A Limited Feedback Joint Precoding for Amplify-and-Forward Relaying

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 3, MARCH 2010 1347 A Limited Feedback Joint Precoding for Amplify--Forward Relaying Yongming Huang, Luxi Yang, Member, IEEE, Mats Bengtsson, Senior Member, IEEE, Björn Ottersten, Fellow, IEEE Abstract This paper deals with the practical precoding design for a dual hop downlink with multiple-input multiple-output (MIMO) amplify--forward relaying. First, assuming that full channel state information (CSI) of the two hop channels is available, a suboptimal dual hop joint precoding scheme, i.e., precoding at both the base station relay station, is investigated. Based on its structure, a scheme of limited feedback joint precoding using joint codebooks is then proposed, which uses a distributed codeword selection to concurrently choose two joint precoders such that the feedback delay is considerably decreased. Finally, the joint codebook design for the limited feedback joint precoding system is analyzed, results reveal that independent codebook designs at the base station relay station using the conventional Grassmannian subspace packing method is able to guarantee that the overall performance of the dual hop joint precoding scheme improves with the size of each of the two codebooks. Simulation results show that the proposed dual hop joint precoding system using distributed codeword selection scheme exhibits a rate or BER performance close to the one using the optimal centralized codeword selection scheme, while having lower computational complexity shorter feedback delay. Index Terms Amplify--forward relaying, dual hop, Grassmannian codebook, joint precoding, limited feedback, multipleinput multiple-output. I. INTRODUCTION T HE introduction of relaying technology in cellular networks shows large promise to increase coverage system capacity at a low cost is therefore considered in Manuscript received November 23, 2008; accepted September 09, 2009. First published November 06, 2009; current version published February 10, 2010. This work was supported in part by the National Basic Research Program of China by Grant 2007CB310603, the National Natural Science Foundation of China by Grants 60902012 60672093, the National High Technology Project of China by Grant 2007AA01Z262, Ph.D. Programs Foundation of the Ministry of Education of China under Grant 20090092120013, the European Research Council under the European Community s Seventh Framework Programme (FP7/2007-2013)/ERC Grant agreement no. 228044, by the Huawei Technologies Corporation. The associate editor coordinating the review of this manuscript approving it for publication was Dr. Shahram Shahbazpanahi. Y. Huang is with the School of Information Science Engineering, Southeast University, Nanjing 210096, China. He is also with the ACCESS Linnaeus Center, KTH Signal Processing Lab, Royal Institute of Technology, SE-100 44 Stockholm, Sweden (e-mail: huangym@seu.edu.cn). L. Yang is with the School of Information Science Engineering, Southeast University, Nanjing 210096, China (e-mail: lxyang@seu.edu.cn). M. Bengtsson is with ACCESS Linnaeus Center, KTH Signal Processing Lab, Royal Institute of Technology, SE-100 44 Stockholm, Sweden (e-mail: mats. bengtsson@ee.kth.se). B. Ottersten is with ACCESS Linnaeus Center, KTH Signal Processing Lab, Royal Institute of Technology, SE-100 44 Stockholm, Sweden. He is also with the securitytrust.lu, University of Luxembourg (e-mail: bjorn.ottersten@ee. kth.se). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSP.2009.2036061 IMT-Advanced stardization work such as 3GPP LTE-Advanced IEEE 802.16m. The same holds for Multiple-Input Multiple-Output (MIMO) technology [1] [7] its application in multiuser environments [8] [14]. As for the combination of MIMO relaying technology, most previous studies focus on the information theoretic limits for multi-antenna relay channels with different protocols. Capacity bounds of relaying channels in a single MIMO relay network have been developed in [15], where a regenerative MIMO relay is considered. For the multiple MIMO relay network, an asymptotical quantitative capacity result is presented in [16], where distributive diversity is achieved through cooperation among all the nonregenerative relays available in the network. This paper focuses on practical signalling design for a dual hop transmission with MIMO relay. Although the use of regenerative relays employing decode--forward (DF) shows advantages over nonregenerative relays using amplify--forward (AF) in many scenarios, it requires much higher delay tolerance may cause security problems, thus here we concentrate on the AF MIMO relaying strategy. For dual hop transmission with a single MIMO AF relay station, the optimal linear transceiver design at the relay-destination link has been developed [17], [18], assuming that the channel state information (CSI) of both the source-relay relay-destination links is available at the relay station. It is revealed that such a dual hop transmission can be transformed into several simultaneous data streams transmitted over orthogonal subchannels. In the case of multiple AF relay stations, a relay selection scheme is presented in [19] to exploit the additional diversity offered by the multiple relay stations available in the network, where the preferred relay station is chosen as a function of CSI to implement a dual hop transmission. Moreover, assuming that the CSI of all the links is available, a quasi-optimal joint design of linear transceivers at both the source-relay the relay-destination links is developed in [20] [21], which achieves very good performance while requiring high computational complexity. Note that the above dual hop transmit schemes all require full CSI of both two hop channels are unfortunately infeasible in practical frequency division duplex (FDD) systems, though they provide considerable performance gains. To overcome this problem, a limited feedback beamforming scheme for MIMO AF relaying was proposed in [22], which employs Grassmannian codebook to reduce the feedback overhead. It can even be extended to the case where the second order statistics of channel vectors are used instead of the limited instantaneous channel knowledge. However, this scheme is only limited in the beamforming case its extension to the precoding case (multiple simultaneous data streams) is nontrivial, which usually results in a rate performance loss especially when all the nodes 1053-587X/$26.00 2010 IEEE 转载

