\documentclass{IEEEtran}[journal]

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\title{}
\author{Aidan Sharpe \& Elise Heim}

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\begin{document}
\maketitle

\begin{abstract}
\end{abstract}

\section{Introduction}

\section{Results \& Discussion}
\subsection{Analysis of Amplitude Modulation}
A message, $m(t)$, with a bandwidth, $B=2[\text{kHz}]$ modulates a cosine carrier with a frequency of 10[kHz]. The combined signal is $s(t) = m(t)\cos(20000\pi t)$. Using a Fourier transform on $s(t)$ reveals a maximum frequency at 12[kHz]. In fact, as seen in figure \ref{fig:AM_Fourier_Whitenoise}, by filling the band that $m(t)$ occupies with white noise, the Fourier transform of $s(t)$ contains white noise centered on the carrier frequency with twice the bandwidth of the original signal. The spike that occurs at 10[kHz] is the result of the original signal having a DC term and the carrier frequency having a value of 10[kHz].

\begin{figure}[h]
	\includegraphics[width=0.48\textwidth]{lab2_1a.png}
	\caption{The Fourier transform of a white noise signal carried at 10[kHz]}
	\label{fig:AM_Fourier_Whitenoise}
\end{figure}

If instead, $m(t)$ had a triangular spectrum of amplitude 1, the spectrum of $s(t)$ will be two triangles touching at the base at 10[kHz] as seen in figure \ref{fig:carried_triangle}.

\begin{figure}[h]
	\includegraphics[width=0.48\textwidth]{lab2_1b.png}
	\caption{The Fourier transform of a signal with a triangular spectrum carried at 10[kHz]}
	\label{fig:carried_triangle}
\end{figure}

\subsection{Periodicity and Sampling Frequency}
Consider the signal $x(t) = \cos(2\pi t/7)$. Given the standard forms, $\cos(2\pi f t)$ and $\cos(\omega t)$, where f is linear frequency and $\omega$ is angular frequency, $f = {1\over7}$ and $\omega = {2\pi \over 7}$. Given a sampling frequency of 1[Hz], the sampling theorem is satisfied. To determine if a sampled signal is periodic, the condition $\omega N = 2 \pi r$, where $r$ is the smallest integer such that $N$ is an integer, must be satisfied.




\section{Conclusions}

\end{document}