofthecodingstrategiesinvolvedintheproposedschemeunderdierent scenarios. Next,werelaxtheassumptionthatthecognitivesourceiscapableoffull-duplex operation,andinsteadsubjectthecognitivesourcetothehalf-duplexconstraint,i.e.it cannotreceiveandtransmitinformationsimultaneouslyoverthesameband.Towards 81

PAGE 82

this,wepresentadiscretememorylesschannelmodelfortheICUCwithhalf-duplexand causalityconstraintsICUC-HDC,andproposeageneralizedcodingschemeforthis channel.Similartothefull-duplexcase,weemployblockMarkovSPCwithbackward coding,conditionalrate-splitting,GPbinning,andcooperativerelayingbythecognitive source.However,forthehalf-duplexcase,thecognitivesourceemploysarandomized listen-transmitschedule[73]toencodeandtransmitinformationviasignaling.Itisalso provedthatthenewrateregioncontainsthepreviouslyknownrateregionof[6]forthe ICUC-HDC. Inthefollowingsection,wepresentthediscretememorylesschannelmodelforthe ICUC-C.ThisisfollowedbySection5.3,wherewepresenttherandomcodingschemeand thecorrespondingachievablerateregionfortheICUC-C.Section5.4detailstheGaussian ICUC-Calongwithnumericalexamplesandadiscussionontheroleofdierentcoding techniquesunderdierentnetworkscenarios.Followingthis,thediscretememoryless ICUC-HDCisintroducedinSection5.5,andinSection5.6,theproposedrandomcoding schemeandthenewachievablerateregionarepresented.TheGaussianICUC-HDCis presentedinSection5.7,followedbyanalyticalandnumericalcomparisonsbetweenthe newachievablerateregionandtheonein[6].Finally,asummaryofthecontributionsin thischapterispresentedinSection5.8. 5.2TheChannelModel ConsiderthecommunicationscenarioasinFig.5-1,whereintheprimarysourcenode S P intendstotransmitinformationtoitsdestinationnode D P .Apartfromtheprimary source-destinationpair,thewirelessnetworkconsistsofasecondaryorcognitivesourcedestinationpair, S C and D C ,with S C havingitsowninformationtotransmitto D C .The primarymessageiscausallyavailableat S C ,andthelattermayusethisknowledgeto assist S P inthetransmissionoftheprimarymessageto D P ,andalsotransmititsown informationto D C 82

PAGE 83

In n channeluses,theprimarysource, S P ,hasamessage w P 2f 1 ; 2 ; ; 2 nR P g totransmitto D P ,whilethesecondarysource S C hasamessage w C 2f 1 ; 2 ; ; 2 nR C g totransmittoitsintendeddestination D C .Let X P ; X C ,and V C ; Y P ; Y C betheinputandoutputalphabetsrespectively.Aratepair R P ;R C isachievableifthere existanencodingfunctionfor S P X n P = f P w P ;f P : f 1 ; 2 ; ; 2 nR P g!X n P andasequenceofencodingfunctionsfor S C X n C = f n C w C ;V n )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 C with X Ci = f Ci w C ;V i )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 C ;f Ci : f 1 ; 2 ; ; 2 nR C gV i )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 C !X C ,andcorrespondingdecodingfunctions ^ w P = g P Y n P ;g P : Y n P !f 1 ; 2 ; ; 2 nR P g and^ w C = g C Y n C ;g C : Y n C !f 1 ; 2 ; ; 2 nR C g suchthattheaverageprobabilityoferror P n e =max f P n e;P ;P n e;C g! 0,where P n e;M = 1 2 n R P + R C X w P ;w C Pr[ g M Y n M 6 = w M j w P ;w C wassent]for M = P;C 5.3AchievableRatesfortheICUC-C Inthissection,wepresentanewachievablerateregionforthediscretememoryless ICUC-C.Westartwithanoverviewofthecodingscheme.Inblock b 2f 1 ; ;B g S P splitsthemessage w P;b intotwoparts w Pco;b and w Ppr;b .Itusessuperpositioncodingto encodethesetwopartsalongwiththemessageforthepreviousblock w Pco;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;w Ppr;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 Thelatteractsasthe resolutioninformation for D P and D C thatusebackwarddecoding todecodetheprimarymessageentirelyorpartially.Incontrasttotherate-splittingtechniquein[71],whereinthetwomessagepartsareencodedindependentlyandsuperposed, S P performsconditionalrate-splitting[65]coupledwithblockMarkovSPC. S C decodes theprimarymessageforthecurrentblockandsimultaneouslyperformsasetofencodingsteps.Inblock b S C splits w C;b intotwoparts w Cco;b and w Cpr;b ,andconditionedon thecodeword T Pco fortheresolutioninformationforthecommonpartoftheprimary message[65],itusesconditionalGPbinning[64]toencode w Cco;b and w Cpr;b as U Cco and U Cpr respectively,againsttheresolutioninformationfortheprivatepartoftheprimary message T Ppr thatcausesinterferenceat D C butisknownat S C .Finally,ittransmitsa combinationoftheabovecodewords,alongwiththeresolutioninformationfortheprimary message,withthelatterpartmanifestingthecooperativerelayingactionby S C .Unlike 83

PAGE 84

thecodingschemein[71], S C doesnotuseblockMarkovSPCtoencodeitsownmessage, therebygivingrisetoasimplercharacterizationoftherateregion. D P waitsuntilthelastblock B andusesbackwarddecodingtojointlydecodeboth commonandprivatepartsoftheprimarymessageandthecommonpartofthesecondary message.Similarly, D C waitsuntilthetransmissionforblock B andthenusesbackward decodingtojointlydecodethecommonpartoftheprimarymessageandbothcommon andprivatepartsofthesecondarymessage.Notethat D C performsbackwarddecoding onlytodecodethecommonpartoftheprimarymessagetotakeadvantageoftheblock MarkovSPCstructureusedtoencodeit.Further,theuseofbackwarddecodingyieldsa muchsimplerrateregioncharacterizationcomparedtotheonein[71].Table5-1liststhe randomvariablesinvolvedinthecodeconstructionalongwiththeirsignicance. Table5-1.DescriptionofRandomVariablesinTheorem5.1 RandomVariableDenition T Pco Resolutioninformationforcommonpartofprimary messageknownto S C T Ppr Resolutioninformationforprivatepartofprimary messageknownto S C X Pco Newinformationforcommonpartofprimarymessage X P Transmittedcodewordby S P ,generatedbysuperposingnewinformationforprivatepartofprimary messageon T Pco T Ppr ,and X Pco pleaserefertoCodebookGenerationintheproofofTheorem5.1 U Cco Commonpartofsecondarymessagegeneratedby conditionalGel'fand-Pinskerbinningagainst T Ppr U Cpr PrivatepartofsecondarymessagegeneratedbyconditionalGel'fand-Pinkserbinningagainst T Ppr X C Transmittedcodewordby S C Theorem5.1. FortheICUC-C,theratetuple R P ;R C ,where R P = R Pco + R Ppr R C = R Cco + R Cpr ,withnon-negativereals R Pco ;R Ppr ;R Cco ;R Cpr satisfying R Ppr min f I X P ; V C j T Pco ;T Ppr ;X Pco ; I T Ppr ;X P ; Y P ;U Cco j T Pco ;X Pco g {1a 84

PAGE 85

R Pco + R Ppr I X P ; V C j T Pco ;T Ppr {1b R Ppr + R Cco I T Ppr ;X P ;U Cco ; Y P j T Pco ;X Pco {1c R Pco + R Ppr + R Cco I T Pco ;T Ppr ;X P ;U Cco ; Y P {1d R Cpr I U Cpr ; Y C ;U Cco j T Pco ;X Pco )]TJ/F21 11.9552 Tf 11.955 0 Td [(I U Cpr ; T Ppr ;U Cco j T Pco {1e R Cco + R Cpr I U Cco ;U Cpr ; Y C j T Pco ;X Pco )]TJ/F21 11.9552 Tf 9.299 0 Td [(I U Cco ;U Cpr ; T Ppr j T Pco {1f R Pco + R Cco + R Cpr I T Pco ;X Pco ;U Cco ;U Cpr ; Y C )]TJ/F21 11.9552 Tf 9.299 0 Td [(I U Cco ;U Cpr ; T Ppr j T Pco {1g isachievableforsomejointdistributionthatfactorsas p t Pco p t Ppr j t Pco p x Pco j t Pco p x P j t Pco ;t Ppr ;x Pco p u Cco j t Pco p u Cpr j t Pco ;u Cco p x C j t Pco ;t Ppr ;u Cco ;u Cpr p v C j x P ;x C p y P j x P ;x C p y C j x P ;x C ; {2 andforwhichtheright-handsidesof 5{1a to 5{1g arenon-negative. Proof. Let A n X;Y denotesetofjointly -typicalsequencesaccordingtothedistribution ofrandomvariables X;Y asinducedbythesamedistributionusedtogeneratethe codebookssee[33].Forconvenience,thedependenceontherandomvariableswillnotbe statedexplicitly,andshouldbeclearfromthecontext. Codebookgeneration: Splittheprimaryandcognitiveusers'ratesas R P = R Pco + R Ppr and R C = R Cco + R Cpr respectively.Thus,inblock b 2f 1 ; ;B g ,theprimary messagecanberepresentedas w P;b = w Pco;b ;w Ppr;b ,andthesecondarymessageas w C;b = w Cco;b ;w Cpr;b ,where co and pr standforthecommonandprivatepartof amessagerespectively.Fixadistribution p t Pco ;t Ppr ;x Pco ;x P ;u Cco ;u Cpr ;x C asin Theorem5.1. 85

PAGE 86

Generate2 nR Pco i.i.d.codewords t n Pco w 0 Pco w 0 Pco 2f 1 ; ; 2 nR Pco g ,accordingto Q n i =1 p t Pcoi Foreachcodeword t n Pco w 0 Pco ,generate2 nR Ppr conditionallyi.i.d.codewords t n Ppr w 0 Pco ;w 0 Ppr w 0 Ppr 2f 1 ; ; 2 nR Ppr g ,accordingto Q n i =1 p t Ppri j t Pcoi w 0 Pco Foreachcodeword t n Pco w 0 Pco ,generate2 nR Pco conditionallyi.i.d.codewords x n Pco w 0 Pco ;w Pco w Pco 2f 1 ; ; 2 nR Pco g ,accordingto Q n i =1 p x Pcoi j t Pcoi w 0 Pco Foreachcodewordtuple )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(t n Pco w 0 Pco ;t n Ppr w 0 Pco ;w 0 Ppr ;x n Pco w 0 Pco ;w Pco ,generate 2 nR Ppr i.i.d.codewords x n P w 0 Pco ;w Pco ;w 0 Ppr ;w Ppr w Ppr 2f 1 ; ; 2 nR Ppr g ,according to Q n i =1 p )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(x Pi j t Pcoi w 0 Pco ;t Ppri )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(w 0 Pco ;w 0 Ppr ;x Pcoi )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(w 0 Pco ;w Pco Foreachcodeword t n Pco w 0 Pco ,generate2 n R Cco + R 0 Cco i.i.d.codewords u n Cco w 0 Pco ;w Cco ;b Cco w Cco 2f 1 ; ; 2 nR Cco g and b Cco 2f 1 ; ; 2 nR 0 Cco g ,accordingto Q n i =1 p u Ccoi j t Pcoi w 0 Pco Foreachcodewordpair )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(t n Pco w 0 Pco ;u n Cco )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(w 0 Pco ;w Cco ;b Cco ,generate2 n R Cpr + R 0 Cpr i.i.d.codewords u n Cpr w 0 Pco ;w Cco ;b Cco ;w Cpr ;b Cpr w Cpr 2f 1 ; ; 2 nR Cpr g and b Cpr 2f 1 ; ; 2 nR 0 Cpr g ,accordingto Q n i =1 p u Cpri j t Pcoi w 0 Pco ;u Ccoi w 0 Pco ;w Cco ;b Cco Generate x n C w 0 Pco ;w 0 Ppr ;w Cco ;b Cco ;w Cpr ;b Cpr where x C isadeterministicfunctionof t Pco ;t Ppr ;u Cco ;u Cpr Encoding: At S P :Inblock b 2f 2 ; ;B )]TJ/F15 11.9552 Tf 12.098 0 Td [(1 g S P transmits x n P w Pco;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;w Pco;b ;w Ppr;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ;w Ppr;b Intherstblock, S P transmits x n P ;w Pco; 1 ; 1 ;w Ppr; 1 ,whileinblock B ,ittransmits x n P w Pco;B )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ; 1 ;w Ppr;B )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ; 1.Notethattheactualratefortheprimarymessageis B )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 B R P butitconvergesto R P asthenumberofblocks B goestoinnity. At S C :Inblock b 2f 1 ; ;B g ,totransmit w Cco;b S C searchesforbinindex b Cco;b suchthat )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(t n Pco ^ w Pco ;u n Cco ^ w Pco ;w Cco;b ;b Cco;b ;t n Ppr ^ w Pco ; ^ w Ppr 2A n ; {3 where ^ w Pco and ^ w Ppr are S C 'sestimatesof w Pco;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 and w Ppr;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 respectivelyfromthe previousblock.Once b Cco;b isdetermined,itsearchesforabinindex b Cpr;b inorderto transmit w Cpr;b suchthat )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(t n Pco ^ w Pco ;u n Cco ^ w Pco ;w Cco;b ;b Cco;b ;u n Cpr )]TJ/F15 11.9552 Tf 7.873 -8.079 Td [(^ w Pco ;w Cco;b ;b Cco;b ;w Cpr;b ;b Cpr;b ; 86

