Correlation Properties of Zadoff-Chu Sequences and Their Relatives

Abstract

Pseudo-random sequences are widely applied in many areas such as direct-sequence code-division multiple-access (DS-CDMA) systems, stream ciphers, radar ranging, channel estimation, and so on. In such applications, it is basically required to employ a family of pseudo-random sequences with low correlation. As a measure of distinguishability, correlations are defined in several different forms depending on various circumstances under which the sequences are employed. Thus it is necessary to consider the most suitable kind of pseudo-random sequences for a specific situation. In this thesis, we analyze and design pseudo-random sequences for many applications.

We first present a systematic approach to partial-period correlations which are an important performance measure of communication systems employing pseudo-random sequences. For a pair of sequences (not necessarily distinct), we introduce the linear phase-shifting sequences of one of them and analyze their full-period correlations, called the induced correlations, with the other one. By exploiting the induced correlations, we obtain the partial-period correlation properties of the given pair. In particular, we investigate Zadoff-Chu sequences and some of their relatives. As a result, we give some theoretical results on their partial-period correlations.

Our second concern is performance of pseudo-random sequences from the cryptographic point of view. For an integer r ≥ 1, the rth-order correlation measure of a sequence was introduced in cryptography as a measure of immunity against correlation attacks. When r = 2, it can be viewed as the maximum value in a subset of magnitudes of partial-period autocorrelation for a given sequence. We determine the second-order correlation measure of Zadoff-Chu sequences of any period and show that it is invariant to the design parameter for them.

In the last part of this thesis, we tackle the correlation properties of generalized chirp-like (GCL) sequences. A GCL sequence of period N is constructed by modulating a Zadoff-Chu sequence of period N with an arbitrary unimodular sequence of period m, where m divides N. Under some specific conditions, the cross-correlations between two GCL sequences are shown to have exactly the same magnitudes as those of their corresponding Zadoff-Chu sequences regardless of the employed unimodular sequences. We investigate the sufficient conditions under which such a relation holds. Using them, we then construct a new class of optimal zero-correlation zone (ZCZ) sequence sets for quasi-synchronous code-division multiple-access (QS-CDMA) systems.