commit 930f7ae4a4c4d31d0566150a15bbe65ca21a544a Author: quirinecker Date: Sun Apr 26 22:23:54 2026 +0200 updated changes so far diff --git a/.direnv/flake-profile b/.direnv/flake-profile new file mode 120000 index 0000000..0c05709 --- /dev/null +++ b/.direnv/flake-profile @@ -0,0 +1 @@ +flake-profile-1-link \ No newline at end of file diff --git a/.direnv/flake-profile-1-link b/.direnv/flake-profile-1-link new file mode 120000 index 0000000..fe02341 --- /dev/null +++ b/.direnv/flake-profile-1-link @@ -0,0 +1 @@ +/nix/store/ha1aav8ld0ym8pr8nmn66np5g4hfnwda-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/.gitignore b/.gitignore new file mode 100644 index 0000000..f877a4d --- /dev/null +++ b/.gitignore @@ -0,0 +1,315 @@ +# 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 + diff --git a/Assignment1_211.pdf b/Assignment1_211.pdf new file mode 100644 index 0000000..7ed5d10 Binary files /dev/null and b/Assignment1_211.pdf differ diff --git a/Assignment1_211.tex b/Assignment1_211.tex new file mode 100644 index 0000000..4f5ef2b --- /dev/null +++ b/Assignment1_211.tex @@ -0,0 +1,351 @@ +\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} diff --git a/Assignment1_211/assignment1_3.m b/Assignment1_211/assignment1_3.m new file mode 100644 index 0000000..79b5e4d --- /dev/null +++ b/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:') diff --git a/fig/Testpicture.eps b/fig/Testpicture.eps new file mode 100644 index 0000000..8191bcf --- /dev/null +++ b/fig/Testpicture.eps @@ -0,0 +1,2021 @@ +%!PS-Adobe-3.0 EPSF-3.0 +%%Creator: (MATLAB, The Mathworks, Inc. Version 9.1.0.441655 \(R2016b\). 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/equal /greater /question /at +/A /B /C /D /E +/F /G /H /I /J +/K /L /M /N /O +/P /Q /R /S /T +/U /V /W /X /Y +/Z /bracketleft /backslash /bracketright /asciicircum +/underscore /quoteleft /a /b /c +/d /e /f /g /h +/i /j /k /l /m +/n /o /p /q /r +/s /t /u /v /w +/x /y /z /braceleft /bar +/braceright /asciitilde /bullet /Euro /bullet +/quotesinglbase /florin /quotedblbase /ellipsis /dagger +/daggerdbl /circumflex /perthousand /Scaron /guilsinglleft +/OE /bullet /Zcaron /bullet /bullet +/quoteleft /quoteright /quotedblleft /quotedblright /bullet +/endash /emdash /asciitilde /trademark /scaron +/guilsinglright /oe /bullet /zcaron /Ydieresis +/space /exclamdown /cent /sterling /currency +/yen /brokenbar /section /dieresis /copyright +/ordfeminine /guillemotleft /logicalnot /sfthyphen /registered +/macron /degree /plusminus /twosuperior /threesuperior +/acute /mu /paragraph /middot /cedilla +/onesuperior /ordmasculine /guillemotright /onequarter /onehalf +/threequarters /questiondown /Agrave /Aacute /Acircumflex +/Atilde /Adieresis /Aring /AE /Ccedilla +/Egrave /Eacute /Ecircumflex /Edieresis /Igrave +/Iacute /Icircumflex /Idieresis /Eth /Ntilde +/Ograve /Oacute /Ocircumflex /Otilde /Odieresis +/multiply /Oslash /Ugrave /Uacute /Ucircumflex +/Udieresis /Yacute /Thorn /germandbls /agrave +/aacute /acircumflex /atilde /adieresis /aring +/ae /ccedilla /egrave /eacute /ecircumflex +/edieresis /igrave /iacute /icircumflex /idieresis +/eth /ntilde /ograve /oacute /ocircumflex +/otilde /odieresis /divide /oslash /ugrave +/uacute /ucircumflex /udieresis /yacute /thorn +/ydieresis +] def +%%EndResource +%FOPBeginFontReencode +/Courier-Bold findfont +dup length dict begin + {1 index /FID ne {def} {pop pop} ifelse} forall + /Encoding