36 research outputs found

    Characterization and computation of canonical tight windows for Gabor frames

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    Let (gnm)n,m∈Z(g_{nm})_{n,m\in Z} be a Gabor frame for L2(R)L_2(R) for given window gg. We show that the window h0=S−1/2gh^0=S^{-1/2} g that generates the canonically associated tight Gabor frame minimizes ∥g−h∥\|g-h\| among all windows hh generating a normalized tight Gabor frame. We present and prove versions of this result in the time domain, the frequency domain, the time-frequency domain, and the Zak transform domain, where in each domain the canonical h0h^0 is expressed using functional calculus for Gabor frame operators. Furthermore, we derive a Wiener-Levy type theorem for rationally oversampled Gabor frames. Finally, a Newton-type method for a fast numerical calculation of \ho is presented. We analyze the convergence behavior of this method and demonstrate the efficiency of the proposed algorithm by some numerical examples

    Approximation of dual Gabor frames, window decay, and wireless communications

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    We consider three problems for Gabor frames that have recently received much attention. The first problem concerns the approximation of dual Gabor frames in L2(R)L_2(R) by finite-dimensional methods. Utilizing Wexler-Raz type duality relations we derive a method to approximate the dual Gabor frame, that is much simpler than previously proposed techniques. Furthermore it enables us to give estimates for the approximation rate when the dimension of the finite model approaches infinity. The second problem concerns the relation between the decay of the window function gg and its dual γ\gamma. Based on results on commutative Banach algebras and Laurent operators we derive a general condition under which the dual γ\gamma inherits the decay properties of gg. The third problem concerns the design of pulse shapes for orthogonal frequency division multiplex (OFDM) systems for time- and frequency dispersive channels. In particular, we provide a theoretical foundation for a recently proposed algorithm to construct orthogonal transmission functions that are well localized in the time-frequency plane

    Rates of convergence for the approximation of dual shift-invariant systems in l2(Z)l_2(Z)

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    A shift-invariant system is a collection of functions {gm,n}\{g_{m,n}\} of the form gm,n(k)=gm(k−an)g_{m,n}(k) = g_m(k-an). Such systems play an important role in time-frequency analysis and digital signal processing. A principal problem is to find a dual system γm,n(k)=γm(k−an)\gamma_{m,n}(k) = \gamma_m(k-an) such that each function ff can be written as f=∑gm,nf = \sum g_{m,n}. The mathematical theory usually addresses this problem in infinite dimensions (typically in L2(R)L_2(R) or l2(Z)l_2(Z)), whereas numerical methods have to operate with a finite-dimensional model. Exploiting the link between the frame operator and Laurent operators with matrix-valued symbol, we apply the finite section method to show that the dual functions obtained by solving a finite-dimensional problem converge to the dual functions of the original infinite-dimensional problem in l2(Z)l_2(Z). For compactly supported gm,ng_{m,n} (FIR filter banks) we prove an exponential rate of convergence and derive explicit expressions for the involved constants. Further we investigate under which conditions one can replace the discrete model of the finite section method by the periodic discrete model, which is used in many numerical procedures. Again we provide explicit estimates for the speed of convergence. Some remarks on tight frames complete the paper

    Parity-check matrix calculation for paraunitary oversampled DFT filter banks

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    International audienceOversampled filter banks, interpreted as error correction codes, were recently introduced in the literature. We here present an efficient calculation and implementation of the parity-check polynomial matrices for oversampled DFT filter banks. If desired, the calculation of the partity-check polynomials can be performed as part of the prototype filter design procedure. We compare our method to those previously presented in the literature

    Designing Gabor windows using convex optimization

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    Redundant Gabor frames admit an infinite number of dual frames, yet only the canonical dual Gabor system, constructed from the minimal l2-norm dual window, is widely used. This window function however, might lack desirable properties, e.g. good time-frequency concentration, small support or smoothness. We employ convex optimization methods to design dual windows satisfying the Wexler-Raz equations and optimizing various constraints. Numerical experiments suggest that alternate dual windows with considerably improved features can be found

    Introduction to frames

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    This survey gives an introduction to redundant signal representations called frames. These representations have recently emerged as yet another powerful tool in the signal processing toolbox and have become popular through use in numerous applications. Our aim is to familiarize a general audience with the area, while at the same time giving a snapshot of the current state-of-the-art

    Superposition frames for adaptive time-frequency analysis and fast reconstruction

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    In this article we introduce a broad family of adaptive, linear time-frequency representations termed superposition frames, and show that they admit desirable fast overlap-add reconstruction properties akin to standard short-time Fourier techniques. This approach stands in contrast to many adaptive time-frequency representations in the extant literature, which, while more flexible than standard fixed-resolution approaches, typically fail to provide efficient reconstruction and often lack the regular structure necessary for precise frame-theoretic analysis. Our main technical contributions come through the development of properties which ensure that this construction provides for a numerically stable, invertible signal representation. Our primary algorithmic contributions come via the introduction and discussion of specific signal adaptation criteria in deterministic and stochastic settings, based respectively on time-frequency concentration and nonstationarity detection. We conclude with a short speech enhancement example that serves to highlight potential applications of our approach.Comment: 16 pages, 6 figures; revised versio
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