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A Coding Theorem for Low-Rate Transform Codes

Transform codes are used to study low-rate quantization of stationary Gaussian sources. The transform decorrelates the source samples and then scalar quantization is applied to the vector of transform coefficients. Two bit allocations are considered: the first permits only zero or one bit to be allocated to each transform coefficient (i.e., the scalar quantizers have only one or two levels), and the second is an optimal bit allocation. For the transform codes with the "0-1" bit allocation, a closed-form, parametric expression is derived for the asymptotic (with dimension) rate vs. distortion performance. This expression is compared to the rate-distortion function, as well as to the performance of transform codes with optimal bit allocations. The principal result is that there is a critical rate, determined by the power spectral density, below which (and only below which) 0-1 allocations are optimal. This is a unique result in that it determines optimal theoretical performance for an important class of vector quantizes at low rates. Quantitative results are presented for Gauss-Markov sources.