Exploiting chaos to suppress spurious tones in general double-loop ΣΔ modulators

Conventional ΣΔ modulators are operated exclusively in the nonchaotic regime in which the internal integrators are stable. Theoretical interest in chaotic ΣΔ modulators recently arose, mainly because the circuits are simple and strongly nonlinear systems with rich behavior, not unlike Chua´s circuit. It has since been recognized that the application of conventional ΣΔ modulators to A/D and D/A conversion may be haunted by spurious “tones” in the output, and it has been proposed to use chaotic modulators to overcome that problem. This paper employs a semi-analytical technique to solve the design problem for general chaotic double-loop modulators with constant inputs. Using the results, optimized scaling factors are easily found for given integrator pole locations, such that an approximate performance measure is maximized subject to bounds on internal state variables. The clarification of this explicit tradeoff between tone rejection through chaos, system stability, and practical performance may help to overcome some understandable scepticism in the community of ΣΔ designers