# 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
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##### References:
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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