1348 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 3, MARCH 2010 are equipped with multiple antennas, due to the fact that the multiplexing gain offered by MIMO channels can not be fully exploited. In this paper we aim to design a practical dual hop transmit scheme which can fully exploit the multiplexing gains offered by multiple antennas. More specifically, we propose a limited feedback joint precoding scheme using the criterion of optimizing the system rate or the BER performance, where the reduction of both feedback overhead feedback delay will be fully considered. The main contributions are listed as follows: 1) We first present a CSI based suboptimal joint precoding scheme for a dual hop downlink with AF, where the overall dual hop MIMO channels can be effectively transformed into several orthogonal subchannels by using the optimal pairing between the eigenmodes of the dual MIMO channels. Based on this, we then propose a codebook based limited feedback joint precoding scheme, where a distributed codeword selection (CS) scheme is further proposed based on the newly derived bounds for the capacity the mean square error (MSE) sum of a dual hop MIMO transmission with a linear minimum mean square error (MMSE) receiver, such that the feedback burden feedback delay are both greatly reduced. 2) Furthermore, we investigate the codebook design for the proposed limited feedback joint precoding scheme, disclose that if the conventional method of Grassmannian subspace packing is separately employed to construct the codebooks at the base station relay station, the overall performance of the dual hop transmit scheme can be guaranteed to improve with the size of each of the two codebooks. The rest of this paper is organized as follows. In the next section we introduce the system model for the dual hop joint precoding. In Section III we investigate the expression of the optimal joint precoders based on full CSI, provide a suboptimal joint precoding scheme which can reduce to a limited feedback scheme. In Section IV we first present a codebook based joint precoding system using a centralized codeword selection scheme, then propose a distributed codeword selection scheme to reduce computational complexity feedback delay. In Section V we analyze the design criterion of the joint codebooks used in the dual hop precoding system. Simulation results are presented in Section VI conclusions are drawn in Section VII. II. SYSTEM MODEL We consider a dual hop downlink model which consists of a base station a relay station transmitting through two time slots. We assume that the base station is equipped with antennas, the relay station is equipped with antennas the user terminal is equipped with antennas. As depicted in Fig. 1, during the first slot, the base station employs linear precoding to transmit simultaneous data streams, i.e., a data vector, to the relay station. Without loss of generality, we assume, with denoting the expectation operator. The received baseb signal at the relay station is written as (1) Fig. 1. The signal model for the dual hop joint precoding system. where denotes the precoding matrix at the base station, without loss of generality, we assume with being the trace operator, denotes the first hop channel matrix between the base station the relay station, denotes the total transmit power at the base station denotes a white Gaussian noise vector with zero mean variance. Keeping in mind that a multiuser downlink can be transformed into several single-user downlinks by employing multiple access techniques such as TDMA OFDMA, here we concentrate on the single-user dual hop downlink. Moreover, we focus on relay deployments intended for coverage expansion, where the direct link between the base station the user terminal can be neglected due to path loss or severe shadowing. To succeed a downlink communication between the base station the user terminal, during the second slot the relay station will forward its received signal using a linear precoding matrix that has to be designed. With the transmit power constraint at the relay station, should satisfy that The received baseb signal at the user terminal during this time slot is written as where denotes the second hop channel matrix between the relay station the user terminal, denotes a white Gaussian noise vector with zero mean variance. Note that in the above system model we can normalize the variances of both, have the effects of large scale fading incorporated into the noise variances of. The key point of the above dual hop joint precoding system lies in the design of two precoders, which commonly requires channel information feedback in FDD systems. Also, the number of simultaneous data streams should be carefully determined. It is well known that a MIMO channel with transmit antennas receive antennas can be transformed into a maximum of orthogonal subchannels via singular value decomposition (SVD). The simultaneous transmission of data streams over orthogonal subchannels can fully utilize the multiplexing gain is thereby capacity-approaching, while the scheme of always transmitting a single data stream in general cannot achieve the (2) (3)

