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This paper presents a simple linear programming (LP) technique for designing high-accuracy low-complexity finite-impulse-response (FIR) variable fractional-delay (VFD) digital filters in the minimax error sense. The objective of the minimax design is to minimize the maximum absolute error of the variable frequency response (VFR) of an FIR VFD filter, which is a nonlinear problem and difficult to solve. This paper shows that the minimax design can be approximately decomposed into a pair of separate LP subproblems by decoupling the minimization of the real-part VFR error from that of the imaginary-part error. As a result, the original nonlinear minimax design problem can be easily solved by solving the two LP subproblems separately. To reduce the VFD filter complexity, we also propose a one-by-one increase scheme for optimizing the subfilter orders in the Farrow structure such that a given design specification (maximum absolute error of VFR) can be exactly satisfied. Both even-order and odd-order design examples are given to illustrate that the decoupling minimax method is not only simple, but also can achieve excellent high-accuracy low-complexity FIR VFD filters.