Construction of filter functions from phase polynomials

This paper shows how a rational transfer function can be constructed from phase polynomials such that, at given frequency points, the transfer function has prespecified phases and perfect power transmission. The frequency points do not have to be equally spaced, and the phase response does not have to be linear. The resulting transfer function is generally realisable, although there are some exceptional cases. Formulas for the reflection and transmission coefficients are given in closed form, which are easily implemented on a desktop computer.