Capacity Gain from Transmitter an Receiver Cooperation Chris T. K. Ng an Anrea J. Golsmith Dept. of Electrical Engineering Stanfor University, Stanfor, CA 90 Email: ngctk, anrea}@wsl.stanfor.eu arxiv:cs/00800v1 [cs.it] Aug 00 Abstract Capacity gain from transmitter an receiver cooperation are compare in a relay network where the cooperating noes are close together. When all noes have equal average transmit power along with full channel state information CSI, it is prove that transmitter cooperation outperforms receiver cooperation, whereas the opposite is true when power is optimally allocate among the noes but only receiver phase CSI is available. In aition, when the noes have equal average power with receiver phase CSI only, cooperation is shown to offer no capacity improvement over a non-cooperative scheme with the same average network power. When the system is uner optimal power allocation with full CSI, the ecoe-an-forwar transmitter cooperation rate is close to its cut-set capacity upper boun, an outperforms compress-an-forwar receiver cooperation. Moreover, it is shown that full CSI is essential in transmitter cooperation, while optimal power allocation is essential in receiver cooperation. I. INTRODUCTION In a-hoc wireless networks, cooperation among noes can be exploite to improve system performance, an the benefits of transmitter an receiver cooperation have been recently investigate by several authors. The iea of cooperative iversity was pioneere in [1], [], where the transmitters cooperate by repeating etecte symbols of other transmitters. In [] the transmitters forwar parity bits of the etecte symbols, instea of the entire message, to achieve cooperation iversity. Cooperative iversity an outage behavior was stuie in []. Multiple-antenna systems an cooperative a-hoc networks were compare in [], [6]. Information-theoretic achievable rate regions an bouns were erive in [7] [11] for channels with transmitter an/or receiver cooperation. In [1] cooperative strategies for relay networks were presente. In this paper, we consier the case in which a relay can be eploye either near the transmitter, or near the receiver. Hence unlike previous works where the channel was assume given, we treat the placement of the relay, an thus the resulting channel, as a esign parameter. Capacity improvement from cooperation is consiere uner system moels of full or partial channel state information CSI, with optimal or equal power allocation. II. SYSTEM MODEL Consier a iscrete-time aitive white Gaussian noise AWGN wireless channel. To exploit cooperation, a relay can be eploye either close to the transmitter to form a transmitter cluster, or close to the receiver to form a receiver cluster, as ge jθ Relay Transmitter e jθ e jθ1 Receiver a Transmitter cooperation Fig. 1. Transmitter e jθ1 Cooperation system moel. Relay e jθ ge jθ b Receiver cooperation Receiver illustrate in Fig. 1. In the transmitter cluster configuration, suppose the channel magnitue between the cluster an the receiver is normalize to unity, while within the cluster it is enote by g. The transmitter cooperation relay network in Fig. 1a is then escribe by y 1 = ge jθ x + n 1, y = e jθ1 x + e jθ x 1 + n, 1 where x, y, n, x 1, y 1, n 1 C, g 0, an θ 1, θ, θ [0, π]: x is the signal sent by the transmitter, y is the signal receive by the receiver, y 1, x 1 are the receive an transmitte signals of the relay, respectively, an n, n 1 are inepenent zero-mean unit-variance complex Gaussian ranom variables. Similarly, the receiver cooperation relay network in Fig. 1b is given by y 1 = e jθ x + n 1, y = e jθ1 x + ge jθ x 1 + n. The output of the relay epens causally on its past inputs, an there is an average network power constraint on the system: E [ x + x 1 ] P, where the expectation is taken over repeate channel uses. We compare the rate achieve by transmitter cooperation versus that by receiver cooperation uner ifferent operational environments. We consier two moels of CSI: i every noe has full CSI; ii only receiver phase CSI is available i.e., the relay knows θ, the receiver knows θ 1, θ, an g is assume to be known to all. In aition, we also consier two moels of power allocation: i power is optimally allocate between the transmitter an the relay, i.e., E [ x ] αp, E [ x 1 ] 1 αp, where α [0, 1] is a parameter to be optimize; ii the network is homogeneous an all noes have equal average power constraints, i.e., E [ x ] = E [ x 1 ] = P/. Power allocation in an AWGN relay network with arbitrary channel gains was treate in [9]; in this paper we only consier the case when the cooperating noes form a cluster. Combining the ifferent consierations of CSI an power allocation moels,
