It is possible to model a channel such that the model combines 4.68 to 4.70. features of the discrete memoryless channel with those of the discrete channel with memory, and this defines a compound chan- (4.68) (4.69) Continuous channels Communications channels may be continuous in two senses. In the first sense, we may allow the input and output alphabets to contain (4.70) an infinite number of elements, but we require that the transmitted material be in the form of a discrete sequence of symbols. This type The information conveyed about the process X by the process Y of channel is called a time-discrete, amplitude continuous channel. is given by Equation 4.71. For the second type of continuous channel, we allow the trans- mission of information to be continuous in time. [XY] 1=H(X)+H(Y)-H(X,Y)YX) = = (4.71) Consider an analog source that emits a message waveform x(t) which is a sample function of a stochastic process X(t). If X(t) is a The capacity of the given channel is defined as the least upper stationary stochastic process with autocorrelation function bound of I [X Y taken over all regular Markov sources, and it can ( t ) and power spectral density function Q (f), and if X(t) is be proved that it is possible to transmit information at any rate less a bandlimited process such that Equation 4.72 is satisfied, then X(t) than the capacity with an arbitrarily small probability of error. has a representation given by the sampling theorem as in Equation The above discussion says nothing about how one calculates the 4.73. channel capacity of a given finite-state channel. This is an extremely complex task, and details may be found in Proakis (1987). = for ISI > W (4.72) 4/12 Information theory (4.73) form of nonuniform quantisation can be achieved by using a com- pressor followed by a uniform quantiser. By cascading this combi- nation with an expander complementary to the compressor, the original signal samples are restored to their correct values apart for quantisation errors. This combination of compressor and ex