Multidimensional companding for lattice vector quantization of circularly symmetric densities

In order to realize a flexible vector quantizer with a large codebook but low complexity we consider a system which consists of a nonlinear mapping (compressor), a lattice vector quantizer, and the inverse of the compressor (expander). In general, the nonlinear expansion of the Voronoi regions of the lattice will increase their normalized second moments and hereby cause a loss. Recently, we derived this loss for the practically important class of optimal lattices [1]. This paper focuses on the special case of a source with a spherically symmetric probability density and a compander (compressor/expander) which is only a function of the magnitude of the input vector. Given the density of the source it is shown how the compander can be determined using high-rate quantization theory. Further, on the basis of a simple compander as an example, it is demonstrated that the loss introduced by the companding operation can be kept very small in practical situations.