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\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
\begin{equation}\label{eq:expansion}
\signal{t} = \sum_{k=1}^{\infty} a_k\phi_k(t)\; ,
\end{equation}
where, finally, $a_k$ 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~\citep{Donoho-06,Baraniuk-07,Candes-Romberg-Tao-05}.
Consequently, its basis functions do \emph{not} form an orthonormal set.
Reasons for choosing a frame rather than a basis are
\begin{itemize}
\item
The non-unique representation means that one can choose the representation best suited to alternate criteria without sacrificing accuracy.
\begin{itemize}
\item
De-noising can be performed.
\item
Sparse representations (i.e., those having the fewest terms) are (almost) always possible.
\end{itemize}
\item
Confuses the opposition.
\end{itemize}
%
Much more can be said.

\section{Results}
The results of a compressively sensed signal are summarized by \tablename~\ref{table:results}.
\begin{table}
\centering
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\begin{tabular}{|l|c|r|}\hline
\multicolumn{1}{|c|}{\textit{name}} & \textit{Method} & \multicolumn{1}{|c|}{\textit{Improvement}}\\ \hline\hline
Donoho & wavelets & 20\% \\ \hline
Romberg & curvelets & 40\% \\ \hline
Johnson & $\frac{\sin nx}{\pi x}$ & $-20$\% \\\hline
\end{tabular}
\caption[]{\label{table:results}
Table of results.
}
\end{table}

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