changed the assignment number
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Assignment5_211.tex
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240
Assignment5_211.tex
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\documentclass[12pt,a4paper,austrian]{article}
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\usepackage{graphicx}
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\usepackage[austrian, english]{babel}
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\usepackage[utf8]{inputenc}
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%\usepackage{listings}
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\usepackage{multirow}
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\usepackage{epstopdf}
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\usepackage{amsmath}
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\usepackage{amssymb} % fuer Mengen \N, Q, C, R
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\graphicspath{{./fig/}}
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%% Satzspiegel
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\setlength{\hoffset}{-1in} \setlength{\textwidth}{18cm}
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\setlength{\oddsidemargin}{1.5cm}
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\setlength{\evensidemargin}{1.5cm}
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\setlength{\marginparsep}{0.7em}
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\setlength{\marginparwidth}{0.5cm}
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\setlength{\voffset}{-1.9in}
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\setlength{\headheight}{12pt}
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\setlength{\topmargin}{2.6cm}
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\addtolength{\topmargin}{-\headheight}
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\setlength{\headsep}{3.5cm}
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\addtolength{\headsep}{-\topmargin}
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\addtolength{\headsep}{-\headheight}
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\setlength{\textheight}{27cm}
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%% How should floats be treated?
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\setlength{\floatsep}{12 pt plus 0 pt minus 8 pt}
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\setlength{\textfloatsep}{12 pt plus 0pt minus 8 pt}
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\setlength{\intextsep}{12 pt plus 0pt minus 8 pt}
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\tolerance2000
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\emergencystretch20pt
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%% Text appearence
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% English text
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\newcommand{\eg}[1]%
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{\selectlanguage{english}\textit{#1}\selectlanguage{austrian}}
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\newcommand{\filename}[1]
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{\begin{small}\texttt{#1}\end{small}}
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\newcommand\IFT{\unitlength1mm\begin{picture}(10,2) \put (1,1)
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{\circle{1.7}} \put(2,1){\line(1,0){5}} \put(8,1)
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{\circle*{1.7}}\end{picture}}
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\newcommand\FT{\unitlength1mm\begin{picture}(10,2) \put (1,1)
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{\circle*{1.7}} \put(2,1){\line(1,0){5}} \put(8,1)
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{\circle{1.7}}\end{picture}}
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% A box for multiple choice problems
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\newcommand{\choicebox}{\fbox{\rule{0pt}{0.5ex}\rule{0.5ex}{0pt}}}
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\newenvironment{wahrfalsch}%
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{\bigskip\par\noindent\makebox[1cm][c]{richtig}\hspace{3mm}\makebox[1cm][c]{falsch}
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\begin{list}%
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{\makebox[1cm][c]{\choicebox}\hspace{3mm}\makebox[1cm][c]{\choicebox}}%
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{\setlength{\labelwidth}{2.31 cm}\setlength{\labelsep}{3mm}
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\setlength{\leftmargin}{2.61 cm}\setlength{\listparindent}{0pt}
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\setlength{\itemindent}{0pt}}%
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}
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{\end{list}}
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\newcounter{theaufgabe}\setcounter{theaufgabe}{1}
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\newenvironment{aufgabe}[1]%
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{\bigskip\par\noindent\begin{nopagebreak}
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\textsf{\textbf{\arabic{theaufgabe}.\thinspace Exercise}}\quad
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\textsf{\textit{#1}}\\*[1ex]%
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\stepcounter{theaufgabe}\hspace{2ex}\end{nopagebreak}}
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{\par\pagebreak[2]}
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% Innerhalb der Aufgaben erfolgt die weitere Unterteilung mittels einer
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% enumerate Umgebung, die allerdings a), b),... zaehlen soll.
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\renewcommand{\labelenumi}{\alph{enumi})}
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\renewcommand{\labelenumii}{\arabic{enumii})}
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% A box to tick for everything which has to done
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\newcommand{\abgabe}{\marginpar{$\Box$}}
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% Margin paragraphs on the left side
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\reversemarginpar
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% Language for listings
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%\lstset{language=Vhdl,
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% basicstyle=\small\tt,
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% keywordstyle=\tt\bf,
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% commentstyle=\sl}
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% No indention
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\setlength{\parindent}{0.0cm}
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% Don't number sections
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\setcounter{secnumdepth}{0}
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%% Beginning of the text
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\begin{document}
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\selectlanguage{austrian}
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\pagestyle{plain}
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%=== This is the header section ============================================================
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\thispagestyle{empty}
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\noindent
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\begin{minipage}[b][4cm]{1.0\textwidth}
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\begin{center}
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\begin{bf}
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\begin{large} Digitale Signalverarbeitung SS 2025/26 -- 4.~Aufgabe\end{large} \\
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\vspace{0.3cm}
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\begin{Large} Reconstruction, DFT, FFT \end{Large} \\
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\vspace{0.3cm}
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\end{bf}
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\begin{large}
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Gruppennummer 211 \\
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Manuel Illmayer, k12308149 \\
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Quirin Ecker, k12310122 \\
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\end{large}
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\end{center}
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\end{minipage}
