Derivation of Z and inverse Z-Transfomr

Derivation of Z-Transform:-

If $x[n]$  is the given sequence then it’s Discrete Time Fourier Transform  is  $X(e^{j\omega&space;})$  .

i.e,    $x[n]\leftrightarrow&space;X(e^{j\omega&space;})$

$x[n]\&space;r^{-n}\leftrightarrow&space;X(r\&space;e^{j\omega&space;})$.

DTFT of $x[n]&space;=&space;\sum_{n=-\infty&space;}^{\infty&space;}&space;x[n]&space;e^{-j\omega&space;n}$ .

DTFT of  $x[n]\&space;r^{-n}&space;=&space;\sum_{n=-\infty&space;}^{\infty&space;}&space;x[n]&space;\&space;r^{-n}e^{-j\omega&space;n}$ .

$X(r\&space;e^{j\omega&space;})&space;=&space;\sum_{n=-\infty&space;}^{\infty&space;}&space;x[n]&space;\&space;\left&space;(&space;r\&space;e^{j\omega&space;}&space;\right&space;)^{-n}$ .

Let    $Z=r\&space;e^{j\omega&space;}$   a complex-variable.

$X(Z)&space;=&space;\sum_{n=-\infty&space;}^{\infty&space;}&space;x[n]&space;\&space;Z&space;^{-n}$   . is the Z-Transform of a sequence  $x[n]$ .

Inverse Z-Transform:-

The inverse DTFT of  $X(e^{j\omega&space;})$  is    $x[n]&space;=&space;\frac{1}{2\pi&space;}&space;\int&space;X(e^{j\omega&space;})&space;\&space;e^{j\omega&space;n}&space;\&space;d\omega$.

$x[n]&space;\&space;r^{-n}&space;=&space;\frac{1}{2\pi&space;}&space;\int&space;X(r\&space;e^{j\omega&space;})&space;\&space;e^{j\omega&space;n}&space;\&space;d\omega$ .

$x[n]&space;=&space;\frac{1}{2\pi&space;}&space;\int&space;X(r\&space;e^{j\omega&space;})&space;\&space;(r\&space;e^{j\omega&space;})^{n}&space;\&space;d\omega$ .

$x[n]&space;=&space;\frac{1}{2\pi\&space;j&space;Z&space;}&space;\int&space;X(Z)&space;\&space;Z^{n}&space;\&space;dz$ .    Let  $r\&space;e^{j\omega&space;}&space;=&space;Z&space;\Rightarrow&space;\j&space;\&space;r&space;\&space;e^{j\omega&space;}\&space;d\omega&space;=&space;dZ$     and  $\&space;j&space;\&space;Z&space;\&space;d\omega&space;=&space;dZ&space;\Rightarrow&space;d\omega&space;=\frac{dz}{j&space;\&space;Z}$ .

$x[n]&space;=&space;\frac{1}{2\pi\&space;j&space;}&space;\int&space;X(Z)&space;\&space;Z^{n-1}&space;\&space;dz$ – represents the inverse Z-Transform.

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