Numerical Methods for Continuous-Time EC Filtering
In Chapter 2, the continuous-time EC filtering problem has been formulated for both a purely analog filter structure and a hybrid filter structure that includes analog and digital signal processing components. Both of these formulations resulted in a semi-infinite programming (SIP) problem with quadratic cost. In this chapter, three methods for solving this SIP problem are presented. These methods can be considered as extensions of the methods developed in Chapter 4 for solving the discrete-time EC filtering problem. One approach is to make use of the active set method by approximating the continuum of constraints with a finite number of constraints. A criterion is derived for selecting the granularity of the discretization to ensure feasibility of the approximate solution. This approach is attractive for its simplicity and efficiency. However, the performance tends to deteriorates with finer discretization. When this happens, we can turn to the primal-dual or penalty method. The primal-dual algorithm can be extended to the continuous-time case by discretizing the infinite dimensional dual problem (the continuum of constraints of primal problem means that the dual problem is infinite dimensional). There is no guarantee that primal solution obtained by this approach is feasible. While the active set method and primal-dual involves discretization of the primal and dual problems respectively, the penalty approach introduces approximation that does not require discretization. Moreover, feasibility of the approximate solutions are guaranteed.
KeywordsDual Problem Magnitude Response Hybrid Filter Steep Descent Linear Interpolator
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