Design of FIR filters with flatness constraints

We treat the problem of designing lowpass FIR digital filters that are very flat at zero frequency, smooth in the passband, and minimax in the stopband. The cases of lowpass and lowpass-differentiator are studied in particular. An effective design algorithm is described based on linear programming with a single equality constraint on the first derivative at the origin plus a concavity constraint in the passband. An empirical design equation for estimating model order in the odd-length lowpass-differentiator case is given.