51 lines
1.3 KiB
TeX
51 lines
1.3 KiB
TeX
\documentclass{article}
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\usepackage{amsmath}
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\usepackage{amsfonts}
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\usepackage{graphicx}
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\usepackage{listings}
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\usepackage{caption}
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\usepackage{subcaption}
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\usepackage{float}
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\usepackage[margin=1in]{geometry}
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\title{}
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\author{Aidan Sharpe \& Elise Heim}
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\DeclareMathOperator{\sinc}{sinc}
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\begin{document}
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\begin{titlepage}
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\maketitle
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\end{titlepage}
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\section{Results \& Discussion}
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\subsection{The Discrete Fourier Transform (DFT)}
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Given a signal, $x[n]$, it's $N$-point DFT is given by
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\begin{equation}
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X_k = \sum_{n=0}^{N-1} x[n] W_N^{kn}
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\label{eqn:DFT_def}
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\end{equation}
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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}, the $N$-point DFT of $x[n]$ is found to be
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\begin{equation}
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X_k = {1 + e^{-j2\pi k} \over 1 + e^{-j2\pi k / N}}.
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\label{eqn:DFT_ex}
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\end{equation}
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\subsection{The Z-Transform}
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Given a discrete signal, $x[n]$, its z-transform is given by
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\begin{equation}
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X(z) = \sum_n x[n] z^{-n}
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\end{equation}
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where $z$ is a complex variable.
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\subsection{The Inverse Z-Transform}
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\section{Conclusions}
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\end{document}
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