# Information, transmission rate of

A quantity characterizing the amount of information that is contained in the output signal of a communication channel relative to the input signal, calculated in a unit of time (cf. Information, amount of). If

are stochastic processes in discrete or continuous time, being the input and output signals of a communication channel, then the quantity

(*) |

is the transmission rate of information (if the limit exists). Here is the amount of information, is the segment of and is analogously defined. The existence of the limit in (*) has been proved for the large class of channels in which the signals and are stationary and stationarily-related stochastic processes. An explicit computation of the transmission rate of information is possible, in particular, for a memoryless channel and a Gaussian channel. E.g., for a Gaussian channel, whose signals and are Gaussian stationary processes forming a joint Gaussian stationary pair of processes, the transmission rate of information is given by

where and are the spectral densities of and , respectively, and is their joint spectral density.

#### References

[1] | R. Gallagher, "Information theory and reliable communication" , Wiley (1968) |

[2] | M.S. Pinsker, "Information and informational stability of random variables and processes" , Holden-Day (1964) (Translated from Russian) |

**How to Cite This Entry:**

Information, transmission rate of.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Information,_transmission_rate_of&oldid=13228