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Robust Uncertainty Principles: Exact Signal Reconstruction From Highly Incomplete Frequency Information
, 2006
"... This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discretetime signal and a randomly chosen set of frequencies. Is it possible to reconstruct from the partial knowledge of its Fourier coefficients on the set? A typical result of this pa ..."
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Cited by 1304 (42 self)
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This paper considers the model problem of reconstructing an object from incomplete frequency samples. Consider a discretetime signal and a randomly chosen set of frequencies. Is it possible to reconstruct from the partial knowledge of its Fourier coefficients on the set? A typical result of this paper is as follows. Suppose that is a superposition of spikes @ Aa @ A @ A obeying @�� � A I for some constant H. We do not know the locations of the spikes nor their amplitudes. Then with probability at least I @ A, can be reconstructed exactly as the solution to the I minimization problem I aH @ A s.t. ” @ Aa ” @ A for all
Asymptotic singular value decay of timefrequency localization operators
 Wavelet Applications in Signal and Image Processing II
, 1994
"... The Weyl correspondence is a convenient way to define a broad class of timefrequency localization operators. Given a region Ω in the timefrequency plane R 2 and given an appropriate µ, the Weyl correspondence can be used to construct an operator L(Ω, µ) which essentially localizes the timefrequen ..."
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Cited by 6 (2 self)
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The Weyl correspondence is a convenient way to define a broad class of timefrequency localization operators. Given a region Ω in the timefrequency plane R 2 and given an appropriate µ, the Weyl correspondence can be used to construct an operator L(Ω, µ) which essentially localizes the timefrequency content of a signal on Ω. Different choices of µ provide different interpretations of localization. Empirically, each such localization operator has the following singular value structure: there are several singular values close to 1, followed by a sharp plunge in values, with a final asymptotic decay to zero. The exact quantification of these qualitative observations is known only for a few specific choices of Ω and µ. In this paper we announce a general result which bounds the asymptotic decay rate of the singular values of any L(Ω, µ) in terms of integrals of χ Ω ∗ ˜µ  2 and (χ Ω ∗ ˜µ) ∧  2 outside squares of increasing radius, where ˜µ(a, b) = µ(−a, −b). More generally, this result applies to all operators L(σ, µ) allowing window functions σ in place of the characteristic functions χ Ω. We discuss the motivation and implications of this result. We also sketch the philosophy of proof, which involves the construction of an approximating operator through the technology of Gabor frames—overcomplete systems which allow basislike expansions and Plancherellike formulas, but which are not bases and are not orthogonal systems.
Slepian Functions and Their Use in Signal Estimation and Spectral Analysis
, 909
"... It is a wellknown fact that mathematical functions that are timelimited (or spacelimited) cannot be simultaneously bandlimited (in frequency). Yet the finite precision of measurement and computation unavoidably bandlimits our observation and modeling scientific data, and we often only have access t ..."
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Cited by 1 (0 self)
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It is a wellknown fact that mathematical functions that are timelimited (or spacelimited) cannot be simultaneously bandlimited (in frequency). Yet the finite precision of measurement and computation unavoidably bandlimits our observation and modeling scientific data, and we often only have access to, or are only interested in, a study area that is temporally or spatially bounded. In the geosciences we may be interested in spectrally modeling a time series defined only on a certain interval, or we may want to characterize a specific geographical area observed using an effectively bandlimited measurement device. It is clear that analyzing and representing scientific data of this kind will be facilitated if a basis of functions can be found that are “spatiospectrally” concentrated, i.e. “localized ” in both domains at the same time. Here, we give a theoretical overview of one particular approach to this “concentration ” problem, as originally proposed for time series by Slepian and coworkers, in the 1960s. We show how this framework leads to practical algorithms and statistically performant methods for the analysis of signals and their power spectra in one and two dimensions, and on the surface of a sphere.