| Line 52: | Line 52: | ||
| <math> z^{-m}\ </math> | | <math> z^{-m}\ </math> | ||
| | | | ||
| − | | <math> All\ z | + | | <math> All\ z,\ except\ </math> |
| + | |- | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | | ||
| + | | | ||
| + | | | ||
| + | | | ||
| + | | | ||
| + | | <math> 0\ (if\ m>0)\ or\ </math> | ||
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | | ||
| + | | | ||
| + | | | ||
| + | | | ||
| + | | | ||
| + | | <math> \infty \ (if\ m<0)\ </math> | ||
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
| Line 155: | Line 170: | ||
| <math> z^{n_{0}}X(z)\ </math> | | <math> z^{n_{0}}X(z)\ </math> | ||
| | | | ||
| − | | <math> R,\ except\ for\ the\ possible\ addition\ or\ deletion\ of\ the\ origin\ </math> | + | | <math> R,\ except\ for\ the\ possible\ addition\ </math> |
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | | ||
| + | | | ||
| + | | | ||
| + | | | ||
| + | | | ||
| + | | <math> or\ deletion\ of\ the\ origin\ </math> | ||
|- | |- | ||
| align="right" style="padding-right: 1em;" | Scaling in the z-domain | | align="right" style="padding-right: 1em;" | Scaling in the z-domain | ||
| Line 176: | Line 198: | ||
| <math> X(a^{-1}z)\ </math> | | <math> X(a^{-1}z)\ </math> | ||
| | | | ||
| − | | <math> Scaled\ version\ of\ R\ (i.e.,\ |a|R=\ the\ set\ of\ points\ {|a|z}\ for\ z\ in\ R)\ </math> | + | | <math> Scaled\ version\ of\ R\ </math> |
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | | ||
| + | | | ||
| + | | | ||
| + | | | ||
| + | | | ||
| + | | <math> (i.e.,\ |a|R=\ the\ set\ of\ points\ {|a|z}\ for\ z\ in\ R)\ </math> | ||
|- | |- | ||
| align="right" style="padding-right: 1em;" | Time reversal | | align="right" style="padding-right: 1em;" | Time reversal | ||
| Line 183: | Line 212: | ||
| <math> X(z^{-1})\ </math> | | <math> X(z^{-1})\ </math> | ||
| | | | ||
| − | | <math> R^{1/k}\ (i.e.,\ the\ set\ of\ points\ z^{1/k},\ where\ z\ is\ in\ R)\ </math> | + | | <math> R^{1/k}\ (i.e.,\ the\ set\ of\ points\ z^{1/k},\ </math> |
| + | |- | ||
| + | | align="right" style="padding-right: 1em;" | | ||
| + | | | ||
| + | | | ||
| + | | | ||
| + | | | ||
| + | | <math> where\ z\ is\ in\ R)\ </math> | ||
|- | |- | ||
| align="right" style="padding-right: 1em;" | Time expansion | | align="right" style="padding-right: 1em;" | Time expansion | ||
| Line 201: | Line 237: | ||
| | | | ||
| | | | ||
| − | | <math> where\ z\ is\ in\ R\ </math> | + | | <math> where\ z\ is\ in\ R)\ </math> |
|- | |- | ||
| align="right" style="padding-right: 1em;" | Conjugation | | align="right" style="padding-right: 1em;" | Conjugation | ||
Revision as of 17:19, 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)\ $ | |||||
| $ \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\ $ | ||
| Single-side sine signal | $ [\sin{\omega_{0}n}]u[n]\ $ | $ \frac{1-[\sin{\omega_{0}}]z^{-1}}{1-[2\cos{\omega_{0}}]z^{-1}+z^{-2}} $ | $ |z| > 1\ $ | ||
| $ [r^{n}\cos{\omega_{0}n}]u[n]\ $ | $ \frac{1-[r\cos{\omega_{0}}]z^{-1}}{1-[2r\cos{\omega_{0}}]z^{-1}+r^{2}z^{-2}} $ | $ |z| > r\ $ | |||
| $ [r^{n}\sin{\omega_{0}n}]u[n]\ $ | $ \frac{1-[r\sin{\omega_{0}}]z^{-1}}{1-[2r\cos{\omega_{0}}]z^{-1}+r^{2}z^{-2}} $ | $ |z| > r\ $ | |||
| Z Transform Properties | |||||
|---|---|---|---|---|---|
| Signal $ x[n] $ | Transform $ X(f) $ | ROC | |||
| $ x[n]\ $ | $ X(z)\ $ | $ R\ $ | |||
| $ x_{1}[n]\ $ | $ X_{1}(z)\ $ | $ R_{1}\ $ | |||
| $ x_{2}[n]\ $ | $ X_{2}(z)\ $ | $ R_{2}\ $ | |||
| Linearity | $ ax_{1}[n]+bx_{2}[n]\ $ | $ aX_{1}(z)+bX_{2}[z]\ $ | $ At\ least\ the\ intersection\ of\ R_{1}\ and\ R_{2}\ $ | ||
| Time shifting | $ x[n-n_{0}]\ $ | $ z^{n_{0}}X(z)\ $ | $ R,\ except\ for\ the\ possible\ addition\ $ | ||
| $ or\ deletion\ of\ the\ origin\ $ | |||||
| Scaling in the z-domain | $ e^{j\omega_{0}n}x[n]\ $ | $ X(e^{j\omega_{0}}z)\ $ | $ R\ $ | ||
| $ z_{0}^{n}x[n]\ $ | $ X(\frac{z}{z_{0}})\ $ | $ z_{0}R\ $ | |||
| $ a^{n}x[n]\ $ | $ X(a^{-1}z)\ $ | $ Scaled\ version\ of\ R\ $ | |||
| $ (i.e.,\ |a|R=\ the\ set\ of\ points\ {|a|z}\ for\ z\ in\ R)\ $ | |||||
| Time reversal | $ x[-n]\ $ | $ X(z^{-1})\ $ | $ R^{1/k}\ (i.e.,\ the\ set\ of\ points\ z^{1/k},\ $ | ||
| $ where\ z\ is\ in\ R)\ $ | |||||
| Time expansion | $ x^{(k)}= \begin{cases} x[r], &n=rk \\ 0, &n\neq rk \end{cases} $ | $ X(z^{k})\ $ | $ R^{1/k}\ (i.e.,\ the\ set\ of\ points\ z^{1/k},\ $ | ||
| $ where\ z\ is\ in\ R)\ $ | |||||
| Conjugation | $ x^{*}[n]\ $ | $ X^{*}(z^{*})\ $ | $ R\ $ | ||
| Convolution | $ x_{1}[n]*x_{2}[n]\ $ | $ X_{1}(z)X_{2}(z)\ $ | $ At\ least\ the\ intersection\ of\ R_{1}\ and\ R_{2}\ $ | ||
| First difference | $ x[n]-x[n-1]\ $ | $ (1-z^(-1))X(z)\ $ | $ At\ least\ the\ intersection\ of\ R\ and\ |z|>0\ $ | ||
| Accumulation | $ \sum_{k=-\infty}^{n}x[k]\ $ | $ \frac{1}{(1-z^(-1))}X(z)\ $ | $ At\ least\ the\ intersection\ of\ R\ and\ |z|>1\ $ | ||
| Differentiation in the z-domain | $ nx[n]\ $ | $ -z\frac{dX(z)}{dz}\ $ | $ R\ $ | ||
| Other Z Transform Properties | |
|---|---|
| Initial Value Theorem | $ If\ x[n]=0\ for\ n<0,\ then\ x[0]=\lim_{z\rightarrow \infty}X(z)\ $ |
