Derivation of Z-Transform/ inverse Z- Transform

Derivation of Z-Transform:-

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

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

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

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

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

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

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

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

Inverse Z-Transform:-

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

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

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

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

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

 

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Author: Lakshmi Prasanna Ponnala

Completed M.Tech in Digital Electronics and Communication Systems.

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