# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2020055

## Optimal maximally decimated M-channel mirrored paraunitary linear phase FIR filter bank design via norm relaxed sequential quadratic programming

 Faculty of Information Engineering, Guangdong University of Technology, Guangzhou, 510006, China

* Corresponding author: Bingo Wing-Kuen Ling

Received  August 2019 Revised  November 2019 Published  March 2020

It is worth noting that the conventional maximally decimated M-channel mirrored paraunitary linear phase finite impulse response condition is defined in the frequency domain. As the frequency domain is a continuous set, it is expressed as a matrix functional (a continuous function of the frequency) equation. On the other hand, this paper expresses the condition as a finite number of discrete (a set of functions of the sampled frequencies) equations. Besides, this paper proposes to sample the magnitude responses of the filters with the total number of the sampled frequencies being more than the filter lengths. Hence, the frequency selectivities of the filters can be controlled more effectively. This filter bank design problem is formulated as an optimization problem in such a way that the total mirrored paraunitary linear phase error is minimized subject to the specifications on the magnitude responses of the filters at these sampling frequencies. However, this optimization problem is highly nonconvex. To address this difficulty, a norm relaxed sequential quadratic programming approach is applied for finding its local optimal solution. By iterating the above procedures using different initial conditions, a near global optimal solution is obtained. Computer numerical simulation results show that our proposed design outperforms the existing designs.

Citation: Qing Liu, Bingo Wing-Kuen Ling, Qingyun Dai, Qing Miao, Caixia Liu. Optimal maximally decimated M-channel mirrored paraunitary linear phase FIR filter bank design via norm relaxed sequential quadratic programming. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020055
##### References:
 [1] Y.-J. Chen, S. Oraintara and K. S. Amaratunga, Dyadic-based factorizations for regular paraunitary filterbanks and $M$-band orthogonal wavelets with structural vanishing moments, IEEE Transactions on Signal Processing, 53 (2005), 193-207.  doi: 10.1109/TSP.2004.838962.  Google Scholar [2] M. T. de Gouvêa and D. Odloak, A new treatment of inconsistent quadratic programs in a sqp-based algorithm, Computers & Chemical Engineering, 22 (1998), 1623-1651.   Google Scholar [3] Y.-T. Fong and C.-W. Kok, Correction to "Iterative least squares design of DC-leakage free paraunitary cosine modulated filter banks'', IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, 50 (2003), 238-243.  doi: 10.1109/TCSII.2007.895968.  Google Scholar [4] X. Q. Gao, T. Q. Nguyen and G. 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Nguyen, Time-domain design and lattice structure of FIR paraunitary filter banks with linear phase, Signal Processing, 80 (2000), 333-342.  doi: 10.1016/S0165-1684(99)00131-0.  Google Scholar [9] M. Ikehara and T. Q. Nguyen, Time-domain design of linear-phase pr filter banks, 1997 IEEE International Conference on Acoustics, Speech, and Signal Processing, 3 (1997), 2077-2080.   Google Scholar [10] J.-Z. Jiang, F. Zhou, S. Ouyang and G. S. Liao, Efficient design of high-complexity interleaved DFT modulated filter bank, Signal Processing, 94 (2014), 130-137.  doi: 10.1016/j.sigpro.2013.06.006.  Google Scholar [11] J.-B. Jian, Q.-J. Xu and D.-L. Han, A strongly convergent norm-relaxed method of strongly sub-feasible direction for optimization with nonlinear equality and inequality constraints, Applied Mathematics and Computation, 182 (2006), 854-870.  doi: 10.1016/j.amc.2006.04.049.  Google Scholar [12] J.-B. Jian, X.-Y. Ke and W.-X. 