The inverse Z –transforms can be obtained by using any one of the following methods.They are
I.Power series method
II. Partial fraction method
III. Inversion Integral method
IV. Long division method

**Inverse Z –Transforms**

The inverse Z –transforms can be obtained by using any one of the following methods.They are

I.Power series method

II. Partial fraction method

III. Inversion Integral method

IV. Long division method

**I. Power series method**

This is the simplest method of finding inverse Z –transform. Here F(z) can the be expanded in a series of ascending powers and the coefficient of z –n will be the of z desired inverse Z- transform.

**Example 8**

**II. Partial Fraction Method**

Here, F(z) is resolved into partial fractions and the inverse transform can be taken directly.

**Example 9**

Find the inverse Z –transform of

**Inversion Integral Method or Residue Method**

The inverse Z-transform of F (z) is given by the formula

Sum of residues of F(z).zn-1 at the poles of F(z) inside the contour C which is drawn according to the given Region of convergence.

**Example 12**

Using the inversion integral method, find the inverse Z-transform of

Its poles are z = 1,2 which are simple poles.

By inversion integral method, we have

\Sum of Residues = -3 + 3.2n = 3 (2n-1).

Thus the required inverse Z-transform is

fn = 3(2n-1), n = 0, 1, 2, …

**Example 13**

Find the inverse z-transform of

The pole of F(z) is z = 1, which is a pole of order 3. By Residue method, we have

**IV. Long Division Method**

If F(z) is expressed as a ratio of two polynomials, namely, F(z) = g(z-1) / h(z-1), which can not be factorized, then divide the numerator by the denominator and the inverse transform can be taken term by term in the quotient.

**Example 14**

Thus F(z) = 1 + 3z-1 + 3z-2 + 3z-3 + . . . . . .

Now, Comparing the quotient with

**Example 15**

Find the inverse Z-transform of

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Mathematics (maths) : Z-Transforms and Difference Equations : Inverse Z-Transforms |

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