Two-Way Denoise-And-Forward Relaying With Non-Coherent Differential Modulation

Transcription

1 Two-Way Denoise-And-Forward Relaying With Non-Coherent Differential Modulation Wei Guan, K. J. Ray Liu Department of Electrical and Computer Engineering University of Maryland, College ark, MD {wguan, Abstract This work focuses on a two-way denoise-andforward relaying system using non-coherent Differential Binary hase-shift Keying DBSK modulation. The relay denoising function and source decoders are designed using Maximum Likelihood ML principles. As the ML denoising function is hard to manipulate, it is approximated as a multi-user detector followed by a physical layer network coding encoder, based on which the closed-form relay decoding error is obtained. It is further shown that the ML source decoder is actually equivalent to the typical DBSK decoder to the relay-source channel, and the exact end-to-end Bit Error Rate BER is derived then. A power allocation problem is also formulated to minimize the average BER at high Signal-to-Noise Ratio SNR. It is shown that the optimal source power is inversely proportional to the square root of the channel gain of the source-relay channel, and the optimal relay power decreases with SNR. I. INTRODUCTION Cooperative communications, which can provide distributed spatial diversity and make a more efficient use of transmitted power [], have gained a lot of research interests recently. The two widely discussed relaying protocols are Amplify-and- Forward AF and Decode-and-Forward DF [], in which the relay node usually works in a half-duplex way. The spectral efficiency is thus low, as two time phases are required to deliver only one information unit, which introduces a pre-log factor on the spectral efficiency [3]. The Two-Way Relaying TWR system are thus proposed to fully recover the rate loss resulted from half-duplexing, as only time phases, i.e., Broadcasting BC phase and Multiple Access MA phase, are required to assist the communication on both directions. Early work on AF-TWR and DF-TWR can be found in [3], which shows great improvements on sumrates. Later, [4] proposes a new Denoise-and-Forward DNF protocol, and a similar scheme called hysical-layer Network Coding LNC is proposed in [5], the condition to guarantee one-to-one mapping is also given. Later work [6] shows that DNF-TWR has higher sum-rates than AF-TWR and DF-TWR, and [7] derives the closed-form BER of DNF- TWR with coherent BSK modulation. While TWR opens a door to improve spectral efficiency, most of the existing work [4]-[8] assume that the terminals have full knowledge of Channel State Information CSI, which is actually hard to acquire in a fast-fading environment []. In a limited number of literatures about TWR using noncoherent modulation, [9] designs the non-coherent decoder for minimum-shift keying signals and validates the throughput gain on a software radio testbed; [0] gives the symbol error rate for AF-TWR with relay selection; and [] designs a set of non-coherent decoders for both AF-TWR and DNF-TWR using differential modulation. As summarized above, DNF-TWR with non-coherent modulation can benefit from both the high spectral efficiency and reduced channel estimation overhead. However, to the best of our knowledge, the performance analysis and resource allocation scheme for such system is still an open problem, which motivates this work. Specifically, we focus on a DNF- TWR system using non-coherent Differential BSK DBSK modulation. We first derive the relay denoising function and source decoder using Maximum Likelihood ML principles, and then proceed to analyze the corresponding decoding errors. As it is hard to manipulate the ML denoising function directly, we approximate it as a Multi-User Detector MUD followed by a LNC encoder and obtain the closed-form relay decoding error. Next, we derive the exact end-to-end BER after showing the equivalence between the ML source decoder and the typical DBSK decoder to the relay-source channel. Finally, we investigate the