# 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

$$\eta = \{ {\eta ( t) } : {- \infty < t < \infty } \} ,\ \ \widetilde \eta = \{ {\widetilde \eta ( t) } : {- \infty < t < \infty } \}$$

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

$$\tag{* } R = \lim\limits _ {T - t \rightarrow \infty } \ \frac{1}{T-} 1 I ( \eta _ {t} ^ {T} , \widetilde \eta {} _ {t} ^ {T} ) ,$$

is the transmission rate of information (if the limit exists). Here $I ( \cdot , \cdot )$ is the amount of information, $\eta _ {t} ^ {T} = \{ {\eta ( s) } : {t < s \leq T } \}$ is the segment $[ t , T ]$ of $\eta$ and $\widetilde \eta {} _ {t} ^ {T}$ is analogously defined. The existence of the limit in (*) has been proved for the large class of channels in which the signals $\eta$ and $\widetilde \eta$ 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 $\eta$ and $\widetilde \eta$ are Gaussian stationary processes forming a joint Gaussian stationary pair of processes, the transmission rate of information is given by

$$R = - \frac{1}{2} \int\limits _ {- \infty } ^ \infty \mathop{\rm log} \ \left ( 1 - \frac{| f _ {\eta \widetilde \eta } ( \lambda ) | ^ {2} }{f _ {\eta \eta } ( \lambda ) f _ {\widetilde \eta \widetilde \eta } ( \lambda ) } \right ) \ d \lambda ,$$

where $f _ {\eta \eta } ( \lambda )$ and $f _ {\widetilde \eta \widetilde \eta } ( \lambda )$ are the spectral densities of $\eta$ and $\widetilde \eta$, respectively, and $f _ {\eta \widetilde \eta } ( \lambda )$ 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=47354
This article was adapted from an original article by R.L. DobrushinV.V. Prelov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article