Structure of round-off noise in single-bit second-order sigma-delta modulators with zero input

Single-bit sigma-delta modulators are used for A-D conversion in a wide variety of systems. Analysis of first-order systems has made much progress, but the analysis of second-order systems is conventionally based on an approximate white-noise model of round-off noise. The present paper applies a new method of analysing second-order systems. It shows that, with zero input, the behaviour for each sample is equivalent to a two-dimensional discontinuous mapping Φ. It derives an invariant form under Φ and describes some of its properties. The motivation is the suggestion that the invariant form is a useful approximation to the probability distribution of round-off noise