Consistent sampling for multi frame generators with distinct sampling rates

Abstract

We consider a general sampling and reconstruction process in the space L<sup>2</sup>(R). For any input signal f from L<sup>2</sup>(R), we take {〈f(t), ψ<inf>i</inf>(t-q<inf>i</inf>k)〉<inf>L</inf><sup>2</sup><inf>(R)</inf>| 1 ≤ i ≤ M; k ∈ Z} as its nonideal generalized measurements, where {ψ<inf>i</inf>}<sup>M</sup><inf>i =1</inf> are suitable pre-filters in L<sup>2</sup>(R) and q<inf>i</inf>'s are rational sampling periods. As a reconstruction space, in which output signals live, we take a shift invariant subspace V (Φ) of L<sup>2</sup>(R), which is generated by multi post-filters F = {f<inf>j</inf>}<sup>N</sup><inf>j=1</inf>. We then seek an approximation f of f in V (Φ), which is consistent with f in the sense that both f and f yield the same generalized measurements. Here we find equivalent conditions for the existence of such consistent sampling in L<sup>2</sup>(R).