finished exercise
This commit is contained in:
@@ -4,4 +4,4 @@
|
||||
\catcode `"\active
|
||||
\babel@aux{english}{}
|
||||
\babel@aux{austrian}{}
|
||||
\gdef \@abspage@last{6}
|
||||
\gdef \@abspage@last{9}
|
||||
|
||||
@@ -1,6 +1,8 @@
|
||||
# Fdb version 4
|
||||
["pdflatex"] 1780937132.21957 "Assignment4_211.tex" "Assignment4_211.pdf" "Assignment4_211" 1780937132.89663 0
|
||||
["pdflatex"] 1781795329.6586 "Assignment5_211.tex" "Assignment5_211.pdf" "Assignment5_211" 1781795330.95715 2
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/enc/dvips/cm-super/cm-super-ts1.enc" 1 2900 1537cc8184ad1792082cd229ecc269f4 ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/map/fontname/texfonts.map" 1 3524 cb3e574dea2d1052e39280babc910dc8 ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/jknappen/ec/tcrm1200.tfm" 1 1536 74b7293ec3713bb7fdca8dd1bd1f469c ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1 1004 54797486969f23fa377b128694d548df ""
|
||||
"/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 ""
|
||||
@@ -24,13 +26,14 @@
|
||||
"/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/fonts/type1/public/amsfonts/cm/cmsy8.pfb" 1 32626 4f5c1b83753b1dd3a97d1b399a005b4b ""
|
||||
"/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/cm-super/sfrm1200.pfb" 1 136101 f533469f523533d38317ab5729d00c8a ""
|
||||
"/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 ""
|
||||
@@ -75,17 +78,20 @@
|
||||
"/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 ""
|
||||
"Assignment5_211.aux" 1781795330.91098 193 cc77e7c564288019cba58ce5da809f9b "pdflatex"
|
||||
"Assignment5_211.tex" 1781795329.47595 8390 ef094b5e34057788696cd94be0b8f051 ""
|
||||
"fig/fig1.1.png" 1781793254.50993 71116 1f126fb584298f937f835f8d954bf41d ""
|
||||
"fig/fig1.2.png" 1781793528.41305 32591 891bb9afa48110749761733491fced84 ""
|
||||
"fig/fig1.3.png" 1781794608.66063 228413 df7b99ac0349942a8b1bf947adbadb24 ""
|
||||
"fig/fig1.4.png" 1781794822.18644 329238 4a4e7e226dbec7d88f92d637aea0e23f ""
|
||||
"fig/fig2.1.png" 1781516725.91893 118408 47d9951b6d31879bf2aefe1aec57080a ""
|
||||
"fig/fig3.1.png" 1781527757.10163 77730 0856adb8cb2337f889f7fc830168ff85 ""
|
||||
"fig/fig3.2.png" 1781535111.49511 72906 1606f6cd0ab181c2597a9e42ff7047bd ""
|
||||
"fig/fig4.1.png" 1781530697.86066 55721 3efcf0273ede31213e0bc0e3dc80a97a ""
|
||||
"fig/fig4.2.png" 1781537172.81547 86203 358749c1ee82ceac054e9852353da34b ""
|
||||
"fig/fig4.3.png" 1781537375.95033 67818 8591679287c4cf93ecefcdafd7bd065f ""
|
||||
(generated)
|
||||
"Assignment4_211.aux"
|
||||
"Assignment4_211.log"
|
||||
"Assignment4_211.pdf"
|
||||
"Assignment5_211.aux"
|
||||
"Assignment5_211.log"
|
||||
"Assignment5_211.pdf"
|
||||
(rewritten before read)
|
||||
|
||||
@@ -1,8 +1,8 @@
|
||||
PWD /home/quirinecker/projects/semester06-signal_processing/assignment04
|
||||
PWD /home/quirinecker/projects/semester06-signal_processing/assignment05
|
||||
INPUT /nix/store/g8bx996bhdrf7gkgbgwh9s5xlpl0qyvv-texlive-2025-r78234-final-env/share/texmf-var/web2c/texmf.cnf
|
||||
INPUT /nix/store/g8bx996bhdrf7gkgbgwh9s5xlpl0qyvv-texlive-2025-r78234-final-env/share/texmf-var/web2c/pdftex/pdflatex.fmt
|
||||
INPUT Assignment4_211.tex
|
||||
OUTPUT Assignment4_211.log
|
||||
INPUT Assignment5_211.tex
|
||||
OUTPUT Assignment5_211.log
|
||||
INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/base/article.cls
|
||||
INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/base/article.cls
|
||||
INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/base/size12.clo
|
||||
@@ -85,10 +85,10 @@ INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-
|
||||
INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/amsfonts/amsfonts.sty
|
||||
INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/l3backend/l3backend-pdftex.def
|
||||
INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/l3backend/l3backend-pdftex.def
|
||||
INPUT ./Assignment4_211.aux
|
||||
INPUT ./Assignment4_211.aux
|
||||
INPUT Assignment4_211.aux
|
||||
OUTPUT Assignment4_211.aux
|
||||
INPUT ./Assignment5_211.aux
|
||||
INPUT ./Assignment5_211.aux
|
||||
INPUT Assignment5_211.aux
|
||||
