updated changes so far

.direnv/flake-profile

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+flake-profile-1-link

.direnv/flake-profile-1-link

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+/nix/store/ha1aav8ld0ym8pr8nmn66np5g4hfnwda-latex-env

.envrc

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+use flake

.gitignore

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+# Created by https://www.toptal.com/developers/gitignore/api/latex
+# Edit at https://www.toptal.com/developers/gitignore?templates=latex
+
+### LaTeX ###
+## Core latex/pdflatex auxiliary files:
+*.aux
+*.lof
+*.log
+*.lot
+*.fls
+*.out
+*.toc
+*.fmt
+*.fot
+*.cb
+*.cb2
+.*.lb
+
+## Intermediate documents:
+*.dvi
+*.xdv
+*-converted-to.*
+# these rules might exclude image files for figures etc.
+# *.ps
+# *.eps
+# *.pdf
+
+## Generated if empty string is given at "Please type another file name for output:"
+.pdf
+
+## Bibliography auxiliary files (bibtex/biblatex/biber):
+*.bbl
+*.bcf
+*.blg
+*-blx.aux
+*-blx.bib
+*.run.xml
+
+## Build tool auxiliary files:
+*.fdb_latexmk
+*.synctex
+*.synctex(busy)
+*.synctex.gz
+*.synctex.gz(busy)
+*.pdfsync
+
+## Build tool directories for auxiliary files
+# latexrun
+latex.out/
+
+## Auxiliary and intermediate files from other packages:
+# algorithms
+*.alg
+*.loa
+
+# achemso
+acs-*.bib
+
+# amsthm
+*.thm
+
+# beamer
+*.nav
+*.pre
+*.snm
+*.vrb
+
+# changes
+*.soc
+
+# comment
+*.cut
+
+# cprotect
+*.cpt
+
+# elsarticle (documentclass of Elsevier journals)
+*.spl
+
+# endnotes
+*.ent
+
+# fixme
+*.lox
+
+# feynmf/feynmp
+*.mf
+*.mp
+*.t[1-9]
+*.t[1-9][0-9]
+*.tfm
+
+#(r)(e)ledmac/(r)(e)ledpar
+*.end
+*.?end
+*.[1-9]
+*.[1-9][0-9]
+*.[1-9][0-9][0-9]
+*.[1-9]R
+*.[1-9][0-9]R
+*.[1-9][0-9][0-9]R
+*.eledsec[1-9]
+*.eledsec[1-9]R
+*.eledsec[1-9][0-9]
+*.eledsec[1-9][0-9]R
+*.eledsec[1-9][0-9][0-9]
+*.eledsec[1-9][0-9][0-9]R
+
+# glossaries
+*.acn
+*.acr
+*.glg
+*.glo
+*.gls
+*.glsdefs
+*.lzo
+*.lzs
+*.slg
+*.slo
+*.sls
+
+# uncomment this for glossaries-extra (will ignore makeindex's style files!)
+# *.ist
+
+# gnuplot
+*.gnuplot
+*.table
+
+# gnuplottex
+*-gnuplottex-*
+
+# gregoriotex
+*.gaux
+*.glog
+*.gtex
+
+# htlatex
+*.4ct
+*.4tc
+*.idv
+*.lg
+*.trc
+*.xref
+
+# hyperref
+*.brf
+
+# knitr
+*-concordance.tex
+# TODO Uncomment the next line if you use knitr and want to ignore its generated tikz files
+# *.tikz
+*-tikzDictionary
+
+# listings
+*.lol
+
+# luatexja-ruby
+*.ltjruby
+
+# makeidx
+*.idx
+*.ilg
+*.ind
+
+# minitoc
+*.maf
+*.mlf
+*.mlt
+*.mtc[0-9]*
+*.slf[0-9]*
+*.slt[0-9]*
+*.stc[0-9]*
+
+# minted
+_minted*
+*.pyg
+
+# morewrites
+*.mw
+
+# newpax
+*.newpax
+
+# nomencl
+*.nlg
+*.nlo
+*.nls
+
+# pax
+*.pax
+
+# pdfpcnotes
+*.pdfpc
+
+# sagetex
+*.sagetex.sage
+*.sagetex.py
+*.sagetex.scmd
+
+# scrwfile
+*.wrt
+
+# svg
+svg-inkscape/
+
+# sympy
+*.sout
+*.sympy
+sympy-plots-for-*.tex/
+
+# pdfcomment
+*.upa
+*.upb
+
+# pythontex
+*.pytxcode
+pythontex-files-*/
+
+# tcolorbox
+*.listing
+
+# thmtools
+*.loe
+
+# TikZ & PGF
+*.dpth
+*.md5
+*.auxlock
+
+# titletoc
+*.ptc
+
+# todonotes
+*.tdo
+
+# vhistory
+*.hst
+*.ver
+
+# easy-todo
+*.lod
+
+# xcolor
+*.xcp
+
+# xmpincl
+*.xmpi
+
+# xindy
+*.xdy
+
+# xypic precompiled matrices and outlines
+*.xyc
+*.xyd
+
+# endfloat
+*.ttt
+*.fff
+
+# Latexian
+TSWLatexianTemp*
+
+## Editors:
+# WinEdt
+*.bak
+*.sav
+
+# Texpad
+.texpadtmp
+
+# LyX
+*.lyx~
+
+# Kile
+*.backup
+
+# gummi
+.*.swp
+
+# KBibTeX
+*~[0-9]*
+
+# TeXnicCenter
+*.tps
+
+# auto folder when using emacs and auctex
+./auto/*
+*.el
+
+# expex forward references with \gathertags
+*-tags.tex
+
+# standalone packages
+*.sta
+
+# Makeindex log files
+*.lpz
+
+# xwatermark package
+*.xwm
+
+# REVTeX puts footnotes in the bibliography by default, unless the nofootinbib
+# option is specified. Footnotes are the stored in a file with suffix Notes.bib.
+# Uncomment the next line to have this generated file ignored.
+#*Notes.bib
+
+### LaTeX Patch ###
+# LIPIcs / OASIcs
+*.vtc
+
+# glossaries
+*.glstex
+
+# End of https://www.toptal.com/developers/gitignore/api/latex
+

