Filters approximating ideal amplitude and arbitrary phase characteristics

The closed-form solution is derived for the scattering transfer function
S_{12}(p)
of a resistively terminated lossless reciprocal two-port prototype network of degree
2n
which satisfies the following constraints:
|S_{12}(j\omega) | \leq 1
|S_{12}({\pm}j{\omega}_i) | = 1 , i = 1 \rightarrow n
\arg S_{12}({\pm}j{\omega}_i) = \pm \psi({\omega}_i), i = 1 \rightarrow n
where the
\omega_i
are arbitrary and
\psi(\omega)
is any odd function. The solution is obtained in terms of the arbitrary phase polynomials of the first and second kinds, which are capable of being generated through simple recurrence formulas.