# Gaussian channel

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A communication channel whose transition function determines a conditional Gaussian distribution. More precisely, a communication channel is called a Gaussian channel on the finite interval if the following conditions hold: 1) the spaces of values of the input and output signals, and , are spaces of real-valued functions and , , with the usual -algebras of measurable sets (that is, the input and output signals of a Gaussian channel are given by stochastic processes and , respectively); 2) for any fixed the transition function of the channel determines a conditional Gaussian distribution (one says that a collection of random variables has a conditional Gaussian distribution if every finite subfamily has a conditional finite-dimensional normal distribution with second moments that are independent of the conditioning); and 3) the restriction is imposed only on the second moment of the random variable .

An example of a Gaussian channel on is a channel whose input signal is given by a stationary random sequence and whose output signal is the stationary random sequence , obtained according to the formulas where is a stationary Gaussian random sequence independent of with , and with spectral density , . The restriction on the input signal has the form where is the spectral density of , is some function and is a constant. The capacity of such a channel is given by the formula where and is determined by the equation 