Design of IIR Lowpass Differentiators Using Parallel Allpass Structure

In this paper, design of infinite impulse response lowpass differentiators that can be realized by parallel connection of two allpass filters whose orders differ by two, where one of the allpass branches is pure delay, is considered. As adopted structure of proposed lowpass differentiators allows formulation of their magnitude and phase responses as functions of allpass filter phase response, a set of nonlinear equations in unknown allpass filter coefficients is derived and iteratively solved in a way that magnitude response approximates the ideal one in weighted Chebyshev sense, in both passband and stopband. On the other hand, passband phase response linearity is shown to be related to the maximum of the magnitude response, which can be directly controlled by an additional design parameter. Design examples reveal that proposed infinite impulse response lowpass differentiators of low order can have very low relative passband magnitude errors and nearly linear phases, while results of comparison with existing lowpass differentiators show that proposed differentiators usually require fewer multiplications.