Consistent sampling with varying sampling rates

Abstract

We will find the approximation of an input signal $f$ in $L^2(\Real)$ with multi pre and post filters $\{\psi_i\}_{i=1}^{M}$ and $\{\phi_j\}_{j=1}^{N}$ respectively. For each $1\leq i \leq M$, we will vary the sampling rate and take $\{\langle f(t),\psi_i(t-q_{i}k)\rangle | k\in\mathbb{Z}\}$ as measurement(of generalized samples). With this samples, we will reconstruct its consistent approximation $\widetilde{f}$ in the reconstruction space. We call $\widetilde{f}$ is a consistent approximation if samples of $f$ and $\widetilde{f}$ are totally same. In this paper we find several equivalent conditions for existence of consistent approximation.