The discrete prolate spheroidal sequences (DPSS's) provide an efficient representation for discrete signals that are perfectly timelimited and nearly bandlimited. Due to the high computational complexity of projecting onto the DPSS basis -also known as the Slepian basis -this representation is often overlooked in favor of the fast Fourier transform (FFT). We show that there exist fast constructions for computing approximate projections onto the leading Slepian basis elements. The complexity of the resulting algorithms is comparable to the FFT, and scales favorably as the quality of the desired approximation is increased. In the process of bounding the complexity of these algorithms, we also establish new nonasymptotic results on the eigenvalue distribution of discrete time-frequency localization operators. We then demonstrate how these algorithms allow us to efficiently compute the solution to certain least-squares problems that arise in signal processing. We also provide simulations comparing these fast, approximate Slepian methods to exact Slepian methods as well as the traditional FFT based methods. arXiv:1611.04950v2 [math.NA]
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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