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/ N
²
for the critical mismatch of ρ = 3/2, and as 1/N
²
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/N
²
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/N
²
)
^1/p
.