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In-Depth Information
On Partial Correlations of Various Z
4
Sequence
Families
Parampalli Udaya
1
and Serdar Boztas
2
1
Department of Computer Science and Software Engineering
University of Melbourne, Melbourne 3010, Australia
udaya@csse.unimelb.edu.au
2
School of Mathematical and Geospatial Sciences, RMIT University,
Melbourne 3001, Australia
serdar.boztas@ems.rmit.edu.au
Abstract.
Galois ring
m−
sequences were introduced in the late 1980s
and early 1990s, and have near-optimal full periodic correlations. They
are related to
Z
4
-linear codes, and are used in CDMA communications.
We consider periodic correlation and obtain algebraic expressions of the
first two partial period correlation moments of the sequences belonging
to families A, B and C. These correlation moments have applications
in synchronisation performance of CDMA systems using Galois ring se-
quences. The use of Association Schemes provides us with a new uniform
technique for analyzing the sequence families A, B and C.
Keywords:
CDMA, spread spectrum sequence design,
Z
4
-sequences, au-
tocorrelation, crosscorrelation, partial period correlation, Galois rings,
coding theory, association schemes.
1
Introduction and Background
The aim of
Code Division Multiple Access
(CDMA) in wireless networks is to
enable wireless transmitters to successfully exchange information in the presence
of potential conflicts which lead to interference. There are two main flavours
of CDMA,
Frequency Hopping
(FH) and
Direct Sequence
(DS). For details of
CDMA networks, we refer the interested reader to the recent survey in the
Spread
Spectrum Communications Handbook
by Simon et al. [8].
In this paper, we shall be concerned with the so-called “spreading codes” in
DS-CDMA, and specifically their performance in synchronisation, when it is con-
venient to use partial period correlations to acquire the correct phase of the chip
sequence which is used for spreading the transmitted signal. For a detailed sur-
vey on pseudorandom sequence design, please see the chapter by Helleseth and
Kumar in the
Handbook of Coding Theory
[4]. Very briefly, a CDMA commu-
nication system with phase-shift keying (PSK) modulation assigns unique-phase
code sequences to each transmitter-receiver pair. The traditional design meth-
ods for sequence families relied on Galois field theory. More recently, Galois rings
have been used (by Sole, Boztas, Hammons, Kumar, Udaya and Siddiqi) to de-
sign CDMA sequence families, both for DS-CDMA [1,5,6] and FH-CDMA [10].
