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| <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;" |  
−
| <math>te^{-at}u(t)\ </math>, where <math>a\in {\mathbb R}, a>0 </math>
+
| <math> \alpha^{n}u[n]\ </math>  
−
|
+
−
| <math>\left( \frac{1}{a+i2\pi f}\right)^2</math>  
+
 
|  
 
|  
 +
| <math> \frac{1}{1-\alpha z^{-1}} </math>
 +
|
 +
| <math> |z| > | \alpha |\ </math>
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | CTFT of a cosine
+
| align="right" style="padding-right: 1em;" |
−
| <math>\cos(\omega_0 t) \ </math>  
+
| <math> -\alpha^{n}u[-n-1]\ </math>  
 
|  
 
|  
−
| <math> \frac{1}{2} \left[\delta (f - \frac{\omega_0}{2\pi}) + \delta (f + \frac{\omega_0}{2\pi})\right] \ </math>  
+
| <math> \frac{1}{1-\alpha z^{-1}} </math>  
 
|  
 
|  
 +
| <math> |z| < | \alpha |\ </math>
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | CTFT of a sine
+
| align="right" style="padding-right: 1em;" |
−
| <math>sin(\omega_0 t)  \ </math>  
+
| <math> n\alpha^{n}u[n]\ </math>  
 
|  
 
|  
−
| <math>\frac{1}{2i} \left[\delta (f - \frac{\omega_0}{2\pi}) - \delta (f + \frac{\omega_0}{2\pi})\right]</math>  
+
| <math> \frac{\alpha z^{-1}}{(1-\alpha z^{-1})^{2}} </math>  
 
|  
 
|  
 +
| <math> |z| > | \alpha |\ </math>
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | CTFT of a rect
+
| align="right" style="padding-right: 1em;" |  
−
| <math>\left\{\begin{array}{ll}1, &  \text{ if }|t|<T,\\ 0, & \text{else.}\end{array} \right. \ </math>  
+
| <math> -n\alpha^{n}u[-n-1]\ </math>  
 
|  
 
|  
−
| <math> \frac{\sin \left(2\pi Tf \right)}{\pi f} \ </math>  
+
| <math> \frac{\alpha z^{-1}}{(1-\alpha z^{-1})^{2}} </math>  
 
|  
 
|  
 +
| <math> |z| < | \alpha |\ </math>
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | CTFT of a sinc
+
| align="right" style="padding-right: 1em;" | Single-side cosine signal
−
| <math>\frac{2 \sin \left( W t  \right)}{\pi t } \ </math>  
+
| <math> [\cos{\omega_{0}n}]u[n] </math>  
 
|  
 
|  
−
| <math>\left\{\begin{array}{ll}1, &  \text{ if }|f| <\frac{W}{2\pi},\\ 0, & \text{else.}\end{array} \right.  \ </math>  
+
| <math> \frac{1-[\cos{\omega_{0}}]z^{-1}}{1-[2\cos{\omega_{0}}]z^{-1}+z^{-2}} </math>  
 
|  
 
|  
 +
| <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 12:07, 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\ $
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 $

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Alumni Liaison

Ph.D. on Applied Mathematics in Aug 2007. Involved on applications of image super-resolution to electron microscopy

Francisco Blanco-Silva