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[[Category:Formulas]]
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[[Category:z-transform]]
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[[Category:inverse z-transform]]
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[[Category:ECE301]]
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[[Category:ECE438]]
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[[Category:ECE]]
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<center><font size= 4>
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'''[[Collective_Table_of_Formulas|Collective Table of Formulas]]'''
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</font size>
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Table of (double-sided)  [[info_z-transform|Z Transform]] Pairs and Properties
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(Used in [[ECE301]], [[ECE438]], [[ECE538]])
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</center>
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----
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{|
 
{|
 
|-
 
|-
−
! style="background: none repeat scroll 0% 0% rgb(228, 188, 126); font-size: 110%;" colspan="2" | Z Transform Pairs and Properties
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! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="3" | (double-sided) [[info_z-transform|Z Transform]] and its Inverse
 
|-
 
|-
−
! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | Definition Z Transform and its Inverse
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| align="right" style="padding-right: 1em;" |   [[info_z-transform|(Double-side)  Z Transform]]
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| <math>X(z)=\mathcal{Z}(x[n])=\sum_{n=-\infty}^{\infty}x[n]z^{-n} \ </math>
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|[[info_z-transform|(info)]]
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" |  Single-side Z Transform  
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| align="right" style="padding-right: 1em;" |  [[info_inverse_z-transform| Inverse Z Transform]]
−
| <math>X(z)=\mathcal{L}(x[n])=\sum^{\infty}_{n=0}x[n]z^{-n}</math>
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| <math>x[n]=\mathcal{Z}^{-1}(X(z))=\frac{1}{2\pi j}\oint_{c}X(z)z^{n-1}dz  \ </math>
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| [[info_inverse_z-transform|(info)]]
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" |  Double-side Z Transform
 
−
| <math>X(z)=\mathcal{L}(x[n])=\sum_{n=-\infty}^{\infty}x[n]z^{-n}</math>
 
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | Inverse Z Transform
 
−
| <math>x[n]=\mathcal{L}^{-1}(X(z))=\frac{1}{2\pi j}\oint_{c}X(z)z^{n-1}dz</math>
 
 
|}
 
|}
  
 
{|
 
{|
 
|-
 
|-
−
! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="4" | Z Transform Pairs  
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! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="6" | (double-sided)  [[info_z-transform|Z Transform]] Pairs  
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" |
 
| align="right" style="padding-right: 1em;" |
−
| Signal <math> x[n] </math>
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| Signal  
 
|   
 
|   
−
| Transform <math> X(z) </math>
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| Transform  
 
|  
 
|  
−
| ROC
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| Region of convergence (ROC)
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" | Unit impulse signal
 
| align="right" style="padding-right: 1em;" | Unit impulse signal
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| <math> 1\ </math>
 
| <math> 1\ </math>
 
|
 
|
−
| <math> All\ z\ </math>
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| All complex <math> z\ </math> including <math>\infty</math>
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" | Unit step signal  
 
| align="right" style="padding-right: 1em;" | Unit step signal  
 
| <math> u[n]\ </math>  
 
| <math> u[n]\ </math>  
 
|  
 
|  
−
| <math> \frac{1}{1-z^{-1}} </math>
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| <math> \frac{1}{1-z^{-1}} \ </math>
 
|
 
|
−
| <math> |z| > 1\ </math>
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| <math> |z| > 1\ </math> [[Compute_z-transform_u_n_ECE301S11|(computation)]]
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
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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>
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| <math> All\ z,\ except\ </math>
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|-
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|-
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| align="right" style="padding-right: 1em;" |
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|
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|
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|
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|
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| <math>  0\ (if\ m>0)\ or\ </math>
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|-
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| align="right" style="padding-right: 1em;" |
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|
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|
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|
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|
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| <math>  \infty \ (if\ m<0)\ </math>
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
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|  
 
|  
 
| <math> |z| > r\ </math>
 
| <math> |z| > r\ </math>
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|-
 
|}
 
|}
  
 
{|
 
{|
 
|-
 
|-
−
! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="4" | CT Fourier Transform Properties
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! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="6" | (double-sided)  [[info_z-transform|Z Transform]] Properties
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|-
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! colspan="6" | Below <math>x[n]</math>, <math>x_1[n]</math> and <math>x_2[n]</math> are DT signals with z-transforms <math>X(z)</math>, <math>X_1(Z)</math>, <math>X_2(z)</math>, and region of convergence (ROC) <math>R</math>, <math>R_1</math>, <math>R_2</math>, respectively.
 
