I read that ROC is a region,which is a set of values where z transform is defined ,that is it converges

Lets say i have a discrete time time signal $x[n]=n^2 u(n)$ and i want to find its ROC(Region of convergence),how can i do that?


1 Answer 1


You need to determine the values of the complex variable $z$ for which the series


converges. So you need to know a few things about infinite series.

For this case a simple test is the ratio test. You take the ratio of two successive elements of the series and compute the limit:


The series converges absolutely for $L<1$, i.e., for $1/|z|<1$, or $|z|>1$. So the ROC is the region $|z|>1$.

Note that the ROC would not change if we used any other power of $n$ in $(1)$.


A step for step explanation of Eq. $(2)$:


  • $\begingroup$ Please kindly elaborate eq(2) right side ,how you reach/arrived at 1/|z|,especillay considering how you removed limit and put limit equal to 1?? $\endgroup$
    – DSP_CS
    Dec 24, 2019 at 16:37
  • 1
    $\begingroup$ @engr: Please see my edited answer. $\endgroup$
    – Matt L.
    Dec 25, 2019 at 9:02
  • $\begingroup$ What if we had a minus sign in eq(1) before n2, ROC will be same that is |z|>1 or will it be opposite |z|<1? $\endgroup$
    – DSP_CS
    Dec 25, 2019 at 16:50
  • $\begingroup$ @engr: The minus sign wouldn't make any difference for the ROC. $\endgroup$
    – Matt L.
    Dec 25, 2019 at 17:26

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