Nonlinear Dynamics of a Sigma-Delta modulator with Integrator Leakage

Sigma-delta (Σ∆) modulators are widely used for analogue-to-digital and digital-to-analogue conversion in both discrete-time and continuous-time implementations [1]. A Σ∆ modulator is an example of a nonlinear system, the nonlinearity being introduced by the presence of the quantiser. In [2] we applied results of our work on driven interval shifts to explain the steady-state dynamics of the first-order ideal discrete-time Σ∆ modulator whose input is a sampled periodic signal. In [3] the Σ∆ modulator with integrator leakage, a common circuit imperfection, was explored. In particular, it was found that under certain conditions the trajectories tend in the limit to lie on two continuous curves. In this paper, an explicit formula for these curves with any periodic input is determined. We will begin by reviewing some of our previous results. In section 3, the new algorithm is presented and then applied to the case of sinusoidal input and a triangular wave input. Finally, the effect of the integrator leakage on the thickness of the limit set [2] is examined.