American Institute of Mathematical Sciences

January  2019, 15(1): 97-112. doi: 10.3934/jimo.2018034

Optimal design of finite precision and infinite precision non-uniform cosine modulated filter bank

 1 School of Electrical Engineering, Computing and Mathematical Sciences, Curtin University, Perth, Australia 2 School of Information Engineering, Guangdong University of Technology, China

Received  April 2017 Revised  December 2017 Published  April 2018

This paper investigates the design of non-uniform cosine modulated filter bank (CMFB) with both finite precision coefficients and infinite precision coefficients. The finite precision filter bank has been designed to reduce the computational complexity related to the multiplication operations in the filter bank. Here, non-uniform filter bank (NUFB) is obtained by merging the appropriate filters of an uniform filter bank. An efficient optimization approach is developed for the design of non-uniform CMFB with infinite precision coefficients. A new procedure based on the discrete filled function is then developed to design the filter bank prototype filter with finite precision coefficients. Design examples demonstrate that the designed filter banks with both infinite precision coefficients and finite precision coefficients have low distortion and better performance when compared with other existing methods.

Citation: Hai Huyen Dam, Wing-Kuen Ling. Optimal design of finite precision and infinite precision non-uniform cosine modulated filter bank. Journal of Industrial & Management Optimization, 2019, 15 (1) : 97-112. doi: 10.3934/jimo.2018034
References:
 [1] F. Argenti, B. Brogelli and E. Del Re, Design of pseudo-QMF banks with rational sampling factors using several prototype filters, IEEE Trans. Signal Process., 46 (1998), 1709-1715. doi: 10.1109/78.678502. [2] H. H. Dam and K. L. Teo, Variable fractional delay filter design with discrete coefficients, Management, 12 (2016), 819-831. [3] H. H. Dam, Design of allpass variable fractional delay filter with powers-of-two coefficients, IEEE Trans. Signal Process., 59 (2011), 6240-6244. doi: 10.1109/TSP.2011.2165951. [4] H. H. Dam, Optimal design of oversampled modulated filter bank, IEEE Signal Processing Letters, 24 (2017), 673-677. doi: 10.1109/LSP.2017.2685641. [5] H. H. Dam and S. Nordholm, Accelerated gradient with optimal step size for second-order blind signal separation, Multidimensional Systems and Signal Processing, (2017), 1-17. doi: 10.1007/s11045-017-0478-8. [6] H. H. Dam, D. Rimantho and S. Nordholm, Second-order blind signal separation with optimal step size, Speech Communication, 55 (2013), 535-543. doi: 10.1016/j.specom.2012.10.003. [7] H. H. Dam, S. Nordholm and A. Cantoni, Uniform FIR filterbank optimization with group delay specifications, IEEE Trans. Signal Process., 53 (2005), 4249-4260. doi: 10.1109/TSP.2005.857008. [8] H. H. Dam, S. Nordholm, A. Cantoni and J. M. de Haan, Iterative method for the design of DFT filter bank, IEEE Transactions on Circuits and Systems Ⅱ: Express Briefs, 51 (2004), 581-586. doi: 10.1109/TCSII.2004.836041. [9] Z. G. Feng and K. L. Teo, A discrete filled function method for the design og FIR filters with signed-powers-of-two coefficients, IEEE Trans. Signal Process., 56 (2008), 134-139. doi: 10.1109/TSP.2007.901164. [10] J. D. Griesbach, T. Bose and D. M. Etter, Non-uniform filterbank bandwidth allocation for system modeling subband adaptive filters, IEEE International Conference on Acoustics, Speech, and Signal Processing, (1999), 1473-1476. doi: 10.1109/ICASSP.1999.756261. [11] S. Kalathil and E. Elias, Efficient design of non-uniform cosine modulated filter banks for digital hearing aids, International Journal of Electronics and Communications, 69 (2015), 1314-1320. doi: 10.1016/j.aeue.2015.05.015. [12] S. Kalathil and E. Elias, Non-uniform cosine modulated filter banks using meta-heuristic algorithms in CSD space, Journal of Advance Research, 6 (2015), 839-849. doi: 10.1016/j.jare.2014.06.008. [13] 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. [14] J. J. Lee and B. Gi Lee, A design of nonuniform cosine modulated filter banks, IEEE Trans. Circuits Syst. Ⅱ: Analog Digit. Signal Process., 42 (1995), 732-737. [15] B. Li, H. H. Dam, A. Cantoni and K. L. Teo, A global optimal zero-forcing beamformer design with signed power-of-two coefficients, J. Ind. Manag. Optim., 12 (2016), 595-607. [16] D. Li, Y. C. Lim, Y. Lian and J. Song, Polynomial-time algorithm for designing FIR filters with power-of-two coefficients, IEEE Trans. Signal Process., 50 (2002), 1935-1941. [17] B. Li, L. Ge and J. Zheng, An efficient channelizer based on nonuniform filter banks in 8th International Conference on Signal Processing, (2006). doi: 10.1109/ICOSP.2006.345902. [18] H. Lin, Y. Wang and L. Fan, A filled function method with one parameter for unconstrained global optimization, Applied Mathematics and Computation, 218 (2011), 3776-3785. doi: 10.1016/j.amc.2011.09.022. [19] B. W. K. Ling, C. Y. F. Ho, K. L. Teo, W. C. Siu, J. Cao and Q. Dai, Optimal design of cosine modulated nonuniform linear phase FIR filter bank via both stretching and shifting frequency response of single prototype filter, IEEE Trans. Signal Process., 62 (2014), 2517-2530. doi: 10.1109/TSP.2014.2312326. [20] W. K. Ling and K. S. Tam, Representation of perfect reconstruction octave decomposition filter banks with set of decimators 2, 4, 4 via tree structure, IEEE Signal Processing Letters, 10 (2003), 184-186. [21] R. C. Nongpiur and D. J. Shpak, Maximizing the signal-to-alias ratio in non-uniform filter banks for acoustic echo cancelation, IEEE Trans Circ-I, 59 (2012), 2315-2325. doi: 10.1109/TCSI.2012.2185333. [22] J. Ogale and S. Ashok, Cosine modulated non-uniform filter banks, Journal of Signal and Information Processing, 2 (2011), 178-183. doi: 10.4236/jsip.2011.23024. [23] P. P. Vaidyanathan, Multirate Systems and Filter Banks, 1993. [24] C. Z. Wu, K. L. Teo, V. Rehbock and H. H. Dam, Global optimum design of uniform FIR filter bank with magnitude constraints, IEEE Trans. Signal Process., 56 (2008), 5478-5486. doi: 10.1109/TSP.2008.927803. [25] C. Yu, K. L. Teo and H. H. Dam, Design of allpass variable fractional delay filter with signed powers-of-two coefficients, Signal Processing, 95 (2014), 32-42.

