Sampling signals with finite rate of innovation

TLDR

The authors define a finite rate of innovation as the number of degrees of freedom per unit time, exemplified by streams of Diracs, nonuniform splines, and piecewise polynomials. They demonstrate that such signals can be uniformly sampled at or above the innovation rate using suitable kernels (sinc, Gaussian, spline) and an annihilating or locator filter to extract the innovative components. They prove sampling theorems that generalize the bandlimited case, provide reconstruction formulas for periodic and finite-length Dirac streams, nonuniform splines, and piecewise polynomials, and validate the approach with experimental results. Applications of these results span signal processing, communications, and biological systems.

Abstract

The authors consider classes of signals that have a finite number of degrees of freedom per unit of time and call this number the rate of innovation. Examples of signals with a finite rate of innovation include streams of Diracs (e.g., the Poisson process), nonuniform splines, and piecewise polynomials. Even though these signals are not bandlimited, we show that they can be sampled uniformly at (or above) the rate of innovation using an appropriate kernel and then be perfectly reconstructed. Thus, we prove sampling theorems for classes of signals and kernels that generalize the classic "bandlimited and sinc kernel" case. In particular, we show how to sample and reconstruct periodic and finite-length streams of Diracs, nonuniform splines, and piecewise polynomials using sinc and Gaussian kernels. For infinite-length signals with finite local rate of innovation, we show local sampling and reconstruction based on spline kernels. The key in all constructions is to identify the innovative part of a signal (e.g., time instants and weights of Diracs) using an annihilating or locator filter: a device well known in spectral analysis and error-correction coding. This leads to standard computational procedures for solving the sampling problem, which we show through experimental results. Applications of these new sampling results can be found in signal processing, communications systems, and biological systems.