Digital filtering and prolate functions

Author

Papoulis, Athanasios ; Bertran, Miguel S.

Volume

Issue

fYear

1972

fDate

11/1/1972 12:00:00 AM

Firstpage

674

Lastpage

681

Abstract

A class of trigonometric polynomials $p(x) = \\sum _{n=-N}^{N} a_{n} e^{j n \\pi x}$ of unit energy is introduced such that their energy concentration $\\alpha = \\int_{-e}^{e} p^{2}(x) dx$ in a specified interval $(- \\epsilon, \\epsilon)$ is maximum. It is shown that the coefficients $a_{n}$ must be the eigenvectors of the system $\\sum _{m=-N}^{N} frac{\\sin (n - m)\\pi \\epsilon}{(n - m)\\epsilon} a_{m} = \\lambda a_{n}$ . corresponding to the maximum eigenvalue X. These polynomials are determined for $N = 1, \\cdots , 10$ and $\\epsilon = 0.025, \\cdots , 0.5$ . The resulting family of periodic functions forms the discrete version of the familiar prolate spheroidal wave functions.

Keywords

Digital networks; Nonrecursive digital filters; Optimization techniques; Polynomials; Prolate functions; Trigonometric polynomials; Continuous time systems; Delay; Difference equations; Digital filters; Eigenvalues and eigenfunctions; Feedback; Filtering; Polynomials; Transfer functions; Wave functions;

fLanguage

English

Journal_Title

Circuit Theory, IEEE Transactions on

Publisher

ieee

ISSN

0018-9324

Type

jour

DOI

10.1109/TCT.1972.1083556

Filename

1083556

Link To Document

https://search.isc.ac/dl/search/defaultta.aspx?DTC=49&DC=1171133