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\title{}

\author{Aidan Sharpe \& Elise Heim}

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\begin{document}
\begin{titlepage}
\maketitle
\end{titlepage}

\section{Results \& Discussion}

\subsection{The Discrete Fourier Transform (DFT)}
Given a signal, $x[n]$, it's $N$-point DFT is given by
\begin{equation}
	X_k = \sum_{n=0}^{N-1} x[n] W_N^{kn}
	\label{eqn:DFT_def}
\end{equation}
where $W_N = e^{-j2\pi/N}$. The discrete Fourier transform is the sampled version of the discrete time Fourier transform (DTFT), which is a continuous function. More specifically, the $N$-point DFT contains $N$ samples from the continuous DTFT.

For example, consider the signal $x[n] = (-1)^n$ for $0 \le n \le N-1$. By evaluating the sum shown in equation \ref{eqn:DFT_def}, the $N$-point DFT of $x[n]$ is found to be
\begin{equation}
	X_k = {1 + e^{-j2\pi k} \over 1 + e^{-j2\pi k / N}}.
	\label{eqn:DFT_ex}
\end{equation}

\subsection{The Z-Transform}
Given a discrete signal, $x[n]$, its z-transform is given by
\begin{equation}
	X(z) = \sum_n x[n] z^{-n}
\end{equation}
where $z$ is a complex variable.

\subsection{The Inverse Z-Transform}

\section{Conclusions}
\end{document}