Properties of scalar quantization with th power distortion and constrained Rényi entropy of order 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 and Gray et al.'s variable-rate mismatch results to general values of the entropy order parameter .

We consider optimal scalar quantization with rth power distortion and constrained Rényi entropy of order α. For sources with absolutely continuous distributions the high rate asymptotics of the quantizer distortion has long been known for α = 0 (fixed-rate quantization) and α = 1 (entropy-constrained quantization). For a large class of absolutely continuous source distributions we determine the sharp asymptotics of the optimal quantization distortion for Rényi entropy constraints of order α ∈ [-∈, 0) ∪ (0; 1). The proof of achievability is based on companding quantization and is thus constructive.

We consider optimal scalar quantization with rth power distortion and constrained Rényi entropy of order α. For sources with absolutely continuous distributions the high rate asymptotics of the quantizer distortion has long been known for α = 0 (fixed-rate quantization) and α = 1 (entropy-constrained quantization). These results have recently been extended to quantization with Rényi entropy constraint of order α ≥r+1. Here we consider the more challenging case α ∈ [-∞,0)∪(0,1) and for a large class of absolutely continuous source distributions we determine the sharp asymptotics of the optimal quantization distortion. The achievability proof is based on finding (asymptotically) optimal quantizers via the companding approach, and is thus constructive.