Reconstruction from Finitely Many Samples in the Presence of Noise

We treat the class of sampling problems in which the underlying function can be specified by a finite set of samples. Our problem is to reconstruct the signal from non-ideal, noisy samples, which are modelled as the inner products of the signal with a set of sampling vectors, contaminated by noise. To mitigate the effect of the noise and the mismatch between the sampling and reconstruction vectors, the samples are linearly transformed prior to reconstruction. Considering a statistical reconstruction framework, we characterize the strategies that are mean-squared error (MSE) admissible, meaning that they are not dominated in terms of MSE by any other linear reconstruction. We also present explicit designs of admissible reconstructions that dominate a given inadmissible method. Adapting several estimation approaches to our problem, we suggest concrete admissible reconstruction methods and compare their performance. The results are then specialized to the case in which the samples are processed by a digital correction filter.