Inverse z transform partial fraction expansion examples

Determine IZT of the following :

X(Z) = 1- 1/2 Z-1 / 1- 1/4 Z-2

Step 1: Find we will whether a given function is the proper form or not.
Here a= 1 M=1 and N=2.

Since M<N the function is in the proper form.

Step 2: The given function is 

X(Z) = 1- 1/2 Z-1 / 1- 1/4 Z-2

Multiply numerator and denominator  Z2

X(Z) = Z- 1/2 Z1 /  Z- 1/4 

Step 3 :  We will obtain the equation of X(Z)/Z as follows :

X(Z) = Z(Z - 1/2)  /  Z- 1/4 

X(Z)/Z = (Z - 1/2)  /  Z- 1/4 

Step 4: We will the root of denominator  Z2- 1/4 

Here Z- 1/4  =  Z- (1/2)2


Root is ( Z - 1/2)  and ( Z + 1/2) 

Thus the poles are   ( Z - 1/2) = (Z - P1) = P1 = 1/2

 ( Z + 1/2) = (Z - P2) = P2 = - 1/2

Step 5: Thus the equation can be written as  


X(Z)/Z = (Z - 1/2)  /  - 1/2) ( Z + 1/2) 

           = 1 / ( Z + 1/2) 

Step 6: The partial fraction expansion form is 

X(Z)/Z = A /  (Z - P1) Thus A=1 we do not have to calculate this value.

Step 7:

X(Z)/Z = 1 / ( Z + 1/2) 


X(Z) = Z /  ( Z + 1/2) 

This function is causal if |Z| > 1/2. The given ROC is  |Z| > 1/2. 

So,

IZT { Z /  Z - PK  } = (PK)n u(n)

x(n) = (-1/2)n u(n)

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