OUTPUT Assignment5_211.aux
|
||||
INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/context/base/mkii/supp-pdf.mkii
|
||||
INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/context/base/mkii/supp-pdf.mkii
|
||||
INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/context/base/mkii/supp-pdf.mkii
|
||||
@@ -103,14 +103,35 @@ INPUT ./fig/fig1.1.png
|
||||
INPUT ./fig/fig1.1.png
|
||||
INPUT ./fig/fig1.1.png
|
||||
INPUT ./fig/fig1.1.png
|
||||
OUTPUT Assignment4_211.pdf
|
||||
OUTPUT Assignment5_211.pdf
|
||||
INPUT ./fig/fig1.1.png
|
||||
INPUT ./fig/fig1.2.png
|
||||
INPUT ./fig/fig1.2.png
|
||||
INPUT ./fig/fig1.2.png
|
||||
INPUT ./fig/fig1.2.png
|
||||
INPUT ./fig/fig1.2.png
|
||||
INPUT ./fig/fig1.3.png
|
||||
INPUT ./fig/fig1.3.png
|
||||
INPUT ./fig/fig1.3.png
|
||||
INPUT ./fig/fig1.3.png
|
||||
INPUT ./fig/fig1.3.png
|
||||
INPUT /nix/store/g8bx996bhdrf7gkgbgwh9s5xlpl0qyvv-texlive-2025-r78234-final-env/share/texmf-var/fonts/map/pdftex/updmap/pdftex.map
|
||||
INPUT ./fig/fig1.4.png
|
||||
INPUT ./fig/fig1.4.png
|
||||
INPUT ./fig/fig1.4.png
|
||||
INPUT ./fig/fig1.4.png
|
||||
INPUT ./fig/fig1.4.png
|
||||
INPUT ./fig/fig2.1.png
|
||||
INPUT ./fig/fig2.1.png
|
||||
INPUT ./fig/fig2.1.png
|
||||
INPUT ./fig/fig2.1.png
|
||||
INPUT ./fig/fig2.1.png
|
||||
INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/jknappen/ec/tcrm1200.tfm
|
||||
INPUT ./fig/fig3.2.png
|
||||
INPUT ./fig/fig3.2.png
|
||||
INPUT ./fig/fig3.2.png
|
||||
INPUT ./fig/fig3.2.png
|
||||
INPUT ./fig/fig3.2.png
|
||||
INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/cm/cmr8.tfm
|
||||
INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/cm/cmr6.tfm
|
||||
INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/cm/cmmi12.tfm
|
||||
@@ -134,16 +155,7 @@ INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-
|
||||
INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm
|
||||
INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm
|
||||
INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/tfm/public/amsfonts/symbols/msbm7.tfm
|
||||
INPUT ./fig/fig1.3.png
|
||||
INPUT ./fig/fig1.3.png
|
||||
INPUT ./fig/fig1.3.png
|
||||
INPUT ./fig/fig1.3.png
|
||||
INPUT ./fig/fig1.3.png
|
||||
INPUT ./fig/fig1.4.png
|
||||
INPUT ./fig/fig1.4.png
|
||||
INPUT ./fig/fig1.4.png
|
||||
INPUT ./fig/fig1.4.png
|
||||
INPUT ./fig/fig1.4.png
|
||||
INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/enc/dvips/cm-super/cm-super-ts1.enc
|
||||
INPUT ./fig/fig3.1.png
|
||||
INPUT ./fig/fig3.1.png
|
||||
INPUT ./fig/fig3.1.png
|
||||
@@ -159,14 +171,20 @@ INPUT ./fig/fig4.2.png
|
||||
INPUT ./fig/fig4.2.png
|
||||
INPUT ./fig/fig4.2.png
|
||||
INPUT ./fig/fig4.2.png
|
||||
INPUT Assignment4_211.aux
|
||||
INPUT ./fig/fig4.3.png
|
||||
INPUT ./fig/fig4.3.png
|
||||
INPUT ./fig/fig4.3.png
|
||||
INPUT ./fig/fig4.3.png
|
||||
INPUT ./fig/fig4.3.png
|
||||
INPUT Assignment5_211.aux
|
||||
INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmbx12.pfb
|
||||
INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmex10.pfb
|
||||
INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmmi12.pfb
|
||||
INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmmi6.pfb
|
||||
INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmmi8.pfb
|
||||
INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmr12.pfb
|
||||
INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmr8.pfb
|
||||
INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmssbx10.pfb
|
||||
INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmssi12.pfb
|
||||
INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmsy10.pfb
|
||||
INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmsy8.pfb
|
||||
INPUT /nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/cm-super/sfrm1200.pfb
|
||||
|
||||
@@ -1,10 +1,10 @@
|
||||
This is pdfTeX, Version 3.141592653-2.6-1.40.27 (TeX Live 2025/nixos.org) (preloaded format=pdflatex 1980.1.1) 8 JUN 2026 18:45
|
||||
This is pdfTeX, Version 3.141592653-2.6-1.40.27 (TeX Live 2025/nixos.org) (preloaded format=pdflatex 1980.1.1) 18 JUN 2026 17:08
|
||||
entering extended mode
|
||||
restricted \write18 enabled.
