\documentclass{article} %twocolumn

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\usepackage{ifthen}

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\newcommand{\signal}[1]{x\left(#1\right)}
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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}
When in two-column mode, you need to format the long equations to fit the column width.
Consider the following.
\ifthenelse{\boolean{@twocolumn}}% @twocolumn
{\begin{multline}
f_{h,\varepsilon}(x,y) = h \int L_{x,z} \varphi(x)\rho_x(dz)\\
\mathrel{\phantom{=}} + 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) \\
 +\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{multline}
}%
{\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}