Convolution

A mathematical operation that combines 2 functions to produce a third expressing how the shape of one is modified by the other. Denoted by $*$.

Slides one function over another, multiplying and integrating at each shift.

Properties

• Commutative
  $f * g = g * f$
• Associative
  $f * (g * h) = (f * g) * h$
• Distributive
  $f * (g + h) = (f * g) + (f * h)$
• Identity
  $f * \delta = f$, where $\delta$ is the Dirac delta function

Convolution Theorem

Laplace

The Laplace transform of a convolution is the product of the individual Laplace transforms.

Suppose $\mathcal{L}\{f\} = F$ and $\mathcal{L}\{g\} = G$:

Useful for solving ODEs: multiply in the $s$-domain, then apply the inverse Laplace transform.

Fourier

Convolution in the time domain equals pointwise multiplication in the frequency domain. Used extensively in signal processing and filter design.