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