# $Z$-transform of a multilinear function/ consecutive multplication of $k$ signals $y_1(n), \ldots, y_k(n)$

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

• does it help that a product in one domain is circular convolution in the other? – Dan Boschen Jun 2 at 12:45
• 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. – Hilmar Jun 2 at 13:42
• yes @Hilmar you are correct. I will omit it anyways because I am not really interested in the computational complexity. – shnnnms Jun 2 at 13:59
• @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 – shnnnms Jun 2 at 14:10
• 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 – Hilmar Jun 2 at 16:54