On the evaluation of the capacity of channels with memory

The determination of the capacity of a binary finite state channel with memory is in general a very difficult task. In this paper we present a new systematic method which amounts to computing the entropy of the channel error sequence represented as the output of a stochastic finite state automaton with state cardinality at most twice the one of the original channel. Each state of the original channel yields a maximum of two states in the automaton state transition diagram according to whether the preceeding error symbol was a one or a zero. The error class E is defined as the class of all states terminating on an error while the remaining class E~ contains states with transitions corresponding to no error. Consequently any path along states in E represents a solid burst of errors and reciprocally all the solid bursts of errors can only result from transitions between states in E and the same property applies to errorless events which can only result from transitions between states in E~. If the channel has K states, the final result is obtained by computing at most 2K series whose elements are the coefficients of the generating functions of the runs of 0's after an error terminating in any of the states of E and of the runs of 1's after an errorless event terminating in any of the states of E~.