Citation |

- Permanent Link:
- http://ufdc.ufl.edu/UFE0041615/00001
## Material Information- Title:
- Cooperative Communication in Wireless Networks Flow-optimized Designs and Information-theoretic Characterizations
- Creator:
- Chatterjee, Debdeep
- Place of Publication:
- [Gainesville, Fla.]
- Publisher:
- University of Florida
- Publication Date:
- 2010
- Language:
- english
- Physical Description:
- 1 online resource (126 p.)
## Thesis/Dissertation Information- Degree:
- Doctorate ( Ph.D.)
- Degree Grantor:
- University of Florida
- Degree Disciplines:
- Electrical and Computer Engineering
- Committee Chair:
- Wong, Tan F.
- Committee Members:
- Shea, John M.
Fang, Yuguang Hager, William W. - Graduation Date:
- 8/7/2010
## Subjects- Subjects / Keywords:
- Broadcasting industry ( jstor )
Channel coding ( jstor ) Cognitive models ( jstor ) Communication systems ( jstor ) Decryption ( jstor ) Multiple access ( jstor ) Power gain ( jstor ) Random variables ( jstor ) Tradeoffs ( jstor ) Transmitters ( jstor ) Electrical and Computer Engineering -- Dissertations, Academic -- UF achievable, bargaining, cognitive, communications, cooperative, discrete, flow, game, interference, memoryless, minimax, multiple, nbs, optimization, rates, relay, wireless - Genre:
- Electronic Thesis or Dissertation
bibliography ( marcgt ) theses ( marcgt ) government publication (state, provincial, terriorial, dependent) ( marcgt ) Electrical and Computer Engineering thesis, Ph.D.
## Notes- Abstract:
- COOPERATIVE COMMUNICATION IN WIRELESS NETWORKS: FLOW-OPTIMIZED DESIGNS AND INFORMATION-THEORETIC CHARACTERIZATIONS The challenges involved in the design of efficient communication systems for the wireless medium can be attributed to the fact that the wireless medium possesses certain unique characteristics, the most important ones being the broadcast nature of the wireless medium, the susceptibility to interference effects, and the effects of path loss and fading on wireless link quality. Cooperation between different transceivers can potentially aid further development of next-generation wireless communication systems that demand high data rates and an excellent quality of service (QoS). This is possible by exploiting the broadcast nature of the wireless medium, and the diversity advantages that a multi-user system offers. We first consider a general single-source-single-destination wireless relay network and propose an information flow-optimized cooperative transmission design that achieves the optimal diversity-multiplexing tradeoff. Next, we apply game-theoretic techniques to the problems of resource allocation and characterization of cooperative behavior in a two-user fading multiple-access channel (MAC), with uncertainty about the channel state information at the transmitters (CSIT). In the third part of the dissertation, a more active form the above cooperative behavior is studied via a two-user fading cooperative multiple-access channel (CMAC), where each user, along with transmitting its own information to the destination, helps the other by forwarding the latter's information. We propose efficient cooperative transmission strategies based on a flow-theoretic approach, and evaluate their performances using numerical simulations. Finally, we consider communication through a two-user interference channel with unidirectional cooperation (ICUC), wherein one source uses its knowledge of the message of the other to reduce the interference to its own transmission, and simultaneously, help the other user-pair via cooperative relaying. We consider a very realistic scenario in which the cooperating source is subjected to a causality constraint. We derive a new achievable rate region for the discrete memoryless version of this form of ICUC, and demonstrate the contributions of the various coding strategies involved via numerical simulations for Gaussian channels. We also study the same channel with the cooperating source being subject to the half-duplex constraint as well. A discrete memoryless channel model incorporating the half-duplex constraint is presented, and a new achievable rate region, that enlarges the largest known rate region for the Gaussian version of this channel, is derived for this channel. The achievable rate region for the proposed coding scheme, specialized for Gaussian channels, is numerically evaluated and the strict inclusion of the previously known largest rate region is demonstrated. ( en )
- 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.
- Thesis:
- Thesis (Ph.D.)--University of Florida, 2010.
- Local:
- Adviser: Wong, Tan F.
- Statement of Responsibility:
- by Debdeep Chatterjee.
## Record Information- Source Institution:
- UFRGP
- Rights Management:
- Applicable rights reserved.
- Embargo Date:
- 10/9/2010
- Resource Identifier:
- 004979565 ( ALEPH )
709593902 ( OCLC ) - Classification:
- LD1780 2010 ( lcc )
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R S 3 D X3 t1 (graph G1) t2 (graph G2) Figure 2-1. Basic graphs G1 and G2 for the three-node relay network with ti + t2 1. R1 f R1 [ D {S :x D > S x7 D (S Figure 2-2. FO protocol for the four-node relay network with tI + + t6 = 1. The flow optimization is performed over all flows x1, X x14, and all time slot lengths tl, t6. where Zs0, Zp, ZDc ~ AV(O, 1) are i.i.d. random variables denoting the additive noise at Sc, Dp, and Dc respectively. gpc, hpc, and hcp are positive reals that denote the channel gains for the links from Sp to Sc, Sp to Dc, and from Sc to Dp respectively. Also, the primary and cognitive sources are subjected to their respective power constraints Pp and Pc: X-ll |ll2 Pp, -XllX|ll2 Pc. (5-17) Let ap, /3p, 7p, 6p, ac, /tc, 7c be real numbers in the interval [0, 1] such that ap + /3p + 7p + 6p < 1. Also, let I 1 TI for I E {ac, c,37c}. We evaluate the rate region of Theorem 5.1 for the case of Gaussian channels with the following transmitted signals in block be {1,... ,B}: Xp(wp,b- 1, Wp,b) = Tpco + Xpo + Tpr + Xppr (5-18) acycPc acycPc' Xc(wp,b- 1, wc,b) = Xco + Xcpr + VTpco \P Tp +TP+p (5-19) aprp 7p p where Tp,, ~ N(O, apPp), Xpo ~ N(O, 0,pPp), Tpr ~ AN(0, 7pPp), Xpp ~ AN(0, pPp), Xcco -~ (0, acicPc), and Xcpr -~ (0, acicPc) are i.i.d. random variables. Xcco and Xcpr are transmitted to communicate the dirty paper coded messages wcco,b and "', I , respectively, with Ucco and Ucpr being the corresponding auxiliary random variables as in [64]. For the coding scheme in Theorem 5.1, it is necessary that global channel state infor- mation (CSI) is available at all nodes of the network. Also, the probability distributions of the different codewords for the two encoders would be required to facilitate efficient rate selection. Moreover, the different coding strategies used in obtaining the achievable rate region of Theorem 5.1 involve certain assumptions regarding the knowledge of the code- book at each node. In particular, cooperative relaying of the primary message by Sc, as well as DPC at Sc, requires the knowledge of the primary codebook at Sc. On the other hand, rate-splitting of the primary message requires Dc to know the primary codebook. LIST OF TABLES Table page 5-1 Description of Random Variables in Theorem 5.1 .... 84 5-2 Description of Random Variables in Theorem 5.2 .... 96 and strong interference at Dp. Moreover, the increase in the difference in the regions for "Coding I" and "Coding IV", as Pp increases from 1.5 to 6, shows the benefit of DPC at Sc when the effective interference at Dc increases with Pp. From these results, it appears that using cooperative relaying with DPC and rate-splitting of the primary message may be practically more suitable strategies (in terms of codebook knowledge requirements at the primary user-pair), except when the primary destination experiences strong interference. 5.5 Discrete Memoryless Channel Model for the ICUC-HDC Till the previous section we have been assuming that the cognitive source can operate in full-duplex mode by performing perfect echo cancelation. Here, we remove the full- duplex assumption, and introduce the discrete memoryless channel model for the ICUC with half-duplex and causality constraints (ICUC-HDC). The ICUC-HDC is depicted in Fig. 5-6, wherein the primary source node Sp intends to transmit information to its destination node Dp. A cognitive (or secondary) source-destination pair, Soc and Dc, wishes to communicate as well, with Soc having its own information to transmit to Dc. As in the case of the ICUC-C, the primary message is only causally available at Sc. To incorporate the half-duplex constraint for the discrete memoryless channel model, we consider a second input at Sc, S, to indicate the state of Sc listening or transmitting. With this, the channel transition probability is determined by the state of the cognitive source as follows: p(yp,yc,vc xpxcif s 1 (5-20) p(yp, | p, xc)6e(vc) ifs t, where e denotes an erasure at Sc, and 6e(vc) 1 if vc = e and 0 otherwise. To incorporate the fact that So cannot transmit when in the listening state, we restrict the joint probability distribution of the inputs as p(xp, xc, s) = p(xp s = 1)66(xc)p(s = 1) + p(xp,xc s t= )p(s = t), where Q is the "null" symbol. Remark 5.4, the increase in the achievable rate region using a randomized listen-transmit schedule over that with a fixed schedule is upper bounded by one bit. Moreover, it should be noted that, for a randomized listen-transmit schedule, the optimal distribution for the random variable Xc may not be Gaussian [73]. For the above model, we consider a fixed listen-transmit schedule and have the following input-output relationships for the ICUC-HDC: Vc, gpcXp,i + Zsc, (5-25) YP,I = Xpl + Zp, (5-26) Yc, = hpcXp,+l ZD, (5-27) YP, t Xp,t + hcpXc + Zp, (5-28) Yc,t = hpcXp,t + Xc + ZD, (5-29) where Zsc, Zp, ZDc A/'(0, 1) are i.i.d. random variables corresponding to the additive noise at Sc, Dp, and Dc respectively. In the above, Xp,i and Xp,t are the transmitted signals during the listen and transmit states respectively. Similar notation is used to describe the received signals at the concerned nodes as well. Finally, it is assumed that the primary and cognitive sources are subject to the power constraints of (5-17) for each state within any communication block. This may be interpreted as the scenario in which both sources are constrained by their respective (listen/transmit) mode power constraints, instead of an average power constraint over a block. It may be noted that average power constraints, of a similar flavor as in [73], may be included along with the mode power constraints in our system model, but is avoided here for the ease of presentation and exposition of the main aspects of the coding scheme. To summarize, the power constraints in this section may be expressed as: S|X 112 < p, Xp,t 112 < Pp, IX | 112 < Pc, (5 30) 17 1Ut 1t with a TS strategy that is optimized to maximize the spectral efficiency (which we call "rate" hereafter for convenience). To avoid interference between concurrent transmissions, a time interval is divided into slots: * During the first slot, the source may BC to all the other nodes in the network, subject to its power constraint P. During the subsequent slots, a relay may BC to all other nodes (except the source node), or it may receive flows from all other nodes (except from the destination) through MA. During the very last slot, the source and the r-el ,- may send information flows to the destination using MA, with the flows in the MA capacity region corresponding to a maximum transmit power of P for each node. Note that the forwarding of information by the rel ,i- is based on the DF approach. For practicality consideration, it is assumed that the phases of the simultaneously transmitted signals from different nodes are not synchronized. In general, for the above transmission protocol, there would be a maximum of 2(N 2) + 2 = 2N 2 time slots of lengths t1, t2, t2N-2 respectively. Next, we describe the optimization problem using a graph-theoretic formulation. Define a graph G (V, E), where V is the set of nodes, E is the set of all links joining the nodes in the graph, and associate the vector r to represent the flow rates associated with each link in E. Thus, the number of elements in r equals the cardinality of E. For convenience, we write G = (V, E, r). Now denote the source by S, the destination by D, and the relay nodes by R7,... ,7RN-2. The slotting of a unit time interval, as described above, yields simpler graphs for each time slot, that we call basic '.i-'l, A basic graph is either one in which a particular node may BC to several nodes, or in which several nodes transmit via MA to a particular node. Thus for a basic graph, we need to include only the links between the nodes that may participate during the concerned time slot. For example, assume that the relay Ri broadcasts to all nodes other than the source, during the i-th time slot. The basic graph is given by Gi = (V, E, r) where V = {S, R1, RN-2, D}, E { = {RiR2, I 2,7 rN - Non-uniform average power gains: Case A Spectral efficiency, R=1bits/s/Hz 0 2 4 6 8 10 12 14 SNR (dB) Figure 2-8. Four-node relay E[Zsg] 2.0, 1.5, E[ZR2v] 10-1 _1-2 10 ----- 10-1 10 1 I network with non-uniform average power gains. Case A: E[Zsg2] 2.0, E[Zsv] = 1.0, E[ZgZ ] 1.0, E[ZD] 1 1.0. Outage probabilities for required rate R l= bit/s/Hz. Non-uniform average power gains: Case A Max-min selection routing : Generalized-link selection : Flow-optimized protocol Lower bound 15 20 25 SNR (dB) 30 35 Figure 2-9. Four-node relay network with non-uniform average power gains. Case A: E[Zs,] 2.0, E[Zsg2] 2.0, E[Zsv] = 1.0, E[ZR1Z] 1.0, E[ZRD] - 1.5, E[Z2D] = 1.0. Outage probabilities for required rate R = 6bit/s/Hz. 100 10-1 -Q o 10- 05 0 10-2 10-4 the outage probability that is obtained by considering the case when the sources have a perfect noiseless channel between them. Figure 4-5 presents the outage performances for the .,i-,ii io. Ii ic situation as in the average rates case, and Figure 4-6 presents the same for the symmetric situation. We see that the OR-C \!.AC suffers a loss of only 0.7 dB and 1.0 dB compared to FO-CM\iAC, at an outage probability of 10-3, for the .,i-i, i,. 1 lic and symmetric situations respectively. The performance of FO-C \! AC is worse than the lower bound by about 2.2 dB and 2.5 dB at the same outage level, for the .,i-iii,,. I lic and symmetric situations respectively. On the other hand, the performance of the DF strategy of [58] is significantly poorer, and even worse than the conventional MAC for the symmetric situation. Another important observation from the outage performance plots is that the slopes of the curves for FO-C \! AC and OR-C \! AC are identical to that for the lower bound curve. Since the latter is identical to the 2 x 1 MISO point-to-point system, it gives a diversity order of two. Hence, the above observation establishes the fact that both the proposed protocols achieve the optimal diversity order of two for the two-user C'\!.AC for the required rate region of interest. 4.4 Summary In this chapter, we proposed flow-theoretic cooperative transmission protocols for the two-user fading C'\ .AC, where the nodes are only capable of half-duplex communication and have access to full CSI. We propose two such protocols, viz. a flow-optimized protocol for the C'\! AC (FO-C \ IAC), and a suboptimal but simpler orthogonal relaying protocol for the C'\ !AC (OR-C \ IAC). Both the proposed protocols are evaluated in terms of achievable rate regions and outage performances. Numerical results show that FO-C'\!.AC yields the largest achievable rate region amongst the different protocols considered here. Although OR-C'\ !AC is clearly suboptimal in terms of the achievable rate region, its outage performance is close (to within 1dB for the scenarios considered) to FO-C'\! AC, and both the proposed protocols achieve the optimal diversity order of two for the required rate region of interest. in [49]. The common outage probability of the MAC is considered, and the bounds on the fading MAC feasible rate region from [47] are used to demonstrate the effect of imperfect CSI on the finite-SNR diversity-multiplexing tradeoff. In this work, we model the resource allocation problem for the two-user fading MAC using a two-person bargaining problem [50], wherein the extent of cooperative behavior is determined by the outcome of the bargaining problem. In this work, we consider the situation when the utility derived by each user is the average rate (over all fading states). When the CSIT is perfect, the solution to the bargaining problem, specified by the NBS, yields the optimal transmission strategy pair for the two users with regard to fairness and efficiency considerations. Here, a transmission strl it; for each user corresponds to a choice of transmission rate and power for a particular fading state. We consider the situation wherein the receiver has access to perfect CSI, but there exists a certain uncertainty regarding the CSIT that may stem from quantization (as, in reality, the feedback channels are likely to have limited capacities) and/or prediction errors. If the available CSIT is inaccurate, the transmission strategy pair -,- -1i ,1 by the NBS may deviate from the true optimum, and thus, lead to considerable performance degradation in terms of the true utilities. To overcome this lack of robustness we propose a scheme in which the conventional two-person bargaining problem is relaxed to acknowl- edge the fact that the NBS may not give the optimal strategy pair. According to this modified bargaining problem formulation, each user independently decides its transmission strategy via a maximin optimization from its respective set of possible strategies. For a particular user, such a set is a range of transmission strategies about a nominal str iI The nominal strategy is obtaining using the NBS to the original bargaining problem and the available CSIT. This is in contrast to conventional bargaining problem formula- tions, wherein, once the p1l -i, rs reach an agreement, they are bound to execute the exact strategy pair 1- .-.: -1. I by the NBS. GLS protocol is slightly worse than that of the FO protocol. This -i-:.-. -1 that the GLS protocol can be used in systems with low-complexity requirements. We also note that the proposed FO and GLS protocols can be used in wireless networks with topologies more complicated that the wireless relay network considered here. For example, application of similar ideas to a parallel relay network in which there is no direct connection between the source and the destination is considered in [37]. listen-transmit schedule for the cognitive source, to derive a new achievable rate region for this channel. For Gaussian channels, we proved the containment of the previously known rate region [6] in the new rate region for the Gaussian ICUC-HDC, and demonstrated this with numerical simulation results. 6.2 Future Directions The flow-theoretic approach introduced in C'!i pter 2 could also be applied to multi- user systems that involve more complex user cooperation than the ones considered in this dissertation. For instance, the flow-theoretic approach would be suitable for the problem of information transmission in a cooperative relay broadcast channel (RBC) [5], wherein one source node broadcasts information to two receiver nodes, who now actively cooperate (fully or partially) with each other in decoding their respective messages. Although achievable rate regions employing block Markov coding along with decode-and-forward and estimate-and-forward [2, Theorem 6] techniques have been proposed in the literature, the appeal of the flow-theoretic approach lies in its simplicity. The flow-theoretic approach essentially breaks down the original channel into much simpler channels for which the capacity regions are known and practical coding schemes that perform close to the random coding scheme have been extensively investigated in the literature. For the RBC, a similar time-slotting approach of C'! Ipter 4 would now involve one BC and two MA with common information time-slots, and it would be interesting to investigate as to how the performance of the flow-optimized solution compares to the more complicated block Markov methods in different channel conditions and SNR regimes. In C'!i pter 5, we presented new achievable rate regions for the ICUC-C and ICUC- HDC for both discrete memoryless and Gaussian channels. Although we have shown the inclusion of previously proposed rate regions both analytically as well as through numerical simulations, we still do not know how close we are to the capacity regions for these channels. In this regard, new outer bounds for these channels that are tighter than the MIMO broadcast channel capacity region [77] would be necessary. For Gaussian At Dc: The cognitive destination Dc also waits until block B, and then performs backward decoding to jointly decode the messages intended for it and the common part of the primary message. For block b {B 1, ... 2}, Dc is assumed to have successfully decoded WPco,b from block b + 1. With this knowledge, it searches for a unique tuple (wpco, wco, bco, cpr, pr, bcpr) such that (tco (Pco), xc ( p 0Pco, WPco,b), U0co wpco, wCco, Cco), U'pr WPco, co, bcco, Oi. bcpr ) ,b) *YA. Error Analysis: Throughout the analysis, we assume, without loss of generality, that all transmitted messages at the primary and cognitive sources, in any block b {1, ... B}, were ones. Encoding errors at Sc: An encoding error occurs at Sc under two circumstances. An error occurs if, in block b, Sc cannot find a bin index bcco,b such that (5-3) is not satisfied for Wpo = 1, wppr 1, and Wcco,b = 1. This event occurs with probability L ] Pr U ( i PTo(1), Uro(1, bCcob), Tpr(1, t)) bCco,b=1 (t Pr [(Trco(t), Uco(t, t, bCco,b),r (tt)) pA < ( -1 c)2-n[I(Ucco;TPpr TP)+co)]0 where e > 0 can be arbitrarily small, and the last two inequalities are due to the joint Asymptotic Equipartition Property (AEP) [33] and the fact that (1 x)" < e-' respectively. Clearly, the above probability can be made arbitrarily small if Rcco > I (Ucco; TPpr Tpco) + 6o. (5-5) considerably increase the achievable rate region, especially in the direction of the primary users' rate, Rp (cf. Remark 5.3). 5.8 Summary In this chapter, a new achievable rate region for the discrete memoryless interference channel with unidirectional cooperation (ICUC), wherein the primary message may only be causally available at the cognitive source, is derived. The coding scheme, specialized for Gaussian channels is also presented and is used to numerically evaluate different coding strategies that are used as building blocks for the proposed coding scheme. These results also demonstrate that the proposed coding scheme significantly enlarges the previously known rate region for various network scenarios. A discrete memoryless channel model for the ICUC-HDC was also presented in this chapter. A random coding scheme, employing block Markov SPC, conditional rate-splitting of primary and secondary messages, conditional inning, and a randomized listen-transmit schedule for the cognitive source, was used to derive a new achievable rate region for this channel. For Gaussian channels, the containment of the previously known rate region [6] in the new rate region was analytically proved, and numerical examples were presented to supplement the analytical comparison. CHAPTER 6 CONCLUSIONS AND FUTURE WORK 6.1 Conclusions In this work, we have studied different levels of cooperation manifested in a variety of multi-user communication system from flow-theoretic and information-theoretic perspectives. For the single-source-single-destination wireless cluster with dedicated rel -i, we proposed cooperative transmission protocols using a flow-theoretic approach. This included the FO protocol and the suboptimal, but much simpler, GLS protocol. Both the protocols are shown to achieve the optimal diversity-multiplexing tradeoff, and the GLS protocol is shown to be a very good candidate for use in systems with low complexity requirements. Simulation results for different cluster sizes, and uniform and non-uniform average power gains indicate that the proposed protocols provide large coding gains by efficiently utilizing the CSI available at all nodes, and that they perform much better than other path selection methods previously proposed in the literature, especially in high data rate requirement situations. We next considered the two-user fading MAC as an example of a multi-source system. We proposed a game-theoretic formulation involving bargaining and maximin games to model the resource allocation problem and develop a characterization of cooperative behavior for this system under uncertainty regarding the accuracy of the CSIT. To improve the robustness of the system, we proposed that the conventional bargaining problem be relaxed so that the users, instead of being bound to execute the strategy pair - .-.- -. ,.1 by the solution to the conventional bargaining problem, may independently choose their transmission strategy from their respective set of strategies defined by the maximum deviation parameters about the nominal strategy pair. This reduces the dependence of the system performance on the solution to the bargaining problem with the (possibly inaccurate) available CSIT. From the development of this formulation, it can be seen that even in the conventional two-user MAC, there exists a certain level of user CHAPTER 3 RESOURCE ALLOCATION AND COOPERATIVE BEHAVIOR IN FADING MULTIPLE-ACCESS CHANNELS UNDER UNCERTAINTY 3.1 Introduction The resource allocation problem for multi-user wireless systems has generated considerable interest in the research community and has been considered from different perspectives with regard to efficiency and fairness issues. The fading MAC is one of the basic amongst such systems and different solutions have been proposed till date to the above-mentioned problem. The throughput capacity region of the fading MAC, which can be achieved by using dynamic power and rate allocation schemes to maximize the average rate, has been characterized in [38]. Using ideas similar to those for long-term power control in [39], the outage capacity region for this channel has been derived in [40], where the users have average power constraints. The authors, in [40], obtain both the common outage capacity region, when all the users have a common outage probability constraint, and also the individual outage capacity region, when users may have different outage requirements. For the latter case, the capacity region is characterized by a channel usage reward vector, which determines the actual operating point for the system. A game-theoretic approach towards solving the resource allocation problem with the average rate utility and users subject to average power constraints, is considered in [41]. They propose a Stackelberg formulation, where the receiver is the game leader and the transmitters p1 iv a water-filling game, where the order of decoding of the users' information, which implies a prioritizing of the users, may be decided by the receiver using an auctioning process as in [42]. A low-complexity dynamic rate allocation policy, that maximizes a general concave utility function of the rates in the throughput capacity region for fixed transmission powers, is presented in [43]. In [44], the optimal power control scheme for maximizing the sum-capacity in the multiple input multiple output (\!I\ 10) fading MAC is characterized. On the other hand, corresponding to Case ii., when t2 > t2max, from (2-18), we have max X(t,t2) < -(1 t2 a)C(ZsS) +t2maxC(Z-DS) {0 C(ZszS) + C(ZKDS + ZSDS) C(ZSDS) where the inequality in (2-20) is obtained from (2-18) by using Zs- > ZsD and t2 > t2max. Hence, from (2-19) and (2-20), we conclude that when ZS-R > ZsD, the maximum achievable rate is given by X(S) maxo t2C(ZSDS + ZvDS). Therefore, the maximum achievable rate of information transmission from the source S to the destination D for different cases can be summarized as under: The maximum information rate from the source S to the destination D for different cases is summarized below: a) ZSD > ZSR: The maximum rate is X(S) = C(ZSDS) with any ti, t2 pair such that t1, t2 > 0 and tj + t2 = 1. This corresponds to directly transmitting all data through the link from the source to the destination, without utilizing the relay. b) ZsD < ZS-R: The maximum rate is X(S) maxo t2C(ZsDS + ZKvS) with t2max C(ZsKS)/ [C(ZSKS) + C(ZKDS + ZsDS) -C(ZSDS) and t\ = 1 t2. Thus for a given power limit (i.e. a given S) at the nodes, relaying is advantageous only when ZSD < ZSR. Further, the optimal solution aliv- allocates a non-zero flow to the direct link. Also, the relay-destination link gain ZKD does not influence the strategy of transmission (i.e. whether to use only the direct link or both the relay and direct links), but only the amount of information through the relay link. 2.3.2 Generalized-link Selection For the general N-node relay network, the flow optimization solution can be compu- tationally demanding even for moderate values of N. Unfortunately, the existing simple channels, one way to approach this problem would be to consider the deterministic version for the ICUC-C (without any randomness in the channels) and model the relationship between the deterministic and their Gaussian counterparts as used in [16]. Yet another research direction that may be pursued in regard to the ICUC-C is the study of the role of transmitter side information at both the primary and cognitive sources. In Section 5.4, we demonstrated the interp-lv, between the different extent of codebook knowledge at the different nodes and the effect of the different coding building blocks like DPC, rate-splitting, and cooperative relaying for various channel and transmit power conditions. This may be considered as a special case of the study of the relationship between a general abstraction of side information at the nodes, the set of coding techniques that may be feasible, and the resulting achievable rate regions. One possible way to model this may be to consider the channels having different states with different levels of information about these states available at the two sources. Related to this, one may also ask the question whether a cognitive source with only signal-level cognition, instead of message-level cognition can help in enlarging the Han-Ko-,-vi -hi region for the traditional two-user interference channel. In other words, the problem would be to determine the coding strategies that may be used by the cognitive source when the level of "cognition" regarding the primary message is at the signal-level instead of the message-level, and if, the resulting achievable rates could improve upon the usual interference avoidance or interference control methods like the interweave or underlay modes [78] of cognitive radio operation. We observe that using the right-most expression of (2-19) instead of Xk(S), for each k c Ko, in (2-24) gives an upper bound on P, rt(r, S). This is utilized in obtain- ing a lower bound on the diversity order of the GLS protocol. Let {S }- 1 be an increasing unbounded sequence of SNRs with S1 > 1. Define the sequence of ran- dom variables {3[} _, {Bk 1 and {Ak 1 with = c(zs -c(zss) n-i n-i nn- C(ZS-DSn+ZR,-DS.) Bk = c(zsRkS ), and A 1 = =S 1 respectively. Note that for all k e Ko, logSI k log XII 3I' -+ 0 a.s. This implies that (B, Ak) -+ 0 a.s. Define A' = maxkeKo A and B' = m--::, Ko B ( B1). Then using the above, it can be seen that (B, A') 0 a.s. Further, lim Pr (B' < rl IKol i) exists, and therefore the above implies that lim Pr (A' < r| I Ko i) lim Pr (B, < r IKIKo i). Using this in (2-22) and (2-24), n-*oo n-*oo the diversity order for the GLS protocol, dgr(r), satisfies dgr) > lira log Pgrt (r, S) -s-oo log S > lim log Pr ZsD < S- =I 0 Pr(|Ko| 0) no log S L Sn. N-2 S s + Pr ZS, < Kol i Pr(|Ko| i) (2-25) -S1 S -1 = lim log Pr max{Zs ZS ,*** ,Z ZS R2 }< Ko- Pr(Ko 0) n-oo log S, SS N-2 (2 2) + Pr max{IZ s, Z ZSRN-} < |Ko| = i Pr(|Ko| = i) (2-26) i= 1 n / - S- log (Pr (max{Zs, zs,, zsRN } < s 1) s-oo log S where (2-26) is obtained from (2-25) by noting that max{ZsD, ZsR, 1 ZsR,_,} = ZSD when Kol = 0, and max{Zs-, ZsRI, 1 ZR_ 2} 1= m-'i --Ko ZsR, = ZsR, when IKol > 0, the first equality in (2-27) is due to the link gains being i.i.d., and the second equality in (2-27) is obtained by using L'Hospital's rule. to formulate a flow theoretic convex optimization problem based on the channel condi- tions. Instead of considering a total power constraint for all the transmitting nodes as in [30], we subject each node to a maximum transmit power constraint. This yields a more reasonable system model for a general wireless relay network, especially when the number of nodes in the relay network is large. The resulting relaying protocol will be referred to as the flow-optimized (FO) protocol. To obtain a more practical cooperative design we develop the generalized-link selection (GLS) protocol, in which we select the best relay node out of the available ones to form an equivalent three-node relay network to transmit the information from the source to the destination. The benefit of this, over other network path selection strategies, becomes evident when the rate requirement is high. It is shown that the simple GLS protocol is optimal in terms of the DMT [31] and yields acceptable performance even when the rate requirement is high. Recently, in [32], the authors have shown that compress-and-forward (CF) relaying achieves the optimal DMT for the three-node, half-duplex network, and that DF relaying can achieve the optimal DMT of the four-node full-duplex network. In this work, we show that the optimal DMT can be achieved for a general N-node (N > 3) half-duplex network using the FO or GLS protocols. Here, it should be clarified that we consider that the wireless links between each node-pair experience independent Rayleigh f ,ii:. and this corresponds to the definition of non-' /.,;/ networked networks in [32]. The performances of the FO and GLS protocols are evaluated numerically in terms of their outage probabilities for four- and five-node relay networks for uniform and non-uniform average power gains. The numerical results motivate the use of the GLS protocol for situations where computation complexity is an issue and show a remarkable improvement over the max-min selection method of [27]. The proposed designs, based on BC and MA alone, are sub-optimal in general. For a fair appraisal of the proposed protocols, we compare the proposed protocols to an upper bound on the maximum rate, derived using the max-flow-min-cut theorem [33, Thm. 14.10.1]. S x 12 t3 t4 Figure 2-3. Transmission strategy to obtain a lower bound on the outage probability for the four-node relay network. Here t1 + + t4 = 1, and the optimization is over xi, x14, and ti, t4, with the application of the max-flow-min-cut theorem for the intermediate slots. definition of diversity order (2-21), we have log Pout (r, 5') d(r) > lim -ilog P(2-22) s-o log S Moreover, the above result from [30] can be directly used to prove the same for the GLS protocol. Using this fact, we derive a lower bound to the diversity-multiplexing tradeoff that can be achieved by the GLS protocol. The sets I and K, used in the sequel, are the sets of indices as described in Section 2.3.2. The outage probability for the GLS protocol is given by Pt (r, S) Pr (maxXk(S) < r logS (2-23) where Xk(S) is the maximum rate achievable by the three-node relay network formed by the source S, the relay Rk and the destination D. We have the following possibilities: * Case A: IKI = 0, i.e. the cardinality of the set K is zero. This corresponds to the case when ZSD > Zs-R for all k E I. Case B: KI =i with i E I. Note that for Case B there are (N -2) possibilities for the set K with cardinality i. Since the link gains are assumed to be i.i.d., and the outage probability depends on the distribu- tion of the maximum of Xk(S) over all k e I (or effectively, over all k e K when IK > 0), only the cardinality of K is significant. Let the (N -2) possible constructions of the set K be represented by a ;, i:, i c" set Ko with cardinality i. Without loss of generality, we describe Ko as the set K corresponding to the case when the indices of the relay nodes are ordered according to their source-relay link gains, i.e. ZsgRz > ZS.R2 > > ZS-Rg 2. Thus, Case B now implies a solitary choice for set K, viz. Ko {1, 2,... i}. Therefore, from (2-23), we have gt(r,S) = Pr(C(Zs-DS) < rlog S IKo 0) Pr(Ko =0) N-2 + Pr maxXk(S) < r logS IKKo = i Pr(|Koo i). (2-24) i 1i ACKNOWLEDGMENTS I would like to thank my advisor Prof. Tan Wong for his guidance, support, patience, and the freedom I enjoi, .1 in choosing my research direction. I consider myself very fortunate to have been able to pursue research under the guidance of someone who encouraged me to try to define my own research problems, and at the same time, was patient enough when a particular idea would fail to bear fruit as expected. I would also like to thank Prof. John Shea for his general guidance and -~i-.-. -if.. regarding pursuing research, and more importantly, those on presenting one's research. I thank Prof. Michael Fang and Prof. William Hager for their time and interest in my work. I would like to take this opportunity to thank Dr. Ozgur Oyman of Intel Research for the stimulating discussions that we had during my stay in Santa Clara, and for his valuable comments and -.-.- -1 i. .'n regarding some of the later parts of this work. My stay in the WING lab almost never had a dull moment, and credit for that is due to my lab-mates, especially Surendra, Ryan, and Byong, who completely changed the atmosphere of the lab ever since the summer of 2006 when I was practically the only person present in the lab. There are way too many people I am indebted to for all the help and support I have received in the past few years. Sridhar, Selvi, Manu, Savya, Vaibhav, and Mallick are just a few people who have endured me over these years, provided me with encouragement and hope (sometimes blatantly false, but they mostly worked) when things have not worked out, and most importantly, been great friends. Finally, I thank my parents, for no achievement, however big or small, may ever be realized without their love and support. transmit Wcpr,b such that ((Ws), 'Plco'. We), uco(^'. I' WCco,b, bcco,b) ,upr '. I WCco,b bco,b, WCpr,b bpr,b) , P1pr (- Wpipr)) A,. (5-24) It sets bCco,b = 1 or bcpr,b 1 if the respective bin index is not found. It can be shown using arguments similar to those in [64] that the probabilities of the events of Sc not able to find a unique bcco,b or bcpr,b satisfying (5-23) and (5-24) can be made arbitrarily small if the following hold true: RCcco > aI(Uco'; TP1pr Tplco, S t) + Co, R!Cpr > aI(Ucpr; TPlpr Uco, TPlco, S = t) + Co, where eo > 0 may be arbitrarily small. Sc transmits x? (ws,b-1, We,b-1, WPIpr,b-1, WCco,b, bcco,b, WCpr,b, bCpr,b). Decoding: At Sc: Assume that decoding till block b 1 has been successful. Then, in block b, Sc knows Wplco,b-1 (Ws,b-1, We,b-1) and Wplpr,b-1. It declares that the pair (wplco,b, wp1pr,b) = wpico, wpipr) was transmitted in block b if there exists a unique pair (wpico, wpipr) such that (S' (Ws,b-1), lpco (Ws,b-1, We,b-1), tlpr (Ws,b-1, We,b-1, Wplpr,b-1) "lco (Wes,b-l, We,b-l, Wplco) "lpr (Ws,b-l, We,b- 1, Wplco, Wplpr,b- lp, ~1pr) b) A Else, an error is declared. It can be shown that the probability of error for this decoding step can be made arbitrarily low if (5-22a) and (5-22b) are satisfied. At Dp: The primary destination Dp waits until block B, and then performs backward decoding. We consider the decoding process using the output in block b C {B 1, ... 