WinAnsiEncoding def + currentdict +end +/Courier-Bold exch definefont pop +/Helvetica findfont +dup length dict begin + {1 index /FID ne {def} {pop pop} ifelse} forall + /Encoding WinAnsiEncoding def + currentdict +end +/Helvetica exch definefont pop +/Courier-BoldOblique findfont +dup length dict begin + {1 index /FID ne {def} {pop pop} ifelse} forall + /Encoding WinAnsiEncoding def + currentdict +end +/Courier-BoldOblique exch definefont pop +/Courier-Oblique findfont +dup length dict begin + {1 index /FID ne {def} {pop pop} ifelse} forall + /Encoding WinAnsiEncoding def + currentdict +end +/Courier-Oblique exch definefont pop +/Times-Roman findfont +dup length dict begin + {1 index /FID ne {def} {pop pop} ifelse} forall + /Encoding WinAnsiEncoding def + currentdict +end +/Times-Roman exch definefont pop +/Helvetica-BoldOblique findfont +dup length dict begin + {1 index /FID ne {def} {pop pop} ifelse} forall + /Encoding WinAnsiEncoding def + currentdict +end +/Helvetica-BoldOblique exch definefont pop +/Helvetica-Bold findfont +dup length dict begin + {1 index /FID ne {def} {pop pop} ifelse} forall + /Encoding WinAnsiEncoding def + currentdict +end +/Helvetica-Bold exch definefont pop +/Helvetica-Oblique findfont +dup length dict begin + {1 index /FID ne {def} {pop pop} ifelse} forall + /Encoding WinAnsiEncoding def + currentdict +end +/Helvetica-Oblique exch definefont pop +/Times-BoldItalic findfont +dup length dict begin + {1 index /FID ne {def} {pop pop} ifelse} forall + /Encoding WinAnsiEncoding def + currentdict +end +/Times-BoldItalic exch definefont pop +/Courier findfont +dup length dict begin + {1 index /FID ne {def} {pop pop} ifelse} forall + /Encoding WinAnsiEncoding def + currentdict +end +/Courier exch definefont pop +/Times-Italic findfont +dup length dict begin + {1 index /FID ne {def} {pop pop} ifelse} forall + /Encoding WinAnsiEncoding def + currentdict +end +/Times-Italic exch definefont pop +/Times-Bold findfont +dup length dict begin + {1 index /FID ne {def} {pop pop} ifelse} forall + /Encoding WinAnsiEncoding def + currentdict +end +/Times-Bold exch definefont pop +%FOPEndFontReencode +%%EndProlog +%%Page: 1 1 +%%PageBoundingBox: 0 0 420 315 +%%BeginPageSetup +[1 0 0 -1 0 315] CT 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index 0000000..b5e8238 Binary files /dev/null and b/fig/aufgabe2.png differ diff --git a/fig/aufgabe3_1.png b/fig/aufgabe3_1.png new file mode 100644 index 0000000..1df6b47 Binary files /dev/null and b/fig/aufgabe3_1.png differ diff --git a/fig/aufgabe3_2.png b/fig/aufgabe3_2.png new file mode 100644 index 0000000..8316ed0 Binary files /dev/null and b/fig/aufgabe3_2.png differ diff --git a/flake.lock b/flake.lock new file mode 100644 index 0000000..b50e3d2 --- /dev/null +++ b/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 +} diff --git a/flake.nix b/flake.nix new file mode 100644 index 0000000..b452254 --- /dev/null +++ b/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 + ]; + }; + }; +} diff --git a/index.pdf b/index.pdf new file mode 100644 index 0000000..955e066 --- /dev/null +++ b/index.pdf @@ -0,0 +1,2553 @@ +%PDF-1.7 +% + +1 0 obj +<< + /Type /Pages + /Count 3 + /Kids [169 0 R 171 0 R 173 0 R] +>> +endobj + +2 0 obj +<< + /Type /Outlines + /First 3 0 R + /Last 3 0 R + /Count 1 +>> +endobj + +3 0 obj +<< + /Parent 2 0 R + /First 4 0 R + /Last 8 0 R + /Count -5 + /Title (Assignment 