367 research outputs found
On Krawtchouk Transforms
Krawtchouk polynomials appear in a variety of contexts, most notably as
orthogonal polynomials and in coding theory via the Krawtchouk transform. We
present an operator calculus formulation of the Krawtchouk transform that is
suitable for computer implementation. A positivity result for the Krawtchouk
transform is shown. Then our approach is compared with the use of the
Krawtchouk transform in coding theory where it appears in MacWilliams' and
Delsarte's theorems on weight enumerators. We conclude with a construction of
Krawtchouk polynomials in an arbitrary finite number of variables, orthogonal
with respect to the multinomial distribution.Comment: 13 pages, presented at 10th International Conference on Artificial
Intelligence and Symbolic Computation, AISC 2010, Paris, France, 5-6 July
201
Time-frequency transforms of white noises and Gaussian analytic functions
A family of Gaussian analytic functions (GAFs) has recently been linked to
the Gabor transform of white Gaussian noise [Bardenet et al., 2017]. This
answered pioneering work by Flandrin [2015], who observed that the zeros of the
Gabor transform of white noise had a very regular distribution and proposed
filtering algorithms based on the zeros of a spectrogram. The mathematical link
with GAFs provides a wealth of probabilistic results to inform the design of
such signal processing procedures. In this paper, we study in a systematic way
the link between GAFs and a class of time-frequency transforms of Gaussian
white noises on Hilbert spaces of signals. Our main observation is a conceptual
correspondence between pairs (transform, GAF) and generating functions for
classical orthogonal polynomials. This correspondence covers some classical
time-frequency transforms, such as the Gabor transform and the Daubechies-Paul
analytic wavelet transform. It also unveils new windowed discrete Fourier
transforms, which map white noises to fundamental GAFs. All these transforms
may thus be of interest to the research program `filtering with zeros'. We also
identify the GAF whose zeros are the extrema of the Gabor transform of the
white noise and derive their first intensity. Moreover, we discuss important
subtleties in defining a white noise and its transform on infinite dimensional
Hilbert spaces. Finally, we provide quantitative estimates concerning the
finite-dimensional approximations of these white noises, which is of practical
interest when it comes to implementing signal processing algorithms based on
GAFs.Comment: to appear in Applied and Computational Harmonic Analysi
A finite oscillator model related to sl(2|1)
We investigate a new model for the finite one-dimensional quantum oscillator
based upon the Lie superalgebra sl(2|1). In this setting, it is natural to
present the position and momentum operators of the oscillator as odd elements
of the Lie superalgebra. The model involves a parameter p (0<p<1) and an
integer representation label j. In the (2j+1)-dimensional representations W_j
of sl(2|1), the Hamiltonian has the usual equidistant spectrum. The spectrum of
the position operator is discrete and turns out to be of the form
, where k=0,1,...,j. We construct the discrete position wave
functions, which are given in terms of certain Krawtchouk polynomials. These
wave functions have appealing properties, as can already be seen from their
plots. The model is sufficiently simple, in the sense that the corresponding
discrete Fourier transform (relating position wave functions to momentum wave
functions) can be constructed explicitly
Fourier-Reflexive Partitions and MacWilliams Identities for Additive Codes
A partition of a finite abelian group gives rise to a dual partition on the
character group via the Fourier transform. Properties of the dual partitions
are investigated and a convenient test is given for the case that the bidual
partition coincides the primal partition. Such partitions permit MacWilliams
identities for the partition enumerators of additive codes. It is shown that
dualization commutes with taking products and symmetrized products of
partitions on cartesian powers of the given group. After translating the
results to Frobenius rings, which are identified with their character module,
the approach is applied to partitions that arise from poset structures
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