Sum-Rate Maximization for Two-Way MIMO Amplify-and-Forward Relaying Systems

Transcription

1 Sum-Rate Maximization for Two-Way MIMO Amplify-and-Forward Relaying Systems Kyoung-Jae Lee, Kwang Won Lee, Hakjea Sung, and Inkyu Lee School of Electrical Eng., Korea University, Seoul, Korea {kyoungjae, kwangwonlee, jaysung, Abstract This paper considers two-way relaying systems with a multiple-input multiple-output MIMO) relay between two MIMO terminal nodes. The two-way relaying can enhance the spectral efficiency compared with the one-way by compensating the loss from half-duplex signaling. In this paper, we propose an iterative scheme to find a relay weighting matrix maximizing the sum-rate for two-way relay channels. Due to the non-convexity of the given problem, the proposed scheme iteratively identifies a local optimal solution by deriving the gradient of the sum-rate and applying the gradient descent algorithm. Simulation results show that the proposed iterative scheme with provable convergence achieves a near-optimal sumrate for the two-way MIMO relay channels. Also, we show that the proposed scheme with a few iterations still outperforms the conventional schemes. I. INTRODUCTION Wireless relaying transmission is a promising technique that has advantages of extending the coverage and increasing the system capacity. Hence, relay based wireless networks have been intensively studied with a lot of interest. The analysis for the capacity of relay systems has been reported in [1], [] and [3]. In order to enhance the capacity of the relay network, multiple antennas can be considered for obtaining a similar performance benefit observed in the point-to-point multipleinput multiple-output MIMO) systems [4]. The capacity of MIMO relay systems have also been studied in [5]. Most relay systems are assumed to be operated in the half-duplex mode where relay nodes do not transmit and receive signals simultaneously to avoid loop interference in the relay nodes. The capacity scaling analysis in [6] shows that such halfduplex relay systems suffer from a substantial loss of spectral efficiency due to the pre-log factor 1/, which dominates the capacity at high signal-to-noise SNR) regime. Two-way relaying has recently been studied to overcome the spectral efficiency loss from the half duplex [7] [8]. In the first time slot, two source terminals transmit their data signals to the relay node at the same time. Subsequently in the second time slot, the relay node transmits the processed signal to the destination terminals as in the broadcast channel. Since each terminal node knows its own transmitted data, the self-interference in the transmitted signal can be eliminated from the received signal [8]. In contrast to the one-way relaying which needs four time slots to exchange the information between two terminal nodes, the two-way relaying requires only two time slots. As a result, the two-way can compensate for a large portion of the capacity loss from the half-duplex relay signaling. Therefore, we focus on the two-way relaying systems in this paper. In practical relay systems, an amplify-and-forward AF) method shows advantages of simple implementation and low computational complexity compared to decode-and-forward DF) systems, since the relay node linearly processes only the received baseband signal without decoding the information. It was shown in [9] that a linear technique based on singular value decomposition SVD) achieves the capacity of MIMO AF relaying systems with the one-way. For the twoway relay channels, zero-forcing ZF) and minimum mean square error MMSE) approaches have been proposed in [], where interferences between two terminals are suppressed at the relay node without the help of the self-interference cancellation. Also, relay networks with multiple-antenna relay and single-antenna terminals have recently been investigated in terms of the sum-rate of the two-way AF [11]. In this paper, we propose a new linear processing scheme which maximizes the sum-rate of two-way MIMO relaying channels where all nodes have multiple antennas. Since the formulated maximization problem is not generally a convex or concave problem, it cannot be solved analytically. Hence, the proposed scheme iteratively finds a linear filter matrix maximizing the sum-rate by deriving the gradient of the sumrate and applying a gradient descent algorithm. Although the gradient descent algorithm may not guarantee the global