Signal reconstruction from nonideal samples on shift-invariant spaces

Abstract

This dissertation handled sampling theorems for nonideal samples on shift-invariant spaces: irregular sampling and consistent sampling.

For irregualr sampling, let $V(\phi)$ be a shift invariant subspace of $L^{2}(\mathbb{R})$ with a Riesz or frame generator $\phi(t)$. We take $\phi(t)$ suitably so that the regular sampling expansion : $f(t) = \sum\limits_{n\in \mathbb{Z}}f(n)S(t-n)$ holds on $V(\phi)$. We then find conditions on the generator $\phi(t)$ and various bounds of the perturbation $\{ \delta _n \}_{n \in \mathbb{Z}}$ under which an irregular sampling expansion \begin{equation*} f(t)=\sum_{n \in \mathbb{Z}} f(n+ \delta_n)S_n(t) \end{equation*} holds on $V(\phi)$.

We also consider the approximate consistent sampling process in the space $L^2(\mathbb{R})$ of signals of finite energy. The consistency means that the original signal and its approximation have the same measurements. We assume that sampling and reconstruction functions $\{\psi_i\}_{i=1}^M$, $\{\phi_j\}_{j=1}^N$ are given as Riesz generators and the measurements $\{\langle f(t),\psi_i(t-qk)\rangle |1\leq i \leq M, k\in\mathbb{Z}\}$ are given as inner-products between the input signal and the sampling functions with rational sampling rate $q=\frac{m}{n}$. We then find an approximation in the reconstruction space $V(\Phi)$, the shift-invariant space generated by the reconstruction functions, which is consistent with the input signal. We also discuss some relevant properties such as the performance analysis of the consistent approximation.