1,672 research outputs found
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
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
Cornerstones of Sampling of Operator Theory
This paper reviews some results on the identifiability of classes of
operators whose Kohn-Nirenberg symbols are band-limited (called band-limited
operators), which we refer to as sampling of operators. We trace the motivation
and history of the subject back to the original work of the third-named author
in the late 1950s and early 1960s, and to the innovations in spread-spectrum
communications that preceded that work. We give a brief overview of the NOMAC
(Noise Modulation and Correlation) and Rake receivers, which were early
implementations of spread-spectrum multi-path wireless communication systems.
We examine in detail the original proof of the third-named author
characterizing identifiability of channels in terms of the maximum time and
Doppler spread of the channel, and do the same for the subsequent
generalization of that work by Bello.
The mathematical limitations inherent in the proofs of Bello and the third
author are removed by using mathematical tools unavailable at the time. We
survey more recent advances in sampling of operators and discuss the
implications of the use of periodically-weighted delta-trains as identifiers
for operator classes that satisfy Bello's criterion for identifiability,
leading to new insights into the theory of finite-dimensional Gabor systems. We
present novel results on operator sampling in higher dimensions, and review
implications and generalizations of the results to stochastic operators, MIMO
systems, and operators with unknown spreading domains
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