Results on Maximally Flat Fractional-Delay Systems

Abstract

The two classes of maximally flat finite-impulse response (FIR) and all-pass infinite-impulse response (IIR) fractional-sample delay systems are thoroughly studied. New expressions for the transfer functions are derived and mathematical properties revealed. Our contributions to the FIR case include a closed-form formula for the Farrow structure, a three-term recurrence relation based on the interpolation algorithm of Neville, a concise operator-based formula using the forward shift operator, and a continued fraction representation. Three types of structures are developed based on these formulas. Our formula for the Farrow structure enhances the existing contributions by Valimaki, and by Vesma and Sarama/spl uml/ki on the subsystems of the structure. For the IIR case, it is rigorously proved, using the theory of Pade approximants, that the continued fraction formulation of Tassart and Depalle yields all-pass fractional delay systems. It is also proved that the maximally flat all-pass fractional-delay systems are closely related to the Lagrange interpolation. It is shown that these IIR systems can be characterized using Thiele's rational interpolation algorithm. A new formula for the transfer function is derived based on the Thiele continued fractions. Finally, a new class of maximally flat FIR fractional-sample delay systems that exhibit an almost all-pass magnitude response is proposed. The systems possess a maximally flat group-delay response at the end frequencies 0 and /spl pi/, and are characterized by a closed-form formula. Their main advantage over the classical FIR Lagrange interpolators is the improved magnitude response characteristics.