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References:
 [1] F. Argenti, B. Brogelli and E. Del Re, Design of pseudo-QMF banks with rational sampling factors using several prototype filters, IEEE Trans. Signal Process., 46 (1998), 1709-1715. doi: 10.1109/78.678502. [2] H. H. Dam and K. L. Teo, Variable fractional delay filter design with discrete coefficients, Management, 12 (2016), 819-831. [3] H. H. Dam, Design of allpass variable fractional delay filter with powers-of-two coefficients, IEEE Trans. Signal Process., 59 (2011), 6240-6244. doi: 10.1109/TSP.2011.2165951. [4] H. H. Dam, Optimal design of oversampled modulated filter bank, IEEE Signal Processing Letters, 24 (2017), 673-677. doi: 10.1109/LSP.2017.2685641. [5] H. H. Dam and S. Nordholm, Accelerated gradient with optimal step size for second-order blind signal separation, Multidimensional Systems and Signal Processing, (2017), 1-17. doi: 10.1007/s11045-017-0478-8. [6] H. H. Dam, D. Rimantho and S. Nordholm, Second-order blind signal separation with optimal step size, Speech Communication, 55 (2013), 535-543. doi: 10.1016/j.specom.2012.10.003. [7] H. H. Dam, S. Nordholm and A. Cantoni, Uniform FIR filterbank optimization with group delay specifications, IEEE Trans. Signal Process., 53 (2005), 4249-4260. doi: 10.1109/TSP.2005.857008. [8] H. H. Dam, S. Nordholm, A. Cantoni and J. M. de Haan, Iterative method for the design of DFT filter bank, IEEE Transactions on Circuits and Systems Ⅱ: Express Briefs, 51 (2004), 581-586. doi: 10.1109/TCSII.2004.836041. [9] Z. G. Feng and K. L. Teo, A discrete filled function method for the design og FIR filters with signed-powers-of-two coefficients, IEEE Trans. Signal Process., 56 (2008), 134-139. doi: 10.1109/TSP.2007.901164. [10] J. D. Griesbach, T. Bose and D. M. Etter, Non-uniform filterbank bandwidth allocation for system modeling subband adaptive filters, IEEE International Conference on Acoustics, Speech, and Signal Processing, (1999), 1473-1476. doi: 10.1109/ICASSP.1999.756261. [11] S. Kalathil and E. Elias, Efficient design of non-uniform cosine modulated filter banks for digital hearing aids, International Journal of Electronics and Communications, 69 (2015), 1314-1320. doi: 10.1016/j.aeue.2015.05.015. [12] S. Kalathil and E. Elias, Non-uniform cosine modulated filter banks using meta-heuristic algorithms in CSD space, Journal of Advance Research, 6 (2015), 839-849. doi: 10.1016/j.jare.2014.06.008. [13] 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. [14] J. J. Lee and B. Gi Lee, A design of nonuniform cosine modulated filter banks, IEEE Trans. Circuits Syst. Ⅱ: Analog Digit. Signal Process., 42 (1995), 732-737. [15] B. Li, H. H. Dam, A. Cantoni and K. L. Teo, A global optimal zero-forcing beamformer design with signed power-of-two coefficients, J. Ind. Manag. Optim., 12 (2016), 595-607. [16] D. Li, Y. C. Lim, Y. Lian and J. Song, Polynomial-time algorithm for designing FIR filters with power-of-two coefficients, IEEE Trans. Signal Process., 50 (2002), 1935-1941. [17] B. Li, L. Ge and J. Zheng, An efficient channelizer based on nonuniform filter banks in 8th International Conference on Signal Processing, (2006). doi: 10.1109/ICOSP.2006.345902. [18] H. Lin, Y. Wang and L. Fan, A filled function method with one