power allocation problem so as to minimize the average system BER using asymptotic analysis. Simulations justify our results. Notations: Boldface lowercase letter a and boldface uppercase letter A represent vector in column form and matrix, respectively. a and A represent the Euclidean norm of a vector a and the determinant of a square matrix A, respectively., T and H stand for conjugate, transpose and conjugate transpose, respectively. We shall use abbreviation i.i.d. for independent and identically distributed, and denote Z CN μ, σ as a circularly symmetric complex Gaussian random variable Z with i.i.d. real part and imaginary part N μ, σ. We define signx= if x>0 and 0 otherwise. Finally, the probability of an event A and the robability Density Function DF of a random variable Z are denoted by A and fz, respectively. II. SYSTEM MODEL Consider a narrow-band DNF-TWR system, the two sources S and S want to exchange information through a single relay node. At the beginning of the MA phase, S i for i=, first generates a sequence of i.i.d uncoded BSK symbols b i n {, } of length L, n=,,..., L is //$ IEEE

2 Σ bn=,b n= Σ bn=,b n= Σ bn=,b n= Σ bn=,b n= = Σ,r = + +I + + Î = Σ,r = + +I + Î = Σ 3,r = + +I + Î = Σ 4,r = + +I + Î 3 the symbol index. These raw symbols are then re-encoded through differential modulation, i.e., x i n=x i n b i n for n=,,..., L with x i 0= being the reference symbol. The two sources then send the whole block of differentially encoded symbols simultaneously to the relay during MA phase. To facilitate demonstrations, we define a sequence of auxiliary symbols bn=b n b n {, } for n=,,..., L to indicate whether the two raw BSK symbols have the same signs or not. Note that because each source knows its own symbol, this common information bn is sufficient for both sources to decode the symbol from the other end. At the end of MA phase, the nth symbol received at the relay node is then yn = s h MA x n+ s h MA x n+w MA n, si =α i is the transmitted power of S i, is the total power and α i [0, ] stands for the corresponding power ratio. h MA i CN 0,σi is the channel coefficient from the ith source to the relay during MA phase, σi is the channel gain. Here we assume that the channels remain unchanged within one block of length L+; however, no terminals know such CSI so as to eliminate the channel estimation overhead. Finally, w MA n CN 0, is the Additive White Gaussian Noise AWGN. With DNF protocol[4][5], the relay just maps the nth receive symbol to another BSK symbol ˆb r n {, } that can be used by each source to uniquely decode the symbol transmitted from the other end. Here ˆb r n {, } can be regarded as an estimate of the auxiliary symbol b n, so the relay denoising function is actually equivalent to the decoder for bn. Asno CSI is available, we use the single-symbol ML decoder similar to that proposed in [] throughout this work, i.e., ˆbr n = arg max bn {,} f yn bn, yn =yn,yn T is the vector of two consecutive received symbols. It is easy to show that given b n and b n, yn bn,b n CN 0, Σ bn,b n, the conditional covariance matrices are given in 3 on the top of this page. Here i = s i σ i =α i σi is the channel SNR from the ith source to the relay, = is defined as the system SNR, and I =[, 0; 0, ] and Î =[0, ;, 0] are two constant matrices. Based on the law of total probability, the conditional DF of yn can be expressed as f yn bn = f yn b n,b n. b n b n=bn 4 After some manipulations, we have ˆbr n =sign ln lrf yn bn, 5 lrf yn bn = g yn, Σ,r+g yn, Σ,r 6 g yn, Σ 3,r +gyn, Σ 4,r is the Likelihood Ratio Function LRF of yn, and g y, Σ = π Σ exp y H Σ y 7 is the DF of y CN 0, Σ. After decoding, the relay re-encodes L {ˆbr n} into tn=tn ˆb r n for n= n=,,..., L through differential modulation with t0=0 being the reference symbol. During BC phase, the relay broadcasts the symbols tn to the source nodes. The received signal at S i is r i n = r h BC i t k n+wi BC n, n=0,,..., L, 8 r =β is the relay power and β [0, ] is the corresponding power ratio. h BC i CN 0,σi is