Application of System Identification Methods for Sigma Delta Modulator Modeling
Konstantinos Touloupas
Abstract
System modeling is a rapidly growing engineering topic, and has been under research for several decades. A model is a mathematical representation of a real system that contributes to its better understanding and to its efficient simulation.
This work proposes a new method for modeling systems when frequency response magnitude data are only available. It is implemented on Matlab and applied to classical Sigma-Delta modulators, as well as Multi-Step Look-Ahead ones.
The methodology includes a technique that enables us to choose properly the excitation of the real system, so as for the data to be informative. These data are passed by a prefilter and decimated, in order to achieve good results, as in classic system identification methods. Furthermore, a phase sequence is inserted in the data, using the complex cepstrum of the output data of the real system. Hence, the absence of phase data is overcome.
The core of the modeling procedure is based on the set of filtered data with the inserted phase sequence. A modification of the Vector Fitting algorithm uses this set to provide us with a discrete time model. System modeling using rational models includes solving nonlinear least squares equations. Vector Fitting linearizes the nonlinear mathematical equations by solving linear problems recursively, and relocating the poles of the model in each iteration. In addition, it ensures good numerical conditioning and it is capable of providing accurate results in wide frequency bands.
The proposed method constitutes a technique for model order reduction as well. More precisely, the model order is selected prior the least squares problem, and in the case that the order chosen is less than the one of the real system, the model retains only the principal components of it. It is proven that it matches the spectral behavior of the larger real system in least-squares sense.
Last, the method is applied to Sigma-Delta modulators, and Multi-Step Look-Ahead ones. In the latter case, there is no closed form input-output mathematical relation proven, due to the inherent nonlinear dynamics of the modulators. Using our method, a relation for the noise transfer function of the modulators is derived. It demonstrates good accuracy and makes use of magnitude data only.