structural:scholar_edition, linguistic:voice_change
Relations with other transforms

Given two functions f (t) {\displaystyle f(t)} and g (t) {\displaystyle g(t)} with bilateral Laplace transforms (two-sided Laplace transform)

and

respectively, the convolution operation (f ∗ g) (t) {\displaystyle (f*g)(t)} can be defined as the inverse Laplace transform of the product of F (s) {\displaystyle F(s)} and G (s) {\displaystyle G(s)}. More precisely,

Let t = u + v {\displaystyle t=u+v} such that

Note that F (s) ⋅ G (s) {\displaystyle F(s)\cdot G(s)} is the bilateral Laplace transform of (f ∗ g) (t) {\displaystyle (f*g)(t)}. A similar derivation can be done using the unilateral Laplace transform (one-sided Laplace transform).

The convolution operation also describes the output (in terms of the input) of an important class of operations known as linear time-invariant (LTI). See LTI system theory for a derivation of convolution as the result of LTI constraints. In terms of the Fourier transforms of the input and output of an LTI operation, no new frequency components are created. The existing ones are only modified (amplitude and/or phase). In other words, the output transform is the pointwise product of the input transform with a third transform (known as a transfer function). See Convolution theorem for a derivation of that property of convolution. Conversely, convolution can be derived as the inverse Fourier transform of the pointwise product of two Fourier transforms.