Stability of high order sigma-delta modulators

Nonlinear analysis of the stability of interpolative sigma-delta modulators is a problem which becomes increasingly difficult as the order of the modulator increases. In this paper we present a technique which, in many cases, greatly simplifies this analysis and, in addition, provides a convenient method of comparison between systems of different architectures. This technique involves a transformation of the state equations of a modulator into a form in which the individual state variables are essentially decoupled and interact only within the quantizer function. This allows for analysis of the system based on results from first order sigma-delta modulators. In this paper we present this transformation technique. We also present the relevant results for first order modulators which are used in the analysis of higher order systems. Finally, we show how these results can be applied to higher order systems and derive a set of sufficient conditions for AC stability for a class of general N´th order modulators