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

摘要

In the last few years, several new methods have been developed for the sampling andudexact 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 andudreconstruction schemes. An example of valid kernels, which we use throughout the thesis,udis given by the family of exponential reproducing functions. These satisfy the generalisedudStrang-Fix conditions, which ensure that proper linear combinations of the kernel with itsudshifted versions reproduce polynomials or exponentials exactly.udThe first contribution of the thesis is to analyse the behaviour of these kernels in theudcase of noisy measurements in order to provide clear guidelines on how to choose the exponentialudreproducing kernel that leads to the most stable reconstruction when estimatingudFRI signals from noisy samples. We then depart from the situation in which we can chooseudthe sampling kernel and develop a new strategy that is universal in that it works with anyudkernel. We do so by noting that meeting the exact exponential reproduction condition isudtoo stringent a constraint. We thus allow for a controlled error in the reproduction formulaudin order to use the exponential reproduction idea with arbitrary kernels and developuda universal reconstruction method which is stable and robust to noise.udNumerical results validate the various contributions of the thesis and in particular showudthat the approximate exponential reproduction strategy leads to more stable and accurateudreconstruction results than those obtained when using the exact recovery methods.