The fast Slepian transform

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

doi:10.1016/j.acha.2017.07.005

Cited by 34 publications

(36 citation statements)

References 35 publications

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“…Slepian [4] first provided an asymptotic expression for the DPSS eigenvalues. In [16], we recently provided a non-asymptotic result for the distribution of the DPSS eigenvalues (which improves upon a previous result in [9]). …”

Section: Introductionmentioning

confidence: 63%

The Eigenvalue Distribution of Discrete Periodic Time-Frequency Limiting Operators

Zhu

Karnik

Davenport

et al. 2018

IEEE Signal Process. Lett.

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Abstract-Bandlimiting and timelimiting operators play a fundamental role in analyzing bandlimited signals that are approximately timelimited (or vice versa). In this paper, we consider a time-frequency (in the discrete Fourier transform (DFT) domain) limiting operator whose eigenvectors are known as the periodic discrete prolate spheroidal sequences (PDPSSs). We establish new nonasymptotic results on the eigenvalue distribution of this operator. As a byproduct, we also characterize the eigenvalue distribution of a set of submatrices of the DFT matrix, which is of independent interest.Keywords-periodic discrete prolate spheroidal sequences, partial discrete Fourier transform matrix, eigenvalue distribution, timefrequency analysis. I. INTRODUCTIONA series of seminal papers by Landau, Pollak, and Slepian explore the degree to which a bandlimited signal can be approximately timelimited [1]- [4]. The key analysis involves a very special class of functions-the prolate spheroidal wave functions (PSWFs) in the continuous case and the discrete prolate spheroidal sequences (DPSSs) in the discrete case. These functions are the eigenvectors of the corresponding composition of bandlimiting and timelimiting operators and provide a natural basis to use in a wide variety of applications involving bandlimiting and timelimiting [1]- [7]; see also [8] and the references therein for applications using PWSFs and see [9] and the references therein for applications using DPSSs.The periodic discrete prolate spheroidal sequences (PDPSSs), introduced by Jain and Ranganath [10] and Grünbaum [11], are the counterparts of the PSWFs in the finite dimensional case. The PDPSSs are the finite-length vectors whose discrete Fourier transform (DFT) is most concentrated in a given bandwidth. Being simultaneously concentrated in the time and frequency domains makes these vectors useful in a number of signal processing applications. For example, Jain and Ranganath used PDPSSs for extrapolation and spectral estimation of periodic discretetime signals [10] . PDPSSs were also used for limited-angle reconstruction in tomography [11], for Fourier extension [12], and in [13], the bandpass PDPSSs were used as a numerical approximation to the bandpass PSWFs for studying synchrony in sampled EEG signals.

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Section: Introductionmentioning

confidence: 63%

The Eigenvalue Distribution of Discrete Periodic Time-Frequency Limiting Operators

Zhu

Karnik

Davenport

et al. 2018

IEEE Signal Process. Lett.

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“…Consequently, it can be shown [2] [2,13,16,17].) For any W ∈ (0, 1 2 ), N ∈ N, and ∈ (0, 1 2 ), we have…”

Section: Definitionsmentioning

confidence: 97%

“…Unfortunately, unlike the DFT which can be computed efficiently with the FFT algorithm, there exists no algorithm that can efficiently compute the DPSS representation for a very large signal. Recently, we proposed [16] a fast Slepian transform (FST), a fast method for computing approximate projections onto the leading DPSS vectors and compressing a signal to the corresponding low dimension. Despite its favorable properties, the fast algorithm presented in [16] did not correspond to an orthogonal projection.…”

Section: Introductionmentioning

confidence: 99%

“…Recently, we proposed [16] a fast Slepian transform (FST), a fast method for computing approximate projections onto the leading DPSS vectors and compressing a signal to the corresponding low dimension. Despite its favorable properties, the fast algorithm presented in [16] did not correspond to an orthogonal projection. In this paper, we illustrate an alternative orthonormal basis that provides an approximate but sufficiently accurate representation of the subspace spanned by the leading DPSS vectors and compactly captures most of the energy in oversampled bandlimited signals.…”

Section: Introductionmentioning

confidence: 99%

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ROAST: Rapid Orthogonal Approximate Slepian Transform

Zhu

Karnik

Wakin

et al. 2018

IEEE Trans. Signal Process.

