\documentclass[a4paper, 11pt]{amsart}

\input{style}
\input{preamble}

\title{Parallel wavelet transform}
\author{Joshua Moerman}

\begin{document}


\tikzstyle{plain_line}=[]
\begin{figure}
	\centering
	\begin{subfigure}[b]{0.5\textwidth}
		\begin{tikzpicture}
		\begin{groupplot}[group style={group size=1 by 4}, clip=false, yticklabels={,,}, height=3cm, width=\textwidth, xmin=0, xmax=128, ymin=-1, ymax=1, domain=0:128]
		\nextgroupplot
		\addplot[plain_line] coordinates {(0,0) (1,0) (1,1) (2,1) (2,0) (128,0)}; \legend{$e_1$}
		\nextgroupplot \addplot[plain_line] coordinates {(0,0) (2,0) (2,1) (3,1) (3,0) (128,0)}; \legend{$e_2$}
		\nextgroupplot \addplot[plain_line] coordinates {(0,0) (3,0) (3,1) (4,1) (4,0) (128,0)}; \legend{$e_3$}
		\nextgroupplot \addplot[plain_line] {0.8*sin(1*360*x/128) + 0.2*sin(3*360*x/128) + 0.08*sin(5*360*x/128)};
		\end{groupplot}
		\end{tikzpicture}
		\caption{Representing a signal on the standard basis.}
	\end{subfigure}~
	\begin{subfigure}[b]{0.5\textwidth}
		\begin{tikzpicture}
		\begin{groupplot}[group style={group size=1 by 4}, yticklabels={,,}, height=3cm, width=\textwidth, xmin=0, xmax=128, ymin=-1, ymax=1, domain=0:128]
		\nextgroupplot \addplot[plain_line] {sin(1*360*x/128)}; \legend{$f_1$}
		\nextgroupplot \addplot[plain_line] {sin(3*360*x/128)}; \legend{$f_3$}
		\nextgroupplot \addplot[plain_line] {sin(5*360*x/128)}; \legend{$f_5$}
		\nextgroupplot \addplot[plain_line] {0.8*sin(1*360*x/128) + 0.2*sin(3*360*x/128) + 0.08*sin(5*360*x/128)};
		\end{groupplot}
		\end{tikzpicture}
		\caption{Representing a signal on the Fourier basis.}
	\end{subfigure}
	\caption{We can represent the same signal on different basis. Note that the Fourier representation is smaller in this case.}
	\label{fig:basicplot}
\end{figure}

$$ 0.088 + 0.174 \times 0.257 $$
$$ 0.798 \times 0.201 + 0.081 $$
$$ = \ldots + $$

\end{document}