The rate-distortion function on classes of sources determined by spectral capacities

Abstract

The quantity\sup_{a \in \cal R} R{a}(D)is considered, where\cal Ris a class of homogeneousn-parameter sources andR_{a}(D)denotes the single-letter mean-square-error (MSE) rate-distortion function for the individual source a. In particular, the case in which the class\cal Ris specified in terms of spectral information is treated for general classes of spectral measures whose upper measures are capacities (in the sense of Choquet) alternating of order two. This type of class includes many common models for spectral uncertainty such as mixture models, spectral band models, and neighborhoods generated by variation and Prohorov metrics. It is shown that each such class contains a worst-case source whose rate-distortion function achieves the supremum over the class for each value of distortion. This source is characterized as having a spectral density that is a derivative (in the sense of Huber and Strassen) of the upper spectral measure with respect to Lebesgue measure on[-\pi,\pi]^{n}. Moreover it is shown that the spectral measure of the worst-case source is closest, in a sense defined by directed divergence, to Lebesgue measure (which corresponds to a memoryless source). Numerical results are presented for the particular case in which the source spectral measure is a mixture of a Gauss-Markov spectrum and an unknown contaminating component.