Piecewise Linear Companders are Robust

This paper considers the design of a robust quantizer for the class of input signal distributions having given quantiles and otherwise arbitrary shape. The quantizer model that consists of a compander and a uniform quantizer is utilized. The case of large number of quantization points is considered, and we use Bennett´s and Gersho´s approximation to the mean rth power distortion measure. We demonstrate that the piecewise linear compander provides robust quantization for the class of all input probability distributions having only their quantiles specified. The optimum robust solution is provided through the determination of all the required parameters. The problem is resolved for the case of block quantizers as well, and we show that the robust solution corresponds to a piecewise constant output point density function. The least favorable input multivariable density function is the piecewise uniform one