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Dai, Finite number of necessary and sufficient discrete condition in frequency domain for maximally decimated m-channel mirrored paraunitary linear phase fir filter bank, 2014 IEEE International Conference on Consumer Electronics-China, (2014), 1–5. Google Scholar [16] B. W.-K. Ling, N. Tian, C. Y.-F. Ho, W.-C. Siu, K.-L. Teo and Q. Y. Dai, Maximally decimated paraunitary linear phase FIR filter bank design via iterative SVD approach, IEEE Transactions on Signal Processing, 63 (2015), 466-481.  doi: 10.1109/TSP.2014.2371779.  Google Scholar [17] T. Q. Nguyen, A quadratic-constrained least-squares approach to the design of digital filter banks, 1992 IEEE International Symposium on Circuits and Systems, 3 (1992), 1344-1347.  doi: 10.1109/ISCAS.1992.230255.  Google Scholar [18] T. Q. Nguyen, A. K. Soman and P. Vaidyanathan, A quadratic-constrained least-squares approach to linear phase orthonormal filter bank design, 1993 IEEE International Symposium on Circuits and Systems, (1993), 383–386. Google Scholar [19] S. Oraintara, T. D. Tran, P. N. Heller and T. Q. Nguyen, Lattice structure for regular paraunitary linear-phase filterbanks and $M$-band orthogonal symmetric wavelets, IEEE Transactions on Signal Processing, 49 (2001), 2659-2672.  doi: 10.1109/78.960413.  Google Scholar [20] S. Patel, R. Dhuli and B. Lall, Design and analysis of matrix wiener synthesis filter for multirate filter bank, Signal Processing, 102 (2014), 256-264.  doi: 10.1016/j.sigpro.2014.03.021.  Google Scholar [21] M. Sangnier, J. Gauthier and A. Rakotomamonjy, Filter bank learning for signal classification, Signal Processing, 113 (2015), 124-137.  doi: 10.1016/j.sigpro.2014.12.028.  Google Scholar [22] A. K. Soman, P. P. Vaidyanathan and T. Q. Nguyen, Linear phase paraunitary filter banks: Theory, factorizations and designs, IEEE Transactions on Signal Processing, 41 (1993), 3480-3496.  doi: 10.1109/78.258087.  Google Scholar [23] A. K. Soman and P. P. Vaidyanathan, A complete factorization of paraunitary matrices with pairwise mirror-image symmetry in the frequency domain, IEEE Transactions on Signal Processing, 43 (1995), 1002-1004.  doi: 10.1109/78.376855.  Google Scholar [24] C. G. Shen, W. J. Xue and X. D. Chen, Global convergence of a robust filter SQP algorithm, European Journal of Operational Research, 206 (2010), 34-45.  doi: 10.1016/j.ejor.2010.02.031.  Google Scholar [25] T. D. Tran and T. Q. Nguyen, On m-channel linear phase fir filter banks and application in image compression, IEEE Transactions on Signal Processing, 45 (1997), 2175-2187.   Google Scholar [26] T. D. Tran, M. Ikehara and T. Q. Nguyen, Linear phase paraunitary filter bank with filters of different lengths and its application in image compression,, IEEE Transactions on Signal Processing, 47 (1999), 2730-2744.  doi: 10.1109/78.790655.  Google Scholar [27] T. D. Tran, R. L. De Queiroz and T. Q. Nguyen, Linear-phase perfect reconstruction filter bank: Lattice structure, design, and application in image coding, IEEE Transactions on Signal Processing, 48 (2000), 133-147.   Google Scholar [28] P. G. Vouras and T. D. Tran, Factorization of paraunitary polyphase matrices using subspace projections, 2008 42nd Asilomar Conference on Signals, Systems and Computers, (2008), 602–605. doi: 10.1109/ACSSC.2008.5074476.  Google Scholar [29] Z. M. Xu and A. Makur, On the arbitrary-length $M$-channel linear phase perfect reconstruction filter banks, IEEE Transactions on Signal Processing, 57 (2009), 4118-4123.  doi: 10.1109/TSP.2009.2024026.  Google Scholar [30] W. J. Xue, C. G. Shen and D. G. Pu, A penalty-function-free line search sqp method for nonlinear programming, Journal of Computational and Applied Mathematics, 228 (2009), 313-325.  doi: 10.1016/j.cam.2008.09.031.  Google Scholar

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##### References:
 [1] Y.-J. Chen, S. Oraintara and K. S. Amaratunga, Dyadic-based factorizations for regular paraunitary filterbanks and $M$-band orthogonal wavelets with structural vanishing moments, IEEE Transactions on Signal Processing, 53 (2005), 193-207.  doi: 10.1109/TSP.2004.838962.  Google Scholar [2] M. T. de Gouvêa and D. Odloak, A new treatment of inconsistent quadratic programs in a sqp-based algorithm, Computers & Chemical Engineering, 22 (1998), 1623-1651.   Google Scholar [3] Y.-T. Fong and C.-W. Kok, Correction to "Iterative least squares design of DC-leakage free paraunitary cosine modulated filter banks'', IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, 50 (2003), 238-243.  doi: 10.1109/TCSII.2007.895968.  Google Scholar [4] X. Q. Gao, T. Q. Nguyen and G. Strang, Theory and lattice structure of complex paraunitary filterbanks with filters of (hermitian-) symmetry/antisymmetry properties, IEEE Transactions on Signal Processing, 49 (2001), 1028-1043.  doi: 10.1109/78.917806.  Google Scholar [5] X. Q. Gao, T. Q. Nguyen and G. Strang, On factorization of $M$-channel paraunitary filterbanks, IEEE Transactions on Signal Processing, 49 (2001), 1433-1446.  doi: 10.1109/78.928696.  Google Scholar [6] L. Gan and K.-K. Ma, A simplified lattice factorization for linear-phase paraunitary filter banks with pairwise mirror image frequency responses, IEEE Transactions on Circuits and Systems II: Express Briefs, 51 (2004), 3-7.  doi: 10.1109/TCSII.2003.821515.  Google Scholar [7] N. Holighaus, Z. Prŭša and P. L. Søndergaard, Reassignment and synchrosqueezing for general time-frequency filter banks, subsampling and processing, Signal Processing, 125 (2016), 1-8.  doi: 10.1016/j.sigpro.2016.01.007.  Google Scholar [8] M. Ikehara, T. Nagai and T. Q. Nguyen, Time-domain design and lattice structure of FIR paraunitary filter banks with linear phase, Signal Processing, 80 (2000), 333-342.  doi: 10.1016/S0165-1684(99)00131-0.  Google Scholar [9] M. Ikehara and T. Q. Nguyen, Time-domain design of linear-phase pr filter banks, 1997 IEEE International Conference on Acoustics, Speech, and Signal Processing, 3 (1997), 2077-2080.   Google Scholar [10] J.-Z. Jiang, F. Zhou, S. Ouyang and G. S. Liao, Efficient design of high-complexity interleaved DFT modulated filter bank, Signal Processing, 94 (2014), 130-137.  doi: 10.1016/j.sigpro.2013.06.006.  Google Scholar [11] J.-B. Jian, Q.-J. Xu and D.-L. Han, A strongly convergent norm-relaxed method of strongly sub-feasible direction for optimization with nonlinear equality and inequality constraints, Applied Mathematics and Computation, 182 (2006), 854-870.  doi: 10.1016/j.amc.2006.04.049.  Google Scholar [12] J.-B. Jian, X.-Y. Ke and W.-X. Cheng, A superlinearly convergent norm-relaxed sqp method of strongly sub-feasible directions for constrained optimization without strict complementarity, Applied Mathematics and Computation, 214 (2009), 632-644.  doi: 10.1016/j.amc.2009.04.022.  Google Scholar [13] C. W. Kok, T. Nagai, M. Ikehara and T. Q. Nguyen, Lattice structures parameterization of linear phase paraunitary matrices with pairwise mirror-image symmetry in the frequency domain with an odd number of rows, IEEE Transactions on Circuits and Systems II: Analog and Digital Signal Processing, 48 (2001), 633-636.  doi: 10.1109/82.943336.  Google Scholar [14] Y.-P. Lin and P. Vaidyanathan, Linear phase cosine modulated maximally decimated filter banks with perfect reconstruction, IEEE Transactions on Signal Processing, 43 (1995), 2525-2539.   Google Scholar [15] C. Liu, B. W. Ling, C. Y. Ho and Q. Dai, Finite number of necessary and sufficient discrete condition in frequency domain for maximally decimated m-channel mirrored paraunitary linear phase fir filter bank, 2014 IEEE International Conference on Consumer Electronics-China, (2014), 1–5. Google Scholar [16] B. W.-K. Ling, N. Tian, C. Y.-F. Ho, W.-C. Siu, K.