I. Introduction. Index Terms Multiuser MIMO, feedback, precoding, beamforming, codebook, quantization, OFDM, OFDMA.

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1 Zero-Forcing Beamforming Codebook Design for MU- MIMO OFDM Systems Erdem Bala, Member, IEEE, yle Jung-Lin Pan, Member, IEEE, Robert Olesen, Member, IEEE, Donald Grieco, Senior Member, IEEE InterDigital Communications, LLC, untington Quadrangle, Melville, New York 77 Abstract-- Zero-forcing (ZF) beamforming, an efficient technique for multiuser MIMO systems, requires perfect channel state information to be available at the base station. This is achieved by quantizing the channel and feeding back the information with a limited number of bits. A challenge of this method is the large downlink signaling overhead, which is due to the large size of the codebook that can be used at the base station for ZF beamforming. In this paper, we first analyze this problem and then present a technique designing codebooks with reduced size and minimal performance degradation. We also define the associated procedures required for the system implementation. Simulation results show that the size of the enodeb codebook can be substantially reduced with only a small ( %) penalty in performance. Index Terms Multiuser MIMO, feedback, precoding, beamforming, codebook, quantization, OFDM, OFDMA. I. Introduction The use of multiple transmit and receive antennas can significantly increase the spectral efficiency of wireless systems [], []. In a wireless system, the base station (BS, or also called as the NodeB) is usually equipped with more antennas than the mobile stations; therefore, the multiplexing gain is limited by the number of antennas at the mobiles, and we cannot get a linear increase in the number of transmit antennas. The full multiplexing gain, however, can be achieved if the base station transmits to multiple users simultaneously in a multiuser MIMO (MU-MIMO) system. In the downlink of a multiuser MIMO system, where the coordination among users is not possible, a receiver generally cannot cancel the interference coming from the data transmitted to the other users []. If the channel state information (CSI) of all the users is available at the BS, multiuser interference can be cancelled or reduced with proper preprocessing. One such approach, called dirty-paper coding (DPC) [7], has been shown to achieve the sum-rate capacity of MIMO broadcast channels [], [], []. Due to the complexity of DPC, suboptimal techniques have also been investigated. One of these approaches is zero-forcing (ZF) beamforming, which has been shown to achieve the same asymptotic gains as with the DPC []. In ZF beamforming, the data streams for users are multiplied by precoding vectors selected to ensure that the interference between the data streams of different users is cancelled. Both DPC and ZF beamforming require perfect CSI to be available at the BS. This is not easy, especially for FDD systems in which the downlink and uplink channels are on different frequency bands. To address this problem, several techniques that require only partial information have been proposed ([], [] study single-user systems). For example, a technique that uses the idea of random beamforming has been proposed in [9]. In this method, the base station generates random orthonormal beamforming vectors; the mobile stations compute an SINR value for each of the vectors, assuming that the others are causing interference. The base station then assigns each beam to the user with the largest corresponding SINR. A similar method was previously proposed in [8]. Another method that relies on partial feedback is based on quantizing the CSI with a few bits and feeding back the bits to the BS, which then uses this information to design a ZF beamforming matrix [], []. In [], the authors assume that the number of users is equal to the number of transmit antennas, and each user receives a single data stream. In [], the authors extend this technique to the case in which there are more users than transmit antennas. ere, in addition to the quantized channel information, each user also feeds back a channel quality indicator (CQI) that is based on the received SINR. The methods summarized above try to minimize the signaling overhead from the mobile station to the BS, i.e., in the uplink direction, by reducing the feedback rate. But the signaling in the other direction, i.e.. the downlink, has not been considered. When zero-forcing precoding is used, for example, the BS needs to inform each user of the precoding matrix used for transmission so that the users can design their receivers. Due to the large number of possible ZF precoding matrices, the required number of signaling bits in the downlink is usually very large and causes significant signaling overhead. Therefore, reducing the number of possible precoding matrices without causing severe performance degradation is a challenge that needs to be addressed. In this paper, we propose a method to reduce the signaling overhead in the downlink without significantly degrading system performance. The method is based on quantizing the ZF precoding codebook, which comprises all possible precoding matrices. We show how the quantized codebook is designed and also describe the associated algorithms: the user-selection, downlink control signaling, and CQI computation. The proposed technique consists of (a) Quantizing the ZF precoding codebook, using an iterative technique based on the Lloyd algorithm []; (b) Excluding combinations of quantized channel indicators that are highly correlated to improve the performance of the quantization process. We also show how the proposed approach could be extended to a DFT based BS codebook. The outline of this paper is as follows: We first explain the system model and review the ZF MU-MIMO approach. Then, we investigate the BS ZF precoding