01) + /Dest 168 0 R +>> +endobj + +4 0 obj +<< + /Parent 3 0 R + /Next 5 0 R + /Title (Exercise 1: Complex numbers) + /Dest 163 0 R +>> +endobj + +5 0 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101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R 101 0 R] +endobj + +11 0 obj +[100 0 R 99 0 R 99 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 98 0 R 97 0 R 97 0 R 97 0 R 94 0 R 96 0 R 96 0 R 96 0 R 96 0 R 94 0 R 95 0 R 95 0 R 95 0 R 95 0 R 94 0 R 94 0 R 93 0 R 92 0 R 91 0 R 91 0 R 91 0 R 91 0 R 91 0 R 91 0 R 91 0 R 91 0 R 91 0 R 91 0 R 91 0 R 91 0 R 91 0 R 91 0 R 91 0 R 91 0 R 91 0 R 91 0 R 91 0 R 91 0 R 91 0 R 91 0 R 91 0 R 88 0 R 90 0 R 90 0 R 90 0 R 90 0 R 88 0 R 89 0 R 89 0 R 89 0 R 89 0 R 88 0 R 86 0 R 87 0 R 87 0 R 87 0 R 87 0 R 87 0 R 86 0 R 85 0 R 85 0 R 85 0 R 85 0 R 85 0 R 85 0 R 85 0 R 85 0 R 85 0 R 85 0 R 85 0 R 85 0 R 85 0 R 85 0 R 85 0 R 85 0 R 85 0 R 85 0 R 85 0 R 85 0 R 85 0 R 84 0 R 83 0 R 83 0 R 83 0 R 83 0 R 83 0 R 83 0 R 83 0 R 83 0 R 83 0 R 83 0 R 83 0 R 83 0 R 83 0 R 83 0 R 83 0 R 83 0 R 83 0 R 83 0 R 81 0 R 82 0 R 82 0 R 81 0 R 81 0 R 79 0 R 79 0 R 79 0 R 79 0 R 80 0 R 79 0 R 78 0 R 75 0 R 77 0 R 77 0 R 77 0 R 77 0 R 75 0 R 76 0 R 76 0 R 76 0 R 76 0 R 75 0 R 75 0 R 75 0 R 74 0 R 73 0 R 73 0 R 73 0 R 73 0 R 73 0 R 73 0 R 73 0 R 73 0 R 73 0 R 73 0 R 73 0 R 73 0 R 69 0 R 68 0 R 68 0 R 68 0 R 68 0 R 68 0 R 68 0 R 68 0 R 68 0 R 68 0 R 68 0 R 68 0 R 68 0 R 68 0 R 68 0 R 68 0 R 63 0 R 61 0 R 62 0 R 61 0 R 61 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R 60 0 R] +endobj + +12 0 obj +[57 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 59 0 R 57 0 R 58 0 R 58 0 R 58 0 R 58 0 R 58 0 R 58 0 R 58 0 R 58 0 R 58 0 R 58 0 R 58 0 R 58 0 R 58 0 R 58 0 R 58 0 R 58 0 R 57 0 R 56 0 R 55 0 R 54 0 R 51 0 R 53 0 R 53 0 R 53 0 R 53 0 R 53 0 R 51 0 R 52 0 R 52 0 R 52 0 R 52 0 R 52 0 R 51 0 R 49 0 R 48 0 R 47 0 R 45 0 R 44 0 R 41 0 R 43 0 R 43 0 R 43 0 R 43 0 R 43 0 R 43 0 R 43 0 R 43 0 R 43 0 R 43 0 R 43 0 R 41 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 42 0 R 41 0 R 39 0 R 38 0 R 36 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 37 0 R 36 0 R 36 0 R 27 0 R 33 0 R 27 0 R 32 0 R 27 0 R 31 0 R 27 0 R 30 0 R 27 0 R 27 0 R 29 0 R 29 0 R 29 0 R 29 0 R 29 0 R 29 0 R 29 0 R 29 0 R 29 0 R 29 0 R 27 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 28 0 R 27 0 R 26 0 R 22 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 25 0 R 22 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 24 0 R 22 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 23 0 R 22 0 R 22 0 R 22 0 R 22 0 R 17 0 R 21 0 R 21 0 R 17 0 R 20 0 R 20 0 R 17 0 R 17 0 R 19 0 R 17 0 R 18 0 R 17 0 R 17 0 R 14 0 R 16 0 R 14 0 R 15 0 R 14 0 R 14 0 R 14 0 R 14 0 R] +endobj + +13 0 obj +<< + /Type /StructElem + /S /Document + /P 9 0 R + /K [134 0 R 133 0 R 132 0 R 131 0 R 130 0 R 129 0 R 128 0 R 103 0 R 102 0 R 101 0 R 100 0 R 99 0 R 98 0 R 97 0 R 94 0 R 93 0 R 92 0 R 91 0 R 88 0 R 86 0 R 85 0 R 84 0 R 83 0 R 81 0 R 79 0 R 78 0 R 75 0 R 64 0 R 63 0 R 61 0 R 60 0 R 57 0 R 56 0 R 34 0 R 27 0 R 26 0 R 22 0 R 17 0 R 14 0 R] +>> +endobj + +14 0 obj +<< + /Type /StructElem + /S /P + /P 13 0 R + /K [193 16 0 R 195 15 0 R 197 198 199 200] + /Pg 173 0 R +>> +endobj + +15 0 obj +<< + /Type /StructElem + /S /Strong + /P 14 0 R + /K [196] + /Pg 173 0 R +>> +endobj + +16 0 obj +<< + /Type /StructElem + /S /Strong + /P 14 0 R + /K [194] + /Pg 173 0 R +>> +endobj + +17 0 obj +<< + /Type /StructElem + /S /P + /P 13 0 R + /K [180 21 0 R 183 20 0 R 186 187 19 0 R 189 18 0 R 191 192] + /Pg 173 0 R +>> +endobj + +18 0 obj +<< + /Type /StructElem + /S /Code + /P 17 0 R + /K [190] + /Pg 173 0 R +>> +endobj + +19 0 obj +<< + /Type /StructElem + /S /Code + /P 17 0 R + /K [188] + /Pg 173 0 R +>> +endobj + +20 0 obj +<< + /Type /StructElem + /S /Formula + /P 17 0 R + /K [184 185] + /Pg 173 0 R +>> +endobj + +21 0 obj +<< + /Type /StructElem + /S /Formula + /P 17 0 R + /K [181 182] + /Pg 173 0 R +>> +endobj + +22 0 obj +<< + /Type /StructElem + /S /P + /P 13 0 R + /K [151 25 0 R 158 24 0 R 165 23 0 R 176 177 178 179] + /Pg 173 0 R +>> +endobj + +23 0 obj +<< + /Type /StructElem + /S /Formula + /P 22 0 R + /K [166 167 168 169 170 171 172 173 174 175] + /Pg 173 0 R +>> +endobj + +24 0 obj +<< + /Type /StructElem + /S /Formula + /P 22 0 R + /K [159 160 161 162 163 164] + /Pg 173 0 R +>> +endobj + +25 0 obj +<< + /Type /StructElem + /S /Formula + /P 22 0 R + /K [152 153 154 155 156 157] + /Pg 173 0 R +>> +endobj + +26 0 obj +<< + /Type /StructElem + /S /P + /P 13 0 R + /K [150] + /Pg 173 0 R +>> +endobj + +27 0 obj +<< + /Type /StructElem + /S /P + /P 13 0 R + /K [118 33 0 R 120 32 0 R 122 31 0 R 124 30 0 R 126 127 29 0 R 138 28 0 R 149] + /Pg 173 0 R +>> +endobj + +28 0 obj +<< + /Type /StructElem + /S /Formula + /P 27 0 R + /K [139 140 141 142 143 144 145 146 147 148] + /Pg 173 0 R +>> +endobj + +29 0 obj +<< + /Type /StructElem + /S /Formula + /P 27 0 R + /K [128 129 130 131 132 133 134 135 136 137] + /Pg 173 0 R +>> +endobj + +30 0 obj +<< + /Type /StructElem + /S /Code + /P 27 0 R + /K [125] + /Pg 173 0 R +>> +endobj + +31 0 obj +<< + /Type /StructElem + /S /Code + /P 27 0 R + /K [123] + /Pg 173 0 R +>> +endobj + +32 0 obj +<< + /Type /StructElem + /S /Code + /P 27 0 R + /K [121] + /Pg 173 0 R +>> +endobj + +33 0 obj +<< + /Type /StructElem + /S /Code + /P 27 0 R + /K [119] + /Pg 173 0 R +>> +endobj + +34 0 obj +<< + /Type /StructElem + /S /L + /P 13 0 R + /A [<< + /O /List + /ListNumbering /Circle + >>] + /K [50 0 R 46 0 R 40 0 R 35 0 R] +>> +endobj + +35 0 obj +<< + /Type /StructElem + /S /LI + /P 34 0 R + /K [39 0 R 36 0 R] +>> +endobj + +36 0 obj +<< + /Type /StructElem + /S /LBody + /P 35 0 R + /K [38 0 R 83 37 0 R 116 117] + /Pg 173 0 R +>> +endobj + +37 0 obj +<< + /Type /StructElem + /S /Formula + /P 36 0 R + /K [84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115] + /Pg 173 0 R +>> +endobj + +38 0 obj +<< + /Type /StructElem + /S /Strong + /P 36 0 R + /K [82] + /Pg 173 0 R +>> +endobj + +39 0 obj +<< + /Type /StructElem + /S /Lbl + /P 35 0 R + /K [81] + /Pg 173 0 R +>> +endobj + +40 0 obj +<< + /Type /StructElem + /S /LI + /P 34 0 R + /K [45 0 R 41 0 R] +>> +endobj + +41 0 obj +<< + /Type /StructElem + /S /LBody + /P 40 0 R + /K [44 0 R 56 43 0 R 68 42 0 R 80] + /Pg 173 0 R +>> +endobj + +42 0 obj +<< + /Type /StructElem + /S /Formula + /P 41 0 R + /K [69 70 71 72 73 74 75 76 77 78 79] + /Pg 173 0 R +>> +endobj + +43 0 obj +<< + /Type /StructElem + /S /Formula + /P 41 0 R + /K [57 58 59 60 61 62 63 64 65 66 67] + /Pg 173 0 R +>> +endobj + +44 0 obj +<< + /Type /StructElem + /S /Strong + /P 41 0 R + /K [55] + /Pg 173 0 R +>> +endobj + +45 0 obj +<< + /Type /StructElem + /S /Lbl + /P 40 0 R + /K [54] + /Pg 173 0 R +>> +endobj + +46 0 obj +<< + /Type /StructElem + /S /LI + /P 34 0 R + /K [49 0 R 47 