Continuous-Time Delta-Sigma Modulator Design Using the Method of Moments

We present a technique for determining the coefficients of the loop filter in a continuous-time delta sigma modulator (CTDSM) from the corresponding discrete time prototype. The method, which is based on computing the moments of the feedback DAC pulse shape, enables the determination of loop filter coefficients to an arbitrary DAC pulse shape without resorting to z-domain analysis. This not only simplifies the algebra, but also gives new insights into modulator operation. The technique helps derive topologies to compensate CTDSMs with FIR feedback DACs, as well as give closed form expressions for the coefficients of the compensated modulator. Finally, we give simple formulae for coefficients to compensate for excess loop delay in CTDSMs with arbitrary DAC pulses.