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The design of an orthogonal FIR quadrature{mirror lter (QMF) bank (H; G) adapted to input signal statistics is considered. The adaptation criterion is maximization of the coding gain and has so far been viewed as a di cult nonlinear constrained optimization problem. In this paper, it is shown that in fact the coding gain depends only upon the product lter P (z) = H (z)H(z ?1), and this transformation leads to a stable class of linear optimization problems having nitely many variables and in nitely many constraints, termed linear semi{ in nite programming (SIP) problems. The sought{for, original lter, H (z), is obtained by de ation and spectral factorization of P (z). With the SIP formulation, every locally optimal solution is also globally optimal and can be computed using reliable numerical algorithms. The natural regularity properties inherent in the SIP formulation enhance the performance of these algorithms. We present a comprehensive theoretical analysis of the SIP problem and its dual, characterize the optimal lters, and analyze uniqueness and sensitivity issues. All these properties are intimately related to those of the input signal and bring considerable insight into the nature of the adaptation process. We present discretization and cutting plane algorithms and apply both methods to several examples.

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By constructing a new infinite dimensional space for which the extreme point—linear independence and opposite sign theorems of Charnes and Cooper continue to hold, and, building on a little-known work of Haar (herein presented), an extended dual theorem comparable in precision and exhaustiveness to the finite space theorem is developed. Building further on this a dual theorem is developed for arbitrary convex programs with convex constraints which subsumes in principle all characterizations of optimality or duality in convex programming. No differentiability or constraint qualifications are involved, and the theorem lends itself to new computational procedures.