Transmit Signal Design for Optimal Estimation of Correlated MIMO Channels

Abstract

Abstract-We address optimal estimation of correlated multiple-input multiple-output (MIMO) channels using pilot signals, assuming knowledge of the second-order channel statistics at the transmitter. Assuming a block fading channel model and minimum mean square error (MMSE) estimation at the receiver, we design the transmitted signal to optimize two criteria: MMSE and the conditional mutual information between the MIMO channel and the received signal. Our analysis is based on the recently proposed virtual channel representation, which corresponds to beamforming in fixed virtual directions and exposes the structure and the true degrees of freedom in the correlated channel. However, our design framework is applicable to more general channel models, which include known channel models, such as the transmit and receive correlated model, as special cases. We show that optimal signaling is in a block form, where the block length depends on the signal-to-noise ratio (SNR) as well as the channel correlation matrix. The block signal corresponds to transmitting beams in successive symbol intervals along fixed virtual transmit angles, whose powers are determined by (nonidentical) water filling solutions based on the optimization criteria. Our analysis shows that these water filling solutions identify exactly which virtual transmit angles are important for channel estimation. In particular, at low SNR, the block length reduces to one, and all the power is transmitted on the beam corresponding to the strongest transmit angle, whereas at high SNR, the block length has a maximum length equal to the number of active virtual transmit angles, and the power is assigned equally to all active transmit angles. Consequently, from a channel estimation viewpoint, a faster fading rate can be tolerated at low SNRs relative to higher SNRs.

Training-based Bayesian MIMO channel and channel norm estimation

“…Remark 1. Observe that the general MMSE channel estimator in Theorem 1 is linear; it has the form vec( HMMSE) = Avec(Y) and coincides with the linear MMSE estimator derived in [4]. Thus, the estimator in (4) is also the linear MMSE estimator for any PDF of vec(H k ) with the same mean and covariance, with (5) as its MSE.…”

“…The structure of the MSE minimizing training matrix was proved in [4], along with somewhat similar asymptotic results. The optimal power allocation for the special case of uncorrelated receive antennas coincides with that proposed in [3].…”

“…The problem of finding the linear MMSE estimator was considered in [3,4], but although they both claim to use a general linear form, we show herein that the structure in [3] is in fact restrictive. The general MMSE estimator of the channel is derived herein, and it coincides with the linear estimator in [4] and achieves a lower mean square error (MSE) than [3]. Herein, we also derive the MMSE estimator of the squared Frobenius norm of the channel.…”

“…Channel estimation techniques are commonly divided into three categories: training-based, semi-blind, and blind. In the former, the estimation is entirely based on transmission of training sequences (known a priori to the receiver) [3,4]. The other extreme is blind estimation, which only exploits some known structure of the received data.…”

Optimal signal design for estimation of correlated MIMO channels

“…The number of active elements in both the columns is one, i.e Q 1 = Q 2 = 1. Both criteria give identical power assignment in this case [8]. Observe that the second transmit beam gets non-zero power only after the SNR increases to about 13 dB (this behavior is similar in the next three figures too).…”