On Super-Resolution in Multirate Sampling Systems

Super-resolution theory aims to estimate the discrete comp onents lying in a continuous space that constitute a sparse signal with optimal precision. This work investiga tes the potential of recent super-resolution techniques for spectral estimation in multirate sampling systems. It i s shown in the first part of this paper that, under the existence of a common supporting grid, the joint frequen cy estimation problem can be interpreted as a spectral recovery from partial uniform observations. More over, under a minimal separation constraint between the frequencies, the sparse spectrum of the observed signal ca 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 co nsiderably smaller than the classic Nyquist rate achieved by an estimation from a standard uniform sampling p attern. The second part addresses the complexity issue arising from the high dimensionality of semidefinite inequalities induced by partial observation systems. An equivalen t compact SDP of minimal dimension is derived by developing the Gram parametrization properties of sparse t rigonometric polynomials. The atomic soft thresholding method for spectrum recovery in presence of noise is extende d to fit in the partial measurements context, and a fast algorithm for sparse spectral estimation is provided t o super-resolve noisy line spectra from the presented framework. Index Terms Sampling theory, spikes model, super-resolution, line spe ctral estimation, sub-Nyquist sampling, multirate sampling, convex optimization. I. I NTRODUCTION Compressed sensing techniques have proven to be of great int rests for detecting, estimating and denoising sparse signals lying on discrete spaces. On the practical si de, the applications of sparse modelling are many: single molecule imaging via fluorescence, blind source sepa ration in speech processing, precise separation of multiple celestial bodies in astronomy, or super-resoluti n radaring, are among those. However, the discrete griding required by the compressed sensing framework weake n the recovery performances, and more precisely the systemresolution: the required minimal separation between two components of the sparse signal to be efficiency distinguished by an observation process. In the recent years, a particular enthusiasm has been placed on solving sparse linear inverse problems over continuous dictionaries. This aims to recover the smallest finite subset of components generating a signal, and 2 lying in a continuous space, by discrete observations of thi s signal, distorted by a kernel function. Considering such approach raises new theoretical and practical concern s, in particular, those problems are commonly infinitely ill-posed. This paper will discuss the spectral spikes recovery proble m, also known as line spectrum estimation problem, which is probably one of the most fundamental and important i llustration of sparse modelling over continuous space. For the spectral case, a complex time signal x is said to follow thes-spikes model if and only if it reads, ∀t ∈ R, x (t) = s ∑ l=1 αle l, (1) whereΞ = {ξl, l ∈ J1, sK} is the ordered set containing the s spectral components generating the signal x, andα = {αl, l ∈ J1, sK} the one of their associated complex amplitudes. The frequen cy stimation problem is naturally defined as building a consistent estimator ( Ξ̄, ᾱ ) of the parameters (Ξ, α) that are supposed to be unknown, byN ∈ N discrete observations y ∈ C of the time signalx. This problem is obviously ill-posed, and since no assumptio n is a priori made on the number of frequencies s to estimate, there are infinitely many pairs ( Ξ̄, ᾱ ) that are consistent with the observation vector y. As for illustration purpose, when drawing uniformly N samplesy from a time signalx such thaty = {x (k/f)}k∈J0,N−1K for some sampling frequency f ∈ R, the discrete Fourier transform of the observation vector y forms a consistent spectral representation of the signal x by N spectral spikes at locations̄ ξk = k N f in the frequency domain. However, this representation has generally no reas on to be sparse, in the sense that time signal x drawn from thes-spikes model will be represented by N > s non-null spectral coefficients; unless the all the elements of Ξ exactly belongs to the spectral grid { k N f, k ∈ Z } . Among all those consistent estimators, the one considered t o be optimal, in the sparse recovery context, is the one returning the sparsest spectral distribution, i.e. , th one that outputs a spectral support Ξ̄0 achieving the smallest cardinalitȳs0. The associated optimal spectral distribution x̂0 can be written as the output of an optimization program taking the form, x̂0 = arg min x̂∈D1 ‖x̂‖0 (2) subject to y = L (x̂) , where x̂ is the spectral distribution of x, ‖·‖0 represents the limit of thep-norm towards0, counting the cardinality of the support. D1 denotes the space of absolutely integrable spectral distri butions, andL denotes a linear operator fully determined by the sampling process an d linking the spectral domain to the measurements. In the previously discussed case of uniform signal sampling , the operatorL will be nothing but the inverse discrete-time Fourier transform operator. Since the generic program given in (2) is an NP-hard combinat orial problem, one cannot reasonably hope to recover the sparsest spectral estimator by a direct subset s election approach. A commonly proposed workaround consists in analysing the output of a relaxed problem, obtai ned by swapping the cardinality cost function ‖·‖0 into a minimization of the total-variation norm over the spe ctral distribution domain‖·‖TV , defined by, ‖x̂‖TV = sup f∈C(R),‖f‖ ∞ ≤1 R [∫