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Wireless Cooperative Networks with Differential Modulation

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Permanent Link: http://ufdc.ufl.edu/UFE0021721/00001

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Title: Wireless Cooperative Networks with Differential Modulation Performance Analysis and Resource Optimization
Physical Description: 1 online resource (102 p.)
Language: english
Creator: Cho, Woong
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: cooperative, differential, error, resource
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: In wireless cooperative networks, virtual antenna arrays formed by distributed network nodes can provide cooperative diversity. Obviating channel estimation, differential schemes have long been appreciated in conventional multi-input multi-output (MIMO) communications. However, distributed differential schemes for general cooperative network setups have not been thoroughly investigated. In this dissertation, we develop and analyze distributed differential schemes using two conventional relaying protocols, decode-and-forward (DF) and amplify-and-forward (AF), and space-time coding (STC)-based relaying protocol with an arbitrary number of relays. For each protocol, we analyze the error performance and consider the resource allocation as a two-dimensional optimization problem: energy optimization, location optimization, and joint energy-location optimization. We first derive an upper bound of the error performance for the DF system, the approximated error performance for the AF system, and an upper bound for the STC-based system at reasonably high SNR, respectively. Based on these results, we then develop the energy optimization and relay location optimization schemes that minimize the average system error. Analytical and simulated comparisons confirm that the optimized systems provide considerable improvement over unoptimized ones, and that the minimum error can be achieved via the joint energy-location optimization. We compare the results of optimization and the effects of different relaying protocols and obtain several interesting results. In addition, the comparison between the conventional system and STC-based system is addressed.
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 Woong Cho.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Yang, Liuqing.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021721:00001

Permanent Link: http://ufdc.ufl.edu/UFE0021721/00001

Material Information

Title: Wireless Cooperative Networks with Differential Modulation Performance Analysis and Resource Optimization
Physical Description: 1 online resource (102 p.)
Language: english
Creator: Cho, Woong
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2007

Subjects

Subjects / Keywords: cooperative, differential, error, resource
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: In wireless cooperative networks, virtual antenna arrays formed by distributed network nodes can provide cooperative diversity. Obviating channel estimation, differential schemes have long been appreciated in conventional multi-input multi-output (MIMO) communications. However, distributed differential schemes for general cooperative network setups have not been thoroughly investigated. In this dissertation, we develop and analyze distributed differential schemes using two conventional relaying protocols, decode-and-forward (DF) and amplify-and-forward (AF), and space-time coding (STC)-based relaying protocol with an arbitrary number of relays. For each protocol, we analyze the error performance and consider the resource allocation as a two-dimensional optimization problem: energy optimization, location optimization, and joint energy-location optimization. We first derive an upper bound of the error performance for the DF system, the approximated error performance for the AF system, and an upper bound for the STC-based system at reasonably high SNR, respectively. Based on these results, we then develop the energy optimization and relay location optimization schemes that minimize the average system error. Analytical and simulated comparisons confirm that the optimized systems provide considerable improvement over unoptimized ones, and that the minimum error can be achieved via the joint energy-location optimization. We compare the results of optimization and the effects of different relaying protocols and obtain several interesting results. In addition, the comparison between the conventional system and STC-based system is addressed.
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 Woong Cho.
Thesis: Thesis (Ph.D.)--University of Florida, 2007.
Local: Adviser: Yang, Liuqing.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2007
System ID: UFE0021721:00001


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WIRELESS COOPERATIVE NETWORKS WITH DIFFERENTIAL MODULATION:
PERFORMANCE ANALYSIS AND RESOURCE OPTIMIZATION



















By

WOONG CHO


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

2007
































2007 Woong Cho


































To my wife and family









ACKNOWLEDGMENTS

I want to express my tremendous gratitude to my supervisor, Dr. Liuqing Yang, for

her tireless effort as well as for providing me with insights, inspiration, and encouragement

without which I could not have performed this research. I deeply thank her for all of the

patient guidance, invaluable advice, and numerous discussions we have shared throughout

my graduate studies.

I would like to thank all the members of my advisory committee, Dr. Jenshan Lin,

Dr. Tao Li and Dr. Shigang C'!, for their valuable time and energy in serving on my

supervisory committee.

I would also like to thank Rui Cao, my friend and colleague at Signal processing,

Communications, and Networking (SCaN) group, for the priceless discussions we shared,

which generated many ideas for this research. I wish to extend my sincere thanks to all

the members of the SCaN group, Huilin Xu, Fengzhong Qu, Dongliang Duan, and Wenshu

Zhang, for their companionship and support throughout our time together.

As ahv-- -~, I want to thank to my parents and parents-in-law, for their unyielding

support and love. Their encouragement and understanding through my studying periods

have meant more than I can ever express.

Last, I would like to express my greatest thanks and adoration to my loving wife. I

want to thank her for supporting and understanding me in innumerable v-- ,i-, particularly

during all our time together in the United States, and throughout my Ph.D studies.









TABLE OF CONTENTS


page

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

LIST OF FIGURES .................................... 7

A BSTRA CT . . . . . . . . . . 10

CHAPTER

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

1.1 Cooperative Networks .................... ........ 12
1.2 Motivation ...................... ........... 13

2 SYSTEM M ODEL .................................. 17

2.1 Relaying Protocols and C('!h i,, I Modeling ........ ......... 17
2.1.1 Decode-and-Forward (DF) ......... ............. 18
2.1.2 Amplify-and-Forward (AF) ........... ........... 19
2.1.3 Distributed Space-Time Coding (DSTC) .............. .. 19
2.2 Differential Demodulation and Decision Rules ............... .. 21
2.2.1 DF Protocol .................. ............ .. 21
2.2.2 AF Protocol .................. ............ .. 23
2.2.3 DSTC Protocol .................. .......... .. 24

3 PERFORMANCE ANALYSIS .................. ......... .. 26

3.1 SER for DF Protocol .................. ........... .. 26
3.2 SER for AF Protocol .................. ........... .. 30
3.3 CER for DSTC Protocol .................. ......... .. 38

4 OPTIMUM RESOURCE ALLOCATION ................. .. 45

4.1 Convexity of SER .................. ............ .. 46
4.2 Energy Optimization ............... ........... ..50
4.3 Relay Location Optimization. .............. ... .. 63
4.4 Joint Energy and Location Optimization ............... .. .. 70

5 SIMULATIONS AND DISCUSSIONS ................ .... 75

5.1 Benefits of Energy and Location Optimizations . . ..... 75
5.2 Benefits of Joint Optimization ................ ... .. 83
5.3 Conventional System vs STC-based System. ............... .. 87

6 CONCLUSIONS AND FUTURE WORK ............. .... .. 92

6.1 Conclusions .................. ................ .. 92
6.2 Future W ork .................. ................ .. 94









LIST OF REFERENCES ..................... .......... 96

BIOGRAPHICAL SKETCH ................... .......... 102










LIST OF FIGURES


Figure

1-1 Simple cooperative network . .........

2-1 Setup of the cooperative network . ......

3-1 SER at different r,s, values (DF, L = 2, M = 2) . .

3-2 SER at different yd,, values (DF, L = 2, M = 2) ..

3-3 SER bound versus ~d,, and -y, (DF, L = 2, M = 2). .

3-4 SER comparison between approximation and simulation (
Yd,r ). . . . . . . .

3-5 SER comparison between approximation and simulation (i
Y d,r ) . . . . . .

3-6 SER comparison between coherent -- -I, i and differential
-7 d,re) . . . . . .

3-7 The CER for the DFSTC protocol (L = 1, 2, and 3, SNR=

3-8 The CER for the AFSTC protocol (L = 1, 2, and 3, SNR-


AF ND,


AF DL,


system


3-9 The effect of unbalanced link SNR for the DFSTC protocol (L = 1, 2, al

3-10 The effect of unbalanced link SNR for the AFSTC protocol (L = 1, 2, ai

4-1 Network topologies: (a) Ellipse case; (b) Line case . .....

4-2 Exact and approximate optimum energy allocations with different path
nents (DF, L = 1, p 10dB) . ...............

4-3 SER versus energy allocation at the given relay location Ds,r (DF, L
D = 1, v = 4) . . . . ... . . .


SNR(dB)


SNR(dB)= 7d,,


(AF DL, SNR=


nd 3).

id 3).


loss expo-


2, p 10=dB,


4-4 Comparison of optimal energy allocation between the numerical search and simulated
results at various L values (DF, p = 10dB, v = 4) .. ................

4-5 Comparison of normalized optimum energy allocation at different p values (DF, L = 1).

4-6 Existence of the optimum solution (AF ND, p = 15dB, v = 3, L=2) .. ......

4-7 SER versus energy allocation at the given relay location Ds,r (AF ND, L = 2, p =
15dB D = 1, v = 4) . . . . . ... . .

4-8 SER versus energy allocation at the given relay location Ds,r (AF DL, L = 2, p
15dB D = 1, v = 4) . . . . . ... . .


page

13

17

29

29

30


36


37


. 37


. 42

. 42

. 43

. 44

. 46


S= d,r),, .

S .. = d,r) .


. '










4-9 Optimum energy allocation (AF, ND and DL, D= 1, p 30dB, v = 4). . ... 62

4-10 Optimum energy allocation (DFSTC and AFSTC, D = 1.2, Ds,d = 1, p=15dB, v = 4). 62

4-11 Optimum location of relays (DF, p = 10dB and L 1). .. . . ...... 65

4-12 SER versus relay location distribution at the given energy allocation ps/p (DF, L =
2, p 10dB, D 1v = 4). ...... .......... .......... 66

4-13 SER versus relay location distribution at the given energy allocation ps/p (AF ND,
L 2, p 15dB, D 1,v = 4) ...... ........... ... ...... 67

4-14 SER versus relay location distribution at the given energy allocation ps/p (AF DL,
L 2, p 15dB, D 1,v = 4) ...... ........... ... ..... 68

4-15 Optimum relay location (AF, ND and DL, D = Ds,d = 1, p 30dB, v =4). ... . 69

4-16 Optimum relay location (DFSTC and AFSTC, D = 1.2, Ds,d = p 15dB, v = 4). 69

4-17 Iterative search: flow chart. .................. ........... .. 71

4-18 Performance surface versus ps/p and Ds,r (DF, p = 10dB, v = 4, L = 3, DBPSK). 72

4-19 Performance surface versus ps/p and Ds,r (AF, ND, p = 15dB, v = 4, L = 3, DBPSK). 72

4-20 Performance surface versus ps/p and Ds,r (DFSTC, p = 15dB, v = 4, L = 2). . 73

4-21 Performance surface versus ps/p and Ds,r (AFSTC, p = 15dB, v = 4, L 2). . 74

5-1 SER comparison between relay systems with and without energy optimization (DF,
p = 15dB, D = 1.2, Ds,d = 1, v = 4) ........... . . 76

5-2 SER comparison between relay systems with and without relay location optimization
(DF, p = 15dB, D = 1.2, Ds,d = 1, v 4). . . . . 76

5-3 SER comparison between relay systems with and without energy optimization (AF
ND, p = 15dB, D = 1.2, Ds,d = 1, v = 4). . .. . . 78

5-4 SER comparison between relay systems with and without energy optimization (AF
DL, p= 15dB, D = 1.2, Ds,d = 1, v = 4). . . . . 78

5-5 SER comparison between relay systems with and without relay location optimization
(AF ND, p 15dB, D= 1.2, Ds,d = 1, v =4). .. . . 79

5-6 SER comparison between relay systems with and without relay location optimization
(AF DL, p = 15dB, D = 1.2, Ds,d = 1, v = 4). .. . . 80

5-7 CER comparison between relay systems with and without energy optimization (DF-
STC, p = 15dB and 25dB, D 1.2, Ds,d 1, v = 4). ... . . 81










5-8 CER comparison between relay systems with and without relay location optimization
(DFSTC, p 15dB and 25dB, D 1.2, Ds,d 1, = 4). . . 81

5-9 CER comparison between relay systems with and without energy optimization (AF-
STC, p 15dB and 25dB, D= 1.2, Ds,d 1, v =4). ... . . 82

5-10 CER comparison between relay systems with and without relay location optimization
(AFSTC, p = 15dB and 25dB, D = 1.2, Ds,d = 1, v = 4). .. . . 82

5-11 The SER contour versus Ds,r and ps/p (AF ND, Ds,d = D = 1, p 15dB, L = 2). 84

5-12 The SER contour versus Ds,r and ps/p (AF DL, Ds,d = D = 1, p 15dB, L = 2). 84

5-13 The SER contour versus Ds,r and ps/p (DF, Ds,d = D = 1, p10dB, L = 3). . 85

5-14 The CER contour versus Ds,r and ps/p (DFSTC, D = 1.2, Ds,d = 1, p 15dB, L = 2). 86

5-15 The CER contour versus Ds,r and ps/p (AFSTC, D = 1.2, Ds,d = 1, p 15dB, L = 2). 86

5-16 Data rate comparison between the conventional system and STC-based system in terms
of the required time slots per information symbol. ................. 87

5-17 BER comparison between the conventional systems and STC-based systems with same
modulation size (DF vs DFSTC, SNR =,,, = 7d,r). .... . . 89

5-18 BER comparison between the conventional systems and STC-based systems with same
modulation size (AF vs AFSTC, SNR 7,,, =- d,r). .... . . 89

5-19 BER comparison between the conventional systems and STC-based systems with equal/similar
transmission rate (DF vs DFSTC, SNR=,, = %d,r)yd. ................ 90

5-20 BER comparison between the conventional systems and STC-based systems with equal/similar
transmission rate (AF vs AFSTC, SNR= -r, = yd,r). ................ 91









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

WIRELESS COOPERATIVE NETWORKS WITH DIFFERENTIAL MODULATION:
PERFORMANCE ANALYSIS AND RESOURCE OPTIMIZATION

By

Woong Cho

December 2007

'C! ir: Liuqing Yang
Major: Electrical and Computer Engineering

In wireless cooperative networks, virtual antenna arrays formed by distributed

network nodes can provide cooperative diversity. Obviating channel estimation, differential

schemes have long been appreciated in conventional multi-input multi-output (MI\ !lO)

communications. However, distributed differential schemes for general cooperative

network setups have not been thoroughly investigated. In this dissertation, we develop

and analyze distributed differential schemes using two conventional relaying protocols,

decode-and-forward (DF) and amplify-and-forward (AF), and space-time coding

(STC)-based relaying protocol with an arbitrary number of relays. For each protocol, we

analyze the error performance and consider the resource allocation as a two-dimensional

optimization problem: energy optimization, location optimization, and joint energy-location

optimization.

We first derive an upper bound of the error performance for the DF system, the

approximated error performance for the AF system, and an upper bound for the

STC-based system at reasonably high SNR, respectively. Based on these results, we

then develop the energy optimization and relay location optimization schemes that

minimize the average system error. Analytical and simulated comparisons confirm that the

optimized systems provide considerable improvement over unoptimized ones, and that the

minimum error can be achieved via the joint energy-location optimization. We compare

the results of optimization and the effects of different relaying protocols and obtain several









interesting results. In addition, the comparison between the conventional system and

STC-based system is addressed.









CHAPTER 1
INTRODUCTION

1.1 Cooperative Networks

Diversity techniques have been widely adopted to combat multipath fading in wireless

communications. In particular, space diversity based on multi-input multi-output (\!l \ O)

has emerged as an attractive research area due to its capacity and error performance

enhancements. However, the antenna packing limitation of the MIMO technique renders

practical implementation clumsy in wireless applications, such as cellular, sensor, or

ad-hoc networks. To overcome this limitation while pertaining the diversity benefits,

distributed modulation schemes have been -i::-I. -1. by using relay node(s). The basic

idea is to create multiple paths over a network by using relay node(s), where each relay

node is equipped with a single antenna. Then, the rJ l ,. .1 signals from each relay node are

combined at the destination node, which provides cooperative diversity gain by forming

the distributed network nodes into virtual antenna arrays. Therefore, by exploiting the

cooperative networks, the antenna packing limitations can be eliminated while the spatial

diversity gain can still be achieved [36, 48, 49]. Thanks to these advantages, cooperative

networks can be applied in various scenarios to enhance the network performance. For

example, wireless personal area network (WPAN) and local area network (WLAN) can

extend their coverage areas by using relay nodes. In home environments, cooperative

networks can help distribute the multimedia data from the central entertainment unit to

devices anywhere in the house, by mounting relay nodes to the wall or even embedding

them inside the walls. Furthermore, cooperative networks also find applications in

intelligent transportation systems including inter-vehicle, intra-vehicle and vehicle-to-road

communications, to enable reliable distribution of the emergency information to certain

groups of drivers via cooperation of multiple vehicles on the road.

To illustrate the basic concept of cooperative networks, Fig. 1-1 represents a simple

cooperative network in which the source node transmits a signal to the destination









relay

source destination



Figure 1-1. Simple cooperative network.


node via the relay node. The source-relay-destination and source-destination links are

commonly referred to as the relay link and the direct link, respectively. There are two

conventional relaying protocols which are widely considered in the existing literature.

One is the decode-and-forward (DF) protocol and the other is the amplify-and-forward

(AF) protocol. With DF relaying protocol, the relay node demodulates the received signal

from the source node and remodulates that signal. Then the relay node transmits the

remodulated signal to the destination node. With AF relaying protocol, the relay amplifies

the signal from the source and then simply forwards it to the destination node. Another

protocol is the distributed space-time coding (DSTC) protocol which can support higher

transmission rate. This protocol can also adopts DF and AF depending on the relay node

operation.

1.2 Motivation

Cooperative networks have received tremendous attention in wireless communications

[9, 38, 42, 46, 53]. Many studies have been carried out to analyze the performance

of cooperative networks. The outage probability of cooperative networks from the

information theoretic perspective is presented in [35]. The error rate and outage

probability of a cooperative network are analyzed in [26] without considering the direct

link. In [28], a later study by the same authors, they consider both relay and direct links

with a fixed gain at the relay. In [10], the multihop relay transmission is considered, and

the exact symbol error rate is derived in [3]. A more general case of the relay networks

is developed in [45]. In [4, 37, 47, 54, 62], the performance of STC-based cooperative

networks is considered. All of these work on cooperative networks focuses on coherent









demodulation based on the availability of the channel state information (CSI) at both the

rel-iv- and the destination.

Accurate estimation of the CSI, however, can induce considerable communication

overhead and transceiver complexity, which increases with the number of relay nodes

employ, .1 In addition, CSI estimation may not be feasible when the channel is rapidly

time-varying. To bypass channel estimation, cooperative networks obviating CSI have

been recently introduced. These cooperative systems rely on noncoherent or differential

modulations, including conventional fre-'ii,-l'-v- hift keying (FSK) and differential

phase-shift keying (DPSK) as well as STC-based ones. In [12, 16], using noncoherent

and differential modulations, it is shown that the log likelihood ratio can be combined

by capturing the detection error at the relay nodes according to the so-termed transition

probability if the partial CSI is known at the reliv-, and destination. The performance of

a single relay system with noncoherent and differential modulations is considered in [6, 13]

and [29, 30, 57, 71-73], respectively. In [16, 17, 56, 63, 64, 70], a distributed STC system

for cooperative networks is introduced by using differential or noncoherent schemes.

To improve the error performance and to enhance the energy efficiency of cooperative

networks, optimum resource allocation recently emerged as an important problem

attracting increasing research interests (see e.g., [5, 22, 27, 39, 65]). These work is based

on different relaying protocols (amplify-and-forward, decode-and-forward and block

Markov coding), under various optimization criteria (signal-to-noise ratio (SNR) gain,

SNR outage probability, energy efficiency and capacity), and with different levels of CSI

(instantaneous CSI and channel statistics). However, all of them only consider the power

allocation and mostly focus on a single-relay setup. In [7, 14, 18, 30], optimum power

allocation for multiple relay links is developed under various relaying protocols. [40] and

[23, 24], respectively, consider the optimum energy and bandwidth allocation in Gaussian

channel and multihop system, while [50] introduces the opportunistic relay selection.









Recently, it has been noticed that the relay location is a critical factor influencing

the relay network performance [38, 67]. However, [38] neglects the path loss effect which

is closely related to the relay location such that the location optimization problem is

erroneously formulated as an energy allocation problem. In [67], it is shown that given

uniform energy allocation there is an optimum relay location which provides the optimum

performance. However, they only consider the optimum location of the rel iv without

appropriate energy allocation. With DF and AF protocols, the joint energy and location

optimization for the cooperative network is introduced in [15, 19-21].

In this research, we develop cooperative networks with an arbitrary number of relays

by employing DF, AF, and DSTC protocols. Equally attractive is that our analysis

is tailored for relay systems with differential modulation, which is known to reduce

the receiver complexity by bypassing channel estimation. Notice that the DF and AF

protocols generalize the standard differential modulation to a distributed scenario with

an arbitrary number of rel- i-. The DSTC protocol relies on an increased level of user

cooperation via the distributed counterpart of the differential space-time codes [31, 32];

that is, for each data block, a space-time codeword encoded across distributed relays is

transmitted over a common relay-destination channel. Using these relaying protocols, we

then derive the error performance and the optimum resource allocation of cooperative

networks. Different from existing works on resource optimization, we tackle the problem

from two angles: i) Optimizing the power allocation across relay and source nodes for

any given source-relay-destination distances; and ii) Optimizing the relay location for any

power distribution and source-destination distance. To the best of our knowledge, we are

among the first to formulate the 2-dimensional optimization problem. In addition, we are

also the first to consider the joint power-and-relay-location optimization for cooperative

networks.

To facilitate the resource optimization, we develop analytical expressions of the error

performance for various relay protocols with arbitrary number of rel i -. We first derive an









upper bound of the overall symbol error rate (SER) for the DF protocol, the approximated

SER for the AF protocol, and an upper bound of the overall codeword error rate (CER)

for the DSTC protocol. Then, the optimum resource allocation which minimizes the

system SER and CER is developed. For the DF and DSTC protocols, the multiple relay

system with no direct link is considered for analytical tractability. For the AF protocol, we

consider the resource allocation of the relay system both with and without the direct link.

The former evaluates a scenario where there is a direct link between the source node and

the destination node, while the latter takes into consideration a situation where obstacles

disable a direct transmission resulting in no direct link. For each protocol, under the

total energy and source-relay-destination distance constraints, we show that the optimum

energy allocation can be achieved given the relay location, the optimum relay location

can be obtained given the source and relay transmit energy, and the optimum error

performance can be achieved through the joint energy and location optimization. The

benefits of optimization are confirmed by numerical examples and simulations. We show

that each optimum resource allocation has a different property depending on the relaying

protocol. In addition, the comparison between the conventional cooperative networks and

STC-based ones is addressed.









CHAPTER 2
SYSTEM MODEL

The system model is based on a network setup with one source node s, L relay nodes

{r}k 1 and one destination node d, as depicted in Fig. 2-1. Each node is equipped with
a switch that controls its transmit/receive mode to enable half-duplex communications.

Multiplexing among the network nodes can be achieved via frequency-division, time-division

or code-division techniques. For notational convenience, we will consider the time-division

multiplexing (TDM). However, the presented analysis and results are readily applicable to

freqi. i -i ---division multiplexing (FDM) and code-division multiplexing (CDM). For this

study, we first consider two conventional relaying protocols, i.e., decode-and-forward (DF)

and amplify-and-forward (AF), and then we develop the distributed space-time coding

(DSTC) protocol.

2.1 Relaying Protocols and Channel Modeling

Let us first consider the conventional relaying protocols. With DF protocol, the relay

nodes demodulate the signal from the source node, then remodulate and forward to the

destination node. With AF protocol, the re1-iv, amplify the signal from its source and then

forward it to the destination. Using TDM, the relay transmission consists of two phases.

In phase I, the source broadcasts a symbol to all rel -i,. In phase II, each relay transmits

the amplified signal to the destination during its distinct time slot. As a result, the L








\\ \ '' destination d



source s
relays {r}

Figure 2-1. Setup of the cooperative network.









source-to-relay (s rk) links share a common channel, while the L relay-to-destination

(rk d) links are mutually orthogonal.
In order to bypass channel estimation and to cope with time variation of the wireless

channel, differential modulation is employ, ,1 at the source node. Specifically, with the

nth phase-shift keying (PSK) symbol being as s, = ej2c,"/M, Cn E {0, 1, ..., M 1}, the

corresponding transmitted signal from the source is:


8 jX7_iSn, n> 1
Xn (21)
1, n = 0.

In phase I, for both DF and AF protocols, the encoded signal is broadcast via a

common channel. The received signals at the kth relay and the destination are given by


yns V-Shs'x8 + zns, k 1, 2, ..., L,
y d,s d s + _Lyd,s, (2-'2)
ysn V hSs'n n (2n2)

where SE is the energy per symbol at the source, and we denote the fading coefficients

of channels s rk and s d during the nth symbol duration as hr'' ~ CA(0, ok )

and h~' ~ CA(0, o),) and the corresponding noise components as z'k, CN(O, rI,s)

and zf'" ~ C.A(O,A/d,,), respectively. Here, CNA'(p, u2) represents the complex Gaussian

distribution with mean p and variance a2

2.1.1 Decode-and-Forward (DF)

In phase II, the received signal from the source is differentially demodulated and

remodulated independently at each relay rk. The demodulation step generates an estimate

s^ from y/'~ in Eq. (2-2), using the decision rule that we will present in the next

subsection. The remodulation step is carried out as in Eq. (2-1), but with s, replaced by

its estimate and x replaced by s^ and xf. Then, the received signal at the destination

corresponding to each relay node is given by


yd,rk h ,r~ x + z ', k 1 2,..., (23)
v/rk, fn + .; k t, 2, ..., L, (2-3)









where S,, is the energy per symbol at the kth relay node, the fading coefficient of the

channel between rk and d during the nth symbol duration is ht'r ~ CA(0, J,, ), and the

noise component is z, ~ CN(0, N,rk).

2.1.2 Amplify-and-Forward (AF)

For the AF protocol, the received signal at the destination corresponding to each relay

is given by


Ynd,rk V ,'S-,rh d n + ,r k Ir d... ^Z ( -)
hdk kr x + zd- k= 1,2,...,L, (24)


where SrC is the energy per symbol and xi denotes the nth transmitted symbol from the

kth relay. The fading coefficients of the rk d channels and the noise component at the

destination are h CA(O, ) and zT ~ CA/'(O,Ad,rA), respectively. At each of the

rel-iv, xi can be represented as


; = A,,y'", k 1, 2, ...,L, (25)

where A, is the amplification factor. To maintain a constant average power at the relay

output, the amplification factor is given by


Ark tk= 1,2,..., L. (2-6)

This A, is reasonable for both differential and noncoherent modulations, since we

can estimate the value of ar by averaging the received signals without knowing the

instantaneous CSI [12, 71].

2.1.3 Distributed Space-Time Coding (DSTC)

In the DSTC protocol, we will consider cooperative networks employing the

differential unitary space time code (DUSTC) which does not require the exact channel

estimation [31, 32]. For simplicity, we will use the diagonal design with the cyclic

construction in [31]. Notice that each diagonal element of the codeword corresponds

to a standard differential phase-shift keying (DPSK) -_ii in.i- where its modulation size









increases as L increases. During the first L time slots of a transmission, the diagonal

entries of the DUSTC symbol block are broadcast to the rel-v-. Then, each relay node

decodes (or amplifies) the corresponding Lth diagonal element of STC signal, and these

signals are transmitted by a common rk d channels during the following L time slots,

which is different from the conventional relaying protocol. With this protocol, both DF

and AF schemes are considered for the cooperative network. We denote DFSTC and

AFSTC as the decode-and-forward space-time coding and amplify-and-forward space-time

coding, respectively.

Denote the nth differentially encoded signal block from the source as X'

X_, V(Q-) with X = IL, where V(Q) is an L x L diagonal unitary matrix,

Q, E {0, 1,..., M 1} with M = 2", and IL is an L x L identity matrix, where r

represents the data rate of the original information which we set to 1. The matrix V(")

has the form V() = V7" with (see [31])

C 6j(27/M)ui o
V,= 0 -. 0 (2-7)

0 .. j(27/M)uL

where uz e {0,..., M 1}; = {1,..., L}. Then, the nth received signal block at the relJ'-,

is given by


Y" = VSSH,"X` + Z' (2-8)


where Qs is the energy per symbol at the source, H'' := diag{h ,', hr2', ...,hr'L'} is

the channel matrix between the source and rel -, -, and ZX := diag{zl'", z,, 2'~, .. z'L}

is the noise matrix at the rel -v. We use diag{al, a2,..., aL} as a diagonal matrix with

[ai, a2, ..., aL] on its diagonal. Let us denote the n-th transmitted signal block from the
rel-~,1- as Xr, then the corresponding received signal block at the destination is given by


yd,= E1/2H,rXr + Z (2-9)
n Er Hn n n+ ,









where Er : diag{S,,,S ...,",} is the energy per symbol matrix at the rel-,v, Hnr :

diag h'l ,hd,"2,...,h'rL/} is the channel matrix between the rel -,v- and destination, and

Z := diag {z', z'2, ..., z rL} is the noise matrix at the destination. Depending on the

relaying protocols, X' has different forms at the rel --.

Throughout this dissertation, all fading coefficients are assumed to be independent.

Without loss of generality, we also assume that all noise components are independent and

identically distributed (i.i.d) with A,,j = Ao, i,j C {s, rk, d}. Accordingly, we can find

the received instantaneous signal-to-noise ratio (SNR) between the transmitter j and the

receiver i as
I h| ,j 12 1
A/2 ; e {srFk,d}.

It then follows that the average received SNR is ; (o jSj)/AMo.

2.2 Differential Demodulation and Decision Rules

In this section, the differential demodulation and decision rules for the relay networks

will be developed. We consider L relay links for the DF and DSTC protocols, and L relay

links and the direct link for the AF protocol.

2.2.1 DF Protocol

As mentioned before, differential demodulation is performed at the relay and

destination nodes. To derive the demodulation, decision and diversity combining rules,

let us begin with the received signal at the relay or the destination node y, = hnx, + zn,

which is extracted from Eqs. (2-2) and (2-3) by dropping the superscripts. Using the

differential encoding in Eq. (2-1), the received signal can be re-expressed as:


yn = hn(xns-lS) + Zn = n-1in + z' ,

where z, = z,,_l,. For M-ary PSK symbols, it follows that L[ s,] = 1. Hence, the

conditional distribution of y, is complex Gaussian with mean ynl-s and variance 2/o0. As









a result, we obtain the log likelihood function (LLF) of y, as:


l(y>) := InpJlx(n s7) {u~ny-1(sn)*} R{(ys)*y- 1si} (2-10)

where i,j e {s, rk, d, s = ej2 m/M and m c {0,1,..., M 1}. We use E[.] for

expectation, (.)* for conjugate, and -{.} for the real part.

