Convolution Theorem

Convolution Theorem:

The convolution theorem of Laplace transform states that, let f1 (t) and f2 (t) are the Laplace transformable functions and F1 (s), F2 (s) are the Laplace transforms of f1 (t) and f2 (t) respectively. Then the product of F1 (s) and F2 (s) is the Laplace transform of f(t) which is obtained from the convolution of f1 (t) and f2 (t). The convolution of f1 (t) and f2 (t) is denoted as f1 (t) * f2 (t) and is obtained by the equation,

Convolution Theorem

where τ is the dummy variable.

Convolution Theorem

where f1 (t) * f2 (t) indicates convolution of f1 (t) and f2 (t).

Proof:

Let

Convolution Theorem

Where

x and y are dummy variables.

As x and y are independent variables we can write,

Convolution Theorem

Consider the new variables t and τ such that,

Let us consider limits of integration interms of t and τ

The smallest value of y is zero hence t ≥ τ.

Convolution Theorem

The equation t = τ is straight line in t – τ plane. So to integrate area between the line t = τ and τ = 0, we get the limits for t as 0 →∞ while limits of τ as 0 to t.

Now

Convolution Theorem

From the definition of Laplace transform the equation (2) represents Laplace transform of integral

Convolution Theorem

which is called convolution integral. So in the equation (1),

Convolution Theorem

Thus explained.

Updated: October 11, 2019 — 11:30 pm