0 R] +>> +endobj + +47 0 obj +<< + /Type /StructElem + /S /LBody + /P 46 0 R + /K [48 0 R 53] + /Pg 173 0 R +>> +endobj + +48 0 obj +<< + /Type /StructElem + /S /Strong + /P 47 0 R + /K [52] + /Pg 173 0 R +>> +endobj + +49 0 obj +<< + /Type /StructElem + /S /Lbl + /P 46 0 R + /K [51] + /Pg 173 0 R +>> +endobj + +50 0 obj +<< + /Type /StructElem + /S /LI + /P 34 0 R + /K [55 0 R 51 0 R] +>> +endobj + +51 0 obj +<< + /Type /StructElem + /S /LBody + /P 50 0 R + /K [54 0 R 38 53 0 R 44 52 0 R 50] + /Pg 173 0 R +>> +endobj + +52 0 obj +<< + /Type /StructElem + /S /Formula + /P 51 0 R + /K [45 46 47 48 49] + /Pg 173 0 R +>> +endobj + +53 0 obj +<< + /Type /StructElem + /S /Formula + /P 51 0 R + /K [39 40 41 42 43] + /Pg 173 0 R +>> +endobj + +54 0 obj +<< + /Type /StructElem + /S /Strong + /P 51 0 R + /K [37] + /Pg 173 0 R +>> +endobj + +55 0 obj +<< + /Type /StructElem + /S /Lbl + /P 50 0 R + /K [36] + /Pg 173 0 R +>> +endobj + +56 0 obj +<< + /Type /StructElem + /S /P + /P 13 0 R + /K [35] + /Pg 173 0 R +>> +endobj + +57 0 obj +<< + /Type /StructElem + /S /P + /P 13 0 R + /K [0 59 0 R 17 58 0 R 34] + /Pg 173 0 R +>> +endobj + +58 0 obj +<< + /Type /StructElem + /S /Formula + /P 57 0 R + /K [18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33] + /Pg 173 0 R +>> +endobj + +59 0 obj +<< + /Type /StructElem + /S /Formula + /P 57 0 R + /K [1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16] + /Pg 173 0 R +>> +endobj + +60 0 obj +<< + /Type /StructElem + /S /Formula + /P 13 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234] + /Pg 171 0 R +>> +endobj + +61 0 obj +<< + /Type /StructElem + /S /P + /P 13 0 R + /K [199 62 0 R 201 202] + /Pg 171 0 R +>> +endobj + +62 0 obj +<< + /Type /StructElem + /S /Code + /P 61 0 R + /K [200] + /Pg 171 0 R +>> +endobj + +63 0 obj +<< + /Type /StructElem + /S /H2 + /P 13 0 R + /T (Exercise 5: Linearity Testing) + /K [198] + /Pg 171 0 R +>> +endobj + +64 0 obj +<< + /Type /StructElem + /S /L + /P 13 0 R + /A [<< + /O /List + /ListNumbering /Circle + >>] + /K [70 0 R 65 0 R] +>> +endobj + +65 0 obj +<< + /Type /StructElem + /S /LI + /P 64 0 R + /K [69 0 R 66 0 R] +>> +endobj + +66 0 obj +<< + /Type /StructElem + /S /LBody + /P 65 0 R + /K [67 0 R] +>> +endobj + +67 0 obj +<< + /Type /StructElem + /S /P + /P 66 0 R + /K [68 0 R] +>> +endobj + +68 0 obj +<< + /Type /StructElem + /S /Formula + /P 67 0 R + /K [183 184 185 186 187 188 189 190 191 192 193 194 195 196 197] + /Pg 171 0 R +>> +endobj + +69 0 obj +<< + /Type /StructElem + /S /Lbl + /P 65 0 R + /K [182] + /Pg 171 0 R +>> +endobj + +70 0 obj +<< + /Type /StructElem + /S /LI + /P 64 0 R + /K [74 0 R 71 0 R] +>> +endobj + +71 0 obj +<< + /Type /StructElem + /S /LBody + /P 70 0 R + /K [72 0 R] +>> +endobj + +72 0 obj +<< + /Type /StructElem + /S /P + /P 71 0 R + /K [73 0 R] +>> +endobj + +73 0 obj +<< + /Type /StructElem + /S /Formula + /P 72 0 R + /K [170 171 172 173 174 175 176 177 178 179 180 181] + /Pg 171 0 R +>> +endobj + +74 0 obj +<< + /Type /StructElem + /S /Lbl + /P 70 0 R + /K [169] + /Pg 171 0 R +>> +endobj + +75 0 obj +<< + /Type /StructElem + /S /P + /P 13 0 R + /K [156 77 0 R 161 76 0 R 166 167 168] + /Pg 171 0 R +>> +endobj + +76 0 obj +<< + /Type /StructElem + /S /Formula + /P 75 0 R + /K [162 163 164 165] + /Pg 171 0 R +>> +endobj + +77 0 obj +<< + /Type /StructElem + /S /Formula + /P 75 0 R + /K [157 158 159 160] + /Pg 171 0 R +>> +endobj + +78 