# 1.1 Optimal encoding

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We shall consider now the encoding of signals on $\left[-T,T\right]$ 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 $\left(E,D\right)$ $E$ 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

$d\left(K,E,D,X\right):=\text{sup}{\phantom{\rule{4.pt}{0ex}}}_{f\in K}\parallel f-D\left(Ef\right){\parallel }_{\overline{\underline{X}}}.$
Let $n\left(K,E\right)=\text{sup}{\phantom{\rule{4.pt}{0ex}}}_{f\in K}#Ef$ where $#Ef$ is the number of bits in the bitstream $Ef$ . Thus $n$ is the maximum length of the bitstreams for the various $f\in K$ . There are two ways we can define optimal encoding:

• Prescribe $ϵ$ , the maximum distortion that we are willing to tolerate. For this $ϵ$ , find the smallest ${n}_{ϵ}\left(K,X\right):=\text{inf}{\phantom{\rule{4.pt}{0ex}}}_{\left(E,D\right)}\left\{n\left(K,E\right):d\left(K,E,D,X\right)\le ϵ\right\}$ . This is the smallest bit budget under which we could encode all elements of $K$ to distortion $ϵ$ .
• Prescribe $N$ : find the smallest distortion $d\left(K,E,D,X\right)$ over all $E,D$ with $n\left(K,E\right)\le N$ . This is the best encoding performance possible with a prescribed bit budget.

There is a simple mathematical solution to these two encoding problems based on the notion of Kolmogorov Entropy.

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