Maximally flat power complementary filters with symmetric bandpass response

The problem of designing three channel power complementary filters that partition the frequency spectrum into nonoverlapping subbands is considered. It is shown that in the rational case maximally flatness, together with the power complementary condition G₁(ω)+G₂(ω)+G ₃(ω)=1, uniquely specifies the desired filter transfer functions in terms of certain Butterworth polynomials. These optimal filters can be chosen to be stable, and they depend solely on the zeros of a key polynomial that is characteristic to the asymptotic behavior of the bandpass gain. Furthermore, if the bandpass gain G ₂(ω) is also required to be symmetric, this key polynomial reduces to a real quadratic equation and its zeros can be used to shape the band responses