Editing Convolution (section)

=== Relations with other transforms ===
Given two functions <math> f(t) </math> and <math> g(t) </math> with [[Two-sided Laplace transform|bilateral Laplace transform]]s (two-sided Laplace transform)

:<math> F(s) = \int_{-\infty}^\infty e^{-su} \ f(u) \ \text{d}u </math>

and

:<math> G(s) = \int_{-\infty}^\infty e^{-sv} \ g(v) \ \text{d}v </math>

respectively, the convolution operation <math> (f * g)(t) </math> can be defined as the [[inverse Laplace transform]] of the product of <math> F(s) </math> and <math> G(s) </math>.<ref>{{cite web |last1=Differential Equations (Spring 2010) |first1=MIT 18.03 |title=Lecture 21: Convolution Formula |url=https://ocw.mit.edu/courses/mathematics/18-03-differential-equations-spring-2010/video-lectures/lecture-21-convolution-formula/ |website=MIT Open Courseware |publisher=MIT |access-date=22 December 2021}}</ref><ref>{{cite web |title=18.03SC Differential Equations Fall 2011 |url=https://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-iii-fourier-series-and-laplace-transform/transfer-system-and-weight-functions-greens-formula/MIT18_03SCF11_s30_5text.pdf |archive-url=https://web.archive.org/web/20150906102242/https://ocw.mit.edu/courses/mathematics/18-03sc-differential-equations-fall-2011/unit-iii-fourier-series-and-laplace-transform/transfer-system-and-weight-functions-greens-formula/MIT18_03SCF11_s30_5text.pdf |archive-date=2015-09-06 |url-status=live |website=Green's Formula, Laplace Transform of Convolution}}</ref> More precisely,

:<math>
\begin{align}
F(s) \cdot G(s) &= \int_{-\infty}^\infty e^{-su} \ f(u) \ \text{d}u \cdot \int_{-\infty}^\infty e^{-sv} \ g(v) \ \text{d}v \\
&= \int_{-\infty}^\infty \int_{-\infty}^\infty e^{-s(u + v)} \ f(u) \ g(v) \ \text{d}u \ \text{d}v
\end{align}
</math>

Let <math> t = u + v </math>, then

:<math>
\begin{align}
F(s) \cdot G(s) &= \int_{-\infty}^\infty \int_{-\infty}^\infty e^{-st} \ f(u) \ g(t - u) \ \text{d}u \ \text{d}t \\
&= \int_{-\infty}^\infty e^{-st} \underbrace{\int_{-\infty}^\infty f(u) \ g(t - u) \ \text{d}u}_{(f * g)(t)} \ \text{d}t \\
&= \int_{-\infty}^\infty e^{-st} (f * g)(t) \ \text{d}t.
\end{align}
</math>

Note that <math> F(s) \cdot G(s) </math> is the bilateral Laplace transform of <math> (f * g)(t) </math>. A similar derivation can be done using the [[Laplace transform|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#Overview|LTI system theory]] for a derivation of convolution as the result of LTI constraints. In terms of the [[Fourier transform]]s 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.