0 obj +<< + /Type /StructElem + /S /H2 + /P 13 0 R + /T (Exercise 4: Linearity and Time Invariance) + /K [155] + /Pg 171 0 R +>> +endobj + +79 0 obj +<< + /Type /StructElem + /S /P + /P 13 0 R + /K [149 150 151 152 80 0 R 154] + /Pg 171 0 R +>> +endobj + +80 0 obj +<< + /Type /StructElem + /S /Code + /P 79 0 R + /K [153] + /Pg 171 0 R +>> +endobj + +81 0 obj +<< + /Type /StructElem + /S /P + /P 13 0 R + /K [144 82 0 R 147 148] + /Pg 171 0 R +>> +endobj + +82 0 obj +<< + /Type /StructElem + /S /Formula + /P 81 0 R + /K [145 146] + /Pg 171 0 R +>> +endobj + +83 0 obj +<< + /Type /StructElem + /S /Formula + /P 13 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143] + /Pg 171 0 R +>> +endobj + +84 0 obj +<< + /Type /StructElem + /S /P + /P 13 0 R + /K [125] + /Pg 171 0 R +>> +endobj + +85 0 obj +<< + /Type /StructElem + /S /Formula + /P 13 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124] + /Pg 171 0 R +>> +endobj + +86 0 obj +<< + /Type /StructElem + /S /P + /P 13 0 R + /K [97 87 0 R 103] + /Pg 171 0 R +>> +endobj + +87 0 obj +<< + /Type /StructElem + /S /Formula + /P 86 0 R + /K [98 99 100 101 102] + /Pg 171 0 R +>> +endobj + +88 0 obj +<< + /Type /StructElem + /S /P + /P 13 0 R + /K [86 90 0 R 91 89 0 R 96] + /Pg 171 0 R +>> +endobj + +89 0 obj +<< + /Type /StructElem + /S /Formula + /P 88 0 R + /K [92 93 94 95] + /Pg 171 0 R +>> +endobj + +90 0 obj +<< + /Type /StructElem + /S /Formula + /P 88 0 R + /K [87 88 89 90] + /Pg 171 0 R +>> +endobj + +91 0 obj +<< + /Type /StructElem + /S /Formula + /P 13 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85] + /Pg 171 0 R +>> +endobj + +92 0 obj +<< + /Type /StructElem + /S /P + /P 13 0 R + /K [62] + /Pg 171 0 R +>> +endobj + +93 0 obj +<< + /Type /StructElem + /S /H2 + /P 13 0 R + /T (Exercise 3: Time Shift and Phase) + /K [61] + /Pg 171 0 R +>> +endobj + +94 0 obj +<< + /Type /StructElem + /S /P + /P 13 0 R + /K [49 96 0 R 54 95 0 R 59 60] + /Pg 171 0 R +>> +endobj + +95 0 obj +<< + /Type /StructElem + /S /Formula + /P 94 0 R + /K [55 56 57 58] + /Pg 171 0 R +>> +endobj + +96 0 obj +<< + /Type /StructElem + /S /Formula + /P 94 0 R + /K [50 51 52 53] + /Pg 171 0 R +>> +endobj + +97 0 obj +<< + /Type /StructElem + /S /P + /P 13 0 R + /K [46 47 48] + /Pg 171 0 R +>> +endobj + +98 0 obj +<< + /Type /StructElem + /S /Formula + /P 13 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45] + /Pg 171 0 R +>> +endobj + +99 0 obj +<< + /Type /StructElem + /S /P + /P 13 0 R + /K [1 2] + /Pg 171 0 R +>> +endobj + +100 0 obj +<< + /Type /StructElem + /S /H2 + /P 13 0 R + /T (Exercise 2: Fourier Transform) + /K [0] + /Pg 171 0 R +>> +endobj + +101 0 obj +<< + /Type /StructElem + /S /Formula + /P 13 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497] + /Pg 169 0 R +>> +endobj + +102 0 obj +<< + /Type /StructElem + /S /P + /P 13 0 R + /K [92 93] + /Pg 169 0 R +>> +endobj + +103 0 obj +<< + /Type /StructElem + /S /L + /P 13 0 R + /A [<< + /O /List + /ListNumbering /Circle + >>] + /K [124 0 R 120 0 R 116 0 R 112 0 R 108 0 R 104 0 R] +>> +endobj + +104 0 obj +<< + /Type /StructElem + /S /LI + /P 103 0 R + /K [107 0 R 105 0 R] +>> +endobj + +105 0 obj +<< + /Type /StructElem + /S /LBody + /P 104 0 R + /K [106 0 R] +>> +endobj + +106 0 obj +<< + /Type /StructElem + /S /Formula + /P 105 0 R + /K [83 84 85 86 87 88 89 90 91] + /Pg 169 0 R +>> +endobj + +107 0 obj +<< + /Type /StructElem + /S /Lbl + /P 104 