# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2020014

## Optimal design of window functions for filter window bank

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

* Corresponding author: Bingo Wing-Kuen Ling

Received  January 2019 Revised  May 2019 Published  January 2020

Fund Project: This paper was supported partly by the National Nature Science Foundation of China.

This paper considers the designs of the periodic window functions in the filter window banks. First, the filter window bank with the constant synthesis periodic window functions is considered. The total number of the nonzero coefficients in the impulse responses of the analysis periodic window functions is minimized subject to the near perfect reconstruction condition. This is an $L_0$ norm optimization problem. To find its solution, the $L_0$ norm optimization problem is approximated by the $L_1$ norm optimization problem. Then, the column of the constraint matrix corresponding to the element in the solution with the smallest magnitude is removed. Next, it is tested whether the feasible set corresponding to the new $L_0$ norm optimization problem is empty or not. By repeating the above procedures, a solution of the $L_0$ norm optimization problem is obtained. Second, the filter window bank with the time varying synthesis periodic window functions is considered. Likewise, the design of the periodic window functions in both the analysis periodic window functions and the synthesis periodic window functions is formulated as an $L_0$ optimization problem. However, this $L_0$ norm optimization problem is subject to a quadratic matrix inequality constraint. To find its solution, the set of the synthesis periodic window functions is initialized. Then, the set of the analysis periodic window functions is optimized based on the initialized set of the synthesis periodic window functions. Next, the set of the synthesis periodic window functions is optimized based on the found set of the analysis periodic window functions. Finally, these two procedures are iterated. It is shown that the iterative algorithm converges. A design example of a filter window bank with the constant synthesis periodic window functions and a design example of a filter window bank with the time varying synthesis periodic window functions are illustrated. It is shown that the near perfect reconstruction condition is satisfied, whereas this is not the cases for the nonuniform filter banks with the conventional samplers and the conventional block samplers.

Citation: Xueling Zhou, Bingo Wing-Kuen Ling, Hai Huyen Dam, Kok-Lay Teo. Optimal design of window functions for filter window bank. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020014
##### References:
 [1] K. D. Abdesselam, Design of stable, causal, perfect reconstruction, IIR uniform DFT filter banks, IEEE Transactions on Signal Processing, 48 (2000), 1110-1119.   Google Scholar [2] T. S. Bindiya and E. Elias, Design of totally multiplier-less sharp transition width tree structured filter banks for non-uniform discrete multitone system, AEU-International Journal of Electronics and Communications, 69 (2015), 655-665.  doi: 10.1016/j.aeue.2014.12.004.  Google Scholar [3] M. Blanco-Velasco, F. Cruz-Roldán, E. Moreno-Martínez, J. I. Godino-Llorente and K. E. Barner, Embedded filter bank-based algorithm for ECG compression, Signal Processing, 88 (2008), 1402-1412.  doi: 10.1016/j.sigpro.2007.12.006.  Google Scholar [4] G. F. Choueiter and J. R. Glass, An implementation of rational wavelets and filter design for phonetic classification, IEEE Transactions on Audio, Speech, and Language Processing, 15 (2007), 939-948.  doi: 10.1109/TASL.2006.889793.  Google Scholar [5] F. Cruz-Roldán, P. Martín-Martín, J. Sáez-Landete, M. Blanco-Velasco and T. Saramäki, A fast windowing-based technique exploiting spline functions for designing modulated filter banks, IEEE Transactions on Circuits and Systems I: Regular Papers, 56 (2009), 168-178.  doi: 10.1109/TCSI.2008.925350.  