Function Convolution

This blog post will just be a quick digest of the basic concept and math of function convolution. A convolution of functions is essentially the mixing or superposition of functions in a given space. The best example is to look at signals in frequency and time domains  using Fourier transforms. Lets start by looking at the inverse Fourier transform of the product of two signals $g(\omega)$ and $f(\omega)$,

$$ h(t) = \int_{-\infty}^{\infty} g(\omega)\,f(\omega) e^{i\omega t} d\omega. $$

we can assume that $g(\omega)$ is some sort of applied signal/filter and $f(\omega)$ is a source signal. The output signal $h(t)$ will be some sort of filtered signal based on what $g(\omega)$ does. The thing to recall is that these signals are actually the Fourier transforms of the time signal:

$$g(\omega) = \int_{-\infty}^{\infty} g(t)\,e^{-i \omega t} dt $$

and

$$f(\omega) = \int_{-\infty}^{\infty} f(t)\,e^{-i \omega t} dt. $$

If we plug these into the first equation for $h(t)$ and simplify we get:

$$\begin{align} h(t) &= \int_{-\infty}^{\infty} g(t) f(t) e^{-i\omega t} e^{-i \omega t} e^{i \omega t} dt d\omega \\ & = \int_{-\infty}^{\infty} \mathscr{F}[g(t)*f(t)] e^{i \omega t} d\omega, \end{align} $$

where $\mathscr{F}[\,]$ denotes the Fourier transform and $*$ indicates the combination/superposition of two functions in the time domain. This is more commonly referred to as the convolution. The intitive picture is that the convolution captures the similarity or correlation between the two functions. and therefore the output function is akin to a inner product of the two functions. This is best seen in the nice wikipedia visualization:

Brian Amberg / CC BY-SA

The quote for this blog post will come from the man himself, Joseph Fourier, who's mathematics are so ubiquitous in math, science, and engineering. 

"Profound study of nature is the most fertile source of mathematical discoveries." --Joseph Fourier in The Analytical Theories of Heat, Ch. 1 p. 7, 1878

References:

[1] The Mathematics Companion, IoP, A. C. Fischer-Cripps.

[2] Advanced Engineering Mathematics, Wiley, 8th ed., E. Kreyszig. 

Reuse and Attribution