0 R + /K [82] + /Pg 169 0 R +>> +endobj + +108 0 obj +<< + /Type /StructElem + /S /LI + /P 103 0 R + /K [111 0 R 109 0 R] +>> +endobj + +109 0 obj +<< + /Type /StructElem + /S /LBody + /P 108 0 R + /K [110 0 R] +>> +endobj + +110 0 obj +<< + /Type /StructElem + /S /Formula + /P 109 0 R + /K [75 76 77 78 79 80 81] + /Pg 169 0 R +>> +endobj + +111 0 obj +<< + /Type /StructElem + /S /Lbl + /P 108 0 R + /K [74] + /Pg 169 0 R +>> +endobj + +112 0 obj +<< + /Type /StructElem + /S /LI + /P 103 0 R + /K [115 0 R 113 0 R] +>> +endobj + +113 0 obj +<< + /Type /StructElem + /S /LBody + /P 112 0 R + /K [114 0 R] +>> +endobj + +114 0 obj +<< + /Type /StructElem + /S /Formula + /P 113 0 R + /K [66 67 68 69 70 71 72 73] + /Pg 169 0 R +>> +endobj + +115 0 obj +<< + /Type /StructElem + /S /Lbl + /P 112 0 R + /K [65] + /Pg 169 0 R +>> +endobj + +116 0 obj +<< + /Type /StructElem + /S /LI + /P 103 0 R + /K [119 0 R 117 0 R] +>> +endobj + +117 0 obj +<< + /Type /StructElem + /S /LBody + /P 116 0 R + /K [118 0 R] +>> +endobj + +118 0 obj +<< + /Type /StructElem + /S /Formula + /P 117 0 R + /K [57 58 59 60 61 62 63 64] + /Pg 169 0 R +>> +endobj + +119 0 obj +<< + /Type /StructElem + /S /Lbl + /P 116 0 R + /K [56] + /Pg 169 0 R +>> +endobj + +120 0 obj +<< + /Type /StructElem + /S /LI + /P 103 0 R + /K [123 0 R 121 0 R] +>> +endobj + +121 0 obj +<< + /Type /StructElem + /S /LBody + /P 120 0 R + /K [122 0 R] +>> +endobj + +122 0 obj +<< + /Type /StructElem + /S /Formula + /P 121 0 R + /K [48 49 50 51 52 53 54 55] + /Pg 169 0 R +>> +endobj + +123 0 obj +<< + /Type /StructElem + /S /Lbl + /P 120 0 R + /K [47] + /Pg 169 0 R +>> +endobj + +124 0 obj +<< + /Type /StructElem + /S /LI + /P 103 0 R + /K [127 0 R 125 0 R] +>> +endobj + +125 0 obj +<< + /Type /StructElem + /S /LBody + /P 124 0 R + /K [126 0 R] +>> +endobj + +126 0 obj +<< + /Type /StructElem + /S /Formula + /P 125 0 R + /K [39 40 41 42 43 44 45 46] + /Pg 169 0 R +>> +endobj + +127 0 obj +<< + /Type /StructElem + /S /Lbl + /P 124 0 R + /K [38] + /Pg 169 0 R +>> +endobj + +128 0 obj +<< + /Type /StructElem + /S /P + /P 13 0 R + /K [37] + /Pg 169 0 R +>> +endobj + +129 0 obj +<< + /Type /StructElem + /S /Formula + /P 13 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [25 26 27 28 29 30 31 32 33 34 35 36] + /Pg 169 0 R +>> +endobj + +130 0 obj +<< + /Type /StructElem + /S /Formula + /P 13 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [12 13 14 15 16 17 18 19 20 21 22 23 24] + /Pg 169 0 R +>> +endobj + +131 0 obj +<< + /Type /StructElem + /S /Formula + /P 13 0 R + /A [<< + /O /Layout + /Placement /Block + >>] + /K [3 4 5 6 7 8 9 10 11] + /Pg 169 0 R +>> +endobj + +132 0 obj +<< + /Type /StructElem + /S /P + /P 13 0 R + /K [2] + /Pg 169 0 R +>> +endobj + +133 0 obj +<< + /Type /StructElem + /S /H2 + /P 13 0 R + /T (Exercise 1: Complex numbers) + /K [1] + /Pg 169 0 R +>> +endobj + +134 0 obj +<< + /Type /StructElem + /S /H1 + /P 13 0 R + /T (Assignment 01) + /K [0] + /Pg 169 0 R +>> +endobj + +135 0 obj +<< + /Type /Font + /Subtype /Type0 + /BaseFont /FOOOQE+LibertinusSerif-Bold-Identity-H + /Encoding /Identity-H + /DescendantFonts [136 0 R] + /ToUnicode 139 0 R +>> +endobj + +136 0 obj +<< + /Type /Font + /Subtype /CIDFontType0 + /BaseFont /FOOOQE+LibertinusSerif-Bold + /CIDSystemInfo << + /Registry (Adobe) + /Ordering (Identity) + /Supplement 0 + >> + /FontDescriptor 138 0 R + /DW 0 + /W [0 0 500 1 1 740 2 2 