Practical implementation of nonlinear time series methods: The TISEAN package

Nonlinear time series analysis is becoming a more and more reliable tool for the study of complicated dynamics from measurements. The concept of low-dimensional chaos has proven to be fruitful in the understanding of many complex phenomena despite the fact that very few natural systems have actually been found to be low dimensional deterministic in the sense of the theory. In order to evaluate the long term usefulness of the nonlinear time series approach as inspired by chaos theory, it will be important that the corresponding methods become more widely accessible. This paper, while not a proper review on nonlinear time series analysis, tries to make a contribution to this process by describing the actual implementation of the algorithms, and their proper usage. Most of the methods require the choice of certain parameters for each specific time series application. We will try to give guidance in this respect. The scope and selection of topics in this article, as well as the implementational choices that have been made, correspond to the contents of the software package TISEAN which is publicly available from http://www.mpipks-dresden.mpg.de/~tisean . In fact, this paper can be seen as an extended manual for the TISEAN programs. It fills the gap between the technical documentation and the existing literature, providing the necessary entry points for a more thorough study of the theoretical background.Comment: 27 pages, 21 figures, downloadable software at http://www.mpipks-dresden.mpg.de/~tisea

Fractal-based models for internet traffic and their application to secure data transmission

This thesis studies the application of fractal geometry to the application of covert communications systems. This involves the process of hiding information in background noise; the information being encrypted or otherwise. Models and methods are considered with regard to two communications systems: (i) wireless communications; (ii) internet communications. In practice, of course, communication through the Internet cannot be disassociated from wireless communications as Internet traffic is 'piped' through a network that can include wireless communications (e.g. satellite telecommunications). However, in terms of developing models and methods for covert communications in general, points (i) and (ii) above require different approaches and access to different technologies. With regard to (i) above, we develop two methods based on fractal modulation and multi-fractal modulation. With regard to (ii), we implement a practical method and associated software for covert transmission of file attachments based on an analysis of Internet traffic noise. In both cases, however, two fractal models are considered; the first is the standard Random Scaling Fractal model and the second is a generalisation of this model that incorporates a greater range of spectral properties than the first—a Generalised Random Scaling Fractal Model. [Continues.

Digital Signal Processing (Second Edition)

This book provides an account of the mathematical background, computational methods and software engineering associated with digital signal processing. The aim has been to provide the reader with the mathematical methods required for signal analysis which are then used to develop models and algorithms for processing digital signals and finally to encourage the reader to design software solutions for Digital Signal Processing (DSP). In this way, the reader is invited to develop a small DSP library that can then be expanded further with a focus on his/her research interests and applications. There are of course many excellent books and software systems available on this subject area. However, in many of these publications, the relationship between the mathematical methods associated with signal analysis and the software available for processing data is not always clear. Either the publications concentrate on mathematical aspects that are not focused on practical programming solutions or elaborate on the software development of solutions in terms of working ‘black-boxes’ without covering the mathematical background and analysis associated with the design of these software solutions. Thus, this book has been written with the aim of giving the reader a technical overview of the mathematics and software associated with the ‘art’ of developing numerical algorithms and designing software solutions for DSP, all of which is built on firm mathematical foundations. For this reason, the work is, by necessity, rather lengthy and covers a wide range of subjects compounded in four principal parts. Part I provides the mathematical background for the analysis of signals, Part II considers the computational techniques (principally those associated with linear algebra and the linear eigenvalue problem) required for array processing and associated analysis (error analysis for example). Part III introduces the reader to the essential elements of software engineering using the C programming language, tailored to those features that are used for developing C functions or modules for building a DSP library. The material associated with parts I, II and III is then used to build up a DSP system by defining a number of ‘problems’ and then addressing the solutions in terms of presenting an appropriate mathematical model, undertaking the necessary analysis, developing an appropriate algorithm and then coding the solution in C. This material forms the basis for part IV of this work. In most chapters, a series of tutorial problems is given for the reader to attempt with answers provided in Appendix A. These problems include theoretical, computational and programming exercises. Part II of this work is relatively long and arguably contains too much material on the computational methods for linear algebra. However, this material and the complementary material on vector and matrix norms forms the computational basis for many methods of digital signal processing. Moreover, this important and widely researched subject area forms the foundations, not only of digital signal processing and control engineering for example, but also of numerical analysis in general. The material presented in this book is based on the lecture notes and supplementary material developed by the author for an advanced Masters course ‘Digital Signal Processing’ which was first established at Cranfield University, Bedford in 1990 and modified when the author moved to De Montfort University, Leicester in 1994. The programmes are still operating at these universities and the material has been used by some 700++ graduates since its establishment and development in the early 1990s. The material was enhanced and developed further when the author moved to the Department of Electronic and Electrical Engineering at Loughborough University in 2003 and now forms part of the Department’s post-graduate programmes in Communication Systems Engineering. The original Masters programme included a taught component covering a period of six months based on two semesters, each Semester being composed of four modules. The material in this work covers the first Semester and its four parts reflect the four modules delivered. The material delivered in the second Semester is published as a companion volume to this work entitled Digital Image Processing, Horwood Publishing, 2005 which covers the mathematical modelling of imaging systems and the techniques that have been developed to process and analyse the data such systems provide. Since the publication of the first edition of this work in 2003, a number of minor changes and some additions have been made. The material on programming and software engineering in Chapters 11 and 12 has been extended. This includes some additions and further solved and supplementary questions which are included throughout the text. Nevertheless, it is worth pointing out, that while every effort has been made by the author and publisher to provide a work that is error free, it is inevitable that typing errors and various ‘bugs’ will occur. If so, and in particular, if the reader starts to suffer from a lack of comprehension over certain aspects of the material (due to errors or otherwise) then he/she should not assume that there is something wrong with themselves, but with the author

LABORATORY SIMULATION OF TURBULENT-LIKE FLOWS

Most turbulence studies up to the present are based on statistical modeling, however, the spatio-temporal flow structure of the turbulence is still largely unexplored. Tur- bulence has been established to have a multi-scale instantaneous streamline structure which influences the energy spectrum and other properties such as dissipation and mixing. In an attempt to further understand the fundamental nature of turbulence and its consequences for efficient mixing, a new class of flows, so called “turbulent-like”, is in- troduced and its spatio-temporal structure of the flows characterised. These flows are generated in the laboratory using a shallow layer of brine and controlled by multi-scale electromagnetic forces resulting from a combination of electric current and a magnetic field created by a fractal permanent magnet distribution. These flows are laminar, yet turbulent-like, in that they have multi-scale streamline topology in the shape of “cat’s eyes” within “cat’s eyes” (or 8’s within 8’s) similar to the known schematic streamline structure of two-dimensional turbulence. Unsteadiness is introduced to the flows by means of time-dependent electrical current. Particle Tracking Velocimetry (PTV) measurements are performed. The technique developed provides highly resolved Eulerian velocity fields in space and time. The analysis focuses on the impact of the forcing frequency, mean intensity and amplitude on various Eulerian and Lagrangian properties of the flows e.g. energy spectrum and fluid element dispersion statistics. Other statistics such as the integral length and time scales are also extracted to characterise the unsteady multi-scale flows. The research outcome provides the analysis of laboratory generated unsteady multi- scale flows which are a tool for the controlled study of complex flow properties related to turbulence and mixing with potential applications as efficient mixers as well as in geophysical, environmental and industrial fields