|
||||
file:line:error style messages enabled.
|
||||
%&-line parsing enabled.
|
||||
**Assignment4_211.tex
|
||||
(./Assignment4_211.tex
|
||||
**Assignment5_211.tex
|
||||
(./Assignment5_211.tex
|
||||
LaTeX2e <2025-11-01>
|
||||
L3 programming layer <2026-01-19>
|
||||
(/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/base/article.cls
|
||||
@@ -201,11 +201,11 @@ LaTeX Font Info: Overwriting math alphabet `\mathfrak' in version `bold'
|
||||
(/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/l3backend/l3backend-pdftex.def
|
||||
File: l3backend-pdftex.def 2025-10-09 L3 backend support: PDF output (pdfTeX)
|
||||
\l__color_backend_stack_int=\count299
|
||||
) (./Assignment4_211.aux
|
||||
) (./Assignment5_211.aux
|
||||
Package babel Info: 'austrian' activates 'german' shorthands.
|
||||
(babel) Reported on input line 6.
|
||||
)
|
||||
\openout1 = `Assignment4_211.aux'.
|
||||
\openout1 = `Assignment5_211.aux'.
|
||||
|
||||
LaTeX Font Info: Checking defaults for OML/cmm/m/it on input line 97.
|
||||
LaTeX Font Info: ... okay on input line 97.
|
||||
@@ -246,103 +246,117 @@ Underfull \hbox (badness 10000) in paragraph at lines 124--125
|
||||
|
||||
[]
|
||||
|
||||
<./fig/fig1.1.png, id=1, 923.45pt x 701.62125pt>
|
||||
File: ./fig/fig1.1.png Graphic file (type png)
|
||||
<use ./fig/fig1.1.png>
|
||||
Package pdftex.def Info: ./fig/fig1.1.png used on input line 127.
|
||||
(pdftex.def) Requested size: 409.72124pt x 311.2944pt.
|
||||
<./fig/fig1.2.png, id=2, 940.51375pt x 568.1225pt>
|
||||
File: ./fig/fig1.2.png Graphic file (type png)
|
||||
<use ./fig/fig1.2.png>
|
||||
Package pdftex.def Info: ./fig/fig1.2.png used on input line 128.
|
||||
(pdftex.def) Requested size: 409.72124pt x 247.49538pt.
|
||||
|
||||
Underfull \hbox (badness 10000) in paragraph at lines 129--130
|
||||
|
||||
[]
|
||||
|
||||
<fig/fig1.1.png, id=1, 1491.5725pt x 814.04124pt>
|
||||
File: fig/fig1.1.png Graphic file (type png)
|
||||
<use fig/fig1.1.png>
|
||||
Package pdftex.def Info: fig/fig1.1.png used on input line 127.
|
||||
(pdftex.def) Requested size: 409.72124pt x 223.60739pt.
|
||||
<fig/fig1.2.png, id=2, 1354.05875pt x 774.895pt>
|
||||
File: fig/fig1.2.png Graphic file (type png)
|
||||
<use fig/fig1.2.png>
|
||||
Package pdftex.def Info: fig/fig1.2.png used on input line 129.
|
||||
(pdftex.def) Requested size: 409.72124pt x 234.46854pt.
|
||||
<fig/fig1.3.png, id=3, 1482.53876pt x 815.045pt>
|
||||
File: fig/fig1.3.png Graphic file (type png)
|
||||
<use fig/fig1.3.png>
|
||||
Package pdftex.def Info: fig/fig1.3.png used on input line 131.
|
||||
(pdftex.def) Requested size: 409.72124pt x 225.2387pt.
|
||||
[1
|
||||
|
||||
{/nix/store/g8bx996bhdrf7gkgbgwh9s5xlpl0qyvv-texlive-2025-r78234-final-env/share/texmf-var/fonts/map/pdftex/updmap/pdftex.map} <./fig/fig1.1.png> <./fig/fig1.2.png>]
|
||||
LaTeX Font Info: Trying to load font information for U+msa on input line 130.
