Anthony J. Lawrance

Performance Analysis and Optimization of Multi-User Differential Chaos-Shift Keying Communication Systems

Bit-error rate (BER) performance of a generalized multi-user differential chaos-shift-keying (DCSK) digital communication system is analyzed and optimized in this paper using baseband models. The generalized version of DSCK involves use of multiple nonrepeating chaotic spreading segments and energy adjustments and allows considerable system optimization. Theoretical BER analysis of correlation decoders based on a mathematical two-stage exact conditional approach is given as the foundation of subsequent performance studies. The consequent fully theoretical BER result gives an odd-symmetric spreading map condition for the BER to be unaffected by the mix of transmitted bits. The basis of the theoretical result is used to improve on the traditional simple Gaussian approximation (SGA) approach to the calculation of BER; this is inaccurate because it does not recognise the chaotic dynamics of DCSK. To overcome this drawback, a dynamically improved Gaussian approximation (DIGA) is developed. The DIGA approach is compared with the SGA approach, with simulation results demonstrating its accuracy and efficiency, certainly for more than five to ten users. Not only does the DIGA approach provide a more accurate BER than the SGA approach, its simple form also makes optimization of the DCSK system practically possible. The BER lower bound for the multi-user DCSK system is deduced from the DIGA approach. It is shown how the parameters and structure of the multi-user DCSK communication system can be optimized to achieve this lower bound. The work gives an exact theoretical basis of DCSK and covers several aspects which have not been treated in earlier research or have been inexactly treated

Chaos Communication Performance: Theory and Computation

In this paper new and existing approaches are developed to compute the bit-error rate for chaos-based communication systems. The multi-user coherent antipodal chaos shift keying system is studied and evaluated in its coherent form, in the sense of perfect synchronisation between transmitted and received chaotic sequences. Transmission is through an additive white Gaussian noise channel. Four methods are interrelated in the paper, three approximate ones and an exact one. The least accurate but most well known is based on simple Gaussian approximation; this is generalised to better reveal its structure. Two accurate and computationally efficient approximate methods are based on conditional Gaussian approximation and the statistical distribution of the typically non-constant bit energy. The most insightful but computationally expensive one is based on exact theory and rests on explicit mathematical results for particular chaotic maps used to spread binary messages. Both upper and lower bounds to the bit-error rate are suggested. The relative advantages of the different approaches are illustrated with plots of bit-error rate against signal to noise ratio.

Paired Bernoulli Circular Spreading: Attaining the BER Lower Bound in a CSK Setting

This paper is concerned with the Paired Bernoulli Circular Spreading (PBCS), a way of generating optimal spreading for the single-user coherent chaos shift-keying (CSK) system. PBCS is optimal spreading in the sense that it attains the Bit Error Rate (BER) lower bound of the system, therefore it has a potential engineering impact on the choice of signal carrier in CSK communications. PBCS optimality is justified theoretically and is further demonstrated through BER simulations. The statistical properties of PBCS are of interest too, as it is an invariant stochastic process with a mixed joint density which allows to sample infinitely many points from a circle.

SENSITIVE PARAMETER DEPENDENCE OF AUTOCORRELATION FUNCTION IN PIECEWISE LINEAR MAPS

This paper is principally concerned with the effect of splitting and permuting, or shifting, the branches of an onto piecewise linear map, with particular regard to the effect on the autocorrelation of the associated chaotic sequence. This is shown to be a chaotic function of the shifting parameter of the map, and its sensitivity with respect to minute changes in this parameter is termed autocorrelation chaos. This paper presents both analytical and computational studies of the phenomenon.

Synchronized Laser Chaos Communication: Statistical Investigation of an Experimental System

The paper is concerned with analyzing data from an experimental antipodal laser-based chaos shift-keying communication system. Binary messages are embedded in a chaotically behaving laser wave, which is transmitted through a fiber-optic cable and are decoded at the receiver using a second laser synchronized with the emitter laser. Instrumentation in the experimental system makes it particularly interesting to be able to empirically analyze both optical noise and synchronization error as well as bit error rate. Both the noise and error are found to significantly depart in distribution from independent Gaussian. The conclusion from bit error rate results is that the antipodal laser chaos shift-keying system can offer a feasible approach to optical communication. The non-Gaussian optical noise and synchronous error results are a challenge to current theoretical modeling.

Nonlinear dynamics of trajectories generated by fully-stretching piecewise linear maps

This paper focuses on the nonlinear dynamical properties of chaotic orbits iteratively generated by maps composed of linear branches which expand across the whole map range. The nonlinear dynamics of such orbits involve both their statistical and chaotic properties. More specifically, analytical expressions are provided for the mean-adjusted quadratic autocorrelation function (ACF) and for the Lyapunov exponent of trajectories produced by the considered collection of piecewise linear maps.

Recent theory and new applications in chaos communications

This paper will begin the Special Session by introducing the topics to be covered and then more particularly will present some of the authors recent work on exact and accurate conditional Gaussian approximations for BER, likelihood decoding, and the communications use of optical chaos. There will be two general themes to the session, one of improving existing theory of BER performance and one covering the latest application developments of chaos-based communications to ultra wide-band radio and cooperative personal networks. The main system used for exemplification will be multi-user antipodal chaos shift-keying, both in coherent and noncoherent forms, and also with some modifications including frequency modulated chaos shift-keying and error correcting. Other topics of the session include highly accurate approximate BER results with channel fading, the role of non-constant bit energy, and optimal spreading.

STATISTICAL DEPENDENCY IN CHAOS

This paper is concerned with the statistical dependency effects in chaotic map processes, both before and after their discretization at branch boundaries. The resulting processes are no longer chaotic but are left with realizable statistical behavior. Such processes have appeared over several years in the electronic engineering literature. Informal but extended mathematical theory that facilitates the practical calculation of autocorrelation of such statistical behavior, is developed. Both the continuous and discretized cases are treated further by using Kohdas notions of equidistribution and constant-sum to maps which are not onto. Some particularly structured chaotic map processes, and also well-known maps are examined for their statistical dependency, with the tailed shift map family from chaotic communications receiving detailed attention. Several parts of the paper form a brief review of existing theory.

The dynamics and statistics of bivariate chaotic maps in communications modeling

Statistical and dynamical properties of bivariate (two-dimensional) maps are less understood than their univariate counterparts. This paper gives a synthesis of extended results with exemplifications by bivariate logistic maps, the bivariate Arnold cat map and a bivariate Chebyshev map. The use of synchronization from bivariate maps in communication modeling is exemplified by an embryonic chaos shift keying system.

Chaos communication performance analysis: Taking advantage of statistical theory

This paper seeks to outline and illustrate the statistical basis of performance modelling of chaos-based communication systems. It is argued that likelihood decoding of message bits and theoretical derivation of decoder error probability leads to exact results and enhanced engineering insights. Moreover, the exact results, replacing earlier Gaussian approximations, suggest ways to optimally design such systems. The main emphasis will be on coherent chaos shift-keying communication, both in its single- and multiple-user versions.