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\noindent \rule[0.8em]{\textwidth}{0.12mm}\\[-0.5em]
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%=======================================================================================
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\begin{aufgabe}{Reconstruction}
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\begin{enumerate}
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\item : \\
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\includegraphics[width=0.8\textwidth]{./fig/fig1.1.png} \\
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\includegraphics[width=0.8\textwidth]{./fig/fig1.2.png}
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\item : \\
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\begin{gather*}
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x(t) = 1 + 0.5 \cos(2\pi f_1 t) + 2 \sin(2\pi f_2 t) + \sin(2\pi f_3 t) \\
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\end{gather*}
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Calculation of $x_1[n]$:
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\begin{gather*}
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x(t) = 1 + 0.5 \cos(2\pi f_1 t) + 2 \sin(2\pi f_2 t) + \sin(2\pi f_3 t) \\
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= 1 + 0.5 \cos(2\pi 2000 t) + 2 \sin(2\pi 4000 t) + \sin(2\pi 6000 t) \\
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x_1[n] = 1 + 0.5 \cos(2\pi \frac{2000}{9000} n) + 2 \sin(2\pi \frac{4000}{9000} n) + \sin(2\pi \frac{6000}{9000} n) \\
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= 1 + 0.5 \cos(\frac{4 \pi}{9} n) + 2 \sin( \frac{8 \pi}{9} n) + \sin( \frac{4 \pi}{3} n) \\
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\\
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\sin(\frac{4 \pi}{3}) = -\sin(\frac{2 \pi}{3})
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\\
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\implies x_1[n]= 1 + 0.5 \cos(\frac{4 \pi}{9} n) + 2 \sin( \frac{8 \pi}{9} n) - \sin( \frac{2 \pi}{3} n)
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\end{gather*}
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Calculation of $x_2[n]$:
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\begin{gather*}
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x_1[n] = 1 + 0.5 \cos(2\pi \frac{2000}{14000} n) + 2 \sin(2\pi \frac{4000}{14000} n) + \sin(2\pi \frac{6000}{14000} n) \\
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= 1 + 0.5 \cos(\frac{2 \pi}{7} n) + 2 \sin( \frac{4 \pi}{7} n) + \sin( \frac{6 \pi}{7} n) \\
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\end{gather*}
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\includegraphics[width=0.8\textwidth]{./fig/fig1.3.png}
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\item: \\
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\[
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f=\frac{\Omega f_s}{2\pi}
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\]
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Calculations for $x_1(t)$:
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\[
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f_1=\frac{\Omega_1 f_s}{2\pi}
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=\frac{\left(\frac{4\pi}{9}\right)9000}{2\pi}
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=2000
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\]
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\[
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f_2 = ... = 4000
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\]
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\[
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f_3=\frac{\Omega_3 f_s}{2\pi}
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=\frac{\left(-\frac{2\pi}{3}\right)9000}{2\pi}
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=-3000
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\]
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Calculations for $x_2(t)$:
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\[
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f_1 = ... = 2000
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\]
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\[
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f_2 = ... = 4000
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\]
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\[
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f_3 = ... = 6000
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\]
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Where '...' is just the formular at the top injected with the $\Omega$ values of b)
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\includegraphics[width=0.8\textwidth]{./fig/fig1.4.png}
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\end{enumerate}
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\end{aufgabe}
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\begin{aufgabe}{DFT Theory}
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\begin{enumerate}
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\item
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\begin{gather*}
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f_s = \frac{1}{T_s} = \frac{1}{0.001} = 1000 \text{Hz} \\
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\Delta f = \frac{f_s}{N} = \frac{1000}{100} = 10 \text{Hz} \\
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\end{gather*}
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\item
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\begin{gather*}
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\text{In terms of samples}: 100 \\
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\text{In terms of frequency}: N \cdot \Delta f_s = 1000 \text{Hz} \\
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\text{In terms of angular frequency}: \omega = 2 \pi \frac{f}{f_s} = 2 \pi \frac{1000}{1000} = 2 \pi
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\end{gather*}
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\item Computers can calculate it more efficently when it is working with powers of two \\
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\item $\Delta f = \frac{f_s}{N} = \frac{1000}{128} = 7.8125 \text{Hz}$ \\
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\item Even though we have not gained new information, we have a finer resolution of the frequency axis. This can make it easier to get closer to the actual value.
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\end{enumerate}
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\end{aufgabe}
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\begin{aufgabe}{FFT in Image Processing}
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Approach: We want to remove the low frequencies (high pass filter) to
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keep the areas of the image where the lightness changes relative to
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nearby pixels \\
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\includegraphics[width=0.8\textwidth]{./fig/fig3.1.png}
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\end{aufgabe}
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\begin{aufgabe}{Window Effects of the DFT}
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\begin{enumerate}
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\item ( see b) )
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\item (see assignment4\_4.m) \\
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\includegraphics[width=0.8\textwidth]{./fig/fig4.1.png}
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\item
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Rectangular window: Clear peaks but also many neighbors have fairly high values
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Hamming window: Still visible peaks, that may be a bit wider, but now clearly separated from neighbors and more visually readable.
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\item We can choose freqencies that align with our 128 steps, for example f1 = 10/N and f2 = 15/N. With this, the signal fits exactly within our window, each cosine has a integer number of cycles it completes \\
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\includegraphics[width=0.8\textwidth]{./fig/fig4.2.png}
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\end{enumerate}
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\end{aufgabe}
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\end{document}
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