PAGE 87

t n Ppr ^ w Pco ; ^ w Ppr 2A n : {4 Itsets b Cco;b =1or b Cpr;b =1iftherespectivebinindexisnotfound.Itcanbeshown usingargumentssimilartothosein[64]thattheprobabilitiesoftheeventsof S C notable tondaunique b Cco;b or b Cpr;b satisfying5{3and5{4canbemadearbitrarilysmallif thefollowingholdtrue: R 0 Cco >I U Cco ; T Ppr j T Pco + 0 ; R 0 Cpr >I U Cpr ; T Ppr j T Pco ;U Cco + 0 ; where 0 > 0maybearbitrarilysmall. S C transmits x n C ^ w Pco;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ; ^ w Ppr;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ;w Cco;b ;b Cco;b ;w Cpr;b ;b Cpr;b ; Decoding: At S C :Assumethatdecodingtillblock b )]TJ/F15 11.9552 Tf 12.45 0 Td [(1hasbeensuccessful. S C declaresthat w Pco;b ;w Ppr;b = ^ w Pco ; ^ w Ppr wastransmittedinblock b ifthereexistsauniquepair ^ w Pco ; ^ w Ppr suchthat )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [(t n Pco w Pco;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;t n Ppr w Pco;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;w Ppr;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ;x n Pco )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [(w Pco;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ; ^ w Pco ; x n P w Pco;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ; ^ w Pco ;w Ppr;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ; ^ w Ppr ;v n C;b 2A n : Else,anerrorisdeclared. At D P :Theprimarydestination D P waitsuntilblock B ,andthenperformsbackward decoding.Weconsiderthedecodingprocessusingtheoutputinblock b 2f B )]TJ/F15 11.9552 Tf 12.093 0 Td [(1 ; ; 2 g Thedecodingfortherstandlastblockscanbeseenasspecialcasesoftheabove.Thus, forblock b 2f B )]TJ/F15 11.9552 Tf 12.312 0 Td [(1 ; ; 2 g ,assumingthatthedecodingforthepair w Pco;b ;w Ppr;b has beensuccessful, D P searchesforauniquepair^ w Pco ; ^ w Ppr andsometuple ^ w Cco ; ^ b Cco for w Cco;b ;b Cco;b suchthat )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(t n Pco ^ w Pco ;t n Ppr ^ w Pco ; ^ w Ppr ;x n Pco ^ w Pco ;w Pco;b ;x n P ^ w Pco ;w Pco;b ; ^ w Ppr ;w Ppr;b ; u n Cco ^ w Pco ; ^ w Cco ; ^ b Cco ;y n P;b 2A n : 87

PAGE 88

At D C :Thecognitivedestination D C alsowaitsuntilblock B ,andthenperforms backwarddecodingtojointlydecodethemessagesintendedforitandthecommonpartof theprimarymessage.Forblock b 2f B )]TJ/F15 11.9552 Tf 12.416 0 Td [(1 ; ; 2 g D C isassumedtohavesuccessfully decoded w Pco;b fromblock b +1.Withthisknowledge,itsearchesforauniquetuple ^ w Pco ; ^ w Cco ; ^ b Cco ; ^ w Cpr ; ^ b Cpr suchthat t n Pco ^ w Pco ;x n Pco ^ w Pco ;w Pco;b ;u n Cco ^ w Pco ; ^ w Cco ; ^ b Cco ; u n Cpr ^ w Pco ; ^ w Cco ; ^ b Cco ; ^ w Cpr ; ^ b Cpr ;y n C;b 2A n : ErrorAnalysis: Throughouttheanalysis,weassume,withoutlossofgenerality,thatalltransmitted messagesattheprimaryandcognitivesources,inanyblock b 2f 1 ; ;B g ,wereones. Encodingerrorsat S C :Anencodingerroroccursat S C undertwocircumstances.An erroroccursif,inblock b S C cannotndabinindex b Cco;b suchthat5{3isnotsatised for ^ w Pco =1, ^ w Ppr =1,and w Cco;b =1.Thiseventoccurswithprobability Pr 2 6 4 2 nR 0 Cco [ b Cco;b =1 )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [(T n Pco ;U n Cco ; 1 ;b Cco;b ;T n Ppr ; 1 = 2A n 3 7 5 = )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(Pr )]TJ/F21 11.9552 Tf 10.461 -9.684 Td [(T n Pco ;U n Cco ; 1 ;b Cco;b ;T n Ppr ; 1 2A n 2 nR 0 Cco )]TJ/F15 11.9552 Tf 5.48 -9.683 Td [(1 )]TJ/F15 11.9552 Tf 11.956 0 Td [( )]TJ/F21 11.9552 Tf 11.955 0 Td [( )]TJ/F22 7.9701 Tf 6.586 0 Td [(n [ I U Cco ; T Ppr j T Pco + 0 ] 2 nR 0 Cco exp )]TJ/F15 11.9552 Tf 9.298 0 Td [( )]TJ/F21 11.9552 Tf 11.956 0 Td [( n [ R 0 Cco )]TJ/F22 7.9701 Tf 6.586 0 Td [(I U Cco ; T Ppr j T Pco )]TJ/F22 7.9701 Tf 6.587 0 Td [( 0 ] where > 0canbearbitrarilysmall,andthelasttwoinequalitiesareduetothejoint AsymptoticEquipartitionPropertyAEP[33]andthefactthat )]TJ/F21 11.9552 Tf 13.174 0 Td [(x n e )]TJ/F22 7.9701 Tf 6.587 0 Td [(nx respectively.Clearly,theaboveprobabilitycanbemadearbitrarilysmallif R 0 Cco >I U Cco ; T Ppr j T Pco + 0 : {5 88

PAGE 89

Anotherpossibilityofanencodingerrorat S C occursif,inblock b ,itcannotndabin index b Cpr;b suchthat5{4isnotsatisedwith ^ w Pco =1, ^ w Ppr =1, w Cco;b =1, b Cco;b =1, and w Cpr;b =1.Proceedingasfortherstkindoferrorevent,itcanbeshownthatthe probabilityofthiseventcanbemadearbitrarilysmallif R 0 Cpr >I U Cpr ; T Ppr j T Pco ;U Cco + 0 : {6 Decodingerrorsat S C :FortheblockMarkovSPCencodedtransmissionfrom S P thecognitivesource S C usesitsknowledgeabouttheinformationinthepreviousblock tojointlydecode w P forthecurrentblock.Forthesuperpositionencodedcommonand privatepartsoftheprimarymessage,itcanbeshownthattheprobabilityoferrorforthis decodingstepcanbemadearbitrarilylowaslongasthefollowinghold: R Ppr I X P ; V C j T Pco ;T Ppr ;X Pco ; R Pco + R Ppr I X P ; V C j T Pco ;T Ppr : Decodingerrorsat D P :Forblock b 2f B )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ; ; 2 g ,let E ijk betheevent t n Pco i ;t n Ppr i;j ;x n Pco i;w Pco;b ;x n P i;w Pco;b ;j;w Ppr;b ;u n Cco i;k; ^ b Cco ;y n P;b 2A n ; wherein,itisassumedthatthepreviousdecodingstep,startingwithdecodingforblock B hasbeensuccessful,i.e.^ w Pco;b ; ^ w Ppr;b = w Pco;b ;w Ppr;b .Notethatweneednotconsider theprobabilityoftheeventsof D P decoding w P;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 correctly,butnot w Cco;b ;b Cco;b .Then theprobabilityoferrorat D P is P e;D P =Pr h E c 111 [ [ i;j 6 = ; 1 E ijk i Pr[ E c 111 ]+2 nR Ppr Pr[ E 121 ]+2 n R Ppr + R Cco + R 0 Cco Pr[ E 122 ] +2 n R Ppr + R Ppr + R Cco + R 0 Cco Pr[ E 222 ]{7 89

PAGE 90

Thersttermin5{7goestozerowith n duetojointAEP.Theprobabilitiesofthelast threeerroreventscanbeshowntobeupperboundedaslistedin5{8a-5{8c. Pr[ E 121 ] 2 )]TJ/F22 7.9701 Tf 6.586 0 Td [(n [ I T Ppr ;X P ; Y P ;U Cco j T Pco ;X Pco +6 ] {8a Pr[ E 122 ] 2 )]TJ/F22 7.9701 Tf 6.586 0 Td [(n [ I T Ppr ;X P ;U Cco ; Y P j T Pco ;X Pco + I U Cco ; T Ppr j T Pco +8 ] {8b Pr[ E 222 ] 2 )]TJ/F22 7.9701 Tf 6.586 0 Td [(n [ I T Pco ;T Ppr ;X P ;U Cco ; Y P + I U Cco ; T Ppr j T Pco +7 ] {8c Thus,theabovesuggeststhat,for n largeenough,^ w Pco ; ^ w Ppr = w Pco;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;w Ppr;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 witharbitrarilysmallprobabilityoferrorif R Ppr islessthanthesecondterminsidethe minimumoperatorinequation5{1a,and5{1c-5{1daresatised. Decodingerrorsat D C :Forblock b 2f B )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ; ; 2 g ,let E D C ijk denotetheevent )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(t n Pco k ;x n Pco k; 1 ;u n Cco k;i;b Cco ;u n Cpr k;i;b Cco ;j;b Cpr ;y n C;b 2A n : Then,thetotalprobabilityoferrorat D C canbeupperboundedas P e;D C Pr h E D C 111 c i +2 n R Cpr + R 0 Cpr Pr[ E D C 121 ]+2 n R Cco + R 0 Cco + R Cpr + R 0 Cpr Pr[ E D C 221 ] +2 n R Pco + R Cco + R 0 Cco + R Cpr + R 0 Cpr Pr[ E D C 222 ] : {9 Notethat,owingtothecodingstructure,if k 6 =1,thenallthetransmittedcodewords wouldbeindependentofthereceivedcodewords.Asaresult, D C needstocorrectly decode w Pco eventhoughitmaynotbeinterestedinthispartoftheprimarymessage. Again,byjointAEP,thersttermof5{9goestozerowith n .Theprobabilitiesofthe otherthreeeventscanbeupperboundedas Pr[ E D C 121 ] 2 )]TJ/F22 7.9701 Tf 6.587 0 Td [(n I U Cpr ; Y C j T Pco ;X Pco ;U Cco +6 {10 Pr[ E D C 221 ] 2 )]TJ/F22 7.9701 Tf 6.587 0 Td [(n I U Cco ;U Cpr ; Y C j T Pco ;X Pco +6 {11 Pr[ E D C 222 ] 2 )]TJ/F22 7.9701 Tf 6.587 0 Td [(n I T Pco ;X Pco ;U Cco ;U Cpr ; Y C +5 {12 90

