Generalized Sampling: Stability and Peformance Analysis

Generalized sampling provides a general mechanism for recovering an unknown input function f(x) # H from the samples of the responses of m linear shift-invariant systems sampled at 1/mth the reconstruction rate. The system can be designed to perform a projection of f(x) onto the reconstruction subspace V (#) = span{#(x - k)} k#Z ; for example, the family of bandlimited signals with #(x) = sinc(x). This implies that the reconstruction will be perfect when the input signal is included in V (#); the traditional framework of Papoulis' generalized sampling theory. Otherwise, one recovers a signal approximation f(x) # V (#) that is consistent with f(x) in the sense that it produces the same measurements. To characterize the stability of the algorithm, we prove that the dual synthesis functions that appear in the generalized sampling reconstruction formula constitute a Riesz basis of V (#) and we use the corresponding Riesz bounds to define the condition number of the system. We then ..