Sampling and reconstruction of operators

We study the recovery of operators with bandlimited Kohn-Nirenberg symbol from the action of such operators on a weighted impulse train, a procedure we refer to as operator sampling. Kailath, and later Kozek and the authors have shown that operator sampling is possible if the symbol of the operator is bandlimited to a set with area less than one. In this paper we develop explicit reconstruction formulas for operator sampling that generalize reconstruction formulas for bandlimited functions. We give necessary and sufficient conditions on the sampling rate that depend on size and geometry of the bandlimiting set. Moreover, we show that under mild geometric conditions, classes of operators bandlimited to an unknown set of area less than one-half permit sampling and reconstruction. A similar result considering unknown sets of area less than one was independently achieved by Heckel and Boelcskei. Operators with bandlimited symbols have been used to model doubly dispersive communication channels with slowly-time-varying impulse response. The results in this paper are rooted in work by Bello and Kailath in the 1960s.Comment: Submitted to IEEE Transactions on Information Theor

A survey of uncertainty principles and some signal processing applications

The goal of this paper is to review the main trends in the domain of uncertainty principles and localization, emphasize their mutual connections and investigate practical consequences. The discussion is strongly oriented towards, and motivated by signal processing problems, from which significant advances have been made recently. Relations with sparse approximation and coding problems are emphasized

Co-compact Gabor systems on locally compact abelian groups

In this work we extend classical structure and duality results in Gabor analysis on the euclidean space to the setting of second countable locally compact abelian (LCA) groups. We formulate the concept of rationally oversampling of Gabor systems in an LCA group and prove corresponding characterization results via the Zak transform. From these results we derive non-existence results for critically sampled continuous Gabor frames. We obtain general characterizations in time and in frequency domain of when two Gabor generators yield dual frames. Moreover, we prove the Walnut and Janssen representation of the Gabor frame operator and consider the Wexler-Raz biorthogonality relations for dual generators. Finally, we prove the duality principle for Gabor frames. Unlike most duality results on Gabor systems, we do not rely on the fact that the translation and modulation groups are discrete and co-compact subgroups. Our results only rely on the assumption that either one of the translation and modulation group (in some cases both) are co-compact subgroups of the time and frequency domain. This presentation offers a unified approach to the study of continuous and the discrete Gabor frames.Comment: Paper (v2) shortened. To appear in J. Fourier Anal. App

Explicit constructions and properties of generalized shift-invariant systems in L2(R)L^2(\mathbb{R})

Generalized shift-invariant (GSI) systems, originally introduced by Hern\'andez, Labate & Weiss and Ron & Shen, provide a common frame work for analysis of Gabor systems, wavelet systems, wave packet systems, and other types of structured function systems. In this paper we analyze three important aspects of such systems. First, in contrast to the known cases of Gabor frames and wavelet frames, we show that for a GSI system forming a frame, the Calder\'on sum is not necessarily bounded by the lower frame bound. We identify a technical condition implying that the Calder\'on sum is bounded by the lower frame bound and show that under a weak assumption the condition is equivalent with the local integrability condition introduced by Hern\'andez et al. Second, we provide explicit and general constructions of frames and dual pairs of frames having the GSI-structure. In particular, the setup applies to wave packet systems and in contrast to the constructions in the literature, these constructions are not based on characteristic functions in the Fourier domain. Third, our results provide insight into the local integrability condition (LIC).Comment: Adv. Comput. Math., to appea

Delay-Coordinates Embeddings as a Data Mining Tool for Denoising Speech Signals

In this paper we utilize techniques from the theory of non-linear dynamical systems to define a notion of embedding threshold estimators. More specifically we use delay-coordinates embeddings of sets of coefficients of the measured signal (in some chosen frame) as a data mining tool to separate structures that are likely to be generated by signals belonging to some predetermined data set. We describe a particular variation of the embedding threshold estimator implemented in a windowed Fourier frame, and we apply it to speech signals heavily corrupted with the addition of several types of white noise. Our experimental work seems to suggest that, after training on the data sets of interest,these estimators perform well for a variety of white noise processes and noise intensity levels. The method is compared, for the case of Gaussian white noise, to a block thresholding estimator

Designing Gabor windows using convex optimization

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