Design of recursive delay equalizers by constrained optimization
Date
2000
Authors
Ko, Nelson Richard
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Abstract
Current design methods for recursive delay equalizers use unconstrained optimization techniques that can lead to unstable designs. Several approaches are taken here to find a better algorithm. To begin this thesis, the unconstrained method is introduced and analyzed. Among the findings of the analysis are regular pole plots for the solutions, a tendency for the poles to be moved to the real axis during the optimization, a strong correlation between the starting point placement for the optimization and the optimization's success, and, due to the unconstrained nature of the algorithm, unstable equalizer designs. Although there are ways to deal with these problems (such as restarting a design with new initial values), there is no guarantee that a stable equalizer which meets prescribed specifications will be obtained. This thesis focuses on achieving a stable design and on making the optimization as efficient as possible. Examination of stable recursive delay equalizers leads to improved design techniques through two main improvements. First, the equalizers are designed using a constrained optimization technique incorporating a known objective function. The constrained technique involves four methods: the Newton method, the primal active set method, the Lagrange method, and the generalized elimination method. Modifications such as a different starting point algorithm were also made to the basic model to make the algorithm more efficient. This new method guarantees a stable design, but is found to require more computations per iteration due to the use of the Hessian matrix. In order to further understand the problem, an analysis of the relationship of the pole positions to the group delay is performed. Delay magnitude and peak positions were found to relate to the pole angle and magnitude. Using this relationship, an explanation of the regular pole patterns found earlier in successful equalizer designs was developed. The objective function is finally reworked to be pole-position based in order to improve the performance and characteristics of the optimization. The new method is shown to be more effective in finding stable and efficient equalizers, and more capable in finding them for difficult filters. The computational complexity is much greater than the unconstrained coefficient-based method, and this thesis mentions some possible solutions to make the algorithm more efficient, including using a mix of unconstrained and constrained optimizations. Some directions on where to further investigate this topic are also given.