the channel coefficient from the relay to S i during BC phase, and we assume h BC i and h MA i are independent but have the same channel gain, which is determined by the distance between two terminals. wi BC n CN 0, is the corresponding AWGN. Each source only needs to detect bn to decode the symbol from the other end. For example, if the decoded symbol for bn at S is ˆb s n=, then b n can be decoded as ˆb,s n=b n, otherwise ˆb,s n= b n if ˆb s n=. AgainweassumeS i uses the single-symbol ML decoder, i.e., ˆbsi n = arg max f r in bn, 9 bn {,} r i n= r i n,r i n T comprises two consecutive received symbols, and r i n ˆbrn C, Σˆbrn,s i with { Σˆbrn=,s i = Σ,si = i +I + i Î, Σˆbrn=,s i = Σ,si = i +I i Î 0 i = rσ i =βσi is the channel SNR from the relay to S i. Now we can rewrite the joint DF in 9 as f r i n bn = f r i n ˆb r n ˆbr n bn, ˆbrn {,} we use the law of total probability and the fact r i n is independent with bn conditioned on ˆb r n. Based on, the ML source decoder 9 can be simplified to ˆbsi n =sign ln lrf r i n bn,

3 lrf yn bn = Σ cosh 3,r Σ,r cosh lrf r i n bn = g r in, Σ,si M,r +g r i n, Σ,si M,r g r i n, Σ,si F,r + g r i n, Σ,si F,r + Σ,r y H nî yn y Σ H nî 3,r yn exp Σ,r Σ 3,r + + yn Σ,r Σ 3,r 3 7 lrf r i n bn given in 3 on the top of this page is the LRF of r i n conditioned on bn, and M,r = ˆbr n = bn = = lrf yn bn bn =, 4 F,r = ˆbr n = bn = = lrf yn bn > bn = 5 are two kinds of conditional decoding error at the relay. The calculation of these two terms is postponed to the next section. Note that as both the relay decoder and source decoder 9 depend only on the second-order statistics of all channels, which remain unchanged over time, the whole system can benefit from a great reduction on channel estimation overheads. III. ERFORMANCE ANALYSIS A. Relay Decoding Error By use of the law of total probability, we can write the relay decoding error as = ˆbr n bn e,r = M,r + F,r, 6 M,r and F,r are two kinds of conditional decoding error defined in 4 and 5, and both of them are related with lrf yn bn. After substituting 7 into 6 and doing some manipulations, we have 7 on the top of this page, coshx= ex +e x is the hyperbolic cosine function. As it is really hard to analyze the error probability based on the above LRF, we use the following approximation coshx max ex,e x = e x, 8 which is quite tight when x is not too small. After such approximation, only exponential terms are left with the exponent being a quadratic form of yn, which is analytically tractable. After substituting 8 back into 7, we will arrive at lrf yn bn max g yn, Σ,r,gyn, Σ,r max g yn, Σ 3,r,gyn, Σ 4,r. 9 Now if we use 9 instead in 5, it is easy to see that this suboptimal decoder is actually a MUD ˆb,r n, ˆb,r n = arg max b in {,} f yn b n,b n 0 followed by a LNC encoder ˆb r n=ˆb,r n ˆb,r n.that is, the relay first jointly decodes the BSK symbols b n and b n, and then maps the decoded symbols to a single BSK symbol ˆb r n as an estimate of the indicator symbol bn. As we shall see in simulations, this suboptimal relay decoder works almost as well as the ML decoder 5 in all cases. After some manipulations, we can show that M,r = h u,u,a,b, th, F,r = h u 3,u 4,a,b, th, u = N, u =, u 3 = u 4 = N, and 0 + +, a = , 3 b = min, + +, th = + + +,k +, 5 h t,t,a,b,= 4abt t exp t+t a ln a t t b t + t. 6 Finally, plugging and back into 6 leads to the closed-form relay decoding error. B. Source Decoding Error After some manipulations, the source decoder can be reduced to ˆbsi n =sign ln lrf r i n bn = = sign ln lrf r i n ˆb r n ˆbr,si n,7 lrf r i n ˆb r n = g r in, Σ,si g r i n, Σ,si 8 is the LRF of r i n conditioned on ˆb r n. Note that the decoder on the second line of 7 is actually a typical noncoherent DBSK decoder [, Eqn.4-4-3] to the relaysource channel, whose output ˆb r,si n is an estimate of the decoded symbol ˆb r n at the relay. Using such equivalence, we can write the source decoding error as ˆbsi n bn = ˆbr,si n bn = e,si = M,s i + F,si, 9

