Peak-constrained least-squares half-band filters and orthogonal wavelets

To recall, Cooklev (1995) made some extensions to the Bernstein polynomial method of Caglar and Akansu (1993) for the design of regular half-band filters leading to orthogonal wavelets. However, the ad hoc methodology of Cooklev had many shortcomings which we eliminate by expressing the problem in the form of a quadratic programming problem with linear inequality constraints. This problem is solved with the Goldfarb-Idnani (1983) algorithm, and the methodology we adopt allows for the minimization of half-band filter stopband energy while simultaneously upper bounding the stopband response. This allows us to make the peak sidelobe level (PSL) and stopband energy (SE) tradeoff explained in Adams and Sullivan (see IEEE Trans. on Signal Proc., vol. 46, p.306-20, 1998). Regular half-band filters designed in this way lead to regular orthogonal wavelets. This paper therefore presents a solution to all difficulties noted in Zarowski (see PACRIM'97, Victoria, BC, Canada, p.477-80, 1997).