Channels and Codes

Abstract

We have considered a random process or source {X _n } as a sequence of random entities, where the object produced at each time could be quite general, e.g., a random variable, vector, or waveform. Hence sequences of pairs of random objects such as {X _n ,Y _n } are included in the general framework. We now focus on the possible interrelations between the two components of such a pair process. In particular, we consider the situation where we begin with one source, say {X _n }, called the input and use either a random or a deterministic mapping to form a new source {Y _n }, called the output. We generally refer to the mapping as a channel if it is random and a code if it is deterministic. Hence a code is a special case of a channel and results for channels will immediately imply the corresponding results for codes. The initial point of interest will be conditions on the structure of the channel under which the resulting pair process {X _n ,Y _n } will inherit stationarity and ergodic properties from the original source {X _n }. We will also be interested in the behavior resulting when the output of one channel serves as the input to another, that is, when we form a new channel as a cascade of other channels. Such cascades yield models of a communication system which typically has a code mapping (called the encoder) followed by a channel followed by another code mapping (called the decoder).