Filters approximating ideal amplitude and arbitrary phase characteristics

The closed-form solution is derived for the scattering transfer functionS₁₂(p)of a resistively terminated lossless reciprocal two-port prototype network of degree2nwhich satisfies the following constraints:|S₁₂(jω) | ≤ 1|S₁₂({±}j{ω}_i) | = 1 , i = 1 → n S₁₂({±}j{ω}_i) = ± ψ({ω}_i), i = 1 → nwhere theω_iare arbitrary andψ(ω)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.