Breaking the feedback loop of a class of ΣΔ A/D converters

ΣΔ modulation is the currently successful technique used to perform high resolution analog-to-digital conversion. In spite of its practical success, its theoretical signal analysis has remained limited because a ΣΔ modulator contains of a feedback loop that includes a nonlinear operation, i.e., the amplitude discretization or quantization. The feedback allows us to use oversampling to compensate for the limitations of the quantizer in resolution and in precision, which are typical of analog circuits. However, because of the lack of signal analysis, it is still not clear how much resolution of conversion can be gained as a function of the oversampling. We show that for a large class of ΣΔ modulators, the feedback loop theoretically yields an equivalent feedforward signal flow graph, at least for constant inputs. This is possible thanks to remarkable modulo properties of these modulators. This equivalence can be asymptotically extrapolated to time-varying inputs with increasing oversampling. Although the exact components of the equivalent graph are not currently known in general, the theoretical structure of the feedforward graph is sufficient to point out misconceptions in the current knowledge on the final resolution of an nth-order ΣΔ modulator. Specifically, except when the modulator is "ideal", the global resolution of conversion increases by n bits per octave of oversampling, instead of the currently believed rate of n+(1/2) bits/octave.