AbstractOversampled sigma‐delta analogue‐to‐digital conversion has been the focus of much research attention in recent years. Converters based on this principle have been found to be particularly suited to integrated circuit implementation owing to their simple structure and robustness to circuit imperfection. the analysis of the sigma‐delta system is complicated by the presence of a discontinuous non‐linear element, with the result that until very recently few rigorous studies of such converters had appeared in the literature.In this paper we study the operation of the single‐loop sigma‐delta modulator, consisting of a quantizer and a discrete‐time integrator in a feedback loop, with constant input. Our analysis uses the tools of non‐linear dynamics to derive rigorous descriptions of system behaviour. In particular we focus on the effects of the discontinuous nonlinear element‐the quantizer. In the most basic case the quantizer has two levels. We study the effect of quantizer offsets in conjunction with leaky integration on the behaviour of this system. We then study the effect on system operation of increasing the number of quantization levels and explain in quantitative terms the structure of the resultant input versus average ou