HUANG et al.: A LIMITED FEEDBACK JOINT PRECODING 1349 potential rate offered by MIMO channels, due to the fact that the multiplexing gain cannot be fully exploited in this case. This result can be easily extended to the dual hop MIMO transmission. Considering that the overall performance of the dual hop downlink is dominated by the worse one of the two hops, it is reasonable to choose the number of simultaneous data streams in our system equal to if possible, instead of always using a single data stream regardless of antenna configuration, such that the overall rate performance can be optimized. where,,, are unitary matrices, are diagonal matrices with their elements being the singular values of, respectively. Obviously, the ordering of the singular values in ( the corresponding ordering of the singular vectors in,, 2) influences the specific decomposition expressions. Here we first assume an arbitrary ordering leave its optimization to be solved later. By substituting (6) in (4) the MSE matrix can be rewritten as III. JOINT PRECODING WITH FULL CSI This section concentrates on the design of two joint precoders assuming that full channel state information of the two hops is available. In difference to the previous related work which aims at the optimal performance by using an iterative approach, we are more interested in the suboptimal scheme which has a simple structure can provide some insight on the design of a limited feedback joint precoding scheme. We consider an MMSE receiver at the user terminal, as shown in [17], [18], the MSE matrix for the dual hop joint precoding can be written as (4), shown at the top of the next page. The diagonalization of can be obtained by (7) (8) (9) where denotes the submatrix formed by the first columns of, are two diagonal matrices with nonnegative elements denoted as, respectively. We partition the matrices,, as (10) where,, all belong to,,,. By substituting (8) (10) in (7), the MSE matrix can be simplified as The sum rate achieved by an MMSE receiver is upper bounded by the instantaneous capacity, which can be expressed as [17], [23] where denotes the th diagonal element of, denotes of the determinant of, the factor 0.5 is due to the two channel uses which are needed by a dual hop downlink, will be omitted henceforth for convenience. Obviously, the equality in (5) holds when is diagonal, which means that the capacity is achieved by an MMSE receiver in this case. Therefore, the design of should first satisfy the condition that the MSE matrix is diagonalized [19]. Let the SVD of be (4) (5) (6) (11) Then, if we further express as, respectively, the achieved sum rate can be easily derived as (12) It is shown that with the above joint precoding, the overall dual hop channel can be transformed into orthogonal subchannels, with their channel gains each represented by the product of a pair of eigenmodes, while the diagonal matrices can be viewed as the power allocation for the joint precoding. Since does not influence the sum rate, it should be set to zero to avoid wasting power. The resulting precoding matrix at the relay station is (13) where denote the submatrices formed by the first columns of, respectively. Aiming to maximize the sum rate of the dual hop transmission, we need to optimize the

1350 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 3, MARCH 2010 power allocation matrices by solving the following optimization problem: (14) where the two constraints are obtained from the power constraints at the base station the relay station. Specifically, the first constrain is obtained by substituting (8) in, while the second constraint is obtained by substituting (8) (9) in (2). We would like to note here that this optimization should be done over the optimal ordering of the singular values at the SVD of, since different ordering will give different values of. Defining new notations,,,, by replacing the notations in (14) with the newly defined notations, the above optimization problem can be simplified as (15) It is clear from the above steps that the ordering of singular values at the SVD of influences both the sum rate the specific expression of the optimal joint precoders. Thus, the joint optimal ordering of singular values at the SVD of needs to be addressed. It is seen from (12) that only singular values of each hop, i.e., eigenmodes of each hop, affect the sum rate. Therefore, the problem reduces to the optimal selection of active eigenmodes from each hop followed by the optimal pairing of active eigenmodes between the two hops. Since the sum rate expressed in (12) monotonically increases with both the eigenmode the eigenmode, the scheme of selecting the largest eigenmodes from each hop will give a maximum sum rate. Moreover, it is found from (15) that the eigenmode pairing problem is equivalent to the subchannel pairing problem of the dual hop MIMO-OFDM systems in [20]. The results in [20] showed that it is optimal to pair the active eigenmodes of the first hop ordered in with the active eigenmodes of the second hop ordered in, which means that the optimal joint precoders should be given by the SVD of both having its singular values arranged in a nonincreasing order. For notation simplicity, henceforth the SVD expressions of refer to a nonincreasing ordering of singular values. It should be noted that although (8) (13) provide a simple expression for the optimal joint precoders, the closed-form solution for the included power allocation matrices are difficult to obtain. Hammerström et al. [20] showed that the optimization problem in (15) cannot be exactly solved but its quasi-optimal solution can be obtained using an iterative method, the optimal power allocation schemes at both the base station relay station are similar to the waterfilling scheme in point-to-point MIMO systems. Since it is well known that an uniform power allocation (UPA) in general only suffers from slight performance loss compared to the optimal waterfilling scheme, while having lower cost reduced feedback burden in FDD systems, we will use UPA to form a suboptimal joint precoding scheme. Next we will show that such a UPA based dual hop joint precoding scheme can reduce to a practical limited feedback joint precoding scheme. IV. LIMITED FEEDBACK PRECODING By employing UPA, it is seen from (8), (13) that the joint precoders with full channel knowledge of can be simplified as (16) (17) where is a common scaling to fulfill the transmit power constraint at the relay station. Since it is reasonable to assume that is available at the relay station available at the user terminal, the above joint precoding solution requires the feedback of to the base station to the relay station. In order to reduce the feedback burden, we use two codebooks to quantize, such that, similar to the precoding for point-to-point MIMO systems, only the indices of the preferred codewords are required to be fed back to the base station relay station, respectively. However, the extension of point-to-point precoding to a dual hop transmission is nontrivial the following problems need to be addressed. 