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! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="4" | Z Transform Pairs | ! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="4" | Z Transform Pairs | ||
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| − | | align="right" style="padding-right: 1em;" | | + | | align="right" style="padding-right: 1em;" | |
| − | | < | + | | Signal <math> x[n] </math> |
| − | | | + | | |
| − | | <math> X( | + | | Transform <math> X(z) </math> |
| + | | | ||
| + | | ROC | ||
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| − | | align="right" style="padding-right: 1em;" | | + | | align="right" style="padding-right: 1em;" | Unit impulse signal |
| − | | <math>\delta | + | | <math> \delta[n]\ </math> |
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| − | | <math> 1 \ | + | | <math> 1\ </math> |
| + | | | ||
| + | | <math> All\ z\ </math> | ||
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| − | | align="right" style="padding-right: 1em;" | | + | | align="right" style="padding-right: 1em;" | Unit step signal |
| − | | <math> | + | | <math> u[n]\ </math> |
| | | | ||
| − | | <math> | + | | <math> \frac{1}{1-z^{-1}} </math> |
| + | | | ||
| + | | <math> |z| > 1\ </math> | ||
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| − | | align="right" style="padding-right: 1em;" | | + | | align="right" style="padding-right: 1em;" | |
| − | | <math> | + | | <math> -u[-n-1]\ </math> |
| | | | ||
| − | | <math> | + | | <math> \frac{1}{1-z^{-1}} </math> |
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| + | | <math> |z| < 1\ </math> | ||
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| − | | align="right" style="padding-right: 1em;" | | + | | align="right" style="padding-right: 1em;" | Shifted unit impulse signal |
| − | | <math> | + | | <math> \delta[n-m]\ </math> |
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| − | | <math> | + | | <math> z^{-m}\ </math> |
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| + | | <math> All\ z, except\ 0\ (if\ m>0)\ or\\ \infty \ (if\ m<0)\ </math> | ||
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| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
Revision as of 12:51, 27 November 2010
| Z Transform Pairs and Properties | |
|---|---|
| Definition Z Transform and its Inverse | |
| Single-side Z Transform | $ X(z)=\mathcal{L}(x[n])=\sum^{\infty}_{n=0}x[n]z^{-n} $ |
| Double-side 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 $ |
| Z Transform Pairs | |||||
|---|---|---|---|---|---|
| Signal $ x[n] $ | Transform $ X(z) $ | ROC | |||
| Unit impulse signal | $ \delta[n]\ $ | $ 1\ $ | $ All\ z\ $ | ||
| Unit step signal | $ u[n]\ $ | $ \frac{1}{1-z^{-1}} $ | $ |z| > 1\ $ | ||
| $ -u[-n-1]\ $ | $ \frac{1}{1-z^{-1}} $ | $ |z| < 1\ $ | |||
| Shifted unit impulse signal | $ \delta[n-m]\ $ | $ z^{-m}\ $ | $ All\ z, except\ 0\ (if\ m>0)\ or\\ \infty \ (if\ m<0)\ $ | ||
| $ te^{-at}u(t)\ $, where $ a\in {\mathbb R}, a>0 $ | $ \left( \frac{1}{a+i2\pi f}\right)^2 $ | ||||
| CTFT of a cosine | $ \cos(\omega_0 t) \ $ | $ \frac{1}{2} \left[\delta (f - \frac{\omega_0}{2\pi}) + \delta (f + \frac{\omega_0}{2\pi})\right] \ $ | |||
| CTFT of a sine | $ sin(\omega_0 t) \ $ | $ \frac{1}{2i} \left[\delta (f - \frac{\omega_0}{2\pi}) - \delta (f + \frac{\omega_0}{2\pi})\right] $ | |||
| CTFT of a rect | $ \left\{\begin{array}{ll}1, & \text{ if }|t|<T,\\ 0, & \text{else.}\end{array} \right. \ $ | $ \frac{\sin \left(2\pi Tf \right)}{\pi f} \ $ | |||
| CTFT of a sinc | $ \frac{2 \sin \left( W t \right)}{\pi t } \ $ | $ \left\{\begin{array}{ll}1, & \text{ if }|f| <\frac{W}{2\pi},\\ 0, & \text{else.}\end{array} \right. \ $ | |||
| CTFT of a periodic function | $ \sum^{\infty}_{k=-\infty} a_{k}e^{ikw_{0}t} $ | $ \sum^{\infty}_{k=-\infty}a_{k}\delta(f-\frac{kw_{0}}{2\pi}) \ $ | |||
| CTFT of an impulse train | $ \sum^{\infty}_{n=-\infty} \delta(t-nT) \ $ | $ \frac{1}{T}\sum^{\infty}_{k=-\infty}\delta(f-\frac{k}{T}) \ $ | |||
| CT Fourier Transform Properties | |||
|---|---|---|---|
| x(t) | $ \longrightarrow $ | $ X(f) $ | |
| multiplication property | $ x(t)y(t) \ $ | $ X(f)*Y(f) =\int_{-\infty}^{\infty} X(\theta)Y(f-\theta)d\theta $ | |
| convolution property | $ x(t)*y(t) \! $ | $ X(f)Y(f) \! $ | |
| time reversal | $ \ x(-t) $ | $ \ X(-f) $ | |
| Other CT Fourier Transform Properties | |
|---|---|
| Parseval's relation | $ \int_{-\infty}^{\infty} |x(t)|^2 dt = \int_{-\infty}^{\infty} |X(f)|^2 df $ |
