Root contours of low-order two-dimensional system functions

This thesis introduces a new approach to the analysis and design of 2-D systems composed of low-order systems used as building blocks. Our approach is to design 2-D low-order system functions with known root contours that may be used as building blocks of systems so that the system characteristics can be inferred from the properties of its components. We have studied unit-circle root contours of low-order 2-D polynomials (i.e., 3 x 3, 3 x 2, 2 x 3, 2 x 2, etc.). Some of the possible geometrical shapes of the root contours have been identified. For the particular cases of linear and elliptical contours, we have used a power series expansion method to derive straightforward relationships between the polynomial coefficients and the parameters of the geometrical shapes of the root contours (slope in the linear case and ellipse parameters in the elliptical case). We have addressed the problem of estimating the polynomial coefficients from the contour parameters and vice versa. Drawing a parallel to the 1-D case, we have introduced the notion of system functions whose magnitude response has a constant minimum along a controllable contour as a counterpart to a 1-D system function with zeros close to the unit circle. We have shown how to construct such system functions for the symmetric 3 x 3 and antisymmetric 2 x 3 cases and have proven the stability of the constructed systems. These low-order system functions are then used as building blocks of higher-order 2-D IIR and FIR filters. 2-D IIR notch filters, 2-D IIR and FIR directional and polygon filters are examples where we have applied our method.;It is shown that these low-order systems are also useful in modeling and processing 2-D signals whose Fourier transforms display a ridge of high energy along the zero contours. 2-D linear prediction is used to model radar clutter signals for both linear and elliptical ridges. The low-order system functions with linear and elliptical unit-circle zero contours were used for nulling the radar clutter ridge of the same shape. Finally, the 2-D IIR notch filter was used to cancel a linear radar clutter ridge.