optimal solution, a locally maximized sum-rate can be found. Consequently, the proposed scheme achieves an enhanced sum-rate performance in two-way MIMO relay channels. Simulation results demonstrate that the proposed scheme with only a few iterations converges to a locally maximized sumrate for the two-way MIMO relay channel and outperforms the conventional schemes. This paper is organized as follows: Section II describes the system model for two-way MIMO relay channels. In Section III, we propose an iterative scheme to maximize the sumrate performance. Section IV presents the simulation results. Finally, the paper is terminated with conclusions in Section V. Throughout this paper, the superscripts ) T, ) and ) stand for transpose, conjugate transpose, and element-wise conjugate, respectively. E denotes the expectation operator. I N /09/$ IEEE

2 1st nd s A A y A A H H T r xfr G G T s B B y B B as ρ R γ { ) } 1) tr F ρ A HH + ρ B GG + I N F with ρ A P A /Mσn), ρ B P B /Mσn) and ρ R P R /σn. The processed signal x is transmitted from the relay node R to two terminal nodes T A and T B as in a broadcasting channel. We assume that the forward channel matrices from R to T A and T B become H T and G T, respectively, due to channel reciprocity in the TDD mode [8]. Then the received signals at two terminal nodes y A and y B can be written by Fig. 1. Schematic diagram of MIMO relaying systems in the two-way and y A γh T FHs A + γh T FGs B + γh T Fn + z A indicates an N N identity matrix. tra) and A represent the trace and the determinant of the matrix A, respectively. II. SYSTEM MODEL In this section, we describe a system model for two-way relay networks shown in Figure 1. We consider that two terminal nodes T A and T B want to communicate to each other, and a single relay node R helps the communication between two terminals. The terminal nodes and the relay node are equipped with M antennas and N antennas, respectively, and operated in time division duplex TDD) mode. In this paper, we assume that a direct link between two terminal nodes can be ignored due to a large path loss. Also, it is assumed that the relay has full channel state information CSI) for both backward and forward channels, and both the terminal nodes know full CSI only at the received time slot. All fading channels are assumed to be frequency-flat. In the first time slot, two terminal nodes T A and T B transmit their signals s A and s B to the relay node R simultaneously as shown in Figure 1. Then, the N dimensional received signal vector r at the relay node is given as r Hs A + Gs B + n where H and G are the N M backward channel matrices from the terminals T A and T B to the relay node R, respectively, n denotes the additive complex Gaussian noise vector with zero mean and Enn )σni N, and s A and s B represent the M dimensional transmitted signal vectors with Es A s PA A ) M I M and Es B s B ) PB M I M. Here, P A and P B indicate the transmission power at T A and T B, respectively. In the second time slot, assuming linear processing at the relay node, the received signal r is multiplied by the N N weighting matrix F. Then, the signal vector x transmitted from the relay node is given by x γfr γfhs A + γfgs B + γfn where γ is the power normalizing coefficient to satisfy the relay power constraint E{trxx )} P R and can be expressed y B γg T FHs A + γg T FGs B + γg T Fn + z B where z A and z B are the complex white Gaussian noise vectors with zero mean and the covariance matrix σ zi M. Since terminal nodes know their own symbols transmitted at the previous time slot and the corresponding channel matrices, the back-propagating self-interference γh T FHs A or γg T FGs B ) can be canceled at each destination. After removing the self-interference terms, received signals ŷ A and ŷ B are then given as and ŷ A γh T FGs B + γh T Fn + z A ) ŷ B γg T FHs A + γg T Fn + z B, 3) respectively. The performance of the two-way relay systems depends on the relay weighting matrix F. Note that F should be simultaneously optimized for bidirectional links in ) and 3) unlike the one-way. Thus, conventional methods for the one-way may fail to fully exploit the potential of two-way channels. In the next section, we investigate the design of a weighting matrix F to maximize the sum-rate performance of the bidirectional MIMO link. III. SUM-RATE MAXIMIZATION In this section, we propose a linear processing scheme for two-way MIMO relaying systems. We first review briefly the conventional scheme optimized for the one-way. Then, we present the proposed iterative algorithm which attempts to identify a relay weighting matrix F which optimizes the sum-rate performance. A. Review of the one-way optimum scheme It was shown in [9] that in one-way relay systems, the SVD-based linear processing achieves the capacity of MIMO relay channels as in point-to-point MIMO systems. However, on the two-way relay, the one-way optimal scheme cannot simultaneously support the bidirectional channels in

