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I know that if I have a closed-form algebraic expression $X(z)$ and I specify the region of convergence, this uniquely identifies exactly time-domain sequence (inverse Z-transform) $x[n]$.

Let's suppose I have a pole-zero plot specifying that $X(z)$ has a single zero at $z=-1$ and four poles at $z=1/2, 2, 3,$ and $4$ (just for the sake of a concrete example). However, I have not specified an algebraic form of $X(z)$. Let's suppose that I then specify the region of convergence as $2<|z|<3$. Am I justified in saying that there is exactly one time-domain sequence $x[n]$ that corresponds to this pole-zero plot and ROC?

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HINT:

Note that the poles and zeros of

$$X(z)=c\cdot\frac{N(z)}{D(z)}$$

only depend on the polynomials $N(z)$ and $D(z)$. Consequently, pole and zero locations determine the transfer function only up to a scalar constant.

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  • $\begingroup$ Oop, I forgot that since the pole-zero plot corresponds to a rational transfer function we can convert it into the form you've written. So there are actually an infinite number of sequences *doh* Many thanks! $\endgroup$
    – marlinTJ
    Feb 14, 2023 at 21:28

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