Maximally linear FIR digital differentiators for midband frequencies

Abstract

Among the known FIR digital differentiators, the minimax relative error (MRE) approximations are the most efficient ones, where the weighting coefficients are derived by using an optimization algorithm. This approximation fails at the frequency ω = π if the order N of the differentiator is odd. A half sample advance (i.e. a delay t = -½) and an even N become necessary to extend the MRE approximation up to ω = χ; these are, however, undesirable features, particularly in large signal processing systems. This paper proposes digital differentiators, which are characterized by maximal linearity of the frequency response at the midband frequency ω = π/2, and gives a mathematical relation for the weighting coefficients. the order N of the differentiator has been ensured to be odd, and no half sample advance has been used. It has been shown that for relative error (RE) 1 per cent, the proposed differentiators are more efficient as compared to those based on the MRE approximation, in terms of multiplications required per sample of the input signal for all N (N odd). the proposed approximations are also capable of giving extremely low RE (10^-3 to 10^-6 per cent), for narrow midband frequencies, with attractively low order N.