FIR/IIR Lattice 필터의 설계를 위한 Circulant Matrix Factorization을 사용한 Spectral Factorization에 관한 연구

Study of Spectral Factorization using Circulant Matrix Factorization to Design the FIR/IIR Lattice Filters

  • 김상태 (아리조나주립대학교 전자공학과) ;
  • 박종원 (한국해양연구원 해양시스템안전연구소)
  • 발행 : 2003.06.01

초록

Circulant Matrix Factorization (CMF)는 covariance 행렬의 spectral factorization된 결과를 얻을 수 있다. 우리는 얻어진 결과를 가지고 일반적으로 잘 알려진 방법인 Schur algorithm을 이용하여 finite impulse response(FIR)와 infinite impulse response (IIR) lattice 필터를 설계하는 방법을 제안하였다. CMF는 기존에 많이 사용되는 root finding을 사용하지 않고 covariance polynomial로부터 minimum phase 특성을 가지는 polynomial을 얻는데 유용한 방법이다. 그리고 Schur algorithm은 toeplitz matrix를 빠르게 Cholesky factorization하기 위한 방법으로 이 방법을 이용하면 FIR/IIR lattice 필터의 계수를 쉽게 찾아낼 수 있다. 본 논문에서는 이러한 방법들을 이용하여 FIR과 IIR lattice 필터의 설계의 계산적인 예제를 제시했으며, 제안된 방법과 다른 기존에 제시되었던 방법 (polynomial root finding과 cepstral deconvolution)들과 성능을 비교 평가하였다.

We propose the methods to design the finite impulse response (FIR) and the infinite impulse response (IIR) lattice filters using Schur algorithm through the spectral factorization of the covariance matrix by circulant matrix factorization (CMF). Circulant matrix factorization is also very powerful tool used fur spectral factorization of the covariance polynomial in matrix domain to obtain the minimum phase polynomial without the polynomial root finding problem. Schur algorithm is the method for a fast Cholesky factorization of Toeplitz matrix, which easily determines the lattice filter parameters. Examples for the case of the FIR Inter and for the case of the IIR filter are included, and performance of our method check by comparing of our method and another methods (polynomial root finding and cepstral deconvolution).

키워드

참고문헌

  1. M. Vetterli, 'A theory of multirate filter banks,' IEEE Trans. Acoust., Speech, Signal Processing, vol. ASSP-35, no. 3, pp. 356-372, 1987
  2. P. P. Vaidyanathan, Multirate Systems and Filter Banks, Prentice-Hall, Englewood Ciffs, NJ, 1993
  3. A. M. Bruckstein and T. Kailath, 'Inverse scattering for discrete transmission-line models,' SIAM Review, vol29, no3, pp.359-389, 1987 https://doi.org/10.1137/1029075
  4. J. D. Markel and A. H. Gray, Jr., Linear Prediction of Speech, Springer-Verlag, NewYork, 1976
  5. E. M. Dowling and D. L. MacFarlane, 'Lightwave lattice filters for optically multiplexed communication systems,' J. Lightwave Technol., vol. 12, no. 3, pp. 471-486, 1994 https://doi.org/10.1109/50.285330
  6. J. Bae, J. Chun, and S. Lee, 'Analysis of the fiber Bragg gratings using the lattice filter model,' Japanese J. Appl. Phys. part 1, vol. 39, no. 4A, pp. 1752-1756, 2000 https://doi.org/10.1143/JJAP.39.1752
  7. H. Schwetlick, 'Inverse methods in the reconstruction of acoustical impedance profilies,' J. Acoust. Soc. Am., vol. 73, no. 4, pp. 754-760, 1983
  8. S.T.Alexander, Adaptive Signal Processing Theory and Applications, SpringerVerlag, NewYork, 1986
  9. T. Kailath, 'Signal processing applications of some moment problems,' in Proc. Symp. Appl. Math., vol. 37, pp. 71-109, 1987
  10. P. P. Vaidyanathan, 'Passive cascaded- lattice structures for low-sensitivity FIR filter design with applications to filter banks,' IEEE Trans. Grcuits Syst., vol. CAS-33, no. 11, pp.1045-1064, 1986 https://doi.org/10.1109/TCS.1986.1085867
  11. J. Chun and T. Kailath, 'A constructive Proof of the Gohberg-Semencul Formula, Linear Algebra and Its Applications,' vol. 121, pp. 475-489, 1989 https://doi.org/10.1016/0024-3795(89)90717-9
  12. X. Chen and T. W. Parks, 'Design of Optimal Minimum Phase FIR Filters By Direct Factoriztion,' Signal Process., vol.10, pp.369-383, 1986 https://doi.org/10.1016/0165-1684(86)90045-9
  13. F. L. Bauer, 'Bin direktes Iterations Verfahren zur Hurwitz-zerlegung eines Polynoms, Arch. Elek. Ubertr.,' vol. 9, no. MR 17, pp. 285-290, 1955
  14. G. Wilson, 'Factorization of the Covariance Generating Function of a Pure Moving Average Process,' SIAM J. Numer. Anal., vol.6, no. 1, pp. 1-7, 1969 https://doi.org/10.1137/0706001
  15. G. A. Mian and A. P. Nainer, 'A Fast Procedure to Design Equiripple Minimum-Phase FIR Filters,' IEEE Trans. Circuits Syst., vol. CAS-29, no. 5, pp. 327-331, 1982 https://doi.org/10.1109/TCS.1982.1085145
  16. R. Boite and H. Leich, 'Comments on A Fast Procedure to Design Equiripple Minimum-Phase FIR Filters,' IEEE Trans. Circuits Syst., vol. CAS-31, no. 5, pp. 503-504, 1984 https://doi.org/10.1109/TCS.1984.1085523
  17. B. D. O. Anderson, K. L. Hitz, and N. D. Diem, 'Recursive Algorithm for Spectral Factorization,' IEEE Trans. Circuits Syst., vol. CAS-21, no. 6, pp. 742-750, 1974 https://doi.org/10.1109/TCS.1974.1083942
  18. P. J. Davis, Circulant Matrices, Wiley, New York, 1979
  19. A. V. Oppenheim and R. W. Schafer, Digital Signal Processing, Prentice-Hall, Englewood Cliffs, NJ, 1975
  20. V. Cizek, 'Discrete Hilbert transform,' IEEE Trans. Audio Electroacoust., vol. AU-18, no. 4, pp. 340-343, 1970 https://doi.org/10.1109/TAU.1970.1162139
  21. O. Herrmann and W. Schuessler, 'Design of nonrecursive digital filters with minimum phase,' Electron. Lett., vol.6, no.11, pp.329-330, 1970 https://doi.org/10.1049/el:19700232
  22. J. H. McClellan and T. W. Parks and L. R. Rabiner, 'A computer program for designing optimum FIR linear phase digital filters,' IEEE Trans. Audio Electroacoust., vol. AU-21, no. 6, pp. 506-526, 1973 https://doi.org/10.1109/TAU.1973.1162525
  23. L. Jackson, Digital Filtering and Signal Processing, Kluwer Academic Publishers, Boston, 1989