2}. The decoding for the first and last blocks can be seen as special cases of the above. Thus, /c, 7c are real numbers in the interval [0, 1]. Sc uses DPC to encode Xcco and Xcpr as Ucco = Xcco + AcoTp1pr, (5-40) Ucpr = Xcpr + prTPlpr, (5-41) with Aco and Apr being non-negative real numbers that denote the correlation between the known interference Tmpr and the auxiliary random variables Ucco and Ucpr respectively, conditioned on Tpico. Note that according to the notation of Theorem 5.2, Xp,i = Xppr, and Xp, = Xp2pr. In the following, we present some numerical examples to compare the achievable rate region corresponding to the transmission scheme proposed in [6] to that for Theorem 5.2, with a fixed listen-transmit schedule and specialized for Gaussian channels. In these examples, the link between the two sources is assumed to be better than the direct link, and we compare the Han-Kc. li, i -,i rate region for the interference channel (without any active cooperation between the user-pairs), the rate region of [6], and that for the proposed coding scheme in this work. In Fig. 5-7, we consider the scenario when both interfering links are weaker than the direct links, while in Fig. 5-8, the interfering link from Sp to Dc is strong and that from Sc to Dp is weak. Comparing the two figures shows that the improvement in the quality of the inter- fering link from Sp to Dc may significantly increase the overall rate region for the two user-pairs. Also, the manner in which the rate region of [6] is enlarged in both examples ~-,p--. -1 that the efficiency of the overall cooperative relaying scheme is the primary contributor to the enlargement of the rate region. The advantage of the coding scheme adapted from Theorem 5.2, and described in equations (5-37) through (5-41), lies in the effective utilization of the direct link for the primary user-pair via the transmission of the codewords corresponding to the message parts wP2 ('t i'I -_, ). Thus, not having the entire primary message being decoded and transmitted through the cognitive source may high, suitably selecting a -i ill" value of AR can ensure that both users back off from the NBS-s,-'-. -1I 1 transmission rate pair of R* to (R* AR, R* AR) with high probability (i.e. both /3* and eo can be made small enough with a choice of small AR and c in (3-8), thereby resulting in p(Z) a 1). Note that, in this case, the modified bargaining problem (that incorporates the maximin criterion) leads to both users backing off in a similar fashion although each user independently chooses its respective strategy. Thus, it can be seen that the modified bargaining problem formulation provides a general framework for resource allocation from an individualistic perspective and a characterization of the optimal strategy pairs in terms of AR. As mentioned at the end of Subsection 3.3.2, the nominal transmission rate pair selected in (3-7) is not the only possible choice, and other jointly randomized transmission rates may also be selected. However, the choice of transmission rates as in (3-7), can be shown to incur no loss of generality. For any choice of jointly randomized nominal transmission rates, the maximin-optimal transmission rates can be derived in terms of their conditional PDFs (with the conditioning on the nominal strategies and fading state) in the same way as above to yield the same maximin-optimal values for the objective functions (i.e. E[R'(R, R*, Z)] of (3-10) and its counterpart for User 2) as for the case considered here. In Fig. 3-1, we present the numerical results for the system described in 3.3.2 with two different models for the error in the CSIT. For this example, the maximum available powers at each user are Ti 100mW and T2 = 10mW respectively. We set e = 0.02 (cf. (3-8)) and consider two simple models for the error in the available CSIT: (i) a 5'. error in CSIT with the true channel power gains less than what the CSIT -,i-.-, -i- and (ii) ,' error in the CSIT with x randomly chosen from a uniform distribution over [-10, 10]. The true utilities are calculated in the following way. For a particular fading state and choice of transmission strategy pair, if the transmission rates lie within the MAC Another possibility of an encoding error at Sc occurs if, in block b, it cannot find a bin index bcpr,b such that (5-4) is not satisfied with wpco = 1, wpp, = 1, Wcco,b = 1, bcco,b and WCpr,b = 1. Proceeding as for the first kind of error event, it can be shown that the probability of this event can be made arbitrarily small if RCpr > I(Ucpr; Tppr TpCo, Uco) + CO. (5-6) Decoding errors at Sc: For the block Markov SPC encoded transmission from Sp, the cognitive source Sc uses its knowledge about the information in the previous block to jointly decode wp for the current block. For the superposition encoded common and private parts of the primary message, it can be shown that the probability of error for this decoding step can be made arbitrarily low as long as the following hold: Rppr < I(Xp; Vc | Tpco, Tppr, Xpco), RpCo + RPpr < I(Xp; Vc ITpco, Tppr). Decoding errors at Dp: For block b c {B 1,... 2}, let Eijk be the event (t"Pco()t"'P( (i, wP'CO Pco,b) (i, WPco,b ,Wppr,b (i, k, beco0Y b) A > I wherein, it is assumed that the previous decoding step, starting with decoding for block B, has been successful, i.e. (wpco,b, Pp,b) (wpco,b, wppr,b). Note that we need not consider the probability of the events of Dp decoding wp,b-1 correctly, but not (wcco,b, bcco,b). Then the probability of error at Dp is Pe,Dp = Pr [E 1 U(iU)#(11,)Eikj < Pr [EcJ] + 2nRP Pr [E121] + 2n(Rpr+Rcco+Rcco) Pr [E122 +2n(Rppr+Rppr+Rcco+R'ccO Pr [ E2221 (5-7) other by forwarding the latter's information. We propose efficient cooperative transmission strategies based on a flow-theoretic approach, and evaluate their performances using numerical simulations. Finally, we consider communication through a two-user interference channel with unidirectional cooperation (ICUC), wherein one source uses its knowledge of the message of the other to reduce the interference to its own transmission, and simultaneously, help the other user-pair via cooperative relaying. We consider a very realistic scenario in which the cooperating source is subjected to a causality constraint. We derive a new achievable rate region for the discrete memoryless version of this form of ICUC, and demonstrate the contributions of the various coding strategies involved via numerical simulations for Gaussian channels. We also study the same channel with the cooperating source being subject to the half-duplex constraint as well. A discrete memoryless channel model incorporating the half-duplex constraint is presented, and a new achievable rate region, that enlarges the largest known rate region for the Gaussian version of this channel, is derived for this channel. The achievable rate region for the proposed coding scheme, specialized for Gaussian channels, is numerically evaluated and the strict inclusion of the previously known largest rate region is demonstrated. user is allotted unique transmission and reception symbol intervals, using a particular scheduling policy, and it transmits a linear combination of its own symbol and the signal observed during its most recent reception symbol interval, thereby creating an artificial inter-symbol interference (ISI) channel. A set of L cooperation frames are combined to form a super-frame, and the assignment of the reception symbol intervals is scheduled for each super-frame, with the lengths of super-frames and codes chosen such that a coherence-interval consists of N 1 consecutive super-frames, and that all codewords span the entire coherence interval. Similar to the above NAF strategy, a cooperative transmission scheme for the two- user C`\ IAC, based on superposition coding, has been proposed in [59]. This scheme uses a time division approach in which a user simultaneously transmits its own information and the other user's information by using the superposition coding (SPC) technique. This scheme is demonstrated to achieve a gain of about 1.5 2 dB over traditional DF approaches for relaying, and at the same level of system complexity of the latter. An extension of this idea to the general N-user C \! AC is presented in [60], wherein the authors prove the optimality of the proposed scheme in achieving the optimal diversity- multiplexing tradeoff for the symmetric rate requirement scenario. In this work, we propose flow-theoretic cooperative transmission protocols for the two-user C\ AC. First, we present an orthogonal relaying protocol for the C I\!AC (OR- C \! AC), wherein each user acts as a dedicated relay for the other in a time-division fashion. The flow-optimized relaying approach of Chi Ipter 2, modified to incorporate coherent combining at the destination is used for the constituent relay channels. This relaying protocol has been shown to achieve the optimal diversity order and provide better coding gains for the relay channel as compared to traditional DF relaying methods, by efficiently utilizing the CSI available at all nodes. Next, we propose the flow-optimized protocol for the C`\ !AC (FO-C \ IAC) that decomposes the C \! AC into two broadcast Sc Sp Figure 5-6. The discrete memoryless ICUC-HDC. Achievable Rate Regions: 1pc=0.55, hcp=0.55, Pp=6, Pc=6 ....... Interference Channel: HK regii - Reference [6] - This work 0.8 1 Rp (bits) Figure 5-7. Achievable cross-links. Rates for the Gaussian ICUC-HDC: Weak interference for both TABLE OF CONTENTS page ACKNOW LEDGMENTS ................................. 4 LIST OF TABLES . . 7 LIST OF FIGURES .. .. .. .. ... .. .. .. .. ... .. .. .. .. ... .. 8 A B ST R A C T . . 10 CHAPTER 1 INTRODUCTION .................................. 12 1.1 Cooperative Transmission in Wireless Relay Networks ........... 13 1.2 Cooperative Behavior in a Fading Multiple-Access C!i Ii, I ......... 14 1.3 Active User Cooperation in a Fading Ci\ AC ................ 15 1.4 Achievable Rates in the ICUC with Causality Constraints ... 15 1.5 Organization of the Dissertation . 17 2 COOPERATIVE TRANSMISSION IN A WIRELESS RELAY NETWORK BASED ON FLOW MANAGEMENT . 19 2.1 Introduction . . 19 2.2 General Design Using A Flow-theoretic Approach ... 22 2.3 Generalized-link Selection and Its Optimality . 26 2.3.1 The Three-node Relay Network . 26 2.3.2 Generalized-link Selection . 32 2.3.3 Diversity-multiplexing tradeoff . 34 2.4 Numerical Examples . 37 2.5 Sum m ary . . 40 3 RESOURCE ALLOCATION AND COOPERATIVE BEHAVIOR IN FADING MULTIPLE-ACCESS CHANNELS UNDER UNCERTAINTY ... 48 3.1 Introduction . . 48 3.2 System Model . . 51 3.3 The Bargaining Problem for the Two-User Fading MAC ... 52 3.3.1 The Disagreement Point . ... 53 3.3.2 The Nash Bargaining Solution (NBS) . 54 3.4 The Modified Bargaining Problem . 58 3.5 Sum m ary . . 62 4 ACTIVE USER COOPERATION IN FADING MULTIPLE-ACCESS CHAN- N E L S . . 64 4.1 Introduction . . 64 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy COOPERATIVE COMMUNICATION IN WIRELESS NETWORKS: FLOW-OPTIMIZED DESIGNS AND INFORMATION-THEORETIC CHARACTERIZATIONS By Debdeep C!I ,I1. ijee August 2010 C!i ,r: Tan F. Wong Major: Electrical and Computer Engineering The challenges involved in the design of efficient communication systems for the wireless medium can be attributed to the fact that the wireless medium possesses certain unique characteristics, the most important ones being the broadcast nature of the wireless medium, the susceptibility to interference effects, and the effects of path loss and fading on wireless link quality. Cooperation between different transceivers can potentially aid further development of next-generation wireless communication systems that demand high data rates and an excellent quality of service (QoS). This is possible by exploiting the broadcast nature of the wireless medium, and the diversity advantages that a multi-user system offers. We first consider a general single-source-single-destination wireless relay network and propose an information flow-optimized cooperative transmission design that achieves the optimal diversity-multiplexing tradeoff. Next, we apply game-theoretic techniques to the problems of resource allocation and characterization of cooperative behavior in a two-user fading multiple-access channel (\!AC), with uncertainty about the channel state information at the transmitters (CSIT). In the third part of the dissertation, a more active form the above cooperative behavior is studied via a two-user fading cooperative multiple-access channel (C \! AC), where each user, along with transmitting its own information to the destination, helps the Remark 5.3. According to the above coding scheme, a part of the primary message (wp2) is not decoded by Sc. This is different from the non-causal case. As Sc cannot receive while it transmits, Sp may improve its rates by transmitting "fresh" information directly to the destination during Sc-transmit states, thereby increasing the achievable rate region. Compared to the situation wherein Sc is capable of full-duplex operation, transmitting a part of the message directly to the destination provides potential gains even when the Sp to So channel is much better than the direct link to Dp. Remark 5.4. Note that the maximum increase in the achievable rates that may be realized by using a random listen-transmit schedule for So is ibit [73]. Remark 5.5. The achievable rate region described in Theorem 5.2 is also convex (cf. Remark 5.2) and consequently, time-sharing is not required to enlarge the rate region. Here, the random variable S in Theorem 5.2 p1 i the role similar to that of U in [74, Lemma 5]. Remark 5.6. For the Gaussian channel model with a fixed listen-transmit schedule, the coding scheme of Theorem 5.2 yields the same rate region as with a time-division strategy with the use of Gaussian parallel channels [75], instead of a block Markov structure, for the decoding of wpp = (wpico, wpipr) at Dp and wpico at Dc. According to this strategy, Sp transmits wpi during the first time-slot while So is in listening mode. In the second time-slot, both Sp and So encode and transmit wpi as a (non-causal) ICUC, and Sp also superposes wp2 on top of wpi (the latter acting as the resolution information for the destinations). Both destinations decode only at the end of the second time-slot and exploit the parallel Gaussian channel structure to decode wpi entirely or partially. 5.7 The Gaussian ICUC-HDC As in Section 5.4, for the Gaussian ICUC-HDC, the direct links for each user-pair is normalized to unity, gpc is the channel gain for the Sp -- Sc link, hpc is that for the Sp -- Dc link, hcp is that for the Sc -- Dp link, and Sc, Dp, and Dc are assumed to experience i.i.d. additive white Gaussian noise (AWGN) of unit-variance. As mentioned in ' 10-1 -Q .-Q 2 0 CD -2 0 10 E[Zs s ]=3.0, E[ZS D]=1.0, E[ZS D]=1.0, K=lbit/s/Hz 1 2 1 2 0 5 10 15 20 25 30 35 40 SNR (dB) Figure 4-6. Outage performance symmetric situation. set of average rates defined by aT' + aT" can be achieved by time sharing between the two agreements. Hence, we may obtain an optimal conditional PDF for the transmission rates and powers using equation (3-4). Thus, we have T* argmax (E[Ri(Z)] T1,d) (E[R2(Z)] T2,d) (3-5) {T: R(Z)E.MAAC(P(Z),Z)} where P(Z) must satisfy the maximum power constraint for each user. It can be easily seen that T* (E C(ZITI) + C (zjj ] E C(Z2) + C Z2 )2 ] (36) and this can be achieved with the following rate and power allocation: 2 czr) + rcz+c Q Z2T )] w. p.1, S 1 C(Z ) + C w.p. 1, (37) A 2 2 G + Z2T2 P* = Ti, w. p. 1, for i 1,2. That is, for this case, employing the NBS for each fading state achieves the optimal solution to the bargaining problem of (3-5). This solution is similar in flavor to the one in [51], with the difference being in the nature of utility functions considered. More specifically, the utility we consider here is an average metric, while the bargaining model in [51] considers the utilities resulting from a single instance of the game for a particular state of nature (the fading state in this work). One important property of the NBS- - -...- -l,1 solution above that is of significance to the development of the modified bargaining problem formulation in Section 3.4 is that the optimal choice of transmission powers is deterministic and independent of the fading state. max z -- 1 o and ao < 1. In essence, this means that the opti- ZSa S (l + Z1 S)t2/t 1a . mal x1 and X2 should lie on the boundary of the degraded BC capacity region. With this, it is obvious then that x3 = min t2C(ZsDS), t2C(ZsDS + ZvRDS) tC ( zs) }s. Therefore the optimization problem of (2-8) can be re-written as: max (xi + x2 + 3) (2-9) subject to max{0, a} < a < 1, xi = tiC(ZsvaS), x2 tC (y- as 1 + ZsnRaS) x3 = min t2C(ZSDS), t2C(ZsVS + ZRS) tic ZSRas . 1 + Zs-RaaS) We observe that X3 t2C(ZSDS) above if and only if 1 1 + ZasS A SS 1+ S t2/t SZ1 + +ZsvS Comparing this to the expression for ao gives ao < a, < 1. Next we consider two possible sub-cases: i I+ n ZR-DS < 1 Z (7S): In this case, we have a > 0 and t2 < C ZRS) A 2max. The maximum a C V)+C(ZsRS) rate can be expressed as X(t,t2) = max{Xi(ti,t2), X2(t1,t2)} where X(t,, t2) = max tC(ZsVaS) + t2C(ZsDS + ZRS) max{, a (1 + ZsVajS)(1 + ZSRS) < t1 log + t2C(Zs-S) 1 + ZSRalSa < C(ZsS) (2- 11) X2(t1,t2) = max t,C(ZsvaS) + tiC ( z s t2Z S) aal<< \1 + Zs-RaS C(ZSDS) (2-12) where the first inequality in (2-11) holds since (2-10) is not satisfied, and the second inequality in (2-11) holds since a1 < 1 and that the first term in the previous step is monotonically increasing in aa when ZSD > ZSR. This way, the last observation regard to the known outer bounds. Very recently, in [16], the authors have shown that a Han-Kc., i, -,i-i type of coding scheme yields a rate region that is within 1 bit of the tightest outer bound for the Gaussian interference channel. Given this backdrop, we direct our focus to the interference channel with an .i-Cm- metric cooperative relationship between the sources. Such a network is known as the interference channel with unidirectional cooperation (ICUC)1 and is the simplest form of general overlay cognitive networks [18]. The ICUC consists of a pair of sources that demonstrate different behaviors. On the one hand, the primary source is solely interested in transmitting information to its respective destination and does not actively cooperate with the other user-pair. On the other hand, the secondary or cognitive source uses its knowledge about the primary message to reduce the interference caused to its own link by the primary transmission, and at the same time, aid the primary user-pair by relaying the primary message to the primary destination, thereby reducing the effect of interference caused by its own transmission to the primary link. Most of the work reported in the lit- erature on this channel consider the scenario in which the cognitive source has non-causal knowledge about the primary message. In this work, we impose a practical restriction that the cognitive source may only obtain the message of the primary source in a causal manner, i.e. the "cognitive" ability of the secondary source needs to be acquired. A new achievable rate region for the discrete memoryless version of the ICUC with causality con- straint (ICUC-C) is derived using block Markov superposition coding (SPC), conditional rate-splitting, conditional Gel'fand-Pinkser (GP) inning, and cooperative relaying. This rate region is evaluated for Gaussian channels, and numerical results are presented to demonstrate the contributions of the various coding strategies used in the proposed coding scheme towards enlarging the achievable rate region. 1 This network is also known in the literature as the cognitive radio channel [6] or inter- ference channel with degraded message sets (IC-DMS) [17] In n channel uses, the primary source, Sp, has message wp c {1, 2,... 2RP } to transmit to Dp, while the secondary source Sc has message wc E {1, 2,... 2nRc} to transmit to Dc. Let Xp, Xc, S, and Vc, Yp, Yc be the input and output alphabets respectively. Further, let S = {1, t}. A rate pair (Rp, Rc) is achievable if there exist an encoding function for Sp, XT = fp(wp), fp {1, 2,... 2R} -} XJ, and a sequence of encoding functions for Sc, (X, S"f) = f(wc, V ) with (Xci, S) = fciwc, VS 1), fci {1, 2,... 2Rc} x V1' -- Xc x S, and corresponding decoding functions wp gp(Y), gp : -> {1,2,--- ,2R 2 } and c = gc(Yc), gc : -Y {1, 2, .. 2nRc } such that the average probability of error P( max{P(-, P)} }- 0, where P() = Pr [g(Y1) /M '(wp, wc) was sent] for M = P, C. (wp wc) 5.6 An Achievable Rate Region for the ICUC-HDC First, we present a brief description of the coding scheme. In block b e {1,... B}, Sp splits the message wp,b as wp,b ( wpi,b, wp2,b) where wi,b ( wpico,b, 'PF.,p,,b) for i = 1, 2. Here, for any block, wpl is the message part that Sc decodes and uses for its cognitive and cooperative actions, whereas wp2 is the message part that Sp directly transmits to Dp when Sc is in transmit mode. As before, the subscripts co and pr indicate the common and private message parts respectively. While the common message parts are decoded by both destinations, the private message parts are decoded only by the intended destination. WPIco,b is further divided into two parts Ws,b, that is forwarded by Sc in the next block using the help of its random listen-transmit schedule [73], and We,b, that is transmitted explicitly using a standard codebook. Conditional rate-splitting [65] and superposition coding are used for the above message splitting step. For block b c {1, B}, Sp transmits WpI,b during the Sc- listen states, and it superposes wp2,b onto WpI,b-1 (using block Markov SPC) during the Sc-transmit states, with wpi,b-1 acting as the resolution information for Dp and Dc to decode wpl entirely or partially. In block b, Sc decodes Wpl,b from the received symbols during the listen-states. In block b, Sc splits wC,b into two parts w co,b and wCpt,b, and with 0 < t1 < 1, is given by (see [30, Lemma 3.1] for proof) Bc 1 (eXlti 1) + 1 e'1 Li(e12-ti 1) for Zs s2 > ZSID, (2 ZL I 1) + Z 621 i(C61i-i 1) for Zs, s2 < ZSID. For t1 = 0, PBC = 0. P1 is the power that Si allocates for the direct transmission of x3 to D, and To denotes the power corresponding to the common information flow (x2), received at D. It can be checked that the above maximization problem belongs to the class of convex optimization problems, and standard numerical techniques can be used to obtain a solution. Let the solution of (4-1) be denoted by X(Z, TI). Then it can be easily seen that X(Z, Ti) = TIX(Z, 1). For the second time slot, an exactly similar optimization problem can be formulated as (4-1) with appropriate changes in the indices. Therefore, for a particular fading state Z, the regular points on the boundary of the achievable rate region can be obtained by maximizing a convex combination of the rates of the two sources during the respective time slots, X(Z, TI) and Y(Z, T2). That is, by solving the following optimization problem, for some 0 < p < 1, max pX(Z, Ti) + (1 p)Y(Z, T2) (4-2) subject to T1, T2 > 0, and T + T2 1. The extreme points of the boundary region, i.e. maximizing only one source's rate are given by T1 1, T2 = 0, etc. Unfortunately, this naive scheme of decomposing the C \! AC into two orthogonal relay channels does not entail the best utilization of resources, and as we shall see in the following subsection, this can be improved upon by a more efficient flow-optimized transmission protocol. 4.2.2 Flow-optimized Protocol for the CMAC (FO-CMAC) Instead of dividing the C \! AC into two separate relay channels, we divide a unit interval into three time slots T1, T2, T3 with no orthogonalization of the relaying actions. [14] M. Gastpar and M. Vetterli, "On the capacity of large Gaussian relay networks," IEEE Trans. In.[..,in The.-'i. vol. 51, no. 3, pp. 765-779, Mar. 2005. [15] T. S. Han and K. Kob- v-l-hi "A new achievable rate region for the interference channel," IEEE Trans. Inform. The.',;/ vol. IT-27, no. 1, pp. 49-60, Jan. 1981. [16] R. H. Etkin, D. N. C. Tse, and H. Wang, "Gaussian interference channel capacity to within one bit," IEEE Trans. Inf[..n The.' ,I vol. 54, no. 12, pp. 5534-5562, Dec. 2008. [17] J. Jiang and Y. Xin, "On the achievable rate regions for interference channels with degraded message sets," IEEE Trans. Inform. Tht'.,i vol. 54, no. 10, pp. 4707-4712, Oct. 2008. [18] A. Goldsmith, S. A. Jafar, I. Maric, and S. Srinivasa, "Breaking spectrum gridlock with cognitive radios: An information theoretic perspective," Proc. 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Theory (ISIT .'/K,), Seattle, U.S.A., Jul. 2006. * the no'.-.' i ll.:.: ,; constraints: xAB, ti > 0 for all cut edges AB and i 1,2,... ,2N- 2, the total-time constraint: t1 + + t2N-2 1, the power (,-'"". /,'/) constraints: for a BC slot the flow rates should lie in the capacity region of the BC channel with the transmitting node having a power constraint of P, for an MA slot the flow rates should lie in the capacity region of the MA channel with a maximum power constraint P for each transmitting node, the flow constraints: considering steady state operation, the total information flow out of a relay should equal the flow into the relay in each unit time interval. Note that the dependence of the objective function on the channel gains and the time slot lengths is implicitly expressed through the capacity constraints. Denote the cut separating S from all the other nodes and the cut separating ED from all nodes as Cs and C-, respectively. Then we observe that the cost function in (2-2) above can be further simplified to max mmin {x(Cs), x(C-)}, where N-2 N-2 x(Cs) = XSD + X and x(CD) = xs, + xzi (2-3) j= 1 i 1 are the total flows across the above-mentioned cuts Cs and CD, respectively. To see this, consider the cut C with V8 = {S, 7RI, R }, and V = {- +l, RN-2, )} for some I c {1, 2,... N 2}. The total flow across this cut is given by N-2 I N-2 x(C) XS-D + x + XK-D + Yx -- (2-4) j =1+1 i= 1 j=1+1 Now, consider node i for i c {1, 2,... 1}. According to the flow constraint for node i, N-2 I N-2 I X-R7iZ + Y X, ,+ > Xg^Za = XSm+ Xi,7 + > XiRzjz (25) j=1+1 k 1,k i j 1+1 k 1,k i Summing (2-5) over all i c {1, 2, .. l} we get I 1 N-2 ) I N-2 ( - X-i= D + f1 Xi= xSK +fxii7. (26) J 1+1 j1+1 Weak Interference: c=1 0, h c=.55, hcp=0.55 ....... Coding . Coding - Coding - Coding ....... Coding . Coding - Coding - Coding I, Pp=1.5, Pc=6 II, Pp=1.5, PC=6 III, Pp=1.5, Pc=6 IV, Pp=1.5, Pc=6 I, Pp=6, Pc=1.5 II, Pp=6, Pc=1.5 III, Pp=6, Pc=1.5 IV, Pp=6, Pc=1.5 Single User Rate (Rp), Pp=6, Pc=1.5 0.5 Single User Rate (Rp), Pp=1.5, Pc=6 I II I I I I 0 0.2 0.8 1 Rp (bits) 1.2 1.4 1.6 1.8 Figure 5-3. Achievable Rates for the Gaussian ICUC-C: Weak interference for both cross-links. I I fI , Non-uniform average power gains: Case B Spectral efficiency, R=1bits/s/Hz 0 2 4 6 8 10 12 14 16 18 SNR (dB) Figure 2-10. Four-node i E[Zsi] l 0.2, E[ZK2I 10 10 - 10 1 10 L- 10 , network with non-uniform average power gains. Case B: E[ZsZ2] 0.75, E[Zsv] = 1.0, E[ZRI2] 3.5, E[ZRD] 3.0. Outage probabilities for required rate R l= bit/s/Hz. Non-uniform average power gains: Case B Spectral efficiency, R=6bits/s/Hz i . 15 20 25 SNR (dB) 30 35 40 Figure 2-11. Four-node relay network with non-uniform average power gains. Case B: E[Zsg] 1.5, E[ZS2] 0.75, E[ZSD] = 1.0, E[ZgIg] 3.5, E[ZgzD] 0.2, E[ZR2D] = 3.0. Outage probabilities for required rate R = 6bit/s/Hz. 100 10 -Q o 10 (D 0 10 -i- Max-min selection routing :\:: - Generalized-link selection:: :: -- Flow-optimized protocol ::::: - Lower bound .. ..... .... .... .... .... ... . . . . ....... .................. 11 . . .. . . . :\ . . ............. ............. ............. ............. 7 .............. ........................ \ . . ........................... . ........................ . . . . . . . . . . . . . . .................. ..... .7 .................... .......................... ........................ CHAPTER 4 ACTIVE USER COOPERATION IN FADING MULTIPLE-ACCESS CHANNELS 4.1 Introduction The growing emphasis on multi-user wireless communication systems and the ever- increasing demand for high data rates have heightened the importance of research in the area of user cooperation. Although wireless systems bring forth design challenges owing to multi-user interference and fading concerns, it also provides potential benefits, like the broadcast nature of the wireless medium and diversity advantages in multi-user systems. The MAC is one of the fundamental kinds of multi-user communication systems. Conventionally, in a MAC, the users transmit directly to the destination, and the capacity region of such a system is well known. Recently, it has been demonstrated [3, 4] that the rate region of the fading MAC can be increased by providing spatial diversity that is achieved by user cooperation in the form of forwarding each other's information to the destination, giving rise to the C \!AC. Although the capacity region for the C \! AC is yet to be determined, there have been a number of different cooperative transmission strategies proposed in the literature. Two broad classes of works reported in this regard can be distinguished based on the transceiver capabilities of the wireless nodes, i.e. whether the nodes can support full- duplex communication or not. In [3], the authors provide a system-level description of the C '\!AC wherein the nodes are capable of full-duplex communication. They present an achievable rate region based on block Markov encoding and backward decoding, and show the potential increase in the rate region as compared to the conventional MAC. It is assumed that the phase of the fading is known to the transmitters and this is exploited to perform coherent combining at the destination node, and obtain beamforming gain. In [4], the CDMA implementation aspects of the scheme in [3] are considered, wherein the authors propose the use of different spreading codes to obtain different channels for simultaneous transmission R1 if R, < min{C(ZiT), max{C(ZiTi +Z2) R2, R'(R, R*, Z) C ( )}},and R, S w. p. (1 ), 0 otherwise. Here, we emphasize that the above choice of transmission rates as in (3-7) is not the only possible solution to the Nash bargaining problem. Unless other constraints are imposed, any choice of jointly randomized transmission rates satisfying (3-6) may be selected as the NBS--,1-.- -i. -l optimal strategy, with the choice of the transmission powers as in (3-7). Specifically, the above choice has been made to facilitate an easy exposition of the modified bargaining problem formulation in Section 3.4 and also to yield a simple solution for practical implementation with less overhead. With regard to the latter point, note that for the solution of (3-7) each user only requires the knowledge of the channel power gains Z (obtained from the CSIT) and can maximize the Nash product of (3-5) independently, whereas for a choice of jointly randomized rates satisfying (3-6) some form of communication between the two users needs to be established to enable the joint randomization of the transmission rates. Note that the NBS, as in (3-6), -,ti-.--i- that no user transmits at its maximum possible transmission rate (i.e. C(ZiTi) for i = 1 or 2) with probability 1, and this "backing off" of each user from its maximum possible rate may be interpreted as the manifestation of its cooperative behavior, motivated by a rational and individualistic evaluation of the benefits of cooperation as against any presumed altruism on its part. Moreover, for any choice of jointly randomized transmission rates satisfying (3-6), the transmission rate pair would aliv--, correspond to a point on the boundary of the MAC capacity region for every fading state, thereby making the solution very sensitive to the CSIT. In the following section, we propose a modified bargaining problem to handle this robustness issue. Sp Figure 5-1. The discrete memoryless ICUC with causality constraint. Figure 5-2. The Gaussian ICUC-C. E[Zs s ]=3.0, E[ZS D]=1.5, E[ZS D]=0.2 1 2 1 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Figure 4-2. Achievable rate regions .i- ,iiiii., I I i situation. E[Zs s ]=3.0, E[ZS D]=1.5, E[ZS D]=0.2 with conventional MA slot 1 2 1 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 R1 Figure 4-3. Achievable rate regions .-iiiii,. I i,: situation with conventional MA slot (without common information). Therefore, for n large enough, (wpco, wcco, bcco, bcpr) = wPco,b-1, WCco,b, bcco,b, WCpr,b, bcpr,b) with arbitrarily low probability of error if (5-le)-(5- g) are satisfied. Thus, the constraints on the rates as given in (5-la) (5- g) ensure that the average probability of error at the two destinations can be driven to zero and thus, they describe an achievable rate region for the ICUC-C. E Remark 5.1. The achievable rates region described in Theorem 5.1 can be expressed explicitly in terms of Rp and Rc using Fourier-Motzkin elimination. Denote the right sides of (5-la)-(5- g) as I,, 12, .. I7. Then the achievable rate region of Theorem 5.1 can be written as Rp < min{J2,4}, (5-13a) Rc < min{I6, 7}, (5-13b) Rp + Rc < min{I4 + 5, 1i + I7}, (5-13c) Rp + 2Rc < I3 + 1 + I7. (5-13d) Remark 5.2. The achievable rate region described in Theorem 5.1 is convex and hence, no time-sharing is required to enlarge the rate region. This can be proved using the Markov chain structure of the code as was used in [74, Lemma 5], with the random variable Tpco in Theorem 5.1 p1 i'ing a role similar to that of U in [74]. 5.4 The Gaussian ICUC-C We apply the result of Theorem 5.1 to the Gaussian ICUC-C. For Gaussian chan- nels, with the direct links' channel gains normalized to unity (cf. [7], etc.), we have the following input-output relationships as shown in Figure 5-2: Vc = gpcXp + Zs, (5-14) Yp = Xp + hcpXc + Zp, (5-15) Yc hpcXp + Xc + ZD, (5-16) Weak PC, Strong CP Interference: c=10, h c=0.55, hcp=1.5 0.5 1 1.5 2 2.5 Rp (bits) Figure 5-5. Achievable Rates for the Gaussian ICUC-C: Weak interference from Sp to Dc and strong interference from Sc to Dp. LIST OF FIGURES Figure page 2-1 Basic graphs Gi and G2 for the three-node relay network with t1 + = 1. 42 2-2 FO protocol for the four-node relay network with tj + + t6 = 1. The flow optimization is performed over all flows xi, ... x14, and all time slot lengths t1, t6 . ... 4 2 2-3 Transmission strategy to obtain a lower bound on the outage probability for the four-node relay network. Here t1 + + t4 = 1, and the optimization is over xa, x14, and ti, t4, with the application of the max-flow-min- cut theorem for the intermediate slots. . .... 43 2-4 Four-node relay network with uniform average power gains: Outage probabili- ties for required rate R l= bit/s/Hz. . 44 2-5 Four-node relay network with uniform average power gains: Outage probabili- ties for required rate R = 6bits/s/Hz. . 44 2-6 Five-node relay network with uniform average power gains: Outage probabili- ties for required rate R l= bits/s/Hz. . 45 2-7 Five-node relay network with uniform average power gains: Outage probabili- ties for required rate R = 6bits/s/Hz. . 45 2-8 Four-node relay network with non-uniform average power gains. Case A: E[Zs-,] 2.0, E[Zs72] 2.0, E[Zs] = 1.0, E[Z7IZ2] 1.0, E[ZD] 1.5, E[Z2D] - 1.0. Outage probabilities for required rate R = Ibit/s/Hz. 46 2-9 Four-node relay network with non-uniform average power gains. Case A: E[Zs-,] 2.0, E[Zsg7] 2.0, E[ZsD] = 1.0, E[ZR1Z] 1.0, E[ZRD] 1.5, E[Zgz] - 1.0. Outage probabilities for required rate R = 6bit/s/Hz. 46 2-10 Four-node relay network with non-uniform average power gains. Case B: E[Zs,] = 1.5, E[Zsg2] 0.75, E[Zs] = 1.0, E[ZR_1] 3.5, E[Zz] 0.2, E[Z2D] - 3.0. Outage probabilities for required rate R = Ibit/s/Hz. 47 2-11 Four-node relay network with non-uniform average power gains. Case B: E[Zs,] = 1.5, E[Zsg] 0.75, E[Zsv] = 1.0, E[ZR_1] 3.5, E[ZD] 0.2, E[Z2D] - 3.0. Outage probabilities for required rate R = 6bit/s/Hz. 47 3-1 Average rates with varying A . 63 4-1 Flow-theoretic transmission protocols for the C' \ AC: (a) OR-C'\!.AC, (b) FO- C AC . .... 75 4-2 Achievable rate regions .,i-viiii 1i lic situation. . 76 bargaining problem solution may not be optimal, thereby resulting in considerable per- formance degradation. To address this robustness issue, we propose a scheme in which the conditions of the bargaining problem are relaxed to reduce the dependence of the system performance on the solution to the bargaining problem. In the process, we develop a game-theoretic framework to characterize the level of cooperation involved from an individualistic viewpoint. 1.3 Active User Cooperation in a Fading CMAC Next, we consider the problem of active user cooperation in a two-user fading C\! AC. This alludes to a system in which each user, apart from transmitting its own information, may cooperate with the other by actively forwarding the other's information to the des- tination. Therefore, the C \! AC can also be considered as a basic example of cooperative communication in a multi-source-single-destination system. Clearly, this involves a higher level of cooperation as compared to the form of cooperation in a conventional MAC. We use a flow-theoretic approach, and propose a flow-optimized solution and a much simpler (but suboptimal) solution that decomposes the C'\! AC into orthogonal relay channels. The performances of the proposed protocols are evaluated in terms of the achievable average rate regions and outage probabilities, and the improvements over conventional MAC sys- tems and a previously proposed method based on decode-and-forward (DF) relaying are demonstrated through simulation results. 1.4 Achievable Rates in the ICUC with Causality Constraints Finally, we study a manifestation of cooperation in a multi-source-multi-destination system. The simplest multi-source-multi-destination system is the two-user interference channel, wherein a pair of sources wish to transmit information to their respective destinations resulting in interference caused to each other. The capacity region for the interference channel, except for the special cases of strong interference, is still unknown, and until recently, the original Han-Kc.1 i, -1i scheme [15] has been known to provide the largest achievable rate region, without a clear idea of tightness of this region with links, and the inter-relay channel is, on average, very good. This situation promotes inter-relay interactions for the FO protocol, and thereby increases the difference between the performances of the FO and GLS protocols. The differences between the outage performances of the FO and GLS protocols, at an outage probability of 10-4, are 1.2dB or 1.0dB, and 2.0dB or 1.3dB (when R l= bits/s/Hz or R = 6bits/s/Hz), for cases A and B respectively. This reduction in the suboptimality of the GLS protocol with increase in the required data rate can be explained by noting that, when the required rate is high, the coding gain offered by a protocol heavily relies on the efficient use of the direct link, and since the usage of the direct link is similar for both the FO and the GLS protocols, the performance gap narrows as the required data rate increases. On the other hand, at the same outage probability, the difference between the outage performance of the FO protocol and the lower bound increases from 1.5dB to 7dB, and from 1.9dB to 6.0dB as the required rate increases from lbit/s/Hz to 6bits/s/Hz, for cases A and B respectively. Overall, these results demonstrate trends similar to the uniform average power gain case, and confirm the generality of the proposed protocols. 