All-pass-based design of nearly-linear phase IIR low-pass differentiators

摘要

A new approach to the design of the nearly linear phase infinite impulse response low-pass differentiators using a parallel all-pass structure is discussed in this paper. The magnitude and phase responses of the proposed low-pass differentiators are first formulated as functions of the phase responses of the corresponding all-pass filters, and a set of equations is derived such that the magnitude response approximates the ideal one in the weighted Chebyshev sense both in the passband and the stopband. The maximum passband phase response linearity error is shown to be related to the maximum passband magnitude error and the value of an additional design parameter. Comparison with the existing nearly-linear phase infinite impulse response low-pass differentiators shows that the low-pass differentiators designed using the proposed method usually require less multiplications, which comes at the cost of a somewhat higher filter order and consequently higher group delay. However, as the reduced number of multiplications lead to lower power consumption if hardware implementation is considered, the proposed low-pass differentiators are an attractive alternative in applications where low group delay is not of crucial importance.