Asymptotic Formulas for Mismatched Fixed-Rate Minimum MSE Laplacian Quantizers

This letter derives asymptotic formulas for the mean square error (MSE) distortion and the signal-to-noise ratio of a mismatched fixed-rate minimum MSE Laplacian quantizer. These closed-form formulas are expressed in terms of the number N of quantization points, the mean displacement, and the ratio p of the standard deviation of the source to that for which the quantizer is optimally designed. Numerical results show that the principal formula is accurate in that, for rate R = log₂ N ges 6, it predicts signal-to-noise ratios within 1 % of the true values for a wide range of variance and mean mismatch. The new findings herein include the fact that, for heavy variance mismatch of p > 3/2, the signal-to-noise ratio increases at the rate of 9/p dB/bit, which is slower than the usual 6 dB/bit, and the fact that an optimal uniform quantizer, though optimally designed, is slightly more than critically mismatched to the source. The derived formulas can be useful in quantization of speech or music signals, which are modeled well as Laplacian sources and have changing short-term variances.