Fine Quantization in Signal Detection and Estimation

The performance lost to data quantization in signal detection and estimation procedures is considered. Assuming that observations lie in R”, a general result giving the f-divergence lost to fine uniform quantization is developed. In particular it is shown that, for a random n-vector, X, a convex function f:R-R, and a smooth function l:Rn-R, we have E{ f(l(X))} -E{f(E{ l(X)|fΔ})} ~ Δ2/24E{||∇l(X)||2fn(X)}, where, for each Δ 0, F is any sub-a-field of Bngenerated by uniform partitioning of Rn into n-dimensional cubes with side Δ This general result is applied to obtain specific asymptotic expressions for the performance lost to uniform data quantization in several signal detection and estimation problems including minimum mean-squared error (mmse) estimation, nonrandom point estimation, and binary signal detection. Numerical results illustrating the closeness of the approximation implied by the above result are also given. An analogous expression for the divergence lost to companded (nonuniform) data quantization also is developed, and the compander that minimizes the divergence loss is derived.