Convolution Integral

If h(t) is the impulse response of a linear network, then the response of the same network y(t) subject to any arbitrary input w(t) is given by the convolution integral as,

Thus, if the impulse response of any linear time-invariant system is known, we can obtain the zero-state response of the system to any other type of input.

Convolution Theorem

If f1(t) and f2(t) are two functions of time which are zero for t < 0, and if their Laplace transforms are F1(s) and F2(s), respectively, then the convolution theorem states that the Laplace transform of the convolution of f1(t) and f2(t) is given by the product F1(sF2(s).

Application of Convolution Theorem

The convolution theorem is used to find the response of a linear system to any arbitrary excitation if the impulse response of the system is known.

We know that the transfer function is defined as the ratio of response transform to excitation transform with zero initial conditions. Thus,

or H(s) =

Thus,