Design of low‐pass and band‐pass infinite impulse response maximally flat stable digital differentiators

Abstract

Maximally flat digital differentiators (MFDDs) are very attractive for differentiation of narrowband signals. The design of infinite impulse response (IIR) MFDDs, however, is very challenging due to the non-convexity of the flatness constraints of the IIR digital differentiator (DD). This study derives a set of linear conditions that are equivalent to the non-convex flatness constraints and then utilises these conditions to design low-pass and band-pass IIR MFDDs with low and nearly constant group delay. In addition, a generalized-positive-realness-based stability constraint is incorporated in the design to ensure the robust stability of the DD. A design algorithm is presented to maximise the total flatness degree subject to the flatness conditions and the stability constraint, resulting in maximally flat stable DDs (MFSDDs). Design examples and comparisons with existing DDs demonstrate that the designed IIR MFSDDs have better magnitude and group-delay responses and lower implementation complexity.