|
||||
<fig/fig1.4.png, id=13, 1309.89375pt x 769.87625pt>
|
||||
File: fig/fig1.4.png Graphic file (type png)
|
||||
<use fig/fig1.4.png>
|
||||
Package pdftex.def Info: fig/fig1.4.png used on input line 134.
|
||||
(pdftex.def) Requested size: 409.72124pt x 240.80893pt.
|
||||
|
||||
./Assignment5_211.tex:135: Undefined control sequence.
|
||||
l.135 ...the following number sequence: \lstinline
|
||||
{004373224685680}
|
||||
The control sequence at the end of the top line
|
||||
of your error message was never \def'ed. If you have
|
||||
misspelled it (e.g., `\hobx'), type `I' and the correct
|
||||
spelling (e.g., `I\hbox'). Otherwise just continue,
|
||||
and I'll forget about whatever was undefined.
|
||||
|
||||
<fig/fig2.1.png, id=14, 2056.68375pt x 856.19875pt>
|
||||
File: fig/fig2.1.png Graphic file (type png)
|
||||
<use fig/fig2.1.png>
|
||||
Package pdftex.def Info: fig/fig2.1.png used on input line 141.
|
||||
(pdftex.def) Requested size: 512.1496pt x 213.21303pt.
|
||||
[2 <./fig/fig1.3.png> <./fig/fig1.4.png>]
|
||||
Underfull \hbox (badness 10000) in paragraph at lines 150--151
|
||||
|
||||
[]
|
||||
|
||||
<fig/fig3.2.png, id=18, 1159.33125pt x 1095.09125pt>
|
||||
File: fig/fig3.2.png Graphic file (type png)
|
||||
<use fig/fig3.2.png>
|
||||
Package pdftex.def Info: fig/fig3.2.png used on input line 154.
|
||||
(pdftex.def) Requested size: 409.72124pt x 387.01395pt.
|
||||
LaTeX Font Info: Trying to load font information for U+msa on input line 157.
|
||||
(/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/amsfonts/umsa.fd
|
||||
File: umsa.fd 2013/01/14 v3.01 AMS symbols A
|
||||
)
|
||||
LaTeX Font Info: Trying to load font information for U+msb on input line 130.
|
||||
LaTeX Font Info: Trying to load font information for U+msb on input line 157.
|
||||
(/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/tex/latex/amsfonts/umsb.fd
|
||||
File: umsb.fd 2013/01/14 v3.01 AMS symbols B
|
||||
)
|
||||
<./fig/fig1.3.png, id=12, 1620.0525pt x 1124.2pt>
|
||||
File: ./fig/fig1.3.png Graphic file (type png)
|
||||
<use ./fig/fig1.3.png>
|
||||
Package pdftex.def Info: ./fig/fig1.3.png used on input line 153.
|
||||
(pdftex.def) Requested size: 409.72124pt x 284.30855pt.
|
||||
|
||||
Underfull \hbox (badness 10000) in paragraph at lines 154--155
|
||||
) [3{/nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/enc/dvips/cm-super/cm-super-ts1.enc} <./fig/fig2.1.png> <./fig/fig3.2.png>]
|
||||
<fig/fig3.1.png, id=23, 1095.09125pt x 1095.09125pt>
|
||||
File: fig/fig3.1.png Graphic file (type png)
|
||||
<use fig/fig3.1.png>
|
||||
Package pdftex.def Info: fig/fig3.1.png used on input line 187.
|
||||
(pdftex.def) Requested size: 409.72124pt x 409.72246pt.
|
||||
[4]
|
||||
Underfull \hbox (badness 10000) in paragraph at lines 192--193
|
||||
|
||||
[]
|
||||
|
||||
[2 <./fig/fig1.3.png>]
|
||||
<./fig/fig1.4.png, id=19, 835.12pt x 772.8875pt>
|
||||
File: ./fig/fig1.4.png Graphic file (type png)
|
||||
<use ./fig/fig1.4.png>
|
||||
Package pdftex.def Info: ./fig/fig1.4.png used on input line 194.
|
||||
(pdftex.def) Requested size: 409.72124pt x 379.18991pt.
|
||||
|
||||
Underfull \hbox (badness 10000) in paragraph at lines 198--199
|
||||
<fig/fig4.1.png, id=33, 964.60374pt x 964.60374pt>
|
||||
File: fig/fig4.1.png Graphic file (type png)
|
||||
<use fig/fig4.1.png>
|
||||
Package pdftex.def Info: fig/fig4.1.png used on input line 196.
|
||||
(pdftex.def) Requested size: 409.72124pt x 409.72304pt.
