State-trajectory behavior in high-order, lowpass sigma-delta modulators with distinct NTF zeros

This paper presents a generic, scalable approach to obtain closed-form state-trajectory expressions for high-order (order > 2) lowpass sigma-delta (ΣΔ) modulators with distinct noise transfer function (NTF) zeros. Constant modulator input is assumed. The techniques of state-space diagonalization, continuous-time embedding, and Poincare map analysis are combined and extended. It is shown that an even-order modulator can be decomposed into individual second-order subsystems with circular trajectories about two half-plane centers, while an odd-order modulator will result in an additional first-order subsystem represented by an oscillating quantity. The trajectory and half-plane transition expressions thus obtained provide effective tools for stability analysis of ΣΔ modulators.