Iterative and One-shot Conferencing in Relay Channels

Transcription

1 Iterative and One-shot onferencin in Relay hannels hris T. K. N, Ivana Maric, Andrea J. Goldsmith, Shlomo Shamai (Shitz) and Roy D. Yates Dept. of Electrical Enineerin, Stanford University, Stanford, A 95 US WINLAB, Ruters University, Piscataway, NJ 885 US Dept. of Electrical Enineerin, Technion - Israel Institute of Technoloy, Haifa, Israel {nctk,andrea}@wsl.stanford.edu, {ivanam,ryates}@winlab.ruters.edu, sshlomo@ee.technion.ac.il Abstract We compare the rates of one-shot and iterative conferencin in a cooperative Gaussian relay channel. The relay and receiver cooperate via a conference, as introduced by Willems, in which they exchane a series of communications over orthoonal links. Under one-shot conferencin, decode-and-forward () is capacity-achievin when the relay has a stron channel. On the other hand, Wyner-Ziv compress-and-forward (F) approaches the cut-set bound when the conference link capacity is lare. To contrast with one-shot conferencin, we consider a two-round iterative conference scheme; it comprises F in the first round, and in the second. When the relay has a weak channel, the iterative scheme is disadvantaeous. However, when the relay channel is stron, iterative cooperation, with optimal allocation of conferencin resources, outperforms one-shot cooperation provided that the conference link capacity is lare. When precise allocation of conferencin resources is not possible, we consider iterative cooperation with symmetric conference links, and show that the iterative scheme still surpasses one-shot cooperation, albeit under more restricted conditions. I. INTRODUTION We consider a relay network where the relay and receiver cooperate via orthoonal conference links with finite capacity. The conference cooperation model was introduced by Willems in [], in which the capacity reion was derived for a multipleaccess channel (MA) with conferencin encoders. The encoders use the links to exchane a series of communications, referred to as a conference, that allows for an iterative jointencodin of a messae. In that work, it was shown that an oneshot conference step was optimal, and iterative conferencin provides no advantae. In [] it was shown that receiver cooperation enlares the broadcast channel (B) rate reion. The achievable rates of a B with conferencin receivers were presented in []. When both receivers wish to decode a common messae, the performance ain from iterative cooperation was illustrated in [] on binary erasure channels with conferencin receivers. In particular, [], [] show that iterative cooperation outperforms one-shot cooperation in the case of a common messae for the receivers. In a relay network, however, when only one of the receivers wishes to decode the messae, it is not clear if iterative cooperation is always advantaeous. In this paper we characterize This work was supported by the US Army under MURI award W9NF- 5--6, and the ONR under award N the relative performance of iterative and one-shot cooperation. We show that one-shot cooperation is sufficient when the relay has a weak channel, but iterative cooperation can be beneficial when the relay has a stron channel, provided that the conference link capacity is lare. We also investiate conferencin rates in comparison to those of a full-duplex relay channel. The rest of the paper is oranized as follows. Section II defines the conferencin relay channel. The one-shot and iterative cooperation strateies are presented in Section III. Section IV contrasts the relative performance of one-shot and iterative cooperation, and Section V considers the iterative scheme with symmetric conference links. Then numerical examples of the cooperation rates are illustrated in Section VI, followed by conclusions in Section VII. II. SYSTEM MODEL We consider a discrete-time memoryless channel where a transmitter is communicatin with a relay and receiver, as illustrated in Fi.. The received sinals are described by y = x + n, y = x + n, () where x, y, y, n, n, R + : x is the sinal sent by the transmitter, y, y are the sinals observed by the relay and receiver, respectively, and n, n are independent zeromean circularly symmetric complex Gaussian (ZMSG) white noise with normalized unit variance. The power ain represents the relative channel strenth of the relay and receiver. The transmitter is under an averae power constraint P, i.e., E[ x ] P, where the expectation is taken over repeated channel uses. We assume a static channel, and every node has perfect channel state information (SI). There is no loss of enerality in restrictin the channel ains to be real, as the receivers can zero-phase the observed sinals. The relay and receiver cooperate by way of a conference, as defined in []. The conference links have finite capacity α and ( α), as shown in Fi., where R + is the total conference link capacity available between the receivers, and α [, ] represents the allocation of conferencin resources in each direction. In practice the conference links may be realized via orthoonal channelization with sufficiently lon codin blocks. A conference is permissible if the total

