637 research outputs found
Mixed-state localization operators: Cohen's class and trace class operators
We study mixed-state localization operators from the perspective of Werner's
operator convolutions which allows us to extend known results from the rank-one
case to trace class operators. The idea of localizing a signal to a domain in
phase space is approached from various directions such as bounds on the
spreading function, probability densities associated to mixed-state
localization operators, positive operator valued measures, positive
correspondence rules and variants of Tauberian theorems for operator
translates. Our results include a rigorous treatment of multiwindow-STFT
filters and a characterization of mixed-state localization operators as
positive correspondence rules. Furthermore, we provide a description of the
Cohen class in terms of Werner's convolution of operators and deduce
consequences on positive Cohen class distributions, an uncertainty principle,
uniqueness and phase retrieval for general elements of Cohen's class.Comment: We call generalized localization operators now mixed-state
localization operators. In addition to a change of title and other parts
involving generalized localization operators. We did a major revision of the
manuscript incorporating suggestions by reviewer
Pulse Shaping, Localization and the Approximate Eigenstructure of LTV Channels
In this article we show the relation between the theory of pulse shaping for
WSSUS channels and the notion of approximate eigenstructure for time-varying
channels. We consider pulse shaping for a general signaling scheme, called
Weyl-Heisenberg signaling, which includes OFDM with cyclic prefix and
OFDM/OQAM. The pulse design problem in the view of optimal WSSUS--averaged SINR
is an interplay between localization and "orthogonality". The localization
problem itself can be expressed in terms of eigenvalues of localization
operators and is intimately connected to the concept of approximate
eigenstructure of LTV channel operators. In fact, on the L_2-level both are
equivalent as we will show. The concept of "orthogonality" in turn can be
related to notion of tight frames. The right balance between these two sides is
still an open problem. However, several statements on achievable values of
certain localization measures and fundamental limits on SINR can already be
made as will be shown in the paper.Comment: 6 pages, 2 figures, invited pape
Two Aspects of the Donoho-Stark Uncertainty Principle
We present some forms of uncertainty principle which involve in a new way
localization operators, the concept of -concentration and the
standard deviation of functions. We show how our results improve the
classical Donoho-Stark estimate in two different aspects: a better general
lower bound and a lower bound in dependence on the signal itself.Comment: 20 page
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