Motivated by recent work on atomic norms in inverse problems, we propose a new approach to line spectral estimation that provides theoretical guarantees for the mean-squared-error (MSE) performance in the presence of noise and without knowledge of the model order. We propose an abstract theory of denoising with atomic norms and specialize this theory to provide a convex optimization problem for estimating the frequencies and phases of a mixture of complex exponentials. We show that the associated convex optimization problem can be solved in polynomial time via semidefinite programming (SDP). We also show that the SDP can be approximated by an l1-regularized least-squares problem that achieves nearly the same error rate as the SDP but can scale to much larger problems. We compare both SDP and l1-based approaches with classical line spectral analysis methods and demonstrate that the SDP outperforms the l1 optimization which outperforms MUSIC, Cadzow's, and Matrix Pencil approaches in terms of MSE over a wide range of signal-to-noise ratios.
@cite_5: We address the problem of estimating spectral lines from irregularly sampled data within the framework of sparse representations. Spectral analysis is formulated as a linear inverse problem, which is solved by minimizing an l1-norm penalized cost function. This approach can be viewed as a basis pursuit de-noising (BPDN) problem using a dictionary of cisoids with high frequency resolution. In the studied case, however, usual BPDN characterizations of uniqueness and sparsity do not apply. This paper deals with the l1-norm penalization of complex-valued variables, that brings satisfactory prior modeling for the estimation of spectral lines. An analytical characterization of the minimizer of the criterion is given and geometrical properties are derived about the uniqueness and the sparsity of the solution. An efficient optimization strategy is proposed. Convergence properties of the iterative coordinate descent (ICD) and iterative reweighted least-squares (IRLS) algorithms are first examined. Then, both strategies are merged in a convergent procedure, that takes advantage of the specificities of ICD and IRLS, considerably improving the convergence speed. The computation of the resulting spectrum estimator can be implemented efficiently for any sampling scheme. Algorithm performance and estimation quality are illustrated throughout the paper using an artificial data set, typical of some astrophysical problems, where sampling irregularities are caused by day night alternation. We show that accurate frequency location is achieved with high resolution. In particular, compared with sequential Matching Pursuit methods, the proposed approach is shown to achieve more robustness regarding sampling artifacts.
@cite_37: Compressive sensing (CS) is a new approach to simultaneous sensing and compression of sparse and compressible signals based on randomized dimensionality reduction. To recover a signal from its compressive measurements, standard CS algorithms seek the sparsest signal in some discrete basis or frame that agrees with the measurements. A great many applications feature smooth or modulated signals that are frequency-sparse and can be modeled as a superposition of a small number of sinusoids; for such signals, the discrete Fourier transform (DFT) basis is a natural choice for CS recovery. Unfortunately, such signals are only sparse in the DFT domain when the sinusoid frequencies live precisely at the centers of the DFT bins; when this is not the case, CS recovery performance degrades signicantly. In this paper, we introduce the spectral CS (SCS) recovery framework for arbitrary frequencysparse signals. The key ingredients are an over-sampled DFT frame and a restricted unionof-subspaces signal model that inhibits closely spaced sinusoids. We demonstrate that SCS signicantly outperforms current state-of-the-art CS algorithms based on the DFT while providing provable bounds on the number of measurements required for stable recovery. We also leverage line spectral estimation methods (specically Thomson’s multitaper method
@cite_2: We present a source localization method based on a sparse representation of sensor measurements with an overcomplete basis composed of samples from the array manifold. We enforce sparsity by imposing penalties based on the spl lscr sub 1 -norm. A number of recent theoretical results on sparsifying properties of spl lscr sub 1 penalties justify this choice. Explicitly enforcing the sparsity of the representation is motivated by a desire to obtain a sharp estimate of the spatial spectrum that exhibits super-resolution. We propose to use the singular value decomposition (SVD) of the data matrix to summarize multiple time or frequency samples. Our formulation leads to an optimization problem, which we solve efficiently in a second-order cone (SOC) programming framework by an interior point implementation. We propose a grid refinement method to mitigate the effects of limiting estimates to a grid of spatial locations and introduce an automatic selection criterion for the regularization parameter involved in our approach. We demonstrate the effectiveness of the method on simulated data by plots of spatial spectra and by comparing the estimator variance to the Crame spl acute r-Rao bound (CRB). We observe that our approach has a number of advantages over other source localization techniques, including increased resolution, improved robustness to noise, limitations in data quantity, and correlation of the sources, as well as not requiring an accurate initialization.
@cite_11: We apply basis pursuit, an atomic decomposition technique, for spectrum estimation. Compared with several modern time series methods, our approach can greatly reduce the problem of power leakage; it is able to superresolve; moreover, it works well with noisy and unevenly sampled signals. We present experiments on bizarrely spaced radial velocity data from one of the newly-discovered extrasolar planetary systems.
In contrast to linear prediction methods, a number of authors @cite_11 @cite_2 @cite_5 have suggested using compressive sensing and viewing the frequency estimation as a sparse approximation problem. For instance, @cite_2 notes that the Lasso based method has better empirical localization performance than the popular MUSIC algorithm. However, the theoretical analysis of this phenomenon is complicated because of the need to replace the continuous frequency space by an oversampled frequency grid. Compressive sensing based results (see, for instance, @cite_37 ) need to carefully control the incoherence of their linear maps to apply off-the-shelf tools from compressed sensing. It is important to note that the performance of our algorithm improves as the grid size increases. But this seems to contradict conventional wisdom in compressed sensing because our design matrix @math becomes more and more coherent, and limits how fine we can grid for the theoretical guarantees to hold.