2 codebook and present the proposed methods. Finally, we present simulation results that quantify the performance of the proposed methods. II. System Model and ZF MU-MIMO We assume the base station has M transmit antennas, and there are L active users each equipped with a single receive antenna. users out of the L active users are selected by the BS for MU-MIMO transmission, and a single data stream is transmitted to each user. Let s k be the data symbol that is to be transmitted to the k th user, w k be the (M x ) beamforming vector that is to be used for transmission to this user, and P k be the power allocated. Then, the transmitted signal from the BS is written as x = Pkw ksk. For the k th user, k = y = P hws + P hws +nk k k k k k j k j j j=, j k is the received signal, where h k denotes the ( x M) channel between user k and the BS. The first part of the received signal is the data stream transmitted to user k, the second part is data transmitted to the other users, i.e., inter-user (or inter-stream) interference, and the third part is the additive noise. In ZF beamforming, the beamforming vectors are selected such that the inter-user interference gets cancelled, i.e., the condition hw k j =, for k j is satisfied. One way of guaranteeing the zero inter-user interference condition is to compute the beamforming vectors from the pseudo-inverse of the composite channel matrix T T T T = [ h h... h ]. Let W = [ w w... w ] be the composite beamforming matrix with normalized vectors w k =. Then, the zero inter-user interference condition is satisfied if W = = ( ). When is poorly conditioned, the effective channel gain might be greatly reduced degrading the performance. For optimal performance, the MU-MIMO users should be selected from the L active users such that the channels of the selected users are nearly orthogonal and have large gains []... Channel Quantization To achieve the optimal performance of the ZF beamforming approach, perfect CSI of all users is required at the BS. The estimated channel is quantized according to a given channel quantization codebook (also called the user equipment [UE] codebook), and then the index from the codebook is transmitted to the BS. Due to the practical limits on the capacity of the feedback channel, the number of bits available to represent the channel is limited. Under these circumstances, the ZF beamforming matrix W computed at the BS does not guarantee zero inter-user interference due to the channel quantization error. Assume that UE codebook consists of N unit-norm vectors and is denoted as C UE = {c, c,, c N }. Each user first normalizes its channel h and then chooses the closest codebook vector to represent the channel. Note that the normalization process loses the amplitude information and that only the channel direction information (CDI) (or spatial signature of the channel is) retained. The amplitude information is transmitted in the CQI feedback. Quantization is done according to the minimum Euclidian distance such that ˆ hk = c n, n = arg max hc % where h % k denotes the i=,..., N k i hˆ k normalized channel, and is the quantized channel. [] investigates the bounds on the expected value of the CQI. After the BS receives the CDI and CQI information from all UEs, it selects users for MU-MIMO transmission. Then, the beamforming matrix W is computed as ( ) / W = ˆ ˆ ˆ diag( p ) where ˆ ˆT ˆT = [ h,..., h ] T is the composite channel matrix, and p = ( p,..., p ) T is the vector of the power allocation coefficients that impose the power constraint on the transmitted signal. Due to the channel quantization error, the condition hw k j =, k j is no longer satisfied, because the beamforming matrix W is computed by using the h but not h. Therefore, the received ˆ k signal contains inter-user interference k Pjhw k jsj. j=, j k III. BS Codebook for ZF MU-MIMO with Quantized Channel After the beamforming matrix W is computed, it has to be signaled to the users so that the they can compute the effective channel W and design their receive filters. The set of all possible beamforming matrices constitutes the BS codebook, and is denoted as C NodeB = {W, W, }. We assume that the channel quantization codebook consists of vectors and that the BS transmits to users in MU-MIMO mode. Then, the composite quantized channel matrix can be denoted as ˆ ˆ T [ ˆ T T ij = hi h j ] ; i, j =..., i < j, i j. In this case, the BS codebook consists of a limited number of beamforming matrices, with each beamforming matrix computed from a possible composite channel matrix. Note that when listing the possible channel matrices, we do not count the cases where i > j because this would result in the same set of beamforming matrices with only columns interchanged. The number of channel matrices with distinct combinations of quantized channel vectors in this case can be computed to be =. For notational convenience, we index the ordered channel matrices as ˆ n, n =.... For each of these channel matrices, there would be a beamforming matrix computed at the BS by using the zero-forcing ( ) condition, i.e., ˆ ˆ ˆ W =. n n n n

