35 research outputs found
Incoherent dictionaries and the statistical restricted isometry property
In this article we present a statistical version of the Candes-Tao restricted
isometry property (SRIP for short) which holds in general for any incoherent
dictionary which is a disjoint union of orthonormal bases. In addition, under
appropriate normalization, the eigenvalues of the associated Gram matrix
fluctuate around 1 according to the Wigner semicircle distribution. The result
is then applied to various dictionaries that arise naturally in the setting of
finite harmonic analysis, giving, in particular, a better understanding on a
remark of Applebaum-Howard-Searle-Calderbank concerning RIP for the Heisenberg
dictionary of chirp like functions.Comment: Key words: Incoherent dictionaries, statistical version of Candes -
Tao RIP, Semi-Circle law, deterministic constructions, Heisenberg-Weil
representatio
Almost Linear Complexity Methods for Delay-Doppler Channel Estimation
A fundamental task in wireless communication is channel estimation: Compute
the channel parameters a signal undergoes while traveling from a transmitter to
a receiver. In the case of delay-Doppler channel, i.e., a signal undergoes only
delay and Doppler shifts, a widely used method to compute delay-Doppler
parameters is the pseudo-random method. It uses a pseudo-random sequence of
length N; and, in case of non-trivial relative velocity between transmitter and
receiver, its computational complexity is O(N^2logN) arithmetic operations. In
[1] the flag method was introduced to provide a faster algorithm for
delay-Doppler channel estimation. It uses specially designed flag sequences and
its complexity is O(rNlogN) for channels of sparsity r. In these notes, we
introduce the incidence and cross methods for channel estimation. They use
triple-chirp and double-chirp sequences of length N, correspondingly. These
sequences are closely related to chirp sequences widely used in radar systems.
The arithmetic complexity of the incidence and cross methods is O(NlogN + r^3),
and O(NlogN + r^2), respectively.Comment: 4 double column pages. arXiv admin note: substantial text overlap
with arXiv:1309.372
Performance Estimates of the Pseudo-Random Method for Radar Detection
A performance of the pseudo-random method for the radar detection is
analyzed. The radar sends a pseudo-random sequence of length , and receives
echo from targets. We assume the natural assumptions of uniformity on the
channel and of the square root cancellation on the noise. Then for , where , the following holds: (i) the probability of
detection goes to one, and (ii) the expected number of false targets goes to
zero, as goes to infinity.Comment: 5 pages, two figures, to appear in Proceedings of ISIT 2014 - IEEE
International Symposium on Information Theory, Honolul
The finite harmonic oscillator and its associated sequences
A system of functions (signals) on the finite line, called the oscillator
system, is described and studied. Applications of this system for discrete
radar and digital communication theory are explained.
Keywords: Weil representation, commutative subgroups, eigenfunctions, random
behavior, deterministic constructionComment: Published in the Proceedings of the National Academy of Sciences of
the United States of America (Communicated by Joseph Bernstein, Tel Aviv
University, Tel Aviv, Israel
A Note on the Diagonalization of the Discrete Fourier Transform
Following the approach developed by S. Gurevich and R. Hadani, an analytical
formula of the canonical basis of the DFT is given for the case where
is a prime number and (mod 4).Comment: 12 pages, accepted by Applied and Computational Harmonic Analysi