3 ) and 3). In order to use the SVD-based scheme in twoway systems, we suppose that one-side of bidirectional links is neglected. To maximize the rate performance for T A T B, the unidirectional optimum weighting matrix F A can be written as [9] F A V A Λ A U A 4) where V A is the right singular matrix of the forward channel G T and U A is the left singular matrix of the backward channel H. Thek-th diagonal element of the diagonal matrix Λ A is given by [ λ k σ ρ ] + z A α k +4ρ Aα k β k ˆμ ρ A α k σ n β k 1 + ρ A α k ) where α k and β k are eigenvalues of HH and G G T, respectively, [x] + max{0,x}, and ˆμ is a unique solution [ N ρ + of k1 1/β k) A α k +4ρ Aα k β k μ ρ A α k ] ρ R σn/σ z [9]. Unlike the one-way, the weight matrix F A may not satisfy the relay power constraint since ρ B is nonzero in 1). Hence, the power normalization for F A is still required. For optimizing a link of T B T A, the weight matrix F B can be found with the corresponding forward and backward channels in the same way. Since the bidirectional link channels are symmetric due to reciprocity, the average sum-rates resulted from F A and F B are the same. B. Sum-rate maximization scheme Assuming that transmitted signals and the total noise terms in ) and 3) have a zero mean circularly symmetric complex Gaussian distribution, the sum-rate of the two-way MIMO relay system can be expressed by R sum 1 {Iŷ B; s A )+Iŷ A ; s B )} 1 log ρ AG T FHH F G + G T FF G + σ z γ σn I M 1 log GT FF G + σ z γ σn I M log ρ BH T FGG F H + H T FF H + σ z γ σn I M 1 log HT FF H + σ z γ σn I M 5) where Ix; y) denotes the mutual information between x and y. It should be noted that the pre-log factor 1/ is caused by the half-duplex mode, and γ is given as a function of the weighting matrix F in 1). Then, to find the optimum relay matrix for maximizing the sum-rate, our problem can be formulated as F opt arg max R sum. 6) F Since the cost function R sum in 5) is not generally a convex or concave function with respect to F, the optimization problem is difficult to solve analytically. Hence, we apply a gradient descent algorithm to solve the unconstrained optimization problem 6). The proposed iterative algorithm exploits a fact that the sum-rate increases fastest when an arbitrary weighting matrix F 0 moves in the direction of the gradient of the sumrate R sum F 0 ). Since the sum-rate R sum is a real-valued function, we have R sum R sum / F ) [1]. To compute the derivative of a function involving matrices, we first compute the differential of the function. For simple representation, denoting Π A ρ A G T FHH F G + G T FF G + σ z γ σn I M, Ω A G T FF G + σ z γ σn I M, Π B ρ B H T FGG F H + H T FF H + σ z γ σn I M and Ω B H T FF H + σ z γ σn I M, the differential of the sum-rate 5) can be written as 7) which is located at the top of the next page. Here, it is taken into account in the computation of the differential that γ is a function of F through 1). In 7), vecx) represents the stacked columns of a matrix X. This result can be verified using some rules such as dln Y ) tr{y 1 dy)}, d{try)} tr{dy)}, vecdx) dvecx), trx T Y)vecX) T vecy) and trx T Y)trXY T ) [13]. Now, the coefficients of dvecf ) in 7) directly lead to the derivative R sum / F, and the gradient of the sum-rate R sum is then derived as ) Rsum R sum F 1 ln G {Π A G T Fρ A HH + I N ) Ω A G T F} ln H {Π B H T Fρ B GG + I N ) Ω B H T F} + σ z/σn ρ R ln trπ A + Π B Ω A Ω B ) Fρ A HH + ρ B GG + I N ). 8) With the derived gradient expression, we propose an iterative algorithm for solving 6) as follows. Initialization: 1) Initialize F as an arbitrary relay weighting matrix. ) Calculate R sum,0 with the initial matrix F and set k 1. Main Loop: 3) Calculate the gradient R sum F). 4) Update F F + δ R sum F). 5) Calculate R sum,k with the updated weighting matrix F. 6) If R sum,k R sum,k 1 <ɛ, stop the iteration. Otherwise k k +1 and go back to step 3).

