A generalized sampling theory without band-limiting constraints

We consider the problem of the reconstruction of a continuous-time function f(x)∈H from the samples of the responses of m linear shift-invariant systems sampled at 1/m the reconstruction rate. We extend Papoulis´ generalized sampling theory in two important respects. First, our class of admissible input signals (typ. H=L₂) is considerably larger than the subspace of band-limited functions. Second, we use a more general specification of the reconstruction subspace V(ψ), so that the output of the system can take the form of a band-limited function, a spline, or a wavelet expansion. Since we have enlarged the class of admissible input functions, we have to give up Shannon and Papoulis´ principle of an exact reconstruction. Instead, we seek an approximation f∈V(ψ) that is consistent in the sense that it produces exactly the same measurements as the input of the system. This leads to a generalization of Papoulis´ sampling theorem and a practical reconstruction algorithm that takes the form of a multivariate filter. In particular, we show that the corresponding system acts as a projector from H onto V(ψ). We then propose two complementary polyphase and modulation domain interpretations of our solution. The polyphase representation leads to a simple understanding of our reconstruction algorithm in terms of a perfect reconstruction filter bank. The modulation analysis, on the other hand, is useful in providing the connection with Papoulis´ earlier results for the band-limited case. Finally, we illustrate the general applicability of our theory by presenting new examples of interlaced and derivative sampling using splines