Line 52: | Line 52: | ||
| <math> z^{-m}\ </math> | | <math> z^{-m}\ </math> | ||
| | | | ||
− | | <math> All\ z, except\ 0\ (if\ m>0)\ or\ \infty \ (if\ m<0)\ </math> | + | | <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;" | | ||
Line 113: | Line 113: | ||
{| | {| | ||
|- | |- | ||
− | ! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="4" | | + | ! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="4" | Z Transform Properties |
|- | |- | ||
| align="right" style="padding-right: 1em;" | | | align="right" style="padding-right: 1em;" | | ||
− | | < | + | | Signal <math> x[n] </math> |
− | | | + | | |
− | | <math> X(f) </math> | + | | Transform <math> X(f) </math> |
+ | | | ||
+ | | ROC | ||
|- | |- | ||
− | | align="right" style="padding-right: 1em;" | | + | | align="right" style="padding-right: 1em;" | |
− | | <math>x | + | | <math> x[n]\ </math> |
| | | | ||
− | | <math> X( | + | | <math> X(z)\ </math> |
+ | | | ||
+ | | <math> R\ </math> | ||
|- | |- | ||
− | | align="right" style="padding-right: 1em;" | | + | | align="right" style="padding-right: 1em;" | |
− | | <math> | + | | <math> x_{1}[n]\ </math> |
| | | | ||
− | | <math> | + | | <math> X_{1}(z)\ </math> |
+ | | | ||
+ | | <math> R_{1}\ </math> | ||
|- | |- | ||
− | | align="right" style="padding-right: 1em;" | | + | | align="right" style="padding-right: 1em;" | |
− | | <math>\ | + | | <math> x_{2}[n]\ </math> |
| | | | ||
− | | <math>\ X(- | + | | <math> X_{2}(z)\ </math> |
+ | | | ||
+ | | <math> R_{2}\ </math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | Linearity | ||
+ | | <math> ax_{1}[n]+bx_{2}[n]\ </math> | ||
+ | | | ||
+ | | <math> aX_{1}(z)+bX_{2}[z]\ </math> | ||
+ | | | ||
+ | | <math> At\ least\ the\ intersection\ of\ R_{1}\ and\ R_{2}\ </math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | Time shifting | ||
+ | | <math> x[n-n_{0}]\ </math> | ||
+ | | | ||
+ | | <math> z^{n_{0}}X(z)\ </math> | ||
+ | | | ||
+ | | <math> R,\ except\ for\ the\ possible\ addition\ or\ deletion\ of\ the\ origin\ </math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | Scaling in the z-domain | ||
+ | | <math> e^{j\omega_{0}n}x[n]\ </math> | ||
+ | | | ||
+ | | <math> X(e^{j\omega_{0}}z)\ </math> | ||
+ | | | ||
+ | | <math> R\ </math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | | ||
+ | | <math> z_{0}^{n}x[n]\ </math> | ||
+ | | | ||
+ | | <math> X(\frac{z}{z_{0}})\ </math> | ||
+ | | | ||
+ | | <math> z_{0}R\ </math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | | ||
+ | | <math> a^{n}x[n]\ </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> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | Time reversal | ||
+ | | <math> x[-n]\ </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> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | Time expansion | ||
+ | | <math> x^{(k)}=\ </math> | ||
+ | | | ||
+ | | <math> \ </math> | ||
+ | | | ||
+ | | <math> Scaled\ version\ of\ R\ (i.e.,\ |a|R=\ the\ set\ of\ points\ {|a|z}\ for\ z\ in\ R)\ </math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | Conjugation | ||
+ | | <math> x^{*}[n]\ </math> | ||
+ | | | ||
+ | | <math> X^{*}(z^{8})\ </math> | ||
+ | | | ||
+ | | <math> R\ </math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | Convolution | ||
+ | | <math> x_{1}[n]*x_{2}[n]\ </math> | ||
+ | | | ||
+ | | <math> X_{1}(z)X_{2}(z)\ </math> | ||
+ | | | ||
+ | | <math> At\ least\ the\ intersection\ of\ R_{1}\ and\ R_{2}\ </math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | First difference | ||
+ | | <math> x[n]-x[n-1]\ </math> | ||
+ | | | ||
+ | | <math> (1-z^(-1))X(z)\ </math> | ||
+ | | | ||
+ | | <math> At\ least\ the\ intersection\ of\ R\ and\ |z|>0\ </math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | Accumulation | ||
+ | | <math> \sum_{k=-\infty}^{n}x[k]\ </math> | ||
+ | | | ||
+ | | <math> \frac{1}{(1-z^(-1))}X(z)\ </math> | ||
+ | | | ||
+ | | <math> At\ least\ the\ intersection\ of\ R\ and\ |z|>1\ </math> | ||
+ | |- | ||
+ | | align="right" style="padding-right: 1em;" | Differentiation in the z-domain | ||
+ | | <math> nx[n]\ </math> | ||
+ | | | ||
+ | | <math> -z\frac{dX(z)}{dz}\ </math> | ||
+ | | | ||
+ | | <math> R\ </math> | ||
|} | |} | ||
{| | {| | ||
|- | |- | ||
− | ! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | Other | + | ! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | Other Z Transform Properties |
|- | |- | ||
− | | align="right" style="padding-right: 1em;" | | + | | align="right" style="padding-right: 1em;" | Initial Value Theorem |
− | | <math>\ | + | | <math> If\ x[n]=0\ for\ n<0,\ then\ x[0]=\lim_{z\rightarrow \infty}X(z)\ </math> |
|} | |} | ||
Revision as of 15:31, 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)}=\ $ | $ \ $ | $ Scaled\ version\ of\ R\ (i.e.,\ |a|R=\ the\ set\ of\ points\ {|a|z}\ for\ z\ in\ R)\ $ | ||
Conjugation | $ x^{*}[n]\ $ | $ X^{*}(z^{8})\ $ | $ 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)\ $ |