2.5 Summary In this chapter, we proposed a cooperative transmission design for a general multi- node half-duplex wireless relay network where channel information is available at the nodes. The proposed design is based on optimizing information flows, using the basic components of BC and MA, to maximize the transmission rate from the source to the destination, subject to maximum power constraints at individual nodes. Motivated by the need for simpler network path selection schemes that perform well even in high- rate scenarios, we developed the generalized-link selection protocol that combines relay selection, and flow optimization for a three-node relay network. The proposed protocols were shown to achieve the optimal diversity-multiplexing tradeoff for a general relay network. Simulation results for the four- and five-node relay networks for uniform and non-uniform average power gains demonstrate that the performance of the much simpler Rpeo + Rpp, < I(Xp; Vc Tpo, Tp,) (5- b) RpP, + Rco < I (Tppr, Xp, Ucco; YP TpCo, Xpco) (5 -c) RPCO + Rppr + RCCO < I (Tpco, Tppr, Xp, Ucco; Yp) (5- d) Rcpr < I(Ucpr; Yc, Ucco Tpco, Xpco) I(Ucpr; Tppr, UCco Tpco) (5 -e) RcCO + Rcpr < I(Ucco, Ucpr; Yc Tpco, XpCo) I (Ucco, Ucpr; Tppr TPco) (5-if) RPCO + RCCO + Rcpr < I (TpCo, Xpco, Ucco, Ucpr; YC) -I(Ucco, Ucpr; Tppr TPco) (5-1g) is achievable for some joint distribution that factors as P(tpco)p(tppr tpco)p(Xpco tpco)P (XpCtpco, tppr, Xpco) x p(Ucco Pco)P UCpr tPco, UCco)p(xc \tpCo, tppr, Ucco, Ucpr) xp(vcIxp,xc)P(ypxp,xc)p('/. p,XC), (5-2) and for which the right-hand sides of (5-la) to (5- g) are non-negative. Proof. Let A"(X, Y) denote set of jointly c-typical sequences according to the distribution of random variables X, Y as induced by the same distribution used to generate the codebooks (see [33]). For convenience, the dependence on the random variables will not be stated explicitly, and should be clear from the context. Codebook generation: Split the primary and cognitive users' rates as Rp = Rpco + Rppr and Rc = RCco + Rcpr respectively. Thus, in block b c {1, ... B}, the primary message can be represented as wp,b (wpco,b, wppr,b), and the secondary message as wc,b (wCco,b, wcpr,b), where co and pr stand for the common and private part of a message respectively. Fix a distribution p(tpotpp, xpp co Cxp UCo,cpr, C) as in Theorem 5.1. User Cooperation: In this case, the users, who themselves wish to transmit or receive information, cooperate with each other by using some of their own resources to relay other users' information. The cooperative multiple-access channel (C'\ AC) [3, 4], the relay broadcast channel (RBC) [5], the interference channel with unidirectional cooperation (ICUC), also known as the cognitive radio channel [6, 7], fall under this class of cooperative communication systems. Both these classes of cooperative communication techniques possess unique benefits and limitations [9] in terms of different performance metrics, implementation cost and complexity. Moreover, hybrid systems, that make use of both classes of cooperative communication, can also be envisioned. For instance, cooperation within multiple cells of an infrastructure network is expected to improve the efficiency and reliability of the overall system [8]. The base stations (BSs) over multiple cells cooperate by jointly processing the transmitted/received signals using high capacity backhaul connections, while the mobile stations (\!SM) cooperate through relaying and "coiIl i- 's, between themselves. Thus, depending on the application requirements and network configuration, different cooperative communication systems demand different design approaches, and manifest different forms of cooperative behavior amongst the participating nodes. 1.1 Cooperative Transmission in Wireless Relay Networks A wireless relay network is a multiuser wireless communication network wherein a wireless link exists between each pair of nodes. In general, it may be conceived as a part of a larger network that itself may not be fully connected. It can also be seen as a special form of a more general relay network. Although a considerable amount of work on relay networks has been reported in the literature ( [10, 11, 12, 13, 14], etc. ), the capacity of even the simplest of all relay networks the general three-node relay channel, is still unknown. In the first part of the dissertation, we are concerned with the problem of transmission of information from a single source node to the single destination node in a wireless relay network, with the help of certain dedicated relay We model the resource allocation problem (i.e. the optimal choice of the transmit powers and rates) as a two-user bargaining problem. This specifies the operating point of the system. Note that a bargaining problem formulation is an appropriate choice to model this problem as it does not presume any inherent cooperation between the two users. Instead, the users negotiate to reach an agreement after evaluating (selfishly and rationally) the potential benefits from cooperation over the event of them not arriving at any mutual agreement. Moreover, it is well known that the NBS can be interpreted as a generalized form of a proportional fairness solution, and coincides with the latter when the p .,ioffs to the two pl -,v rs in the event of disagreement equal zero. Thus, the NBS provides a fair and efficient (i.e. it is not possible to improve one user's performance without degrading the performance of the other) solution to the resource allocation problem. Unfortunately, owing to the dependence of the operating point on the fading state, if the CSIT is not accurate, the operation point obtained with the erroneous CSIT may not be optimal. In order to obtain a more robust solution to the resource allocation problem under uncertainty, we propose a relaxed bargaining problem formulation in this work. 3.3 The Bargaining Problem for the Two-User Fading MAC In this section, we solve the two-user bargaining problem to obtain the optimal strategy for the two users using the available CSIT. Thus, for the two utility function choices, we need to solve the two-person bargaining problem, defined as (T, Td). Here, T is the set of feasible utilities, i.e. the achievable average rates for the two users, and Td (Td, T2,d) represents the I. ireement point, i.e. the utility each user will derive if they do not cooperate. Thus, the two users negotiate to reach an agreement regarding the optimal transmission rates and powers, given the fading state. Moreover, it is assumed that the users can agree to jointly randomized strategies regarding the their transmission rates and powers. The disagreement points for this bargaining problem, followed by the NBS, are derived next. 100 10 - 10-1 -Q o 2 10-4 0D -3 0 10-4 0 Spectral efficiency, R=1bits/s/Hz 5 10 SNR (dB) Figure 2-4. Four-node relay network with uniform average power gains: Outage probabilities for required rate R l= bit/s/Hz. Spectral efficiency, R=6bits/s/Hz 100 1011 10-3 Max-min selection routing * -- Generalized-link selection Flow-optimized protocol S- -. Low er bound. ....... ............... . .............. 10 15 20 25 SNR (dB) 30 35 Figure 2-5. Four-node relay network with uniform average power gains: Outage probabilities for required rate R = 6bits/s/Hz. --- Max-min selection routing - Generalized-link selection -- Flow-optimized protocol - -Lower bound over flow allocations xi, x2, x3, x4, and time slot lengths ti, t2, subject to: * no, -t,' ii/,;, ,i;/ constraints: x1, X2, X3, x4 > 0, t1, t2 > 0, * total-time constraint: t1 + t2 1, * power constraints: SBC < S, x1 < tlC(ZSDS), X2 < tlC(Zs-RS) for the BC slot, x3 < t2C(Zs-DS), X4 <_ t2C(ZzS), x3 + X4 <_ t2C(ZSDS + ZTDS) for the MA slot, flow constraint: x2 4, where C(x) = log(1 + x), and SBC, the minimum SNR required for the source to broadcast at rates xl/ti and x2/tl to the destination and the relay, respectively, in the first time slot with 0 < t1 < 1, is given by (see [30, Lemma 3.1] for proof) S Z xl(elt 1) + Z e 1 1,ti 1) for ZsK > ZsD, SBC \ { ZS 2/2 1) + ZSD t (eli -L 1) for Zsg < ZsD. For t1 = 0, SBC 0. Note that for the BC slot, the last two constraints are redundant when t1 > 0, and complement the first constraint when t1 = 0. As mentioned in Section 2.1, the above optimization problem formulation is different from that in [30] wherein the sum of the source and relay powers, required to achieve a certain data rate, is minimized. More specifically, when considering individual power constraints for each node, we cannot use part 2 of [30, Lemma 3.1] to describe the power constraints for the MA slot. This is because doing so would restrict the flows x2 and x3 such that the sum of powers expended at S and 7R is minimized. On the other hand, in the present problem, the power constraints only dictate that the flow-rates should lie in the MA capacity region specified by the maximum power available at each transmitting node, for the particular fading state. With this modification in the constraint for the MA slot, the solution approach to the above problem needs to be markedly different from that in [30]. To solve the optimization problem in (2-7) we consider two cases with regard to the link gains: (a) Zsv > ZS-R, and (b) Zsv < ZS-R. For both cases, we solve the optimization In n channel uses, the primary source, Sp, has a message wp E {1, 2,... 2"RP} to transmit to Dp, while the secondary source Sc has a message wc E {1, 2,... 2nRc} to transmit to its intended destination Dc. Let Xp, Xc, and Vc, Yp, Yc be the in- put and output alphabets respectively. A rate pair (Rp, Rc) is achievable if there exist an encoding function for Sp, X' fp(wp), fp : {1,2,-.. ,2nRP } XJ, and a sequence of encoding functions for Sc, X5 = fj(wc, V ,-) with Xci = fci(wcV, V1), fci {1, 2, ... 2R} x VS --1 Xc, and corresponding decoding functions Wp gp(pY), gp yP {1, 2,... 2' } and c = gc (Yf'), gc : Y {1, 2,... 2nRc} such that the average probability of error Pe = max{P(,) P)} -> 0, where P2) = 1t 1 Pr [gM (Y) /t I (Wp, Wc) was sent] for M = P, C. S 2n(Rp+Rc) tw 5.3 Achievable Rates for the ICUC-C In this section, we present a new achievable rate region for the discrete memoryless ICUC-C. We start with an overview of the coding scheme. In block b e {1, B}, Sp splits the message Wp,b into two parts Wpco,b and Wppr,b. It uses superposition coding to encode these two parts along with the message for the previous block (wpco,b-1, WPpr,b-1). The latter acts as the resolution 'f. f., in.il. n for Dp and Dc that use backward decoding to decode the primary message entirely or partially. In contrast to the rate-splitting tech- nique in [71], wherein the two message parts are encoded independently and superposed, Sp performs conditional rate-splitting [65] coupled with block Markov SPC. Sc decodes the primary message for the current block and simultaneously performs a set of encod- ing steps. In block b, Sc splits wc,b into two parts Wcco,b and WCpr,b, and conditioned on the codeword (Tpco) for the resolution information for the common part of the primary message [65], it uses conditional GP inning [64] to encode wcco,b and wcpr,b as Ucco and Ucpr respectively, against the resolution information for the private part of the primary message (Tppr) that causes interference at DC but is known at Sc. Finally, it transmits a combination of the above codewords, along with the resolution information for the primary message, with the latter part manifesting the cooperative relaying action by Sc. Unlike channels (BC) and a multiple access ( \!A) channel with common information. The bound- aries of the achievable rate regions are characterized by means of convex optimization formulations. The improvement provided, in terms of the achievable rate region, by OR- C \! AC and FO-C \! AC, over conventional MAC capacity and the DF strategy of [58] without power control, increases as the amount of disparity between the channels from the two sources to the destination increases. The outage performances of the proposed protocols indicate that although the much simpler OR-C \! AC is suboptimal in terms of the achievable rate region, it provides outage performance that is within 1 dB of that of FO-C \ IAC. Moreover, both the proposed protocols achieve a diversity of order two for the required rate region of interest. The rest of the chapter is organized as follows. In Section 4.2, the flow-theoretic pro- tocols of OR-C\ !.AC and FO-C \! AC are presented, and the boundaries of the achievable rate regions are characterized by convex optimization formulations. This is followed by nu- merical results in presenting the achievable average rate regions and outage performances for different scenarios in Section 4.3. Finally, the primary contributions in this chapter are summarized in Section 4.4. 4.2 Flow-theoretic Transmission Protocols for the Cooperative Multiple-Access Channel Consider a two-user C'\! AC where the two sources (S1 and S2) may actively coop- erate to transmit information to a common destination (D). We use the phrase active cooperation to distinguish between the cooperation involved in transmission strategies in which one source may forward the other's information, and that in the conventional MAC, wherein a user transmits at a rate lower than the maximum single-user rate possible for the particular channel state and power expended. The quantification of this type of coop- erative behavior was studied in C! Ipter 3. Thus, as a higher level of cooperative behavior, the users may relay the information of each other by utilizing the broadcast advantage of involves a maximum of 2-level SPC/DPC and one IC operation for the BC and MA slots respectively, for any N > 3. 2.3.3 Diversity-multiplexing tradeoff In this section, we show that both the FO and GLS protocols achieve the optimal DMT. As in [31], the multiplexing gain is defined as r = lims,_ R(S) where S is the SNR log S and R(S) is the rate at an SNR level of S. Following [31], we parameterize the system, in terms of the SNR S and the multiplexing gain, 0 < r < 1, with the rate increasing with the SNR as R r log(S). With the parameterization (r, S), the diversity order achieved by the transmission scheme is given by 1(r) lira log P, (r, S) (2 t) S-oo log S where P (r, S) is the average probability of error when the SNR is S and multiplexing gain r. With the above definitions, we evaluate the performance of the proposed protocols in terms of their diversity-multiplexing tradeoffs. The following theorem establishes the optimality of the the GLS protocol (and hence the FO protocol) in terms of the diversity- multiplexing tradeoff: Theorem 2.1. The GLS protocol, and hence the FO protocol, achieve the optimal diver.-,,iinllJ'..1, i.:,i tradeoff d(r) = (N 1)(1 r) for all 0 < r < 1, for the N-node half-duplex wireless i/.,;/ network. Proof. Here, we sketch the proof of the theorem. Part 3) of Theorem 4.2 of [30] can be generalized for the N-node relay network to prove that, as the block length goes to infinity (during any particular time interval), the average error probability for the FO protocol is upper bounded by its corresponding outage probability. Here, the outage probability denotes the probability that the data rate R cannot be supported by the system when the SNR is S, i.e., Pou0t(r, S) = Pr[X(S) < rlog S] where X(S) denotes the maximum rate possible for the given channel gain realizations when the SNR is S. Thus, from the for the two users conditioned on the fading state belonging to one of these sets. Next, we make the following important observation: Z e Z' == R* + AR > C(ZT) for i = 1, 2, (3- 11) ( Z1TZ2 ) with a relation analogous to (3-12) being true for User 2 as well. For p(Z) > 0, let co = min {1, c/p(Z)}. For any choice of fRpIR,z, the choice of User 2's transmission rate that minimizes the average rate of User 1 in (3-10) can be derived as 2 (r2 C(Z2T2))+( o) (r2 (R2 +AR)) if Z e Z, ftRt,z(r2) (r2C(3 13) 6 (2 + AR)) otherwise. Substituting the above solution in (3-10), the maximin-optimal distribution for the transmission rate can be shown to be the following mixed strategy: \6 r( C ( {Z2 ( )) if Z e Z .6 r( C+ (l ) 6 (ri (R A)) otherwise, where eo if C ( )>(- co) (R AR), 0 otherwise. The maximin-optimal transmission rates for User 2 can be derived analogously. Intuitively, when the channel power gains belong to the set Z', AR is comparatively "1 i5, and the restriction imposed through (3-8) loses its effectiveness as the window of strategies about the nominal strategy pair becomes "too wide". Hence, both users select transmission rates corresponding to the disagreement point as in Section 3.3.1. On the other hand, when Z e Z, the maximin-optimal strategy for each user is to respond to the (worst case) mixed strategy of the other user (as in (3-13)) while satisfying constraint (3-8). Hence, when the error in the CSIT (in terms of the power gains) is not To the reader and reception, without the use of complicated echo cancelation techniques. The power allocation problem for the C \! AC with full-duplex nodes and full CSI available at all nodes has been addressed in [55] and [56]. In [55], the authors consider average power constraints and characterize the optimal power allocation policies that maximize the set of ergodic rates achievable by block Markov encoding and backward decoding technique as in [3], by a dimensionality reduction approach, i.e. by noting that some of the power allocations are zero for every fading state. A more direct approach to solve the similar problem of optimal power allocation, with an almost closed-form solution, is presented in [56]. It has been established in [57] that windowed decoding is sufficient to achieve the same sum-rate as backward decoding for the block Markov superposition encoding scheme for the C \!AC. The optimal power and resource allocation problem for the C\ I\AC with nodes capable of half-duplex communication is considered in [58], where it is assumed that full CSI is available at all the nodes, and the transmitters cooperate by relaying each other's information over orthogonal frequency bands or time slots. The solution to the problem is presented as a two-step convex optimization problem formulation: first, for a particular bandwidth (or time) sharing parameter value, the optimal power allocation is characterized by a convex optimization problem, and then, the optimal resource sharing (time or bandwidth) parameter is obtained as a solution to the quasi-concave problem of maximizing the rate of one user, given a target rate for the other. All the works mentioned above use a DF approach for the relaying of information to the destination. In [19], the authors present a cooperative transmission scheme, based on the non- orthogonal amplify-and-forward (NAF) technique, that is proved to achieve the optimal diversity-multiplexing tradeoff of N(1 r) for the N-user half-duplex C \! AC, with symmetric data rate requirement and CSI available only at the receiving node of any link. According to the proposed strategy, time is divided into cooperation frames of length N cooperation symbols, and each user transmits only once during a cooperation frame. Every 2010 Debdeep C'!I ili. ijee * For each codeword pair (s"(w'), tpo(,w'> w)), generate 2nRP2, conditionally i.i.d. codewords x2c (w'w, ,wp2co), wpco C {1, 2 E 2n'P2o}, according to I(X P2coi Si, tp Ioi). For each codeword tuple (s~(w),t w,x2~co(w',, w e ,P),t ,(W we, wP1r)), generate 2nRP2p, conditionally i.i.d. codewords x"2pr (s, We, p2co, W pr, I'i ) ,-,, _, {1,... ,2"P2p-}, according to I p(xp2pri Si,tPlcoi, 2coitplpri). For each codeword pair (s"(w'),tpIco(w> ,w2)), generate 2n(Rcco+Rcco) i.i.d. codewords ', ,(w', wcco, bcco), wcco C {1,... 2nRcco} and bcco c {1,... 2ncco}, according to HT1 P' UCcoi i, tpilcoi) For each codeword tuple (s"(w>), t Ico(wl, w'), uT'co (w', wcco, bcco)), gen- erate 2n(Rc p+R cp) i.i.d. codewords u w'r (w, w, wcco, bcco, wcpr, bcpr), wcpr c { 1,... 2Rcp,} and bcpr e {1,... 2nRcp~}, according to ]iJ1 P(UCpri Si, tplcoi, uccoi). Generate ,' (w', wse, W'lpr, wPlco, wp pr, WP2co, ', _- ) where xp is a deterministic function of s, tPl tpIpr, XPlco, XpIpr, XP2co, Xp2pr. Generate x'(w', w w'p1p, wcco, bcco, wcpr, bcpr) where xc is a deterministic function of s, tPlco, tplpr, UCco, UCpr such that Xc if s 1. Encoding: At Sp: Sp transmits x" (ws,b-1, We,b-1, Plpr,b- wPlco,b, wPlpr,b, WP2co,b, b' -; ,b) in block b {2, ... B 1}. In the first block, there is no resolution information to transmit, and Sp transmits xT (1,, 1, co1, Wppr,i, Wp2co,1, t't -; ,1), while in block B, it transmits x"7 (ws,B-1, We,B_-1, Plpr,B-l 1,1, lWP2co,B, "'/ ,B). Note that the actual rate for the primary message is B(R + Re + Rppr) + Rp2co + 2pr, but it converges to Rp as the number of blocks B goes to infinity. At Sc: In block b E {1,.-- B}, to transmit WCco,b, Sc searches for bin index bCco,b such that (s"(ws), tlco(t'. We), ". ,( "'. '. WCco,b, bCco,b),t1pr(,. P W pr)) C A, (5-23) where i'. we and wplpr are Sc's estimates of Ws,b-1, We,b-1 and WP1pr,b-1 respectively from the previous block. Once bCco,b is determined, it searches for a bin index bcpr,b in order to for User 1, the optimal transmission strategy is f plz = argmax min E[R'], (3-2) fRllPlZ fR2P2 \Z where E[.] denotes the expectation operator and the expectation is over all fading states. Also, fRpplz is the joint conditional PDF of User i's transmission rate and power such that Pr(Pi > Ti) = 0 for i = 1, 2, with the latter probabilities computed with respect to the PDFs fpxz and fp2z, respectively. Since R' in (3-1) is a monotonically non-increasing function of R2, a solution of the minimization problem in (3-2) is given by fpZ ,P2) 6 (r2 C (Z2T2)) 6 (P2 T2). (3-3) Using (3-3) in (3-2) gives filpz argmax E[R[], such that Pr(Pi > TI) 0. In the fRIPI|Z above, R' is Ri if R1 < C 1+2T 0 otherwise. Clearly, it can be seen that the solution to the above problem is given by fAlPlz(rlPl) = 6 r- C Z621T2 1i T1). Therefore, when there is no agreement between the two users, the maximin optimal achievable average rates are T1,d E C (ZT1 T2,d = E )221 for Users 1 and 2 respectively. Since the operation point (for every fading state Z) is in the interior of the MAC capacity region, a small perturbation in Z may not cause a significant degradation in the actual achievable utilities. 3.3.2 The Nash Bargaining Solution (NBS) We use the NBS to obtain the optimal utility allocation for this bargaining problem using the available CSIT. The optimal allocation of the usage probabilities, using the NBS, 0 < Pi < Pi; 0 To < ZS,D(PI Pi) + ZSDP2 P2) + 2ZD(P Pl)ZSD(P2 P2). As in the previous subsection, PBc and PFc are the minimum powers required by S1 and S2 respectively for the two BC slots. PFc is defined as in (4-4) and PFc is defined similarly. Also, Pi and P2 are the powers allocated by S1 and S2 to transmit x3 and Y3 respectively, and To denotes the received power at D corresponding to the common information x22 + Y122- 1 (e(X+Y21)/T 1) + e(xl+w)/T(e/ 1) for Zss > ZSD, (44) FPC S[D ZS1 S2 1 S SS (X/TI 1) + eX2I ((1+Y21)1 1) for Zs1s2 < ZSID-. For T1i 0, Pc 0. Once again, it can be checked that the above maximization is a convex optimization problem that can be solved using standard numerical optimization methods. Thus, FO- C \! AC addresses the half-duplex limitation of the nodes by dividing the C \! AC into two BC and one MAC with common information, and provides a more efficient utilization of system resources as compared to protocols using two separate relay channels. 4.3 Numerical Results In this section, we present some numerical results to demonstrate the performance of the proposed protocols and compare them to the conventional MAC and the DF-based strategy proposed in [58]. Figures 4-2 through 4-4 show the achievable average rate regions for the various schemes for different scenarios. We consider different means of the fading gains as stated in the figures. Considerable variations in the statistics of the fading gains for the channels ZSID and ZS2D can occur in practical situations owing to different path loss and shadowing effects and different amounts of scattering for the two direct links. Thus, the .,i- ii ii. Ii ic situation corresponds to the case when the direct link from one source to the destination is much worse than the other. routing schemes based on different network path selection methods fail to provide ac- ceptable outage performance in high-rate situations. The GLS protocol described below provides a simple sub-optimal design to address this complexity issue. In essence, the GLS protocol identifies the best relay path out of the possible N 2 relay paths and considers only the chosen relay along with the source and destination to form a three-node relay net- work, which we call a generalized-link from the source to the destination, for information transmission. In other words, the aim is to choose the best relay such that the equivalent three-node relay network obtained (containing the source, destination and the chosen relay) gives the maximum rate over all possible equivalent three-node networks containing the source and destination. More precisely, we need to consider the following possibilities: * ZsD > Zs-Ri for all i c I {1, 2, .. N 2}: From the results of the optimization problem (2-7), it is clear that the maximum rate would be C(ZSDS) with direct transmission of all data from the source to the destination without using any relay. There exists a k C I such that Zs-, > ZSD: Let the set of all such node indices be K and for all i E I \ K, ZsD > ZS-,. For this case, choose the node 7R' as the relay such that k' arg rn ::K =Xk (S), where Xk(S) is the maximum rate for the three-node relay network with the source S, the relay )Rk and destination D. In terms of the worst-case computational complexities for the FO and GLS protocols, it can be seen that, for an N-node relay network with N > 3, the FO protocol involves a max-min optimization over 2(N2 2N + 2) variables (all possible flows and time slot lengths), subject to N-1 non-linear and 2(N2-N+1) linear constraints, whereas the GLS protocol involves a maximum of N 2 maximizations of a non-linear concave function over two variables, subject to two linear constraints, followed by finding the maximum of N 2 real numbers with a worst-case complexity of O(N 2). Moreover, for N > 3, for the FO protocol, the BC slots potentially involve (N 1)- and (N 2)-level superposition coding (SPC) or dirty paper coding (DPC) implementations for S and the rel-,,-, respectively, while the MA slots at the rel'-iv, and D may involve a maximum of (N 3) and (N 2) interference cancelation (IC) operations respectively. On the other hand, the GLS protocol BIOGRAPHICAL SKETCH Debdeep C!i i.1, ijee received the B.Tech. degree in electrical engineering in 2004 from the Indian Institute of Technology (IIT), Klio '-pur, India; and the M.S. and Ph.D. degrees in electrical and computer engineering in 2006 and 2010 respectively from the University of Florida, Gainesville. From August 2008 until July 2009, and from May 2010 until August 2010, he was with the Wireless Standards and Technologies team, Intel Corporation, Santa Clara, California. His research interests include cooperative communi- cations, multi-user information theory, game theory, and design of next-generation wireless systems. this, we present a discrete memoryless channel model for the ICUC with half-duplex and causality constraints (ICUC-HDC), and propose a generalized coding scheme for this channel. Similar to the full-duplex case, we employ block Markov SPC with backward coding, conditional rate-splitting, GP inning, and cooperative relaying by the cognitive source. However, for the half-duplex case, the cognitive source employs a randomized listen-transmit schedule [73] to encode and transmit information (via signaling). It is also proved that the new rate region contains the previously known rate region of [6] for the ICUC-HDC. In the following section, we present the discrete memoryless channel model for the ICUC-C. This is followed by Section 5.3, where we present the random coding scheme and the corresponding achievable rate region for the ICUC-C. Section 5.4 details the Gaussian ICUC-C along with numerical examples and a discussion on the role of different coding techniques under different network scenarios. Following this, the discrete memoryless ICUC-HDC is introduced in Section 5.5, and in Section 5.6, the proposed random coding scheme and the new achievable rate region are presented. The Gaussian ICUC-HDC is presented in Section 5.7, followed by analytical and numerical comparisons between the new achievable rate region and the one in [6]. Finally, a summary of the contributions in this chapter is presented in Section 5.8. 5.2 The Channel Model Consider the communication scenario as in Fig. 5-1, wherein the primary source node Sp intends to transmit information to its destination node Dp. Apart from the primary source-destination pair, the wireless network consists of a secondary (or cognitive) source- destination pair, Sc and Dc, with Sc having its own information to transmit to Dc. The primary message is causally available at Sc, and the latter may use this knowledge to assist Sp in the transmission of the primary message to Dp, and also transmit its own information to Dc. CHAPTER 5 INTERFERENCE CHANNELS WITH UNIDIRECTIONAL COOPERATION AND CAUSALITY CONSTRAINTS 5.1 Introduction As mentioned in Ch ipter 1, the interference channel with unidirectional coopera- tion (ICUC) is essentially the simplest realization of an overlay cognitive radio network. Cognitive radios have a considerable potential in facilitating an efficient use of the li- censed spectrum that is currently under-utilized [18]. The overlay paradigm for cognitive radios not only provides an efficient technique for cognitive radio deployments but also yields newer insights towards the understanding of interference channels and cooperative behavior in multi-terminal networks, through the different manifestations of cognition, co- operation and competition levels amongst different users or user groups [63]. In the overlay form, the simplest of which being the ICUC (also known as cognitive radio channel [6, 7]), the cognitive (or cooperating) radio utilizes the same spectrum as the primary user-pair for its own data transmission. Whereas this would generally cause interference to the primary link, the cognitive source may exploit its knowledge about the primary message to improve its own transmission rates by preceding its information against the known interference from the primary transmission and simultaneously alleviate the detrimental effects of the interference to the primary destination owing to the secondary transmission by cooperative relaying of the primary message. Of late there has been a considerable body of work reported in the literature that have helped improve the achievable rate region for the ICUC proposed in [6]. In [64], the authors present inner bounds to the capacity region for joint and sequential decoding, and encoding strategies that include rate-splitting for both primary and secondary messages, conditional Gel'fand-Pinkser (GP) inning and cooperative relaying. They also present a general outer bound that is very similar to an outer bound for the broadcast channel, and a much simpler outer bound for the case wherein the primary destination experiences strong interference. A slightly different coding scheme has been proposed in [17], wherein Now, the first two time slots are BC slots and the last one is an MA slot with common information, as shown in Figure 4-1. During the first time slot, S1 transmits two indepen- dent flows: xz + Y21 and x2 to D and S2 respectively using BC. Similarly, S2 transmits two independent flows: yi + X21 and Y2 to D and $1 respectively using BC. Finally, Si and S2 send two flows X3 + Y22 and y3 + a22 to D using MA with common information. Here Y21 and Y22 are two parts of the information flow Y2 that Si received from S2 during the previous unit interval. Similarly, x21 and X22 constitute the amount of information that S2 reh-,,-, for SI. Thus, for this scheme, the flow constraints imply that x2 x 21 + x22, and Y2 21 + Y22. Also, for the last time slot, x22 and Y22 are known to both the sources, and hence, they form the common information to be transmitted to D. Hence, the total transmission rates from sources Si and S2 are given by X = xi + X21 + X22 + X3 and Y = yi + y21 + Y22 + Y3 respectively. For this scheme, the boundary of the achievable rate region can be characterized as follows: the regular points on the boundary can be obtained by maximizing a convex combination of the rates X and Y, and the extreme points correspond to the C \! AC degenerating into relay channels with one source solely acting as a relay for the other. For compactness, let the information flows corresponding to the two sources be represented by the vectors x = (Xa a21 22 a3) and y = (yi Y21 Y22 Y3). The maximization problem that needs to be solved to obtain the regular points can be formally stated as given below: max IpX + (1 p)Y for p (0, 1) over TI, T2, T3, x, y, P, P2, To, (4-3) subject to non-negativity constraints: x, y > 0; TI, T2, T3 > 0; total-time constraint: Ti + T2 + T3 1; capacity (power) constraints: P1c < Pi; Pc < P2; x3 < T3C(ZSDP); 3 < T3C(ZS2DP2); X22 + X3 + Y22 + Y3 COOPERATIVE COMMUNICATION IN WIRELESS NETWORKS: FLOW-OPTIMIZED DESIGNS AND INFORMATION-THEORETIC CHARACTERIZATIONS By DEBDEEP CHATTERJEE A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2010 We then impose yet another very practical constraint on the cognitive source. It is assumed that the cognitive source may only operate in a half-duplex fashion. A discrete memoryless channel model for the ICUC with half-duplex and causality constraint (ICUC- HDC) is presented, and a new achievable rate region is derived for this channel. The random coding scheme used to obtain the rate region involves block Markov superposition coding (SPC), conditional rate-splitting, cooperative relaying, conditional Gel'fand-Pinkser (GP) inning, and a randomization of the listen-transmit schedule for the cognitive source. We also prove that this rate region contains the largest achievable rate region for the Gaussian ICUC-HDC previously reported in the literature, and further demonstrate this using numerical results for the case of Gaussian channels. 1.5 Organization of the Dissertation The rest of the dissertation is organized as follows. In C'!i pter 2, we present the proposed protocols for efficient information transmission in a wireless relay network. We first present the general design of the flow-optimized transmission protocol, and then, the generalized-link selection protocol. Next, the optimality of the proposed protocol in terms of the diversity-multiplexing tradeoff is established, and finally, we present the performance evaluation of the proposed protocols for the finite signal-to-noise ratio (SNR) regime through numerical results. In C'!i pter 3, we present the modified bargaining problem formulation to model the resource allocation problem in a two-user fading MAC under uncertainty. Solutions to these problems for the two choices of utility functions are also presented, and numerical examples are shown to highlight various aspects of the problem and the proposed solutions. We consider the problem of active user cooperation in a two-user fading MAC in Ch! Ipter 4, and present the flow-optimized and orthogonal relaying protocols for the C \! AC. Simulation results for different scenarios are then presented for the performance evaluation of the proposed protocols. In C!h Ipter 5 we study the problem of communicating through an ICUC with causality constraints, and present new achievable rate regions for these networks first, for the scenario in which the [40] L. Li, N. Jindal, and A. Goldsmith, "Outage capacities and optimal power allocation for fading multiple-access channels," IEEE Trans. hr.[..,,, Ti ..' .; vol. 51, no. 4, pp. 1326 1347, Apr. 2005. [41] L. Lai and H. E. Gamal, "The water-filling game in fading multiple access channels," submitted to IEEE Trans. Inform. The.. ,. Nov. 2005. [Online]. Available:http: //arxiv.org/abs/cs/0512013. [42] J. Sun, L. Zi, i.- and E. Modiano, \\ i. I. -- channel allocation using an auction algorithm," in Proc. Allerton Conf. Commun., Control and Con,,il','!, pp. 1114 - 1123, Oct. 2003. [43] A. ParandehGheibi, A. Eryilmaz, A. Ozdaglar, and M. 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Shamai (Shitz), : ,l!iii channels: How perfect need "perfect side information" be?," IEEE Trans. I, f[..' Th(.,, vol. 48, no. 5, pp. 1118 1134, M, ,l- 2002. [49] R. N o i-ii, ili,, "Effect of channel estimation errors on diversity-multiplexing trade- off in multiple access channels," Proc. IEEE Global Commun. Conf., (GLOBECOM .'ii,), pp. 1-5, Nov. 2006. [50] M. J. Osborne and A. Rubinstein, "A course in game theory," MIT Press, 1994. [51] W. C. Riddell, "Bargaining under uncertainty," Amer. Econ. Rev., vol. 71, no. 4, pp. 579 590, Sep. 1981. [52] W. Bossert, E. Nosal, and V. Sadanand, "Bargaining under uncertainty and the monotone path solutions," Games and Econ. Behavior, vol. 14, no. 2, pp. 173 189, 1996. U T S2 ... RIN -2 X )- where XAB is the flow from node A to node B during the i-th time slot. In general, the proposed design involves TS between the basic graphs to yield the following equivalent graph G corresponding to a unit interval (see [34] for a similar idea): G = V, [JE, tir) =tG, + t2G2 + ... 2N-2G2N-2. (2-1) where the number of elements in each vector ri is extended to | U Ei by inserting zeros appropriately. The second equality in (2-1) implies that G can be viewed as a linear combination of the basic graphs Gis, with the equivalent set of edges given by the union of the sets Ei, and the equivalent flow rate vector given by the linear combination of the individual flow rate vectors ri. Further, this results in G being fully connected. To maximize the data rate from the source to the destination through the relay network, we need to consider each cut that partitions V into sets V8 and Vd with S E VS and D) E Vd resulting cut sets are such that one set contains the source node (S) and the other, the destination node (D). Clearly, there can be 2N-2 such possible cuts for the N- node relay network. Let these cuts and the corresponding cut sets be denoted by Ck, Vk, and Vkd, respectively, for k = 1, 2,... 2N-2. Further, for the graph G, for any two nodes A c Vk and B c Vkd, there exists a cut edge AB that crosses the cut. Denote the total flow through cut edge AB in a unit time interval by XAB Z= 2Ni- AB. Now recall from network flow theory [35] that the maximum flow rate from the source to the destination is specified by the minimal cut of the equivalent graph (2-1). Consequently, we arrive at the following convex flow optimization problem that can be solved using standard optimization techniques: max min 5 XAB, S XAB, s XAB (2-2) AEVf,BEVd AEV2,BEV2d AeV2_ ,BEVd / over all flow allocations xB and all time slot lengths ti, subject to: [27] A. Bletsas, A. Khisti, D. P. Reed, and A. Lippman, "A simple cooperative diversity method based on network path selection," IEEE J. Select. Areas Commun., vol. 24, No. 3, pp. 659-672, Mar. 2006. [28] E. Beres and R. S. Adve, "On selection cooperation in distributed networks," in Proc. Conf. Inr.[.., i Sciences and Syst. (C'ISS .:W'), pp. 1056-1061, Mar. 2006. [29] J. N. Laneman and G. W. Wornell, "Distributed space-time-coded protocols for exploiting cooperative diversity in wireless networks," IEEE Trans. Inform. Th(., vol. 49, no. 10, pp. 2415-2425, Oct. 2003. [30] T. F. Wong, T. M. Lok, and J. M. Shea, "Flow-optimized Cooperative Transmission for the Relay C'!i ,i,,. I" submitted to IEEE Trans. Inform. Th(.. -; Dec. 2006. [Online]. Available: http: //arxiv. org/PS_cache/cs/pdf/0701/0701019v3. pdf [31] L. Zheng and D. N. C. Tse, "Diversity and multiplexing: A fundamental tradeoff in multiple antenna channels," IEEE Trans. Inform. Th(.., vol. 49, no. 5, pp. 1073-1096, May 2003. [32] M. Yuksel and E. Erkip, \!ili ipl! .-antenna cooperative wireless systems: A diversity-multiplexing tradeoff perspective," IEEE Trans. Inr. ..t The.., ,/ vol. 53, no. 10, pp. 3371-3393, Oct. 2007. [33] T. M. Cover and J. A. Thomas, Elements of information Ji,, .,;, Wiley, 1991. [34] Y. Wu, P. A. Chou, and S.-Y. Kung, \Iiiiiiiiiiiii-energy multicast in mobile ad hoc networks using network coding," IEEE Trans. Commun., vol. 53, no. 11, Nov. 2005. [35] R. K. Alli T. L. Magnanti, and J. B. Orlin, Network flows: Th(..,; l1J..,':thms, and applications, Prentice Hall, 1993. [36] C. T. Lawrence, J. L. Zhou, and A. L. Tits, "User's guide for CFSQP version 2.5: A C code for solving (large scale) constrained nonlinear (minimax) optimization, generating iterates satisfying all inequality constraints," Technical Report TR-94- 16rl, University of Maryland, College Park, 1997. [37] W. P. Tam, T. M. Lok, and T. F. Wong, "Flow optimization in parallel relay networks with cooperative relaying," IEEE Trans. Wireless Commun., vol. 8, no. 1, pp. 278-287, 2009. [38] D. Tse and S. Hanly, \!!li i-access fading channels. Part I. Polymatroid structure, optimal resource allocation and throughput capacities," IEEE Trans. I.i i[. Ti .., -,; vol. 44, No. 7, pp. 2796 2815, Nov. 1998. [39] G. Caire, G. Taricco, and E. Biglieri, "Optimum power control over fading chan- nels," IEEE Trans. It. [..,, The..,,l vol. 45, no. 5, pp. 1468-1489, Jul. 1999. as follows: pAtPlco 1 P) 6(tpilco), (5-21a) PtpPlpr tplco, S 1) 6(Plpr), (5-21b) P(x"P2co tPlco, s 1) = (XP2co), (5-21c) p(XP2pr Xp2co, tPpr, tplco, S 1) 6(XP2pr), (5-21d) P(uccotplco, s 1) 60(ucco), (5-21e) p(UCpr lCcotpico, S 1) O 60(ucpr), (5-21f) p(xpico tplco, s t) 0 (XPlico), (5-21g) p(XP1pr Xpico, tp1pr, tp1co, S t) 0 6("xplpr). (5-21h) Theorem 5.2. For the discrete memornl' ICUC-HDC, all rate tuples (Rp, Rc), where Rp Rp1 + RP2 RPlco + RPpr + RP2co + RP2pr, RPIco Rs + R Re C RCco + Rcr, with non-negative reals Rs, Re, RPpr, RP2co, P2pr, Cco, Rpr f.' Rpipr < al (Xpipr; Vc XpIco, S 1) (5-22a) Rpi < aI(Xpipr; Vc IS 1) (5-22b) RP2pr < aI (Xp2pr; Yp, Ucco Xp2co, TpIpr, TpIco, S t) (5-22c) RP2 < aI (Xp2pr; YP, UccoI TPpr, Tlco, S t ) (5-22d) P2pr + RCco aI (XP2pr, UCco; Yp Xp2co, Tp1pr, Tplco, S = t) (5-22e) RP2 + RCco < a I (XP2pr, Ucco; YP Tppr, T lco, S = t) (5-22f) RPlpr + RP2pr < OaI (Xpipr; Yp|XpIco, S = 1) +aI (Tplpr, Xp2pr; Yp, UCcoXp2co, Tpco, S =t) (5-22g) RPlpr + RP2 < aI (Xplpr; Yp XpIco, S = 1) +aI (Tppr, Xp2pr; Yp, Ucco TPlco, S = t) (5-22h) RPIpr + RP2pr + RCco < aI (Xplpr; Yp XpIco, S = 1) The rest of the chapter is organized as follows. In the next section, we present the design of the proposed flow-optimized protocol for a general N-node wireless relay network. In Section 2.3, we use the flow-optimized solution to the three-node relay network and use it to develop the GLS protocol and establish its optimality in terms of the diversity-multiplexing tradeoff. Numerical examples comparing the performances of the FO and GLS protocols to that corresponding to the work in [27] are presented in Section 2.4. Lastly, the main contributions in this chapter are summarized in Section 2.5. 