The Capacity of Communication Channels with Memory
Shaohua Yang · 2004
For a state machine channel, a simple form of the feedback-capacity-achieving source distribution is revealed. A Markov source, whose memory length equals the channel memory length, achieves the feedback capacity. Given the posterior channel-state distribution, the optimal source Markov transition probabilities become independent of the whole history of past channel outputs. Further, when the feedback is delayed, the delayed feedback capacity is achieved by a Markov source whose memory length equals the sum of the channel memory length and the feedback delay. The Markov source optimization is formulated as a standard stochastic control problem and is solved by dynamic programming. The (delayed) feedback capacity is an upper-bound on the feed-forward channel capacity, and this bound can be made tight by increasing the feedback delay. The linear Gaussian channel with an average input power constraint can be equivalently modelled as a state machine channel. When the channel has feedback, by following similar procedures as developed for the state machine channel, it is shown that Gauss-Markov sources achieve the feedback capacity and a Kalman-Bucy filter is optimal for processing