On suboptimal multidimensional companding

A vector quantizer (VQ) consisting of a nonlinear mapping (compressor), a lattice VQ, and the inverse of the compressor (expander) is considered. While it was previously pointed out that in dimensions k>2 except for linear transformations and translations only reflections through reciprocal radii can preserve optimality in terms of the lattice cells' normalized second moments, we consider the suboptimal case and provide a method to determine the loss introduced by companding. Using a spherically symmetric compander as an example, it is demonstrated that the loss can be kept very small in practical situations, especially when large VQ dimensions are chosen.