(New page: Category:ECE301 Category:ECE438 Category:ECE438Fall2011Boutin Category:problem solving = Properties of the Z-transform = Prove the following scaling property of the z-trans...) |
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===Answer 1=== | ===Answer 1=== | ||
| − | + | I think there is a mistake, it should be <math>z_0^n</math> instead of <math>z_0^2</math>. | |
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| + | proof: | ||
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| + | <math>x'[n]=z_0^n x[n]</math> | ||
| + | |||
| + | <math>Z[x'[n]]=\sum_{n=-\infty}^{\infty}x'[n]z^{-n}=\sum_{n=-\infty}^{\infty}z_0^n x[n]z^{-n}=\sum_{n=-\infty}^{\infty}x[n](\frac{z}{z_0})^{-n}</math> | ||
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| + | <math>let k=\frac{z}{z_0}</math> | ||
| + | |||
| + | <math>Z[z_0^n x[n]]=\sum_{n=-\infty}^{\infty}x[n]k^{-n}=X(k)=X(\frac{z}{z_0})</math> | ||
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=== Answer 2=== | === Answer 2=== | ||
Write it here. | Write it here. | ||
Revision as of 09:09, 10 September 2011
Properties of the Z-transform
Prove the following scaling property of the z-transform:
$ z_0^2 x[n] \rightarrow X \left( \frac{z}{z_0}\right) $
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
I think there is a mistake, it should be $ z_0^n $ instead of $ z_0^2 $.
proof:
$ x'[n]=z_0^n x[n] $
$ Z[x'[n]]=\sum_{n=-\infty}^{\infty}x'[n]z^{-n}=\sum_{n=-\infty}^{\infty}z_0^n x[n]z^{-n}=\sum_{n=-\infty}^{\infty}x[n](\frac{z}{z_0})^{-n} $
$ let k=\frac{z}{z_0} $
$ Z[z_0^n x[n]]=\sum_{n=-\infty}^{\infty}x[n]k^{-n}=X(k)=X(\frac{z}{z_0}) $
Answer 2
Write it here.
