36 research outputs found
Characterization and computation of canonical tight windows for Gabor frames
Let be a Gabor frame for for given window .
We show that the window that generates the canonically
associated tight Gabor frame minimizes among all windows
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
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
We consider three problems for Gabor frames that have recently received much
attention. The first problem concerns the approximation of dual Gabor frames in
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 and its dual . Based on results on
commutative Banach algebras and Laurent operators we derive a general condition
under which the dual inherits the decay properties of . 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
A shift-invariant system is a collection of functions of the
form . Such systems play an important role in
time-frequency analysis and digital signal processing. A principal problem is
to find a dual system such that each
function can be written as . The
mathematical theory usually addresses this problem in infinite dimensions
(typically in or ), 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 . For compactly supported (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
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
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
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
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