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%
% A more standardized way of making paper titles
%
\author{Don H. Johnson\\[4pt]
Electrical \& Computer Engineering Department\\
Rice University\\
\url{dhj@rice.edu}
}
\title{Representation of Signals}

\begin{document}
\sloppy

\maketitle

\section{Introduction}\label{sec:intro}

In simple terms, a signal $\signal{t}$ can be decomposed as a superposition---a sum---of simpler signals in most cases of interest.
In particular, we can define a \emph{basis} as a family of functions $\{\phi_k(\cdot)\}$ that represents a signal according to
\[
\signal{t} = \sum_{k=1}^{\infty} a_k\phi_k(t)\; ,
\]
where $a_k$ is a sequence of scalars defined both by the basis and the signal.

In simple terms, a signal $\signal{t}$ can be decomposed as a superposition---a sum---of simpler signals in most cases of interest.
In particular, we can define a \emph{basis} as a family of functions $\{\phi_k(\cdot)\}$ that represents a signal according to
\begin{equation}\label{eq:expansion}
\signal{t} = \sum_{k=1}^{\infty} a_k\phi_k(t)\; ,
\end{equation}
where $a_k$ is a sequence of scalars defined both by the basis and the signal.

When the basis consists of an orthonormal set of functions, the coefficients are found by calculating
\[
a_k = \int \!\signal{t} \phi_k(t)\,dt
\]

\section{Frames}\label{sec:frames}
As we saw in \S\ref{sec:intro}, signals can be represented by an orthonormal basis%
\footnote{``Orthonormal'' means $\int \!\phi_i(t)\phi_j(t)\,dt = \delta_{ij}$.}
as in \equationname~\eqref{eq:expansion}.
In this case, the expansion fits ``just right:''
it produces exactly the precise number of terms needed to represent the signal without error.
A \emph{frame} is ``fat:''
it is over-complete in that it typically employs more terms than needed.
Consequently, its basis functions do \emph{not} form an orthonormal set.
Reasons for choosing a frame rather than a basis are

\section{Long equations}
Alignment of multiple-line equations can be accomplished in several ways.
The \texttt{align} environment uses a single alignment operator and wants to number each line.
\begin{align}
\iiiint\limits_{Q}f(w,x,y,z)\,dw\,dx\,dy\,dz 
&\le \oint_{\bm{\partial Q}} f'\left(\max\left\{\frac{\|w\|}{|w^2+x^2|}; \frac{\|z\|}{|y^2+z^2|}; \frac{\|w\oplus z\|}{\|x\oplus y\|}\right\}\right)\\
&\precapprox \biguplus_{\mathbb{Q}\Subset\bar{\bm{Q}}}
\left[
f*\left(\frac{\left\lmoustache\mathbb{Q}(t)\right\rmoustache}{\sqrt{1-t^2}}\right)
\right]_{t=\alpha}^{t=\vartheta}
\end{align}
The \texttt{multline} environment allows a very long equation to be split across several lines and results in only the last line being numbered.
\begin{multline}
\sum_{t \in \mathbf{T}} \int_a^t \bigg\lbrace \int_a^t f(t-x)^2\, g(y)^2\,dx\bigg\rbrace\,dy
+
\int_a^t \bigg\lbrace \int_a^t f(t-x)^2\, g(y)^2\,dx\bigg\rbrace\,dy
\\
= \sum_{t \notin \mathbf{T}} \int_t^a \bigg\lbrace g(y)^2 \int_t^a f(x)^2 \,dx\bigg\rbrace\,dy
-
\int_t^a \bigg\lbrace g(y)^2 \int_t^a f(x)^2 \,dx\bigg\rbrace\,dy
\end{multline}
The \texttt{split} environment is contained within the \texttt{equation} environment.
\begin{equation}
\begin{split}
f_{h,\varepsilon}(x,y) &= h \int L_{x,z} \varphi(x)\rho_x(dz)
 + h \bigg[
\frac{1}{t_\varepsilon}\left(\mathbf{E}_y \int_0^{t_\varepsilon} L_{x,y^x (s)} \varphi(x) \,ds
-t_{\varepsilon} \int L_{x,z} \varphi(x) \rho_x(dz)\right) \\
&\mathrel{\phantom{=}} +
\frac{1}{t_\varepsilon} \left( \mathbf{E}_{y} \int_0^{t_\varepsilon} L_{x,y^x (s)} \varphi(x)\, ds
   -\mathbf{E}_{x,y} \int_0^{t_\varepsilon} L_{x, y_{\varphi}(\varepsilon s)} \varphi(x)\,ds \right)\bigg]
\end{split}
\end{equation}
\end{document}