NONLINEAR DYNAMICS OF A CLASS OF ANALOG-TO-DIGITAL CONVERTERS

Oversampled sigma-delta modulators are finding widespread use in audio and other signal processing applications, due to their simple structure and robustness to circuit imperfections. Exact analyses of the system are complicated by the presence of a discontinuous nonlinear element—a one-bit quantizer. In this paper, we study the dynamics of the one-dimensional mapping which models the behavior of the single-loop modulator. This mapping has a discontinuity at the origin and constant slope at all other points. With slope one, the dynamics in the region of interest reduce to those of the rotation of the circle. With slope less than one, almost all system inputs give rise to globally asymptotically stable periodic orbits. We emphasize the case with slope greater than one, and explain the structure of the resultant bifurcation diagram. A symbolic dynamics based study allows us to explain the self-similarity of the dynamics and the nature of chaos in the system.