Short-period oscillations from a sigma-delta modulator

A method is given to derive existence conditions for short-period oscillations which can occur in the zero-input behaviour of a sigma-delta modulator structure incorporating a third order direct-form digital-filter. The approach is limited in practice to short periods and low-order digital filters because of the complexity of the symbolic and numerical analysis involved. However, short-period oscillations give rise to narrowband `tones´ in the output spectrum of ΣΔ modulators, and their understanding and elimination is important for practical applications. The method is explained with simple examples