Sigma delta modulation of a chaotic signal
Sigma delta modulation SDM has become a widespread method of analogue to digital conversion, however its operation has not been completely defined. 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 configuration 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 differential equations is formulated to represent SDM; these equations are subsequently utilised in a stability study of the sigma delta modulator. Of more interest, and more uncertainty, is the effect sigma delta modulation may have on a chaotic signal. If SDM 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 SDM and direct quantisation, and it is shown that SDM does not increase the chaos of the signal. Indeed, it is shown that SDM has no more effect 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 SDM 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.