Case Case 1 Case Case Case Description Optimal power allocation with full CSI Equal power allocation with full CSI Optimal power allocation with receiver phase CSI Equal power allocation with receiver phase CSI TABLE I COOPERATION UNDER DIFFERENT OPERATIONAL ENVIRONMENTS Notation Description Transmitter cooperation cut-set boun Decoe-an-forwar transmitter cooperation rate Receiver cooperation cut-set boun Compress-an-forwar receiver cooperation rate Non-cooperative channel capacity TABLE II NOTATIONS FOR THE UPPER BOUNDS AND ACHIEVABLE RATES Table I enumerates the four cases uner which the benefits of transmitter an receiver cooperation are investigate in the next section. III. COOPERATION STRATEGIES The three-terminal networks shown in Fig. 1 are relay channels [1], [1], an their capacity is not known in general. The cut-set boun escribe in [1], [1] provies a capacity upper boun. Achievable rates obtaine by two coing strategies were also given in [1]. The first strategy [1, Thm. 1] has become known as along with other slightly varie nomenclature ecoe-an-forwar [], [7], [1], an the secon one [1, Thm. 6] compress-an-forwar [8], [9], [1]. In particular, it was shown in [1] that ecoean-forwar approaches capacity an achieves capacity uner certain conitions when the relay is near the transmitter, whereas compress-an-forwar is close to optimum when the relay is near the receiver. Therefore, in our analysis ecoe-an-forwar is use in transmitter cooperation, while compress-an-forwar is use in receiver cooperation. Notations for the upper bouns an achievable rates are summarize in Table II. A superscript is use, when applicable, to enote which case liste in Table I is uner consieration; e.g., Ct 1 correspons to the transmitter cut-set boun in Case 1. For comparison, represents the noncooperative channel capacity when the relay is not available an the transmitter has average power P ; hence = C1, where Cx log 1 + xp. Suppose that the transmitter is operating uner an average power constraint αp, 0 α 1, an the relay uner constraint 1 αp. Then for the transmitter cooperation configuration epicte in Fig. 1a, the cut-set boun is = max min C αg + 11 ρ, C 1 + ρ } α1 α, where ρ represents the correlation between the transmitte signals of the transmitter an the relay. With optimal power allocation in Case 1 an Case, α is to be further optimize, whereas α = 1/ in Case an Case uner equal power allocation. In the ecoe-an-forwar transmitter cooperation strategy, transmission is one in blocks: the relay first fully ecoes the transmitter s message in one block, then in the ensuing block the relay an the transmitter cooperatively sen the message to the receiver. The following rate can be achieve: = max min C αg1 ρ, C 1 + ρ α1 α }, where ρ an α carry similar interpretations as escribe above in. Note that R g t = C g 1 t for g 1, which can be use to ai the calculation of in the subsequent sections. For the receiver cooperation configuration shown in Fig. 1b, the cut-set boun is = max min C α1 ρ, C α + 1 αg + ρ } α1 αg. In the compress-an-forwar receiver cooperation strategy, the relay sens a compresse version of its observe signal to the receiver. The compression is realize using Wyner-Ziv source coing [16], which exploits the correlation between the receive signal of the relay an that of the receiver. The following rate is achievable: α1 αg = C 1 αg+α+1/p. + α 6 Likewise, in an 6 α nees to be optimize in Case 1 an Case, an α = 1/ in Case an Case. Case 1: Optimal power allocation with full CSI Consier the transmitter cooperation cut-set boun in. Recognizing the first term insie min } is a ecreasing function of ρ, while the secon one is an