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" | CT Fourier Transform Properties
+
! 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;" |  
−
| <span class="texhtml">''x''(''t'')</span>  
+
| Signal <math> x[n] </math>  
−
| <math>\longrightarrow</math>
+
|
−
| <math> X(f) </math>
+
| Transform <math> X(f) </math>
 +
|
 +
| ROC
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | multiplication property
+
| align="right" style="padding-right: 1em;" |  
−
| <math>x(t)y(t) \ </math>  
+
| <math> x[n]\ </math>  
 
|  
 
|  
−
| <math> X(f)*Y(f) =\int_{-\infty}^{\infty} X(\theta)Y(f-\theta)d\theta</math>
+
| <math> X(z)\ </math>
 +
|
 +
| <math> R\ </math>
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | convolution property
+
| align="right" style="padding-right: 1em;" |  
−
| <math>x(t)*y(t) \!</math>  
+
| <math> x_{1}[n]\ </math>  
 
|  
 
|  
−
| <math> X(f)Y(f) \!</math>
+
| <math> X_{1}(z)\ </math>
 +
|
 +
| <math> R_{1}\ </math>
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | time reversal
+
| align="right" style="padding-right: 1em;" |  
−
| <math>\ x(-t) </math>  
+
| <math> x_{2}[n]\ </math>  
 
|  
 
|  
−
| <math>\ X(-f)</math>
+
| <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 CT Fourier Transform Properties
+
! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | Other Z Transform Properties
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | Parseval's relation
+
| align="right" style="padding-right: 1em;" | Initial Value Theorem
−
| <math>\int_{-\infty}^{\infty} |x(t)|^2 dt = \int_{-\infty}^{\infty} |X(f)|^2 df</math>
+
| <math> If\ x[n]=0\ for\ n<0,\ then\ x[0]=\lim_{z\rightarrow \infty}X(z)\ </math>
 
|}
 
|}
  

Revision as of 14:31, 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)\ $

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