|-
 
|-
 
| align="right" style="padding-right: 1em;" |  
 
| align="right" style="padding-right: 1em;" |  
−
| <span class="texhtml">''x''(''t'')</span>
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| Signal 
−
| <math>\longrightarrow</math>
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|
−
| <math> X(f) </math>
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| Z-Transform
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|
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| ROC
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | multiplication property
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| align="right" style="padding-right: 1em;" | Linearity
−
| <math>x(t)y(t) \ </math>  
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| <math> ax_{1}[n]+bx_{2}[n]\ </math>  
 
|  
 
|  
−
| <math> X(f)*Y(f) =\int_{-\infty}^{\infty} X(\theta)Y(f-\theta)d\theta</math>
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| <math> aX_{1}(z)+bX_{2}[z]\ </math>
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|
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| <math> At\ least\ the\ intersection\ of\ R_{1}\ and\ R_{2}\ </math>
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | convolution property
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| align="right" style="padding-right: 1em;" | Time shifting
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| <math>x(t)*y(t) \!</math>  
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| <math> x[n-n_{0}]\ </math>  
 
|  
 
|  
−
| <math> X(f)Y(f) \!</math>
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| <math> z^{-n_{0}}X(z)\ </math>
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|
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| <math> R,\ except\ for\ the\ possible\ addition\ </math>
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|-
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| align="right" style="padding-right: 1em;" |
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|
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|
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|
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|
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| <math> or\ deletion\ of\ the\ origin\ </math>
 
|-
 
|-
−
| align="right" style="padding-right: 1em;" | time reversal
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| align="right" style="padding-right: 1em;" | Scaling in the z-domain
−
| <math>\ x(-t) </math>  
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| <math> e^{j\omega_{0}n}x[n]\ </math>  
 
|  
 
|  
−
| <math>\ X(-f)</math>
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| <math> X(e^{j\omega_{0}}z)\ </math>
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|
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| <math> R\ </math>
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|-
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| align="right" style="padding-right: 1em;" | Modulation [[Practice_prove_modulation_property_z_transform| (proof)]]
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| <math> a^{n}x[n]\ </math>
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|
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| <math> X(a^{-1}z)\ </math>
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|
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| <math> |a_0| R </math> (Scaled version of) <math> R\ </math>
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|-
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| align="right" style="padding-right: 1em;" |
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|
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|
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|
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|
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| <math> (i.e.,\ |a|R=\ the\ set\ of\ points\ {|a|z}\ for\ z\ in\ R)\ </math>
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|-
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| align="right" style="padding-right: 1em;" | Time reversal
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| <math> x[-n]\ </math>
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|
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| <math> X(z^{-1})\ </math>
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|
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| <math> R^{1/k}\ (i.e.,\ the\ set\ of\ points\ z^{1/k},\ </math>
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|-
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| align="right" style="padding-right: 1em;" |
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|
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|
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|
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|
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| <math> where\ z\ is\ in\ R)\ </math>
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|-
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| align="right" style="padding-right: 1em;" | Time expansion
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| <math> x^{(k)}=
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\begin{cases}
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x[r],  &n=rk \\
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0, &n\neq rk
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\end{cases}  </math>
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|
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| <math> X(z^{k})\ </math>
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|
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| <math> R^{1/k}\ (i.e.,\ the\ set\ of\ points\ z^{1/k},\ </math>
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|-
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| align="right" style="padding-right: 1em;" |
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|
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|
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|
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|
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| <math> where\ z\ is\ in\ R)\ </math>
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|-
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| align="right" style="padding-right: 1em;" | Conjugation
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| <math> x^{*}[n]\ </math>
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|
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| <math> X^{*}(z^{*})\ </math>
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|
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| <math> R\ </math>
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|-
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| align="right" style="padding-right: 1em;" | Convolution
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| <math> x_{1}[n]*x_{2}[n]\ </math>
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|
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| <math> X_{1}(z)X_{2}(z)\ </math>
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|
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| <math> At\ least\ the\ intersection\ of\ R_{1}\ and\ R_{2}\ </math>
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|-
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| align="right" style="padding-right: 1em;" | First difference
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| <math> x[n]-x[n-1]\ </math>
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|
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| <math> (1-z^{-1})X(z)\ </math>
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|
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| <math> At\ least\ the\ intersection\ of\ R\ and\ |z|>0\  </math>
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|-
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| align="right" style="padding-right: 1em;" | Accumulation
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| <math> \sum_{k=-\infty}^{n}x[k]\ </math>
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|
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| <math> \frac{1}{(1-z^{-1})}X(z)\ </math>
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|
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| <math> At\ least\ the\ intersection\ of\ R\ and\ |z|>1\  </math>
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|-
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| align="right" style="padding-right: 1em;" | Differentiation in the z-domain
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| <math> nx[n]\ </math>
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|
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| <math> -z\frac{dX(z)}{dz}\ </math>
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|
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| <math> R\  </math>
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|-
 
