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Definition (Optimal Quantizer)
Let QN\mathcal{Q}_{N}QN denote the family of all NNN-level quantizers. Q∗∈QNQ^{*}\in\mathcal{Q}_{N}Q∗∈QN is an optimal quantizer if E[d(X,Q∗(X))]=minQ∈QNE[d(X,Q(X))]E[d(X,Q^{*}(X))]=\min_{Q\in\mathcal{Q}_{N}}E[d(X,Q(X))]E[d(X,Q∗(X))]=Q∈QNminE[d(X,Q(X))]
Remark
Q∗Q^{*}Q∗ is not necessarily unique. Q∗Q^{*}Q∗ exists for all NNN and “reasonable” ddd if E[d(X,y)]<∞E[d(X,y)]<\inftyE[d(X,y)]<∞ for some y∈Ry\in\mathbb{R}y∈R.
Lloyd-Max Algorithm
Optimal Compressor
Performance Analysis of Quantizers