396 research outputs found
Irregular and multi--channel sampling of operators
The classical sampling theorem for bandlimited functions has recently been
generalized to apply to so-called bandlimited operators, that is, to operators
with band-limited Kohn-Nirenberg symbols. Here, we discuss operator sampling
versions of two of the most central extensions to the classical sampling
theorem. In irregular operator sampling, the sampling set is not periodic with
uniform distance. In multi-channel operator sampling, we obtain complete
information on an operator by multiple operator sampling outputs
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
Sampling of operators
Sampling and reconstruction of functions is a central tool in science. A key
result is given by the sampling theorem for bandlimited functions attributed to
Whittaker, Shannon, Nyquist, and Kotelnikov. We develop an analogous sampling
theory for operators which we call bandlimited if their Kohn-Nirenberg symbols
are bandlimited. We prove sampling theorems for such operators and show that
they are extensions of the classical sampling theorem
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
Estimation of Overspread Scattering Functions
In many radar scenarios, the radar target or the medium is assumed to possess
randomly varying parts. The properties of a target are described by a random
process known as the spreading function. Its second order statistics under the
WSSUS assumption are given by the scattering function. Recent developments in
operator sampling theory suggest novel channel sounding procedures that allow
for the determination of the spreading function given complete statistical
knowledge of the operator echo from a single sounding by a weighted pulse
train.
We construct and analyze a novel estimator for the scattering function based
on these findings. Our results apply whenever the scattering function is
supported on a compact subset of the time-frequency plane. We do not make any
restrictions either on the geometry of this support set, or on its area. Our
estimator can be seen as a generalization of an averaged periodogram estimator
for the case of a non-rectangular geometry of the support set of the scattering
function
On the invertibility of "rectangular" bi-infinite matrices and applications in time--frequency analysis
Finite dimensional matrices having more columns than rows have no left
inverses while those having more rows than columns have no right inverses. We
give generalizations of these simple facts to bi--infinite matrices and use
those to obtain density results for --frames of time--frequency molecules in
modulation spaces and identifiability results for operators with bandlimited
Kohn--Nirenberg symbols
Spherical Slepian functions and the polar gap in geodesy
The estimation of potential fields such as the gravitational or magnetic
potential at the surface of a spherical planet from noisy observations taken at
an altitude over an incomplete portion of the globe is a classic example of an
ill-posed inverse problem. Here we show that the geodetic estimation problem
has deep-seated connections to Slepian's spatiospectral localization problem on
the sphere, which amounts to finding bandlimited spherical functions whose
energy is optimally concentrated in some closed portion of the unit sphere.
This allows us to formulate an alternative solution to the traditional damped
least-squares spherical harmonic approach in geodesy, whereby the source field
is now expanded in a truncated Slepian function basis set. We discuss the
relative performance of both methods with regard to standard statistical
measures as bias, variance and mean-square error, and pay special attention to
the algorithmic efficiency of computing the Slepian functions on the region
complementary to the axisymmetric polar gap characteristic of satellite
surveys. The ease, speed, and accuracy of this new method makes the use of
spherical Slepian functions in earth and planetary geodesy practical.Comment: 14 figures, submitted to the Geophysical Journal Internationa
Sampling of stochastic operators
We develop sampling methodology aimed at determining stochastic operators
that satisfy a support size restriction on the autocorrelation of the operators
stochastic spreading function. The data that we use to reconstruct the operator
(or, in some cases only the autocorrelation of the spreading function) is based
on the response of the unknown operator to a known, deterministic test signal
Measurement of time--varying Multiple--Input Multiple--Output Channels
We derive a criterion on the measurability / identifiability of
Multiple--Input Multiple--Output (MIMO) channels based on the size of the
so-called spreading support of its subchannels. Novel MIMO transmission
techniques provide high-capacity communication channels in time-varying
environments and exact knowledge of the transmission channel operator is of key
importance when trying to transmit information at a rate close to channel
capacity
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