Design Guidelines for Training-based MIMO Systems with Feedback

In this paper, we study the optimal training and data transmission strategies for block fading multiple-input multiple-output (MIMO) systems with feedback. We consider both the channel gain feedback (CGF) system and the channel covariance feedback (CCF) system. Using an accurate capacity lower bound as a figure of merit, we investigate the optimization problems on the temporal power allocation to training and data transmission as well as the training length. For CGF systems without feedback delay, we prove that the optimal solutions coincide with those for non-feedback systems. Moreover, we show that these solutions stay nearly optimal even in the presence of feedback delay. This finding is important for practical MIMO training design. For CCF systems, the optimal training length can be less than the number of transmit antennas, which is verified through numerical analysis. Taking this fact into account, we propose a simple yet near optimal transmission strategy for CCF systems, and derive the optimal temporal power allocation over pilot and data transmission.Comment: Submitted to IEEE Trans. Signal Processin

MIMO Transceiver Optimization With Linear Constraints on Transmitted Signal Covariance Components

This correspondence revisits the joint transceiver optimization problem for multiple-input multiple-output (MIMO) channels. The linear transceiver as well as the transceiver with linear precoding and decision feedback equalization are considered. For both types of transceivers, in addition to the usual total power constraint, an individual power constraint on each antenna element is also imposed. A number of objective functions including the average bit error rate, are considered for both of the above systems under the generalized power constraint. It is shown that for both types of systems the optimization problem can be solved by first solving a class of MMSE problems (AM-MMSE or GM-MMSE depending on the type of transceiver), and then using majorization theory. The first step, under the generalized power constraint, can be formulated as a semidefinite program (SDP) for both types of transceivers, and can be solved efficiently by convex optimization tools. The second step is addressed by using results from majorization theory. The framework developed here is general enough to add any finite number of linear constraints to the covariance matrix of the input

Optimal Transmit Covariance for Ergodic MIMO Channels

In this paper we consider the computation of channel capacity for ergodic multiple-input multiple-output channels with additive white Gaussian noise. Two scenarios are considered. Firstly, a time-varying channel is considered in which both the transmitter and the receiver have knowledge of the channel realization. The optimal transmission strategy is water-filling over space and time. It is shown that this may be achieved in a causal, indeed instantaneous fashion. In the second scenario, only the receiver has perfect knowledge of the channel realization, while the transmitter has knowledge of the channel gain probability law. In this case we determine an optimality condition on the input covariance for ergodic Gaussian vector channels with arbitrary channel distribution under the condition that the channel gains are independent of the transmit signal. Using this optimality condition, we find an iterative algorithm for numerical computation of optimal input covariance matrices. Applications to correlated Rayleigh and Ricean channels are given.Comment: 22 pages, 14 figures, Submitted to IEEE Transactions on Information Theor

Optimization of Training and Feedback Overhead for Beamforming over Block Fading Channels

We examine the capacity of beamforming over a single-user, multi-antenna link taking into account the overhead due to channel estimation and limited feedback of channel state information. Multi-input single-output (MISO) and multi-input multi-output (MIMO) channels are considered subject to block Rayleigh fading. Each coherence block contains $L$ symbols, and is spanned by $T$ training symbols, $B$ feedback bits, and the data symbols. The training symbols are used to obtain a Minimum Mean Squared Error estimate of the channel matrix. Given this estimate, the receiver selects a transmit beamforming vector from a codebook containing $2^B$ {\em i.i.d.} random vectors, and sends the corresponding $B$ bits back to the transmitter. We derive bounds on the beamforming capacity for MISO and MIMO channels and characterize the optimal (rate-maximizing) training and feedback overhead ($T$ and $B$) as $L$ and the number of transmit antennas $N_t$ both become large. The optimal $N_t$ is limited by the coherence time, and increases as $L/\log L$. For the MISO channel the optimal $T/L$ and $B/L$ (fractional overhead due to training and feedback) are asymptotically the same, and tend to zero at the rate $1/\log N_t$. For the MIMO channel the optimal feedback overhead $B/L$ tends to zero faster (as $1/\log^2 N_t$).Comment: accepted for IEEE Trans. Info. Theory, 201