Orthogonal frequency division multiplexing (OFDM) has gained considerable interest as an efficient technology for high-date-data transmission over wireless channels. Standard OFDM systems are associated with a rectangular grid in the time-frequency plane. However such a setup is in general not optimum for pulse shaping OFDM systems for doubly dispersive channels. We introduce lattice-OFDM systems (LOFDM), which are OFDM systems constructed with respect to general lattices in the time-frequency plane. We show how to design optimum pulse shaping LOFDM system. Our analysis is based on results from Heisenberg groups, Gabor frames, and sphere coverings. Numerical simulations confirm that LOFDM systems outperform ordinary OFDM systems with regard to robustness against intersymbol interference and interchannel interference.
Orthogonal frequency division multiplexing (OFDM) has gained considerable interest as an efficient technology for high- date-data transmission over wireless channels. The design of pulse shapes that are well-localized in the time-frequency plane is of great importance in order to combat intersymbol interference and interchannel interference caused by the mobile radio channel. Recently proposed methods to construct such well-localized functions are utilizing the link between OFDM and Gabor systems. We derive a theoretical framework that shows why and under which conditions these methods will yield well-localized pulse shapes. In our analysis we exploit the connection between Gabor systems, Laurent operators and the classical work of Gelfand, Raikov, and Shilov on commutative Banach algebras. In the language of Gabor analysis we derive a general condition under which the dual window and the canonical tight window inherit the decay properties of the analysis window.
We analyze the relation between infinite-dimensional frame theory and finite-dimensional models for frames as they are used for numerical algorithms. Special emphasis in this paper is on perfect reconstruction oversampled filter banks, also known as shift-invariant frames. For certain finite- dimensional models it is shown that the corresponding finite dual frame provides indeed an approximation of the dual frame of the original infinite-dimensional dual frame. For filter banks on l2 (Z) we derive error estimates for the approximation of the synthesis filter bank when the analysis filter bank satisfies certain decay conditions. We show how one has to design the finite-dimensional model to preserve important structural properties of filter banks, such as polyphase representation. Finally an efficient regularization method is presented to solve the ill-posed problem arising when approximating the dual frame on L2(R) via truncated Gram matrix.
The Gabor transform yields a discrete representation of a signal in the phase space. Since the Gabor transform is non-orthogonal, efficient reconstruction of a signal from its phase space samples is not straightforward and involves the computation of the so- called dual Gabor function. We present a unifying approach to the derivation of numerical algorithms for discrete Gabor analysis, based on unitary matrix factorization. The factorization point of view is notably useful for the design of efficient numerical algorithms. This presentation is the first systematic account of its kind. In particular, it is shown that different algorithms for the computation of the dual window correspond to different factorizations of the frame operator. Simple number theoretic conditions on the time-frequency lattice parameters imply additional structural properties of the frame operator.
Various generalizations of the classical Gabor expansion are considered. Studying the frame operator via the Kohn- Nirenberg correspondence allows to obtain straightforward structural results for the situation of (i) nonseparable prototypes, and/or (ii) nonseparable time-frequency sampling lattices, and/or (iii) multi-prototypes. For such general Weyl-Heisenberg frames, it is shown how to reformulate the Janssen representation of the frame operator and the Wexler- Raz result. Moreover, an analysis of the analysis operator is performed that leads to quantitative results about the variety of admissible analysis/synthesis prototypes.
KEYWORDS: Chemical species, Fourier transforms, Iterative methods, Signal processing, Detection theory, Reconstruction algorithms, Algorithm development, Time-frequency analysis, Signal to noise ratio, Information technology
The short-time Fourier transform (STFT) leads to a highly redundant linear time-frequency signal representation. In order to remove this redundancy it is usual to sample the STFT on a rectangular grid. For such regular sampling the basic features of the reconstruction problem are well understood. In this paper, we consider the problem of reconstructing a signal from irregular samples of its STFT. It may happen that certain samples of the STFT from a regular grid are lost or that the STFT has been purposely sampled in an irregular way. We investigate that problem using Weyl-Heisenberg frames, which are generated from a single atom by time- frequency-shifts (along the sampling set). We compare various iterative methods and present typical numerical experiments. Whereas standard frame iterations are doing not very well it turns out that for many reasons the conjugate gradient algorithm behaves best, most often even better than one might expect from the observations made for general frame operators.
