Recursive Halfband-Filters

Summary The paper describes the properties and the design of recursive halfband-filters.The two possibilities of being complementary are introduced. The lowpass with the transfer function H Lp ( z )and the corresponding highpass, described by H Hp ( z ) = H Lp (- z )can either be strictly complementary or power complementary. According to the respective symmetry, the impulse responses, transfer functions and frequency responses possess certain characteristic properties, which are described in section 2. It turns out that these resulting symmetries of the frequency response reduce the number of the choosable design parameters. We can only prescribe the cutoff frequency and the tolerated deviation either for the passband or the stopband. In the third section we treat the design of halfband-filters with approximately linear phase. By coupling an appropriately designed allpass of even degree n A with a delay of order m = n A ±1 we obtain the desired solution by solving a corresponding approximation problem for the phase of the allpass. The resulting lowpass and highpass are strictly as well as power complementary!The kind of approximation will be done in the sense of maximal flatness, where a closed form solution exists [8], or in the sense of Chebychev, where the solution is obtained iteratively [13]. The design of systems with minimum phase is presented in section 4. The resulting lowpass and highpass are power complementary. Closed form solutions yield Butterworthand Cauer filters, if a maximal flat or a Chebychev approximation is desired. In all cases a fixed relation exists between the passband frequency Ω P and the tolerated deviation δ P in the passband when the degree n has been chosen.