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This paper considers the design of linear-phase finite impulse response digital filters using an L1 optimality criterion. The motivation for using such filters as well as a mathematical framework for their design is introduced. It is shown that L1 filters possess flat passbands and stopbands while keeping the transition band comparable to that of least-squares filters. The uniqueness of L1-based filters is explored, and an alternation type theorem for the optimal frequency response is derived. An efficient algorithm for calculating the optimal filter coefficients is proposed, which may be viewed as the analogue of the celebrated Remez exchange method. A comparison with other design techniques is made, demonstrating that the L1 approach may be a good alternative in several applications.