Assignment1_211.tex

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+\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 WS 2025/26 -- 1.~Aufgabe\end{large} \\
+\vspace{0.3cm}
+\begin{Large} Analog Signals and Systems \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}{}
+
+\begin{align*}
+c_1 &= -5 + 3j
+= \sqrt{(-5)^2 + 3^2}\, e^{j \arctan\left(\frac{3}{-5}\right)}
+= \sqrt{34}\, e^{-j\,0.54} \\
+\end{align*}
+
+\begin{align*}
+c_2 &= \frac{\sqrt{2}}{2} e^{-j \frac{3\pi}{4}}
+= \frac{\sqrt{2}}{2} \cos\left(-\frac{3\pi}{4}\right)
++ j \frac{\sqrt{2}}{2} \sin\left(-\frac{3\pi}{4}\right)
+= -\frac{1}{2} - \frac{1}{2}j \\
+\end{align*}
+
+\begin{align*}
+c_3 &= \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}j
+= \sqrt{\left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2}
+\, e^{j \arctan\left(\frac{\frac{1}{\sqrt{2}}}{\frac{1}{\sqrt{2}}}\right)}
+= e^{j \frac{\pi}{4}} \\
+\end{align*}
+
+\begin{align*}
+c_4 &= c_1 + c_2
+= (-5 + 3j) + \left(-\frac{1}{2} - \frac{1}{2}j\right)
+= -\frac{11}{2} + \frac{5}{2}j \\
+\end{align*}
+
+\begin{align*}
+c_5 &= c_1 \cdot c_2
+= \left(\sqrt{34}\, e^{-j\,0.54}\right)\left(\frac{\sqrt{2}}{2} e^{-j \frac{3 \pi}{4}}\right)
+\approx \frac{\sqrt{70}}{2} e^{-j\,2.8961} \\
+\end{align*}
+
+\begin{align*}
+c_6 &= |c_3|^2
+= \left(\sqrt{\left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2}\right)^2
+= 1 \\
+\end{align*}
+
+\begin{align*}
+c_7 &= \arg(c_3) = \frac{\pi}{4} \\
+\end{align*}
+
+\begin{align*}
+c_8 &= \frac{c_1}{c_2}
+= \frac{-5 + 3j}{-\frac{1}{2} - \frac{1}{2}j}
+\cdot \frac{-\frac{1}{2} + \frac{1}{2}j}{-\frac{1}{2} + \frac{1}{2}j} \\
+&= \frac{(-5)(-\frac{1}{2}) + 3(-\frac{1}{2}) + j\left(3(-\frac{1}{2}) - (-5)(-\frac{1}{2})\right)}{\left(-\frac{1}{2}\right)^2 + \left(-\frac{1}{2}\right)^2} \\
+&= 2 - 8j \\
+\end{align*}
+
+\begin{align*}
+c_9 &= c_1 \cdot c_1^*
+= \sqrt{34}\, e^{-j\,0.54} \cdot \sqrt{34}\, e^{j\,0.54}
+= 34
+\end{align*}
+\end{aufgabe}
+
+\begin{aufgabe}{}
+
+
+\[
+    x(t) = \hat{X} \cos(2\pi f_0 t) \xleftrightarrow{} \iff X(f) = \frac{\hat{X}}{2} \delta(f - f_0) + \frac{\hat{X}}{2} \delta(f + f_0) \\
+\]
+
+\[
+	x(t) = \hat{X}( \frac{1}{2} \cdot e^{j2\pi f_0 t} + \frac{1}{2} \cdot e^{-j2\pi f_0 t} )
+\]
+
+\[
+	X(f) = \hat{X}( \frac{1}{2} \cdot \delta (f-f_0) + \frac{1}{2} \cdot \delta (f+f_0) )
+\]
+
+\[
+	X(f) = \frac{\hat{X}}{2} \cdot \delta (f-f_0) + \frac{\hat{X}}{2} \cdot \delta (f+f_0)
+\]
+
+\begin{center}
+	\includegraphics[width=0.95\textwidth]{fig/aufgabe2.png}
+\end{center}
+
+
+
+\end{aufgabe}
+
+\begin{aufgabe}{}
+
+\begin{enumerate}
+	\item
+\[
