Inverse z-transform

INVERSE Z-TRANSFORM

Def. Inverse Z-transform

If Z[x(n)] = X(z) then Z-1[X (z)] = [x (n)]

Z-1[X (z)] can be found out by any one of the following methods.

1. (METHOD I). PARTIAL FRACTIONS METHOD

X. Problems based on Inverse Z-transform

Find the inverse Z-transform of

1. Find  

Solution: 

2. Find

Solution: 

3. Find

Solution: 

4. Find

Solution: 

5. Find

Solution :

6. Find

Solution: 

Change the second term in terms of negative powers.

7. Find using partial fraction.

Solution: 

8. Evaluate

Solution :

2. (Method: II). Inverse of Z-transform by Inverse integral method. (Cauchy's residue theorem)

From the relation between the Z-transform and Fourier transform of a sequence we get

By Cauchy's residue theorem

Note: Take the contour C such that all the poles of the function X(z) zn-1 lie within the contour.

1. Find

Solution: 

z = 1 is a simple pole and z = 2 is a simple pole

Let us consider a contour in [z] > 2

2. Find

Solution :

z = a is a simple pole and z = b is a simple pole.

Let us consider the contour C, sufficiently large.

3. Find

Solution: 

z = 1 is a simple pole

z = -1 is a pole of order 2

Let us consider the contour in |z| > 1

4. Find

Solution :

To get singularities put Dr = 0

z2 + 2z + 2 = 0

z = -1 + i is a simple pole

z = -1 - i is a simple pole

5. Find

Solution :

z = 1 is a pole of order 3.

6. Find

Solution : 

⸫ z = -1 is a simple pole

z = 2 is a pole of order 2.

Let us consider the contour C in |z| > 2

7. Find

Solution :

8. Find

Solution :

z = 2 is a simple pole

Let us consider the contour C in |z| > 2