Digital Communication over Fixed Timecontinuous Channels with Memory, with Special Application to Telephone Channels

摘要

The channel model assumed is a linear, time-invariant filter followed by additive, colored Gaussian noise. A general problem formulation is introduced which involves use of this channel once for T seconds to communicate one of M signals. Two questions are considered: (1) given a set of signals, what is the probability of error. and (2) how should these signals be selected to minimize the probability of error. It is shown that answers to these questions are possible when a suitable vector space representation is used, and the basis functions required for this representation are presented. Using this representation and the random coding technique, a bound on the probability of error for a random ensemble of signals is determined and the structure of the ensemble of signals yielding a minimum error bound is derived. The interrelation of coding and modulation in this analysis is discussed and it is concluded that: (1) the optimum ensemble of signals involves an impractical modulation technique, and (2) the error bound for the optimum ensemble of signals provides a 'best possible' result against which more practical modulation techniques may be compared. (Author)