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How should one go about calculating the $Z$-transform of a signal that is the multiplication of $k$ signals (i.e. a multilinear function with regards to signals $y_1(n) \ldots y_k(n)$ ? Namely, $\mathcal{Z}\{ y_1(n) \cdots y_k(n) \}$ ?

Is there a way to bound the coefficients of the resulting $z$ polynomial?

Would the complexity be $O( {\text{length-of-$y(\cdot)$} } ^k)$?

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  • $\begingroup$ does it help that a product in one domain is circular convolution in the other? $\endgroup$ – Dan Boschen Jun 2 at 12:45
  • $\begingroup$ I think there is some context missing. What is $n$? It looks like a time index $y(n)$ but then the expression $O(n^k)$ makes no sense since $n$ goes from minus infinity to plus infinity. $\endgroup$ – Hilmar Jun 2 at 13:42
  • $\begingroup$ yes @Hilmar you are correct. I will omit it anyways because I am not really interested in the computational complexity. $\endgroup$ – shnnnms Jun 2 at 13:59
  • $\begingroup$ @DanBoschen, I am aware of this property but I am unsure about moving from $k=2$ to $k \geq 2$. $y_1(n) \cdot y_2(n) \cdot y_3(n)$. How does associativity of multiplication carry over to the $z$-domain $\endgroup$ – shnnnms Jun 2 at 14:10
  • $\begingroup$ This depends a lot on whether your original sequence finite, infinite aperiodic, or infinite periodic, i.e.. whether the z-transform is continuous or discrete. Good treatment for the continuous case: youtube.com/watch?v=MgjH5dWLJyo $\endgroup$ – Hilmar Jun 2 at 16:54

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