427 3 3 322 4 4 521 5 5 616 6 6 905 7 7 489 8 8 358 9 9 250 10 11 514 12 12 609 13 13 561 14 14 428 15 15 456 16 16 256 17 17 706 18 18 551 19 19 581 20 20 325 21 21 598 22 22 542 23 23 514 24 24 545 25 25 652 26 26 505.99997 27 27 391 28 28 514 29 29 504 30 30 619 31 31 688 32 32 561 33 33 614 34 34 514 35 35 577 36 36 558 37 37 367 38 38 529 39 39 514 40 40 730] +>> +endobj + +137 0 obj +<< + /Length 14 + /Filter /FlateDecode +>> +stream +x r| +endstream +endobj + +138 0 obj +<< + /Type /FontDescriptor + /FontName /FOOOQE+LibertinusSerif-Bold + /Flags 131078 + /FontBBox [0 -238 893 700] + /ItalicAngle 0 + /Ascent 894 + /Descent -246 + /CapHeight 645 + /StemV 168.6 + /CIDSet 137 0 R + /FontFile3 140 0 R +>> +endobj + +139 0 obj +<< + /Length 1170 + /Type /CMap + /WMode 0 +>> +stream +%!PS-Adobe-3.0 Resource-CMap +%%DocumentNeededResources: procset CIDInit +%%IncludeResource: procset CIDInit +%%BeginResource: CMap Custom +%%Title: (Custom Adobe Identity 0) +%%Version: 1 +%%EndComments +/CIDInit /ProcSet findresource begin +12 dict begin +begincmap +/CIDSystemInfo 3 dict dup begin + /Registry (Adobe) def + /Ordering (Identity) def + /Supplement 0 def +end def +/CMapName /Custom def +/CMapVersion 1 def +/CMapType 0 def +/WMode 0 def +1 begincodespacerange +<0000> +endcodespacerange +40 beginbfchar +<0001> <0041> +<0002> <0073> +<0003> <0069> +<0004> <0067> +<0005> <006E> +<0006> <006D> +<0007> <0065> +<0008> <0074> +<0009> <0020> +<000A> <0030> +<000B> <0031> +<000C> <0045> +<000D> <0078> +<000E> <0072> +<000F> <0063> +<0010> <003A> +<0011> <0043> +<0012> <006F> +<0013> <0070> +<0014> <006C> +<0015> <0075> +<0016> <0062> +<0017> <0032> +<0018> <0046> +<0019> <0054> +<001A> <0061> +<001B> <0066> +<001C> <0033> +<001D> <0053> +<001E> <0068> +<001F> <00660074> +<0020> <0064> +<0021> <0050> +<0022> <0034> +<0023> <004C> +<0024> <0079> +<0025> <0049> +<0026> <0076> +<0027> <0035> +<0028> <004F> +endbfchar +endcmap +CMapName currentdict /CMap defineresource pop +end +end +%%EndResource +%%EOF +endstream +endobj + +140 0 obj +<< + /Length 4735 + /Filter /FlateDecode + /Subtype /CIDFontType0C +>> +stream +xWyxSeO(MԠZ{+`A*ʀ^htKҽIl77[6WچP[E"00Μ2D6:~=s9ѣy|>+.JA'ZҞh߇eM2P5G wIE&OxGO㞘c߫ x +_N&ťe.MJLM yp7go̚3{ !Ґ9MI-7M/^/-)@G[YM +Qec :B8r<~<>U55X_]f\~֨.__`IhFtq1c+;0n׸M~KB|vDTce'DO8 +dn ܴ~?hNw{R8,,F6;C'O Q bXo7EwZ)M0ͽuK,Vs XId"FMbj!ݪyدَ!0j5TLQ 䬉%QU cَ1RSixO/29 ͿXRQ Pܼag~ZXVK'Ir[ꊊƶյT:Eڃ-Ti|#ܴEO)Wjjmuf:cRm=T֊(BέW& ?1ۄ!,RI +j¨ j . SWkb+[ڬ4]&%kvwv׵5HO KQA@'  +5pް(ݱnDpdʍ]tM{)=6) S#Žs]nhkg}qFK s(}HꐤF)~L6vwZ\i&GJZ:w[71k^V Awݦݙtz)"h馵}%u3*pBy}w"J|BΉ. Xk@ۅr)s= +eD~nS +0*E;ᔮMd \L%{B{ZtYeLᇟ* ߄IإnZ  Q(t{ZYbʺn-9\Άke$C9ƵdRiz5 v_J՗00oHDBPa7㻢6*$ۺY"Ah:ݘCU>M>Gv.є(T 8f9Xw8zޅjC^O۪b?L inUщB ˘PMB*3rdCvJ ~ +f;,4 +VYSL +^$SeYEQQ6y!!FCh;7TxM߄ϠhP(}A(m>;Do , +#;iq{̠͆us~)8`;nJ +,2ZbجRvc)Ybna6mR  |0a5T";ܴ|{͊ >X') +>.]J"nh4m"#(]V!J^a*7.4aĝi°,';k _G.UYӍy:RE%d =KኌHmU7VTlD{ j(\'/XNmVNX/ &F<]S_-&I3wvYJ'An4;p72k\I|\x2Xc3Lk nhmw5F;`0XEUk:RqqN1h#? 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+ /Info 175 0 R + /ID [(smMdLJyVtDKFoN9sTuN6hQ==) (smMdLJyVtDKFoN9sTuN6hQ==)] +>> +startxref +86294 +%%EOF \ No newline at end of file diff --git a/index.typ b/index.typ new file mode 100644 index 0000000..c44de9f --- /dev/null +++ b/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. +] + diff --git a/spec.pdf b/spec.pdf new file mode 100644 index 0000000..c596c26 Binary files /dev/null and b/spec.pdf differ