A simple derivation of Lloyd's classical result for the optimum scalar quantizer (Corresp.)

The classical result of Lloyd for the optimum scalar quantizer in the asymptotic case of fine quantization is derived from first principles. The derivation is offered as a simple alternative to Lloyd's original and elegant piece of analysis, and the result is used to derive the optimum compander. We then show why a compander that presents uniformly distributed random variables to the quantizer is not a good idea.