56 lines
1.9 KiB
TeX
56 lines
1.9 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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\begin{abstract}
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\end{abstract}
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\section{Introduction}
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\section{Results \& Discussion}
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\subsection{Uniform Convergence}
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Consider the discrete time signal, $x[n] = a^n u[n]$, where $u[n]$ is the unit-step and $a=0.9$. Let $X(e^{j\omega})$ be the discrete time Fourier transform (DTFT) of $x[n]$. For the DTFT of a signal to exist, it must be absolutely summable over all $n \in \mathbb{Z}$. If the signal is absolutely summable, then it must also be bounded, meaning $|x[n]| < \infty, \forall n \in \mathbb{Z}$. In other words, $x[n]$ must have a finite maximum value.
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\\
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\\
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In this case, $x[n]$ is bounded because $a^n$ grows with smaller values of $n$, and $u[n]$ is zero when $n<0$. Therefore, the maximum value of $x[n] = a^n u[n]$ is $x[0] = a$.
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\\
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\\
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Additionally, $x[n]$ takes the form of a geometric series, so its sum is given by
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\begin{equation}
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\sum_{n=m}^\infty s^n = {s^m \over 1-s}, |s| < 1.
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\end{equation}
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In this case, $m=0$, because the signal has no value for $n<0$. Since $a=0.9$, the sum evaluates to is ${1 \over 1-0.9} = 10$. Considering that $x[n]$ is always a positive real number, each term is its own absolute value, so the sum and the absolute sum are equivalent. By taking the absolute sum of the first 200 terms of $x[n]$, it becomes clear that it approaches 10 in the limit, seen in figure \ref{fig:abs_sum_anun}.
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\begin{figure}[h]
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\center
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\includegraphics[width=\textwidth]{abs_sum_anun.png}
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\caption{The signal value of $x[n]=a^n u[n]$ and its absolute sum}
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\label{fig:abs_sum_anun}
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\end{figure}
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\section{Conclusions}
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\end{document}
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