commit 4daf219dd6f3b08eeab8731dc5c7f488b8d5b94d Author: quirinecker Date: Mon Jun 15 11:10:31 2026 +0200 initial commit 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 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(max. 500000) + 36 words of extra memory for PDF output out of 10000 (max. 10000000) + diff --git a/Assignment4_211.pdf b/Assignment4_211.pdf new file mode 100644 index 0000000..215db01 Binary files /dev/null and b/Assignment4_211.pdf differ diff --git a/Assignment4_211.synctex.gz b/Assignment4_211.synctex.gz new file mode 100644 index 0000000..4b911c1 Binary files /dev/null and b/Assignment4_211.synctex.gz differ diff --git a/Assignment4_211.tex b/Assignment4_211.tex new file mode 100644 index 0000000..0ab76f7 --- /dev/null +++ b/Assignment4_211.tex @@ -0,0 +1,240 @@ +\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} diff --git a/Assignment4_211.zip b/Assignment4_211.zip new file mode 100644 index 0000000..cf47a3e Binary files /dev/null and b/Assignment4_211.zip differ diff --git a/Assignment4_211/assignment4.4.m b/Assignment4_211/assignment4.4.m new file mode 100644 index 0000000..7eae517 --- /dev/null +++ b/Assignment4_211/assignment4.4.m @@ -0,0 +1,33 @@ +clear; close all; clc; + +N = 128; +n = 0:N-1; +w1 = 2*pi*0.1; +w2 = 2*pi*0.15; +x = cos(w1*n) + cos(w2*n); + +% a) + +X = fft(x); +magX = abs(X); + +figure; +stem(n, magX, 'filled'); +xlabel('n'); +ylabel('|X[n]|'); +title('Magnitude Spectrum of x[n]'); +grid on; + +% b) + +y = hamming(N)' .* x; + +Y = fft(y); +magY = abs(Y); + +figure; +stem(n, magY, 'filled'); +xlabel('n'); +ylabel('|Y[n]|'); +title('Magnitude Spectrum of y[n]'); +grid on; diff --git a/Assignment4_211/assignment4_1.m b/Assignment4_211/assignment4_1.m new file mode 100644 index 0000000..5c88e67 --- /dev/null +++ b/Assignment4_211/assignment4_1.m @@ -0,0 +1,33 @@ +clc; +clear; +close all; + +%% Parameters +f1 = 2000; +f2 = 4000; +f3 = 6000; + +f3_rec = -3000 + +%% Time vectors +t=0:1e-6:1e-3; + +%% Signals +x = 1+0.5*cos(2*pi*f1*t)+2*sin(2*pi*f2*t)+sin(2*pi*f3*t); +x1 = 1+0.5*cos(2*pi*f1*t)+2*sin(2*pi*f2*t)+sin(2*pi*f3_rec*t); +x2 = 1+0.5*cos(2*pi*f1*t)+2*sin(2*pi*f2*t)+sin(2*pi*f3*t); + +%% Plot +figure +hold on + +plot(t, x, 'b', 'LineWidth', 1.5) +plot(t, x1, 'r', 'LineWidth', 1.5) +plot(t, x2, '--g', 'LineWidth', 1.5) + +grid on + +xlabel('Time (ms)') +ylabel('x(t)') + +legend('x(t)', 'x1(t)', 'x2(t)'); diff --git a/Assignment4_211/assignment4_3.m b/Assignment4_211/assignment4_3.m new file mode 100644 index 0000000..55d9e47 --- /dev/null +++ b/Assignment4_211/assignment4_3.m @@ -0,0 +1,51 @@ +clear; close all; clc; + +img = imread('pavian.png'); % Choose any picture you like here +img = rgb2gray(img); + +%% Display image +figure; subplot(131); +imshow(img); +title('Ausgangsbild'); + +%% FFT +fft2d = fft2(img); + +% Display the FFT image. +subplot(132); +imshow(log(fftshift(abs(fft2d))), []); +title('2D-FFT'); + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% + +% 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 + +% Shift FFT so low frequencies are in the center +F = fftshift(fft2d); + +% Get image size +[M, N] = size(F); + +% Radius of how much to remove +r = 25; + +% We cut out the center of the fft low frequencies +H = ones(M, N); +H(M/2-r:M/2+r, N/2-r:N/2+r) = 0; + +% Apply filter +F_filtered = F .* H; + +% Shift back +fft2d = ifftshift(F_filtered); + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% + +%% IFFT to obtain filtered image +img_filtered = ifft2(fft2d); +subplot(133); +imshow(real(img_filtered), []); +title('Gefiltertes Ausgangssignal'); diff --git a/fig/fig1.1.png b/fig/fig1.1.png new file mode 100644 index 0000000..e59c39f Binary files /dev/null and b/fig/fig1.1.png differ diff --git a/fig/fig1.2.png b/fig/fig1.2.png new file mode 100644 index 0000000..69d6c6f Binary files /dev/null and b/fig/fig1.2.png differ diff --git a/fig/fig1.3.png b/fig/fig1.3.png new file mode 100644 index 0000000..cee577c Binary files /dev/null and b/fig/fig1.3.png differ diff --git a/fig/fig1.4.png b/fig/fig1.4.png new file mode 100644 index 0000000..7f193fa Binary files /dev/null and b/fig/fig1.4.png differ diff --git a/fig/fig3.1.png b/fig/fig3.1.png new file mode 100644 index 0000000..91617eb Binary files /dev/null and b/fig/fig3.1.png differ diff --git a/fig/fig4.1.png b/fig/fig4.1.png new file mode 100644 index 0000000..36a5936 Binary files /dev/null and b/fig/fig4.1.png differ diff --git a/fig/fig4.2.png b/fig/fig4.2.png new file mode 100644 index 0000000..0277f32 Binary files /dev/null and b/fig/fig4.2.png differ diff --git a/flake.lock b/flake.lock new file mode 100644 index 0000000..a617285 --- /dev/null +++ b/flake.lock @@ -0,0 +1,26 @@ +{ + "nodes": { + "nixpkgs": { + "locked": { + "lastModified": 1779017031, + "narHash": "sha256-bSFtTjXkh80yk4RfdQXQ9mCtHTG7mwJrVXpolJeXqsQ=", + "owner": "NixOS", + "repo": "nixpkgs", + "rev": "71b4e08640c2222a2aad7777b6936edb2ea1ccb7", + "type": "github" + }, + "original": { + "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 + ]; + }; + }; +}