Inverse Z Transform

INVERSE Z TRANSFORM

Inverse z transform can be obtained by

(i) Power series expansion

(ii) Partial fraction expansion

(iii) Contour integration

Inverse z Transform using Power Series Expansion

In inverse x transform x(n) is obtained from X(z). From X(z), x(n) can be obtained as

From above expansion, the sequence x(n) can be obtained as

x(n) = {..., x(-3), x(-2), x(-1), x(0), x(1), x(2), x(3)...}

Power series expansion can be obtained directly (or) by long division method.

Problem 1:

Determine inverse z transform of X(z) = 

Solution:

Problem 2:

Determine inverse z transform of X(z) = 

Solution:

Taking inverse z transform, x(n) = {1, a, a², a³,...} = aⁿ u(n)

Hint: When ROC is |z|>|a| (or) |z| > 1, we should expand, X(z) in negative powers of z.

Inverse z - transform using partial fraction expansion

Following steps are performed for partial fraction expansion case I

Case I:

I Step: Arrange the given 

II Step: 

Then find A₁, A₂, A₃ ... A_N values using partial fraction expansion.

Case II

III Step: Equation (1) can be written as

IV Step: Depending upon ROC, following standard z transform pairs must be used.

Problem 1:

Determine the inverse z transform of X(z) =  ROC is z > 0.6.

Solution:

By partial fraction technique we get

Taking inverse z transform using standard relation, we get

Problem 2:

Determine the inverse z transform of

Solution:

Convert X(z) to positive powers of z

Taking inverse z transform using standard relations

Problem 3:

Find the inverse z transform of X(z) = 

Solution:

Divide the above equation by z in order to get X(z)/z

Taking inverse z transform using standard relations

Problem 4:

Find the inverse z transform of

 May 04 – Marks 10

Solution:

Taking inverse z transform using standard relation.

Problem 5:

Find the inverse z transform of

 May 12-Marks 8

Solution:

Taking inverse z transform using standard relation and time shifting property.

Problem 6:

Find the inverse z transform of X(z) = 

Solution:

Divide above equation by z in orders to get X(z)/z

By taking inverse z transform using standard relations

Problem 7:

Determine inverse z transform of the following function

Solution:

Roots of quadratic equation

By partial fraction expansion,

After substituting the values of A and A*, X(z)/z becomes

Taking inverse z transform using standard relations,

Problem 8:

Determine the inverse z transform of X(z) = log (1-2z), |z| < 1/2 by using the power series log (1 − x) = , |x| < 1 and by first differentiating X(z) and then using this to recover x(n).

Solution:

Inverse z transform using power series

Inverse z transform using differentiation

Taking inverse z transform on both sides

Problem 9:

Find the inverse z transform of the following.

Solution:

Taking inverse z transform on both sides.

Problem 10:

Find the inverse z transform of the following. Dec-10, 5marks

Solution:

Taking inverse z transform using standard relationship

Problem 11:

Determine inverse z transform of X(z) =  ROC:|z|>1

Solution:

Taking inverse z transform using standard relation,

Problem 12:

Find the initial and final values of the function.

 (May-97)

Solution:

Initial value is given as

Final value is given as

Problem 13:

Find the final value of the given signal.

 Dec-03/3 marks

Solution:

Final value is given as