# Properties of Z-transforms

1.Linearity Property:-

$x_{1}[n]\leftrightarrow&space;X_{1}(Z)&space;\&space;\&space;\&space;ROC&space;:a_{1}<&space;\left&space;|&space;Z&space;\right&space;|&space;

$x_{2}[n]\leftrightarrow&space;X_{2}(Z)&space;\&space;\&space;\&space;ROC&space;:a_{2}<&space;\left&space;|&space;Z&space;\right&space;|&space;

$a\&space;x_{1}[n]+b\&space;x_{2}[n]\leftrightarrow&space;\&space;\&space;?$.

we know that  Bi-lateral Z- Transform of a signal  $x[n]$  is $X(Z)&space;=&space;\sum_{n=-\infty&space;}^{\infty&space;}&space;x[n]&space;\&space;Z^{-n}$ .

$Z\left&space;\{a\&space;x_{1}[n]+b\&space;x_{2}[n]\right&space;\}&space;=&space;\sum_{n=-\infty&space;}^{\infty&space;}&space;\&space;\&space;\left&space;\{&space;a\&space;x_{1}[n]+b\&space;x_{2}[n]&space;\right&space;\}&space;Z^{-n}$.

$Z\left&space;\{a\&space;x_{1}[n]+b\&space;x_{2}[n]\right&space;\}&space;=&space;\sum_{n=-\infty&space;}^{\infty&space;}&space;\&space;\&space;\left&space;\{&space;a\&space;x_{1}[n]\&space;Z^{-n}+b\&space;x_{2}[n]&space;\&space;Z^{-n}\right&space;\}$

$Z\left&space;\{a\&space;x_{1}[n]+b\&space;x_{2}[n]\right&space;\}&space;=&space;\&space;a\&space;\sum_{n=-\infty&space;}^{\infty&space;}&space;x_{1}[n]\&space;Z^{-n}+b&space;\&space;\sum_{n=-\infty&space;}^{\infty&space;}&space;x_{2}[n]&space;\&space;Z^{-n}$.

$Z\left&space;\{a\&space;x_{1}[n]+b\&space;x_{2}[n]\right&space;\}&space;=&space;\&space;a&space;\&space;X_{1}(Z)+b&space;\&space;X_{2}(Z)$ .          $ROC:&space;R_{1}&space;\cap&space;R_{2}$.

2.Time-shifting Property:-

$x[n]\leftrightarrow&space;X(Z)&space;\&space;\&space;\&space;ROC&space;:&space;R$

$x[n-k]\leftrightarrow&space;\&space;?$.

we know that  $X(Z)&space;=&space;\sum_{n=-\infty&space;}^{\infty&space;}&space;x[n]&space;\&space;Z^{-n}$ .

$Z\left&space;\{&space;x[n-k]&space;\right&space;\}&space;=&space;\sum_{n=-\infty&space;}^{\infty&space;}&space;x[n-k]&space;\&space;Z^{-n}$.    Let     $n-k=m&space;\Rightarrow&space;m=&space;n+k$

Here n is a variable and k is a constant.

$Z\left&space;\{&space;x[n-k]&space;\right&space;\}&space;=&space;\sum_{m\&space;=-\infty&space;}^{\infty&space;}&space;x[m]&space;\&space;Z^{-(m+k)}$ .

$Z\left&space;\{&space;x[n-k]&space;\right&space;\}&space;=&space;Z^{-k}&space;\sum_{m\&space;=-\infty&space;}^{\infty&space;}&space;x[m]&space;\&space;Z^{-m}$.

$Z\left&space;\{&space;x[n-k]&space;\right&space;\}&space;=&space;Z^{-k}&space;X(Z)$

$x[n-k]\leftrightarrow&space;\&space;Z^{-k}&space;X(Z)\&space;,&space;\&space;\&space;ROC:R$.

from the above equation $x[n-k]$  forms Z Transform pair with $Z^{-k}&space;X(Z)$.

3. Scaling  in-Z-domain property:-

$x[n]\leftrightarrow&space;X(Z)&space;\&space;\&space;\&space;ROC&space;:r_{1}<&space;\left&space;|&space;Z&space;\right&space;|&space;

$a^{n}x[n]\leftrightarrow&space;\&space;\&space;\&space;?&space;,$

we know that  $X(Z)&space;=&space;\sum_{n=-\infty&space;}^{\infty&space;}&space;x[n]&space;\&space;Z^{-n}$ .

$Z\left&space;\{&space;a^{n}\&space;x[n]&space;\right&space;\}&space;=&space;\sum_{n=-\infty&space;}^{\infty&space;}a^{n}\&space;x[n]&space;\&space;Z^{-n}$.

$Z\left&space;\{&space;a^{n}\&space;x[n]&space;\right&space;\}&space;=&space;\sum_{n=-\infty&space;}^{\infty&space;}\&space;x[n]&space;\&space;(a^{-1}Z)^{-n}$ .

$Z\left&space;\{&space;a^{n}\&space;x[n]&space;\right&space;\}&space;=&space;X(a^{-1}Z)$

$a^{n}x[n]\leftrightarrow&space;\&space;X(\frac{Z}{a}),&space;\&space;\&space;\&space;if&space;\&space;a>0$.   $\&space;\&space;ROC&space;\&space;of&space;\&space;a^{n}x[n]&space;\&space;is&space;:\left&space;|&space;a&space;\right&space;|\&space;r_{1}<&space;\left&space;|&space;Z&space;\right&space;|&space;<\left&space;|&space;a&space;\right&space;|\&space;r_{2}$ .

