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Many filter and filterbank design problems can be posed as the optimization of linear or convex quadratic objectives over trigonometric semi-infinite constraints. Recent advances in design methodology are based on various linear matrix inequality (LMI) characterizations of the semi-infinite constraints, and semidefinite programming (SDP) solutions. Despite these advances, the design of filters of several hundredth order, which typically arise in multicarrier communication and signal compression, cannot be accommodated. This hurdle is due mainly to the large number of additional variables incurred in the LMI characterizations. This paper proposes a novel LMI characterization of the semi-infinite constraints that involves additional variables of miminal dimensions. Consequently, the design of high-order filters required in practical applications can be achieved. Examples of designs of up to 1200-tap filters are presented to verify the viability of the proposed approach.