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Hamming Distortion

Definition (Hamming distortion)

Assume X=X^\mathcal{X}=\hat{\mathcal{X}}. The Hamming Distortion is defined with a distortion measure such that d(x,x^)={0x=x^1x≠x^d(x,\hat{x})=\begin{cases} 0&x=\hat{x} \\ 1&x\not=\hat{x} \end{cases}

Theorem (Hamming Rate Distortion Function)

For a binary source with P(X=1)=pP(X=1)=p and Hamming Distortion R(D)={hb(p)−hb(D)0≤D≤min⁡{p,1−p}0D≥min⁡{p,1−p}R(D)=\begin{cases} h_{b}(p)-h_{b}(D)&0\le D\le \min\{ p,1-p \} \\ 0&D\ge\min\{ p,1-p \} \end{cases}

Remark

The Shannon limit for a BSC with crossover probability ϵ\epsilon is DSL=ϵD_{SL}=\epsilon

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Hamming Distortion