-L. Teo and Q. Y. Dai, Maximally decimated paraunitary linear phase FIR filter bank design via iterative SVD approach, IEEE Transactions on Signal Processing, 63 (2015), 466-481.  doi: 10.1109/TSP.2014.2371779.  Google Scholar [17] T. Q. Nguyen, A quadratic-constrained least-squares approach to the design of digital filter banks, 1992 IEEE International Symposium on Circuits and Systems, 3 (1992), 1344-1347.  doi: 10.1109/ISCAS.1992.230255.  Google Scholar [18] T. Q. Nguyen, A. K. Soman and P. Vaidyanathan, A quadratic-constrained least-squares approach to linear phase orthonormal filter bank design, 1993 IEEE International Symposium on Circuits and Systems, (1993), 383–386. Google Scholar [19] S. Oraintara, T. D. Tran, P. N. Heller and T. Q. Nguyen, Lattice structure for regular paraunitary linear-phase filterbanks and $M$-band orthogonal symmetric wavelets, IEEE Transactions on Signal Processing, 49 (2001), 2659-2672.  doi: 10.1109/78.960413.  Google Scholar [20] S. Patel, R. Dhuli and B. Lall, Design and analysis of matrix wiener synthesis filter for multirate filter bank, Signal Processing, 102 (2014), 256-264.  doi: 10.1016/j.sigpro.2014.03.021.  Google Scholar [21] M. Sangnier, J. Gauthier and A. Rakotomamonjy, Filter bank learning for signal classification, Signal Processing, 113 (2015), 124-137.  doi: 10.1016/j.sigpro.2014.12.028.  Google Scholar [22] A. K. Soman, P. P. Vaidyanathan and T. Q. Nguyen, Linear phase paraunitary filter banks: Theory, factorizations and designs, IEEE Transactions on Signal Processing, 41 (1993), 3480-3496.  doi: 10.1109/78.258087.  Google Scholar [23] A. K. Soman and P. P. Vaidyanathan, A complete factorization of paraunitary matrices with pairwise mirror-image symmetry in the frequency domain, IEEE Transactions on Signal Processing, 43 (1995), 1002-1004.  doi: 10.1109/78.376855.  Google Scholar [24] C. G. Shen, W. J. Xue and X. D. Chen, Global convergence of a robust filter SQP algorithm, European Journal of Operational Research, 206 (2010), 34-45.  doi: 10.1016/j.ejor.2010.02.031.  Google Scholar [25] T. D. Tran and T. Q. Nguyen, On m-channel linear phase fir filter banks and application in image compression, IEEE Transactions on Signal Processing, 45 (1997), 2175-2187.   Google Scholar [26] T. D. Tran, M. Ikehara and T. Q. Nguyen, Linear phase paraunitary filter bank with filters of different lengths and its application in image compression,, IEEE Transactions on Signal Processing, 47 (1999), 2730-2744.  doi: 10.1109/78.790655.  Google Scholar [27] T. D. Tran, R. L. De Queiroz and T. Q. Nguyen, Linear-phase perfect reconstruction filter bank: Lattice structure, design, and application in image coding, IEEE Transactions on Signal Processing, 48 (2000), 133-147.   Google Scholar [28] P. G. Vouras and T. D. Tran, Factorization of paraunitary polyphase matrices using subspace projections, 2008 42nd Asilomar Conference on Signals, Systems and Computers, (2008), 602–605. doi: 10.1109/ACSSC.2008.5074476.  Google Scholar [29] Z. M. Xu and A. Makur, On the arbitrary-length $M$-channel linear phase perfect reconstruction filter banks, IEEE Transactions on Signal Processing, 57 (2009), 4118-4123.  doi: 10.1109/TSP.2009.2024026.  Google Scholar [30] W. J. Xue, C. G. Shen and D. G. Pu, A penalty-function-free line search sqp method for nonlinear programming, Journal of Computational and Applied Mathematics, 228 (2009), 313-325.  doi: 10.1016/j.cam.2008.09.031.  Google Scholar
$20{\log _{10}}\left( {{{\left| {{H_m}\left( \omega \right)} \right|} \over {\sqrt M }}} \right)$ for $m = 0, \dotsc, 3$ of the analysis filters in decibels designed by our proposed method as well as those designed by the methods discussed in both [14] and [30]
