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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

Hai Huyen Dam 1, and  Wing-Kuen Ling 2,

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  January 2019 Early access  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.

Keywords: Non-uniform cosine modulated filter bank, optimal design, optimal design, finite precision.
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.  Google Scholar

[2]

H. H. Dam and K. L. Teo, Variable fractional delay filter design with discrete coefficients, Management, 12 (2016), 819-831.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

P. P. Vaidyanathan, Multirate Systems and Filter Banks, 1993. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

show all references

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.  Google Scholar

[2]

H. H. Dam and K. L. Teo, Variable fractional delay filter design with discrete coefficients, Management, 12 (2016), 819-831.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

P. P. Vaidyanathan, Multirate Systems and Filter Banks, 1993. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

Figure 1.  Uniform CMFB with $M$ subbands
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Figure 2.  Non-uniform CMFB with $\bar{M}$ subbands
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Figure 3.  Magnitude response for the 5-channel non-uniform CMFB with decimation factor (4, 4, 8, 8, 4) and infinite precision coefficients
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Figure 4.  Amplitude distortion for the 5-channel non-uniform CMFB with decimation factor (4, 4, 8, 8, 4) and infinite precision coefficients
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Figure 5.  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
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Figure 6.  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
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Figure 7.  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
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Figure 8.  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
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Figure 9.  Magnitude response for the 4-channel non-uniform CMFB with decimation factor (8, 8, 4, 2) and finite precision coefficients
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Figure 10.  Amplitude distortion for the 4-channel non-uniform CMFB with decimation factor (8, 8, 4, 2) and finite precision coefficients
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Table 1.  Non-uniform (4, 4, 8, 8, 4) CMFB with infinite precision coefficients and N = 154
MethodsAmplitude
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=650.0067-69.85 dB
Window method [13] with As=65 as the initial to (12) with a constraint of -65 dB for prototype filter stopband0.0014-65.00 dB
Proposed method with As=65 and a constraint of -65 dB for prototype filter stopband0.00048-78.23 dB
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MethodsAmplitude
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=650.0067-69.85 dB
Window method [13] with As=65 as the initial to (12) with a constraint of -65 dB for prototype filter stopband0.0014-65.00 dB
Proposed method with As=65 and a constraint of -65 dB for prototype filter stopband0.00048-78.23 dB
Table 2.  Non-uniform (8, 8, 4, 2) CMFB with infinite precision coefficients
NMethodsAmplitude dist.Stopband att.
154Weighted Chebyshev [12]0.0039-60.65 dB
WCLS [12]0.0028-61.49 dB
Optimal solution with As=650.00061-71.44 dB
198 Method in [13] as quoted in [12]0.0025-79.65 dB
Method in [13] with As=800.0021-89.95 dB
Proposed method with restriction of -85 dB for prototype filter stopband0.0011-85.00 dB
Proposed method with restriction of -90 dB for prototype filter stopband0.0012-90.01 dB
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NMethodsAmplitude dist.Stopband att.
154Weighted Chebyshev [12]0.0039-60.65 dB
WCLS [12]0.0028-61.49 dB
Optimal solution with As=650.00061-71.44 dB
198 Method in [13] as quoted in [12]0.0025-79.65 dB
Method in [13] with As=800.0021-89.95 dB
Proposed method with restriction of -85 dB for prototype filter stopband0.0011-85.00 dB
Proposed method with restriction of -90 dB for prototype filter stopband0.0012-90.01 dB
Table 3.  Non-uniform (8, 8, 4, 2) CMFB with finite precision coefficients and N = 154
SPTMethodsAmplitude dist.Stopband att.Total adders
Inf. precision sol. As=900.0017-90.00 dB-
Q=250Quantized solution0.0018-81.57 dB250
$ \epsilon_{d}$=-58 dBLocal optimal0.000696-72.63 dB250
Optimal solution0.000318-66.52 dB250
Q=260Quantized solution0.0019-81.63 dB256
$\epsilon_{d} $=-58 dBLocal optimal0.000684-71.56 dB263
Optimal solution0.000312-67.84 dB268
Method in [12] using GA0.0058-56.25 dB266
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SPTMethodsAmplitude dist.Stopband att.Total adders
Inf. precision sol. As=900.0017-90.00 dB-
Q=250Quantized solution0.0018-81.57 dB250
$ \epsilon_{d}$=-58 dBLocal optimal0.000696-72.63 dB250
Optimal solution0.000318-66.52 dB250
Q=260Quantized solution0.0019-81.63 dB256
$\epsilon_{d} $=-58 dBLocal optimal0.000684-71.56 dB263
Optimal solution0.000312-67.84 dB268
Method in [12] using GA0.0058-56.25 dB266
Table 4.  Non-uniform (8, 8, 4, 2) CMFB with finite precision coefficients and N = 198
SPTMethodsAmplitude dist.Stopband att.Total adders
Inf. precision sol. $A_s $=900.0012-90.00 dB-
Q=290Quantized solution0.0013-77.28 dB290
$\epsilon_{d} $=-75 dBLocal optimal0.000715-75.12 dB289
Optimal solution0.000498-75.30 dB290
Q=300Quantized solution0.0014-77.14 dB300
$\epsilon_{d} $=-75 dBLocal optimal0.00078-76.16 dB300
Optimal solution0.000505-75.10 dB300
Method in [12] using GA0.003-62.30 dB315
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SPTMethodsAmplitude dist.Stopband att.Total adders
Inf. precision sol. $A_s $=900.0012-90.00 dB-
Q=290Quantized solution0.0013-77.28 dB290
$\epsilon_{d} $=-75 dBLocal optimal0.000715-75.12 dB289
Optimal solution0.000498-75.30 dB290
Q=300Quantized solution0.0014-77.14 dB300
$\epsilon_{d} $=-75 dBLocal optimal0.00078-76.16 dB300
Optimal solution0.000505-75.10 dB300
Method in [12] using GA0.003-62.30 dB315
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