Closed-form design of maximally flat FIR differentiators with fractional delay using interlaced sampling method

In this paper, the closed-form design of finite impulse response (FIR) digital differentiators (DD) with fractional delay (FD) using interlaced sampling (IS) method is proposed. Typically, design of FIR DDs with real impulse responses suffers from the difficulty of approximation within the high frequency band. The issue arises from the fact that the desired frequency response is complex and the designed one is real at ω = π. The IS structure provides a possible solution to this issue. We solve the closed-form expressions of the coefficients in the IS structure in maximally flat (MF) error criterion. In addition, we also propose another design method in time domain, called the polynomial invariance (PI) method.We show that the PI design in time domain is equivalent to the MF design in frequency domain, and accordingly provide an interpretation of the IS MF FD DD. Several design examples are provided.