Design of Variable Fractional-Delay FIR Differentiators
In this paper, the least-squares design of variable fractional-delay (VFD) finite impulse response (FIR) digital differentiators is proposed. The used transfer function is formulated so that Farrow structure can be applied to realize the designed system. Also, the symmetric characteristics of filter coefficients are derived, which leads to the complexity reduction by saving almost a half of the number of coefficients. Moreover, all the elements of related vectors or matrices for the optimal process can be represented in closed forms, which make the design easier. Design example is also presented to illustrate the effectiveness of the proposed method.
[1] S. Usui and I. Amidror, "Digital low-pass differentiation for biological
signal processing,-- IEEE Trans. Biomed. Eng., vol.BME-29, no.10,
pp.686-693, Oct. 1982.
[2] R. C. Kavanagh, "FIR differentiators for quantized signals,-- IEEE Trans.
Signal Process., vol.49, no.11, pp.2713-2720, Nov. 2001.
[3] J. Luo, J. Bai, P. He and K. Ying, "Axial strain calculation using a
low-pass digital differentiator in ultrasound elastography,-- IEEE Trans.
Ultrason., Ferroelect., Freq. Contr., vol.51, no.9, pp.1119-1127, Sep.
2004.
[4] S. L. Lin and S. Mourad, "On-chip rise-time measurement,-- IEEE Trans.
Instrum. Meas., vol.53, no.6, pp.1510-1516, Dec. 2004.
[5] S.-C. Pei and J.-J. Shyu, "Eigenfilter design of high-order digital
differentiators,-- IEEE Trans. Acoust., Speech, Signal Process., vol.37,
no.4, pp.505-511, Apr. 1989.
[6] S. Sunder, Y. Su, A. Antoniou and W.-S. Lu , "Design of digital
differentiators satisfying prescribed specifications using optimization
techniques,-- in Proc. ISCAS 1989, May 1989, pp.1652-1655.
[7] B. Kumar, S. C. D. Roy and H. Shah, "On the design of FIR digital
differentiators which are maximally linear at the frequency
¶Çüô / p , p¶ÇéŶÇü×positive integers¶Çüá ,-- IEEE Trans. Signal Process., vol.40, no.9,
pp.2334-2338, Sep. 1992.
[8] E. Hermanowicz and M. Rojewski, "Design of FIR first order digital
differentiators of variable fractional sample delay using maximally flat
error criterion,-- Electron. Lett., vol.30, no.1, pp.17-18, Jan. 1994.
[9] M. A. Al-Alaoui, "Novel IIR differentiator from the Simpon integration
rule,-- IEEE Trans. Circuits Syst. I, Fundament. Theory Appl., vol.41,
no.2, pp.186-187, Feb. 1994.
[10] X. Zhang and T. Yoshikawa, "Design of full band IIR digital
differentiators,-- in Proc. IEEE Int. Conf. Neural Net. Signal Process.,
Dec. 2003, pp.652-655.
[11] C.-C. Tseng, "Digital differentiator design using fractional delay filter
and limit computation,-- IEEE Trans. Circuits Syst. I, Reg. Papers, vol.52,
no.10, pp.2248-2259, Oct. 2005.
[12] N. Q. Ngo, "A new approach for the design of wideband digital integrator
and differentiator,-- IEEE Trans. Circuits Syst. II, Exp. Briefs, vol.53, no.9,
pp.936-940, Sep. 2006.
[13] M. A. Al-Alaoui, "Linear phase low-pass IIR digital differentiators,--
IEEE Trans. Signal Process., vol.55, no.2, pp.697-706, Feb. 2007.
[14] C.-C. Tseng and S.-L. Lee, "Design of digital differentiator using
difference formula and Richardson extrapolation,-- vol.2, no.2,
pp.177-188, Jun. 2008.
[15] C. Y.-F. Ho, B. W.-K. Ling, Y.-Q. Liu, P. K.-S. Tam and K.-L. Teo,
"Optimum design of discrete-time differentiators via semi-infinite
programming approach,-- IEEE Trans. Instrum. Meas., vol.57, no.10,
pp.2226-2230, Oct. 2008.
[16] C. W. Farrow, "A continuously variable digital delay element,-- in Proc.
ISCAS 1988, May 1988, pp.2641-2645.
[17] T. I. Laakso, V. Välimäki, M. Karjalainen and U. K. Laine, "Splitting the
unit delay: Tools for fractional delay filter design,-- IEEE Signal Process.