At the kth relay node, the differential demodulator is then straightforward:

s e j2f n/M :h argmax l'rs(y,) argmax R{ (yS) *y s"5}. (2-11)


At the destination node, however, there are L different LLF's corresponding to the L

transmitted signals from the relays:


lrk (y-) Inpy1l (y ) {(y *y ds} k 1,2,...,L (2-12)

If the channel state information is known at the rel-Ji and the destination node, then

it is possible to combine the LLF's by weighting them accordingly. However, keeping

in mind that the differential modulation is considered in the first place because of its

capability of bypassing channel estimation, we will focus on the scenario where no channel

state information is available. In this case, the LLF's in Eq. (2-12) have to be combined

with equal weights. As a result, the decision rule at the destination node can be readily

obtained as:
L
S, ej2/ M : r argmax l '(y,). (2 13)
k= 1
With no channel information assumed at either the relays or the destination node, this

decision rule turns out to be the differential detection with equal gain combining (EGC)

[55, C'!h Ipter 6.6].









2.2.2 AF Protocol

Now consider the AF protocol. Similar to the DF protocol, we can represent the

received signal at the destination corresponding to each relay node as


yt -d hrkx, + zr = ysr + r)', k = 1, 2, ..., L,

where hd,fr = A h hrk, A = -zA, fh z + zT, and ( )r' =

zT zr'(_s,. Then, conditioned on the channels, the received signal is y? ~
C(y drkl~", a2 ), where the variance of the .,.:--regate noise is given by


hk,e-f = 2Aro(TAr K d, + 1), k= 1, 2,..., L. (2-14)

The received signal at the destination corresponding to the source can be represented

as y's = yis, + (z with y ~ C(yss,, 2f/o). As a result, we obtain the LLF
corresponding to the L transmitted signals from the rel -1v and the transmitted signal from

the source, given that x' is transmitted by the source:
Idr(yr: In p (y d,rk m )ydrn,rk d },k rk2,..,

Id, (y ) ds n- n ,
S(yp: Inp (yd'|s') = RI(yds)*y }, (2-15)

where s = ej2 r/M and m e {0, 1,..., M 1}. Notice that, although the LLF, ldr(y,),

in Eqs. (2-12) and (2-15) has the same form, the one in (2-15) is obtained from a

different transmitted signal. For the DF protocol, the LLF is obtained from the given

relay transmitted signal which is a demodulated signal from the source transmitted signal.

However, the LLF for the AF protocol is obtained from the source transmitted signal

which is simply amplified and forwarded at the rel i without any demodulation. At the

destination node, these (L + 1) signals can be combined to estimate the transmitted signal

from the source. Using the multichannel communication results in [44, ('! Ilpter. 12] and









the above LLFs, the decision rule at the destination node can be obtained as (see [30, 72]):

L
1drrenr (2 16)(
=ej2m/M : arg max Wd,s T un) + Wd,r n (2-16)
me{0,1,...,M-1} -
k/ 1

where Wd,s and Wd,,r are combining weights which are given by 1/A/o and 2/o a ,h,

respectively, and it is assumed that the variances of channels are available at the

destination node. Interestingly, this decision rule is the same as the DF protocol in

Eq. (2-13) except for the weighing terms.

2.2.3 DSTC Protocol

Since the transmission signal is based on the differential space-time code, we can

apply the corresponding space-time differential demodulation. Then, the maximum

likelihood (il I) differential demodulation rule [31], given X' = Xn, is


Qn = arg max |r Y i + YV m (2-17)
nme{0,1,...,M-l}

where || || represents the Frobenius norm. This decision rule is the general structure for

DUSTC. Depending on the relaying protocol, the Frobenius norm part can have different

values.

In the DFSTC protocol, the received signal at the re!-i, Y"', is decoded. Since each

diagonal entry of the codeword X' is a DPSK signal and the kth relay demodulates and

remodulates .:4,./ I". .l ';./.l, the corresponding kth entry of Y"', we can re-encode X'

using standard differential demodulation. The received signal block for the given relay

transmitted signal X = X7' is

ydr Hd,rXr V(m') + Zd yd,r V(m') 'd (218
n Hn n-1 n + n n-ln + n,(2 o)

where Z't = Z Z _V m '). Since VX"') is a unitary matrix, Z' has twice the variance

of Z%. Then, given X' = X7', we can apply the ML decision rule in Eq. (2-17).

For the AFSTC protocol, each entry of the received signal from the source, Y,", is

amplified and forwarded to the destination. Therefore, the amplified signal block at the









re1,i- can be represented as


Xr = AY", (2-19)


where A:= diag{Al,, A2, ..., A, } is the amplification matrix, and Ar is the amplification

factor which we defined in Eq. (2-6). Then, using the differential modulation, the received

signal block at the destination can be represented as

Yd,r HX +Z = Yr V(m) -+ (220)


where f, = -E l /2AH drHs, = E /2AH d,rZ + Z and d, = ,V(')
rZn Ei AH*Z* + Zn and Z Zn

The ML decision rule is the same as Eq. (2-17) given X, = X". Notice that the ML

decision rule of both relaying protocols has the same form. However, the value of the

Frobenius norm is different depending on the protocols.









CHAPTER 3
PERFORMANCE ANALYSIS

To facilitate our resource optimization, we will derive the analytical expressions of the

error performance for the cooperative systems described in the preceding section. Symbol

error probability of cooperative networks with relay transmissions has been derived in

[2, 7, 45] for coherent detection, and in [29, 71, 72] for a differential scheme with a single

relay, both employing the AF protocol. The performance of traditional STC systems is

well analyzed in [59-61, 68]. The differential and distributed STC systems are considered

in [31, 32, 58] and [66], respectively. All these existing work considered the performance

of cooperative systems to some extent. However, the error performance of general cases

with differential modulation has not been thoroughly investigated. We will consider

the error rate for a general L-relay setup in the distributed scenario. First, we derive

an upper bound of the symbol error rate (SER) for the DF protocol. Then, under high

signal-to-noise ratio (SNR) approximation, an approximated SER for the AF protocol is

derived. Finally, we derive upper bounds of the codeword error rate (CER) for the DFSTC

and AFSTC protocols.

3.1 SER for DF Protocol

Let us denote the average SER at the kth relay node as P',r. For differential A-ary

PSK (DMPSK) Jii, ii- the s rk link SER pD can be obtained as [52, C!i pter 8.2]

pDF V9PSK / M (-[ gpSK cos 0]) "
1e,r2 dO, 9PSK = sin (3-1)
27r J2/7 1 V/ PSK cos 0 M

where MA(x) = 1/(1 x), Vx > 0, and y represents the average SNR. In particular, for

M = 2 (DBPSK), Eq. (3-1) can be simplified as

PDF 32)
Cr" 2(1 + (3

At the destination, the signals from the L re1 i-. are combined to make a decision.

Conditioned on that the symbol s, is correctly demodulated and remodulated at all relay








nodes, the conditional SER DdF can be obtained by applying the results for L-diversity
branch reception of M-phase signals in [44, Appendix C] as:

pDF ) L- 2)L (L- I t 71-M )1
e,d r(L 1)! 9bL-1 b b- 2 M
p sin(7/M) ot cos(7/M) })13)
b p2 COS2(T/M) b p_2 COS2(/M) b1

where = 7d,, /(1 + Yd,rc). For DBPSK, Eq. (3-3) can be simplified as

PeDdF iF (1-4)
d 2 1l 4 (3-4)

Using the unconditional SER PDF at the relays and the conditional SER PDF at
the destination, we formulate an upper bound on the overall average error performance,
namely the unconditional SER pDF at the destination, as follows:
Proposition 1. With PDF and P9DdF given by Eqs. (3-1) and (3-3), "1'.. ,:;, ; an upper
bound on pDF can be found as:
L
pDF < pDF (_ p ,)(1 Pe,d). (3-5)
k=
Proof. To prove that Eq. (3-5) provides an upper bound on the exact SER pDF let us
start with the probability of correct detection pDF 1 _- pDF. Counting the events that
lead to the correct detection, pDF can be obtained as

PDF Pr{[( k -S, Vrk) n ( Sn)] U [(s^ :/ S for some k) n (s sn)]}

= Pr{< s = Sn, Vrk} Pr{j Sn, Vrk
+ Pr{< SnS^ / S for some k} Pr{sf / sT, for some k}. (3-6)

where s^ and s^ are the symbol estimates formed at the relay rk and the destination d,
respectively. The first summand in Eq. (3-6) turns out to be C (1 PF)(F dF),
which leads to the upper bound in Eq. (3-5). D









Several remarks are due here on the second summand in Eq. (3-6), which corresponds

to the gap between the true SER and its upper bound APDF : pF PDF and

determines the tightness of the error bound in Proposition 1. For DBPSK with a single

relay (L = 1, M = 2), this gap can be easily obtained as:


ApDF pDFpDF (3 7)


For practical pDF and pDF values (e.g., < 10-3), APF is negligible compared with

pDF = PfF + P 2PDF pDF However, for L > 2, all possible errors have to be

considered for both the s- r and r- d links, which renders AP[F analytically untractable.

But intuitively, as L increases, APDF also increases since there is an increasing chance

that detection errors at the relay nodes do not lead to a detection error at the destination

node. In addition to this effect, the performance bound PDF and the gap APDF also

depends on the quality of the s r and r d links.

These effects are evident from the simulated examples in Figs. 3-1 and 3-2, where a

relay network with L = 2 relay nodes using DBPSK signaling is considered at various

7,,, and 7d,r levels. In these simulated examples, the channels between the source and

all relays have identical powers a2, = Vk, which implies that = r,, Vk.
Accordingly, we have PDF pDF, Vk, and pDF < PDF 1 (1 _- Pf)L( PDF) from

Proposition 1. Likewise, the SNRs between all the relay nodes and the destination have

the same value, 7d,rm = 7d,r,Vk. In both Figs. 3-1 and 3-2, the bound P, closely captures

the dependency of the system SER on the SNR levels r7,, and 'd,r. Specifically, we have

the following observations:

Fig. 3-1 reveals that, at any given value of 7,,,, the system SER exhibits an error

floor as 7d,r increases. Intuitively, this error floor comes from the detection error at

the re- v, which heavily relies on the s r link quality 7r,s and can only be reduced

by imposing sufficiently high 7,*,.



















0 O a


rW 10-2 >, '*-.
w :: !! :-
L,


-3 y,,,=10dB (bound)
< 10 ,,=20dB (bound)
...... y ,,=30dB (bound)
,, =40dB (bound
104 0 Y=10dB (true)
> 7,=20dB (ture)
Y 7,,=30dB (true)
7,=40dB(true)


5 10 15
Y, (dB)


Figure 3-1. SER at different r,,, values (DF, L


10


UJ

S10




10


Figure 3-2. SER at different 7y,r values (DF, L = 2, M


* On the contrary, Fig. 3-2 shows that, at medium-to-high 7d,r levels, the overall SER


can alv--,v be reduced by increasing the SNR of the s r link 7,,, and does not


exhibit any error floor.


20 25 30


2, M =2).


5 10 15 20 25 30
r, (dB)


E' F E
: : : : :: : : : : :













1000 " ":" .
10 .





. .
.. .....



. :.. ... ..
L 10

10-
10
20 20
(dB) 30 30
40 Yd,. (dB)


Figure 3-3. SER bound versus yd,, and 7r,s (DF, L = 2, M = 2).


To better illustrate these points, we plot the SER bound PDF as a function of

both Yd,r and y7,s in Fig. 3-3. Notice that the surface flattens along the Yd,r axis, but

keeps descending along the ,,s axis. These observations si r-.- -I that the overall error

performance of the DF based cooperative system depends more on the s r link than the

r d link. Such unbalanced effects of the relay links confirm that appropriate resource

allocation is critical in achieving the optimum error performance.

3.2 SER for AF Protocol

In this section, we will derive the analytical expression of the error performance for

the system under high SNR approximation. While the SER upper bound for an L-relay

system using differential (de-)modulation is considered in [30], we provide a more general

yet, simple expression for the average SER. The SER is developed in two cases, the system

with no direct link and the one with a direct transmission. The two cases are denoted as

"ND" and "DL," respectively.









At the destination, we evaluate the equivalent SNR from the source through the kth

relay node as


Yeq d,er (3-8)
.1 + Yd,rk + 1

This allows us to deduce the SER using a multichannel model. We will first investigate the

SER of the relay systems with no direct link, the result can be easily extended to the case

with a direct link.

In a L-relay system with no direct link between the source and the destination, the

received SNR is:
L
7ND = e7q,rk. (3-9)
k= 1

Proposition 2. At high SNR, with the SNR given by Eq. (3-9), the average SER PA FD

can be found as:

k 1 1
PND D C(L) -7J + In(7a,)j (3-10)
k=1 7d,rke

where C(L) is a constant depending on the number of relays, which is given by


C(L) 1 1 Lk L-) (3-11)
22L-1 L- t k
n=0 k-0

Proof. The conditional SER for multiple independent channel DMPSK is given in [51]. For

the binary case, the SER conditioned on 7ND is given by [44, Chap. 12]

L-1
pAF 1 -7 ( 1
PAFD e -7ND n D, (3-12)
n=0

where 7ND is defined in Eq. (3-9), and

1 ln 2L (3
k (3 13)
k=0 2 1)










In Eq. (3-12), 77VD can be expressed as follows by expanding 7ND:

L-2
/ T n2--i C 1 / '--= L-1
7N D E n1 lri M2 1 2.. r L-1
m=-0 m2=0 mLl =0
L-1

where L = n E ~ mi, then Eq. (3-12) becomes:


2
9i Te r L (3-14)
)7eq,rL 1 ,eq,rL^ ;\-
^^^^^ ^^^^ ^^^^^ ^^^^


L
pAF q,rlkTk
eANoD q 1 e1 '-, er,rk,
n,m k=0


(3-15)


where


L-l n n-m /
-1 T, n .mi .
S22L-1 C2 M2
n,m n=0 m =0 mn20


L-2
n-f mi,
n

mL1=0


The average SER can be obtained by averaging the conditional SER with respect to

the probability density function (PDF) of %Yq,rl, namely p(%Yq,r~), which is given as


AF
eND -' j / |N P( eqrk) eqri l eqlr2l Teqr

L


where the PDF of Yq,rk is derived in [72], and which is given as


P(7eq,rk) exp -e Ko(/ Y ),r


+ 2 eqr + '7dr exp ( K qk),
I *7d,r V


(3 17)


(3-18)


where 2 := 2 ,- 'a Ko(.) and K1(.) denote the zeroth-order modified Bessel

function of the second kind and the first order modified Bessel function of the second

kind, respectively.

By substituting (3-15) into (3-17), we can get:


(3-19)


L
eND -eq,,r k P7q,rk) ) q,k.
n,m k=1 0


EL-2 )
EL- 1 )


(316)









Let us first evaluate the integral in Eq. (3-19), denoted as In,m,k,


0/oo


+ T --qC e q,r, 2P+
=k -- e qr k rk,! 2 qKo) d7eq,

2 m oo
2 Ye ae (r/3o q ,r )d7eq,rq


0
-e,'3e j 7-yIjg e aJK1 (/ '/7eq-r-) d'7eq,r _, (320)


where 3 2 /(1 + 1/ /7d,rk and a = 1 1/ .. Each integration term in Eq. (3 20)

can be computed by using the integration property of Bessel function, [25, Eqs. 6.631.3]
0 2 1 1i 1 +p+ V _-2+p W 02 )
x -"e K,(Ox)dx-a 2- P ) eo ) w '('2 )8 (3 21)
JO 2-3 2 2 (4a

where Wm,,Q() is the Whittaker function


Wm, (Z) e-z/2+1/2 (1/2 + n T, 1 + 2n, z),


with U(., .) denoting confluent hypergeometric function of the second kind. By further

using the properties U(a, 1, l/x) U(a, 2, /x) a (see [1, Eqs. 13.5.9 and

13.5.7]), and a 1 + 1/, a 1 at high SNR, we can simplify Eq. (3 20) as


In,m,k n! 1 + in 7d,] (3 22)
7d,rl

Plugging the above result back into PCoAD, we get


PAF L F 1
eA/rD nk InY 7,d,r (3-23)
n,m k 1 7d,rk

Now, we move to the coefficient part of Eq. (3 23), which is given by

Z. 1 a n L 2 L. (-2
M k =C E2 L T2 ,1 l n k !.(3 2 4 )
n,mk 1 n=0 mi=0 m2=0 m -0 k 1
L-1









By using the fact


(3-25)


) j ) ( L-2 L
n 1 (T, i=1 i k!
(M M2 TL-1I k-I
L-l

the coefficient can be simplified as


L

n,mn k i


L-l n n-m\ n-L mY


22L-L-
n-0 m 0 m2 20 mL-1 0

L-i
L-l L-l-n / n-1m n- mi
-2L t- i
22L-n= k=I =0 m k
n O k-0 m'1 m 0 m2=0 m =L 1 0
11J


(3-26)


Denote the coefficient in Eq. (3-26) as C(L). By using mathematical induction, we can

prove that:


We know that, for L


n + L 1
L-t )'


1, (1) =1 = ("ni ). Suppose for an arbitrary L


(3-27)



1 > 1,


((1) (n" 1i), then, we have (1+ 1)


that:


nm


E"o (n- -1) when L 1+ 1. We can also show


(3-28)


m+l-1
1-1


( + 1).


Therefore, equality (3-27) holds for any L > 1. Using the result in Eq. (3-27), we can

simplify Eq. (3-26) to Eq. (3-11). Finally, the average SER yields Eq. (3-10).


n n-m1 n-' Z=1 2m,
(L) -:= .- 15
m1=0 m2=0 mL =0


(n +1 (n +1)! (n +1- 1>)! (n +1- 1>)!



(n +1 1)! (n 1 + 1 )! (1- )!
n!(1 1)! (n 1)!(1 1)! 0!(1 1)!









In the relay system with a direct link between the source and the destination, the

received SNR will be calculated by adding the direct link SNR:

L
7DL + 7eq,rk +7d,s. (3-29)
k= 1

Following the same steps as the no-direct-link case, the average SER can be similarly

evaluated by calculating the integral for the direct link as in Eq. (3-20):


Id,s d= e-e dyds = m + 1)!. (3 30)
0J 7d,s 7d,s

Combining the result above with (3-22), we can evaluate the average SER for a system

with a direct link as:

L1 1I
PeL CLzC(L +1)1 ln(7yd< (3-31)
Yd,skl1 r.
where C(.) is the the same function as in (3-11), which depends on the number of relays.

For example, for L = 1, 2, 3, and 4, we have C(1) = 1/2, C(2) = 3/4, C(3) = 5/4, C(4)

35/16. It is worth stressing that the SER expressions in Eqs. (3-10) and (3-31) coincide

with the average SER of the coherent system in [45] except for the log term, which leads

to the coding gain loss compared with the coherent system.

When = 7d,rk = 7d,s = V, Vk, and as y -- oo, Eqs. (3-10) and (3-31) give rise to


P, ND C L and C (L+1) (332)

where C' and C" are both constants. From Eq. (3-32), it is clear that the diversity gain

can be obtained using a differential scheme with AF protocol for sufficiently large SNR.

In Figs. 3-4 and 3-5, we compare the approximated and simulated SER when

L = 1, 2, and 3 for the systems with and without a direct link. The figures confirm

that the diversity benefit increases in direct proportion to the number of relays, and

demonstrate that the approximations are very tight compared with the simulations,

especially when the SNR is high and for small L. From Fig. 3-5, it is certain that a direct

















10 2 : i ...
. . . . . . .. . .
I : ::: .. .o .. : : . .



S. . .- . . . .. . . .

1 0 3 ',
Approximation L=l : : '. : .
-Approximation L=2 .
-4 ...... Approximation L=3. ... .
S D Simulation L=; -
O Simulation L=2: : : : : : : : -: : : : : : :
o Simulation L=3
Sim ulation L=3 .. .. .... .* .. ..
10-5 i i
0 5 10 15 20 25 30
SNR (dB)


Figure 3-4. SER comparison between approximation and simulation (AF ND,
SNR(dB) = d,r).


transmission contributes to the diversity gain compared with the system with no direct

link as in Fig. 3-4. As L increases, the quality of approximation decreases, since more

approximation errors are accumulated as the number of relays increases.

In Fig. 3-6, we compare the SER of the systems with coherent modulation and

differential modulation when a direct link is present. We use the result in [45] for the

coherent system. The only difference between the coherent system and differential system

is the log term in Eqs. (3 10) and (3 31). The figure shows that there are coding gain

differences between the systems, and these differences increase as the SNR increases,

i.e., the required SER decreases. These increasing differences are due to the log term in

the differential system. For example, approximately 2.2dB, 2.8dB, and 3.2dB more SNR

are required in differential system to achieve 10-3, 10-4, and 10-s of SER, respectively,

compared with the coherent system.
















100
.: : Approximation L=1
S.:: : : -! -Approximation L=2
S'.... Approximation L=3
10 ..... .. Simulation L=1
O : : Simulation L=2
: : : : : : : + Simulation L=3




. . .. . . .. . .. .
C* 10-2

. . + . . . .. . . .

S103









0 5 10 15 20 25 30
SNR (dB)


Figure 3-5. SER comparison between approximation and simulation (AF DL,
SNR(dB)= 7, .. d ).
~ ~ ~ ~ .N.. . .. . . .



0 5 10 15. 20 25 30 .... .....
~ ~ ~ ~ ~ ~ ~ N (dB)........ .. ...

Figur 3-. SE comarso bewe aprxmto an siuato (A ..DL, ........ .....
SN IcB .'Y .....d ..s.


0 5 10 15 20 25 30
SNR (dB)


Figure 3-6. SER comparison between coherent -, -.i, and differential system (AF DL,
SNR= ., d,r).









3.3 CER for DSTC Protocol

In this section, we will analyze the error performance of the cooperative system

employing the DUSTC. Under high SNR assumption, the upper bound of CER will be

derived depending on the relaying protocols.

Let us first consider the DFSTC protocol. Due to independent demodulation and

remodulation of the corresponding diagonal entry at each relay, the s rk link SER, pDF

can be obtained using Eq. (3-1). Since one symbol error at each relay can induce the

codeword error, the CER PDF at s rk links is given by

L
PDFSTc PDF (3-33)
k=1
At the destination, the received signals from the L-relays reconstruct the transmitted STC

signal. Conditioned on that the source transmitted signal block V, is correctly decoded at

the rel -i-, and by dropping the superscripts for notational brevity, the CER at rk d links

is given by


P[V, VQY,_I] = Q d2(V, V,)/4 )

< exp [-d2(V,, V,)/8Vo] (3 34)

where

d2(V,, V ) | [V V]Y _12

Str{Y,1(V,-V')(VT-V')-Y ,}. (3-35)

At high SNR, we can make the following assumption

Y, n E /2H H X,, (3-36)


where where Er := diag{S,, ,2,,...,, L} is the energy per symbol matrix at the rel-,v-,

H, := diag{h'dr, h'2,..., hr'rL} is the channel matrix between the relays and destination,

and XX is the n-th transmitted signal block from the rel ~1i. Then, Eq. (3-35) can be









approximated as


d2(V,, V') trf{Hd,rADF(H',)}, (3-37)


where ADF = E /2X,_1(V,- V')(V,- V')Xf E 1/2. Since Af 7 is Hermitian, we can

express Eq. (3-37) as


d2(V,, V,) tr {H rU DF U(Hdr) }, (3-38)

where U is a unitary matrix and DeF is diag{ADe f, A,..., ADF}. Each diagonal entry
DF, f k 1,2, ..., L, represents the eigenvalue of ADF. Therefore, we can obtain the

CER by averaging Eq. (3-34) with respect to the channel Hd,r. For simplicity, by

assuming that the fading coefficient has unit variance, the conditional CER PDdFSTCat the

destination is given by

pDFSTC P[V V < 1 DF (39)
c^d n k ( n\ e,k

and under high SNR condition, this equation can be further simplified as
L /DF\-1
PDFSTC P[V, V ] <--1 8./V) (3-40)


Finally, using Eqs. (3-33) and (3-40), we can formulate the unconditional CER for

DFSTC protocol as :

PDFSTC < _(_ PDFSTC) DFSTC). (341)
)e,r ( e,d (3

It is worth mentioning that if there is no error between the source and re- i, the above

equation boils down to the CER of multi-input single-output (\ ISO) system employing

the DUSTC. However, as L increases, the CER of s rk links becomes worse because of

the increasing modulation size at each diagonal entry. This will induce the performance

degradation of the DF-based system. To provide better performance and pertain the

diversity gain, s rk links have to maintain lower CER.









For the AFSTC protocol, similar to the DFSTC protocol at rk d links, the CER can

be found by calculating the pairwise CER between the source and the destination. The
id
covariance matrix of the .,.-- regate noise Z, in Eq. (2 20) is diag{o ,, j 2,h, --- L, }

where the kth diagonal entry of the covariance matrix is given by Eq. (2-14). To

normalize the .,.-- regate noise variance, let us define the matrix G diag{gl,, g, ..., gL

with gk A (S.rA 2 cr, + 1)-1/2. Then, by multiplying G with the received signal block at

the destination, we can rewrite Eq. (2-20) as

yd,r dr d
YrG = Y'V(')G + ZTG, (3-42)

or equivalently,

-d,r d,r y (') (
Yn Yn-I n + Z~, (3-43)

where Y =YG, V = VG, and Z = ZdG. Then, the CER for the AFSTC protocol can

be achieved using Eq. (3-43). Following the same steps as Eq. (3-34) to (3-37), the CER

can be obtained as:

P[ V, i I Y, ] < exp -d2(3VT V')/8A]0 (3 44)

where


d n Y-V,1(V,- VT)GG (VT,-VT )Y,- 1}. (3-45)

At high SNR, the code distance can be approximated as

d2(V,, V') tr (Hd,r Hr,)AAF (H r Hr)}, (3-46)

where AAF EE /2X,_1(V, V')(AG)(AG)H(V, V')X-H YzE /2. Similar to the

DFSTC protocol, we can express AAF as


eAF U'eDAU'FU (3 47)









where U' is a unitary matrix, and DAF is the L x L diagonal matrix in which each

diagonal entry is AA, k = 1, 2, ..., L. Then, the CER for the DFSTC protocol can be

achieved by averaging Eq. (3-44) with respect to the combined channel H H rH,". Let us

define h : hd'h'"'8, then the PDF of a = Ih is given by [43]

Pa Ko 4a (2 a2
P(a) Ko (2 2 2, (348)
d,rk rks d,rk rk s

where Ko(.) is the zeroth order modified Bessel function of the second kind. By assuming

that each fading coefficient has unit variance, and using the properties of Bessel function

and confluent hypergeometric function (see [25, Eq. 6.631.3] and [1, Eqs. 13.5.9]) at high

SNR, the CER can be simplified as

L AF AF-1
PFSTc = P[Vrn V'] < ] ek (-49)
k=- 1 \I /
Notice that the CER of the AFSTC protocol has almost the same form as its counterpart

of the DFSTC protocol at rk d links except for the log term which reflects the effect

of the amplification and .r.- .-egate noise and this leads to coding gain loss. Eq. (3-49)

confirms that AFSTC protocol provides full diversity gain.

In Figs. 3-7 and 3-8, we plot the bounds and simulated CERs for the systems with

DFSTC and AFSTC, respectively, when L = 1, 2, and 3. When L = 1, the STC-based

cooperative system is reduced to the conventional cooperative network, thus we can use

the SER formulas derived in [20, 21] as the CER bound of the STC-based system. Fig. 3-7

shows that the bounds are tight to the simulations, especially when L is small. Notice that

the cardinality of the signal block at the rel- i equals to ML because of the independent

decoding at each relay. However, the bound at rk d links only considers M signals.

Thus, as L increases, the gap between the bound and simulation increases. Fig. 3-7 also

shows that no diversity gain is obtained by DFSTC protocol, since the CER at s rk links

increases in direct proportion to the number of rel -v, which induces the degradation of

the overall error performance of DF-based system. These results confirm our analysis in
























10






10 -m lto : : : :.:s::m




L=2, simulation i
-310-






0 5 10 15
SNR(



Figure 3-7. The CER for the DFSTC protocol (L
1 0 -1 . . . . .


















10
--L=1, bound
2 L=2, bound
-4
10 '.. L=3, bound
L=1, simulation
L=2, simulation
'0 *O L=3, simulation
10- 3
0 5 10
SNR(



Figure 3-7. The CER for the DFSTC protocol (L









w\-....-.-.-.-.-..




10-2








--L=1, bound
L=2, bound
104 ....... L=3, bound
: : --- L=1, simulation : :::
; 6 L=2, simulation I ;;
0 L=3, simulation

10-51
15 20


dB)


1, 2, and 3, SNR=


SNR(dB)


25 30


Figure 3-8. The CER for the AFSTC protocol (L 1, 2, and 3, SNR=















10' -"s,,


102
1 0 :-. : : : : : ..: : : : : : :: : : : : : :




10-3 L=1 .
-L=2 : : : : :: : : :
. L=3
10-4 --,s=dr=SNR *
+: Yrs=SNRydr=SNR+20dB : :: :
S-d-r=SNR, yrs=SNR+20dB '
10-5
0 5 10 15 20 25 30
SNR(dB)


Figure 3-9. The effect of unbalanced link SNR for the DFSTC protocol (L 1, 2, and 3).


the preceding section. In Fig. 3-8, though the bounds for AFSTC protocol are inaccurate

when SNR is low because of the log term in the analytical expression, the bounds and

simulations have tight values at high SNR. Furthermore, it is clear that AFSTC protocol

provides full diversity gain.

As we mentioned above, the link quality between the source and relays is critical to

the performance of the DF-based cooperative system. To capture the effect of unbalanced

link quality, we consider different average SNRs at s rk and rk d links for both DFSTC

and AFSTC protocols in Figs. 3-9 and 3-10, respectively. We assume that = 7,

and d,rr = 7d,r, Vk, and consider i) equal SNR for both s rk and rk d links, ii) higher

SNR is assigned at s rk, and iii) higher SNR is assigned at rk d links with L = 1, 2

and 3. As shown in Fig. 3-9, when we assign high SNRs at s rk links, the overall CER

decreases and the diversity gain begins to appear. For the extreme case, i.e, infinite SNR

is assigned at s rk links, the DFSTC cooperative system behaves like a MISO system.

The figure also shows that the coding gain is achieved by assigning higher SNR at rk d

links. However, the diversity gain is dominant compared with the coding gain especially














10 -1 --- ------
10 2' :: :: l ._* ~ .. ..., : :: : .: : :: : :: : ::..
... : . % .. .