|
||||
[5 <./fig/fig3.1.png>] [6 <./fig/fig4.1.png>]
|
||||
Underfull \hbox (badness 10000) in paragraph at lines 215--216
|
||||
|
||||
[]
|
||||
|
||||
[3 <./fig/fig1.4.png>]
|
||||
Underfull \hbox (badness 10000) in paragraph at lines 211--212
|
||||
<fig/fig4.2.png, id=40, 886.31125pt x 1213.53375pt>
|
||||
File: fig/fig4.2.png Graphic file (type png)
|
||||
<use fig/fig4.2.png>
|
||||
Package pdftex.def Info: fig/fig4.2.png used on input line 220.
|
||||
(pdftex.def) Requested size: 409.72124pt x 560.99135pt.
|
||||
[7]
|
||||
<fig/fig4.3.png, id=45, 2191.18625pt x 849.1725pt>
|
||||
File: fig/fig4.3.png Graphic file (type png)
|
||||
<use fig/fig4.3.png>
|
||||
Package pdftex.def Info: fig/fig4.3.png used on input line 223.
|
||||
(pdftex.def) Requested size: 409.72124pt x 158.7789pt.
|
||||
[8 <./fig/fig4.2.png>]
|
||||
Underfull \hbox (badness 10000) in paragraph at lines 227--228
|
||||
|
||||
[]
|
||||
|
||||
|
||||
Underfull \hbox (badness 10000) in paragraph at lines 212--213
|
||||
|
||||
[]
|
||||
|
||||
<./fig/fig3.1.png, id=25, 1236.62pt x 467.7475pt>
|
||||
File: ./fig/fig3.1.png Graphic file (type png)
|
||||
<use ./fig/fig3.1.png>
|
||||
Package pdftex.def Info: ./fig/fig3.1.png used on input line 221.
|
||||
(pdftex.def) Requested size: 409.72124pt x 154.97807pt.
|
||||
|
||||
Underfull \hbox (badness 10000) in paragraph at lines 224--225
|
||||
|
||||
[]
|
||||
|
||||
<./fig/fig4.1.png, id=26, 839.135pt x 860.21375pt>
|
||||
File: ./fig/fig4.1.png Graphic file (type png)
|
||||
<use ./fig/fig4.1.png>
|
||||
Package pdftex.def Info: ./fig/fig4.1.png used on input line 228.
|
||||
(pdftex.def) Requested size: 409.72124pt x 420.01208pt.
|
||||
[4 <./fig/fig3.1.png>]
|
||||
<./fig/fig4.2.png, id=31, 836.12375pt x 851.18pt>
|
||||
File: ./fig/fig4.2.png Graphic file (type png)
|
||||
<use ./fig/fig4.2.png>
|
||||
Package pdftex.def Info: ./fig/fig4.2.png used on input line 233.
|
||||
(pdftex.def) Requested size: 409.72124pt x 417.10782pt.
|
||||
[5 <./fig/fig4.1.png>] [6 <./fig/fig4.2.png>] (./Assignment4_211.aux)
|
||||
[9 <./fig/fig4.3.png>] (./Assignment5_211.aux)
|
||||
***********
|
||||
LaTeX2e <2025-11-01>
|
||||
L3 programming layer <2026-01-19>
|
||||
***********
|
||||
)
|
||||
Here is how much of TeX's memory you used:
|
||||
4667 strings out of 467535
|
||||
82020 string characters out of 5388798
|
||||
491699 words of memory out of 5000000
|
||||
33503 multiletter control sequences out of 15000+600000
|
||||
634545 words of font info for 65 fonts, out of 8000000 for 9000
|
||||
4693 strings out of 467535
|
||||
82446 string characters out of 5388798
|
||||
494699 words of memory out of 5000000
|
||||
33527 multiletter control sequences out of 15000+600000
|
||||
634882 words of font info for 66 fonts, out of 8000000 for 9000
|
||||
1141 hyphenation exceptions out of 8191
|
||||
57i,10n,65p,280b,256s stack positions out of 10000i,1000n,20000p,200000b,200000s
|
||||
</nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmbx12.pfb></nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmex10.pfb></nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmmi12.pfb></nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmmi6.pfb></nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmmi8.pfb></nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmr12.pfb></nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmr8.pfb></nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmssbx10.pfb></nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmssi12.pfb></nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmsy10.pfb>
|
||||
Output written on Assignment4_211.pdf (6 pages, 713525 bytes).
|
||||
57i,9n,65p,594b,260s stack positions out of 10000i,1000n,20000p,200000b,200000s
|
||||
</nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmbx12.pfb></nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmex10.pfb></nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmmi12.pfb></nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmmi8.pfb></nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmr12.pfb></nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmr8.pfb></nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmssbx10.pfb></nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmssi12.pfb></nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmsy10.pfb></nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/amsfonts/cm/cmsy8.pfb></nix/store/56vq9r21wl8m731rsri7nh3l4ipih7jj-texlive-2025-r78234-final-env-texmfdist/fonts/type1/public/cm-super/sfrm1200.pfb>
|
||||
Output written on Assignment5_211.pdf (9 pages, 1128302 bytes).
|
||||
PDF statistics:
|
||||
80 PDF objects out of 1000 (max. 8388607)
|
||||
44 compressed objects within 1 object stream
|
||||
100 PDF objects out of 1000 (max. 8388607)
|
||||
56 compressed objects within 1 object stream
|
||||
0 named destinations out of 1000 (max. 500000)
|
||||
36 words of extra memory for PDF output out of 10000 (max. 10000000)
|
||||
51 words of extra memory for PDF output out of 10000 (max. 10000000)
|
||||
|
||||
|
||||
Binary file not shown.
