Optimal design of window functions for filter window bank

Xueling Zhou; Bingo Wing-Kuen Ling; Hai Huyen Dam; Kok-Lay Teo

doi:10.3934/jimo.2020014

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2021, Volume 17, Issue 3: 1119-1145. Doi: 10.3934/jimo.2020014

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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
^* Corresponding author: Bingo Wing-Kuen Ling

Received: January 2019

Revised: May 2019

Early access: January 2020

Published: May 2021

This paper was supported partly by the National Nature Science Foundation of China

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Abstract

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.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Figure 1. Block diagram of a filter window bank

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Figure 2. Nonuniform filter bank with the conventional samplers

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Figure 3. Nonuniform block filter bank with the block samplers

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Figure 4. (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

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Table 1. 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	$-23.9515dB$	$-0.7191dB$	$23.2325dB$

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