__10 .. i ii ..
-L=1
104 L=2
10 .
..... L=3 : :: :: ::: :
Yrs dr=SNR
10o5 Trs=SNR, yd,r=SNR+20dB .. ; '
yd, r=SNR, yr,s=SNR+20dB : :
10-6
0 5 10 15 20 25 30
SNR(dB)


Figure 3-10. The effect of unbalanced link SNR for the AFSTC protocol (L 1, 2, and 3).


at medium to high SNR, which confirms that the s rk links are more critical to provide

better performance. Fig. 3-10 shows that the coding gain is obtained by assigning higher

SNR at both links. Increasing SNR at s rk and rk d links leads to decreasing the effect

of .,.-:-regate noise and increasing SNR at the destination, respectively. Both scenarios

induce the enhancement of coding gain. Notice that the effect of SNR at rk d links

provides more coding gain than rk d links, which implies that increasing average power

at the relay output is more crucial than reducing the effect of noise at the s rk links.









CHAPTER 4
OPTIMUM RESOURCE ALLOCATION

In this chapter, we will investigate the effects of resource allocation on the error

performance. We will show that an optimum allocation of the limited resources is possible,

and it achieves the optimum system error performance. The resource allocation which

minimizes the average error rate will be addressed from three perspectives:

1) Given the relative distances among the source, relay and destination nodes, the path

loss exponent of the wireless channel, and the total available energy per symbol,

determine the optimum energy allocation among the source and relay nodes.

2) Given the source-destination distance, the path loss exponent of the wireless channel,

and the energy per symbol at the source and relay nodes, determine the optimum

location of the relay nodes.

3) Given the source-destination distance and the total available energy per symbol,

determine the joint energy and location optimization.

For analytical tractability, we consider an idealized L-relay system with all relay

nodes located at the same distance from the source and destination nodes; that is,

Ds,r = Ds,r and Drm,d Dr,d, Vk. It is then reasonable to assign equal energies at all relay

nodes 8,r = S, Vk. To carry out the optimization in the ensuing subsections, we will also

make use of the relationship between the average power of channel fading coefficient ohij

and the inter-node distance Dj, as follows:


a = C D i,j e{s, r, d} (4-1)


where v is the path loss exponent of the wireless channel and C is a constant which we

henceforth set to 1 without loss of generality. For the conventional DF and AF protocols,

we will present the analytical results of optimizations as well as simulated examples. For

the DSTC protocol, due to the analytical intractability, the optimization results will be

shown by simulations and compared with conventional systems.









Relay(s)
/D

Source s'd Destination!


(a)

Ds,r (- Dr,d

D D----
s,d
(b)

Figure 4-1. Network topologies: (a) Ellipse case; (b) Line case.


We will consider two network topologies as depicted in Fig. 4-1. One is the ellipse

case and the other one is the line case. For the ellipse case, Ds,r + Dr,d = D > Ds,d.

The line case can be regarded as a special case of the ellipse case, i.e., D = D,d. By

changing the value of D, we can solve the optimization problem at any point on a 2-D

plane. Therefore, optimum resource allocation for these idealized topologies can provide

useful insights for understanding the effect of resource allocation in relay networks.

4.1 Convexity of SER

Let us first consider the convergence of the error performance. The convergence

of our optimization is guaranteed by showing the convexity of the error rate. With

DF and DSTC protocols, it is cumbersome to prove the convexity analytically, due to

their complex expressions of the error performance (see Eqs. (3-5), (3-41), and (3-49)).

However, our simulations will show that the error rate is generally convex, which ensures

convergence of the optimization. With the AF protocol, we can prove the convexity

analytically and confirm by simulations. The proof of convexity for the AF protocol is

given as below, which guarantees the convergence of the error performance as a function of

energy and location.









Lemma 1. Under high SNR, the average SERs in Eqs. (3-10) and (3-31) are convex

functions of the energy and location, i'..1/,. ;

Proof. Given location Dr,d, i.e. ar,, and energy constraint ps + Lpr p, the SER can be

written as:

AF 1 1
'eDN C(L) L 2 l(p,) (4-
(p Lpr)2rs Prca,r


L
peL = C(L + 1)-- In (4-
p Lpr (p Lpr)r,7 p dr,

which are functions of the single variable pr e (0, p/L).

The second derivatives of pFD and pAF are given as follows:

For L > 2,

2 pAF
e,ND C'(L) {(L 1)[f(pr)]L-2 [g(r)]2 + [f(r )]L- h(r) (4-
apr


2 pAF
e,DL
ap


SC"(L) [f ()]L-2[g(p,)]2 {A()}
(p Lp)3
1
+ C"(L) [f(p,)]L-l[h(pr)],
P Lpr


for L = 1,


a2 AF
e,ND
ap


2 pAF
e,DL C"1 1
,DL C1) (t )3 {2 f(p,) + 2pg(p,) + p h(p)} ,
where C'L) LCL) and C ) LCL + 1), and
where C'(L) = LC(L) and C"(L) = LC(L + t), and


f(Pr)


2)


3)


4)


(4-5)


(4-6)


(4-7)




(4 8)


1 1
( Lpr) 'r pr
(p Lp,)u, Pr2+ ,r


C'(t) h (p,),









g(pr) =
Jr,s (P


1 1
Lp)2 2+ 2 (pr),
Lpr) P "ar


2L2 1 1
h(pr) = 2 )3 + -321n(prr) 3],
,s (P Lpr) Pra


A(p) 2L P [ "p 2 + 2L ()l + (L -
p Lp [(P)] p- Lpr [g(Pr)
2
Under high SNR approximation, pr dr > 1, we have f(pr) > 0 and

notice that A(pr) is a quadratic function of 1 [ (P, and its quadratic

A (2L)2 4(2L)(L 1) 4L(2 L) < 0, for L > 2.

Thus, we have A(pr) > 0 when L > 2. Therefore,
02 pAF 2 pAF
0 "eND eDL
'- > 0 and > 0, for L > 2.
Op2 Op2


(4-11)


h(pr) > 0. We
discriminant is:


(4-12)


(4-13)


When L = 1, the convexity of the system with no direct link is readily obtained from

Eq. (4-6). For the system with a direct link, after some manipulation, Eq. (4-7) can be

reexpressed as:

= c (() pP {2[P (P + 1 ln(pr,1 )- 3


+ C"//(1) P 2 p) + 6PS (4-14)
(p Pr)3 pr pr s

Using the inequality, (pr/ps)2 (r/ps) + 1 > 3/4, we have the lower bound of 2pDL/AF L 2.

ad2 pAF1 13 2
2pe > C/(1) 3p ( )ln (Pr) 2} (4-15)
Op, (p pr)3 7r 2 pr,

On the condition of high SNR, In(prf) > 2, we have '2AF P /p > 0 for L = 1.

Finally, both PAeND and PAFDL are convex functions of pr, and accordingly ps for any

L> 1.


(4-9)



(4-10)









Now, consider the average SER as a function of location. Plug -,, = Dr' and
2r = D into the formula of PAF, and use the constraint Ds,r + Dr,d = D. It follows
ad,r r,d iNDu
that:

'eND ( CL) [(D- Dr) Dd In(prDr]) (4-16)
(L (- Lpr) pr

which is a function of a single variable Dr,d e (0, D). The second derivative of PAJD is
2 pAF
2,ND C'(L) {(L )[u(Dr,)]L-2[v(Dr,)]2 + [u(Dr,d)]-I [w(Dr,)]} (4-17)
aD2r,d

where

(D D )V DV
u(Dr,d) ( +D r, ln(prD ), (4-18)
(p Lpr) Pr

D"v
v(Dr,d) ( -(D Dr,d)- + rd [ln(prDr) 1] (4-19)
Ps Pr


w(Dr,d) (= 1)(D D,) v-2 + Dd {(- 1) [ln(prD,)- 2] 1}. (4-20)
p Lpr Pr

Under high SNR approximation, i.e., prd > 1, and v > 1, we have u(Dr,d) > 0,

V(Dr,d) > 0 and w(Dr,d) > 0. Thus:


aD2
ODr,d

Similarly, for the system with a direct link, we have

PL L = C(L 1 + )ln(prD ) (4 22)
p Lpr (p Lpr) Pr pD (4

This equation is the same as Eq. (4-16) except for the constant term, C(L + l)/(p Lpr),

therefore its convexity can be readily proved using the same steps in the above.

D









4.2 Energy Optimization

Now let us consider the energy optimization given the relative distances among the

nodes.

Problem Statement 1. For i,. ; given source, ,. Ir;, and destination node locations (Ds,r

and Dr, or ('I;'. nlj.i and a,), and the total energy per symbol S, determine the

optimum energy allocation S, and S, which minimize the average SER in Eq. (3-5) for DF

and Eq. (3-10) or Eq. (3-31) for AF while .,7/i.f,,

L
s, + r, = S, + LS, S. (4-23)
k=l

By defining the total SNR, p := S/Ao, the transmit SNR at the source node

ps := s/AMo and the transmit SNR at the relay nodes pr := S/A/o, the energy constraint

can be re-expressed as the SNR constraint:


p = ps + Lpr. (4-24)


Using Eq. (4-1), the average received SNRs at the relay and destination nodes can be

expressed in terms of the transmit SNRs as:
-2 p -D
ir,s = Ps =h, PsD

and 7d,, pr hl,, = prD (4-25)


As a result, the total energy constraint, Eq. (4-24), can be further rewritten as


P = + L7d/ ,, 7r,sD + L7d,rD"d (4-26)

Let us consider the DF protocol first. To gain some insights, we start from a

single-relay setup and establish the following result:

Proposition 3. With DF protocol, for a single-,, AI.r; setup with L = 1, at given s r

and r d distances Ds,r and Dr,d, and under the total energy constraint in Eq. (4-23), the









optimum energy allocation s should -if1' fy:


rfo- V~r1 Vi- 5) n -2,(2D-2vp2 in-- 1
s 2- o 2~~ 2 + D'ysDr (6Dr p + 5) + 2Ds,r (2Dr P2 + 3D-rp +
PS
roV 4D-5D-(D-D D-"4)2
s,r r,d 8,r r,d
2Dr"p + 3
r' (4-27)
2(D-v Dr ')
8\s,r r,di

and ,,, -i, ,'.,,.:,i,, S = 7 so.

Proof. Treating the SER bound pDF as a function of 7r,s and 7d,r, we have the first order

conditions for the optimum solution

8PDF
apF AD_ = 0, (4-28)

QPDF
7d ADd = 0, (4-29)

p (r,sDr + 7d,rDf,) 0, (4-30)

where A is the Lagrange multiplier. Solving Eqs. (4-28) and (4-29), we obtain

(1 + 27d,r)(1 + 7r,,) Drv
rd (4-31)
(1 + 27r,)(1 + 7,) D, '

which leads to the following relationship between 7d,r and 7,,

-3DS,/2 + D S + 8D-' (1 + 37r, + 7,)
7d,r 4Dv 2 (4-32)

Substituting Eq. (4-32) into Eq. (4-30), we find po as in Eq. (4-27). E

Although Eq. (4-27) is accurate for all S and A/o values and for all s r and r d

distances, its complex form does not provide much intuition. Fortunately, for several

special cases, it can be simplified without much loss of accuracy. Next we will consider

some of such cases.

Special Case 1 [Centered Relays]: When D,r = Dr,d, Eq. (4-31) simplifies to

27d2, + 37d,r + 1
S1.
2~2, + 3, + 1









By symmetry, we obtain 7r,, = 7d,r. Therefore, the optimum energy allocation amounts to

assigning equal energies to both the source and relay nodes; that is, S = f = 8/2.

Special Case 2 [High SNR]: In this case, expanding Eq. (4-31) as

27d,t + 37d,r + 1 Dr
272, + 3r, + 1 D,-

and neglecting the constant terms in both the numerator and the denominator, we obtain

an approximate solution for the optimal energy allocation

S-(2D p + 3) + (2D-p + 3)(2Drp + 3)
o o 2(DV, Dr) (433)

This solution can be further simplified by neglecting its constant terms to

D-v/2 D-v/2
0 r,d r ,d (4-34)
SDv2 v+D 2 1 D8f2 (434)
D,,/2 + Dr-d Ds,/ + Dr,d

Interestingly, this solution coincides with the optimum power allocation obtained by

minimizing the outage probability [27, (8)] with a single-relay transmission. From Eq.

(4-34), it readily follows that the energy allocation ratio between the source and the relay

nodes is


co D /2


Eq. (4-35) reveals explicitly that the optimum energy allocation heavily hinges upon the

inter-node distances. In addition, the path loss exponent of the wireless channel, v also

affects the optimal energy allocation. Interestingly, the S/7o ratio is linear in Ds,r/Dr,d

only when v = 2. The optimum energy allocation favors the link with a larger node

separation if v > 2 and vice versa, as we will show next with an example.

Fig. 4-2 depicts the transmit SNR ps obtained from the optimum energy allocation.

A one-dimensional setup is considered; that is, Ds,r + Dr,d = D,d = D. The system

parameters are: p = 10dB, L = 1, Ds,d = D = 1 and v = (1, 2, 3, 4). In Fig. 4-2, the

















E 7

.9
6v=4
6 v=3




S. Simulated=

0 0.2 0.4 0.6 0.8 1
E 4
E









Figure 4-2. Exact and approximate optimum energy allocations with different path loss
exponents (DF, L= p =ldB).
simulated optimum is plotted together with the exact (4-6)analytical value in Eq. (4








and the approximations in Eqs. (4-33) and (4-34). These results are nearly identical

in all cases with various v values. By closely inspecting the figure, we find that the
Spproximation in Eq. (433) provides more accurate curves than (4-12)the one in Eq. (434), as
0
0 0.2 0.4 0.6 0.8 1
D
s,r

Figure 4-2. Exact and approximate optimum energy allocations with different path loss
exponents (DF, L 1, p 10dB).


simulated optimum p, is plotted together with the exact analytical value in Eq. (4-27)

and the approximations in Eqs. (4-33) and (4-34). These results are nearly identical

in all cases with various v values. By closely inspecting the figure, we find that the

approximation in Eq. (4-33) provides more accurate curves than the one in Eq. (4-34), as

expected. Although the approximate expressions in Eqs. (4-33) and (4-34) are obtained

under high SNR assumption, they remain very accurate even at medium SNR of 10dB.

From Fig. 4-2, we also observe that, for all v values, the source node energy E,

increases as the relay moves towards the destination node. With v = 2, S, increases

linearly with Ds,r. At higher values of the path loss exponent, v > 2, we observe that


p < Ds,r/Ds,, when Ds,r < Ds,a/2 ,
(4-36)
P > Ds,r/Ds,d, when Ds,r > D,d/2 .

In other words, the optimum energy allocation favors the link with irj inter-node

distance. When the path loss exponent v = 1, Fig. 4-2 shows the opposite of Eq. (4-36).









So far, we have been focusing on the single-relay case, where an analytical solution

(4-27) can be obtained and a very accurate and insightful approximation is available

under high SNR assumption. For L > 2, however, the first order conditions obtained

by differentiating the SER bound PeDF have complicated forms, which render analytical

solutions impossible. Fortunately, the SER bound PeDF as in Proposition 1 still allows for

a numerical search, as opposed to Monte Carlo simulations needed otherwise.

For example, with DBPSK and L = 2 we have

pDF 1 3 ( d,r 1 d,r 37)
e,d 2 4 1 + ,r 4 &1+. (,r/

and, accordingly, the SER bound is given by

pDF 1 (2 +7,yd)(1+ 27d,)2
p = 1 (4-38)
4(1 + r,s)2( + 7d, )3

By using the first order conditions in Eq. (4-28) and the high SNR approximation, the

optimum 7r,s and 7d,r should satisfy

4(1 + 7y,r)(2 + 7d,))(1 + 27y,r) Dj
--r (4 39)
37ry,(1 + 7q,<) (D4,-

Although an analytical solution is not readily available, one can resort to the numerical

search.

Let us consider some examples of optimum energy allocation. Fig. 4-3 represents the

average SER for various energy allocations at the fixed relay location. Total transmit SNR

p = 10dB and L = 2 are considered with D,r = 0.25, 0.5, and 0.75. For each case, the

SER has one minimum point, and the corresponding energy allocation is the optimum

value, i.e., pO/p. The figure shows that the p /p increases as the re-i- move towards the

destination. This coincides with our analytical results and simulations in Fig. 4-2.

The optimum energy allocation obtained from the numerical search is plotted in

Fig. 4-4 and compared with the simulated results. The total SNR value of p = 10dB

and a path loss exponent of v = 4 are considered with various L values. The results




























10 -






10-3
0


0.2 0.4 0.6 0.8


Figure 4-3. SER versus energy allocation at the given relay location Ds,r (DF, L = 2, p = 10dB,
D = 1, v =4).


0.2 0.4 0.6 0.8


Figure 4-4. Comparison of optimal energy allocation between the numerical search and
simulated results at various L values (DF, p = 10dB, v = 4).


















0.7
0
0.6 / v=4

2v=2
0.4-
E
E
Z 0.3- p=OdB
O p=5dB
0.2 -- p=lOdB

0.1 p=15dB
p=20dB
0 1
0 0.2 0.4 0.6 0.8 1
D
s,r

Figure 4-5. Comparison of normalized optimum energy allocation at different p values (DF,
L 1).


show that the analytical values and simulated ones are nearly identical for L = 1. As

the number of rel i- L increases beyond 2, a gap between the numerical search and the

simulated results can be observed from Fig. 4-4. This discrepancy arises from the fact that

the numerical search is based on the SER bound, whereas the simulations generate the

true SER, and that the SER bound is looser for larger L values, as we mentioned in the

preceding section. Nevertheless, the numerical results still closely indicate the trend and

relative distances corresponding to various L values. Notice that these curves exhibit a

converging tendency as L increases. This implies that the optimum energy allocation curve

may achieve an .,-i-'i:!l, 1 ic limit as the number of re!-,v L grows.

We have seen that the exact expression of the optimum energy allocation in Eq.

(4-27) will give rise to a ps/p ratio that depends on the actual value of the total SNR p

with DF protocol. However, the high-SNR approximation in Eq. (4-34) results in a ps/p

ratio which is independent of p (see Eq. (4-35)). As a result, the approximate solutions

(4-33) and (4-34) are expected to differ from the exact solution (4-27), depending on









different values of p. Fig. 4-2 shows that these solutions agree very well at p = 10dB.

In Fig. 4-5, the optimum energy ratio ps/p obtained from the exact solution (4-27) is

depicted at various SNR values p = (0, 5, 10, 15, 20)dB, and with two values of v (2 and

4). We observe that when v = 4, the curves corresponding to different p values are almost

identical; whereas when v = 2, all curves coincide except for the p = OdB one. These

observations confirm that the approximations (4-33) and (4-34) are both very accurate

even for p as low as 5dB when v = 2 and for all SNR levels when v = 4. In other words,

the optimum energy allocation ratio ps/p is almost independent of the actual energy

level except for very low p values. It only depends on the location of the relays as in Eq.

(4-35). Likewise, a similar result can be deduced for the optimum distance allocation;

that is, the optimum distance allocation ratio Ds,r/D,,d is nearly independent of the actual

source-destination distance Ds,d, as we derived in Eq. (4-60).

Now let us consider the AF protocol. Similar to the DF protocol, by treating the

approximated SER PD or P' DLS a function of ps and pr,, we can find an optimum

solution.

Proposition 4. With AF protocol, at given s r and r d distances Ds,r and Dr,d, and

under the total energy constraint in Eq. (4-24), the optimum energy allocation pg and pO

should -,/-;f,


a ,[ln(p ) l]p + porp ppo ,r 0 (4-40)

and

(L + 1) r, ln(p odr) L P + P 0'r L- n(po ,r) P PP7 ,r } 0(4 )

for the system with no direct link and with a direct link, ,, -"i' /, *;. /;









Proof. We have the following first order conditions for the system with no direct link:

8pAF
S,ND O, (4-42)
OPs
8PAF
ND A =0, (4-43)
aprk
p (p, + Lpr) = 0, (4-44)

where A is the Lagrange multiplier. Eqs. (4-42) and (4-43) give us

LpAF 2
,ND Pr d,r o- A 0, (4-45)
Ps Pr ,r PsP ,s dpr ,r)


LPAFD Ps2,s PJr) 1]
,ND [- P--- ln(p LA 0. (4-46)
Pr Prr + Psrsaln(prr)

Then, by substituting ps and pr in Eqs. (4-45) and (4-46) for the total energy constraint

in Eq. (4-44), we have
LpAF 2 p2lnp2)
ePND Prad,r + r,s P ,r
2p. (4-47)
A Pr ,r Ps lnrs (pr,7,r)- 1]

With Eqs. (4-45) and (4-47), we arrive at Eq. (4-40). Similarly, the first order conditions

of the system with a direct link are given by

,DL 1 + L 2 Pd,r A 0, (4-48)
Ps p Prd Pr,sPrdr)


LPfL Pss [ln(pdr) 1]
DL -In n(pr,-- 1) LA 0. (4-49)
Pr Prd ,r Ps ,sn prdr)

By using the same steps as shown, we have Eq. (4-41). D

In Eq. (4-41), the p71,,n(pi ,rd)/(L + 1) term mainly affects the energy allocation

compared with the system with no direct link. This effect is obvious especially when the

reh-.v are located close to the the source. Notice that the log term in Eqs. (4-40) and

(4-41) renders a closed-form solution incalculable. Although an analytical solution is not



















-"
150 ,-
S,o'
100- -



00

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Pr/P


Figure 4-6. Existence of the optimum solution (AF ND, p = 15dB, v = 3, L=2).


readily available, one could resort to the numerical search using Eqs. (4-40) and (4-41).

The following Lemma 2 shows that optimum values have only one solution, which allows

us the numerical search.

Lemma 2. The optimum solutions in Eqs. (4-40) and (4-41) have one solution of p, and

accordingly pO.

Proof. Since the Eqs. (4-40) and (4-41) cannot be solved algebraically, we represent a

solution graphically. Using Eq. (4-24), Eq. (4-40) is given by


2 o 2 + _2 l o ,o o o.
,s[.l(p,.o-) ]?p2+ po-,rp (p + Lp)pco-,r =0. (4-50)


This can be represented as


o,[ln(por) 1]p Lp o ,r. (4-51)


By squaring root both sides, we have


v2 [n(p-,r) 1]p pdr, (4-52)






































i ...... .. 1-- -
0.2 0.4 0.6 0.8
Ps/P


Figure 4-7. SER versus energy allocation
p = 15dB, D= 1, v 4).










10 \


'-. iii\

ng -. .. .. ... ...

g, 10 : ::

g : : : :'.. : ::: --^-
10-2










10-5
10





10-4


io .. .


at the given relay location Ds,r (AF ND, L = 2,


0.2 0.4 0.6 0.8


Figure 4-8. SER versus energy allocation at the given relay location Ds,r (AF DL, L = 2,
p = 15dB, D= 1, v 4).


10-2


10-3
0









since pi > 0 and oaj > 0,Vi,j. Let us denote yl and y2 as the left-hand side and the

right-hand side of Eq. (4-52), respectively. Given that domain of ln(poa2,r) > 1 and using

Eq. (4-1), we plot yl and Y2 in Fig. 4-6 for various relay locations Ds,r with D,d = 1. The

line with no marker and with circle marker represent yl and y2, respectively. The figure

shows only one crossing point, and that point provides us the optimum solution. Similar

trends are observed in the system with a direct link. E

Figs. 4-7 and 4-8 depict the average SER for various energy allocations at fixed relay

location for the system with no direct link and with a direct link. We locate rel i at

0.25, 0.5, and 0.75 with p = 15dB and L = 2. Both figures show that the SER has one

minimum point, and the corresponding energy allocation is the optimum value, which

confirms Lemma 2. For the system with no direct link the optimum energy allocation

increases as the relays move toward the destination. However, for the system with a direct

link, optimum energy allocation -l i,- at the middle value, i.e., uniform energy allocation

when the rel i-,- are located close to the source (see Fig. 4-8 with D,,r = 0.25 and 0.5).

This result will be verified by the following numerical search results. For all cases, the

SERs in Fig. 4-8 show better performance in Fig. 4-7 due to the direct link.

The optimum energy allocation obtained from the numerical search for both the

system with and without a direct link is plotted in Fig. 4-9. We consider the total

SNR value of p = 30dB and a path loss exponent of v = 4 with various L values and

D = Ds,d = 1. In the system with no direct transmission, for all L values, the optimum

energy allocation at the source increases as the relay moves towards the destination.

However, for the system with a direct link, a uniform energy allocation is optimum

when the rel i-, are located close to the source. Intuitively, this is because the direct

transmission is present that the diversity gain is dominant over the coding gain. When the

rel i are located close to the destination, much of the energy is assigned at the source to

assure that a transmitted signal can reach the r el-;: it is the same as in the system with

no direct link.





































0.2 0.4 0.6 0.8


Figure 4-9. Optimum energy allocation (AF, ND and DL, D


0.7 P


0.4 P


0 0.2 0.4 0.6
D
s,r


1, p 30dB, v


0.8 1 1.2


Figure 4-10. Optimum energy allocation (DFSTC and
v 4).


AFSTC, D =


1.2, Ds,d 1, p 15dB,


L=2
L=3









Fig. 4-10 represents the optimum energy allocation for both DFSTC and AFSTC

protocols when L = 2 and 3. We use the total SNR value of p = 15dB and a path loss

exponent of v = 4. with D = 1.2 and D,d = 1. The figure shows that the optimum energy

allocation at the source increases as the rel i-, move toward the destination for both cases,

which is the same as the conventional cooperative networks with no direct link. The figure

also shows that, in general, DFSTC protocol requires more energy than AFSTC protocol

to decrease the error rate at the rel i-,v.

4.3 Relay Location Optimization

Consider now the optimum relay location with the sum distance between the

source-reli.-,] and relays-destination being D, and the source-destination distance Ds,d.

If the transmit energies at the source and reli-,- are preset, where is the optimum location

to place the rel i-v? To answer this question, we treat the distance D as a fixed resource

and formulate an optimization problem as follows:

Problem Statement 2. For iwq ; given transmit energies at the source and luI.; nodes

(S and or (,;it'',. ,i': ;1 ps and pr), and the path loss exponent v of the wireless

channel, determine the optimum location of the relays, DQ,r, which minimizes the average

SER in Eq. (3-5) for DF and Eq. (3-10) or Eq. (3-31) for AF while -,il flying:


Ds,r + Dr,d = D, where 0 < D,r < D. (4-53)


Starting with the single-relay (L = 1) setup and applying the high-SNR approximation

with DF protocol, we establish the following result:

Proposition 5. For DF protocol, with a single-,' AI.r; setup where L = 1 and the source-

relay(s)-destination distance D, and $s and S, denote the prescribed transmit energy levels

at the source and ', 1.r; nodes, ', /'.. /i 1.:, the optimum location of the ', Ir;, is
1/(V-1)
Do, P D (4-54)
PS + pr

and accordingly, D,d = D Ds,r.









Proof. Treating the SER bound pDF as a function of Ds,r and Dr,d, we have the first order

conditions for the optimum solution

8pDF -^
aP 0 X A =0, (4-55)
07r,s ODs,r
aPF ,r A 0, (4-56)
07d,r ODr,d
D (D,,r + Dr,d) = 0 (4-57)


where we used the fact that 7,,s (or 7d,r) is independent of Ds,r (or Dr,d). Using the

definition of the average receiver SNRs in Eq. (4-25), we can re-express Eqs. (4-55) and

(4-56) as

aPDF -
07,s D,+1

aPEF -V
P_ r -.- A= 0, (4-58)
0d,r r+,d


which leads to

pr Dy,+1 PDF/d7r,s (1 + 27y,r)(1 + 7,) (459)
Ps D+I PpDF/ d,r (1 + 27r,s)(1 + 7r,s)

At high SNR, the constant terms on the right-hand side of Eq. (4-59) can be ndglected.

Consequently, we have [c.f. (4-25)]

pr D 81 r p2 D 2v
,D+1 2 2 2V
Pr_ s,r d,r Pr Ds,r
s -r,d r,s s r,d

As a result, the optimum distances Ds,r and Dr,d should satisfy

D-_ (p)1( -1-) (4-60)
Dr,d Pr Sr

which, together with the condition in Eq. (4-57), concludes the proof of Proposition 5. E

Interestingly, Eq. (4-60) bears a very similar form as its counterpart for the optimum

energy allocation in Eq. (4-35). In fact, when the path loss exponent v = 2, Eq. (4-60) is

essentially identical to Eq. (4-35). For general v values, however, these two relationships



















0.6

o 0.5
Sv=2 v3 v=4
E 0.4 -
0.3

0.2- Numerical search
S- Approxitmation
.1....... Simulated
01
0 0.2 0.4 0.6 0.8 1
ps/P


Figure 4-11. Optimum location of relays (DF, p = 10dB and L 1).


are quite different. Such a discrepancy is actually very reasonable, because Eqs. (4-35)

and (4-60) result from two distinct optimization problems: the former is obtained for

arbitrary distances Ds,, and Dr,d under a total energy constraint; whereas the latter is

obtained for prescribed S, and S, under a total distance constraint. With the SER bound

pDF being a two-dimensional function, the energy and location optimizations are carried

out on uncorrelated dimensions.

For general L values, the optimum location can be determined in a similar manner

as we discussed in the above. Essentially, the path loss exponent v renders it impossible

to derive an analytical solution for the optimum location problem, even with the high

SNR approximation. One can resort to the numerical search using the SER bound in

Proposition 1. In Fig. 4-11, the optimum distances obtained from the numerical search

and the simulations are compared for different v values, at total SNR p = 10dB and with

L = 1 relay node. Notice that, as its counterpart in Fig. 4-2, the optimum relay location is

linear in Es/S only when v = 2.

















10- : : : :
'. ... .. . . ..
0) . . .. .. .... .... . ... .









0 0.2 0.4 0.6 0.8 1




L = 2, p = 10dB, D = 1 v = 4).


Fig. 4-12 depicts the average SER versus relay locations at the given energy allocation

when p = 10dB and L = 2. Notice that at the given prescribed energy, there exists only

one minimum SER point, and the corresponding Ds,r is the optimum relay location. This

figure is the counterpart of the optimum energy allocation in Fig. 4-3. We can see that

the range of optimum relay location is smaller than the optimum energy allocation, which

results in the flatness of location optimization. We will verify this phenomenon with a

numerical example.

Let us now consider the optimum relay location for the AF protocol.

Lemma 3. For rn; given energy at the source, the optimum ', .r';X location which mini-

mizes the average SER is independent of a direct link between the source and the destina-

tion.

