Sigma Delta Modulation Of A Chaotic Signal

Sigma delta modulation M has become a widespread method of analogue to digital conversion however its operation has not been completely de ned The majority of the analysis carried out on the circuit has been from a linear standpoint with non linear analysis hinting at hidden complexities in the modulator s operation The sigma delta modulator itself is a non linear system consisting as it does of a number of integrators and a one bit quantiser in a feedback loop This con guration can be generalised as a non linearity within a feedback path which is a classic route to chaotic behaviour This initially raises the prospect that a sigma delta modulator may be capable of chaotic modes of operation with a non chaotic input In fact the problem does not arise and we show why not To facilitate this investigation a set of di erential equations is formulated to represent M these equations are subsequently utilised in a stability study of the sigma delta modulator Of more interest and more uncertainty is the e ect sigma delta modulation may have on a chaotic signal If M makes a chaotic signal more chaotic then this will have serious repercussions on the predictability of that signal In the past analysis of the circuit has tended to be based around a steady state input or a slowly moving non chaotic input such as a low frequency sine wave This has greatly eased the complexity of such analyses but it does not address the problem at hand In this thesis we present the results of comparing the sigma delta modulation of a chaotic signal to a direct quantisation of the same signal The tool we use to investigate this is the Lyapunov spectrum of the time series measured using an algorithm developed at Edinburgh University The Lyapunov exponents of a chaotic signal are presented before and after both M and direct quantisation and it is shown that M does not increase the chaos of the signal Indeed it is shown that M has no more e ect on the predictability of the signal as measured by the Lyapunov spectrum than direct quantisation As such we conclude that sigma delta modulation provides a reliable method for analogue to digital conversion of chaotic signals It should be pointed out that due to the incompleteness of rigorous analysis of M and the complex processes involved in applying such analysis to a chaotic signal the results of this thesis are largely based upon experimentation and observation from a simulation of a sigma delta modulator Declaration of originality I hereby declare that this thesis and the work reported herein was composed and originated entirely by myself in the Department of Electrical Engineering at the University of Edinburgh except for the development of the Lyapunov extraction algorithm described in chapter which was carried out in collaboration with Mike Banbrook