Google Scholar [6] H. H. Dam, S. Nordholm and A. Cantoni, Uniform FIR filter bank optimization with gGroup delay specifications, IEEE Transactions on Signal Processing, 53 (2005), 4249-4260.  doi: 10.1109/TSP.2005.857008.  Google Scholar [7] G. Doblinger, A fast design method for perfect-reconstruction uniform cosine-modulated filter banks, IEEE Transactions on Signal Processing, 60 (2012), 6693-6697.  doi: 10.1109/TSP.2012.2217139.  Google Scholar [8] B. Farhang-Boroujeny, Filter bank spectrum sensing for cognitive radios, IEEE Transactions on Signal Processing, 56 (2008), 1801-1811.  doi: 10.1109/TSP.2007.911490.  Google Scholar [9] C. Gu, J. Zhao, W. Xu and D. Sun, Design of linear-phase notch filters based on the OMP scheme and the chebyshev window, IEEE Transactions on Circuits and Systems Ⅱ: Express Briefs, 59 (2012), 592-596.   Google Scholar [10] C. Y. F. Ho, B. W. K. Ling and P. K. S. Tam, Representations of linear dual-rate system via single SISO LTI filter, conventional sampler and block sampler, IEEE Transactions on Circuits and Systems Ⅱ: Express Briefs, 55 (2008), 168-172.  doi: 10.1109/TCSII.2007.910803.  Google Scholar [11] A. Kumar, G. K. Singh and S. Anurag, An optimized cosine-modulated nonuniform filter bank design for subband coding of ECG signal, Journal of King Saud University-Engineering Science, 27 (2015), 158-169.  doi: 10.1016/j.jksues.2013.10.001.  Google Scholar [12] B. W. K. Ling, C. Y. F. Ho, J. Cao and Q. Dai, Necessary and sufficient condition for a set of maximally decimated integers to be incompatible, Necessary and Sufficient Condition for a Set of Maximally Decimated Integers to be Incompatible, 9 (2013), 564-566.   Google Scholar [13] B. W. K. Ling, C. Y.-F. Ho, K. L. Teo, W. C. Siu, J. Z. Cao and Q. Y. Dai, Optimal design of cosine modulated nonuniform linear phase FIR filter bank via both stretching and shifting frequency response of single prototype filter, IEEE Transactions on Signal Processing, 62 (2014), 2517-2530.  doi: 10.1109/TSP.2014.2312326.  Google Scholar [14] Q. Liu, X. Z. Zhang, W. K. Ling, M. Wang and Q. Dai, Exact perfect reconstruction of filter window bank with application to incompatible nonuniform filter banks, IEEE/IET International Symposium on Communication Systems, Networks and Digital Signal Processing, CSNDSP, (2016), 20–22. Google Scholar [15] M. Narendar, A. P. Vinod, A. S. Madhukumar and A. K. Krishna, A tree-structured DFT filter bank based spectrum detector for estimation of radio channel edge frequencies in cognitive radios, Physical Communication, 9 (2013), 45-60.  doi: 10.1016/j.phycom.2013.06.001.  Google Scholar [16] R. C. Nongpiur and D. J. Shpak, Maximizing the signal-to-alias ratio in non-uniform filter banks for acoustic echo cancellation, IEEE Transactions on Circuits and Systems Ⅰ: Regular Papers, 59 (2012), 2315-2325.  doi: 10.1109/TCSI.2012.2185333.  Google Scholar [17] G. W. Ou, D. P. K. Lun and B. W. K. Ling, Compressive sensing of images based on discrete periodic Radon transform, IET Electronics Letters, 50 (2014), 591-593.  doi: 10.1049/el.2014.0770.  Google Scholar [18] A. Pandharipande and S. Dasgupta, On biorthogonal nonuniform filter banks and tree structures, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 49 (2002), 1457-1467.  doi: 10.1109/TCSI.2002.803248.  Google Scholar [19] S. Rahimi and B. Champagne, Oversampled perfect reconstruction DFT modulated filter banks for multi-carrier transceiver systems, Signal Processing, 93 (2013), 2942-2955.  doi: 10.1016/j.sigpro.2013.05.003.  Google Scholar [20] A. K. Soman and P. P. Vaidyanathan, On orthonormal wavelets and paraunitary filter banks, IEEE Transactions on Signal Processing, 41 (1993), 1170-1183.  doi: 10.1109/78.205722.  Google Scholar [21] R. Soni, A. Jain and R. Saxena, An optimized design of nonuniform filter bank using variable-combinational window function, AEU-International Journal of Electronics and Communications, 67 (2013), 595-601.  doi: 10.1016/j.aeue.2013.01.003.  Google Scholar [22] K. Swaminathan and P. Vaidyanathan, Theory and design of uniform DFT, parallel, quadrature mirror filter banks, IEEE Transactions on Circuits and Systems, 33 (1986), 1170-1191.  