Optimum error quantization for LMS adaptation

The authors remove restrictions upon the transition and reconstruction levels of an L-level quantizer and examine error quantization as a design variable. The authors present a general performance factor expressing the misadjustment ratio of the quantized error algorithm with respect to the infinite-precision least-mean-squares (LMS) adaptation for a given convergence rate for Gaussian inputs. It is shown that maximizing this performance factor an an L-level quantizer with arbitrary transition levels and reconstruction levels produces the nonlinear system of equations defining the optimum Lloyd-Max quantizer for the Gaussian error. Simulations verify the theoretical performance loss predicted. The optimum-scaled uniform quantizer is shown to perform at most 0.013 dB worse than the optimum Lloyd-Max quantizer, indicating that in terms of both minimizing complexity and performance loss, uniform error quantization is close to the best for Gaussian inputs