Exact and approximate Strang-Fix conditions to reconstruct signals with finite rate of innovation from samples taken with arbitrary kernels

Abstract

In the last few years, several new methods have been developed for the sampling and exact reconstruction of specific classes of non-bandlimited signals known as signals with finite rate of innovation (FRI). This is achieved by using adequate sampling kernels and reconstruction schemes. An example of valid kernels, which we use throughout the thesis, is given by the family of exponential reproducing functions. These satisfy the generalised Strang-Fix conditions, which ensure that proper linear combinations of the kernel with its shifted versions reproduce polynomials or exponentials exactly. The first contribution of the thesis is to analyse the behaviour of these kernels in the case of noisy measurements in order to provide clear guidelines on how to choose the exponential reproducing kernel that leads to the most stable reconstruction when estimating FRI signals from noisy samples. We then depart from the situation in which we can choose the sampling kernel and develop a new strategy that is universal in that it works with any kernel. We do so by noting that meeting the exact exponential reproduction condition is too stringent a constraint. We thus allow for a controlled error in the reproduction formula in order to use the exponential reproduction idea with arbitrary kernels and develop a universal reconstruction method which is stable and robust to noise. Numerical results validate the various contributions of the thesis and in particular show that the approximate exponential reproduction strategy leads to more stable and accurate reconstruction results than those obtained when using the exact recovery methods.