Robust Analytical Design of Optimal Equiripple Lowpass FIR Filters

For the first time, an analytical design of optimal equiripple lowpass finite impulse response filters is presented. An analytical filter design, which is based on formulas, stands in contrast to the Parks-McClellan approach which is based on a numerical optimization. An advantage of evaluating impulse response coefficients using formulas over a numerical optimization is the robustness of the analytical design. Equiripple filters are optimal in terms of a minimal filter length for an arbitrary filter specification. The novel design is based on an equiripple approximating polynomial which approximates two constants in two disjoint intervals in optimal equiripple sense. A recursive formula for evaluating the impulse response of the filter is also introduced. The algorithm provides not only robust formulas for evaluating the impulse response, but also an analytical view on its coefficients. An example demonstrates the efficiency of the design. Its superiority in terms of robustness over the Parks-McClellan approach is emphasized.