increasing one, the optimal ρ can be foun by equating the two terms or maximizing the lesser term if they o not equate. Next the optimal α can be calculate by setting its erivative to zero. The other upper bouns an achievable rates, unless otherwise note, can be optimize using similar techniques; thus in the following sections they will be state without repeating the analogous arguments. The transmitter cooperation cut-set boun is foun to be C 1 t = C g+1 g+, 7 with ρ = g/g +, α = g + /g +. The ecoean-forwar transmitter cooperation rate is Rt 1 = C g g+1 if g 1, Cg if g < 1, 8 with ρ = g 1/g +, α = g+/g+ if g 1, an ρ = 0, α = 1 otherwise. It can be observe that the transmitter cooperation rate Rt 1 in 8 is close to its upper boun Ct 1 in 7 when g 1. For receiver cooperation, the cut-set boun is given by C 1 r = C g+1 g+, 9
.... Rate bps Rate bps.. R r. 0 0. 1 1. Fig.. Cut-set bouns an achievable rates in Case 1. with ρ = 1/ g + g +, α = g +g+/g +g+. The expression of the optimal value α for the compressan-forwar receiver cooperation rate in 6 is complicate, an oes not facilitate straightforwar comparison of Rr 1 with the other upper bouns an achievable rates. A simpler upper boun to Rr 1, however, can be obtaine by omitting the term 1/P in the enominator in 6 as follows: Rr 1 = max C α1 αg 0 α 1 1 αg+α+1/p + α 10 < max 0 α 1 C α1 αg 1 αg+α + α R r. 11 Since the term 1 αg + α in the enominator in 10 ranges between an g, the upper boun in 11 is tight when g > an P 1. Specifically, for g >, the receiver cooperation rate upper boun is foun to be = C g g 1 1g 1 g 1 g 1g, 1 with the upper boun s optimal α = gg 1 g 1 g g+. Note that the transmitter an receiver cut-set bouns C 1 t an C 1 r are ientical. However, for > g > 1, it can be shown that the ecoe-an-forwar transmitter cooperation rate R 1 t outperforms the compress-an-forwar receiver cooperation upper boun R r. Moreover, the ecoe-an-forwar rate is close to the cut-set bouns when g ; therefore, transmitter cooperation is the preferable strategy when the system is uner optimal power allocation with full CSI. Numerical examples of the upper bouns an achievable rates are shown in Fig.. In all plots of the numerical results, we assume the channel has unit banwith, the system has an average network power constraint P = 0, an is the istance between the relay an its cooperating noe. We assume a pathloss power attenuation exponent of, an hence g = 1/. The vertical otte lines mark = 1/ an = 1, which correspon to g = an g = 1, respectively. We are intereste in capacity improvement when the cooperating noes are close together, an < 1/ or g > is the region of our main focus.. 0 0. 1 1. Fig.. Cut-set bouns an achievable rates in Case. Case : Equal power allocation with full CSI With equal power allocation, both the transmitter an the relay are uner an average power constraint of P/, an so α is set to 1/. For transmitter cooperation, the cut-set capacity upper boun is foun to be C t = C g g+1 if g 1, C 1+g if g < 1, 1 with ρ = g 1/g+1 if g 1, an ρ = 0 otherwise. Incientally, the boun Ct in 1 coincies with the transmitter cooperation rate Rt 1 in 8 obtaine in Case 1 for g 1. Next, the ecoe-an-forwar transmitter cooperation rate is given by Rt = C g 1 g if g, C g if g <, 1 with ρ = g /g if g, an ρ = 0 otherwise. Similar to Case 1, the transmitter cooperation rate Rt in 1 is close to its upper boun Ct in 1 when g 1. For receiver cooperation, the corresponing cut-set boun resolves to Cr 1+ g g = C1 if g 1, C if g < 1, 1 with ρ = 0 for g 1, an ρ = g g/ otherwise. Lastly, the compress-an-forwar receiver cooperation rate is g Rr = C g++/p + 1. 16 It can be observe that if the cooperating noes are close together such that g >, the transmitter cooperation rate R t is strictly higher than the receiver cooperation cut-set boun C r ; therefore, transmitter cooperation conclusively outperforms receiver cooperation when the system is uner equal power allocation with full CSI. Fig. illustrates the transmitter an receiver cooperation upper bouns an achievable rates. Case : Optimal power allocation with receiver phase CSI When remote phase information is not available, it was erive in [9], [1] that it is optimal to set ρ = 0 in the cutset bouns,, an the ecoe-an-forwar transmitter cooperation rate. Intuitively, with only receiver phase CSI,
