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===Answer 1=== | ===Answer 1=== | ||
| − | + | alec green | |
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| + | [[Image:Green26_ece438_hmwrk3_power_series.png| 480x320px]] | ||
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| + | <math>X(z) = \sum_{n=-\infty}^{+\infty} x[n]z^{-n}</math> | ||
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| + | <math>= \sum_{n=-3}^{+\infty} 3^{n}z^{-n}</math> | ||
| + | |||
| + | <math>= \sum_{n=-3}^{+\infty} (\frac{3}{z})^{n}</math> | ||
| + | |||
| + | Let k = n+3: | ||
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| + | <math>= \sum_{k=0}^{+\infty} (\frac{3}{z})^{k-3}</math> | ||
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| + | Using the geometric series property: | ||
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| + | <math> | ||
| + | X(z) = \left\{ | ||
| + | \begin{array}{l l} | ||
| + | (\frac{z}{3})^3 \frac{1}{1-\frac{3}{z}} & \quad |z| > 3\\ | ||
| + | \text{diverges} & \quad \text{else} | ||
| + | \end{array} \right. | ||
| + | </math> | ||
| + | |||
=== Answer 2=== | === Answer 2=== | ||
Write it here. | Write it here. | ||
Revision as of 12:45, 12 September 2013
Contents
Practice Problem on Z-transform computation
Compute the compute the z-transform (including the ROC) of the following DT signal:
$ x[n]=3^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
alec green
$ X(z) = \sum_{n=-\infty}^{+\infty} x[n]z^{-n} $
$ = \sum_{n=-3}^{+\infty} 3^{n}z^{-n} $
$ = \sum_{n=-3}^{+\infty} (\frac{3}{z})^{n} $
Let k = n+3:
$ = \sum_{k=0}^{+\infty} (\frac{3}{z})^{k-3} $
Using the geometric series property:
$ X(z) = \left\{ \begin{array}{l l} (\frac{z}{3})^3 \frac{1}{1-\frac{3}{z}} & \quad |z| > 3\\ \text{diverges} & \quad \text{else} \end{array} \right. $
Answer 2
Write it here.
Answer 3
Write it here.
Answer 4
Write it here.
