On the condition number of Vandermonde matrices with pairs of nearly-colliding nodes

Kunis, Stefan; Nagel, Dominik

doi:10.1007/s11075-020-00974-x

Cited by 15 publications

(5 citation statements)

References 19 publications

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“…Moreover, we also require that the nonequispaced Fourier matrix A has full rank. Eigenvalue estimates in [22,6,9,45,43] indeed confirm that this condition is satisfied for sufficiently nice sampling sets.…”

Section: Introductionsupporting

confidence: 53%

Direct inversion of the nonequispaced fast Fourier transform

Melanie¹,

Potts²

2019

Linear Algebra and its Applications

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Various applications such as MRI, solution of PDEs, etc. need to perform an inverse nonequispaced fast Fourier transform (NFFT), i. e., compute M Fourier coefficients from given N nonequispaced data. In the present paper we consider direct methods for the inversion of the NFFT. We introduce algorithms for the setting M = N as well as for the underdetermined and overdetermined cases. For the setting M = N a direct method of complexity O(N log N ) is presented, which utilizes Lagrange interpolation and the fast summation. For the remaining cases, we use the matrix representation of the NFFT to deduce our algorithms. Thereby, we are able to compute an inverse NFFT up to a certain accuracy by means of a modified adjoint NFFT in O(M log M + N ) arithmetic operations. Finally, we show that these approaches can also be explained by means of frame approximation.

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Section: Introductionsupporting

confidence: 53%

Direct inversion of the nonequispaced fast Fourier transform

Melanie¹,

Potts²

2019

Linear Algebra and its Applications

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“…A key quantity in the mathematics of super-resolution is σ S (Φ), which crucially depends on the support geometry. Explicit lower bounds for σ S (Φ) were derived in [2], [3], [9], [20], [21], [25], [27], [28] for various support models. This paper uses the lower bound under the separated clumps model in [25].…”

Section: Comparison and Connection To Other Workmentioning

confidence: 99%

Stability and Super-resolution of MUSIC and ESPRIT for Multi-snapshot Spectral Estimation

Zhu

Gao

et al. 2021

Preprint

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This paper studies the spectral estimation problem of estimating the locations of a fixed number of point sources given multiple snapshots of Fourier measurements collected by a uniform array of sensors. We prove novel non-asymptotic stability bounds for MUSIC and ESPRIT as a function of the noise standard deviation, number of snapshots, source amplitudes, and support. Our most general result is a perturbation bound of the signal space in terms of the minimum singular value of Fourier matrices. When the point sources are located in several separated clumps, we provide an explicit upper bound of the noise-space correlation perturbation error in MUSIC and the support error in ESPRIT in terms of a Super-Resolution Factor (SRF). The upper bound for ESPRIT is then compared with a new Cramér-Rao lower bound for the clumps model. As a result, we show that ESPRIT is comparable to that of the optimal unbiased estimator(s) in terms of the dependence on noise, number of snapshots and SRF. As a byproduct of our analysis, we discover several fundamental differences between the single-snapshot and multi-snapshot problems. Our theory is validated by numerical experiments.

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“…Wireless performance studies usually employ Vandermonde matrices as mathematical channel models in applications such as system identification, harmonic analysis, direction-finding and precoding [ 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 ]. Specifically, given a particular wireless environment, the sum-rate channel capacity is better modelled by the Vandermonde matrix approach as presented in [ 20 , 21 , 22 , 23 ].…”

Section: Introductionmentioning

confidence: 99%

Sum-Rate Channel Capacity for Line-of-Sight Models

Dias

Figueiredo

Lima

et al. 2021

Sensors

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This work considers a base station equipped with an M-antenna uniform linear array and L users under line-of-sight conditions. As a result, one can derive an exact series expansion necessary to calculate the mean sum-rate channel capacity. This scenario leads to a mathematical problem where the joint probability density function (JPDF) of the eigenvalues of a Vandermonde matrix WWH are necessary, where W is the channel matrix. However, differently from the channel Rayleigh distributed, this joint PDF is not known in the literature. To circumvent this problem, we employ Taylor’s series expansion and present a result where the moments of mn are computed. To calculate this quantity, we resort to the integer partition theory and present an exact expression for mn. Furthermore, we also find an upper bound for the mean sum-rate capacity through Jensen’s inequality. All the results were validated by Monte Carlo numerical simulation.

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On the condition number of Vandermonde matrices with pairs of nearly-colliding nodes

Cited by 15 publications

References 19 publications

Direct inversion of the nonequispaced fast Fourier transform

Direct inversion of the nonequispaced fast Fourier transform

Stability and Super-resolution of MUSIC and ESPRIT for Multi-snapshot Spectral Estimation

Sum-Rate Channel Capacity for Line-of-Sight Models

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