MSE behavior and centroid function of mth-order asymptotic ΣΔ modulators

We say that a ΣΔ modulator is asymptotic when all the zeros of its noise transfer function are located at z = 1 (zero frequency). It is usually understood from the classical white-noise model that the output of a stable mth-order asymptotic modulator includes a baseband mean squared error (MSE) that decays with the oversampling ratio R in O(1/R^2m+1). While this is indeed always true with constant inputs, we show numerically that with time-varying inputs, this result is in general not true, except in the particular case of the standard nonoverloading multibit multiloop configuration. The general MSE decay rate is in fact in O (1/R^2m ) instead. We give mixed analytical and experimental explanations for this phenomenon thanks to the notion of "centroid function" of a modulator. We show that, besides the standard multibit multiloop configuration, it is actually possible to force an mth-order ΣΔ modulator to yield an MSE decay rate of O(1/R^2m+1 ) with time-varying inputs, but this necessitates the introduction of a new species of EA modulators that include additional nonlinear memoryless operations