4 M,si F,si = ˆbr,si n = bn =, 30 = ˆbr,si n = bn = 3 are two kinds of conditional decoding error at S i, and we use the relation ˆb si n=ˆb r,si n. After expanding 30 by use of the law of total probability, we have M,si = D,si M,r + D,si M,r, 3 we use the fact that ˆb r,si n is independent of bn conditioned on ˆb r n, and that the two kinds of conditional decoding error of a typical non-coherent DBSK decoder are equal and are given by [, Eqn.4-4-6] ˆbr,si n = ˆb r n = = ˆbr,si n = ˆb r n = In a similar way, we can derive = D,si = i F,si = D,si F,r + D,si F,r. 34 lugging 3 and 34 back into 9 we have e,si = D,si e,r + D,si e,r, 35 which is the end-to-end BER at S i. IV. OWER ALLOCATION Now we are about to investigate the power allocation among the two sources and the single relay so as to minimize the average BER, which can be formulated as min e = e,s + e,s s.t. α + α + β =, 0 α,α,β. 36 However, it is hard to manipulate the exact BER expression 35 directly, and the optimal solution can only be derived through exhaustive search. In order to obtain one simple closed-form solution, we examine the asymptotic BER at high SNRs i.e., and have M,r cm,r,c M,r = minα σ,ασ F,r df,r ln, = ασ +ασ. 37 α α σ σ D,si qd,s i,q D,si =,i=, βσi After plugging these approximations back into 35, we have c M,r + ln, 38 e + q D,s + q D,s we neglect the higher-order terms. There are several observations here. Firstly, it is easy to see that the BER is dominated by F,r, which scales as ln at high SNRs. Therefore, more power should be allocated to the sources in order to reduce the relay decoding error. Secondly, the BER of the direct transmission with non-coherent DBSK modulation scales as [, Eqn.4-4-8], which decreases faster than the dominant error term F,r at high SNRs. In other words, the DNF-TWR is comparatively not preferred than direct transmission when SNR is increasing. Finally, it can be observed that F,r > M,r when source power is fixed and SNR is sufficiently high. This is because it is relatively easier to decode bn when the two source symbols have the same signs, in which case the two consecutive observations yn and yn would have similar envelopes at high SNRs. Note that the first two terms in 38 depend only on α and α while the last two terms depend only on β. So 36 can be resolved by two steps. In the first step, we fix β and seek to find the optimal source power, i.e., min c M,r + ln s.t. α + α = β, ln 0 α,α β. 39 we neglect the term c M,r because it is much smaller than ln at high SNRs. Note that the function φ x =x ln x is increasing when x<e, which is the case for sufficiently large. Therefore, it is equivalent to minimizing instead, whose optimizer is { α opt σ = β σ +σ α opt σ = β. 40 σ +σ Clearly, the optimal source power is inversely proportional to the square root of the channel gain of the corresponding source-relay channel. That is, more power should be allocated to the source that is far away from the relay, otherwise its signal would be shadowed by that from the other end during MA phase, which increases the relay decoding error. Therefore, the above source power allocation scheme actually provides an elegant way to resolve the near-far problem. Next, if we plug 40 into 38, it leads to an optimization problem that only involves the relay power coefficient β, i.e., min η = η β + η,s.t.0 β, 4 β σ + σ 4σ σ min σ,σ + σ + σ 4σ σ ln σ σ σ + σ, 4 η = σ + σ 4σ. 43 σ Note that we neglect the term β within the log function in 4 when deriving the objective function in 4, as it is generally much smaller than at high SNRs. The optimizer of 4 can be easily derived as β opt = η η + η. 44