4 dr sum 1 { trπa dπ 1 A ln ) trω AdΩ 1 A )+trπ BdΠ 1 B ) trω BdΩ 1 B )} 1 ln vecρ AG Π A G T FHH + G Π A G T F G Ω A G T F) T dvecf ) ln vecρ BH Π B H T FGG + H Π B H T F H Ω B H T F) T dvecf ) + σ z/σn ρ R ln trπ A + Π B Ω A Ω B )vec{fρ A HH + ρ B GG + I N )} T dvecf ) ln vecρ AGΠ T AG F H H T + GΠ T AG F GΩ T AG F ) T dvecf) ln vecρ BHΠ T BH F G G T + HΠ T BH F HΩ T BH F ) T dvecf) + σ z/σn ρ R ln trπ A + Π B Ω A Ω B )vec{f ρ A HH + ρ B GG + I N ) T } T dvecf) 7) In this algorithm, ɛ is the tolerance factor for terminating the iteration. Several line search methods are introduced in [14] to efficiently determine the step size. We employ a line search method called Armijo s Rule which provides provable convergence [14]. According to the Armijo s Rule, the step size δ is determined as δ ν m where m is the smallest integer such that R sum F + ν m R sum F)) R sum F) μν m tr { R sum F) R sum F) }. Here ν and μ are fixed scalars between zero and one. The proposed algorithm then obtains a non-decreasing sum-rate value with respect to the number of iterations. Due to non-convexity of the maximization problem in 6), the above algorithm cannot guarantee the global optimal solution. Thus, in order to increase the probability that the proposed iterative algorithm achieves the global optimal solution, we can repeat the proposed algorithm for randomly chosen multiple initial matrices and select one corresponding to the largest sum-rate. Then, our solution converges to the best of the local optimum solutions. Consequently, the proposed scheme can provide a lower bound of the capacity for the two-way MIMO AF relaying system. IV. SIMULATION RESULTS In this section, we present simulation results for the proposed iterative scheme on the two-way MIMO channels in terms of the sum-rate. In our simulation, we assume that P A P B P/4, P R P/ and σn σz to make a fair comparison with the one-way, where P denotes the total transmission power of the network. Then, the SNR is defined as P/σn. Also, elements of channel matrices H and G have independent and identically distributed i.i.d.) complex Gaussian distribution with zero mean and unit variance. In Figure, we plot the results of the proposed sum-rate maximizing scheme with respect to the number of iterations. In this figure, we can see that the sum-rate of the proposed iterative algorithm almost converges to the maximum value Average Sum Rate [bps/hz] Fig.. N M, N4 SNR30dB SNR0dB SNRdB Number of Iterations k Convergence of the proposed iterative algorithm for M and with less than iterations. Here, the near-optimal sum-rates dashed lines) are plotted by the proposed iterative algorithm with 00 iterations for 0 different initial matrices. Figure 3 plots the average sum-rate curve as a function of SNR for the two-way MIMO systems with M and N 4. The proposed scheme with the near-optimum sumrate outperforms the unidirectional optimum scheme 4) in [9], the naive AF scheme in [8], which employs no relay weighting matrix F I N ), and the MMSE filter proposed in [] by about 4 db, 6 db and 7 db at 15 bps/hz, respectively. The sum-rate of the MMSE filtering method in [] degrades compared to even the naive AF scheme since the self-interference cancellation is not exploited. Also, it is interesting to see that the proposed scheme with 3 iterations and one initial matrix achieves the most of the near-optimal sum-rate obtained by 00 iterations and 0 initial matrices.

