EUSIPCO

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1 EUSIPCO SINGLE-CARRIER EQUALIZATION AND DISTRIBUTED BEAMFORMING FOR ASYNCHRONOUS TWO-WAY RELAY NETWORKS Reza Vahidnia and Shahram Shahbazpanahi Department of Electrical, Computer, and Software Engineering, University of Ontario Institute of Technology ABSTRACT In this paper, we consider a single-carrier communication scheme, where two transceivers exchange information with the help of multiple relays. The propagation delay in each relaying path is assumed to be different from those of the other paths. As such, the end-to-end channel is frequency selective, and hence, it produces inter-symbol-interference ISI at the two transceivers. The simple amplify-and-forward scheme is used at the relays and channel equalization is employed at both transceivers to combat ISI. We minimize the mean square error MSE of the total estimated received signals at the both transceivers, subject to a total power budget constraint, by optimizing the channel equalizers, the relay beamforming weights, and the transceivers powers. We show that our proposed approach leads to a relay selection method which transforms the end-to-end channel into a frequency flat channel. We also present a semi-closed-form solution for the optimal relay beamforming weight. Index Terms Two-way relay network, single-carrier equalization, distributed beamforming. INTRODUCTION In a wireless two-way relay network, two transceivers communicate with each other with the help of one or more intermediate relays. There are different relaying approaches among which, because of its simplicity, amplify-and-forward technique is of particular interest and has been widely studied in the literature [ 4]. In almost all these publications, the authors consider perfectly time-synchronized relays and transceivers. However, since the propagation delay of each relaying path can be different from that of the other paths, the signals transmitted by the relays arrive at the transceivers at different times, thereby leading to the frequency selectivity of the end-to-end channel. Therefore, even if the relaytransceiver channels are frequency flat, ISI is inevitable at the transceivers. There are two different approaches to combat such an ISI. The first approach is based on multi-carrier communication, where all the nodes of the network are equipped with orthogonal frequency division multiplexing OFDM transmission and reception to transform the end-to-end frequency selective channel into multiple parallel flat fading channels [5, 6]. In the second approach, which is a single-carrier scheme, block or symbol channel equalization is used at the nodes of the network to eliminate ISI [7 0]. In this work, we consider a single-carrier scheme in an asynchronous two-way relay network and develop our system model for such a scheme. The relays are assumed to simply amplify their received signals, while the burden of equalization is on the shoulders of two block channel equalizers implemented at the two transceivers. We optimally obtain the transceivers powers and the relay beamforming weight vector as well as the block equalizers in order to minimize the total MSE of the estimated received signals at both transceivers. We prove that such an optimization problem leads to a relay selection scheme, where only the relays contributing to one tap of the end-to-end channel are turned on and the remaining relays have to be turned off. Moreover, we present a semiclosed-form solution to find the optimum relay weights for such a relay selection method. Notations: The statistical expectation is represented as E{ } andtr denotes the trace of a matrix. Complex conjugate, transpose, and Hermitian transpose are denoted as, T, and H, respectively. We represent then N identity matrix and the M N all-zero matrix as I N and 0 M N, respectively. We let diaga stand for a diagonal matrix whose diagonal entries are the elements of the vectora. 