1) Though the optimal depend on, respectively, the codebook based choice of the precoder at the base station or the relay station is in general a function of both. In practical FDD systems, however, only the user terminal may know the channel of both two hops without feedback. If both two precoders are selected by the user, it will suffer from a severe feedback delay due to the fact that the communication between the base station the user terminal has to be forwarded by the relay station. Therefore, the precoder selection feedback scheme should be carefully designed to reduce the feedback delay. 2) The criterion for precoding codebook design has been widely studied in point-to-point MIMO communication systems. However, it is an open problem whether these developed codebook design criteria can be directly employed in the dual hop joint precoding systems. In order to address the first problem, we first present a centralized codeword selection scheme which provides the optimal performance but a high feedback delay. Then, we propose a suboptimal distributed codeword selection scheme where feedback delay complexity are both greatly reduced. A. Centralized Codeword Selection We employ precoding according to (16) (17) assume that two codebooks for have been designed denoted as, respectively. In order to maximize the ca-

HUANG et al.: A LIMITED FEEDBACK JOINT PRECODING 1351 pacity expressed in (5), the codeword selection for can be written as terminal needs), the distributed codeword selection for should be merely based on, respectively, such that the feedback overhead feedback delay can be considerably reduced. To this end, a new objective function, either from the capacity or the error rate perspective, should be designed. In this section we will derive bounds for the capacity the MSE-trace, then use them as the objective functions. By replacing with its SVD expression, the MSE matrix in (4) can be simplified as (18) Alternatively, considering that the minimization of the trace of MSE matrix means to some degree the optimization of the error rate performance of an MMSE receiver, an MSE-trace selection scheme aiming to minimize the error rate may be employed is expressed as (20) Based on this, the capacity of the dual hop transmission can be lower bounded by (19) Obviously, the codeword selection either from the sum rate or the error rate perspective is a function of both, which requires the selection operator to know full CSI of both two hops, thereby is called a centralized codeword selection scheme. Due to the fact that each calculation of the objective function includes one or two matrix inversions, this centralized selection scheme requires a high computational complexity. Moreover, since full knowledge of the two hop channels may only be available at the user terminal without feedback in practical FDD systems, the codeword selection for should be both conducted by the user. Unfortunately, the feedback of selection result for from the user terminal to the base station has to be forwarded by the relay station, which results in a high delay. B. Distributed Codeword Selection In order to reduce the feedback latency, we propose a distributed codeword selection scheme where the codeword selections for can be concurrently conducted by the relay station the user terminal, respectively. Since in practical systems only can be available at the relay station without feedback, while only can be easily available at the user terminal ( should be fed forward by the relay station if the user (21) where,,,, are the eigenvalues of the Hermitian matrix arranged in a nonincreasing order. For a proof, refer to Appendix A. Note that this capacity lower bound increases with both, namely, the lower bound increases if is increased, for any value of, or increases if is increased, for any value of. Since merely depend on respectively, the following distributed codeword selection scheme for, will maximize the lower bound of the capacity (22) In order that the proposed distributed codeword selection scheme can minimize the error rate of the dual hop transmission, we derive two upper bounds for the MSE trace. Both decrease with two decoupled functions of, can

1352 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 3, MARCH 2010 be utilized as the codeword selection criteria. Based on (20), upper bounds of the MSE trace can be expressed as (23) (24) See Appendix B for proofs. Obviously, minimization of the upper bound in (23) is equivalent to the maximization of the lower bound in (21). Thus, the distributed codeword selection scheme of (22) also works from the perspective of minimizing the error rate. In addition, since the upper bound in (24) is formed by a sum of two functions of, an alternative distributed codeword selection scheme, to optimize the error rate performance, is given as (25) V. CODEBOOK DESIGN CRITERIA We have derived codeword selection schemes for the dual hop joint precoding system, it is important that the codebook pair of are designed specifically for the chosen selection schemes. Love et al. [5] have shown that the criterion of maximizing the minimum Grassmannian subspace distance between any pair of codewords is quasi-optimal for point-to-point precoding systems. In dual hop precoding systems using the proposed distributed codeword selection scheme, our following analysis shows that a separate design for using the conventional Grassmannian subspace packing method is able to guarantee that the overall performance increases with the size of each of the two codebooks. To define a notion of an optimal codebook, we need a distortion measure with which to measure the average distortion. It is seen from (21), (23), (24) that when