doi:10.1016/j.sigpro.2005.06.004 
Copyright © 2005 Elsevier B.V. All rights reserved.
Linear phase FIR differentiator design based on maximum signal-to-noise ratio criterion
Chien-Cheng Tsenga, 
, 
and Su-Ling Leeb
aDepartment of Computer and Communication Engineering, National Kaohsiung First University of Science and Technology, Kaohsiung, Taiwan
bInternet and Multimedia Application Technical Laboratory, Telecommunication Laboratories, Chunghwa Telecom Co., Ltd. Chung-Li, Taiwan
Received 4 August 2000;
revised 1 June 2001;
accepted 3 June 2005.
Available online 20 July 2005.
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Abstract
In this paper, a new approach to the design of digital FIR differentiators is presented. The differentiator is linear phase and has zero derivatives at DC frequency. The design is based on the maximization of signal-to-noise ratio (SNR) at the output of the differentiator. The optimal filter coefficients are obtained from the generalized eigenvector associated with the maximum eigenvalue of a pair of symmetric matrices. Estimation of the time derivative of polynomial signal, sinusoidal signal and handwritten Chinese signature is used to demonstrate that the proposed method provides better accuracy and higher SNR than the conventional differentiator methods.
Keywords: Differentiator; Signal-to-noise ratio
Fig. 1. The experimental results of a noisy polynomial signal case: (a) the amplitude response of a differentiator designed by the eigenfilter method; (b) the output signal of the differentiator designed in (a); (c) the amplitude response of a differentiator designed by Vainio–Renfors–Saramaki method; (d) the output signal of the differentiator designed in (c); (e) the amplitude response of a differentiator designed by the proposed method and (f) the output signal of the differentiator designed in (e).
Fig. 2. The experimental results of a noisy sinusoidal signal case: (a) the amplitude response of a differentiator designed by the Vainio–Renfors–Saramaki method; (b) the output signal of the differentiator designed in (b); (c) the amplitude response of a differentiator designed by the proposed method and (d) the output signal of the differentiator designed in (c).
Fig. 3. An online handwritten Chinese signature captured by a digitizer.
Fig. 4. The position signal of Chinese signature: (a) the component px(n) and (b) the component py(n).
Fig. 5. The experimental results of the Chinese signature case: (a) the amplitude response of a differentiator designed by the eigenfilter method; (b) the output velocity signal of the differentiator designed in (a); (c) the amplitude response of a differentiator designed by the Vainio–Renfors–Saramaki method; (d) the output velocity signal of the differentiator designed in (c); (e) the amplitude response of a differentiator designed by the proposed method and (f) the output velocity signal of the differentiator designed in (e).
Table 1.
Performance comparison of three design methods of differentiator
Table 2.
Performance comparison of the proposed method and Vainio–Renfors–Saramaki method for a noisy sinusoidal signal
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