.... Rate bps Rate bps.. R r. 0 0. 1 1. Fig.. Cut-set bouns an achievable rates in Case. the relay an the transmitter, being unable to realize the gain from coherent combining, resort to sening uncorrelate signals. The receiver cooperation strategy of compress-an-forwar, on the other han, i not make use of remote phase information [1], an so the receiver cooperation rate is still given by 6 with the power allocation parameter α optimally chosen. Uner the transmitter cooperation configuration, the cut-set boun is foun to be C t = C1, 17 where α is any value in the range [1/g + 1, 1]. When the relay is close to the transmitter g 1, the ecoe-anforwar strategy is capacity achieving, as reporte in [1]. Specifically, the transmitter cooperation rate is given by Rt = C1 if g 1, Cg if g < 1, 18 where α is any value in the range [1/g, 1] if g 1, an α = 1 otherwise. For receiver cooperation, the cut-set boun is Cr = C g g+1 if g 1, C1 if g < 1, 19 where α = g/g+1 if g > 1, α = 1 if g < 1, an α is any value in the range [g/g + 1, 1] if g = 1. Since compressan-forwar oes not require remote phase information, the receiver cooperation rate is the same as 10 given in Case 1: R r = R 1 r. Note that the argument insie C in 10 is 1 when α = 1, an hence R r C1. In contrast to Case, the receiver cooperation rate R r in equals or outperforms the transmitter cooperation cut-set boun C t ; consequently receiver cooperation is the superior strategy when the system is uner optimal power allocation with only receiver phase CSI. Numerical examples of the upper bouns an achievable rates are shown in Fig.. Case : Equal power allocation with receiver phase CSI With equal power allocation, α is set to 1/. With only receiver phase CSI, similar to Case, ρ = 0 is optimal for the cut-set bouns an ecoe-an-forwar rate. Therefore, in. 0 0. 1 1. Fig.. Cut-set bouns an achievable rates in Case. this case no optimization is necessary, an the bouns an achievable rates can be reaily evaluate. For transmitter cooperation, the cut-set boun an the ecoe-an-forwar rate, respectively, are Ct = C1 if g 1, C 1+g R t = C1 if g, if g < 1, 0 C g if g <. 1 For receiver cooperation, the cut-set boun is Cr = C1 if g 1, C 1+g if g < 1, an the compress-an-forwar rate is the same as 16 in Case : Rr = Rr. Parallel to Case 1, the transmitter an receiver cooperation cut-set bouns Ct an Cr are ientical. Note that the noncooperative capacity meets the cut-set bouns when g 1, an even beats the bouns when g < 1. Hence it can be conclue cooperation offers no capacity improvement when the system is uner equal power allocation with only receiver phase CSI. Numerical examples are plotte in Fig.. IV. IMPLEMENTATION STRATEGIES In the previous section, for each given operational environment we erive the most avantageous cooperation strategy. The available moe of cooperation is sometimes ictate by practical system constraints, however. For instance, in a wireless sensor network collecting measurements for a single remote base station, only transmitter cooperation is possible. In this section, for a given transmitter or receiver cluster, the trae-off between cooperation capacity gain an implementation complexity is investigate. The upper bouns an achievable rates from the previous section are summarize, an orere, in Table III: the rate of an upper row is at least as high as that of a lower one. It is assume that the cooperating noes are close together such that g >. The transmitter cooperation rates are plotte in Fig. 6. It can be observe that optimal power allocation contributes only marginal aitional capacity gain over equal power allocation, while having full CSI is essential to achieving any cooperative capacity gain. Accoringly, in transmitter
