Role of linear semi-infinite programming in signal-adapted QMF bank design

We consider the problem of designing a perfect-reconstruction, FIR, quadrature-mirror filter (QMF) bank (H, G) adapted to input signal statistics, with coding gain as the adaptation criterion. Maximization of the coding gain has so far been viewed as a difficult nonlinear constrained optimization problem. In this paper, it is shown that the coding gain depends only on the product filter P(z) = H(z)H(z^-1), and this transformation leads to a stable class of linear optimization problems having finitely many variables and infinitely many constraints, termed linear semi-infinite programming (SIP) problems. The sought-for, original filter H(z) is obtained by deflation 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 filters, 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.