We describe new methods to obtain nonorthogonal Gabor expansions of discrete and finite signals and reconstruction of signals from regularly sampled short time Fourier transform (STFT) values by series expansions. By this we understand the expansion of a signal of a given length n into a (finite) series of coherent building blocks, obtained from a Gabor atom through discrete time- and frequency-shift operators. Although bump-type atoms are natural candidates, the approach is not restricted to such building blocks. Also the set of time- and frequency-shift operators does not have to be a (product) lattice, but just an ordinary (additive) subgroup of the time/frequency plane, which is naturally identified with the 2-D n x n cyclic group. In contrast, other nonseparable subgroups turn out to be more interesting for our task: the efficient determination of a suitable set of coefficients for the coherent expansion. It is sufficient to determine the so-called dual Gabor atom. The existence and basic properties of this dual atom are well known in the case of lattice groups. It is shown that this is true for general groups. But more importantly, we demonstrate that the conjugate gradient method reduces the computational complexity drastically.
This is a paper on the discrete Gabor transform. We discuss the calculation of dual and tight Gabor atoms, for Gabor atoms which satisfy certain support restrictions related to the relevant time- and frequency lattice constants. These conditions imply that the Gabor frame operator is just a pointwise multiplication operator and therefore computing the inverse or the square root of the inverse frame operator are computationally inexpensive.
The so-called irregular sampling problem is concerned with the problem of reconstruction of an image from irregularly located samples. We have developed new algorithms for efficient reconstruction of signals from arbitrary sampling sets, which are guaranteed to converge as long as the image is bandlimited and the sampling set does not have holes that are too big. However, experimental evidence does exist that for a typical real-world image very satisfactory reconstructions can be obtained even if it is not bandlimited in the strict sense. Only a certain type of irregularities is considered, i.e., missing lines or missing rectangles in a given image. This problem occurs in practical situations where such parts may be lost during the transmission of an image or may be badly recorded by a camera. The basic idea is to solve the problem by iterative solution of 1-D subproblems, i.e., by interpolation along horizontal and then vertical lines or vice versa. The use of fast and efficient 1-D reconstruction methods is the basis for a very fast and highly parallelizable algorithm. We demonstrate the efficiency of the product ACT algorithm by typical applications.
The by now well-known matching pursuit method of S. Mallat as well as the recently proposed orthogonal matching pursuit work purely sequential and are based on the idea of choosing only one single atom at a given time. Pursuing ideas which are related to modifications of the POCS method, we suggest a new type of orthogonalization procedure which allows to operate in parallel at different 'levels'. More precisely, we assume that we have a dictionary which consists of similar 'pages', i.e. those pages are collection of functions, generated from a single function by translations along a subgroup which is the same for all such pages. Based on a certain hierarchical structure (preference of pages) we apply an appropriate Gram-Schmidt type orthogonalization procedure which allows to deal with the matching pursuit problem in parallel at different levels. After carrying out appropriate approximations at the different, now orthogonal levels, one comes back to a representation based on the given family of atoms, in a straight-forward way.
The POCS-method (projection onto convex subsets) has been proposed as an efficient way of recovering a band-limited signal from irregular sampling values. However, both the ordinary POCS-method (which uses one sampling point at a given time, i.e. consists of a succession of projections onto affine hyperplanes) and the one-step method (which uses all sampling values at the same time) become extremely slow if the number of sampling points gets large. Already for midsize 2D-problems (e.g. 128 X 128 images) one may easy run into memory problems. Based on the theory of pseudo-inverse matrices new efficient variants of the POCS- method (so to say intermediate versions) are described, which make use of a finite number of sampling points at each step. Depending on the computational environment appropriate strategies of designing those families of sampling points (either many families with few points, or few families with many points, overlapping families or disjoint ones...) have to be found. We also report on numerical results for these algorithms.
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