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.direnv/flake-profile
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.direnv/flake-profile
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flake-profile-2-link
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.direnv/flake-profile-2-link
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.direnv/flake-profile-2-link
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/nix/store/99fjzvv2y3g37jbqi28q8080db1183xk-latex-env
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7
Assignment4_211.aux
Normal file
7
Assignment4_211.aux
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||||
\relax
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\providecommand \babel@aux [2]{\global \let \babel@toc \@gobbletwo }
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\@nameuse{bbl@beforestart}
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\catcode `"\active
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\babel@aux{english}{}
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\babel@aux{austrian}{}
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\gdef \@abspage@last{6}
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91
Assignment4_211.fdb_latexmk
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Assignment4_211.fdb_latexmk
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# Fdb version 4
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["pdflatex"] 1780937132.21957 "Assignment4_211.tex" "Assignment4_211.pdf" "Assignment4_211" 1780937132.89663 0
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/map/fontname/texfonts.map" 1 3524 cb3e574dea2d1052e39280babc910dc8 ""
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||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1 1004 54797486969f23fa377b128694d548df ""
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||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/amsfonts/symbols/msam10.tfm" 1 916 f87d7c45f9c908e672703b83b72241a3 ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/amsfonts/symbols/msam7.tfm" 1 928 2dc8d444221b7a635bb58038579b861a ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm" 1 908 2921f8a10601f252058503cc6570e581 ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/amsfonts/symbols/msbm7.tfm" 1 940 228d6584342e91276bf566bcf9716b83 ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/cm/cmbx12.tfm" 1 1324 c910af8c371558dc20f2d7822f66fe64 ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/cm/cmex10.tfm" 1 992 662f679a0b3d2d53c1b94050fdaa3f50 ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/cm/cmmi12.tfm" 1 1524 4414a8315f39513458b80dfc63bff03a ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/cm/cmmi6.tfm" 1 1512 f21f83efb36853c0b70002322c1ab3ad ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/cm/cmmi8.tfm" 1 1520 eccf95517727cb11801f4f1aee3a21b4 ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/cm/cmr12.tfm" 1 1288 655e228510b4c2a1abe905c368440826 ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/cm/cmr6.tfm" 1 1300 b62933e007d01cfd073f79b963c01526 ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/cm/cmr8.tfm" 1 1292 21c1c5bfeaebccffdb478fd231a0997d ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/cm/cmss12.tfm" 1 1324 37b971caf729d7edd9cbb9f9b0ea76eb ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/cm/cmssbx10.tfm" 1 1272 e2d13f0df30bf3ad990bb9d028e37f34 ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/cm/cmssi12.tfm" 1 1532 48589e216c7fb787d0f3ee1a3047705d ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/cm/cmsy10.tfm" 1 1124 6c73e740cf17375f03eec0ee63599741 ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/cm/cmsy6.tfm" 1 1116 933a60c408fc0a863a92debe84b2d294 ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/cm/cmsy8.tfm" 1 1120 8b7d695260f3cff42e636090a8002094 ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmbx12.pfb" 1 32080 340ef9bf63678554ee606688e7b5339d ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmex10.pfb" 1 30251 6afa5cb1d0204815a708a080681d4674 ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmmi12.pfb" 1 36741 fa121aac0049305630cf160b86157ee4 ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmmi6.pfb" 1 37166 8ab3487cbe3ab49ebce74c29ea2418db ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmmi8.pfb" 1 35469 70d41d2b9ea31d5d813066df7c99281c ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmr12.pfb" 1 32722 d7379af29a190c3f453aba36302ff5a9 ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmr8.pfb" 1 32726 0a1aea6fcd6468ee2cf64d891f5c43c8 ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmssbx10.pfb" 1 28902 2f5c04fd2884d1878057baa5aad22765 ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmssi12.pfb" 1 25095 0ff1f6d531b320ecbbd6ea84d875816c ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmsy10.pfb" 1 32569 5e5ddc8df908dea60932f3c484a54c0d ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/context/base/mkii/supp-pdf.mkii" 1 71627 94eb9990bed73c364d7f53f960cc8c5b ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/generic/babel-english/english.ldf" 1 7008 9ff5fdcc865b01beca2b0fe4a46231d4 ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/generic/babel-german/austrian.ldf" 