On Super-Resolution in Multirate Sampling Systems

Super-resolution theory aims to estimate the discrete components lying in a continuous space that constitute a sparse signal with optimal precision. This work investigates the potential of recent super-resolution techniques for spectral estimation in multirate sampling systems. It is shown in the first part of this paper that, under the existence of a common supporting grid, the joint frequency estimation problem can be interpreted as a spectral recovery from partial uniform observations. Moreover, under a minimal separation constraint between the frequencies, the sparse spectrum of the observed signal can be exactly jointly recovered up to an aliasing factor, by solving a semidefinite program (SDP). It is proven that multirate sampling allows sub-Nyquist recovery of sparse spectra, and that the aliasing factor can be made considerably smaller than the classic Nyquist rate achieved by an estimation from a standard uniform sampling pattern. The second part addresses the complexity issue arising from the high dimensionality of semidefinite inequalities induced by partial observation systems. An equivalent compact SDP of minimal dimension is derived by developing the Gram parametrization properties of sparse trigonometric polynomials. The atomic soft thresholding method for spectrum recovery in presence of noise is extended to fit in the partial measurements context, and a fast algorithm for sparse spectral estimation is provided to super-resolve noisy line spectra from the presented framework. Index Terms Sampling theory, spikes model, super-resolution, line spectral estimation, sub-Nyquist sampling, multirate sampling, convex optimization.