Transmission Strategies for Full Duplex Multiuser MIMO Systems

Transcription

1 International Workshop on Small Cell Wireless Networks 2012 Transmission Strategies for Full Duplex Multiuser MIMO Systems Dan Nguyen, Le-Nam Tran, Pekka Pirinen, and Matti Latva-aho Centre for Wireless Communications and Dept. Commun. Eng., University of Oulu, Oulu, Finland Abstract We introduce a full duplex multiuser multiple-input multiple-output (FD MU-MIMO) system, and consider the total throughput maximization problem under a sum power constraint in the downlink (DL) channel and per-user power constraints in the uplink (UL) channel. Due to the nature of asymmetric DL/UL capacity, a trivial method to this problem is to optimize the DL and UL channels sequentially. However, when the selfinterference (SI) is large, the sum rate of the UL channel in this sequential design is dramatically degraded. Herein, a joint design is proposed, in which the DL and UL channels are optimized simultaneously. Since the objective function of the throughput maximization problem is non-convex, it is difficult to find the optimal solution. Thus, we propose a joint iterative algorithm to find a suboptimal design, using a local optimization strategy. Simulation results demonstrate that the iterative joint design outperforms the sequential design, and the FD MU-MIMO system is superior to the conventional half duplex () system in terms of the total system throughput when the SI is sufficiently small. This makes the FD MU-MIMO techniques promising for small cell deployments where the transmit power is relatively small. I. INTRODUCTION Multiple-input multiple-output (MIMO) communications techniques have received large attention over the last decade, due to the capability of boosting the link reliability and spectral efficiency without the need for additional power [1]. Pioneer works in the area focused on the single-user MIMO (SU-MIMO) technique, i.e., point-to-point transmission, while recent studies have been concerned with the multiuser MIMO (MU-MIMO) technique. In cellular networks, MU-MIMO transmissions can be realized in the DL and UL channels. Currently, the DL and UL channels are designed to operate in the half-duplex () mode. Since the transmission systems split the radio resources for the DL and UL channels, their spectral efficiency is not maximal. Recently, the interest in developing the full duplex (FD) transmission to improve the spectral efficiency of the mode has increased [2], [3]. Obviously, the FD technique is able to greatly improve the system capacity over systems when the self interference (SI) from the transmit antennas to the receive antennas at a base station (BS) is efficiently canceled. More recently, FD has been considered in the context of MU- MIMO relay systems, where FD relays are employed to extend This research work has been funded in part by the Finnish Funding Agency for Technology and Innovation (Tekes), Nokia Siemens Networks, Renesas Mobile, and Elektrobit. the cell coverage and enhance the cell-edge throughput [4]. In this paper, we are interested in designing a FD MU- MIMO system, where a BS is allowed to communicate with the several users in the DL and UL channels at the same time and frequencies. The motivation for our interest is due to recent progresses on the SI cancellation techniques. For example, [3] demonstrated the capability of suppressing the SI up to 73 db, making the system model of consideration applicable at least to microcell or femtocell deployment scenarios. The design of the transmission strategies is challenged by the fact that the DL and UL channels are coupled, and thus it is difficult to find the optimal transmission strategy. Consequently, suboptimal designs are of great practical importance. A trivial approach is to sequentially design the DL and UL channels. Specifically, the DL channel is first optimized, and the UL channel is then designed with the result of the DL optimization. Clearly, this method is efficient if the SI is negligible. However, as the SI increases, the throughput of the UL channel is very small. To maintain the operability of the UL channel in the FD transmission, we propose a joint design approach, which accounts for the DL and UL channels simultaneously. The problem of interest is to maximize the total system throughput under the sum power constraint (SPC) in the DL channel and per-user power constraints (PUPCs) in the UL channel. The design of linear precoders (for the DL channel) and input covariance matrices (for the UL channel) is formulated as a