Relationship between Z Transform and DIFT

Inverse z Transform Using Contour Integration

Steps to calculate inverse z transform using contour integration.

I Step: Define the function X₀(z). X₀(z) is rational

Here m ⇒ order of the pole

II Step: (i) If order of the pole is m = 1, residue of X₀(z) at pole Pi is gives 

(ii) If order of the pole is m = 2, residue of X(z) can be calculated as

III Step: By using residue theorem, calculate x(n) for poles inside the unit circle

(ii) To calculate poles outside of contour integration

Problem 1:

Find the inverse z transform using contour integral method.

Solution:

I Step:

Hence the order of pole (m = 1)

II step:

III Step:

RELATIONSHIP BETWEEN Z TRANSFORM AND DTFT

z transform of discrete time signal is given as

By comparing (1) & (2), we find that X(z) indicates the Fourier transform of x(n)r^-n.

If X(z) of (1) is evaluated on unit circle then [z] = r = 1. Hence

R.H.S of (3) is DTFT of x(n). Hence