|}
 
|}
  
 
{|
 
{|
 
|-
 
|-
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! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | Other CT Fourier Transform Properties
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! style="background: none repeat scroll 0% 0% rgb(238, 238, 238);" colspan="2" | Other [[info_z-transform|Z Transform]] Properties
 
|-
 
|-
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| align="right" style="padding-right: 1em;" | Parseval's relation
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|-
−
| <math>\int_{-\infty}^{\infty} |x(t)|^2 dt = \int_{-\infty}^{\infty} |X(f)|^2 df</math>
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| align="right" style="padding-right: 1em;" | Initial Value Theorem
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| <math> If\ x[n]=0\ for\ n<0,\ then\ x[0]=\lim_{z\rightarrow \infty}X(z)\ </math>
 
|}
 
|}
  
 
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[[Collective_Table_of_Formulas|Back to Collective Table]]
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[[Collective_Table_of_Formulas|Back to Collective Table]]
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[[Category:Formulas]]
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Latest revision as of 07:55, 6 March 2015


Collective Table of Formulas

Table of (double-sided) Z Transform Pairs and Properties

(Used in ECE301, ECE438, ECE538)


(double-sided) Z Transform and its Inverse
(Double-side) Z Transform $ X(z)=\mathcal{Z}(x[n])=\sum_{n=-\infty}^{\infty}x[n]z^{-n} \ $ (info)
Inverse Z Transform $ x[n]=\mathcal{Z}^{-1}(X(z))=\frac{1}{2\pi j}\oint_{c}X(z)z^{n-1}dz \ $ (info)
(double-sided) Z Transform Pairs
Signal Transform Region of convergence (ROC)
Unit impulse signal $ \delta[n]\ $ $ 1\ $ All complex $ z\ $ including $ \infty $
Unit step signal $ u[n]\ $ $ \frac{1}{1-z^{-1}} \ $ $ |z| > 1\ $ (computation)
$ -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\ $
(double-sided) Z Transform Properties
Below $ x[n] $, $ x_1[n] $ and $ x_2[n] $ are DT signals with z-transforms $ X(z) $, $ X_1(Z) $, $ X_2(z) $, and region of convergence (ROC) $ R $, $ R_1 $, $ R_2 $, respectively.
Signal Z-Transform ROC
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\ $
Modulation (proof) $ a^{n}x[n]\ $ $ X(a^{-1}z)\ $ $ |a_0| R $ (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)\ $

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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