2 fra replacements x Transmitter Fi.. α Relay Receiver onferencin relay and receiver. y ( α) cardinality of the conference communications (possibly sent over multiple rounds) does not exceed that allowed by the capacity of the conference link. Formally, a K-round conference is iven by two sets of K communicatin functions {h t,,..., h t,k }, t =,. Each function h t,k causally maps the received sinal y t and the sequence of k previously received communications from the other encoder into the kth communication V t,k, where V t,k ranes over a finite alphabet V t,k. The conference is permissible if in N channel uses, N K lo ( V,k ) α, k= N y K lo ( V,k ) ( α) k= for all N sufficiently lare. The lo is base, and has units of bits per channel use. This settin can be extended to a B with cooperatin receivers, where each of the receivers decodes its own independent messae, possibly alon with a common messae both wish to decode. Our model is a special case in which only one receiver wishes to decode the messae sent by the transmitter, while the other acts as a relay. III. OOPERATION STRATEGIES For the three-terminal cooperative network shown in Fi., its capacity is not known in eneral, and the optimal cooperation stratey remains an open problem. To study the performance of the network, we consider the capacity upper bounds and achievable rates under one-shot and iterative conferencin. The cut-set bound described in [5, Thm...] provides a capacity upper bound for any eneral multiterminal network. For the achievable rates, we consider conferencin schemes based on the decode-and-forward () [6, Thm. ] and compress-and-forward (F) [6, Thm. 6] codin strateies as they form the basis of the most common cooperative codin techniques for eneral multiuser channels. A. One-shot cooperation In one-shot cooperation, we require the receiver to decode the messae sent by the transmitter after at most one round of conference communication (i.e., K = ). Since only the receiver wishes to decode the messae, we naturally set α =. Hence the conferencin relay network reduces to a standard relay channel with an orthoonal link between the relay and the receiver. One-shot conferencin is a special case of the decentralized aents model investiated in [7], when one of the aents has an infinite capacity link to the destination. The cut-set (S) bound was presented in [5], [6]. Under a Gaussian channel with conferencin relay and receiver the one-shot cooperation cut-set bound evaluates to os,s = min { lo ( + ( + )P ), lo( + P ) + }. () In the cooperation stratey [6], [8] [], transmission is done in blocks: the relay first fully decodes the transmitter s messae in one block, then in the ensuin block the relay and the transmitter cooperatively send the messae to the receiver. The achievable rate is R os, = min { lo( + P ), lo( + P ) + }. () In the F cooperation stratey [6], [9], [], [], the relay sends a compressed version of its observed sinal to the receiver. The compression is realized usin Wyner-Ziv source codin [], which exploits the correlation between the received sinal of the relay and that of the receiver. The receiver then performs maximal-ratio combinin (MR) of the compressed sinal and its own observation; the followin rate is achieved: R os,f = lo ( + P + P/( + ˆN os ) ), () where ˆN os is the variance of the Gaussian-distributed Wyner- Ziv compression noise iven by ˆN os = B. Iterative cooperation ( + )P + ( )(P + ). (5) Under iterative cooperation, we allow the relay and receiver to perform multiple rounds of conference communications (i.e., K > ) before decodin. Furthermore, the parameter α can be optimized to determine the best allocation of conferencin resources between the relay and receiver. The cut-set bound is maximized at α =, thus the iterative cooperation cut-set bound is the same as that for one-shot cooperation: i,s = os,s. Intuitively, allowin feedback from a destination node to a source node does not increase the cut-set bound; therefore, it is optimal to simply maximize the capacity of the conference link to the destination. The and F strateies have been applied to iterative cooperation [] []. With Wyner-Ziv source codin of Gaussian codewords, havin the side information at the encoder does not improve the source codin rate []. Thus we consider the followin two-round scheme: In the first round, the receiver sends its observation to the relay usin Wyner-Ziv F. Then the relay decodes the messae, and provides feedback to the receiver in the second round via. The iteration achieves the minimum of the F and rates: = max α min{ lo ( + P + P/( + ˆN i ) ), lo( + P ) + α }, where ˆN i is the Wyner-Ziv compression noise variance in the first round: ( + )P + ˆN i = ( ( α) )(P + ). (7) (6)

