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I'm learning about poles and zeros of transfer functions in my signals class and we have just covered the effects of poles outside the unit circle on the pole-zero plot. Intuitively, it makes sense that a pole outside the circle would cause the system to become unstable. However, I am a bit confused about the effect of a zero outside the unit circle. I was told that zeros do not affect the stability of a system, even if they are found outside the unit circle. Is there a reason why this is true? Mathematically, how would this be different from a pole?

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    $\begingroup$ Do you know the influence of poles and zeros on the behavior of the system response? $\endgroup$
    – Jdip
    Dec 4, 2023 at 17:25
  • $\begingroup$ I think I understand the base concepts. A zero will zero out the frequency it is located at while a pole will amplify its location frequency. $\endgroup$
    – marco
    Dec 4, 2023 at 17:52

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Instability refers to a situation where the output of a system grows without bound over time. This can happen when one or more poles lie outside the unit circle.

As you correctly observed in your question's comment section, zeros will "zero out" the output, so the output can never "grow" no matter where this zero is located.

Zeros inside/outside the unit circle is also an important concept, but they do not affect stability.

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  • $\begingroup$ "This can happen when one or more poles lie outside the unit circle.": if you need detail on that, happy to give you some, but you said you already understood this. $\endgroup$
    – Jdip
    Dec 4, 2023 at 19:02
  • $\begingroup$ No, I understand it now, thank you for your help! $\endgroup$
    – marco
    Dec 4, 2023 at 19:08

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