[Tex/LaTex] How to write a convolution and a Fourier transform

Question

I want to write the following equation in LaTeX:

\begin{equation}
x(t) \ast h(t) = y(t) 

X(f) H(f) = Y(f) 
\end{equation}

I want \ast to denote the convolution. I know there is also the \star command. Does it matter which one I use to represent convolution? Then I want a Fourier-transform symbol, I mean the line with a coloured and an empty circle on either side, to connect the x(t) and X(f), h(t) and H(f), y(t) and Y(f) respectively. Is there a way of doing this ?

Answer 1

The \circledast symbol from amssymb package is usually used to denote the circular convolution process.

\documentclass{article}
\usepackage{amsmath}
\usepackage{amssymb}
\begin{document}

\begin{align*}
x(t) \circledast h(t) &= y(t) \\
X(f) H(f) &= Y(f) 
\end{align*}

\end{document}

For linear convolution, a simple * is more appropriate:

\begin{align*}
x(t)*h(t) &= y(t) \\
X(f) H(f) &= Y(f) 
\end{align*}

To draw connections between parts of the equations, TikZ package can be used with its tikzmark library to mark locations to begin and end your lines.

\documentclass{article}
\usepackage{amsmath,amssymb,tikz}
\usetikzlibrary{arrows.meta,tikzmark}
\begin{document}

\begin{align*}
x\tikzmark{x}(t)*h\tikzmark{h}(t) &= y\tikzmark{y}(t) \\[2em]
X(f) \, H(f) &= Y(f) 
\end{align*}

\begin{tikzpicture}[overlay,remember picture, > = {Circle[open,blue]}]
  \draw [<->] ([yshift=-.7ex]pic cs:x) -- ++(0,-2.2em);
  \draw [<->] ([yshift=-.7ex]pic cs:h) -- ++(0,-2.2em);
  \draw [<->] ([yshift=-.7ex]pic cs:y) -- ++(0,-2.2em);
\end{tikzpicture}

\end{document}