Optimal partial feedback design for SISO block fading channels

In this paper, we shall investigate the problem of optimal transmission and feedback strategies design when feedback link has a capacity constraint. Optimization is based on forward channel capacity and forward error exponent. Channel state information is estimated at the receiver and feedback to the transmitter. The feedback is assumed to be noiseless and casual with a feedback capacity constraint, C_fb. We show that the design of the optimal feedback scheme is identical to the design of scalar quantizer (Lloyd´s algorithm with a modified distortion measure). The feedback optimization algorithm could be applied to both the forward capacity and error exponent with modifications of distortion measure. The optimal tradeoff of forward channel capacity and forward channel error exponent versus the feedback link capacity could be clearly illustrated. It is shown that the optimal transmission strategy also has a general form of temporal water-filling. Furthermore, while feedback enhances the forward channel capacity more effectively in low SNR region compared with high SNR region, the enhancement in error exponent is significant in both high and low SNR region. This indicates that significant gain in practical coding design could be expected with partial feedback.