the term is maximized, the lower bound of capacity will be maximized, the upper bound of MSE trace will be minimized as well. Thus, we utilize this term as a performance metric define the following error difference: C. Distributed Beamforming Selection In general, the proposed distributed codeword selection schemes for the joint precoding system are able to reduce both the overall feedback delay the computational complexity, while they may suffer from a performance loss compared to the centralized selection scheme, due to the fact that the employed selection objective functions are not the exact capacity or the MSE trace, but their bounds. However, our following brief analysis shows that the proposed distributed selection scheme in the special case of beamforming (it happens when ) will suffer from no performance loss as compared with the centralized one, which is consistent with the result found in [22], though different analyzing methods are used. For the beamforming case, the MSE matrix in (20) reduces into a scalar can be written as (26) As now both reduce to scalars, their eigenvalues are equal to themselves. Also, it follows from (2), (16), (17) that Substituting (27) in (26), yields (27) (28) (29) which is nonnegative for any choices of, since the first term is the performance metric obtained by the optimal precoders of. Furthermore, we will design our codebook pair to minimize the average distortion (30) where denotes the expectation with respect to. If we define the minimum distances of the codebook pair, as (31) namely, the so-called projection two-norm distance between two subspaces is employed, the average distortion can be upper bounded as where. It can be easily derived that the MSE is minimized when both are maximized, which means that the proposed distributed codeword selection schemes are optimal from the perspective of both the capacity the error rate. (32)

HUANG et al.: A LIMITED FEEDBACK JOINT PRECODING 1353 where denote the sizes of the codebooks, respectively. For a proof, refer to Appendix C. Similar to the conclusion in [5], assuming that, we always have that the average distortion is decreased with both. Thus, we can design the codebook pair separately, with each codebook constructed to maximize the minimum projection two-norm distance between any pair of codewords. VI. SIMULATION RESULTS Monte Carlo simulations are performed to illustrate the performance of the proposed dual hop joint precoding system with distributed centralized codeword selection schemes. A block fading flat MIMO channel model is used throughout the simulations. The two hop channel matrices are both assumed to have entries independently identically distributed with, with the large scale factors of channels incorporated into the effective noise variances. The antenna configurations are focused on,. The Grassmannian codebook provided in [24] is employed in our simulations, we use the same codebook at the base station relay station, with its size shown in figures in terms of the number of feedback bits. The average SNR at the relay station the user terminal are defined as, respectively. For comparison, some optimal or suboptimal dual hop precoding systems based on full channel state information are also simulated, where the hereinafter mentioned joint optimal scheme denotes the precoding system in (8) (13), the suboptimal scheme denotes the precoding system in (16) (17) with uniform power allocation, the relay side optimal scheme denotes the system in [17], [18], where only the precoding matrix at the relay station is optimized based on full CSI, its rate performance is calculated as the information theoretic instantaneous capacity of an equivalent open-loop MIMO system. Note that in the case of, the joint optimal precoding can not be analytically solved since the objective function in (15) is not concave with respect to. Here we use the alternating optimization method presented in [20] to find the global or local optimum repeat it with 50 romly generated starting vectors, using the maximum one in comparison. A. Dual Hop Joint Beamforming This section focuses on the configurations. Since the receiver is only equipped with single antenna, a joint beamforming, i.e.,, should be employed. As disclosed in Section IV-C, in this case the proposed distributed codeword selection scheme will not result in any performance loss as compared with the centralized codeword selection scheme, it reduces to the same scheme as the one presented in [22]. Fig. 2 shows that the proposed dual hop joint beamforming scheme using distributed CS exhibits slight rate loss as compared with the full CSI based joint optimal beamforming scheme, especially for the case of. Fig. 3 illustrates the cumulative distribution function of the rate achieved by the dual hop joint beamforming, the results also show a slight gap between the proposed limited feedback joint beamforming scheme the Fig. 2. The rate of the dual hop joint beamforming system with, 15 db SNR at the relay station. Fig. 3. The cumulative distribution functions of the rate achieved by the proposed dual hop joint beamforming system, with, 15 db SNR at both the receiver relay station. joint optimal scheme. Fig. 4 illustrates the BER performance of the proposed dual hop joint beamforming using QPSK modulation. Similar results are also observed. B. Dual Hop Joint Precoding This section focuses on the configurations. Fig. 5 shows the sum rate of the dual hop joint precoding system using two different codeword selection schemes. It is seen that the performances of the dual hop joint precoding schemes using distributed centralized CS both increase with the codebook size. Compared with the centralized CS, the distributed CS suffers from a slight rate loss. This is because the distributed CS is based on a bound but not an exact rate metric. However, the distributed CS has a shorter feedback delay requires much lower computational complexity. Also, it is reasonable to see that even the scheme using centralized CS has a gap from the full CSI based suboptimal scheme, due to the quantization of the optimal joint precoders.