+\sin\left(2\pi f_i (t - 0.1)\right) = \sin\left(2\pi f_i t + \phi_i\right)
+\]
+
+\[
+2\pi f_i (t - 0.1) = 2\pi f_i t + \phi_i
+\]
+
+\[
+2\pi f_i (t - 0.1) - 2\pi f_i t = \phi_i
+\]
+
+\[
+2\pi f_i \big((t - 0.1) - t\big) = \phi_i
+\]
+
+\[
+\phi_i = 2\pi f_i (-0.1) = -0.1 \cdot 2\pi f_i
+\]
+
+The fourier transform shift theorem states $x(t - T) \;\longleftrightarrow\; X(f)\, e^{-j 2\pi f T}$, where the angle is $2\pi fT$ and $T = 0.1$ in this case. Combining everything, we get $-0.1 \cdot 2\pi f_i$ which is equivalent to the result we got.
+
+\item Figures:
+		\begin{center}
+			\includegraphics[width=0.95\textwidth]{fig/aufgabe3_1.png}
+			\includegraphics[width=0.95\textwidth]{fig/aufgabe3_2.png}
+		\end{center}
+\end{enumerate}
+
+\end{aufgabe}
+
+
+\begin{aufgabe}{}
+
+\begin{itemize}
+	\item $y(t) = (x(t))^2$
+
+\[
+y(t) = (x(t))^2
+\]
+
+\[
+y_1(t) = (\alpha x_1(t))^2 + (\beta x_2(t))^2
+\]
+
+\[
+= \alpha^2 (x_1(t))^2 + \beta^2 (x_2(t))^2
+\]
+
+\[
+y_2(t) = \alpha (x_1(t))^2 + \beta (x_2(t))^2
+\]
+
+\[
+y_1(t) \ne y_2(t)
+\]
+
+\[
+\implies \text{Not linear}
+\]
+
+
+\[
+    y_1(t) &= \left(x(t - \alpha)\right)^2 \\
+\]
+
+\[
+    y_2(t) &= y_1(t - \Delta) = \left(x(t - \alpha)\right)^2 \\
+\]
+
+\[
+    y_1 &= y_2 \\
+\]
+
+\[
+\implies \text{Time invariant}
+\]
+	\item $y(t) = w(t) sin(\Omega_0 t)$
+
+\[
+	y(t) &= x(t) \sin(n \Omega_0 t) \\
+\]
+
+\[
+	y_1(t) &= \left( \alpha \, x(t) \sin(n_1 \omega t) \right) + \left( \beta \, x(t) \sin(n_0 t) \right) \\
+\]
+
+\[
+	y_t(t) &= \alpha \left( x(t) \sin(n_1 \omega t) \right) + \beta \left( x(t) \sin(n_0 t) \right) \\
+\]
+
+\[
+	y_1 &= y_2
+\]
+
+\[
+	\implies \text{linear}
+\]
+
+
+
+
+\[
+    y_1(t) &\coloneqq x(t - \alpha) \sin(\omega_0 t - \alpha) \\
+\]
+
+\[
+    y_2(t) &= y_1(t - \alpha) = x(t - \alpha) \sin(\omega_0 t - \alpha) \\
+\]
+
+\[
+    y_1 &= y_2 \\
+\]
+
+\[
+	\implies \text{Time invariant}
+\]
+\end{itemize}
+
+\end{aufgabe}
+
+\begin{aufgabe}
+
+	\begin{enumerate}
+		\item System A is linear because there is no difference between the plot $T [x_1(t) + x_2(t)]$ and $T [x_1(t)] + T [x_2(t)]$.
+			You can also see in the error plot that the error is small.
+			It is not exactyly zero becuase rounding errors.
+		\item Input and output are not proportional in a non linear system and
+			therefore the error change is greater than the one of the linear system
+		\item The output sound of System B does differ when comparing $T [x_1(t) + x_2(t)]$ and $T [x_1(t)] + T [x_2(t)]$.
+			But is the same when using System A. This matches the differences in the graphs.
+	\end{enumerate}
+
+\end{aufgabe}
+
+\end{document}

