On the behavior of the double-loop sigma-delta modulator

The authors prove a stability theorem for the integrator states of the double-loop sigma-delta modulator with inputs chosen from a class of functions that satisfy certain boundedness conditions. This input class includes functions formed from the sum of a limited magnitude constant term and a finite sum of sinusoids of arbitrary amplitude and of nonzero frequency. To demonstrate the quality of the state bounds derived in the stability theorem, they construct external periodic solutions for the modulator equations with pure sinusoidal inputs. Some observations on the case of constant (DC) inputs are also presented. It is shown that for rational DC inputs and for rational initial states, the output of the double-loop modulator becomes periodic in a finite number of iterations. The length of this period is dependent on the DC value and on the initial conditions of the modulator.>