# Erroneous decoding, probability of

One of the possible measures for characterizing the accuracy of information reproduction over a communication channel (see also Information, exactness of reproducibility of). For the transmission of information $\xi$ from a source which has $M$ different possible values $1 \dots M$ with probability distribution $p _ {m} = {\mathsf P} \{ \xi = m \}$, $m = 1 \dots M$, a certain communication channel is used. Then for fixed methods of coding and decoding (see also Information, transmission of), the probability of erroneous decoding $P _ {e,m}$, $m = 1 \dots M$, is defined by

$$P _ {e,m} = {\mathsf P} \{ \xi \neq m \mid \xi = m \} ,$$

where $\widetilde \xi$ is the decoded message at the channel output. The variable quantity

$$P _ {e} = {\mathsf P} \{ \widetilde \xi \neq \xi \} = \sum _ { m= } 1 ^ { M } p _ {m} P _ {e,m}$$

is called the mean probability of erroneous decoding. The study of the optimal probability of erroneous decoding, $P _ {e} ^ { \mathop{\rm opt} }$, defined by

$$P _ {e} ^ { \mathop{\rm opt} } = \inf P _ {e} ,$$

is of particular interest. The infimum is taken over all possible methods of coding and decoding. Owing to the difficulty of obtaining an accurate expression for $P _ {e} ^ { \mathop{\rm opt} }$, which is caused by the fact that optimal coding methods are usually unknown, the study of the asymptotic behaviour of $P _ {e} ^ { \mathop{\rm opt} }$ when the length of the message in the channel increases is of interest. More precisely, the following situation is studied. Let a segment of length $N$ in a communication channel with discrete time be used, and put $R = ( \mathop{\rm ln} M)/N$. The asymptotic behaviour of $P _ {e} ^ { \mathop{\rm opt} }$ has to be studied when $N \rightarrow \infty$ and $R = \textrm{ const }$( this means that the length of the message increases, while its transmission rate remains constant). If the channel is a homogeneous memoryless channel with discrete time and if the spaces $Y$ and $\widetilde{Y}$ of values of the components of the input and output signals are finite, then the following upper and lower bounds for $P _ {e} ^ { \mathop{\rm opt} } = P _ {e} ^ { \mathop{\rm opt} } ( N, R)$ are known:

$$\tag{1 } \mathop{\rm exp} \{ - N[ E _ {sp} ( R - \alpha ( N)) + \beta ( N)] \} \leq$$

$$\leq \ P _ {e} ^ { \mathop{\rm opt} } ( N, R) \leq \mathop{\rm exp} \{ - NE _ {r} ( R) \} ,$$

where $\alpha ( N)$ and $\beta ( N)$ tend to zero with increasing $N$, while the functions $E _ {r} ( R)$ and $E _ {sp} ( R)$ are defined as follows:

$$\tag{2 } E _ {r} ( R) = \max _ {0 \leq \rho \leq 1 } \max _ {q( \cdot ) } [ E _ {0} ( \rho , q( \cdot )) - \rho R],$$

$$\tag{3 } E _ {sp} ( R) = \sup _ {\rho > 0 } \max _ {q( \cdot ) } [ E _ {0} ( \rho , q( \cdot )) - \rho R],$$

where

$$E _ {0} ( \rho , q( \cdot )) = \ - \mathop{\rm ln} \sum _ {\widetilde{y} \in \widetilde{Y} } \left [ \sum _ {y \in Y } q( y) q( y, \widetilde{y} ) ^ {1/( 1+ \rho ) } \right ] ^ {1+ \rho } .$$

Here, $q( \cdot ) = \{ {q( y) } : {y \in Y } \}$ is an arbitrary probability distribution on $Y$, $q( y, \widetilde{y} ) = {\mathsf P} \{ \eta _ {k} = \widetilde{y} \mid \widetilde \eta _ {k} = y \}$, $y \in Y$, $\widetilde{y} \in \widetilde{Y}$, while $\eta _ {k}$ and $\widetilde \eta _ {k}$, $k = 1 \dots N$, are the components of the input and output signals of the memoryless channel being considered. It is known that $E _ {r} ( R)$ and $E _ {sp} ( R)$, defined by the formulas (2) and (3), are positive, convex from below, monotone decreasing functions of $R$ when $0 \leq R < C$, where $C$ is the transmission rate of the channel (cf. Transmission rate of a channel), and that $E _ {r} ( R) = E _ {sp} ( R)$ when $R \geq R _ {cr}$; here $R _ {cr}$ is a quantity defined by the transition matrix of the channel and called the critical speed of the given memoryless channel. Thus, for values $R$, $R _ {cr} \leq R < C$, the main terms of the upper and lower bounds (1) for $P _ {e} ^ { \mathop{\rm opt} }$ coincide asymptotically, hence for this range of $R$ the exact value of the reliability function of the channel, $E( R)$, defined by

$$E( R) = \overline{\lim\limits}\; _ {N \rightarrow \infty } \frac{- \mathop{\rm ln} P _ {e} ^ { \mathop{\rm opt} } ( N, R) }{N} ,$$

is known. For values of $R$, $0 \leq R < R _ {cr}$, the exact value of $E( R)$ for arbitrary memoryless channels is still unknown (1983), although the bounds (1) can still be improved. The exponential character of decrease of $P _ {e} ^ { \mathop{\rm opt} } ( N, R)$ has been proved for a very broad class of channels.

How to Cite This Entry:
Erroneous decoding, probability of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Erroneous_decoding,_probability_of&oldid=46850
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