Pseudo-inverse reconstruction of bandlimited signals from nonuniform generalized samples with orthogonal kernels

Contrary to the traditional pursuit of research on nonuniform sampling of bandlimited signals, the objective of the present paper is not to find sampling conditions that permit perfect reconstruction, but to perform the best possible signal recovery from any given set of nonuniform samples, whether it is finite as in practice, or infinite to achieve the possibility of unique reconstruction in L² (R). This leads us to consider the pseudo-inverse of the whole sampling map as a linear operator of Hilbert spaces. We propose in this paper an iterative algorithm that systematically performs this pseudo-inversion under the following conditions: (i) the input lies in some closed space A (such as a space of bandlimited functions) ; (ii) the samples are formed by inner product of the input with given kernel functions; (iii) these functions are orthogonal at least in a Hilbert space H that is larger than A. This situation turns out to appear in certain time encoders that are part of the increasingly important area of event-based sampling. As a result of pseudo-inversion, we systematically achieve perfect reconstruction whenever the samples uniquely characterize the input, we obtain minimal-norm estimates when the sampling is insufficient, and the reconstruction errors are controlled in the case of noisy sampling. The algorithm consists in alternating two projections according to the general method of projections onto convex sets (POCS) and can be implemented by iterating time-varying discrete-time filtering. We finally show that our signal and sampling assumptions appear in a nontrivial manner in other existing problems of data acquisition. This includes multi-channel time encoding where H is of the type L² (R) M, and traditional point sampling with the adoption of a Sobolev space H.