Solving Overdetermined Eigenvalue Problems

Das, Saptarshi; Neumaier, Arnold

doi:10.1137/110828514

Cited by 12 publications

(16 citation statements)

References 23 publications

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“…In general, parametric methods also require prior knowledge of the model order. Widely used parametric methods assuming a multi-exponential model include MUSIC [6], ESPRIT [7], the matrix pencil algorithm [8], simultaneous QR factorization [9] or a generalized overdetermined eigenvalue solver [10] and the approximate Prony method APM [11,12,13].…”

Section: Introductionmentioning

confidence: 99%

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How to get high resolution results from sparse and coarsely sampled data

Cuyt

Lee

2020

Applied and Computational Harmonic Analysis

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Sampling a signal below the Shannon-Nyquist rate causes aliasing, meaning different frequencies to become indistinguishable. It is also wellknown that recovering spectral information from a signal using a parametric method can be ill-posed or ill-conditioned and therefore should be done with caution.We present an exponential analysis method to retrieve high-resolution information from coarse-scale measurements, using uniform downsampling. We exploit rather than avoid aliasing. While we loose the unicity of the solution by the downsampling, it allows to recondition the problem statement and increase the resolution.Our technique can be combined with different existing implementations of multi-exponential analysis (matrix pencil, MUSIC, ESPRIT, APM, generalized overdetermined eigenvalue solver, simultaneous QR factorization, . . .) and so is very versatile. It seems to be especially useful in the presence of clusters of frequencies that are difficult to distinguish from one another.

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Section: Introductionmentioning

confidence: 99%

“…• that it can be combined with an existing implementation of a multiexponential spectral analysis (we used ESPRIT [7] and oeig [10]).…”

Section: Introductionmentioning

confidence: 99%

How to get high resolution results from sparse and coarsely sampled data

Cuyt

Lee

2020

Applied and Computational Harmonic Analysis

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“…For a general matrix pencil (A, B) this problem is known as the rectangular generalized eigenvalue problem [14] [32]. Recently, several efficient algorithms have been proposed for finding all d local minima of (A − x B) y 2 [3] [5]. Here, we compare these algorithms to our RQ methods.…”

Section: Comparison To Rectangular Generalized Eigenvalue Methodsmentioning

confidence: 99%

Rayleigh Quotient Methods for Estimating Common Roots of Noisy Univariate Polynomials

Stegeman

Lathauwer

2018

Computational Methods in Applied Mathematics

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The problem is considered of approximately solving a system of univariate polynomials with one or more common roots and its coefficients corrupted by noise. The goal is to estimate the underlying common roots from the noisy system. Symbolic algebra methods are not suitable for this. New Rayleigh quotient methods are proposed and evaluated for estimating the common roots. Using tensor algebra, reasonable starting values for the Rayleigh quotient methods can be computed. The new methods are compared to Gauss–Newton, solving an eigenvalue problem obtained from the generalized Sylvester matrix, and finding a cluster among the roots of all polynomials. In a simulation study it is shown that Gauss–Newton and a new Rayleigh quotient method perform best, where the latter is more accurate when other roots than the true common roots are close together.

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“…In the presence of noise, the linear system (11) is solved in the least squares sense for the α i . We can also solve the generalized eigenvalue problem (10) in a least squares sense [21], where the Hankel matrices defined by (8) are enlarged to dimension (M − n) × n to utilize all the available samples f 0 , . .…”

Section: A Generalized Eigenvalue Approachmentioning

confidence: 99%

“…Note that any one-dimensional exponential analysis solver could be used on the first ULA. Examples are MUSIC [4], ESPRIT [5], the matrix pencil method [6], simultaneous QR factorization [20], a generalized overdetermined eigenvalue solver [21] and the approximate Prony method [22]- [24].…”

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confidence: 99%

Regular Sparse Array Direction of Arrival Estimation in One Dimension

Knaepkens

Cuyt

Lee

et al. 2020

IEEE Trans. Antennas Propagat.

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Traditionally regularly spaced antenna arrays follow the spatial Nyquist criterion to guarantee an unambiguous analysis. We present a novel technique that makes use of two sparse non-Nyquist regularly spaced antenna arrays, where one of the arrays is just a shifted version of the other. The method offers several advantages over the use of traditional dense Nyquistspaced arrays, while maintaining a comparable algorithmic complexity for the analysis. Among the advantages we mention: an improved resolution for the same number of receivers and reduced mutual coupling effects between the receivers, both due to the increased separation between the antennas. Because of a shared structured linear system of equations between the two arrays, as a consequence of the shift between the two, the analysis of both is automatically paired, thereby avoiding a computationally expensive matching step as is required in the use of so-called co-prime arrays. In addition, an easy validation step allows to automatically detect the precise number of incoming signals, which is usually considered a difficult issue. At the same time, the validation step improves the accuracy of the retrieved results and eliminates unreliable results in the case of noisy data. The performance of the proposed method is illustrated with respect to the influence of noise as well to the effect of mutual coupling. Index Terms-Array Antennas, Direction of Arrival Estimation, Sparse Arrays. I. INTRODUCTION D IRECTION of arrival (DOA) estimation, using array antenna systems, is a topic of increasing interest in a variety of applications including radar, remote sensing, radio frequency interference mitigation, and smart wireless networks [1]-[3]. One of the most well-known limitations in regularly spaced antenna array systems is, arguably, the requirement that the elements should have spacing closer than a half-wavelength (the spatial Nyquist criterion) in order to avoid aliasing resulting in ambiguous arrival angle estimates. Unique results can be obtained for larger spacings if a limited range of near-broadside receiving angles are considered, or if the antenna element patterns exhibit zeros in the end-fire directions, but in general this still limits the allowable spacing to distances close to the Nyquist limit.

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Solving Overdetermined Eigenvalue Problems

Cited by 12 publications

References 23 publications

How to get high resolution results from sparse and coarsely sampled data

How to get high resolution results from sparse and coarsely sampled data

Rayleigh Quotient Methods for Estimating Common Roots of Noisy Univariate Polynomials

Regular Sparse Array Direction of Arrival Estimation in One Dimension

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