# Gaussian channel

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A communication channel whose transition function determines a conditional Gaussian distribution. More precisely, a communication channel $( Q , V )$ is called a Gaussian channel on the finite interval $[ 0 , T ]$ if the following conditions hold: 1) the spaces of values of the input and output signals, $( {\mathcal Y} , {\mathcal S} _ {\mathcal Y} )$ and $( \widetilde {\mathcal Y} , {\mathcal S} _ {\widetilde {\mathcal Y} } )$, are spaces of real-valued functions $y ( t)$ and $\widetilde{y} ( t)$, $t \in [ 0 , T ]$, with the usual $\sigma$- algebras of measurable sets (that is, the input and output signals of a Gaussian channel are given by stochastic processes $\eta = \{ {\eta ( t) } : {t \in [ 0 , T ] } \}$ and $\widetilde \eta = \{ {\widetilde \eta ( t) } : {t \in [ 0 , T ] } \}$, respectively); 2) for any fixed $y \in Y$ the transition function $Q ( y , \cdot )$ 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 $V$ is imposed only on the second moment of the random variable $\eta$.

An example of a Gaussian channel on $( - \infty , \infty )$ is a channel whose input signal is given by a stationary random sequence $\eta = (\dots, \eta _ {-} 1 , \eta _ {0} , \eta _ {1} ,\dots )$ and whose output signal is the stationary random sequence $\widetilde \eta = ( \dots, \widetilde \eta _ {-} 1 , \widetilde \eta _ {0} , \widetilde \eta _ {1} ,\dots )$, obtained according to the formulas

$$\widetilde \eta _ {i} = \ \sum _ {k = - \infty } ^ \infty a _ {k} \eta _ {i-} k + \zeta _ {i} ,\ \ i = 0 , \pm 1 , \pm 2 \dots$$

where $\zeta = ( \dots, \zeta _ {-} 1 , \zeta _ {0} , \zeta _ {1} ,\dots )$ is a stationary Gaussian random sequence independent of $\eta$ with ${\mathsf E} \zeta _ {i} = 0$, $i = \pm 1 , \pm 2 \dots$ and with spectral density $f _ \zeta ( \lambda )$, $- 1 / 2 \leq \lambda \leq 1 / 2$. The restriction on the input signal has the form

$$\int\limits _ {- 1 / 2 } ^ { {1 } / 2 } | \Phi ( \lambda ) | ^ {2} f _ \eta ( \lambda ) d \lambda \leq S ,$$

where $f _ \eta ( \lambda )$ is the spectral density of $\eta$, $\phi ( \lambda )$ is some function and $S$ is a constant. The capacity of such a channel is given by the formula

$$C = \frac{1}{2} \int\limits _ {- 1 / 2 } ^ { {1 } / 2 } { \mathop{\rm log} \max } \ \left [ \left | \frac{a ( \lambda ) }{\Phi ( \lambda ) } \right | ^ {2} \cdot \frac \mu {f _ \zeta ( \lambda ) } , 1 \right ] \ d \lambda = S ,$$

where $a ( \lambda ) = \sum _ {k = - \infty } ^ \infty e ^ {- 2 \pi i k \lambda }$ and $\mu$ is determined by the equation

$$\int\limits _ { - } 1/2 ^ { 1/2 } \max \left [ \mu - \left | \frac{\Phi ( \lambda ) }{a ( \lambda ) } \ \right | ^ {2} f _ \zeta ( \lambda ) , 0 \right ] d \lambda = S .$$