Variable fractional delay filter design with discrete coefficients

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July 2016, 12(3): 819-831. doi: 10.3934/jimo.2016.12.819

Variable fractional delay filter design with discrete coefficients

1.	Dept. of Mathematics and Statistics, Curtin University of Technology, Perth, Australia, Australia

Received May 2014 Revised June 2014 Published September 2015

Abstract
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This paper investigates the optimal design of variable fractional delay (VFD) filter with discrete coefficients as a means of achieving low complexity and efficient hardware implementation. The filter coefficients are expressed as the sum of signed power-of-two (SPT) terms with a restriction on the total number of power-of-two terms. An optimization problem with least squares criterion is formulated as a mixed-integer programming problem. An optimal scaling factor quantization scheme is applied to the problem resulting in an optimal scaling factor quantized solution. This solution is then improved further by applying a discrete filled function, that has been extended for a mixed integer optimization problem. To apply the discrete filled function method, it requires multiple calculations of the objective function around the neighborhood of a searched point. Thus, an updating scheme is developed to efficiently calculate the objective function in a neighborhood of a point. Design examples demonstrate the effectiveness of the proposed optimization approach.

Keywords: discrete filled function, Variable fractional delay, signed power-of-two (SPT) coefficients..

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.

Citation: Hai Huyen Dam, Kok Lay Teo. Variable fractional delay filter design with discrete coefficients. Journal of Industrial & Management Optimization, 2016, 12 (3) : 819-831. doi: 10.3934/jimo.2016.12.819

References:

[1]	H. H. Dam, A. Cantoni, K. L. Teo and S. Nordholm, Variable digital filter with least square criterion and peak gain constraints,, IEEE Trans. Circuits Systems II, 54 (2007), 24. Google Scholar
[2]	H. H. Dam, A. Cantoni, K. L. Teo and S. Nordholm, Variable digital filter with group delay flatness specification or phase constraints,, IEEE Trans. Circuits Systems II, 55 (2008), 442. Google Scholar
[3]	H. H. Dam, A. Cantoni, K. L. Teo and S. Nordholm, FIR variable digital filter with signed power-of-two coefficients,, IEEE Trans. Circuits Systems I, 54 (2007), 1348. doi: 10.1109/TCSI.2007.897775. Google Scholar
[4]	H. H. Dam, Design of allpass variable fractional delay filter with powers-of-two coefficients,, IEEE Signal Processing Letters, 22 (2015), 1643. doi: 10.1109/LSP.2015.2420652. Google Scholar
[5]	H. H. Dam, Design of variable fractional delay filter with fractional delay constraints,, IEEE Signal Processing Letters, 21 (2014), 1361. doi: 10.1109/LSP.2014.2336662. Google Scholar
[6]	H. H. Dam and K. L. Teo, Allpass VFD filter design,, IEEE Trans. Signal Processing, 58 (2010), 4432. doi: 10.1109/TSP.2010.2048316. Google Scholar
[7]	H. H. Dam, Variable Fractional Delay Filter with Sub-Expressions Coefficients,, International Journal of Innovative Computing, 9 (2013), 2995. Google Scholar
[8]	T.-B. Deng and S. Chivapreecha, Bi-minimax design of even-order variable fractional-delay FIR digital filters,, IEEE Trans. Circuits Systems I: Reg. Paper, 59 (2012), 1766. doi: 10.1109/TCSI.2011.2180431. Google Scholar
[9]	T.-B. Deng and W. Qin, Coefficient relation-based minimax design and low-complexity structure of variable fractional-delay digital filters,, Signal Processing, 93 (2013), 923. doi: 10.1016/j.sigpro.2012.11.004. Google Scholar
[10]	T.-B. Deng, Decoupling minimax design of low-complexity variable fractional-delay FIR digital filters,, IEEE Trans. Circuits Syst. I: Reg. Papers, 58 (2011), 2398. doi: 10.1109/TCSI.2011.2123510. Google Scholar
[11]	C. W. Farrow, A continuously variable digital delay element,, in Proc. IEEE Int. Symp. Circuits Syst., (1988), 2641. doi: 10.1109/ISCAS.1988.15483. Google Scholar
[12]	Y.-D. Huang, S.-C. Pei and J.-J. Shyu, WLS design of variable fractional-delay FIR filters using coefficient relationship,, IEEE Trans. Circuits Systems II: Express Brief, 56 (2009), 220. Google Scholar
[13]	D. Li, Y. C. Lim and Y. Lian, A polynomial-time algorithm for designing FIR filters with power-of-two coefficients,, IEEE Trans. Signal Processing, 50 (2002), 1935. Google Scholar
[14]	Y. C. Lim, Design of discrete-coefficient-value linear phase FIR filters with optimum normalized peak ripple magnitude,, in IEEE Trans. Circuits Systems, 37 (1990), 1480. doi: 10.1109/31.101268. Google Scholar
[15]	H. Lin, Y. Wang and X. Wang, An auxiliary function method for global minimization in integer programming,, Mathematical Problems in Engineering, 2011 (2011), 1. doi: 10.1155/2011/402437. Google Scholar
[16]	C. K. S. Pun, Y. C. Wu, S. C. Chan and K. L. Ho, On the design and efficient implementation of the Farrow structure,, IEEE Signal Processing Letters, 10 (2003), 189. doi: 10.1109/LSP.2003.813681. Google Scholar

