From ae0ccf2c56e31b0146354b8fb663ec2284aa1dc6 Mon Sep 17 00:00:00 2001 From: quirinecker Date: Thu, 18 Jun 2026 17:10:52 +0200 Subject: [PATCH] finished exercise --- Assignment5_211.aux | 2 +- Assignment5_211.fdb_latexmk | 34 +++--- Assignment5_211.fls | 58 +++++---- Assignment5_211.log | 172 ++++++++++++++------------ Assignment5_211.pdf | Bin 713525 -> 1128302 bytes Assignment5_211.synctex.gz | Bin 15239 -> 20559 bytes Assignment5_211.tex | 206 +++++++++++++++----------------- Assignment5_211/assignment5.2.m | 43 +++++++ Assignment5_211/assignment5.4.m | 47 ++++++++ fig/fig1.1.png | Bin 0 -> 71116 bytes fig/fig1.2.png | Bin 0 -> 32591 bytes fig/fig1.3.png | Bin 0 -> 228413 bytes fig/fig1.4.png | Bin 0 -> 329238 bytes fig/fig2.1.png | Bin 0 -> 118408 bytes fig/fig3.1.png | Bin 0 -> 77730 bytes fig/fig3.2.png | Bin 0 -> 72906 bytes fig/fig4.1.png | Bin 0 -> 55721 bytes fig/fig4.2.png | Bin 0 -> 86203 bytes fig/fig4.3.png | Bin 0 -> 67818 bytes 19 files changed, 340 insertions(+), 222 deletions(-) create mode 100644 Assignment5_211/assignment5.2.m create mode 100644 Assignment5_211/assignment5.4.m create mode 100644 fig/fig1.1.png create mode 100644 fig/fig1.2.png create mode 100644 fig/fig1.3.png create mode 100644 fig/fig1.4.png create mode 100644 fig/fig2.1.png create mode 100644 fig/fig3.1.png create mode 100644 fig/fig3.2.png create mode 100644 fig/fig4.1.png create mode 100644 fig/fig4.2.png create mode 100644 fig/fig4.3.png diff --git a/Assignment5_211.aux b/Assignment5_211.aux index eb30ef6..a9f6350 100644 --- a/Assignment5_211.aux +++ b/Assignment5_211.aux @@ -4,4 +4,4 @@ \catcode `"\active \babel@aux{english}{} \babel@aux{austrian}{} -\gdef \@abspage@last{6} +\gdef \@abspage@last{9} diff --git a/Assignment5_211.fdb_latexmk b/Assignment5_211.fdb_latexmk index 72090ce..f4b977b 100644 --- a/Assignment5_211.fdb_latexmk +++ b/Assignment5_211.fdb_latexmk @@ -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) diff --git a/Assignment5_211.fls b/Assignment5_211.fls index d6c5197..c489598 100644 --- a/Assignment5_211.fls +++ b/Assignment5_211.fls @@ -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 diff --git a/Assignment5_211.log b/Assignment5_211.log index c73830c..5dd0efe 100644 --- a/Assignment5_211.log +++ b/Assignment5_211.log @@ -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. 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\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} + \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] \\ + \\ + Y(z) &= X(z) - \frac{1}{15} z^{-1} Y(z) + \frac{2}{5} z^{-2} Y(z) \\ + \\ + 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*} + 0 = {z^2 + \frac{1}{15} z - \frac{2}{5}} \\ + z_1 = 0.6 \\ + z_2 = -0.6\overline{6} + \end{gather*} + 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 : \\ - \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*} + \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*} - Calculation of $x_1[n]$: + This was for the $z^{+i}$ case. Now follows the $z^{-i}$ case. - \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} + \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} diff --git a/Assignment5_211/assignment5.2.m b/Assignment5_211/assignment5.2.m new file mode 100644 index 0000000..cc9e9d6 --- /dev/null +++ b/Assignment5_211/assignment5.2.m @@ -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') diff --git a/Assignment5_211/assignment5.4.m b/Assignment5_211/assignment5.4.m new file mode 100644 index 0000000..89c0009 --- /dev/null +++ b/Assignment5_211/assignment5.4.m @@ -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 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