Proof. This can be proven by the average SER expression in Eq. (3-31). The 1/Ys =

D s/Ps has a fixed value given the s d distance Dsd and prescribed energy at the

source. Therefore, a direct link does not affect location optimization. Hence, the location
10'
0 0.2 0.4 0.6 0.8 1
D
s,r

Figure 4-12. SER versus relay location distribution at the given energy allocation Ps/P (DF,
L 2, p 0OdB, D v 4).


Fig. 4-12 depicts the average SER versus relay locations at the given energy allocation

when p 10dB and L 2. Notice that at the given prescribed energy, there exists only

one minimum SER point, and the corresponding D,,, is the optimum relay location. This

figure is the counterpart of the optimum energy allocation in Fig. 4-3. We can see that

the range of optimum relay location is smaller than the optimum energy allocation, which

results in the flatness of location optimization. We will verify this phenomenon with a

numerical example.

Let us now consider the optimum relay location for the AF protocol.

Lemma 3. For ,,.,. given energy at the source, the optimum ,i location which mini-

mizes the average SER is iZndependent of a direct link between the source and the destina-

tion.

Proof. This can be proven by the average SER expression in Eq. (3 31). The 1/7d,,

Dv,dp/p has a fixed value given the s d distance D8,d and prescribed energy at the

source. Therefore, a direct link does not affect location optimization. Hence, the location


III


















102

. . . .. ., .. .. ... . .
. . . .. . .. . .

o : :::: p:: p .::: ::: : ::^ : ::: : :,, : : : :::: :":

S. .... . .... .... . .. : .. .


S ps/p=0.8
10-4
10-4----------------------------------------


0 0.2 0.4 0.6 0.8 1
D
s,r

Figure 4-13. SER versus relay location distribution at the given energy allocation ps/p (AF ND,
L = 2, p = 15dB, D = 1, v = 4).


optimization can be achieved without considering the direct link; i.e., the results of

location optimization are the same in both systems with and without a direct link. D


The optimum location can be found by treating the SER as a function of distance and

solving the first order conditions.

Proposition 6. For the AF protocol, with the given source-destination distance Ds,d,

source-relay(s)-destination distance D, and the prescribed transmit energy levels ps and pr

Sthe optimum location of the '. 1.,;/ should ,.ri/fy

vD,;1 r -,(D -- o I- I -06

vD pr vD D )-p{ln[p(D D,)-"] 1} 0, (4-61)


and accordingly, Dd = D D .

This solution can be obtained by using Lagrange multiplier as we discussed in the

previous propositions; therefore, we omit the detailed derivation. Again, the log term

and path loss exponent v make it difficult to find the closed form solution. By applying

Lemma 2, one could resort to numerical search for the optimum solution.


I I I I



















10- '' 3


C, N
S. .... .
.o . .. . . .. . .. . .
.. ...p./.p.




10- : : : : : :
1 4 .. .... .. .. ... . .. .:'. :.:::: : :
-- ps/p=0.2 ... . .... .. ^ .... ..... .
S p /p=0. .. . ... -. . .. . .. .

10-5----------------------------------------
..... ps/p=0.8 .. ..


10'5
0 0.2 0.4 0.6 0.8 1
D
s,r

Figure 4-14. SER versus relay location distribution at the given energy allocation ps/p (AF DL,
L =2, p 15dB, D =1,v = 4).



Figs. 4-13 and 4-14 present the average SER for the various relay locations at

the given energy allocation. Both figures confirm that there exists only one minimum

point which provides the optimum relay location for relay networks. It is interesting

that the minimum points are the same for both figures although the SER is different.

Hence, the existence of a direct link is independent on the optimum relay location, which

confirms the Lemma 3. In both figures, the range of optimum locations is smaller than the

energy optimization cases (see Figs. 4-7 and 4-8), which is the same as the DF case (see

Fig. 4-12).

Fig. 4-15 depicts the optimum relay location which is applicable to systems with

and without a direct link. We consider the total SNR value of p = 30dB and a path loss

exponent of v = 4 with various L values. One dimensional setup, Ds,r + D,d = D = Ds,d =

1, is considered. As more transmit energy is assigned at the source, the optimum location

moves toward the destination. The figure shows that the optimized values change slowly

compared with the energy optimization.


I I I I










































0.2 0.4 0.6 0.8
P,/P


Figure 4-15. Optimum relay location (AF, ND and DL, D


0.9 -


0.6 H


Ds,a = 1, p 30dB, v = 4).


0 0.2 0.4 0.6 0.8


Figure 4-16. Optimum relay location (DFSTC and AFSTC, D = 1.2, Ds,d = p 15dB, v = 4).


o 0.8
C
O
8 0.7


0.6-


. 0.5


-- L=1
- L=2
S.... L=3


L=2
SL=3 I









Fig. 4-16 depicts the optimum relay location of DFSTC and AFSTC protocols when

L = 2 and 3. We use the total SNR value of p = 15dB and a path loss exponent of v = 4.

with D = 1.2 and D,d = 1. The figure shows that the optimum relay locations move

toward the destination as the transmit energy at the source increases for both cases. The

figure also shows that, in comparison with AFSTC protocol, the relay locates closer to the

source for the DFSTC protocol. From Figs. 4-10 and 4-16, we can see that the location

optimizations are much flatter than the energy optimizations. These results are the same

as those of the conventional cooperative systems with no direct link.

4.4 Joint Energy and Location Optimization

So far, we have been focusing on the energy optimization and location optimization

separately. Now let us consider the joint optimization which satisfies both the energy and

location optimization. The analytical solution can be obtained by using Eqs. (3-5), (3-10)

or (3-31). First, by treating each equation as the function of transmit energy, we can

find the solution for the first order conditions. Then, the same step is proceeded by the

transmit energy that is replaced with the location. Finally, by equating two solutions, we

can find the common solution. Consequently, this solution provides the global optimization

which minimizes the error rate. With DF protocol, for L = 1, we can readily obtain

the global solution from Eqs. (4-35) and (4-60), which gives us Ds, = Dr,d = 0.5 with

ps/P = Pr/P = 0.5, V. However, the analytical solution cannot be easily obtained
even with the idealized case as we have seen in the previous section. In general, the

joint optimization can be obtained by carrying out a two-dimensional numerical search

iteratively. The searching steps are as follows :

Step 1. (Initialization) Set the uniform energy allocation as the optimum, i.e. p

p/(L + 1).

Step 2. (Location Optimization) For a given energy allocation, find the optimum relay

location, D,,new, which is error rate-minimizing. If the difference between new

optimum location and the original one is smaller than the threshold distance, ED,



































Figure 4-17. Iterative search: flow chart.


i.e., ID,.new D ,r < ED, stop; otherwise, set the optimum location to the new

one, Dr = -Dre and continue to Step 3.

Step 3. (Energy Optimization) For a given relay location, find the optimum energy

allocation, ponw. If the difference between new optimum energy allocation and

the original one is smaller than the threshold energy, Ep, i.e., |pOnew PO < Ep,

stop; otherwise set the optimum energy to the new one, pO = pOn and go back to

Step 2.

These iterative searching can be illustrated as the flow chart in Fig 4-17.

Without considering a direct link, Figs. 4-18 and 4-19 depict the SER performance

surface when L = 3 with Ds,d = D = 1, and v = 4, for both the DF and AF protocol.

We use p = 10dB and p = 15dB for the DF and AF protocol, respectively. We can obtain

the energy optimization and the location optimization by taking minimum value along the















-!




LU 10 -





10
1 00.8 0. 0.5
0 0.2 0 0
Ds, r P/P


Figure 4-20. Performance surface versus ps/p and Ds,r (DFSTC, p = 15dB, v = 4, L 2).


Ds,r axis and ps/p axis, respectively. Using the above steps, the global minimum can be

obtained. This point provides the joint energy and location optimization.

Similar to the conventional protocols, we plot the CER versus ps/p and Ds,r for

DFSTC and AFSTC protocols in Figs. 4-20 and 4-21, respectively, when D = 1.2, Ds,d

1, and v = 4. These figures exhibit the same trends as in the DF and AF protocols. Notice

that the two figures show almost the same shape. However, the axis values of the two

figures are opposite. This implies that the systems employing DF and AF protocol have

quite different relationships for the optimum values. Notice that above searching steps

for joint energy and location optimization can be applicable for the DFSTC and AFSTC

protocols by replace SER with CER though the analytical solutions are intractable. More

detailed examples and comparisons are given in the following chapter. Notice that our

simulations (Figs. 4-3, 4-12, 4-18, 4-20, and 4-21) confirm that the error rate is generally

convex, which ensures convergence of the iterative strategy.









CHAPTER 5
SIMULATIONS AND DISCUSSIONS

In this chapter, we will discuss the performance of relay systems combined with

differential demodulation and the optimum resource allocation. We will compare the

performance of the systems with and without optimization. The benefits of the joint

energy and location as well as the resource allocation comparison of different protocols are

addressed.

5.1 Benefits of Energy and Location Optimizations

To verify the advantages of the optimum energy allocation and relay location

selection, the SERs of the relay systems with and without optimization are depicted

in Fig. 5-1 through Fig. 5-6. We use the following system parameters: p = 15dB, D = 1.2,

v = 4, and L = (1,2, 3) with DBPSK. In the system without energy optimization, a

uniform energy allocation is emplo,-' d: that is, ps = pr = p/(L + 1) at any Ds,,. In

the system without location optimization, the re -i,, are placed at the midpoint of the

source-destination link.

Figs. 5-1 and 5-2 illustrate the benefits of optimization of the DF system. In Fig. 5-1,

we observe that, as L increases, the SER performance can get even worse unless the

energy optimization is performed, and that the energy-optimized system universally

outperforms the unoptimized one. These observations confirm our discussions in the

preceding chapter. Interestingly, notice that the minima of the energy-optimized SER

curves almost coincide with the unoptimized ones. This implies that the near-optimum

SER can be achieved even with the uniform energy allocation across the source and relay

nodes, provided that the relay location is carefully selected. As shown in Fig. 5-1, the

optimum relay location corresponding to the uniform energy allocation shifts from the

midpoint for L = 1 to the source node as L increases. Intuitively, this is because the

overall SER is more sensitive to the source-relay link quality, as we mentioned in C'! plter

3.

















- /
P


- r: Unoptimized (ND)

S p =po: Optimized (ND)


0.2 0.4 0.6
D
s,r


0.8 1 1.2


Figure 5-1. SER comparison between relay systems with and without energy optimization (DF,
p = 15dB, D = 1.2, Ds,d = 1, v = 4).


- =Dr,d: Unoptimized (ND)

-- D =D : Optimized (ND) ......
s,r s,r

0.2 0.4 0.6 0.8
ps/P


Figure 5-2. SER comparison between relay systems with and without relay location optimization
(DF, p = 15dB, D = 1.2, Ds,d = v = 4).


10-4
0


L=21 : : : ,
L..2 /









In Fig. 5-2, we verify the advantage of the optimum relay location by comparing

the SER with and without location optimization. Similar to the energy optimization

case, Fig. 5-2 confirms the advantages of the location optimization, in which the

location-optimized system universally outperforms the unoptimized system. Different

from the energy optimization case, however, as L increases, the SER performance ah--,v-

improves even without any location optimization.

The curves in Fig. 5-2 also exhibit more flatness compared with the ones in Fig. 5-1.

This implies that the system SER is more sensitive to the location distribution than the

energy distribution. In addition, the minima of the location-optimized SER curves are

far from those of the unoptimized ones, except for the L = 1 case (see Fig. 5-2). This

indicates that placing the relay nodes at the midpoint cannot achieve the minimum SER

even with careful allocation of the source and relay energies, for any L > 1. This is to

be distinguished from the uniform energy case depicted in Fig. 5-1, as well as from the

coherent relay systems in [45].

Figs. 5-3 and 5-4 depict the benefits of energy optimization for the AF system with no

direct link and with a direct link, respectively. From Figs. 5-3 and 5-4, we observe that the

energy-optimized system universally outperforms the unoptimized system as we expected.

We also observe that, in the system with a direct link, the SERs of the optimized system

and unoptimized system are almost identical when the rel -i- are located close to the

source, since a uniform energy allocation is optimum. These observations coincide with our

analysis in the preceding chapter. Both figures show that the unoptimized systems have

the minimum SER almost at the midpoint, coinciding with the results in [38, 45, 67].

Notice that the minimum points of the energy-optimized SER curves move towards

the destination except in the system without a direct link for L = 1, which is opposite

compared with the DF case. From Figs. 5-3 and 5-4, it is clear that we cannot achieve

optimum SER value without energy optimization except for the system L = 1 with no

direct link. This is different from the DF case, as in Fig. 5-1. It is worth mentioning






















10-1



Lu
U)
, 10-2
10





10-3





10-4
0


0.8 1 1.2


Figure 5-3. SER comparison between relay systems with and without energy optimization (AF
ND, p = 15dB, D = 1.2, Ds,d = 1, v = 4).


ps Pr: Unoptimized (DL)

ps=ps: Optimized (DL)

0 0.2 0.4 0.6
D
s.r


0.8 1 1.2


Figure 5-4. SER comparison between relay systems with and without energy optimization (AF
DL, p = 15dB, D = 1.2, Ds,d 1, v = 4).


- ps=Pr: Unoptimized (ND)

Sp=p: Optimized (ND)


0.2 0.4 0.6
D
s,r


S . .. .. / .


10-4


I.. .I











. ... D =D : Unoptimized (ND)
.. . .. r r, d
...... .... D =Do :Optimized (ND)
.s,r s,r


69I~~~~l iI ... .. .... .. .... .. - ---- : : ..... /
10-2 \:


. .... ... . .... .. ..... ... ..


0- -------------------------------------------- --



0 0.2 0.4 0.6 0.8 1
Ps/P

Figure 5-5. SER comparison between relay systems with and without relay location optimization
(AF ND, p = 15dB, D = 1.2, Ds,d = 1, v = 4).


that the optimum relay location corresponding to the uniform energy allocation alh--,v-

keeps the midpoint regardless of the number of rel -1v-, which is the same for the coherent

systems in [45] with AF protocol, but different from the DF case.

Next, let us consider the benefits of location optimization for the AF systems.

Figs. 5-5 and 5-6 verify the advantages by comparing the SERs of the systems with and

without location optimization. Similar to the energy optimization case, the location-optimized

system universally outperforms the unoptimized system. The figures show that the

optimum SER can be achieved by assigning more energy to the source except for the

system without a direct link with L = 1, which is the opposite of the DF case. The curves

in Figs. 5-5 and 5-6 also show more flatness compared with the energy optimized curves,

as we have observed for the DF case. Similar to the location optimization of the DF and

the energy optimization of the AF, Figs. 5-5 and 5-6 show that optimum SER cannot be

obtained without relay location optimization except for the system L = 1 with no direct


.. L=1
/











10
S. .... : ....... D ,r=Dr,d: Unoptimized (DL)
SD =D : Optimized (DL)
s,r s,r







10-1 3




:L-3
. . . . .. . . .










10-5
0 0.2 0.4 0.6 0.8 1
Ps/P

Figure 5-6. SER comparison between relay systems with and without relay location optimization
(AF DL, p = 15dB, D = 1.2, Ds,d = 1, v = 4).


link. The benefit of a direct transmission is obvious in the SERs from Figs. 5-4 and 5-6

compared with Figs. 5-3 and 5-5, respectively.

Figs. 5-7, 5-8, 5-9, and 5-10 depict the benefits of energy and location optimizations

for DFSTC (Figs. 5-7 and 5-8) AFSTC (Figs. 5-9 and 5-10) protocols. For all cases, we

consider D = 1.2, Ds,d 1, and v = 4 with p = 15dB and 25dB when L = 2 and 3 since

the single relay setup (L = 1) is the same as the conventional cooperative system. Similar

to the conventional case, we plot the CER for the system with and without resource

allocations. We use the same parameters for the unoptimized systems. The figures confirm

that the minimum CER can be achieved by the optimum energy and relay location

selection. The figures also show that the trends of optimizations for DFSTC and AFSTC

protocols are the same as those for DF and AF protocols, respectively. Notice that, at

low p, the systems with more L may underperform the systems with less L. However, at

high p, the error performance improves as L increases except some cases of DFSTC-based

system as we have seen in DF-based system.






































0.2 0.4 0.6
D
s,r


0.8 1 1.2


Figure 5-7. CER comparison between relay systems with and without energy optimization
(DFSTC, p = 15dB and 25dB, D = 1.2, D,d = 1, v = 4).


0.2 0.4 0.6 0.8


Figure 5-8. CER comparison between relay systems with and without relay location
optimization (DFSTC, p = 15dB and 25dB, D = 1.2, Ds,d = v = 4).


10-2


10-5L
0


p=15dB



. . ___-_






t -- -L=2,p up. UnoptinizEd

S. .- L=3, p=p:Unoptimized
S p 25 dB -e-L=2,ps=Ps: Optimized

. ... . ... . + L=3, ps=p: Optimized







































0.2 0.4 0.6
D
s,r


- L=2,p =p: Optimized

- + L=3, ps=ps: Optimized

0.8 1 1.2


Figure 5-9. CER comparison between relay systems with and without energy optimization
(AFSTC, p = 15dB and 25dB, D = 1.2, Ds,d = 1, v = 4).


10



10


Cr
w 10



10



10



10
10-


0.2 0.4 0.6 0.8


Figure 5-10. CER comparison between relay systems with and without relay location
optimization (AFSTC, p = 15dB and 25dB, D = 1.2, D,d = 1, v = 4).


10-1



10-2



S103
C)


10-6
0


,* _-* p-15dB













L=3, ps =Pr:Unoptimized
SL=3, Ps=Pr:Unoptimized









Summarizing, we confirm the advantages of energy and relay location optimization

which minimize the error rate of the cooperative systems. With uniform energy allocation,

the DF-based systems and AF-based systems exhibit an unbalanced and balanced effect

on the error performance, respectively. Interestingly, for the DF-based systems, relay

location optimization may be more critical than energy optimization. In other words,

near-optimum performance can be achieved by allocating a uniform energy at each node,

but not by locating the re 1 i, at the midpoint between source-destination distance. For

the AF-based systems, both energy and location optimization are critical in the sense that

the optimum SER cannot be achieved without any optimization. Our results show that

the optimum resource allocation has different optimum values depending on the protocols,

which is confirmed in the next section.

5.2 Benefits of Joint Optimization

In this section we will consider the joint energy and location optimization. Figs. 5-11

and 5-12 depict the SER contour of the relay systems for the AF protocol with no direct

link and with a direct link, respectively, and Fig. 5-13 depicts the SER contour of the

relay system for the DF protocol. With system parameter D,d = D = 1, L 2 is

considered for the system with no direct link and with a direct link. We use p = 15dB

and 10dB for the AF and DF protocol, respectively. In all figures, the vertical line and

horizontal line represent the SERs of the systems with a uniform energy allocation and

mid-distance allocation, respectively, i.e., unoptimized systems. We also plot lines for the

energy optimization and location optimization. Notice that the crossing point of the two

optimizations is the minimum SER of the system; accordingly, this point corresponds to

the joint energy and location optimization.

With AF protocol in Figs. 5-11 and 5-12, it is clear that the minimum SER of the

unoptimized systems is far from the optimized ones; this indicates that we cannot obtain

the minimum SER with a uniform energy allocation or mid-distance allocation. To

achieve the minimum SER, we have to adapt a system via the energy optimization or


























0.5 -

0.4 sr./ oc mized

0.3 s,=D rd: Unoptimized

0.2 PsPr: Unoptimized
p: =p: Energy optimized
0.1 .

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
PS/P


Figure 5-11. The SER contour versus Ds,r and ps/p (AF ND, Ds,d









S=D oca opti
0 i9 sr r

08

0-

06

0 .5

0.4-

0.3 energy optimize

0.2
----- Prp: Unoptimized
0.1 -

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
PS/P


Figure 5-12. The SER contour versus Ds,r and ps/p (AF DL, Ds,d


0.9


D = 1, p 15dB, L= 2).


0.9


D = 1, p 15dB, L 2).





















0.4


0.2
1 s Unoptimized
0.1 -ps =p : Energy Cpli~mn'iz.3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Ps/P

Figure 5-13. The SER contour versus Ds,r and ps/p (DF, D,d = D = 1, p10dB, L = 3).


location optimization. However, the DF-based system in Fig. 5-13 shows that the near

optimum performance can be achieved with a uniform energy allocation, but cannot with

a mid-distance allocation. All these results coincide with the simulation results in the

previous section and theoretical analysis in C'! lpter 4. From Fig. 5-13, we can see that the

uniform energy allocation is a very good starting point for the iterative optimization.

Figs. 5-14 and 5-15 depict the CER contour for the DFSTC and AFSTC protocols,

together with the optimum energy allocation and relay location optimization curves. We

consider D = 1.2, Ds,d 1, and p=15dB when L = 2. These figures show that the CER

contours for the DFSTC and AFSTC protocols have the same trends as the SER contours

for the DF and AF protocols, respectively.

It is interesting that the optimum resource allocation is different in the two protocols.

For the AF-based protocol, we can achieve the minimum error rate by locating the

rel-,i,- closer to the destination and assigning more energy at the source than the rel-,i- .

This is due to the fact that the error rate decreases by allowing the source to transmit

signals with more energy while reducing the effect of noise on the amplification factor,

































0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9


Figure 5-14. The CER contour versus Ds,r and ps/p (DFSTC, D = 1.2, Ds,d 1, p=15dB,
L 2).


0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
P,/P


Figure 5-15. The CER contour versus Ds,r and ps/p (AFSTC, D = 1.2, Ds,d = p 15dB,
L 2).














0 Conventional system
7 -+ STC-based system


-0 6 .0'
E



S- .0
0

E
-o
5 3 .0
z

2 ---- + -+- ---- -+- .- .. +- .


1 2 3 4 5 6
Number of relays


Figure 5-16. Data rate comparison between the conventional -, -I, and STC-based system in
terms of the required time slots per information symbol.


and enhancing the error performance of relay-destination links. On the contrary, for the

DF-based protocol, the minimum error rate can be achieved by locating the relays closer

to the source while assigning less energy at the source than the relays. This is due to the

fact that the error of the source-relay links decreases, thereby causing a decreasing of the

entire system error.

5.3 Conventional System vs STC-based System

In this section, we will compare the conventional cooperative system and the

STC-based cooperative systems. We first compare the data rate of two systems, and then

the error rates of two systems are compared in two scenarios, i.e., the same modulation

size is adopted and same/similar data rate is used.

Assume that the original information symbols are equi-probable binary signal

(q = 1) and there are two relay nodes (L = 2). In the conventional system, one symbol

transmission takes three time slots: one for broadcasting the symbol to the relay nodes,

and two time slots are used to transmit the remodulated/amplified signal from each relay


I I I I









to the destination. For the STC-based system, 2 consecutive symbols can be packed

into each 2 x 2 unitary matrix. Since we use the diagonal design for STC, four time

slots are needed to transmit 2 symbols from the source node: two time slots are used to

transmit between the source and each relay, and two time slots for the transmission of the

2 x 2 matrix to the destination over the common relay-destination channel. Accordingly,

the number of time slots required for the transmission of each symbol further decreases

for the STC-based system, as the number of r1eliv- increases. This trend is depicted in

Fig. 5-16. We observe that the number of required time slots increases with the number

of relays for the conventional system, but remains constant for the STC-based system.

Therefore, we can increase transmission rate by using STC-based system. This implies

that the STC-based system can potentially provide the differential benefit as well as high

transmission rate regardless of the number of relays.

For the STC-based system, the modulation size increases as the number of re- v -

increases. Whereas, we can choose the modulation size for the conventional system. By

adopting the same modulation size, the comparison of bit error rate (BER) between

the conventional system and STC-based system is depicted in Figs. 5-17 and 5-18 with

DF/DFSTC and AF/AFSTC protocols, respectively. We consider DQPSK, D8PSK, and

D16PSK for L=2, 3, and 4, respectively, for the conventional system. The figures show

that STC-based system performs comparably with or better than the conventional system

especially when L > 3. For the conventional system, the error performance decreases as

L increases for both DF and AF protocols. However, for the STC-based system, the BER

is not affected dramatically regardless of L and the diversity gain is alv--bv- guaranteed by

decreasing error rate for the DFSTC and AFSTC protocol, respectively.

As we mentioned in the above, the STC-based system provides higher transmission

rate. If the conventional system adopts the same date rate as in the STC-based system,

how does this affect the error performance? To answer this question, we compare the BER

of the cooperative systems which have the same or similar data rate in Figs. 5-19 and 5-20






















S"10i "t. .. .
10-2




-2
wU 10



S L=2, DQPSK

S10. L=4, D16PSK ... . ...
,,- .......... L=4, D 16PS K :: : : : : : : :: : : : : : : ::: : : : : : ::4I-
--- L=2, DUSTC
8 L=3, DUSTC
+ L=4, DUSTC

10-4 i i
0 5 10 15 20 25 30
SNR


Figure 5-17. BER comparison between the conventional systems and STC-based systems with
same modulation size (DF vs DFSTC, SNR=Br,s = d,r).


5 10 15
SNR


20 25 30


Figure 5-18. BER comparison between the conventional systems and STC-based systems with
same modulation size (AF vs AFSTC, SNR ,r, = yd,r).


100



10-1



10-2



S10-3



10-4



10-5



10-6



















> '''-',,



-- L=2, DQPSK(2/3)
L=3, DQPSK(1/2) . ''* ,
10- .... L=4, D8PSK(3/5). .. .. -...
.. . ..L=2, DUSTC(1/2) : : :'
: e L=3, DUSTC(1/2). : : : :: : : : : : : :.
S*+ L=4, DUSTC(1/2)

10-4
0 5 10 15 20 25 30
SNR


Figure 5-19. BER comparison between the conventional systems and STC-based systems with
equal/similar transmission rate (DF vs DFSTC, SNR=7r,s = 7d,r).


for the DF/DFSTC and AF/AFSTC protocols, respectively. To keep equal data rate, we

use DQPSK when L=2 and 3, and D8PSK for L=4. These modulation sizes correspond

to the data rate of 2/3, 1/2, and 3/5 (bits/time slot) for L = 2, 3 and 4, respectively. The

STC-based system a-lv-- keeps data rate as 1/2. The figures show that the STC-based

system does not necessarily outperform the conventional system. The conventional system

may provide better performance by properly choosing the modulation size. From Fig. 5-17

to Fig. 5-20, we can see that the modulation size dramatically affects the performance of

the conventional system.

Summarizing, we compared two cooperative systems with respect to the transmission

rate and BER. Our results show that the STC-based system provides higher transmission

rate compared with the conventional system. It is also shown that the conventional system

may provide better performance than the STC-based one by keeping the same/higher

data rate. In addition, the BER comparisons reveal that the modulation size is critical to

determine the error performance of conventional system.








































10 2




S10-32.



10-4 L=2, DQPSK(2/3)
L=3, DQPSK(1/2)
S : : L=4, D8PSK(3/5): : ::
S L=2, DUSTC(1/2)
e L=3, DUSTC(1/2) !
*+ L=4, DUSTC(1/2) : : : :

10-6----
0 5 10 15 20 25 30
SNR


Figure 5-20. BER comparison between the conventional systems and STC-based systems with
equal/similar transmission rate (AF vs AFSTC, SNR-,,, = yd,r).









CHAPTER 6
CONCLUSIONS AND FUTURE WORK

6.1 Conclusions

In this research, we investigated cooperative networks with an arbitrary number of

rel -i- employing differential (de)modulation. Two conventional relaying protocols, i.e.,

decode-and-forward (DF) and amplify-and-forward (AF), and two distributed space-time

coding (DSTC) protocols, i.e., decode-and-forward space-time coding (DFSTC) and

AFSTC, are considered for the relay systems. We analyzed the error performances of

cooperative networks, and based on these, we developed the optimum resource allocation

which minimizes the error performance.

We derived the upper bound of symbol error rate (SER) for the DF protocol,

approximated SER for the AF protocol, and the upper bound of the codeword error

rate (CER) for STC-based systems. The DF-based (DF and DFSTC) protocols showed

an unbalanced error performance depending on the relay locations, and our analytical

results and simulations sI .-.- --I that it is mainly the source-relay links that determine

the overall system performance. For the AF protocol, the SER was derived for two cases:

the system with no direct link and the system with a direct link. This SER expression had

a general but very simple form. The AFSTC protocol showed the same trends as the AF

protocol with no direct link.

Based on the error performance, the average SER and CER for the conventional

DF/AF protocols and the DSTC protocol, respectively, we explored the resource allocation

as a two-dimension problem. We showed that: i) given the source, relay and destination

locations, the average error rate can be minimized by appropriately distributing the

prescribed total energy per symbol across the source and the r-e1 i-: ii) given the source

and relay energy levels, there is an optimum relay location which minimizes the average

error rate; and iii) given the source, relay and destination locations, and total transmit

energy, the minimum error rate can be achieved by the joint energy-location optimization.









With DF and DFSTC protocols, for any L values, the optimum energy allocation at

the source increases as the relay moves toward the destination. This is the same as the

AF protocol with no direct link and the AFSTC protocol. However, a uniform energy

allocation is optimum for the AF relay system with a direct link when the rel -1- are

located near to the source. Our analysis reveals that the optimum energy allocation

depends on a direct link, but the optimum relay location does not. For all cases, the

optimum relay locations move towards the destination as more energy is assigned to the

source, and the optimum relay locations show more flatness than the optimum energy

allocations.

Our simulations and numerical examples confirm that both the energy and location

optimizations provide considerable error performance advantages. We have observed the

following results for the DF and DFSTC protocols.

(1) Without energy optimization, performance degradation is observed when more

re -1 i- are included in the system especially when the re1- i,- are located close to the

destination node.

(2) For all cases, the optimized systems universally outperform the unoptimized ones.

(3) The location optimization is more critical than the energy optimization. In other

words, the differential relay system with uniform energy distribution can achieve

near-optimum error performance by appropriately choosing the relay location; while

a system with re- i.- sitting at the midpoint between the source and the destination

cannot approach the optimum error performance even with optimized energy

distribution.

For the AF and AFSTC protocols, we have observed following results.

(1) For the system based on AF protocol with a direct link, the error rates of the system

with a uniform energy allocation and optimum energy allocation are almost identical

when the rel-i ,- are located near to the source, since the uniform energy allocation is

optimum in such cases.









(2) For all cases, the optimized systems universally outperform the unoptimized ones.

(3) Energy and location optimizations are equally important, since minimum error

performance cannot be achievable without either of them.

We have also shown that the minimum error performance can be achieved by the joint

energy-location optimization.