4. Time-reversal property:-

$x[n]\leftrightarrow&space;X(Z)&space;\&space;\&space;\&space;ROC&space;:r_{1}<&space;\left&space;|&space;Z&space;\right&space;|&space;

$x[-n]\leftrightarrow&space;\&space;\&space;?$.

we know that  $X(Z)&space;=&space;\sum_{n=-\infty&space;}^{\infty&space;}&space;x[n]&space;\&space;Z^{-n}$ .

$Z\left&space;\{&space;x[-n]&space;\right&space;\}&space;=&space;\sum_{n=-\infty&space;}^{\infty&space;}&space;x[-n]&space;\&space;Z^{-n}$ .  Let  $-n=\&space;m$ ,

$Z\left&space;\{&space;x[-n]&space;\right&space;\}&space;=&space;\sum_{m=-\infty&space;}^{\infty&space;}&space;x[m]&space;\&space;(Z^{-1})^{-m}\&space;\&space;\&space;ROC&space;:r_{1}<&space;\left&space;|&space;Z^{-1}&space;\right&space;|&space;.

$x[-n]\leftrightarrow&space;\&space;X(Z^{-1})\&space;\&space;\&space;ROC&space;:\frac{1}{\left&space;|&space;r_{2}&space;\right&space;|}<&space;\left&space;|&space;Z&space;\right&space;|&space;<\frac{1}{\left&space;|&space;r_{1}&space;\right&space;|}$.

from the above equation $x[-n]$  forms Z Transform pair with $X(\frac{1}{Z})$.

5. Differentiation in Z-domain:-

$x[n]\leftrightarrow&space;X(Z)&space;\&space;\&space;\&space;ROC&space;:r_{1}<&space;\left&space;|&space;Z&space;\right&space;|&space;

$n\&space;x[n]\leftrightarrow&space;\&space;\&space;?$.

we know that  $X(Z)&space;=&space;\sum_{n=-\infty&space;}^{\infty&space;}&space;x[n]&space;\&space;Z^{-n}$ .

$\frac{d(X(Z))}{dZ}&space;=&space;\sum_{n=-\infty&space;}^{\infty&space;}&space;x[n]&space;\frac{d&space;(Z^{-n})}{dZ}$ .

$\frac{d(X(Z))}{dZ}&space;=&space;\sum_{n=-\infty&space;}^{\infty&space;}&space;\&space;-n&space;\&space;x[n]&space;Z^{(-n-1)}$ .

$\frac{d(X(Z))}{dZ}&space;=&space;-\left&space;[&space;\sum_{n=-\infty&space;}^{\infty&space;}&space;\&space;-n&space;\&space;x[n]&space;\&space;Z^{-n}\right&space;]&space;Z^{-1}$ .

$\frac{d(X(Z))}{dZ}&space;=&space;-&space;Z^{-1}&space;\&space;Z\left&space;\{&space;n\&space;x[n]&space;\right&space;\}$ .

from the above equation $\frac{d(X(Z))}{dZ}$  forms Z-Transform pair with $n\&space;x[n]$  and the ROC is same as that of the original sequence x[n].

6. Convolution in Time-domain:-

$x_{1}[n]\leftrightarrow&space;X_{1}(Z)&space;\&space;\&space;\&space;ROC&space;:a_{1}<&space;\left&space;|&space;Z&space;\right&space;|&space;

$x_{2}[n]\leftrightarrow&space;X_{2}(Z)&space;\&space;\&space;\&space;ROC&space;:a_{2}<&space;\left&space;|&space;Z&space;\right&space;|&space;

$x_{1}[n]*&space;x_{2}[n]\leftrightarrow&space;\&space;\&space;?$.

we know that  $X(Z)&space;=&space;\sum_{n=-\infty&space;}^{\infty&space;}&space;x[n]&space;\&space;Z^{-n}$ .

$Z\left&space;\{&space;x_{1}[n]&space;*x_{2}[n]\right&space;\}&space;=&space;\sum_{n=-\infty&space;}^{\infty&space;}(x_{1}[n]&space;*x_{2}[n])&space;\&space;Z^{-n}$ .

$Z\left&space;\{&space;x_{1}[n]&space;*x_{2}[n]\right&space;\}&space;=&space;\sum_{k=-\infty&space;}^{\infty&space;}(x_{1}[k]&space;x_{2}[n-k])&space;\sum_{n=-\infty&space;}^{\infty&space;}\&space;Z^{-n}$ .

$Z\left&space;\{&space;x_{1}[n]&space;*x_{2}[n]\right&space;\}&space;=&space;\sum_{k=-\infty&space;}^{\infty&space;}x_{1}[k]&space;\sum_{n=-\infty&space;}^{\infty&space;}x_{2}[n-k]&space;\&space;Z^{-n}$ .

$Z\left&space;\{&space;x_{1}[n]&space;*x_{2}[n]\right&space;\}&space;=&space;\sum_{k=-\infty&space;}^{\infty&space;}x_{1}[k]&space;\&space;Z^{-k}\&space;X_{2}(Z)$ .

$Z\left&space;\{&space;x_{1}[n]&space;*x_{2}[n]\right&space;\}&space;=&space;\left&space;[&space;\sum_{k=-\infty&space;}^{\infty&space;}x_{1}[k]&space;\&space;Z^{-k}&space;\right&space;]\&space;X_{2}(Z)$ .

$Z\left&space;\{&space;x_{1}[n]&space;*x_{2}[n]\right&space;\}&space;=&space;X_{1}(Z)&space;\&space;X_{2}(Z)$ .

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## Author: Lakshmi Prasanna Ponnala

Completed M.Tech in Digital Electronics and Communication Systems.

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