2. SIGNAL MODEL We consider a two-way relay network which includes two single-antenna transceivers which communicate with each other with the help of L single-antenna relay nodes. For the sake of simplicity, the relays are assumed to simply amplify and forward their received signals. We use block transmission to send a vector of N s symbols. As it is shown in Fig., the transmitted symbols go through the serial-to-parallel block represented as. The ith transmitted block from Transceiver p is represented as s p i = [ sp [in s ] s p [in s +N s ] ] T, for p {,2}. Due to the frequency selectivity of the channel, inter-blockinterference IBI arises between successive transmitted blocks which renders the received signal dependent on both

2 s p i ands p i. In order to mitigate the IBI thereby preventing the consecutive blocks from overlapping with each other, a cyclic prefix insertion matrix, represented as T cp, is pre-multiplied by s p i to transmit the data in blocks with the size N t = N + N s. Hence, the ith transmitted block is defined as s p i T cp s p i = [ s p [i+n s N] s p [i+n s ] s p [in s ] s p [i+n s ] ] T. Here,N is the length of the vector of the equivalent discretetime end-to-end channel taps and T cp = [I T cp I T N s ] T is an N t N s matrix which is the concatenation of the last N rows of the identity matrix I Ns denoted as I cp, and the identity matrix I Ns. Then, using the parallel-to-serial block denoted as P/S, the serial signal goes through the channel. After the self-interference-cancelation block, denoted as SIC, the firstn entries of the received block are simply discarded by pre-multiplying it with the cyclic removal matrix R cp [0 Ns N I Ns ]. The burden of channel equalization is on the shoulders of the block equalizers, which are implemented at Transceivers and 2 and are denoted as F r and F r2, respectively. Assuming reciprocal and frequency flat channels between each [ transceiver and] each relay, the 2 2 matrix h [n] h H[n] = 2 [n] represents the end-to-end channel between the two transceivers. Note that for p {,2}, h 2 [n] h 22 [n] h pp [n] represents the linear time-invariant LTI channel corresponding to the so-called self-interference signal, which is transmitted by Transceiver p and is relayed back to the same transceiver. Let T s be the sampling time and τ lpq be the propagation delay of the signal path from Transceiver p to Transceiver q which is going through the lth relay, for p, q {, 2}. Therefore, the impulse response of the endto-end channel from Transceiver p to Transceiver q, can be written as h pq [n] = b lpq δ[n n l ] for p,q {,2} l= where n l is an integer number satisfying n l T s τ lpq n l T s, whereas b lpq w l g lp g lq is the total amplification factor applied to the signal going through the lth relay, w l is the complex weight of the lth relay, and g lp is the flat fading channel coefficient between Transceiver p and the lth relay, for p,q {,2}. The vector of channel impulse response coefficients can be represented as h pq = [ h pq [0] h pq [] h pq [N ] ] T. The contribution of different relays to different taps of h pq [ ] is shown byn L matrixb pq, whosen,lth element is defined as B pq n,l { glp g lq, n T s τ lpq nt s 0, otherwise. 2 Denoting the vector of complex relay weights as w [w,...,w L ] T, the vector of channel impulse response coefficients can be represented as h pq = B pq w. 3 The reciprocity of the end-to-end channel implies that h 2 = h 2 h [ h[0] h[] h[n ] ] T. We assume that the propagation delay between the lth relay and Transceiver q, denoted as τ lq, satisfies n l T s τ lq n l T s. Let υ l [n] represent the temporally and spatially zeromean white Gaussian noise at the lth relay with variance of σ 2. This noise is multiplied by w l, is delayed by n lq, and arrives at the block equalizers. Hence, the relay noise received