non-convex optimization problem. To solve this problem, we propose an iterative algorithm based on a local optimization method. First, a first-order approximation of the objective function, which turns out to be convex, is derived. Then, in each iteration, we find the linear precoders and the input covariance matrices that maximize the lower bound. Since the lower bound is non-decreasing after each iteration, the algorithm surely converges to a locally optimal solution. The numerical results show that the joint optimization mechanism reduces the transmit power of the DL channel to improve the sum rate of the UL channel when the SI is high. Notation: Standard notations are used in this paper. Bold lower and upper case letters represent vectors and matrices, respectively; H H and H T are Hermitian and standard transpose of H, respectively; tr(h) and H are the trace and determinant of H, respectively; H 0 means that H is a positive semidefinite matrix /12/$ IEEE 6825

2 s 1,dl s,dl M x 1,dl x,dl Fig. 1. N t N r y ul z ul H 1,dl H,dl Downlink channel z 1,dl z,dl y 1,dl y,dl H SI (Self interference) H 1,ul H Kul,ul Uplink channel FD MU-MIMO system model. x 1,ul x Kul,ul II. SYSTEM MODEL AND PROBLEM FORMULATION A. System Model We consider a FD MU-MIMO system, shown in Fig. 1, where the BS operates in FD transmission mode, i.e., transmits and receives data at the same time on the same frequency. The BS is equipped with N t transmit antennas and different N r receive antennas to communicate with K dl users in the DL channel and to K ul users in the UL channel, respectively. We denote by N and N i,ul the number of antennas at the ith user in the DL and UL channels, respectively. For simplicity, we assume that the UL and DL users are assumed to be geographically separated such that only the SI from downlink to uplink channels is taken into account. First, consider the transmission from the BS to the users in the DL channel. Assuming flat fading channels, the received signal y at the ith user is given by y = H x k,dl z = H x,k i H x k,dl z, where H C N N t is the channel matrix, x C Nt 1 is the vector of transmitted symbols of the ith user, respectively. The background noise z C N 1 is assumed to be additive white Gaussian (AWGN) with distribution CN(0, I). It is well known that the optimal transmission strategy for the DL channel is the dirty paper coding (DPC) [5]. Although DPC is optimal, it may be too complex for a practical application. In this paper, we consider the linear precoding for the DL channel, i.e., we can write (1) x = M s, (2) where M C Nt mi is the linear precoder of the ith user, s is the vector of data symbols, and m i is the number of data streams. We assume that E[s s H ]=I. Accordingly, we can rewrite (1) as y = H M s,k i H M k,dl s k,dl z. (3) Next, for the UL channel, the received signal vector y ul at the BS is given by y ul = H i,ul x i,ul H SI M k,dl s k,dl z ul, (4) } {{ } self interference where H i,ul C Nr N i,ul and x i,ul C Ni,ul 1 are the channel matrix and the vector of data symbols of the ith user, respectively; z ul C Nr 1 is assumed to be an AWGN vector with distribution CN(0, I). H SI C Nr Nt characterizes the SI channel between the DL and UL channels. The value of H SI depends on the placement of antennas and the SI cancellation techniques in the hardware implementation. We can see from (4) that the received signal in the UL channel suffers from the interference from the transmitted symbols in the DL channel. This couples the design of linear precoders and input covariance matrices. B. Problem Formulation In this paper, we are interested in maximizing the total throughput, i.e., the sum rate of DL and UL channels of the considered FD MU-MIMO system. For the DL channel, assuming perfect channel state information (CSI) at the BS, the sum rate of the DL channel is given by I K dl j=1 R dl = log H M j,dl M H j,dl HH I K dl j=1,j i H (5) M j,dl M H j,dl HH. For the UL channel, we also assume that the BS can perfectly estimate H i,ul. By using minimum mean square error with successive interference cancellation (MMSE-SIC) and treating the SI caused by the DL transmissions as the background noise, we can write the achievable sum rate of the UL channel as (6), shown on the top of the following page, where Q i,ul = E[x i,ul x H i,ul ] is the input covariance matrix of the vector of data symbols of the ith user. Herein, we solve the problem of finding M k,dl, and Q i,ul for maximizing R total = R dl R ul under the SPC in the DL channel and PUPCs in the UL channel, which is formulated as maximize M,Q i,ul subject to R total tr(m M H ) P dl, tr(q j,ul ) P j,ul,j=1, 2,...,K ul, Q j,ul 0. where P dl is the total transmit at the BS and P i,ul is the power constraint of the ith user. Note that due to the multiuser inference in the DL channel, the optimal solution to the linear precoder design that maximizes the sum rate for the DL channel in (5) itself is NP-hard [6]. The precoder design for the considered system model in (7) 6826

