Analysis of ΔΣ modulators with zero mean stochastic inputs

In this paper, a new framework for the analysis of ΔΣ modulators with stochastic inputs is proposed. The framework is based on assuming that the input to the one-hit quantizer is a Gaussian random process with zero mean and is thus able to interrelate the autocorrelation and cross-correlation of different signals of the modulator. Two main equations describing the behavior of two different ΔΣ topologies are derived. These two equations are generally nonlinear and can be of arbitrary order, hence approximations are used to study some interesting cases. First, the nonlinear equations are linearized and solved analytically for first-order modulator with white inputs and numerically for colored inputs both with and without dithering. Also, a numerical iterative approach is used for second and fourth order modulators with white and colored inputs. In these cases, the variance of the one-bit quantizer input is found as a function of modulator input power. Next, the variance of the one-bit quantizer input is calculated when large amplitude oscillations are present assuming a large amplitude limit cycle to have a sinusoidal autocorrelation. Finally, an attempt is made to estimate the modulator´s critical input power level beyond which these large amplitude limit cycles start