Ieee transactions on antennas and propagation

We derive a new generalization of Prony?s method to reconstruct M-sparse expansions of (generalized) eigenfunctions of linear operators from only suitable values in a deterministic way. The proposed method covers the well-known reconstruction methods for M-sparse sums of exponentials as well as for the interpolation of M-sparse polynomials by using special linear operators in . Further, we can derive new reconstruction formulas for M-sparse expansions of orthogonal polynomials using the Sturm?Liouville operator. The method is also applied to the recovery of M-sparse vectors in finite-dimensional vector spaces.

/pdf/a-generalized-prony-method-for-reconstruction-of-sparse-sums-4n86lc97jk.pdf

A generalized Prony method for reconstruction of sparse sums of eigenfunctions of linear operators

Koopman operator is a composition operator defined for a dynamical system described by nonlinear differential or difference equation. Although the original system is nonlinear and evolves on a finite-dimensional state space, the Koopman operator itself is linear but infinite-dimensional (evolves on a function space). This linear operator captures the full information of the dynamics described by the original nonlinear system. In particular, spectral properties of the Koopman operator play a crucial role in analyzing the original system. In the first part of this paper, we review the so-called Koopman operator theory for nonlinear dynamical systems, with emphasis on modal decomposition and computation that are direct to wide applications. Then, in the second part, we present a series of applications of the Koopman operator theory to power systems technology. The applications are established as data-centric methods, namely, how to use massive quantities of data obtained numerically and experimentally, through spectral analysis of the Koopman operator: coherency identification of swings in coupled synchronous generators, precursor diagnostic of instabilities in the coupled swing dynamics, and stability assessment of power systems without any use of mathematical models. Future problems of this research direction are identified in the last concluding part of this paper.

/pdf/applied-koopman-operator-theory-for-power-systems-technology-4sghz21jwr.pdf

Applied Koopman operator theory for power systems technology

http://sigproc.eng.cam.ac.uk/foswiki/pub/SPARS2015/Program/talk20_UlrichVon-der-Ohe_slides.pdf

A multivariate generalization of Prony's method

We consider the problem of stable recovery of sparse signals of the form $$F(x)=\sum_{j=1}^d a_j\delta(x-x_j),\quad x_j\in\mathbb{R},\;a_j\in\mathbb{C}, $$ from their spectral measurements, known in a bandwidth $\Omega$ with absolute error not exceeding $\epsilon>0$. We consider the case when at most $p\le d$ nodes $\{x_j\}$ of $F$ form a cluster whose extent is smaller than the Rayleigh limit ${1\over\Omega}$, while the rest of the nodes are well separated. Provided that $\epsilon \lessapprox SRF^{-2p+1}$, where $SRF=(\Omega\Delta)^{-1}$ and $\Delta$ is the minimal separation between the nodes, we show that the minimax error rate for reconstruction of the cluster nodes is of order ${1\over\Omega}SRF^{2p-1}\epsilon$, while for recovering the corresponding amplitudes $\{a_j\}$ the rate is of the order $SRF^{2p-1}\epsilon$. Moreover, the corresponding minimax rates for the recovery of the non-clustered nodes and amplitudes are ${\epsilon\over\Omega}$ and $\epsilon$, respectively. These results suggest that stable super-resolution is possible in much more general situations than previously thought. Our numerical experiments show that the well-known Matrix Pencil method achieves the above accuracy bounds.

pdf/super-resolution-of-near-colliding-point-sources-3cmpjucsyr.pdf

Super-resolution of near-colliding point sources

In this work we present two sparse deconvolution methods for nondestructive testing. The first method is a special matching pursuit (MP) algorithm in order to deconvolve the mixed data (signal and noise), and thus to remove the unwanted noise. The second method is based on the approximate Prony method (APM). Both methods employ the sparsity assumption about the measured ultrasonic signal as prior knowledge. The MP algorithm is used to derive a sparse representation of the measured data by a deconvolution and subtraction scheme. An orthogonal variant of the algorithm (OMP) is presented as well. The APM technique also relies on the assumption that the desired signals are sparse linear combinations of (reflections of) the transmitted pulse. For blind deconvolution, where the transducer impulse response is unknown, we offer a general Gaussian echo model whose parameters can be iteratively adjusted to the real measurements. Several test results show that the methods work well even for high noise levels. Further, an outlook for possible applications of these deconvolution methods is given.

/pdf/sparse-deconvolution-methods-for-ultrasonic-ndt-1wqvvikz40.pdf

Sparse Deconvolution Methods for Ultrasonic NDT

Prony's method is a prototypical eigenvalue analysis based method for the reconstruction of a finitely supported complex measure on the unit circle from its moments up to a certain degree. In this note, we give a generalization of this method to the multivariate case and prove simple conditions under which the problem admits a unique solution. Provided the order of the moments is bounded from below by the number of points on which the measure is supported as well as by a small constant divided by the separation distance of these points, stable reconstruction is guaranteed. In its simplest form, the reconstruction method consists of setting up a certain multilevel Toeplitz matrix of the moments, compute a basis of its kernel, and compute by some method of choice the set of common roots of the multivariate polynomials whose coefficients are given in the second step. All theoretical results are illustrated by numerical experiments.

Assume we only know equidistant measurements f(k), k = 0, . . . , N , of the signal f(x) = ∑M j=1 cje j, tj , cj ∈ C. The task is to recover the parameters tj and the corresponding coefficients cj . Methods that accomplish this problem include super-resolution [1], Prony’s Method [4], ESPRIT [9], MUSIC [10], Matrix-Pencil-Method [2], or the Annihilating Filter Method for signals with finite rate of innovation [11], where all but the first method can be seen as Prony-like methods [7]. In [1], it is demonstrated that minimizing total variation is also applicable for multivariate signals. Another way to analyze multivariate signals is to reduce the problem to a number of one-dimensional problems via projections of the data to multiple lines through the origin and to apply a Prony-like method to each projection. Recent recovering results in [6, 8] using this approach work only if the coefficients have equal sign, i.e., cj ∈ R+, since the inherited projection might otherwise cause cancellation. In contrast, we established a new, fully multidimensional Prony method that is applicable for arbitrary cj ∈ C. This advantage comes at the price of calculating common zeros of multivariate polynomials, which is a challenging task itself.

https://onlinelibrary.wiley.com/doi/pdf/10.1002/pamm.201510322

Prony's Method for Multivariate Signals

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Papers

A generalized Prony method for reconstruction of sparse sums of eigenfunctions of linear operators

A multivariate generalization of Prony's method

Sparse Deconvolution Methods for Ultrasonic NDT

A multivariate generalization of Prony's method

Prony's Method for Multivariate Signals