1 2238 5a1f6203e82197c5a4435155488ffdcf ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/generic/babel-german/babel-german.def" 1 32946 cb91c8893d96faf3f2fd433dbf43f3f1 ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/generic/babel/babel.sty" 1 147019 f2ca437186bfeb562e1066ad59426a04 ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/generic/babel/locale/de/babel-austrian.tex" 1 670 ed2363bc8191e44da866315c2a7eb96c ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/generic/babel/locale/de/babel-de-AT-1901.ini" 1 4996 a7449054008fbf7583054d19176ec406 ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/generic/babel/locale/en/babel-en.ini" 1 3936 fef897bbcfe8e9d18c93a3d2d53cf9d8 ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/generic/babel/locale/en/babel-english.tex" 1 374 a9c05f002d9437e6de45ac1cc23d56c2 ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/generic/babel/txtbabel.def" 1 5231 c1599e5bf7d2ee42743212a7f8d20a56 ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/generic/iftex/iftex.sty" 1 7984 7dbb9280f03c0a315425f1b4f35d43ee ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/generic/infwarerr/infwarerr.sty" 1 8356 7bbb2c2373aa810be568c29e333da8ed ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/generic/kvdefinekeys/kvdefinekeys.sty" 1 5412 d5a2436094cd7be85769db90f29250a6 ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/generic/ltxcmds/ltxcmds.sty" 1 17865 1a9bd36b4f98178fa551aca822290953 ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/generic/pdftexcmds/pdftexcmds.sty" 1 20089 80423eac55aa175305d35b49e04fe23b ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/amsfonts/amsfonts.sty" 1 5949 3f3fd50a8cc94c3d4cbf4fc66cd3df1c ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/amsfonts/amssymb.sty" 1 13829 94730e64147574077f8ecfea9bb69af4 ""
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||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/amsfonts/umsa.fd" 1 961 6518c6525a34feb5e8250ffa91731cff ""
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||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/amsfonts/umsb.fd" 1 961 d02606146ba5601b5645f987c92e6193 ""
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"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/amsmath/amsbsy.sty" 1 2222 27db7d52163edae53881b71ff62e754e ""
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||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/amsmath/amsgen.sty" 1 4173 1b3e76addfb8afcb47db4811d66e1dc6 ""
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||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/amsmath/amsmath.sty" 1 88471 b1bb09142edddebd46ba986341b867bd ""
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||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/amsmath/amsopn.sty" 1 4474 c510a88aa5f51b8c773b50a7ee92befd ""
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"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/amsmath/amstext.sty" 1 2444 9983e1d0683f102e3b190c64a49313aa ""
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||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/base/article.cls" 1 20144 b966087dda3b194755eb460d32e2ef75 ""
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"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/base/inputenc.sty" 1 5048 0270515b828149155424600fd2d58ac5 ""
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||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/base/size12.clo" 1 8449 8dc66c6c313c8eb2d774af83bca435dd ""
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"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/epstopdf-pkg/epstopdf-base.sty" 1 13886 d1306dcf79a944f6988e688c1785f9ce ""
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||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/epstopdf-pkg/epstopdf.sty" 1 4393 47f27fd4d95928d20b1885ba77de11d2 ""
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||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/graphics-cfg/graphics.cfg" 1 1224 978390e9c2234eab29404bc21b268d1e ""
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||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/graphics-def/pdftex.def" 1 19626 23e2822b9b2b5005f4c549ca98b9334d ""
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||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/graphics/graphics.sty" 1 18363 69bb4f5538964bfea50d1e6d89cbe69f ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/graphics/graphicx.sty" 1 8118 43b99e52946c33a23f5f43b52d5cc5ec ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/graphics/keyval.sty" 1 2671 d9941f4bf4750e9b0603c9a2ec54693b ""
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||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/graphics/trig.sty" 1 4023 e66acf578d6b564c4670fb57ff336a7a ""
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||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/grfext/grfext.sty" 1 7133 b94bbacbee6e4fdccdc7f810b2aec370 ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/kvoptions/kvoptions.sty" 1 22555 6d8e155cfef6d82c3d5c742fea7c992e ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/kvsetkeys/kvsetkeys.sty" 1 13815 760b0c02f691ea230f5359c4e1de23a7 ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/l3backend/l3backend-pdftex.def" 1 30662 bfd6e864f4ffc5018b0e2b6260c3181c ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/latexconfig/epstopdf-sys.cfg" 1 678 4792914a8f45be57bb98413425e4c7af ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/multirow/multirow.sty" 1 6696 886c9f3087d0b973ed2c19aa79cb3023 ""
