Convolution

3.5 CONVOLUTION

Concept of convolution

Convolution is a mathematical operation which is used to express the input- output relationship of an LTI system. It is a most important operation in LTI continuous -time system. It relates input and impulse response of the system to output.

This is called convolution integral, or simply convolution. The convolution of two signals x(t) and h(t) can be represented as:

Properties of Convolution

Let us consider two signals x₁(t) and x₂(t). The convolution of two signals x₁(t) and x₂(t) is given by

The properties of convolution are as follows:

Commutative Property:

The commutative property of convolution states that x₁(t) * x₂(t) = x₂(t) * x₁(t).

Distributive Property:

The Distributive property of convolution states that

Associative Property:

The associative property of convolution states that

Shift Property:

The Shift property of convolution states that if x₁(t) * x₂(t) = z(t)

(ii) Given x₁(t) = t.u(t); x₂(t) = t.u(t)

We know that 

(iii) Given 

We know that

Problem 2:

Find the convolution of the signals  using Fourier transform.

Solution:

Problem 3:

Find the convolution of the signals x₁(t) = 2 e^-2t u(t) and x2(t) = u(t) using Fouier transform.

Solution:

Problem 4:

Find the convolution of signals using Fourier transform. x₁(t) = e^t u(t) and x₂(t) = e^-tu(t).

Solution:

Problem 5:

Find the convolution of the following signal.  

Solution:

Convolution of two signal is given as

Problem 6:

Evaluate continuous time (CT) convolution integral given below.  [Dec-05, May 10 - Marks 8]

Solution:

Depends upon the overlap between x(τ) and h(t-τ) two cases are possible.

Case I:

Figure 3.23 shows x(τ) and h(t-τ) for different values of t. In this case there is no overlap between x(τ) and h(t-τ). Hence products of x(τ) and h(t-τ) is zero. y(t) = 0, for t < -2

Case II:

Figure 3.24 shows that there is overlap between x(τ) and h(t-τ) for t ≥ - 2. Hence convolution becomes.