The Roundoff Noise Analysis Of Digital Filters Under Two-Dimensional Frequency Transformation

A system with two independent variables is said to be the two-dimensional (2-D) system, which has been widely used in many engineering fields, e.g.. control theory, digital filters, image processing, so that the design of2-D digital filters with good performance plays a significant important role in2-D systems theory.Generally speaking, the frequency transformation is a simple and useful method to design one-dimensional (1-D) and2-D filters, i.e., transform a given prototype filter into a new filter by frequency transformation that has certain desired frequency characteristic such as low-pass, high-pass, band-pass, and band-stop. However, the roundoff noise in the digital filter, which is measured by the so-called second-order modes, can reduce the stability and the closed-loop performance of the system. Therefore, the miniaturization of roundoff noise can not only reduce the practical costs of digital systems, but also improve the overall performance. Under such a background, the invariance of second-order modes under frequency transformation has attracted considerable attention, e.g., Mullis and Roberts have shown the second-order modes invariance under1-D frequency transformation.This thesis first introduces the design method of1-D and2-D frequency transformation. Then, the invariance of the second-order modes under2-D frequency transformation is investigated in detail based on the state-space formulation of2-D frequency transformation. Four numerical examples are given to illustrate the second-order modes invariance of2-D separable denominator digital filters under1-D frequency transformation can be extended to the separable and non-separable2-D digital filters under2-D frequency transformation case, i.e., if one of the all-pass filters used in2-D frequency transformation is strictly proper, the second-order modes are invariant in one direction; and if the all-pass filters used in2-D frequency transformation are both not strictly proper, the second-order modes cannot keep invariant. Moreover, a new proof for the invariance of the second-order modes under1-D frequency transformation is put forward to amend the problems in the existing one, which may provide a novel way for the proof of the second-order modes invariance under the2-D frequency transformation. Finally, a rough proof of the second-order modes invariance of2-D separable denominator digital filters under2-D frequency transformation is given.