Assignment1_211/assignment1_3.m

@@ -0,0 +1,21 @@
+figure;
+t = 0:0.001:1
+y = sin(2*pi*1*t)
+plot(t,y,'b')
+hold on
+y = sin(2*pi*1*(t-0.1))
+plot(t,y,'r')
+y = sin(2*pi*1*t-2*pi*0.1)
+plot(t,y,'g:')
+
+
+
+figure;
+t = 0:0.001:1
+y = sin(2*pi*3*t)
+plot(t,y,'b')
+hold on
+y = sin(2*pi*3*(t-0.1))
+plot(t,y,'r')
+y = sin(2*pi*3*t-2*pi*0.3)
+plot(t,y,'g:')

flake.lock

@@ -0,0 +1,26 @@
+{
+  "nodes": {
+    "nixpkgs": {
+      "locked": {
+        "lastModified": 1775122944,
+        "narHash": "sha256-ZNsGK5DbLo1D3zGRTVQltVfhkQG+tpYFOHcXtofTxDc=",
+        "owner": "NixOS",
+        "repo": "nixpkgs",
+        "rev": "8d4ac3a959da88fa03784bddcb64f007f931e7d1",
+        "type": "github"
+      },
+      "original": {
+        "owner": "NixOS",
+        "repo": "nixpkgs",
+        "type": "github"
+      }
+    },
+    "root": {
+      "inputs": {
+        "nixpkgs": "nixpkgs"
+      }
+    }
+  },
+  "root": "root",
+  "version": 7
+}

flake.nix

@@ -0,0 +1,22 @@
+{
+  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
+        ];
+      };
+    };
+}

