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when n=0,1,2, x[n] is n; otherwise x[n]=0. So: | when n=0,1,2, x[n] is n; otherwise x[n]=0. So: | ||
| − | <math>x(z)=0z^{-0}+1z^{-1}+2z^{-2}=\frac{1}{z}+\frac{2}{z^2}</math> | + | <math>x(z)=0z^{-0}+1z^{-1}+2z^{-2}=\frac{1}{z}+\frac{2}{z^2}</math> with ROC=all finite complex number. |
| + | |||
| + | test for infinity: | ||
| + | |||
| + | <math>X(\frac{1}{z})=z+z^2</math> | ||
| + | |||
| + | when z=0,<math>X(\frac{1}{z})</math>converges | ||
| + | |||
| + | X(z) converges at <math>z=\infty</math> | ||
| + | |||
| + | so ROC of X(z) is all complex number. | ||
---- | ---- | ||
[[2011_Fall_ECE_438_Boutin|Back to ECE438 Fall 2011 Prof. Boutin]] | [[2011_Fall_ECE_438_Boutin|Back to ECE438 Fall 2011 Prof. Boutin]] | ||
Revision as of 08:24, 10 September 2011
Z-transform computation
Compute the compute the z-transform (including the ROC) of the following DT signal:
$ x[n]= n u[n]-n u[n-3] $
(Write enough intermediate steps to fully justify your answer.)
You will receive feedback from your instructor and TA directly on this page. Other students are welcome to comment/discuss/point out mistakes/ask questions too!
Answer 1
Begin with the definition of a Z-Transform.
$ X(z) = \sum_{n=-\infty}^{\infty}(n u[n]-n u[n-3])z^{-n} $
Simplify a little. (pull out the n and realize $ u[n]-u[n-3] $ is only non-zero for 0, 1, and 2.)
$ X(z) = \sum_{n=0}^{2}n z^{-n} $
Then we have a simple case of evaluating for 3 points.
$ \begin{align} X(z) &= 0 z^{-0} + 1 z^{-1} + 2 z^{-2} \\ &= \frac{z+2}{z^2} \end{align} $
Answer 2
$ Z(x[n])= \sum_{n=-\infty}^{\infty}x[n]z^{-n}= \sum_{n=-\infty}^{\infty}n(u[n]- u[n-3])z^{-n} $
when n=0,1,2, x[n] is n; otherwise x[n]=0. So:
$ x(z)=0z^{-0}+1z^{-1}+2z^{-2}=\frac{1}{z}+\frac{2}{z^2} $ with ROC=all finite complex number.
test for infinity:
$ X(\frac{1}{z})=z+z^2 $
when z=0,$ X(\frac{1}{z}) $converges
X(z) converges at $ z=\infty $
so ROC of X(z) is all complex number.
