On Optimal Training and Beamforming in Uncorrelated MIMO Systems with Feedback Francisco Rubio * , Dongning Guo † , Michael L. Honig † and Xavier Mestre * * Centre Tecnol` ogic de Telecomunicacions de Catalunya (CTTC) (PMT) Av. Canal Ol´ ımpic s/n, 08860 Castelldefels (Barcelona), Spain E-mail:{francisco.rubio,xavier.mestre}@cttc.es † Dept. of Electrial Engineering and Computer Science, Northwestern University 2145 Sheridan Road, Evanston, IL 60208, USA E-mail:{dguo,mh}@northwestern.edu Abstract—This paper studies the design and analysis of optimal training-based beamforming in uncorrelated multiple- input multiple-output (MIMO) channels with known Gaussian statistics. First, given the response of the MIMO channel to a finite sequence of training vectors, the beamforming vector which maximizes the average received signal-to-noise ratio (SNR) over all channel realizations is found. Secondly, the question of what consists of optimal training for a given amount of training is addressed. Upper and lower bounds for the maximum achievable SNR using beamforming are established. Furthermore, optimal training sequences are conjectured to satisfy the Welch bound. The conjecture is supported by the evidence that such sequences achieve close to the upper bound with moderate to large amount of trainings. I. I NTRODUCTION The performance of multiple-input multiple-output (MIMO) channels can be significantly enhanced if the channel state is known to the transmitter, the receiver, or both. Alternative blind techniques applied in order to avoid channel training may often incur in a nonnegligible loss of performance and a fairly increased computational complexity. In practice, the coefficients of a MIMO channel often vary over time and need to be estimated. The impact of the realistic availability of an imprecise channel state information in the capacity gains predicted for MIMO systems are summarized in [1]. If the channel state varies slowly, one may carry out some measurements in order to learn the channel statistics and estimate (or predict) its instantaneous realization. Typically, the channel coefficients are measured at the receiver by having the transmitter send known training vectors. Knowledge of the channel at the receiver can be sent to the transmitter via feedback channels [2]. The tradeoff between the time and the power allocated to training operation and data transmission was evaluated in [3]. In particular, they provided the optimum number of pilots and training power allocation of a training-based MIMO system in the sense of maximizing a lower-bound on the Shannon capacity over the class of ergodic block-fading (memoryless This work was supported by the Catalan Government (DURSI) under grant BE-2007, the U.S. Army Research Office under grant DAAD19-99-1-0288, the NSF CAREER Award under grant CCF-0644344, and the DARPA IT- MANET program Grant W911NF-07-1-0028. and uncorrelated) channels, as a function of the number of transmit and receive antennas, the received signal-to-noise ratio (SNR) and the length of the fading coherence time. Earlier related contributions include [4], as well as [5], [6], where the number of channel uses available for training and the optimal input distribution achieving capacity at high SNR over unknown block-fading uncorrelated MIMO channels with a finite coherence time interval is investigated. Furthermore, the effects of pilot-assisted channel estimation on achievable data rates over frequency-flat time-varying channels is analyzed in [7]. Along with the time-division multiplexing training scheme considered in the previous works, a tight lower-bound on the maximum mutual information of a MIMO system using superimposed pilots is derived in [8]. One simple use of the channel state information at the transmitter is to modulate the transmitted symbol onto a beam- forming vector matched to the channel in order to improve the received SNR. In this paper, we focus on the previous beamforming approach with the aim of exploiting the diversity gain achieved over the MIMO channel. If the MIMO channel is completely known to the transmitter, the evident choice of the beamforming vector is the right eigenvector of the channel matrix corresponding to the maximum singular value in amplitude, which maximizes the received SNR. Consider training-based beamforming for uncorrelated MIMO channels with known statistics. We assume that a sequence of known training vectors are sent by the transmitter so that the receiver can estimate the channel. In [9], the capacity of beamforming over a block-fading MIMO channel is maximized with respect to the finite amount of training available for channel estimation as well as of limited feed- back. Instead of following the generally suboptimal approach consisting of obtaining an intermediate estimate of the channel matrix to be used in further post-processing, we pursue the direct estimation of the optimal (channel-adapted) beamformer vector that needs to be fed back. In particular, the following questions are addressed in this paper: 1) Given the channel response to training, what is the optimal beamforming vector which maximizes the re- ceived SNR averaged over all possible realizations of the channel?