Driven interval shift dynamics in sigma-delta modulators and phase-locked loops

In this paper, we show how consideration of the first-order sigma-delta ((sigma)(delta)) modulator with sampled periodic input motivates study of the maps known mathematically as driven interval shifts. We derive new results concerning the behavior of these maps, and apply these results to provide new insight into the behavior of (sigma)(delta) modulators and phase-locked loops. We will see that viewing these systems through the mechanism of the driven interval shift allows us to identify common features in their behavior.