Convolution integral

Convolution integral


A convolution integral is defined as:

(fgf*g)(tt) = otf(tτ)g(τ)dτ\int_{o}^{t}f(t - \tau)g(\tau)d\tau

Where ff, and gg are both functions.

Note that

(fgf*g)(tt) = (gfg*f)(tt)


otf(tτ)g(τ)dτ\int_{o}^{t}f(t - \tau)g(\tau)d\tau = otf(τ)g(tτ)dτ\int_{o}^{t}f(\tau)g(t - \tau)d\tau

Laplace Transforms of Convolution Integrals:

LL{(fgf*g)(tt)} = F(s)F(s)G(s)G(s)

  • Introduction
    What is the Convolution Integral?

    What is the Laplace Transform of a Convolution Integral?

  • 1.
    Determining Convolution Integrals

    Solve the following convolution,


    If ff(tt) = 2 + t2t^{2} and gg(tt) = 2tt

    What about,


  • 2.
    Determining the inverse Laplace Transform of functions using convolution integrals

    Find the inverse Laplace Transform of the following function using convolution integrals

    H(s) = 2s(s2+1)2\frac{2s}{(s^{2} + 1)^{2}}