2.2 General Design Using A Flow-theoretic Approach We propose a general design for the transmission of messages from a source to a particular destination through a relay network using the idea of network flows with the optimal application of BC, MA and TS techniques. This cooperative transmission scheme is developed for a relay network of N nodes, with a wireless link between each pair of nodes. We consider an N-node wireless relay network with a link joining each pair of nodes. Each such wireless link is described by a bandpass Gaussian channel with bandwidth W and one-sided noise spectral density No. We denote the power gain of the link from node i to node j as Zi. The link power gains are assumed to be independent and identically distributed (i.i.d.) exponential random variables with unit mean. This corresponds to the case of independent Rayleigh fading channels with unit average power gains. Moreover, we assume that each node has a maximum power limit of P and can only support half-duplex transmission. Note that this model can be easily generalized to the case where channels may have non-uniform average power gains (for which numerical examples are presented in Section 2.4), and where different nodes may have different maximum power constraints. More specifically, the latter case can be converted into the uniform maximum power constraints case by absorbing the non-uniformity in the transmit powers into the average power gains of the corresponding links. In the sequel, we characterize the system in terms of the transmit signal-to-noise ratio (SNR), S = P at the input of the links. Time is divided into unit intervals, and BC and MA are applied CHAPTER 2 COOPERATIVE TRANSMISSION IN A WIRELESS RELAY NETWORK BASED ON FLOW MANAGEMENT 2.1 Introduction A wireless relay network is one in which a set of relay nodes assist a source node transmit information to a destination node. Practically the wireless nodes can only sup- port half-duplex communication [11], i.e., no nodes can receive and transmit information simultaneously on the same frequency band. Different cooperative transmission schemes for systems with half-duplex nodes have been proposed in the literature. Fundamentally, these schemes consist of two basic steps. First, the source transmits to the destination, and the relay listens and "< .i ni. [12] the transmission from the source at the same time. Next, the relays send processed source information to the destination while the source may still transmit to the destination directly. Variants of these techniques have been proposed and have been shown to yield good performance under different circum- stances [20, 19, 11]. Assuming channel state information (CSI) at the nodes, an opportunistic decode-and- forward (DF) protocol for half-duplex relay channels is proposed in [21]. The maximum delay-limited rate for this protocol is obtained by minimizing the average power over all feasible resource allocation functions such that the required rate is achieved. In [22], the authors present routing algorithms to optimize the rate from a source to a destination, based on the DF technique that uses regular block Markov encoding and windowed decoding [23, 24], for the Gaussian full-duplex multiple-relay channel. The achievable rate of [23] for the Gaussian physically degraded full-duplex multi-relay channel has been established as the capacity of this channel in [25]. In [26], it is shown that the cut-set bound on the capacity of the Gaussian single source-multiple re! -v-,i-:.l' destination mesh network can be achieved using the compress-and-forward (CF) method, as the relay powers go to infinity. 3.4 The Modified Bargaining Problem 2 The sensitivity of the operating point of the system to the uncertainty in CSIT may be reduced by decreasing its dependence on the NBS--,-.;.; -I. I1 strategy pair as described in this section. According to the modified bargaining problem, the NBS- ~-,-.; -1. .1 strategy pair (henceforth, the nominal strl' i'11 pair) is obtained using the available CSIT as in Subsection 3.3.2, but instead of the two users being constrained to implement these strategies, they get the flexibility of h.i. /*,p i'.1 ,.1;:i choosing their transmission strategies from a certain set of strategies about this nominal strategy pair. Let Ri(P*, R*, Z) and Pi(P*,R*, Z) (for i = 1, 2) be the actual transmission rates and powers respectively, given the nominal transmission strategies and available CSIT. Next, we utilize the fact that the proposed solution to (3-7) -,-i.; -1 using the maximum available powers for all fading states, and hence, we set Pi(P*, R*, Z) = Ti w. p. 1. Consequently, in what follows, we shall not represent the dependence of the transmission rates on P* explicitly. Also, we define the transmission strategies for the two users by the conditional PDF of only the transmission rates, with the conditioning on the fading state and the nominal transmission rates R*. The sets of allowable transmission strategies specify certain limited deviations from their nominal values to account for the uncertainty regarding the CSIT. Define the sets of allowed transmission rates for the two users as S, = {R : R, e [cR* AR, R* + AR] } for i = 1, 2. Then the choice of the actual transmission strategy of User i (i = 1, 2) is subject to the following constraint: RL(R*, Z) e Si w. p. (1 e), (3-8) 2 As would become clear from the subsequent discussion, we do not modify the bargain- ing problem as such. Only the implications of reaching an agreement are modified. Thus, the bargaining problem still stands valid as described in Section 3.3. Achievable Rate Regions: cgc=10, h C=1.5, hcP=0.55, Pp=6, Pc=6 ' I Interference Channel: HK region - Reference [6] - This work A \ \\ \ 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Rp (bits) Figure 5-8. Achievable Rates for the Gaussian ICUC-HDC: Dc and weak interference from Sc to Dp. Strong interference from Sp to 0.5 0 0 for block b C {B 1,... 2}, assuming that the decoding for the pair (wpIco,b, wP1pr,b) has been successful from block b+1, Dp searches for a unique tuple (,,. i,, wpipt, "/. "' .- ) and some tuple (wcco, bcco) such that (S"(w ), tjit'.O. We),tp,('. i', wpip,) xp(r) c i WPlco,b) Ppr (," 'I. WPlco,b, Wppr, WP1pr,b) xco ("'. p2co) ,x 2pr ( '. It' P2co, Wplpr, i' ) co (''. It'. Cco, co ) c A The error analysis for this decoding step shows that, for n large enough, (t'. i'. Wplpr, wP2co, "' ) (= Ws,b- 1, We,b-1, Wplpr,b-1, Wp2co,b, "' .- ,b) with arbitrarily small probability of error if (5-22c)-(5-221) are satisfied. At DC: The cognitive destination Dc also waits until block B, and then performs backward decoding to jointly decode the messages intended for it and the common part of the primary message. For block b e {B 1, ... 2}, Dc is assumed to have successfully decoded Wplco,b from block b + 1. With this knowledge, it searches for a unique tuple (" wcco, bcco, cpr, bcpr) and some it', such that s( ws), th '. We), x 1p ( '. ', w' wpico,b ), 2co co Cco, b co), '*. wcco, Cco, Cpr,bcpr), 1y c Ab . Again, using the properties of joint typicality, it can be established that, for n large enough, (n. co, bcc, ccpr, bcpr) (Ws,b- ,We,b- ,WCco,b, b, WCpr,b, bCpr,b) with an arbitrarily low probability of error if (5-22m)-(5-22r) are satisfied. Thus, the constraints on the rates as given in (5-22a) (5-22r) ensure that the average probability of error at the two destinations can be driven to zero and thus, they describe an achievable rate region for the ICUC-HDC. E E[Zs s ]=3.0, E[ZS D]=1.0, E[ZS D]=1.0 1 2 1 2 -. ---MAC --- FO-CMAC '' OR-CMAC S- Reference [6] . I I I I I i 0.1 0.2 0.3 0.4 R 0.5 0.6 0.7 0.8 Figure 4-4. Achievable rate regions symmetric situation. E[Zs s ]=3.0, E[ZS D]=1.5, E[ZS D]=0.2, K=1bit/s/Hz 1 2 1 2 0 5 10 15 20 SNR (dB) O 0.4 100 . 10-1 -Q g 0 -2 Figure 4-5. Outage performance .,-vmmetric situation. 25 30 35 40 helps avoid the non-convexity issue mentioned before. The value of X2(t1, 2) in (2-12) is obtained using similar arguments. ii. c(zs) > tIC(ZsHS): In this case, we have aa < 0 and 2 > t2max. Thus the maximum rate is given by X(tl,t2) max tlC(Zs-DaS)+tC 1 zsa S S}+t2C(ZSDS) = C(ZSDS). (2-13) o Hence, (2-11)-(2-13) imply that the solution to (2-8), for any ti, t2 pair, occurs at a = 1, and the solution to the original problem of (2-7), when ZSD > ZS-, is given by max{0_t,,t2 : tl+t2 1} X(tl, t2) = C(ZSDS) with any t1, t2 pair such that t1, t2 > 0 and t + t2 1. ZSD < ZS-R. In this case, the source-relay link is better than the source-destination link. Again, we first fix time slot lengths t, and t2 and solve for the optimal values of xa, X2, a3, and a, and then maximize the objective function of (2-7) over all feasible time slot lengths. Following similar arguments as in Case a), the optimization problem of (2-7) can be re-written as: max (xa + x2 + X3) (2-14) subject to 0 < a < min{1, a}, x = t1C ZSv-aS X2 tiC(ZsgZaS), b1 + ZSvaS ) x3 min {t2C(ZsDS), t2C(ZsDS + ZDS) tiC(ZsgaS)} where a 1 S [(1 + ZKRS)ts/tL 1] is an upper bound on a such that tiC(Zsg-aS) < t2C(ZRDS). Note that a > 0. The optimization problem in (2-14) is non-convex in a, but this technicality can be overcome by using the approach as in the previous case. As before, X3 t2C(ZsvS) if and only if a < 1 + ai'.S ijtl A Also, 0 < a' < a. Again, we consider two possible sub-cases: i. t2 -S nodes. Thus, the relay nodes, along with the source node, cooperate to form a virtual multiple-input-single-output (\! ISO) system in order to achieve transmit diversity benefits. In this work, we subject all the nodes in the relay network to the half-duplex constraint. Previously proposed solutions to this problem include different path selection methods and distributed space-time coding methods. Unfortunately, these methods fail to achieve the optimal diversity-multiplexing tradeoff for the wireless relay network, and fail to be efficient in the high data rate requirement regime. Under the assumption that the channel state information (CSI) is available at all nodes, we develop a flow-optimized protocol, and a suboptimal, but much simpler, generalized-link selection protocol, that are shown to be optimal in terms of the diversity-multiplexing tradeoff, and that provide large coding gains over direct transmission, even for high data rate requirements. 1.2 Cooperative Behavior in a Fading Multiple-Access Channel In the previous section, we introduced a problem involving a single-source-single- destination scenario. Next, we consider the simplest multi-source network: the multiple- access channel (\!AC), wherein two users wish to transmit information to a single desti- nation node. As the multiple transmitters share the same communication medium, unless there exists a certain amount of cooperation or understanding between the users, they could cause mutual interference to each other. Thus, even though it does not belong to the class of cooperative communication systems that typically involve cooperation in the form of active forwarding of information, there exists a certain level of cooperation between the users as would be made explicit in C'!i pter 3. Towards this, we consider the resource allocation problem for the two-user fading MAC. We consider the case in which there exists some uncertainty in the channel state information at the transmitters (CSIT). Under assumptions of perfect CSIT, the transmis- sion strategies 1- -.-- -1. 1 by the solution to the bargaining problem yield optimal operation points. On the other hand, owing to inaccuracies in the available CSIT, the conventional Next given an N-node relay network, consider the multiple access cut that separates the destination from all the other nodes. Clearly, the total flow across this cut gives an upper bound on the maximum rate achievable in the N-node relay network. Consequently, a lower bound on the outage probability PJ)t1(r, S) can be obtained using the maximum sum-rate across this cut: Put(r, S) > Pr [C((ZsD + Z, + + Z+Z 2D_)S) < r logS] = Pr [(ZSv + Z,vD + ---+ Z _,2D) P(N- 1) where 7(a, x) -= f' 1et dt is the lower incomplete gamma function and F(a) = fo ta- letdt is the complete gamma function. The result in part 1) of Theorem 4.2 of [30] can be extended to show that the diversity order of any transmission scheme over the wireless relay network must satisfy log (N-,(s"-)/s) d(r) < lim log (Plut (r, S)) < lim log (N-2) d~r) < hm -- ^ -- -< hm ----- -- ' s--oo log S s--oo log S im6 xN-2[- (1 -- )- (N- 1)(1 -r). (2-29) s- J0f ttN-2e tdt Finally, from (2-27) and (2-29), we see that the GLS protocol, and hence the FO protocol, achieve the optimal diversity-multiplexing tradeoff of (N 1)(1 r) for all 0 < r < 1. E 2.4 Numerical Examples To demonstrate the performance of the proposed protocols, we consider the four- and five-node relay networks, wherein information is to be sent from a source to a destination with the help of two and three relay nodes, respectively. Using outage probability as the performance metric, we compare the FO and GLS protocols against the max-min selection method of [27], as it provides the best performance amongst previously proposed path selection methods, and an outage probability lower bound derived using the max-flow-min- cut theorem of [33, Thm. 14.10.1]. Thus, we effectively have two relay channels over two orthogonal time slots. We use the flow-optimized transmission scheme of C'!i pter 2 for the three-node relay channel. According to the protocol presented therein, each time slot T1 (resp. T2) is further divided into two sub-slots of lengths t1 and t2 (resp. t' and t'). Consider the time slot TI. During the first sub-slot, S1 sends two independent flows of information xz and x2 to D and S2 respectively using a broadcast channel (BC), and in the second sub-slot, S2 forwards X2 to D and at the same time S1 sends out another information flow x3 to D, and X2 and X3 are received at D via multiple-access ('\ A). To improve the achievable rates even further, we modify the second sub-slot as follows. Since the flow X2, that S2 forwards to D originated at SI, the latter is aware of it, and hence, we modify the second sub-slot from a conventional MAC to a MAC with common information [61], where x2 forms the common information between S1 and S2, X3 is the independent information from SI, and S2 does not have any independent information to transmit. For this relaying scheme, maximizing the overall transmission rate from Si to D can be formulated as the following optimization problem: max (xi + 2 + 3) overt 1, t2, Xl, x2, x3, P1, To (4 1) subject to non-negativity constraints: xI,x2, 3 > 0; tI,t2 > 0; total-time constraint: t1 + t2 Ti; capacity (power) constraints: PBc < PI; x3 < t2C(ZsDPi); 2 + x3 0 < P I < P; To < ZS,D(PI Pi) + ZS2DP2 + 2ZS D(P P)ZS2DP2; where C(x) = log(1 +x) and PBC, the minimum power required for the source to broadcast at rates xz/tl and x2/ti to the destination and the relay, respectively, in the first sub-slot capacity region defined by the transmission powers and the true power gains, then these rates are considered achievable. Else, both the users suffer outages. For both error models, as AR increases from Obps/Hz to 2.5bps/Hz, the true utilities increase from the minimum utility points (when the error in the CSIT is unaccounted for), reach their respective maxima, that are very close to the achievable rates for the perfect CSIT scenario, and then decrease to eventually converge at the disagreement points. Hence, it can be concluded that with a proper choice of AR, the proposed solution can provide the necessary robustness against inaccuracies in the available CSIT. Moreover, as expected, it can be observed that the disagreement point is not affected significantly by the error in CSIT as it corresponds to an interior point of the MAC capacity region for any fading state. 3.5 Summary Optimal resource allocation and cooperative behavior in a two-user fading MAC with uncertainty regarding the CSIT are considered from a game-theoretic perspective. The resource allocation problem is modeled as a two-user bargaining problem with the average rate utility function. Owing to possible inaccuracies in the CSIT, the solution to the bargaining problem (for instance, the NBS), that depends on the CSIT, may not be optimal, and may cause system outages. To address this lack of robustness a modification to the bargaining problem formulation is proposed. Numerical results demonstrate that the proposed solution can be used to provide significant robustness without an explicit modeling of the error in the CSIT. * Generate 2nRP-o i.i.d. codewords tcoPc(w o), wpo e {1 ... 2nRP~}, according to i^ p(t Pcoi). For each codeword tGco(Wpco), generate 2nRpP" conditionally i.i.d. codewords tPr"(', "'1~.,), "',, E {1, C 2TRPP-}, according to Hi 1P(tPpri tpcoi(WP'o)). For each codeword tcoP(wco), generate 2nRP,0 conditionally i.i.d. codewords xco(,,' wpco), wpco e {1, ... 2nRP}, according to J p IP(xpcoi tpcoi(w'po)) For each codeword tuple (tPco(WP'co), pr(,', 'p,, ), x pco(po, wPco)), generate 2T'Rp i.i.d. codewords ,' (w'p, wpco, wpp., wppr), wppr e {, 2RPP }, according to =H lp (Xpi tcoi(wPc), tpPri (WP, co Ppr) (coi w/p co))- For each codeword t0co(w co), generate 2n(Rcc+R'cco) i.i.d. codewords ,, ,(w'co, wcco, bcco), wcco e {1,... 2nRco} and bcco {1, ... 2TRC'o}, according to 7iJ lp(uccoi tpcoi(w'co)). For each codeword pair (t'co(w co) ". (w'co, Cco, bcco)), generate 2T(Rc+pr+ pr) i.i.d. codewords ,,, (wpo, wcco, bcco, WCpr, bcpr), wcpr {1,- -...- 2Rcp} and br {1, ... 2nRc }, according to Y[ ip(UCpri tpcoi(w'pco) uccoi(,, wco, bcco)). Generate x(, -', ,tr'p, wcco, bcco, ~ pr, bcpr) where xc is a deterministic function of tPco, tPpr, UCco, UCpr. Encoding: At Sp: In block b {2, ... B 1}, Sp transmits i, (Wpco,b-1, WPco,b, WPpr,b-1, Wppr,b). In the first block, Sp transmits xT"(1, Wpco, 1, wppr,1), while in block B, it transmits x"'(wpco,B- 1, Wppr,B-, 1). Note that the actual rate for the primary message is BBRp, but it converges to Rp as the number of blocks B goes to infinity. At Sc: In block b E {1,.-- B}, to transmit WCco,b, Sc searches for bin index bcco,b such that (tco(wpo), u" o(wpo, WCco,b, bCco,b, pr (Pco, WPpr)) c AX, (5-3) where Wpco and wpp, are Sc's estimates of WPco,b-1 and Wppr,b-1 respectively from the previous block. Once bCco,b is determined, it searches for a bin index bcpr,b in order to transmit WCpr,b such that (tco((wpo)), aUo(wpco, WCco,b, bCco,b) ,~ r b( o, bCco,b, bCco,b, WCpr,b, bCpr,b), R at lbit/s/Hz and 6bits/s/Hz respectively, for the four-node relay network. Figs. 2-6 and 2-7 present the same for the five-node relay network. When compared to the FO protocol, the GLS protocol suffers a loss of around 1.0dB, and around 1.5dB (when R is either lbit/s/Hz or 6bits/s/Hz), at an outage probability of 10-4, for the four- and five-node relay networks respectively. On the other hand, the performance degradation for the max-min selection method of [27], as compared to the FO protocol or even the GLS protocol, is more than 12dB at an outage probability of 7.0 x 10-2, when R = 6bits/s/Hz for the four-node relay network, and an exactly similar situation can be observed for the five-node relay network. Moreover, for the four-node relay network, the FO protocol is within 2.14dB (when R l= bit/s/Hz) to within 7.05dB (when R = 6bits/s/Hz) of the lower bound on the outage probability when the outage probability is 10-4. For the five-node relay network, the corresponding differences are approximately 3dB and 9.6dB respectively. Thus, we see that as the number of nodes in the relay network increases, the GLS protocol becomes more suboptimal, whereas the gap between the outage performance of the FO protocol and the lower bound widens. With regard to the latter observation, it should be kept in mind that the lower bound obtained using the max-flow-min-cut theorem is, in general, not a tight bound. The performances of the different protocols for the four-node relay network with non-uniform average power gains are presented in Figs. 2-8 and 2-9, and Figs. 2-10 and 2-11 for cases A and B respectively, with the average power gains as stated in the figures. Case A represents the situation when both the source-relay links are, on average, better than the direct link, and one relay-destination link (the link between Ri and D)is, on average, better than the other, resulting in relay Ri being a better candidate to forward the information than the other relay. On the other hand, case B represents the situation when no one relay has very good source-relay and relay-destination links. In this case, one source-relay link is, on average, better than the direct link, which, in turn, is better than the other source-relay link. The reverse is true for the relay-destination 3.3.1 The Disagreement Point Here, we consider the case when there is no agreement between the two users with regard to their transmission strategies. Let Ri and Pi be the transmission rate and power, respectively, of User i. Also, given a particular fading state and the transmis- sion strategies for the two users, the achievable transmission rate for User i is given by R'(P(Z), R(Z), Z). That is, if, for the fading state Z, the actions for the two users are specified by the transmission powers and rates P(Z) = (Pi(Z), P2(Z)) and R(Z) = (Ri(Z), R2(Z)) respectively, then the p ,ioff received by User i is given by R'(P, R, Z), where the dependence of P and R on Z is not made explicit for brevity. Particularly, these utilities correspond to the rates for the two users at which reliable transmission can be supported with an arbitrarily small probability of error. Due to the symmetric nature of the problem, it is sufficient to consider any one user's opera- tion, iw User 1. In the absence of any agreement between the two users, the achievable transmission rate for User 1, R', is given by (3-1), where C(x) = log(1 + x). RI if R, < min{C(ZiPi), max{C(ZiPi + Z2P2)- R (P, R, Z) < ( PR2, (3-t) 0 otherwise. Note that for the scenario wherein the users fail to reach an agreement, there is no restriction on the choice of the transmission strategies, as against the scenario wherein the users reach a mutual agreement whereby each user's choice of the transmission strategy is restricted in some specific way. Since each user is unaware of the strategy of the other, we derive the optimal strategy of each user as the solution to a maximin problem, wherein each user's aim is to maximize its own worst-case usage probability. Note that the users being able to i,.../. p .J ;./i. decide from the set of all randomized strategies implies that, in general, the users may use mixed strategies for the maximin games in this model. Thus, cognitive source operates in full-duplex manner, and then, for the situation wherein this requirement is removed and instead a half-duplex constraint is imposed on the cognitive source. Finally, in Ch'! pter 6, we conclude the dissertation, and discuss the possible directions for future work. comparing (5-31)-(5-33) to (5-34)-(5-36) establishes that the region corresponding to Protocol 1 is contained in R7V. The inclusion of the rate region corresponding to Protocol 2 can be easily proved by considering the same coding structure and input distribution as used to obtain R", with one further restriction the input distribution at Sp for the first time-slot is given by p(xplcos = 1)p(xpip, s = 1). This yields an achievable rate region R7' (C 7), that has exactly the same bounds as that for Protocol 2, except that the achievable rate region for the NC-CRC (during the second time-slot) is 7'NC 7-DMT, thereby proving the above inclusion. The rate region for Protocol 3 can be obtained by setting S = t w.p. 1, Xppco = XPIpr = TPIco = TPpr = in Theorem 5.2. Finally, the rate pair corresponding to Protocol 4 may be obtained by using a fixed listen-transmit schedule, and by setting Tpico = XIco = Xp2co XP2pr Ucco Ucpr = As the four rate regions of [6] are contained in 7?, the convex hull of these regions (7Ro) is also contained in 7R (cf. Remark 5.5). 5.7.2 Numerical Results For a fixed listen-transmit schedule and considering parallel Gaussian channels (as in Remark 5.6), the transmitted signals in any communication block, corresponding to the coding scheme of Theorem 5.2, can be expressed as: Xp,i(wp,) = XPI + XP+p,r (5-37) Xp,t(wpl, Wp2) Tp1co + XP2o + TP1pr + Xppr, (5-38) Xc(Wpi, Wc) = Xcco ~ XCpr ++ p TpCo + a TPc1pr (5-39) apypPp apppPp where Xpico -~ V(0, ]pPp), Xp'p, A~ (O, ]pPp), TpIco ~ /V(0 apypPp), Xp2co f(O,ap7ypPp), T'p,- ~ Af(0,apPpPp), X'2p ~ A(0,apP3pPp), Xcco (O0,acf3cPc), and Xcp, ~ A/(0, accPc) are i.i.d. random variables. In the above, Tp, ap, /3, 7p, ac, In this case we have a' < 1 and t2 < C(ZSRS) A t2max. The maximum S-C ( V) +C(Zs S) ) rate X(ti,t2) can be expressed as X(ti, t2) max{Xi(ti,t2), X2(t1,t2)} where XI(tl,t2) = max t1c 1ZSaS) + t1C(ZS aS) + t2C(ZsDS) o 1- (1 ZsDab'S + t1C(ZsRalS) + t2C(ZSDS) (2-15) 1 + ZAS-DaS X2(ti,t2) max tiC ZDaS + 2C(Z-DS + ZRDS) a tiC 1 ZsDabS + t2C(ZSDS + ZRDS), (2-16) 1 + ZsvaS and both the maxima in (2-15) and (2-16) are attained at a a'. Substituting the expression for a' in (2-15) and (2-16), we obtain XI(t1, t2) X21(t,t2) and X(t ,t2) tilog 10 +ZS-DS ] + t2C(ZS-S + ZRvS). (2-17) 1 + z"S ( + z- s t 1 SIctC Zs SZS In this case we have a' > 1 and t2 > t2max (with t2max as in sub-case i). Thus the maximum rate is given by X(t,,t2) = max ti Z S 1C(ZsaS) + t2C(ZSDS) o = tlC(ZSR) + t2C(ZsDS) (2-18) where the maximum occurs at a = 1, as the function to be maximized is monoton- ically increasing in a when ZSR > ZSD. Again, the apparent non-convexity of the optimization problem (2-14) in a is avoided by considering the sum of Xi and x2 together, and utilizing the last observation regarding the monotonicity property of the objective function in (2-18). Finally, we optimize the above solution to (2-14) over all possible time slot lengths to obtain the solution to the original problem in (2-7) when ZSD < ZSR. Corresponding to Case i. above, when t2 t2max, we note that max X(t,,t2) > X(1 t2,t2) t,=t2max {O C(ZsRS) + C(ZRDS + ZSDS) C(ZsDS) 4.2 Flow-theoretic Transmission Protocols for the Cooperative Multiple-Access C i i . .. 67 4.2.1 Orthogonal Relaying Protocol for the C'\! AC (OR-C \!AC) 68 4.2.2 Flow-optimized Protocol for the C\ 1. AC (FO-C \! AC) ... 70 4.3 Numerical Results . . 72 4.4 Sum m ary . . 74 5 INTERFERENCE CHANNELS WITH UNIDIRECTIONAL COOPERATION AND CAUSALITY CONSTRAINTS . 79 5.1 Introduction . . 79 5.2 The CI, ,ii,, I M odel . 82 5.3 Achievable Rates for the ICUC-C . ... 83 5.4 The Gaussian ICUC-C ...... . 91 5.5 Discrete Memoryless CI, .i,,, I Model for the ICUC-HDC .. 94 5.6 An Achievable Rate Region for the ICUC-HDC ... 95 5.7 The Gaussian ICUC-HDC . 103 5.7.1 Inclusion Of Causal Achievable Region of [6] 105 5.7.2 Numerical Results . 107 5.8 Sum m ary . . 109 6 CONCLUSIONS AND FUTURE WORK . 116 6.1 Conclusions . . 116 6.2 Future Directions . . 118 REFERENCES . .. .. .. ... .. 120 BIOGRAPHICAL SKETCH . . 126 conditioned on the codeword pair (S, Tpico) for the resolution information for the common part of wpi,b-1, it uses conditional GP inning [64] to encode wcco,b and wcpr,b as Ucco and Ucpr respectively, against the resolution information for the private part of wpI,b-1 (TPpr). It transmits a combination of the above codewords, along with the resolution information, during the Sc-transmit states. Both Dp and DC wait until the transmission in block B, and then use backward decoding [72] to jointly decode both common and private parts of its intended message and the common message parts) from the interfering transmission. Note that Dc performs backward decoding only to decode WPlco,b-1 in order to take advantage of the block Markov SPC structure used to encode it. Table 5-2 lists the random variables involved in the code construction along with their significance. Table 5-2. Description of Random Variables in Theorem 5.2 Random Variable Definition S Listen-transmit state for Sc Tpico Resolution information for common part of primary message wpi (known to Sc) Tplpr Resolution information for private part of primary message wpi (known to Sc) XPIco New information for common part of primary message wpi (decoded by Sc) Xp1pr New information for private part of primary message wpi (decoded by Sc) Xp2co Common part of primary message Wp2 (not decoded by Sc) Xp2pr Private part of primary message Wp2 (not decoded by Sc) Ucco Common part of secondary message (generated by conditional Gel'fand-Pinsker inning) Ucpr Private part of secondary message (generated by con- ditional Gel'fand-Pinkser inning) Xp Transmitted codeword by Sp Xc Transmitted codeword by Sc Let a = Pr[S = 1], and a = 1 a. Owing to the half-duplex constraint to the channel model, we restrict the distributions for the codewords used in the codebook construction Strong PC, Weak CP Interference: c=10, h c=1.5, hcp=0.55 I I I II !' Single User RatE 4; PP-1.5, Pc=f I ..,Coding 1, PP=1.5, P =6 .....Coding I, Pp=1.5, PC=6 - Coding III, Pp=1.5, PC=6 I Ok - Coding IV, Pp=1.5, Pc=6 ....... Coding I, Pp=6, P =6 ..... Coding II, Pp=6, P =6 : - Coding III, Pp=6, PC=6 - Coding IV, Pp=6, Pc=6 0 0.2 0.4 0.6 0.8 Figure 5-4. 1 Rp (bits) Single User Rate (Rp Pp=6, Pc=6 / 1.2 1.4 1.6 1.8 Achievable Rates for the Gaussian ICUC-C: Strong interference from Sp to Dc and weak interference from Sc to Dp. S i s x3+Y22 S2 y3+x22 T3 (b) Figure 4-1. Flow-theoretic transmission protocols for the C \! AC: (a) OR-C\ A!.C, (b) FO-C \! AC. x+y21 S, x2 S2 yi+x21 T2 problem in two stages: first, we fix ti, t2 > 0 such that tl + t2 = 1 and find the optimal flows x1, X2, X3 in terms of t1, t2, and then, find the optimal values for t1, t2 to maximize the objective function. ZSD > ZS-R. For this case, the source-destination link is at least as good as the source-relay link. To obtain an analytical solution to the optimization problem and better insight into the nature of the solution to the flow optimization problem, we modify the representation of the BC slot power constraint from that in (2-7) to the one that is more conventionally used to describe the capacity region of the Gaussian BC channel, as presented in (2-8). Using the flow constraint in (2-7), we first solve (2-8) for fixed ti, t2. max(xi + X2 + X3) over x1, X2, X3, a, subject to (2-8) x1, X2, X3 > 0, 0 < a < 1, x X2 < t2C(ZZDS), X3 < min t2C(ZsS), t2C(ZSDS + ZDS) x2} - Here, a C [0, 1] is the fraction of total power spent at the source to transmit x, directly to the destination during the BC slot, and a = 1 a. Although, this modification of the BC slot power constraint apparently makes the optimization problem non-convex owing to the non-convexity in a, as we shall see in the sequel, this issue can be handled easily by utilizing the monotonicity of the logarithm function. Denote the optimal solution by (x*, x*, x*, a*) and the corresponding maximum rate by X(tl,t2). It is clear that x = tlC(ZsDa*S). Suppose that x* < tiC (l+Zs~i s Since t1C ZsRas ) is a decreasing function of a, we can increase a from a* to ao such that x tiC(ZsDvaS) > x and x tiC ( zs' ). Thus the objective function becomes (xo + x< + x*) > (x* + x' + x*) at a This contradicts the optimality of (x, x, x ,*a*). As a consequence, we have x* = t1C lZsa*s) <- t2C(ZKZS). This implies that a* > |

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PAGE 1 COOPERATIVECOMMUNICATIONINWIRELESSNETWORKS: FLOW-OPTIMIZEDDESIGNSANDINFORMATION-THEORETIC CHARACTERIZATIONS By DEBDEEPCHATTERJEE ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2010 1 PAGE 2 c 2010DebdeepChatterjee 2 PAGE 3 Tothereader 3 PAGE 4 ACKNOWLEDGMENTS IwouldliketothankmyadvisorProf.TanWongforhisguidance,support,patience, andthefreedomIenjoyedinchoosingmyresearchdirection.Iconsidermyselfvery fortunatetohavebeenabletopursueresearchundertheguidanceofsomeonewho encouragedmetotrytodenemyownresearchproblems,andatthesametime,was patientenoughwhenaparticularideawouldfailtobearfruitasexpected. IwouldalsoliketothankProf.JohnSheaforhisgeneralguidanceandsuggestions regardingpursuingresearch,andmoreimportantly,thoseonpresentingone'sresearch.I thankProf.MichaelFangandProf.WilliamHagerfortheirtimeandinterestinmywork. IwouldliketotakethisopportunitytothankDr.OzgurOymanofIntelResearchforthe stimulatingdiscussionsthatwehadduringmystayinSantaClara,andforhisvaluable commentsandsuggestionsregardingsomeofthelaterpartsofthiswork. MystayintheWINGlabalmostneverhadadullmoment,andcreditforthatis duetomylab-mates,especiallySurendra,Ryan,andByong,whocompletelychanged theatmosphereofthelabeversincethesummerof2006whenIwaspracticallythe onlypersonpresentinthelab.TherearewaytoomanypeopleIamindebtedtofor allthehelpandsupportIhavereceivedinthepastfewyears.Sridhar,Selvi,Manu, Savya,Vaibhav,andMallickarejustafewpeoplewhohaveenduredmeovertheseyears, providedmewithencouragementandhopesometimesblatantlyfalse,buttheymostly workedwhenthingshavenotworkedout,andmostimportantly,beengreatfriends. Finally,Ithankmyparents,fornoachievement,howeverbigorsmall,mayeverbe realizedwithouttheirloveandsupport. 4 PAGE 5 TABLEOFCONTENTS page ACKNOWLEDGMENTS.................................4 LISTOFTABLES.....................................7 LISTOFFIGURES....................................8 ABSTRACT........................................10 CHAPTER 1INTRODUCTION..................................12 1.1CooperativeTransmissioninWirelessRelayNetworks............13 1.2CooperativeBehaviorinaFadingMultiple-AccessChannel.........14 1.3ActiveUserCooperationinaFadingCMAC.................15 1.4AchievableRatesintheICUCwithCausalityConstraints.........15 1.5OrganizationoftheDissertation........................17 2COOPERATIVETRANSMISSIONINAWIRELESSRELAYNETWORK BASEDONFLOWMANAGEMENT.......................19 2.1Introduction...................................19 2.2GeneralDesignUsingAFlow-theoreticApproach..............22 2.3Generalized-linkSelectionandItsOptimality................26 2.3.1TheThree-nodeRelayNetwork....................26 2.3.2Generalized-linkSelection........................32 2.3.3Diversity-multiplexingtradeo.....................34 2.4NumericalExamples..............................37 2.5Summary....................................40 3RESOURCEALLOCATIONANDCOOPERATIVEBEHAVIORINFADING MULTIPLE-ACCESSCHANNELSUNDERUNCERTAINTY..........48 3.1Introduction...................................48 3.2SystemModel..................................51 3.3TheBargainingProblemfortheTwo-UserFadingMAC..........52 3.3.1TheDisagreementPoint........................53 3.3.2TheNashBargainingSolutionNBS.................54 3.4TheModiedBargainingProblem.......................58 3.5Summary....................................62 4ACTIVEUSERCOOPERATIONINFADINGMULTIPLE-ACCESSCHANNELS.........................................64 4.1Introduction...................................64 5 PAGE 6 4.2Flow-theoreticTransmissionProtocolsfortheCooperativeMultiple-Access Channel.....................................67 4.2.1OrthogonalRelayingProtocolfortheCMACOR-CMAC.....68 4.2.2Flow-optimizedProtocolfortheCMACFO-CMAC........70 4.3NumericalResults................................72 4.4Summary....................................74 5INTERFERENCECHANNELSWITHUNIDIRECTIONALCOOPERATION ANDCAUSALITYCONSTRAINTS........................79 5.1Introduction...................................79 5.2TheChannelModel...............................82 5.3AchievableRatesfortheICUC-C.......................83 5.4TheGaussianICUC-C.............................91 5.5DiscreteMemorylessChannelModelfortheICUC-HDC..........94 5.6AnAchievableRateRegionfortheICUC-HDC...............95 5.7TheGaussianICUC-HDC...........................103 5.7.1InclusionOfCausalAchievableRegionof[6].............105 5.7.2NumericalResults............................107 5.8Summary....................................109 6CONCLUSIONSANDFUTUREWORK......................116 6.1Conclusions...................................116 6.2FutureDirections................................118 REFERENCES.......................................120 BIOGRAPHICALSKETCH................................126 6 PAGE 7 LISTOFTABLES Table page 5-1DescriptionofRandomVariablesinTheorem5.1.................84 5-2DescriptionofRandomVariablesinTheorem5.2.................96 7 PAGE 8 LISTOFFIGURES Figure page 2-1Basicgraphs G 1 and G 2 forthethree-noderelaynetworkwith t 1 + t 2 =1....42 2-2FOprotocolforthefour-noderelaynetworkwith t 1 + + t 6 =1.Theow optimizationisperformedoverallows x 1 ; ;x 14 ,andalltimeslotlengths t 1 ; ;t 6 .......................................42 2-3Transmissionstrategytoobtainalowerboundontheoutageprobabilityfor thefour-noderelaynetwork.Here t 1 + + t 4 =1,andtheoptimizationis over x 1 ; ;x 14 ,and t 1 ; ;t 4 ,withtheapplicationofthemax-ow-mincuttheoremfortheintermediateslots........................43 2-4Four-noderelaynetworkwithuniformaveragepowergains:Outageprobabilitiesforrequiredrate R =1bit/s/Hz.........................44 2-5Four-noderelaynetworkwithuniformaveragepowergains:Outageprobabilitiesforrequiredrate R =6bits/s/Hz.........................44 2-6Five-noderelaynetworkwithuniformaveragepowergains:Outageprobabilitiesforrequiredrate R =1bits/s/Hz.........................45 2-7Five-noderelaynetworkwithuniformaveragepowergains:Outageprobabilitiesforrequiredrate R =6bits/s/Hz.........................45 2-8Four-noderelaynetworkwithnon-uniformaveragepowergains.CaseA: E [ Z SR 1 ]= 2 : 0 ; E [ Z SR 2 ]=2 : 0 ; E [ Z SD ]=1 : 0 ; E [ Z R 1 R 2 ]=1 : 0 ; E [ Z R 1 D ]=1 : 5 ; E [ Z R 2 D ]= 1 : 0.Outageprobabilitiesforrequiredrate R =1bit/s/Hz.............46 2-9Four-noderelaynetworkwithnon-uniformaveragepowergains.CaseA: E [ Z SR 1 ]= 2 : 0 ; E [ Z SR 2 ]=2 : 0 ; E [ Z SD ]=1 : 0 ; E [ Z R 1 R 2 ]=1 : 0 ; E [ Z R 1 D ]=1 : 5 ; E [ Z R 2 D ]= 1 : 0.Outageprobabilitiesforrequiredrate R =6bit/s/Hz.............46 2-10Four-noderelaynetworkwithnon-uniformaveragepowergains.CaseB: E [ Z SR 1 ]= 1 : 5 ; E [ Z SR 2 ]=0 : 75 ; E [ Z SD ]=1 : 0 ; E [ Z R 1 R 2 ]=3 : 5 ; E [ Z R 1 D ]=0 : 2 ; E [ Z R 2 D ]= 3 : 0.Outageprobabilitiesforrequiredrate R =1bit/s/Hz.............47 2-11Four-noderelaynetworkwithnon-uniformaveragepowergains.CaseB: E [ Z SR 1 ]= 1 : 5 ; E [ Z SR 2 ]=0 : 75 ; E [ Z SD ]=1 : 0 ; E [ Z R 1 R 2 ]=3 : 5 ; E [ Z R 1 D ]=0 : 2 ; E [ Z R 2 D ]= 3 : 0.Outageprobabilitiesforrequiredrate R =6bit/s/Hz.............47 3-1Averagerateswithvarying R ............................63 4-1Flow-theoretictransmissionprotocolsfortheCMAC:aOR-CMAC,bFOCMAC.........................................75 4-2Achievablerateregions-asymmetricsituation...................76 8 PAGE 9 4-3Achievablerateregions-asymmetricsituationwithconventionalMAslotwithoutcommoninformation...............................76 4-4Achievablerateregions-symmetricsituation....................77 4-5Outageperformance-asymmetricsituation.....................77 4-6Outageperformance-symmetricsituation......................78 5-1ThediscretememorylessICUCwithcausalityconstraint..............110 5-2TheGaussianICUC-C.................................110 5-3AchievableRatesfortheGaussianICUC-C:Weakinterferenceforbothcrosslinks...........................................111 5-4AchievableRatesfortheGaussianICUC-C:Stronginterferencefrom S P to D C andweakinterferencefrom S C to D P ........................112 5-5AchievableRatesfortheGaussianICUC-C:Weakinterferencefrom S P to D C andstronginterferencefrom S C to D P ........................113 5-6ThediscretememorylessICUC-HDC.........................114 5-7AchievableRatesfortheGaussianICUC-HDC:Weakinterferenceforbothcrosslinks...........................................114 5-8AchievableRatesfortheGaussianICUC-HDC:Stronginterferencefrom S P to D C andweakinterferencefrom S C to D P ......................115 9 PAGE 10 AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy COOPERATIVECOMMUNICATIONINWIRELESSNETWORKS: FLOW-OPTIMIZEDDESIGNSANDINFORMATION-THEORETIC CHARACTERIZATIONS By DebdeepChatterjee August2010 Chair:TanF.Wong Major:ElectricalandComputerEngineering Thechallengesinvolvedinthedesignofecientcommunicationsystemsforthe wirelessmediumcanbeattributedtothefactthatthewirelessmediumpossessescertain uniquecharacteristics,themostimportantonesbeingthebroadcastnatureofthewireless medium,thesusceptibilitytointerferenceeects,andtheeectsofpathlossandfadingon wirelesslinkquality.Cooperationbetweendierenttransceiverscanpotentiallyaidfurther developmentofnext-generationwirelesscommunicationsystemsthatdemandhighdata ratesandanexcellentqualityofserviceQoS.Thisispossiblebyexploitingthebroadcast natureofthewirelessmedium,andthediversityadvantagesthatamulti-usersystem oers. Werstconsiderageneralsingle-source-single-destinationwirelessrelaynetwork andproposeaninformationow-optimizedcooperativetransmissiondesignthatachieves theoptimaldiversity-multiplexingtradeo.Next,weapplygame-theoretictechniques totheproblemsofresourceallocationandcharacterizationofcooperativebehaviorina two-userfadingmultiple-accesschannelMAC,withuncertaintyaboutthechannelstate informationatthetransmittersCSIT. Inthethirdpartofthedissertation,amoreactiveformtheabovecooperative behaviorisstudiedviaatwo-userfadingcooperativemultiple-accesschannelCMAC, whereeachuser,alongwithtransmittingitsowninformationtothedestination,helpsthe 10 PAGE 11 otherbyforwardingthelatter'sinformation.Weproposeecientcooperativetransmission strategiesbasedonaow-theoreticapproach,andevaluatetheirperformancesusing numericalsimulations. Finally,weconsidercommunicationthroughatwo-userinterferencechannelwith unidirectionalcooperationICUC,whereinonesourceusesitsknowledgeofthemessage oftheothertoreducetheinterferencetoitsowntransmission,andsimultaneously,help theotheruser-pairviacooperativerelaying.Weconsideraveryrealisticscenarioinwhich thecooperatingsourceissubjectedtoacausalityconstraint.Wederiveanewachievable rateregionforthediscretememorylessversionofthisformofICUC,anddemonstrate thecontributionsofthevariouscodingstrategiesinvolvedvianumericalsimulations forGaussianchannels.Wealsostudythesamechannelwiththecooperatingsource beingsubjecttothehalf-duplexconstraintaswell.Adiscretememorylesschannelmodel incorporatingthehalf-duplexconstraintispresented,andanewachievablerateregion, thatenlargesthelargestknownrateregionfortheGaussianversionofthischannel,is derivedforthischannel.Theachievablerateregionfortheproposedcodingscheme, specializedforGaussianchannels,isnumericallyevaluatedandthestrictinclusionofthe previouslyknownlargestrateregionisdemonstrated. 