90 deg phase difference all-pass filter pairs... Lemme wave my hands a bit:
It's been years, but I recall I first tried a structure with complex conjugate pairs of poles (and their companion poles to make the filters all-pass). Globally optimizing that using Differential Evolution, the poles "wanted to be" near the real line. The ripples in the phase response difference of the optimized filters got finer by having two real poles in place of each complex conjugate pair of poles. The real poles and real zeros of the two filter paths alternate on the real line like a zebra stripe black pearl white pearl necklace. One of the paths is delayed (z^-1) by one sample, which gives the "center pearl" pole. I think the alternating order reflects the fact that a first-order all-pass filter gives a 180 deg phase shift end-to-end, and alternating the poles of the two paths corrects the phase response difference up and down so that we can stay approximately half-way between 0 deg and 180 deg, at 90 deg. One optimization here was to make the filter requirements symmetric around frequency pi/4, so that each positive pole can have a negative companion pole. This symmetry simplifies the all-pass filter computation. Artur Krukowski's papers were always over my head. -olli
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