# Channel with a finite number of states

*finite-state channel*

A communication channel for which the statistical properties of the output signal at a time are determined by the input signal at this moment and the state of the channel at the previous moment, and where the set of possible states of the channel is finite. A channel with a finite number of states can also be defined by a finite probabilistic automaton (cf. Automaton, probabilistic). Below a rigorous definition is given of a discrete-time homogeneous channel with a finite number of states and with finite spaces of values, and , for the components of the input and output signals. Suppose that functions , , , , are given, where is a finite set, called the set of states of the channel, as well as a probability distribution . Intuitively, the function defines the conditional probability that at a time the signal appears at the output and the channel goes over to the state under the condition that the signal was transmitted and that at the previous moment the channel was in state . The distribution can be regarded as the probability distribution of the initial state of the channel (that is, the state of the channel at the initial moment). Let the functions be recursively defined by the formulas

where , , , , , . Let

Then the transition function

of segments of length of the channel with a finite number of states, for any , is, by definition, equal to

where and are the segments of length of the input and the output of the channel.

For references see ,

in the article Communication channel.

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Channel with a finite number of states.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Channel_with_a_finite_number_of_states&oldid=18715