PAGE 91

Therefore,for n largeenough,^ w Pco ; ^ w Cco ; ^ b Cco ; ^ w Cpr ; ^ b Cpr = w Pco;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ;w Cco;b ;b Cco;b ;w Cpr;b ;b Cpr;b witharbitrarilylowprobabilityoferrorif5{1e-5{1garesatised. Thus,theconstraintsontheratesasgivenin5{1a-5{1gensurethattheaverage probabilityoferroratthetwodestinationscanbedriventozeroandthus,theydescribe anachievablerateregionfortheICUC-C. Remark 5.1 TheachievableratesregiondescribedinTheorem5.1canbeexpressed explicitlyintermsof R P and R C usingFourier-Motzkinelimination.Denotetheright sidesof5{1a-5{1gas I 1 ;I 2 ; ;I 7 .ThentheachievablerateregionofTheorem5.1can bewrittenas R P min f I 2 ;I 4 g ; {13a R C min f I 6 ;I 7 g ; {13b R P + R C min f I 4 + I 5 ;I 1 + I 7 g ; {13c R P +2 R C I 3 + I 5 + I 7 : {13d Remark 5.2 TheachievablerateregiondescribedinTheorem5.1isconvexandhence,no time-sharingisrequiredtoenlargetherateregion.ThiscanbeprovedusingtheMarkov chainstructureofthecodeaswasusedin[74,Lemma5],withtherandomvariable T Pco in Theorem5.1playingarolesimilartothatof U in[74]. 5.4TheGaussianICUC-C WeapplytheresultofTheorem5.1totheGaussianICUC-C.ForGaussianchannels,withthedirectlinks'channelgainsnormalizedtounitycf.[7],etc.,wehavethe followinginput-outputrelationshipsasshowninFigure5-2: V C = g PC X P + Z S C ; {14 Y P = X P + h CP X C + Z P ; {15 Y C = h PC X P + X C + Z D C ; {16 91

PAGE 92

where Z S C ;Z P ;Z D C N ; 1arei.i.d.randomvariablesdenotingtheadditivenoise at S C D P ,and D C respectively. g PC h PC ,and h CP arepositiverealsthatdenotethe channelgainsforthelinksfrom S P to S C S P to D C ,andfrom S C to D P respectively. Also,theprimaryandcognitivesourcesaresubjectedtotheirrespectivepowerconstraints P P and P C : 1 n jj X n P jj 2 P P ; 1 n jj X n C jj 2 P C : {17 Let P ; P ; P ; P ; C ; C ; C berealnumbersintheinterval[0 ; 1]suchthat P + P + P + P 1.Also,let =1 )]TJ/F21 11.9552 Tf 12.566 0 Td [( for 2f C ; C ; C g .Weevaluatetherateregionof Theorem5.1forthecaseofGaussianchannelswiththefollowingtransmittedsignalsin block b 2f 1 ; ;B g : X P w P;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;w P;b = T Pco + X 0 Pco + T 0 Ppr + X 0 Ppr {18 X C w P;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ;w C;b = X Cco + X Cpr + r C C P C P P P T Pco + s C C P C P P P T 0 Ppr {19 where T Pco N ; P P P X 0 Pco N ; P P P T 0 Ppr N ; P P P X 0 Ppr N ; P P P X Cco N ; C C P C ,and X Cpr N ; C C P C arei.i.d.randomvariables. X Cco and X Cpr aretransmittedtocommunicatethedirtypapercodedmessages w Cco;b and w Cpr;b respectively,with U Cco and U Cpr beingthecorrespondingauxiliaryrandomvariablesas in[64]. ForthecodingschemeinTheorem5.1,itisnecessarythatglobalchannelstateinformationCSIisavailableatallnodesofthenetwork.Also,theprobabilitydistributionsof thedierentcodewordsforthetwoencoderswouldberequiredtofacilitateecientrate selection.Moreover,thedierentcodingstrategiesusedinobtainingtheachievablerate regionofTheorem5.1involvecertainassumptionsregardingtheknowledgeofthecodebookateachnode.Inparticular,cooperativerelayingoftheprimarymessageby S C ,as wellasDPCat S C ,requirestheknowledgeoftheprimarycodebookat S C .Ontheother hand,rate-splittingoftheprimarymessagerequires D C toknowtheprimarycodebook. 92

PAGE 93

Lastly,rate-splittingforthesecondarymessagerequires D P toknowthesecondarycodebook.WhereastheCSIandrateselectioninformationmaybeprovidedthroughlow-rate controlchannelsbetweenthenodes,providingcodebookknowledgeatthenon-pairing nodesmaybemorediculttoachieveinpractice.Thiscanbeseentobeespeciallytrue forthelastcaseifweconsiderthefundamentalphilosophyofcognitiveradionetworks thattheyshouldbedeployedsuchthatthealreadyexistingprimaryuser-pairsshouldbe asobliviousaspossibletotheexistenceofthesecondaryuser-pairs. Figs.5-3to5-5presentsomenumericalresultsforthedierentcodingstrategies forvariouschoicesofavailablepowersandchannelgainsasgiveninthegures.Inthese gures,CodingI"referstorate-splittingforbothmessageswithoutuseofDPCat S C thiscodingschemeisthesameasthatproposedin[71],CodingII"referstothe strategyofusingDPCat S C butwithoutanyrate-splitting,CodingIII"standsforthe schemeofusingrate-splittingforonlytheprimarymessagealongwithDPCat S C ,and nally,CodingIV"referstothecodingstrategydescribedinthebeginningofthissection i.e.Theorem5.1specializedforGaussianchannelsasoutlinedatthebeginningofthis section,andinvolvesrate-splittingforbothmessagesaswellasDPCat S C .Notethat cooperativerelayingisusedinalloftheabovecodingschemes. Fig.5-3demonstratesthatthecodingschemeof[71]isstrictlysub-optimalwhenboth destinationsexperienceweakinterferencewhereasusingDPCandcooperativeingresultin alargerrateregion.Inlightoftheabovediscussiononcodebookknowledgerequirements, itturnsoutthatforscenariosasthis,usingDPCandcooperativerelayingwithoutany rate-splittingisthebestandmostfeasiblecodingscheme.Thegainfromrate-splitting fortheprimarymessagewhen D C experiencesstronginterferenceand D P experiences weakinterferenceisevidentfromFig.5-4.Inthiscase,astheinterferenceat D P isweak, rate-splittingforthesecondarymessagedoesnotappeartoprovideanybenettowards enlargingtherateregion.Finally,Fig.5-5illustratesthegainsfromrate-splittingfor thesecondarymessagecoupledwithDPCat S C forthecaseofweakinterferenceat D C 93

PAGE 94

andstronginterferenceat D P .Moreover,theincreaseinthedierenceintheregions forCodingI"andCodingIV",as P P increasesfrom1.5to6,showsthebenetof DPCat S C whentheeectiveinterferenceat D C increaseswith P P .Fromtheseresults, itappearsthatusingcooperativerelayingwithDPCandrate-splittingoftheprimary messagemaybepracticallymoresuitablestrategiesintermsofcodebookknowledge requirementsattheprimaryuser-pair,exceptwhentheprimarydestinationexperiences stronginterference. 5.5DiscreteMemorylessChannelModelfortheICUC-HDC Tilltheprevioussectionwehavebeenassumingthatthecognitivesourcecanoperate infull-duplexmodebyperformingperfectechocancelation.Here,weremovethefullduplexassumption,andintroducethediscretememorylesschannelmodelfortheICUC withhalf-duplexandcausalityconstraintsICUC-HDC.TheICUC-HDCisdepicted inFig.5-6,whereintheprimarysourcenode S P intendstotransmitinformationtoits destinationnode D P .Acognitiveorsecondarysource-destinationpair, S C and D C wishestocommunicateaswell,with S C havingitsowninformationtotransmitto D C AsinthecaseoftheICUC-C,theprimarymessageisonlycausallyavailableat S C .To incorporatethehalf-duplexconstraintforthediscretememorylesschannelmodel,we considerasecondinputat S C S ,toindicatethestateof S C -listeningortransmitting. Withthis,thechanneltransitionprobabilityisdeterminedbythestateofthe cognitivesourceasfollows: p y P ;y C ;v C j x P ;x C ;s = 8 > > < > > : p y P ;y C ;v C j x P if s = l p y P ;y C j x P ;x C e v C if s = t {20 where e denotesanerasureat S C ,and e v C =1if v C = e and0otherwise.To incorporatethefactthat S C cannottransmitwheninthelisteningstate,werestrictthe jointprobabilitydistributionoftheinputsas p x P ;x C ;s = p x P j s = l x C p s = l + p x P ;x C j s = t p s = t ,where isthenull"symbol. 94

PAGE 95

In n channeluses,theprimarysource, S P ,hasmessage w P 2f 1 ; 2 ; ; 2 nR P g to transmitto D P ,whilethesecondarysource S C hasmessage w C 2f 1 ; 2 ; ; 2 nR C g totransmitto D C .Let X P ; X C ; S ,and V C ; Y P ; Y C betheinputandoutputalphabets respectively.Further,let S = f l;t g .Aratepair R P ;R C isachievableifthereexist anencodingfunctionfor S P X n P = f P w P ;f P : f 1 ; 2 ; ; 2 nR P g!X n P ,anda sequenceofencodingfunctionsfor S C X n C ;S n = f n C w C ;V n )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 C with X Ci ;S i = f Ci w C ;V i )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 C ;f Ci : f 1 ; 2 ; ; 2 nR C gV i )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 C !X C S ,andcorrespondingdecoding functions^ w P = g P Y n P ;g P : Y n P !f 1 ; 2 ; ; 2 nR P g and^ w C = g C Y n C ;g C : Y n C f 1 ; 2 ; ; 2 nR C g suchthattheaverageprobabilityoferror P n e =max f P n e;P ;P n e;C g! 0, where P n e;M = 1 2 n R P + R C X w P ;w C Pr[ g M Y n M 6 = w M j w P ;w C wassent]for M = P;C 5.6AnAchievableRateRegionfortheICUC-HDC First,wepresentabriefdescriptionofthecodingscheme.Inblock b 2f 1 ; ;B g S P splitsthemessage w P;b as w P;b = w P 1 ;b ;w P 2 ;b where w Pi;b = w Pico;b ;w Pipr;b for i =1 ; 2. Here,foranyblock, w P 1 isthemessagepartthat S C decodesandusesforitscognitiveand cooperativeactions,whereas w P 2 isthemessagepartthat S P directlytransmitsto D P when S C isintransmitmode.Asbefore,thesubscripts co and pr indicatethecommon andprivatemessagepartsrespectively.Whilethecommonmessagepartsaredecodedby bothdestinations,theprivatemessagepartsaredecodedonlybytheintendeddestination. w P 1 co;b isfurtherdividedintotwopartsw s;b ,thatisforwardedby S C inthenextblock usingthehelpofitsrandomlisten-transmitschedule[73],and w e;b ,thatistransmitted explicitlyusingastandardcodebook. Conditionalrate-splitting[65]andsuperpositioncodingareusedfortheabove messagesplittingstep.Forblock b 2f 1 ; ;B g S P transmits w P 1 ;b duringthe S C listenstates,anditsuperposes w P 2 ;b onto w P 1 ;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 usingblockMarkovSPCduringthe S C -transmitstates,with w P 1 ;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 actingasthe resolutioninformation for D P and D C to decode w P 1 entirelyorpartially.Inblock b S C decodes w P 1 ;b fromthereceivedsymbols duringthelisten-states.Inblock b S C splits w C;b intotwoparts w Cco;b and w Cpr;b ,and 95

