From 4daf219dd6f3b08eeab8731dc5c7f488b8d5b94d Mon Sep 17 00:00:00 2001 From: quirinecker Date: Mon, 15 Jun 2026 11:10:31 +0200 Subject: [PATCH] initial commit --- .direnv/flake-profile | 1 + .direnv/flake-profile-2-link | 1 + .envrc | 1 + Assignment4_211.aux | 7 + Assignment4_211.fdb_latexmk | 91 +++++++++ Assignment4_211.fls | 172 ++++++++++++++++ Assignment4_211.log | 348 ++++++++++++++++++++++++++++++++ Assignment4_211.pdf | Bin 0 -> 713525 bytes Assignment4_211.synctex.gz | Bin 0 -> 15239 bytes Assignment4_211.tex | 240 ++++++++++++++++++++++ Assignment4_211.zip | Bin 0 -> 1803 bytes Assignment4_211/assignment4.4.m | 33 +++ Assignment4_211/assignment4_1.m | 33 +++ Assignment4_211/assignment4_3.m | 51 +++++ fig/fig1.1.png | Bin 0 -> 29540 bytes fig/fig1.2.png | Bin 0 -> 23905 bytes fig/fig1.3.png | Bin 0 -> 103888 bytes fig/fig1.4.png | Bin 0 -> 50673 bytes fig/fig3.1.png | Bin 0 -> 362615 bytes fig/fig4.1.png | Bin 0 -> 31524 bytes fig/fig4.2.png | Bin 0 -> 29167 bytes flake.lock | 26 +++ flake.nix | 23 +++ 23 files changed, 1027 insertions(+) create mode 120000 .direnv/flake-profile create mode 120000 .direnv/flake-profile-2-link create mode 100644 .envrc create mode 100644 Assignment4_211.aux create mode 100644 Assignment4_211.fdb_latexmk create mode 100644 Assignment4_211.fls create mode 100644 Assignment4_211.log create mode 100644 Assignment4_211.pdf create mode 100644 Assignment4_211.synctex.gz create mode 100644 Assignment4_211.tex create mode 100644 Assignment4_211.zip create mode 100644 Assignment4_211/assignment4.4.m create mode 100644 Assignment4_211/assignment4_1.m create mode 100644 Assignment4_211/assignment4_3.m create mode 100644 fig/fig1.1.png create mode 100644 fig/fig1.2.png create mode 100644 fig/fig1.3.png create mode 100644 fig/fig1.4.png create mode 100644 fig/fig3.1.png create mode 100644 fig/fig4.1.png create mode 100644 fig/fig4.2.png create mode 100644 flake.lock create mode 100644 flake.nix diff --git a/.direnv/flake-profile b/.direnv/flake-profile new file mode 120000 index 0000000..c7ae5b7 --- /dev/null +++ b/.direnv/flake-profile @@ -0,0 +1 @@ +flake-profile-2-link \ No newline at end of file diff --git a/.direnv/flake-profile-2-link b/.direnv/flake-profile-2-link new file mode 120000 index 0000000..cd899a3 --- /dev/null +++ b/.direnv/flake-profile-2-link @@ -0,0 +1 @@ +/nix/store/99fjzvv2y3g37jbqi28q8080db1183xk-latex-env \ No newline at end of file diff --git a/.envrc b/.envrc new file mode 100644 index 0000000..3550a30 --- /dev/null +++ b/.envrc @@ -0,0 +1 @@ +use flake diff --git a/Assignment4_211.aux b/Assignment4_211.aux new file mode 100644 index 0000000..eb30ef6 --- /dev/null +++ b/Assignment4_211.aux @@ -0,0 +1,7 @@ +\relax +\providecommand \babel@aux [2]{\global \let \babel@toc \@gobbletwo } +\@nameuse{bbl@beforestart} +\catcode `"\active +\babel@aux{english}{} +\babel@aux{austrian}{} +\gdef \@abspage@last{6} diff --git a/Assignment4_211.fdb_latexmk b/Assignment4_211.fdb_latexmk new file mode 100644 index 0000000..72090ce --- /dev/null +++ b/Assignment4_211.fdb_latexmk @@ -0,0 +1,91 @@ +# Fdb version 4 +["pdflatex"] 1780937132.21957 "Assignment4_211.tex" "Assignment4_211.pdf" "Assignment4_211" 1780937132.89663 0 + "/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/map/fontname/texfonts.map" 1 3524 cb3e574dea2d1052e39280babc910dc8 "" + "/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1 1004 54797486969f23fa377b128694d548df "" + "/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 "" + "/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/amsfonts/symbols/msam10.tfm" 1 916 f87d7c45f9c908e672703b83b72241a3 "" + 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\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} diff --git a/Assignment4_211.zip b/Assignment4_211.zip new file mode 100644 index 0000000000000000000000000000000000000000..cf47a3eb09e4fabe59bd323c0448aba887099268 GIT binary patch literal 1803 zcmWIWW@Zs#00EWGqcLCxln`K0U~nuh&P>nCP0cGYi8nGd)Gw{zW?*D_!OXzGk^)o~ z8p6xKKINHG*5z|*Ss*$9Y#Ijx2gCFlRMX_Kn3jlSy0Km^HpAatP|CXgQ7sEZuMIw! zciVtx?{Do2PNn?ZZM$wPJ;ryG%iMZPd+q@~*T558y4(|0{U^;%jkotId$VY-nS+q= z!|MJg%B^niyaaAlv8*et5lNl4((h1sz+C5j$*qrKbMs3Ral04^y#n~sPblX;nt^b$! zt)QIkd;b(^<*;5$`yYP1D~mp{?Q-05TEocw3>r8NA=S{=8z4xlhmdw$Qbp z%Q~wjU7L2dR_okd(b~F_FTq^zk|zW|dtS83Rem{V`qj%n8Vc`f-Kc5E-O023S(3f} z;#s@n9p3M?_{gx&-8;i(8{ZtA8Ly@LyVpHTJ*hSK#L2#=B6`z<7c9N_M)l2A-aC(e zxGUWDXMZ`>B>Sg5R&4&y$iU#u!oYw?Yl}%tYlhg<8q-BcS_9E@19s;f 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"owner": "NixOS", + "repo": "nixpkgs", + "type": "github" + } + }, + "root": { + "inputs": { + "nixpkgs": "nixpkgs" + } + } + }, + "root": "root", + "version": 7 +} diff --git a/flake.nix b/flake.nix new file mode 100644 index 0000000..46331fd --- /dev/null +++ b/flake.nix @@ -0,0 +1,23 @@ +{ + description = "LaTeX paper build environment"; + + inputs = { + nixpkgs.url = "github:NixOS/nixpkgs"; + }; + + outputs = + { self, nixpkgs }: + let + system = "x86_64-linux"; + pkgs = import nixpkgs { inherit system; }; + in + { + devShells.${system}.default = pkgs.mkShell { + name = "latex"; + buildInputs = with pkgs; [ + texliveFull + octave + ]; + }; + }; +}

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