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   -> Volume 3, Issue 17


Preprint: preprints from Shidong Li available
 
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Shidong Li (Math GS) (shidong@math.umbc.edu)
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PostPosted: Mon Dec 02, 2002 1:27 pm    
Subject: Preprint: preprints from Shidong Li available
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Preprint: preprints from Shidong Li available

The following preprints are available by anonymous ftp to
"ftp.math.umbc.edu" under the subdirectory "/pub/shidong", or by request
from email to "shidong@math.umd.edu"

Title: General Frame Decompositions, Pseudo-duals and its Application
to Weyl-Heisenberg Frames. (submitted to Numerical
Functional Analysis and Optimization, Sept.'94)
Author: Shidong Li
Filename: pseudoframe.ps
Abstract:

We consider frame decompositions other than merely using the dual frame.
We provide a characterization of all possible frame decompositions
with $l^2$-coefficients. We show that this class of frame
decompositions can be generated using the theory of pseudo frame
decompositions ({it Section 2}). We demonstrate that generating the
pseudo-duals ({it Section 2}) for a given frame amounts to find left
inverses of an operator similar to the usual frame operator. A general
(parametric) formula for computing the pseudo-duals is provided.
The usual dual frame decomposition is shown to be a special
case.

A successful application of the theory to Weyl-Heisenberg (WH) frames is
discussed. Besides the (usual) dual frame that preserves
the translation and modulation structure, we construct a class of
infinite many pseudo-duals that preserve such a structure. We discuss the
issue of selecting the best localized analysis waveforms (pseudo-duals) with
the concept of a ``good" Gabor expansion. We also show
constructively that there are a whole set of pseudo-duals which do not have
the translation and modulation structure, yet still constitute a mathematical
decomposition with a given WH frame.


Title: On Pseudo Frame Decompositions and a Class of Discrete Gabor
Expansions. (submitted to IEEE Trans on Signal Processing,
April'94)
Authors: Shidong Li and Dennis M. Healy, Jr.
Filename: gabor.ps
Abstract:

A new approach to the study of the discrete Gabor expansion (DGE)
is introduced and analyzed in detail. We show that, in general,
the DGE goes well beyond the usual biorthogonal decomposition.
Rather, it belongs to a more general decomposition scheme which we
call {it pseudo frame decomposition}. This takes the form, in general,
[
forall xinH, ab x=sum_n<x,x^*_n>x_n,
]
in which ${x^*_n}$ does not have to be the dual frame even if ${x_n}$
is a frame ({it Section 2}). It includes the usual DGE scheme, e.g.,
cite{WR}, as a special case. The standard dual frame decomposition
is also a special case.

Algorithms are derived to compute the ``pseudo-dual" sequence ${x^*_n}$
through a matrix representation. This allows us to
generate various pseudo-dual window sequences with different features
by simply solving a system of linear equations, as in cite{WR}.


Title: On Dimension Invariance of Discrete Gabor Expansions.
(submitted to IEEE Signal Processing Letters, July'94)
Author: Shidong Li
Filename: DimInvariance.ps
Abstract:

We reveal a dimension invariance property of the (discrete)
Gabor expansion (DGE). This is an important and very useful result in
applications where signal dimension is large, or signals are
on-going with infinite length. As a consequence of this result,
a DGE scheme developed in a given (relatively small) dimension may
be applied in spaces of arbitrary large dimension by
simply adjusting the number of translations accordingly.
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