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References:

[1]	H. H. Dam, A. Cantoni, K. L. Teo and S. Nordholm, Variable digital filter with least square criterion and peak gain constraints,, IEEE Trans. Circuits Systems II, 54 (2007), 24. Google Scholar
[2]	H. H. Dam, A. Cantoni, K. L. Teo and S. Nordholm, Variable digital filter with group delay flatness specification or phase constraints,, IEEE Trans. Circuits Systems II, 55 (2008), 442. Google Scholar
[3]	H. H. Dam, A. Cantoni, K. L. Teo and S. Nordholm, FIR variable digital filter with signed power-of-two coefficients,, IEEE Trans. Circuits Systems I, 54 (2007), 1348. doi: 10.1109/TCSI.2007.897775. Google Scholar
[4]	H. H. Dam, Design of allpass variable fractional delay filter with powers-of-two coefficients,, IEEE Signal Processing Letters, 22 (2015), 1643. doi: 10.1109/LSP.2015.2420652. Google Scholar
[5]	H. H. Dam, Design of variable fractional delay filter with fractional delay constraints,, IEEE Signal Processing Letters, 21 (2014), 1361. doi: 10.1109/LSP.2014.2336662. Google Scholar
[6]	H. H. Dam and K. L. Teo, Allpass VFD filter design,, IEEE Trans. Signal Processing, 58 (2010), 4432. doi: 10.1109/TSP.2010.2048316. Google Scholar
[7]	H. H. Dam, Variable Fractional Delay Filter with Sub-Expressions Coefficients,, International Journal of Innovative Computing, 9 (2013), 2995. Google Scholar
[8]	T.-B. Deng and S. Chivapreecha, Bi-minimax design of even-order variable fractional-delay FIR digital filters,, IEEE Trans. Circuits Systems I: Reg. Paper, 59 (2012), 1766. doi: 10.1109/TCSI.2011.2180431. Google Scholar
[9]	T.-B. Deng and W. Qin, Coefficient relation-based minimax design and low-complexity structure of variable fractional-delay digital filters,, Signal Processing, 93 (2013), 923. doi: 10.1016/j.sigpro.2012.11.004. Google Scholar
[10]	T.-B. Deng, Decoupling minimax design of low-complexity variable fractional-delay FIR digital filters,, IEEE Trans. Circuits Syst. I: Reg. Papers, 58 (2011), 2398. doi: 10.1109/TCSI.2011.2123510. Google Scholar
[11]	C. W. Farrow, A continuously variable digital delay element,, in Proc. IEEE Int. Symp. Circuits Syst., (1988), 2641. doi: 10.1109/ISCAS.1988.15483. Google Scholar
[12]	Y.-D. Huang, S.-C. Pei and J.-J. Shyu, WLS design of variable fractional-delay FIR filters using coefficient relationship,, IEEE Trans. Circuits Systems II: Express Brief, 56 (2009), 220. Google Scholar
[13]	D. Li, Y. C. Lim and Y. Lian, A polynomial-time algorithm for designing FIR filters with power-of-two coefficients,, IEEE Trans. Signal Processing, 50 (2002), 1935. Google Scholar
[14]	Y. C. Lim, Design of discrete-coefficient-value linear phase FIR filters with optimum normalized peak ripple magnitude,, in IEEE Trans. Circuits Systems, 37 (1990), 1480. doi: 10.1109/31.101268. Google Scholar
[15]	H. Lin, Y. Wang and X. Wang, An auxiliary function method for global minimization in integer programming,, Mathematical Problems in Engineering, 2011 (2011), 1. doi: 10.1155/2011/402437. Google Scholar
[16]	C. K. S. Pun, Y. C. Wu, S. C. Chan and K. L. Ho, On the design and efficient implementation of the Farrow structure,, IEEE Signal Processing Letters, 10 (2003), 189. doi: 10.1109/LSP.2003.813681. Google Scholar

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