index.typ

@@ -0,0 +1,144 @@
+#import "@preview/codly:1.3.0": *
+#import "@preview/codly-languages:0.1.1": *
+#import "@preview/catppuccin:1.0.1": catppuccin, flavors
+
+#show: codly-init.with()
+#show: catppuccin.with(flavors.latte)
+
+#codly(languages: codly-languages)
+
+#let question(body) = block(
+  fill: luma(255),
+  inset: 8pt,
+  radius: 4pt
+)[
+	#body
+]
+
+= Assignment 01
+
+== Exercise 1: Complex numbers
+
+#question[
+	  These complex numbers are given:
+
+  $
+  c_1 = -5 + j 3,
+  $
+  $
+  c_2 = frac(sqrt(2), 2) e^(-j 3 pi/4),
+  $
+  $
+  c_3 = frac(1, sqrt(2)) + j frac(1, sqrt(2)).
+  $
+
+  Calculate the following numbers and show the calculations/derivations in the report.
+
+  - $c_4 = c_1 + c_2$
+  - $c_5 = c_1 dot c_2$
+  - $c_6 = |c_3|^2$
+  - $c_7 = arg(c_3)$
+  - $c_8 = frac(c_1, c_2)$
+  - $c_9 = c_1 dot c_1^*$
+
+  Subsequently check your results with Matlab. Useful functions in this context are abs and angle.
+You do not need to add those checks to your report, or show the corresponding code.
+
+]
+
+$
+c_1 = -5 + 3j = sqrt((-5)^2 + 3^2) dot e^ (j dot arctan(3/(-5))) = sqrt(35) dot e ^ (-j dot 0.54) \
+c_2 = sqrt(2)/2 dot e ^ (-j dot 3 dot pi/4) = sqrt(2)/2 * cos(-(3 pi)/ 4) + j dot sqrt(2)/2 * sin(-(3 pi)/ 4) = -1/2 - 1/2 j \
+c_3 = 1 / sqrt(2) + 1 / sqrt(2) j = sqrt((1/sqrt(2)^2 + (1/sqrt(2)^2)))^(arctan(1/sqrt(2)/(1/sqrt(2)))) = e ^ (pi/4j) \
+c_4 = c_1 + c_2 = (-5 + 3j) + (-1/2 - 1/2 j) = -11/2 + 5/2 j \
+c_5 = c_1 * c_2 = (sqrt(35) dot e ^ (-j dot 0.54)) dot (sqrt(2)/2 dot e ^ (-j dot 3 dot pi/3)) approx sqrt(70)/2 dot e ^ (-2.8961) \
+c_6 = |c_3|^2 = sqrt((1/sqrt(2))^2 + (1/sqrt(2))^2) = 1 ^ 2 = 1 \
+c_7 = arg(c_3) = pi/4 \
+c_8 = c_1 / c_2 =  (-5 + 3j)/(-1/2 - 1/2j) - (-1/2 + 1/2j)/(-1/2 + 1/2j) = ((-5) * (-1/2) + 3 * (-1/2) + j ( 3 * (-1/2) - ( -5 ) * ( -1/2 )))/ ((-1/2)^2 + (-1/2)^2) \ \
+ = 1/1/2 - 4/1/2 j = 2 - 8j \
+c_9 = c_1 dot c_2^* = sqrt(34) dot e^(-j dot 0.54) dot sqrt(45) dot e ^ (j dot 0.54) = sqrt(34) * sqrt(34) = 34
+$
+
+== Exercise 2: Fourier Transform
+
+#question[
+  The lecture notes show the following Fourier transform pair for the cosine wave (DSP_2.pdf, page 38):
+
+  $
+  x(t) = hat(X) cos(2 pi f_0 t)
+  arrow.l.r.double.long
+  X(f) = frac(hat(X), 2) delta(f - f_0) + frac(hat(X), 2) delta(f + f_0).
+  $
+
+  Mathematically prove this relation. To do so, use Euler's formula to express the cosine in the time domain as a sum of complex exponentials and the Fourier transform of a complex exponential function from DSP_2.pdf, page 39.
+
+  Add a diagram of $X(f)$ in the report (draw the real and the imaginary part of $X(f)$ in the same diagram).
+]
+
+== Exercise 3: Time Shift and Phase
+
+#question[
+	Given are two sines according to
+
+$
+x_i(t) = sin(2 pi f_i t), i in {1, 2},
+$
+
+with $f_1 = 1$ Hz and $f_2 = 3$ Hz.
+
+Both sines are time delayed by $tau = 0.1$ s to yield
+
+$
+y_i(t) = sin(2 pi f_i (t - 0.1)).
+$
+