Cooperation Scheme C 1 t, C1 r R 1 t,, C r R t R r R 1 r, R r, C r, C t, R t,,, R r, Rate bps...8.6...8 Rate C g+1 g+ C g g+1 g 1 C g g g 1 1g 1 g 1 C g 1g max C α1 αg 0 α 1 1 αg+α+1/p + α C1 g C g++/p + 1 TABLE III COOPERATION RATES COMPARISON C 1 R 1 C R C, C, R, R 0 0.1 0. 0. 0. 0. 0.6 0.7 Fig. 6. Transmitter Cooperation cooperation, homogeneous noes with common battery an amplifier specifications can be employe to simplify network eployment, but synchronous-carrier shoul be consiere necessary. On the other han, in receiver cooperation, the compressan-forwar scheme oes not require full CSI, but optimal power allocation is crucial in attaining cooperative capacity gain, as illustrate in Fig. 7. When remote phase information is not utilize i.e., ρ = 0, as note in [7], carrier-level synchronization is not require between the relay an the transmitter; implementation complexity is thus significantly reuce. It is important, however, to allow for the network noes to have ifferent power requirements an power allocation be optimize among them. V. CONCLUSION We have stuie the capacity improvement from transmitter an receiver cooperation when the cooperating noes form a cluster in a relay network. It was shown that electing the proper cooperation strategy base on the operational environment is a key factor in realizing the benefits of cooperation in an a-hoc wireless network. When full CSI is available, transmitter cooperation is the preferable strategy. On the other han, when remote phase information is not available but power can be optimally allocate, the superior strategy is receiver cooperation. Finally, when the system is uner equal Rate bps...8.6...8 C 1 C R 1, R C, C R, R 0 0.1 0. 0. 0. 0. 0.6 0.7 Fig. 7. Receiver Cooperation power allocation with receiver phase CSI only, cooperation offers no capacity improvement over a non-cooperative singletransmitter single-receiver channel uner the same average network power constraint. REFERENCES [1] A. Senonaris, E. Erkip, an B. Aazhang, User cooperation iversity Part I: System escription, IEEE Trans. Commun., vol. 1, no. 11, pp. 197 198, Nov. 00. [], User cooperation iversity Part II: Implementation aspects an performance analysis, IEEE Trans. Commun., vol. 1, no. 11, pp. 199 198, Nov. 00. [] T. E. Hunter an A. Nosratinia, Cooperation iversity through coing, in Proc. IEEE Int. Symp. Inform. Theory, 00. [] J. N. Laneman, D. N. C. Tse, an G. W. Wornell, Cooperative iversity in wireless networks: Efficient protocols an outage behavior, IEEE Trans. Inform. Theory, vol. 0, no. 1, pp. 06 080, Dec. 00. [] N. Jinal, U. Mitra, an A. J. Golsmith, Capacity of a-hoc networks with noe cooperation, in Proc. IEEE Int. Symp. Inform. Theory, 00, also in preparation for IEEE Trans. Inform. Theory. [6] C. T. K. Ng an A. J. Golsmith, Transmitter cooperation in a-hoc wireless networks: Does irty-payer coing beat relaying? in Proc. IEEE Inform. Theory Workshop, San Antonio, Texas, Oct. 00. [7] A. Host-Masen, Capacity bouns for cooperative iversity, IEEE Trans. Inform. Theory, submitte for publication. [Online]. Available: http://www-ee.eng.hawaii.eu/ masen/papers [8] M. A. Khojastepour, A. Sabharwal, an B. Aazhang, Improve achievable rates for user cooperation an relay channels, in Proc. IEEE Int. Symp. Inform. Theory, 00. [9] A. Host-Masen an J. Zhang, Capacity bouns an power allocation for wireless relay channel, IEEE Trans. Inform. Theory, submitte for publication. [Online]. Available: http://www-ee.eng.hawaii.eu/ masen/papers [10] A. Host-Masen, On the achievable rate for receiver cooperation in a-hoc networks, in Proc. IEEE Int. Symp. Inform. Theory, 00. [11], A new achievable rate for cooperative iversity base on generalize writing on irty paper, in Proc. IEEE Int. Symp. Inform. Theory, 00. [1] G. Kramer, M. Gastpar, an P. Gupta, Cooperative strategies an capacity theorems for relay networks, IEEE Trans. Inform. Theory, Feb. 00, to be publishe. [Online]. Available: http://cm.bell-labs.com/cm/ms/who/gkr/pub.html [1] E. C. van er Meulen, Three-terminal communication channels, Av. Appl. Prob., vol., pp. 10 1, 1971. [1] T. M. Cover an A. A. El Gamal, Capacity theorems for the relay channel, IEEE Trans. Inform. Theory, vol., no., 1979. [1] T. M. Cover an J. A. Thomas, Elements of Information Theory. Wiley- Interscience, 1991. [16] A. D. Wyner an J. Ziv, The rate-istortion function for source coing with sie information at the ecoer, IEEE Trans. Inform. Theory, vol., no. 1, pp. 1 10, Jan. 1976.