iBet uBet web content aggregator. Adding the entire web to your favor.
iBet uBet web content aggregator. Adding the entire web to your favor.



Link to original content: https://doi.org/10.1155/2007/92953
Subspace-Based Noise Reduction for Speech Signals via Diagonal and Triangular Matrix Decompositions: Survey and Analysis | EURASIP Journal on Advances in Signal Processing | Full Text
Skip to main content
EURASIP Journal on Advances in Signal Processing
Download PDF
  • Research Article
  • Open access
  • Published:

Subspace-Based Noise Reduction for Speech Signals via Diagonal and Triangular Matrix Decompositions: Survey and Analysis

EURASIP Journal on Advances in Signal Processing volume 2007, Article number: 092953 (2007) Cite this article

Abstract

We survey the definitions and use of rank-revealing matrix decompositions in single-channel noise reduction algorithms for speech signals. Our algorithms are based on the rank-reduction paradigm and, in particular, signal subspace techniques. The focus is on practical working algorithms, using both diagonal (eigenvalue and singular value) decompositions and rank-revealing triangular decompositions (ULV, URV, VSV, ULLV, and ULLIV). In addition, we show how the subspace-based algorithms can be analyzed and compared by means of simple FIR filter interpretations. The algorithms are illustrated with working Matlab code and applications in speech processing.

References

  1. Doclo S, Moonen M: GSVD-based optimal filtering for single and multimicrophone speech enhancement. IEEE Transactions on Signal Processing 2002,50(9):2230-2244. 10.1109/TSP.2002.801937

    Article  Google Scholar 

  2. Golub GH, Van Loan CF: Matrix Computations. 3rd edition. The Johns Hopkins University Press, Baltimore, Md, USA; 1996.

    MATH  Google Scholar 

  3. Stewart GW: Matrix Algorithms. Volume 1: Basic Decompositions. SIAM Press, Philadelphia, Pa, USA; 1998.

    Book  MATH  Google Scholar 

  4. Stewart GW: The decompositional approach to matrix computation. Computing in Science and Engineering 2000,2(1):50-59. 10.1109/5992.814658

    Article  Google Scholar 

  5. Scharf LL, Tufts DW: Rank reduction for modeling stationary signals. IEEE Transactions on Acoustics, Speech, and Signal Processing 1987,35(3):350-355. 10.1109/TASSP.1987.1165136

    Article  Google Scholar 

  6. Tufts DW, Kumaresan R: Singular value decomposition and improved frequency estimation using linear prediction. IEEE Transactions on Acoustics, Speech, and Signal Processing 1982,30(4):671-675. 10.1109/TASSP.1982.1163927

    Article  Google Scholar 

  7. Cadzow JA: Signal enhancement-a composite property mapping algorithm. IEEE Transactions on Acoustics, Speech, and Signal Processing 1988,36(1):49–62. 10.1109/29.1488

    Article  MathSciNet  MATH  Google Scholar 

  8. De Moor B: The singular value decomposition and long and short spaces of noisy matrices. IEEE Transactions on Signal Processing 1993,41(9):2826-2838. 10.1109/78.236505

    Article  MATH  Google Scholar 

  9. Scharf LL: The SVD and reduced rank signal processing. Signal Processing 1991,25(2):113-133. 10.1016/0165-1684(91)90058-Q

    Article  MATH  Google Scholar 

  10. Dendrinos M, Bakamidis S, Carayannis G: Speech enhancement from noise: a regenerative approach. Speech Communication 1991,10(1):45-57. 10.1016/0167-6393(91)90027-Q

    Article  Google Scholar 

  11. Van Huffel S: Enhanced resolution based on minimum variance estimation and exponential data modeling. Signal Processing 1993,33(3):333-355. 10.1016/0165-1684(93)90130-3

    Article  Google Scholar 

  12. Ephraim Y, Van Trees HL: A signal subspace approach for speech enhancement. IEEE Transactions on Speech and Audio Processing 1995,3(4):251-266. 10.1109/89.397090