5 0 0 d r :d r =0.: d r :d r =0.: BER 0 - d r :d r =0.5:0.5 ML Suboptimal T heoretical A pproximation SNR db BER 0 - Optimal Suboptimal Equal d r :d r =0.4: SNR db Fig.. BER performances versus SNR. Fig.. BER performances with power allocation versus SNR. It can be shown that β opt is a decreasing function with SNR, which coincides with our previous analysis that more power should be allocated to the source as SNR is increasing. V. NUMERICAL RESULTS In this section, we shall present some simulation results for the considered system. We use the path loss model σ = d 4, σ is the channel gain and d is the distance between two terminals. For simplicity, we normalize the distance between two sources to, and we always place the relays on the line connecting two sources. In all cases, BER refers to the average decoding error at source and source. Without special explanation, the transmitted power is always equally split among all terminals. We first examine the BER performance in Fig., d,r and d,r are the distances between the relay and two sources, respectively. We compare the simulated BER of different relay decoders with the theoretical results. The suboptimal relay decoder refers to the MUD followed by a LNC encoder. It can be observed that there is almost no difference between the ML decoder and the suboptimal one, and both of them coincide with our theoretical results. Besides, the asymptotic BER is tight when SNR is sufficiently high, e.g., when 5dB for d,r :d,r =0.:0.8 and when 5dB for d,r :d,r =0.5:0.5. The tightness for the latter case is due to the high channel gains of both of the two source-relay channels, which makes it easier to satisfy the high SNR assumption. Then in Fig., we proceed to study the performance gain of power allocation. The optimal scheme is found through exhaustive search, and the suboptimal one refers to that derived through asymptotic analysis. Compared with equal power allocation, about db SNR gain can be observed in Fig. when d,r :d,r =0.:0.9. Such performance gain is diminishing as the relay moves to the halfway between two sources, in which case the equal power allocation is near-optimal. VI. CONCLUSION AND FUTURE WORK In this work, we have analyzed the BER performances of the DNF-TWR system using non-coherent DBSK modulation. A near-optimal power allocation scheme is also derived based on the asymptotic analysis at high SNRs. Future work may focus on the system with multiple relays. One may also investigate the denoising function design using higher-order non-coherent modulations. REFERENCES [] K. J. R. Liu, A. K. Sadek, W. Su, and A. Kwasinski, Cooperative Communications and Networking, Cambridge Univ. ress, 008. [] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, Cooperative Diversity in Wireless Networks: Efficient rotocols and Outage Behavior, IEEE Trans. Inf. Theory, vol. 50, no., pp , Dec [3] B. Rankov and A. Wittneben, Spectral Efficient rotocols for Half- Duplex Fading Relay Channels, IEEE J. Sel. Areas Commun., vol. 5, no., pp , Feb [4]. opovski and H. Yomo, The Anti-ackets Can Increase the Achievable Throughput of a Wireless Multi-Hop Network, in roc. IEEE International Conference on Communication ICC 006, Istanbul, Turkey, pp , Jun [5] S. Zhang, S. C. Liew, and.. Lam, hysical-layer Network Coding, in ACM MOBICOM 006, Los Angeles, Sept [6]. opovski and H. Yomo, hysical Network Coding in Two-Way Wireless Relay Channels, in roc. IEEE International Conference on Communication ICC 007, Glasgow, Scotland, pp , June, 007. [7] E. C. Y. eh, Y.-C. Liang, and Y. L. Guan, ower Control for hysical-layer Network Coding in Fading Environments, in roc. IEEE ersonal, Indoor and Mobile Radio Commun. IMRC, Cannes, France, pp. -5, Sep.008. [8]. opovski and H. Yomo, Wireless Network Coding by Amplify-and- Forward for Bi-directional Traffic Flows, IEEE Commun. Lett., Vol., No., pp. 6-9, Jan [9] S. Katti, S. Gollakota, and D. Katabi, Embracing wireless interference: Analog network coding, in ACM SIGCOMM 007, Aug [0] L. Y. Song, G. Hong, B. L. Jiao, and M. Debbah, Joint Relay Selection and Analog Network Coding Using Differential Modulation in Two-Way Relay Channels, IEEE Trans. Veh. Technol., vol. 59, no. 6, pp , July 00. [] T. Cui, F. F. Gao, and C. Tellambura, Differential Modulation for Two- Way Wireless Communications: A erspective of Differential Network Coding at the hysical Layer, IEEE Trans. Commun., vol. 57, no. 0, pp , Oct [] J. roakis, Digital Communications, 4th ed. New York: McGraw-Hill, 00.