1354 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 3, MARCH 2010 Fig. 4. station. The BER of the proposed dual hop joint beamforming system with, 15 db SNR at the relay Fig. 6. The cumulative distribution functions of the rate achieved by the proposed dual hop joint precoding system, with, 15dB SNR at both the receiver relay station. Fig. 5. The rate of the dual hop joint precoding system with 15 db SNR at the relay station. Fig. 7. The BER of the proposed dual hop joint precoding system with 15 db SNR at the relay station. Moreover, the results reveal that the proposed joint precoding scheme with distributed CS shows obvious advantage over the beamforming scheme presented in [22] in terms of the rate performance, especially in medium-to-high SNR regions. This is due to the fact that our proposed precoding scheme employs multiple simultaneous data streams thus can fully exploit the multiplexing gain offered by the dual hop MIMO channels. Fig. 6 shows the cumulative distribution function of the rate achieved by the dual hop joint precoding system. Similar results are seen as in Fig. 5. Interestingly, it is also found from Fig. 5 Fig. 6 that the full CSI based suboptimal scheme with UPA shows slight performance loss as compared with the joint optimal scheme, only in the range of medium-to-high SNRs. And, the relay side optimal scheme shows the worst performance among three full CSI based schemes, especially in high SNR region. This is due to the fact that the precoder at the base station is not optimized. It should also be noted that, though it seems from the curves that the relay side optimal scheme outperforms the proposed scheme in most of the SNR region, this is a result of unfair comparison, where the performance of the relay side optimal scheme is calculated as the instantaneous capacity, but not the sum rate achieved by an MMSE receiver. Fig. 7 shows the BER performance of the dual hop joint precoding scheme using QPSK MMSE receiver. Both the proposed two distributed CS schemes, i.e., (22) (25), are simulated. It is seen that the BER performance of these two schemes (denoted as distributed CS #1 #2) are very close, they both increase with the codebook size. Compared with the centralized CS scheme, a loss of less than 2 db is observed in the proposed two distributed CS schemes. VII. CONCLUSION In this paper we have presented a limited feedback joint precoding for the dual hop downlink with amplify--forward relaying. The proposed scheme employs a distributed codeword selection thus has lower computational complexity feedback delay. Also, we have analyzed the joint codebook

HUANG et al.: A LIMITED FEEDBACK JOINT PRECODING 1355 design for the joint precoding system, revealed that a separate codebook design for the base station the relay station using Grassmannian subspace packing method can guarantee that the overall performance of the proposed scheme improves with the size of each of the two codebooks. Finally, computer simulations have confirmed the advantage of the proposed scheme in terms of the tradeoff between performance complexity, as compared with the limited feedback joint precoding with a centralized codeword selection. APPENDIX A PROOF OF (21) We first present the following matrix inequalities [25]: Given two positive semidefinite Hermitian matrices with eigenvalues arranged in nonincreasing order, respectively, we have (33) Since the matrix determinant equals the product of the eigenvalues, the capacity of the dual hop transmission with precoders can be rewritten as Assuming that the relay station transmit signal with full power, it is derived from (2) that Thus, we further have (36) (37) This concludes the proof. APPENDIX B PROOF OF (23) AND (24) We first prove the first upper bound of the MSE trace in (23) (38) (34) where By applying the inequality in (33), this yields the inequality in (a) comes from the lower bound of, which has been derived in Appendix A. Similar to the above derivation, the second upper bound of the MSE trace in (24) can be obtained as follows: (39) (35) This concludes the proof.