3 Once the beamforming matrix to be used for the downlink transmission is selected, the index of that matrix has to be signaled to the scheduled users. For the given system model, signaling the index of the selected beamforming matrix from a codebook of size would require 7 bits. This overhead would increase significantly with an increasing number of scheduled users or a larger channel quantization codebook. To reduce the signaling overhead, the size of the BS codebook C NodeB = {W, W,, W ) needs to be reduced, and the degradation in system performance due to this reduction need to be limited. In the next sections, an approach to achieve a reduced size BS codebook is presented, and the performance of the proposed approach is analyzed with simulations.. Design of Reduced size BS Codebook In this section, we describe a method to reduce the size of the BS codebook. The proposed method is based on quantizing the codebook with respect to some optimality criterion in order to minimize the performance degradation that would occur due to the quantization. The new codebook is designed off-line and then used at both the BS and the UEs. The quantization is achieved with a method based on the Generalized Lloyd Algorithm []. We firstly need to define the optimality criterion, such as signal-to-interference ratio (SIR), where the interference is due to the BS codebook quantization error, or capacity. Let us denote the quantized version of the beamforming matrix W as Ŵ. When Ŵ is used to compute W ˆ ˆ, the off-diagonal coefficients are no longer zero, due to the quantization of the beamforming matrix; as a result, we get ˆ ˆ α β W = β α. In this case, the received data y α β s α s + β s is y =. From this, y = β α s + n = α s β + n + s we can see that the off-diagonal coefficients which represent the inter-user interference are no longer zero, due to the quantization of W. One possible criterion is the total capacity, which can be written as W ˆ ˆ ˆ ˆ i W, i, C = log + + log + N ˆ ˆ ˆ ˆ + W i N + W i,, Another possible criterion is the total signal to interference ratio (SIR): SIR= ˆ ˆ / ˆ ˆ i W W mm, i. Note that, mn, m= m, n= ; m n when defining the criterion, we use the quantized channel information available at the BS. After defining an appropriate criterion, we list all possible channel pairings ˆ n, n =,..., which are computed from the UE codebook. Then, an initial BS codebook of size N C NodeB = Wˆ ˆ ˆ, W,..., WN { } is selected. This codebook can be chosen, for example, from the beamforming matrices in the original BS codebook or it can be randomly generated. In the following, we assume that N is set to make the sizes of the BS and UE codebooks the same. The proposed algorithm works as follows: Step : In the first step of the algorithm, we associate each of the channel pairings ˆ n, i =,... with one of the beamforming matrices in the BS codebook. The set of all channel pairings associated to a given beamforming matrix Wˆ i is called the region of that beamforming matrix and is denoted by R i. This association is done so that the defined optimality criterion is maximized. For example, for the capacity criterion, the region would be defined as R i { ˆ ˆ ˆ ˆ ˆ } R = : C( W ) C( W ), i j, i, j =,..., N i i j Step : In the second step of the algorithm, Wˆ i, i =,... are updated by using the channel pairings that were associated with the matrices in Step. The new beamforming matrices are computed as L i ( ) ˆ W ˆ ˆ ˆ, ˆ R, i =,..., N, where L i i = n n n n i Li n= denotes the number of channel matrices in R i. Step : With the new beamforming matrices, we go back to Step and continue the iteration until a convergence criterion is met. For example, after a few iterations the quantized beamforming matrices converge and do not get updated, so the algorithm can be terminated. Note that the final BS codebook designed with this method would be different for different optimality criteria, different parameters in the algorithm, or a different initial codebook. At the end of the algorithm, we end up with a BS codebook of C NodeB = { Wˆ, Wˆ,..., Wˆ N } and a mapping from each possible channel pairing to one of the beamforming matrices in C NodeB, i.e. ˆ.. Improving the Performance of Quantization As mentioned previously, the user selection process in a ZF MU-MIMO system tries to pair users whose channels are as orthogonal as possible; i.e., users with highly correlated channels are not selected for transmission. This implies that discarding the beamforming matrices that correspond to highly correlated channel pairs from the original BS codebook can improve the quantization performance. When a DFT based UE codebook is used, the correlations = hˆ h ˆ of all possible channel pairings take one of the possible values ρ =,.8,.,.7,.8,.,.9. If we discard the channel pairings with correlation. and.9, and do not include the associated beamforming ρ ij i j

