Syncing to work on interview with entrepreneur essay

6th-Semester-Spring-2024/DSP/Labs/Lab-04/lab-4.tex

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+\documentclass{article}
+
+\usepackage{amsmath}
+\usepackage{amsfonts}
+\usepackage{graphicx}
+\usepackage{listings}
+\usepackage{caption}
+\usepackage{subcaption}
+\usepackage{float}
+\usepackage[margin=1in]{geometry}
+
+
+\title{}
+
+\author{Aidan Sharpe \& Elise Heim}
+
+\DeclareMathOperator{\sinc}{sinc}
+
+\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}