Analysis of a model of a sigma-delta modulator with an arbitrary input signal

The work of this thesis focuses on providing an analysis of a model of a multi-stage sigma-delta modulator using an input signal {dollar}beta(t){dollar} for which the only constraint is {dollar}vertbeta(t)vert <{dollar} 1 for all time t. The model which is used involves a sequence of nonlinear difference equations. Forms for the binary quantization error and quantization noise are defined for an input signal {dollar}beta(t).{dollar} A closed form solution for the equation is derived and used in the analysis of the characteristics of the noise.; Using results from ergodic theory and Fourier analysis and employing techniques used in previous works, the spectral density of the noise is computed for the specific case of an irrational DC (constant) input. It is shown that after two stages, the autocorrelation with lag k of the noise is 0 for {dollar}k >{dollar} 2, and the spectral density of the noise is independent of the input.; A theorem is then presented and proved which permits the analysis to be extended to the case of an arbitrary bounded input signal for almost all initial conditions for the first stage equation. Under this condition, it is demonstrated that for the three stage modulator, the autocorrelation with lag k of the noise is zero for {dollar}k >{dollar} 3, and the spectral density may be expressed as the sum of a finite number of harmonically related cosines.; It is also shown that when the output from the third stage is put through a sinc{dollar}sp{lcub}j{rcub}{dollar} filter (with {dollar}jleq{dollar} 3), the noise power is proportional to the reciprocal of the oversampling ratio raised to the 2j power, consistent with the earlier results in which DC and sinusoidal signals were used as inputs. Computer simulation results are given to support the theoretical results.