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\title{Discrete Fourier Transforms and Z-Transforms}

\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.
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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} as a truncated geometric series, the $N$-point DFT of $x[n]$ can be found. All truncated geometric series are evaluated as
\begin{equation}
    \sum_{k=0}^{n-1} a r^k = 
    \begin{cases}
        an & r = 1 \\
        a\left({1 - r^n \over 1 - r}\right) & r \ne 1
    \end{cases},
\end{equation}
where $r$ is the common ratio between adjacent terms. For the $N$-point DFT of $x[n]$, the common ratio is $-W_N^k$, which takes a value of 1 for $k = {N\over2}$. Therefore, the $N$-point DFT of $x[n]$ is
\begin{equation}
    X[k] = \begin{cases}
        N & k = {N \over 2} \\
        \left({1 - (-W_N^k)^N \over 1 - (-W_N^k)}\right) & k \ne {N \over 2}
    \end{cases}.
    \label{eqn:DFT_N_point}
\end{equation}
The $N$-point DFT of $x[n]$, where $N=8$ is seen in figure \ref{fig:N_point_DFT}. It only has a non-zero value for $k={N\over2}=4$. This is the case for all even-number-point DFTs. Therefore only odd-number-point DFTs should be used.
\begin{figure}[h]
    \center
    \includegraphics[width=0.5\textwidth]{N8_point_DFT.png}
    \caption{The $N$-point DFT of $x[n]$, where $N=8$}
    \label{fig:N_point_DFT}
\end{figure}
For example, the 9-point DFT of $x[n]$, where $N=8$ is seen in figure \ref{fig:9_point_DFT}. While  equation \ref{eqn:DFT_N_point} cannot be used because there are a different number of samples for the DFT and the input signal, the overall DFT is more useful than the 8-point DFT.
\begin{figure}[h]
        \center
    \includegraphics[width=0.5\textwidth]{Q9_point_DFT.png}
    \label{fig:9_point_DFT}
\end{figure}



\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}
