A swiss army knife for finite rate of innovation sampling theory

Abstract

Finite-Rate-of-Innovation (FRI) sampling theory prescribes a procedure for exact recovery of Dirac impulses from linear measurements in the form of orthogonal projections of streams of Dirac impulses onto the subspace of Fourier—bandlimited functions. This enables recovery of a continuous time sparse signals at sub-Nyquist rates. In many cases, the transform domain of interest may be more general than the Fourier domain. Recent work has extended FRI sampling theory to the spherical Fourier Transform, fractional Fourier Transform and the Laplace Transform. In this paper, we develop a broad FRI framework applicable to a general class of transformations that includes Fourier, Laplace, Fresnel, fractional Fourier, Bargmann and Gauss—Weierstrass transforms, among others. For this purpose, we consider the Special Affine Fourier Transform (SAFT) which parametrically generalizes a number of well known unitary transforms linked with signal processing and optics. We first derive a version of Shannon's sampling theory based on the convolution structure tailored for the SAFT domain. Having identified the subspace of SAFT—bandlimited functions, we apply FRI sampling theory to the SAFT and study recovery of sparse signals, thus providing a unified view of FRI sampling theory for a large class of disparately studied operations.