3 .8.5 achievin α *.6. = =. =.5 F Fi.. The best one-shot cooperation stratey as a function of and. IV. ITERATIVE VS. ONE-SHOT OOPERATION In this section we first present the one-shot cooperation stratey, then we derive the optimal allocation of conferencin resources for the iterative scheme. Their relative performance is then characterized under different channel conditions. A. Best one-shot cooperation stratey We assume the conferencin relay and receiver can choose between the and F stratey based on the channel condition and available conferencin resources. We note that is limited by the rate the relay can reliably decode, but F improves with the conference link capacity. A comparison of () and () yields that one-shot outperforms one-shot F when P (P + ) + >. (8) P In particular, meets the cut-set bound when ( (P + ) )/P. (9) Therefore, for lare relative to, the one-shot stratey is capacity-achievin. On the other hand, outside of the reion defined by (8), F achieves a hiher rate. Fi. shows the best one-shot cooperation stratey for different values of and. We set P = in the examples in this paper. B. Weak relay channel When the relay has a weak channel (i.e., ), we show that iterative cooperation is not necessary; in particular, oneshot F outperforms the iterative scheme. To prove R os,f for, it suffices to show that for all α [, ], R os,f lo ( + P + P/( + ˆN i ) ). () We rewrite the condition () as which follows from ˆN i ˆN os /( + ˆN os ( )), () ˆN i ˆN os ˆN os /( + ˆN os ( )) () when. Thus the iterative scheme provides no advantae when the relay has a weak channel. Fi.. Optimal conference link capacity allocation α for the two-round iterative cooperation scheme..5.5 One shot achievin Iterative One shot F 5 6 Fi.. The best cooperative stratey as a function of and.. Stron relay channel When the relay has a stron channel (i.e., > ), the best cooperation stratey depends on the channel condition and the available conferencin resources. First we derive the optimal conference capacity allocation α for the iterative scheme. In reion (9) where is optimal, the iterative stratey reduces to a one-shot cooperation scheme (the receiver does not communicate in the first round; the relay performs in the second round), hence α =. Outside of the reion in (9), we observe that the first expression inside of the min{ } in (6) is monotonically decreasin in α while the second is increasin. The optimal α is obtained by equatin the two expressions: α = ( ) A(A + B) A lo, () P ( + P ) where A (P +)(P +) and B P (P +P +). The optimal conference capacity link allocation α is illustrated in Fi. for selected values of. Under optimal conference link capacity allocation, the iterative scheme outperforms one-shot cooperation for lare and. Fi. shows the best one-shot or iterative cooperation stratey in different reions of and. The boundary between the iterative and the one-shot F reion is numerically computed, as the expression for the iterative rate with the optimal α is rather tedious.

4 One shot achievin One shot Iterative (α=.5).5.5 One shot > Iterative (α=.5) S bound Rate (bps) One shot F 5 6 Fi. 5. Iterative (α =.5) vs. one-shot cooperation. The best stratey is labeled for each reion. V. SYMMETRI ONFERENE LINKS In Section IV- we show that the optimal conference link capacity allocation α depends on the system SNR P, the relative channel condition, and the total conferencin resources. In the case when capacity allocation is not possible, we consider if iterative cooperation can still outperform one-shot cooperation. Specifically, we now assume that the cooperatin receivers have symmetric conference links, i.e., the conference links have equal capacity (α =.5). Under this condition the cut-set bound becomes i,s α=.5 = min { lo ( + ( + )P ), lo( + P ) + / }. Similarly, the two-round iterative scheme achieves the rate α=.5 = min { lo ( + P + P/( + ˆN i α=.5 ) ), lo( + P ) + / }. () (5) omparin (5) to the rate (), we note that now the iterative scheme overtakes one-shot only when ( / (P + ) )/P, (6) which represents a smaller reion over which iterative cooperation is beneficial. Furthermore, outside of reion (6) the one-shot rate is strictly hiher than the symmetric conference link cut-set bound (). Hence for such and, one-shot cooperation conclusively outperforms any iterative scheme with symmetric conference links. When we compare the iterative scheme to the F rate (), we note that the iterative scheme becomes superior when / P (P + ) +. (7) P The best one-shot or iterative cooperation stratey for different values of and is presented in Fi. 5. As expected, without optimal allocation of α the reion where iteration is beneficial shrinks. Nevertheless, the iterative scheme still surpasses oneshot cooperation for lare and. os,s, i,s (α=.5) i,s R (α=.5) i R os,f R os, (bps) Fi. 6. Weak relay channel ( = ). One-shot F outperforms the iterative cooperation scheme. Rate (bps) , os,s i,s.8 i,s (α=.5).6 (α=.5) R os,f R os,..5.5 (bps) Fi. 7. Stron relay channel ( = ). The best cooperation stratey depends on and. VI. AHIEVABLE RATES In this section we present numerical examples to illustrate the rates achievable by the one-shot and iterative cooperation strateies. We assume the channel has unit bandwidth, and P =. In Fi. 6 we consider the case when the relay has a weak channel ( = ). Under one-shot cooperation, fails to take advantae of the conference link, since it is limited by the rate at which the relay can decode. Moreover, as one-shot F achieves a hiher rate than the iterative scheme, iterative cooperation is disadvantaeous. When, on the other hand, the relay has a stron channel ( = ) as shown in Fi. 7, F has the worst performance. For small, the rate meets the cut-set bound; however, as increases the rate plateaus but the iterative cooperation rate continues to improve. When we restrict the iterative scheme to symmetric conference links, the iterative performance derades as expected. Nevertheless, the symmetric conference link iterative scheme still manaes to surpass one-shot cooperation as becomes lare.

