Analysis framework for the digital inversion of nonlinear deterministic characteristics in ΣΔ modulation

The classical noise-shaping analysis provides a framework to invert any existing distortion in the transfer function of a ΣΔ modulator, provided that the distortion has the form of a time-invariant and linear operation. We establish here a more general algebraic framework for the analysis of ΣΔ modulation, which can explicitly take into account any nonlinear deterministic distortion. We show in this paper that this framework permits the algebraic inversion of nonlinear characteristics in the specific case of static ADC-DAC nonuniformity, which can be seen as a generalization of the linear noise-shaping inversion formula. From this algebraic inversion formula, the classical qualitative knowledge of the behavior of ΣΔ modulation with regard to ADC-DAC nonuniformity can be reexplained in a concise and formal manner, with some new insights and results