In addition, we compared the cooperative systems with different protocols by

considering both the energy distribution and the relay location selection. It is interesting

that the optimum location and energy allocations are very different in the two protocols.

In general, we can achieve the minimum error performance of the cooperative networks by

locating relays closer to the source node with a less amount of the source transmit energy

for the DF-based system, and by locating r1el i- closer to the destination node with a large

amount of the source transmit energy for the AF-based system.

Finally, we compared the conventional system with STC-based system. In general,

STC-based system can support higher transmission rates. However, the conventional

system can achieve comparable performance in comparison with STC-based one by

choosing appropriate modulation size.

6.2 Future Work

In our research, we consider the conventional cooperative networks which are the

distributed counterpart of standard differential scheme for a single-input single-output

(SISO) channel, and STC-based cooperative networks which are the distributed counterpart

of the differential space-time codes. We showed that the cooperative networks have

different properties in their performance and optimum resource allocation depending on

the relaying protocol.

Recently, the hybrid scheme which selects the advantages of DF and AF protocols

is -i--.-. -I. 1 for coherent system [8, 11, 33]. It will be interesting to develop a hybrid

cooperative system and analyze its performance and resource optimizing schemes. It will

be also valuable to consider multihop cooperative networks which can support reliable









communications (see [34, 41, 69]). Finally, in this research, we mainly focused on physical

l v-r analysis. It will be helpful to consider higher l v-r issues, i.e, medium access control

(! AC) or network lv-. -i~, for improving the overall networking performance.









LIST OF REFERENCES


[1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formu-
las, Graphs, and Mathematical Tables. Dover, New York, 1972.

[2] P. A. Anghel and M. Kaveh, "Exact symbol error probability of a cooperative
network in a rayleigh-fading environment," IEEE Trans. on Wireless Communica-
tions, vol. 3, no. 5, pp. 1416-1421, September 2004.

[3] P. A. Anghel and M. Kaveh, "On the diversity of cooperative systems," in Proc. of
Intl. Conf. on Acoustics, Speech and S.:g,,Ll Processing, vol. 4, Montreal, Quebec,
Canada, M .i 17-21, 2004, pp. 577-580.

[4] P. A. Anghel and M. Kaveh, "On the performance of distributed space-time coding
systems with one and two non-regeneratvie relays," IEEE Trans. on Wireless Com-
munications, vol. 5, no. 3, pp. 682-692, March 2006.

[5] P. A. Anghel, M. Kaveh, and Z. Q. Luo, "Optimal rml i, I power allocation in
interfernce-free non-regenerative cooperative systems," in Proc. of S.':,,rl Proc.
Workshop on Advances in Wireless Communications, Lisbon, Portugal, July 11-14,
2004, pp. 21-25.

[6] R. Annavajjala, P. C. Cosman, and L. B. Milstein, "On the performance of optimum
noncoherent amplify-and-forward reception for cooperative diversity," in Proc. of
MILCOM Conf., vol. 5, Atlantic city, NJ, oct 17-20, 2005, pp. 3280-3288.

[7] R. Annavajjala, P. C. Cosman, and L. B. Milstein, "Statistical channel
knowledge-based optimum power allocation for relayig protocols in the high snr
regime," IEEE Journal on Selected Areas in Communications, vol. 25, no. 2, pp.
292-305, February 2007.

[8] X. Bao and J. Li, "Decode-amplify-forward (daf): A new class of forwarding strategy
for wireless relay channels," in Proc. of S.:,1i'l Proc. Workshop on Advances in
Wireless Communications, New York, NY, June 5-8, 2005, pp. 816-820.

[9] A. Bletsas and A. Lippman, Ii,!, i i i, i iiig cooperative diversity antenna arrays with
commodity hardware," IEEE Communications M rl. ..:,. vol. 44, no. 12, pp. 33-49,
December 2006.

[10] J. Bc.V r, D. D. Falconer, and H. Yanikomeroglu, '\!,ull !i'p diversity in wireless
relaying channels," IEEE Trans. on Communications, vol. 52, no. 10, pp. 1820-1830,
October 2004.

[11] B. Can, H. Yomo, and E. D. Carvalho, "Hybrid forwarding scheme for cooperative
relaying in ofdm based networks," in Proc. of International Conf. on Communica-
tions, vol. 10, Istanbul, Turkey, June 11-15, 2006, pp. 4520-4525.









[12] D. C'!, i, and J. N. Laneman, "Cooperative diversity for wireless fading channels
without channel state information," in Proc. of Asilomar Conf. on S.:i,.l. S',
and Computers, Monterey, CA, November 7-10, 2004, pp. 1307-1312.

[13] D. C'!, i1 and J. N. Laneman, "Modulation and demodulation for cooperative diversity
in wireless systems," IEEE Trans. on Wireless Communications, vol. 5, no. 7, pp.
1785-1794, July 2006.

[14] W. Cho, R. Cao, and L. Yang, "Optimum energy allocation in cooperative networks:
A comparative study," in Proc. of MILCOM Conf., Orlando, FL, Oct 29-31, 2007.

[15] W. Cho, R. Cao, and L. Yang, "Optimum resource allocation for amplify-and-forward
relay networks with differential modulation," IEEE Trans. on S.:,i'..l Processing, June
2007 (submitted).

[16] W. Cho and L. Yang, "Differential modulation schemes for cooperative diversity,"
in Proc. of IEEE International Conference on Networking, Sensing and Control, Ft.
Lauderdale, FL, April 23-25, 2006, pp. 813-818.

[17] W. Cho and L. Yang, "Distributed differential schemes for cooperative wireless
networks," in Proc. of Intl. Conf. on Acoustics, Speech and S.: ji..l Processing, vol. 4,
Toulouse, France, May 15-19, 2006, pp. 61-64.

[18] W. Cho and L. Yang, "Optimum energy allocation for cooperative networks with
differential modulation," in Proc. of MILCOM Conf., Washington, DC, Oct 23-25,
2006.

[19] W. Cho and L. Yang, "Joint energy and location optimization for relay networks
with differential modulation," in Proc. of Intl. Conf. on Acoustics, Speech and S.:,,..ld
Processing, vol. 3, Honolulu, Hawaii, apr 15-20, 2007, pp. 153-156.

[20] W. Cho and L. Yang, "Optimum resource allocation for relay networks with
differential modulation," IEEE Trans. on Communications, 2007 (To appear).

[21] W. Cho and L. Yang, "Resource allocation for amplify-and-forward relay networks
with differential modulation," in Proc. of Global Telecommunications Conf.,
Washington, D.C., November 26-30, 2007 (To appear).

[22] X. Deng and A. M. Haimovich, "Power allocation for cooperative relaying in wireless
networks," IEEE Communications Letters, vol. 9, no. 11, pp. 994-996, November
2005.

[23] M. Dohler, A. Gkelias, and H. Aghvami, "Resource allocation for fdma-based
regenerative multihop links," IEEE Trans. on Wireless Communications, vol. 3,
no. 6, pp. 1989-1993, November 2004.

[24] M. Dohler, A. Gkelias, and H. Aghvami, "A resource allocation strategy for
distributed mimo multi-hop communication systems," IEEE Communications Letters,
vol. 8, no. 2, pp. 99-101, February 2004.









[25] I. S. Gradshteyn and I. M. Ryzhik, Table of Integradls, Series, and Products, 6th ed.
Academic Press, 2000.

[26] M. O. Hasna and M. Alouini, "End-to-end performance of transmsiion systems with
rel ,i over rayleigh-fading channels," IEEE Trans. on Wireless Communications,
vol. 2, no. 6, pp. 1126-1131, November 2003.

[27] M. O. Hasna and M. Alouini, "Optimal power allocation for r~l '1iv transmissions
over rayleigh-fading channels," IEEE Trans. on Wireless Communications, vol. 3,
no. 6, pp. 1999-2004, November 2004.

[28] M. O. Hasna and M. Alouini, "A performance study of dual-hop transmissions with
fixed gain relays," IEEE Trans. on Wireless Communications, vol. 3, no. 6, pp.
1963-1968, November 2004.

[29] T. Himsoon, W. Su, and K. J. R. Liu, "Differential transmission for
amplify-and-forward cooperative communications," IEEE S.:,ii/l Processing Let-
ters, vol. 12, no. 9, pp. 597-600, September 2005.

[30] T. Himsoon, W. Su, and K. J. R. Liu, "Differential modulation for multi-node
amplify-and-forward wireless relay networks," in Proc. of Wireless Communications
and Networking Conf., vol. 2, Las Vegas, NV, April 3-6, 2006, pp. 1195-1200.

[31] B. M. Hochwald and W. Sweldens, "Differential unitary space-time modulation,"
IEEE Trans. on Communications, vol. 48, no. 12, pp. 2041-2052, December 2000.

[32] B. L. Hughes, "Differential space-time modulation," IEEE Trans. on Information
The(. .; vol. 46, no. 7, pp. 2567-2578, November 2000.

[33] A. Kannan and J. R. Barry, "Space-divison relay: a high-rate cooperation scheme
for fading multiple-access channels," in Proc. of Global Telecommunications Conf.,
Washington, D.C., November 26-30, 2007 (To appear).

[34] G. K. Karagiannidis, "Performance bounds of multihop wireless communications with
blind rel i.- over generalized fadin channels," IEEE Trans. on Wireless Communica-
tions, vol. 5, no. 3, pp. 498-503, March 2006.

[35] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, "Cooperative diversity in wireless
networks: Efficient protocols and outage behavior," IEEE Trans. on Information
The(.':; vol. 50, no. 12, pp. 3062-3080, December 2004.

[36] J. N. Laneman and G. W. Wornell, "Energy-efficient antenna sharing and relaying
for wireless networks," in Proc. of Wireless Communications and Networking Conf.,
vol. 1, Chicago, IL, September 23-28, 2000, pp. 7-12.

[37] J. N. Laneman and G. W. Wornell, "Distributed space-time-coded protocols for
exploiting cooperative diverstiy in wireless networks," IEEE Trans. on Information
The(.-;, vol. 49, no. 10, pp. 2415-2425, October 2003.









[38] H. Li and Q. Zhao, "Distributed modulation for cooperative wireless
communications," IEEE S.:..:'il Processing _I.ii..:,. vol. 23, no. 5, pp. 30-36,
September 2006.

[39] Y. Liang and V. V. Veeravalli, "Gaussian orthogonal relay channels: Optimal resource
allocation and capacity," IEEE Trans. on Information Th(. -,; vol. 51, no. 9, pp.
3284-3289, September 2005.

[40] I. Maric and R. D. Yates, ,. i.-- v-ding strategies for gaussian rarallel-relay networks,"
in Proc. of Conf. on Info. Sciences and Si-l1' i- Princeton, NJ, March 17-19, 2004.

[41] T. Miyano, H. Murata, and K. Araki, "Space time coded cooperative relaying
technique for multihop communications," in Proc. of Vehicular T .- I, i. i/; Conf.,
vol. 7, Los Angeles, CA, September 26-29, 2004, pp. 5140-5144.

[42] R. Pabst, B. H. Walke, D. C. Schultz, R. Herhold, H. Yanikomeroglu, S. Mukherjee,
H. Viswanathan, M. Lott, W. Zirwas, M. Dohler, H. Aghvami, D. D. Falconer, and
G. P. Gettweis, "Relay-based deployment concepts for wireless and mobile broadband
radio," IEEE Communications M''i .,, vol. 42, no. 9, pp. 80-89, September 2004.

[43] C. S. Patel, G. L. Stiiber, and T. G. Pratt, "Statistical properties of amplify and
forward relay fading channel," IEEE Trans. on Vehicular Tech., vol. 55, no. 1, pp.
1-9, January 2006.

[44] J. Proakis, D.:.l:l.d Communications, 4th ed. McGraw-Hill, New York, 2001.

[45] A. Ribeiro, X. Cai, and G. B. Giannakis, "Symbol error probabilities for general
cooperative links," IEEE Trans. on Wireless Communications, vol. 4, no. 3, pp.
1264-1273, A li- 2005.

[46] A. Scaglione, D. L. Goeckel, and J. N. Laneman, "Cooperative communications in
mobile ad hoc networks," IEEE S.:l,.rl Processing MI rr..:,. vol. 23, no. 5, pp. 18-29,
September 2006.

[47] G. Scutari and S. Barbarossa, "Distributed space-time coding for regeneratvie relay
networks," IEEE Trans. on Wireless Communications, vol. 4, no. 5, pp. 2387-2399,
September 2005.

[48] A. Sendonaris, E. Erkip, and B. A .. i i.i "User cooperation diversity, part I: system
description," IEEE Trans. on Communications, vol. 51, no. 11, pp. 1927-1938,
November 2003.

[49] A. Sendonaris, E. Erkip, and B. A .. l-i i.i "User cooperation diversity, part II:
implementation aspect and performance ,i1 ,iev-i- IEEE Trans. on Communications,
vol. 51, no. 11, pp. 1939-1948, November 2003.

[50] A. B. H. Shin and M. Z. Win, "Outage-optimal cooperative communications with
regenerative rel-,," in Proc. of Conf. on Info. Sciences and Sl'.-. ii- Princenton, NJ,
Mar. 22-24, 2006, pp. 632-637.









[51] M. K. Simon and M. S. Alouini, "A unified approach to the probability of error
for noncoherent and differentially coherent modulations over generalized fading
channels," IEEE Trans. on Communications, vol. 46, no. 12, pp. 1625-1638,
December 1998.

[52] M. K. Simon and M. S. Alouini, D.:,:l,1 Communication over Fading C'lh,,,.
2nd ed. Wiley, 2004.

[53] V. Stankovi<, A. Host-Madsen, and Z. Xiong, "Cooperative diversity for wireless
ad hoc networks," IEEE S.:,j.irl Processing M'rLj' ..:,. vol. 23, no. 5, pp. 37-49,
September 2006.

[54] A. Stefanov and E. Erkip, "Cooperative space-time coding for wireless networks,"
IEEE Trans. on Communications, vol. 53, no. 11, pp. 1804-1809, November 2005.

[55] G. L. Stiiber, Principles of Mobile Communication, 2nd ed. Springer, 2001.

[56] K. T and B. S. R ii in "Partially-cohernet distributed space-time codes with
differential encoder and decoder," IEEE Journal on Selected Areas in Communi-
cations, vol. 25, no. 2, pp. 426-433, February 2007.

[57] P. Tarasak, H. Minn, and V. K. Bhargava, "Differential modulation for two-user
cooperative diversity systems," IEEE Journal on Selected Areas in Communications,
vol. 23, no. 9, pp. 1891-1900, September 2005.

[58] V. Tarokh and H. Jafarkhani, "A differential detection scheme for transmit diversity,"
IEEE Journal on Selected Areas in Communications, vol. 18, no. 7, pp. 1169-1174,
July 2000.

[59] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, "Space-time block codes from
orthogonal designs," IEEE Trans. on In f.'i i,,l.:.n The(..,; vol. 45, no. 5, pp.
1456-1467, July 1999.

[60] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, "Space-time block coding for
wireless communications: performance results," IEEE Journal on Selected Areas in
Communications, vol. 17, no. 3, pp. 451-460, March 1999.

[61] V. Tarokh, N. Seshadri, and A. R. Calderbank, "Space-time codes for high data rate
wireless communication: Performance criterion and code construction," IEEE Trans.
on Information Th(..,i vol. 44, no. 2, pp. 744-765, March 1998.

[62] M. Uysal, O. Canpolat, and M. M. Fareed, "Asymptotic performance analysis of
distributed space-time codes," IEEE Communications Letters, vol. 10, no. 11, pp.
775-777, November 2006.

[63] G. Wang, Y. Z!i ii.- and M. Amin, "Differential distributed space-time modulation
for cooperative networks," IEEE Trans. on Wireless Communications, vol. 5, no. 11,
pp. 3097-3180, November 2006.









[64] T. Wang, Y. Yao, and G. B. Giannakis, "Non-coherent distributed space-time
processing for multiuser cooperative transmissions," in Proc. of Global Telecommuni-
cations Conf., vol. 6, St. Louis, MO, November 28-December 2, 2005, pp. 3738-3742.

[65] Y. Yao, X. Cai, and G. B. Giannakis, "On energy efficiency and optimum resource
allocation of relay transmissions in the low-power regime," IEEE Trans. on Wireless
Communications, vol. 4, no. 6, pp. 2917-2927, November 2005.

[66] S. Yiu, R. Schober, and L. Lampe, "Performance and design of space-time coding in
fading channels," IEEE Trans. on Communications, vol. 54, no. 7, pp. 1195-1206,
July 2006.

[67] M. Yu, J. Li, and H. q ,li ilpour, "Amplify-forward and decode-forward: The impact
of location and capacity contour," in Proc. of MILCOM Conf., vol. 3, Atlantic city,
NJ, October 17-20, 2005, pp. 1609-1615.

[68] J. Yuan, Z. C'!. i1 B. Vucetic, and W. Firmanto, "Performance and design of
space-time coding in fading channels," IEEE Trans. on Communications, vol. 51,
no. 12, pp. 1991-1996, December 2003.

[69] J. Zhang and T. M. Lok, "Performance comparison of conventional and cooperaitve
multihop transmission," in Proc. of Wireless Communications and Networking Conf.,
vol. 2, Las Vegas, NV, April 3-6, 2006, pp. 897-901.

[70] Y. Zhli ii, "Differential modulation schemes for decode-and-forward cooperative
diversity," in Proc. of Intl. Conf. on Acoustics, Speech and S.':,,',l Processing, vol. 4,
Philadelphia, PA, March 19-23, 2005, pp. 917-920.

[71] Q. Zhao and H. Li, "Performance of a differential modulation scheme with wireless
reliv-, in rayleigh fading channels," in Proc. of Asilomar Conf. on S.:j,'l, S, 1.
and Computers, vol. 1, Monterey, CA, November 7-10, 2004, pp. 1198-1202.

[72] Q. Zhao and H. Li, "Performance of differential modulation with wireless rel i.,- in
rayleigh fading channels," IEEE Communications Letters, vol. 9, no. 4, pp. 343-345,
April 2005.

[73] Q. Zhao and H. Li, "Differential modulation for cooperative wireless systems," IEEE
Trans. on S.:j.i',1 Processing, vol. 55, no. 5, pp. 2273-2283, May 2007.









BIOGRAPHICAL SKETCH

Woong Cho was born in Tongyoung, South Korea. He received his B.S. degree in

electronics engineering from University of Ulsan, Ulsan, South Korea, in 1997 and his M.S.

degree in electronic communications engineering from Hanyang University, Seoul, South

Korea, in 1999. From March 1999 to August 2000, we was a research engineer in division

of mobile telecommunication at Hyundai Electronics. He received his M.S. degree form

electrical engineering from University of Southern California, Los Angeles, CA, in 2003.

Since August 2003, he has been a Ph.D student in electrical and computer engineering

at University of Florida, Gainesville, FL. His research interests include communications,

signal processing, and networking. He received his Ph.D degree in 2007.





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Iwanttoexpressmytremendousgratitudetomysupervisor,Dr.LiuqingYang,forhertirelesseortaswellasforprovidingmewithinsights,inspiration,andencouragementwithoutwhichIcouldnothaveperformedthisresearch.Ideeplythankherforallofthepatientguidance,invaluableadvice,andnumerousdiscussionswehavesharedthroughoutmygraduatestudies.Iwouldliketothankallthemembersofmyadvisorycommittee,Dr.JenshanLin,Dr.TaoLiandDr.ShigangChen,fortheirvaluabletimeandenergyinservingonmysupervisorycommittee.IwouldalsoliketothankRuiCao,myfriendandcolleagueatSignalprocessing,Communications,andNetworking(SCaN)group,forthepricelessdiscussionsweshared,whichgeneratedmanyideasforthisresearch.IwishtoextendmysincerethankstoallthemembersoftheSCaNgroup,HuilinXu,FengzhongQu,DongliangDuan,andWenshuZhang,fortheircompanionshipandsupportthroughoutourtimetogether.Asalways,Iwanttothanktomyparentsandparents-in-law,fortheirunyieldingsupportandlove.TheirencouragementandunderstandingthroughmystudyingperiodshavemeantmorethanIcaneverexpress.Last,Iwouldliketoexpressmygreatestthanksandadorationtomylovingwife.Iwanttothankherforsupportingandunderstandingmeininnumerableways,particularlyduringallourtimetogetherintheUnitedStates,andthroughoutmyPh.Dstudies. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFFIGURES .................................... 7 ABSTRACT ........................................ 10 CHAPTER 1INTRODUCTION .................................. 12 1.1CooperativeNetworks ............................. 12 1.2Motivation .................................... 13 2SYSTEMMODEL .................................. 17 2.1RelayingProtocolsandChannelModeling .................. 17 2.1.1Decode-and-Forward(DF) ....................... 18 2.1.2Amplify-and-Forward(AF) ....................... 19 2.1.3DistributedSpace-TimeCoding(DSTC) ............... 19 2.2DierentialDemodulationandDecisionRules ................ 21 2.2.1DFProtocol ............................... 21 2.2.2AFProtocol ............................... 23 2.2.3DSTCProtocol ............................. 24 3PERFORMANCEANALYSIS ............................ 26 3.1SERforDFProtocol .............................. 26 3.2SERforAFProtocol .............................. 30 3.3CERforDSTCProtocol ............................ 38 4OPTIMUMRESOURCEALLOCATION ...................... 45 4.1ConvexityofSER ................................ 46 4.2EnergyOptimization .............................. 50 4.3RelayLocationOptimization .......................... 63 4.4JointEnergyandLocationOptimization ................... 70 5SIMULATIONSANDDISCUSSIONS ....................... 75 5.1BenetsofEnergyandLocationOptimizations ............... 75 5.2BenetsofJointOptimization ......................... 83 5.3ConventionalSystemvsSTC-basedSystem .................. 87 6CONCLUSIONSANDFUTUREWORK ...................... 92 6.1Conclusions ................................... 92 6.2FutureWork ................................... 94 5

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................................. 96 BIOGRAPHICALSKETCH ................................ 102 6

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Figure page 1-1Simplecooperativenetwork. ............................. 13 2-1Setupofthecooperativenetwork. .......................... 17 3-1SERatdierentr;svalues(DF,L=2;M=2). 29 3-2SERatdierentd;rvalues(DF,L=2;M=2). 29 3-3SERboundversusd;randr;s(DF,L=2;M=2). 30 3-4SERcomparisonbetweenapproximationandsimulation(AFND,SNR(dB)=rk;s=d;rk). 36 3-5SERcomparisonbetweenapproximationandsimulation(AFDL,SNR(dB)=d;s=rk;s=d;rk). 37 3-6SERcomparisonbetweencoherentsystemanddierentialsystem(AFDL,SNR=rk;s=d;rk). 37 3-7TheCERfortheDFSTCprotocol(L=1;2;and3;SNR=rk;s=d;rk). 42 3-8TheCERfortheAFSTCprotocol(L=1;2;and3;SNR=rk;s=d;rk). 42 3-9TheeectofunbalancedlinkSNRfortheDFSTCprotocol(L=1;2;and3). 43 3-10TheeectofunbalancedlinkSNRfortheAFSTCprotocol(L=1;2;and3). 44 4-1Networktopologies:(a)Ellipsecase;(b)Linecase. 46 4-2Exactandapproximateoptimumenergyallocationswithdierentpathlossexpo-nents(DF,L=1;=10dB). 53 4-3SERversusenergyallocationatthegivenrelaylocationDs;r(DF,L=2,=10dB,D=1,=4). 55 4-4ComparisonofoptimalenergyallocationbetweenthenumericalsearchandsimulatedresultsatvariousLvalues(DF,=10dB,=4). 55 4-5Comparisonofnormalizedoptimumenergyallocationatdierentvalues(DF,L=1). 4-6Existenceoftheoptimumsolution(AFND,=15dB,=3,L=2). 59 4-7SERversusenergyallocationatthegivenrelaylocationDs;r(AFND,L=2,=15dB,D=1,=4). 60 4-8SERversusenergyallocationatthegivenrelaylocationDs;r(AFDL,L=2,=15dB,D=1,=4). 60 7

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62 4-10Optimumenergyallocation(DFSTCandAFSTC,D=1:2;Ds;d=1,=15dB,=4). 4-11Optimumlocationofrelays(DF,=10dBandL=1). 65 4-12SERversusrelaylocationdistributionatthegivenenergyallocations=(DF,L=2,=10dB,D=1=4). 66 4-13SERversusrelaylocationdistributionatthegivenenergyallocations=(AFND,L=2,=15dB,D=1,=4). 67 4-14SERversusrelaylocationdistributionatthegivenenergyallocations=(AFDL,L=2,=15dB,D=1,=4). 68 4-15Optimumrelaylocation(AF,NDandDL,D=Ds;d=1,=30dB,=4). 69 4-16Optimumrelaylocation(DFSTCandAFSTC,D=1:2;Ds;d=1,=15dB,=4). 69 4-17Iterativesearch:owchart. 71 4-18Performancesurfaceversuss=andDs;r(DF,=10dB,=4;L=3,DBPSK). 72 4-19Performancesurfaceversuss=andDs;r(AF,ND,=15dB,=4;L=3,DBPSK). 4-20Performancesurfaceversuss=andDs;r(DFSTC,=15dB,=4;L=2). 73 4-21Performancesurfaceversuss=andDs;r(AFSTC,=15dB,=4;L=2). 74 5-1SERcomparisonbetweenrelaysystemswithandwithoutenergyoptimization(DF,=15dB,D=1:2,Ds;d=1,=4). 76 5-2SERcomparisonbetweenrelaysystemswithandwithoutrelaylocationoptimization(DF,=15dB,D=1:2,Ds;d=1,=4). 76 5-3SERcomparisonbetweenrelaysystemswithandwithoutenergyoptimization(AFND,=15dB,D=1:2,Ds;d=1,=4). 78 5-4SERcomparisonbetweenrelaysystemswithandwithoutenergyoptimization(AFDL,=15dB,D=1:2,Ds;d=1,=4). 78 5-5SERcomparisonbetweenrelaysystemswithandwithoutrelaylocationoptimization(AFND,=15dB,D=1:2,Ds;d=1,=4). 79 5-6SERcomparisonbetweenrelaysystemswithandwithoutrelaylocationoptimization(AFDL,=15dB,D=1:2,Ds;d=1,=4). 80 5-7CERcomparisonbetweenrelaysystemswithandwithoutenergyoptimization(DF-STC,=15dBand25dB,D=1:2,Ds;d=1,=4). 81 8

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81 5-9CERcomparisonbetweenrelaysystemswithandwithoutenergyoptimization(AF-STC,=15dBand25dB,D=1:2,Ds;d=1,=4). 82 5-10CERcomparisonbetweenrelaysystemswithandwithoutrelaylocationoptimization(AFSTC,=15dBand25dB,D=1:2,Ds;d=1,=4). 82 5-11TheSERcontourversusDs;rands=(AFND,Ds;d=D=1,=15dB,L=2). 84 5-12TheSERcontourversusDs;rands=(AFDL,Ds;d=D=1,=15dB,L=2). 84 5-13TheSERcontourversusDs;rands=(DF,Ds;d=D=1,=10dB,L=3). 85 5-14TheCERcontourversusDs;rands=(DFSTC,D=1:2;Ds;d=1,=15dB,L=2). 5-15TheCERcontourversusDs;rands=(AFSTC,D=1:2;Ds;d=1,=15dB,L=2). 5-16DataratecomparisonbetweentheconventionalsystemandSTC-basedsystemintermsoftherequiredtimeslotsperinformationsymbol. 87 5-17BERcomparisonbetweentheconventionalsystemsandSTC-basedsystemswithsamemodulationsize(DFvsDFSTC,SNR=r;s=d;r). 89 5-18BERcomparisonbetweentheconventionalsystemsandSTC-basedsystemswithsamemodulationsize(AFvsAFSTC,SNR=r;s=d;r). 89 5-19BERcomparisonbetweentheconventionalsystemsandSTC-basedsystemswithequal/similartransmissionrate(DFvsDFSTC,SNR=r;s=d;r). 90 5-20BERcomparisonbetweentheconventionalsystemsandSTC-basedsystemswithequal/similartransmissionrate(AFvsAFSTC,SNR=r;s=d;r). 91 9

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Inwirelesscooperativenetworks,virtualantennaarraysformedbydistributednetworknodescanprovidecooperativediversity.Obviatingchannelestimation,dierentialschemeshavelongbeenappreciatedinconventionalmulti-inputmulti-output(MIMO)communications.However,distributeddierentialschemesforgeneralcooperativenetworksetupshavenotbeenthoroughlyinvestigated.Inthisdissertation,wedevelopandanalyzedistributeddierentialschemesusingtwoconventionalrelayingprotocols,decode-and-forward(DF)andamplify-and-forward(AF),andspace-timecoding(STC)-basedrelayingprotocolwithanarbitrarynumberofrelays.Foreachprotocol,weanalyzetheerrorperformanceandconsidertheresourceallocationasatwo-dimensionaloptimizationproblem:energyoptimization,locationoptimization,andjointenergy-locationoptimization. WerstderiveanupperboundoftheerrorperformancefortheDFsystem,theapproximatederrorperformancefortheAFsystem,andanupperboundfortheSTC-basedsystematreasonablyhighSNR,respectively.Basedontheseresults,wethendeveloptheenergyoptimizationandrelaylocationoptimizationschemesthatminimizetheaveragesystemerror.Analyticalandsimulatedcomparisonsconrmthattheoptimizedsystemsprovideconsiderableimprovementoverunoptimizedones,andthattheminimumerrorcanbeachievedviathejointenergy-locationoptimization.Wecomparetheresultsofoptimizationandtheeectsofdierentrelayingprotocolsandobtainseveral 10

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36 48 49 ].Thankstotheseadvantages,cooperativenetworkscanbeappliedinvariousscenariostoenhancethenetworkperformance.Forexample,wirelesspersonalareanetwork(WPAN)andlocalareanetwork(WLAN)canextendtheircoverageareasbyusingrelaynodes.Inhomeenvironments,cooperativenetworkscanhelpdistributethemultimediadatafromthecentralentertainmentunittodevicesanywhereinthehouse,bymountingrelaynodestothewallorevenembeddingtheminsidethewalls.Furthermore,cooperativenetworksalsondapplicationsinintelligenttransportationsystemsincludinginter-vehicle,intra-vehicleandvehicle-to-roadcommunications,toenablereliabledistributionoftheemergencyinformationtocertaingroupsofdriversviacooperationofmultiplevehiclesontheroad. Toillustratethebasicconceptofcooperativenetworks,Fig. 1-1 representsasimplecooperativenetworkinwhichthesourcenodetransmitsasignaltothedestination 12

