Stochastic analysis of the ΣΔ modulator and differential pulse code modulator

One of the most popular systems for performing high resolution analog to digital conversion is the ΣΔ modulator. Though common in applications, theoretical analysis of the ΣΔ modulator is difficult due to the presence of a discontinuous quantizer in the modulator. This paper presents asymptotic results regarding the statistical behavior of the ΣΔ modulator when inside the loop dithering is utilized. In some recent papers examining the stochastic behavior of the ΣΔ, it was shown (via simulations) that the input to the quantizer can be accurately modeled as a stationary Gaussian process. Our analysis shows that both the input to the quantizer and the quantization noise are asymptotically stationary Gaussian processes, under mild assumptions on the input and the dither process. The results of this paper are derived by letting the quantizer stepsize approach zero, and the analytical approach is related to the stochastic analysis of adaptive filtering algorithms. Our analysis is valid for a large collection of stochastic input signals, including ARMA processes. Furthermore, previous stochastic analysis assumed that the quantizer never overloaded, while the present analysis does not make this assumption. It is also shown that analysis of the Differential Pulse Code Modulator is in fact analogous to the analysis of the ΣΔ modulator. Simulation results presented for the first-order ΣΔ modulator and two second-order ΣΔ modulators demonstrate the practicality of the analysis