11 PAGE 12 CHAPTER1 INTRODUCTION Overthelastdecade,wirelesscommunicationsystemshavegainedpopularityata veryfastpace,andhavecurrentlybecomeanintegralpartofourdailylives.Withthe growingdemandsforhigherdataratesandbetterqualityofserviceQoSguarantees tosupportnext-generationwirelesssystems,thedemandforthedesignanddevelopmentoffasterandmorereliablemultiusercommunicationsystems,ascomparedto theexistingsolutions,ismorethaneverbefore.Itiswellknownthatmultiple-inputmultiple-outputMIMOsystemsoerconsiderableperformanceimprovementsover single-input-single-outputSISOsystemsbyecientexploitationofthediversitybenets offadingchannels[1].However,variouspracticalconsiderationsrelatedtothecost,form factorlimitationsandhardwareimplementationrestricttheadvantagesofMIMOsystems, especiallytransmitdiversitybenets,tobasestationsandaccesspointsofwirelessnetworks.Inthisregard, cooperativecommunication oersagoodalternativeinproviding similarbenetsbythesharingofantennaresourcesofmultiplesingle-antennanodesto formvirtualMIMOsystems.Further,cooperativecommunicationtechniquescanbeemployedforMIMOsystemsaswelltobolstertheoverallsystemeciencyandreliability.In general,cooperativecommunicationcomprisesofspecialsignaltransmissionandreception schemesthateectivelyexploitthebroadcastanddiversityadvantagesofthewireless medium,withdueconsiderationtothedetrimentaleectsofinterference. Theaspectofcooperationinwirelesssystemscanbebroadlycharacterizedintotwo dierentmanifestationsofcooperativebehavior: RelayCooperation: Dedicatedrelaynodesareavailabletohelpthesourcenodes transmitinformationtothedestinationnodes.Forsuchasystem,thesource anddestinationnodeswouldconstitutetheclassofusers,whiletherelaynodes wouldbepartofthesystem'sinfrastructure.Thethree-noderelaychannel[2]isthe simplestexampleofsuchasystem. 12 PAGE 13 UserCooperation: Inthiscase,theusers,whothemselveswishtotransmitorreceive information,cooperatewitheachotherbyusingsomeoftheirownresourcestorelay otherusers'information.Thecooperativemultiple-accesschannelCMAC[3,4], therelaybroadcastchannelRBC[5],theinterferencechannelwithunidirectional cooperationICUC,alsoknownasthecognitiveradiochannel[6,7],fallunderthis classofcooperativecommunicationsystems. Boththeseclassesofcooperativecommunicationtechniquespossessuniquebenets andlimitations[9]intermsofdierentperformancemetrics,implementationcostand complexity.Moreover,hybridsystems,thatmakeuseofbothclassesofcooperative communication,canalsobeenvisioned.Forinstance,cooperationwithinmultiplecellsof aninfrastructurenetworkisexpectedtoimprovetheeciencyandreliabilityoftheoverall system[8].ThebasestationsBSsovermultiplecellscooperatebyjointlyprocessing thetransmitted/receivedsignalsusinghighcapacitybackhaulconnections,whilethe mobilestationsMSscooperatethroughrelayingandconferencing"betweenthemselves. Thus,dependingontheapplicationrequirementsandnetworkconguration,dierent cooperativecommunicationsystemsdemanddierentdesignapproaches,andmanifest dierentformsofcooperativebehavioramongsttheparticipatingnodes. 1.1CooperativeTransmissioninWirelessRelayNetworks Awirelessrelaynetworkisamultiuserwirelesscommunicationnetworkwherein awirelesslinkexistsbetweeneachpairofnodes.Ingeneral,itmaybeconceivedas apartofalargernetworkthatitselfmaynotbefullyconnected.Itcanalsobeseen asaspecialformofamoregeneralrelaynetwork.Althoughaconsiderableamount ofworkonrelaynetworkshasbeenreportedintheliterature[10,11,12,13,14], etc.,thecapacityofeventhesimplestofallrelaynetworks-thegeneralthree-node relaychannel,isstillunknown.Intherstpartofthedissertation,weareconcerned withtheproblemoftransmissionofinformationfromasinglesourcenodetothesingle destinationnodeinawirelessrelaynetwork,withthehelpofcertaindedicatedrelay 13 PAGE 14 nodes.Thus,therelaynodes,alongwiththesourcenode,cooperatetoformavirtual multiple-input-single-outputMISOsysteminordertoachievetransmitdiversitybenets. Inthiswork,wesubjectallthenodesintherelaynetworktothehalf-duplexconstraint. Previouslyproposedsolutionstothisproblemincludedierentpathselectionmethods anddistributedspace-timecodingmethods.Unfortunately,thesemethodsfailtoachieve theoptimaldiversity-multiplexingtradeoforthewirelessrelaynetwork,andfailtobe ecientinthehighdataraterequirementregime.Undertheassumptionthatthechannel stateinformationCSIisavailableatallnodes,wedevelopaow-optimizedprotocol,and asuboptimal,butmuchsimpler,generalized-linkselectionprotocol,thatareshowntobe optimalintermsofthediversity-multiplexingtradeo,andthatprovidelargecodinggains overdirecttransmission,evenforhighdataraterequirements. 1.2CooperativeBehaviorinaFadingMultiple-AccessChannel Intheprevioussection,weintroducedaprobleminvolvingasingle-source-singledestinationscenario.Next,weconsiderthesimplestmulti-sourcenetwork:themultipleaccesschannelMAC,whereintwouserswishtotransmitinformationtoasingledestinationnode.Asthemultipletransmitterssharethesamecommunicationmedium,unless thereexistsacertainamountofcooperationorunderstandingbetweentheusers,they couldcausemutualinterferencetoeachother.Thus,eventhoughitdoesnotbelongtothe classofcooperativecommunicationsystemsthattypicallyinvolvecooperationintheform ofactiveforwardingofinformation,thereexistsacertainlevelofcooperationbetweenthe usersaswouldbemadeexplicitinChapter3. Towardsthis,weconsidertheresourceallocationproblemforthetwo-userfading MAC.Weconsiderthecaseinwhichthereexistssomeuncertaintyinthechannelstate informationatthetransmittersCSIT.UnderassumptionsofperfectCSIT,thetransmissionstrategiessuggestedbythesolutiontothebargainingproblemyieldoptimaloperation points.Ontheotherhand,owingtoinaccuraciesintheavailableCSIT,theconventional 14 PAGE 15 bargainingproblemsolutionmaynotbeoptimal,therebyresultinginconsiderableperformancedegradation.Toaddressthisrobustnessissue,weproposeaschemeinwhich theconditionsofthebargainingproblemarerelaxedtoreducethedependenceofthe systemperformanceonthesolutiontothebargainingproblem.Intheprocess,wedevelop agame-theoreticframeworktocharacterizethelevelofcooperationinvolvedfroman individualisticviewpoint. 1.3ActiveUserCooperationinaFadingCMAC Next,weconsidertheproblemof active usercooperationinatwo-userfadingCMAC. Thisalludestoasysteminwhicheachuser,apartfromtransmittingitsowninformation, maycooperatewiththeotherbyactivelyforwardingtheother'sinformationtothedestination.Therefore,theCMACcanalsobeconsideredasabasicexampleofcooperative communicationinamulti-source-single-destinationsystem.Clearly,thisinvolvesahigher levelofcooperationascomparedtotheformofcooperationinaconventionalMAC.We useaow-theoreticapproach,andproposeaow-optimizedsolutionandamuchsimpler butsuboptimalsolutionthatdecomposestheCMACintoorthogonalrelaychannels.The performancesoftheproposedprotocolsareevaluatedintermsoftheachievableaverage rateregionsandoutageprobabilities,andtheimprovementsoverconventionalMACsystemsandapreviouslyproposedmethodbasedondecode-and-forwardDFrelayingare demonstratedthroughsimulationresults. 1.4AchievableRatesintheICUCwithCausalityConstraints Finally,westudyamanifestationofcooperationinamulti-source-multi-destination system.Thesimplestmulti-source-multi-destinationsystemisthetwo-userinterference channel,whereinapairofsourceswishtotransmitinformationtotheirrespective destinationsresultingininterferencecausedtoeachother.Thecapacityregionforthe interferencechannel,exceptforthespecialcasesofstronginterference,isstillunknown, anduntilrecently,theoriginalHan-Kobayashischeme[15]hasbeenknowntoprovide thelargestachievablerateregion,withoutaclearideaoftightnessofthisregionwith 15 PAGE 16 regardtotheknownouterbounds.Veryrecently,in[16],theauthorshaveshownthat aHan-Kobayashitypeofcodingschemeyieldsarateregionthatiswithin1bitofthe tightestouterboundfortheGaussianinterferencechannel. Giventhisbackdrop,wedirectourfocustotheinterferencechannelwithanasymmetriccooperativerelationshipbetweenthesources.Suchanetworkisknownasthe interferencechannelwithunidirectionalcooperationICUC 1 ,andisthesimplestform ofgeneraloverlaycognitivenetworks[18].TheICUCconsistsofapairofsourcesthat demonstratedierentbehaviors.Ontheonehand,theprimarysourceissolelyinterested intransmittinginformationtoitsrespectivedestinationanddoesnotactivelycooperate withtheotheruser-pair.Ontheotherhand,thesecondaryor cognitive sourceusesits knowledgeabouttheprimarymessagetoreducetheinterferencecausedtoitsownlink bytheprimarytransmission,andatthesametime,aidtheprimaryuser-pairbyrelaying theprimarymessagetotheprimarydestination,therebyreducingtheeectofinterference causedbyitsowntransmissiontotheprimarylink.Mostoftheworkreportedintheliteratureonthischannelconsiderthescenarioinwhichthecognitivesourcehasnon-causal knowledgeabouttheprimarymessage.Inthiswork,weimposeapracticalrestriction thatthecognitivesourcemayonlyobtainthemessageoftheprimarysourceinacausal manner,i.e.thecognitive"abilityofthesecondarysourceneedstobe acquired .Anew achievablerateregionforthediscretememorylessversionoftheICUCwithcausalityconstraintICUC-CisderivedusingblockMarkovsuperpositioncodingSPC,conditional rate-splitting,conditionalGel'fand-PinkserGPbinning,andcooperativerelaying.This rateregionisevaluatedforGaussianchannels,andnumericalresultsarepresentedto demonstratethecontributionsofthevariouscodingstrategiesusedintheproposedcoding schemetowardsenlargingtheachievablerateregion. 1 Thisnetworkisalsoknownintheliteratureasthecognitiveradiochannel[6]orinterferencechannelwithdegradedmessagesetsIC-DMS[17] 16 PAGE 17 Wethenimposeyetanotherverypracticalconstraintonthecognitivesource.Itis assumedthatthecognitivesourcemayonlyoperateinahalf-duplexfashion.Adiscrete memorylesschannelmodelfortheICUCwithhalf-duplexandcausalityconstraintICUCHDCispresented,andanewachievablerateregionisderivedforthischannel.The randomcodingschemeusedtoobtaintherateregioninvolvesblockMarkovsuperposition codingSPC,conditionalrate-splitting,cooperativerelaying,conditionalGel'fand-Pinkser GPbinning,andarandomizationofthelisten-transmitscheduleforthecognitivesource. Wealsoprovethatthisrateregioncontainsthelargestachievablerateregionforthe GaussianICUC-HDCpreviouslyreportedintheliterature,andfurtherdemonstratethis usingnumericalresultsforthecaseofGaussianchannels. 1.5OrganizationoftheDissertation Therestofthedissertationisorganizedasfollows.InChapter2,wepresentthe proposedprotocolsforecientinformationtransmissioninawirelessrelaynetwork.We rstpresentthegeneraldesignoftheow-optimizedtransmissionprotocol,andthen, thegeneralized-linkselectionprotocol.Next,theoptimalityoftheproposedprotocol intermsofthediversity-multiplexingtradeoisestablished,andnally,wepresentthe performanceevaluationoftheproposedprotocolsforthenitesignal-to-noiseratioSNR regimethroughnumericalresults.InChapter3,wepresentthemodiedbargaining problemformulationtomodeltheresourceallocationprobleminatwo-userfadingMAC underuncertainty.Solutionstotheseproblemsforthetwochoicesofutilityfunctions arealsopresented,andnumericalexamplesareshowntohighlightvariousaspectsofthe problemandtheproposedsolutions.Weconsidertheproblemofactiveusercooperation inatwo-userfadingMACinChapter4,andpresenttheow-optimizedandorthogonal relayingprotocolsfortheCMAC.Simulationresultsfordierentscenariosarethen presentedfortheperformanceevaluationoftheproposedprotocols.InChapter5westudy theproblemofcommunicatingthroughanICUCwithcausalityconstraints,andpresent newachievablerateregionsforthesenetworks|rst,forthescenarioinwhichthe 17 PAGE 18 cognitivesourceoperatesinfull-duplexmanner,andthen,forthesituationwhereinthis requirementisremovedandinsteadahalf-duplexconstraintisimposedonthecognitive source.Finally,inChapter6,weconcludethedissertation,anddiscussthepossible directionsforfuturework. 18 PAGE 19 CHAPTER2 COOPERATIVETRANSMISSIONINAWIRELESSRELAYNETWORKBASEDON FLOWMANAGEMENT 2.1Introduction Awirelessrelaynetworkisoneinwhichasetofrelaynodesassistasourcenode transmitinformationtoadestinationnode.Practicallythewirelessnodescanonlysupporthalf-duplexcommunication[11],i.e.,nonodescanreceiveandtransmitinformation simultaneouslyonthesamefrequencyband.Dierentcooperativetransmissionschemes forsystemswithhalf-duplexnodeshavebeenproposedintheliterature.Fundamentally, theseschemesconsistoftwobasicsteps.First,thesourcetransmitstothedestination, andtherelaylistensandcaptures"[12]thetransmissionfromthesourceatthesame time.Next,therelayssendprocessedsourceinformationtothedestinationwhilethe sourcemaystilltransmittothedestinationdirectly.Variantsofthesetechniqueshave beenproposedandhavebeenshowntoyieldgoodperformanceunderdierentcircumstances[20,19,11]. AssumingchannelstateinformationCSIatthenodes,anopportunisticdecode-andforwardDFprotocolforhalf-duplexrelaychannelsisproposedin[21].Themaximum delay-limitedrateforthisprotocolisobtainedbyminimizingtheaveragepoweroverall feasibleresourceallocationfunctionssuchthattherequiredrateisachieved.In[22],the authorspresentroutingalgorithmstooptimizetheratefromasourcetoadestination, basedontheDFtechniquethatusesregularblockMarkovencodingandwindowed decoding[23,24],fortheGaussianfull-duplexmultiple-relaychannel.Theachievable rateof[23]fortheGaussianphysicallydegradedfull-duplexmulti-relaychannelhasbeen establishedasthecapacityofthischannelin[25].In[26],itisshownthatthecut-set boundonthecapacityoftheGaussiansinglesource-multiplerelay-singledestination meshnetworkcanbeachievedusingthecompress-and-forwardCFmethod,astherelay powersgotoinnity. 19 PAGE 20 Cooperativediversitymethodsbasedonnetworkpathselectionhavebeenproposed in[27,28].Theseselectionmethodsinclude:ithemax-minselectionmethod[27], whereintherelaynodewiththemaximumoftheminimumofthesource-relayand relay-destinationchannelgainsisselected;iitheharmonicmeanselectionmethod[27], whereintherelaynodewiththehighestharmonicmeanofthesource-relayandrelaydestinationchannelgainsisselected;andiiitheselectionschemeof[28],inwhichthe relaythatcancorrectlydecodetheinformationfromthesourceandhasthebestrelaydestinationchannelisselected.ThesemethodsachieveaDMTof d r = N )]TJ/F15 11.9552 Tf 12.188 0 Td [(1 )]TJ/F15 11.9552 Tf 12.187 0 Td [(2 r foran N noderelaynetworkandmultiplexinggain0 PAGE 21 toformulateaowtheoreticconvexoptimizationproblembasedonthechannelconditions.Insteadofconsideringatotalpowerconstraintforallthetransmittingnodesas in[30],wesubjecteachnodetoamaximumtransmitpowerconstraint.Thisyieldsamore reasonablesystemmodelforageneralwirelessrelaynetwork,especiallywhenthenumber ofnodesintherelaynetworkislarge.Theresultingrelayingprotocolwillbereferredto astheow-optimizedFOprotocol.Toobtainamorepracticalcooperativedesignwe developthegeneralized-linkselectionGLSprotocol,inwhichweselectthebestrelay nodeoutoftheavailableonestoformanequivalentthree-noderelaynetworktotransmit theinformationfromthesourcetothedestination.Thebenetofthis,overothernetwork pathselectionstrategies,becomesevidentwhentheraterequirementishigh.Itisshown thatthesimpleGLSprotocolisoptimalintermsoftheDMT[31]andyieldsacceptable performanceevenwhentheraterequirementishigh. Recently,in[32],theauthorshaveshownthatcompress-and-forwardCFrelaying achievestheoptimalDMTforthethree-node,half-duplexnetwork,andthatDFrelaying canachievetheoptimalDMTofthefour-nodefull-duplexnetwork.Inthiswork,we showthattheoptimalDMTcanbeachievedforageneral N -node N 3half-duplex networkusingtheFOorGLSprotocols.Here,itshouldbeclariedthatweconsider thatthewirelesslinksbetweeneachnode-pairexperienceindependentRayleighfading, andthiscorrespondstothedenitionof non-relaynetworked networksin[32].The performancesoftheFOandGLSprotocolsareevaluatednumericallyintermsoftheir outageprobabilitiesforfour-andve-noderelaynetworksforuniformandnon-uniform averagepowergains.ThenumericalresultsmotivatetheuseoftheGLSprotocolfor situationswherecomputationcomplexityisanissueandshowaremarkableimprovement overthemax-minselectionmethodof[27].Theproposeddesigns,basedonBCand MAalone,aresub-optimalingeneral.Forafairappraisaloftheproposedprotocols,we comparetheproposedprotocolstoanupperboundonthemaximumrate,derivedusing themax-ow-min-cuttheorem[33,Thm.14.10.1]. 21 PAGE 22 Therestofthechapterisorganizedasfollows.Inthenextsection,wepresent thedesignoftheproposedow-optimizedprotocolforageneral N -nodewirelessrelay network.InSection2.3,weusetheow-optimizedsolutiontothethree-noderelay networkanduseittodeveloptheGLSprotocolandestablishitsoptimalityintermsof thediversity-multiplexingtradeo.Numericalexamplescomparingtheperformancesof theFOandGLSprotocolstothatcorrespondingtotheworkin[27]arepresentedin Section2.4.Lastly,themaincontributionsinthischapteraresummarizedinSection2.5. 2.2GeneralDesignUsingAFlow-theoreticApproach Weproposeageneraldesignforthetransmissionofmessagesfromasourcetoa particulardestinationthrougharelaynetworkusingtheideaofnetworkowswith theoptimalapplicationofBC,MAandTStechniques.Thiscooperativetransmission schemeisdevelopedforarelaynetworkof N nodes,withawirelesslinkbetweeneach pairofnodes.Weconsideran N -nodewirelessrelaynetworkwithalinkjoiningeach pairofnodes.EachsuchwirelesslinkisdescribedbyabandpassGaussianchannelwith bandwidth W andone-sidednoisespectraldensity N 0 .Wedenotethepowergainofthe linkfromnode i tonode j as Z ij .Thelinkpowergainsareassumedtobeindependent andidenticallydistributedi.i.d.exponentialrandomvariableswithunitmean.This correspondstothecaseofindependentRayleighfadingchannelswithunitaveragepower gains.Moreover,weassumethateachnodehasamaximumpowerlimitof P andcan onlysupporthalf-duplextransmission.Notethatthismodelcanbeeasilygeneralizedto thecasewherechannelsmayhavenon-uniformaveragepowergainsforwhichnumerical examplesarepresentedinSection2.4,andwheredierentnodesmayhavedierent maximumpowerconstraints.Morespecically,thelattercasecanbeconvertedinto theuniformmaximumpowerconstraintscasebyabsorbingthenon-uniformityinthe transmitpowersintotheaveragepowergainsofthecorrespondinglinks.Inthesequel,we characterizethesystemintermsofthetransmitsignal-to-noiseratioSNR, S = P N 0 W attheinputofthelinks.Timeisdividedintounitintervals,andBCandMAareapplied 22 PAGE 23 withaTSstrategythatisoptimizedtomaximizethespectraleciencywhichwecall rate"hereafterforconvenience.Toavoidinterferencebetweenconcurrenttransmissions, atimeintervalisdividedintoslots: Duringtherstslot,thesourcemayBCtoalltheothernodesinthenetwork, subjecttoitspowerconstraint P Duringthesubsequentslots,arelaymayBCtoallothernodesexceptthesource node,oritmayreceiveowsfromallothernodesexceptfromthedestination throughMA. Duringtheverylastslot,thesourceandtherelaysmaysendinformationowsto thedestinationusingMA,withtheowsintheMAcapacityregioncorrespondingto amaximumtransmitpowerof P foreachnode. NotethattheforwardingofinformationbytherelaysisbasedontheDFapproach.For practicalityconsideration,itisassumedthatthephasesofthesimultaneouslytransmitted signalsfromdierentnodesarenotsynchronized.Ingeneral,fortheabovetransmission protocol,therewouldbeamaximumof2 N )]TJ/F15 11.9552 Tf 12.713 0 Td [(2+2=2 N )]TJ/F15 11.9552 Tf 12.713 0 Td [(2timeslotsoflengths t 1 ;t 2 ; ;t 2 N )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 respectively. Next,wedescribetheoptimizationproblemusingagraph-theoreticformulation. Deneagraph G = V;E ,where V isthesetofnodes, E isthesetofalllinksjoining thenodesinthegraph,andassociatethevectorr torepresenttheowratesassociated witheachlinkin E .Thus,thenumberofelementsinr equalsthecardinalityof E Forconvenience,wewrite G = V;E; r .Nowdenotethesourceby S ,thedestination by D ,andtherelaynodesby R 1 ;:::; R N )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 .Theslottingofaunittimeinterval,as describedabove,yieldssimplergraphsforeachtimeslot,thatwecall basicgraphs .A basicgraphiseitheroneinwhichaparticularnodemayBCtoseveralnodes,orin whichseveralnodestransmitviaMAtoaparticularnode.Thusforabasicgraph, weneedtoincludeonlythelinksbetweenthenodesthatmayparticipateduringthe concernedtimeslot.Forexample,assumethattherelay R 1 broadcaststoallnodes otherthanthesource,duringthe i -thtimeslot.Thebasicgraphisgivenby G i = V;E i ; r i where V = fS ; R 1 ; ; R N )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 ; Dg ;E i = fR 1 R 2 ; ; R 1 R N )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 ; R 1 Dg ; r i = 23 PAGE 24 x i R 1 R 2 t i x i R 1 R N )]TJ/F18 5.9776 Tf 5.757 0 Td [(2 t i x i R 1 D t i T ,where x i AB istheowfromnode A tonode B duringthe i -thtimeslot. Ingeneral,theproposeddesigninvolvesTSbetweenthebasicgraphstoyieldthe followingequivalentgraph G correspondingtoaunitintervalsee[34]forasimilaridea: G = V; [ i E i ; X i t i r i = t 1 G 1 + t 2 G 2 + ::: + t 2 N )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 G 2 N )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 : {1 wherethenumberofelementsineachvectorr i isextendedto j S i E i j byinsertingzeros appropriately.Thesecondequalityin2{1impliesthat G canbeviewedasalinear combinationofthebasicgraphs G i s,withtheequivalentsetofedgesgivenbytheunion ofthesets E i ,andtheequivalentowratevectorgivenbythelinearcombinationofthe individualowratevectorsr i .Further,thisresultsin G beingfullyconnected. Tomaximizethedataratefromthesourcetothedestinationthroughtherelay network,weneedtoconsidereachcutthatpartitions V intosets V s and V d with S2 V s and D2 V d resultingcutsetsaresuchthatonesetcontainsthesourcenode S andthe other,thedestinationnode D .Clearly,therecanbe2 N )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 suchpossiblecutsforthe N noderelaynetwork.Letthesecutsandthecorrespondingcutsetsbedenotedby C k V s k and V d k ,respectively,for k =1 ; 2 ; ; 2 N )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 .Further,forthegraph G ,foranytwonodes A 2 V s k and B 2 V d k ,thereexistsa cutedge AB thatcrossesthecut.Denotethetotal owthroughcutedge AB inaunittimeintervalby x AB = P 2 N )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 i =1 x i AB .Nowrecallfrom networkowtheory[35]thatthemaximumowratefromthesourcetothedestinationis speciedbytheminimalcutoftheequivalentgraph2{1.Consequently,wearriveatthe followingconvexowoptimizationproblemthatcanbesolvedusingstandardoptimization techniques: maxmin 0 B @ X A 2 V s 1 ;B 2 V d 1 x AB ; X A 2 V s 2 ;B 2 V d 2 x AB ; ; X A 2 V s 2 N )]TJ/F18 5.9776 Tf 5.756 0 Td [(2 ;B 2 V d 2 N )]TJ/F18 5.9776 Tf 5.756 0 Td [(2 x AB 1 C A {2 overallowallocations x i AB andalltimeslotlengths t i ,subjectto: 24 PAGE 25 the non-negativityconstraints : x i AB t i 0forallcutedges AB and i = 1 ; 2 ; ; 2 N )]TJ/F15 11.9552 Tf 11.955 0 Td [(2, the total-timeconstraint : t 1 + ::: + t 2 N )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 =1, the powercapacityconstraints : {foraBCslottheowratesshouldlieinthecapacityregionoftheBCchannel withthetransmittingnodehavingapowerconstraintof P {foranMAslottheowratesshouldlieinthecapacityregionoftheMA channelwithamaximumpowerconstraint P foreachtransmittingnode, the owconstraints :consideringsteadystateoperation,thetotalinformationow outofarelayshouldequaltheowintotherelayineachunittimeinterval. Notethatthedependenceoftheobjectivefunctiononthechannelgainsandthetime slotlengthsisimplicitlyexpressedthroughthecapacityconstraints.Denotethecut separating S fromalltheothernodesandthecutseparating D fromallnodesas C S and C D ,respectively.Thenweobservethatthecostfunctionin2{2abovecanbefurther simpliedtomaxmin f x C S ;x C D g ,where x C S = x SD + N )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 X j =1 x SR j and x C D = x SD + N )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 X i =1 x R i D {3 arethetotalowsacrosstheabove-mentionedcuts C S and C D ,respectively.Toseethis, considerthecut C with V s = fS ; R 1 ; ; R l g ,and V d = fR l +1 ; ; R N )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 ; Dg forsome l 2f 1 ; 2 ; ;N )]TJ/F15 11.9552 Tf 11.955 0 Td [(2 g .Thetotalowacrossthiscutisgivenby x C = x SD + N )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 X j = l +1 x SR j + l X i =1 x R i D + N )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 X j = l +1 x R i R j : {4 Now,considernode i for i 2f 1 ; 2 ; ;l g .Accordingtotheowconstraintfornode i x R i D + N )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 X j = l +1 x R i R j + l X k =1 ;k 6 = i x R i R k = x SR i + N )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 X j = l +1 x R j R i + l X k =1 ;k 6 = i x R k R i : {5 Summing2{5overall i 2f 1 ; 2 ; ;l g weget l X i =1 x R i D + N )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 X j = l +1 x R i R j = l X i =1 x SR i + N )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 X j = l +1 x R j R i : {6 25 PAGE 26 Since P l i =1 P N )]TJ/F20 7.9701 Tf 6.587 0 Td [(2 j = l +1 x R j R i 0,combining2{3,2{4and2{6gives x C x C S Similarly,wehave x C x C D .Thusthecostfunctionin2{2reducestotheabovementionedform. 2.3Generalized-linkSelectionandItsOptimality Inthissection,wepresenttheGLSprotocolandestablishtheoptimalityoftheFO andGLSprotocolsintermsoftheDMT.Thisisaccomplishedinthreesteps.First,we applytheFOprotocoltothethree-noderelaynetwork.Next,weproposetheGLSprotocolbasedonaselectionstrategythatissub-optimaltotheFOprotocolofSection2.2. Finally,theoptimalityoftheGLSprotocol,andthereby,thatoftheFOprotocol,is established. 2.3.1TheThree-nodeRelayNetwork Thethree-noderelaynetworkconsistsofasource S ,arelay R ,andadestination D .Wespecializethegeneraldesigndescribedintheprevioussectiontothisthree-node relaynetwork.Aunittimeintervalisdividedintotwotimeslotsoflengths t 1 and t 2 with t 1 + t 2 =1,andinformationisdividedinto3owsofdata x 1 x 2 ,and x 3 .Duringtherst timeslot, S sendsviaBCtwoowsofrates x 1 SD =t 1 = x 1 =t 1 and x 1 SR =t 1 = x 2 =t 1 to D and R ,respectively,resultinginthebasicgraph G 1 asinFig.2-1.Duringthesecondtime slot, R and S sendviaMAtwoowsofrates x 2 RD =t 2 = x 4 =t 2 and x 2 SD =t 2 = x 3 =t 2 to D respectively,resultinginthebasicgraph G 2 asinFig.2-1. Combiningthetwobasicgraphsyieldstheequivalentgraphas G = t 1 G 1 + t 2 G 2 .Note thattheinformationowofrate x 4 =t 2 sentby R duringtheMAtimeslotisfromtheow ofrate x 2 =t 1 itreceivedduringtheBCtimeslot.Thisgivesrisetotheowconstraintfor R ,i.e, x 4 = x 2 .Thus,wehavetheowconstraint x 4 = x 2 .Therateforthisnetworkis speciedbythemin-cutwhichisclearlymin f x 1 + x 2 + x 3 ; x 1 + x 4 + x 3 g .Hence,the owoptimizationproblemisgivenby: maxmin f x 1 + x 2 + x 3 ; x 1 + x 4 + x 3 g {7 26 PAGE 27 overowallocations x 1 ;x 2 ;x 3 ;x 4 ,andtimeslotlengths t 1 ;t 2 ,subjectto: non-negativityconstraints: x 1 ;x 2 ;x 3 ;x 4 0 ;t 1 ;t 2 0, total-timeconstraint: t 1 + t 2 =1, powerconstraints: S BC S;x 1 t 1 C Z SD S ;x 2 t 1 C Z SR S fortheBCslot, x 3 t 2 C Z SD S ;x 4 t 2 C Z RD S ;x 3 + x 4 t 2 C Z SD S + Z RD S fortheMAslot, owconstraint: x 2 = x 4 where C x =log+ x ,and S BC ,theminimumSNRrequiredforthesourcetobroadcast atrates x 1 =t 1 and x 2 =t 1 tothedestinationandtherelay,respectively,inthersttimeslot with0 PAGE 28 problemintwostages:rst,wex t 1 ;t 2 0suchthat t 1 + t 2 =1andndtheoptimal ows x 1 ;x 2 ;x 3 intermsof t 1 ;t 2 ,andthen,ndtheoptimalvaluesfor t 1 ;t 2 tomaximize theobjectivefunction. Z SD Z SR .Forthiscase,thesource-destinationlinkisatleastasgoodasthe source-relaylink.Toobtainananalyticalsolutiontotheoptimizationproblemandbetter insightintothenatureofthesolutiontotheowoptimizationproblem,wemodifythe representationoftheBCslotpowerconstraintfromthatin2{7totheonethatis moreconventionallyusedtodescribethecapacityregionoftheGaussianBCchannel,as presentedin2{8.Usingtheowconstraintin2{7,werstsolve2{8forxed t 1 ;t 2 max x 1 + x 2 + x 3 over x 1 ;x 2 ;x 3 ;; subjectto{8 x 1 ;x 2 ;x 3 0 ; 0 1 ; x 1 t 1 C Z SD S ;x 2 t 1 C Z SR S 1+ Z SR S ; x 2 t 2 C Z RD S ;x 3 min f t 2 C Z SD S ;t 2 C Z SD S + Z RD S )]TJ/F21 11.9552 Tf 11.956 0 Td [(x 2 g : Here, 2 [0 ; 1]isthefractionoftotalpowerspentatthesourcetotransmit x 1 directly tothedestinationduringtheBCslot,and =1 )]TJ/F21 11.9552 Tf 12.252 0 Td [( .Although,thismodicationofthe BCslotpowerconstraintapparentlymakestheoptimizationproblemnon-convexowingto thenon-convexityin ,asweshallseeinthesequel,thisissuecanbehandledeasilyby utilizingthemonotonicityofthelogarithmfunction. Denotetheoptimalsolutionby x 1 ;x 2 ;x 3 ; andthecorrespondingmaximumrate by X t 1 ;t 2 .Itisclearthat x 1 = t 1 C Z SD S .Supposethat x 2 PAGE 29 max 1 Z SR S 1+ Z SR S + Z RD S t 2 =t 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 | {z } a 0 ; 0 ; and 0 a 1 : Inessence,thismeansthattheoptimal x 1 and x 2 shouldlieontheboundaryofthedegradedBCcapacityregion.Withthis, itisobviousthenthat x 3 =min n t 2 C Z SD S ;t 2 C Z SD S + Z RD S )]TJ/F21 11.9552 Tf 11.955 0 Td [(t 1 C Z SR S 1+ Z SR S o Thereforetheoptimizationproblemof2{8canbere-writtenas: max x 1 + x 2 + x 3 {9 subjecttomax f 0 ; 0 a g 1 ;x 1 = t 1 C Z SD S ;x 2 = t 1 C Z SR S 1+ Z SR S ; x 3 =min t 2 C Z SD S ;t 2 C Z SD S + Z RD S )]TJ/F21 11.9552 Tf 11.955 0 Td [(t 1 C Z SR S 1+ Z SR S : Weobservethat x 3 = t 2 C Z SD S aboveifandonlyif 1 Z SR S 2 6 4 1+ Z SR S 1+ Z RD S 1+ Z SD S t 2 =t 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 3 7 5 1 a : {10 Comparingthistotheexpressionfor 0 a gives 0 a 1 a 1. Nextweconsidertwopossiblesub-cases: i. t 2 C Z RD S 1+ Z SD S t 1 C Z SR S : Inthiscase,wehave 1 a 0and t 2 C Z SR S C Z RD S 1+ Z SD S + C Z SR S t 2max .Themaximum ratecanbeexpressedas X t 1 ;t 2 =max f X 1 t 1 ;t 2 ;X 2 t 1 ;t 2 g where X 1 t 1 ;t 2 =max max f 0 ; 0 a g 1 a t 1 C Z SD S + t 2 C Z SD S + Z RD S = t 1 C Z SD 1 a S + t 2 C Z SD S + Z RD S t 1 log + Z SD 1 a S + Z SR S 1+ Z SR 1 a S + t 2 C Z SD S C Z SD S {11 X 2 t 1 ;t 2 =max 1 a 1 t 1 C Z SD S + t 1 C Z SR S 1+ Z SR S + t 2 C Z SD S = C Z SD S {12 wheretherstinequalityin2{11holdssince2{10isnotsatised,andthesecond inequalityin2{11holdssince 1 a 1andthattherstterminthepreviousstep ismonotonicallyincreasingin 1 a when Z SD Z SR .Thisway,thelastobservation 29 PAGE 30 helpsavoidthenon-convexityissuementionedbefore.Thevalueof X 2 t 1 ;t 2 in2{12isobtainedusingsimilararguments. ii. t 2 C Z RD S 1+ Z SD S >t 1 C Z SR S : Inthiscase,wehave 1 a < 0and t 2 >t 2max .Thusthemaximumrateisgivenby X t 1 ;t 2 =max 0 1 t 1 C Z SD S + t 1 C Z SR S 1+ Z SR S + t 2 C Z SD S = C Z SD S : {13 Hence,2{11{2{13implythatthesolutionto2{8,forany t 1 ;t 2 pair,occursat =1,andthesolutiontotheoriginalproblemof2{7,when Z SD Z SR ,isgivenby max f 0 t 1 ;t 2 : t 1 + t 2 =1 g X t 1 ;t 2 = C Z SD S withany t 1 ;t 2 pairsuchthat t 1 ;t 2 0and t 1 + t 2 =1. Z SD PAGE 31 Inthiscasewehave 1 b 1and t 2 C Z SR S C Z RD S 1+ Z SD S + C Z SR S t 2max .Themaximum rate X t 1 ;t 2 canbeexpressedas X t 1 ;t 2 =max f X 1 t 1 ;t 2 ;X 2 t 1 ;t 2 g where X 1 t 1 ;t 2 =max 0 1 b t 1 C Z SD S 1+ Z SD S + t 1 C Z SR S + t 2 C Z SD S = t 1 C Z SD 1 b S 1+ Z SD 1 b S + t 1 C Z SR 1 b S + t 2 C Z SD S {15 X 2 t 1 ;t 2 =max 1 b min f 1 ; 0 b g t 1 C Z SD S 1+ Z SD S + t 2 C Z SD S + Z RD S = t 1 C Z SD 1 b S 1+ Z SD 1 b S + t 2 C Z SD S + Z RD S ; {16 andboththemaximain2{15and2{16areattainedat = 1 b .Substitutingthe expressionfor 1 b in2{15and2{16,weobtain X 1 t 1 ;t 2 = X 2 t 1 ;t 2 and X t 1 ;t 2 = t 1 log 0 B B @ 1+ Z SD S 1+ Z SD Z SR 1+ Z RD S 1+ Z SD S t 2 =t 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 1 C C A + t 2 C Z SD S + Z RD S : {17 ii. t 2 C Z RD S 1+ Z SD S >t 1 C Z SR S : Inthiscasewehave 1 b > 1and t 2 >t 2max with t 2max asinsub-casei.Thusthe maximumrateisgivenby X t 1 ;t 2 =max 0 1 t 1 C Z SD S 1+ Z SD S + t 1 C Z SR S + t 2 C Z SD S = t 1 C Z SR S + t 2 C Z SD S {18 wherethemaximumoccursat =1,asthefunctiontobemaximizedismonotonicallyincreasingin when Z SR >Z SD .Again,theapparentnon-convexityofthe optimizationproblem2{14in isavoidedbyconsideringthesumof x 1 and x 2 together,andutilizingthelastobservationregardingthemonotonicitypropertyof theobjectivefunctionin2{18. Finally,weoptimizetheabovesolutionto2{14overallpossibletimeslotlengthsto obtainthesolutiontotheoriginalproblemin2{7when Z SD PAGE 32 Ontheotherhand,correspondingtoCaseii.,when t 2 >t 2max ,from2{18,wehave max f 0 t 1 ;t 2 : t 1 + t 2 =1 g X t 1 ;t 2 < )]TJ/F21 11.9552 Tf 11.955 0 Td [(t 2max C Z SR S + t 2max C Z SD S = C Z SR S C Z SD S + Z RD S C Z SR S + C Z RD S + Z SD S )]TJ/F21 11.9552 Tf 11.955 0 Td [(C Z SD S {20 wheretheinequalityin2{20isobtainedfrom2{18byusing Z SR >Z SD and t 2 >t 2max Hence,from2{19and2{20,weconcludethatwhen Z SR >Z SD ,themaximum achievablerateisgivenby X S =max 0 t 2 t 2max t 1 log 1+ Z SD S 1+ Z SD Z SR 1+ Z RD S 1+ Z SD S t 2 =t 1 )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 + t 2 C Z SD S + Z RD S Therefore,themaximumachievablerateofinformationtransmissionfromthesource S tothedestination D fordierentcasescanbesummarizedasunder: Themaximuminformationratefromthesource S tothedestination D fordierent casesissummarizedbelow: a Z SD Z SR :Themaximumrateis X S = C Z SD S withany t 1 ;t 2 pairsuchthat t 1 ;t 2 0and t 1 + t 2 =1.Thiscorrespondstodirectlytransmittingalldatathroughthe linkfromthesourcetothedestination,withoututilizingtherelay. b Z SD PAGE 33 routingschemesbasedondierentnetworkpathselectionmethodsfailtoprovideacceptableoutageperformanceinhigh-ratesituations.TheGLSprotocoldescribedbelow providesasimplesub-optimaldesigntoaddressthiscomplexityissue.Inessence,theGLS protocolidentiesthe best relaypathoutofthepossible N )]TJ/F15 11.9552 Tf 12.226 0 Td [(2relaypathsandconsiders onlythechosenrelayalongwiththesourceanddestinationtoformathree-noderelaynetwork,whichwecalla generalized-link fromthesourcetothedestination,forinformation transmission.Inotherwords,theaimistochoosethebestrelaysuchthattheequivalent three-noderelaynetworkobtainedcontainingthesource,destinationandthechosen relaygivesthemaximumrateoverallpossibleequivalentthree-nodenetworkscontaining thesourceanddestination.Moreprecisely,weneedtoconsiderthefollowingpossibilities: Z SD Z SR i forall i 2 I = f 1 ; 2 ; ;N )]TJ/F15 11.9552 Tf 12.259 0 Td [(2 g :Fromtheresultsoftheoptimization problem2{7,itisclearthatthemaximumratewouldbe C Z SD S withdirect transmissionofalldatafromthesourcetothedestinationwithoutusinganyrelay. Thereexistsa k 2 I suchthat Z SR k >Z SD :Letthesetofallsuchnodeindices be K andforall i 2 I n K Z SD Z SR i .Forthiscase,choosethenode R 0 k asthe relaysuchthat k 0 =argmax k 2 K X k S ,where X k S isthemaximumrateforthe three-noderelaynetworkwiththesource S ,therelay R k anddestination D Intermsoftheworst-casecomputationalcomplexitiesfortheFOandGLSprotocols, itcanbeseenthat,foran N -noderelaynetworkwith N> 3,theFOprotocolinvolves amax-minoptimizationover2 N 2 )]TJ/F15 11.9552 Tf 12.47 0 Td [(2 N +2variablesallpossibleowsandtimeslot lengths,subjectto N )]TJ/F15 11.9552 Tf 10.153 0 Td [(1non-linearand2 N 2 )]TJ/F21 11.9552 Tf 10.154 0 Td [(N +1linearconstraints,whereastheGLS protocolinvolvesamaximumof N )]TJ/F15 11.9552 Tf 11.51 0 Td [(2maximizationsofanon-linearconcavefunctionover twovariables,subjecttotwolinearconstraints,followedbyndingthemaximumof N )]TJ/F15 11.9552 Tf 12.006 0 Td [(2 realnumberswithaworst-casecomplexityof O N )]TJ/F15 11.9552 Tf 12.148 0 Td [(2.Moreover,for N> 3,fortheFO protocol,theBCslotspotentiallyinvolve N )]TJ/F15 11.9552 Tf 12.102 0 Td [(1-and N )]TJ/F15 11.9552 Tf 12.102 0 Td [(2-levelsuperpositioncoding SPCordirtypapercodingDPCimplementationsfor S andtherelaysrespectively, whiletheMAslotsattherelaysand D mayinvolveamaximumof N )]TJ/F15 11.9552 Tf 12.304 0 Td [(3and N )]TJ/F15 11.9552 Tf 12.304 0 Td [(2 interferencecancelationICoperationsrespectively.Ontheotherhand,theGLSprotocol 33 PAGE 34 involvesamaximumof2-levelSPC/DPCandoneICoperationfortheBCandMAslots respectively,forany N> 3. 