The Nyquist Criterion: A Useful Tool for the Robust Design of Continuous-Time $\Sigma\Delta$ Modulators

In this brief, we introduce a figure of merit for the robustness of continuous-time (CT) ΣΔ modulators. It is based on the Nyquist criterion for the equivalent discrete-time loop filter. We show how CT modulators can be designed by optimizing this figure of merit. This way, we obtain modulators with increased robustness against variations in the noise-transfer-function (NTF) parameters. This is particularly useful for constrained systems, where the system order exceeds the number of design parameters. This situation occurs, for example, due to the effect of the excess loop delay or the finite gain bandwidth of the operational amplifiers. Additionally, it is shown that the optimization is equivalent to the minimization of H
_∞
, which is the maximum out-of-band gain of the NTF. This explains why conventional design strategies that are based on H
_∞
, such as Schreier's approach, provide quite robust modulator designs in the case of unconstrained architectures.