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Simplecooperativenetwork. nodeviatherelaynode.Thesource-relay-destinationandsource-destinationlinksarecommonlyreferredtoastherelaylinkandthedirectlink,respectively.Therearetwoconventionalrelayingprotocolswhicharewidelyconsideredintheexistingliterature.Oneisthedecode-and-forward(DF)protocolandtheotheristheamplify-and-forward(AF)protocol.WithDFrelayingprotocol,therelaynodedemodulatesthereceivedsignalfromthesourcenodeandremodulatesthatsignal.Thentherelaynodetransmitstheremodulatedsignaltothedestinationnode.WithAFrelayingprotocol,therelayampliesthesignalfromthesourceandthensimplyforwardsittothedestinationnode.Anotherprotocolisthedistributedspace-timecoding(DSTC)protocolwhichcansupporthighertransmissionrate.ThisprotocolcanalsoadoptsDFandAFdependingontherelaynodeoperation. 9 38 42 46 53 ].Manystudieshavebeencarriedouttoanalyzetheperformanceofcooperativenetworks.Theoutageprobabilityofcooperativenetworksfromtheinformationtheoreticperspectiveispresentedin[ 35 ].Theerrorrateandoutageprobabilityofacooperativenetworkareanalyzedin[ 26 ]withoutconsideringthedirectlink.In[ 28 ],alaterstudybythesameauthors,theyconsiderbothrelayanddirectlinkswithaxedgainattherelay.In[ 10 ],themultihoprelaytransmissionisconsidered,andtheexactsymbolerrorrateisderivedin[ 3 ].Amoregeneralcaseoftherelaynetworksisdevelopedin[ 45 ].In[ 4 37 47 54 62 ],theperformanceofSTC-basedcooperativenetworksisconsidered.Alloftheseworkoncooperativenetworksfocusesoncoherent 13

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AccurateestimationoftheCSI,however,caninduceconsiderablecommunicationoverheadandtransceivercomplexity,whichincreaseswiththenumberofrelaynodesemployed.Inaddition,CSIestimationmaynotbefeasiblewhenthechannelisrapidlytime-varying.Tobypasschannelestimation,cooperativenetworksobviatingCSIhavebeenrecentlyintroduced.Thesecooperativesystemsrelyonnoncoherentordierentialmodulations,includingconventionalfrequency-shiftkeying(FSK)anddierentialphase-shiftkeying(DPSK)aswellasSTC-basedones.In[ 12 16 ],usingnoncoherentanddierentialmodulations,itisshownthattheloglikelihoodratiocanbecombinedbycapturingthedetectionerrorattherelaynodesaccordingtotheso-termedtransitionprobabilityifthepartialCSIisknownattherelaysanddestination.Theperformanceofasinglerelaysystemwithnoncoherentanddierentialmodulationsisconsideredin[ 6 13 ]and[ 29 30 57 71 { 73 ],respectively.In[ 16 17 56 63 64 70 ],adistributedSTCsystemforcooperativenetworksisintroducedbyusingdierentialornoncoherentschemes. Toimprovetheerrorperformanceandtoenhancetheenergyeciencyofcooperativenetworks,optimumresourceallocationrecentlyemergedasanimportantproblemattractingincreasingresearchinterests(seee.g.,[ 5 22 27 39 65 ]).Theseworkisbasedondierentrelayingprotocols(amplify-and-forward,decode-and-forwardandblockMarkovcoding),undervariousoptimizationcriteria(signal-to-noiseratio(SNR)gain,SNRoutageprobability,energyeciencyandcapacity),andwithdierentlevelsofCSI(instantaneousCSIandchannelstatistics).However,allofthemonlyconsiderthepowerallocationandmostlyfocusonasingle-relaysetup.In[ 7 14 18 30 ],optimumpowerallocationformultiplerelaylinksisdevelopedundervariousrelayingprotocols.[ 40 ]and[ 23 24 ],respectively,considertheoptimumenergyandbandwidthallocationinGaussianchannelandmultihopsystem,while[ 50 ]introducestheopportunisticrelayselection. 14

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38 67 ].However,[ 38 ]neglectsthepathlosseectwhichiscloselyrelatedtotherelaylocationsuchthatthelocationoptimizationproblemiserroneouslyformulatedasanenergyallocationproblem.In[ 67 ],itisshownthatgivenuniformenergyallocationthereisanoptimumrelaylocationwhichprovidestheoptimumperformance.However,theyonlyconsidertheoptimumlocationoftherelayswithoutappropriateenergyallocation.WithDFandAFprotocols,thejointenergyandlocationoptimizationforthecooperativenetworkisintroducedin[ 15 19 { 21 ]. Inthisresearch,wedevelopcooperativenetworkswithanarbitrarynumberofrelaysbyemployingDF,AF,andDSTCprotocols.Equallyattractiveisthatouranalysisistailoredforrelaysystemswithdierentialmodulation,whichisknowntoreducethereceivercomplexitybybypassingchannelestimation.NoticethattheDFandAFprotocolsgeneralizethestandarddierentialmodulationtoadistributedscenariowithanarbitrarynumberofrelays.TheDSTCprotocolreliesonanincreasedlevelofusercooperationviathedistributedcounterpartofthedierentialspace-timecodes[ 31 32 ];thatis,foreachdatablock,aspace-timecodewordencodedacrossdistributedrelaysistransmittedoveracommonrelay-destinationchannel.Usingtheserelayingprotocols,wethenderivetheerrorperformanceandtheoptimumresourceallocationofcooperativenetworks.Dierentfromexistingworksonresourceoptimization,wetackletheproblemfromtwoangles:i)Optimizingthepowerallocationacrossrelayandsourcenodesforanygivensource-relay-destinationdistances;andii)Optimizingtherelaylocationforanypowerdistributionandsource-destinationdistance.Tothebestofourknowledge,weareamongthersttoformulatethe2-dimensionaloptimizationproblem.Inaddition,wearealsothersttoconsiderthejointpower-and-relay-locationoptimizationforcooperativenetworks. Tofacilitatetheresourceoptimization,wedevelopeanalyticalexpressionsoftheerrorperformanceforvariousrelayprotocolswitharbitrarynumberofrelays.Werstderivean 15

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16

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Thesystemmodelisbasedonanetworksetupwithonesourcenodes,LrelaynodesfrkgLk=1andonedestinationnoded,asdepictedinFig. 2-1 .Eachnodeisequippedwithaswitchthatcontrolsitstransmit/receivemodetoenablehalf-duplexcommunications.Multiplexingamongthenetworknodescanbeachievedviafrequency-division,time-divisionorcode-divisiontechniques.Fornotationalconvenience,wewillconsiderthetime-divisionmultiplexing(TDM).However,thepresentedanalysisandresultsarereadilyapplicabletofrequency-divisionmultiplexing(FDM)andcode-divisionmultiplexing(CDM).Forthisstudy,werstconsidertwoconventionalrelayingprotocols,i.e.,decode-and-forward(DF)andamplify-and-forward(AF),andthenwedevelopthedistributedspace-timecoding(DSTC)protocol. Setupofthecooperativenetwork. 17

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Inordertobypasschannelestimationandtocopewithtimevariationofthewirelesschannel,dierentialmodulationisemployedatthesourcenode.Specically,withthenthphase-shiftkeying(PSK)symbolbeingassn=ej2cn=M,cn2f0;1;:::;M1g,thecorrespondingtransmittedsignalfromthesourceis: InphaseI,forbothDFandAFprotocols,theencodedsignalisbroadcastviaacommonchannel.Thereceivedsignalsatthekthrelayandthedestinationaregivenby whereEsistheenergypersymbolatthesource,andwedenotethefadingcoecientsofchannelssrkandsdduringthenthsymboldurationashrk;snCN(0;2rk;s)andhd;snCN(0;2d;s),andthecorrespondingnoisecomponentsaszrk;snCN(0;Nrk;s)andzd;snCN(0;Nd;s),respectively.Here,CN(;2)representsthecomplexGaussiandistributionwithmeanandvariance2. 2{2 ),usingthedecisionrulethatwewillpresentinthenextsubsection.TheremodulationstepiscarriedoutasinEq.( 2{1 ),butwithsnreplacedbyitsestimateandxsnreplacedby^srknandxrkn.Then,thereceivedsignalatthedestinationcorrespondingtoeachrelaynodeisgivenby 18

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whereErkistheenergypersymbolandxrkndenotesthenthtransmittedsymbolfromthekthrelay.Thefadingcoecientsoftherkdchannelsandthenoisecomponentatthedestinationarehd;rknCN(0;2d;rk)andzd;rknCN(0;Nd;rk),respectively.Ateachoftherelays,xrkncanberepresentedas whereArkistheamplicationfactor.Tomaintainaconstantaveragepowerattherelayoutput,theamplicationfactorisgivenby ThisArkisreasonableforbothdierentialandnoncoherentmodulations,sincewecanestimatethevalueof2rk;sbyaveragingthereceivedsignalswithoutknowingtheinstantaneousCSI[ 12 71 ]. 31 32 ].Forsimplicity,wewillusethediagonaldesignwiththecyclicconstructionin[ 31 ].Noticethateachdiagonalelementofthecodewordcorrespondstoastandarddierentialphase-shiftkeying(DPSK)signaling,whereitsmodulationsize 19

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DenotethenthdierentiallyencodedsignalblockfromthesourceasXsn:=Xsn1V(Qn)withXs0=IL,whereV(Qn)isanLLdiagonalunitarymatrix,Qn2f0;1;:::;M1gwithM=2L,andILisanLLidentitymatrix,whererepresentsthedatarateoftheoriginalinformationwhichwesetto1.ThematrixV(Qn)hastheformV(Qn)=VQn1with(see[ 31 ]) whereul2f0;:::;M1g;l=f1;:::;Lg.Then,thenthreceivedsignalblockattherelaysisgivenby whereEsistheenergypersymbolatthesource,Hr;sn:=diagfhr1;sn;hr2;sn;:::;hrL;sngisthechannelmatrixbetweenthesourceandrelays,andZrn:=diagfzr1;sn;zr2;sn;:::;zrL;sngisthenoisematrixattherelays.Weusediagfa1;a2;:::;aLgasadiagonalmatrixwith[a1;a2;:::;aL]onitsdiagonal.Letusdenotethen-thtransmittedsignalblockfromtherelaysasXrn,thenthecorrespondingreceivedsignalblockatthedestinationisgivenby 20

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Throughoutthisdissertation,allfadingcoecientsareassumedtobeindependent.Withoutlossofgenerality,wealsoassumethatallnoisecomponentsareindependentandidenticallydistributed(i.i.d)withNi;j=N0;i;j2fs;rk;dg.Accordingly,wecanndthereceivedinstantaneoussignal-to-noiseratio(SNR)betweenthetransmitterjandthereceiveriasi;j=jhi;jnj2Ej 2{2 )and( 2{3 )bydroppingthesuperscripts.UsingthedierentialencodinginEq.( 2{1 ),thereceivedsignalcanbere-expressedas: 21

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wherei;j2fs;rk;dg,smn=ej2m=Mandm2f0;1;:::;M1g.WeuseE[]forexpectation,()forconjugate,and
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Esp Thereceivedsignalatthedestinationcorrespondingtothesourcecanberepresentedasyd;sn=yd;sn1sn+(~zd;sn)0withyd;snCN(yd;sn1sn;2N0).Asaresult,weobtaintheLLFcorrespondingtotheLtransmittedsignalsfromtherelaysandthetransmittedsignalfromthesource,giventhatxmnistransmittedbythesource: wheresmn=ej2m=Mandm2f0;1;:::;M1g.Noticethat,althoughtheLLF,ld;rkm(yn),inEqs.( 2{12 )and( 2{15 )hasthesameform,theonein( 2{15 )isobtainedfromadierenttransmittedsignal.FortheDFprotocol,theLLFisobtainedfromthegivenrelaytransmittedsignalwhichisademodulatedsignalfromthesourcetransmittedsignal.However,theLLFfortheAFprotocolisobtainedfromthesourcetransmittedsignalwhichissimplyampliedandforwardedattherelayswithoutanydemodulation.Atthedestinationnode,these(L+1)signalscanbecombinedtoestimatethetransmittedsignalfromthesource.Usingthemultichannelcommunicationresultsin[ 44 ,Chapter.12]and 23

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30 72 ]): ^sn=ej2^m=M:^m=argmaxm2f0;1;:::;M1g"wd;sld;sm(yn)+LXk=1wd;rkld;rkm(yn)#; wherewd;sandwd;rkarecombiningweightswhicharegivenby1=N0and2=2hk;eff,respectively,anditisassumedthatthevariancesofchannelsareavailableatthedestinationnode.Interestingly,thisdecisionruleisthesameastheDFprotocolinEq.( 2{13 )exceptfortheweighingterms. 31 ],givenXsn=Xmn,is ^Qn=argmaxm2f0;1;:::;M1gkYd;rn1+Yd;rnV(m)nHk; wherekkrepresentstheFrobeniusnorm.ThisdecisionruleisthegeneralstructureforDUSTC.Dependingontherelayingprotocol,theFrobeniusnormpartcanhavedierentvalues. IntheDFSTCprotocol,thereceivedsignalattherelays,Yr;sn,isdecoded.SinceeachdiagonalentryofthecodewordXsnisaDPSKsignalandthekthrelaydemodulatesandremodulatesindependentlythecorrespondingkthentryofYr;sn,wecanre-encodeXrnusingstandarddierentialdemodulation.ThereceivedsignalblockforthegivenrelaytransmittedsignalXrn=Xm0nis whereZ0dn=ZdnZdn1V(m0)n.SinceV(m0)nisaunitarymatrix,Z0dnhastwicethevarianceofZdn.Then,givenXrn=Xm0n,wecanapplytheMLdecisionruleinEq.( 2{17 ). FortheAFSTCprotocol,eachentryofthereceivedsignalfromthesource,Yr;sn,isampliedandforwardedtothedestination.Therefore,theampliedsignalblockatthe 24

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whereA:=diagfAr1;Ar2;:::;ArLgistheamplicationmatrix,andArkistheamplicationfactorwhichwedenedinEq.( 2{6 ).Then,usingthedierentialmodulation,thereceivedsignalblockatthedestinationcanberepresentedas where~Hn=p 2{17 )givenXsn=Xmn.NoticethattheMLdecisionruleofbothrelayingprotocolshasthesameform.However,thevalueoftheFrobeniusnormisdierentdependingontheprotocols. 25

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Tofacilitateourresourceoptimization,wewillderivetheanalyticalexpressionsoftheerrorperformanceforthecooperativesystemsdescribedintheprecedingsection.Symbolerrorprobabilityofcooperativenetworkswithrelaytransmissionshasbeenderivedin[ 2 7 45 ]forcoherentdetection,andin[ 29 71 72 ]foradierentialschemewithasinglerelay,bothemployingtheAFprotocol.TheperformanceoftraditionalSTCsystemsiswellanalyzedin[ 59 { 61 68 ].ThedierentialanddistributedSTCsystemsareconsideredin[ 31 32 58 ]and[ 66 ],respectively.Alltheseexistingworkconsideredtheperformanceofcooperativesystemstosomeextent.However,theerrorperformanceofgeneralcaseswithdierentialmodulationhasnotbeenthoroughlyinvestigated.WewillconsidertheerrorrateforageneralL-relaysetupinthedistributedscenario.First,wederiveanupperboundofthesymbolerrorrate(SER)fortheDFprotocol.Then,underhighsignal-to-noiseratio(SNR)approximation,anapproximatedSERfortheAFprotocolisderived.Finally,wederiveupperboundsofthecodeworderrorrate(CER)fortheDFSTCandAFSTCprotocols. 52 ,Chapter8.2] 1p M; whereM(x)=1=(1x),8x>0,andrepresentstheaverageSNR.Inparticular,forM=2(DBPSK),Eq.( 3{1 )canbesimpliedas 2(1+rk;s): Atthedestination,thesignalsfromtheLrelaysarecombinedtomakeadecision.Conditionedonthatthesymbolsniscorrectlydemodulatedandremodulatedatallrelay 26

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44 ,AppendixC]as: M(M1)sin(=M) where=d;rk=(1+d;rk).ForDBPSK,Eq.( 3{3 )canbesimpliedas 2"1L1Xk=02kk12 UsingtheunconditionalSERPDFe;rkattherelaysandtheconditionalSERPDFe;datthedestination,weformulateanupperboundontheoverallaverageerrorperformance,namelytheunconditionalSERPDFeatthedestination,asfollows: 3{1 )and( 3{3 ),respectively,anupperboundonPDFecanbefoundas: 3{5 )providesanupperboundontheexactSERPDFe,letusstartwiththeprobabilityofcorrectdetectionPDFc=1PDFe.Countingtheeventsthatleadtothecorrectdetection,PDFccanbeobtainedas where^srknand^sdnarethesymbolestimatesformedattherelayrkandthedestinationd,respectively.TherstsummandinEq.( 3{6 )turnsouttobeQLk=1(1PDFe;rk)(1PDFe;d),whichleadstotheupperboundinEq.( 3{5 ). 27

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3{6 ),whichcorrespondstothegapbetweenthetrueSERanditsupperboundPDFe:=PDFePDFeanddeterminesthetightnessoftheerrorboundinProposition1.ForDBPSKwithasinglerelay(L=1,M=2),thisgapcanbeeasilyobtainedas: PDFe=PDFe;rPDFe;d: ForpracticalPDFe;randPDFe;dvalues(e.g.,<103),PDFeisnegligiblecomparedwithPDFe=PDFe;r+PDFe;d2PDFe;rPDFe;d.However,forL2,allpossibleerrorshavetobeconsideredforboththesrandrdlinks,whichrendersPDFeanalyticallyuntractable.Butintuitively,asLincreases,PDFealsoincreasessincethereisanincreasingchancethatdetectionerrorsattherelaynodesdonotleadtoadetectionerroratthedestinationnode.Inadditiontothiseect,theperformanceboundPDFeandthegapPDFealsodependsonthequalityofthesrandrdlinks. TheseeectsareevidentfromthesimulatedexamplesinFigs. 3-1 and 3-2 ,wherearelaynetworkwithL=2relaynodesusingDBPSKsignalingisconsideredatvariousr;sandd;rlevels.Inthesesimulatedexamples,thechannelsbetweenthesourceandallrelayshaveidenticalpowers2rk;s=2r;s,8k,whichimpliesthatrk;s=r;s,8k.Accordingly,wehavePDFe;rk=PDFe;r,8k,andPDFePDFe=1(1PDFe;r)L(1PDFe;d)fromProposition1.Likewise,theSNRsbetweenalltherelaynodesandthedestinationhavethesamevalue,d;rk=d;r;8k.InbothFigs. 3-1 and 3-2 ,theboundPecloselycapturesthedependencyofthesystemSERontheSNRlevelsr;sandd;r.Specically,wehavethefollowingobservations: 3-1 revealsthat,atanygivenvalueofr;s,thesystemSERexhibitsanerroroorasd;rincreases.Intuitively,thiserroroorcomesfromthedetectionerrorattherelays,whichheavilyreliesonthesrlinkqualityr;sandcanonlybereducedbyimposingsucientlyhighr;s. 28

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3-2 showsthat,atmedium-to-highd;rlevels,theoverallSERcanalwaysbereducedbyincreasingtheSNRofthesrlinkr;sanddoesnotexhibitanyerroroor. 29

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3-3 .Noticethatthesurfaceattensalongthed;raxis,butkeepsdescendingalongther;saxis.TheseobservationssuggestthattheoverallerrorperformanceoftheDFbasedcooperativesystemdependsmoreonthesrlinkthantherdlink.Suchunbalancedeectsoftherelaylinksconrmthatappropriateresourceallocationiscriticalinachievingtheoptimumerrorperformance. 30 ],weprovideamoregeneralyet,simpleexpressionfortheaverageSER.TheSERisdevelopedintwocases,thesystemwithnodirectlinkandtheonewithadirecttransmission.Thetwocasesaredenotedas\ND"and\DL,"respectively. 30

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ThisallowsustodeducetheSERusingamultichannelmodel.WewillrstinvestigatetheSERoftherelaysystemswithnodirectlink,theresultcanbeeasilyextendedtothecasewithadirectlink. InaL-relaysystemwithnodirectlinkbetweenthesourceandthedestination,thereceivedSNRis: 3{9 ),theaverageSERPAFe;NDcanbefoundas: rk;s+1 d;rkln(d;rk); 22L1L1Xn=0n+L1L1L1nXk=02L1k: 51 ].Forthebinarycase,theSERconditionedonNDisgivenby[ 44 ,Chap.12] 22L1eNDL1Xn=0cnnND; whereNDisdenedinEq.( 3{9 ),and 31

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3{12 ),nNDcanbeexpressedasfollowsbyexpandingND: {z }L1; wheremL=nPL1i=1mi,thenEq.( 3{12 )becomes: where 22L1L1Xn=0cnnXm1=0nm1nm1Xm2=0nm1m2nPL2i=1miXmL1=0nPL2i=1mimL1| {z }L1: TheaverageSERcanbeobtainedbyaveragingtheconditionalSERwithrespecttotheprobabilitydensityfunction(PDF)ofeq;rk,namelyp(eq;rk),whichisgivenas PAFe;ND=Z10Z10Z10| {z }LPejNDLYk=1p(eq;rk)deq;r1deq;r2deq;rL; wherethePDFofeq;rkisderivedin[ 72 ],andwhichisgivenas (3{18) +2 rk;sd;rks Bysubstituting( 3{15 )into( 3{17 ),wecanget: Pe;ND=Xn;mLYk=1Z10eeq;rkmkeq;rkP(eq;rk)deq;rk: 32

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3{19 ),denotedasIn;m;k, 2eq;rkeeq;rkK1(p where=2p 3{20 )canbecomputedbyusingtheintegrationpropertyofBesselfunction,[ 25 ,Eqs.6.631.3] 21 211++ 8W whereWm;n()istheWhittakerfunction (a),U(a;2;1=x)x 1 ,Eqs.13.5.9and13.5.7]),and=1+1=rk;s1athighSNR,wecansimplifyEq.( 3{20 )as rk;s+1 d;rklnd;rk: PluggingtheaboveresultbackintoPAFe;ND,weget PAFe;ND=Xn;mLYk=1mk!1 rk;s+1 d;rklnd;rk: Now,wemovetothecoecientpartofEq.( 3{23 ),whichisgivenby 22L1L1Xn=0cnnXm1=0nm1nm1Xm2=0nm1m2nPL2i=1miXmL1=0nPL2i=1mimL1| {z }L1LYk=1mk!: 33

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{z }L1LYk=1mk!=n!; thecoecientcanbesimpliedas 22L1L1Xn=0cnnXm1=0nm1Xm2=0nPL2i=1miXmL1=0| {z }L1n!=1 22L1L1Xn=0L1nXk=02L1knXm1=0nm1Xm2=0nPL2i=1miXmL1=01| {z }L1: DenotethecoecientinEq.( 3{26 )asC(L).Byusingmathematicalinduction,wecanprovethat: {z }L1=n+L1L1: Weknowthat,forL=1,(1)=1=n+1111.SupposeforanarbitraryL=l1,(l)=n+l1l1,then,wehave(l+1)=Pnm=0nm+l1l1whenL=l+1.Wecanalsoshowthat: (n1)!(l1)!...=(n+l1)! (n1)!(l1)!++(l1)! 0!(l1)!=nXm=0nm+l1l1=(l+1): Therefore,equality( 3{27 )holdsforanyL1.UsingtheresultinEq.( 3{27 ),wecansimplifyEq.( 3{26 )toEq.( 3{11 ).Finally,theaverageSERyieldsEq.( 3{10 ). 34

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Followingthesamestepsastheno-direct-linkcase,theaverageSERcanbesimilarlyevaluatedbycalculatingtheintegralforthedirectlinkasinEq.( 3{20 ): d;s(m+1)!: Combiningtheresultabovewith( 3{22 ),wecanevaluatetheaverageSERforasystemwithadirectlinkas: PAFe;DLC(L+1)1 d;sLYk=11 rk;s+1 d;rkln(d;rk); whereC()isthethesamefunctionasin( 3{11 ),whichdependsonthenumberofrelays.Forexample,forL=1;2;3;and4,wehaveC(1)=1=2;C(2)=3=4;C(3)=5=4;C(4)=35=16.ItisworthstressingthattheSERexpressionsinEqs.( 3{10 )and( 3{31 )coincidewiththeaverageSERofthecoherentsystemin[ 45 ]exceptforthelogterm,whichleadstothecodinggainlosscomparedwiththecoherentsystem. Whenrk;s=d;rk=d;s=~;8k,andas~!1,Eqs.( 3{10 )and( 3{31 )giveriseto PAFe;ND/C0~LandPAFe;DL/C00~(L+1); whereC0andC00arebothconstants.FromEq.( 3{32 ),itisclearthatthediversitygaincanbeobtainedusingadierentialschemewithAFprotocolforsucientlylargeSNR. InFigs. 3-4 and 3-5 ,wecomparetheapproximatedandsimulatedSERwhenL=1;2;and3forthesystemswithandwithoutadirectlink.Theguresconrmthatthediversitybenetincreasesindirectproportiontothenumberofrelays,anddemonstratethattheapproximationsareverytightcomparedwiththesimulations,especiallywhentheSNRishighandforsmallL.FromFig. 3-5 ,itiscertainthatadirect 35

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3-4 .AsLincreases,thequalityofapproximationdecreases,sincemoreapproximationerrorsareaccumulatedasthenumberofrelaysincreases. InFig. 3-6 ,wecomparetheSERofthesystemswithcoherentmodulationanddierentialmodulationwhenadirectlinkispresent.Weusetheresultin[ 45 ]forthecoherentsystem.TheonlydierencebetweenthecoherentsystemanddierentialsystemisthelogterminEqs.( 3{10 )and( 3{31 ).Thegureshowsthattherearecodinggaindierencesbetweenthesystems,andthesedierencesincreaseastheSNRincreases,i.e.,therequiredSERdecreases.Theseincreasingdierencesareduetothelogterminthedierentialsystem.Forexample,approximately2.2dB,2.8dB,and3.2dBmoreSNRarerequiredindierentialsystemtoachieve103,104,and105ofSER,respectively,comparedwiththecoherentsystem. 36

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LetusrstconsidertheDFSTCprotocol.Duetoindependentdemodulationandremodulationofthecorrespondingdiagonalentryateachrelay,thesrklinkSER,PDFe;rk,canbeobtainedusingEq.( 3{1 ).Sinceonesymbolerrorateachrelaycaninducethecodeworderror,theCERPDFe;ratsrklinksisgivenby Atthedestination,thereceivedsignalsfromtheL-relaysreconstructthetransmittedSTCsignal.ConditionedonthatthesourcetransmittedsignalblockVniscorrectlydecodedattherelays,andbydroppingthesuperscriptsfornotationalbrevity,theCERatrkdlinksisgivenby where AthighSNR,wecanmakethefollowingassumption wherewhereEr:=diagfEr1;Er2;:::;ErLgistheenergypersymbolmatrixattherelays,Hd;rn:=diagfhd;r1n;hd;r2n;:::;hd;rLngisthechannelmatrixbetweentherelaysanddestination,andXrnisthen-thtransmittedsignalblockfromtherelays.Then,Eq.( 3{35 )canbe 38

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whereDFe=E1=2rXn1(VnV0n)(VnV0n)XHn1E1=2r.SinceDFeisHermitian,wecanexpressEq.( 3{37 )as whereUisaunitarymatrixandDDFeisdiagfDFe;1;DFe;2;:::;DFe;Lg.EachdiagonalentryDFe;k;k=1;2;:::;L;representstheeigenvalueofDFe.Therefore,wecanobtaintheCERbyaveragingEq.( 3{34 )withrespecttothechannelHd;r.Forsimplicity,byassumingthatthefadingcoecienthasunitvariance,theconditionalCERPDFSTCe;datthedestinationisgivenby 8N0DFe;k1; andunderhighSNRcondition,thisequationcanbefurthersimpliedas Finally,usingEqs.( 3{33 )and( 3{40 ),wecanformulatetheunconditionalCERforDFSTCprotocolas: Itisworthmentioningthatifthereisnoerrorbetweenthesourceandrelays,theaboveequationboilsdowntotheCERofmulti-inputsingle-output(MISO)systememployingtheDUSTC.However,asLincreases,theCERofsrklinksbecomesworsebecauseoftheincreasingmodulationsizeateachdiagonalentry.ThiswillinducetheperformancedegradationoftheDF-basedsystem.Toprovidebetterperformanceandpertainthediversitygain,srklinkshavetomaintainlowerCER. 39

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2{20 )isdiagf2h1;eff;2h2;eff;:::;2hL;effg,wherethekthdiagonalentryofthecovariancematrixisgivenbyEq.( 2{14 ).Tonormalizetheaggregatenoisevariance,letusdenethematrixG:=diagfg1;g2;:::;gLgwithgk=(ErkA2rk2d;rk+1)1=2.Then,bymultiplyingGwiththereceivedsignalblockatthedestination,wecanrewriteEq.( 2{20 )as orequivalently, ~Yd;rn=Yd;rn1~V(m0)n+~Zn; where~Y=YG;~V=VG,and~Z=~Z0dG.Then,theCERfortheAFSTCprotocolcanbeachievedusingEq.( 3{43 ).FollowingthesamestepsasEq.( 3{34 )to( 3{37 ),theCERcanbeobtainedas: where AthighSNR,thecodedistancecanbeapproximatedas whereAFe=EsE1=2rXn1(VnV0n)(AG)(AG)H(VnV0n)HXHn1E1=2r.SimilartotheDFSTCprotocol,wecanexpressAFeas AFe=U0HDAFeU0; 40

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3{44 )withrespecttothecombinedchannelHd;rnHr;sn.Letusdeneh:=hd;rkhrk;s,thenthePDFof=jhjisgivenby[ 43 ] 2d;rk2rk;sK02s whereK0()isthezerothordermodiedBesselfunctionofthesecondkind.Byassumingthateachfadingcoecienthasunitvariance,andusingthepropertiesofBesselfunctionandconuenthypergeometricfunction(see[ 25 ,Eq.6.631.3]and[ 1 ,Eqs.13.5.9])athighSNR,theCERcanbesimpliedas NoticethattheCERoftheAFSTCprotocolhasalmostthesameformasitscounterpartoftheDFSTCprotocolatrkdlinksexceptforthelogtermwhichreectstheeectoftheamplicationandaggregatenoiseandthisleadstocodinggainloss.Eq.( 3{49 )conrmsthatAFSTCprotocolprovidesfulldiversitygain. InFigs. 3-7 and 3-8 ,weplottheboundsandsimulatedCERsforthesystemswithDFSTCandAFSTC,respectively,whenL=1;2;and3.WhenL=1,theSTC-basedcooperativesystemisreducedtotheconventionalcooperativenetwork,thuswecanusetheSERformulasderivedin[ 20 21 ]astheCERboundoftheSTC-basedsystem.Fig. 3-7 showsthattheboundsaretighttothesimulations,especiallywhenLissmall.NoticethatthecardinalityofthesignalblockattherelaysequalstoMLbecauseoftheindependentdecodingateachrelay.However,theboundatrkdlinksonlyconsidersMsignals.Thus,asLincreases,thegapbetweentheboundandsimulationincreases.Fig. 3-7 alsoshowsthatnodiversitygainisobtainedbyDFSTCprotocol,sincetheCERatsrklinksincreasesindirectproportiontothenumberofrelays,whichinducesthedegradationoftheoverallerrorperformanceofDF-basedsystem.Theseresultsconrmouranalysisin 41