    Article  Google Scholar 

  13. Huang J, Zhao Y: Energy-constrained signal subspace method for speech enhancement and recognition. IEEE Signal Processing Letters 1997,4(10):283-285. 10.1109/97.633769

    Article  Google Scholar 

  14. Rezayee A, Gazor S: An adaptive KLT approach for speech enhancement. IEEE Transactions on Speech and Audio Processing 2001,9(2):87-95. 10.1109/89.902276

    Article  Google Scholar 

  15. Jensen SH, Hansen PC, Hansen SD, Sørensen JAa: Reduction of broad-band noise in speech by truncated QSVD. IEEE Transactions on Speech and Audio Processing 1995,3(6):439–448. 10.1109/89.482211

    Article  MATH  Google Scholar 

  16. Hu Y, Loizou PC: A subspace approach for enhancing speech corrupted by colored noise. IEEE Signal Processing Letters 2002,9(7):204-206. 10.1109/LSP.2002.801721

    Article  Google Scholar 

  17. Hu Y, Loizou PC: A generalized subspace approach for enhancing speech corrupted by colored noise. IEEE Transactions on Speech and Audio Processing 2003,11(4):334-341. 10.1109/TSA.2003.814458

    Article  Google Scholar 

  18. Lev-Ari H, Ephraim Y: Extension of the signal subspace speech enhancement approach to colored noise. IEEE Signal Processing Letters 2003,10(4):104-106. 10.1109/LSP.2003.808544

    Article  Google Scholar 

  19. Mittal U, Phamdo N: Signal/noise KLT based approach for enhancing speech degraded by colored noise. IEEE Transactions on Speech and Audio Processing 2000,8(2):159–167. 10.1109/89.824700

    Article  Google Scholar 

  20. Moonen M, Van Dooren P, Vandewalle J: A singular value decomposition updating algorithm for subspace tracking. SIAM Journal on Matrix Analysis and Applications 1992,13(4):1015-1038. 10.1137/0613061

    Article  MathSciNet  MATH  Google Scholar 

  21. Stewart GW: An updating algorithm for subspace tracking. IEEE Transactions on Signal Processing 1992,40(6):1535-1541. 10.1109/78.139256

    Article  Google Scholar 

  22. Stewart GW: Updating a rank-revealing ULV decomposition. SIAM Journal on Matrix Analysis and Applications 1993,14(2):494-499. 10.1137/0614034

    Article  MathSciNet  MATH  Google Scholar 

  23. Adams G, Griffin MF, Stewart GW: Direction-of-arrival estimation using the rank-revealing URV decomposition. Proceedings of the International Conference on Acoustics, Speech, and Signal Processing (ICASSP '91), May 1991, Toronto, Ontario, Canada 2: 1385–1388.

    Google Scholar 

  24. Liu KJR, O'Leary DP, Stewart GW, Wu Y-JJ: URV ESPRIT for tracking time-varying signals. IEEE Transactions on Signal Processing 1994,42(12):3441-3448. 10.1109/78.340778

    Article  Google Scholar 

  25. Van Huffel S, Zha H: An efficient total least squares algorithm based on a rank-revealing two-sided orthogonal decomposition. Numerical Algorithms 1993,4(1-2):101-133.

    Article  MathSciNet  MATH  Google Scholar 

  26. Luk FT, Qiao S: A new matrix decomposition for signal processing. Automatica 1994,30(1):39–43. 10.1016/0005-1098(94)90227-5

    Article  MathSciNet  MATH  Google Scholar 

  27. Luk FT, Qiao S: Adaptive algorithm for interference canceling in array processing. In Advanced Signal Processing Algorithms, Architectures, and Implementations VI, August 1996, Denver, Colo, USA, Proceedings of SPIE Edited by: Luk FT. 2846: 151–161.

