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Design of two-channel causal-stable IIR filter banks by spectral factorization

  • Dept. of Electronic and Information Engineering, Anhui university of Finance & Economic, Bengbu, 233041, China

Funds:  Supported by the National Natural Science Foundation of China (61102118).
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https://doi.org/10.3969/j.issn.0253-2778.2015.01.005
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  • Corresponding author: YIN Shishu (corresponding author), female, born in 1973, PhD. Research field: multirate signal processing, filter banks, wavelets, image signal processing. E-mail: ssyin@eee.hku.hk
  • Received Date: 17 December 2013
  • Accepted Date: 09 October 2014
  • Rev Recd Date: 09 October 2014
  • Publish Date: 30 January 2015
    Abstract

  • Abstract

    A new method was proposed for designing two-channel filter banks (FBs) with causal-stable IIR filters. By using IIR filters with a cosine-rolloff transition band, the flatness condition required for two-channel NPR FB was automatically satisfied. Instead of designing the frequency magnitude responses of the analysis filters, the power spectra of the desired filters were designed by solving a quasi-convex problem. When the solution was found, the analysis filters desired can be obtained by spectral factorization. The polyphase components of the analysis filters were assumed to have an identical denominator to simplify the PR condition. Two-channel NPR IIR FB so obtained has a reasonably low reconstruction error and can be employed as the initial guess to constrained nonlinear optimization software for designing the PR IIR FB.

