Direct design of orthogonal filter banks and wavelets

Abstract

Abstract-This paper presents a new method for the design of two-channel conjugate quadrature (CQ) filter banks in which halfband filter and spectrum factorization are not required. Instead, a CQ filter is directly optimized subject to the perfect reconstruction and possibly other constraints (such as number of vanishing moments (VM)). We develop a design strategy in that the solution is approached sequentially with each update confined to within a small vicinity of the current iterate where the problem at hand behaves like a convex one, thus the update can be obtained as a solution of a convex problem. Four design scenarios are considered, namely the least squares designs with or without VM requirement, and equiripple designs with or without VM requirement. The simulation studies demonstrate that the proposed method is reliable to design high-order CQ filters with improved performance.

Towards global design of orthogonal filter banks and wavelets

“…A recent addition to this rich field of research is a direct design method [1] that deals with problems (5) and (6) by local convex approximations in a sequential manner, and the method is shown to produce satisfactory results.…”

“…Apply the method of [1] with h wi as its initial point (impulse response) to design an optimal CQ filter of length N w + 2. Denote the impulse response obtained by h w .…”

“…Although no claim about the global optimality of the above design methodology can be made for large N , the designs obtained are quite likely to be globally optimal because in each round of iteration the initial point is sufficiently close to the global minimizer and the algorithm in [1] is known to converge to a nearby minimizer. In the next section, we provide experimental evidences that support this analysis.…”

“…The design methods proposed in this paper are made possible by virtue of recent progress in global polynomial optimization [6], [9] and direct design techniques for the CQ filters [1] in conjunction with our observations on a common pattern shared among globally optimal impulse responses of loworder CQ filters and a progressive design procedure in terms of filter length. Two design scenarios are considered, namely the least squares designs with vanishing moment (VM) requirement, and equiripple (i.e.…”

“…Despite the fact that many algorithms for the design of CQ filters have been proposed since 1980's, see e.g. references [1], [2], [10]- [14] and the work cited therein, to date only locally optimal designs can be claimed. From a mathematical point of view, this is primarily because the design problems are inherently nonconvex, admitting many local solutions.…”