Simplified Analysis and Simulation of the STF, NTF, and Noise in Continuous-Time $\Delta\Sigma$ Modulators

Determining the signal transfer function (STF), noise transfer function, and in-band thermal noise spectral density of a continuous-time delta-sigma modulator (CTDSM) is challenging due to the periodically time-varying nature of the system. This is particularly so when the loop filter is also time-varying, like in the case of a modulator using a switched-capacitor (SC) feedback digital-to-analog converter (DAC). In this brief, we use the adjoint network to greatly simplify the analysis and simulation of the STF and noise in CTDSMs with SC feedback DACs. First- and third-order modulators are used to illustrate the efficacy of our techniques.