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We shall consider now the encoding of signals on where is fixed. Ultimately we shall be interested in encoding classes of bandlimited signals like the class However, we begin the story by considering the more general setting of encoding the elements of any given compact subset of a normed linear space . One can determine the best encoding of by what is known as the Kolmogorov entropy of in .
To begin, let us consider an encoder-decoder pair maps to a finite stream of bits. maps a stream of bits to a signal in . This is illustrated in . Note that many functions can be mapped onto the same bitstream.
Define the distortion for this encoder-decoder by
There is a simple mathematical solution to these two encoding problems based on the notion of Kolmogorov Entropy.
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