Optimal partial feedback design for MIMO block fading channels with causal noiseless feedback

In this paper, we focus on investigating the optimal MIMO transmission strategy for forward channel capacity and forward error exponent of the MIMO block fading channels when the feedback link is causal and has a capacity constraint. In this paper, we assume a forward MIMO block fading channel where the channel state information is estimated at the receiver and partially feedback to the transmitter. The feedback link is assumed to be noiseless and causal with a feedback capacity constraint in terms of maximum number of feedback bits per fading block. We show that the design of the optimum feedback scheme is identical to the design of vector quantizer (Lloyd´s algorithm [S.P. Lloyd, 1982]) with a modified distortion measure. It is shown that in the general case, the optimal feedback strategy has a general form of power water-filling cascaded with beamforming matrix as well. Furthermore, we show that the SNR gain with feedback is contributed by focusing transmission power on active eigen-channel and temporal power water-filling. The former factor contributed at most log₁₀(n_T)n_R dB SNR gain when n_T > n_R in all SNR region while the latter contribution is significant only at low SNR region. Finally, MMSE receiver could be used to achieve the optimal capacity in the general case of partial feedback.