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We shall consider now the encoding of signals on $[-T,T]$ where $T>0$ is fixed. Ultimately we shall be interested in encoding classes of bandlimited signals like the class ${B}_{A}$ However, we begin the story by considering the more general setting of encoding the elements of any given compact subset $K$ of a normed linear space $X$ . One can determine the best encoding of $K$ by what is known as the Kolmogorov entropy of $K$ in $X$ .
To begin, let us consider an encoder-decoder pair $(E,D)$ $E$ maps $K$ to a finite stream of bits. $D$ maps a stream of bits to a signal in $X$ . This is illustrated in . Note that many functions can be mapped onto the same bitstream.
Define the distortion $d$ 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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