parameter for unconstrained global optimization, Applied Mathematics and Computation, 218 (2011), 3776-3785. doi: 10.1016/j.amc.2011.09.022. [19] B. W. K. Ling, C. Y. F. Ho, K. L. Teo, W. C. Siu, J. Cao and Q. Dai, Optimal design of cosine modulated nonuniform linear phase FIR filter bank via both stretching and shifting frequency response of single prototype filter, IEEE Trans. Signal Process., 62 (2014), 2517-2530. doi: 10.1109/TSP.2014.2312326. [20] W. K. Ling and K. S. Tam, Representation of perfect reconstruction octave decomposition filter banks with set of decimators 2, 4, 4 via tree structure, IEEE Signal Processing Letters, 10 (2003), 184-186. [21] R. C. Nongpiur and D. J. Shpak, Maximizing the signal-to-alias ratio in non-uniform filter banks for acoustic echo cancelation, IEEE Trans Circ-I, 59 (2012), 2315-2325. doi: 10.1109/TCSI.2012.2185333. [22] J. Ogale and S. Ashok, Cosine modulated non-uniform filter banks, Journal of Signal and Information Processing, 2 (2011), 178-183. doi: 10.4236/jsip.2011.23024. [23] P. P. Vaidyanathan, Multirate Systems and Filter Banks, 1993. [24] C. Z. Wu, K. L. Teo, V. Rehbock and H. H. Dam, Global optimum design of uniform FIR filter bank with magnitude constraints, IEEE Trans. Signal Process., 56 (2008), 5478-5486. doi: 10.1109/TSP.2008.927803. [25] C. Yu, K. L. Teo and H. H. Dam, Design of allpass variable fractional delay filter with signed powers-of-two coefficients, Signal Processing, 95 (2014), 32-42.
Uniform CMFB with $M$ subbands
Non-uniform CMFB with $\bar{M}$ subbands
Magnitude response for the 5-channel non-uniform CMFB with decimation factor (4, 4, 8, 8, 4) and infinite precision coefficients
Amplitude distortion for the 5-channel non-uniform CMFB with decimation factor (4, 4, 8, 8, 4) and infinite precision coefficients
Magnitude response for the 4-channel non-uniform CMFB with decimation factor (8, 8, 4, 2) and infinite precision coefficients for -85 dB restriction in the prototype filter stopband
Amplitude distortion for the 4-channel non-uniform CMFB with decimation factor (8, 8, 4, 2) and infinite precision coefficients for -85 dB restriction in the prototype filter stopband
Magnitude response for the 4-channel non-uniform CMFB with decimation factor (8, 8, 4, 2) and infinite precision coefficients for -90 dB restriction in the prototype filter stopband
Amplitude distortion for the 4-channel non-uniform CMFB with decimation factor (8, 8, 4, 2) and infinite precision coefficients for -90 dB restriction in the prototype filter stopband
Magnitude response for the 4-channel non-uniform CMFB with decimation factor (8, 8, 4, 2) and finite precision coefficients
Amplitude distortion for the 4-channel non-uniform CMFB with decimation factor (8, 8, 4, 2) and finite precision coefficients
Non-uniform (4, 4, 8, 8, 4) CMFB with infinite precision coefficients and N = 154
 Methods Amplitude distortion Stopband attenuation Weighted Chebyshev in [12] 0.0042 -60.65 dB WCLS approach in [12] 0.0029 -61.49 dB Window method [13] with As=65 0.0067 -69.85 dB Window method [13] with As=65 as the initial to (12) with a constraint of -65 dB for prototype filter stopband 0.0014 -65.00 dB Proposed method with As=65 and a constraint of -65 dB for prototype filter stopband 0.00048 -78.23 dB