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3-8 ,thoughtheboundsforAFSTCprotocolareinaccuratewhenSNRislowbecauseofthelogtermintheanalyticalexpression,theboundsandsimulationshavetightvaluesathighSNR.Furthermore,itisclearthatAFSTCprotocolprovidesfulldiversitygain. Aswementionedabove,thelinkqualitybetweenthesourceandrelaysiscriticaltotheperformanceoftheDF-basedcooperativesystem.Tocapturetheeectofunbalancedlinkquality,weconsiderdierentaverageSNRsatsrkandrkdlinksforbothDFSTCandAFSTCprotocolsinFigs. 3-9 and 3-10 ,respectively.Weassumethatrk;s=r;s,andd;rk=d;r;8k,andconsideri)equalSNRforbothsrkandrkdlinks,ii)higherSNRisassignedatsrk,andiii)higherSNRisassignedatrkdlinkswithL=1;2and3.AsshowninFig. 3-9 ,whenweassignhighSNRsatsrklinks,theoverallCERdecreasesandthediversitygainbeginstoappear.Fortheextremecase,i.e,inniteSNRisassignedatsrklinks,theDFSTCcooperativesystembehaveslikeaMISOsystem.ThegurealsoshowsthatthecodinggainisachievedbyassigninghigherSNRatrkdlinks.However,thediversitygainisdominantcomparedwiththecodinggainespecially 43

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3-10 showsthatthecodinggainisobtainedbyassigninghigherSNRatbothlinks.IncreasingSNRatsrkandrkdlinksleadstodecreasingtheeectofaggregatenoiseandincreasingSNRatthedestination,respectively.Bothscenariosinducetheenhancementofcodinggain.NoticethattheeectofSNRatrkdlinksprovidesmorecodinggainthanrkdlinks,whichimpliesthatincreasingaveragepowerattherelayoutputismorecrucialthanreducingtheeectofnoiseatthesrklinks. 44

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Inthischapter,wewillinvestigatetheeectsofresourceallocationontheerrorperformance.Wewillshowthatanoptimumallocationofthelimitedresourcesispossible,anditachievestheoptimumsystemerrorperformance.Theresourceallocationwhichminimizestheaverageerrorratewillbeaddressedfromthreeperspectives: 1) Giventherelativedistancesamongthesource,relayanddestinationnodes,thepathlossexponentofthewirelesschannel,andthetotalavailableenergypersymbol,determinetheoptimumenergyallocationamongthesourceandrelaynodes. 2) Giventhesource-destinationdistance,thepathlossexponentofthewirelesschannel,andtheenergypersymbolatthesourceandrelaynodes,determinetheoptimumlocationoftherelaynodes. 3) Giventhesource-destinationdistanceandthetotalavailableenergypersymbol,determinethejointenergyandlocationoptimization. Foranalyticaltractability,weconsideranidealizedL-relaysystemwithallrelaynodeslocatedatthesamedistancefromthesourceanddestinationnodes;thatis,Ds;rk=Ds;randDrk;d=Dr;d,8k.ItisthenreasonabletoassignequalenergiesatallrelaynodesErk=Er,8k.Tocarryouttheoptimizationintheensuingsubsections,wewillalsomakeuseoftherelationshipbetweentheaveragepowerofchannelfadingcoecient2hi;jandtheinter-nodedistanceDj;iasfollows: whereisthepathlossexponentofthewirelesschannelandCisaconstantwhichwehenceforthsetto1withoutlossofgenerality.FortheconventionalDFandAFprotocols,wewillpresenttheanalyticalresultsofoptimizationsaswellassimulatedexamples.FortheDSTCprotocol,duetotheanalyticalintractability,theoptimizationresultswillbeshownbysimulationsandcomparedwithconventionalsystems. 45

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(b) 4-1 .Oneistheellipsecaseandtheotheroneisthelinecase.Fortheellipsecase,Ds;r+Dr;d=DDs;d.Thelinecasecanberegardedasaspecialcaseoftheellipsecase,i.e.,D=Ds;d.BychangingthevalueofD,wecansolvetheoptimizationproblematanypointona2-Dplane.Therefore,optimumresourceallocationfortheseidealizedtopologiescanprovideusefulinsightsforunderstandingtheeectofresourceallocationinrelaynetworks. 3{5 ),( 3{41 ),and( 3{49 )).However,oursimulationswillshowthattheerrorrateisgenerallyconvex,whichensuresconvergenceoftheoptimization.WiththeAFprotocol,wecanprovetheconvexityanalyticallyandconrmbysimulations.TheproofofconvexityfortheAFprotocolisgivenasbelow,whichguaranteestheconvergenceoftheerrorperformanceasafunctionofenergyandlocation. 46

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3{10 )and( 3{31 )areconvexfunctionsoftheenergyandlocation,respectively. Proof. PAFe;ND=C(L)"1 (Lr)2r;s+1 PAFe;DL=C(L+1)1 (Lr)2r;s+1 whicharefunctionsofthesinglevariabler2(0;=L). ThesecondderivativesofPAFe;NDandPAFe;DLaregivenasfollows: ForL2, (Lr)3[f(r)]L2[g(r)]2f(r)g+C00(L)1 forL=1, (r)32f(r)+2sg(r)+2sh(s); whereC0(L)=LC(L)andC00(L)=LC(L+1),and (Lr)2r;s+1 47

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2r;s1 (Lr)2+1 (Lr)3+1 [g(r)]2+2L1 [g(r)]+(L1): UnderhighSNRapproximation,r2d;r1,wehavef(r)>0andh(r)>0.Wenoticethat(r)isaquadraticfunctionof1 [g(r)],anditsquadraticdiscriminantis: =(2L)24(2L)(L1)=4L(2L)0;forL2: Thus,wehave(r)0whenL2.Therefore, WhenL=1,theconvexityofthesystemwithnodirectlinkisreadilyobtainedfromEq.( 4{6 ).Forthesystemwithadirectlink,aftersomemanipulation,Eq.( 4{7 )canbereexpressedas: (r)31 (r)32 Usingtheinequality,(r=s)2(r=s)+13=4,wehavethelowerboundof@2PAFe;DL=@2r: (r)31 2s OntheconditionofhighSNR,ln(r2r)>2,wehave@2PAFe;DL=@2r>0forL=1. Finally,bothPAFe;NDandPAFe;DLareconvexfunctionsofr,andaccordinglysforanyL1. 48

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PAFe;ND=C(L)(DDr;d) whichisafunctionofasinglevariableDr;d2(0;D).ThesecondderivativeofPAFe;NDis where s(DDr;d)1+D1r;d UnderhighSNRapproximation,i.e.,r2rd1,and>1,wehaveu(Dr;d)>0,v(Dr;d)>0andw(Dr;d)>0.Thus: Similarly,forthesystemwithadirectlink,wehave PAFe;DL=C(L+1)1 ThisequationisthesameasEq.( 4{16 )exceptfortheconstantterm,C(L+1)=(Lr),thereforeitsconvexitycanbereadilyprovedusingthesamestepsintheabove. 49

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3{5 )forDFandEq.( 3{10 )orEq.( 3{31 )forAFwhilesatisfying: BydeningthetotalSNR,:=E=N0,thetransmitSNRatthesourcenodes:=Es=N0andthetransmitSNRattherelaynodesr:=Er=N0,theenergyconstraintcanbere-expressedastheSNRconstraint: UsingEq.( 4{1 ),theaveragereceivedSNRsattherelayanddestinationnodescanbeexpressedintermsofthetransmitSNRsas: r;s=s2hr;s=sDs;randd;r=r2hd;r=rDr;d: Asaresult,thetotalenergyconstraint,Eq.( 4{24 ),canbefurtherrewrittenas LetusconsidertheDFprotocolrst.Togainsomeinsights,westartfromasingle-relaysetupandestablishthefollowingresult: 4{23 ),the

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4Ds;rDr;d(Ds;rDr;d)22Dr;d+3 2(Ds;rDr;d); Proof. whereistheLagrangemultiplier.SolvingEqs.( 4{28 )and( 4{29 ),weobtain (1+2d;r)(1+d;r) (1+2r;s)(1+r;s)=Dr;d whichleadstothefollowingrelationshipbetweend;randr;s: d;r=3D=2s;r+q SubstitutingEq.( 4{32 )intoEq.( 4{30 ),wendosasinEq.( 4{27 ). AlthoughEq.( 4{27 )isaccurateforallEandN0valuesandforallsrandrddistances,itscomplexformdoesnotprovidemuchintuition.Fortunately,forseveralspecialcases,itcanbesimpliedwithoutmuchlossofaccuracy.Nextwewillconsidersomeofsuchcases. 4{31 )simpliesto22d;r+3d;r+1 22r;s+3r;s+1=1:

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4{31 )as22d;r+3d;r+1 22r;s+3r;s+1=Dr;d 2(Ds;rDr;d): Thissolutioncanbefurthersimpliedbyneglectingitsconstanttermsto Interestingly,thissolutioncoincideswiththeoptimumpowerallocationobtainedbyminimizingtheoutageprobability[ 27 ,(8)]withasingle-relaytransmission.FromEq.( 4{34 ),itreadilyfollowsthattheenergyallocationratiobetweenthesourceandtherelaynodesis Eq.( 4{35 )revealsexplicitlythattheoptimumenergyallocationheavilyhingesupontheinter-nodedistances.Inaddition,thepathlossexponentofthewirelesschannel,,alsoaectstheoptimalenergyallocation.Interestingly,theEos=EorratioislinearinDs;r=Dr;donlywhen=2.Theoptimumenergyallocationfavorsthelinkwithalargernodeseparationif>2andviceversa,aswewillshownextwithanexample. Fig. 4-2 depictsthetransmitSNRsobtainedfromtheoptimumenergyallocation.Aone-dimensionalsetupisconsidered;thatis,Ds;r+Dr;d=Ds;d=D.Thesystemparametersare:=10dB,L=1,Ds;d=D=1and=(1;2;3;4).InFig. 4-2 ,the 52

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4{27 )andtheapproximationsinEqs.( 4{33 )and( 4{34 ).Theseresultsarenearlyidenticalinallcaseswithvariousvalues.Bycloselyinspectingthegure,wendthattheapproximationinEq.( 4{33 )providesmoreaccuratecurvesthantheoneinEq.( 4{34 ),asexpected.AlthoughtheapproximateexpressionsinEqs.( 4{33 )and( 4{34 )areobtainedunderhighSNRassumption,theyremainveryaccurateevenatmediumSNRof10dB. FromFig. 4-2 ,wealsoobservethat,forallvalues,thesourcenodeenergyEsincreasesastherelaymovestowardsthedestinationnode.With=2,EsincreaseslinearlywithDs;r.Athighervaluesofthepathlossexponent,>2,weobservethat Inotherwords,theoptimumenergyallocationfavorsthelinkwithlargerinter-nodedistance.Whenthepathlossexponent=1,Fig. 4-2 showstheoppositeofEq.( 4{36 ). 53

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4{27 )canbeobtainedandaveryaccurateandinsightfulapproximationisavailableunderhighSNRassumption.ForL2,however,therstorderconditionsobtainedbydierentiatingtheSERboundPDFehavecomplicatedforms,whichrenderanalyticalsolutionsimpossible.Fortunately,theSERboundPDFeasinProposition1stillallowsforanumericalsearch,asopposedtoMonteCarlosimulationsneededotherwise. Forexample,withDBPSKandL=2,wehave 23 4d;r 4d;r and,accordingly,theSERboundisgivenby PDFe=12r;s(2+d;r)(1+2d;r)2 ByusingtherstorderconditionsinEq.( 4{28 )andthehighSNRapproximation,theoptimumr;sandd;rshouldsatisfy 4(1+d;r)(2+d;r)(1+2d;r) 3r;s(1+r;s)=Dr;d Althoughananalyticalsolutionisnotreadilyavailable,onecanresorttothenumericalsearch. Letusconsidersomeexamplesofoptimumenergyallocation.Fig. 4-3 representstheaverageSERforvariousenergyallocationsatthexedrelaylocation.TotaltransmitSNR=10dBandL=2areconsideredwithDs;r=0:25;0:5;and0.75.Foreachcase,theSERhasoneminimumpoint,andthecorrespondingenergyallocationistheoptimumvalue,i.e.,os=.Thegureshowsthattheos=increasesastherelaysmovetowardsthedestination.ThiscoincideswithouranalyticalresultsandsimulationsinFig. 4-2 TheoptimumenergyallocationobtainedfromthenumericalsearchisplottedinFig. 4-4 andcomparedwiththesimulatedresults.ThetotalSNRvalueof=10dBandapathlossexponentof=4areconsideredwithvariousLvalues.Theresults 54

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4-4 .ThisdiscrepancyarisesfromthefactthatthenumericalsearchisbasedontheSERbound,whereasthesimulationsgeneratethetrueSER,andthattheSERboundislooserforlargerLvalues,aswementionedintheprecedingsection.Nevertheless,thenumericalresultsstillcloselyindicatethetrendandrelativedistancescorrespondingtovariousLvalues.NoticethatthesecurvesexhibitaconvergingtendencyasLincreases.ThisimpliesthattheoptimumenergyallocationcurvemayachieveanasymptoticlimitasthenumberofrelaysLgrows. WehaveseenthattheexactexpressionoftheoptimumenergyallocationinEq.( 4{27 )willgiverisetoas=ratiothatdependsontheactualvalueofthetotalSNRwithDFprotocol.However,thehigh-SNRapproximationinEq.( 4{34 )resultsinas=ratiowhichisindependentof(seeEq.( 4{35 )).Asaresult,theapproximatesolutions( 4{33 )and( 4{34 )areexpectedtodierfromtheexactsolution( 4{27 ),dependingon 56

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4-2 showsthatthesesolutionsagreeverywellat=10dB.InFig. 4-5 ,theoptimumenergyratios=obtainedfromtheexactsolution( 4{27 )isdepictedatvariousSNRvalues=(0;5;10;15;20)dB,andwithtwovaluesof(2and4).Weobservethatwhen=4,thecurvescorrespondingtodierentvaluesarealmostidentical;whereaswhen=2,allcurvescoincideexceptforthe=0dBone.Theseobservationsconrmthattheapproximations( 4{33 )and( 4{34 )arebothveryaccurateevenforaslowas5dBwhen=2andforallSNRlevelswhen=4.Inotherwords,theoptimumenergyallocationratios=isalmostindependentoftheactualenergylevelexceptforverylowvalues.ItonlydependsonthelocationoftherelaysasinEq.( 4{35 ).Likewise,asimilarresultcanbededucedfortheoptimumdistanceallocation;thatis,theoptimumdistanceallocationratioDs;r=Ds;disnearlyindependentoftheactualsource-destinationdistanceDs;d,aswederivedinEq.( 4{60 ). NowletusconsidertheAFprotocol.SimilartotheDFprotocol,bytreatingtheapproximatedSERPAFe;NDorPAFe;DLasafunctionofsandrk,wecanndanoptimumsolution. 4{24 ),theoptimumenergyallocationosandorshouldsatisfy: (4{40) L+1o2s+or2d;r2r;s (4{41)

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whereistheLagrangemultiplier.Eqs.( 4{42 )and( 4{43 )giveus Then,bysubstitutingsandrinEqs.( 4{45 )and( 4{46 )forthetotalenergyconstraintinEq.( 4{44 ),wehave WithEqs.( 4{45 )and( 4{47 ),wearriveatEq.( 4{40 ).Similarly,therstorderconditionsofthesystemwithadirectlinkaregivenby Byusingthesamestepsasshown,wehaveEq.( 4{41 ). InEq.( 4{41 ),the2r;sln(or2d;r)=(L+1)termmainlyaectstheenergyallocationcomparedwiththesystemwithnodirectlink.Thiseectisobviousespeciallywhentherelaysarelocatedclosetothethesource.NoticethatthelogterminEqs.( 4{40 )and( 4{41 )rendersaclosed-formsolutionincalculable.Althoughananalyticalsolutionisnot 58

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4{40 )and( 4{41 ).ThefollowingLemma2showsthatoptimumvalueshaveonlyonesolution,whichallowsusthenumericalsearch. 4{40 )and( 4{41 )haveonesolutionofos,andaccordinglyor. Proof. 4{40 )and( 4{41 )cannotbesolvedalgebraically,werepresentasolutiongraphically.UsingEq.( 4{24 ),Eq.( 4{40 )isgivenby Thiscanberepresentedas Bysquaringrootbothsides,wehave 59

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4{52 ),respectively.Giventhatdomainofln(or2d;r)>1andusingEq.( 4{1 ),weploty1andy2inFig. 4-6 forvariousrelaylocationsDs;rwithDs;d=1.Thelinewithnomarkerandwithcirclemarkerrepresenty1andy2,respectively.Thegureshowsonlyonecrossingpoint,andthatpointprovidesustheoptimumsolution.Similartrendsareobservedinthesystemwithadirectlink. Figs. 4-7 and 4-8 depicttheaverageSERforvariousenergyallocationsatxedrelaylocationforthesystemwithnodirectlinkandwithadirectlink.Welocaterelaysat0:25;0:5;and0:75with=15dBandL=2.BothguresshowthattheSERhasoneminimumpoint,andthecorrespondingenergyallocationistheoptimumvalue,whichconrmsLemma2.Forthesystemwithnodirectlinktheoptimumenergyallocationincreasesastherelaysmovetowardthedestination.However,forthesystemwithadirectlink,optimumenergyallocationstaysatthemiddlevalue,i.e.,uniformenergyallocationwhentherelaysarelocatedclosetothesource(seeFig. 4-8 withDs;r=0:25and0.5).Thisresultwillbeveriedbythefollowingnumericalsearchresults.Forallcases,theSERsinFig. 4-8 showbetterperformanceinFig. 4-7 duetothedirectlink. TheoptimumenergyallocationobtainedfromthenumericalsearchforboththesystemwithandwithoutadirectlinkisplottedinFig. 4-9 .WeconsiderthetotalSNRvalueof=30dBandapathlossexponentof=4withvariousLvaluesandD=Ds;d=1.Inthesystemwithnodirecttransmission,forallLvalues,theoptimumenergyallocationatthesourceincreasesastherelaymovestowardsthedestination.However,forthesystemwithadirectlink,auniformenergyallocationisoptimumwhentherelaysarelocatedclosetothesource.Intuitively,thisisbecausethedirecttransmissionispresentthatthediversitygainisdominantoverthecodinggain.Whentherelaysarelocatedclosetothedestination,muchoftheenergyisassignedatthesourcetoassurethatatransmittedsignalcanreachtherelays;itisthesameasinthesystemwithnodirectlink. 61

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4-10 representstheoptimumenergyallocationforbothDFSTCandAFSTCprotocolswhenL=2and3.WeusethetotalSNRvalueof=15dBandapathlossexponentof=4.withD=1:2andDs;d=1.Thegureshowsthattheoptimumenergyallocationatthesourceincreasesastherelaysmovetowardthedestinationforbothcases,whichisthesameastheconventionalcooperativenetworkswithnodirectlink.Thegurealsoshowsthat,ingeneral,DFSTCprotocolrequiresmoreenergythanAFSTCprotocoltodecreasetheerrorrateattherelays. 3{5 )forDFandEq.( 3{10 )orEq.( 3{31 )forAFwhilesatisfying: Startingwiththesingle-relay(L=1)setupandapplyingthehigh-SNRapproximationwithDFprotocol,weestablishthefollowingresult:

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whereweusedthefactthatr;s(ord;r)isindependentofDs;r(orDr;d).UsingthedenitionoftheaveragereceiverSNRsinEq.( 4{25 ),wecanre-expressEqs.( 4{55 )and( 4{56 )as D+1s;rs=0;@PDFe D+1r;dr=0; whichleadsto (1+2r;s)(1+r;s): AthighSNR,theconstanttermsontheright-handsideofEq.( 4{59 )canbendglected.Consequently,wehave[c.f.( 4{25 )]r which,togetherwiththeconditioninEq.( 4{57 ),concludestheproofofProposition5. Interestingly,Eq.( 4{60 )bearsaverysimilarformasitscounterpartfortheoptimumenergyallocationinEq.( 4{35 ).Infact,whenthepathlossexponent=2,Eq.( 4{60 )isessentiallyidenticaltoEq.( 4{35 ).Forgeneralvalues,however,thesetworelationships 64

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4{35 )and( 4{60 )resultfromtwodistinctoptimizationproblems:theformerisobtainedforarbitrarydistancesDs;randDr;dunderatotalenergyconstraint;whereasthelatterisobtainedforprescribedEsandErunderatotaldistanceconstraint.WiththeSERboundPDFebeingatwo-dimensionalfunction,theenergyandlocationoptimizationsarecarriedoutonuncorrelateddimensions. ForgeneralLvalues,theoptimumlocationcanbedeterminedinasimilarmanneraswediscussedintheabove.Essentially,thepathlossexponentrendersitimpossibletoderiveananalyticalsolutionfortheoptimumlocationproblem,evenwiththehighSNRapproximation.OnecanresorttothenumericalsearchusingtheSERboundinProposition1.InFig. 4-11 ,theoptimumdistancesobtainedfromthenumericalsearchandthesimulationsarecomparedfordierentvalues,attotalSNR=10dBandwithL=1relaynode.Noticethat,asitscounterpartinFig. 4-2 ,theoptimumrelaylocationislinearinEs=Eonlywhen=2. 65

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4-12 depictstheaverageSERversusrelaylocationsatthegivenenergyallocationwhen=10dBandL=2.Noticethatatthegivenprescribedenergy,thereexistsonlyoneminimumSERpoint,andthecorrespondingDs;ristheoptimumrelaylocation.ThisgureisthecounterpartoftheoptimumenergyallocationinFig. 4-3 .Wecanseethattherangeofoptimumrelaylocationissmallerthantheoptimumenergyallocation,whichresultsintheatnessoflocationoptimization.Wewillverifythisphenomenonwithanumericalexample. LetusnowconsidertheoptimumrelaylocationfortheAFprotocol. Proof. 3{31 ).The1=d;s=Ds;d=shasaxedvaluegiventhesddistanceDs;dandprescribedenergyatthesource.Therefore,adirectlinkdoesnotaectlocationoptimization.Hence,thelocation 66

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TheoptimumlocationcanbefoundbytreatingtheSERasafunctionofdistanceandsolvingtherstorderconditions. 67

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4-13 and 4-14 presenttheaverageSERforthevariousrelaylocationsatthegivenenergyallocation.Bothguresconrmthatthereexistsonlyoneminimumpointwhichprovidestheoptimumrelaylocationforrelaynetworks.ItisinterestingthattheminimumpointsarethesameforbothguresalthoughtheSERisdierent.Hence,theexistenceofadirectlinkisindependentontheoptimumrelaylocation,whichconrmstheLemma3.Inbothgures,therangeofoptimumlocationsissmallerthantheenergyoptimizationcases(seeFigs. 4-7 and 4-8 ),whichisthesameastheDFcase(seeFig. 4-12 ). Fig. 4-15 depictstheoptimumrelaylocationwhichisapplicabletosystemswithandwithoutadirectlink.WeconsiderthetotalSNRvalueof=30dBandapathlossexponentof=4withvariousLvalues.Onedimensionalsetup,Ds;r+Dr;d=D=Ds;d=1,isconsidered.Asmoretransmitenergyisassignedatthesource,theoptimumlocationmovestowardthedestination.Thegureshowsthattheoptimizedvalueschangeslowlycomparedwiththeenergyoptimization. 68

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4-16 depictstheoptimumrelaylocationofDFSTCandAFSTCprotocolswhenL=2and3.WeusethetotalSNRvalueof=15dBandapathlossexponentof=4.withD=1:2andDs;d=1.Thegureshowsthattheoptimumrelaylocationsmovetowardthedestinationasthetransmitenergyatthesourceincreasesforbothcases.Thegurealsoshowsthat,incomparisonwithAFSTCprotocol,therelaylocatesclosertothesourcefortheDFSTCprotocol.FromFigs. 4-10 and 4-16 ,wecanseethatthelocationoptimizationsaremuchatterthantheenergyoptimizations.Theseresultsarethesameasthoseoftheconventionalcooperativesystemswithnodirectlink. 3{5 ),( 3{10 )or( 3{31 ).First,bytreatingeachequationasthefunctionoftransmitenergy,wecanndthesolutionfortherstorderconditions.Then,thesamestepisproceededbythetransmitenergythatisreplacedwiththelocation.Finally,byequatingtwosolutions,wecanndthecommonsolution.Consequently,thissolutionprovidestheglobaloptimizationwhichminimizestheerrorrate.WithDFprotocol,forL=1,wecanreadilyobtaintheglobalsolutionfromEqs.( 4{35 )and( 4{60 ),whichgivesusDs;r=Dr;d=0:5withs==r==0:5;8.However,theanalyticalsolutioncannotbeeasilyobtainedevenwiththeidealizedcaseaswehaveseenintheprevioussection.Ingeneral,thejointoptimizationcanbeobtainedbycarryingoutatwo-dimensionalnumericalsearchiteratively.Thesearchingstepsareasfollows: Step1. (Initialization)Settheuniformenergyallocationastheoptimum,i.e.os==(L+1). Step2. (LocationOptimization)Foragivenenergyallocation,ndtheoptimumrelaylocation,Dos;r;new,whichiserrorrate-minimizing.Ifthedierencebetweennewoptimumlocationandtheoriginaloneissmallerthanthethresholddistance,"D, 70

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Step3. (EnergyOptimization)Foragivenrelaylocation,ndtheoptimumenergyallocation,os;new.Ifthedierencebetweennewoptimumenergyallocationandtheoriginaloneissmallerthanthethresholdenergy,",i.e.,jos;newosj<",stop;otherwisesettheoptimumenergytothenewone,os=os;newandgobacktoStep2. TheseiterativesearchingcanbeillustratedastheowchartinFig 4-17 Withoutconsideringadirectlink,Figs. 4-18 and 4-19 depicttheSERperformancesurfacewhenL=3withDs;d=D=1,and=4,forboththeDFandAFprotocol.Weuse=10dBand=15dBfortheDFandAFprotocol,respectively.Wecanobtaintheenergyoptimizationandthelocationoptimizationbytakingminimumvaluealongthe 71

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Similartotheconventionalprotocols,weplottheCERversuss=andDs;rforDFSTCandAFSTCprotocolsinFigs. 4-20 and 4-21 ,respectively,whenD=1:2;Ds;d=1,and=4.TheseguresexhibitthesametrendsasintheDFandAFprotocols.Noticethatthetwoguresshowalmostthesameshape.However,theaxisvaluesofthetwoguresareopposite.ThisimpliesthatthesystemsemployingDFandAFprotocolhavequitedierentrelationshipsfortheoptimumvalues.NoticethatabovesearchingstepsforjointenergyandlocationoptimizationcanbeapplicablefortheDFSTCandAFSTCprotocolsbyreplaceSERwithCERthoughtheanalyticalsolutionsareintractable.Moredetailedexamplesandcomparisonsaregiveninthefollowingchapter.Noticethatoursimulations(Figs. 4-3 4-12 4-18 4-20 ,and 4-21 )conrmthattheerrorrateisgenerallyconvex,whichensuresconvergenceoftheiterativestrategy. 73

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Inthischapter,wewilldiscusstheperformanceofrelaysystemscombinedwithdierentialdemodulationandtheoptimumresourceallocation.Wewillcomparetheperformanceofthesystemswithandwithoutoptimization.Thebenetsofthejointenergyandlocationaswellastheresourceallocationcomparisonofdierentprotocolsareaddressed. 5-1 throughFig. 5-6 .Weusethefollowingsystemparameters:=15dB,D=1:2,=4,andL=(1;2;3)withDBPSK.Inthesystemwithoutenergyoptimization,auniformenergyallocationisemployed;thatis,s=r==(L+1)atanyDs;r.Inthesystemwithoutlocationoptimization,therelaysareplacedatthemidpointofthesource-destinationlink. Figs. 5-1 and 5-2 illustratethebenetsofoptimizationoftheDFsystem.InFig. 5-1 ,weobservethat,asLincreases,theSERperformancecangetevenworseunlesstheenergyoptimizationisperformed,andthattheenergy-optimizedsystemuniversallyoutperformstheunoptimizedone.Theseobservationsconrmourdiscussionsintheprecedingchapter.Interestingly,noticethattheminimaoftheenergy-optimizedSERcurvesalmostcoincidewiththeunoptimizedones.Thisimpliesthatthenear-optimumSERcanbeachievedevenwiththeuniformenergyallocationacrossthesourceandrelaynodes,providedthattherelaylocationiscarefullyselected.AsshowninFig. 5-1 ,theoptimumrelaylocationcorrespondingtotheuniformenergyallocationshiftsfromthemidpointforL=1tothesourcenodeasLincreases.Intuitively,thisisbecausetheoverallSERismoresensitivetothesource-relaylinkquality,aswementionedinChapter3. 75

