Optimal implicit channel estimation for finite state Markov communication channels

This paper shows the existence of the optimal training, in terms of achievable mutual information rate, for an output feedback implicit estimator for finite-state Markov communication channels. A proper quantification of source redundancy information, implicitly used for channel estimation, is performed. This enables an optimal training rate to be determined as a tradeoff between input signal entropy rate reduction (source redundancy) and channel process entropy rate reduction (channel estimation). The maximal mutual information rate, assuming the optimal implicit training and the presence of channel noise, is shown to be strictly below the ergodic channel information capacity. It is also shown that this capacity penalty, caused by noisy time-varying channel process estimation, vanishes only if the channel process is known or memoryless (channel estimation cannot improve system performance)