DC stability analysis of high-order, lowpass ΣΔ modulators with distinct unit circle NTF zeros

This paper presents an analytical approach to the investigation of the dc stability of high-order (order>2), lowpass (LP) ΣΔ modulators with distinct noise transfer function (NTF) zeros on the unit circle. The techniques of state-space diagonalization and decomposition, continuous-time embedding and Poincare map analysis are combined and extended. It is revealed that high-order ΣΔ modulators can be transformed and decomposed into second- and first-order subsystems. The investigation, coupled with efficient numerical methods, generalizes itself to different types of transition flow and provides theoretical insight into the state trajectory and limit cycle behavior. It is shown that estimation of dc input bounds based solely on the boundary transition flow is inadequate. A procedure utilizing the information from different transition flow assumptions and the discrete nature of a modulator is introduced for locating the stable dc input bounds of practical, discrete-time ΣΔ modulators.