5 Average Sum Rate [bps/hz] Average sum rate of two way relaying systems M, N4) Proposed scheme w/ iter. Proposed scheme w/ 3 iter. Unidirectional optimum in [9] Naive AF in [8] MMSE filter in [] in one way One way capacity in [9] two way one way SNR [db] Average Sum Rate [bps/hz] Average sum rate of two way relaying systems SNR0dB) Proposed scheme w/ iter. Proposed scheme w/ 3 iter. Unidirectional optimum in [9] Naive AF in [8] M4 M Number of Relay Antennas N Fig. 3. Sum-rate comparison as a function of SNR in M and N 4 Fig. 4. Sum-rate comparison for the number of relay antennas N For a sanity check, the proposed scheme with P A P/ and P B 0is plotted by a dotted line in Figure 3 which attempts to maximize the rate for T A T B on the oneway. We can see that the proposed scheme achieves the capacity of the one-way systems in [9] without any loss. Hence, we conclude that the proposed iterative scheme can also provide a tight capacity lower bound of AF systems in two-way MIMO relay channels. In Figure 4, we compare sum-rates with various numbers of relay antennas N at the SNR of 0 db. The sum-rate gain of the proposed algorithm becomes much larger as the number of relay antennas increases. With 3 iterations and one initial matrix, the proposed scheme still shows better sumrate performance than the conventional two-way schemes for various antenna configurations. The small number of iterations indicates that the proposed scheme can be implemented in practical relaying environments. V. CONCLUSION In this paper, we have proposed new MIMO relaying schemes in the two-way systems. By exploiting the gradient descent algorithm, we obtain the relay weighting matrix which maximizes the sum-rate. The proposed scheme converges to a local maximum of the sum-rate with a small number of iterations. From the proposed iterative scheme, a near-optimum sum-rate capacity of two-way AF MIMO systems can be obtained. The simulation results demonstrate that the proposed scheme is quite effective in maximizing the sum-rate for two-way MIMO relay channels. ACKNOWLEDGMENT This research was supported by the MKE Ministry of Knowledge Economy), Korea, under the ITRC Information Technology Research Center) support program supervised by the IITA Institute for Information Technology Advancement) IITA-008- C ). REFERENCES [1] T. M. Cover and A. A. El Gamal, Capacity theorems for the relay channels, IEEE Trans. Inform. Theory, vol. 5, no. 5, pp , Sep [] M. Gastpar and M. Vetterli, On the capacity of wireless networks: The relay case, in Proc. IEEE INFOCOM, vol. 3, pp , Jun. 00. [3] G. Kramer, M. Gastpar, and P. Guptar, Capacity theorems for wireless relay channels, in Proc. Allerton Conf. Comm., Control and Comput., pp , Oct [4] G. J. Foschini and M. Gans, On limits of wireless communications in a fading environment when using multiple antennas, Wireless Personal Comm., vol. 6, pp , Mar [5] B. Wang, J. Zhang, and A. Host-Madsen, On the capacity of MIMO relay channels, IEEE Trans. Inform. Theory, vol. 51, no.1, pp. 9 43, Jan [6] H. Bolcskei, R. U. Nabar, O. Oyman, and A. J. Paulraj, Capacity scaling laws in MIMO relay networks, IEEE Trans. Wireless Comm., vol. 5, no. 6, pp , Jun [7] B. Rankov and A. Wittneben, Spectral efficient signaling for halfduplex relay channels, in Proc. IEEE Asilomar Conf. on Signals, Systems and Comput., Nov [8] B. Rankov and A. Wittneben, Spectral efficient s for halfduplex fading relay channels, IEEE J Select. Areas Comm., vol. 5, no., pp , Feb [9] X. Tang and Y. Hua, Optimal design of non-regenerative MIMO wireless relays, IEEE Trans. Wireless Comm., vol. 6, no. 4, pp , Apr [] T. Unger and A. Klein, On the performance of two-way relaying with multiple-antenna relay stations, in Proc. IST Mobile and Wireless Comm. Summit, Jul [11] Y.-C. Liang and R. Zhang, Optimal analogue relaying with multiantennas for physical layer network coding, in Proc. IEEE ICC, pp , May 008. [1] S. Haykin, Adaptive Filter Theory. New Jersey: Prentice Hall, 4th ed., 001. [13] J. R. Magnus and H. Neudecker, Matrix Differential Calculus with Applications in Statistics and Econometrics. John Wiley & Sons, revised ed., 00. [14] M. S. Bazaraa, H. D. Sherali, and C. M. Shetty, Nonlinear Programming: Theory and Algorithms. New York: John Wiley & Sons, 3rd ed., 006.