 
Summary: Using fractional delay to control the magnitudes
and phases of integrators and differentiators
M.A. AlAlaoui
Abstract: The use of fractional delay to control the magnitudes and phases of integrators and dif
ferentiators has been addressed. Integrators and differentiators are the basic building blocks of
many systems. Often applications in controls, waveshaping, oscillators and communications
require a constant 908 phase for differentiators and 2908 phase for integrators. When the design
neglects the phase, a phase equaliser is often needed to compensate for the phase error or a
phase lock loop should be added. Applications to the firstorder, AlAlaoui integrator and differen
tiator are presented. A fractional delay is added to the integrator leading to an almost constant phase
response of 2908. Doubling the sampling rate improves the magnitude response. Combining the
two actions improves both the magnitude and phase responses. The same approach is applied to
the differentiator, with a fractional sample advance leading to an almost constant phase response
of 908. The advance is, in fact, realised as the ratio of two delays. Filters approximating the frac
tional delay, the finite impulse response (FIR) Lagrange interpolator filters and the Thiran allpass
infinite impulse response (IIR) filters are employed. Additionally, a new hybrid filter, a combi
nation of the FIR Lagrange interpolator filter and the Thiran allpass IIR filter, is proposed.
Methods to reduce the approximation error are discussed.
1 Introduction
Rabiner and Steiglitz in [1] noted that introducing a half