Binary file not shown.
@@ -105,9 +105,9 @@
|
||||
\begin{minipage}[b][4cm]{1.0\textwidth}
|
||||
\begin{center}
|
||||
\begin{bf}
|
||||
\begin{large} Digitale Signalverarbeitung SS 2025/26 -- 4.~Aufgabe\end{large} \\
|
||||
\begin{large} Digitale Signalverarbeitung SS 2025/26 -- 5.~Aufgabe\end{large} \\
|
||||
\vspace{0.3cm}
|
||||
\begin{Large} Reconstruction, DFT, FFT \end{Large} \\
|
||||
\begin{Large} STFT, z-Transform, Digital Filters \end{Large} \\
|
||||
\vspace{0.3cm}
|
||||
\end{bf}
|
||||
\begin{large}
|
||||
@@ -121,120 +121,110 @@ Quirin Ecker, k12310122 \\
|
||||
\noindent \rule[0.8em]{\textwidth}{0.12mm}\\[-0.5em]
|
||||
%=======================================================================================
|
||||
|
||||
\begin{aufgabe}{Reconstruction}
|
||||
\begin{aufgabe}{FFT in Audio Signal Processing – Short Time Fourier Transform}
|
||||
\begin{enumerate}
|
||||
\item No, because you can't make out the frequencies in the signals without a forier transform
|
||||
\includegraphics[width=0.8\textwidth]{fig/fig1.1.png}
|
||||
\item : \\
|
||||
\includegraphics[width=0.8\textwidth]{fig/fig1.2.png}
|
||||
\item : \\
|
||||
\includegraphics[width=0.8\textwidth]{fig/fig1.3.png}
|
||||
\item The actual frequencies are more readable with the hamming windows.
|
||||
Wihtout them, the borders aren't as clear. Also known as Spectral Leakage \\
|
||||
\includegraphics[width=0.8\textwidth]{fig/fig1.4.png}
|
||||
\item Decoding it with the figure in (c) gives us the following number sequence: \lstinline{004373224685680}
|
||||
\item The diagram in (b) contains all frequencies of the entire signal, and we can not determine which number sequence it is. The signal in (c) looks at the fourier transform at different locations along the signal to localize what frequencies are present at certain times, and we can now easily see the encoding. An example for this is the landline phone system where pressing numbers will play different combinations of tones into the phone line, letting the system on the other side determine which number you have dialed.
|
||||
\end{enumerate}
|
||||
\end{aufgabe}
|
||||
|
||||
\begin{aufgabe}{Signal Distortion and Group Delay}
|
||||
\includegraphics[width=1\textwidth]{fig/fig2.1.png}
|
||||
\begin{itemize}
|
||||
\item \textbf{FIR Filter}: The FIR filter kept its shape but is a little bit delayed on the phase.
|
||||
\item \textbf{IIR Filter}: The IIR filter lost some of the shape similarity to the original signal
|
||||
and is also a little delayed, but uses fewer coefficients,
|
||||
which means that it also takes less memory and time to compute
|
||||
\end{itemize}
|
||||
\end{aufgabe}
|
||||
|
||||
\begin{aufgabe}{z-Transform}
|
||||
|
||||
\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) \\
|
||||
\includegraphics[width=0.8\textwidth]{fig/fig3.2.png}
|
||||
\item IIR, because its a recursive LTI System.
