A method for searching the limit cycles of high order sigma-delta modulators

In this paper an approach for description and validation of potential limit cycles of high order sigma-delta modulators is presented. The approach is based on a parallel decomposition of the modulator. In this representation, the general N-th order modulator is transformed into a decomposition of low order, generally complex modulators, which interact only through the quantizer function. The decomposition considered helps to describe easily the time domain behavior of the modulator. Based on this, the conditions for the existence of limit cycles in the high order modulator for constant inputs, are obtained. They are determined by the periodicity conditions for the states of the first order modulators. In this case, the state variables are uncoupled and the obtained conditions are very easy to be checked. Limit cycles correspond to periodic output sequences and the proposed method includes description and validation of possible sequences.