Closed-form design of generalized maximally flat low-pass FIR filters using generating functions

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In this paper, we propose a closed-form design of the generalized maximally flat low-pass FIR filters. By representing the transfer function as the Bernstein polynomial form, a simple generating function of the weighting coefficients is derived. We show that the linear phase maximally flat FIR filters and the nonlinear phase maximally flat half-band FIR filters are both special cases of the proposed filter. Closed-form expressions of both linear and nonlinear phase maximally flat FIR filters with prescribed degrees of flatness and delay can be easily obtained by the proposed method.