Editing Z-transform (section)

==Properties==
{| class="wikitable"
 |+ '''Properties of the z-transform'''
 !
Property	
! Time domain
	! Z-domain
	! Proof
 	! ROC
 
|-
!Definition of Z-transform
|<math>x[n]</math>
|<math>X(z)</math>
|<math>X(z)=\mathcal{Z}\{x[n]\}</math> (definition of the z-transform)
<math>x[n]=\mathcal{Z}^{-1}\{X(z)\}</math> (definition of the inverse z-transform)
|<math>r_2<|z|<r_1</math>
 |-
 ! [[Linearity]]
	| <math>a_1 x_1[n] + a_2 x_2[n]</math>
	| <math>a_1 X_1(z) + a_2 X_2(z)</math>
	| <math>\begin{align}X(z) &= \sum_{n=-\infty}^{\infty} (a_1x_1[n]+a_2x_2[n])z^{-n} \\
         &= a_1\sum_{n=-\infty}^{\infty} x_1[n] \, z^{-n} + a_2\sum_{n=-\infty}^{\infty}x_2[n] \, z^{-n} \\
         &= a_1X_1(z) + a_2X_2(z) \end{align} </math>
	| Contains ROC<sub>1</sub> ∩ ROC<sub>2</sub>
 |-
 ! [[Upsampling|Time expansion]]
	| <math>x_K[n] = \begin{cases} x[r], & n = Kr \\ 0, & n \notin K\mathbb{Z} \end{cases}</math>
with <math>K\mathbb{Z} := \{Kr: r \in \mathbb{Z}\}</math>
	| <math>X(z^K)</math>
	| <math>\begin{align} X_K(z) &=\sum_{n=-\infty}^{\infty} x_K[n]z^{-n} \\
&= \sum_{r=-\infty}^{\infty}x[r]z^{-rK}\\
&= \sum_{r=-\infty}^{\infty}x[r](z^{K})^{-r}\\
&= X(z^{K}) \end{align}</math>
	| <math>R^{\frac{1}{K}}</math>
 |-
 ! [[Downsampling|Decimation]]
	| <math>x[Kn]</math>
	| <math>\frac{1}{K} \sum_{p=0}^{K-1} X\left(z^{\tfrac{1}{K}} \cdot e^{-i \tfrac{2\pi}{K} p}\right)</math>
	| [http://www2.ece.ohio-state.edu/~schniter/ee700/handouts/multirate.pdf ohio-state.edu] or [http://www.ee.ic.ac.uk/hp/staff/dmb/courses/DSPDF/01100_Multirate.pdf ee.ic.ac.uk]
	| 
 |-
 ! Time delay
	| <math>x[n-k]</math>
with <math>k>0</math> and <math>x : x[n]=0\ \forall \, n<0</math>
	| <math>z^{-k}X(z)</math>
	| <math>\begin{align} \mathcal{Z}\{x[n-k]\} &= \sum_{n=0}^{\infty} x[n-k]z^{-n}\\
&= \sum_{j=-k}^{\infty} x[j]z^{-(j+k)}&& j = n-k \\
&= \sum_{j=-k}^{\infty} x[j]z^{-j}z^{-k} \\
&= z^{-k}\sum_{j=-k}^{\infty}x[j]z^{-j}\\
&= z^{-k}\sum_{j=0}^{\infty}x[j]z^{-j} && x[\beta] = 0,  \beta < 0\\
&= z^{-k}X(z)\end{align} </math>
	| ROC, except <math>z{=}0</math> if <math>k > 0</math> and <math>z {=} \infty</math> if <math>k < 0</math>
 |-
 ! Time advance
	| <math>x[n+k]</math>
with <math>k>0</math>
	| Bilateral Z-transform:
<math display="block">z^kX(z)</math>
Unilateral Z-transform:<ref>{{cite book |last1=Bolzern |first1=Paolo |last2=Scattolini |first2=Riccardo |last3=Schiavoni |first3=Nicola |title=Fondamenti di Controlli Automatici |language=it |publisher=MC Graw Hill Education |isbn=978-88-386-6882-1|year=2015 }}</ref>
<math display="block">z^k \, X(z)-z^k\sum^{k-1}_{n=0}x[n] \, z^{-n}</math>
	| 
	|
 |-
 ! First difference backward
	|  <math>x[n] - x[n-1]</math>
with <math>x[n]{=}0 </math> for <math>n < 0 </math>
	| <math> (1-z^{-1}) \, X(z)</math>
	| 
	| Contains the intersection of ROC of <math>X_1(z)</math> and <math>z \neq 0</math>
 |-
 ! First difference forward
|  <math>x[n+1] - x[n]</math>
	| <math> (z-1) \, X(z)-z \, x[0]</math>
	| 
	|
 |-
 ! Time reversal
	| <math>x[-n]</math>
	| <math>X(z^{-1})</math>
	| <math>\begin{align} \mathcal{Z}\{x(-n)\} &= \sum_{n=-\infty}^{\infty} x[-n]z^{-n} \\
&= \sum_{m=-\infty}^{\infty} x[m]z^{m}\\
&= \sum_{m=-\infty}^{\infty} x[m]{(z^{-1})}^{-m}\\
&= X(z^{-1}) \\
\end{align} </math>
	| <math>\tfrac{1}{r_1}<|z|<\tfrac{1}{r_2}</math>
 |-
 ! Scaling in the z-domain
	| <math>a^n x[n]</math>
	| <math>X(a^{-1}z)</math>
	| <math>\begin{align}\mathcal{Z} \left \{a^n x[n] \right \} &=  \sum_{n=-\infty}^{\infty} a^{n}x[n]z^{-n} \\
&= \sum_{n=-\infty}^{\infty} x[n](a^{-1}z)^{-n} \\
&= X(a^{-1}z)
\end{align} </math>
	| <math>|a|r_2 < |z|< |a|r_1</math> 
 |-