 Methods Amplitude distortion Stopband attenuation Weighted Chebyshev in [12] 0.0042 -60.65 dB WCLS approach in [12] 0.0029 -61.49 dB Window method [13] with As=65 0.0067 -69.85 dB Window method [13] with As=65 as the initial to (12) with a constraint of -65 dB for prototype filter stopband 0.0014 -65.00 dB Proposed method with As=65 and a constraint of -65 dB for prototype filter stopband 0.00048 -78.23 dB
Non-uniform (8, 8, 4, 2) CMFB with infinite precision coefficients
 N Methods Amplitude dist. Stopband att. 154 Weighted Chebyshev [12] 0.0039 -60.65 dB WCLS [12] 0.0028 -61.49 dB Optimal solution with As=65 0.00061 -71.44 dB 198 Method in [13] as quoted in [12] 0.0025 -79.65 dB Method in [13] with As=80 0.0021 -89.95 dB Proposed method with restriction of -85 dB for prototype filter stopband 0.0011 -85.00 dB Proposed method with restriction of -90 dB for prototype filter stopband 0.0012 -90.01 dB
 N Methods Amplitude dist. Stopband att. 154 Weighted Chebyshev [12] 0.0039 -60.65 dB WCLS [12] 0.0028 -61.49 dB Optimal solution with As=65 0.00061 -71.44 dB 198 Method in [13] as quoted in [12] 0.0025 -79.65 dB Method in [13] with As=80 0.0021 -89.95 dB Proposed method with restriction of -85 dB for prototype filter stopband 0.0011 -85.00 dB Proposed method with restriction of -90 dB for prototype filter stopband 0.0012 -90.01 dB
Non-uniform (8, 8, 4, 2) CMFB with finite precision coefficients and N = 154
 SPT Methods Amplitude dist. Stopband att. Total adders Inf. precision sol. As=90 0.0017 -90.00 dB - Q=250 Quantized solution 0.0018 -81.57 dB 250 $\epsilon_{d}$=-58 dB Local optimal 0.000696 -72.63 dB 250 Optimal solution 0.000318 -66.52 dB 250 Q=260 Quantized solution 0.0019 -81.63 dB 256 $\epsilon_{d}$=-58 dB Local optimal 0.000684 -71.56 dB 263 Optimal solution 0.000312 -67.84 dB 268 Method in [12] using GA 0.0058 -56.25 dB 266
 SPT Methods Amplitude dist. Stopband att. Total adders Inf. precision sol. As=90 0.0017 -90.00 dB - Q=250 Quantized solution 0.0018 -81.57 dB 250 $\epsilon_{d}$=-58 dB Local optimal 0.000696 -72.63 dB 250 Optimal solution 0.000318 -66.52 dB 250 Q=260 Quantized solution 0.0019 -81.63 dB 256 $\epsilon_{d}$=-58 dB Local optimal 0.000684 -71.56 dB 263 Optimal solution 0.000312 -67.84 dB 268 Method in [12] using GA 0.0058 -56.25 dB 266
Non-uniform (8, 8, 4, 2) CMFB with finite precision coefficients and N = 198
 SPT Methods Amplitude dist. Stopband att. Total adders Inf. precision sol. $A_s$=90 0.0012 -90.00 dB - Q=290 Quantized solution 0.0013 -77.28 dB 290 $\epsilon_{d}$=-75 dB Local optimal 0.000715 -75.12 dB 289 Optimal solution 0.000498 -75.30 dB 290 Q=300 Quantized solution 0.0014 -77.14 dB 300 $\epsilon_{d}$=-75 dB Local optimal 0.00078 -76.16 dB 300 Optimal solution 0.000505 -75.10 dB 300 Method in [12] using GA 0.003 -62.30 dB 315
 SPT Methods Amplitude dist. Stopband att. Total adders Inf. precision sol. $A_s$=90 0.0012 -90.00 dB - Q=290 Quantized solution 0.0013 -77.28 dB 290 $\epsilon_{d}$=-75 dB Local optimal 0.000715 -75.12 dB 289 Optimal solution 0.000498 -75.30 dB 290 Q=300 Quantized solution 0.0014 -77.14 dB 300 $\epsilon_{d}$=-75 dB Local optimal 0.00078 -76.16 dB 300 Optimal solution 0.000505 -75.10 dB 300 Method in [12] using GA 0.003 -62.30 dB 315
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