Design of FIR Digital Differentiators Using Maximal Linearity Constraints

Ishtiaq Rasool KHAN  Masahiro OKUDA  Ryoji OHBA  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E87-A   No.8   pp.2010-2017
Publication Date: 2004/08/01
Online ISSN: 
Print ISSN: 0916-8508
Type of Manuscript: Special Section PAPER (Special Section on Digital Signal Processing)
Category: Filter Design
maximally linear,  MAXFLAT,  digital differentiators,  fullband,  digital filters,  

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Classical designs of maximally flat finite impulse response digital filters need to perform inverse discrete Fourier transformation of the frequency responses, in order to calculate the impulse response coefficients. Several attempts have been made to simplify the designs by obtaining explicit formulas for the impulse response coefficients. Such formulas have been derived for digital differentiators having maximal linearity at zero frequency, using different techniques including interpolating polynomials and the Taylor series etc. We show that these formulas can be obtained directly by application of maximal linearity constraints on the frequency response. The design problem is formulated as a system of linear equations, which can be solved to achieve maximal linearity at an arbitrary frequency. Certain special characteristics of the determinant of the coefficients matrix of these equations are explored for designs centered at zero frequency, and are used in derivation of explicit formulas for the impulse response coefficients of digital differentiators of both odd and even lengths.