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This paper proposes a novel method to design exactly linear phase infinite impulse response half-band filters with arbitrary regularity. Broadly speaking, the design problem is formulated as a semi-infinite program, which is then turned into a semidefinite program of minimal order via a new linear matrix inequality characterization of convex hulls of trigonometric polynomials. In contrast to maximally flat approach, the proposed method allows direct control of various design parameters, which in turn enables the synthesis of filters with better transition response. The viability of the proposed method is demonstrated through several numerical examples.