A general formula for impulse-invariant transformation for continuous-time Delta-Sigma Modulators

This paper presents a generalised new formula for impulse-invariant transformation which can be used to convert an nth-order Discrete-Time (DT) ΔΣ modulator to an nth-order equivalent Continuous-Time (CT) ΔΣ modulator. Impulse-invariant transformation formulas have been published in many open literature articles for s-domain to z-domain conversion and vice-versa. However, some of the published works contain omissions and oversights. To verify the newly derived formulas, very many designs of varying orders have been tested and a representative 4th-order single-loop DT ΔΣ modulator converted to an equivalent CT ΔΣ modulator through the new formulas are presented in this paper. The simulation results confirm that the CT ΔΣ modulator which has been derived by these formulas works in accordance with the initial DT specifications without any noticeable degradation in performance in comparison to its original DT ΔΣ modulator prototype.