|
||||
\item
|
||||
\begin{align*}
|
||||
y[n] &= x[n] - \frac{1}{15} y[n-1] + \frac{2}{5} y[n-2] \\
|
||||
\\
|
||||
\sin(\frac{4 \pi}{3}) = -\sin(\frac{2 \pi}{3})
|
||||
Y(z) &= X(z) - \frac{1}{15} z^{-1} Y(z) + \frac{2}{5} z^{-2} Y(z) \\
|
||||
\\
|
||||
\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 $x_2[n]$:
|
||||
|
||||
Y(z) + \frac{1}{15} z^{-1} Y(z) - \frac{2}{5} z^{-2} Y(z) &= X(z) \\
|
||||
\\
|
||||
Y(z) \left(1 + \frac{1}{15} z^{-1} - \frac{2}{5} z^{-2}\right) &= X(z) \\
|
||||
\\
|
||||
Y(z) &= \frac{X(z)}{1 + \frac{1}{15} z^{-1} - \frac{2}{5} z^{-2}} \\
|
||||
\\
|
||||
Y(z) &= \frac{X(z)}{1 + \frac{1}{15} z^{-1} - \frac{2}{5} z^{-2}} \cdot \frac{z^2}{z^2} \\
|
||||
\\
|
||||
Y(z) &= \frac{X(z) z^2}{z^2 + \frac{1}{15} z - \frac{2}{5}} \\
|
||||
\\
|
||||
H(z) &= \frac{Y(z)}{X(z)} = \frac{z^2}{z^2 + \frac{1}{15} z - \frac{2}{5}}
|
||||
\end{align*}
|
||||
\item
|
||||
Poles:
|
||||
\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) \\
|
||||
0 = {z^2 + \frac{1}{15} z - \frac{2}{5}} \\
|
||||
z_1 = 0.6 \\
|
||||
z_2 = -0.6\overline{6}
|
||||
\end{gather*}
|
||||
\includegraphics[width=0.8\textwidth]{./fig/fig1.3.png}
|
||||
Zeros:
|
||||
\begin{gather*}
|
||||
0 = z^2 \\
|
||||
z_{1,2} = 0
|
||||
\end{gather*}
|
||||
Plot: \\
|
||||
\includegraphics[width=0.8\textwidth]{fig/fig3.1.png}
|
||||
\item The system is BIBO stable because $|p_i| < 1$ holds for all poles.
|
||||
\end{enumerate}
|
||||
\end{aufgabe}
|
||||
|
||||
\begin{aufgabe}{Recursive Filter}
|
||||
\begin{enumerate}
|
||||
\item Yes the coefficents are real valued, because all non real poles and zeros appear in complex conjugate pairs
|
||||
\item : \\
|
||||
\includegraphics[width=0.8\textwidth]{fig/fig4.1.png}
|
||||
\item
|
||||
\begin{align*}
|
||||
H(z) &= b_0 z^{M - N} \cdot \frac{(z + 1) (z - j) (z + j)}{(z - 0) (z - (0.75 + 0.25j)) (z - (0.75 - 0.25j))} \\
|
||||
H(z) &= b_0 z^{3 - 3} \cdot \frac{(z + 1) (z - j) (z + j)}{(z - 0) (z - (0.75 + 0.25j)) (z - (0.75 - 0.25j))} \\
|
||||
H(z) &= b_0 z^{0} \cdot \frac{(z + 1) (z - j) (z + j)}{(z - 0) (z - (0.75 + 0.25j)) (z - (0.75 - 0.25j))} \\
|
||||
H(z) &= b_0 \cdot \frac{(z + 1) (z - j) (z + j)}{(z - 0) (z - (0.75 + 0.25j)) (z - (0.75 - 0.25j))} \\
|
||||
H(z) &= b_0 \cdot \frac{(z + 1) (z - j) (z + j)}{(z - 0) (z - (0.75 + 0.25j)) (z - (0.75 - 0.25j))} \\
|
||||
H(z) &= b_0 \cdot \frac{z^3 + z^2 + z + 1}{z^3 - 1.5z^2 + 0.625z}
|
||||
\end{align*}
|
||||
|
||||
\[
|
||||
f=\frac{\Omega f_s}{2\pi}
|
||||
\]
|
||||
This was for the $z^{+i}$ case. Now follows the $z^{-i}$ case.
|
||||
|
||||
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}
|
||||
\begin{align*}
|
||||
H(z) &= b_0 \cdot \frac{z^3 + z^2 + z + 1}{z^3 - 1.5z^2 + 0.625z} \\
|
||||
H(z) &= b_0 \cdot \frac{z^3 + z^2 + z + 1}{z^3 - 1.5z^2 + 0.625z} \cdot \frac{z^{-3}}{z^{-3}} \\
|
||||
H(z) &= b_0 \cdot \frac{1 + z^{-1} + z^{-2} + z^{-3}}{1 - 1.5z^{-1} + 0.625z^{-2}} \\
|
||||
\end{align*}
|
||||
\item
|
||||
With the Direct Form I and the $H(z)$ from above, we receive the following LTI System: \\
|
||||
\begin{align*}
|
||||
y[n] &= b_0 x[n] + b_0 x[n-1] + b_0 x[n-2] + b_0 x[n-3] + 1.5 y[n-1] - 0.625 y[n-2] \\
|
||||
\end{align*}
|
||||
See the following block diagram: \\
|
||||
\includegraphics[width=0.8\textwidth]{fig/fig4.2.png}
|
||||
\item (see f)
|
||||
\item : \\
|
||||
\includegraphics[width=0.8\textwidth]{fig/fig4.3.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}
|
||||
\begin{aufgabe}{Lowpass Filter Design}
|
||||
\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}
|
||||
|
||||
43
Assignment5_211/assignment5.2.m
Normal file
43
Assignment5_211/assignment5.2.m
Normal file
@@ -0,0 +1,43 @@
|
||||
clear; close all; clc;
|
||||
|
||||
n = 0:599; % 3 periods (3 × 200 samples)
|
||||
|
||||
x = sin(2*pi*0.005*n) ...