    Chapter  Google Scholar 

  28. Bojanczyk AW, Lebak JM: Downdating a ULLV decomposition of two matrices. In Proceedings of the 5th SIAM Conference on Applied Linear Algebra, June 1994, Snowbird, Utah, USA. Edited by: Lewis JG. SIAM; 261–265.

    MATH  Google Scholar 

  29. Zhong W, Li S, Tai H-M: Signal subspace approach for narrowband noise reduction in speech. IEE Proceedings: Vision, Image and Signal Processing 2005,152(6):800-805. 10.1049/ip-vis:20045246

    Google Scholar 

  30. Hansen PC: Rank-deficient prewhitening with quotient SVD and ULV decompositions. BIT Numerical Mathematics 1998,38(1):34-43. 10.1007/BF02510915

    Article  MathSciNet  MATH  Google Scholar 

  31. Hansen PC, Jensen SH: Prewhitening for rank-deficient noise in subspace methods for noise reduction. IEEE Transactions on Signal Processing 2005,53(10):3718-3726.

    Article  MathSciNet  MATH  Google Scholar 

  32. Dologlou I, Carayannis G: Physical interpretation of signal reconstruction from reduced rank matrices. IEEE Transactions on Signal Processing 1991,39(7):1681-1682. 10.1109/78.134407

    Article  Google Scholar 

  33. Hansen PC, Jensen SH: FIR filter representations of reduced-rank noise reduction. IEEE Transactions on Signal Processing 1998,46(6):1737-1741. 10.1109/78.678511

    Article  Google Scholar 

  34. Hansen PC: Rank-Deficient and Discrete Ill-Posed Problems: Numerical Aspects of Linear Inversion. SIAM, Philadelphia, Pa, USA; 1998.

    Book  Google Scholar 

  35. Stewart GW: Rank degeneracy. SIAM Journal on Scientific and Statistical Computing 1984,5(2):403-413. 10.1137/0905030

    Article  MathSciNet  MATH  Google Scholar 

  36. Konstantinides K, Yao K: Statistical analysis of effective singular values in matrix rank determination. IEEE Transactions on Acoustics, Speech, and Signal Processing 1988,36(5):757-763. 10.1109/29.1585

    Article  MATH  Google Scholar 

  37. Jabloun F, Champagne B: Incorporating the human hearing properties in the signal subspace approach for speech enhancement. IEEE Transactions on Speech and Audio Processing 2003,11(6):700-708. 10.1109/TSA.2003.818031

    Article  Google Scholar 

  38. You CH, Koh SN, Rahardja S: An invertible frequency eigendomain transformation for masking-based subspace speech enhancement. IEEE Signal Processing Letters 2005,12(6):461-464.

    Article  Google Scholar 

  39. McAulay RJ, Quateri TF: Speech analysis/synthesis based on a sinusoidal representation. IEEE Transactions on Acoustics, Speech, and Signal Processing 1986,34(4):744-754. 10.1109/TASSP.1986.1164910

    Article  Google Scholar 

  40. Vanhamme L, Fierro RD, Van Huffel S, de Beer R: Fast removal of residual water in proton spectra. Journal of Magnetic Resonance 1998,132(2):197-203. 10.1006/jmre.1998.1425

    Article  Google Scholar 

  41. Vanhamme L, Sundin T, van Hecke P, Van Huffel S: MR spectroscopy quantitation: a review of time-domain methods. NMR in Biomedicine 2001,14(4):233-246. 10.1002/nbm.695

    Article  Google Scholar 

  42. De Moor B: Total least squares for affinely structured matrices and the noisy realization problem. IEEE Transactions on Signal Processing 1994,42(11):3104-3113. 10.1109/78.330370

    Article  Google Scholar 

  43. Hansen PC: The truncated SVD as a method for regularization. BIT Numerical Mathematics 1987,27(4):534-553. 10.1007/BF01937276

    Article  MathSciNet  MATH  Google Scholar 

  44. Björck Å: Numerical Methods for Least Squares Problems. SIAM, Philadelphia, Pa, USA; 1996.

    Book  MATH  Google Scholar 

  45. Thorpe AJ, Scharf LL: Data adaptive rank-shaping methods for solving least squares problems. IEEE Transactions on Signal Processing 1995,43(7):1591-1601. 10.1109/78.398720