at Transceiverq can be written as ξ q [n] = w l g lq υ l [n n lq] = υ T n,qg q w 4 l= where υ n,q [ υ [n n q ] υ L[n n Lq ] ] T and G q = diag{g lq,g 2q,,g Lq }, for q {,2}. Considering γ q [n] as the vector of noise added to the signal at Transceiver q and before the serial-to-parallel block, the total noise vector received at Transceiver q and before the cyclic removal matrix can be written as γ q i = ξ q i+γ i = Υ q ig q w+γ qi 5 where γ q i [ γ q [in t ] γ q [in t +N t ] ] T, ξ q i [ ξ q [in t ] ξ q [in t +N t ] ] T, γ q i [ γ q [in t] γ q [in t +N t ] ] T, and Υ q i [ υ int,q υ int+,q υ int+n t,q is an N t L matrix. Assuming E{ s p [k] 2 } = for p {, 2}, we can write the ith transmitted block received at the output of self-interference-cancelation block at Transceiverq as r q i = P p H 0 s p i+ P p H s p i +γ q i 6 where P p is the transmit power of Transceiverp, and the following definitions are used: h[0] h[0] 0 0 H 0. h[n ] h[n ] h[0] 0 h[n ] h[] H. 0.. h[n ] ] T 2

3 R w s [.] s s s 2 s 2 w T cp P/S 2 P/S T cp s 2 [.] ŝ 2 r Fr cp SIC + + SIC r 2 R cp F r2 ŝ noise w L noise Fig.. System model After removing the cyclic prefix and discarding the first N entries from the received block, the received block can be written as r q i = R cp r q i = P p Hsp i+ γ q i 9 where γ q i R cp γ q i and H R cp H 0 T cp is an N s N s circulant matrix with its k,lth entry given by h[k l mod N s ], where h[ ] is obtained by zero padding h[ ] withn s N zeros. It can be verified thatr cp H = 0, and therefore, the IBI-inducing matrix H is eliminated. Hence, the estimated transmitted signal block at Transceiver q can be represented as ŝ p i F rq r q i = P p F rq Hsp i+f rq γ q i 0 where ŝ p i is an N s vector. Considering υ l i as the noise added to the ith signal block at the lth relay, the total relayed signal at thelth relay can be represented as [ x l i = w l P g l s i+ ] P 2 g l2 s 2 i+υ l i wherex l i [ x l [in t ] x l [in t +N t ] ] T, while x l [t] is the signal transmitted by the lth relay at time t, and υ l i [ υ l [in t ] υ l [in t +N t ] ] T is the ith block of measurement noise of the lth relay. The average transmit power of the lth, Pl = N s E { x H l ix l i } relay is calculated as P l = w l 2 g l 2 P + g l2 2 P 2 +σ 2 2 andp total = P +P 2 + L l= P l is the total consumed power. 3. PROPOSED DESIGN APPROACH In this section, we aim to optimally obtain the block channel equalizers at both transceivers as well as the relay weight vector and the transceivers powers in order to minimize the sum- MSE of the estimated received signals at both transceiver. The vector of signal estimation error at Transceiver q can be Here, we assume that the transmission length is much longer than the difference between the times of arrivals of the transceiver s signals at the relay. written ase q i = ŝ p i s p i = F rq r q i s p i. We now formulate our design problem as P 0 w min F r,f r2 E { e q i 2} s.t. P total P max q= 3 where P max is the total available power in the network. Assuming thate{s p i} = 0 and thats p i and γ q i are statistically independent, we can write E { e H q ie qi } = tr F rq R q F H rq where P p tr F rq H+ HH F H rq +N s 4 R q E { r q ir H q i } { = P p HE sp is H p i } HH + } E { γ q i γ H q i = P p H HH +σ 2 G q w 2 + I Ns is the correlation matrix of the received block. The optimal value of F r can be obtained by taking the derivative of 4 with respect to F r and equating it to zero. This yields F opt r = P 2 HH R. Similarly, we can obtain Fopt r2 = P HH R 2 and write λw,p,p 2 min F r,f r2 E { e H q ie qi } q= HH = 2N s tr P 2 R +P R 2 H. 5 TheN s N s circulant matrix H can be diagonalized by preand post-multiplying it with N s -point DFT and IDFT matrices as represented below F HF = D H diag{he j0,,he j 2πNs Ns } 6 where He j2πf = N n=0 h[n]exp j2πf n is the frequency response of the LTI channel at the normalized frequency f, and the elements of the DFT matrix are defined as Fk, n = N 2 s exp j2πkn/n s. Using 6, we deduce that H = 3

