Calculating transfer functions of CT sigma-delta modulators with arbitrary DAC waveforms

In this paper, a new method of calculating the transfer functions of continuous-time (CT) sigma-delta modulators is presented. The common approach of calculating the transfer functions is to analyze a discrete-time (DT) equivalent of the CT loop filter together with the CT input filter path. The DT equivalent is most commonly obtained via either the impulse invariant or the modified Z-transformation. This work provides a numerical alternative to these methods, based on DT simulation models, which can be obtained by a previously published method of DT simulating CT sigma-delta modulators, called lifting. This allows incorporating all linear-time-invariant non-idealities, such as finite DC gains and gain-bandwidths in the integrator stages, as well as excess-loop-delay and any waveform shape in the digital-to-analog converter.