3 R ul =log I K dl H SIM k,dl M H k,dl HH SI K ul H i,ulq i,ul H H i,ul I K dl H SIM k,dl M H k,dl HH SI. (6) this paper is even more difficult since the linear precoders and the input covariance matrices are coupled. Thus, it is practically useful to find low-complexity suboptimal solutions, which are presented in the next section. III. SUBOPTIMAL TRANSMISSION STRATEGIES We notice that the considered model is often asymmetric in reality, i.e., the sum rate in the DL channel is greatly larger than that of the UL channel. In other words, the total throughput of the system is dominantly contributed by the sum rate of the DL channel. Consequently, a trivial method is to maximize the sum rate of the DL and UL channels sequentially as described in the following. A. Sequential Design The sequential design first finds the linear precoders for the DL channel without caring for the interference that they cause to the UL channel. Since finding the optimal linear precoders of the DL channel is still an open problem, we focus on the suboptimal alternatives, which are easier to be solved. For simplicity, we adopt the block diagonalization (BD) scheme, proposed in [7], which is a zero-forcing (ZF) precoding method for multiple-antenna receivers. Basically, the BD scheme completely eliminates the multiuser interference by designing the precoders such that H M j,dl = 0, for all i j. This decomposes a DL MU-MIMO system into a group of K dl parallel SU-MIMO channels. To be specific, for the ith user, define H as H = [ H T 1,dl... H T i 1,dl H T i1,dl... H T K dl,dl], (8) and consider a singular value decomposition (SVD) of H as H = Ũi Λ [Ṽ(1) ] i i Ṽ (0) i, (9) where the columns of Ṽ(0) i form an orthogonal basis for the null space of H. To satisfy the ZF constraints in the BD scheme, we can write M = Ṽ(0) i ˆM. (10) Now, the sum rate of the DL channel in (5) is given by R BD = log I 1 H ˆM ˆM H N H H (11) 0 where H = H Ṽ (0) i is called the effective channel matrix for the ith user. The problem of computing ˆM that maximizes R BD in (11) can be easily solved using the water-filling algorithm with the total power P dl to the nonzero eigenvalues of H HH (cf. [7] for more details). Next, we consider finding the input covariance matrices for the UL channel with the obtained linear precoders from the DL channel. For notational convenience, let Φ = I H SIM k,dl M H k,dl HH SI. Then, (6) is equivalent to R ul =log I Φ 1 H i,ul Q i,ul H H i,ul =log I Φ 1/2 H i,ul Q i,ul H H i,ulφ 1/2. (12) Consequently, the optimal covariance matrices are the solution to the optimization problem maximize log I H i,ul Q i,ul HH i,ul Q i,ul (13) subject to tr(q i,ul ) P i,ul,, 2,...,K ul, Q i,ul 0, where H i,ul = Φ 1/2 H i,ul. Problem (13) is actually the problem of finding the capacity of a MIMO multiple-access channel, which can be solved efficiently using the iterative water-filling algorithm [8]. In terms of the total system throughput, this sequential design method, although simple, can achieve a good performance because it attempts to optimize the DL channel, which mostly determines the total system sum rate. However, when transmit power in the DL channel is high, and/or H SI is large, the sum rate of the UL channel obtained by this method is dramatically decreased, making the UL channel highly inefficient. In the sequel, we propose an approach that jointly designs the linear precoders in the DL channel and the input covariance matrices in the UL channel. B. Proposed Joint Design The joint design for the DL and UL channels aims at providing an improvement on the sum rate of the UL channel. Since the objective function in (7) is non-convex, its optimal solution is difficult to find. Herein, we propose to find a suboptimal design using a convex relaxation method. Note that, due to the concavity of the log det function, it follows that log det(ix) log det(ix 0 )tr ( (I X 0 ) 1 (X X 0 ) ) for X 0. This is the first-order approximation of the log det function [9]. In fact, this is an estimator of log det(i X) at X 0. The proposed design is based on an iterative method as follows. Suppose after n iterations, linear precoders and input covariance matrices are denoted by M (n) and Q (n) i,ul, respectively. For ease of description, let Q j,dl = M j,dl M H j,dl. Then the sum rates of the DL and UL channels are lower bounded by (14) and (15), respectively, shown on the top of the following page. In (14) and (15), Ω (n) i = I 1 j=1,j i H Q (n) ( j,dl HH, γ(n) i = (Ω log Ω (n) i 1 (n)) 1 ) tr i j=1,j i H Q (n) j,dl H, 6827

