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.
The classical methods of line spectral estimation, often called linear prediction methods, are built upon the seminal interpolation method of Prony @cite_9 . In the noiseless case, with as little as @math measurements, Prony's technique can identify the frequencies exactly, no matter how close the frequencies are. However, Prony's technique is known to be sensitive to noise due to instability of polynomial rooting @cite_33 . Following Prony, several methods have been employed to robustify polynomial rooting method including the Matrix Pencil algorithm @cite_10 , which recasts the polynomial rooting as a generalized eigenvalue problem and cleverly uses extra observations to guard against noise. The MUSIC @cite_47 and ESPRIT @cite_42 algorithms exploit the low rank structure of the autocorrelation matrix.
@cite_33 @cite_9 @cite_42 @cite_47 @cite_10
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Abstract Modifications of Prony's classical technique for estimating rate constants in exponential fitting problems have many contemporary applications. In this article the consistency of Prony's method and of related algorithms based on maximum likelihood is discussed as the number of observations n → ∞ by considering the simplest possible models for fitting sums of exponentials to observed data. Two sampling regimes are relevant, corresponding to transient problems and problems of frequency estimation, each of which is associated with rather different kinds of behavior. The general pattern is that the stronger results are obtained for the frequency estimation problem. However, the algorithms considered are all scaling dependent and consistency is not automatic. A new feature that emerges is the importance of an appropriate choice of scale in order to ensure consistency of the estimates in certain cases. The tentative conclusion is that algorithms referred to as Objective function Reweighting Algorithms ... A study of a matrix pencil method for estimating frequencies and damping factors of exponentially damped and or undamped sinusoids in noise is presented. Comparison of this method to a polynomial method (SVD-Prony method) shows that the matrix pencil method and the polynomial method are two special cases of a matrix prediction approach and that the pencil method is more efficient in computation and less restrictive about signal probes. It is found through perturbation analysis and simulation that, for signals with unknown damping factors, the pencil method is less sensitive to noise than the polynomial method. An expression of the Cramer-Rao bound for the exponential signals is presented. > We consider the problem of sparse interpolation of an approximate multivariate black-box polynomial in floating point arithmetic. That is, both the inputs and outputs of the black-box polynomial have some error, and all numbers are represented in standard, fixed-precision, floating point arithmetic. By interpolating the black box evaluated at random primitive roots of unity, we give efficient and numerically robust solutions. We note the similarity between the exact Ben-Or Tiwari sparse interpolation algorithm and the classical Prony's method for interpolating a sum of exponential functions, and exploit the generalized eigenvalue reformulation of Prony's method. We analyse the numerical stability of our algorithms and the sensitivity of the solutions, as well as the expected conditioning achieved through randomization. Finally, we demonstrate the effectiveness of our techniques in practice through numerical experiments and applications. Abstract This paper studies the problem of line spectral estimation in the continuum of a bounded interval with one snapshot of array measurement. The single-snapshot measurement data are turned into a Hankel data matrix which admits the Vandermonde decomposition and is suitable for the MUSIC algorithm. The MUSIC algorithm amounts to finding the null space (the noise space) of the adjoint of the Hankel matrix, forming the noise-space correlation function and identifying the s smallest local minima of the noise-space correlation as the frequency set. In the noise-free case exact reconstruction is guaranteed for any arbitrary set of frequencies as long as the number of measurement data is at least twice the number of distinct frequencies to be recovered. In the presence of noise the stability analysis shows that the perturbation of the noise-space correlation is proportional to the spectral norm of the noise matrix as long as the latter is smaller than the smallest (nonzero) singular value of the noiseless Hankel data matrix. Under the assumption that the true frequencies are separated by at least twice the Rayleigh Length (RL), the stability of the noise-space correlation is proved by means of novel discrete Ingham inequalities which provide bounds on the largest and smallest nonzero singular values of the noiseless Hankel data matrix. The numerical performance of MUSIC is tested in comparison with other algorithms such as BLO-OMP and SDP (TV-min). While BLO-OMP is the stablest algorithm for frequencies separated above 4 RL, MUSIC becomes the best performing one for frequencies separated between 2 RL and 3 RL. Also, MUSIC is more efficient than other methods. MUSIC truly shines when the frequency separation drops to 1 RL or below when all other methods fail. Indeed, the resolution length of MUSIC decreases to zero as noise decreases to zero as a power law with an exponent smaller than an upper bound established by Donoho.