Convolution

Colorized Definition

\newcommand{\integral}{\color{c1}}
\newcommand{\signal}{\color{c2}}
\newcommand{\flip}{\color{c3}}
\newcommand{\kernel}{\color{c4}}
\newcommand{\slide}{\color{c5}}
\newcommand{\convolve}{\color{c6}}

$$
 (\kernel f
  \convolve *
  \signal g
\plain )( \slide t \plain )
\ \ \stackrel{\mathrm{def}}{=}\
\integral  \int_{-\infty}^\infty
\kernel f(\tau)
\signal g(\slide t \plain \flip - \tau \signal )\
\integral d\tau
$$

\plain     To
\convolve  convolve
\kernel    a kernel
\plain     with an
\signal    input signal:
\\\
\flip     flip the signal,
\slide    move to the desired time,
\\
\integral and accumulate every interaction
\kernel with the kernel