4 R dl {log I 1N0 H Q j,dl H H 1 ( (Ω (n)) 1 tr j=1 i j=1,j i R ul log I 1 H SI Q k,dl H H SI 1 H i,ul Q i,ul H H N i,ul 1 ( (Ψ (n) tr ) 1 K dl 0 ) } H Q j,dl H γ (n) i H SI Q k,dl H H SI (14) ) ξ (n) (15) Ψ (n) = I 1 H SIQ (n) k,dl HH SI and ξ(n) =log Ψ (n) (Ψ 1 (n) tr( ) 1 ) H SIQ (n) k,dl HH SI. In each iteration of the proposed design method, Q k,dl and Q k,ul are updated to maximize the lower bound of the total system sum rate. Mathematically, in the (n 1)th iteration, Q k,dl and Q k,ul are found to be the solution to the following problem maximize f 1 (Q )f 2 (Q, Q j,ul ) Q,Q j,ul subject to tr(q ) P dl (16) tr(q j,ul ) P j,ul,j=1, 2,...,K ul Q 0, Q j,ul 0. where f 1 (Q ) and f 2 (Q, Q j,ul ) are the right hand sides of (14) and (15), excluding the constants, which do not affect the optimization of the lower bound. Since the objective function of (16) is concave with respect to Q and Q j,ul, its optimal solution can be found efficiently using standard convex optimization packages, e.g., CVX [10]. The proposed design method is summarized in Algorithm 1. Algorithm 1 Proposed design for FD MU-MIMO systems 1: Randomly initialize: {Q (0) }K dl ; {Q(0) j,ul }K ul j=1. 2: Set: n := 1. 3: repeat 4: Solve (16) to find optimal solutions {Q }K dl {Q j,ul }K ul j=1. 5: Update covariance matrices: Q (n) and = Q ; Q(n) j,ul = Q j,ul. 6: Increase the number of iterations: n := n 1. 7: until convergence. 8: Apply the Cholesky decomposition to Q (n) to find the precoder M for the ith user in the DL channel. The proposed algorithm is guaranteed to converge to a local optimum of (7), which can be proved, following the same arguments as in [11]. After every iteration, the lower bound is increased. Moreover, the total sum rate of the system is bounded above. Thus, the proposed algorithm surely converges to a locally optimal solution. For the linear precoders Q k,dl to be feasible, we have to make sure that rank(q k,dl ) N k,dl. Lemma 1. Let Q k,dl be the optimal solutions to (16). Then it follows that rank(q k,dl ) N k,dl for all k =1, 2,..., K dl. Proof: Basically, the proof is based on embracing the Karush-Kuhn-Tucker (KKT) conditions of (16). The details of the proof is omitted due to the space limitation. IV. NUMERICAL RESULTS In this section, we numerically evaluate the performance of the proposed joint design, sequential design, and conventional scheme. A quasi-static fading channel is assumed, i.e, the channel matrices are constant during a transmission, but vary independently over transmissions. In the simulation setting, the entries of H and H i,ul are independently generated as zeromean complex Gaussian random variables with unit variance. Currently, to the best of our knowledge, there is no reference channel model for H SI. In this paper, we generate the elements of H SI as CN(0,σ 2 ), where σ 2 is varied to represent different degrees of the capability of the SI cancellation technique. We consider an exemplary system of two users in the DL channel and two users in the UL channel, each with two antennas. The number of transmit and receive antennas at the BS is 4 and 4, respectively. The initial values for Q and Q j,ul in the proposed iterative method is chosen randomly. The SNR dl in the DL channel, defined as SNR dl = P dl /, is set to be 15 db. For the UL channel, we simply assume P i,ul = P ul and define the SNR ul in the UL channel as SNR ul = P ul /, which is set to be 5 db. For a fair comparison, the BS in the system uses all the antennas, i.e., N t N r, to communicate with its users. We plot the sum rate of the DL and UL channels versus in Figs. 2 and 3, respectively. Conclusively, the FD transmission system can achieve a total system throughput greatly higher than the conventional system. In Fig. 3, when σ 2 10 db, the sum rate of the UL channel of the FD system is smaller than that of the system. In other words, the FD system is only efficient when the SI is small, for which the FD system had been considered impractical in the past due to the lack of efficient SI cancellation techniques. For the sequential design method, the sum rate of the DL channel is independent of σ 