Resolution limits in signal recovery

Abstract

Resolution analysis for the problem of signal recovery from finitely many linear measurements is the subject of this paper. The classical Rayleigh limit serves only as a lower bound on resolution since it does not assume any recovery strategy and is based only on observed data. We show that details finer than the Rayleigh limit can be recovered by simple linear processing that incorporates prior information. We first define a measure of resolution based on allowable levels of error that is more appropriate for current signal recovery strategies than the Rayleigh definition. In the practical situation in which only finitely many noisy observations are available, we have to restrict the class of signals in order to make the resolution measure meaningful. We consider the set of bandlimited and essentially timelimited signals since it describes most signals encountered in practice. For this set, we show how to precompute resolution limits from knowledge of measurement functionals, signal-to-noise ratio, passband, energy concentration regions, energy concentration factor, and a prescribed level of error tolerance. In the process, we also derive an algorithm for high-resolution signal recovery. We illustrate the results with examples in one and two dimensions.