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The aim of this work is to present several topics in time-frequency analysis as subjects in abelian group theory. The algebraic point of view pre­ dominates as questions of convergence are not considered. Our approach emphasizes the unifying role played by group structures on the development of theory and algorithms. This book consists of two main parts. The first treats Weyl-Heisenberg representations over finite abelian groups and the second deals with mul­ tirate filter structures over free abelian groups of finite rank. In both, the methods are dimensionless and coordinate-free and apply to one and multidimensional problems. The selection of topics is not motivated by mathematical necessity but rather by simplicity. We could have developed Weyl-Heisenberg theory over free abelian groups of finite rank or more generally developed both topics over locally compact abelian groups. However, except for having to dis­ cuss conditions for convergence, Haar measures, and other standard topics from analysis the underlying structures would essentially be the same. A re­ cent collection of papers [17] provides an excellent review of time-frequency analysis over locally compact abelian groups. A further reason for limiting the scope of generality is that our results can be immediately applied to the design of algorithms and codes for time­ frequency processing.

Signal processing --- Time-series analysis --- Frequency spectra --- Mathematics --- 517.518 --- 681.3*G12 --- Metric theory of functions --- Approximation: chebyshev; elementary function; least squares; linear approximation; minimax approximation and algorithms; nonlinear and rational approximation; spline and piecewise polynomial approximation (Numerical analysis) --- 681.3*G12 Approximation: chebyshev; elementary function; least squares; linear approximation; minimax approximation and algorithms; nonlinear and rational approximation; spline and piecewise polynomial approximation (Numerical analysis) --- 517.518 Metric theory of functions --- Abelian groups --- Analysis of time series --- Autocorrelation (Statistics) --- Harmonic analysis --- Mathematical statistics --- Probabilities --- Spectra, Frequency --- Spectrum, Frequency --- Spectrum analysis --- Commutative groups --- Group theory --- Computer mathematics. --- Signal processing. --- Image processing. --- Speech processing systems. --- Applied mathematics. --- Engineering mathematics. --- Computational Mathematics and Numerical Analysis. --- Signal, Image and Speech Processing. --- Computational Science and Engineering. --- Applications of Mathematics. --- Processing, Signal --- Information measurement --- Signal theory (Telecommunication) --- Engineering --- Engineering analysis --- Mathematical analysis --- Computational linguistics --- Electronic systems --- Information theory --- Modulation theory --- Oral communication --- Speech --- Telecommunication --- Singing voice synthesizers --- Pictorial data processing --- Picture processing --- Processing, Image --- Imaging systems --- Optical data processing --- Computer mathematics --- Electronic data processing --- Abelian groups. --- Frequency spectra. --- Signal processing - Mathematics

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This self-contained book develops theory and algorithms leading to systematic sequence design in time-frequency space. The primary tool used is the Zak transform, which provides sparse representation for the Fourier transform, convolution, and correlation. Using this multi-dimensional representation, the authors construct a large class of sequence sets satisfying pairwise ideal correlation. The complex algebraic analysis of sequences is replaced by an elegant and efficient geometric analysis of images, whose advantage is realized as an N to N! increase in the number of ideal sequence sets. Topics and features: * Mathematical development of the theory is illustrated with many examples. * Standard communication theory and Zak space methods are numerically compared. * Application areas covered include pulse radar and sonar, multi-beam radar and sonar imaging systems, remote identification of dielectrics, and code division multiple-access communication. * Background is provided in two introductory chapters on matrix algebra, tensor products, and permutation groups. * A list of open problems is presented and directions for further research are discussed. Ideal Sequence Design in Time-Frequency Space is an excellent reference text for graduate students, researchers, and engineers interested in radar, sonar, and communication systems. The work may also be used as a supplementary textbook for a graduate course or seminar on sequence design in time-frequency space.

Fourier transformations. --- Frequencies of oscillating systems. --- Radar. --- Sequency theory. --- Sonar. --- Telecommunication systems --Design and construction. --- Sequency theory --- Fourier transformations --- Frequencies of oscillating systems --- Radar --- Sonar --- Telecommunication systems --- Telecommunications --- Electrical & Computer Engineering --- Engineering & Applied Sciences --- Design and construction --- Asdic --- Echo ranging --- Sound navigation ranging --- Frequencies of vibrating systems --- Frequency analysis (Dynamics) --- Frequency of oscillation --- Frequency of vibration --- Vibration frequencies --- Transformations, Fourier --- Transforms, Fourier --- Design and construction. --- Engineering. --- Harmonic analysis. --- Fourier analysis. --- Applied mathematics. --- Engineering mathematics. --- Algorithms. --- Electrical engineering. --- Signal, Image and Speech Processing. --- Fourier Analysis. --- Communications Engineering, Networks. --- Applications of Mathematics. --- Abstract Harmonic Analysis. --- Detectors --- Electronics in navigation --- Signals and signaling, Submarine --- Ultrasonic equipment --- Underwater acoustics --- Electronic systems --- Pulse techniques (Electronics) --- Radio --- Remote sensing --- Oscillations --- Vibration --- Fourier analysis --- Transformations (Mathematics) --- Signal theory (Telecommunication) --- Telecommunication --- Walsh functions --- Telecommunication. --- Mathematics. --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Banach algebras --- Calculus --- Mathematical analysis --- Mathematics --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Math --- Science --- Electric communication --- Mass communication --- Telecom --- Telecommunication industry --- Communication --- Information theory --- Telecommuting --- Algorism --- Algebra --- Arithmetic --- Analysis, Fourier --- Foundations --- Signal processing. --- Image processing. --- Speech processing systems. --- Engineering --- Engineering analysis --- Electric engineering --- Computational linguistics --- Modulation theory --- Oral communication --- Speech --- Singing voice synthesizers --- Pictorial data processing --- Picture processing --- Processing, Image --- Imaging systems --- Optical data processing --- Processing, Signal --- Information measurement

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Fourier transformations --- Fourier analysis --- Numerical analysis --- Fourier, Transformations de. --- Fourier, Analyse de. --- Analyse numérique --- Fourier transformations.