Sampling Approximations for Time- and Bandlimiting

Hogan, Jeffrey A.; Izu, Scott; Lakey, Joseph D.

doi:10.1007/bf03549526

Cited by 7 publications

(3 citation statements)

References 30 publications

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“…In order to achieve a similar bound, but for more general regions, in Proposition 4.4 in the next section, we consider the projection of a function f onto the best-concentrated eigenfunctions of a localization operator and derive the following estimate. We note that approximations of bandlimited functions via projections onto eigenspaces of approximately time-and bandlimited functions were presented in [27,17].…”

Section: Time-frequency Concentration Via the Stftmentioning

confidence: 99%

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Sampling time-frequency localized functions and constructing localized time-frequency frames

Dörfler

Velasco

2016

Eur. J. Appl. Math

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We study functions whose time-frequency content are concentrated in a compact region in phase space using time-frequency localization operators as a main tool. We obtain approximation inequalities for such functions using a finite linear combination of eigenfunctions of these operators, as well as a local Gabor system covering the region of interest. These would allow the construction of modified time-frequency dictionaries concentrated in the region.

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Section: Time-frequency Concentration Via the Stftmentioning

confidence: 99%

“…In [10], the authors presented an approximate reconstruction of f from the given samples using approximate projection H µ Nµ in Remark 4.7(b). In particular, the following error (17) f…”

Section: 2mentioning

confidence: 99%

Sampling time-frequency localized functions and constructing localized time-frequency frames

Dörfler

Velasco

2016

Eur. J. Appl. Math

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“…Interpolation with sinc functions is then required to obtain sampled values of the PSWF at points other than the Shannon sampling points. Hogan and Lakey extended the work of Walter and Shen to establish error estimates of the eigenfunction samples and on the matrices used to generate these sample approximations [39].…”

Section: Introductionmentioning

confidence: 99%

Evaluation of the Prolate Spheroidal Wavefunctions via a Discrete-Time Fourier Transform Based Approach

Baddour,

Sun

2023

Symmetry

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Computation of prolate spheroidal wavefunctions (PSWFs) is notoriously difficult and time consuming. This paper applies operator theory to the discrete Fourier transform (DFT) to address the problem of computing PSWFs. The problem is turned into an infinite dimensional matrix operator eigenvalue problem, which we recognize as being the definition of the DPSSs. Truncation of the infinite matrix leads to a finite dimensional matrix eigenvalue problem which in turn yields what is known as the Slepian basis. These discrete-valued Slepian basis vectors can then be used as (approximately) discrete time evaluations of the PSWFs. Taking an inverse Fourier transform further demonstrates that continuous PSWFs can be reconstructed from the Slepian basis. The feasibility of this approach is shown via theoretical derivations followed by simulations to consider practical aspects. Simulations demonstrate that the level of errors between the reconstructed Slepian basis approach and true PSWFs are low when the orders of the eigenvectors are low but can become large when the orders of the eigenvectors are high. Accuracy can be increased by increasing the number of points used to generate the Slepian basis. Users need to balance accuracy with computational cost. For large time-bandwidth product PSWFs, the number of Slepian basis points required increases for a reconstruction to reach the same error as for low time-bandwidth products. However, when the time-bandwidth products increase and reach maximum concentration, the required number of points to achieve a given error level achieves steady state values. Furthermore, this method of reconstructing the PSWF from the Slepian basis can be more accurate when compared to the Shannon sampling approach and traditional quadrature approach for large time-bandwidth products. Finally, since the Slepian basis represents the (approximate) sampled values of PSWFs, when the number of points is sufficiently large, the reconstruction process can be omitted entirely so that the Slepian vectors can be used directly, without a reconstruction step.

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An Overview of Time and Multiband Limiting

Lakey

2012

Applied and Numerical Harmonic Analysis

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Sampling Approximations for Time- and Bandlimiting

Cited by 7 publications

References 30 publications

Sampling time-frequency localized functions and constructing localized time-frequency frames

Sampling time-frequency localized functions and constructing localized time-frequency frames

Evaluation of the Prolate Spheroidal Wavefunctions via a Discrete-Time Fourier Transform Based Approach

An Overview of Time and Multiband Limiting

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