2.3.3Diversity-multiplexingtradeo Inthissection,weshowthatboththeFOandGLSprotocolsachievetheoptimal DMT.Asin[31],themultiplexinggainisdenedas r =lim S !1 R S log S where S istheSNR and R S istherateatanSNRlevelof S .Following[31],weparameterizethesystem,in termsoftheSNR S andthemultiplexinggain,0 PAGE 35 denitionofdiversityorder2{21,wehave d r lim S !1 )]TJ/F15 11.9552 Tf 11.291 0 Td [(log P out r;S log S : {22 Moreover,theaboveresultfrom[30]canbedirectlyusedtoprovethesamefortheGLS protocol.Usingthisfact,wederivealowerboundtothediversity-multiplexingtradeo thatcanbeachievedbytheGLSprotocol.Thesets I and K ,usedinthesequel,arethe setsofindicesasdescribedinSection2.3.2.TheoutageprobabilityfortheGLSprotocol isgivenby P gr out r;S =Pr max k 2 I X k S PAGE 36 Weobservethatusingtheright-mostexpressionof2{19insteadof X k S ,foreach k 2 K 0 ,in2{24givesanupperboundon P gr out r;S .ThisisutilizedinobtainingalowerboundonthediversityorderoftheGLSprotocol.Let f S n g 1 n =1 bean increasingunboundedsequenceofSNRswith S 1 > 1.Denethesequenceofrandomvariables f M k n g 1 n =1 f B k n g 1 n =1 and f A k n g 1 n =1 with M k n = C Z SR k S n )]TJ/F22 7.9701 Tf 6.587 0 Td [(C Z SD S n C Z SD S n + Z R k D S n B k n = C Z SR k S n log S n ,and A k n = X k S n log S n = B k n 1+ M k n ,respectively.Notethatforall k 2 K 0 M k n 0a.s.Thisimpliesthat B k n )]TJ/F21 11.9552 Tf 13.013 0 Td [(A k n 0a.s.Dene A 0 n =max k 2 K 0 A k n and B 0 n =max k 2 K 0 B k n = B 1 n .Thenusingtheabove,itcanbeseenthat B 0 n )]TJ/F21 11.9552 Tf 12.641 0 Td [(A 0 n 0 a.s.Further,lim n !1 Pr B 0 n PAGE 37 Nextgivenan N -noderelaynetwork,considerthemultipleaccesscutthatseparates thedestinationfromalltheothernodes.Clearly,thetotalowacrossthiscutgivesan upperboundonthemaximumrateachievableinthe N -noderelaynetwork.Consequently, alowerboundontheoutageprobability P l out r;S canbeobtainedusingthemaximum sum-rateacrossthiscut: P l out r;S Pr C Z SD + Z R 1 D + + Z R N )]TJ/F18 5.9776 Tf 5.756 0 Td [(2 D S PAGE 38 Forthefour-noderelaynetwork,therecanbe6possibletimeslotsintheFOprotocol asshowninFig.2-2:threeBCslotsforthesourceandthetworelaystotransmitinformation,andthreeMAslotsforthetworelaysandthedestinationtoreceiveinformation respectively.Toderiveanupperboundontheachievablerateandtherebyalowerbound ontheoutageprobability,weusemax-ow-min-cuttypeboundsforhalf-duplexcommunication.TherearefourpossibletimeslotsasshowninFig.2-3,withtherstBCslotand thelastMAslotatthedestinationsameasintheFOprotocol,butnow,thesourceand arelaymaytransmitsimultaneouslytotheotherrelayandthedestinationduringeachof theintermediateslotsoverinterferencechannels.Weusethemax-ow-min-cuttheoremto upperboundthemaximuminformationowinthesetwotimeslots. Fortheve-noderelaynetwork,therecanbe8possibletimeslotsintheFOprotocol -fourBCslotsforthesourceandthethreerelaystotransmitinformation,andfourMA slotsforthethreerelaysandthedestinationtoreceiveinformationrespectively.Similar tothefour-noderelaynetwork,forthemax-ow-min-cutbound,thereare8possible timeslotswiththerstBCslotandthelastMAslotatthedestinationbeingthesame asfortheFOprotocol,andmulti-source-multi-destinationtransmissionsduringthesix intermediateslots.Ingeneral,thefollowingmayoccurduringthesixintermediatetime slots:thesourceandarelaymaytransmitsimultaneouslytotheotherrelaysandthe destinationduringthesecond,thirdandfourthslots,andthesourceandtworelaysmay transmitsimultaneouslytotheremainingrelayandthedestinationduringthefth,sixth andseventhslots.Asinthecaseoffour-noderelaynetwork,weusethemax-ow-min-cut theoremtoupperboundthemaximumowofinformationduringtheintermediatetime slots. Withtheabovedivisionoftimeslots,theformalizationoftheproblemisdoneasin theprevioussections,andweusetheoptimizationroutineof[36]toobtainthemaximum achievableratesandupperboundsfordierentvaluesofrequiredrates.InFigs.2-4 and2-5,weplottheoutageprobabilitiesofthevariousschemeswiththerequiredrate 38 PAGE 39 R at1bit/s/Hzand6bits/s/Hzrespectively,forthefour-noderelaynetwork.Figs.2-6 and2-7presentthesamefortheve-noderelaynetwork.WhencomparedtotheFO protocol,theGLSprotocolsuersalossofaround1 : 0dB,andaround1 : 5dBwhen R iseither1bit/s/Hzor6bits/s/Hz,atanoutageprobabilityof10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(4 ,forthefour-and ve-noderelaynetworksrespectively.Ontheotherhand,theperformancedegradationfor themax-minselectionmethodof[27],ascomparedtotheFOprotocoloreventheGLS protocol,ismorethan12dBatanoutageprobabilityof7 : 0 10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(2 ,when R =6bits/s/Hz forthefour-noderelaynetwork,andanexactlysimilarsituationcanbeobservedfor theve-noderelaynetwork.Moreover,forthefour-noderelaynetwork,theFOprotocol iswithin2 : 14dBwhen R =1bit/s/Hztowithin7 : 05dBwhen R =6bits/s/Hzof thelowerboundontheoutageprobabilitywhentheoutageprobabilityis10 )]TJ/F20 7.9701 Tf 6.586 0 Td [(4 .Forthe ve-noderelaynetwork,thecorrespondingdierencesareapproximately3dBand9 : 6dB respectively.Thus,weseethatasthenumberofnodesintherelaynetworkincreases,the GLSprotocolbecomesmoresuboptimal,whereasthegapbetweentheoutageperformance oftheFOprotocolandthelowerboundwidens.Withregardtothelatterobservation, itshouldbekeptinmindthatthelowerboundobtainedusingthemax-ow-min-cut theoremis,ingeneral,notatightbound. Theperformancesofthedierentprotocolsforthefour-noderelaynetworkwith non-uniformaveragepowergainsarepresentedinFigs.2-8and2-9,andFigs.2-10 and2-11forcasesAandBrespectively,withtheaveragepowergainsasstatedinthe gures.CaseArepresentsthesituationwhenboththesource-relaylinksare,onaverage, betterthanthedirectlink,andonerelay-destinationlinkthelinkbetween R 1 and D is,onaverage,betterthantheother,resultinginrelay R 1 beingabettercandidateto forwardtheinformationthantheotherrelay.Ontheotherhand,caseBrepresentsthe situationwhennoonerelayhasverygoodsource-relayandrelay-destinationlinks.In thiscase,onesource-relaylinkis,onaverage,betterthanthedirectlink,which,inturn, isbetterthantheothersource-relaylink.Thereverseistruefortherelay-destination 39 PAGE 40 links,andtheinter-relaychannelis,onaverage,verygood.Thissituationpromotes inter-relayinteractionsfortheFOprotocol,andtherebyincreasesthedierencebetween theperformancesoftheFOandGLSprotocols.Thedierencesbetweentheoutage performancesoftheFOandGLSprotocols,atanoutageprobabilityof10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(4 ,are1 : 2dBor 1 : 0dB,and2 : 0dBor1 : 3dBwhen R =1bits/s/Hzor R =6bits/s/Hz,forcasesAand Brespectively.ThisreductioninthesuboptimalityoftheGLSprotocolwithincreasein therequireddataratecanbeexplainedbynotingthat,whentherequiredrateishigh, thecodinggainoeredbyaprotocolheavilyreliesontheecientuseofthedirectlink, andsincetheusageofthedirectlinkissimilarforboththeFOandtheGLSprotocols, theperformancegapnarrowsastherequireddatarateincreases.Ontheotherhand,at thesameoutageprobability,thedierencebetweentheoutageperformanceoftheFO protocolandthelowerboundincreasesfrom1 : 5dBto7dB,andfrom1 : 9dBto6 : 0dBas therequiredrateincreasesfrom1bit/s/Hzto6bits/s/Hz,forcasesAandBrespectively. Overall,theseresultsdemonstratetrendssimilartotheuniformaveragepowergaincase, andconrmthegeneralityoftheproposedprotocols. 2.5Summary Inthischapter,weproposedacooperativetransmissiondesignforageneralmultinodehalf-duplexwirelessrelaynetworkwherechannelinformationisavailableatthe nodes.Theproposeddesignisbasedonoptimizinginformationows,usingthebasic componentsofBCandMA,tomaximizethetransmissionratefromthesourcetothe destination,subjecttomaximumpowerconstraintsatindividualnodes.Motivatedby theneedforsimplernetworkpathselectionschemesthatperformwelleveninhighratescenarios,wedevelopedthegeneralized-linkselectionprotocolthatcombinesrelay selection,andowoptimizationforathree-noderelaynetwork.Theproposedprotocols wereshowntoachievetheoptimaldiversity-multiplexingtradeoforageneralrelay network.Simulationresultsforthefour-andve-noderelaynetworksforuniformand non-uniformaveragepowergainsdemonstratethattheperformanceofthemuchsimpler 40 PAGE 41 GLSprotocolisslightlyworsethanthatoftheFOprotocol.ThissuggeststhattheGLS protocolcanbeusedinsystemswithlow-complexityrequirements.Wealsonotethatthe proposedFOandGLSprotocolscanbeusedinwirelessnetworkswithtopologiesmore complicatedthatthewirelessrelaynetworkconsideredhere.Forexample,applicationof similarideastoaparallelrelaynetworkinwhichthereisnodirectconnectionbetweenthe sourceandthedestinationisconsideredin[37]. 41 PAGE 42 Figure2-1.Basicgraphs G 1 and G 2 forthethree-noderelaynetworkwith t 1 + t 2 =1. Figure2-2.FOprotocolforthefour-noderelaynetworkwith t 1 + + t 6 =1.Theow optimizationisperformedoverallows x 1 ; ;x 14 ,andalltimeslotlengths t 1 ; ;t 6 42 PAGE 43 Figure2-3.Transmissionstrategytoobtainalowerboundontheoutageprobabilityfor thefour-noderelaynetwork.Here t 1 + + t 4 =1,andtheoptimizationis over x 1 ; ;x 14 ,and t 1 ; ;t 4 ,withtheapplicationofthe max-ow-min-cuttheoremfortheintermediateslots. 43 PAGE 44 Figure2-4.Four-noderelaynetworkwithuniformaveragepowergains:Outage probabilitiesforrequiredrate R =1bit/s/Hz. Figure2-5.Four-noderelaynetworkwithuniformaveragepowergains:Outage probabilitiesforrequiredrate R =6bits/s/Hz. 44 PAGE 45 Figure2-6.Five-noderelaynetworkwithuniformaveragepowergains:Outage probabilitiesforrequiredrate R =1bits/s/Hz. Figure2-7.Five-noderelaynetworkwithuniformaveragepowergains:Outage probabilitiesforrequiredrate R =6bits/s/Hz. 45 PAGE 46 Figure2-8.Four-noderelaynetworkwithnon-uniformaveragepowergains.CaseA: E [ Z SR 1 ]=2 : 0 ; E [ Z SR 2 ]=2 : 0 ; E [ Z SD ]=1 : 0 ; E [ Z R 1 R 2 ]=1 : 0 ; E [ Z R 1 D ]= 1 : 5 ; E [ Z R 2 D ]=1 : 0.Outageprobabilitiesforrequiredrate R =1bit/s/Hz. Figure2-9.Four-noderelaynetworkwithnon-uniformaveragepowergains.CaseA: E [ Z SR 1 ]=2 : 0 ; E [ Z SR 2 ]=2 : 0 ; E [ Z SD ]=1 : 0 ; E [ Z R 1 R 2 ]=1 : 0 ; E [ Z R 1 D ]= 1 : 5 ; E [ Z R 2 D ]=1 : 0.Outageprobabilitiesforrequiredrate R =6bit/s/Hz. 46 PAGE 47 Figure2-10.Four-noderelaynetworkwithnon-uniformaveragepowergains.CaseB: E [ Z SR 1 ]=1 : 5 ; E [ Z SR 2 ]=0 : 75 ; E [ Z SD ]=1 : 0 ; E [ Z R 1 R 2 ]=3 : 5 ; E [ Z R 1 D ]= 0 : 2 ; E [ Z R 2 D ]=3 : 0.Outageprobabilitiesforrequiredrate R =1bit/s/Hz. Figure2-11.Four-noderelaynetworkwithnon-uniformaveragepowergains.CaseB: E [ Z SR 1 ]=1 : 5 ; E [ Z SR 2 ]=0 : 75 ; E [ Z SD ]=1 : 0 ; E [ Z R 1 R 2 ]=3 : 5 ; E [ Z R 1 D ]= 0 : 2 ; E [ Z R 2 D ]=3 : 0.Outageprobabilitiesforrequiredrate R =6bit/s/Hz. 47 PAGE 48 CHAPTER3 RESOURCEALLOCATIONANDCOOPERATIVEBEHAVIORINFADING MULTIPLE-ACCESSCHANNELSUNDERUNCERTAINTY 3.1Introduction Theresourceallocationproblemformulti-userwirelesssystemshasgenerated considerableinterestintheresearchcommunityandhasbeenconsideredfromdierent perspectiveswithregardtoeciencyandfairnessissues.ThefadingMACisoneofthe basicamongstsuchsystemsanddierentsolutionshavebeenproposedtilldatetothe above-mentionedproblem.ThethroughputcapacityregionofthefadingMAC,which canbeachievedbyusingdynamicpowerandrateallocationschemestomaximizethe averagerate,hasbeencharacterizedin[38].Usingideassimilartothoseforlong-term powercontrolin[39],theoutagecapacityregionforthischannelhasbeenderivedin[40], wheretheusershaveaveragepowerconstraints.Theauthors,in[40],obtainboththe commonoutagecapacityregion,whenalltheusershaveacommonoutageprobability constraint,andalsotheindividualoutagecapacityregion,whenusersmayhavedierent outagerequirements.Forthelattercase,thecapacityregionischaracterizedbyachannel usagerewardvector,whichdeterminestheactualoperatingpointforthesystem. Agame-theoreticapproachtowardssolvingtheresourceallocationproblemwith theaveragerateutilityanduserssubjecttoaveragepowerconstraints,isconsidered in[41].TheyproposeaStackelbergformulation,wherethereceiveristhegameleader andthetransmittersplayawater-llinggame,wheretheorderofdecodingoftheusers' information,whichimpliesaprioritizingoftheusers,maybedecidedbythereceiverusing anauctioningprocessasin[42].Alow-complexitydynamicrateallocationpolicy,that maximizesageneralconcaveutilityfunctionoftheratesinthethroughputcapacityregion forxedtransmissionpowers,ispresentedin[43].In[44],theoptimalpowercontrol schemeformaximizingthesum-capacityinthemultipleinputmultipleoutputMIMO fadingMACischaracterized. 48 PAGE 49 Intheaboveworks,theavailabilityofperfectchannelstateinformationCSIat thetransmittersandthereceiverisassumed.Theoptimalmediumaccessandresource allocationscheme,thatmaximizesthetotalthroughputforthefadingMACwithpartial CSI,isproposedin[45].ThepartialCSIiseitherintheformofathreshold-based1-bit quantizedfeedback,orthebest-userfeedback.Theoutagerateregionswithconstraints onindividualuseroutageprobabilitiesandnoCSIatthetransmitters,arederivedin[46], andtheyareusedtoobtainthebesttargetratevectortomaximizethesumthroughput atthereceiver. Aninformation-theoreticanalysisoftheachievablerateforthesingle-userfading channelandthefeasiblerateregionforthefadingMAC,withimperfectCSIatboth transmittersandthereceiver,ispresentedin[47].Thisisdonebyconsideringthe dierenceinthemutualinformationbetweentheinputsandtheoutputwithperfect andimperfectCSI,byassumingoptimaldecodingatthereceiverbasedontheknowledge regardingthedistributionofthefadingprocess.In[48],theauthorshavequantiedthe notionofimperfect"CSIbyshowingthatifthesideinformationissuchthatthesecond momentoftheerrorisnegligiblecomparedtothereciprocalofthesignal-to-noiseratio SNR,thenitcanbeconsideredtobeperfect",whereasotherwise,theachievablerates maybereducedasaresultoftheerrorsinthefadingstateestimation,andtheuseof Gaussiancodebooksandnearestneighbordecodingmaynotyieldgoodperformance. ThisisestablishedbyconsideringthegeneralizedmutualinformationGMI,whichgives thehighestrateforwhichtheaverageerrorprobability,averagedovertheensembleof Gaussiancodebooks,convergestozero.Itisprovedin[48]thatintheabsenceofCSI, theperformanceispoorforbothlow-andhigh-SNRoperations.Ontheotherhand, withpartialCSIavailable,forlow-SNRregimes,theuseofGaussiancodebooksand maximumlikelihoodMLdecodingperformsclosewithinaconstantfactortothe channelcapacity,butnotsointhehigh-SNRregime.TheeectofimperfectCSIonthe nite-SNRdiversity-multiplexingtradeoforthequasi-staticfadingMACisanalyzed 49 PAGE 50 in[49].ThecommonoutageprobabilityoftheMACisconsidered,andtheboundsonthe fadingMACfeasiblerateregionfrom[47]areusedtodemonstratetheeectofimperfect CSIonthenite-SNRdiversity-multiplexingtradeo. Inthiswork,wemodeltheresourceallocationproblemforthetwo-userfadingMAC usingatwo-personbargainingproblem[50],whereintheextentofcooperativebehavior isdeterminedbytheoutcomeofthebargainingproblem.Inthiswork,weconsiderthe situationwhentheutilityderivedbyeachuseristhe averagerate overallfadingstates. WhentheCSITisperfect,thesolutiontothebargainingproblem,speciedbytheNBS, yieldstheoptimaltransmissionstrategypairforthetwouserswithregardtofairness andeciencyconsiderations.Here,a transmissionstrategy foreachusercorrespondstoa choiceoftransmissionrateandpowerforaparticularfadingstate. WeconsiderthesituationwhereinthereceiverhasaccesstoperfectCSI,butthere existsacertainuncertaintyregardingtheCSITthatmaystemfromquantizationas, inreality,thefeedbackchannelsarelikelytohavelimitedcapacitiesand/orprediction errors.IftheavailableCSITisinaccurate,thetransmissionstrategypairsuggestedby theNBSmaydeviatefromthetrueoptimum,andthus,leadtoconsiderableperformance degradationintermsofthe trueutilities .Toovercomethislackofrobustnesswepropose aschemeinwhichtheconventionaltwo-personbargainingproblemisrelaxedtoacknowledgethefactthattheNBSmaynotgivetheoptimalstrategypair.Accordingtothis modiedbargainingproblemformulation,eachuserindependentlydecidesitstransmission strategyviaamaximinoptimizationfromitsrespectivesetofpossiblestrategies.Fora particularuser,suchasetisarangeoftransmissionstrategiesabouta nominalstrategy ThenominalstrategyisobtainingusingtheNBStotheoriginalbargainingproblem andtheavailableCSIT.Thisisincontrasttoconventionalbargainingproblemformulations,wherein,oncetheplayersreachanagreement,theyareboundtoexecutetheexact strategypairsuggestedbytheNBS. 50 PAGE 51 Inthefollowingsection,wepresentthesystemmodel.Thisisfollowedbythe descriptionofthebargainingproblemformulationinSection3.3.Themodiedbargaining problemisproposedinSection3.4alongwithnumericalresults.Finally,weconcludethe chapterinSection3.5. 3.2SystemModel Inthissection,wegiveabriefdescriptionofthesystemmodelforthefadingMAC andintroducetheformulationoftheresourceallocationproblemtocharacterizethe cooperativebehaviorbetweentwousersUsers1and2whowishtotransmitinformation toasinglereceiver.Notethattheformofcooperativebehaviorconsideredinthepresent workisnotthesameasusuallyinterpretedintheliterature,whereinoneusermay actuallyforwardtheinformationofanothertothedestination,andsuchasituationis studiedinChapter4.Asweshallseeinthesequel,fortheconventionaltwo-userfading MACconsideredhere,thelevelofcooperativebehaviorismanifestedbyhowmuchauser backso"fromitsmaximumpossibletransmissionrateforacertaintransmissionpower choice. Consideradiscrete-timetwo-userfadingMACwithunitbandwidth,inthepresence ofunit-varianceGaussiannoise,withthefadingstatedescribedbythepowergainvector Z = Z 1 Z 2 .Thepowergainsareassumedtobeindependentandidenticallydistributed i.i.d.exponentialrandomvariableswithunitmean. Let i bethemaximumtransmissionpoweravailabletothe i thuser.Weassume thatperfectCSIisavailableatthereceiver,whereastheCSIatthetransmittersmaynot beaccurate.Thetransmissionstrategiesaredeterminedintermsofthejointconditional probabilitydensityfunctionsPDFs 1 ofthetransmissionrateandpower,wherethe conditioningisonthefadingstate. 1 Inthiswork,weusethetermPDFtorefertobothcontinuousanddiscreteprobability functions. 51 PAGE 52 Wemodeltheresourceallocationproblemi.e.theoptimalchoiceofthetransmit powersandratesasatwo-userbargainingproblem.Thisspeciestheoperatingpoint ofthesystem.Notethatabargainingproblemformulationisanappropriatechoiceto modelthisproblemasitdoesnotpresumeanyinherentcooperationbetweenthetwo users.Instead,theusersnegotiatetoreachanagreementafterevaluatingselshlyand rationallythepotentialbenetsfromcooperationovertheeventofthemnotarrivingat anymutualagreement.Moreover,itiswellknownthattheNBScanbeinterpretedasa generalizedformofaproportionalfairnesssolution,andcoincideswiththelatterwhenthe payostothetwoplayersintheeventofdisagreementequalzero.Thus,theNBSprovides afairand ecient i.e.itisnotpossibletoimproveoneuser'sperformancewithout degradingtheperformanceoftheothersolutiontotheresourceallocationproblem. Unfortunately,owingtothedependenceoftheoperatingpointonthefadingstate,ifthe CSITisnotaccurate,theoperationpointobtainedwiththeerroneousCSITmaynot beoptimal.Inordertoobtainamorerobustsolutiontotheresourceallocationproblem underuncertainty,weproposearelaxedbargainingproblemformulationinthiswork. 3.3TheBargainingProblemfortheTwo-UserFadingMAC Inthissection,wesolvethetwo-userbargainingproblemtoobtaintheoptimal strategyforthetwousersusingtheavailableCSIT.Thus,forthetwoutilityfunction choices,weneedtosolvethetwo-personbargainingproblem,denedas T ; T d .Here, T isthesetoffeasibleutilities,i.e.theachievableaverageratesforthetwousers,and T d = T 1 ; d ;T 2 ; d representsthe disagreementpoint ,i.e.theutilityeachuserwillderive iftheydonotcooperate.Thus,thetwousersnegotiatetoreachanagreementregarding theoptimaltransmissionratesandpowers,giventhefadingstate.Moreover,itisassumed thattheuserscanagreetojointlyrandomizedstrategiesregardingthetheirtransmission ratesandpowers.Thedisagreementpointsforthisbargainingproblem,followedbythe NBS,arederivednext. 52 PAGE 53 3.3.1TheDisagreementPoint Here,weconsiderthecasewhenthereisnoagreementbetweenthetwouserswith regardtotheirtransmissionstrategies.Let R i and P i bethetransmissionrateand power,respectively,ofUser i .Also,givenaparticularfadingstateandthetransmissionstrategiesforthetwousers,theachievabletransmissionrateforUser i isgiven by R r i P Z ; R Z ; Z .Thatis,if,forthefadingstate Z ,theactionsforthetwo usersarespeciedbythetransmissionpowersandrates P Z = P 1 Z ;P 2 Z and R Z = R 1 Z ;R 2 Z respectively,thenthepayoreceivedbyUser i isgivenby R r i P ; R ; Z ,wherethedependenceof P and R on Z isnotmadeexplicitforbrevity. Particularly,theseutilitiescorrespondtotheratesforthetwousersatwhichreliable transmissioncanbesupportedwithanarbitrarilysmallprobabilityoferror.Dueto thesymmetricnatureoftheproblem,itissucienttoconsideranyoneuser'soperation,sayUser1.Intheabsenceofanyagreementbetweenthetwousers,theachievable transmissionrateforUser1, R r 1 ,isgivenby3{1,where C x =log+ x R r 1 P ; R ; Z = 8 > > > > > > < > > > > > > : R 1 if R 1 min f C Z 1 P 1 ; max f C Z 1 P 1 + Z 2 P 2 )]TJ/F21 11.9552 Tf -90.726 -28.69 Td [(R 2 ;C Z 1 P 1 1+ Z 2 P 2 oo ; 0otherwise : {1 Notethatforthescenariowhereintheusersfailtoreachanagreement,thereisno restrictiononthechoiceofthetransmissionstrategies,asagainstthescenariowhereinthe usersreachamutualagreementwherebyeachuser'schoiceofthetransmissionstrategy isrestrictedinsomespecicway.Sinceeachuserisunawareofthestrategyoftheother, wederivetheoptimalstrategyofeachuserasthesolutiontoamaximinproblem,wherein eachuser'saimistomaximizeitsownworst-caseusageprobability.Notethattheusers beingableto independently decidefromthesetofallrandomizedstrategiesimpliesthat, ingeneral,theusersmayusemixedstrategiesforthemaximingamesinthismodel.Thus, 53 PAGE 54 forUser1,theoptimaltransmissionstrategyis f R 1 P 1 j Z =argmax f R 1 P 1 j Z min f R 2 P 2 j Z E [ R r 1 ] ; {2 where E [ ]denotestheexpectationoperatorandtheexpectationisoverallfadingstates. Also, f R i P i j Z isthejointconditionalPDFofUser i 'stransmissionrateandpowersuchthat Pr P i > i =0for i =1 ; 2,withthelatterprobabilitiescomputedwithrespecttothe PDFs f P 1 Z and f P 2 Z ,respectively. Since R r 1 in3{1isamonotonicallynon-increasingfunctionof R 2 ,asolutionofthe minimizationproblemin3{2isgivenby f y R 2 P 2 j Z r 2 ;p 2 = r 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(C Z 2 2 p 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 2 : {3 Using3{3in3{2gives f R 1 P 1 j Z =argmax f R 1 P 1 j Z E [ ^ R r 1 ],suchthatPr P 1 > 1 =0.Inthe above, ^ R r 1 is ^ R r 1 = 8 > > < > > : R 1 if R 1 C Z 1 P 1 1+ Z 2 2 ; 0otherwise : Clearly,itcanbeseenthatthesolutiontotheaboveproblemisgivenby f R 1 P 1 j Z r 1 ;p 1 = r 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(C Z 1 1 1+ Z 2 2 p 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [( 1 : Therefore,whenthereisnoagreementbetweenthetwousers,themaximinoptimal achievableaverageratesare T 1 ; d = E h C Z 1 1 1+ Z 2 2 i and T 2 ; d = E h C Z 2 2 1+ Z 1 1 i for Users1and2respectively.Sincetheoperationpointforeveryfadingstate Z isinthe interioroftheMACcapacityregion,asmallperturbationin Z maynotcauseasignicant degradationintheactualachievableutilities. 3.3.2TheNashBargainingSolutionNBS WeusetheNBStoobtaintheoptimalutilityallocationforthisbargainingproblem usingtheavailableCSIT.Theoptimalallocationoftheusageprobabilities,usingtheNBS, 54 PAGE 55 isgivenby[50] Y =argmax Y 2Y Y 1 )]TJ/F21 11.9552 Tf 11.956 0 Td [(Y 1 ; d Y 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(Y 2 ; d ; {4 where Y denotesthecorrespondingutilityofinterest,and Y isthefeasiblesetofutility allocations.TheNBSyieldstheoptimalallocationofutilities Y betweenthetwo playersinthebargaininggame.Thus,forthepresentproblem,theNBSmaybeusedto obtaintheoptimalstrategypair f P R j Z i.e.theoptimaljointconditionalPDFforthe powers P Z = )]TJ/F15 11.9552 Tf 8.114 -6.662 Td [( P 1 Z ; P 2 Z andrates R Z = )]TJ/F15 11.9552 Tf 8.033 -6.662 Td [( R 1 Z ; R 2 Z ,wheretheuniqueness of f P R j Z isuptothecorrespondingachievableaverageratepair T ,thatis,weidentify anytwoagreementswhichyieldthesameutilityallocation.Asaresult,thepossible non-uniquenessofthetransmissionstrategiesdoesnotaecttheoptimalityoftheresource allocationsolution. BeforeproceedingwiththederivationoftheNBSweverifythatthenecessary conditions[50]ofatwo-personbargainingproblemaresatisedfortheaveragerateutility function.Thatthesetoffeasibleutilities, T ,iscompactcanbeprovedbyusingthefact thatthecapacityregionoftheMAC,foraparticularfadingstate,isclosedandbounded in R 2 .Also,thedisagreementpoint,denedby T d ,isin T .Thedenitionoftheutility functionasanaverageofthepayosreceivedineachfadingstate,andthefactthatthe MACcapacityregionismonotonicallyincreasinginthetransmissionpowersimplythat theusershavecontinuouspreferencerelationsonthesetofallconditionalPDFsforthe nominaltransmissionratesandpowers.Moreover,thesepreferencerelationssatisfythe vonNeumann-MorgensternVNMaxiomsofindependence,continuityandofbeing completeandtransitive.Further,thattheset f T 2T : T T d g isnonemptycanbe concludedfromtheobservationfromtheprevioussubsectionthatthedisagreementpoint correspondtothesystemoperatingstrictlyinsidethecapacityregionofthetwo-userMAC foranyfadingstate.Theconvexityof T canbeseenasfollows:if T 0 = T 0 1 ;T 0 2 canbe achievedbythejointconditionalPDF f 0 P R j Z ,and T 00 by f 00 P R j Z ,then,forany 2 [0 ; 1],the 55 PAGE 56 setofaverageratesdenedby T 0 + T 00 canbeachievedbytimesharingbetweenthe twoagreements. Hence,wemayobtainanoptimalconditionalPDFforthetransmissionratesand powersusingequation3{4.Thus,wehave T =argmax f T : R Z 2MAC P Z ; Z g )]TJ/F32 11.9552 Tf 5.48 -9.684 Td [(E [ R 1 Z ] )]TJ/F21 11.9552 Tf 11.955 0 Td [(T 1 ; d )]TJ/F32 11.9552 Tf 12.952 -9.684 Td [(E [ R 2 Z ] )]TJ/F21 11.9552 Tf 11.955 0 Td [(T 2 ; d ; {5 where P Z mustsatisfythemaximumpowerconstraintforeachuser.Itcanbeeasily seenthat T = 1 2 E C Z 1 1 + C Z 1 1 1+ Z 2 2 ; 1 2 E C Z 2 2 + C Z 2 2 1+ Z 1 1 ; {6 andthiscanbeachievedwiththefollowingrateandpowerallocation: R 1 = 1 2 C Z 1 1 + C Z 1 1 1+ Z 2 2 w.p.1 ; R 2 = 1 2 C Z 2 2 + C Z 2 2 1+ Z 1 1 w.p.1 ; {7 P i = i ; w.p.1 ; for i =1 ; 2 : Thatis,forthiscase,employingtheNBSforeachfadingstateachievestheoptimal solutiontothebargainingproblemof3{5.Thissolutionissimilarinavortothe onein[51],withthedierencebeinginthenatureofutilityfunctionsconsidered.More specically,theutilityweconsiderhereisanaveragemetric,whilethebargainingmodel in[51]considerstheutilitiesresultingfromasingleinstanceofthegameforaparticular stateofnature thefadingstateinthiswork.OneimportantpropertyoftheNBSsuggestedsolutionabovethatisofsignicancetothedevelopmentofthemodied bargainingproblemformulationinSection3.4isthattheoptimalchoiceoftransmission powersisdeterministicandindependentofthefadingstate. 56 PAGE 57 R r 1 R ; R ; Z = 8 > > > > > > < > > > > > > : R 1 if R 1 min f C Z 1 1 ; max f C Z 1 1 + Z 2 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(R 2 ; C Z 1 1 1+ Z 2 2 oo ; and R 1 2S 1 w.p. )]TJ/F21 11.9552 Tf 11.955 0 Td [( ; 0otherwise : Here,weemphasizethattheabovechoiceoftransmission rates asin3{7isnot theonlypossiblesolutiontotheNashbargainingproblem.Unlessotherconstraintsare imposed,anychoiceofjointlyrandomizedtransmissionratessatisfying3{6maybe selectedastheNBS-suggestedoptimalstrategy,withthechoiceofthetransmissionpowers asin3{7.Specically,theabovechoicehasbeenmadetofacilitateaneasyexposition ofthemodiedbargainingproblemformulationinSection3.4andalsotoyieldasimple solutionforpracticalimplementationwithlessoverhead.Withregardtothelatterpoint, notethatforthesolutionof3{7eachuseronlyrequirestheknowledgeofthechannel powergains Z obtainedfromtheCSITandcanmaximizetheNashproductof3{5 independently,whereasforachoiceofjointlyrandomizedratessatisfying3{6some formofcommunicationbetweenthetwousersneedstobeestablishedtoenablethejoint randomizationofthetransmissionrates. NotethattheNBS,asin3{6,suggeststhatnousertransmitsatitsmaximum possibletransmissionratei.e. C Z i i for i =1or2withprobability1,andthis backingo"ofeachuserfromitsmaximumpossibleratemaybeinterpretedasthe manifestationofitscooperativebehavior,motivatedbyarationalandindividualistic evaluationofthebenetsofcooperationasagainstanypresumedaltruismonitspart. Moreover,foranychoiceofjointlyrandomizedtransmissionratessatisfying3{6,the transmissionratepairwouldalwayscorrespondtoapointontheboundaryoftheMAC capacityregionforeveryfadingstate,therebymakingthesolutionverysensitivetothe CSIT.Inthefollowingsection,weproposeamodiedbargainingproblemtohandlethis robustnessissue. 57 PAGE 58 3.4TheModiedBargainingProblem 2 ThesensitivityoftheoperatingpointofthesystemtotheuncertaintyinCSIT maybereducedbydecreasingitsdependenceontheNBS-suggestedstrategypairas describedinthissection.Accordingtothemodiedbargainingproblem,theNBSsuggestedstrategypairhenceforth,the nominalstrategypair isobtainedusingthe availableCSITasinSubsection3.3.2,butinsteadofthetwousersbeingconstrained toimplementthesestrategies,theygettheexibilityof independently choosingtheir transmissionstrategiesfromacertainsetofstrategiesaboutthisnominalstrategypair. Let R i P ; R ; Z and P i P ; R ; Z for i =1 ; 2betheactualtransmissionrates andpowersrespectively,giventhenominaltransmissionstrategiesandavailableCSIT. Next,weutilizethefactthattheproposedsolutionto3{7suggestsusingthemaximum availablepowersforallfadingstates,andhence,weset P i P ; R ; Z = i w.p.1. Consequently,inwhatfollows,weshallnotrepresentthedependenceofthetransmission rateson P explicitly.Also,wedenethetransmissionstrategiesforthetwousersby theconditionalPDFofonlythetransmissionrates,withtheconditioningonthefading stateandthenominaltransmissionrates R .Thesetsofallowabletransmissionstrategies specifycertainlimiteddeviationsfromtheirnominalvaluestoaccountfortheuncertainty regardingtheCSIT.Denethesetsofallowedtransmissionratesforthetwousersas S i = R i : R i 2 R i )]TJ/F15 11.9552 Tf 11.955 0 Td [( R ; R i + R for i =1 ; 2.Thenthechoiceoftheactual transmissionstrategyofUser i i =1 ; 2issubjecttothefollowingconstraint: R i R ; Z 2S i w.p.1 )]TJ/F21 11.9552 Tf 11.955 0 Td [( ; {8 2 Aswouldbecomeclearfromthesubsequentdiscussion,wedonotmodifythebargainingproblemassuch.Onlytheimplicationsofreachinganagreementaremodied.Thus, thebargainingproblemstillstandsvalidasdescribedinSection3.3. 58 PAGE 59 forsome > 0thatcanbearbitrarilysmall.Here R apre-determinednon-negative constantdenotesthemaximumdeviationw.p. )]TJ/F21 11.9552 Tf 11.955 0 Td [( fromtherespectivenominal values 3 .ThisallowsustorobustlyhandletheinaccuracyintheCSITbyreducingthe sensitivityoftheoperatingpointandhence,thatofthetrueutilities,withrespecttothe CSIT.Notethattheprobabilitiesin3{8arecomputedwithrespecttothejointPDF f R i P RZ forUser i Withthisexibilityinchoosingthetransmissionrates,eachusercannomorebe certainoftheother'sexacttransmissionrate,i.e.forUser1, R 2 R ; Z isunknown,and vice-versa.Hence,anaturaloptionforeachuserwouldbeselectthetransmissionrate usingamaximincriteriontohandletheuncertaintyregardingtheexacttransmission rateoftheother.Moreover,foraparticularfadingstate,theactualpayoreceivedby User1,undertheconstraintsdenedby3{8,canbeinterpretedinthesamespiritas R r 1 P ; R ; Z in3{1,andisgivenby3{8.Accordingtothemaximincriterion,the optimalchoiceoftransmissionrateforUser1isgivenas: f R 1 j R ; Z =argmax f R 1 j R ; Z min f R 2 j R ; Z E R r 1 R ; R ; Z : {10 Here,weemphasizethatthismaximinoptimizationiscarriedoutusingtheavailable CSIT. Usingthevaluesof R Z from3{7,dene R Z C Z 1 1 )]TJ/F15 11.9552 Tf 14.508 3.022 Td [( R 1 = C Z 2 2 )]TJ/F15 11.9552 Tf 14.509 3.022 Td [( R 2 = 1 2 [ C Z 1 1 + C Z 2 2 )]TJ/F21 11.9552 Tf 11.956 0 Td [(C Z 1 1 + Z 2 2 ] : Wepartitiontheentirespaceofpowergains R 2 + intotwosets Z = f Z : R < R Z g and itscomplement Z c .Withthispartitioning,wecanderivethemaximin-optimalstrategies 3 Althoughweassumeequaluncertaintyvaluesforthetwousers,thiscaneasilybegeneralizedtothecasewhereintheyaredierent. 59 PAGE 60 forthetwousersconditionedonthefadingstatebelongingtooneofthesesets.Next,we makethefollowingimportantobservation: Z 2Z c = R i + R C Z i i for i =1 ; 2 ; {11 and Z 2Z = R 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [( R >C Z 1 1 1+ Z 2 2 ; {12 witharelationanalogousto3{12beingtrueforUser2aswell.For Z > 0,let 0 =min f 1 ;= Z g .Foranychoiceof f R 1 j R ; Z ,thechoiceofUser2'stransmissionrate thatminimizestheaveragerateofUser1in3{10canbederivedas f y R 2 j R ; Z r 2 = 8 > > < > > : 0 r 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(C Z 2 2 + )]TJ/F21 11.9552 Tf 11.955 0 Td [( 0 )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [(r 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( R 2 + R if Z 2Z ; )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(r 2 )]TJ/F15 11.9552 Tf 11.955 0 Td [( R 2 + R otherwise : {13 Substitutingtheabovesolutionin3{10,themaximin-optimaldistributionforthe transmissionratecanbeshowntobethefollowingmixedstrategy: f R 1 j R ; Z r 1 = 8 > > < > > : r 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(C Z 1 1 1+ Z 2 2 if Z 2Z c ; 1 r 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(C Z 1 1 1+ Z 2 2 + )]TJ/F21 11.9552 Tf 11.955 0 Td [( 1 )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(r 1 )]TJ/F15 11.9552 Tf 11.956 0 Td [( R 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [( R otherwise ; where 1 = 8 > > < > > : 0 if C Z 1 1 1+ Z 2 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [( 0 )]TJ/F15 11.9552 Tf 8.033 -6.662 Td [( R 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [( R ; 0otherwise : Themaximin-optimaltransmissionratesforUser2canbederivedanalogously. Intuitively,whenthechannelpowergainsbelongtotheset Z c R iscomparatively large",andtherestrictionimposedthrough3{8losesitseectivenessasthewindow ofstrategiesaboutthenominalstrategypairbecomestoowide".Hence,bothusers selecttransmissionratescorrespondingtothedisagreementpointasinSection3.3.1.On theotherhand,when Z 2Z ,themaximin-optimalstrategyforeachuseristorespond totheworstcasemixedstrategyoftheotheruserasin3{13whilesatisfying constraint3{8.Hence,whentheerrorintheCSITintermsofthepowergainsisnot 60 PAGE 61 high,suitablyselectingasmall"valueof R canensurethatbothusersbackofromthe NBS-suggestedtransmissionratepairof R to R 1 )]TJ/F15 11.9552 Tf 12.073 0 Td [( R ; R 2 )]TJ/F15 11.9552 Tf 12.074 0 Td [( R withhighprobability i.e.both i and 0 canbemadesmallenoughwithachoiceofsmall R and in3{8, therebyresultingin Z 1.Notethat,inthiscase,themodiedbargainingproblem thatincorporatesthemaximincriterionleadstobothusersbackingoinasimilar fashionalthougheachuserindependentlychoosesitsrespectivestrategy.Thus,itcan beseenthatthemodiedbargainingproblemformulationprovidesageneralframework forresourceallocationfromanindividualisticperspectiveandacharacterizationofthe optimalstrategypairsintermsof R AsmentionedattheendofSubsection3.3.2,thenominaltransmissionratepair selectedin3{7isnottheonlypossiblechoice,andotherjointlyrandomizedtransmission ratesmayalsobeselected.However,thechoiceoftransmissionratesasin3{7,can beshowntoincurnolossofgenerality.Foranychoiceofjointlyrandomizednominal transmissionrates,themaximin-optimaltransmissionratescanbederivedintermsof theirconditionalPDFswiththeconditioningonthenominalstrategiesandfadingstate inthesamewayasabovetoyieldthesamemaximin-optimalvaluesfortheobjective functionsi.e. E [ R r 1 R ; R ; Z ]of3{10anditscounterpartforUser2asforthecase consideredhere. InFig.3-1,wepresentthenumericalresultsforthesystemdescribedin3.3.2with twodierentmodelsfortheerrorintheCSIT.Forthisexample,themaximumavailable powersateachuserare 1 =100mWand 2 =10mWrespectively.Weset =0 : 02 cf.3{8andconsidertwosimplemodelsfortheerrorintheavailableCSIT:ia 5%errorinCSITwiththetruechannelpowergainslessthanwhattheCSITsuggests, andii x %errorintheCSITwith x randomlychosenfromauniformdistributionover [ )]TJ/F15 11.9552 Tf 9.298 0 Td [(10 ; 10].The trueutilities arecalculatedinthefollowingway.Foraparticularfading stateandchoiceoftransmissionstrategypair,ifthetransmissionratesliewithintheMAC 61 PAGE 62 capacityregiondenedbythetransmissionpowersandthe true powergains,thenthese ratesareconsideredachievable.Else,boththeuserssueroutages. Forbotherrormodels,as R increasesfrom0bps/Hzto2 : 5bps/Hz,thetrueutilities increasefromtheminimumutilitypointswhentheerrorintheCSITisunaccountedfor, reachtheirrespectivemaxima,thatareveryclosetotheachievableratesfortheperfect CSITscenario,andthendecreasetoeventuallyconvergeatthedisagreementpoints. Hence,itcanbeconcludedthatwithaproperchoiceof R ,theproposedsolutioncan providethenecessaryrobustnessagainstinaccuraciesintheavailableCSIT.Moreover,as expected,itcanbeobservedthatthedisagreementpointisnotaectedsignicantlyby theerrorinCSITasitcorrespondstoaninteriorpointoftheMACcapacityregionfor anyfadingstate. 3.5Summary Optimalresourceallocationandcooperativebehaviorinatwo-userfadingMAC withuncertaintyregardingtheCSITareconsideredfromagame-theoreticperspective. Theresourceallocationproblemismodeledasatwo-userbargainingproblemwiththe averagerateutilityfunction.OwingtopossibleinaccuraciesintheCSIT,thesolutionto thebargainingproblemforinstance,theNBS,thatdependsontheCSIT,maynotbe optimal,andmaycausesystemoutages.Toaddressthislackofrobustnessamodication tothebargainingproblemformulationisproposed.Numericalresultsdemonstratethat theproposedsolutioncanbeusedtoprovidesignicantrobustnesswithoutanexplicit modelingoftheerrorintheCSIT. 62 PAGE 63 Figure3-1.Averagerateswithvarying R 63 PAGE 64 CHAPTER4 ACTIVEUSERCOOPERATIONINFADINGMULTIPLE-ACCESSCHANNELS 4.1Introduction Thegrowingemphasisonmulti-userwirelesscommunicationsystemsandtheeverincreasingdemandforhighdatarateshaveheightenedtheimportanceofresearchin theareaofusercooperation.Althoughwirelesssystemsbringforthdesignchallenges owingtomulti-userinterferenceandfadingconcerns,italsoprovidespotentialbenets, likethebroadcastnatureofthewirelessmediumanddiversityadvantagesinmulti-user systems.TheMACisoneofthefundamentalkindsofmulti-usercommunicationsystems. Conventionally,inaMAC,theuserstransmitdirectlytothedestination,andthecapacity regionofsuchasystemiswellknown.Recently,ithasbeendemonstrated[3,4]that therateregionofthefadingMACcanbeincreasedbyprovidingspatialdiversitythatis achievedbyusercooperationintheformofforwardingeachother'sinformationtothe destination,givingrisetotheCMAC. AlthoughthecapacityregionfortheCMACisyettobedetermined,therehave beenanumberofdierentcooperativetransmissionstrategiesproposedintheliterature. Twobroadclassesofworksreportedinthisregardcanbedistinguishedbasedonthe transceivercapabilitiesofthewirelessnodes,i.e.whetherthenodescansupportfullduplexcommunicationornot. In[3],theauthorsprovideasystem-leveldescriptionoftheCMACwhereinthenodes arecapableoffull-duplexcommunication.Theypresentanachievablerateregionbased onblockMarkovencodingandbackwarddecoding,andshowthepotentialincreaseinthe rateregionascomparedtotheconventionalMAC.Itisassumedthatthephaseofthe fadingisknowntothetransmittersandthisisexploitedtoperformcoherentcombining atthedestinationnode,andobtainbeamforminggain.In[4],theCDMAimplementation aspectsoftheschemein[3]areconsidered,whereintheauthorsproposetheuseof dierentspreadingcodestoobtaindierentchannelsforsimultaneoustransmission 64 PAGE 65 andreception,withouttheuseofcomplicatedechocancelationtechniques.Thepower allocationproblemfortheCMACwithfull-duplexnodesandfullCSIavailableatall nodeshasbeenaddressedin[55]and[56].In[55],theauthorsconsideraveragepower constraintsandcharacterizetheoptimalpowerallocationpoliciesthatmaximizetheset ofergodicratesachievablebyblockMarkovencodingandbackwarddecodingtechnique asin[3],byadimensionalityreductionapproach,i.e.bynotingthatsomeofthepower allocationsarezeroforeveryfadingstate.Amoredirectapproachtosolvethesimilar problemofoptimalpowerallocation,withanalmostclosed-formsolution,ispresented in[56].Ithasbeenestablishedin[57]thatwindoweddecodingissucienttoachievethe samesum-rateasbackwarddecodingfortheblockMarkovsuperpositionencodingscheme fortheCMAC. TheoptimalpowerandresourceallocationproblemfortheCMACwithnodes capableofhalf-duplexcommunicationisconsideredin[58],whereitisassumedthat fullCSIisavailableatallthenodes,andthetransmitterscooperatebyrelayingeach other'sinformationoverorthogonalfrequencybandsortimeslots.Thesolutiontothe problemispresentedasatwo-stepconvexoptimizationproblemformulation:rst,fora particularbandwidthortimesharingparametervalue,theoptimalpowerallocationis characterizedbyaconvexoptimizationproblem,andthen,theoptimalresourcesharing timeorbandwidthparameterisobtainedasasolutiontothequasi-concaveproblemof maximizingtherateofoneuser,givenatargetratefortheother.Alltheworksmentioned aboveuseaDFapproachfortherelayingofinformationtothedestination. In[19],theauthorspresentacooperativetransmissionscheme,basedonthenonorthogonalamplify-and-forwardNAFtechnique,thatisprovedtoachievetheoptimal diversity-multiplexingtradeoof N )]TJ/F21 11.9552 Tf 13.085 0 Td [(r forthe N -userhalf-duplexCMAC,with symmetricdataraterequirementandCSIavailableonlyatthereceivingnodeofanylink. Accordingtotheproposedstrategy,timeisdividedintocooperationframesoflength N cooperationsymbols,andeachusertransmitsonlyonceduringacooperationframe.Every 65 PAGE 66 userisallotteduniquetransmissionandreceptionsymbolintervals,usingaparticular schedulingpolicy,andittransmitsalinearcombinationofitsownsymbolandthesignal observedduringitsmostrecentreceptionsymbolinterval,therebycreatinganarticial inter-symbolinterferenceISIchannel.Asetof L cooperationframesarecombinedto formasuper-frame,andtheassignmentofthereceptionsymbolintervalsisscheduled foreachsuper-frame,withthelengthsofsuper-framesandcodeschosensuchthata coherence-intervalconsistsof N )]TJ/F15 11.9552 Tf 12.016 0 Td [(1consecutivesuper-frames,andthatallcodewordsspan theentirecoherenceinterval. SimilartotheaboveNAFstrategy,acooperativetransmissionschemeforthetwouserCMAC,basedonsuperpositioncoding,hasbeenproposedin[59].Thisschemeuses atimedivisionapproachinwhichausersimultaneouslytransmitsitsowninformation andtheotheruser'sinformationbyusingthesuperpositioncodingSPCtechnique. Thisschemeisdemonstratedtoachieveagainofabout1 : 5 )]TJ/F15 11.9552 Tf 12.531 0 Td [(2dBovertraditionalDF approachesforrelaying,andatthesamelevelofsystemcomplexityofthelatter.An extensionofthisideatothegeneral N -userCMACispresentedin[60],whereinthe authorsprovetheoptimalityoftheproposedschemeinachievingtheoptimaldiversitymultiplexingtradeoforthesymmetricraterequirementscenario. Inthiswork,weproposeow-theoreticcooperativetransmissionprotocolsforthe two-userCMAC.First,wepresentanorthogonalrelayingprotocolfortheCMACORCMAC,whereineachuseractsasadedicatedrelayfortheotherinatime-division fashion.Theow-optimizedrelayingapproachofChapter2,modiedtoincorporate coherentcombiningatthedestinationisusedfortheconstituentrelaychannels.This relayingprotocolhasbeenshowntoachievetheoptimaldiversityorderandprovidebetter codinggainsfortherelaychannelascomparedtotraditionalDFrelayingmethods,by ecientlyutilizingtheCSIavailableatallnodes.Next,weproposetheow-optimized protocolfortheCMACFO-CMACthatdecomposestheCMACintotwobroadcast 66 PAGE 67 channelsBCandamultipleaccessMAchannelwithcommoninformation.Theboundariesoftheachievablerateregionsarecharacterizedbymeansofconvexoptimization formulations.Theimprovementprovided,intermsoftheachievablerateregion,byORCMACandFO-CMAC,overconventionalMACcapacityandtheDFstrategyof[58] withoutpowercontrol,increasesastheamountofdisparitybetweenthechannelsfrom thetwosourcestothedestinationincreases.Theoutageperformancesoftheproposed protocolsindicatethatalthoughthemuchsimplerOR-CMACissuboptimalintermsof theachievablerateregion,itprovidesoutageperformancethatiswithin1dBofthatof FO-CMAC.Moreover,boththeproposedprotocolsachieveadiversityofordertwoforthe requiredrateregionofinterest. Therestofthechapterisorganizedasfollows.InSection4.2,theow-theoreticprotocolsofOR-CMACandFO-CMACarepresented,andtheboundariesoftheachievable rateregionsarecharacterizedbyconvexoptimizationformulations.Thisisfollowedbynumericalresultsinpresentingtheachievableaveragerateregionsandoutageperformances fordierentscenariosinSection4.3.Finally,theprimarycontributionsinthischapterare summarizedinSection4.4. 