A minimum squared-error framework for generalized sampling

We treat the problem of reconstructing a signal from its nonideal samples where the sampling and reconstruction spaces as well as the class of input signals can be arbitrary subspaces of a Hilbert space. Our formulation is general, and includes as special cases reconstruction from finitely many samples as well as uniform-sampling of continuous-time signals, which are not necessarily bandlimited. To obtain a good approximation of the signal in the reconstruction space from its samples, we suggest two design strategies that attempt to minimize the squared-norm error between the signal and its reconstruction. The approaches we propose differ in their assumptions on the input signal: If the signal is known to lie in an appropriately chosen subspace, then we propose a method that achieves the minimal squared error. On the other hand, when the signal is not restricted, we show that the minimal-norm reconstruction cannot generally be obtained. Instead, we suggest minimizing the worst-case squared error between the reconstructed signal, and the best possible (but usually unattainable) approximation of the signal within the reconstruction space. We demonstrate both theoretically and through simulations that the suggested methods can outperform the consistent reconstruction approach previously proposed for this problem.