Estimating the optimal support and the rate of convergence to the Panter-Dite formula for a Laplacian source

For a Laplacian source, we estimate the support threshold for a minimum mean-squared error (MSE), fixed-rate scalar quantizer. We provide upper and lower bounds to the support threshold as a function of the number of quantization levels N and observe that the optimal support threshold grows as 3/√2 log (N/2)+O_N (1). An upper bound is given by the support threshold for a nonuniform scalar quantizer designed around the asymptotically optimal companding function c(x). A lower bound is constructed by examining the decay rate of the smallest lower half step as a function of N. Using these bounds, we derive an upper bound to the convergence rate of N²D*_N to the Panter-Dite constant, where D*_N is the least MSE of any even, N-level scalar quantizer.