    Abstract

    A new method was proposed for designing two-channel filter banks (FBs) with causal-stable IIR filters. By using IIR filters with a cosine-rolloff transition band, the flatness condition required for two-channel NPR FB was automatically satisfied. Instead of designing the frequency magnitude responses of the analysis filters, the power spectra of the desired filters were designed by solving a quasi-convex problem. When the solution was found, the analysis filters desired can be obtained by spectral factorization. The polyphase components of the analysis filters were assumed to have an identical denominator to simplify the PR condition. Two-channel NPR IIR FB so obtained has a reasonably low reconstruction error and can be employed as the initial guess to constrained nonlinear optimization software for designing the PR IIR FB.
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    • References(16)
  • [1]
    Vaidyanathan P P. Multirate Systems and Filter Banks [M]. NJ, USA: Prentice Hall, 1993.
    [2]
    Vaidyanathan P P. Multirate digital filters, filter banks, polyphase networks and applications: A tutorial [J]. Proceedings of IEEE, 1990, 78(1):56-93.
    [3]
    Basu S, Chiang C H, Choi H N. Wavelets and perfect reconstruction subband coding with causal stable IIR filters [J]. IEEE Transactions on Circuits and Systems-II, 1995, 42(1): 24-38.
    [4]
    Phoong S M, Kim C W, Vaidyanathan P P, etc al. A new class of two-channel biorthogonal filter banks and wavelet bases[J]. IEEE Transactions on Signal Processing, 1995, 43(3): 649-665.
    [5]
    Tsui K M, Chan S C. Multi-plet two-channel perfect reconstruction FIR and causal stable IIR filter banks[J]. IEEE Transactions on Circuits and Systems-I, 2006, 53(12): 2 804-2 817.
    [6]
    Bregovicc′ R, Saramki T. Design of causal stable perfect reconstruction two-channel IIR filter banks[C]// Proceedings of the 2nd International Symposium on Image and Signal Processing and Analysis. Pula, Yugoslavia: IEEE Press, 2001: 545-550.
    [7]
    Okuda M, Fukuoka T, Ikehara M, et al. Design of causal IIR filter banks satisfying perfect reconstruction[C]// Proceedings of the International Symposium on Circuits and Systems. Hongkong, China: IEEE Press, 1997, 4: 2 433-2 436.
    [8]
    Mao J S, Chan S C, Ho K L. Theory and design of causal stable IIR PR cosine-modulated filter banks [C] // Proceedings of the IInternational Symposium on Circuits and Systems. Orlando, USA: IEEE Press, 1999, 3: 427-430.
    [9]
    Yin S S, Chan S C. Design and factorization of two-channel perfect reconstruction filter banks with causal-stable IIR filters[J]. Journal of Signal Processing Systems, 2010, 68(3): 353-366.
    [10]
    Yin S S. New method for designing of two-channel causal-stable IIR filter banks[C] // Proceedings of the 10th International Conference on Signal Processing. Beijing, China: IEEE Press, 2010, 1: 22-26.
    [11]
    Yin S S, Chan S C, Tsui K M. On the design of nearly-PR and PR FIR cosine modulated filter banks having approximate cosine-rolloff transition band[J]. IEEE Transactions on Circuits and Systems-II, 2008, 55(6): 571-575.
    [12]
    Wu S P, Boyd S, Vandenberghe L. FIR filter design via semidefinite programming and spectral factorization[C]// Proceeding of the 35th IEEE Conference on Decision and Control. Boston, USA: Birkhuser, 1996, 1:271-276.
    [13]
    Rabiner L R, Graham N Y, Helms H D. Maximize stopband attenuation of a lowpass IIR filter[EB/OL]. http://cvxr.com/cvx/examples/filter_design/iir_mag_design_lowpass_max_atten.m.
    [14]
    Quelhas M F, Petraglia A, Petraglia M R. Linear programming for the design of IIR filters[C]// Proceedings of the 3rd Latin American Symposium on Circuits and Systems. Playa del Carmen, Mexico: IEEE Press, 2012: 1-4.
    [15]
    Jiang A M, Kwan H K. Minimax design of IIR digital filters using SDP relaxation technique[J]. IEEE Transactions on Circuits and Systems-I, 2010, 57(2): 378-390.
    [16]
    Lai X P, Lin Z P. Minimax phase error design of IIR digital filters with prescribed magnitude and phase responses[J]. IEEE Transactions on Signal Processing, 2012, 60(2): 980-986.
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    [1]
    Vaidyanathan P P. Multirate Systems and Filter Banks [M]. NJ, USA: Prentice Hall, 1993.
    [2]
    Vaidyanathan P P. Multirate digital filters, filter banks, polyphase networks and applications: A tutorial [J]. Proceedings of IEEE, 1990, 78(1):56-93.
    [3]
    Basu S, Chiang C H, Choi H N. Wavelets and perfect reconstruction subband coding with causal stable IIR filters [J]. IEEE Transactions on Circuits and Systems-II, 1995, 42(1): 24-38.
    [4]
    Phoong S M, Kim C W, Vaidyanathan P P, etc al. A new class of two-channel biorthogonal filter banks and wavelet bases[J]. IEEE Transactions on Signal Processing, 1995, 43(3): 649-665.
    [5]
    Tsui K M, Chan S C. Multi-plet two-channel perfect reconstruction FIR and causal stable IIR filter banks[J]. IEEE Transactions on Circuits and Systems-I, 2006, 53(12): 2 804-2 817.
    [6]
    Bregovicc′ R, Saramki T. Design of causal stable perfect reconstruction two-channel IIR filter banks[C]// Proceedings of the 2nd International Symposium on Image and Signal Processing and Analysis. Pula, Yugoslavia: IEEE Press, 2001: 545-550.
    [7]
    Okuda M, Fukuoka T, Ikehara M, et al. Design of causal IIR filter banks satisfying perfect reconstruction[C]// Proceedings of the International Symposium on Circuits and Systems. Hongkong, China: IEEE Press, 1997, 4: 2 433-2 436.
    [8]
    Mao J S, Chan S C, Ho K L. Theory and design of causal stable IIR PR cosine-modulated filter banks [C] // Proceedings of the IInternational Symposium on Circuits and Systems. Orlando, USA: IEEE Press, 1999, 3: 427-430.
    [9]
    Yin S S, Chan S C. Design and factorization of two-channel perfect reconstruction filter banks with causal-stable IIR filters[J]. Journal of Signal Processing Systems, 2010, 68(3): 353-366.
    [10]
    Yin S S. New method for designing of two-channel causal-stable IIR filter banks[C] // Proceedings of the 10th International Conference on Signal Processing. Beijing, China: IEEE Press, 2010, 1: 22-26.
    [11]
    Yin S S, Chan S C, Tsui K M. On the design of nearly-PR and PR FIR cosine modulated filter banks having approximate cosine-rolloff transition band[J]. IEEE Transactions on Circuits and Systems-II, 2008, 55(6): 571-575.
    [12]
    Wu S P, Boyd S, Vandenberghe L. FIR filter design via semidefinite programming and spectral factorization[C]// Proceeding of the 35th IEEE Conference on Decision and Control. Boston, USA: Birkhuser, 1996, 1:271-276.
    [13]
    Rabiner L R, Graham N Y, Helms H D. Maximize stopband attenuation of a lowpass IIR filter[EB/OL]. http://cvxr.com/cvx/examples/filter_design/iir_mag_design_lowpass_max_atten.m.
    [14]
    Quelhas M F, Petraglia A, Petraglia M R. Linear programming for the design of IIR filters[C]// Proceedings of the 3rd Latin American Symposium on Circuits and Systems. Playa del Carmen, Mexico: IEEE Press, 2012: 1-4.
    [15]
    Jiang A M, Kwan H K. Minimax design of IIR digital filters using SDP relaxation technique[J]. IEEE Transactions on Circuits and Systems-I, 2010, 57(2): 378-390.
    [16]
    Lai X P, Lin Z P. Minimax phase error design of IIR digital filters with prescribed magnitude and phase responses[J]. IEEE Transactions on Signal Processing, 2012, 60(2): 980-986.
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