From 930f7ae4a4c4d31d0566150a15bbe65ca21a544a Mon Sep 17 00:00:00 2001 From: quirinecker Date: Sun, 26 Apr 2026 22:23:54 +0200 Subject: [PATCH] updated changes so far --- .direnv/flake-profile | 1 + .direnv/flake-profile-1-link | 1 + .envrc | 1 + .gitignore | 315 ++++ Assignment1_211.pdf | Bin 0 -> 244874 bytes Assignment1_211.tex | 351 +++++ Assignment1_211/assignment1_3.m | 21 + fig/Testpicture.eps | 2021 ++++++++++++++++++++++++ fig/aufgabe2.png | Bin 0 -> 17711 bytes fig/aufgabe3_1.png | Bin 0 -> 44727 bytes fig/aufgabe3_2.png | Bin 0 -> 62783 bytes flake.lock | 26 + flake.nix | 22 + index.pdf | 2553 +++++++++++++++++++++++++++++++ index.typ | 144 ++ spec.pdf | Bin 0 -> 270434 bytes 16 files changed, 5456 insertions(+) create mode 120000 .direnv/flake-profile create mode 120000 .direnv/flake-profile-1-link create mode 100644 .envrc create mode 100644 .gitignore create mode 100644 Assignment1_211.pdf create mode 100644 Assignment1_211.tex create mode 100644 Assignment1_211/assignment1_3.m create mode 100644 fig/Testpicture.eps create mode 100644 fig/aufgabe2.png create mode 100644 fig/aufgabe3_1.png create mode 100644 fig/aufgabe3_2.png create mode 100644 flake.lock create mode 100644 flake.nix create mode 100644 index.pdf create mode 100644 index.typ create mode 100644 spec.pdf 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. 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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 +>> 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/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 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"@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. 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