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_18: In this introduction we briefly describe the methodology of the Singular Spectrum Analysis (SSA), some versions and extensions of the basic version of SSA as well as connections between SSA and subspace-based methods in signal processing. We also briefly touch upon some history of SSA and mention some areas of application of SSA.
@cite_14: This paper concerns the construction of a structured low rank matrix that is nearest to a given matrix. The notion of structured low rank approximation arises in various applications, ranging from signal enhancement to protein folding to computer algebra, where the empirical data collected in a matrix do not maintain either the specified structure or the desirable rank as is expected in the original system. The task to retrieve useful information while maintaining the underlying physical feasibility often necessitates the search for a good structured lower rank approximation of the data matrix. This paper addresses some of the theoretical and numerical issues involved in the problem. Two procedures for constructing the nearest structured low rank matrix are proposed. The procedures are flexible enough that they can be applied to any lower rank, any linear structure, and any matrix norm in the measurement of nearness. The techniques can also be easily implemented by utilizing available optimization packages. The special case of symmetric Toeplitz structure using the Frobenius matrix norm is used to exemplify the ideas throughout the discussion. The concept, rather than the implementation details, is the main emphasis of the paper.
@cite_8: Abstract The spectral line analysis (SLA), also called line spectrum analysis, deals with the problem of extracting information on sinusoidal signals from noise-corrupted measurements. It has numerous applications in sonar, radar, underwater surveillance, communications, geophysical exploration, speech analysis, nuclear physics and other fields. The sinusoids-in-noise model, called the hidden periodicity model in the time series literature, also has a significant theoretical relevance as a typical example of a composite spectrum consisting of both continuous and discrete (or line) components, and it is frequently used in theoretical and empirical performance studies of spectral estimation methods. Owing to its practical and theoretical interest, the SLA is a ubiquituous topic in the signal processing literature. When seen from outside by a newcomer, the SLA may appear as a rather narrow field of research. Actually, it contains a myriad of techniques and analysis methods, many of them borrowed from other fields, such as time-series analysis, system identification, matrix algebra, multivariate statistical analysis, complexity theory and others. In the following, we provide a comprehensive list of basic and recent references on SLA. While this list is not, and cannot be, exhaustive, it is believed that it includes most of the key references. The list should be useful to all those interested in the SLA problem, particularly to the newcomers but also to the researchers trying to find the roots of a specific result or recent references for a particular topic. The field of SLA has many ramifications in other fields of signal processing. Many SLA methods can, for example, be used to estimate the parameters of a transient signal, such as a damped sine wave, or the direction-of-arrival of the plane waves impinging on an array of sensors. In general, references on the methods used to solve signal processing problems related to SLA are not provided, unless those methods are strictly relevant to the SLA problem.
@cite_48: 
@cite_29: Abstract Singular-spectrum analysis (SSA) is developed further, based on experience with applications to geophysical time series. It is shown that SSA provides a crude but robust approximation of strange attractors by tori, in the presence of noise. The method works well for short, noisy time series. The lagged-covariance matrix of the processes studied is the basis of SSA. We select subsets of eigenelements and associated principal components (PCs) in order to provide (i) a noise-reduction algorithm, (ii) a detrending algorithm, and (iii) an algorithm for the identification of oscillatory components. Reconstructed components (RCs) are developed to provide optimal reconstruction of a dynamic process at precise epochs, rather than averaged over the window length of the analysis. SSA is combined with advanced spectral-analysis methods - the maximum entropy method (MEM) and the multi-taper method (MTM) - to refine the interpretation of oscillatory behavior. A combined SSA-MEM method is also used for the prediction of selected subsets of RCs. The entire toolkit is validated against a set of four prescribed time series generated by known processes, quasi-periodic or chaotic. It is also applied to a time series of global surface air temperatures, 130 years long, which has attracted considerable attention in the context of the global warming issue and provides a severe test for noise reduction and prediction.
@cite_45: Sparse sampling of continuous-time sparse signals is addressed. In particular, it is shown that sampling at the rate of innovation is possible, in some sense applying Occam's razor to the sampling of sparse signals. The noisy case is analyzed and solved, proposing methods reaching the optimal performance given by the Cramer-Rao bounds. Finally, a number of applications have been discussed where sparsity can be taken advantage of. The comprehensive coverage given in this article should lead to further research in sparse sampling, as well as new applications. One main application to use the theory presented in this article is ultra-wide band (UWB) communications.
@cite_49: A signal enhancement algorithm is developed that seeks to recover a signal from noise-contaminated distorted measurements made on that signal. This object is achieved by utilizing a set of properties which the signal is known or is hypothesized as possessing. The measured signal is modified to the smallest degree necessary to sequentially possess each of the individual properties. Conditions for the algorithm's convergence are established in which the primary requirement is that a composite property mapping be closed. This is a relatively unrestricted condition in comparison to that required of most existing signal-enhancement algorithms. >
@cite_25: Part I. Basic Concepts: 1. Introduction: why nonlinear methods? 2. Linear tools and general considerations 3. Phase space methods 4. Determinism and predictability 5. Instability: Lyapunov exponents 6. Self-similarity: dimensions 7. Using nonlinear methods when determinism is weak 8. Selected nonlinear phenomena Part II. Advanced Topics: 9. Advanced embedding methods 10. Chaotic data and noise 11. More about invariant quantities 12. Modeling and forecasting 13. Chaos control 14. Other selected topics Appendix 1. Efficient neighbour searching Appendix 2. Program listings Appendix 3. Description of the experimental data sets.
@cite_20: 1. Basic Concepts. 2. Nonparametric Methods. 3. Parametric Methods for Rational Spectra. 4. Parametric Methods for Line Spectra. 5. Filter Bank Methods. 6. Spatial Methods. Appendix A: Linear Algebra and Matrix Analysis Tools. Appendix B: Cramer-Rao Bound Tools. Appendix C: Model Order Selection Tools. Appendix D: Answers to Selected Exercises. Bibliography. References Grouped by Subject. Subject Index.
Cadzow @cite_49 proposed a heuristic that improves over MUSIC by exploiting the Toeplitz structure of the matric of moments by alternately projecting between the linear space of Toeplitz matrices and the space of rank @math matrices where @math is the desired model order. Cadzow's technique is very similar @cite_18 to a popular technique in time series literature @cite_25 @cite_48 called Singular Spectrum Analysis @cite_29 , which uses autocorrelation matrix instead of the matrix of moments for projection. Both these techniques may be viewed as instances of structured low rank approximation @cite_14 which exploit additional structure beyond low rank structure used in subspace based methods such as MUSIC and ESPRIT. Cadzow's method has been identified as a fruitful preprocessing step for linear prediction methods @cite_45 . A survey of classical linear prediction methods can be found in @cite_45 @cite_20 and an extensive list of references is given in @cite_8 .