1356 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 3, MARCH 2010 (41) APPENDIX C PROOF OF (32) Before the proof of (32), we first give the following inequality. Given arbitrary nonnegative variables,,, we have [22, Lemma 1] (40) With that, the average distortion can now be upper bounded as shown in (41) at the top of the page, where the inequality is a result of direct use of (40). Based on the results in [5, eq. 29, 30], the two terms in the right-h side (RHS) can be further upper bounded as (42) (43) Thus, the upper bound of the average distortion can modified as This concludes the proof. (44) ACKNOWLEDGMENT The authors would like to thank all the anonymous reviewers the editor for their valuable comments that have helped to improve the quality of this paper. REFERENCES [1] G. J. Foschini M. J. Gans, On limits of wireless communications in a fading environment when using multiple antennas, Wireless Pers. Commun., vol. 6, no. 3, pp. 311 335, 1998. [2] G. J. Foschini, Layered space-time architecture for wireless communication in fading environment when using multielement antennas, Bell Labs Tech. J., vol. 1, no. 2, pp. 41 59, Aug. 1996. [3] A. Scaglione, P. Stoica, S. Barbarossa, G. B. Giannakis, H. Sampath, Optimal designs for space-time linear precoders decoders, IEEE Trans. Signal Process., vol. 50, no. 5, pp. 1051 1064, May 2002. [4] H. Sampath, P. Stoica, A. Paulraj, Generalized linear precoder decoder design for MIMO channels using the weighted MMSE criterion, IEEE Trans. Commun., vol. 49, no. 12, pp. 2198 2206, Dec. 2001. [5] D. J. Love R. W. Heath, Jr., Limited feedback unitary precoding for spatial multiplexing systems, IEEE Trans. Inf. Theory, vol. 51, no. 8, pp. 2967 2976, Aug. 2005. [6] Y. Huang, D. Xu, L. Yang, W. P. Zhu, A limited feedback precoding system with hierarchical codebook linear receiver, IEEE Trans. Wireless Commun., vol. 7, no. 12, pp. 4843 4848, Dec. 2008. [7] S. M. Alamouti, A simple transmit diversity technique for wireless communications, IEEE J. Sel. Areas Commun., vol. 16, no. 8, pp. 1451 1458, Oct. 1998. [8] H. Weingarten, Y. Stenberg, S. Shamai, The capacity region of the Gaussian multiple-input multiple-output broadcast channel, IEEE Trans. Inf. Theory, vol. 52, no. 9, pp. 3936 3964, Sep. 2006. [9] Q. H. Spencer, A. L. Swindelhurst, M. Haardt, Zero forcing methods for downlink spatial multiplexing in multiuser MIMO channels, IEEE Trans. Signal Process., vol. 52, pp. 461 471, Feb. 2004. [10] K. K. Wong, R. Murch, K. B. Letaief, A joint-channel diagonalization for multiuser MIMO antenna systems, IEEE Trans. Wireless Commun., vol. 2, pp. 773 786, July 2003. [11] N. Jindal, MIMO broadcast channels with finite-rate feedback, IEEE Trans. Inf. Theory, vol. 52, pp. 5045 5060, Nov. 2006. [12] M. Sharif B. Hassibi, On the capacity of MIMO broadcast channels with partial side information, IEEE Trans. Inf. Theory, vol. 51, pp. 506 522, Feb. 2005. [13] D. Xu, Y. Huang, L. Yang, B. Li, Linear transceiver design for multiuser MIMO downlink, in Proc. IEEE Int. Conf. Commun., May 2008, pp. 761 765. [14] Y. Huang, L. Yang, J. Liu, A limited feedback SDMA for downlink of multiuser MIMO communication system, EURASIP J. Adv. Signal Process., vol. 2008, Oct. 2008. [15] B. Wang, J. Zhang, A. Høst-Madsen, On the capacity of MIMO relay channels, IEEE Trans. Inf. Theory, vol. 51, no. 1, pp. 29 43, Jan. 2005.