PAGE 96

conditionedonthecodewordpair S;T P 1 co fortheresolutioninformationforthecommon partof w P 1 ;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ,itusesconditionalGPbinning[64]toencode w Cco;b and w Cpr;b as U Cco and U Cpr respectively,againsttheresolutioninformationfortheprivatepartof w P 1 ;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 T P 1 pr Ittransmitsacombinationoftheabovecodewords,alongwiththeresolutioninformation, duringthe S C -transmitstates. Both D P and D C waituntilthetransmissioninblock B ,andthenusebackward decoding[72]tojointlydecodebothcommonandprivatepartsofitsintendedmessage andthecommonmessagepartsfromtheinterferingtransmission.Notethat D C performsbackwarddecodingonlytodecode w P 1 co;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 inordertotakeadvantageof theblockMarkovSPCstructureusedtoencodeit.Table5-2liststherandomvariables involvedinthecodeconstructionalongwiththeirsignicance. Table5-2.DescriptionofRandomVariablesinTheorem5.2 RandomVariableDenition S Listen-transmitstatefor S C T P 1 co Resolutioninformationforcommonpartofprimary message w P 1 knownto S C T P 1 pr Resolutioninformationforprivatepartofprimary message w P 1 knownto S C X P 1 co Newinformationforcommonpartofprimarymessage w P 1 decodedby S C X P 1 pr Newinformationforprivatepartofprimarymessage w P 1 decodedby S C X P 2 co Commonpartofprimarymessage w P 2 notdecoded by S C X P 2 pr Privatepartofprimarymessage w P 2 notdecodedby S C U Cco Commonpartofsecondarymessagegeneratedby conditionalGel'fand-Pinskerbinning U Cpr PrivatepartofsecondarymessagegeneratedbyconditionalGel'fand-Pinkserbinning X P Transmittedcodewordby S P X C Transmittedcodewordby S C Let =Pr[ S = l ],and =1 )]TJ/F21 11.9552 Tf 12.021 0 Td [( .Owingtothehalf-duplexconstrainttothechannel model,werestrictthedistributionsforthecodewordsusedinthecodebookconstruction 96

PAGE 97

asfollows: p t P 1 co j s = l = t P 1 co ; {21a p t P 1 pr j t P 1 co ;s = l = t P 1 pr ; {21b p x P 2 co j t P 1 co ;s = l = x P 2 co ; {21c p x P 2 pr j x P 2 co ;t P 1 pr ;t P 1 co ;s = l = x P 2 pr ; {21d p u Cco j t P 1 co ;s = l = u Cco ; {21e p u Cpr j u Cco ;t P 1 co ;s = l = u Cpr ; {21f p x P 1 co j t P 1 co ;s = t = x P 1 co ; {21g p x P 1 pr j x P 1 co ;t P 1 pr ;t P 1 co ;s = t = x P 1 pr : {21h Theorem5.2. ForthediscretememorylessICUC-HDC,allratetuples R P ;R C ,where R P = R P 1 + R P 2 = R P 1 co + R P 1 pr + R P 2 co + R P 2 pr R P 1 co = R s + R e R C = R Cco + R Cpr withnon-negativereals R s ;R e ;R P 1 pr ;R P 2 co ;R P 2 pr ;R Cco ;R Cpr satisfying R P 1 pr I X P 1 pr ; V C j X P 1 co ;S = l {22a R P 1 I X P 1 pr ; V C j S = l {22b R P 2 pr I X P 2 pr ; Y P ;U Cco j X P 2 co ;T P 1 pr ;T P 1 co ;S = t {22c R P 2 I X P 2 pr ; Y P ;U Cco j T P 1 pr ;T P 1 co ;S = t {22d R P 2 pr + R Cco I X P 2 pr ;U Cco ; Y P j X P 2 co ;T P 1 pr ;T P 1 co ;S = t {22e R P 2 + R Cco I X P 2 pr ;U Cco ; Y P j T P 1 pr ;T P 1 co ;S = t {22f R P 1 pr + R P 2 pr I X P 1 pr ; Y P j X P 1 co ;S = l + I T P 1 pr ;X P 2 pr ; Y P ;U Cco j X P 2 co ;T P 1 co ;S = t {22g R P 1 pr + R P 2 I X P 1 pr ; Y P j X P 1 co ;S = l + I T P 1 pr ;X P 2 pr ; Y P ;U Cco j T P 1 co ;S = t {22h R P 1 pr + R P 2 pr + R Cco I X P 1 pr ; Y P j X P 1 co ;S = l 97

PAGE 98

+ I T P 1 pr ;X P 2 pr ;U Cco ; Y P j X P 2 co ;T P 1 co ;S = t {22i R P 1 pr + R P 2 + R Cco I X P 1 pr ; Y P j X P 1 co ;S = l + I T P 1 pr ;X P 2 pr ;U Cco ; Y P j T P 1 co ;S = t {22j R e + R P 1 pr + R P 2 + R Cco I X P 1 pr ; Y P j S = l + I T P 1 co ;T P 1 pr ;X P 2 pr ;U Cco ; Y P j S = t {22k R P + R Cco I S ; Y P + I X P 1 pr ; Y P j S = l + I T P 1 co ;T P 1 pr ;X P 2 pr ;U Cco ; Y P j S = t {22l R Cpr [ I U Cpr ; Y C ;U Cco j X P 2 co ;T P 1 co ;S = t )]TJ/F21 11.9552 Tf 9.299 0 Td [(I U Cpr ; T P 1 pr ;U Cco j T P 1 co ;S = t ]{22m R C [ I U Cco ;U Cpr ; Y C j X P 2 co ;T P 1 co ;S = t )]TJ/F21 11.9552 Tf 9.298 0 Td [(I U Cco ;U Cpr ; T P 1 pr j T P 1 co ;S = t ]{22n R P 2 co + R Cpr [ I X P 2 co ;U Cpr ; Y C ;U Cco j T P 1 co ;S = t )]TJ/F21 11.9552 Tf 9.298 0 Td [(I U Cpr ; T P 1 pr ;U Cco j T P 1 co ;S = t ]{22o R P 2 co + R C [ I X P 2 co ;U Cco ;U Cpr ; Y C j T P 1 co ; S = t )]TJ/F21 11.9552 Tf 11.955 0 Td [(I U Cco ;U Cpr ; T P 1 pr j T P 1 co ;S = t ]{22p R e + R P 2 co + R C I X P 1 co ; Y C j S = l + [ I T P 1 co ;X P 2 co ;U Cco ;U Cpr ; Y C j S = t )]TJ/F21 11.9552 Tf 9.298 0 Td [(I U Cco ;U Cpr ; T P 1 pr j T P 1 co ;S = t ]{22q R P 1 co + R P 2 co + R C I S ; Y C + I X P 1 co ; Y C j S = l + [ I T P 1 co ;X P 2 co ;U Cco ;U Cpr ; Y C j S = t )]TJ/F21 11.9552 Tf 9.298 0 Td [(I U Cco ;U Cpr ; T P 1 pr j T P 1 co ;S = t ]{22r areachievableforsomejointdistributionthatfactorsas p s p t P 1 co j s p t P 1 pr j t P 1 co ;s p x P 1 co j t P 1 co ;s p x P 1 pr j x P 1 co ;t P 1 pr ;t P 1 co ;s 98

PAGE 99

p x P 2 co j t P 1 co ;s p x P 2 pr j x P 2 co ;t P 1 pr ;t P 1 co ;s p x P j x P 2 pr ;x P 2 co ;x P 1 pr ; x P 1 co ;t P 1 pr ;t P 1 co ;s p u Cco j t P 1 co ;s p u Cpr j u Cco ;t P 1 co ;s p x C j u Cpr ;u Cco ;t P 1 pr ;t P 1 co ;s p v C j x P ;x C ;s p y P j x P ;x C ;s p y C j x P ;x C ;s ; andsatises 5{21a 5{21h ,andforwhichtheright-handsidesof 5{22a 5{22r are non-negative. Proof. Let A n X;Y denotesetofjointly -typicalsequencesaccordingtothedistribution ofrandomvariables X;Y asinducedbythesamedistributionusedtogeneratethe codebooks.AsintheproofforTheorem5.1,thedependenceontherandomvariableswill notbestatedexplicitly,andshouldbeclearfromthecontext.Toavoidrepetition,the erroranalysisfortherandomcodingschemeisnotpresentedhere,andcanbederivedina mannersimilartotheanalysisintheproofofTheorem5.1. Codebookgeneration: Splittheprimaryandcognitiveusers'ratesas R P = R s + R e + R P 1 pr + R P 2 co + R P 2 pr ,and R C = R Cco + R Cpr respectively.Fixadistribution p s;t P 1 co ;t P 1 pr ;x P 1 co ;x P 1 pr ;x P 2 co ;x P 2 pr ;x P ;u Cco ;u Cpr ;x C asinTheorem5.2. Generate2 nR s i.i.d.codewords s n w 0 s 2S n w 0 s 2f 1 ; ; 2 nR s g ,accordingto Q n i =1 p s i Foreachcodeword s n w 0 s ,generate2 nR e conditionallyi.i.d.codewords t n P 1 co w 0 s ;w 0 e w 0 e 2f 1 ; ; 2 nR e g ,accordingto Q n i =1 p t P 1 coi j s i Foreachcodewordpair s n w 0 s ;t n P 1 co w 0 s ;w 0 e ,generate2 nR P 1 pr conditionally i.i.d.codewords t n P 1 pr w 0 s ;w 0 e ;w 0 P 1 pr w 0 P 1 pr 2f 1 ; ; 2 nR P 1 pr g ,accordingto Q n i =1 p t P 1 pri j s i ;t P 1 coi Foreachcodewordpair s n w 0 s ;t n P 1 co w 0 s ;w 0 e ,generate2 nR P 1 co conditionally i.i.d.codewords x n P 1 co w 0 s ;w 0 e ;w P 1 co w P 1 co 2f 1 ; ; 2 nR P 1 co g ,accordingto Q n i =1 p x P 1 coi j s i ;t P 1 coi Foreachcodewordtuple )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [(s n w 0 s ;t n P 1 co w 0 s ;w 0 e ;x n P 1 co w 0 s ;w 0 e ;w P 1 co ;t n P 1 pr w 0 s ;w 0 e ;w 0 P 1 pr generate2 nR P 1 pr conditionallyi.i.d.codewords x n P 1 pr )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(w 0 s ;w 0 e ;w P 1 co ;w 0 P 1 pr ;w P 1 pr w P 1 pr 2 1 ; ; 2 nR P 1 pr ,accordingto Q n i =1 p x P 1 pri j s i ;t P 1 coi ;x P 1 coi ;t P 1 pri 99

PAGE 100

Foreachcodewordpair s n w 0 s ;t n P 1 co w 0 s ;w 0 e ,generate2 nR P 2 co conditionally i.i.d.codewords x n P 2 co w 0 s ;w 0 e ;w P 2 co w P 2 co 2f 1 ; ; 2 nR P 2 co g ,accordingto Q n i =1 p x P 2 coi j s i ;t P 1 coi Foreachcodewordtuple )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(s n w 0 s ;t n P 1 co w 0 s ;w 0 e ;x n P 2 co w 0 s ;w 0 e ;w P 2 co ;t n P 1 pr w 0 s ;w 0 e ;w 0 P 1 pr generate2 nR P 2 pr conditionallyi.i.d.codewords x n P 2 pr )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [(w 0 s ;w 0 e ;w P 2 co ;w 0 P 1 pr ;w P 2 pr w P 2 pr 2 1 ; ; 2 nR P 2 pr ,accordingto Q n i =1 p x P 2 pri j s i ;t P 1 coi ;x P 2 coi ;t P 1 pri Foreachcodewordpair s n w 0 s ;t n P 1 co w 0 s ;w 0 e ,generate2 n R Cco + R 0 Cco i.i.d.codewords u n Cco w 0 s ;w 0 e ;w Cco ;b Cco w Cco 2f 1 ; ; 2 nR Cco g and b Cco 2f 1 ; ; 2 nR 0 Cco g ,according to Q n i =1 p u Ccoi j s i ;t P 1 coi Foreachcodewordtuple )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(s n w 0 s ;t n P 1 co w 0 s ;w 0 e ;u n Cco )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(w 0 s ;w 0 e ;w Cco ;b Cco ,generate2 n R Cpr + R 0 Cpr i.i.d.codewords u n Cpr w 0 s ;w 0 e ;w Cco ;b Cco ;w Cpr ;b Cpr w Cpr 2 f 1 ; ; 2 nR Cpr g and b Cpr 2f 1 ; ; 2 nR 0 Cpr g ,accordingto Q n i =1 p u Cpri j s i ;t P 1 coi ;u Ccoi Generate x n P w 0 s ;w 0 e ;w 0 P 1 pr ;w P 1 co ;w P 1 pr ;w P 2 co ;w P 2 pr where x P isadeterministic functionof s;t P 1 co ;t P 1 pr ;x P 1 co ;x P 1 pr ;x P 2 co ;x P 2 pr Generate x n C w 0 s ;w 0 e ;w 0 P 1 pr ;w Cco ;b Cco ;w Cpr ;b Cpr where x C isadeterministicfunction of s;t P 1 co ;t P 1 pr ;u Cco ;u Cpr suchthat x C = if s = l Encoding: At S P : S P transmits x n P w s;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;w e;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ;w P 1 pr;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;w P 1 co;b ;w P 1 pr;b ;w P 2 co;b ;w P 2 pr;b in block b 2f 2 ; ;B )]TJ/F15 11.9552 Tf 12.032 0 Td [(1 g .Intherstblock,thereisnoresolutioninformationtotransmit, and S P transmits x n P ; 1 ; 1 ;w P 1 co; 1 ;w P 1 pr; 1 ;w P 2 co; 1 ;w P 2 pr; 1 ,whileinblock B ,ittransmits x n P w s;B )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ;w e;B )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ;w P 1 pr;B )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ; 1 ; 1 ;w P 2 co;B ;w P 2 pr;B .Notethattheactualrateforthe primarymessageis B )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 B R s + R e + R P 1 pr + R P 2 co + R P 2 pr ,butitconvergesto R P asthe numberofblocks B goestoinnity. At S C :Inblock b 2f 1 ; ;B g ,totransmit w Cco;b S C searchesforbinindex b Cco;b suchthat )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(s n ^ w s ;t n P 1 co ^ w s ; ^ w e ;u n Cco ^ w s ; ^ w e ;w Cco;b ;b Cco;b ;t n P 1 pr ^ w s ; ^ w e ; ^ w P 1 pr 2A n ; {23 where ^ w s ; ^ w e and ^ w P 1 pr are S C 'sestimatesof w s;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;w e;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 and w P 1 pr;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 respectivelyfrom thepreviousblock.Once b Cco;b isdetermined,itsearchesforabinindex b Cpr;b inorderto 100

