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Algebraic
theory of signal processing (SMART) Algebraic theory
of signal processing (SMART) as an area was started by Markus
Püschel, whose goal is to formulate an algebraic framework for signal
processing. Our current work focuses on understanding and formulating such a
framework for filter banks and multiresolution transforms. |
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Overview Sponsors Some of this material is based upon work
supported by the National Science Foundation under 310941 and 9988296 and the
PA State Tobacco Settlement, Kamlet-Smith Bioinformatics Grant. Any opinions, findings, and conclusions or
recommendations expressed in this material are those of the author(s) and do
not necessarily reflect the views of the sponsor(s). Collaborators Markus Püschel,
Aliaksei Sandryhaila Research Corner Teaching Corner Links |
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Sampling theorems
for trigonometric transforms One way of deriving the discrete Fourier transform
(DFT) is by equispaced sampling of periodic signals or signals on a
circle. In this paper, we show
that an analogous derivation can be used to obtain the DCT (type 2). To
achieve this goal, we replace the circle by a line graph with symmetric boundary
conditions, and define signal space, filter space, and filtering operation
appropriately. Further, we derive
the corresponding sampling theorem including the proper notions of
``bandlimited'' and ``sinc function.''
The results show that, in a rigorous sense, the DCT is closely related
to the DFT, and can be introduced without concepts from statistical signal
processing as is the current practice. J.
Kovačević and M. Püschel, ''Sampling theorem associated with the discrete
cosine transform'', Proc. IEEE Int.
Conf. Acoust., Speech, and Signal Proc., |
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Under construction |
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Under construction |
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Frames Frames with maximal
robustness to erasures Motivated by the use of frames for robust transmission
over the Internet, we present a first systematic construction of real tight
frames with maximum robustness to erasures. We approach the problem in steps:
we first construct maximally robust frames by using polynomial transforms. We
then add tightness as an additional property with the help of orthogonal
polynomials. Finally, we impose
the last requirement of equal norm and construct, to our best knowledge, the
first real, tight, equal-norm frames maximally robust to erasures. M. Püschel and
J. Kovačević, ''Real,
tight frames with maximal robustness to erasures'', Proc. Data Compr. Conf., Snowbird, UT, March 2005,
pp. 63-72. |
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J.
Kovačević and M. Püschel, ''Sampling theorem associated with the discrete
cosine transform'', Proc. IEEE Int.
Conf. Acoust., Speech, and Signal Proc., Toulouse, France, May 2006. M. Püschel and
J. Kovačević, ''Real,
tight frames with maximal robustness to erasures'', Proc. Data Compr. Conf., Snowbird, UT, March 2005,
pp. 63-72. |
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Sampling theorem associated with the discrete cosine transform Real,
tight frames with maximal robustness to erasures |