Design of three channel power complementary filters with maximally flat characteristics

Abstract

This paper considers the problem of designing filters that partition the frequency spectrum into nonoverlapping subbands for preferential processing. The simplest scheme for this purpose consists of a low- and high-pass two-channel with gain functions G 1( ω) and G 2( ω) and if the low-pass response is required to be as flat as possible (maximally flat) in the passband region, then for a given order it is well known that the Butterworth filter is optimum. Further, the power complementary condition G 1( ω) +_ G 2( ω) = 1, for all ω, gives the Butterworth low- and high-pass pair as the optimal solution. Often two channels are inadequate and for further refinement, in this paper we consider the three subband channel case: a low-, a band- and a high-pass situation with respective gain functions G 1( ω), G 2( ω) and G 3( ω) and analyze their optimal forms. It is shown here that by requiring the passband region to be maximally flat together with the power complementary condition G 1( ω) + G 2( ω) + G 3( ω) = 1, explicit solutions can be derived for the filter transfer functions in terms of certain Butterworth polynomial functions. These optimal filters so generated are shown to be stable and interestingly, they depend solely on the zeros of a key polynomial that is characteristic to the asymptotic behavior of th3 band-pass gain. Further, if the bandpass gain G 2( ω) is also required to be symmetric, this key polynomial reduces to a real quadratic equation and details of implementing these filters in the z-domain as well as their robustness issues are also discussed in this paper.