PAGE 101

transmit w Cpr;b suchthat )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(s n ^ w s ;t n P 1 co ^ w s ; ^ w e ;u n Cco ^ w s ; ^ w e ;w Cco;b ;b Cco;b ;u n Cpr )]TJ/F15 11.9552 Tf 7.873 -8.079 Td [(^ w s ; ^ w e ;w Cco;b ;b Cco;b ;w Cpr;b ;b Cpr;b ; t n P 1 pr ^ w s ; ^ w e ; ^ w P 1 pr 2A n : {24 Itsets b Cco;b =1or b Cpr;b =1iftherespectivebinindexisnotfound.Itcanbeshown usingargumentssimilartothosein[64]thattheprobabilitiesoftheeventsof S C notable tondaunique b Cco;b or b Cpr;b satisfying5{23and5{24canbemadearbitrarilysmall ifthefollowingholdtrue: R 0 Cco > I U Cco ; T P 1 pr j T P 1 co ;S = t + 0 ; R 0 Cpr > I U Cpr ; T P 1 pr j U Cco ;T P 1 co ;S = t + 0 ; where 0 > 0maybearbitrarilysmall. S C transmits x n C w s;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;w e;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;w P 1 pr;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;w Cco;b ;b Cco;b ;w Cpr;b ;b Cpr;b Decoding: At S C :Assumethatdecodingtillblock b )]TJ/F15 11.9552 Tf 13.534 0 Td [(1hasbeensuccessful.Then,in block b S C knows w P 1 co;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 = w s;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ;w e;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 and w P 1 pr;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 .Itdeclaresthatthepair w P 1 co;b ;w P 1 pr;b = ^ w P 1 co ; ^ w P 1 pr wastransmittedinblock b ifthereexistsauniquepair ^ w P 1 co ; ^ w P 1 pr suchthat )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(s n w s;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;t n P 1 co w s;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ;w e;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ;t n P 1 pr w s;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;w e;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ; w P 1 pr;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;x n P 1 co )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(w s;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ;w e;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ; ^ w P 1 co ;x n P 1 pr w s;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ; w e;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ; ^ w P 1 co ;w P 1 pr;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ; ^ w P 1 pr ;v n C;b 2A n : Else,anerrorisdeclared.Itcanbeshownthattheprobabilityoferrorforthisdecoding stepcanbemadearbitrarilylowif5{22aand5{22baresatised. At D P :Theprimarydestination D P waitsuntilblock B ,andthenperformsbackward decoding.Weconsiderthedecodingprocessusingtheoutputinblock b 2f B )]TJ/F15 11.9552 Tf 12.093 0 Td [(1 ; ; 2 g Thedecodingfortherstandlastblockscanbeseenasspecialcasesoftheabove.Thus, 101

PAGE 102

forblock b 2f B )]TJ/F15 11.9552 Tf 12.112 0 Td [(1 ; ; 2 g ,assumingthatthedecodingforthepair w P 1 co;b ;w P 1 pr;b has beensuccessfulfromblock b +1, D P searchesforauniquetuple^ w s ; ^ w e ; ^ w P 1 pr ; ^ w P 2 co ; ^ w P 2 pr andsometuple ^ w Cco ; ^ b Cco suchthat )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(s n ^ w s ;t n P 1 co ^ w s ; ^ w e ;t n P 1 pr ^ w s ; ^ w e ; ^ w P 1 pr ;x n P 1 co ^ w s ; ^ w e ; w P 1 co;b ;x n P 1 pr ^ w s ; ^ w e ;w P 1 co;b ; ^ w P 1 pr ;w P 1 pr;b ;x n P 2 co ^ w s ; ^ w e ; ^ w P 2 co ;x n P 2 pr ^ w s ; ^ w e ; ^ w P 2 co ; ^ w P 1 pr ; ^ w P 2 pr ;u n Cco ^ w s ; ^ w e ; ^ w Cco ; ^ b Cco ;y n P;b 2A n : Theerroranalysisforthisdecodingstepshowsthat,for n largeenough, ^ w s ; ^ w e ; ^ w P 1 pr ; ^ w P 2 co ; ^ w P 2 pr = w s;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;w e;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;w P 1 pr;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;w P 2 co;b ;w P 2 pr;b witharbitrarilysmallprobabilityoferrorif5{22c-5{22laresatised. At D C :Thecognitivedestination D C alsowaitsuntilblock B ,andthenperforms backwarddecodingtojointlydecodethemessagesintendedforitandthecommonpartof theprimarymessage.Forblock b 2f B )]TJ/F15 11.9552 Tf 12.416 0 Td [(1 ; ; 2 g D C isassumedtohavesuccessfully decoded w P 1 co;b fromblock b +1.Withthisknowledge,itsearchesforauniquetuple ^ ^ w s ; ^ ^ w e ; ^ w Cco ; ^ b Cco ; ^ w Cpr ; ^ b Cpr andsome ^ ^ w P 2 co suchthat s n ^ ^ w s ;t n P 1 co ^ ^ w s ; ^ ^ w e ;x n P 1 co ^ ^ w s ; ^ ^ w e ;w P 1 co;b ;x n P 2 co ^ ^ w s ; ^ ^ w e ; ^ ^ w P 2 co ;u n Cco ^ ^ w s ; ^ ^ w e ; ^ w Cco ; ^ b Cco ; u n Cpr ^ ^ w s ; ^ ^ w e ; ^ w Cco ; ^ b Cco ; ^ w Cpr ; ^ b Cpr ;y n C;b 2A n : Again,usingthepropertiesofjointtypicality,itcanbeestablishedthat,for n large enough, ^ ^ w s ; ^ ^ w e ; ^ w Cco ; ^ b Cco ; ^ w Cpr ; ^ b Cpr = w s;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ;w e;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;w Cco;b ;b Cco;b ;w Cpr;b ;b Cpr;b withan arbitrarilylowprobabilityoferrorif5{22m-5{22raresatised. Thus,theconstraintsontheratesasgivenin5{22a-5{22rensurethatthe averageprobabilityoferroratthetwodestinationscanbedriventozeroandthus,they describeanachievablerateregionfortheICUC-HDC. 102

PAGE 103

Remark 5.3 Accordingtotheabovecodingscheme,apartoftheprimarymessage w P 2 isnotdecodedby S C .Thisisdierentfromthenon-causalcase.As S C cannotreceive whileittransmits, S P mayimproveitsratesbytransmittingfresh"informationdirectly tothedestinationduring S C -transmitstates,therebyincreasingtheachievablerateregion. Comparedtothesituationwherein S C iscapableoffull-duplexoperation,transmittinga partofthemessagedirectlytothedestinationprovidespotentialgainsevenwhenthe S P to S C channelismuchbetterthanthedirectlinkto D P Remark 5.4 Notethatthemaximumincreaseintheachievableratesthatmayberealized byusingarandomlisten-transmitschedulefor S C is1bit[73]. Remark 5.5 TheachievablerateregiondescribedinTheorem5.2isalsoconvexcf. Remark5.2andconsequently,time-sharingisnotrequiredtoenlargetherateregion. Here,therandomvariable S inTheorem5.2playstherolesimilartothatof U in[74, Lemma5]. Remark 5.6 FortheGaussianchannelmodelwithaxedlisten-transmitschedule,the codingschemeofTheorem5.2yieldsthesamerateregionaswithatime-divisionstrategy withtheuseofGaussianparallelchannels[75],insteadofablockMarkovstructure,for thedecodingof w P 1 = w P 1 co ;w P 1 pr at D P and w P 1 co at D C .Accordingtothisstrategy, S P transmits w P 1 duringthersttime-slotwhile S C isinlisteningmode.Inthesecond time-slot,both S P and S C encodeandtransmit w P 1 asanon-causalICUC,and S P alsosuperposes w P 2 ontopof w P 1 thelatteractingastheresolutioninformationforthe destinations.Bothdestinationsdecodeonlyattheendofthesecondtime-slotandexploit theparallelGaussianchannelstructuretodecode w P 1 entirelyorpartially. 5.7TheGaussianICUC-HDC AsinSection5.4,fortheGaussianICUC-HDC,thedirectlinksforeachuser-pairis normalizedtounity, g PC isthechannelgainforthe S P S C link, h PC isthatforthe S P D C link, h CP isthatforthe S C D P link,and S C D P ,and D C areassumedto experiencei.i.d.additivewhiteGaussiannoiseAWGNofunit-variance.Asmentionedin 103

PAGE 104

Remark5.4,theincreaseintheachievablerateregionusingarandomizedlisten-transmit scheduleoverthatwithaxedscheduleisupperboundedbyonebit.Moreover,itshould benotedthat,forarandomizedlisten-transmitschedule,theoptimaldistributionfor therandomvariable X C maynotbeGaussian[73].Fortheabovemodel,weconsidera xedlisten-transmitscheduleandhavethefollowinginput-outputrelationshipsforthe ICUC-HDC: V C;l = g PC X P;l + Z S C ; {25 Y P;l = X P;l + Z P ; {26 Y C;l = h PC X P;l + Z D C ; {27 Y P;t = X P;t + h CP X C + Z P ; {28 Y C;t = h PC X P;t + X C + Z D C ; {29 where Z S C ;Z P ;Z D C N ; 1arei.i.d.randomvariablescorrespondingtotheadditive noiseat S C D P ,and D C respectively.Intheabove, X P;l and X P;t arethetransmitted signalsduringthelistenandtransmitstatesrespectively.Similarnotationisusedto describethereceivedsignalsattheconcernednodesaswell.Finally,itisassumedthat theprimaryandcognitivesourcesaresubjecttothepowerconstraintsof5{17foreach statewithinanycommunicationblock.Thismaybeinterpretedasthescenarioinwhich bothsourcesareconstrainedbytheirrespectivelisten/transmitmodepowerconstraints, insteadofanaveragepowerconstraintoverablock.Itmaybenotedthataveragepower constraints,ofasimilaravorasin[73],maybeincludedalongwiththemodepower constraintsinoursystemmodel,butisavoidedherefortheeaseofpresentationand expositionofthemainaspectsofthecodingscheme.Tosummarize,thepowerconstraints inthissectionmaybeexpressedas: 1 n l jj X n l P;l jj 2 P P ; 1 n t jj X n t P;t jj 2 P P ; 1 n t jj X n t C jj 2 P C ; {30 104

