Generalized sampling without bandlimiting constraints
Michael A. Unser, Josiane B. Zerubia · 2002
We investigate the problem of the reconstruction of a continuous-time function f(x)/spl isin//spl Hscr/ from the responses of m linear shift-invariant systems sampled at 1/m the reconstruction rate, extending Papoulis' (1977) generalized sampling theory in two important respects. First, we allow for arbitrary (non-bandlimited) input signals (typ. /spl Hscr/=L/sub 2/). Second, we use a more general specification of the reconstruction subspace V(/spl phi/), so that the output of the system can take the form of a bandlimited function, a spline, or a wavelet expansion. The system that we describe yields an approximation f/spl isin/V(/spl phi/) that is consistent with the input f(x) in the sense that it produces exactly the same measurements. We show that this solution can be computed by multivariate filtering. We also characterize the stability of the system (condition number). Finally, we illustrate the theory by presenting a new example of interlaced sampling using splines.