Robust memoryless quantization for minimum signal distortion

Abstract

Robust quantizers are designed for situations where there is only an incomplete statistical description of the quantizer input. The goal of the design is to closely approximate quantizer inputs by quantizer outputs without using more than a specified number of quantization levels. The exact probability distribution of the input is unknown, but this distribution is known to belong to some set <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</tex> . The primary, set <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</tex> considered is the set of all unimodal probability distributions which satisfy generalized moment constraint (e.g., mean-square value less than or equal to a constant). A quantizer is derived which minimizes over all quantizers the maximum distortion over all distributions in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">C</tex> . This robust quantizer guarantees a significanfiy lower worst case distortion than the classical Gaussian-optimal quantizer, while performing nearly as well as the Gaussian-optimal quantizer when the input is, in fact, Gaussian.