4.2Flow-theoreticTransmissionProtocolsfortheCooperative Multiple-AccessChannel Consideratwo-userCMACwherethetwosources S 1 and S 2 mayactivelycooperatetotransmitinformationtoacommondestination D .Weusethephrase active cooperation todistinguishbetweenthecooperationinvolvedintransmissionstrategiesin whichonesourcemayforwardtheother'sinformation,andthatintheconventionalMAC, whereinausertransmitsataratelowerthanthemaximumsingle-userratepossiblefor theparticularchannelstateandpowerexpended.ThequanticationofthistypeofcooperativebehaviorwasstudiedinChapter3.Thus,asahigherlevelofcooperativebehavior, theusersmayrelaytheinformationofeachotherbyutilizingthebroadcastadvantageof 67 PAGE 68 thewirelessmedium,givingrisetotheCMACmodel.Weconsideradiscrete-timetwouserfadingMACwithunitbandwidth,inthepresenceofunit-varianceGaussiannoise, withthefadingstatedescribedbythepowergainvector Z = Z S 1 S 2 Z S 1 D Z S 2 D .Note that,owingtothereciprocityofchannels,weassume Z S 1 S 2 = Z S 2 S 1 .Thepowergainsfor thewirelesslinksaremodeledasindependentexponentialrandomvariables.Weconsider twotypesoffadingmodelsinthiswork.First,inSection4.3,weconsiderthesituation inwhichthechannelsareergodicwithinatransmissionblock,forwhichweevaluatethe transmissionprotocolsusingaverageratesastheperformancemetric.Then,wepresent theoutageperformanceoftheproposedprotocolsforthemodelinwhichthefadingisnot fastenoughandhence,thechannelsmaynotbeergodicduringatransmissionblock.This systemmodelcanbeeasilygeneralizedtothecasewherethebandwidth W 6 =1. Let P i bethemaximumtransmissionpoweravailabletothesource S i .Inthiswork, weconsidershorttermpowerconstraintsonly,andthisprecludesanypotentialadvantage ofpowerallocation.WeassumethatfullCSIisavailableatallnodesofthesystem. Moreover,asapracticalconsideration,weassumethatthenodesarenotcapableof transmittingandreceivinginformationsimultaneouslyoverthesamefrequency,i.e.they aresubjectedtoahalf-duplexconstraint.Inthefollowingsubsections,wepresenttwo protocolsbasedonow-theoreticdesignstodevelopcooperativetransmissionschemesfor informationtransmissionfromsources S 1 and S 2 todestination D 4.2.1OrthogonalRelayingProtocolfortheCMACOR-CMAC Inthissubsection,wepresentasimplecooperativetransmissionprotocolbasedon theconventionalrelayingapproach.Timeisdividedintounitintervals.Owingtothe half-duplexlimitationofthesources,thetwosourcescannotrelayeachother'sinformation atthesametime.Toaddressthis,wedivideeachunitintervalintotimeslotsoflengths T 1 and T 2 .Duringtimeslot T 1 ,source S 2 solelyassistssource S 1 ,byactingasadedicated relayto S 1 ,totransmitthelatter'sinformationtothedestination D .Thereversehappens duringtimeslot T 2 .ThisisdepictedinFigure4-1. 68 PAGE 69 Thus,weeectivelyhavetworelaychannelsovertwoorthogonaltimeslots.Weuse theow-optimizedtransmissionschemeofChapter2forthethree-noderelaychannel. Accordingtotheprotocolpresentedtherein,eachtimeslot T 1 resp. T 2 isfurtherdivided intotwosub-slotsoflengths t 1 and t 2 resp. t 0 1 and t 0 2 .Considerthetimeslot T 1 .During therstsub-slot, S 1 sendstwoindependentowsofinformation x 1 and x 2 to D and S 2 respectivelyusingabroadcastchannelBC,andinthesecondsub-slot, S 2 forwards x 2 to D andatthesametime S 1 sendsoutanotherinformationow x 3 to D ,and x 2 and x 3 arereceivedat D viamultiple-accessMA.Toimprovetheachievablerateseven further,wemodifythesecondsub-slotasfollows.Sincetheow x 2 ,that S 2 forwardsto D originatedat S 1 ,thelatterisawareofit,andhence,wemodifythesecondsub-slotfroma conventionalMACtoaMACwithcommoninformation[61],where x 2 formsthecommon informationbetween S 1 and S 2 x 3 istheindependentinformationfrom S 1 ,and S 2 does nothaveanyindependentinformationtotransmit.Forthisrelayingscheme,maximizing theoveralltransmissionratefrom S 1 to D canbeformulatedasthefollowingoptimization problem: max x 1 + x 2 + x 3 over t 1 ;t 2 ;x 1 ;x 2 ;x 3 ;P 1 ; 0 {1 subjectto non-negativityconstraints: x 1 ;x 2 ;x 3 0; t 1 ;t 2 0; total-timeconstraint: t 1 + t 2 = T 1 ; capacitypowerconstraints: P BC P 1 ; x 3 t 2 C Z S 1 D P 1 ; x 2 + x 3 t 2 C Z S 1 D P 1 + 0 ; 0 P 1 P 1 ; 0 Z S 1 D P 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(P 1 + Z S 2 D P 2 +2 q Z S 1 D P 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(P 1 Z S 2 D P 2 ; where C x =log+ x and P BC ,theminimumpowerrequiredforthesourcetobroadcast atrates x 1 =t 1 and x 2 =t 1 tothedestinationandtherelay,respectively,intherstsub-slot 69 PAGE 70 with0 PAGE 71 Now,thersttwotimeslotsareBCslotsandthelastoneisanMAslotwithcommon information,asshowninFigure4-1.Duringthersttimeslot, S 1 transmitstwoindependentows: x 1 + y 21 and x 2 to D and S 2 respectivelyusingBC.Similarly, S 2 transmitstwo independentows: y 1 + x 21 and y 2 to D and S 1 respectivelyusingBC.Finally, S 1 and S 2 sendtwoows x 3 + y 22 and y 3 + x 22 to D usingMAwithcommoninformation.Here y 21 and y 22 aretwopartsoftheinformationow y 2 that S 1 receivedfrom S 2 duringthe previousunitinterval.Similarly, x 21 and x 22 constitutetheamountofinformationthat S 2 relaysfor S 1 .Thus,forthisscheme,theowconstraintsimplythat x 2 = x 21 + x 22 ,and y 2 = y 21 + y 22 .Also,forthelasttimeslot, x 22 and y 22 areknowntoboththesources,and hence,theyformthecommoninformationtobetransmittedto D Hence,thetotaltransmissionratesfromsources S 1 and S 2 aregivenby X = x 1 + x 21 + x 22 + x 3 and Y = y 1 + y 21 + y 22 + y 3 respectively.Forthisscheme,theboundary oftheachievablerateregioncanbecharacterizedasfollows:theregularpointsonthe boundarycanbeobtainedbymaximizingaconvexcombinationoftherates X and Y ,and theextremepointscorrespondtotheCMACdegeneratingintorelaychannelswithone sourcesolelyactingasarelayfortheother.Forcompactness,lettheinformationows correspondingtothetwosourcesberepresentedbythevectors x = x 1 x 21 x 22 x 3 and y = y 1 y 21 y 22 y 3 .Themaximizationproblemthatneedstobesolvedtoobtainthe regularpointscanbeformallystatedasgivenbelow: max X + )]TJ/F21 11.9552 Tf 11.956 0 Td [( Y for 2 ; 1over T 1 ;T 2 ;T 3 ; x ; y ;P 1 ;P 2 ; 0 ; {3 subjectto non-negativityconstraints: x ; y 0 ; T 1 ;T 2 ;T 3 0; total-timeconstraint: T 1 + T 2 + T 3 =1; capacitypowerconstraints: P 1 BC P 1 ; P 2 BC P 2 ; x 3 T 3 C Z S 1 D P 1 ; y 3 T 3 C Z S 2 D P 2 ; x 22 + x 3 + y 22 + y 3 T 3 C Z S 1 D P 1 + Z S 2 D P 2 + 0 ; 71 PAGE 72 0 P 1 P 1 ;0 P 2 P 2 ; 0 Z S 1 D P 1 )]TJ/F21 11.9552 Tf 11.956 0 Td [(P 1 + Z S 2 D P 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(P 2 +2 q Z S 1 D P 1 )]TJ/F21 11.9552 Tf 11.955 0 Td [(P 1 Z S 2 D P 2 )]TJ/F21 11.9552 Tf 11.955 0 Td [(P 2 : Asintheprevioussubsection, P 1 BC and P 2 BC aretheminimumpowersrequiredby S 1 and S 2 respectivelyforthetwoBCslots. P 1 BC isdenedasin4{4and P 2 BC isdened similarly.Also, P 1 and P 2 arethepowersallocatedby S 1 and S 2 totransmit x 3 and y 3 respectively,and 0 denotesthereceivedpowerat D correspondingtothecommon information x 22 + y 22 P 1 BC = 8 > < > : 1 Z S 1 D )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(e x 1 + y 21 =T 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 + 1 Z S 1 S 2 e x 1 + y 21 =T 1 )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(e x 2 =T 1 )]TJ/F15 11.9552 Tf 11.956 0 Td [(1 for Z S 1 S 2 >Z S 1 D ; 1 Z S 1 S 2 )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(e x 2 =T 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 + 1 Z S 1 D e x 2 =T 1 )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(e x 1 + y 21 =T 1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 for Z S 1 S 2 Z S 1 D : {4 For T 1 =0 ;P 1 BC =0 : Onceagain,itcanbecheckedthattheabovemaximizationisaconvexoptimization problemthatcanbesolvedusingstandardnumericaloptimizationmethods.Thus,FOCMACaddressesthehalf-duplexlimitationofthenodesbydividingtheCMACintotwo BCandoneMACwithcommoninformation,andprovidesamoreecientutilizationof systemresourcesascomparedtoprotocolsusingtwoseparaterelaychannels. 4.3NumericalResults Inthissection,wepresentsomenumericalresultstodemonstratetheperformance oftheproposedprotocolsandcomparethemtotheconventionalMACandtheDF-based strategyproposedin[58].Figures4-2through4-4showtheachievableaveragerateregions forthevariousschemesfordierentscenarios.Weconsiderdierentmeansofthefading gainsasstatedinthegures.Considerablevariationsinthestatisticsofthefadinggains forthechannels Z S 1 D and Z S 2 D canoccurinpracticalsituationsowingtodierentpath lossandshadowingeectsanddierentamountsofscatteringforthetwodirectlinks. Thus,theasymmetricsituationcorrespondstothecasewhenthedirectlinkfromone sourcetothedestinationismuchworsethantheother. 72 PAGE 73 InFigure4-2,thechannelfrom S 2 to D ismuchworsecomparedtothatfrom S 1 to D ,andthelargeincreaseintheachievablerateregionfortheFO-CMACrateregion isevident.Moreover,weseethatthemuchsimplerOR-CMACperformsclosetoFOCMACforthisscenario.Ontheotherhand,theperformanceofthestrategyof[58]is muchpoorerthantheabovetwo.Recallthat,inthiswork,weconsideronlyshortterm powerconstraint,andunlike[58],wedonotconsiderthepowerconstraintovertheentire unitinterval.Thiseliminatesanypotentialgainsfromoptimizingthepowerallocations forthedierenttransmissions.So,forthepresentsystem,thecooperativetransmission strategyof[58]isclearlysuboptimal.Figure4-3ispresentedtohighlighttheincreasein theachievablerateregionresultingfromtheuseofMAslotswithcommoninformationas inFigure4-2insteadoftheconventionalMAslots.AscanbeseenfromFigure4-3,with theuseofconventionalMAslots,themaximumsum-ratefortheFO-CMACcoincides withthatoftheconventionalMACwithoutactivecooperationbetweenthetransmitters. TheimprovementinFigure4-2canbeinterpretedastheeectofthebeamforminggain asinatwo-transmitterone-receiverMISOsystem.Figure4-4presentsthesymmetric situation,whenthetwodirectlinksarestatisticallyidentical.Weseethatinthiscase,the increaseintherateregionisnotaspronouncedasintheprevioussituation.Moreover, theachievablerateregionforthestrategyof[58]liesstrictlyinsidethatforthebaseline systemofMACwithoutactivecooperationbetweenthetransmitters. AsmentionedinSection4.2,forthesituationinwhichthefadingisnotfastenough sothattheergodicpropertiesofthechannelsareobserved,theoutageperformanceof thetransmissionstrategiesareamorereasonableperformancemetricasagainstaverage rates.Theoutageperformancesoftheproposedprotocolsareevaluatedwhenthedata raterequirementissymmetricat K =1bit/s/Hzforbothusers.Theoutageeventis denedsimilartothedenitionin[62],i.e.itistheunionoftheeventsthateitherone orbothoftheuserssueranoutage.Theoutageperformancesoftheproposedprotocols arecomparedtothatoftheconventionalMAC,thestrategyof[58],andalowerboundon 73 PAGE 74 theoutageprobabilitythatisobtainedbyconsideringthecasewhenthesourceshavea perfectnoiselesschannelbetweenthem.Figure4-5presentstheoutageperformancesfor theasymmetricsituationasintheaverageratescase,andFigure4-6presentsthesame forthesymmetricsituation.WeseethattheOR-CMACsuersalossofonly0 : 7dBand 1 : 0dBcomparedtoFO-CMAC,atanoutageprobabilityof10 )]TJ/F20 7.9701 Tf 6.587 0 Td [(3 ,fortheasymmetric andsymmetricsituationsrespectively.TheperformanceofFO-CMACisworsethanthe lowerboundbyabout2 : 2dBand2 : 5dBatthesameoutagelevel,fortheasymmetric andsymmetricsituationsrespectively.Ontheotherhand,theperformanceoftheDF strategyof[58]issignicantlypoorer,andevenworsethantheconventionalMACforthe symmetricsituation.Anotherimportantobservationfromtheoutageperformanceplotsis thattheslopesofthecurvesforFO-CMACandOR-CMACareidenticaltothatforthe lowerboundcurve.Sincethelatterisidenticaltothe2 1MISOpoint-to-pointsystem,it givesadiversityorderoftwo.Hence,theaboveobservationestablishesthefactthatboth theproposedprotocolsachievetheoptimaldiversityorderoftwoforthetwo-userCMAC fortherequiredrateregionofinterest. 4.4Summary Inthischapter,weproposedow-theoreticcooperativetransmissionprotocolsforthe two-userfadingCMAC,wherethenodesareonlycapableofhalf-duplexcommunication andhaveaccesstofullCSI.Weproposetwosuchprotocols,viz.aow-optimizedprotocol fortheCMACFO-CMAC,andasuboptimalbutsimplerorthogonalrelayingprotocol fortheCMACOR-CMAC.Boththeproposedprotocolsareevaluatedintermsof achievablerateregionsandoutageperformances.NumericalresultsshowthatFO-CMAC yieldsthelargestachievablerateregionamongstthedierentprotocolsconsideredhere. AlthoughOR-CMACisclearlysuboptimalintermsoftheachievablerateregion,its outageperformanceisclosetowithin1dBforthescenariosconsideredtoFO-CMAC, andboththeproposedprotocolsachievetheoptimaldiversityorderoftwoforthe requiredrateregionofinterest. 74 PAGE 75 Figure4-1.Flow-theoretictransmissionprotocolsfortheCMAC:aOR-CMAC,b FO-CMAC. 75 PAGE 76 Figure4-2.Achievablerateregions-asymmetricsituation. Figure4-3.Achievablerateregions-asymmetricsituationwithconventionalMAslot withoutcommoninformation. 76 PAGE 77 Figure4-4.Achievablerateregions-symmetricsituation. Figure4-5.Outageperformance-asymmetricsituation. 77 PAGE 78 Figure4-6.Outageperformance-symmetricsituation. 78 PAGE 79 CHAPTER5 INTERFERENCECHANNELSWITHUNIDIRECTIONALCOOPERATIONAND CAUSALITYCONSTRAINTS 5.1Introduction AsmentionedinChapter1,theinterferencechannelwithunidirectionalcooperationICUCisessentiallythesimplestrealizationofanoverlaycognitiveradionetwork. Cognitiveradioshaveaconsiderablepotentialinfacilitatinganecientuseofthelicensedspectrumthatiscurrentlyunder-utilized[18].Theoverlayparadigmforcognitive radiosnotonlyprovidesanecienttechniqueforcognitiveradiodeploymentsbutalso yieldsnewerinsightstowardstheunderstandingofinterferencechannelsandcooperative behaviorinmulti-terminalnetworks,throughthedierentmanifestationsofcognition,cooperationandcompetitionlevelsamongstdierentusersorusergroups[63].Intheoverlay form,thesimplestofwhichbeingtheICUCalsoknownascognitiveradiochannel[6,7], thecognitiveorcooperatingradioutilizesthesamespectrumastheprimaryuser-pair foritsowndatatransmission.Whereasthiswouldgenerallycauseinterferencetothe primarylink,thecognitivesourcemayexploititsknowledgeabouttheprimarymessage toimproveitsowntransmissionratesbyprecodingitsinformationagainsttheknown interferencefromtheprimarytransmissionandsimultaneouslyalleviatethedetrimental eectsoftheinterferencetotheprimarydestinationowingtothesecondarytransmission bycooperativerelayingoftheprimarymessage. Oflatetherehasbeenaconsiderablebodyofworkreportedintheliteraturethat havehelpedimprovetheachievablerateregionfortheICUCproposedin[6].In[64],the authorspresentinnerboundstothecapacityregionforjointandsequentialdecoding,and encodingstrategiesthatincluderate-splittingforbothprimaryandsecondarymessages, conditionalGel'fand-PinkserGPbinningandcooperativerelaying.Theyalsopresent ageneralouterboundthatisverysimilartoanouterboundforthebroadcastchannel, andamuchsimplerouterboundforthecasewhereintheprimarydestinationexperiences stronginterference.Aslightlydierentcodingschemehasbeenproposedin[17],wherein 79 PAGE 80 thereisnorate-splittingfortheprimarymessageandthetwopartsofthesecondary messagearebinnedindependentlyagainsttheinterferencefromprimarytransmission. Veryrecently,anothercodingschemeforthediscretememorylessnon-causalICUC hasbeenproposedin[65].Accordingtothisscheme,conditionalrate-splittingisapplied tobothprimaryandsecondarymessages,andthecognitivesourceusesGPbinningand atwo-waybinningstrategyasin[67],conditionedonthecodewordforthecommonpart oftheprimarymessage,totransmitthecommonpartofthesecondarymessage,and theprivatepartsoftheprimaryandsecondarymessagesrespectively.Theauthorsalso considerthecaseinwhichtheprimarymessagemaybeavailableatthecognitivesource inacausalfashion,butthechannelmodelismodiedtothatforaZinterferencechannel ZIC,whereintheprimarydestinationdoesnotexperienceanyinterferenceduetothe secondarytransmission.Anotheruniedcodingscheme,verysimilartotheonein[65], thatyieldsarateregionthatincludestheregionof[65],hasbeenproposedin[66]. Cognitiveradionetworksthataremoreinvolvedthanthetwo-userICUChavealso beenstudiedinsomerecentworks.ThesenetworkscombinetheICUCwithothermultiterminalnetworkslikethemultiple-accesschannels[68],broadcastchannels[69],orrelay channels[70]. Cognitivemultipleaccessnetworks,inwhichasetofcognitiveusersisprovided withafunctionofthemessagesofthesetofnon-cognitiveusersandeachsethasits correspondingreceiver,havebeenstudiedintermsoftheirachievablerateregionsin[68]. Achievablerateregionsforone-sidedinterferencechannelswithacognitiverelay,thathas non-causalmessage-orsignal-levelinformationfrombothsourcesandalinktoonlythe destinationexperiencinginterference,havebeenobtainedin[70].In[69],anachievable rateregionforthecaseoftwointerferingbroadcastchannels,withonesourcehaving non-causalknowledgeaboutthemessageoftheother,hasbeenderived. Mostoftheaboveworksconsiderthenon-causalformofICUCwhereinitisassumed thattheprimarymessageisavailablenon-causallyatthecognitivesource.Inreality, 80 PAGE 81 someresourcesintimeorfrequencyneedtobeexpendedbythesystemforthecognitive sourcetoacquiretheprimarymessage.Thescenarioinwhichthecognitivesourcesneed toobtaintheinformationcausallyfromtheprimarysourceexplicitlymodelsthisoverhead andhasbeenconsideredforthecaseofthetwo-userICUCin[6]and[71].In[6],the authorsconsiderhalf-duplexoperationofthesecondarysource,andproposeatwo-phase protocol.Therstphaseisusedtotransmittheprimarymessagetothecognitivesource andapartoftheprimarymessagetotheprimarydestinationviaabroadcastchannel, andoncethecognitivesourcesuccessfullydecodestheprimarymessage,theoperation proceedsasforthenon-causalcase.Ontheotherhand,in[71],afull-duplexoperation ofthesecondarysourceisassumed,andblockMarkovSPCalongwithsliding-window decoding,andrate-splittingforthetwomessagesareusedtoobtainanachievablerate region. Inthiswork,weconsiderthetwo-userICUCwithcausalityconstraintICUC-C. Itisassumedthatthecognitivesourceiscapableofperfectechocancellation,thereby makingfull-duplexoperationofthecognitivesourcefeasible.Wepresentageneralized codingschemeforthediscretememorylessICUC-C.Theproposedachievablerateregion isbasedonblockMarkovSPCwithbackwarddecoding[72]fortheprimarymessage transmission,conditionalrate-splittingfortheprimaryandsecondarymessagesto facilitatepartialdecodingatthenon-pairingdestinations,GPbinningatthecognitive source,andcooperativerelayingoftheprimarymessagebythecognitivesource.This rateregionisthenevaluatedforthecaseofGaussianchannelsandnumericalresultsfor dierentvaluesofnetworkparametersarepresented.Theresultsareusedtoshedlighton theperformanceofthecodingstrategiesinvolvedintheproposedschemeunderdierent scenarios. Next,werelaxtheassumptionthatthecognitivesourceiscapableoffull-duplex operation,andinsteadsubjectthecognitivesourcetothehalf-duplexconstraint,i.e.it cannotreceiveandtransmitinformationsimultaneouslyoverthesameband.Towards 81 PAGE 82 this,wepresentadiscretememorylesschannelmodelfortheICUCwithhalf-duplexand causalityconstraintsICUC-HDC,andproposeageneralizedcodingschemeforthis channel.Similartothefull-duplexcase,weemployblockMarkovSPCwithbackward coding,conditionalrate-splitting,GPbinning,andcooperativerelayingbythecognitive source.However,forthehalf-duplexcase,thecognitivesourceemploysarandomized listen-transmitschedule[73]toencodeandtransmitinformationviasignaling.Itisalso provedthatthenewrateregioncontainsthepreviouslyknownrateregionof[6]forthe ICUC-HDC. Inthefollowingsection,wepresentthediscretememorylesschannelmodelforthe ICUC-C.ThisisfollowedbySection5.3,wherewepresenttherandomcodingschemeand thecorrespondingachievablerateregionfortheICUC-C.Section5.4detailstheGaussian ICUC-Calongwithnumericalexamplesandadiscussionontheroleofdierentcoding techniquesunderdierentnetworkscenarios.Followingthis,thediscretememoryless ICUC-HDCisintroducedinSection5.5,andinSection5.6,theproposedrandomcoding schemeandthenewachievablerateregionarepresented.TheGaussianICUC-HDCis presentedinSection5.7,followedbyanalyticalandnumericalcomparisonsbetweenthe newachievablerateregionandtheonein[6].Finally,asummaryofthecontributionsin thischapterispresentedinSection5.8. 5.2TheChannelModel ConsiderthecommunicationscenarioasinFig.5-1,whereintheprimarysourcenode S P intendstotransmitinformationtoitsdestinationnode D P .Apartfromtheprimary source-destinationpair,thewirelessnetworkconsistsofasecondaryorcognitivesourcedestinationpair, S C and D C ,with S C havingitsowninformationtotransmitto D C .The primarymessageiscausallyavailableat S C ,andthelattermayusethisknowledgeto assist S P inthetransmissionoftheprimarymessageto D P ,andalsotransmititsown informationto D C 82 PAGE 83 In n channeluses,theprimarysource, S P ,hasamessage w P 2f 1 ; 2 ; ; 2 nR P g totransmitto D P ,whilethesecondarysource S C hasamessage w C 2f 1 ; 2 ; ; 2 nR C g totransmittoitsintendeddestination D C .Let X P ; X C ,and V C ; Y P ; Y C betheinputandoutputalphabetsrespectively.Aratepair R P ;R C isachievableifthere existanencodingfunctionfor S P X n P = f P w P ;f P : f 1 ; 2 ; ; 2 nR P g!X n P andasequenceofencodingfunctionsfor S C X n C = f n C w C ;V n )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 C with X Ci = f Ci w C ;V i )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 C ;f Ci : f 1 ; 2 ; ; 2 nR C gV i )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 C !X C ,andcorrespondingdecodingfunctions ^ w P = g P Y n P ;g P : Y n P !f 1 ; 2 ; ; 2 nR P g and^ w C = g C Y n C ;g C : Y n C !f 1 ; 2 ; ; 2 nR C g suchthattheaverageprobabilityoferror P n e =max f P n e;P ;P n e;C g! 0,where P n e;M = 1 2 n R P + R C X w P ;w C Pr[ g M Y n M 6 = w M j w P ;w C wassent]for M = P;C 5.3AchievableRatesfortheICUC-C Inthissection,wepresentanewachievablerateregionforthediscretememoryless ICUC-C.Westartwithanoverviewofthecodingscheme.Inblock b 2f 1 ; ;B g S P splitsthemessage w P;b intotwoparts w Pco;b and w Ppr;b .Itusessuperpositioncodingto encodethesetwopartsalongwiththemessageforthepreviousblock w Pco;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;w Ppr;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 Thelatteractsasthe resolutioninformation for D P and D C thatusebackwarddecoding todecodetheprimarymessageentirelyorpartially.Incontrasttotherate-splittingtechniquein[71],whereinthetwomessagepartsareencodedindependentlyandsuperposed, S P performsconditionalrate-splitting[65]coupledwithblockMarkovSPC. S C decodes theprimarymessageforthecurrentblockandsimultaneouslyperformsasetofencodingsteps.Inblock b S C splits w C;b intotwoparts w Cco;b and w Cpr;b ,andconditionedon thecodeword T Pco fortheresolutioninformationforthecommonpartoftheprimary message[65],itusesconditionalGPbinning[64]toencode w Cco;b and w Cpr;b as U Cco and U Cpr respectively,againsttheresolutioninformationfortheprivatepartoftheprimary message T Ppr thatcausesinterferenceat D C butisknownat S C .Finally,ittransmitsa combinationoftheabovecodewords,alongwiththeresolutioninformationfortheprimary message,withthelatterpartmanifestingthecooperativerelayingactionby S C .Unlike 83 PAGE 84 thecodingschemein[71], S C doesnotuseblockMarkovSPCtoencodeitsownmessage, therebygivingrisetoasimplercharacterizationoftherateregion. D P waitsuntilthelastblock B andusesbackwarddecodingtojointlydecodeboth commonandprivatepartsoftheprimarymessageandthecommonpartofthesecondary message.Similarly, D C waitsuntilthetransmissionforblock B andthenusesbackward decodingtojointlydecodethecommonpartoftheprimarymessageandbothcommon andprivatepartsofthesecondarymessage.Notethat D C performsbackwarddecoding onlytodecodethecommonpartoftheprimarymessagetotakeadvantageoftheblock MarkovSPCstructureusedtoencodeit.Further,theuseofbackwarddecodingyieldsa muchsimplerrateregioncharacterizationcomparedtotheonein[71].Table5-1liststhe randomvariablesinvolvedinthecodeconstructionalongwiththeirsignicance. Table5-1.DescriptionofRandomVariablesinTheorem5.1 RandomVariableDenition T Pco Resolutioninformationforcommonpartofprimary messageknownto S C T Ppr Resolutioninformationforprivatepartofprimary messageknownto S C X Pco Newinformationforcommonpartofprimarymessage X P Transmittedcodewordby S P ,generatedbysuperposingnewinformationforprivatepartofprimary messageon T Pco T Ppr ,and X Pco pleaserefertoCodebookGenerationintheproofofTheorem5.1 U Cco Commonpartofsecondarymessagegeneratedby conditionalGel'fand-Pinskerbinningagainst T Ppr U Cpr PrivatepartofsecondarymessagegeneratedbyconditionalGel'fand-Pinkserbinningagainst T Ppr X C Transmittedcodewordby S C Theorem5.1. FortheICUC-C,theratetuple R P ;R C ,where R P = R Pco + R Ppr R C = R Cco + R Cpr ,withnon-negativereals R Pco ;R Ppr ;R Cco ;R Cpr satisfying R Ppr min f I X P ; V C j T Pco ;T Ppr ;X Pco ; I T Ppr ;X P ; Y P ;U Cco j T Pco ;X Pco g {1a 84 PAGE 85 R Pco + R Ppr I X P ; V C j T Pco ;T Ppr {1b R Ppr + R Cco I T Ppr ;X P ;U Cco ; Y P j T Pco ;X Pco {1c R Pco + R Ppr + R Cco I T Pco ;T Ppr ;X P ;U Cco ; Y P {1d R Cpr I U Cpr ; Y C ;U Cco j T Pco ;X Pco )]TJ/F21 11.9552 Tf 11.955 0 Td [(I U Cpr ; T Ppr ;U Cco j T Pco {1e R Cco + R Cpr I U Cco ;U Cpr ; Y C j T Pco ;X Pco )]TJ/F21 11.9552 Tf 9.299 0 Td [(I U Cco ;U Cpr ; T Ppr j T Pco {1f R Pco + R Cco + R Cpr I T Pco ;X Pco ;U Cco ;U Cpr ; Y C )]TJ/F21 11.9552 Tf 9.299 0 Td [(I U Cco ;U Cpr ; T Ppr j T Pco {1g isachievableforsomejointdistributionthatfactorsas p t Pco p t Ppr j t Pco p x Pco j t Pco p x P j t Pco ;t Ppr ;x Pco p u Cco j t Pco p u Cpr j t Pco ;u Cco p x C j t Pco ;t Ppr ;u Cco ;u Cpr p v C j x P ;x C p y P j x P ;x C p y C j x P ;x C ; {2 andforwhichtheright-handsidesof 5{1a to 5{1g arenon-negative. Proof. Let A n X;Y denotesetofjointly -typicalsequencesaccordingtothedistribution ofrandomvariables X;Y asinducedbythesamedistributionusedtogeneratethe codebookssee[33].Forconvenience,thedependenceontherandomvariableswillnotbe statedexplicitly,andshouldbeclearfromthecontext. Codebookgeneration: Splittheprimaryandcognitiveusers'ratesas R P = R Pco + R Ppr and R C = R Cco + R Cpr respectively.Thus,inblock b 2f 1 ; ;B g ,theprimary messagecanberepresentedas w P;b = w Pco;b ;w Ppr;b ,andthesecondarymessageas w C;b = w Cco;b ;w Cpr;b ,where co and pr standforthecommonandprivatepartof amessagerespectively.Fixadistribution p t Pco ;t Ppr ;x Pco ;x P ;u Cco ;u Cpr ;x C asin Theorem5.1. 85 PAGE 86 Generate2 nR Pco i.i.d.codewords t n Pco w 0 Pco w 0 Pco 2f 1 ; ; 2 nR Pco g ,accordingto Q n i =1 p t Pcoi Foreachcodeword t n Pco w 0 Pco ,generate2 nR Ppr conditionallyi.i.d.codewords t n Ppr w 0 Pco ;w 0 Ppr w 0 Ppr 2f 1 ; ; 2 nR Ppr g ,accordingto Q n i =1 p t Ppri j t Pcoi w 0 Pco Foreachcodeword t n Pco w 0 Pco ,generate2 nR Pco conditionallyi.i.d.codewords x n Pco w 0 Pco ;w Pco w Pco 2f 1 ; ; 2 nR Pco g ,accordingto Q n i =1 p x Pcoi j t Pcoi w 0 Pco Foreachcodewordtuple )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(t n Pco w 0 Pco ;t n Ppr w 0 Pco ;w 0 Ppr ;x n Pco w 0 Pco ;w Pco ,generate 2 nR Ppr i.i.d.codewords x n P w 0 Pco ;w Pco ;w 0 Ppr ;w Ppr w Ppr 2f 1 ; ; 2 nR Ppr g ,according to Q n i =1 p )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(x Pi j t Pcoi w 0 Pco ;t Ppri )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(w 0 Pco ;w 0 Ppr ;x Pcoi )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(w 0 Pco ;w Pco Foreachcodeword t n Pco w 0 Pco ,generate2 n R Cco + R 0 Cco i.i.d.codewords u n Cco w 0 Pco ;w Cco ;b Cco w Cco 2f 1 ; ; 2 nR Cco g and b Cco 2f 1 ; ; 2 nR 0 Cco g ,accordingto Q n i =1 p u Ccoi j t Pcoi w 0 Pco Foreachcodewordpair )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(t n Pco w 0 Pco ;u n Cco )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(w 0 Pco ;w Cco ;b Cco ,generate2 n R Cpr + R 0 Cpr i.i.d.codewords u n Cpr w 0 Pco ;w Cco ;b Cco ;w Cpr ;b Cpr w Cpr 2f 1 ; ; 2 nR Cpr g and b Cpr 2f 1 ; ; 2 nR 0 Cpr g ,accordingto Q n i =1 p u Cpri j t Pcoi w 0 Pco ;u Ccoi w 0 Pco ;w Cco ;b Cco Generate x n C w 0 Pco ;w 0 Ppr ;w Cco ;b Cco ;w Cpr ;b Cpr where x C isadeterministicfunctionof t Pco ;t Ppr ;u Cco ;u Cpr Encoding: At S P :Inblock b 2f 2 ; ;B )]TJ/F15 11.9552 Tf 12.098 0 Td [(1 g S P transmits x n P w Pco;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;w Pco;b ;w Ppr;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ;w Ppr;b Intherstblock, S P transmits x n P ;w Pco; 1 ; 1 ;w Ppr; 1 ,whileinblock B ,ittransmits x n P w Pco;B )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ; 1 ;w Ppr;B )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ; 1.Notethattheactualratefortheprimarymessageis B )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 B R P butitconvergesto R P asthenumberofblocks B goestoinnity. At S C :Inblock b 2f 1 ; ;B g ,totransmit w Cco;b S C searchesforbinindex b Cco;b suchthat )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(t n Pco ^ w Pco ;u n Cco ^ w Pco ;w Cco;b ;b Cco;b ;t n Ppr ^ w Pco ; ^ w Ppr 2A n ; {3 where ^ w Pco and ^ w Ppr are S C 'sestimatesof w Pco;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 and w Ppr;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 respectivelyfromthe previousblock.Once b Cco;b isdetermined,itsearchesforabinindex b Cpr;b inorderto transmit w Cpr;b suchthat )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(t n Pco ^ w Pco ;u n Cco ^ w Pco ;w Cco;b ;b Cco;b ;u n Cpr )]TJ/F15 11.9552 Tf 7.873 -8.079 Td [(^ w Pco ;w Cco;b ;b Cco;b ;w Cpr;b ;b Cpr;b ; 86 PAGE 87 t n Ppr ^ w Pco ; ^ w Ppr 2A n : {4 Itsets b Cco;b =1or b Cpr;b =1iftherespectivebinindexisnotfound.Itcanbeshown usingargumentssimilartothosein[64]thattheprobabilitiesoftheeventsof S C notable tondaunique b Cco;b or b Cpr;b satisfying5{3and5{4canbemadearbitrarilysmallif thefollowingholdtrue: R 0 Cco >I U Cco ; T Ppr j T Pco + 0 ; R 0 Cpr >I U Cpr ; T Ppr j T Pco ;U Cco + 0 ; where 0 > 0maybearbitrarilysmall. S C transmits x n C ^ w Pco;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ; ^ w Ppr;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ;w Cco;b ;b Cco;b ;w Cpr;b ;b Cpr;b ; Decoding: At S C :Assumethatdecodingtillblock b )]TJ/F15 11.9552 Tf 12.45 0 Td [(1hasbeensuccessful. S C declaresthat w Pco;b ;w Ppr;b = ^ w Pco ; ^ w Ppr wastransmittedinblock b ifthereexistsauniquepair ^ w Pco ; ^ w Ppr suchthat )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [(t n Pco w Pco;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;t n Ppr w Pco;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;w Ppr;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ;x n Pco )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [(w Pco;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ; ^ w Pco ; x n P w Pco;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ; ^ w Pco ;w Ppr;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ; ^ w Ppr ;v n C;b 2A n : Else,anerrorisdeclared. At D P :Theprimarydestination D P waitsuntilblock B ,andthenperformsbackward decoding.Weconsiderthedecodingprocessusingtheoutputinblock b 2f B )]TJ/F15 11.9552 Tf 12.093 0 Td [(1 ; ; 2 g Thedecodingfortherstandlastblockscanbeseenasspecialcasesoftheabove.Thus, forblock b 2f B )]TJ/F15 11.9552 Tf 12.312 0 Td [(1 ; ; 2 g ,assumingthatthedecodingforthepair w Pco;b ;w Ppr;b has beensuccessful, D P searchesforauniquepair^ w Pco ; ^ w Ppr andsometuple ^ w Cco ; ^ b Cco for w Cco;b ;b Cco;b suchthat )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(t n Pco ^ w Pco ;t n Ppr ^ w Pco ; ^ w Ppr ;x n Pco ^ w Pco ;w Pco;b ;x n P ^ w Pco ;w Pco;b ; ^ w Ppr ;w Ppr;b ; u n Cco ^ w Pco ; ^ w Cco ; ^ b Cco ;y n P;b 2A n : 87 PAGE 88 At D C :Thecognitivedestination D C alsowaitsuntilblock B ,andthenperforms backwarddecodingtojointlydecodethemessagesintendedforitandthecommonpartof theprimarymessage.Forblock b 2f B )]TJ/F15 11.9552 Tf 12.416 0 Td [(1 ; ; 2 g D C isassumedtohavesuccessfully decoded w Pco;b fromblock b +1.Withthisknowledge,itsearchesforauniquetuple ^ w Pco ; ^ w Cco ; ^ b Cco ; ^ w Cpr ; ^ b Cpr suchthat t n Pco ^ w Pco ;x n Pco ^ w Pco ;w Pco;b ;u n Cco ^ w Pco ; ^ w Cco ; ^ b Cco ; u n Cpr ^ w Pco ; ^ w Cco ; ^ b Cco ; ^ w Cpr ; ^ b Cpr ;y n C;b 2A n : ErrorAnalysis: Throughouttheanalysis,weassume,withoutlossofgenerality,thatalltransmitted messagesattheprimaryandcognitivesources,inanyblock b 2f 1 ; ;B g ,wereones. Encodingerrorsat S C :Anencodingerroroccursat S C undertwocircumstances.An erroroccursif,inblock b S C cannotndabinindex b Cco;b suchthat5{3isnotsatised for ^ w Pco =1, ^ w Ppr =1,and w Cco;b =1.Thiseventoccurswithprobability Pr 2 6 4 2 nR 0 Cco [ b Cco;b =1 )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [(T n Pco ;U n Cco ; 1 ;b Cco;b ;T n Ppr ; 1 = 2A n 3 7 5 = )]TJ/F15 11.9552 Tf 5.48 -9.684 Td [(1 )]TJ/F15 11.9552 Tf 11.955 0 Td [(Pr )]TJ/F21 11.9552 Tf 10.461 -9.684 Td [(T n Pco ;U n Cco ; 1 ;b Cco;b ;T n Ppr ; 1 2A n 2 nR 0 Cco )]TJ/F15 11.9552 Tf 5.48 -9.683 Td [(1 )]TJ/F15 11.9552 Tf 11.956 0 Td [( )]TJ/F21 11.9552 Tf 11.955 0 Td [( )]TJ/F22 7.9701 Tf 6.586 0 Td [(n [ I U Cco ; T Ppr j T Pco + 0 ] 2 nR 0 Cco exp )]TJ/F15 11.9552 Tf 9.298 0 Td [( )]TJ/F21 11.9552 Tf 11.956 0 Td [( n [ R 0 Cco )]TJ/F22 7.9701 Tf 6.586 0 Td [(I U Cco ; T Ppr j T Pco )]TJ/F22 7.9701 Tf 6.587 0 Td [( 0 ] where > 0canbearbitrarilysmall,andthelasttwoinequalitiesareduetothejoint AsymptoticEquipartitionPropertyAEP[33]andthefactthat )]TJ/F21 11.9552 Tf 13.174 0 Td [(x n e )]TJ/F22 7.9701 Tf 6.587 0 Td [(nx respectively.Clearly,theaboveprobabilitycanbemadearbitrarilysmallif R 0 Cco >I U Cco ; T Ppr j T Pco + 0 : {5 88 PAGE 89 Anotherpossibilityofanencodingerrorat S C occursif,inblock b ,itcannotndabin index b Cpr;b suchthat5{4isnotsatisedwith ^ w Pco =1, ^ w Ppr =1, w Cco;b =1, b Cco;b =1, and w Cpr;b =1.Proceedingasfortherstkindoferrorevent,itcanbeshownthatthe probabilityofthiseventcanbemadearbitrarilysmallif R 0 Cpr >I U Cpr ; T Ppr j T Pco ;U Cco + 0 : {6 Decodingerrorsat S C :FortheblockMarkovSPCencodedtransmissionfrom S P thecognitivesource S C usesitsknowledgeabouttheinformationinthepreviousblock tojointlydecode w P forthecurrentblock.Forthesuperpositionencodedcommonand privatepartsoftheprimarymessage,itcanbeshownthattheprobabilityoferrorforthis decodingstepcanbemadearbitrarilylowaslongasthefollowinghold: R Ppr I X P ; V C j T Pco ;T Ppr ;X Pco ; R Pco + R Ppr I X P ; V C j T Pco ;T Ppr : Decodingerrorsat D P :Forblock b 2f B )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ; ; 2 g ,let E ijk betheevent t n Pco i ;t n Ppr i;j ;x n Pco i;w Pco;b ;x n P i;w Pco;b ;j;w Ppr;b ;u n Cco i;k; ^ b Cco ;y n P;b 2A n ; wherein,itisassumedthatthepreviousdecodingstep,startingwithdecodingforblock B hasbeensuccessful,i.e.^ w Pco;b ; ^ w Ppr;b = w Pco;b ;w Ppr;b .Notethatweneednotconsider theprobabilityoftheeventsof D P decoding w P;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 correctly,butnot w Cco;b ;b Cco;b .Then theprobabilityoferrorat D P is P e;D P =Pr h E c 111 [ [ i;j 6 = ; 1 E ijk i Pr[ E c 111 ]+2 nR Ppr Pr[ E 121 ]+2 n R Ppr + R Cco + R 0 Cco Pr[ E 122 ] +2 n R Ppr + R Ppr + R Cco + R 0 Cco Pr[ E 222 ]{7 89 PAGE 90 Thersttermin5{7goestozerowith n duetojointAEP.Theprobabilitiesofthelast threeerroreventscanbeshowntobeupperboundedaslistedin5{8a-5{8c. Pr[ E 121 ] 2 )]TJ/F22 7.9701 Tf 6.586 0 Td [(n [ I T Ppr ;X P ; Y P ;U Cco j T Pco ;X Pco +6 ] {8a Pr[ E 122 ] 2 )]TJ/F22 7.9701 Tf 6.586 0 Td [(n [ I T Ppr ;X P ;U Cco ; Y P j T Pco ;X Pco + I U Cco ; T Ppr j T Pco +8 ] {8b Pr[ E 222 ] 2 )]TJ/F22 7.9701 Tf 6.586 0 Td [(n [ I T Pco ;T Ppr ;X P ;U Cco ; Y P + I U Cco ; T Ppr j T Pco +7 ] {8c Thus,theabovesuggeststhat,for n largeenough,^ w Pco ; ^ w Ppr = w Pco;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;w Ppr;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 witharbitrarilysmallprobabilityoferrorif R Ppr islessthanthesecondterminsidethe minimumoperatorinequation5{1a,and5{1c-5{1daresatised. Decodingerrorsat D C :Forblock b 2f B )]TJ/F15 11.9552 Tf 11.955 0 Td [(1 ; ; 2 g ,let E D C ijk denotetheevent )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(t n Pco k ;x n Pco k; 1 ;u n Cco k;i;b Cco ;u n Cpr k;i;b Cco ;j;b Cpr ;y n C;b 2A n : Then,thetotalprobabilityoferrorat D C canbeupperboundedas P e;D C Pr h E D C 111 c i +2 n R Cpr + R 0 Cpr Pr[ E D C 121 ]+2 n R Cco + R 0 Cco + R Cpr + R 0 Cpr Pr[ E D C 221 ] +2 n R Pco + R Cco + R 0 Cco + R Cpr + R 0 Cpr Pr[ E D C 222 ] : {9 Notethat,owingtothecodingstructure,if k 6 =1,thenallthetransmittedcodewords wouldbeindependentofthereceivedcodewords.Asaresult, D C needstocorrectly decode w Pco eventhoughitmaynotbeinterestedinthispartoftheprimarymessage. Again,byjointAEP,thersttermof5{9goestozerowith n .Theprobabilitiesofthe otherthreeeventscanbeupperboundedas Pr[ E D C 121 ] 2 )]TJ/F22 7.9701 Tf 6.587 0 Td [(n I U Cpr ; Y C j T Pco ;X Pco ;U Cco +6 {10 Pr[ E D C 221 ] 2 )]TJ/F22 7.9701 Tf 6.587 0 Td [(n I U Cco ;U Cpr ; Y C j T Pco ;X Pco +6 {11 Pr[ E D C 222 ] 2 )]TJ/F22 7.9701 Tf 6.587 0 Td [(n I T Pco ;X Pco ;U Cco ;U Cpr ; Y C +5 {12 90 PAGE 91 Therefore,for n largeenough,^ w Pco ; ^ w Cco ; ^ b Cco ; ^ w Cpr ; ^ b Cpr = w Pco;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ;w Cco;b ;b Cco;b ;w Cpr;b ;b Cpr;b witharbitrarilylowprobabilityoferrorif5{1e-5{1garesatised. Thus,theconstraintsontheratesasgivenin5{1a-5{1gensurethattheaverage probabilityoferroratthetwodestinationscanbedriventozeroandthus,theydescribe anachievablerateregionfortheICUC-C. Remark 5.1 TheachievableratesregiondescribedinTheorem5.1canbeexpressed explicitlyintermsof R P and R C usingFourier-Motzkinelimination.Denotetheright sidesof5{1a-5{1gas I 1 ;I 2 ; ;I 7 .ThentheachievablerateregionofTheorem5.1can bewrittenas R P min f I 2 ;I 4 g ; {13a R C min f I 6 ;I 7 g ; {13b R P + R C min f I 4 + I 5 ;I 1 + I 7 g ; {13c R P +2 R C I 3 + I 5 + I 7 : {13d Remark 5.2 TheachievablerateregiondescribedinTheorem5.1isconvexandhence,no time-sharingisrequiredtoenlargetherateregion.ThiscanbeprovedusingtheMarkov chainstructureofthecodeaswasusedin[74,Lemma5],withtherandomvariable T Pco in Theorem5.1playingarolesimilartothatof U in[74]. 5.4TheGaussianICUC-C WeapplytheresultofTheorem5.1totheGaussianICUC-C.ForGaussianchannels,withthedirectlinks'channelgainsnormalizedtounitycf.[7],etc.,wehavethe followinginput-outputrelationshipsasshowninFigure5-2: V C = g PC X P + Z S C ; {14 Y P = X P + h CP X C + Z P ; {15 Y C = h PC X P + X C + Z D C ; {16 91 PAGE 92 where Z S C ;Z P ;Z D C N ; 1arei.i.d.randomvariablesdenotingtheadditivenoise at S C D P ,and D C respectively. g PC h PC ,and h CP arepositiverealsthatdenotethe channelgainsforthelinksfrom S P to S C S P to D C ,andfrom S C to D P respectively. Also,theprimaryandcognitivesourcesaresubjectedtotheirrespectivepowerconstraints P P and P C : 1 n jj X n P jj 2 P P ; 1 n jj X n C jj 2 P C : {17 Let P ; P ; P ; P ; C ; C ; C berealnumbersintheinterval[0 ; 1]suchthat P + P + P + P 1.Also,let =1 )]TJ/F21 11.9552 Tf 12.566 0 Td [( for 2f C ; C ; C g .Weevaluatetherateregionof Theorem5.1forthecaseofGaussianchannelswiththefollowingtransmittedsignalsin block b 2f 1 ; ;B g : X P w P;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;w P;b = T Pco + X 0 Pco + T 0 Ppr + X 0 Ppr {18 X C w P;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ;w C;b = X Cco + X Cpr + r C C P C P P P T Pco + s C C P C P P P T 0 Ppr {19 where T Pco N ; P P P X 0 Pco N ; P P P T 0 Ppr N ; P P P X 0 Ppr N ; P P P X Cco N ; C C P C ,and X Cpr N ; C C P C arei.i.d.randomvariables. X Cco and X Cpr aretransmittedtocommunicatethedirtypapercodedmessages w Cco;b and w Cpr;b respectively,with U Cco and U Cpr beingthecorrespondingauxiliaryrandomvariablesas in[64]. ForthecodingschemeinTheorem5.1,itisnecessarythatglobalchannelstateinformationCSIisavailableatallnodesofthenetwork.Also,theprobabilitydistributionsof thedierentcodewordsforthetwoencoderswouldberequiredtofacilitateecientrate selection.Moreover,thedierentcodingstrategiesusedinobtainingtheachievablerate regionofTheorem5.1involvecertainassumptionsregardingtheknowledgeofthecodebookateachnode.Inparticular,cooperativerelayingoftheprimarymessageby S C ,as wellasDPCat S C ,requirestheknowledgeoftheprimarycodebookat S C .Ontheother hand,rate-splittingoftheprimarymessagerequires D C toknowtheprimarycodebook. 