Convolution Theorem

CONVOLUTION THEOREM

The convolution theorem plays an important role in the solution of difference equations and in probability problems involving sums of two independent random variables.

Definition: Convolution of sequences :

1. The convolution of two sequences

{x(n)} and {y (n)} is defined as

2. The convolution of two functions f(t) and g(t) is defined as

State and prove convolution theorem on Z-transform.

Statement :

convolution operation.

Proof :

We know that,

XI. Z-transform of f (n) * g (n) type.

1. Find the Z-transform of the convolution of

x(n) = u(n) and y(n) = an u (n)

Solution :

The Z-transform of the convolution of x(n) and y(n) is

2. Find the Z-transform of f(n) * g(n), where f(n) = u(n) and

Solution : By convolution theorem,

3. Find the Z-transform of f(n) * g(n) where f(n) = and g(n) = cos nπ

Solution : By convolution theorem,

XII. Use convolution theorem to find the inverse Z-transform of

(a) Find

Solution : 

1b. Find

Solution : 

1(c) Find

Solution :

1d. Using convolution theorem evaluate inverse Z-transform of

Solution :

Here, a = 1, b = 3

1(e). Find

Solution :

2. Find

Solution : 

3(a). Find

Solution : 

3(b). Find

Solution: 

Replace a by -a in 3 (a), we get