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5-2 ,weverifytheadvantageoftheoptimumrelaylocationbycomparingtheSERwithandwithoutlocationoptimization.Similartotheenergyoptimizationcase,Fig. 5-2 conrmstheadvantagesofthelocationoptimization,inwhichthelocation-optimizedsystemuniversallyoutperformstheunoptimizedsystem.Dierentfromtheenergyoptimizationcase,however,asLincreases,theSERperformancealwaysimprovesevenwithoutanylocationoptimization. ThecurvesinFig. 5-2 alsoexhibitmoreatnesscomparedwiththeonesinFig. 5-1 .ThisimpliesthatthesystemSERismoresensitivetothelocationdistributionthantheenergydistribution.Inaddition,theminimaofthelocation-optimizedSERcurvesarefarfromthoseoftheunoptimizedones,exceptfortheL=1case(seeFig. 5-2 ).ThisindicatesthatplacingtherelaynodesatthemidpointcannotachievetheminimumSERevenwithcarefulallocationofthesourceandrelayenergies,foranyL>1.ThisistobedistinguishedfromtheuniformenergycasedepictedinFig. 5-1 ,aswellasfromthecoherentrelaysystemsin[ 45 ]. Figs. 5-3 and 5-4 depictthebenetsofenergyoptimizationfortheAFsystemwithnodirectlinkandwithadirectlink,respectively.FromFigs. 5-3 and 5-4 ,weobservethattheenergy-optimizedsystemuniversallyoutperformstheunoptimizedsystemasweexpected.Wealsoobservethat,inthesystemwithadirectlink,theSERsoftheoptimizedsystemandunoptimizedsystemarealmostidenticalwhentherelaysarelocatedclosetothesource,sinceauniformenergyallocationisoptimum.Theseobservationscoincidewithouranalysisintheprecedingchapter.BothguresshowthattheunoptimizedsystemshavetheminimumSERalmostatthemidpoint,coincidingwiththeresultsin[ 38 45 67 ]. Noticethattheminimumpointsoftheenergy-optimizedSERcurvesmovetowardsthedestinationexceptinthesystemwithoutadirectlinkforL=1,whichisoppositecomparedwiththeDFcase.FromFigs. 5-3 and 5-4 ,itisclearthatwecannotachieveoptimumSERvaluewithoutenergyoptimizationexceptforthesystemL=1withnodirectlink.ThisisdierentfromtheDFcase,asinFig. 5-1 .Itisworthmentioning 77

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45 ]withAFprotocol,butdierentfromtheDFcase. Next,letusconsiderthebenetsoflocationoptimizationfortheAFsystems.Figs. 5-5 and 5-6 verifytheadvantagesbycomparingtheSERsofthesystemswithandwithoutlocationoptimization.Similartotheenergyoptimizationcase,thelocation-optimizedsystemuniversallyoutperformstheunoptimizedsystem.TheguresshowthattheoptimumSERcanbeachievedbyassigningmoreenergytothesourceexceptforthesystemwithoutadirectlinkwithL=1,whichistheoppositeoftheDFcase.ThecurvesinFigs. 5-5 and 5-6 alsoshowmoreatnesscomparedwiththeenergyoptimizedcurves,aswehaveobservedfortheDFcase.SimilartothelocationoptimizationoftheDFandtheenergyoptimizationoftheAF,Figs. 5-5 and 5-6 showthatoptimumSERcannotbeobtainedwithoutrelaylocationoptimizationexceptforthesystemL=1withnodirect 79

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5-4 and 5-6 comparedwithFigs. 5-3 and 5-5 ,respectively. Figs. 5-7 5-8 5-9 ,and 5-10 depictthebenetsofenergyandlocationoptimizationsforDFSTC(Figs. 5-7 and 5-8 )AFSTC(Figs. 5-9 and 5-10 )protocols.Forallcases,weconsiderD=1:2,Ds;d=1,and=4with=15dBand25dBwhenL=2and3sincethesinglerelaysetup(L=1)isthesameastheconventionalcooperativesystem.Similartotheconventionalcase,weplottheCERforthesystemwithandwithoutresourceallocations.Weusethesameparametersfortheunoptimizedsystems.TheguresconrmthattheminimumCERcanbeachievedbytheoptimumenergyandrelaylocationselection.TheguresalsoshowthatthetrendsofoptimizationsforDFSTCandAFSTCprotocolsarethesameasthoseforDFandAFprotocols,respectively.Noticethat,atlow,thesystemswithmoreLmayunderperformthesystemswithlessL.However,athigh,theerrorperformanceimprovesasLincreasesexceptsomecasesofDFSTC-basedsystemaswehaveseeninDF-basedsystem. 80

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5-11 and 5-12 depicttheSERcontouroftherelaysystemsfortheAFprotocolwithnodirectlinkandwithadirectlink,respectively,andFig. 5-13 depictstheSERcontouroftherelaysystemfortheDFprotocol.WithsystemparameterDs;d=D=1,L=2isconsideredforthesystemwithnodirectlinkandwithadirectlink.Weuse=15dBand10dBfortheAFandDFprotocol,respectively.Inallgures,theverticallineandhorizontallinerepresenttheSERsofthesystemswithauniformenergyallocationandmid-distanceallocation,respectively,i.e.,unoptimizedsystems.Wealsoplotlinesfortheenergyoptimizationandlocationoptimization.NoticethatthecrossingpointofthetwooptimizationsistheminimumSERofthesystem;accordingly,thispointcorrespondstothejointenergyandlocationoptimization. WithAFprotocolinFigs. 5-11 and 5-12 ,itisclearthattheminimumSERoftheunoptimizedsystemsisfarfromtheoptimizedones;thisindicatesthatwecannotobtaintheminimumSERwithauniformenergyallocationormid-distanceallocation.ToachievetheminimumSER,wehavetoadaptasystemviatheenergyoptimizationor 83

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5-13 showsthatthenearoptimumperformancecanbeachievedwithauniformenergyallocation,butcannotwithamid-distanceallocation.AlltheseresultscoincidewiththesimulationresultsintheprevioussectionandtheoreticalanalysisinChapter4.FromFig. 5-13 ,wecanseethattheuniformenergyallocationisaverygoodstartingpointfortheiterativeoptimization. Figs. 5-14 and 5-15 depicttheCERcontourfortheDFSTCandAFSTCprotocols,togetherwiththeoptimumenergyallocationandrelaylocationoptimizationcurves.WeconsiderD=1:2;Ds;d=1,and=15dBwhenL=2.TheseguresshowthattheCERcontoursfortheDFSTCandAFSTCprotocolshavethesametrendsastheSERcontoursfortheDFandAFprotocols,respectively. Itisinterestingthattheoptimumresourceallocationisdierentinthetwoprotocols.FortheAF-basedprotocol,wecanachievetheminimumerrorratebylocatingtherelaysclosertothedestinationandassigningmoreenergyatthesourcethantherelays.Thisisduetothefactthattheerrorratedecreasesbyallowingthesourcetotransmitsignalswithmoreenergywhilereducingtheeectofnoiseontheamplicationfactor, 85

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Assumethattheoriginalinformationsymbolsareequi-probablebinarysignal(=1)andtherearetworelaynodes(L=2).Intheconventionalsystem,onesymboltransmissiontakesthreetimeslots:oneforbroadcastingthesymboltotherelaynodes,andtwotimeslotsareusedtotransmittheremodulated/ampliedsignalfromeachrelay 87

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5-16 .Weobservethatthenumberofrequiredtimeslotsincreaseswiththenumberofrelaysfortheconventionalsystem,butremainsconstantfortheSTC-basedsystem.Therefore,wecanincreasetransmissionratebyusingSTC-basedsystem.ThisimpliesthattheSTC-basedsystemcanpotentiallyprovidethedierentialbenetaswellashightransmissionrateregardlessofthenumberofrelays. FortheSTC-basedsystem,themodulationsizeincreasesasthenumberofrelaysincreases.Whereas,wecanchoosethemodulationsizefortheconventionalsystem.Byadoptingthesamemodulationsize,thecomparisonofbiterrorrate(BER)betweentheconventionalsystemandSTC-basedsystemisdepictedinFigs. 5-17 and 5-18 withDF/DFSTCandAF/AFSTCprotocols,respectively.WeconsiderDQPSK,D8PSK,andD16PSKforL=2,3,and4,respectively,fortheconventionalsystem.TheguresshowthatSTC-basedsystemperformscomparablywithorbetterthantheconventionalsystemespeciallywhenL3.Fortheconventionalsystem,theerrorperformancedecreasesasLincreasesforbothDFandAFprotocols.However,fortheSTC-basedsystem,theBERisnotaecteddramaticallyregardlessofLandthediversitygainisalwaysguaranteedbydecreasingerrorratefortheDFSTCandAFSTCprotocol,respectively. Aswementionedintheabove,theSTC-basedsystemprovideshighertransmissionrate.IftheconventionalsystemadoptsthesamedaterateasintheSTC-basedsystem,howdoesthisaecttheerrorperformance?Toanswerthisquestion,wecomparetheBERofthecooperativesystemswhichhavethesameorsimilardatarateinFigs. 5-19 and 5-20 88

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5-17 toFig. 5-20 ,wecanseethatthemodulationsizedramaticallyaectstheperformanceoftheconventionalsystem. Summarizing,wecomparedtwocooperativesystemswithrespecttothetransmissionrateandBER.OurresultsshowthattheSTC-basedsystemprovideshighertransmissionratecomparedwiththeconventionalsystem.ItisalsoshownthattheconventionalsystemmayprovidebetterperformancethantheSTC-basedonebykeepingthesame/higherdatarate.Inaddition,theBERcomparisonsrevealthatthemodulationsizeiscriticaltodeterminetheerrorperformanceofconventionalsystem. 90

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Wederivedtheupperboundofsymbolerrorrate(SER)fortheDFprotocol,approximatedSERfortheAFprotocol,andtheupperboundofthecodeworderrorrate(CER)forSTC-basedsystems.TheDF-based(DFandDFSTC)protocolsshowedanunbalancederrorperformancedependingontherelaylocations,andouranalyticalresultsandsimulationssuggestedthatitismainlythesource-relaylinksthatdeterminetheoverallsystemperformance.FortheAFprotocol,theSERwasderivedfortwocases:thesystemwithnodirectlinkandthesystemwithadirectlink.ThisSERexpressionhadageneralbutverysimpleform.TheAFSTCprotocolshowedthesametrendsastheAFprotocolwithnodirectlink. Basedontheerrorperformance,theaverageSERandCERfortheconventionalDF/AFprotocolsandtheDSTCprotocol,respectively,weexploredtheresourceallocationasatwo-dimensionproblem.Weshowedthat:i)giventhesource,relayanddestinationlocations,theaverageerrorratecanbeminimizedbyappropriatelydistributingtheprescribedtotalenergypersymbolacrossthesourceandtherelays;ii)giventhesourceandrelayenergylevels,thereisanoptimumrelaylocationwhichminimizestheaverageerrorrate;andiii)giventhesource,relayanddestinationlocations,andtotaltransmitenergy,theminimumerrorratecanbeachievedbythejointenergy-locationoptimization. 92

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Oursimulationsandnumericalexamplesconrmthatboththeenergyandlocationoptimizationsprovideconsiderableerrorperformanceadvantages.WehaveobservedthefollowingresultsfortheDFandDFSTCprotocols. (1) Withoutenergyoptimization,performancedegradationisobservedwhenmorerelaysareincludedinthesystemespeciallywhentherelaysarelocatedclosetothedestinationnode. (2) Forallcases,theoptimizedsystemsuniversallyoutperformtheunoptimizedones. (3) Thelocationoptimizationismorecriticalthantheenergyoptimization.Inotherwords,thedierentialrelaysystemwithuniformenergydistributioncanachievenear-optimumerrorperformancebyappropriatelychoosingtherelaylocation;whileasystemwithrelayssittingatthemidpointbetweenthesourceandthedestinationcannotapproachtheoptimumerrorperformanceevenwithoptimizedenergydistribution. FortheAFandAFSTCprotocols,wehaveobservedfollowingresults. (1) ForthesystembasedonAFprotocolwithadirectlink,theerrorratesofthesystemwithauniformenergyallocationandoptimumenergyallocationarealmostidenticalwhentherelaysarelocatedneartothesource,sincetheuniformenergyallocationisoptimuminsuchcases. 93

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Forallcases,theoptimizedsystemsuniversallyoutperformtheunoptimizedones. (3) Energyandlocationoptimizationsareequallyimportant,sinceminimumerrorperformancecannotbeachievablewithouteitherofthem. Wehavealsoshownthattheminimumerrorperformancecanbeachievedbythejointenergy-locationoptimization. Inaddition,wecomparedthecooperativesystemswithdierentprotocolsbyconsideringboththeenergydistributionandtherelaylocationselection.Itisinterestingthattheoptimumlocationandenergyallocationsareverydierentinthetwoprotocols.Ingeneral,wecanachievetheminimumerrorperformanceofthecooperativenetworksbylocatingrelaysclosertothesourcenodewithalessamountofthesourcetransmitenergyfortheDF-basedsystem,andbylocatingrelaysclosertothedestinationnodewithalargeamountofthesourcetransmitenergyfortheAF-basedsystem. Finally,wecomparedtheconventionalsystemwithSTC-basedsystem.Ingeneral,STC-basedsystemcansupporthighertransmissionrates.However,theconventionalsystemcanachievecomparableperformanceincomparisonwithSTC-basedonebychoosingappropriatemodulationsize. Recently,thehybridschemewhichselectstheadvantagesofDFandAFprotocolsissuggestedforcoherentsystem[ 8 11 33 ].Itwillbeinterestingtodevelopahybridcooperativesystemandanalyzeitsperformanceandresourceoptimizingschemes.Itwillbealsovaluabletoconsidermultihopcooperativenetworkswhichcansupportreliable 94

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34 41 69 ]).Finally,inthisresearch,wemainlyfocusedonphysicallayeranalysis.Itwillbehelpfultoconsiderhigherlayerissues,i.e,mediumaccesscontrol(MAC)ornetworklayers,forimprovingtheoverallnetworkingperformance. 95

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[1] M.AbramowitzandI.A.Stegun,HandbookofMathematicalFunctionswithFormu-las,Graphs,andMathematicalTables.Dover,NewYork,1972. [2] P.A.AnghelandM.Kaveh,\Exactsymbolerrorprobabilityofacooperativenetworkinarayleigh-fadingenvironment,"IEEETrans.onWirelessCommunica-tions,vol.3,no.5,pp.1416{1421,September2004. [3] P.A.AnghelandM.Kaveh,\Onthediversityofcooperativesystems,"inProc.ofIntl.Conf.onAcoustics,SpeechandSignalProcessing,vol.4,Montreal,Quebec,Canada,May17-21,2004,pp.577{580. [4] P.A.AnghelandM.Kaveh,\Ontheperformanceofdistributedspace-timecodingsystemswithoneandtwonon-regeneratvierelays,"IEEETrans.onWirelessCom-munications,vol.5,no.3,pp.682{692,March2006. [5] P.A.Anghel,M.Kaveh,andZ.Q.Luo,\Optimalrelayedpowerallocationininterfernce-freenon-regenerativecooperativesystems,"inProc.ofSignalProc.WorkshoponAdvancesinWirelessCommunications,Lisbon,Portugal,July11-14,2004,pp.21{25. [6] R.Annavajjala,P.C.Cosman,andL.B.Milstein,\Ontheperformanceofoptimumnoncoherentamplify-and-forwardreceptionforcooperativediversity,"inProc.ofMILCOMConf.,vol.5,Atlanticcity,NJ,oct17-20,2005,pp.3280{3288. [7] R.Annavajjala,P.C.Cosman,andL.B.Milstein,\Statisticalchannelknowledge-basedoptimumpowerallocationforrelayigprotocolsinthehighsnrregime,"IEEEJournalonSelectedAreasinCommunications,vol.25,no.2,pp.292{305,February2007. [8] X.BaoandJ.Li,\Decode-amplify-forward(daf):Anewclassofforwardingstrategyforwirelessrelaychannels,"inProc.ofSignalProc.WorkshoponAdvancesinWirelessCommunications,NewYork,NY,June5-8,2005,pp.816{820. [9] A.BletsasandA.Lippman,\Im;lementingcooperativediverisityantennaarrayswithcommodityhardware,"IEEECommunicationsMagazine,vol.44,no.12,pp.33{49,December2006. [10] J.Boyer,D.D.Falconer,andH.Yanikomeroglu,\Multihopdiversityinwirelessrelayingchannels,"IEEETrans.onCommunications,vol.52,no.10,pp.1820{1830,October2004. [11] B.Can,H.Yomo,andE.D.Carvalho,\Hybridforwardingschemeforcooperativerelayinginofdmbasednetworks,"inProc.ofInternationalConf.onCommunica-tions,vol.10,Istanbul,Turkey,June11-15,2006,pp.4520{4525. 96

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D.ChenandJ.N.Laneman,\Cooperativediversityforwirelessfadingchannelswithoutchannelstateinformation,"inProc.ofAsilomarConf.onSignals,Systems,andComputers,Monterey,CA,November7-10,2004,pp.1307{1312. [13] D.ChenandJ.N.Laneman,\Modulationanddemodulationforcooperativediversityinwirelesssystems,"IEEETrans.onWirelessCommunications,vol.5,no.7,pp.1785{1794,July2006. [14] W.Cho,R.Cao,andL.Yang,\Optimumenergyallocationincooperativenetworks:Acomparativestudy,"inProc.ofMILCOMConf.,Orlando,FL,Oct29-31,2007. [15] W.Cho,R.Cao,andL.Yang,\Optimumresourceallocationforamplify-and-forwardrelaynetworkswithdierentialmodulation,"IEEETrans.onSignalProcessing,June2007(submitted). [16] W.ChoandL.Yang,\Dierentialmodulationschemesforcooperativediversity,"inProc.ofIEEEInternationalConferenceonNetworking,SensingandControl,Ft.Lauderdale,FL,April23-25,2006,pp.813{818. [17] W.ChoandL.Yang,\Distributeddierentialschemesforcooperativewirelessnetworks,"inProc.ofIntl.Conf.onAcoustics,SpeechandSignalProcessing,vol.4,Toulouse,France,May15-19,2006,pp.61{64. [18] W.ChoandL.Yang,\Optimumenergyallocationforcooperativenetworkswithdierentialmodulation,"inProc.ofMILCOMConf.,Washington,DC,Oct23-25,2006. [19] W.ChoandL.Yang,\Jointenergyandlocationoptimizationforrelaynetworkswithdierentialmodulation,"inProc.ofIntl.Conf.onAcoustics,SpeechandSignalProcessing,vol.3,Honolulu,Hawaii,apr15-20,2007,pp.153{156. [20] W.ChoandL.Yang,\Optimumresourceallocationforrelaynetworkswithdierentialmodulation,"IEEETrans.onCommunications,2007(Toappear). [21] W.ChoandL.Yang,\Resourceallocationforamplify-and-forwardrelaynetworkswithdierentialmodulation,"inProc.ofGlobalTelecommunicationsConf.,Washington,D.C.,November26-30,2007(Toappear). [22] X.DengandA.M.Haimovich,\Powerallocationforcooperativerelayinginwirelessnetworks,"IEEECommunicationsLetters,vol.9,no.11,pp.994{996,November2005. [23] M.Dohler,A.Gkelias,andH.Aghvami,\Resourceallocationforfdma-basedregenerativemultihoplinks,"IEEETrans.onWirelessCommunications,vol.3,no.6,pp.1989{1993,November2004. [24] M.Dohler,A.Gkelias,andH.Aghvami,\Aresourceallocationstrategyfordistrubutedmimomulti-hopcommuniationsystems,"IEEECommunicationsLetters,vol.8,no.2,pp.99{101,February2004. 97

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I.S.GradshteynandI.M.Ryzhik,TableofIntegradls,Series,andProducts,6thed.AcademicPress,2000. [26] M.O.HasnaandM.Alouini,\End-to-endperformanceoftransmsiionsystemswithrelaysoverrayleigh-fadingchannels,"IEEETrans.onWirelessCommunications,vol.2,no.6,pp.1126{1131,November2003. [27] M.O.HasnaandM.Alouini,\Optimalpowerallocationforrelayedtransmissionsoverrayleigh-fadingchannels,"IEEETrans.onWirelessCommunications,vol.3,no.6,pp.1999{2004,November2004. [28] M.O.HasnaandM.Alouini,\Aperformancestudyofdual-hoptransmissionswithxedgainrelays,"IEEETrans.onWirelessCommunications,vol.3,no.6,pp.1963{1968,November2004. [29] T.Himsoon,W.Su,andK.J.R.Liu,\Dierentialtransmissionforamplify-and-forwardcooperativecommunications,"IEEESignalProcessingLet-ters,vol.12,no.9,pp.597{600,September2005. [30] T.Himsoon,W.Su,andK.J.R.Liu,\Dierentialmodulationformulti-nodeamplify-and-forwardwirelessrelaynetworks,"inProc.ofWirelessCommunicationsandNetworkingConf.,vol.2,LasVegas,NV,April3{6,2006,pp.1195{1200. [31] B.M.HochwaldandW.Sweldens,\Dierentialunitaryspace-timemodulation,"IEEETrans.onCommunications,vol.48,no.12,pp.2041{2052,December2000. [32] B.L.Hughes,\Dierentialspace-timemodulation,"IEEETrans.onInformationTheory,vol.46,no.7,pp.2567{2578,November2000. [33] A.KannanandJ.R.Barry,\Space-divisonrelay:ahigh-ratecooperationschemeforfadingmultiple-accesschannels,"inProc.ofGlobalTelecommunicationsConf.,Washington,D.C.,November26-30,2007(Toappear). [34] G.K.Karagiannidis,\Performanceboundsofmultihopwirelesscommunicationswithblindrelaysovergeneralizedfadinchannels,"IEEETrans.onWirelessCommunica-tions,vol.5,no.3,pp.498{503,March2006. [35] J.N.Laneman,D.N.C.Tse,andG.W.Wornell,\Cooperativediversityinwirelessnetworks:Ecientprotocolsandoutagebehavior,"IEEETrans.onInformationTheory,vol.50,no.12,pp.3062{3080,December2004. [36] J.N.LanemanandG.W.Wornell,\Energy-ecientantennasharingandrelayingforwirelessnetworks,"inProc.ofWirelessCommunicationsandNetworkingConf.,vol.1,Chicago,IL,September23-28,2000,pp.7{12. [37] J.N.LanemanandG.W.Wornell,\Distributedspace-time-codedprotocolsforexploitingcooperativediverstiyinwirelessnetworks,"IEEETrans.onInformationTheory,vol.49,no.10,pp.2415{2425,October2003. 98

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H.LiandQ.Zhao,\Distributedmodulationforcooperativewirelesscommunications,"IEEESignalProcessingMagazine,vol.23,no.5,pp.30{36,September2006. [39] Y.LiangandV.V.Veeravalli,\Gaussianorthogonalrelaychannels:Optimalresourceallocationandcapacity,"IEEETrans.onInformationTheory,vol.51,no.9,pp.3284{3289,September2005. [40] I.MaricandR.D.Yates,\Forwardingstrategiesforgaussianrarallel-relaynetworks,"inProc.ofConf.onInfo.SciencesandSystems,Princeton,NJ,March17-19,2004. [41] T.Miyano,H.Murata,andK.Araki,\Spacetimecodedcooperativerelayingtechniqueformultihopcommunications,"inProc.ofVehicularTechnologyConf.,vol.7,LosAngeles,CA,September26{29,2004,pp.5140{5144. [42] R.Pabst,B.H.Walke,D.C.Schultz,R.Herhold,H.Yanikomeroglu,S.Mukherjee,H.Viswanathan,M.Lott,W.Zirwas,M.Dohler,H.Aghvami,D.D.Falconer,andG.P.Gettweis,\Relay-baseddeploymentconceptsforwirelessandmobilebroadbandradio,"IEEECommunicationsMagazine,vol.42,no.9,pp.80{89,September2004. [43] C.S.Patel,G.L.Stuber,andT.G.Pratt,\Statisticalpropertiesofamplifyandforwardrelayfadingchannel,"IEEETrans.onVehicularTech.,vol.55,no.1,pp.1{9,January2006. [44] J.Proakis,DigitalCommunications,4thed.McGraw-Hill,NewYork,2001. [45] A.Ribeiro,X.Cai,andG.B.Giannakis,\Symbolerrorprobabilitiesforgeneralcooperativelinks,"IEEETrans.onWirelessCommunications,vol.4,no.3,pp.1264{1273,May2005. [46] A.Scaglione,D.L.Goeckel,andJ.N.Laneman,\Cooperativecommunicationsinmobileadhocnetworks,"IEEESignalProcessingMagazine,vol.23,no.5,pp.18{29,September2006. [47] G.ScutariandS.Barbarossa,\Distributedspace-timecodingforregeneratvierelaynetworks,"IEEETrans.onWirelessCommunications,vol.4,no.5,pp.2387{2399,September2005. [48] A.Sendonaris,E.Erkip,andB.Aazhang,\Usercooperationdiversity,partI:systemdescription,"IEEETrans.onCommunications,vol.51,no.11,pp.1927{1938,November2003. [49] A.Sendonaris,E.Erkip,andB.Aazhang,\Usercooperationdiversity,partII:implementationaspectandperformancealanysis,"IEEETrans.onCommunications,vol.51,no.11,pp.1939{1948,November2003. [50] A.B.H.ShinandM.Z.Win,\Outage-optimalcooperativecommunicationswithregenerativerelays,"inProc.ofConf.onInfo.SciencesandSystems,Princenton,NJ,Mar.22-24,2006,pp.632{637. 99

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M.K.SimonandM.S.Alouini,\Auniedapproachtotheprobabilityoferrorfornoncoherentanddierentiallycoherentmodulationsovergeneralizedfadingchannels,"IEEETrans.onCommunications,vol.46,no.12,pp.1625{1638,December1998. [52] M.K.SimonandM.S.Alouini,DigitalCommunicationoverFadingChannels,2nded.Wiley,2004. [53] V.Stankovic,A.Hst-Madsen,andZ.Xiong,\Cooperativediversityforwirelessadhocnetworks,"IEEESignalProcessingMagazine,vol.23,no.5,pp.37{49,September2006. [54] A.StefanovandE.Erkip,\Cooperativespace-timecodingforwirelessnetworks,"IEEETrans.onCommunications,vol.53,no.11,pp.1804{1809,November2005. [55] G.L.Stuber,PrinciplesofMobileCommunication,2nded.Springer,2001. [56] K.TandB.S.Rajan,\Partially-cohernetdistributedspace-timecodeswithdierentialencoderanddecoder,"IEEEJournalonSelectedAreasinCommuni-cations,vol.25,no.2,pp.426{433,February2007. [57] P.Tarasak,H.Minn,andV.K.Bhargava,\Dierentialmodulationfortwo-usercooperativediversitysystems,"IEEEJournalonSelectedAreasinCommunications,vol.23,no.9,pp.1891{1900,September2005. [58] V.TarokhandH.Jafarkhani,\Adierentialdetectionschemefortransmitdiversity,"IEEEJournalonSelectedAreasinCommunications,vol.18,no.7,pp.1169{1174,July2000. [59] V.Tarokh,H.Jafarkhani,andA.R.Calderbank,\Space-timeblockcodesfromorthogonaldesigns,"IEEETrans.onInformationTheory,vol.45,no.5,pp.1456{1467,July1999. [60] V.Tarokh,H.Jafarkhani,andA.R.Calderbank,\Space-timeblockcodingforwirelesscommunications:performanceresults,"IEEEJournalonSelectedAreasinCommunications,vol.17,no.3,pp.451{460,March1999. [61] V.Tarokh,N.Seshadri,andA.R.Calderbank,\Space-timecodesforhighdataratewirelesscommunication:Performancecriterionandcodeconstruction,"IEEETrans.onInformationTheory,vol.44,no.2,pp.744{765,March1998. [62] M.Uysal,O.Canpolat,andM.M.Fareed,\Asymptoticperformanceanalysisofdistrubutedspace-timecodes,"IEEECommunicationsLetters,vol.10,no.11,pp.775{777,November2006. [63] G.Wang,Y.Zhang,andM.Amin,\Dierentialdistributedspace-timemodulationforcooperativenetworks,"IEEETrans.onWirelessCommunications,vol.5,no.11,pp.3097{3180,November2006. 100

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T.Wang,Y.Yao,andG.B.Giannakis,\Non-coherentdistributedspace-timeprocessingformultiusercooperativetransmissions,"inProc.ofGlobalTelecommuni-cationsConf.,vol.6,St.Louis,MO,November28-December2,2005,pp.3738{3742. [65] Y.Yao,X.Cai,andG.B.Giannakis,\Onenergyeciencyandoptimumresourceallocationofrelaytransmissionsinthelow-powerregime,"IEEETrans.onWirelessCommunications,vol.4,no.6,pp.2917{2927,November2005. [66] S.Yiu,R.Schober,andL.Lampe,\Performanceanddesignofspace-timecodinginfadingchannels,"IEEETrans.onCommunications,vol.54,no.7,pp.1195{1206,July2006. [67] M.Yu,J.Li,andH.Sadjadpour,\Amplify-forwardanddecode-forward:Theimpactoflocationandcapacitycontour,"inProc.ofMILCOMConf.,vol.3,Atlanticcity,NJ,October17-20,2005,pp.1609{1615. [68] J.Yuan,Z.Chen,B.Vucetic,andW.Firmanto,\Performanceanddesignofspace-timecodinginfadingchannels,"IEEETrans.onCommunications,vol.51,no.12,pp.1991{1996,December2003. [69] J.ZhangandT.M.Lok,\Performancecomparionofconventionalandcooperaitvemultihoptransmission,"inProc.ofWirelessCommunicationsandNetworkingConf.,vol.2,LasVegas,NV,April3-6,2006,pp.897{901. [70] Y.Zhang,\Dierentialmodulationschemesfordecode-and-forwardcooperativediversity,"inProc.ofIntl.Conf.onAcoustics,SpeechandSignalProcessing,vol.4,Philadelphia,PA,March19-23,2005,pp.917{920. [71] Q.ZhaoandH.Li,\Performanceofadierentialmodulationschemewithwirelessrelaysinrayleighfadingchannels,"inProc.ofAsilomarConf.onSignals,Systems,andComputers,vol.1,Monterey,CA,November7-10,2004,pp.1198{1202. [72] Q.ZhaoandH.Li,\Performanceofdierentialmodulationwithwirelessrelaysinrayleighfadingchannels,"IEEECommunicationsLetters,vol.9,no.4,pp.343{345,April2005. [73] Q.ZhaoandH.Li,\Dierentialmodulationforcooperativewirelesssystems,"IEEETrans.onSignalProcessing,vol.55,no.5,pp.2273{2283,May2007. 101

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WoongChowasborninTongyoung,SouthKorea.HereceivedhisB.S.degreeinelectronicsengineeringfromUniversityofUlsan,Ulsan,SouthKorea,in1997andhisM.S.degreeinelectroniccommunicationsengineeringfromHanyangUniversity,Seoul,SouthKorea,in1999.FromMarch1999toAugust2000,wewasaresearchengineerindivisionofmobiletelecommunicationatHyundaiElectronics.HereceivedhisM.S.degreeformelectricalengineeringfromUniversityofSouthernCalifornia,LosAngeles,CA,in2003.SinceAugust2003,hehasbeenaPh.DstudentinelectricalandcomputerengineeringatUniversityofFlorida,Gainesville,FL.Hisresearchinterestsincludecommunications,signalprocessing,andnetworking.HereceivedhisPh.Ddegreein2007. 102