• The Journal of Korean Institute of Communications and Information Sciences
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• v.36 no.6C
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• pp.384-390
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• 2011

The paper derives formulas for the mean-squared error distortion and resulting signal-to-noise (SNR) ratio of a fixed-rate scalar quantizer designed optimally in the minimum mean-squared error sense for a Gaussian density with the standard deviation ${\sigma}_q$ when it is mismatched to a Laplacian density with the standard deviation ${\sigma}_q$. The SNR formulas, based on the key parameter and Bennett's integral, are found accurate for a wide range of $p\({\equiv}\frac{\sigma_p}{\sigma_q}\){\geqq}0.25$. Also an upper bound to the SNR is derived, which becomes tighter with increasing rate R and indicates that the SNR behaves asymptotically as $\frac{20\sqrt{3{\ln}2}}{{\rho}{\ln}10}\;{\sqrt{R}}$ dB.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.25 no.10A
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• pp.1566-1575
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• 2000

This paper studies mismatched scalar quantization of a generalized gamma source by a quantizer that is optimally (in the mean square error sense) designed for another generalized gamma source. Specifically, it considers variance-mismatched quantization which occurs when the variance of the source to be quantized differs from tat of the designed-for source. The main result is the two distortion formulas derived from Bennett's integral. The first formula is an approximation expression that uses the outermost threshold of an optimum scalar quantizer, and the second formula, in turn, uses an approximation formula for this outermost threshold. Numerical results are obtained for Laplacian sources, which are example of a generalized gamma source, and comparisons are made between actual mismatched distortions and the two formulas. These numerical results show that the two formulas become more accurate, as the number of quantization points gets larger and the ratio of the source variance to that of the designed-for source gets bigger. For example, the formulas are within 2~4% of the actual distortion for approximately 64 quantization points or more. In conclusion, the proposed approximation formulas are considered to have contribution as closed formulas and for their accuracy.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.33 no.5C
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• pp.413-421
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• 2008

The paper derives asymptotic formulas for the 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 $\mu$, and the ratio $\rho$ 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=$log_2N{\geq}6$, it predicts signal-to-noise ratios within 1% of the true values for a wide range of $\mu$, and $\rho$. The new findings herein include the fact that, for heavy variance mismatch of ${\rho}>3/2$, the signal-to-noise ratio increases at the rate of $9/\rho$ 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. It is also found that signal-to-noise ratio loss due to $\mu$ is moderate. 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.