Wideband fading channel capacity with training and partial feedback

We consider the capacity of a wideband fading channel with partial feedback, subject to an average power constraint. The channel is modeled as a set of parallel independent block Rayleigh fading subchannels with finite coherence time (L channel uses). The transmitter probes a subset of subchannels during each coherence time by transmitting pilot sequences for channel estimation. For each subchannel probed, one bit of feedback indicates whether or not the channel gain exceeds a threshold allowing transmission. Our problem is to optimize jointly the training (both length and power), number of subchannels probed (probing bandwidth), and feedback threshold to maximize the achievable rate (lower bound on ergodic capacity) taking into account the subchannel estimation error. Optimizing the probing bandwidth balances diversity against the quality of the subchannel estimate. We show that the achievable rate increases as S log L, where S is the signal-to-noise ratio, and exceeds the capacity with impulsive signaling (given by S) when L exceeds a (positive) threshold value. Moreover, the optimal probing bandwidth scales as S L/log ²L. In contrast, without feedback the optimal probing bandwidth for the probing scheme scales as SL^1/3 and the achievable rate converges to S, where the gap diminishes as SL^-1/3.