Mag., vol.13, pp.30-60, Jan. 1996.
[18] W.-S. Lu and T.-B. Deng, "An improved weighted least-squares design
for variable fractional delay FIR filters,-- IEEE Trans. Circuits Syst. I,
Analog Digit. Signal Process., vol.46, no.8, pp.1035-1040, Aug. 1999.
[19] H. Johansson and P. Löwenborg, "On the design of adjustable fractional
delay FIR filters,-- IEEE Trans. Circuits Syst. II, Analog Digit. Signal
Process., vol.50, no.4, pp.164-169, Apr. 2003.
[20] C.-C. Tseng, "Design of variable fractional delay FIR filter using
symmetry,-- in Proc. ISCAS 2004, May 2004, pp.III/477-III/480.
[21] H. Zhao and J. Yu, "A simple and efficient design of variable fractional
delay FIR filters,-- IEEE Trans. Circuits Syst. II, Exp. Briefs, vol.53, no.2,
pp.157-160, Feb. 2006.
[22] T.-B. Deng, and Y. Lian, "Weighted-least-squares design of variable
fractional-delay FIR filters using coefficient symmetry,-- IEEE Trans.
Signal Process., vol.54, no.8, pp.3023-3038, Aug. 2006.
[23] E. Hermanowice and H. Johansson, "A complex variable fractional-delay
FIR filter structure,-- IEEE Trans. Circuits Syst. II, Exp. Briefs, vol.54,
no.9, pp.785-789, Sep. 2007.
[24] T.-B. Deng, "Symmetric structures for odd-order maximally flat and
weighted-least- squares variable fractional-delay filters,-- IEEE Trans.
Circuits Syst. I, Reg. Papers, vol.54, no.12, pp.2718-2732, Dec. 2007
[25] C. Y. Chi and Y. T. Kou, "A new self-initiated optimum WLS
approximation method for the design of linear phase FIR digital filters,--
in Proc. ISCAS 1991, June 1991, pp.168-171.
[1] S. Usui and I. Amidror, "Digital low-pass differentiation for biological
signal processing,-- IEEE Trans. Biomed. Eng., vol.BME-29, no.10,
pp.686-693, Oct. 1982.
[2] R. C. Kavanagh, "FIR differentiators for quantized signals,-- IEEE Trans.
Signal Process., vol.49, no.11, pp.2713-2720, Nov. 2001.
[3] J. Luo, J. Bai, P. He and K. Ying, "Axial strain calculation using a
low-pass digital differentiator in ultrasound elastography,-- IEEE Trans.
Ultrason., Ferroelect., Freq. Contr., vol.51, no.9, pp.1119-1127, Sep.
2004.
[4] S. L. Lin and S. Mourad, "On-chip rise-time measurement,-- IEEE Trans.
Instrum. Meas., vol.53, no.6, pp.1510-1516, Dec. 2004.
[5] S.-C. Pei and J.-J. Shyu, "Eigenfilter design of high-order digital
differentiators,-- IEEE Trans. Acoust., Speech, Signal Process., vol.37,
no.4, pp.505-511, Apr. 1989.
[6] S. Sunder, Y. Su, A. Antoniou and W.-S. Lu , "Design of digital
differentiators satisfying prescribed specifications using optimization
techniques,-- in Proc. ISCAS 1989, May 1989, pp.1652-1655.
[7] B. Kumar, S. C. D. Roy and H. Shah, "On the design of FIR digital
differentiators which are maximally linear at the frequency
¶Çüô / p , p¶ÇéŶÇü×positive integers¶Çüá ,-- IEEE Trans. Signal Process., vol.40, no.9,
pp.2334-2338, Sep. 1992.
[8] E. Hermanowicz and M. Rojewski, "Design of FIR first order digital
differentiators of variable fractional sample delay using maximally flat
error criterion,-- Electron. Lett., vol.30, no.1, pp.17-18, Jan. 1994.
[9] M. A. Al-Alaoui, "Novel IIR differentiator from the Simpon integration
rule,-- IEEE Trans. Circuits Syst. I, Fundament. Theory Appl., vol.41,
no.2, pp.186-187, Feb. 1994.
[10] X. Zhang and T. Yoshikawa, "Design of full band IIR digital
differentiators,-- in Proc. IEEE Int. Conf. Neural Net. Signal Process.,
Dec. 2003, pp.652-655.