doi: 10.1109/TCS.1986.1085876.  Google Scholar [23] G. Wang, Time-varying discrete-time signal expansions as time-varying filter banks, IET Signal Processing, 3 (2009), 353-367.  doi: 10.1049/iet-spr.2008.0049.  Google Scholar [24] X. G. Xia and B. W. Suter, Multirate filter banks with block sampling, IEEE Transactions on Signal Processing, 44 (1996), 484-496.   Google Scholar [25] X. M. Xie, S. C. Chan and T. I. Yuk, Design of perfect-reconstruction nonuniform recombination filter banks with flexible rational sampling factors, IEEE Transactions on Circuits and Systems Ⅰ: Regular Papers, 52 (2005), 1965-1981.  doi: 10.1109/TCSI.2005.852009.  Google Scholar [26] H. Xiong, L. Zhu, N. Ma and Y. F. Zheng, Scalable video compression framework with adaptive orientational multiresolution transform and nonuniform directional filterbank design, IEEE Transactions on Circuits and Systems for Video Technology, 21 (2011), 1085-1099.   Google Scholar [27] Z. Xiong, K. Ramchandran, C. Herley and M. T. Orchard, Flexible tree-structured signal expansions using time-varying wavelet packets, IEEE Transactions on Signal Processing, 45 (1997), 333-345.   Google Scholar [28] W. Xu, J. X. Zhao and C. Gu, Design of linear-phase FIR multiple-notch filters via an iterative reweighted OMP scheme, IEEE Transactions on Circuits and Systems Ⅱ: Express Briefs, 61 (2014), 813-817.  doi: 10.1109/TCSII.2014.2345299.  Google Scholar [29] C. Q. Yang, J. Xiao, Y. F. Zeng, B. W. Deng and W.-K. Ling, Design of periodic window functions in filter window filter banks for harsh environment, International Conference on Industrial Informatics, INDIN, (2016), 18–21. doi: 10.1109/INDIN.2016.7819305.  Google Scholar [30] Z. Yang, B. W. K. Ling and C. Bingham, Approximate affine linear relationship between $L_1$ norm objective function values and $L_2$ norm constraint bounds, IET Signal Processing, 9 (2015), 670-680.   Google Scholar [31] K. C. Zangi and R. D. Koilpillai, Software radio issues in cellular base stations, IEEE Journal on Selected Areas in Communications, 17 (1999), 561-573.  doi: 10.1109/49.761036.  Google Scholar [32] Y. Zhang, S. Negahdaripour and Q. Z. Li, Low bit-rate compression of underwater imagery based on adaptive hybrid Wavelets and directional filter banks, Signal Processing: Image Communication, 47 (2016), 96-114.  doi: 10.1016/j.image.2016.06.001.  Google Scholar

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##### References:
 [1] K. D. Abdesselam, Design of stable, causal, perfect reconstruction, IIR uniform DFT filter banks, IEEE Transactions on Signal Processing, 48 (2000), 1110-1119.   Google Scholar [2] T. S. Bindiya and E. Elias, Design of totally multiplier-less sharp transition width tree structured filter banks for non-uniform discrete multitone system, AEU-International Journal of Electronics and Communications, 69 (2015), 655-665.  doi: 10.1016/j.aeue.2014.12.004.  Google Scholar [3] M. Blanco-Velasco, F. Cruz-Roldán, E. Moreno-Martínez, J. I. Godino-Llorente and K. E. Barner, Embedded filter bank-based algorithm for ECG compression, Signal Processing, 88 (2008), 1402-1412.  doi: 10.1016/j.sigpro.2007.12.006.  Google Scholar [4] G. F. Choueiter and J. R. Glass, An implementation of rational wavelets and filter design for phonetic classification, IEEE Transactions on Audio, Speech, and Language Processing, 15 (2007), 939-948.  doi: 10.1109/TASL.2006.889793.  Google Scholar [5] F. Cruz-Roldán, P. Martín-Martín, J. Sáez-Landete, M. Blanco-Velasco and T. Saramäki, A fast windowing-based technique exploiting spline functions for designing modulated filter banks, IEEE Transactions on Circuits and Systems I: Regular Papers, 56 (2009), 168-178.  doi: 10.1109/TCSI.2008.925350.  Google Scholar [6] H. H. Dam, S. Nordholm and A. Cantoni, Uniform FIR filter bank optimization with gGroup delay specifications, IEEE Transactions on Signal Processing, 53 (2005), 4249-4260.  doi: 10.1109/TSP.2005.857008.  Google Scholar [7] G. Doblinger, A fast design method for perfect-reconstruction uniform cosine-modulated filter banks, IEEE Transactions on Signal Processing, 60 (2012), 6693-6697.  