2 research outputs found
Exact and approximate Strang-Fix conditions to reconstruct signals with finite rate of innovation from samples taken with arbitrary kernels
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.Open Acces