5 Rate (bps).5.5 S R R F,TD SNR P T (db) Fi. 8. Performance loss from orthoonal channelization. We note that when and are very lare, the performance ap between the different cooperation strateies vanishes, as they all approach the performance limit of the cut-set bound. In addition, for lare the cut-set bound () evaluates to the sinle-input-multiple-output (SIMO) channel capacity lo( + ( + )P ), which is a capacity upper bound on the sum-rate of a B with cooperatin receivers. onsequently, electin the proper cooperation stratey is most relevant at moderate values of and, where efficient application of system power and conferencin resources is sinificant. In the system model we assume the conference channel capacity does not affect system power or bandwidth. However, if the conference links are realized via channel orthoonalization, the capacity ain from iterative cooperation could be erased by suboptimal spectral efficiency. Fi. 8 shows the cut-set bound,, and F rates [9] of a full-duplex relay channel, where =, with unit channel ain between the relay and receiver, and the transmitter and relay each under power constraint P T /. In the iterative scheme, the conference links are realized throuh equal-duration time-division (TD) amon the transmitter, relay and receiver, where each node is under power constraint P T durin its time slot. We observe that the orthoonal scheme suffers a considerable performance loss, which is aravated as the system SNR P T increases. This lare performance loss was exhibited over a wide rane of channel ains and SNRs. These results indicate that usin system power and bandwidth to perform conferencin does not provide capacity benefit. VII. SUMMARY We compare iterative and one-shot conferencin in relay channels. Under one-shot conferencin we show that outperforms F when the relay has a stron channel. In particular, when is sufficiently lare in comparison to, is capacity-achievin. By contrast, when is lare relative to, F is superior; expressly, the F rate approaches the capacity cut-set bound as increases. For the iterative conferencin scheme, we show that it underperforms one-shot F when the relay has a weak channel (i.e., ). Under a stron relay channel (i.e., > ), however, the iterative scheme outperforms one-shot cooperation when is lare. In addition, we consider iterative cooperation with symmetric conference links when precise allocation of conferencin resources is not possible. With symmetric conference links (i.e, α =.5), the performance of the iterative scheme derades as expected. Specifically, when is lare relative to we show that one-shot exceeds the cut-set bound of symmetric conference link iterative cooperation. Notwithstandin, the symmetric conference link iterative scheme still surpasses one-shot cooperation as becomes lare. We consider a two-round iterative conferencin scheme that comprises F in the first round, and in the second. Further work includes allowin partial decodin under multiple rounds (K > ) of iteration, and investiatin the optimal number of rounds of iteration. Moreover, conferencin can be applied as a boundin tool in the more eneral settin of a B with cooperatin receivers. Lettin the conferencin capacity o to infinity would yield the capacity of a multiple antenna system, while lettin it o to zero would reduce the system to a standard B. With finite conference link capacity, iterative conferencin can be applied as a cooperation stratey by the receivers. AKNOWLEDGMENT The authors would like to thank Stark Draper for helpful discussions on iterative cooperation schemes. REFERENES [] F. M. J. Willems, The discrete memoryless multiple access channel with partially cooperatin encoders, IEEE Trans. Inform. Theory, vol. 9, no., pp. 5, May 98. [] Y. Lian and V. V. Veeravalli, ooperative relay broadcast channels, IEEE Trans. Inform. Theory, submitted. [] R. Dabora and S. D. 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