+This corresponds to a phase shift. Thus, the delayed sines may also be written as
+
+$
+y_i(t) = sin(2 pi f_i t + phi_i).
+$
+
+a) Calculate the phase shifts $phi_i$ for each sine, and verify that this corresponds to the Shift Theorem of the Fourier Transform (DSP_2.pdf, page 42).
+
+b) For both sines in separated plots: Plot the original signal, the time delayed signal and the phase shifted signal. Since the latter two are identical, show this by plotting the first with a solid line and the overlaid one in a different colour with a dashed line. Plot each signal from 0 to 1 s. In Matlab use the time vector: `t = 0:0.001:1`.
+]
+
+== Exercise 4: Linearity and Time Invariance
+
+#question[
+	Examine the following systems (input $x(t)$ and output $y(t)$) for linearity and time invariance. Clearly show the mathematical derivations and state if the systems are linear and/or time-invariant.
+
+	- $y(t) = (x(t))^2$
+
+	- $y(t) = x(t) sin(Omega_0 t)$
+]
+
+== Exercise 5: Linearity Testing
+
+#question[
+	Run the provided Matlab script `Assignment1_Linearity_Testing.m`. It opens an interactive GUI that compares
+
+	$
+	T[x_1(t) + x_2(t)] quad #text[with] quad T[x_1(t)] + T[x_2(t)]
+	$
+
+	for two unknown systems (System A and System B) using sinusoidal inputs $x_1(t) = A_1 sin(2 pi f_1 t)$ and $x_2(t) = A_2 sin(2 pi f_2 t)$.
+
+	The GUI provides the following controls:
+
+	- *Signal Parameters* – text fields to set $f_1, f_2$ (in Hz) and $A_1, A_2$.
+	- *Select System* – dropdown to switch between System A and System B.
+	- *Play Inputs* – buttons to listen to $x_1(t), x_2(t)$, and $x_1(t) + x_2(t)$.
+	- *Play Outputs* – buttons to listen to $T[x_1(t) + x_2(t)], T[x_1(t)] + T[x_2(t)]$, and the error signal.
+
+	Students who do not use Matlab should inspect the provided PDF plots `SystemA_Setting1.pdf`, `SystemB_Setting1.pdf`, `SystemA_Setting2.pdf`, and `SystemB_Setting2.pdf`. Here, Setting 1 corresponds to $A_1 = A_2 = 0.15$ and Setting 2 corresponds to $A_1 = A_2 = 0.30$.
+
+	Perform the following tasks:
+
+	a) Using the default parameters ($f_1 = 440$ Hz, $f_2 = 480$ Hz, $A_1 = A_2 = 0.15$), inspect the superposition comparison and error plots for both systems. Identify which system is approximately linear and which is clearly nonlinear, and justify your answer. For System A, look carefully at the y-axis scaling of the error plot and explain why the error might appear like that.
+
+	b) In the GUI, increase $A_1$ and $A_2$ to 0.30 and observe how the error plot changes for each system. This corresponds to the plots in `SystemA_Setting2.pdf` and `SystemB_Setting2.pdf`. Explain why a larger amplitude makes nonlinear behaviour more visible.
+
+	c) Use the *Play Inputs* and *Play Outputs* buttons for both systems. Describe in one or two sentences what you hear and how it relates to the superposition results. Students using only the PDF plots should instead comment briefly on how the visual differences between Setting 1 and Setting 2 support their conclusions.
+]
+