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I came across this root locus diagram in my DSP class. It has a pole at the origin, another pole within the unit circle and a zero just outside the unit circle. I understand that the poles and zeros should be complex conjugates for an all pass filter. So will a pole or a zero at the origin prevent the system from being an all pass?My professor tells me that the pole at the origin is allowed. So that is basically leaving me confused.Can anyone explain how this is possible?

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a delay is also an APF of sorts. it passes the same magnitude and it affects the phase (in a linear manner w.r.t. frequency). a delay of one sample has transfer function

$$ H(z) = z^{-1} = \frac{1}{z} = \frac{1}{z-0} $$

which means there is a pole at $z=0$. and it is true that $\left|H\left( e^{j \omega} \right)\right| = 1$, so it's an APF.

so, if the remaining pole/zero pairs are at reciprocal values this extra delay changes the phase of your APF a little, but not the magnitude response, so i think, strictly speaking, that it's still an APF.

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  • $\begingroup$ ok.What happens in the case of zeros at the origin.If H(z)=z(z-0.5)/z-2;theres a zero at the origin and the other pole/zero pair is reciprocal.I assume the system still should be an APF.right? $\endgroup$ – sacjohn Sep 29 '14 at 22:09
  • $\begingroup$ @sacjohn: Yes, but it's not causal anymore. A causal and stable allpass filter has all its poles inside the unit circle and all its zeros outside the unit circle. $\endgroup$ – Matt L. Sep 30 '14 at 6:51

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