Skip to main content
SpringerLink
Log in

Complete Factorization of the 2m-Band Paraunitary Polyphase Matrix with Multiple Centers of Symmetry

  • Published:
Circuits, Systems, and Signal Processing Aims and scope Submit manuscript

A Correction to this article was published on 30 October 2019

This article has been updated

Abstract

The design of filter banks with multiple centers of symmetry is very difficult. In this paper, a space decomposition of an orthogonal projection matrix is studied. This decomposition plays a key role in a new complete factorization theory. In addition, the concept of a minimal starting block matrix is proposed and is used to establish a new factorization of a 2m-band paraunitary polyphase matrix with multiple centers of symmetry. This factorization has the completeness property. The different possible forms of the minimal starting block matrix, which lead to the different types of filter banks, are obtained. Through different combinations of minimal starting block matrices and orthogonal projection matrices, the general solutions of a 2m-band paraunitary system with multiple centers are obtained theoretically. The four-band issue is discussed in detail as an example.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Change history

  • 30 October 2019

    The original version of the article unfortunately contained an error in article.

References

  1. J.Z. Chen, W.X. Gao, Z.W. Ju, T. Xia, H.F. Ling, C.N. Chen, X. Wang, A new design method of 9-7 biorthogonal filter banks based on odd harmonic function. Circuit Syst. Signal Process. 31(3), 1245–1255 (2012)

    Article  MathSciNet  Google Scholar 

  2. J.Z. Chen, Z.W. Ju, H. Cao, P. Xie, C.N. Chen, R. Li, T. Xia, L.H. Qin, A lifting scheme of symmetric-antisymmetric multiwavelet transform for image coding. Circuit Syst. Signal Process. 31(5), 1887–1900 (2012)

    Article  MathSciNet  Google Scholar 

  3. Y.J. Chen, S. Oraintara, K.S. Amaratunga, Dyadic-based factorizations for regular paraunitary filter banks and M-band orthogonal wavelets with structural vanishing moments. IEEE Trans. Signal Process. 53(1), 193–207 (2003)

    Article  Google Scholar 

  4. A. Cohen, I. Daubechies, J.C. Feauveau, Biorthogonal bases of compactly supported wavelets. Commun. Pure Appl. Math. 45, 485–560 (1992)

    Article  MathSciNet  Google Scholar 

  5. I. Daubechies, W. Sweldens, Factoring wavelet transform into lifting steps. J. Fourier Anal. Appl. 4(3), 245–269 (1998)

    Article  MathSciNet  Google Scholar 

  6. R.L. de Queiroz, T.Q. Nguyen, K.R. Rao, The Genlot: generalized linear-phase lapped orthogonal transform. IEEE Trans. Signal Process. 44(3), 497–507 (1996)

    Article  Google Scholar 

  7. L. Gan, K.K. Ma, A simplified lattice factorization for linear-phase perfect reconstruction filter bank. IEEE Signal Process. Lett. 8(7), 207–209 (2001)

    Article  Google Scholar 

  8. X. Gao, T.Q. Nguyen, G. Strang, On factorization of M-band paraunitary filter banks. IEEE Trans. Signal Process. 49(7), 1433–1446 (2001)

    Article  MathSciNet  Google Scholar 

  9. C.W. Kok, T. Nagai, M. Ikehara, T.Q. Nguyen, Structures and factorizations of linear-phase filter banks, in Proceedings of ISCASS (1996)

  10. C.W. Kok, T. Nagai, M. Ikehara, T.Q. Nguyen, Lattice structures parameterization of linear phase paraunitary matrices with pairwise mirrorimage symmetry in the frequency domain with an odd number of rows. IEEE Trans. Circuits Syst. II Analog Digit. Signal Process. 48(6), 633–636 (2001)

    Article  Google Scholar 

  11. B. Li, X. Gao, F. Xiao, A new design method of the starting block in lattice structure of arbitrary-length linear phase paraunitary filter bank by combining two polyphase matrices. IEEE Trans. Circuits Syst. II Exp. Briefs 59(2), 118–122 (2012)

    Article  Google Scholar 

  12. S. Oraintara, T.D. Tran, P.N. Heller, Lattice structure for regular paraunitary linear-phase filter banks and M-band orthogonal symmetric wavelet. IEEE Trans. Signal Process. 49(1), 2659–2671 (2001)

    Article  MathSciNet  Google Scholar 

  13. T.A. Ramstad, S.O. Aase, J.H. Husoy, Subband Compression of Images: Principles and Examples (Elsevier, New York, 1995)

    Google Scholar 

  14. A.K. Soman, P.P. Vaidyanathan, T.Q. Nguyen, Linear phase paraunitary filter banks: theory, factorizations and designs. IEEE Trans. Signal Process. 41(12), 3480–3496 (1993)

    Article  Google Scholar 

  15. G. Strang, T.Q. Nguyen, Wavelets and Filter banks (Wellesley-Cambridge, Wellesley, MA, 1996)

    MATH  Google Scholar 

  16. T.D. Tran, M. Ikehara, T.Q. Nguyen, Linear phase paraunitary filterbank with filters of different lengths and its application in image compression. IEEE Trans. Signal Process. 47(10), 2730–2744 (1999)

    Article  Google Scholar 

  17. T.D. Tran, R.L. de Queiroz, T.Q. Nguyen, Linear-phase perfect reconstruction filter banks: lattice structure, design, and application in image compression. IEEE Trans. Signal Process. 48(1), 133–147 (2000)

    Article  Google Scholar 

  18. P.P. Vaidyanathan, Multirate Systems and Filter Banks (Prentice-Hall, Englewood Cliffs, NJ, 1993)

    MATH  Google Scholar 

  19. G.Q. Wang, Four-band compactly supported bisymmetric orthonormal wavelets bases. Opt. Eng. 43(10), 2362–2368 (2004)

    Article  Google Scholar 

  20. Z. Xu, A. Makur, On the arbitrary length M-channel linear phase perfect reconstruction filter banks. IEEE Trans. Signal Process. 57(10), 4118–4123 (2009)

    Article  MathSciNet  Google Scholar 

  21. H. Zou, A.H. Tewfik, Discrete orthogonal M-band wavelet decompositions, in Proceedings of ICASSP, pp. 605–608 (1992)

Download references

Author information

Authors and Affiliations

  1. Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha, 410081, China

    Guoqiu Wang, Yong Chen & Dingxun Yi

  2. School of Mathematics and Statistics, Anhui Normal University, Wuhu, 241002, China

    Yong Chen

Authors
  1. Guoqiu Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yong Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Dingxun Yi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yong Chen.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The original version of the article was revised. Article title was corrected.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, G., Chen, Y. & Yi, D. Complete Factorization of the 2m-Band Paraunitary Polyphase Matrix with Multiple Centers of Symmetry. Circuits Syst Signal Process 39, 2533–2549 (2020). https://doi.org/10.1007/s00034-019-01273-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00034-019-01273-0

Keywords

Search

Navigation