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 a0 = 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) = Z2 - 1/2 Z1 / Z2 - 1/4
Step 3 : We will obtain the equation of X(Z)/Z as follows :
X(Z) = Z(Z - 1/2) / Z2 - 1/4
X(Z)/Z = (Z - 1/2) / Z2 - 1/4
Step 4: We will the root of denominator Z2- 1/4
Here Z2 - 1/4 = Z2 - (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) / ( Z - 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)