Consistent Sampling and Reconstruction of Signals in Noisy Under-Determined Case

We propose a sampling theorem that reconstructs a consistent signal from noisy under-determined samples. Consistency in the context of sampling theory means that the reconstructed signal yields the same measurements as the original ones. The conventional consistent sampling theorems for the underdetermined case reconstruct a signal from noiseless samples in a complementary subspace L in the reconstruction space of the intersection between the reconstruction space and the orthogonal complement of the sampling space. To obtain a good reconstruction result, one has to determine L effectively. To this end, we first extend the sampling theorem for a noisy case. Since the reconstructed signal is scattered in L, we propose to reconstruct a signal so that the variance is minimized provided that the average of the signals agrees with the noiseless reconstruction. Note that the minimum variance depends on the subspace L. Therefore, we next propose to determine L so that the minimum variance is further minimized in terms of L. We show that such L can be chosen if and only if L includes a subspace determined by the noise covariance matrix. By computer simulations, we demonstrate that there is a considerable difference between the minimum and non-minimum variance reconstructions.