2. Consequently, when the SI increases, the sum rate of the UL channel is rapidly decayed. In contrast, the proposed joint design is more robust than the sequential design. Particularly, when the SI is small (σ 2 10 db), the proposed joint design method offers larger sum rate in both the DL and the UL channels. Another observation is that the sum rate of the UL channel of the joint design decreases slowly as the SI increases. Recall that, for the DL channel, the BD scheme is adopted in the sequential design method. In [11], it was shown that the precoder design using the lower bound in (14) yields a higher sum rate than BD. This explains the improvement of the joint design over σ

5 Downlink Sum Rate (b/s/hz) Total Sum Rate of the system (b/s/hz) Fig. 2. The sum rates of DL channels, SNR dl =15dB, SNR ul =5dB. Fig. 4. The total system sum rates of the UL and DL channels, SNR dl =15 db, SNR ul =5dB. Uplink Sum Rate (b/s/hz) Fig. 3. The sum rates of UL channels, SNR dl =15dB, SNR ul =5dB. the sequential method for the case of the small interference. On the other hand, when σ 2 is large, the sum rate of the DL channel in the joint design becomes smaller than that in the sequential design, but the sum rate of the UL channel is significantly improved, compared to the sequential method. This is because the joint design attempts to optimize the total system throughput. Thus, when the SI is large, the better strategy is to reduce the total transmit power in the DL channel to increase the capacity of the UL channel, which then results in a reduction on the sum rate of the DL channel and an increase on that of the UL channel. Interestingly, however, the total system throughput of the joint design method is still higher than that of the sequential design method, as illustrated in Fig. 4. That is, the decrease on the sum rate of the DL channel is smaller than the increase on that of the UL channel. V. CONCLUSIONS In this paper, we have introduced a FD transmission system, where the DL and UL channels are designed to operate over the same resources, and considered the design of transmission strategies for maximizing the total system throughput. For this problem, we have proposed a joint optimization approach that simultaneously optimizes the DL and UL channels using a convex relaxation method based on the iterative algorithm in which the linear precoders of the DL channel and a power allocation strategy of the UL channel are found. The simulation results indicate the superior performance of the FD MU-MIMO system over the system. We also hope to draw attention to the FD MU-MIMO system since, as demonstrated in this paper, it is a promising technique for small cell deployments where the transmit power is relatively small. REFERENCES [1] E. Telatar, Capacity of multi-antenna gaussian channels, Eur. Trans. Telecommun, vol. 10, pp , Nov [2] J. I. Choi, M. Jain, K. Srinivasan, P. Levis, and S. Katti, Achieving single channel, full duplex wireless communication, in Proceedings of the sixteenth annual international conference on Mobile computing and networking, ser. MobiCom 10, 2010, pp [3] M. Jain, J. I. Choi, T. Kim, D. Bharadia, S. Seth, K. Srinivasan, P. Levis, S. Katti, and P. Sinha, Practical, real-time, full duplex wireless, in Proceedings of the 17th annual international conference on Mobile computing and networking, ser. MobiCom 11, 2011, pp [4] J. Sangiamwong, T. Asai, J. Hagiwara, Y. Okumura, and T. Ohya, Joint multi-filter design for full-duplex MU-MIMO relaying, in Vehicular Technology Conference, VTC Spring IEEE 69th, Apr. 2009, pp [5] H. Weingarten, Y. Steinberg, and S. Shamai, The capacity region of the Gaussian multiple-input multiple-output broadcast channel, IEEE Trans. Inf. Theory, vol. 52, no. 9, pp , Sep [6] Z.-Q. Luo and S. Zhang, Dynamic spectrum management: Complexity and duality, IEEE J. Sel. Topics Signal Process., vol. 2, no. 1, pp , Feb [7] Q. Spencer, A. Swindlehurst, and M. Haardt, Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels, IEEE Trans. Signal Process., vol. 52, no. 2, pp , Feb [8] W. Yu, W. Rhee, S. Boyd, and J. Cioffi, Iterative water-filling for Gaussian vector multiple-access channels, IEEE Trans. Inf. Theory, vol. 50, no. 1, pp , Jan [9] S. Bold and L. Vandenberghe, Convex Optimization. Cambridge University Press, [10] M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming, version 1.21, Apr [11] Chris T. K. Ng and H. Huang, Linear precoding in cooperative MIMO cellular networks with limited coordination clusters, IEEE J. Sel. Areas Commun., vol. 28, no. 9, pp , Dec