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In-Depth Information
Group Representation Design of Digital Signals
and Sequences
Shamgar Gurevich
1
, Ronny Hadani
2
, and Nir Sochen
3
1
Department of Mathematics, University of California, Berkeley, CA 94720, USA
shamgar@math.berkeley.edu
2
Department of Mathematics, University of Chicago, IL 60637, USA
hadani@math.uchicago.edu
3
School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
sochen@post.tau.ac.il
Abstract.
In this survey a novel system, called the
oscillator system
,
consisting of order of
p
3
functions (signals) on the finite field
F
p
,
is
described and studied. The new functions are proved to satisfy good
auto-correlation, cross-correlation and low peak-to-average power ratio
properties. Moreover, the oscillator system is closed under the operation
of discrete Fourier transform. Applications of the oscillator system for
discrete radar and digital communication theory are explained. Finally,
an explicit algorithm to construct the oscillator system is presented.
Keywords:
Weil representation, commutative subgroups, eigen-
functions, good correlations, low supremum, Fourier invariance, explicit
algorithm.
1
Introduction
One-dimensional
analog signals
are complex valued functions on the real line
.
In the same spirit, one-dimensional
digital signals,
also called
sequences,
might
be considered as complex valued functions on the finite line
R
F
p
, i.e., the finite
field with
p
elements, where
p
isanoddprime.Inbothsituationstheparam-
eter of the line is denoted by
t
andisreferredtoas
time.
In this survey, we
will consider digital signals only, which will be simply referred to as signals.
The space of signals
H
=
C
(
F
p
) is a Hilbert space with the Hermitian product
given by
=
t∈
F
p
φ, ϕ
φ
(
t
)
ϕ
(
t
)
.
A central problem is to construct interesting and useful systems of signals.
Given a system
, there are various desired properties which appear in the
engineering wish list. For example, in various situations [1,2] one requires that
the signals will be weakly correlated, i.e., that for every
φ
S
=
ϕ
∈
S
|
φ, ϕ
|
1
.