4 F D H F, and hence, rewrite 5 as λw,p,p 2 = 2N s P 2 tr [ P 2 I+σ 2 G w 2 + D 2 H ] P tr [ P I+σ 2 G 2 w 2 + D 2 H ]. 7 [ Let f k Ns e j 2πk Ns e j 2N k π Ns ] T, we can write D H = diag{ N s f H Bw,, N s f H N s Bw}. Therefore,λw,P,P 2 can be written as λw,p,p 2 = P p tr N s q= diag{p p + σ2 G q w 2 + N s f H k Bw 2 } k=ns k=. 8 Using 8 and the fact that the trace of a diagonal matrix can be written as the sum of its diagonal entries, we can rewrite 3 as P 0 w N s q= k= φ k,q w s.t. P total P max 9 where φ k,q w = + PpNs fh k Bw 2 σ 2 G qw 2 +. In [], it has been shown that optimization problem 9 can be written as P 0 w subject to q= N s P pw H B H Bw σ 2 G qw w l 2 g l 2 P + g l2 2 P 2 +σ 2 l= +P +P 2 P max and w U n. 20 whereu n is the set of relay weight vectors such that only the nth tap of the channel is non-zero and the remaining taps are zero [6]. Indeed, the constraintw U n, implies that only relays which contribute to the nth tap of the end-to-end channel are turned on and the remaining of the relays have to turned off. Letw n represent any vector of relay weights corresponding to the nth tap of the channel. Hence, if w U n then, w H B H Bw = wn Hb nb H n w n, where b H n captures the nonzero entries of the n+th row of B. In order to minimize the error, we need to find the index of the tap which yields the minimum value for the objective function. Therefore, the optimization problem 20 can be written as min 0 n N P 0 w n q= N s ψ n,q w for p q subject to P +P 2 +P G n w n 2 +P 2 G n 2 w n 2 +σ 2 w H n w n P max 2 where ψ n,q w P pw H n bnbh n wn σ 2 G n q w n and G n q, for q {,2}, is a diagonal matrix whose diagonal entries are a subset of those diagonal entries of G q which correspond to the relays that contribute to the nth tap of the end-to-end channel impulse response. Without loss of optimality, we can assume that ψ n, w = ψ n,2 w. Otherwise, if for example, ψ n, w > ψ n,2 w we can reduce the optimal value of P 2 to make ψ n, w and ψ n,2 w equal, thereby contradicting optimality. Therefore, the optimization problem 2 can be written as min 0 n N P 0 w n 2N s P 2wn HbnbH n wn σ 2 G n w n subject to P + G n w n 2 +P 2 + G n 2 wn 2 +σ 2 wn H w n P max P G n w n 2 + = P 2 2 w n G n In [], it is shown that the constrained optimization problem 22 can be written as max max 0 n N w n subject to P max σ 2 wn Hw nwn Hb nb H n w n 2σ 2 G n w n 2 + G n 2 w n 2 + w H n w n P max σ In [2], a semi-closed-form solution to 23 is given as w o n µ n = σ 2 κµ n 2ν n 2µ n Q n +2ν nq n 2 +σ 2 I bn where Q n q = G nh q κµ n is defined as 24 q, ν n 0.5P max /σ 2 µ n, and G n κµ n σ b H n σ2 I+2µ n Q n σ 2 I+2µ n Q n +2ν n Q n 2 2 b n 2 25 andµ n is the unique solution to the following equation: σ 2 P max /σ 2 4µ n b H n 2µ n Q n +P max /σ 2 2µ n Q n 2 +I b n µ n P max /σ 2 2µ n b H n 2µ n Q n +P max /σ 2 2µ n Q n 2 +I 2 2Q n 2Q n 2 b n = 0 26 which satisfies 0 µ n 0.5P max /σ 2. A simple bisection method can be used to solve 26 and to find the value of µ n in the interval [0 0.5P max /σ 2 ]. The optimal value of n is determined by evaluating the objective function for w n for n = 0,,,N and choosing the value of n which leads to the maximum objective function. In other words, the 4

