Dynamics of a bandpass sigma-delta modulator as a piecewise isometry

We examine a model for a bandpass ΣΔ modulator introduced by Feely and co-workers (1996). This is shown to have the dynamics of a piecewise isometry of a union of convex polygons on the plane by an appropriate transformation of the linearised parts into normal form. Using these we show that the periodically coded regions form a packing of the phase space by circles and we link this system to a number of similar ones. In particular we conjecture that for typical values of the parameter θ there is a positive measure set of points that have aperiodic codings