A Closed-Form Description for the Continuous Frequency Response of Lagrange Interpolators

In this letter, we present a closed formula for the continuous frequency response of Lagrange interpolators. The continuous frequency response is one of the most useful characterizations of an interpolation system. The proposed formula holds for arbitrary interpolation orders and enables efficient and numerically stable evaluation. Until now, several properties of Lagrange interpolators such as zero locations were described only empirically. Due the concise form of the proposed formula, these properties can now be characterized in an exact way.

[1]  I Iiiii.  PERFORMANCE AND DESIGN CONSIDERATIONS OF THE FARROW FILTER WHEN USED FOR ARBITRARY RESAMPLING OF SAMPLED TIME SERIES , 1998 .

[2]  F. Harris.  Performance and design considerations of the Farrow filter when used for arbitrary resampling of sampled time series , 1997, Conference Record of the Thirty-First Asilomar Conference on Signals, Systems and Computers (Cat. No.97CB36136).

[3]  Vesa Vlimki,et al.  Discrete-Time Modeling of Acoustic Tubes Using Fractional Delay Filters , 1998 .

[4]  Gerhard Fettweis,et al.  Sample rate conversion for software radio , 2000 .

[5]  K. W. Cattermole.  The Fourier Transform and its Applications , 1965 .

[6]  L. Rabiner,et al.  A digital signal processing approach to interpolation , 1973 .

[7]  N. E. Nörlund.  Vorlesungen über Differenzenrechnung , 1924 .

[8]  I. J. Schoenberg.  Contributions to the Problem of Approximation of Equidistant Data by Analytic Functions , 1988 .

[9]  T. Saramaki,et al.  Polynomial-Based Interpolation Filters—Part I: Filter Synthesis , 2007 .

[10]  S. Biyiksiz,et al.  Multirate digital signal processing , 1985, Proceedings of the IEEE.

[11]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[12]  Michael Unser,et al.  Splines: a perfect fit for signal and image processing , 1999, IEEE Signal Process. Mag..

[13]  Vesa V Alim Aki.  Discrete-Time Modeling of Acoustic Tubes Using Fractional Delay Filters , 1995 .

[14]  Andreas Franck,et al.  Efficient Algorithms and Structures for Fractional Delay Filtering Based on Lagrange Interpolation , 2009 .

[15]  Lars Erup,et al.  Interpolation in digital modems. II. Implementation and performance , 1993, IEEE Trans. Commun..