Analysis and Problem Characterization
The previous chapter addresses the envelope constrained (EC) filtering problem from the general view point of a convex programming problem by examining properties of the cost functional and the feasible region. This chapter examines the primal-dual nature of the EC filtering problem. In particular, generalized versions of the Kuhn-Tucker Theorem and Duality Theorem (Appendix B.5) are used to convert the quadratic programming problem in Hilbert space into its dual in the space of regular Borel measures. Furthermore, by exploiting the mutual singularity of the dual variables, we can rewrite the dual problem as an unconstrained one. This form, however, does not retain the desirable smooth feature of the original dual. The relationship between the primal and dual solutions suggests an interesting interpretation of the structure of optimal EC filters. For the case in which the decision variable is finite dimensional, we have a semi-infinite programming (SIP) problem, i.e. finite number of degrees of freedom but infinite number of constraints. In this case, Carathéodor’s Dimensionality Theorem (Appendix B.3) can be applied to convert the convex but infinite dimensional dual problem into a non-convex finite dimensional problem.
KeywordsCovariance Autocorrelation Convolution
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