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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 |