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

Hai Huyen Dam; Wing-Kuen Ling

doi:10.3934/jimo.2018034

Article Contents

2019, Volume 15, Issue 1: 97-112. Doi: 10.3934/jimo.2018034

This issue Previous Article Optimal stopping investment with non-smooth utility over an infinite time horizon Next Article A novel modeling and smoothing technique in global optimization

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: March 31, 2017

Revised: November 30, 2017

Early access: April 2018

Published: December 2018

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Abstract

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:

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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

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 A_s=65	0.0067	-69.85 dB
Window method [13] with A_s=65 as the initial to (12) with a constraint of -65 dB for prototype filter stopband	0.0014	-65.00 dB
Proposed method with A_s=65 and a constraint of -65 dB for prototype filter stopband	0.00048	-78.23 dB

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Table 2. 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 A_s=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

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Table 3. 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. A_s=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

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Table 4. 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

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