Low-Pass Filters Using Ultraspherical Polynomials

The problem of approximating the ideal normalized amplitude response of a low-pass filter by the use of a set of ultraspherical polynomials is considered. The amplitude response obtained is more general than the analogous response of the Chebyshev filter because of an additional parameter available with the ultraspherical polynomials. It is shown that this additional parameter may be used to obtain a response having either less ripple or sharper cutoff than the Chebyshev response. The ultraspherical polynomial filter is shown to include as special cases the Chebyshev filter, the Butterworth filter, and a filter recently developed utilizing modified Legendre polynomials.