New sampling expansions for bandlimited signals based on chromatic derivatives

Abstract

Shannon's sampling theorem for bandlimited signals has been generalized in many directions in the last few decades. These extensions lead to various types of signal representations having different sets of basis functions. One particular extension proposed by Papoulis (1977) and later developed further by Brown (1981) can be interpreted in terms of a continuous time minimally sampled filter bank. In this paper we take a second look at these filter banks and use a continuous time version of the familiar biorthogonality property to obtain further insights into these sampling theorems. This viewpoint also makes a natural connection to the theory of orthogonal polynomials. We then elaborate on an elegant representation called the chromatic derivative expansion based on the use of Chebyshev polynomials. Using this expansion, the analysis/synthesis system can be described with a Chebyshev/Bessel pair of functions.