HUANG et al.: A LIMITED FEEDBACK JOINT PRECODING 1357 [16] H. Bolcskei, R. U. Nabar, O. Oyman, A. J. Paulraj, Capacity scaling laws in MIMO relay networks, IEEE Trans. Wireless Commun., vol. 5, no. 6, Jun. 2006. [17] O. Munoz-Medina, J. Vidal, A. Agustin, Linear transceiver design in nonregenerative relays with channel state information, IEEE Trans. Signal Process., vol. 55, no. 6, pp. 2593 2604, Jun. 2007. [18] X. Tang Y. Hua, Optimal design of non-regenerative MIMO wireless relays, IEEE Trans. Wireless Commun., vol. 6, no. 4, pp. 1398 1407, Apr. 2007. [19] Y. Fan J. Thompson, MIMO configurations for relay channels: Theory Practice, IEEE Trans. Wireless Commun., vol. 5, no. 5, pp. 1774 1786, May 2007. [20] I. Hammerström A. Wittneben, Power allocation schemes for amplify--forward MIMO-OFDM relay links, IEEE Trans. Wireless Commun., vol. 6, no. 8, pp. 2798 2802, Aug. 2007. [21] Z. Fang, Y. Hua, J. C. Koshy, Joint source relay optimization for a non-regenerative MIMO relay, in Proc. IEEE Workshop Sens. Array Multichannel Signal Process., Jul. 2006, pp. 239 243. [22] B. Khoshnevis, W. Yu, R. Adve, Grassmannian beamforming for MIMO amplify--forward relaying, IEEE J. Sel. Areas. Commun., vol. 26, no. 8, pp. 1397 1407, Oct. 2008. [23] R. W. Heath, Jr., S. Shu, A. Paulraj, Antenna selection for spatial multiplexing systems with linear receivers, IEEE Commun. Lett., vol. 5, no. 4, pp. 142 144, Apr. 2001. [24] D. J. Love, Personal Webpage on Grassmannian Subspace Packing [Online]. Available: http://dynamo.ecn.purdue.edu/djlove/grass.html [25] H. Sha, Estimation of the eigenvalues of for,, Linear Algebra Its Appl., vol. 73, pp. 147 150, 1986. Yongming Huang received the B.S. M.S. degrees from Nanjing University, China, in 2000 2003, the Ph.D. degree from the School of Information Science Engineering, Southeast University, China, 2007, respectively. Since 2007, he has been an Assistant Professor with the School of Information Science Engineering, Southeast University. In December 2008, he joined in the Signal Processing Lab, Electrical Engineering, Royal Institute of Technology (KTH), Stockholm, Sweden, as a Postdoctoral Researcher. His current research interest includes MIMO communication systems, multiuser MIMO communications, cooperative communications. Luxi Yang (M 96) received the M.S. Ph.D. degree in electrical engineering, from the Southeast University, Nanjing, China, in 1990 1993, respectively. Since 1993, he has been with the Department of Radio Engineering, Southeast University, where he is currently a professor of Information Systems Communications, the Director of Digital Signal Processing Division. His current research interests include signal processing for wireless communications, MIMO communications, cooperative relaying systems, statistical signal processing. He is the author or coauthor of two published books more than 100 journal papers, holds 10 patents. Prof. Yang received the first- second-class prizes of Science Technology Progress Awards of the State Education Ministry of China in 1998 2002. He is currently a member of Signal Processing Committee of Chinese Institute of Electronics. Mats Bengtsson (M 00 SM 06) received the M.S. degree in computer science from Linköping University, Linköping, Sweden, in 1991 the Tech. Lic. Ph.D. degrees in electrical engineering from the Royal Institute of Technology (KTH), Stockholm, Sweden, in 1997 2000, respectively. From 1991 to 1995, he was with Ericsson Telecom AB Karlstad. He currently holds a position as Associate Professor with the Signal Processing Laboratory, School of Electrical Engineering, KTH. His research interests include statistical signal processing its applications to antenna-array processing communications, radio resource management, propagation channel modeling. Dr. Bengtsson served as Associate Editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING during 2007 2009 is a member of the IEEE SPCOM Technical Committee. Björn Ottersten (S 87-M 89-SM 99-F 04) was born in Stockholm, Sweden, in 1961. He received the M.S. degree in electrical engineering applied physics from Linköping University, Linköping, Sweden, in 1986. In 1989, he received the Ph.D. degree in electrical engineering from Stanford University, Stanford, CA. He has held research positions with the Department of Electrical Engineering, Linköping University, the Information Systems Laboratory, Stanford University, the Katholieke Universiteit Leuven, Leuven, the University of Luxembourg. During 1996 1997, he was Director of Research at ArrayComm, Inc., a start-up company in San Jose, CA, based on Ottersten s patented technology. In 1991, he was appointed Professor of Signal Processing at the Royal Institute of Technology (KTH), Stockholm. From 1992 to 2004, he was head of the Department for Signals, Sensors, Systems at KTH from 2004 to 2008 he was dean of the School of Electrical Engineering at KTH. Currently, he is Director for the Interdisciplinary Centre for Security, Reliability Trust at the University of Luxembourg. His research interests include security trust, reliable wireless communications, statistical signal processing. Dr. Ottersten has served as Associate Editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING on the editorial board of the IEEE Signal Processing Magazine. He is currently Editor-in-Chief of the EURASIP Signal Processing Journal a member of the editorial board of the EURASIP Journal of Applied Signal Processing. He has coauthored papers that received the IEEE Signal Processing Society Best Paper Award in 1993, 2001, 2006. He is a Fellow of the EURASIP. He is a first recipient of the European Research Council advanced research grant.