A GENERALIZED SAMPLING THEORY WITHOUT BAND-LIMITING CONSTRAINTS

We consider the problem of the reconstruction of a continuous-time fun ction f(x) is an element of H from the samples of the responses of m l inear shift-invariant Systems sampled at 1/m the reconstruction rate. We extend Papoulis' generalized sampling theory in two important respe cts, First, our class of admissible input signals (typ. H = L-2). is c onsiderably larger than the subspace of band-limited functions. Second , we Use a more general specification of the reconstruction subspace V (phi), so that the output of the system can take the form of a bandlim ited function, a spline, or a wavelet expansion, Since we have enlarge d the class of admissible input functions, we have to give up Shannon and Papoulis' principle of an exact reconstruction, Instead, we seek a n approximation (f) over tilde is an element of V(phi) that is consist ent in the sense that it produces exactly the same measurements as the input of the system, This leads to a generalization of Papoulis' samp ling theorem and a practical reconstruction algorithm that takes the f orm of a multivariate filter. In particular, we show that the correspo nding system acts as a projector from H onto V(phi), We then propose t wo complementary polyphase and modulation domain interpretations of ou r solution. The polyphase representation leads to a simple understandi ng of our reconstruction algorithm in terms of a perfect reconstructio n filter bank, The modulation analysis, on the other hand, is useful i n providing the connection,vith 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 sampli ng using splines.