51,883 research outputs found

    Band-limited functions and the sampling theorem

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    The definition of band-limited functions (and random processes) is extended to include functions and processes which do not possess a Fourier integral representation. This definition allows a unified approach to band-limited functions and band-limited (but not necessarily stationary) processes. The sampling theorem for functions and processes which are band-limited under the extended definition is derived

    Sampling Theorem and Discrete Fourier Transform on the Riemann Sphere

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    Using coherent-state techniques, we prove a sampling theorem for Majorana's (holomorphic) functions on the Riemann sphere and we provide an exact reconstruction formula as a convolution product of NN samples and a given reconstruction kernel (a sinc-type function). We also discuss the effect of over- and under-sampling. Sample points are roots of unity, a fact which allows explicit inversion formulas for resolution and overlapping kernel operators through the theory of Circulant Matrices and Rectangular Fourier Matrices. The case of band-limited functions on the Riemann sphere, with spins up to JJ, is also considered. The connection with the standard Euler angle picture, in terms of spherical harmonics, is established through a discrete Bargmann transform.Comment: 26 latex pages. Final version published in J. Fourier Anal. App

    Error Reduction In Two-Dimensional Pulse-Area Modulation, With Application To Computer-Generated Transparencies

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    The paper deals with the analysis of computer-generated half-tone transparencies that are realised as a regular array of area-modulated unit-height pulses and with the help of which we want to generate [via low-pass filtering] band-limited space functions by optical means. The mathematical basis for such transparencies is, of course, the well-known sampling theorem [1], which says that a band-limited func-tion y(x), say, [with x a two-dimensional spatial column vector] can be generated by properly low-pass filtering a regular array of Dirac functions whose weights are proportional to the required sample values yey(Xm) [with X the sampling matrix and m=(mi,m2)t an integer-valued column vector; the superscript t denotes transposition]

    Exact solutions for a class of matrix Riemann-Hilbert problems

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    Consider the matrix Riemann-Hilbert problem. In contrast to scalar Riemann-Hilbert problems, a general matrix Riemann-Hilbert problem cannot be solved in term of Sokhotskyi-Plemelj integrals. As far as the authors know,  the only known exact solutions known are for a class of  matrix Riemann-Hilbert problems with commutative and factorable kernel, and a class of homogeneous problems. This article employs the well known Shannon sampling theorem to provide exact solutions for a class of matrix Riemann-Hilbert problems. We consider matrix Riemann-Hilbert problems in which all the partial indices are zero and the logarithm of the components of the kernels and their nonhomogeneous vectors are functions of exponential type (equivalently, band-limited functions). Then, we develop exact solutions for such matrix Riemann-Hilbert problems. Several well known examples along with a remark on the case of functions not of exponential type  are given.

    Average sampling of band-limited stochastic processes

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    We consider the problem of reconstructing a wide sense stationary band-limited process from its local averages taken either at the Nyquist rate or above. As a result, we obtain a sufficient condition under which average sampling expansions hold in mean square and for almost all sample functions. Truncation and aliasing errors of the expansion are also discussed

    A novel sampling theorem on the rotation group

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    We develop a novel sampling theorem for functions defined on the three-dimensional rotation group SO(3) by connecting the rotation group to the three-torus through a periodic extension. Our sampling theorem requires 4L34L^3 samples to capture all of the information content of a signal band-limited at LL, reducing the number of required samples by a factor of two compared to other equiangular sampling theorems. We present fast algorithms to compute the associated Fourier transform on the rotation group, the so-called Wigner transform, which scale as O(L4)O(L^4), compared to the naive scaling of O(L6)O(L^6). For the common case of a low directional band-limit NN, complexity is reduced to O(NL3)O(N L^3). Our fast algorithms will be of direct use in speeding up the computation of directional wavelet transforms on the sphere. We make our SO3 code implementing these algorithms publicly available.Comment: 5 pages, 2 figures, minor changes to match version accepted for publication. Code available at http://www.sothree.or
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