|
||||
"/nix/store/g8bx996bhdrf7gkgbgwh9s5xlpl0qyvv-texlive-2025-r78234-final-env/share/texmf-var/fonts/map/pdftex/updmap/pdftex.map" 1 5541489 98dcaa26546008c59b74ed5f7e93078c ""
|
||||
"/nix/store/g8bx996bhdrf7gkgbgwh9s5xlpl0qyvv-texlive-2025-r78234-final-env/share/texmf-var/web2c/pdftex/pdflatex.fmt" 1 3637270 2f5a0a831f2ff9f2929d06d36e122866 ""
|
||||
"/nix/store/g8bx996bhdrf7gkgbgwh9s5xlpl0qyvv-texlive-2025-r78234-final-env/share/texmf-var/web2c/texmf.cnf" 1 44263 cffdb522e8ff03abb028d8b9b1d3dcaf ""
|
||||
"Assignment4_211.aux" 1780937132.85779 193 347950d4fe37faf4cda436926310d6c9 "pdflatex"
|
||||
"Assignment4_211.tex" 1780937131.07975 7340 5eda71aa5204acd3ada880aeb21beff0 ""
|
||||
"fig/fig1.1.png" 1780661010.16298 29540 fbe16c23e73dd5b574352e19d95cefab ""
|
||||
"fig/fig1.2.png" 1780661025.82539 23905 f6bafcade54f5f0c79bd69f01e99ef22 ""
|
||||
"fig/fig1.3.png" 1780661037.90572 103888 fc80510949e033cc1ea8097918da30c8 ""
|
||||
"fig/fig1.4.png" 1780664950.44617 50673 8282f7d7a9a6bbb198f3f8088612d58d ""
|
||||
"fig/fig3.1.png" 1780666655.93179 362615 411378ab0ed29697877a5e0d95a522f0 ""
|
||||
"fig/fig4.1.png" 1780672164.80068 31524 5546400b68ed83ec746009978f8e732a ""
|
||||
"fig/fig4.2.png" 1780672272.76268 29167 847b86c140a9ad57ffea3fee13be55d0 ""
|
||||
(generated)
|
||||
"Assignment4_211.aux"
|
||||
"Assignment4_211.log"
|
||||
"Assignment4_211.pdf"
|
||||
(rewritten before read)
|
||||
172
Assignment4_211.fls
Normal file
172
Assignment4_211.fls
Normal file
@@ -0,0 +1,172 @@
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||||
PWD /home/quirinecker/projects/semester06-signal_processing/assignment04
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|
||||
INPUT Assignment4_211.tex
|
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OUTPUT Assignment4_211.log
|
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||||
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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|
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/kvoptions/kvoptions.sty
|
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|
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|
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/kvsetkeys/kvsetkeys.sty
|
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/kvsetkeys/kvsetkeys.sty
|
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/generic/iftex/iftex.sty
|
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|
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|
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/epstopdf-pkg/epstopdf-base.sty
|
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/latexconfig/epstopdf-sys.cfg
|
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/latexconfig/epstopdf-sys.cfg
|
||||
INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/latexconfig/epstopdf-sys.cfg
|
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|
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/amsmath/amsopn.sty
|
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/amsmath/amstext.sty
|
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/amsmath/amstext.sty
|
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/amsmath/amsgen.sty
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/amsmath/amsgen.sty
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/amsmath/amsopn.sty
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/amsfonts/amssymb.sty
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/l3backend/l3backend-pdftex.def
|
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/l3backend/l3backend-pdftex.def
|
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INPUT ./Assignment4_211.aux
|
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INPUT ./Assignment4_211.aux
|
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INPUT Assignment4_211.aux
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OUTPUT Assignment4_211.aux
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/context/base/mkii/supp-pdf.mkii
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/context/base/mkii/supp-pdf.mkii
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/context/base/mkii/supp-pdf.mkii
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/cm/cmbx12.tfm
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/cm/cmbx12.tfm
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/cm/cmbx12.tfm
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/cm/cmr12.tfm
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/cm/cmss12.tfm
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/cm/cmssbx10.tfm
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/cm/cmssi12.tfm
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INPUT ./fig/fig1.1.png
|
||||
INPUT ./fig/fig1.1.png
|
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INPUT ./fig/fig1.1.png
|