92 PAGE 93 Lastly,rate-splittingforthesecondarymessagerequires D P toknowthesecondarycodebook.WhereastheCSIandrateselectioninformationmaybeprovidedthroughlow-rate controlchannelsbetweenthenodes,providingcodebookknowledgeatthenon-pairing nodesmaybemorediculttoachieveinpractice.Thiscanbeseentobeespeciallytrue forthelastcaseifweconsiderthefundamentalphilosophyofcognitiveradionetworks thattheyshouldbedeployedsuchthatthealreadyexistingprimaryuser-pairsshouldbe asobliviousaspossibletotheexistenceofthesecondaryuser-pairs. Figs.5-3to5-5presentsomenumericalresultsforthedierentcodingstrategies forvariouschoicesofavailablepowersandchannelgainsasgiveninthegures.Inthese gures,CodingI"referstorate-splittingforbothmessageswithoutuseofDPCat S C thiscodingschemeisthesameasthatproposedin[71],CodingII"referstothe strategyofusingDPCat S C butwithoutanyrate-splitting,CodingIII"standsforthe schemeofusingrate-splittingforonlytheprimarymessagealongwithDPCat S C ,and nally,CodingIV"referstothecodingstrategydescribedinthebeginningofthissection i.e.Theorem5.1specializedforGaussianchannelsasoutlinedatthebeginningofthis section,andinvolvesrate-splittingforbothmessagesaswellasDPCat S C .Notethat cooperativerelayingisusedinalloftheabovecodingschemes. Fig.5-3demonstratesthatthecodingschemeof[71]isstrictlysub-optimalwhenboth destinationsexperienceweakinterferencewhereasusingDPCandcooperativeingresultin alargerrateregion.Inlightoftheabovediscussiononcodebookknowledgerequirements, itturnsoutthatforscenariosasthis,usingDPCandcooperativerelayingwithoutany rate-splittingisthebestandmostfeasiblecodingscheme.Thegainfromrate-splitting fortheprimarymessagewhen D C experiencesstronginterferenceand D P experiences weakinterferenceisevidentfromFig.5-4.Inthiscase,astheinterferenceat D P isweak, rate-splittingforthesecondarymessagedoesnotappeartoprovideanybenettowards enlargingtherateregion.Finally,Fig.5-5illustratesthegainsfromrate-splittingfor thesecondarymessagecoupledwithDPCat S C forthecaseofweakinterferenceat D C 93 PAGE 94 andstronginterferenceat D P .Moreover,theincreaseinthedierenceintheregions forCodingI"andCodingIV",as P P increasesfrom1.5to6,showsthebenetof DPCat S C whentheeectiveinterferenceat D C increaseswith P P .Fromtheseresults, itappearsthatusingcooperativerelayingwithDPCandrate-splittingoftheprimary messagemaybepracticallymoresuitablestrategiesintermsofcodebookknowledge requirementsattheprimaryuser-pair,exceptwhentheprimarydestinationexperiences stronginterference. 5.5DiscreteMemorylessChannelModelfortheICUC-HDC Tilltheprevioussectionwehavebeenassumingthatthecognitivesourcecanoperate infull-duplexmodebyperformingperfectechocancelation.Here,weremovethefullduplexassumption,andintroducethediscretememorylesschannelmodelfortheICUC withhalf-duplexandcausalityconstraintsICUC-HDC.TheICUC-HDCisdepicted inFig.5-6,whereintheprimarysourcenode S P intendstotransmitinformationtoits destinationnode D P .Acognitiveorsecondarysource-destinationpair, S C and D C wishestocommunicateaswell,with S C havingitsowninformationtotransmitto D C AsinthecaseoftheICUC-C,theprimarymessageisonlycausallyavailableat S C .To incorporatethehalf-duplexconstraintforthediscretememorylesschannelmodel,we considerasecondinputat S C S ,toindicatethestateof S C -listeningortransmitting. Withthis,thechanneltransitionprobabilityisdeterminedbythestateofthe cognitivesourceasfollows: p y P ;y C ;v C j x P ;x C ;s = 8 > > < > > : p y P ;y C ;v C j x P if s = l p y P ;y C j x P ;x C e v C if s = t {20 where e denotesanerasureat S C ,and e v C =1if v C = e and0otherwise.To incorporatethefactthat S C cannottransmitwheninthelisteningstate,werestrictthe jointprobabilitydistributionoftheinputsas p x P ;x C ;s = p x P j s = l x C p s = l + p x P ;x C j s = t p s = t ,where isthenull"symbol. 94 PAGE 95 In n channeluses,theprimarysource, S P ,hasmessage w P 2f 1 ; 2 ; ; 2 nR P g to transmitto D P ,whilethesecondarysource S C hasmessage w C 2f 1 ; 2 ; ; 2 nR C g totransmitto D C .Let X P ; X C ; S ,and V C ; Y P ; Y C betheinputandoutputalphabets respectively.Further,let S = f l;t g .Aratepair R P ;R C isachievableifthereexist anencodingfunctionfor S P X n P = f P w P ;f P : f 1 ; 2 ; ; 2 nR P g!X n P ,anda sequenceofencodingfunctionsfor S C X n C ;S n = f n C w C ;V n )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 C with X Ci ;S i = f Ci w C ;V i )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 C ;f Ci : f 1 ; 2 ; ; 2 nR C gV i )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 C !X C S ,andcorrespondingdecoding functions^ w P = g P Y n P ;g P : Y n P !f 1 ; 2 ; ; 2 nR P g and^ w C = g C Y n C ;g C : Y n C f 1 ; 2 ; ; 2 nR C g suchthattheaverageprobabilityoferror P n e =max f P n e;P ;P n e;C g! 0, where P n e;M = 1 2 n R P + R C X w P ;w C Pr[ g M Y n M 6 = w M j w P ;w C wassent]for M = P;C 5.6AnAchievableRateRegionfortheICUC-HDC First,wepresentabriefdescriptionofthecodingscheme.Inblock b 2f 1 ; ;B g S P splitsthemessage w P;b as w P;b = w P 1 ;b ;w P 2 ;b where w Pi;b = w Pico;b ;w Pipr;b for i =1 ; 2. Here,foranyblock, w P 1 isthemessagepartthat S C decodesandusesforitscognitiveand cooperativeactions,whereas w P 2 isthemessagepartthat S P directlytransmitsto D P when S C isintransmitmode.Asbefore,thesubscripts co and pr indicatethecommon andprivatemessagepartsrespectively.Whilethecommonmessagepartsaredecodedby bothdestinations,theprivatemessagepartsaredecodedonlybytheintendeddestination. w P 1 co;b isfurtherdividedintotwopartsw s;b ,thatisforwardedby S C inthenextblock usingthehelpofitsrandomlisten-transmitschedule[73],and w e;b ,thatistransmitted explicitlyusingastandardcodebook. Conditionalrate-splitting[65]andsuperpositioncodingareusedfortheabove messagesplittingstep.Forblock b 2f 1 ; ;B g S P transmits w P 1 ;b duringthe S C listenstates,anditsuperposes w P 2 ;b onto w P 1 ;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 usingblockMarkovSPCduringthe S C -transmitstates,with w P 1 ;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 actingasthe resolutioninformation for D P and D C to decode w P 1 entirelyorpartially.Inblock b S C decodes w P 1 ;b fromthereceivedsymbols duringthelisten-states.Inblock b S C splits w C;b intotwoparts w Cco;b and w Cpr;b ,and 95 PAGE 96 conditionedonthecodewordpair S;T P 1 co fortheresolutioninformationforthecommon partof w P 1 ;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ,itusesconditionalGPbinning[64]toencode w Cco;b and w Cpr;b as U Cco and U Cpr respectively,againsttheresolutioninformationfortheprivatepartof w P 1 ;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 T P 1 pr Ittransmitsacombinationoftheabovecodewords,alongwiththeresolutioninformation, duringthe S C -transmitstates. Both D P and D C waituntilthetransmissioninblock B ,andthenusebackward decoding[72]tojointlydecodebothcommonandprivatepartsofitsintendedmessage andthecommonmessagepartsfromtheinterferingtransmission.Notethat D C performsbackwarddecodingonlytodecode w P 1 co;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 inordertotakeadvantageof theblockMarkovSPCstructureusedtoencodeit.Table5-2liststherandomvariables involvedinthecodeconstructionalongwiththeirsignicance. Table5-2.DescriptionofRandomVariablesinTheorem5.2 RandomVariableDenition S Listen-transmitstatefor S C T P 1 co Resolutioninformationforcommonpartofprimary message w P 1 knownto S C T P 1 pr Resolutioninformationforprivatepartofprimary message w P 1 knownto S C X P 1 co Newinformationforcommonpartofprimarymessage w P 1 decodedby S C X P 1 pr Newinformationforprivatepartofprimarymessage w P 1 decodedby S C X P 2 co Commonpartofprimarymessage w P 2 notdecoded by S C X P 2 pr Privatepartofprimarymessage w P 2 notdecodedby S C U Cco Commonpartofsecondarymessagegeneratedby conditionalGel'fand-Pinskerbinning U Cpr PrivatepartofsecondarymessagegeneratedbyconditionalGel'fand-Pinkserbinning X P Transmittedcodewordby S P X C Transmittedcodewordby S C Let =Pr[ S = l ],and =1 )]TJ/F21 11.9552 Tf 12.021 0 Td [( .Owingtothehalf-duplexconstrainttothechannel model,werestrictthedistributionsforthecodewordsusedinthecodebookconstruction 96 PAGE 97 asfollows: p t P 1 co j s = l = t P 1 co ; {21a p t P 1 pr j t P 1 co ;s = l = t P 1 pr ; {21b p x P 2 co j t P 1 co ;s = l = x P 2 co ; {21c p x P 2 pr j x P 2 co ;t P 1 pr ;t P 1 co ;s = l = x P 2 pr ; {21d p u Cco j t P 1 co ;s = l = u Cco ; {21e p u Cpr j u Cco ;t P 1 co ;s = l = u Cpr ; {21f p x P 1 co j t P 1 co ;s = t = x P 1 co ; {21g p x P 1 pr j x P 1 co ;t P 1 pr ;t P 1 co ;s = t = x P 1 pr : {21h Theorem5.2. ForthediscretememorylessICUC-HDC,allratetuples R P ;R C ,where R P = R P 1 + R P 2 = R P 1 co + R P 1 pr + R P 2 co + R P 2 pr R P 1 co = R s + R e R C = R Cco + R Cpr withnon-negativereals R s ;R e ;R P 1 pr ;R P 2 co ;R P 2 pr ;R Cco ;R Cpr satisfying R P 1 pr I X P 1 pr ; V C j X P 1 co ;S = l {22a R P 1 I X P 1 pr ; V C j S = l {22b R P 2 pr I X P 2 pr ; Y P ;U Cco j X P 2 co ;T P 1 pr ;T P 1 co ;S = t {22c R P 2 I X P 2 pr ; Y P ;U Cco j T P 1 pr ;T P 1 co ;S = t {22d R P 2 pr + R Cco I X P 2 pr ;U Cco ; Y P j X P 2 co ;T P 1 pr ;T P 1 co ;S = t {22e R P 2 + R Cco I X P 2 pr ;U Cco ; Y P j T P 1 pr ;T P 1 co ;S = t {22f R P 1 pr + R P 2 pr I X P 1 pr ; Y P j X P 1 co ;S = l + I T P 1 pr ;X P 2 pr ; Y P ;U Cco j X P 2 co ;T P 1 co ;S = t {22g R P 1 pr + R P 2 I X P 1 pr ; Y P j X P 1 co ;S = l + I T P 1 pr ;X P 2 pr ; Y P ;U Cco j T P 1 co ;S = t {22h R P 1 pr + R P 2 pr + R Cco I X P 1 pr ; Y P j X P 1 co ;S = l 97 PAGE 98 + I T P 1 pr ;X P 2 pr ;U Cco ; Y P j X P 2 co ;T P 1 co ;S = t {22i R P 1 pr + R P 2 + R Cco I X P 1 pr ; Y P j X P 1 co ;S = l + I T P 1 pr ;X P 2 pr ;U Cco ; Y P j T P 1 co ;S = t {22j R e + R P 1 pr + R P 2 + R Cco I X P 1 pr ; Y P j S = l + I T P 1 co ;T P 1 pr ;X P 2 pr ;U Cco ; Y P j S = t {22k R P + R Cco I S ; Y P + I X P 1 pr ; Y P j S = l + I T P 1 co ;T P 1 pr ;X P 2 pr ;U Cco ; Y P j S = t {22l R Cpr [ I U Cpr ; Y C ;U Cco j X P 2 co ;T P 1 co ;S = t )]TJ/F21 11.9552 Tf 9.299 0 Td [(I U Cpr ; T P 1 pr ;U Cco j T P 1 co ;S = t ]{22m R C [ I U Cco ;U Cpr ; Y C j X P 2 co ;T P 1 co ;S = t )]TJ/F21 11.9552 Tf 9.298 0 Td [(I U Cco ;U Cpr ; T P 1 pr j T P 1 co ;S = t ]{22n R P 2 co + R Cpr [ I X P 2 co ;U Cpr ; Y C ;U Cco j T P 1 co ;S = t )]TJ/F21 11.9552 Tf 9.298 0 Td [(I U Cpr ; T P 1 pr ;U Cco j T P 1 co ;S = t ]{22o R P 2 co + R C [ I X P 2 co ;U Cco ;U Cpr ; Y C j T P 1 co ; S = t )]TJ/F21 11.9552 Tf 11.955 0 Td [(I U Cco ;U Cpr ; T P 1 pr j T P 1 co ;S = t ]{22p R e + R P 2 co + R C I X P 1 co ; Y C j S = l + [ I T P 1 co ;X P 2 co ;U Cco ;U Cpr ; Y C j S = t )]TJ/F21 11.9552 Tf 9.298 0 Td [(I U Cco ;U Cpr ; T P 1 pr j T P 1 co ;S = t ]{22q R P 1 co + R P 2 co + R C I S ; Y C + I X P 1 co ; Y C j S = l + [ I T P 1 co ;X P 2 co ;U Cco ;U Cpr ; Y C j S = t )]TJ/F21 11.9552 Tf 9.298 0 Td [(I U Cco ;U Cpr ; T P 1 pr j T P 1 co ;S = t ]{22r areachievableforsomejointdistributionthatfactorsas p s p t P 1 co j s p t P 1 pr j t P 1 co ;s p x P 1 co j t P 1 co ;s p x P 1 pr j x P 1 co ;t P 1 pr ;t P 1 co ;s 98 PAGE 99 p x P 2 co j t P 1 co ;s p x P 2 pr j x P 2 co ;t P 1 pr ;t P 1 co ;s p x P j x P 2 pr ;x P 2 co ;x P 1 pr ; x P 1 co ;t P 1 pr ;t P 1 co ;s p u Cco j t P 1 co ;s p u Cpr j u Cco ;t P 1 co ;s p x C j u Cpr ;u Cco ;t P 1 pr ;t P 1 co ;s p v C j x P ;x C ;s p y P j x P ;x C ;s p y C j x P ;x C ;s ; andsatises 5{21a 5{21h ,andforwhichtheright-handsidesof 5{22a 5{22r are non-negative. Proof. Let A n X;Y denotesetofjointly -typicalsequencesaccordingtothedistribution ofrandomvariables X;Y asinducedbythesamedistributionusedtogeneratethe codebooks.AsintheproofforTheorem5.1,thedependenceontherandomvariableswill notbestatedexplicitly,andshouldbeclearfromthecontext.Toavoidrepetition,the erroranalysisfortherandomcodingschemeisnotpresentedhere,andcanbederivedina mannersimilartotheanalysisintheproofofTheorem5.1. Codebookgeneration: Splittheprimaryandcognitiveusers'ratesas R P = R s + R e + R P 1 pr + R P 2 co + R P 2 pr ,and R C = R Cco + R Cpr respectively.Fixadistribution p s;t P 1 co ;t P 1 pr ;x P 1 co ;x P 1 pr ;x P 2 co ;x P 2 pr ;x P ;u Cco ;u Cpr ;x C asinTheorem5.2. Generate2 nR s i.i.d.codewords s n w 0 s 2S n w 0 s 2f 1 ; ; 2 nR s g ,accordingto Q n i =1 p s i Foreachcodeword s n w 0 s ,generate2 nR e conditionallyi.i.d.codewords t n P 1 co w 0 s ;w 0 e w 0 e 2f 1 ; ; 2 nR e g ,accordingto Q n i =1 p t P 1 coi j s i Foreachcodewordpair s n w 0 s ;t n P 1 co w 0 s ;w 0 e ,generate2 nR P 1 pr conditionally i.i.d.codewords t n P 1 pr w 0 s ;w 0 e ;w 0 P 1 pr w 0 P 1 pr 2f 1 ; ; 2 nR P 1 pr g ,accordingto Q n i =1 p t P 1 pri j s i ;t P 1 coi Foreachcodewordpair s n w 0 s ;t n P 1 co w 0 s ;w 0 e ,generate2 nR P 1 co conditionally i.i.d.codewords x n P 1 co w 0 s ;w 0 e ;w P 1 co w P 1 co 2f 1 ; ; 2 nR P 1 co g ,accordingto Q n i =1 p x P 1 coi j s i ;t P 1 coi Foreachcodewordtuple )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [(s n w 0 s ;t n P 1 co w 0 s ;w 0 e ;x n P 1 co w 0 s ;w 0 e ;w P 1 co ;t n P 1 pr w 0 s ;w 0 e ;w 0 P 1 pr generate2 nR P 1 pr conditionallyi.i.d.codewords x n P 1 pr )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(w 0 s ;w 0 e ;w P 1 co ;w 0 P 1 pr ;w P 1 pr w P 1 pr 2 1 ; ; 2 nR P 1 pr ,accordingto Q n i =1 p x P 1 pri j s i ;t P 1 coi ;x P 1 coi ;t P 1 pri 99 PAGE 100 Foreachcodewordpair s n w 0 s ;t n P 1 co w 0 s ;w 0 e ,generate2 nR P 2 co conditionally i.i.d.codewords x n P 2 co w 0 s ;w 0 e ;w P 2 co w P 2 co 2f 1 ; ; 2 nR P 2 co g ,accordingto Q n i =1 p x P 2 coi j s i ;t P 1 coi Foreachcodewordtuple )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(s n w 0 s ;t n P 1 co w 0 s ;w 0 e ;x n P 2 co w 0 s ;w 0 e ;w P 2 co ;t n P 1 pr w 0 s ;w 0 e ;w 0 P 1 pr generate2 nR P 2 pr conditionallyi.i.d.codewords x n P 2 pr )]TJ/F21 11.9552 Tf 5.48 -9.683 Td [(w 0 s ;w 0 e ;w P 2 co ;w 0 P 1 pr ;w P 2 pr w P 2 pr 2 1 ; ; 2 nR P 2 pr ,accordingto Q n i =1 p x P 2 pri j s i ;t P 1 coi ;x P 2 coi ;t P 1 pri Foreachcodewordpair s n w 0 s ;t n P 1 co w 0 s ;w 0 e ,generate2 n R Cco + R 0 Cco i.i.d.codewords u n Cco w 0 s ;w 0 e ;w Cco ;b Cco w Cco 2f 1 ; ; 2 nR Cco g and b Cco 2f 1 ; ; 2 nR 0 Cco g ,according to Q n i =1 p u Ccoi j s i ;t P 1 coi Foreachcodewordtuple )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(s n w 0 s ;t n P 1 co w 0 s ;w 0 e ;u n Cco )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(w 0 s ;w 0 e ;w Cco ;b Cco ,generate2 n R Cpr + R 0 Cpr i.i.d.codewords u n Cpr w 0 s ;w 0 e ;w Cco ;b Cco ;w Cpr ;b Cpr w Cpr 2 f 1 ; ; 2 nR Cpr g and b Cpr 2f 1 ; ; 2 nR 0 Cpr g ,accordingto Q n i =1 p u Cpri j s i ;t P 1 coi ;u Ccoi Generate x n P w 0 s ;w 0 e ;w 0 P 1 pr ;w P 1 co ;w P 1 pr ;w P 2 co ;w P 2 pr where x P isadeterministic functionof s;t P 1 co ;t P 1 pr ;x P 1 co ;x P 1 pr ;x P 2 co ;x P 2 pr Generate x n C w 0 s ;w 0 e ;w 0 P 1 pr ;w Cco ;b Cco ;w Cpr ;b Cpr where x C isadeterministicfunction of s;t P 1 co ;t P 1 pr ;u Cco ;u Cpr suchthat x C = if s = l Encoding: At S P : S P transmits x n P w s;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;w e;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ;w P 1 pr;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;w P 1 co;b ;w P 1 pr;b ;w P 2 co;b ;w P 2 pr;b in block b 2f 2 ; ;B )]TJ/F15 11.9552 Tf 12.032 0 Td [(1 g .Intherstblock,thereisnoresolutioninformationtotransmit, and S P transmits x n P ; 1 ; 1 ;w P 1 co; 1 ;w P 1 pr; 1 ;w P 2 co; 1 ;w P 2 pr; 1 ,whileinblock B ,ittransmits x n P w s;B )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ;w e;B )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ;w P 1 pr;B )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ; 1 ; 1 ;w P 2 co;B ;w P 2 pr;B .Notethattheactualrateforthe primarymessageis B )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 B R s + R e + R P 1 pr + R P 2 co + R P 2 pr ,butitconvergesto R P asthe numberofblocks B goestoinnity. At S C :Inblock b 2f 1 ; ;B g ,totransmit w Cco;b S C searchesforbinindex b Cco;b suchthat )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(s n ^ w s ;t n P 1 co ^ w s ; ^ w e ;u n Cco ^ w s ; ^ w e ;w Cco;b ;b Cco;b ;t n P 1 pr ^ w s ; ^ w e ; ^ w P 1 pr 2A n ; {23 where ^ w s ; ^ w e and ^ w P 1 pr are S C 'sestimatesof w s;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;w e;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 and w P 1 pr;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 respectivelyfrom thepreviousblock.Once b Cco;b isdetermined,itsearchesforabinindex b Cpr;b inorderto 100 PAGE 101 transmit w Cpr;b suchthat )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(s n ^ w s ;t n P 1 co ^ w s ; ^ w e ;u n Cco ^ w s ; ^ w e ;w Cco;b ;b Cco;b ;u n Cpr )]TJ/F15 11.9552 Tf 7.873 -8.079 Td [(^ w s ; ^ w e ;w Cco;b ;b Cco;b ;w Cpr;b ;b Cpr;b ; t n P 1 pr ^ w s ; ^ w e ; ^ w P 1 pr 2A n : {24 Itsets b Cco;b =1or b Cpr;b =1iftherespectivebinindexisnotfound.Itcanbeshown usingargumentssimilartothosein[64]thattheprobabilitiesoftheeventsof S C notable tondaunique b Cco;b or b Cpr;b satisfying5{23and5{24canbemadearbitrarilysmall ifthefollowingholdtrue: R 0 Cco > I U Cco ; T P 1 pr j T P 1 co ;S = t + 0 ; R 0 Cpr > I U Cpr ; T P 1 pr j U Cco ;T P 1 co ;S = t + 0 ; where 0 > 0maybearbitrarilysmall. S C transmits x n C w s;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;w e;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;w P 1 pr;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;w Cco;b ;b Cco;b ;w Cpr;b ;b Cpr;b Decoding: At S C :Assumethatdecodingtillblock b )]TJ/F15 11.9552 Tf 13.534 0 Td [(1hasbeensuccessful.Then,in block b S C knows w P 1 co;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 = w s;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ;w e;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 and w P 1 pr;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 .Itdeclaresthatthepair w P 1 co;b ;w P 1 pr;b = ^ w P 1 co ; ^ w P 1 pr wastransmittedinblock b ifthereexistsauniquepair ^ w P 1 co ; ^ w P 1 pr suchthat )]TJ/F21 11.9552 Tf 5.479 -9.684 Td [(s n w s;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;t n P 1 co w s;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ;w e;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ;t n P 1 pr w s;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;w e;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ; w P 1 pr;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;x n P 1 co )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(w s;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ;w e;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ; ^ w P 1 co ;x n P 1 pr w s;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ; w e;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ; ^ w P 1 co ;w P 1 pr;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ; ^ w P 1 pr ;v n C;b 2A n : Else,anerrorisdeclared.Itcanbeshownthattheprobabilityoferrorforthisdecoding stepcanbemadearbitrarilylowif5{22aand5{22baresatised. At D P :Theprimarydestination D P waitsuntilblock B ,andthenperformsbackward decoding.Weconsiderthedecodingprocessusingtheoutputinblock b 2f B )]TJ/F15 11.9552 Tf 12.093 0 Td [(1 ; ; 2 g Thedecodingfortherstandlastblockscanbeseenasspecialcasesoftheabove.Thus, 101 PAGE 102 forblock b 2f B )]TJ/F15 11.9552 Tf 12.112 0 Td [(1 ; ; 2 g ,assumingthatthedecodingforthepair w P 1 co;b ;w P 1 pr;b has beensuccessfulfromblock b +1, D P searchesforauniquetuple^ w s ; ^ w e ; ^ w P 1 pr ; ^ w P 2 co ; ^ w P 2 pr andsometuple ^ w Cco ; ^ b Cco suchthat )]TJ/F21 11.9552 Tf 5.48 -9.684 Td [(s n ^ w s ;t n P 1 co ^ w s ; ^ w e ;t n P 1 pr ^ w s ; ^ w e ; ^ w P 1 pr ;x n P 1 co ^ w s ; ^ w e ; w P 1 co;b ;x n P 1 pr ^ w s ; ^ w e ;w P 1 co;b ; ^ w P 1 pr ;w P 1 pr;b ;x n P 2 co ^ w s ; ^ w e ; ^ w P 2 co ;x n P 2 pr ^ w s ; ^ w e ; ^ w P 2 co ; ^ w P 1 pr ; ^ w P 2 pr ;u n Cco ^ w s ; ^ w e ; ^ w Cco ; ^ b Cco ;y n P;b 2A n : Theerroranalysisforthisdecodingstepshowsthat,for n largeenough, ^ w s ; ^ w e ; ^ w P 1 pr ; ^ w P 2 co ; ^ w P 2 pr = w s;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;w e;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;w P 1 pr;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;w P 2 co;b ;w P 2 pr;b witharbitrarilysmallprobabilityoferrorif5{22c-5{22laresatised. At D C :Thecognitivedestination D C alsowaitsuntilblock B ,andthenperforms backwarddecodingtojointlydecodethemessagesintendedforitandthecommonpartof theprimarymessage.Forblock b 2f B )]TJ/F15 11.9552 Tf 12.416 0 Td [(1 ; ; 2 g D C isassumedtohavesuccessfully decoded w P 1 co;b fromblock b +1.Withthisknowledge,itsearchesforauniquetuple ^ ^ w s ; ^ ^ w e ; ^ w Cco ; ^ b Cco ; ^ w Cpr ; ^ b Cpr andsome ^ ^ w P 2 co suchthat s n ^ ^ w s ;t n P 1 co ^ ^ w s ; ^ ^ w e ;x n P 1 co ^ ^ w s ; ^ ^ w e ;w P 1 co;b ;x n P 2 co ^ ^ w s ; ^ ^ w e ; ^ ^ w P 2 co ;u n Cco ^ ^ w s ; ^ ^ w e ; ^ w Cco ; ^ b Cco ; u n Cpr ^ ^ w s ; ^ ^ w e ; ^ w Cco ; ^ b Cco ; ^ w Cpr ; ^ b Cpr ;y n C;b 2A n : Again,usingthepropertiesofjointtypicality,itcanbeestablishedthat,for n large enough, ^ ^ w s ; ^ ^ w e ; ^ w Cco ; ^ b Cco ; ^ w Cpr ; ^ b Cpr = w s;b )]TJ/F20 7.9701 Tf 6.587 0 Td [(1 ;w e;b )]TJ/F20 7.9701 Tf 6.586 0 Td [(1 ;w Cco;b ;b Cco;b ;w Cpr;b ;b Cpr;b withan arbitrarilylowprobabilityoferrorif5{22m-5{22raresatised. Thus,theconstraintsontheratesasgivenin5{22a-5{22rensurethatthe averageprobabilityoferroratthetwodestinationscanbedriventozeroandthus,they describeanachievablerateregionfortheICUC-HDC. 102 PAGE 103 Remark 5.3 Accordingtotheabovecodingscheme,apartoftheprimarymessage w P 2 isnotdecodedby S C .Thisisdierentfromthenon-causalcase.As S C cannotreceive whileittransmits, S P mayimproveitsratesbytransmittingfresh"informationdirectly tothedestinationduring S C -transmitstates,therebyincreasingtheachievablerateregion. Comparedtothesituationwherein S C iscapableoffull-duplexoperation,transmittinga partofthemessagedirectlytothedestinationprovidespotentialgainsevenwhenthe S P to S C channelismuchbetterthanthedirectlinkto D P Remark 5.4 Notethatthemaximumincreaseintheachievableratesthatmayberealized byusingarandomlisten-transmitschedulefor S C is1bit[73]. Remark 5.5 TheachievablerateregiondescribedinTheorem5.2isalsoconvexcf. Remark5.2andconsequently,time-sharingisnotrequiredtoenlargetherateregion. Here,therandomvariable S inTheorem5.2playstherolesimilartothatof U in[74, Lemma5]. Remark 5.6 FortheGaussianchannelmodelwithaxedlisten-transmitschedule,the codingschemeofTheorem5.2yieldsthesamerateregionaswithatime-divisionstrategy withtheuseofGaussianparallelchannels[75],insteadofablockMarkovstructure,for thedecodingof w P 1 = w P 1 co ;w P 1 pr at D P and w P 1 co at D C .Accordingtothisstrategy, S P transmits w P 1 duringthersttime-slotwhile S C isinlisteningmode.Inthesecond time-slot,both S P and S C encodeandtransmit w P 1 asanon-causalICUC,and S P alsosuperposes w P 2 ontopof w P 1 thelatteractingastheresolutioninformationforthe destinations.Bothdestinationsdecodeonlyattheendofthesecondtime-slotandexploit theparallelGaussianchannelstructuretodecode w P 1 entirelyorpartially. 5.7TheGaussianICUC-HDC AsinSection5.4,fortheGaussianICUC-HDC,thedirectlinksforeachuser-pairis normalizedtounity, g PC isthechannelgainforthe S P S C link, h PC isthatforthe S P D C link, h CP isthatforthe S C D P link,and S C D P ,and D C areassumedto experiencei.i.d.additivewhiteGaussiannoiseAWGNofunit-variance.Asmentionedin 103 PAGE 104 Remark5.4,theincreaseintheachievablerateregionusingarandomizedlisten-transmit scheduleoverthatwithaxedscheduleisupperboundedbyonebit.Moreover,itshould benotedthat,forarandomizedlisten-transmitschedule,theoptimaldistributionfor therandomvariable X C maynotbeGaussian[73].Fortheabovemodel,weconsidera xedlisten-transmitscheduleandhavethefollowinginput-outputrelationshipsforthe ICUC-HDC: V C;l = g PC X P;l + Z S C ; {25 Y P;l = X P;l + Z P ; {26 Y C;l = h PC X P;l + Z D C ; {27 Y P;t = X P;t + h CP X C + Z P ; {28 Y C;t = h PC X P;t + X C + Z D C ; {29 where Z S C ;Z P ;Z D C N ; 1arei.i.d.randomvariablescorrespondingtotheadditive noiseat S C D P ,and D C respectively.Intheabove, X P;l and X P;t arethetransmitted signalsduringthelistenandtransmitstatesrespectively.Similarnotationisusedto describethereceivedsignalsattheconcernednodesaswell.Finally,itisassumedthat theprimaryandcognitivesourcesaresubjecttothepowerconstraintsof5{17foreach statewithinanycommunicationblock.Thismaybeinterpretedasthescenarioinwhich bothsourcesareconstrainedbytheirrespectivelisten/transmitmodepowerconstraints, insteadofanaveragepowerconstraintoverablock.Itmaybenotedthataveragepower constraints,ofasimilaravorasin[73],maybeincludedalongwiththemodepower constraintsinoursystemmodel,butisavoidedherefortheeaseofpresentationand expositionofthemainaspectsofthecodingscheme.Tosummarize,thepowerconstraints inthissectionmaybeexpressedas: 1 n l jj X n l P;l jj 2 P P ; 1 n t jj X n t P;t jj 2 P P ; 1 n t jj X n t C jj 2 P C ; {30 104 PAGE 105 where,foranychoiceofpositive n t and n l n l + n t = n isthetotalnumberofchanneluses inacommunicationblock. Next,weprovideananalyticalprooffortheinclusionoftheachievablerateregionfor theGaussianICUC-HDCpresentedin[6].Weshowthatanouterboundnotnecessarily achievabletotherateregionpresentedin[6]iscontainedinasubspaceoftheachievable rateregionofTheorem5.2.Wedescribethespecializationofthecodingschemeof Theorem5.2totheGaussiancaseinfurtherdetailinSubsection5.7.2,alongwithsome numericalexamples. 5.7.1InclusionOfCausalAchievableRegionof[6] In[6],anachievablerateregionfortheGaussianICUC-HDCwaspresented.The authorsproposedfourprotocolsandtheoverallachievablerateregion R 0 isgivenbythe convexhullofthefourrateregions[6,Theorem5].Inthissection,weshowthattherate regionofTheorem5.2, R ,contains R 0 Forthenon-causalICUC,thecontainmentoftheregionof[6,Corollary2], R DMT intheregion R D of[76,Theorem1]isclear.Itisshownin[66]that R RTD [66,Theorem 1]contains R D .Morespecically,[66]showsthat R D R out D R in RTD R RTD ,where R out D isobtainedfrom R D byremovingcertainrateconstraints,and R in RTD isobtained from R RTD byrestrictingtheinputdistributiontomatchthatfor R D .Thecodingscheme ofTheorem5.2maybespecializedtoyieldarateregionfortheICUC.Towardsthis,we set S = t w.p.1 ;X P 2 co = X P 2 pr = ,andassumethatagenieprovides S C with w P Thisgivesusanachievablerateregion R NC fortheICUC.Moreover,byrestrictingthe inputdistributiontoindependentrate-splittingandindependentbinningofthesecondary messagesasin[6,76]insteadofconditionalrate-splittingandconditionalbinningat S C itcanbeshownusinganappropriatemappingofthecodebookrandomvariablesomitted duetolackofspace,thattheresultingregion R in NC isidenticalto R in RTD ,andhence, R DMT R D R in NC 105 PAGE 106 Next,weshowthattherateregionsobtainedviaeachoftheprotocolsproposedin[6] arecontainedin R .Notethatforalltheseprotocols, w P = w P 1 ;w P = w Pco ;w Ppr ,with rates R P = R Pco + R Ppr ,etc.Tocomparethetworateregions,westartwithanalternate descriptionfortheachievablerateregionforProtocol1[6,Lemma3].Accordingto Protocol1,foranychoiceof ,theratepair R P ;R C isachievableif R P 2 log+ g PC P P +log 1+ P P 1+ P P ; {31 R 0 P ;R 0 C 2R DMT ;R C = R 0 C ;R Pco = R 0 Pco ; {32 R Ppr 2 log 1+ P P 1+ P P + R 0 Ppr ; {33 where 2 [0 ; 1]isthepowerfractionallocatedfortransmittingapartsameas in[6]of w Ppr Considertheregioncorrespondingtothexedlisten-transmitscheduleandusingparallelGaussianchannelsasinRemark5.6.Forthersttime-slot,settheinputdistribution at S P as p x P 1 co j s = l p x P 1 pr j x P 1 co ;s = l .FortheequivalentICUCduringthesecond time-slot,set X P 2 co = X P 2 pr = ,andrestricttheinputdistributiontocorrespondto independentrate-splittingandbinningasin[66,]tomatchthedistributioncorrespondingto R D .Lettheoverallrateregiontherebyobtainedbe R in 1 .Clearly, R in 1 R UsingtheresultforparallelGaussianchannels,itcanbeshownthatforanychoiceof theratepair R P ;R C isachievableif R P 2 log+ g PC P P ; R 0 P ;R 0 C 2R in NC ; {34 R C = R 0 C ;R Pco =min 2 log 1+ P P P 1+ P P P ; 2 log 1+ h PC P P P 1+ h PC P P P + R 0 Pco ; {35 R Ppr 2 log+ P P P + R 0 Ppr ; {36 where P 2 [0 ; 1]isthepowerfractionallocatedfortransmitting w Pco inthersttimeslot.Notethat,givenan value, P maybechosensuchthat 1+ P P P 1.Then, 106 PAGE 107 comparing5{31-5{33to5{34-5{36establishesthattheregioncorrespondingto Protocol1iscontainedin R in 1 TheinclusionoftherateregioncorrespondingtoProtocol2canbeeasilyprovedby consideringthesamecodingstructureandinputdistributionasusedtoobtain R in 1 ,with onefurtherrestriction-theinputdistributionat S P forthersttime-slotisgivenby p x P 1 co j s = l p x P 1 pr j s = l .Thisyieldsanachievablerateregion R in 2 R ,thathas exactlythesameboundsasthatforProtocol2,exceptthattheachievablerateregionfor theNC-CRCduringthesecondtime-slotis R in NC R DMT ,therebyprovingtheabove inclusion. TherateregionforProtocol3canbeobtainedbysetting S = t w.p.1 ;X P 1 co = X P 1 pr = T P 1 co = T P 1 pr = inTheorem5.2.Finally,theratepaircorrespondingto Protocol4maybeobtainedbyusingaxedlisten-transmitschedule,andbysetting T P 1 co = X P 1 co = X P 2 co = X P 2 pr = U Cco = U Cpr = .Asthefourrateregionsof[6] arecontainedin R ,theconvexhulloftheseregions R 0 isalsocontainedin R cf. Remark5.5. 5.7.2NumericalResults Foraxedlisten-transmitscheduleandconsideringparallelGaussianchannelsasin Remark5.6,thetransmittedsignalsinanycommunicationblock,correspondingtothe codingschemeofTheorem5.2,canbeexpressedas: X P;l w P 1 = X P 1 co + X 0 P 1 pr ; {37 X P;t w P 1 ;w P 2 = T P 1 co + X 0 P 2 co + T 0 P 1 pr + X 0 P 2 pr ; {38 X C w P 1 ;w C = X Cco + X Cpr ++ s C C P C P P P P T P 1 co + s C C P C P P P P T 0 P 1 pr ; {39 where X P 1 co N ; P P P X 0 P 1 pr N ; P P P T P 1 co N ; P P P P X 0 P 2 co N ; P P P P T 0 P 1 pr N ; P P P P X 0 P 2 pr N ; P P P P X Cco N ; C C P C and X Cpr N ; C C P C arei.i.d.randomvariables.Intheabove, P P P P C 107 PAGE 108 C C arerealnumbersintheinterval[0 ; 1]. S C usesDPCtoencode X Cco and X Cpr as U Cco = X Cco + co T P 1 pr ; {40 U Cpr = X Cpr + pr T P 1 pr ; {41 with co and pr beingnon-negativerealnumbersthatdenotethecorrelationbetweenthe knowninterference T P 1 pr andtheauxiliaryrandomvariables U Cco and U Cpr respectively, conditionedon T P 1 co .NotethataccordingtothenotationofTheorem5.2, X P;l = X P 1 pr and X P;t = X P 2 pr Inthefollowing,wepresentsomenumericalexamplestocomparetheachievablerate regioncorrespondingtothetransmissionschemeproposedin[6]tothatforTheorem5.2, withaxedlisten-transmitscheduleandspecializedforGaussianchannels.Inthese examples,thelinkbetweenthetwosourcesisassumedtobebetterthanthedirectlink, andwecomparetheHan-Kobayashirateregionfortheinterferencechannelwithout anyactivecooperationbetweentheuser-pairs,therateregionof[6],andthatforthe proposedcodingschemeinthiswork.InFig.5-7,weconsiderthescenariowhenboth interferinglinksareweakerthanthedirectlinks,whileinFig.5-8,theinterferinglink from S P to D C isstrongandthatfrom S C to D P isweak. Comparingthetwoguresshowsthattheimprovementinthequalityoftheinterferinglinkfrom S P to D C maysignicantlyincreasetheoverallrateregionforthetwo user-pairs.Also,themannerinwhichtherateregionof[6]isenlargedinbothexamples suggeststhattheeciencyoftheoverallcooperativerelayingschemeistheprimary contributortotheenlargementoftherateregion.Theadvantageofthecodingscheme adaptedfromTheorem5.2,anddescribedinequations5{37through5{41,liesinthe eectiveutilizationofthedirectlinkfortheprimaryuser-pairviathetransmissionofthe codewordscorrespondingtothemessageparts w P 2 = w P 2 co ;w P 2 pr .Thus,nothavingthe entireprimarymessagebeingdecodedandtransmittedthroughthecognitivesourcemay 108 PAGE 109 considerablyincreasetheachievablerateregion,especiallyinthedirectionoftheprimary users'rate, R P cf.Remark5.3. 5.8Summary Inthischapter,anewachievablerateregionforthediscretememorylessinterference channelwithunidirectionalcooperationICUC,whereintheprimarymessagemayonly becausallyavailableatthecognitivesource,isderived.Thecodingscheme,specialized forGaussianchannelsisalsopresentedandisusedtonumericallyevaluatedierent codingstrategiesthatareusedasbuildingblocksfortheproposedcodingscheme. Theseresultsalsodemonstratethattheproposedcodingschemesignicantlyenlarges thepreviouslyknownrateregionforvariousnetworkscenarios.Adiscretememoryless channelmodelfortheICUC-HDCwasalsopresentedinthischapter.Arandomcoding scheme,employingblockMarkovSPC,conditionalrate-splittingofprimaryandsecondary messages,conditionalbinning,andarandomizedlisten-transmitscheduleforthecognitive source,wasusedtoderiveanewachievablerateregionforthischannel.ForGaussian channels,thecontainmentofthepreviouslyknownrateregion[6]inthenewrateregion wasanalyticallyproved,andnumericalexampleswerepresentedtosupplementthe analyticalcomparison. 109 PAGE 110 Figure5-1.ThediscretememorylessICUCwithcausalityconstraint. Figure5-2.TheGaussianICUC-C. 110 PAGE 111 Figure5-3.AchievableRatesfortheGaussianICUC-C:Weakinterferenceforboth cross-links. 111 PAGE 112 Figure5-4.AchievableRatesfortheGaussianICUC-C:Stronginterferencefrom S P to D C andweakinterferencefrom S C to D P 112 PAGE 113 Figure5-5.AchievableRatesfortheGaussianICUC-C:Weakinterferencefrom S P to D C andstronginterferencefrom S C to D P 113 PAGE 114 Figure5-6.ThediscretememorylessICUC-HDC. Figure5-7.AchievableRatesfortheGaussianICUC-HDC:Weakinterferenceforboth cross-links. 114 PAGE 115 Figure5-8.AchievableRatesfortheGaussianICUC-HDC:Stronginterferencefrom S P to D C andweakinterferencefrom S C to D P 115 PAGE 116 CHAPTER6 CONCLUSIONSANDFUTUREWORK 6.1Conclusions Inthiswork,wehavestudieddierentlevelsofcooperationmanifestedinavariety ofmulti-usercommunicationsystemfromow-theoreticandinformation-theoretic perspectives.Forthesingle-source-single-destinationwirelessclusterwithdedicatedrelays, weproposedcooperativetransmissionprotocolsusingaow-theoreticapproach.This includedtheFOprotocolandthesuboptimal,butmuchsimpler,GLSprotocol.Boththe protocolsareshowntoachievetheoptimaldiversity-multiplexingtradeo,andtheGLS protocolisshowntobeaverygoodcandidateforuseinsystemswithlowcomplexity requirements.Simulationresultsfordierentclustersizes,anduniformandnon-uniform averagepowergainsindicatethattheproposedprotocolsprovidelargecodinggainsby ecientlyutilizingtheCSIavailableatallnodes,andthattheyperformmuchbetterthan otherpathselectionmethodspreviouslyproposedintheliterature,especiallyinhighdata raterequirementsituations. Wenextconsideredthetwo-userfadingMACasanexampleofamulti-sourcesystem. Weproposedagame-theoreticformulationinvolvingbargainingandmaximingames tomodeltheresourceallocationproblemanddevelopacharacterizationofcooperative behaviorforthissystemunderuncertaintyregardingtheaccuracyoftheCSIT.To improvetherobustnessofthesystem,weproposedthattheconventionalbargaining problemberelaxedsothattheusers,insteadofbeingboundtoexecutethestrategypair suggestedbythesolutiontotheconventionalbargainingproblem,mayindependently choosetheirtransmissionstrategyfromtheirrespectivesetofstrategiesdenedby themaximumdeviationparametersaboutthenominalstrategypair.Thisreducesthe dependenceofthesystemperformanceonthesolutiontothebargainingproblemwiththe possiblyinaccurateavailableCSIT.Fromthedevelopmentofthisformulation,itcan beseenthatevenintheconventionaltwo-userMAC,thereexistsacertainlevelofuser 116 PAGE 117 cooperation.Usingnumericalexamples,wedemonstratedtheeectsofuncertaintyonthe achievableaverageratesandtheimprovementinthesystemrobustnessprovidedbythe proposeddesign. TheCMAC,thatinvolvesahigherlevelofcooperationbetweentheusersintheform ofactiveforwardingofeachother'sinformation,wasstudiedinChapter4.Again,using theow-theoreticapproach,wedevelopedtwocooperativetransmissionprotocols,based onDFrelaying,forcooperativetransmissionintheCMAC.WeproposedtheOR-CMAC thatdecomposestheCMACintotwoorthogonalrelaychannels,andtheFO-CMACthat decomposestheCMACintotwobroadcastchannelsandoneMAC.Moreover,withthe assumptionoftheavailabilityofphasesynchronization,weproposedthemodicationof theMAslotsofChapter2toMAwithcommoninformationforfurtherperformancegain. Simulationresultsfordierentscenariosindicatethepotentialperformanceimprovements overpreviouslyproposedtransmissionstrategies,intermsofaverageratesandoutage probabilities. Finally,weaddressedtheproblemofcommunicatingthroughtheICUCwitha causalityconstraint.Thishelpsusavoidthesomewhatunrealisticassumptionofthe cognitivesourcehavingnon-causalknowledgeoftheprimarymessagethatisconsidered inmostrelatedworksintheliterature.Wederivedanewachievablerateregionforthe discretememorylessversionofthischannel,withanassumptionoffull-duplexoperation atthecognitivesource.WespecializedthecodingschemeforGaussianchannelsand usedittonumericallyevaluatedierentcodingstrategiesthatareusedasbuilding blocksfortheproposedcodingscheme.Theseresultsalsodemonstratethattheproposed codingschemesignicantlyenlargesthepreviouslyknownrateregionforvariousnetwork scenarios.Followingthis,weremovedtheassumptionoffull-duplexcapabilityatthe cognitivesource,andpresentedadiscretememorylesschannelmodelfortheICUC-HDC. Wedevelopedarandomcodingscheme,employingblockMarkovSPC,conditionalratesplittingofprimaryandsecondarymessages,conditionalbinning,andarandomized 117 PAGE 118 listen-transmitscheduleforthecognitivesource,toderiveanewachievablerateregionfor thischannel.ForGaussianchannels,weprovedthecontainmentofthepreviouslyknown rateregion[6]inthenewrateregionfortheGaussianICUC-HDC,anddemonstratedthis withnumericalsimulationresults. 6.2FutureDirections Theow-theoreticapproachintroducedinChapter2couldalsobeappliedtomultiusersystemsthatinvolvemorecomplexusercooperationthantheonesconsideredinthis dissertation.Forinstance,theow-theoreticapproachwouldbesuitablefortheproblem ofinformationtransmissioninacooperativerelaybroadcastchannelRBC[5],wherein onesourcenodebroadcastsinformationtotworeceivernodes,whonowactivelycooperate fullyorpartiallywitheachotherindecodingtheirrespectivemessages.Although achievablerateregionsemployingblockMarkovcodingalongwithdecode-and-forwardand estimate-and-forward[2,Theorem6]techniqueshavebeenproposedintheliterature,the appealoftheow-theoreticapproachliesinitssimplicity.Theow-theoreticapproach essentiallybreaksdowntheoriginalchannelintomuchsimplerchannelsforwhichthe capacityregionsareknownandpracticalcodingschemesthatperformclosetothe randomcodingschemehavebeenextensivelyinvestigatedintheliterature.FortheRBC, asimilartime-slottingapproachofChapter4wouldnowinvolveoneBCandtwoMA withcommoninformationtime-slots,anditwouldbeinterestingtoinvestigateastohow theperformanceoftheow-optimizedsolutioncomparestothemorecomplicatedblock MarkovmethodsindierentchannelconditionsandSNRregimes. InChapter5,wepresentednewachievablerateregionsfortheICUC-CandICUCHDCforbothdiscretememorylessandGaussianchannels.Althoughwehaveshown theinclusionofpreviouslyproposedrateregionsbothanalyticallyaswellasthrough numericalsimulations,westilldonotknowhowclosewearetothecapacityregions forthesechannels.Inthisregard,newouterboundsforthesechannelsthataretighter thantheMIMObroadcastchannelcapacityregion[77]wouldbenecessary.ForGaussian 118 PAGE 119 channels,onewaytoapproachthisproblemwouldbetoconsiderthedeterministicversion fortheICUC-Cwithoutanyrandomnessinthechannelsandmodeltherelationship betweenthedeterministicandtheirGaussiancounterpartsasusedin[16]. YetanotherresearchdirectionthatmaybepursuedinregardtotheICUC-Cis thestudyoftheroleoftransmittersideinformationatboththeprimaryandcognitive sources.InSection5.4,wedemonstratedtheinterplaybetweenthedierentextent ofcodebookknowledgeatthedierentnodesandtheeectofthedierentcoding buildingblockslikeDPC,rate-splitting,andcooperativerelayingforvariouschanneland transmitpowerconditions.Thismaybeconsideredasaspecialcaseofthestudyofthe relationshipbetweenageneralabstractionofsideinformationatthenodes,thesetof codingtechniquesthatmaybefeasible,andtheresultingachievablerateregions.One possiblewaytomodelthismaybetoconsiderthechannelshavingdierentstateswith dierentlevelsofinformationaboutthesestatesavailableatthetwosources.Related tothis,onemayalsoaskthequestionwhetheracognitivesourcewithonlysignal-level cognition,insteadofmessage-levelcognitioncanhelpinenlargingtheHan-Kobayashi regionforthetraditionaltwo-userinterferencechannel.Inotherwords,theproblemwould betodeterminethecodingstrategiesthatmaybeusedbythecognitivesourcewhen thelevelofcognition"regardingtheprimarymessageisatthesignal-levelinsteadof themessage-level,andif,theresultingachievableratescouldimproveupontheusual interferenceavoidanceorinterferencecontrolmethodsliketheinterweaveorunderlay modes[78]ofcognitiveradiooperation. 119 PAGE 120 REFERENCES [1]V.Tarokh,N.Seshadri,andA.R.Calderbank,Space-timecodesforhighdatarate wirelesscommunication:performancecriterionandcodeconstruction," IEEETrans. 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