5 BER cyclic insertion and removal matrices at the transceivers to combat IBI. The channel equalizers, the relay beamformer weights and the transceivers powers were optimized to minimize the MSE of the total estimated transmitted signals under a total power budget constraint. We proved that at the optimum, single-carrier equalization leads to a relay selection scheme and presented a semi-closed-form solution for the optimal beamformer BER of our proposed algorithm BER of the EPA algorithm Total Transmit power dbw Fig. 2. Bit error rate versus total transmit power set of relays which contribute to one tap of the finite impulse response channel with the smallest total MSE, is selected and the remaining relays have to be turned off. In [2], it is shown that the solution to 24 requires all participating relays to use half of the total available transmit power. 4. SIMULATION RESULTS We consider an asynchronous bi-directional relay network with L = 8 single-antenna relays. The transceivers are assumed to transmit signal blocks of N s = 64 symbols with a cyclic prefix length of N = 6. The zero-mean Gaussian white noise has a variance σ 2 =. The flat fading channel coefficients are zero-mean independent and identically distributed i.i.d complex Gaussian random variables with unit variance. For sampling time T s =, we model the propagation delay of each relay path as a uniformly distributed random variable uniformly distributed in the interval[0,8t s ]. Assuming QPSK modulation, in Fig. 2 we show the endto-end BER curves versus the total transmit power P max for our proposed algorithm and compare it with the one for equal power allocation EPA method where the total transmit power is equally allocated to the all nodes of the network. As it is illustrated in this figure, our proposed approach yields a better performance in comparison with the EPA scheme. That is for a given total transmit power, our proposed algorithm yields less error compared to the EPA method. Moreover, for transmit powers higher than 0 db, in our proposed algorithm, the bit error rate decreases drastically. 5. CONCLUSION In this paper, we considered two transceivers which aim to communicate using an amplify-and-forward relay network with different propagation delays in different relaying paths. Since the end-to-end channel is frequency selective, we used 6. REFERENCES [] M. Zaeri-Amirnai and S. ShahbazPanahi, Performance comparison in amplify-and-forward bi-directional network beamforming, IEEE Transactions, Signal Processing, vol. revised, Sep. 20. [2] V. Havary-Nassab, S. Shahbazpanahi, and A. Grami, Optimal distributed beamforming for two-way relay networks, IEEE Trans. Signal Process., vol. 58, pp , Mar [3] R. Vaze and R. W. Heath, Optimal amplify and forward strategy for two-way relay channel with multiple relays, in Proc. IEEE ITW 2009, Volos, Greece, Jun. 2 0, 2009, pp [4] Y. Zhao, R. Adve, and T. Lim, Improving amplify-andforward relay networks: optimal power allocation versus selection, IEEE Trans. Wireless Commun., vol. 6, pp , Aug [5] F. Gao, R. Zhang, and Y.-C. Liang, Channel estimation for ofdm modulated two-way relay networks, IEEE Transactions on Signal Processing, vol. 57, pp , Nov [6] R. Vahidnia and S. Shahbazpanahi, Multi-carrier asynchronous bi-directional relay networks: Joint subcarrier power allocation and network beamforming, To be appeared, IEEE Trans. Wireless Communication., 203. [7] A. Schad, H. Chen, A. Gershman, and S. Shahbazpanahi, Filter-and-forward multiple peer-to-peer beamforming in relay networkswith frequency selective channels, in Proc. ICASSP 0, 200, pp [8] Y. wen Liang, A. Ikhlef, W. Gerstacker, and R. Schober, Cooperative filter-and-forward beamforming for frequencyselective channels with equalization, IEEE Trans. Wireless Commun., vol. 0, pp , Jan 20. [9] M. Hasna and M.-S. Alouini, Optimal power allocation for relayed transmissions over rayleigh-fading channels, IEEE Trans. Wireless Commun., vol. 3, pp , Nov [0] P. Wu and R. Schober, Cooperative beamforming for singlecarrier frequency-domain equalization systems with multiple relays, IEEE Trans. Wireless Communication., vol., no. 6, pp , 202. [] R. Vahidnia and S. Shahbazpanahi, Single-carrier equalization for asynchronous two-way relay networks, Submitted to IEEE Trans. Signal Proc, 203. [2] S. Shahbazpanahi and M. Dong, Achievable rate region and sum-rate maximization for network beamforming for bidirectional relay networks, in Proc. ICASSP 0, Mar. 200, pp