Variance-Mismatched Fixed-Rate Scalar Quantization of Laplacian Sources

The paper investigates the mean-squared error (MSE) distortion of a variance-mismatched fixed-rate scalar quantizer that is optimal or asymptotically optimal in the minimum MSE sense for a Laplacian density with standard deviation σ_q but is applied to another with standard deviation σ_p. A Nitadori-like formula is discovered for an optimal quantizer when ρ(^△= σ_p/σ_q) = 1/2. Also it is shown that the distortion expressions derived rigorously for asymptotically optimal quantile quantizers essentially confirm the heuristically obtained previous result that the distortion decreases as 1/ N^3/ρ for the heavy mismatch of ρ >; 3/2, as lnN/ N² for the critical mismatch of ρ = 3/2, and as 1/N² for the mild mismatch of ρ <; 3/2, where N is the number of quantization points. These asymptotic behaviors agree surprisingly well with Bennett´s integral even in the critical and heavy mismatched cases. Thus, in the case of nonuniform quantizers, the long-conjectured convergence of the MSE distortion to zero at a rate other than 1/N² is confirmed rigorously. In addition, optimal uniform quantizers are found to be heavily mismatched when ρ >; 1 and the resulting distortion decreases as (In N/N²)^1/p.