Application of non-linear analysis techniques to sigma-delta modulators

In the past sigma-delta modulators have been analysed using linear analysis techniques. In these approaches the quantiser is replaced by an additive white noise source and then standard linear systems analysis is applied. This approach can provide good estimates of noise performance but it is unable to explain much of the behaviour of sigma-delta modulators, especially such phenomena as instability, integrator spans, quantiser switching frequency, idle tones, strong limit cycle behaviour and chaos, which are inherently non-linear. To examine these problems it will be necessary to utilise some of the many non-linear analysis techniques now available. A full rigorous non-linear analysis of these systems using any one technique would be very difficult, if not impossible. An alternative is to identify specific problems, such as instability, and to apply the most suitable non-linear analysis technique to the problem. Three main approaches have been applied to the non-linear analysis of sigma-delta modulators: spectral analysis (noise and signal performances), geometric analysis (stability analysis and integrator spans) and non-linear dynamics (limit cycle behaviour)