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This paper investigates several design issues concerning two-channel conjugate quadrature (CQ) filter banks and orthogonal wavelets. It is well known that optimal designs of CQ filters and wavelets in the least squares (LS) or minimax sense are nonconvex problems and to date only local solutions can be claimed. By virtue of recent progress in global polynomial optimization and a direct design technique for CQ filters, we in this paper present a design strategy that may be viewed as our endeavors towards global solutions for CQ filters. Two design scenarios are considered, namely the least squares designs with vanishing moment (VM) requirement, and equiripple (i.e. minimax) designs with VM requirement. Simulation studies are presented to verify our design concept for both LS and minimax designs of low-order CQ filters, and to evaluate and compare the proposed algorithms with existing design algorithms for high-order CQ filters.