(a) $10{\log _{10}}\left( {\left| {\left| {{1 \over M}\sum\limits_{m = 0}^{M - 1} {{H_m}\left( \omega \right){{\tilde H}_m}\left( \omega \right)} } \right| - 1} \right|} \right)$ and (b) $10{\log _{10}}\left( {\left| {{1 \over M}\sum\limits_{k = 1}^{M - 1} {\sum\limits_{m = 0}^{M - 1} {{H_m}\left( {\omega - {{2\pi k} \over M}} \right){{\tilde H}_m}\left( \omega \right)} } } \right|} \right)$ of the filter banks in decibels designed by our proposed method as well as those designed by the methods discussed in both [14] and [30]
$\mathop {\max }\limits_{\omega \in B_m^p \cup B_m^s} 20{\log _{10}}\left( {\big| {\left| {{H_m}\left( \omega \right)} \right| - \left| {{D_m}\left( \omega \right)} \right|} \big|} \right)$ for $m = 0, \dotsc, 3$ of the analysis filters in decibels designed by our proposed method as well as those designed by the methods discussed in both [14] and [30]
 The maximum ripple magnitude of the first analysis filter in decibel The maximum ripple magnitude of the second analysis filter in decibel The maximum ripple magnitude of the third analysis filter in decibel The maximum ripple magnitude of the fourth analysis filter in decibel Method discussed in [14] -0.3251dB -11.3525dB -11.3525dB -0.3251dB Method discussed in [30] -0.7366dB -13.0137dB -12.9932dB -0.7372dB Our proposed method -6.1628dB -6.1930dB -6.1930dB -6.1628dB
 The maximum ripple magnitude of the first analysis filter in decibel The maximum ripple magnitude of the second analysis filter in decibel The maximum ripple magnitude of the third analysis filter in decibel The maximum ripple magnitude of the fourth analysis filter in decibel Method discussed in [14] -0.3251dB -11.3525dB -11.3525dB -0.3251dB Method discussed in [30] -0.7366dB -13.0137dB -12.9932dB -0.7372dB Our proposed method -6.1628dB -6.1930dB -6.1930dB -6.1628dB
${\log_{10}}\left( {err_{para}\left( l \right)} \right)$ for $l = 0, \dotsc, L-1$ of the filter banks in decibels designed by our proposed method as well as those designed by the methods discussed in both [14] and [30]
 Method discussed in [22] Method discussed in [16] Our proposed method ${\log_{10}}\left( {err_{para}\left( 0 \right)} \right)$ -66.7193dB -141.8743dB 1.1415dB ${\log_{10}}\left( {err_{para}\left( l \right)} \right)$ -141.4721dB -140.8206dB -0.8275dB
 Method discussed in [22] Method discussed in [16] Our proposed method ${\log_{10}}\left( {err_{para}\left( 0 \right)} \right)$ -66.7193dB -141.8743dB 1.1415dB ${\log_{10}}\left( {err_{para}\left( l \right)} \right)$ -141.4721dB -140.8206dB -0.8275dB
$\mathop {\max }\limits_\omega 10{\log _{10}}\left( {\left| {\left| {{1 \over M}\sum\limits_{m = 0}^{M - 1} {{H_m}\left( \omega \right){{\tilde H}_m}\left( \omega \right)} } \right| - 1} \right|} \right)$ and $\mathop {\max }\limits_\omega 10{\log _{10}}\left( {\left| {{1 \over M}\sum\limits_{k = 1}^{M - 1} {\sum\limits_{m = 0}^{M - 1} {{H_m}\left( {\omega - {{2\pi k} \over M}} \right){{\tilde H}_m}\left( \omega \right)} } } \right|} \right)$ of the filter banks in decibels designed by our proposed method as well as those designed by the methods discussed in both [14] and [30]
 Method discussed in [14] Method discussed in [30] Our proposed method $\mathop {\max }\limits_\omega 10{\log _{10}}( | | {{1 \over M}\sum\limits_{m = 0}^{M - 1} {{H_m}(\omega){{\tilde H}_m}(\omega)} } | - 1 | )$ -72.7399dB -142.8249dB -2.9505dB $\mathop {\max }\limits_\omega 10{\log _{10}}( {| {{1 \over M}\sum\limits_{k = 1}^{M - 1} {\sum\limits_{m = 0}^{M - 1} {{H_m}( {\omega - {{2\pi k} \over M}} ){{\tilde H}_m}( \omega )} } } |})$ -146.7204dB -145.1184dB -6.0642dB
 Method discussed in [14] Method discussed in [30] Our proposed method $\mathop {\max }\limits_\omega 10{\log _{10}}( | | {{1 \over M}\sum\limits_{m = 0}^{M - 1} {{H_m}(\omega){{\tilde H}_m}(\omega)} } | - 1 | )$ -72.7399dB -142.8249dB -2.9505dB $\mathop {\max }\limits_\omega 10{\log _{10}}( {| {{1 \over M}\sum\limits_{k = 1}^{M - 1} {\sum\limits_{m = 0}^{M - 1} {{H_m}( {\omega - {{2\pi k} \over M}} ){{\tilde H}_m}( \omega )} } } |})$ -146.7204dB -145.1184dB -6.0642dB
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