|
||||
+ (1/3)*sin(2*pi*0.005*3*n) ...
|
||||
+ (1/5)*sin(2*pi*0.005*5*n) ...
|
||||
+ (1/7)*sin(2*pi*0.005*7*n);
|
||||
|
||||
stem(n,x)
|
||||
grid on
|
||||
xlabel('n')
|
||||
ylabel('x[n]')
|
||||
title('Signal over 3 periods')
|
||||
|
||||
load("Filter_coefficients.mat","-mat")
|
||||
|
||||
y_FIR = filter(b1,a1,x);
|
||||
y_IIR = filter(b2,a2,x);
|
||||
|
||||
figure
|
||||
|
||||
plot(n,x,'y','LineWidth',1.5)
|
||||
hold on
|
||||
|
||||
plot(n,y_FIR,'b')
|
||||
plot(n,y_IIR,'r')
|
||||
|
||||
legend('Original','FIR','IIR')
|
||||
grid on
|
||||
title('Comparison of Signals')
|
||||
|
||||
figure
|
||||
|
||||
subplot(2,1,1)
|
||||
freqz(b1,a1)
|
||||
title('FIR Frequency Response')
|
||||
|
||||
figure
|
||||
|
||||
subplot(2,1,2)
|
||||
freqz(b2,a2)
|
||||
title('IIR Frequency Response')
|
||||
47
Assignment5_211/assignment5.4.m
Normal file
47
Assignment5_211/assignment5.4.m
Normal file
@@ -0,0 +1,47 @@
|
||||
clear; close all; clc;
|
||||
% Define filter coefficients
|
||||
b0 = 1;
|
||||
|
||||
b = b0 * [1 1 1 1];
|
||||
a = [1 -1.5 0.625];
|
||||
|
||||
% Frequency response
|
||||
[H,w] = freqz(b,a,1024);
|
||||
|
||||
% Magnitude and phase plots
|
||||
figure;
|
||||
|
||||
subplot(2,1,1);
|
||||
plot(w/pi, 20*log10(abs(H)), 'LineWidth', 1.5);
|
||||
grid on;
|
||||
xlabel('Normalized Frequency (\times\pi rad/sample)');
|
||||
ylabel('Magnitude (dB)');
|
||||
title('Magnitude Response');
|
||||
|
||||
subplot(2,1,2);
|
||||
plot(w/pi, unwrap(angle(H)), 'LineWidth', 1.5);
|
||||
grid on;
|
||||
xlabel('Normalized Frequency (\times\pi rad/sample)');
|
||||
ylabel('Phase (rad)');
|
||||
title('Phase Response');
|
||||
|
||||
% Define filter coefficients
|
||||
b0 = 1;
|
||||
|
||||
b = b0 * [1 1 1 1];
|
||||
a = [1 -1.5 0.625];
|
||||
|
||||
% Generate impulse signal
|
||||
n = 0:50;
|
||||
x = [1 zeros(1,50)];
|
||||
|
||||
% Compute impulse response
|
||||
h = filter(b,a,x);
|
||||
|
||||
% Plot impulse response
|
||||
figure;
|
||||
stem(n,h,'filled');
|
||||
grid on;
|
||||
xlabel('n');
|
||||
ylabel('h[n]');
|
||||
title('Impulse Response (0 \leq n \leq 50)');
|
||||
BIN
fig/fig1.1.png
Normal file
BIN
fig/fig1.1.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 69 KiB |
BIN
fig/fig1.2.png
Normal file
BIN
fig/fig1.2.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 32 KiB |
BIN
fig/fig1.3.png
Normal file
BIN
fig/fig1.3.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 223 KiB |
BIN
fig/fig1.4.png
Normal file
BIN
fig/fig1.4.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 322 KiB |
BIN
fig/fig2.1.png
Normal file
BIN
fig/fig2.1.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 116 KiB |
BIN
fig/fig3.1.png
Normal file
BIN
fig/fig3.1.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 76 KiB |
BIN
fig/fig3.2.png
Normal file
BIN
fig/fig3.2.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 71 KiB |
BIN
fig/fig4.1.png
Normal file
BIN
fig/fig4.1.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 54 KiB |
BIN
fig/fig4.2.png
Normal file
BIN
fig/fig4.2.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 84 KiB |
BIN
fig/fig4.3.png
Normal file
BIN
fig/fig4.3.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 66 KiB |
Reference in New Issue
Block a user