Entropy Density and Mismatch in High-Rate Scalar Quantization with Renyi Entropy Constraint

Properties of scalar quantization with $r$th power distortion and constrained Rényi entropy of order $α\in (0,1)$ are investigated. For an asymptotically (high-rate) optimal sequence of quantizers, the contribution to the Rényi entropy due to source values in a fixed interval is identified in terms of the "entropy density" of the quantizer sequence. This extends results related to the well-known point density concept in optimal fixed-rate quantization. A dual of the entropy density result quantifies the distortion contribution of a given interval to the overall distortion. The distortion loss resulting from a mismatch of source densities in the design of an asymptotically optimal sequence of quantizers is also determined. This extends Bucklew's fixed-rate ($α=0$) and Gray \emph{et al.}'s variable-rate ($α=1$) mismatch results to general values of the entropy order parameter $α$.

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