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INPUT ./fig/fig1.1.png
|
||||
OUTPUT Assignment4_211.pdf
|
||||
INPUT ./fig/fig1.1.png
|
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INPUT ./fig/fig1.2.png
|
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INPUT ./fig/fig1.2.png
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INPUT ./fig/fig1.2.png
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INPUT ./fig/fig1.2.png
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/cm/cmmi12.tfm
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/cm/cmmi8.tfm
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/cm/cmmi6.tfm
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/cm/cmsy10.tfm
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/cm/cmsy8.tfm
|
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/cm/cmsy6.tfm
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/cm/cmex10.tfm
|
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm
|
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm
|
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/amsfonts/umsa.fd
|
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/amsfonts/umsa.fd
|
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/amsfonts/umsa.fd
|
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/amsfonts/symbols/msam10.tfm
|
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/amsfonts/symbols/msam10.tfm
|
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/amsfonts/symbols/msam7.tfm
|
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INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/amsfonts/umsb.fd
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[5 <./fig/fig4.1.png>] [6 <./fig/fig4.2.png>] (./Assignment4_211.aux)
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***********
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LaTeX2e <2025-11-01>
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L3 programming layer <2026-01-19>
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***********
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\newenvironment{aufgabe}[1]%
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{\bigskip\par\noindent\begin{nopagebreak}
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\textsf{\textbf{\arabic{theaufgabe}.\thinspace Exercise}}\quad
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\textsf{\textit{#1}}\\*[1ex]%
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\stepcounter{theaufgabe}\hspace{2ex}\end{nopagebreak}}
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{\par\pagebreak[2]}
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% Innerhalb der Aufgaben erfolgt die weitere Unterteilung mittels einer
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% enumerate Umgebung, die allerdings a), b),... zaehlen soll.
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%% Beginning of the text
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\begin{document}
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\selectlanguage{austrian}
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%=== This is the header section ============================================================
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\thispagestyle{empty}
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\noindent
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\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}
|
||||
BIN
Assignment4_211.zip
Normal file
BIN
Assignment4_211.zip
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Binary file not shown.
33
Assignment4_211/assignment4.4.m
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33
Assignment4_211/assignment4.4.m
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@@ -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;
|
||||
33
Assignment4_211/assignment4_1.m
Normal file
33
Assignment4_211/assignment4_1.m
Normal file
@@ -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)');
|
||||
51
Assignment4_211/assignment4_3.m
Normal file
51
Assignment4_211/assignment4_3.m
Normal file
@@ -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');
|
||||
BIN
fig/fig1.1.png
Normal file
BIN
fig/fig1.1.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 29 KiB |
BIN
fig/fig1.2.png
Normal file
BIN
fig/fig1.2.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 23 KiB |
BIN
fig/fig1.3.png
Normal file
BIN
fig/fig1.3.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 102 KiB |
BIN
fig/fig1.4.png
Normal file
BIN
fig/fig1.4.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 50 KiB |
BIN
fig/fig3.1.png
Normal file
BIN
fig/fig3.1.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 354 KiB |
BIN
fig/fig4.1.png
Normal file
BIN
fig/fig4.1.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 31 KiB |
BIN
fig/fig4.2.png
Normal file
BIN
fig/fig4.2.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 28 KiB |
26
flake.lock
generated
Normal file
26
flake.lock
generated
Normal file
@@ -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
|
||||
}
|
||||
23
flake.nix
Normal file
23
flake.nix
Normal file
@@ -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
|
||||
];
|
||||
};
|
||||
};
|
||||
}
|
||||
Reference in New Issue
Block a user