The Journal of Korean Institute of Communications and Information Sciences (한국통신학회논문지)

The Korean Institute of Commucations and Information Sciences (한국통신학회)
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On the Support of Minimum Mean-Square Error Scalar Quantizers for a Laplacian Source

라플라스 신호원에 대한 최소평균제곱오차 홑 양자기의 지지역에 관하여

  • 김성민 (아주대학교 전자공학부 부호화 연구실) ;
  • 나상신 (아주대학교 전자공학부 부호화 연구실)
  • Published : 2006.10.31
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Abstract

This paper shows that the support growth of an optimum (minimum mean square-error) scalar quantizer for a Laplacian density is logarithmic with the number of quantization points. Specifically, it is shown that, for a unit-variance Laplacian density, the ratio of the support-determining threshold of an optimum quantizer to $\frac 3{\sqrt{2}}1n\frac N 2$ converges to 1, as the number of quantization points grows. Also derived is a limiting upper bound that says that the optimum support cannot exceed the logarithmic growth by more than a constant. These results confirm the logarithmic growth of the optimum support that has previously been derived heuristically.

이 논문은 라플라스 밀도 함수에 대한 최적 흩 양자기의 지지역의 증가는 양자점의 개수와 대수적인 관계가 있음을 보여준다. 구체적으로, 분산이 1인 라플라스 밀도함수에 대해서 양자정의 개수 N이 증가할 때 최적 양자기의 경계값에 의해 결정되는 지지역과 $\frac 3{\sqrt{2}}1n\frac N 2$의 비율이 1로 수렴함을 보여준다. 또한 극한 상한값을 유도하여 최적 지지역의 로그적 증가가 그 값을 초과하지 않음을 보였다. 이 결과들로부터 이전부터 경험적으로 연구되어 온 최적 지지역의 로그 증가를 확인 할 수 있다.

Keywords

References

  1. K. Nitadori, 'Statistical analysis of $\Delta$PCM, ' Electron, Commun. in Japan, vol. 48, pp.17-26, Feb. 1965
  2. P.E. Fleischer, 'Sufficient conditions for achieving minimum distortion in a quantizer,' int.Conv.Rec., Part 1, pp. 104-111, 1964
  3. H. Lanfer. 'Maximum signal-to-noise-ratio quantization for Laplacian-distributed signals,' Information and System Theory in Digital Communications, NTG-Report vol.65 VDE-Verlag GmbH Berlin, Germany, p. 52, 1978
  4. P. Nol and Zelinski, 'comments on 'quantizing characteristics for signals Laplacian amplitude probability density function',' IEEE Trans. on Comm., vol. COM-27,n. 8, pp. 1259-1260, Aug. 1979
  5. J.A. Bucklew and N.C. Gallagher, Jr.,'Note on the computation of optimal minimum mean-square error quantizers,' IEEE Trans. on Commun, COM-30, pp. 298-301, Jan. 1982
  6. F.S. Lu and G.L. Wise, 'A further investigation of Max's algorithm for optimum quantization,' IEEE Trans. on Comm., COM-33, pp. 746-750, Jul. 1985
  7. S. Na and D.L. Neuhoff, 'On the support region of asymptotically optimum mean-square error scalar quantizers for a generalized gamma source,' Submitted to IEEE Trans. on Inform Theory for publication
  8. P.F Panter and W. Dite, 'Quantization distortion in pulse count modulation with nonuniform spacing of levels,' Proc. IRE, pp. 44-48, Jan. 1951
  9. J.A. Bucklew and G.L. Wise, 'A note on multidimensional asymptoti quantization theory,' Proc. Eighteen Ann. Allerton Conf. Commun. Contr. Comput, pp. 454-463, Oct. 1980
  10. E.J. Purcell and D. Varberg, Calculus with Analytic Geometry, 4th edition, Prentice-Hall, 1984
  11. S. Cambanis and N.L. Gerr, 'Simple class of asymptotically optimal quantizers,' IEEE Trans. on Inform. Theory, IT-29, pp.664-676, Sept. 1983
  12. J.W. Harris and H. Stocker. Handbook of Mathematics and Computational Science, Springer, 1998