α-Spline design of finite impulse response digital filters

Abstract

The use of transition functions to connect the passband and the stopband in the ideal amplitude frequency response of an FIR filter reduces the oscillations near the band edges caused by the Gibbs phenomenon in filters designed by windowing. To this end, several methods based on β-splines have been proposed, in which an integer order of the spline is used to properly shape the transition band of the filter. However, it has been reported in the literature that the optimal value of the spline order that minimizes the approximation׳s mean square error may take non-integer values. Therefore, there is a lack of a precise formulation that allows the use of real positive numbers to properly design spline-based digital filters. The aim of this paper is to solve this problem, presenting a new formulation based on α-splines. By means of the proposed approach, non-integer values for the spline order can be employed in an appropriate mathematical framework, allowing its use as a real variable in any design procedure. Design examples demonstrate that smaller approximation errors than other approaches based on conventional windows can be obtained.