Near-Optimal Codebook Constructions for Limited Feedback Beamforming in Correlated MIMO Channels with Few Antennas Vasanthan Raghavan, Akbar M. Sayeed, Nigel Boston University of Wisconsin-Madison, Madison, WI, USA raghavan@cae.wisc.edu, akbar@engr.wisc.edu, boston@engr.wisc.edu Abstract— Transmit beamforming with receive combining is a low-complexity solution that achieves the full diversity afforded by a multi-antenna channel. Building on our recent result which shows that even channel statistics are sufficient to achieve perfect feedback performance (in the limit of antenna dimensions) with beamforming and combining, we propose near-optimal codebook designs for correlated channels with a focus on few antennas at the transmitter and the receiver. In the process, we refine the answer to the question: When are channel statistics sufficient to achieve near perfect feedback performance? We show that the condition number of the transmit and receive covariance matrices hold the key to this question. We partition the transmit and receive covariance spaces into 4 regions based on well and ill-conditioning of the covariance matrices and show that the number of bits required for near perfect feedback performance is dependent on the condition numbers of these matrices. I. I NTRODUCTION Multiple antennas at the transmitter and the receiver provide a mechanism to increase the reliability of signal reception, or rate of information transfer, or a combination of both these aspects. In this paper we focus on achieving the highest level of reliability (full diversity) by employing multiple antennas at both the ends. A simple low-complexity solution towards this goal is transmit beamforming and receive combining. This technique however requires perfect channel state information (CSI) at both the transmitter and the receiver. While perfect CSI at the receiver maybe a reasonable assumption, such knowledge at the transmitter is infeasible in most practical situations. Recent works have therefore focused attention on enhancing the performance of multi-input multi-output (MIMO) systems by exploiting the limited channel knowledge that is usually available at the transmitter [1], [2], [3], [4], [5]. Towards this goal, various solutions have been studied. These solutions range from pure first/second-order statistical feedback to feedback of quantized channel information. In particular, it had been assumed that significant benefits can be achieved when reliable channel information is available at the transmitter than with pure statistical feedback. A recent result of ours [6] shows that in the limit of antenna dimensions, even second-order channel statistics are sufficient to achieve the same performance gains as perfect feedback. Building on [6], in this work, we focus on two inter-related problems: 1) What are the factors that influence the rate of convergence to the asymptotics?, and 2) What insights can be acquired from the asymptotics in the context of a practical system deploying few transmit and receive antennas? To answer the first question, we refine our results in [6] and show that the conditioning of the transmit and receive covari- ance matrices control the rate of convergence of performance with pure statistical feedback to that of perfect feedback. In the non-asymptotic case of few transmit and receive antennas, we propose a codebook design methodology that exploits the phenomenon of eigenvector hardening (convergence of the dominant right singular vector of the channel to the statistical direction) in correlated MIMO channels. Numerical studies show that the proposed codebook constructions are near- optimal even at low error probabilities for a variety of corre- lated channel statistics. Furthermore, the number of feedback bits required for a certain level of diversity performance is determined by the level of ill and well-conditioning of the transmit and receive covariance matrix respectively, and we use this criterion to partition the covariance spaces into 4 regions where limited feedback performance can be clustered. In this context, our work is closely related to [5]. However, our work differs from [5] in two fundamental aspects: 1) In contrast to a rotation-based Grassmannian design that exploits separability in channel statistics and takes care of only the dominant eigen-directions, we propose a systematic design which is explicitly tailored to the conditioning of the transmit and receive covariance matrices. As a consequence of our design methodology, we show that the design in [5] is near- optimal only if there is a strictly dominant eigen-direction and the number of antenna dimensions is large. In other cases, our design incorporates codevectors that quantize the rare events associated with local and global perturbations of the dominant eigen-directions, and 2) Our codebook design is naturally extendable to more realistic channel models like the canonical statistical model that do not have a separable correlation structure. II. SYSTEM MODEL We consider a single user communication system employing transmit beamforming and receive combining, and assume that signalling is done using N T transmit and N R receive antennas. The input-output relationship of this system is y = z H Hw x + z H n (1) where H is the N R × N T channel matrix, z and w are the combining and beamforming vectors respectively, x is the transmitted symbol from a chosen constellation (QPSK, 16-QAM etc.), and n is the independent receiver noise. We also assume a low-rate, error-free feedback channel and a block fading model where the channel fades independently (depending on channel statistics) from block to block. We describe the statistical channel modeling framework now. Ideal channel modeling assumes that the entries of H are independent and identically distributed (i.i.d.) Gaussian random variables. The i.i.d. channel assumption makes the problem studied mathematically tractable, but is unrealistic in applications where either large antenna spacings or a rich