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In this paper, we provide a Rapid Orthogonal Approximate Slepian Transform (ROAST) for the discrete vector that one obtains when collecting a finite set of uniform samples from a baseband analog signal. The ROAST offers an orthogonal projection which is an approximation to the orthogonal projection onto the leading discrete prolate spheroidal sequence (DPSS) vectors (also known as Slepian basis vectors). As such, the ROAST is guaranteed to accurately and compactly represent not only oversampled bandlimited signals but also the leading DPSS vectors themselves. Moreover, the subspace angle between the ROAST subspace and the corresponding DPSS subspace can be made arbitrarily small. The complexity of computing the representation of a signal using the ROAST is comparable to the FFT, which is much less than the complexity of using the DPSS basis vectors. We also give non-asymptotic results to guarantee that the proposed basis not only provides a very high degree of approximation accuracy in a mean squared error sense for bandlimited sample vectors, but also that it can provide high-quality approximations of all sampled sinusoids within the band of interest. Z. Zhu is with the Center for

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“…It makes sense only if ε is not too small. Recently, in [11], the authors have given the following asymptotic estimate of N (W, ε), which is valid for small values of ε,…”

Section: The Spectrum Associated With the Dpswf's: Behaviour And Decamentioning

confidence: 99%

Discrete Prolate Spheroidal Wave Functions: Further Spectral Analysis and Some Related Applications

2020

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For fixed W ∈ 0, 1 2 and positive integer N ≥ 1, the discrete prolate spheroidal wave functions (DPSWFs), denoted by U N k,W , 0 ≤ k ≤ N − 1 form the set of the eigenfunctions of the positive and finite rank integral operator Q N,W , defined on L 2 (−1/2, 1/2), with kernel K N (x, y) = sin(N π(x−y)) sin(π(x−y)) 1 [−W,W ] (y). It is well known that the DPSWF's have a wide range of classical as well as recent signal processing applications. These applications rely heavily on the properties of the DPSWFs as well as the behaviour of their eigenvalues λ k,N (W ). In his pioneer work [17], D. Slepian has given the properties of the DPSWFs, their asymptotic approximations as well as the asymptotic behaviour and asymptotic decay rate of these eigenvalues. In this work, we give further properties as well as new non-asymptotic decay rates of the spectrum of the operator Q N,W . In particular, we show that each eigenvalue λ k,N (W ) is up to a small constant bounded above by the corresponding eigenvalue, associated with the classical prolate spheroidal wave functions (PSWFs). Then, based on the well established results concerning the distribution and the decay rates of the eigenvalues associated with the PSWFs, we extend these results to the eigenvalues λ k,N (W ). Also, we show that the DPSWFs can be used for the approximation of classical band-limited functions and they are well adapted for the approximation of functions from periodic Sobolev spaces. Finally, we provide the reader with some numerical examples that illustrate the different results of this work.2010 Mathematics Subject Classification. Primary 42A38, 15B52. Secondary 60F10, 60B20.Key words and phrases. Band-limited sequences, eigenvalues and eigenfunctions, discrete prolate spheroidal wave functions and sequences, eigenvalues distribution and decay rate.

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The fast Slepian transform

Cited by 34 publications

References 35 publications

The Eigenvalue Distribution of Discrete Periodic Time-Frequency Limiting Operators

The Eigenvalue Distribution of Discrete Periodic Time-Frequency Limiting Operators

ROAST: Rapid Orthogonal Approximate Slepian Transform

Discrete Prolate Spheroidal Wave Functions: Further Spectral Analysis and Some Related Applications

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