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| + | = Examples<br> = | ||
| − | = | + | == 1. Unit step == |
| + | When <math>x[n]=1 (n{\ge}0)</math> | ||
| + | <span class="texhtml">''x''[''n''] = 0(''n'' < 0)</span> | ||
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| + | <math>X(z)=\sum_{n=0}^{\infty}x[n]z^{n}=\sum_{n=0}^{\infty}1\cdot z^{-n}=\frac{1}{1-z^{-1}}</math> | ||
| − | + | == 2. Power series == | |
| + | <math>X(z)=\sum_{n=0}^{\infty}x[n]z^{n}=\sum_{n=0}^{\infty}a^{n} z^{-n}=\frac{1}{1-az^{-1}}</math> | ||
| + | == 3. Exponential funtion == | ||
| + | <math>X(z)=\sum_{n=0}^{\infty}x[n]z^{n}=\sum_{n=0}^{\infty}e^{-an} z^{-n}=\sum_{n=0}^{\infty}[e^{-a} z^{-1}]^{n}=\frac{1}{1-e^{-a}z^{-n}}</math> | ||
| − | [[ Z transform ECE438F10|Back to Z transform ECE438F10]] | + | == 4. Sinusoidal function == |
| + | |||
| + | <math>X(z)=\sum_{n=0}^{\infty}x[n]z^{n}=\sum_{n=0}^{\infty}\frac{e^{jn{\omega}} -e^{-jn{\omega}}} {2j} z^{-n}</math> <math> =\frac{1}{2j} (\frac{1}{1-e^{j\omega}z^{-1}}-\frac{1}{1-e^{-j\omega}z^{-1}})</math> <math> =\frac{1}{2j} (\frac{-e^{-j\omega}z^{-1}+e^{j\omega}z^{-1}}{1-e^{-j\omega}z^{-1}-e^{j\omega}z^{-1}+z^{-2}})</math> <math> =\frac{z^{-1}sin(\omega)}{1-2z^{-1}cos(\omega)+z^{-2}}</math> | ||
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| + | <br>[[Z transform ECE438F10|Back to Z transform ECE438F10]] | ||
Latest revision as of 17:52, 16 December 2010
Contents
Examples
1. Unit step
When $ x[n]=1 (n{\ge}0) $ x[n] = 0(n < 0)
$ X(z)=\sum_{n=0}^{\infty}x[n]z^{n}=\sum_{n=0}^{\infty}1\cdot z^{-n}=\frac{1}{1-z^{-1}} $
2. Power series
$ X(z)=\sum_{n=0}^{\infty}x[n]z^{n}=\sum_{n=0}^{\infty}a^{n} z^{-n}=\frac{1}{1-az^{-1}} $
3. Exponential funtion
$ X(z)=\sum_{n=0}^{\infty}x[n]z^{n}=\sum_{n=0}^{\infty}e^{-an} z^{-n}=\sum_{n=0}^{\infty}[e^{-a} z^{-1}]^{n}=\frac{1}{1-e^{-a}z^{-n}} $
4. Sinusoidal function
$ X(z)=\sum_{n=0}^{\infty}x[n]z^{n}=\sum_{n=0}^{\infty}\frac{e^{jn{\omega}} -e^{-jn{\omega}}} {2j} z^{-n} $ $ =\frac{1}{2j} (\frac{1}{1-e^{j\omega}z^{-1}}-\frac{1}{1-e^{-j\omega}z^{-1}}) $ $ =\frac{1}{2j} (\frac{-e^{-j\omega}z^{-1}+e^{j\omega}z^{-1}}{1-e^{-j\omega}z^{-1}-e^{j\omega}z^{-1}+z^{-2}}) $ $ =\frac{z^{-1}sin(\omega)}{1-2z^{-1}cos(\omega)+z^{-2}} $
