Consistent Sampling and Signal Reconstruction in Noisy Under-Determined Case

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We present sampling theorems that reconstruct consistent signals from noisy underdetermined measurements. The consistency criterion requires that the reconstructed signal yields the same measurements as the original one. The main issue in underdetermined cases is a choice of a complementary subspace L in the reconstruction space of the intersection between the reconstruction space and the orthogonal complement of the sampling space because signals are reconstructed in L. Conventional theorems determine L without taking noise in measurements into account. Hence, the present paper proposes to choose L such that variance of reconstructed signals due to noise is minimized. We first arbitrarily fix L and compute the minimum variance under the condition that the average of the reconstructed signals agrees with the noiseless reconstruction. The derived expression clearly shows that the minimum variance depends on L and leads us to a condition for L to further minimize the minimum value of the variance. This condition indicates that we can choose such an L if and only if L includes a subspace determined by the noise covariance matrix. Computer simulations show that the standard deviation for the proposed sampling theorem is improved by 8.72% over that for the conventional theorem.

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