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Of course it's valid for real coefficients because real number is a subset of complex number and the proof doesn't make any assumption whether the coefficient is complex or real. For a complex $d$ \begin{aligned} |A(z)|^2 &= A(z)A^*(z) = \frac{1-d^*z}{z-d} \frac{1-dz^*}{z^*-d^*}\\ &=\frac{1-(d^*z+dz^*)+|d|^2|z|^2}{|z|^2 - (d^*z+dz^*) + |d|^2} \end{... 1 As far as I know this only works with odd-order Butterworth filters. You don't have a choice of all pass filters: they are simply determined by the pole locations of the Butterworth. An adjustable delay line won't help here. 1 The Audio EQ Cookbook has the general formulae for all-pass filters (APF). You can maybe cascade four of these APFs together and get 12° at 1 kHz and 360°+4° at 2 kHz. But it's a wild APF and will definitely have some "resonance" (of a sort) around 1.5 kHz. 1 This is a tricky problem. First you need to decide whether you need a "true" all pass filter or can do with an approximate one. The difference being|H_\text{trueAP}(\omega) = 1| \\ | |H_\text{apprAP}(\omega)| -1 | < \epsilon, \omega_1 < \omega < \omega_2  i.e. do you need the magnitude to be exactly one for all frequencies or do you ...