doi: 10.1109/TSP.2012.2217139.  Google Scholar [8] B. Farhang-Boroujeny, Filter bank spectrum sensing for cognitive radios, IEEE Transactions on Signal Processing, 56 (2008), 1801-1811.  doi: 10.1109/TSP.2007.911490.  Google Scholar [9] C. Gu, J. Zhao, W. Xu and D. Sun, Design of linear-phase notch filters based on the OMP scheme and the chebyshev window, IEEE Transactions on Circuits and Systems Ⅱ: Express Briefs, 59 (2012), 592-596.   Google Scholar [10] C. Y. F. Ho, B. W. K. Ling and P. K. S. Tam, Representations of linear dual-rate system via single SISO LTI filter, conventional sampler and block sampler, IEEE Transactions on Circuits and Systems Ⅱ: Express Briefs, 55 (2008), 168-172.  doi: 10.1109/TCSII.2007.910803.  Google Scholar [11] A. Kumar, G. K. Singh and S. Anurag, An optimized cosine-modulated nonuniform filter bank design for subband coding of ECG signal, Journal of King Saud University-Engineering Science, 27 (2015), 158-169.  doi: 10.1016/j.jksues.2013.10.001.  Google Scholar [12] B. W. K. Ling, C. Y. F. Ho, J. Cao and Q. Dai, Necessary and sufficient condition for a set of maximally decimated integers to be incompatible, Necessary and Sufficient Condition for a Set of Maximally Decimated Integers to be Incompatible, 9 (2013), 564-566.   Google Scholar [13] B. W. K. Ling, C. Y.-F. Ho, K. L. Teo, W. C. Siu, J. Z. Cao and Q. Y. Dai, Optimal design of cosine modulated nonuniform linear phase FIR filter bank via both stretching and shifting frequency response of single prototype filter, IEEE Transactions on Signal Processing, 62 (2014), 2517-2530.  doi: 10.1109/TSP.2014.2312326.  Google Scholar [14] Q. Liu, X. Z. Zhang, W. K. Ling, M. Wang and Q. Dai, Exact perfect reconstruction of filter window bank with application to incompatible nonuniform filter banks, IEEE/IET International Symposium on Communication Systems, Networks and Digital Signal Processing, CSNDSP, (2016), 20–22. Google Scholar [15] M. Narendar, A. P. Vinod, A. S. Madhukumar and A. K. Krishna, A tree-structured DFT filter bank based spectrum detector for estimation of radio channel edge frequencies in cognitive radios, Physical Communication, 9 (2013), 45-60.  doi: 10.1016/j.phycom.2013.06.001.  Google Scholar [16] R. C. Nongpiur and D. J. Shpak, Maximizing the signal-to-alias ratio in non-uniform filter banks for acoustic echo cancellation, IEEE Transactions on Circuits and Systems Ⅰ: Regular Papers, 59 (2012), 2315-2325.  doi: 10.1109/TCSI.2012.2185333.  Google Scholar [17] G. W. Ou, D. P. K. Lun and B. W. K. Ling, Compressive sensing of images based on discrete periodic Radon transform, IET Electronics Letters, 50 (2014), 591-593.  doi: 10.1049/el.2014.0770.  Google Scholar [18] A. Pandharipande and S. Dasgupta, On biorthogonal nonuniform filter banks and tree structures, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 49 (2002), 1457-1467.  doi: 10.1109/TCSI.2002.803248.  Google Scholar [19] S. Rahimi and B. Champagne, Oversampled perfect reconstruction DFT modulated filter banks for multi-carrier transceiver systems, Signal Processing, 93 (2013), 2942-2955.  doi: 10.1016/j.sigpro.2013.05.003.  Google Scholar [20] A. K. Soman and P. P. Vaidyanathan, On orthonormal wavelets and paraunitary filter banks, IEEE Transactions on Signal Processing, 41 (1993), 1170-1183.  doi: 10.1109/78.205722.  Google Scholar [21] R. Soni, A. Jain and R. Saxena, An optimized design of nonuniform filter bank using variable-combinational window function, AEU-International Journal of Electronics and Communications, 67 (2013), 595-601.  doi: 10.1016/j.aeue.2013.01.003.  Google Scholar [22] K. Swaminathan and P. Vaidyanathan, Theory and design of uniform DFT, parallel, quadrature mirror filter banks, IEEE Transactions on Circuits and Systems, 33 (1986), 1170-1191.  doi: 10.1109/TCS.1986.1085876.  Google Scholar [23] G. Wang, Time-varying discrete-time signal expansions as time-varying filter banks, IET Signal Processing, 3 (2009), 353-367.  doi: 10.1049/iet-spr.2008.0049.  