PAGE 105

where,foranychoiceofpositive n t and n l n l + n t = n isthetotalnumberofchanneluses inacommunicationblock. Next,weprovideananalyticalprooffortheinclusionoftheachievablerateregionfor theGaussianICUC-HDCpresentedin[6].Weshowthatanouterboundnotnecessarily achievabletotherateregionpresentedin[6]iscontainedinasubspaceoftheachievable rateregionofTheorem5.2.Wedescribethespecializationofthecodingschemeof Theorem5.2totheGaussiancaseinfurtherdetailinSubsection5.7.2,alongwithsome numericalexamples. 5.7.1InclusionOfCausalAchievableRegionof[6] In[6],anachievablerateregionfortheGaussianICUC-HDCwaspresented.The authorsproposedfourprotocolsandtheoverallachievablerateregion R 0 isgivenbythe convexhullofthefourrateregions[6,Theorem5].Inthissection,weshowthattherate regionofTheorem5.2, R ,contains R 0 Forthenon-causalICUC,thecontainmentoftheregionof[6,Corollary2], R DMT intheregion R D of[76,Theorem1]isclear.Itisshownin[66]that R RTD [66,Theorem 1]contains R D .Morespecically,[66]showsthat R D R out D R in RTD R RTD ,where R out D isobtainedfrom R D byremovingcertainrateconstraints,and R in RTD isobtained from R RTD byrestrictingtheinputdistributiontomatchthatfor R D .Thecodingscheme ofTheorem5.2maybespecializedtoyieldarateregionfortheICUC.Towardsthis,we set S = t w.p.1 ;X P 2 co = X P 2 pr = ,andassumethatagenieprovides S C with w P Thisgivesusanachievablerateregion R NC fortheICUC.Moreover,byrestrictingthe inputdistributiontoindependentrate-splittingandindependentbinningofthesecondary messagesasin[6,76]insteadofconditionalrate-splittingandconditionalbinningat S C itcanbeshownusinganappropriatemappingofthecodebookrandomvariablesomitted duetolackofspace,thattheresultingregion R in NC isidenticalto R in RTD ,andhence, R DMT R D R in NC 105

PAGE 106

Next,weshowthattherateregionsobtainedviaeachoftheprotocolsproposedin[6] arecontainedin R .Notethatforalltheseprotocols, w P = w P 1 ;w P = w Pco ;w Ppr ,with rates R P = R Pco + R Ppr ,etc.Tocomparethetworateregions,westartwithanalternate descriptionfortheachievablerateregionforProtocol1[6,Lemma3].Accordingto Protocol1,foranychoiceof ,theratepair R P ;R C isachievableif R P 2 log+ g PC P P +log 1+ P P 1+ P P ; {31 R 0 P ;R 0 C 2R DMT ;R C = R 0 C ;R Pco = R 0 Pco ; {32 R Ppr 2 log 1+ P P 1+ P P + R 0 Ppr ; {33 where 2 [0 ; 1]isthepowerfractionallocatedfortransmittingapartsameas in[6]of w Ppr Considertheregioncorrespondingtothexedlisten-transmitscheduleandusingparallelGaussianchannelsasinRemark5.6.Forthersttime-slot,settheinputdistribution at S P as p x P 1 co j s = l p x P 1 pr j x P 1 co ;s = l .FortheequivalentICUCduringthesecond time-slot,set X P 2 co = X P 2 pr = ,andrestricttheinputdistributiontocorrespondto independentrate-splittingandbinningasin[66,]tomatchthedistributioncorrespondingto R D .Lettheoverallrateregiontherebyobtainedbe R in 1 .Clearly, R in 1 R UsingtheresultforparallelGaussianchannels,itcanbeshownthatforanychoiceof theratepair R P ;R C isachievableif R P 2 log+ g PC P P ; R 0 P ;R 0 C 2R in NC ; {34 R C = R 0 C ;R Pco =min 2 log 1+ P P P 1+ P P P ; 2 log 1+ h PC P P P 1+ h PC P P P + R 0 Pco ; {35 R Ppr 2 log+ P P P + R 0 Ppr ; {36 where P 2 [0 ; 1]isthepowerfractionallocatedfortransmitting w Pco inthersttimeslot.Notethat,givenan value, P maybechosensuchthat 1+ P P P 1.Then, 106

PAGE 107

comparing5{31-5{33to5{34-5{36establishesthattheregioncorrespondingto Protocol1iscontainedin R in 1 TheinclusionoftherateregioncorrespondingtoProtocol2canbeeasilyprovedby consideringthesamecodingstructureandinputdistributionasusedtoobtain R in 1 ,with onefurtherrestriction-theinputdistributionat S P forthersttime-slotisgivenby p x P 1 co j s = l p x P 1 pr j s = l .Thisyieldsanachievablerateregion R in 2 R ,thathas exactlythesameboundsasthatforProtocol2,exceptthattheachievablerateregionfor theNC-CRCduringthesecondtime-slotis R in NC R DMT ,therebyprovingtheabove inclusion. TherateregionforProtocol3canbeobtainedbysetting S = t w.p.1 ;X P 1 co = X P 1 pr = T P 1 co = T P 1 pr = inTheorem5.2.Finally,theratepaircorrespondingto Protocol4maybeobtainedbyusingaxedlisten-transmitschedule,andbysetting T P 1 co = X P 1 co = X P 2 co = X P 2 pr = U Cco = U Cpr = .Asthefourrateregionsof[6] arecontainedin R ,theconvexhulloftheseregions R 0 isalsocontainedin R cf. Remark5.5. 5.7.2NumericalResults Foraxedlisten-transmitscheduleandconsideringparallelGaussianchannelsasin Remark5.6,thetransmittedsignalsinanycommunicationblock,correspondingtothe codingschemeofTheorem5.2,canbeexpressedas: X P;l w P 1 = X P 1 co + X 0 P 1 pr ; {37 X P;t w P 1 ;w P 2 = T P 1 co + X 0 P 2 co + T 0 P 1 pr + X 0 P 2 pr ; {38 X C w P 1 ;w C = X Cco + X Cpr ++ s C C P C P P P P T P 1 co + s C C P C P P P P T 0 P 1 pr ; {39 where X P 1 co N ; P P P X 0 P 1 pr N ; P P P T P 1 co N ; P P P P X 0 P 2 co N ; P P P P T 0 P 1 pr N ; P P P P X 0 P 2 pr N ; P P P P X Cco N ; C C P C and X Cpr N ; C C P C arei.i.d.randomvariables.Intheabove, P P P P C 107

PAGE 108

C C arerealnumbersintheinterval[0 ; 1]. S C usesDPCtoencode X Cco and X Cpr as U Cco = X Cco + co T P 1 pr ; {40 U Cpr = X Cpr + pr T P 1 pr ; {41 with co and pr beingnon-negativerealnumbersthatdenotethecorrelationbetweenthe knowninterference T P 1 pr andtheauxiliaryrandomvariables U Cco and U Cpr respectively, conditionedon T P 1 co .NotethataccordingtothenotationofTheorem5.2, X P;l = X P 1 pr and X P;t = X P 2 pr Inthefollowing,wepresentsomenumericalexamplestocomparetheachievablerate regioncorrespondingtothetransmissionschemeproposedin[6]tothatforTheorem5.2, withaxedlisten-transmitscheduleandspecializedforGaussianchannels.Inthese examples,thelinkbetweenthetwosourcesisassumedtobebetterthanthedirectlink, andwecomparetheHan-Kobayashirateregionfortheinterferencechannelwithout anyactivecooperationbetweentheuser-pairs,therateregionof[6],andthatforthe proposedcodingschemeinthiswork.InFig.5-7,weconsiderthescenariowhenboth interferinglinksareweakerthanthedirectlinks,whileinFig.5-8,theinterferinglink from S P to D C isstrongandthatfrom S C to D P isweak. Comparingthetwoguresshowsthattheimprovementinthequalityoftheinterferinglinkfrom S P to D C maysignicantlyincreasetheoverallrateregionforthetwo user-pairs.Also,themannerinwhichtherateregionof[6]isenlargedinbothexamples suggeststhattheeciencyoftheoverallcooperativerelayingschemeistheprimary contributortotheenlargementoftherateregion.Theadvantageofthecodingscheme adaptedfromTheorem5.2,anddescribedinequations5{37through5{41,liesinthe eectiveutilizationofthedirectlinkfortheprimaryuser-pairviathetransmissionofthe codewordscorrespondingtothemessageparts w P 2 = w P 2 co ;w P 2 pr .Thus,nothavingthe entireprimarymessagebeingdecodedandtransmittedthroughthecognitivesourcemay 108

PAGE 109

considerablyincreasetheachievablerateregion,especiallyinthedirectionoftheprimary users'rate, R P cf.Remark5.3. 5.8Summary Inthischapter,anewachievablerateregionforthediscretememorylessinterference channelwithunidirectionalcooperationICUC,whereintheprimarymessagemayonly becausallyavailableatthecognitivesource,isderived.Thecodingscheme,specialized forGaussianchannelsisalsopresentedandisusedtonumericallyevaluatedierent codingstrategiesthatareusedasbuildingblocksfortheproposedcodingscheme. Theseresultsalsodemonstratethattheproposedcodingschemesignicantlyenlarges thepreviouslyknownrateregionforvariousnetworkscenarios.Adiscretememoryless channelmodelfortheICUC-HDCwasalsopresentedinthischapter.Arandomcoding scheme,employingblockMarkovSPC,conditionalrate-splittingofprimaryandsecondary messages,conditionalbinning,andarandomizedlisten-transmitscheduleforthecognitive source,wasusedtoderiveanewachievablerateregionforthischannel.ForGaussian channels,thecontainmentofthepreviouslyknownrateregion[6]inthenewrateregion wasanalyticallyproved,andnumericalexampleswerepresentedtosupplementthe analyticalcomparison. 109

PAGE 110

Figure5-1.ThediscretememorylessICUCwithcausalityconstraint. Figure5-2.TheGaussianICUC-C. 110

PAGE 111

Figure5-3.AchievableRatesfortheGaussianICUC-C:Weakinterferenceforboth cross-links. 111

PAGE 112

Figure5-4.AchievableRatesfortheGaussianICUC-C:Stronginterferencefrom S P to D C andweakinterferencefrom S C to D P 112

PAGE 113

Figure5-5.AchievableRatesfortheGaussianICUC-C:Weakinterferencefrom S P to D C andstronginterferencefrom S C to D P 113

PAGE 114

Figure5-6.ThediscretememorylessICUC-HDC. Figure5-7.AchievableRatesfortheGaussianICUC-HDC:Weakinterferenceforboth cross-links. 114

PAGE 115

Figure5-8.AchievableRatesfortheGaussianICUC-HDC:Stronginterferencefrom S P to D C andweakinterferencefrom S C to D P 115

PAGE 116

CHAPTER6 CONCLUSIONSANDFUTUREWORK 6.1Conclusions Inthiswork,wehavestudieddierentlevelsofcooperationmanifestedinavariety ofmulti-usercommunicationsystemfromow-theoreticandinformation-theoretic perspectives.Forthesingle-source-single-destinationwirelessclusterwithdedicatedrelays, weproposedcooperativetransmissionprotocolsusingaow-theoreticapproach.This includedtheFOprotocolandthesuboptimal,butmuchsimpler,GLSprotocol.Boththe protocolsareshowntoachievetheoptimaldiversity-multiplexingtradeo,andtheGLS protocolisshowntobeaverygoodcandidateforuseinsystemswithlowcomplexity requirements.Simulationresultsfordierentclustersizes,anduniformandnon-uniform averagepowergainsindicatethattheproposedprotocolsprovidelargecodinggainsby ecientlyutilizingtheCSIavailableatallnodes,andthattheyperformmuchbetterthan otherpathselectionmethodspreviouslyproposedintheliterature,especiallyinhighdata raterequirementsituations. Wenextconsideredthetwo-userfadingMACasanexampleofamulti-sourcesystem. Weproposedagame-theoreticformulationinvolvingbargainingandmaximingames tomodeltheresourceallocationproblemanddevelopacharacterizationofcooperative behaviorforthissystemunderuncertaintyregardingtheaccuracyoftheCSIT.To improvetherobustnessofthesystem,weproposedthattheconventionalbargaining problemberelaxedsothattheusers,insteadofbeingboundtoexecutethestrategypair suggestedbythesolutiontotheconventionalbargainingproblem,mayindependently choosetheirtransmissionstrategyfromtheirrespectivesetofstrategiesdenedby themaximumdeviationparametersaboutthenominalstrategypair.Thisreducesthe dependenceofthesystemperformanceonthesolutiontothebargainingproblemwiththe possiblyinaccurateavailableCSIT.Fromthedevelopmentofthisformulation,itcan beseenthatevenintheconventionaltwo-userMAC,thereexistsacertainlevelofuser 116

