63 research outputs found

    Sampling innovations

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    Sampling theory has prospered extensively in the last century. The elegant mathematics and the vast number of applications are the reasons for its popularity. The applications involved in this thesis are in signal processing and communications and call out to mathematical notions in Fourier theory, spectral analysis, basic linear algebra, spline and wavelet theory. This thesis is divided in two parts. Chapters 2 and 3 consider uniform sampling of non-bandlimited signals and Chapters 4, 5, and 6 treat different irregular sampling problems. In the first part we address the problem of sampling signals that are not bandlimited but are characterized as having a finite number of degrees of freedom per unit of time. These signals will be called signals with a finite rate of innovation. We show that these signals can be uniquely represented given a sufficient number of samples obtained using an appropriate sampling kernel. The number of samples must be greater or equal to the degrees of freedom of the signal; in other words, the sampling rate must be greater or equal to the rate of innovation of the signal. In particular, we derive sampling schemes for periodic and finite length streams of Diracs and piecewise polynomial signals using the sinc, the differentiated sinc and the Gaussian kernels. Sampling and reconstruction of piecewise bandlimited signals and filtered piecewise polynomials is also considered. We also derive local reconstruction schemes for infinite length piecewise polynomials with a finite "local" rate of innovation using compact support kernels such as splines. Numerical experiments on all of the reconstruction schemes are shown. The first topic of the second part of this thesis is the irregular sampling problem of bandlimited signals with unknown sampling instances. The locations of the irregular set of samples are found by treating the problem as a combinatorial optimization problem. Search methods for the locations are described and numerical simulations on a random set and a jittery set of locations are made. The second topic is the periodic nonuniform sampling problem of bandlimited signals. The irregular set of samples involved has a structure which is irregular yet periodic. We develop a fast scheme that reduces the complexity of the problem by exploiting the special pattern of the locations. The motivation for developing a fast scheme originates from the fact that the periodic nonuniform set was also considered in the sampling with unknown locations problem and that a fast search method for the locations was sought. Finally, the last topic is the irregular sampling of signals that are linearly and nonlinearly approximated using Fourier and wavelet bases. We present variants of the Papoulis Gerchberg algorithm which take into account the information given in the approximation of the signal. Numerical experiments are presented in the context of erasure correction

    Reconstruction of irregularly sampled discrete-time bandlimited signals with unknown sampling locations

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    The purpose of this paper is to develop methods that can reconstruct a bandlimited discrete-time signal from an irreg- ular set of samples at unknown locations. We define a solution to the problem using first a geometric and then an algebraic point of view. We find the locations of the irregular set of samples by treating the problem as a combinatorial optimization problem. We employ an exhaustive method and two descent methods: the random search and cyclic coordinate methods. The numerical simulations were made on three types of irregular sets of locations: random sets; sets with jitter around a uniform set; and periodic nonuniform sets. Furthermore, for the periodic nonuniform set of locations, we de- velop a fast scheme that reduces the computational complexity of the problem by exploiting the periodic nonuniform structure of the sample locations in the DFT

    A sampling theorem for periodic piecewise polynomial signals

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    We consider the problem of sampling signals which are not bandlimited, but still have a finite number of degrees of freedom per unit of time, such as, for example, piecewise polynomials. We demonstrate that by using an adequate sampling kernel and a sampling rate greater or equal to the number of degrees of freedom per unit of time, one can uniquely reconstruct such signals. This proves a sampling theorem for a wide class of signals beyond bandlimited signals. Applications of this sampling theorem can be found in signal processing, communication systems and biological systems

    Sampling and exact reconstruction of bandlimited signals with additive shot noise

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    In this correspondence, we consider sampling continuous-time periodic bandlimited signals which contain additive shot noise. The classical sampling scheme does not perfectly recover these particular nonbandlimited signals but only reconstructs a lowpass filtered approximation. By modeling the shot noise as a stream of Dirac pulses, we first show that the sum of a bandlimited signal with a stream of Dirac pulses falls into the class of signals that contain a finite rate of innovation, that is, a finite number of degrees of freedom. Second, by taking into account the degrees of freedom of the bandlimited signal in the sampling and reconstruction scheme developed previously for streams of Dirac pulses, we derive a sampling and perfect reconstruction scheme for the bandlimited signal with additive shot noise

    Robust Video Watermarking of H.264/AVC

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