All pass systems- Poles and Zeros

Question

If an All Pass system has poles at $0.8\exp\left(-j \dfrac{\pi}{4}\right)$ and $0.8\exp\left(j \dfrac{\pi}{4}\right)$, There would be $2\pi$ jumps in the phase (principal value) at angular frequency $\omega= \dfrac{\pi}{4}$ and $\omega= \dfrac{7\pi}{4}$, but why would there be $2\pi$ jumps also at $\omega= \dfrac{3\pi}{4}$ and $\omega= \dfrac{5\pi}{4}$ (Figure 5.24 in Oppenheim Schafer and Buck "Discrete Time Signal Processing 2nd Edition)?

I understand that in all pass systems with real coefficient frequency response, a pole at $z= z_1$ would mean a pole at $z=z_1^*$ and zeros at $z=\dfrac{1}{z_1^*}$, $z= \dfrac{1}{z_1}$. With this reasoning a pole at $0.8\exp\left(-j \dfrac{\pi}{4}\right)$ would mean zeros at $1.25\exp\left(j \dfrac{\pi}{4}\right)$ and $1.25\exp\left(-j \dfrac{\pi}{4}\right)$. But this still does not explain the $2\pi$ jumps in phase at $\dfrac{3\pi}{4}$ and $\dfrac{3\pi}{4}$.

Please Help. Thank you very much in advance.

Answer 1

There are no phase jumps. The discontinuities in the phase plot result from the computation of the principal value of the phase, which requires the phase to be in the range $[-\pi,\pi)$. But this is just a convention and has nothing to do with the physical system. Apart from issues with the principal value, the phase of allpass systems does not jump. Instead, the phase always decreases monotonically, because the group delay of an allpass is non-negative (Eq. (5.98) in Oppenheim's book). Also, if you look closely, the first "jump" is actually not at $\omega=\pi/4$ but a bit before, so it is not at all related to the pole angle.

'Real' phase jumps occur with systems whose frequency response has zeros. At these frequencies the phase jumps by a value of $\pi$. But obviously this can't happen with allpass systems because their frequency response is constant and doesn't have zeros. (Don't get confused: an allpass transfer function does have zeros in the complex $z$-plane, but not on the unit circle; this is why its frequency response, i.e. its transfer function evaluated on the unit circle, has no zeros).