PAGE 117

cooperation.Usingnumericalexamples,wedemonstratedtheeectsofuncertaintyonthe achievableaverageratesandtheimprovementinthesystemrobustnessprovidedbythe proposeddesign. TheCMAC,thatinvolvesahigherlevelofcooperationbetweentheusersintheform ofactiveforwardingofeachother'sinformation,wasstudiedinChapter4.Again,using theow-theoreticapproach,wedevelopedtwocooperativetransmissionprotocols,based onDFrelaying,forcooperativetransmissionintheCMAC.WeproposedtheOR-CMAC thatdecomposestheCMACintotwoorthogonalrelaychannels,andtheFO-CMACthat decomposestheCMACintotwobroadcastchannelsandoneMAC.Moreover,withthe assumptionoftheavailabilityofphasesynchronization,weproposedthemodicationof theMAslotsofChapter2toMAwithcommoninformationforfurtherperformancegain. Simulationresultsfordierentscenariosindicatethepotentialperformanceimprovements overpreviouslyproposedtransmissionstrategies,intermsofaverageratesandoutage probabilities. Finally,weaddressedtheproblemofcommunicatingthroughtheICUCwitha causalityconstraint.Thishelpsusavoidthesomewhatunrealisticassumptionofthe cognitivesourcehavingnon-causalknowledgeoftheprimarymessagethatisconsidered inmostrelatedworksintheliterature.Wederivedanewachievablerateregionforthe discretememorylessversionofthischannel,withanassumptionoffull-duplexoperation atthecognitivesource.WespecializedthecodingschemeforGaussianchannelsand usedittonumericallyevaluatedierentcodingstrategiesthatareusedasbuilding blocksfortheproposedcodingscheme.Theseresultsalsodemonstratethattheproposed codingschemesignicantlyenlargesthepreviouslyknownrateregionforvariousnetwork scenarios.Followingthis,weremovedtheassumptionoffull-duplexcapabilityatthe cognitivesource,andpresentedadiscretememorylesschannelmodelfortheICUC-HDC. Wedevelopedarandomcodingscheme,employingblockMarkovSPC,conditionalratesplittingofprimaryandsecondarymessages,conditionalbinning,andarandomized 117

PAGE 118

listen-transmitscheduleforthecognitivesource,toderiveanewachievablerateregionfor thischannel.ForGaussianchannels,weprovedthecontainmentofthepreviouslyknown rateregion[6]inthenewrateregionfortheGaussianICUC-HDC,anddemonstratedthis withnumericalsimulationresults. 6.2FutureDirections Theow-theoreticapproachintroducedinChapter2couldalsobeappliedtomultiusersystemsthatinvolvemorecomplexusercooperationthantheonesconsideredinthis dissertation.Forinstance,theow-theoreticapproachwouldbesuitablefortheproblem ofinformationtransmissioninacooperativerelaybroadcastchannelRBC[5],wherein onesourcenodebroadcastsinformationtotworeceivernodes,whonowactivelycooperate fullyorpartiallywitheachotherindecodingtheirrespectivemessages.Although achievablerateregionsemployingblockMarkovcodingalongwithdecode-and-forwardand estimate-and-forward[2,Theorem6]techniqueshavebeenproposedintheliterature,the appealoftheow-theoreticapproachliesinitssimplicity.Theow-theoreticapproach essentiallybreaksdowntheoriginalchannelintomuchsimplerchannelsforwhichthe capacityregionsareknownandpracticalcodingschemesthatperformclosetothe randomcodingschemehavebeenextensivelyinvestigatedintheliterature.FortheRBC, asimilartime-slottingapproachofChapter4wouldnowinvolveoneBCandtwoMA withcommoninformationtime-slots,anditwouldbeinterestingtoinvestigateastohow theperformanceoftheow-optimizedsolutioncomparestothemorecomplicatedblock MarkovmethodsindierentchannelconditionsandSNRregimes. InChapter5,wepresentednewachievablerateregionsfortheICUC-CandICUCHDCforbothdiscretememorylessandGaussianchannels.Althoughwehaveshown theinclusionofpreviouslyproposedrateregionsbothanalyticallyaswellasthrough numericalsimulations,westilldonotknowhowclosewearetothecapacityregions forthesechannels.Inthisregard,newouterboundsforthesechannelsthataretighter thantheMIMObroadcastchannelcapacityregion[77]wouldbenecessary.ForGaussian 118

PAGE 119

channels,onewaytoapproachthisproblemwouldbetoconsiderthedeterministicversion fortheICUC-Cwithoutanyrandomnessinthechannelsandmodeltherelationship betweenthedeterministicandtheirGaussiancounterpartsasusedin[16]. YetanotherresearchdirectionthatmaybepursuedinregardtotheICUC-Cis thestudyoftheroleoftransmittersideinformationatboththeprimaryandcognitive sources.InSection5.4,wedemonstratedtheinterplaybetweenthedierentextent ofcodebookknowledgeatthedierentnodesandtheeectofthedierentcoding buildingblockslikeDPC,rate-splitting,andcooperativerelayingforvariouschanneland transmitpowerconditions.Thismaybeconsideredasaspecialcaseofthestudyofthe relationshipbetweenageneralabstractionofsideinformationatthenodes,thesetof codingtechniquesthatmaybefeasible,andtheresultingachievablerateregions.One possiblewaytomodelthismaybetoconsiderthechannelshavingdierentstateswith dierentlevelsofinformationaboutthesestatesavailableatthetwosources.Related tothis,onemayalsoaskthequestionwhetheracognitivesourcewithonlysignal-level cognition,insteadofmessage-levelcognitioncanhelpinenlargingtheHan-Kobayashi regionforthetraditionaltwo-userinterferencechannel.Inotherwords,theproblemwould betodeterminethecodingstrategiesthatmaybeusedbythecognitivesourcewhen thelevelofcognition"regardingtheprimarymessageisatthesignal-levelinsteadof themessage-level,andif,theresultingachievableratescouldimproveupontheusual interferenceavoidanceorinterferencecontrolmethodsliketheinterweaveorunderlay modes[78]ofcognitiveradiooperation. 119

PAGE 120

REFERENCES [1]V.Tarokh,N.Seshadri,andA.R.Calderbank,Space-timecodesforhighdatarate wirelesscommunication:performancecriterionandcodeconstruction," IEEETrans. Inform.Theory ,vol.44,no.2,pp.744{765,Feb.1998. [2]T.CoverandA.ElGamal,Capacitytheoremsfortherelaychannel," IEEETrans. Inform.Theory ,vol.25,no.5,pp.572{584,May1979. [3]A.Sendonaris,E.Erkip,andB.Aazhang,Usercooperationdiversity.PartI. Systemdescription," IEEETrans.Commun. ,vol.51,no.11,pp.1927{1938,Nov. 2003. [4]A.Sendonaris,E.Erkip,andB.Aazhang,Usercooperationdiversity.PartII. Implementationaspectsandperformanceanalysis," IEEETrans.Commun. ,vol.51, no.11,pp.1939{1948,Nov.2003. [5]Y.LiangandV.V.Veeravalli,Cooperativerelaybroadcastchannels," IEEETrans. Inform.Theory ,vol.53,no.3,pp.900{928,Mar.2007. [6]N.Devroye,P.Mitran,andV.Tarokh,Achievableratesincognitiveradiochannels," IEEETrans.Inform.Theory ,vol.52,no.5,pp.1813{1827,May2006. [7]A.JovicicandP.Viswanath,Cognitiveradio:Aninformationtheoreticperspective,"submittedto IEEETrans.Inform.Theory ,July2006. [8]S.ShamaiShitz,O.Somekh,O.Simeone,A.Sanderovich,B.M.Zaidel,andH. V.Poor,Cooperativemulti-cellnetworks:Impactoflimited-capacitybackhauland inter-userslinks,"in Proc.JointWorkshopCodingandCommun.JWCC2007 Durnstein,Austria,2007. [9]L.Sankaranarayan,G.Kramer,andN.B.Mandayam,Cooperationvs.hierarchy: aninformation-theoreticcomparison,"in Proc.IEEEInternationalSymp.Inform. TheoryISIT2005 ,Adelaide,Australia,Sep.2005. [10]P.GuptaandP.Kumar,Thecapacityofwirelessnetworks," IEEETrans.Inform. Theory ,vol.46,no.2,pp.388{404,Feb.2000. [11]J.N.Laneman,D.N.C.Tse,andG.W.Wornell,Cooperativediversityinwireless networks:Ecientprotocolsandoutagebehavior," IEEETrans.Inform.Theory vol.50,no.12,pp.3062{3080,Dec.2004. [12]A.Host-MadsenandJ.Zhang,Capacityboundsandpowerallocationforwireless relaychannels," IEEETrans.Inform.Theory ,vol.51,no.6,pp.2020{2040,Jun. 2005. [13]G.Kramer,M.Gastpar,andP.Gupta,Cooperativestrategiesandcapacity theoremsforrelaynetworks," IEEETrans.Inform.Theory ,vol.51,no.9,pp. 3037{3063,Sep.2005. 120

PAGE 121

[14]M.GastparandM.Vetterli,OnthecapacityoflargeGaussianrelaynetworks," IEEETrans.Inform.Theory ,vol.51,no.3,pp.765{779,Mar.2005. [15]T.S.HanandK.Kobayashi,Anewachievablerateregionfortheinterference channel," IEEETrans.Inform.Theory ,vol.IT-27,no.1,pp.49{60,Jan.1981. [16]R.H.Etkin,D.N.C.Tse,andH.Wang,Gaussianinterferencechannelcapacityto withinonebit," IEEETrans.Inform.Theory ,vol.54,no.12,pp.5534{5562,Dec. 2008. [17]J.JiangandY.Xin,Ontheachievablerateregionsforinterferencechannelswith degradedmessagesets," IEEETrans.Inform.Theory ,vol.54,no.10,pp.4707{4712, Oct.2008. [18]A.Goldsmith,S.A.Jafar,I.Maric,andS.Srinivasa,Breakingspectrumgridlock withcognitiveradios:Aninformationtheoreticperspective," Proc.IEEE ,vol.97, issue5,pp.894{914,May2009. [19]K.Azarian,H.ElGamal,andP.Schniter,Ontheachievablediversity-multiplexing tradeoinhalf-duplexcooperativechannels," IEEETrans.Inform.Theory ,vol.51, no.12,pp.4152{4172,Dec.2005. [20]A.S.AvestimehrandD.N.C.Tse,Outagecapacityofthefadingrelaychannelin low-SNRregime," IEEETrans.Inform.Theory ,vol.53,no.4,pp.1401{1415,Apr. 2007. [21]D.GunduzandE.Erkip,Opportunisticcooperationbydynamicresourceallocation," IEEETrans.WirelessCommun. ,vol.6,no.4,pp.1446{1454,Apr.2007. [22]L.OngandM.Motani,OptimalroutingfortheGaussianmultiple-relaychannel withdecode-and-forward,"in Proc.IEEECommun.Soc.Conf.Sensor,Mesh,and AdHocCommun.andNetw.SECON2007 ,SanDiego,U.S.A.,Jun.2007. [23]L.-L.XieandP.R.Kumar,Anetworkinformationtheoryforwirelesscommunication:ScalinglawsandoptimalOperation," IEEETrans.Inform.Theory ,vol.50,no. 5,pp.748{767,May2004. [24]G.Kramer,M.Gastpar,andP.Gupta,Capacitytheoremsforwirelessrelay channels," IEEETrans.Inform.Theory ,vol.51,no.9,pp.3037{3063,Sep.2005. [25]A.Reznik,S.R.Kulkarni,andS.Verdu,DegradedGaussianmultirelaychannel: CapacityandoptimalpowerAllocation," IEEETrans.Inform.Theory ,vol.50,no. 12,pp.3037{3046,Dec.2004. [26]L.OngandM.Motani,Thecapacityofthesinglesourcemultiplerelaysingle destinationmeshnetwork,"in Proc.IEEEInternationalSymp.Inform.Theory ISIT2006 ,Seattle,U.S.A.,Jul.2006. 121

