(New page: Category:slecture Category:ECE438Fall2014Boutin Category:ECE Category:ECE438 Category:signal processing <center><font size= 5> Inverse Z Transform </font size></cen...) |
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<math>x[n]=\mathcal{L}^{-1}(X(z))=\frac{1}{2\pi j}\oint_{c}X(z)z^{n-1}dz</math> | <math>x[n]=\mathcal{L}^{-1}(X(z))=\frac{1}{2\pi j}\oint_{c}X(z)z^{n-1}dz</math> | ||
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| + | 1. Example Problems of the Inverse Z Transform | ||
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| + | :We will find the Inverse Z transform of various signals by manipulation and then using direct Inversion | ||
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| + | Ex. 1 Find the Inverse Z transform of the following signal | ||
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| + | <math>X(z)=\frac{1}{1-z}, \text{ ROC } |z|<1 </math> | ||
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| + | :<font size = 2>note: It is important to realize that we are not going to try to use the direct formula for an inverse Z transform, Instead our approach will be to manipulate the signal so that we can directly compare it with the Z transform equation and by inspection obtain the Inverse Z transform.</font size> | ||
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| + | <font size= 2>First we need to manipulate the given ROC equation to be in the following form, with 'A' being some expression that contains z</font size> | ||
| + | <center><math>|A| < 1</math></center> | ||
Revision as of 14:50, 13 December 2014
Inverse Z Transform
Overview
- The purpose of this page is to...
- 1. Define the Z Transform and Inverse Z Transform
- 2. Provide Example Problems of the Inverse Z Transform
1. Definitions
- Z Transform
$ X(z)=\mathcal{L}(x[n])=\sum_{n=-\infty}^{\infty}x[n]z^{-n} $
- Inverse Z Transform
$ x[n]=\mathcal{L}^{-1}(X(z))=\frac{1}{2\pi j}\oint_{c}X(z)z^{n-1}dz $
1. Example Problems of the Inverse Z Transform
- We will find the Inverse Z transform of various signals by manipulation and then using direct Inversion
Ex. 1 Find the Inverse Z transform of the following signal
$ X(z)=\frac{1}{1-z}, \text{ ROC } |z|<1 $
- note: It is important to realize that we are not going to try to use the direct formula for an inverse Z transform, Instead our approach will be to manipulate the signal so that we can directly compare it with the Z transform equation and by inspection obtain the Inverse Z transform.
First we need to manipulate the given ROC equation to be in the following form, with 'A' being some expression that contains z
