A New Class of Explicit Interpolatory Splines and Related Measurement Estimation

Abstract

This paper proposes a new and rich class of cardinal splines called generalized MK-splines (GMK-splines), which manifest several beneficial properties (especially the interpolation property) and the ability to obtain a construction with a desired frequency response. The GMK-spline basis functions of given degree can be constructed by a linear combination of a finite number of shifted B-splines of that degree in a unified framework, the GMK-spline bases form Riesz bases of their generating spaces, and the GMK-spline frequency responses meet Strang-Fix conditions. Furthermore, we provide search methods for minimal-support GMK-splines and local optimal GMK-spline frequency responses under different given passband frequencies. Several related estimates are theoretically assessed, including energy magnitude, maximum, and approximation error of GMK-spline interpolated signals. In signal processing applications, the GMK-splines behave as interpolating FIR filters with simple implementation, and some of them attain lower error than other traditional interpolation system responses, such as cubic cardinal B-spline and linear spline, when approximating to an ideal lowpass filter. Experimental results show that the GMK-spline filters can be implemented efficiently and have significant characteristics in the process of bandlimited signal reconstruction.