PAGE 122

[27]A.Bletsas,A.Khisti,D.P.Reed,andA.Lippman,Asimplecooperativediversity methodbasedonnetworkpathselection," IEEEJ.Select.AreasCommun. ,vol.24, No.3,pp.659{672,Mar.2006. [28]E.BeresandR.S.Adve,Onselectioncooperationindistributednetworks,"in Proc.Conf.Inform.SciencesandSyst.CISS2006 ,pp.1056{1061,Mar.2006. [29]J.N.LanemanandG.W.Wornell,Distributedspace-time-codedprotocolsfor exploitingcooperativediversityinwirelessnetworks," IEEETrans.Inform.Theory vol.49,no.10,pp.2415{2425,Oct.2003. [30]T.F.Wong,T.M.Lok,andJ.M.Shea,Flow-optimizedCooperativeTransmission fortheRelayChannel,"submittedto IEEETrans.Inform.Theory ,Dec.2006. [Online].Available: http://arxiv.org/PS_cache/cs/pdf/0701/0701019v3.pdf [31]L.ZhengandD.N.C.Tse,Diversityandmultiplexing:Afundamentaltradeo inmultipleantennachannels," IEEETrans.Inform.Theory ,vol.49,no.5,pp. 1073{1096,May2003. [32]M.YukselandE.Erkip,Multiple-antennacooperativewirelesssystems:A diversity-multiplexingtradeoperspective," IEEETrans.Inform.Theory ,vol. 53,no.10,pp.3371{3393,Oct.2007. [33]T.M.CoverandJ.A.Thomas, Elementsofinformationtheory ,Wiley,1991. [34]Y.Wu,P.A.Chou,andS.-Y.Kung,Minimum-energymulticastinmobileadhoc networksusingnetworkcoding," IEEETrans.Commun. ,vol.53,no.11,Nov.2005. [35]R.K.Ahuja,T.L.Magnanti,andJ.B.Orlin, Networkows:Theory,algorithms, andapplications ,PrenticeHall,1993. [36]C.T.Lawrence,J.L.Zhou,andA.L.Tits,User'sguideforCFSQPversion2.5: ACcodeforsolvinglargescaleconstrainednonlinearminimaxoptimization, generatingiteratessatisfyingallinequalityconstraints," TechnicalReportTR-9416r1 ,UniversityofMaryland,CollegePark,1997. [37]W.P.Tam,T.M.Lok,andT.F.Wong,Flowoptimizationinparallelrelay networkswithcooperativerelaying," IEEETrans.WirelessCommun. ,vol.8,no.1, pp.278{287,2009. [38]D.TseandS.Hanly,Multi-accessfadingchannels.PartI.Polymatroidstructure, optimalresourceallocationandthroughputcapacities," IEEETrans.Inform. Theory ,vol.44,No.7,pp.2796-2815,Nov.1998. [39]G.Caire,G.Taricco,andE.Biglieri,Optimumpowercontroloverfadingchannels," IEEETrans.Inform.Theory ,vol.45,no.5,pp.1468{1489,Jul.1999. 122

PAGE 123

[40]L.Li,N.Jindal,andA.Goldsmith,Outagecapacitiesandoptimalpowerallocation forfadingmultiple-accesschannels," IEEETrans.Inform.Theory ,vol.51,no.4,pp. 1326-1347,Apr.2005. [41]L.LaiandH.E.Gamal,Thewater-llinggameinfadingmultipleaccesschannels," submittedto IEEETrans.Inform.Theory ,Nov.2005.[Online].Available: http: //arxiv.org/abs/cs/0512013 [42]J.Sun,L.Zheng,andE.Modiano,Wirelesschannelallocationusinganauction algorithm,"in Proc.AllertonConf.Commun.,ControlandComputing ,pp.11141123,Oct.2003. [43]A.ParandehGheibi,A.Eryilmaz,A.Ozdaglar,andM.Medard,Dynamicrateallocationinfadingmultipleaccesschannels,"in Proc.Inform.TheoryandApplications ITAWorkshop ,2008. [44]W.Yu,W.Rhee,J.Cio,Optimalpowercontrolinmultipleaccessfading channelswithmultipleantennas,"in Proc.IEEEInternationalConf.Commun., ICC2001 ,vol.2,pp.575-579,Jun.2001. [45]M.Mecking,Resourceallocationforfadingmultiple-accesschannelswithpartial channelstateinformation,"in Proc.IEEEInternationalConf.Commun.,ICC 2002 ,vol.3,pp.1419-1423,2002. [46]R.Narasimhan,Individualoutagerateregionsforfadingmultipleaccesschannels," in Proc.IEEEInternationalSymp.Inform.TheoryISIT2007 ,pp.1571{1575,Jun. 2007. [47]M.Medard,Theeectuponchannelcapacityinwirelesscommunicationsofperfect andimperfectknowledgeofthechannel," IEEETrans.Inform.Theory ,vol.46,no. 3,pp.933-946,May2000. [48]A.Lapidoth,andS.ShamaiShitz,Fadingchannels:Howperfectneedperfect sideinformation"be?," IEEETrans.Inform.Theory ,vol.48,no.5,pp.1118-1134, May2002. [49]R.Narasimhan,Eectofchannelestimationerrorsondiversity-multiplexingtradeoinmultipleaccesschannels," Proc.IEEEGlobalCommun.Conf.,GLOBECOM 2006 ,pp.1-5,Nov.2006. [50]M.J.OsborneandA.Rubinstein,Acourseingametheory,"MITPress,1994. [51]W.C.Riddell,Bargainingunderuncertainty," Amer.Econ.Rev. ,vol.71,no.4,pp. 579-590,Sep.1981. [52]W.Bossert,E.Nosal,andV.Sadanand,Bargainingunderuncertaintyandthe monotonepathsolutions," GamesandEcon.Behavior ,vol.14,no.2,pp.173-189, 1996. 123

PAGE 124

[53]W.Thomson,andR.B.Myerson,Monotonicityandindependenceaxioms," InternationalJ.GameTheory ,vol.9,no.1,pp.37-49,Mar.1980. [54]E.Kalai,Proportionalsolutionstobargainingsituations:Interpersonalutility comparisons," Econometrica ,vol.45,no.7,pp.1623-1630,1977. [55]O.KayaandS.Ulukus,Powercontrolforfadingcooperativemultipleaccess channels," IEEETrans.Commun. ,vol.6,no.8,pp.2915{2923,Aug.2007. [56]W.MesbahandT.N.Davidson,Optimalpowerallocationforfull-duplexcooperativemultipleaccess,"in Proc.IEEEInternationalConf.Acoust.,Speech,andSignal Process.ICASSP2006 ,Toulouse,France,vol.IV,pp.689{692,May2006. [57]O.Kaya,WindowandbackwardsdecodingachievethesamesumrateforthefadingcooperativeGaussianmultipleaccesschannel,"in Proc.IEEEGlobalCommun. Conf.,GLOBECOM2006 ,SanFrancisco,CA,Nov.2006. [58]W.MesbahandT.N.Davidson,Optimalpowerandresourceallocationforhalfduplexcooperativemultipleaccess,"in Proc.IEEEInternationalConf.Commun. ICC2006 ,vol.10,Istanbul,Turkey,Jun.2006. [59]E.G.LarssonandB.R.Vojcic,Cooperativetransmitdiversitybasedonsuperpositioncoding," IEEECommun.Lett. ,vol.9,no.9,pp.778{780,Sep.2005. [60]Z.Ding,T.Ratnarajah,andC.C.F.Cowan,Onthediversity-multiplexingtradeo forwirelesscooperativemultipleaccesssystems," IEEETrans.SignalProcess. ,vol. 55,no.9,pp.4627{4638,Sep.2007. [61]N.Liu,andS.Ulukus,CapacityRegionandOptimumPowercontrolstrategies forfadingGaussianmultipleaccesschannelswithcommondata," IEEETrans. Commun. ,vol.54,no.10,pp.1815-1826,Oct.2006. [62]D.N.C.Tse,P.Viswanath,andL.Zheng,Diversity-multiplexingtradeoin multiple-accesschannels," IEEETrans.Inform.Theory ,vol.50,no.9,pp.1859{ 1874,Sep.2004. [63]N.Devroye,P.Mitran,andV.Tarokh,Cognitivedecompositionofwireless networks,"in Proc.1stInternationalSymp.CognitiveRadioOrientedWireless Netw.andCommun.CROWNCOM2006 ,pp.1{5,2006. [64]I.Maric,A.Goldsmith,G.Kramer,andS.ShamaiShitz,Onthecapacityof interferencechannelswithonecooperatingtransmitter," EuroTrans.Telecommunications ,vol.19,no.4,pp.405{420,2008. [65]Y.CaoandB.Chen,Interferencechannelswithonecognitivetransmitter," submittedto IEEETrans.Inform.Theory ,Oct.2009. 124

PAGE 125

[66]S.Rini,D.Tuninetti,andN.Devroye,Stateofthecognitiveinterferencechannel:anewuniedinnerbound,andcapacitytowithin1 : 87bits," Arxivpreprint arXiv:0910.3028v1 ,2009. [67]K.Marton,Acodingtheoremforthediscretememorylessbroadcastchannel," IEEETrans.Inform.Theory ,vol.25,no.3,pp.306{311,May1979. [68]N.Devroye,P.Mitran,andV.Tarokh,Cognitivemultipleaccessnetworks,"in Proc.InternationalSymp.Inform.TheoryISIT2005 ,pp.57{61,2005. [69]H.CharmchiandM.N.-Kenari,Achievableratesfortwointerferingbroadcast channelswithacognitivetransmitter,"in Proc.InternationalSymp.Inform.Theory ISIT2008 ,pp.1358{1362,July2008. [70]O.SahinandE.Erkip,Cognitiverelayingwithone-sidedinterference,"in Proc. AsilomarConf.Signals,Syst.andComput. ,2008. [71]S.H.Seyedmehdi,Y.Xin,andY.Lian,Anachievablerateregionforthecausal cognitiveradio,"in Proc.AllertonConf.Commun.Control,andComputing ,pp. 783{790,Sep.2007. [72]C.-M.Zeng,F.Kuhlmann,andA.Buzo,Achievabilityproofofsomemultiuser channelcodingtheoremsusingbackwarddecoding," IEEETrans.Inform.Theory vol.35,no.6,pp.1160{1165,Nov.1989. [73]G.Kramer,Modelsandtheoryforrelaychannelswithreceiveconstraints,"in Proc. AllertonConf.Commun.,Control,andComputing ,pp.783{790,Sep.2007. [74]I.CsiszarandJ.Korner,Broadcastchannelswithcondentialmessages," IEEE Trans.Inform.Theory ,vol.IT-24,no.3,pp.339{348,May1978. [75]A.Host-MadsenandJ.Zhang,Capacityboundsandpowerallocationforwireless relaychannels," IEEETrans.Inform.Theory ,vol.51,no.6,pp.2020{2040,Jun. 2005. [76]N.Devroye,Informationtheoreticlimitsofcognitionandcooperationinwireless networks,"Ph.D.dissertation,HarvardUniversity,2007. [77]H.Weingarten,Y.Steinberg,andS.ShamaiShitz,Thecapacityregionofthe GaussianMIMObroadcastchannel," IEEETrans.Inform.Theory ,vol.52,no.9, pp.3936{3964,Sep.2006. [78]E.Hossain,L.B.Le,N.Devroye,andM.Vu,Cognitiveradio:Fromtheoryto practicalnetworkengineering,"in AdvancesinWirelessCommunications ,V.Tarokh andI.Blake,Eds.,Springer,2009,pp.251{289. 125

PAGE 126

BIOGRAPHICALSKETCH DebdeepChatterjeereceivedtheB.Tech.degreeinelectricalengineeringin2004 fromtheIndianInstituteofTechnologyIIT,Kharagpur,India;andtheM.S.andPh.D. degreesinelectricalandcomputerengineeringin2006and2010respectivelyfromthe UniversityofFlorida,Gainesville.FromAugust2008untilJuly2009,andfromMay 2010untilAugust2010,hewaswiththeWirelessStandardsandTechnologiesteam,Intel Corporation,SantaClara,California.Hisresearchinterestsincludecooperativecommunications,multi-userinformationtheory,gametheory,anddesignofnext-generationwireless systems. 126