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Optimal Encoding

Module by: Ronald DeVore, Shriram Sarvotham. E-mail the authors

We shall consider now the encoding of signals on [-T,T][-T,T] where T>0T>0 is fixed. Ultimately we shall be interested in encoding classes of bandlimited signals like the class B A B A However, we begin the story by considering the more general setting of encoding the elements of any given compact subset KK of a normed linear space XX. One can determine the best encoding of KK by what is known as the Kolmogorov entropy of KK in XX.

To begin, let us consider an encoder-decoder pair (E,D)(E,D) EE maps KK to a finite stream of bits. DD maps a stream of bits to a signal in XX. This is illustrated in Figure 1. Note that many functions can be mapped onto the same bitstream.

Figure 1: Illustration of encoding and decoding.
Figure 1 (figure3.png)

Define the distortion dd for this encoder-decoder by

d(K,E,D,X):=sup fK f-D(Ef) X ̲ ¯ .d(K,E,D,X):=sup fK f-D(Ef) X ̲ ¯ .
(1)
Let n(K,E)=sup fK #Efn(K,E)=sup fK #Ef where #Ef#Ef is the number of bits in the bitstream EfEf. Thus nn is the maximum length of the bitstreams for the various fKfK. There are two ways we can define optimal encoding:

  1. Prescribe ϵϵ, the maximum distortion that we are willing to tolerate. For this ϵϵ, find the smallest n ϵ (K,X):=inf (E,D) {n(K,E):d(K,E,D,X)ϵ}n ϵ (K,X):=inf (E,D) {n(K,E):d(K,E,D,X)ϵ}. This is the smallest bit budget under which we could encode all elements of KK to distortion ϵϵ.
  2. Prescribe NN : find the smallest distortion d(K,E,D,X)d(K,E,D,X) over all E,DE,D with n(K,E)Nn(K,E)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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