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This paper studies the design of complex-valued variable digital filters (CVDFs) and their applications to the efficient arbitrary sample rate conversion for complex signals in software radio receivers. The design of CVDFs using either the minimax or least squares criteria is formulated as a convex optimization problem and solved using the second order cone programming (SOCP) or semidefinite programming (SDP). In addition, linear and convex quadratic inequality constraints can be readily incorporated. Design examples are given to demonstrate the effectiveness of the proposed approach.