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}_{0 }= 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 }Z

^{2}

X(Z) = Z

^{2 }- 1/2 Z^{1}/ Z^{2 }- 1/4**Step 3 :**We will obtain the equation of X(Z)/Z as follows :

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

^{2 }- 1/4
X(Z)/Z = (Z - 1/2) / Z

^{2 }- 1/4**Step 4:**We will the root of denominator Z

^{2}- 1/4

Here Z

^{2 }- 1/4 = Z^{2 }- (1/2)2
Root is ( Z - 1/2) and ( Z + 1/2)

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

_{1}) = P_{1 =}1/2_{}

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

_{2}) = P_{2 = }- 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 - P

_{1}) 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 - P

_{K}} = (P_{K})n u(n)
x(n) = (-1/2)n u(n)