    Article  Google Scholar 

  46. Fierro RD, Hansen PC: Low-rank revealing UTV decompositions. Numerical Algorithms 1997,15(1):37-55. 10.1023/A:1019254318361

    Article  MathSciNet  MATH  Google Scholar 

  47. Fierro RD, Hansen PC, Hansen PSK: UTV Tools: Matlab templates for rank-revealing UTV decompositions. Numerical Algorithms 1999,20(2-3):165-194.

    Article  MathSciNet  MATH  Google Scholar 

  48. Fierro RD, Hansen PC: UTV Expansion Pack: special-purpose rank-revealing algorithms. Numerical Algorithms 2005,40(1):47-66. 10.1007/s11075-005-2263-2

    Article  MathSciNet  MATH  Google Scholar 

  49. Golub GH: Numerical methods for solving least squares problems. Numerische Mathematik 1965,7(3):206-216. 10.1007/BF01436075

    Article  MathSciNet  MATH  Google Scholar 

  50. Hansen PSK, Hansen PC, Hansen SD, Sørensen JAa: Noise reduction of speech signals using the rank-revealing ULLV decomposition. Signal Processing VIII. Theories and Applications, Proceedings of the 8th European Signal Processing Conference (EUSIPCO '96), September 1996, Trieste, Italy 2: 967–970.

    Google Scholar 

  51. Hansen PSK, Hansen PC, Hansen SD, Sørensen JAa: ULV-based signal subspace methods for speech enhancement. Proceedings of International Workshop on Acoustic Echo and Noise Control (IWAENC '97), September 1997, London, UK 9–12.

    Google Scholar 

  52. Luk FT, Qiao S: A symmetric rank-revealing Toeplitz matrix decomposition. The Journal of VLSI Signal Processing 1996,14(1):19–28. 10.1007/BF00925265

    Article  MATH  Google Scholar 

  53. Hansen PC, Yalamov PY: Computing symmetric rank-revealing decompositions via triangular factorization. SIAM Journal on Matrix Analysis and Applications 2001,23(2):443-458. 10.1137/S0895479800370068

    Article  MathSciNet  MATH  Google Scholar 

  54. Jensen J, Hendriks RC, Heusdens R, Jensen SH: Smoothed subspace based noise suppression with application to speech enhancement. Proceedings of the 13th European Signal Processing Conference (EUSIPCO '05), September 2005, Antalya, Turkey

    Google Scholar 

  55. Bunch JR, Nielsen CP: Updating the singular value decomposition. Numerische Mathematik 1978,31(2):111-129. 10.1007/BF01397471

    Article  MathSciNet  MATH  Google Scholar 

  56. Gu M, Eisenstat SC: Downdating the singular value decomposition. SIAM Journal on Matrix Analysis and Applications 1995,16(3):793-810. 10.1137/S0895479893251472

    Article  MathSciNet  MATH  Google Scholar 

  57. Björck Å, Golub GH: Numerical methods for computing angles between linear subspaces. Mathematics of Computation 1973,27(123):579–594.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Informatics and Mathematical Modelling, Technical University of Denmark, Building 321, Lyngby, 2800, Denmark

    Per Christian Hansen

  2. Department of Electronic Systems, Aalborg University, Niels Jernes Vej 12, Aalborg, 9220, Denmark

    Søren Holdt Jensen

Authors
  1. Per Christian Hansen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Søren Holdt Jensen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Per Christian Hansen.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://doi.org/creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Hansen, P.C., Jensen, S.H. Subspace-Based Noise Reduction for Speech Signals via Diagonal and Triangular Matrix Decompositions: Survey and Analysis. EURASIP J. Adv. Signal Process. 2007, 092953 (2007). https://doi.org/10.1155/2007/92953

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1155/2007/92953

Keywords