Bernoulli Low-Pass Filters

Selective filters are obtained by the approximation of the rectangular magnitude. Classic approximation methods employ polynomials or rational functions. Modern methods are based on numerical optimization. The optimization-based approach is effective and gives the designer much freedom. However, the polynomial methods are still attractive because they result in closed-form expressions and simple design procedures. This brief presents a class of low-pass filters that approximate the rectangular magnitude by using the Bernoulli polynomials. The presented filters have equiripple magnitude responses in the lower parts of the passbands, nonequiripple responses at the frequencies approaching the cutoff, and steep transition bands. Furthermore, they have low quality factors of the poles.