Line 52: | Line 52: | ||
| <math> z^{-m}\ </math> | | <math> z^{-m}\ </math> | ||
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− | | <math> All\ z, except\ 0\ (if\ m>0)\ or | + | | <math> All\ z, except\ 0\ (if\ m>0)\ or\ \infty \ (if\ m<0)\ </math> |
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
− | | <math> | + | | <math> \alpha^{n}u[n]\ </math> |
− | + | ||
− | + | ||
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+ | | <math> \frac{1}{1-\alpha z^{-1}} </math> | ||
+ | | | ||
+ | | <math> |z| > | \alpha |\ </math> | ||
|- | |- | ||
− | | align="right" style="padding-right: 1em;" | | + | | align="right" style="padding-right: 1em;" | |
− | | <math>\ | + | | <math> -\alpha^{n}u[-n-1]\ </math> |
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− | | <math> \frac{1}{ | + | | <math> \frac{1}{1-\alpha z^{-1}} </math> |
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+ | | <math> |z| < | \alpha |\ </math> | ||
|- | |- | ||
− | | align="right" style="padding-right: 1em;" | | + | | align="right" style="padding-right: 1em;" | |
− | | <math> | + | | <math> n\alpha^{n}u[n]\ </math> |
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− | | <math>\frac{1}{ | + | | <math> \frac{\alpha z^{-1}}{(1-\alpha z^{-1})^{2}} </math> |
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+ | | <math> |z| > | \alpha |\ </math> | ||
|- | |- | ||
− | | align="right" style="padding-right: 1em;" | | + | | align="right" style="padding-right: 1em;" | |
− | | <math>\ | + | | <math> -n\alpha^{n}u[-n-1]\ </math> |
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− | | <math> \frac{\ | + | | <math> \frac{\alpha z^{-1}}{(1-\alpha z^{-1})^{2}} </math> |
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+ | | <math> |z| < | \alpha |\ </math> | ||
|- | |- | ||
− | | align="right" style="padding-right: 1em;" | | + | | align="right" style="padding-right: 1em;" | Single-side cosine signal |
− | | <math>\ | + | | <math> [\cos{\omega_{0}n}]u[n] </math> |
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− | | <math>\ | + | | <math> \frac{1-[\cos{\omega_{0}}]z^{-1}}{1-[2\cos{\omega_{0}}]z^{-1}+z^{-2}} </math> |
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+ | | <math> |z| > 1\ </math> | ||
|- | |- | ||
| align="right" style="padding-right: 1em;" | CTFT of a periodic function | | align="right" style="padding-right: 1em;" | CTFT of a periodic function |
Revision as of 13:07, 27 November 2010
Z Transform Pairs and Properties | |
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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)\ $ | ||
$ \alpha^{n}u[n]\ $ | $ \frac{1}{1-\alpha z^{-1}} $ | $ |z| > | \alpha |\ $ | |||
$ -\alpha^{n}u[-n-1]\ $ | $ \frac{1}{1-\alpha z^{-1}} $ | $ |z| < | \alpha |\ $ | |||
$ n\alpha^{n}u[n]\ $ | $ \frac{\alpha z^{-1}}{(1-\alpha z^{-1})^{2}} $ | $ |z| > | \alpha |\ $ | |||
$ -n\alpha^{n}u[-n-1]\ $ | $ \frac{\alpha z^{-1}}{(1-\alpha z^{-1})^{2}} $ | $ |z| < | \alpha |\ $ | |||
Single-side cosine signal | $ [\cos{\omega_{0}n}]u[n] $ | $ \frac{1-[\cos{\omega_{0}}]z^{-1}}{1-[2\cos{\omega_{0}}]z^{-1}+z^{-2}} $ | $ |z| > 1\ $ | ||
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 | |
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Parseval's relation | $ \int_{-\infty}^{\infty} |x(t)|^2 dt = \int_{-\infty}^{\infty} |X(f)|^2 df $ |