Unified design procedure for digital filters in the complex domain

In this thesis, digital filter design as a constrained norm problem is studied with the intent of producing a unified design algorithm for all types of filters—FIR, IIR and 2-D FIR—with complex specifications. The design problem is treated as an “Approximation” or “Optimization” problem where the theory and design techniques are developed. A general filter design norm is proposed to allow easy trade-offs between the relative importance of error energy and worst-case error. The Chebyshev, least-squares, and constrained least-squares problems become special cases because this norm uses a convex combination of the least-squares norm and the Chebyshev norm. The primary benefit of this new problem formulation is that a single efficient multiple exchange algorithm (similar to Remez) has been developed to cover all the different filter types for magnitude and phase approximation. In the new algorithm, a small subproblem is formed at each step and is solved with an iterative reweighted least squares (IRLS) technique. The IRLS technique was chosen because it can handle the design of complex filters easily and it is very simple to implement.; This thesis presents a detailed implementation as well as the convergence theory of the new algorithm. Thousands of filters were designed to test the robustness of the new algorithm. Design examples of complex FIR, IIR and 2-D FIR filters are provided and the speed of the new algorithm is compared to other algorithms. The characteristics of IIR versus FIR digital filters are also studied to illustrate the benefits of the new design approach.