 ! [[Complex conjugation]]
	| <math>x^*[n]</math>
	| <math>X^*(z^*)</math>
	| <math>\begin{align} \mathcal{Z} \{x^*(n)\} &= \sum_{n=-\infty}^{\infty} x^*[n]z^{-n}\\
&= \sum_{n=-\infty}^{\infty} \left [x[n](z^*)^{-n} \right ]^*\\
&= \left [ \sum_{n=-\infty}^{\infty} x[n](z^*)^{-n}\right ]^*\\
&= X^*(z^*)
\end{align} </math>
	|
 |-
 ! [[Real part]]
	| <math>\operatorname{Re}\{x[n]\}</math>
	| <math>\tfrac{1}{2}\left[X(z)+X^*(z^*) \right]</math>
	|
	|
 |-
 ! [[Imaginary part]]
	| <math>\operatorname{Im}\{x[n]\}</math>
	| <math>\tfrac{1}{2j}\left[X(z)-X^*(z^*) \right]</math>
	|
	| 
 |-
 ! [[Differentiation (calculus)|Differentiation]] in the z-domain
	| <math>n \, x[n]</math>
	| <math> -z \frac{dX(z)}{dz}</math>
	| <math>\begin{align} \mathcal{Z}\{n \, x(n)\} &= \sum_{n=-\infty}^{\infty} n \, x[n]z^{-n}\\
&= z \sum_{n=-\infty}^{\infty} n \, x[n]z^{-n-1}\\
&= -z \sum_{n=-\infty}^{\infty} x[n](-n \, z^{-n-1})\\
&= -z \sum_{n=-\infty}^{\infty} x[n]\frac{d}{dz}(z^{-n}) \\
&= -z \frac{dX(z)}{dz}
\end{align} </math>
	| ROC, if <math>X(z)</math> is rational;
ROC possibly excluding the boundary, if <math>X(z)</math> is irrational<ref name = forouzan>{{cite journal  | journal = Electronics Letters| title = Region of convergence of derivative of Z transform  | author = A. R. Forouzan  | volume = 52  | issue = 8  | pages = 617–619  | year = 2016| doi = 10.1049/el.2016.0189| bibcode = 2016ElL....52..617F | s2cid = 124802942 }}</ref>
 |-
 ! [[Convolution]]
	| <math>x_1[n] * x_2[n]</math>
	| <math>X_1(z) \, X_2(z)</math>
	| <math>\begin{align} \mathcal{Z}\{x_1(n)*x_2(n)\} &= \mathcal{Z} \left \{\sum_{l=-\infty}^{\infty} x_1[l]x_2[n-l] \right \} \\
                                   &= \sum_{n=-\infty}^{\infty} \left [\sum_{l=-\infty}^{\infty} x_1[l]x_2[n-l] \right ]z^{-n}\\
                                   &=\sum_{l=-\infty}^{\infty} x_1[l] \left [\sum_{n=-\infty}^{\infty} x_2[n-l]z^{-n} \right ]\\
                                   &= \left [\sum_{l=-\infty}^{\infty} x_1(l)z^{-l} \right ] \! \!\left [\sum_{n=-\infty}^{\infty} x_2[n]z^{-n} \right ] \\
                                   &=X_1(z)X_2(z)
\end{align} </math>
	| Contains ROC<sub>1</sub> ∩ ROC<sub>2</sub>
 |-
 ! [[Cross-correlation]]
	| <math>r_{x_1,x_2}=x_1^*[-n] * x_2[n]</math>
	| <math>R_{x_1,x_2}(z)=X_1^*(\tfrac{1}{z^*})X_2(z)</math>
	|
	| Contains the intersection of ROC of <math>X_1(\tfrac{1}{z^*})</math> and <math>X_2(z)</math>
 |-
 ! Accumulation
	|<math>\sum_{k=-\infty}^{n} x[k]</math>
	|<math> \frac{1}{1-z^{-1}}X(z)</math>
	|<math>\begin{align}
\sum_{n=-\infty}^{\infty}\sum_{k=-\infty}^{n} x[k] z^{-n}&=\sum_{n=-\infty}^{\infty}(x[n]+\cdots)z^{-n}\\
        &=X(z) \left (1+z^{-1}+z^{-2}+\cdots \right )\\
        &=X(z) \sum_{j=0}^{\infty}z^{-j} \\
        &=X(z) \frac{1}{1-z^{-1}}\end{align}</math>
	|
 |-
 ! [[Multiplication]]
	|  <math>x_1[n] \, x_2[n]</math>
	| <math>\frac{1}{j2\pi}\oint_C X_1(v)X_2(\tfrac{z}{v})v^{-1}\mathrm{d}v</math>
	|
	| At least <math>r_{1l}r_{2l}<|z|<r_{1u}r_{2u}</math> |-
|}

'''[[Parseval's theorem]]'''
:<math>\sum_{n=-\infty}^{\infty} x_1[n]x^*_2[n] \quad = \quad \frac{1}{j2\pi}\oint_C X_1(v)X^*_2(\tfrac{1}{v^*})v^{-1}\mathrm{d}v</math>

'''[[Initial value theorem]]''': If <math>x[n]</math> is causal, then
:<math>x[0]=\lim_{z\to \infty}X(z).</math>

'''[[Final value theorem]]''': If the poles of <math>(z - 1) X(z)</math> are inside the unit circle, then
:<math>x[\infty]=\lim_{z\to 1}(z-1)X(z).</math>