Using fractional delay to control the magnitudes and phases of integrators and differentiators

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Using fractional delay to control the magnitudes and phases of integrators and differentiators

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The use of fractional delay to control the magnitudes and phases of integrators and differentiators has been addressed. Integrators and differentiators are the basic building blocks of many systems. Often applications in controls, wave-shaping, oscillators and communications require a constant 90° phase for differentiators and −90° 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 first-order, Al-Alaoui integrator and differentiator are presented. A fractional delay is added to the integrator leading to an almost constant phase response of −90°. 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 90°. The advance is, in fact, realised as the ratio of two delays. Filters approximating the fractional 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 combination of the FIR Lagrange interpolator filter and the Thiran allpass IIR filter, is proposed. Methods to reduce the approximation error are discussed.

Inspec keywords: FIR filters; IIR filters; delays

Other keywords: fractional delay; sampling rate; hybrid filter; finite impulse response Lagrange interpolator filters; Thiran allpass IIR filters; Al-Alaoui integrator; FIR Lagrange interpolator filters; Thiran allpass infinite impulse response filters; magnitude response; phase response; Al-Alaoui differentiator

Subjects: Filtering methods in signal processing; Signal processing theory

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