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)