Google Scholar [24] X. G. Xia and B. W. Suter, Multirate filter banks with block sampling, IEEE Transactions on Signal Processing, 44 (1996), 484-496.   Google Scholar [25] X. M. Xie, S. C. Chan and T. I. Yuk, Design of perfect-reconstruction nonuniform recombination filter banks with flexible rational sampling factors, IEEE Transactions on Circuits and Systems Ⅰ: Regular Papers, 52 (2005), 1965-1981.  doi: 10.1109/TCSI.2005.852009.  Google Scholar [26] H. Xiong, L. Zhu, N. Ma and Y. F. Zheng, Scalable video compression framework with adaptive orientational multiresolution transform and nonuniform directional filterbank design, IEEE Transactions on Circuits and Systems for Video Technology, 21 (2011), 1085-1099.   Google Scholar [27] Z. Xiong, K. Ramchandran, C. Herley and M. T. Orchard, Flexible tree-structured signal expansions using time-varying wavelet packets, IEEE Transactions on Signal Processing, 45 (1997), 333-345.   Google Scholar [28] W. Xu, J. X. Zhao and C. Gu, Design of linear-phase FIR multiple-notch filters via an iterative reweighted OMP scheme, IEEE Transactions on Circuits and Systems Ⅱ: Express Briefs, 61 (2014), 813-817.  doi: 10.1109/TCSII.2014.2345299.  Google Scholar [29] C. Q. Yang, J. Xiao, Y. F. Zeng, B. W. Deng and W.-K. Ling, Design of periodic window functions in filter window filter banks for harsh environment, International Conference on Industrial Informatics, INDIN, (2016), 18–21. doi: 10.1109/INDIN.2016.7819305.  Google Scholar [30] Z. Yang, B. W. K. Ling and C. Bingham, Approximate affine linear relationship between $L_1$ norm objective function values and $L_2$ norm constraint bounds, IET Signal Processing, 9 (2015), 670-680.   Google Scholar [31] K. C. Zangi and R. D. Koilpillai, Software radio issues in cellular base stations, IEEE Journal on Selected Areas in Communications, 17 (1999), 561-573.  doi: 10.1109/49.761036.  Google Scholar [32] Y. Zhang, S. Negahdaripour and Q. Z. Li, Low bit-rate compression of underwater imagery based on adaptive hybrid Wavelets and directional filter banks, Signal Processing: Image Communication, 47 (2016), 96-114.  doi: 10.1016/j.image.2016.06.001.  Google Scholar
Block diagram of a filter window bank
Nonuniform filter bank with the conventional samplers
Nonuniform block filter bank with the block samplers
(a) The transfer functional distortions, (b) the first aliasing distortion components, (c) the second aliasing distortion components, (d) the third aliasing distortion components, (e) the fourth aliasing distortion components and (f) the fifth aliasing distortion components of both the filter window bank and the block filter bank
Both the transfer functional distortions and the aliasing distortions of both the filter window bank and the block filter bank
 Filter window bank Block filter bank Improvement The maximum absolute value of the transfer functional distortion $-17.3157dB$ $0.9315dB$ $18.2472dB$ The maximum absolute value of the first aliasing distortion $-24.8282dB$ $-2.0172dB$ $22.8110dB$ The maximum absolute value of the second aliasing distortion $-18.6761dB$ $-5.4446dB$ $13.2316dB$ The maximum absolute value of the third aliasing distortion $-27.5018dB$ $-9.3399dB$ $18.1618dB$ The maximum absolute value of the fourth aliasing distortion $-18.7578dB$ $-5.1906$ $13.5672dB$ The maximum absolute value of the fifth aliasing distortion $-23.9515dB$ $-0.7191dB$ $23.2325dB$
 Filter window bank Block filter bank Improvement The maximum absolute value of the transfer functional distortion $-17.3157dB$ $0.9315dB$ $18.2472dB$ The maximum absolute value of the first aliasing distortion $-24.8282dB$ $-2.0172dB$ $22.8110dB$ The maximum absolute value of the second aliasing distortion $-18.6761dB$ $-5.4446dB$ $13.2316dB$ The maximum absolute value of the third aliasing distortion $-27.5018dB$ $-9.3399dB$ $18.1618dB$ The maximum absolute value of the fourth aliasing distortion $-18.7578dB$ $-5.1906$ $13.5672dB$ The maximum absolute value of the fifth aliasing distortion $-23.9515dB$ $-0.7191dB$ $23.2325dB$
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