Mathematical model for quantitative analysis of sigma-delta converters

An exact mathematical model in the time domain for a first order sigma-delta modulator based on the solution of a nonlinear difference equation is given and discussed. It provides an explicit relation between the analog input signal and the output bitstream and allows quantitative determination of certain system parameters. Subsequently, an alternative time domain modeling strategy is introduced based on Taylor series expansion and the Euler-McLaurin sum formula. It is shown, that this model, unlike the difference equation model, can be easily derived for modulators of arbitrary order