4 matrices in the original BS codebook, the size of the original codebook can be reduced from to CQI Computation The CQI information is used to select users for transmission and possibly for adaptive modulation and coding. The CQI is usually an estimate of the SINR and, in the MU-MIMO case, has to consider the inter-user interference. If the k th UE knew the quantized channel of the other simultaneous user, it would be able to compute the exact SINR. The UE, however, does not have any information about the interfering user s channel. But it knows that the interfering user s quantized channel can only be one of the remaining different vectors in the channel quantization codebook. (The scheduler would not pair users with the same channel). For each of these M = possibilities, the UE can compute the pk k k, m SINR as SINR km, =, m=,..., M. * σ + hwhw p i k * i k i, m where and can easily be determined from the BS w km, w im, codebook and the ˆ mapping. The CQI, for example, can be determined as the average of these SINRs... User Selection and Downlink Control Signaling The BS has to select users need for transmission. Selecting the set of optimal users is computationally very complex, but suboptimal algorithms exist []. We use the following algorithm: Choose the two users with the largest CQI values. If the correlation between the quantized channels of the selected users is above the threshold, select another pair of users with the next largest CQI s and continue similarly. Assume the BS selects two users whose quantized channel ˆ = [ hˆ ] indexes are m and n, respectively. Given T ˆ T T mhn, the Wˆ index of the beamforming matrix i is easily found from the ˆ mapping and is transmitted in the downlink control channel... Design of a DFT Based BS Codebook The proposed method can also be applied to design codebooks that have a well-defined structure. As an example, we consider the design of a BS codebook that is based on a DFT codebook. The iterative algorithm introduced above starts with an initial codebook. Let us assume that this initial codebook is produced from the first four rows of the x DFT matrix, where we assume the number of transmit antennas is four. From this matrix we can have different combinations of beamforming matrices (also considering the column ordering), which constitute the initial BS codebook. We first associate each of the channel pairs to one of the matrices in the BS codebook. If we assume that channel pairs with high correlation are discarded as before, we end up with 88 composite channel matrices Ĥ. At the end of the first step, we can see that the possible composite channels are mapped to 7 different matrices in the initial BS codebook; i.e., we can discard the rest of the matrices from the initial codebook. After the first step, the size of the BS codebook is still large, and continuing with Step would destroy the DFT property of the codebook. The optimal set of beamforming matrices out of the total 7 can be found by exhaustive search, but this would require considerable computational complexity. Instead of comparing all possible combinations, here we select the N matrices suboptimally by comparing several possible combinations and choosing the best one. Once the codebook of size N is designed, all the other steps are similar to what has been described above. IV. Numerical Results and Discussions The performance of the BS codebooks designed by using the proposed techniques is analyzed with simulations. In the simulations, we use the GPP Long Term Evolution system, which is based on a Mz OFDMA system []. We assume that the BS has transmit antennas and transmits two data streams, each one to a user. There are active UEs, and each UE is equipped with one receive antenna. We assume that the channel is TU- and the speed is km/h. The UEs quantize their channels and compute the CQI. This information is then transmitted to the BS with a feedback delay of subframes. The granularity for channel quantization and CQI computation is resource block (RB), where one RB consists of subcarriers. The BS selects the two users for each RB and chooses the corresponding beamforming matrix from the BS codebook to use in the transmission. We then compute the received SINR on every subcarrier for each of the data streams and compute the total achievable capacity on that subcarrier. The computed capacities are averaged over all subcarriers and subframes. The channel quantization codebook is the first rows of a x FFT matrix. The figures illustrate the average capacities that are achieved with different BS codebooks. The optimal performance is achieved when the BS codebook is not quantized, i.e., when we use all possible beamforming matrices. The performance of the other codebooks is compared to this optimal case. Figure compares the average capacities achieved by BS codebooks computed by using different quantization methods. Method refers to the case in which channel pairings with correlation. and.9 are discarded, and method refers to the case in which all possible channel pairings are used. From this figure, we can see that the codebook designed by using the capacity criterion performs better than the one designed with the SIR criterion, and also that discarding the highly correlated channel pairs improves the performance. The loss in capacity compared to the unquantized BS codebook is less than %.Figure illustrates the performance of the DFT based BS codebook. The optimal performance of the DFT codebook is achieved when all possible 7 beamforming matrices are used. The size of the codebook is then reduced to by sub-optimally selecting of these 7 beamforming matrices. Figure compares the performance of the best BS codebook from Figure and the DFT based codebook from Figure. From the comparison we can see that the performance of the quantized codebook is close to the unquantized codebook, and that the DFT based codebook also performs well.

