The Eigenvalue Distribution of Discrete Periodic Time-Frequency Limiting Operators

Zhu, Zhihui; Karnik, Santhosh; Davenport, Mark A.; Romberg, Justin; Wakin, Michael B.

doi:10.1109/lsp.2017.2751578

Cited by 18 publications

(12 citation statements)

References 18 publications

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“…The eigenvalue spectrum of space-andfrequency limited Fourier operators has been studied, beginning with a series of papers by Slepian, Landau, and Pollak [26][27][28][29][30]. In the discrete case, the eigenvalue and singular value spectrum of space-and-frequency limited Discrete Fourier Transform (DFT) matrices have been studied; such matrices are submatrices formed by consecutive rows and columns of a DFT matrix [31][32][33]. The singular values of a space-and-frequency limited DFT matrix are divided into three distinct regions: (1) a region wherein the singular values are near one; (2) a transition region where the singular values decay exponentially; and (3) the remaining singular values are nearly zero.…”

Section: Preliminaries: the Single Species Casementioning

confidence: 99%

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Composition-Aware Spectroscopic Tomography

Pfister,

Bhargava,

Bresler

et al. 2019

Preprint

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Chemical imaging provides information about the distribution of chemicals within a target. When combined with structural information about the target, in situ chemical imaging opens the door to applications ranging from tissue classification to industrial process monitoring. The combination of infrared spectroscopy and optical microscopy is a powerful tool for chemical imaging of thin targets. Unfortunately, extending this technique to targets with appreciable depth is prohibitively slow.We combine confocal microscopy and infrared spectroscopy to provide chemical imaging in three spatial dimensions. Interferometric measurements are acquired at a small number of focal depths, and images are formed by solving a regularized inverse scattering problem. A low-dimensional signal model is key to our approach: we assume the target comprises a finite number of distinct chemical species. We establish conditions on the constituent spectra and the number of measurements needed for unique recovery of the target. Simulations illustrate imaging of cellular phantoms and sub-wavelength targets from noisy measurements.

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Section: Preliminaries: the Single Species Casementioning

confidence: 99%

“…The number of singular values in the first region is called the effective rank and is written r e . A direct application of Slepian-Pollak theory predicts [29,33]…”

Section: Preliminaries: the Single Species Casementioning

confidence: 99%

Composition-Aware Spectroscopic Tomography

Pfister,

Bhargava,

Bresler

et al. 2019

Preprint

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“…As we explained before, the time-frequency limiting operators in the context of the classical groups where G are the real-line, Z, and ZN were first studied by Landau, Pollak, and Slepian who wrote a series of papers regarding the dimensionality of time-limited signals that are approximately band-limited (or vice versa) [36,37,56,58,60] (see also [57,59] for concise overviews of this body of work). After that, a set of results concerning the number of eigenvalues within the transition region (0, 1) have been established in [13,26,31,38,44,76]. which will be reviewed in detail in the following remarks.…”

Section: Eigenvalue Distribution Of Time-frequency Limiting Operatorsmentioning

confidence: 99%

Time-Limited Toeplitz Operators on Abelian Groups: Applications in Information Theory and Subspace Approximation

Zhu,

Wakin

2017

Preprint

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Toeplitz operators are fundamental and ubiquitous in signal processing and information theory as models for linear, time-invariant (LTI) systems. Due to the fact that any practical system can access only signals of finite duration, time-limited restrictions of Toeplitz operators are naturally of interest. To provide a unifying treatment of such systems working on different signal domains, we consider timelimited Toeplitz operators on locally compact abelian groups with the aid of the Fourier transform on these groups. In particular, we survey existing results concerning the relationship between the spectrum of a time-limited Toeplitz operator and the spectrum of the corresponding non-time-limited Toeplitz operator. We also develop new results specifically concerning the eigenvalues of time-frequency limiting operators on locally compact abelian groups. Applications of our unifying treatment are discussed in relation to channel capacity and in relation to representation and approximation of signals.

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“…The associated properties of these sequences are studied in [6,[36][37][38][39]. For most of the practical applications of digital signal processing where we often encounter with finite duration signals, it is desired to have simultaneously concentrated discrete window with finite support.…”

Section: Introductionmentioning

confidence: 99%

“…These P‐DPSSs are band‐limited in the interval

false(- L, L false)

while maximally energy concentrated in the time interval

false(- M / 2, M / 2 - 1 false)

. Similar to PSWFs and DPSSs, P‐DPSSs can be obtained by finding the eigenvector of associated eigenfunction equation [35]

\sum_{m = - M / 2}^{M / 2 - 1} \frac{\sin false(false[n - m false] false(2 L + 1 false) π / N false)}{N \sin false(false[n - m false] π / N false)} ϕ_{M} [m] = λ (M, B) ϕ_{M} [n]

The associated properties of these sequences are studied in [6, 36–39].…”

Section: Introductionmentioning

confidence: 99%

Achievable simultaneous time and frequency domain energy concentration for finite length sequences

Singh

Pradhan

2019

IET signal process.

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For numerous applications in the field of signal processing, it is desired to design a compact window that can simultaneously concentrate maximum energy in finite time interval and frequency band. Although zero-order discrete prolate spheroidal sequence (DPSS) meets this requirement, it is of infinite support. Limiting this sequence to finite support no longer guarantees the optimality property. This study aims at designing a discrete finite length window that can maximise the energy simultaneously in narrow time interval and frequency band. A multi-objective optimisation approach is adopted to obtain the upper bound of maximum achievable time and frequency domain energy concentrations for finite length sequences. The optimal sequence thus obtained is termed as the optimal window with finite support (OWFS), and its various associated properties are discussed. It is shown analytically that as the support of OWFS approaches infinity, it converges to zero-order DPSS. In order to illustrate its optimality, the compactness of the proposed OWFS is compared with those of various window functions. Extending the proposed OWFS, this study also discusses the formulation and associated properties of the zero-order periodic DPSS. Further, the closed form expression for the upper bound of achievable time and frequency domain energy concentrations is also derived.

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The Eigenvalue Distribution of Discrete Periodic Time-Frequency Limiting Operators

Cited by 18 publications

References 18 publications

Composition-Aware Spectroscopic Tomography

Composition-Aware Spectroscopic Tomography

Time-Limited Toeplitz Operators on Abelian Groups: Applications in Information Theory and Subspace Approximation

Achievable simultaneous time and frequency domain energy concentration for finite length sequences

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