Design of Digital Differentiator to Optimize Relative Error

It is observed that the Weighted least-square (WLS) technique, including the modifications, results in equiripple error curve. The resultant error as a percent of the ideal value is highly non-uniformly distributed over the range of frequencies for which the differentiator is designed. The present paper proposes a modification to the technique so that the optimization procedure results in lower maximum relative error compared to the ideal values. Simulation results for first order as well as higher order differentiators are given to illustrate the excellent performance of the proposed method.

Authors:

• Vinita V. Sondur
• Vilas B. Sondur
• Narasimha H. Ayachit

Keywords:

• Differentiator
• equiripple
• error distribution
• relative error.

References:

[1] C.A.Rahenkamp and B.V.K.Vijay Kumar, "Modifications to the
McCellan, Parks and Rabiner computer Program for Designing Higher
order Differentiating FIR Filter", IEEE Transactions Acoust Speech
Signal Processing , vol Assp-34, p 1671, December 1986.
[2] S.C.Pei and J.J.Shyu, "Eigen filter Design of Higher order Digital
Differentiators", IEEE Transactions Acoust Speech Signal Processing ,
vol 37, p 505, April 1989.
[3] M.R.R.Reddy, B.Kumar and S.C.Dutta Roy, " Design Efficient Second
and Higher Order FIR Digital Differentiators for Low Frequencies",
Signal Processing, vol 20, p 219, July 1990.
[4] S.Sunder and R.P.Ramachandran, "Least-squares Design of Higher
Order Nonrecursive Differentiators", IEEE Transactions on Signal
Processing, vol 42, No. 4, p 956, April 1994.
[5] Shian-Tang, Tzeng, Hung-Ching Lu, " Genetic Algorithm Approach for
Designing Higher order Digital Differentiators," Signal Processing, vol
79, p 175, 1999.
[6] S.C.Pei and J.J.Shyu, "Analytic closed-formed matrix for designing
higher order digital differentiators using eigen-approach," ", IEEE
Transactions on Signal Processing, vol 44, No. 3, pp 698-701,
March1996.
[7] G.S.Mollova and R.Unbehauen, "Analytical design of higher order
differentiators using least-squares technique," Electron. Lett.,, vol 37,
No. 22, pp 1098-1099, 2001.
[8] D.W.Tank and J.Hopfield, "Simple neural optimization networks: an
A/D converter, signal decision circuit, and a linear programming
circuit," IEEE Trans.Circuits Syst., vol. CAS-33, No. 4, pp 533-541,
April 1986.
[9] Yue-Dar Jou, "Neural Least-Squares Design of Higher Order Digital
Differentiators," IEEE Signal Processing Letters, vol 12, No. 1, pp9-12,
Jan 2005.
[10] S.Sunder and V.Ramachandran, "Design of Equiripple Nonrecursive
Digital Differentiators and Hilbert Transformers Using a Weighted
Least-squares Technique", IEEE Transactions on Signal Processing, vol
42, No. 9, p 2504, Sep 1994.
[11] V.V.Sondur, V.B.Sondur, D.H.Rao, M.V.Latte, N.H.Ayachit, "Issues in
the Design of Equiripple FIR Higher Order Digital Differentiators using
Weighted Least-squares Technique," in: Proceedings of the 2007 IEEE
International Conference on Signal Processing and Communications,
Nov 24-27, 2007, Dubai, UAE, pp 189-192-
[12] V.V.Sondur, V.B.Sondur, N.H.Ayachit, "Design of FIR Digital
Differentiators using Modified Weighted Least-squares Technique,"
communicated to IEEE Signal Processing Letters.