A Study Of Nonlinear Dynamics Of Bandpass Sigma-Delta Modulation Systems

As the powerful technique of analog-to-digital and digital-to analog conversion with low circuit complexity and large robustness against circuit imperfection, Sigma-Delta modulators are finding widespread use in audio and other signal processing applications. Despite the popularity of the technique, understanding of the operation of Sigma-Delta modulation is far from complete due to the presence of a nonlinear element, a one-bit quantizer, in the feedback loop.In this thesis, we apply the theory from the field of nonlinear dynamics to provide an analytical description of the dynamics of a class of Bandpass Sigma-Delta modulators under imperfect circuit. The forms of the periodic points are investigated, and the diversity of the periodic trajectories is demonstrated from two different views: difference equation and expanded system, through which the existence of the periodic trajectories is equivalent with the existence of the solutions to the constructed high order system of linear equations. Meanwhile, the dynamics of the non-periodic trajectories and the basin of attraction of the periodic points are studied. It is shown that arbitrary order periodic trajectories of the systems under considered could exist with certain conditions, and the periodic trajectories may coexist. The results obtained in the thesis allow us to discuss in quantitative rather than qualitative terms the affection of the parameters to the systems under considered.