[11] C.-C. Tseng, "Digital differentiator design using fractional delay filter
and limit computation,-- IEEE Trans. Circuits Syst. I, Reg. Papers, vol.52,
no.10, pp.2248-2259, Oct. 2005.
[12] N. Q. Ngo, "A new approach for the design of wideband digital integrator
and differentiator,-- IEEE Trans. Circuits Syst. II, Exp. Briefs, vol.53, no.9,
pp.936-940, Sep. 2006.
[13] M. A. Al-Alaoui, "Linear phase low-pass IIR digital differentiators,--
IEEE Trans. Signal Process., vol.55, no.2, pp.697-706, Feb. 2007.
[14] C.-C. Tseng and S.-L. Lee, "Design of digital differentiator using
difference formula and Richardson extrapolation,-- vol.2, no.2,
pp.177-188, Jun. 2008.
[15] C. Y.-F. Ho, B. W.-K. Ling, Y.-Q. Liu, P. K.-S. Tam and K.-L. Teo,
"Optimum design of discrete-time differentiators via semi-infinite
programming approach,-- IEEE Trans. Instrum. Meas., vol.57, no.10,
pp.2226-2230, Oct. 2008.
[16] C. W. Farrow, "A continuously variable digital delay element,-- in Proc.
ISCAS 1988, May 1988, pp.2641-2645.
[17] T. I. Laakso, V. Välimäki, M. Karjalainen and U. K. Laine, "Splitting the
unit delay: Tools for fractional delay filter design,-- IEEE Signal Process.
Mag., vol.13, pp.30-60, Jan. 1996.
[18] W.-S. Lu and T.-B. Deng, "An improved weighted least-squares design
for variable fractional delay FIR filters,-- IEEE Trans. Circuits Syst. I,
Analog Digit. Signal Process., vol.46, no.8, pp.1035-1040, Aug. 1999.
[19] H. Johansson and P. Löwenborg, "On the design of adjustable fractional
delay FIR filters,-- IEEE Trans. Circuits Syst. II, Analog Digit. Signal
Process., vol.50, no.4, pp.164-169, Apr. 2003.
[20] C.-C. Tseng, "Design of variable fractional delay FIR filter using
symmetry,-- in Proc. ISCAS 2004, May 2004, pp.III/477-III/480.
[21] H. Zhao and J. Yu, "A simple and efficient design of variable fractional
delay FIR filters,-- IEEE Trans. Circuits Syst. II, Exp. Briefs, vol.53, no.2,
pp.157-160, Feb. 2006.
[22] T.-B. Deng, and Y. Lian, "Weighted-least-squares design of variable
fractional-delay FIR filters using coefficient symmetry,-- IEEE Trans.
Signal Process., vol.54, no.8, pp.3023-3038, Aug. 2006.
[23] E. Hermanowice and H. Johansson, "A complex variable fractional-delay
FIR filter structure,-- IEEE Trans. Circuits Syst. II, Exp. Briefs, vol.54,
no.9, pp.785-789, Sep. 2007.
[24] T.-B. Deng, "Symmetric structures for odd-order maximally flat and
weighted-least- squares variable fractional-delay filters,-- IEEE Trans.
Circuits Syst. I, Reg. Papers, vol.54, no.12, pp.2718-2732, Dec. 2007
[25] C. Y. Chi and Y. T. Kou, "A new self-initiated optimum WLS
approximation method for the design of linear phase FIR digital filters,--
in Proc. ISCAS 1991, June 1991, pp.168-171.
@article{"International Journal of Electrical, Electronic and Communication Sciences:57087", author = "Jong-Jy Shyu and Soo-Chang Pei and Min-Han Chang", title = "Design of Variable Fractional-Delay FIR Differentiators", abstract = "In this paper, the least-squares design of variable fractional-delay (VFD) finite impulse response (FIR) digital differentiators is proposed. The used transfer function is formulated so that Farrow structure can be applied to realize the designed system. Also, the symmetric characteristics of filter coefficients are derived, which leads to the complexity reduction by saving almost a half of the number of coefficients. Moreover, all the elements of related vectors or matrices for the optimal process can be represented in closed forms, which make the design easier. Design example is also presented to illustrate the effectiveness of the proposed method.
", keywords = "Differentiator, variable fractional-delay filter, FIR filter, least-squares method, Farrow structure.", volume = "3", number = "6", pages = "1325-5", }