5 Average total capacity.... NodeBCodeBook not quantized NodeBCodeBook quantized method with SIR criterion NodeBCodeBook quantized method with SIR criterion NodeBCodeBook quantized method with capacity criterion NodeBCodeBook quantized method with capacity criterion 8 8 Figure : Comparison of the different quantization methods Average total capacity.... NodeBCodeBook not quantized NodeBCodeBook of size 7 based on FFT NodeBCodeBook of size based on FFT 8 8 Figure : Performance of the DFT based BS codebook Average total capacity.... NodeBCodeBook not quantized NodeBCodeBook quantized to NodeBCodeBook quantized to based on FFT 8 8 Figure : Comparison of the DFT based BS codebook V. Conclusions In this paper, we studied the ZF beamforming method for MU-MIMO with quantized channel information. We observed that a disadvantage of this method is large downlink signaling overhead, which is due to the large size of the ZF beamforming codebook. We presented a procedure for the design of codebooks with reduced size for MU-MIMO, which preserves the performance. We also provided formulas for SINR and related CQI computations. We demonstrated that a codebook quantized to matrices is sufficient for reasonable performance. We also showed that these methods result in efficient user-selection and downlink control signaling overhead. Simulation results showed that the size of the NodeB codebook can be reduced with only a small ( %) penalty in performance. References [] T. Yoo, N. Jindal and A. Goldsmith, Multi-Antenna Broadcast Channels with Limited Feedback and User Selection, IEEE Journal Sel. Areas in Communications, pp. 78-9, Sep. 7. [] T. Yoo and A. Goldsmith, On the optimality of multiantenna broadcast scheduling using zero-forcing beamforming, IEEE J. Select. Areas Commun., vol., pp. 8, Mar.. [] C. R. Murthy, and B. D. Rao, Quantization methods for equal gain transmission with finite rate feedback IEEE Trans. Signal Proc., vol., pp. -, Jan. 7. [] G. J. Foschini and M. J. Gans, On limits of wireless communications in a fading environment when using multiple antennas, Wireless Personal Commun., vol., pp., Mar [] E. Telatar, Capacity of multi-antenna Gaussian channels, Eur. Trans. Telecomm. ETT, vol., pp. 8 9, Nov [] Q.. Spencer, C. B. Peel, A. L. Swindlehurst, and M. aardt, An introduction to the multi-user MIMO downlink, IEEE Commun. Mag., vol., pp. 7, Oct. [7] M. Costa, Writing on dirty paper, IEEE Trans. Inform. Theory, vol. 9, pp. 9, May 98. [8] P. Viswanath, D. N. C. Tse, and R. Laroia, Opportunistic beamforming using dumb antennas, IEEE Trans. Inf. Theory, vol. 8, pp. 77 9, Jun.. [9] M. Sharif and B. assibi, On the capacity of MIMO broadcast channels with partial side information, IEEE Trans. On Inf. Theory, vol., pp., Feb.. [] D. J. Love and R. W. eath., Jr., Limited feedback unitary precoding for spatial multiplexing systems, IEEE Trans. Inf. Theory, vol., pp , Aug.. [] D. J. Love, R. W. eath., Jr., W. Santipach, and M. L. onig, What is the value of limited feedback for MIMO channels?, IEEE Commun. Mag., vol., pp. 9, Oct.. [] S. Vishwanath, N. Jindal, and A. Goldsmith, Duality, achievable rates, and sum-rate capacity of gaussian MIMO broadcast channels, IEEE Trans. Inf. Theory, vol. 9, pp. 8 8, Oct.. []. Weingarten, Y. Steinberg, and S. S. (Shitz), The capacity region of the gaussian MIMO broadcast channel, in Proc. IEEE Int. Symp. Inf. Theory, p. 7, Jun.. [] R. D. Wesel, and J. M. Cioffi, Achievable Rates for Tomlinson arashima Precoding, IEEE Trans. Inf. Theory, vol., pp. 8-8, Oct.. [] N. Jindal, MIMO broadcast channels with finite rate feedback, in Proc. IEEE GLOBECOM, pp., Nov.. [] GPP TS. V.., rd Generation Partnership Project; Technical Specification Group Radio Access Network; Evolved Universal Terrestrial Radio Access (E-UTRA); Physical Channels and Modulation (Release 8).