Nonlinear dynamical analysis of a bandpass sigma-delta modulator in ideal, imperfect and chaotic regimes

Many systems in signal processing are of the form of discrete-time feedback loops containing a quantizer, and are modelled by discontinuous, piecewise linear and noninvertible maps. This paper aims to show the applicability of piecewise-continuous models to such systems, taking as an example a system from signal processing, the bandpass sigma-delta modulator, which is analysed in the case of ideal, imperfect and chaotic regimes for zero and sinusoidal inputs. It is shown that the phase plane consists of regions with different numbers of rank one backward iterates, separated by critical lines. Noninvertible map theory is later used to explain aspects of the complex behaviour of the sigma-delta modulator