Syncing to work on interview with entrepreneur essay
This commit is contained in:
Aidan Sharpe
Binary file not shown.
|
Before Width: | Height: | Size: 37 KiB After Width: | Height: | Size: 20 KiB |
BIN
6th-Semester-Spring-2024/DSP/Labs/Lab-03/abs_sum_nanun-1.png
Normal file
BIN
6th-Semester-Spring-2024/DSP/Labs/Lab-03/abs_sum_nanun-1.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 25 KiB |
@ -1,9 +1,45 @@
|
||||
\relax
|
||||
\@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{1}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {section}{\numberline {2}Results \& Discussion}{1}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {subsection}{\numberline {2.1}Uniform Convergence}{1}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {section}{\numberline {1}Results \& Discussion}{1}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {subsection}{\numberline {1.1}Simple Example of Uniform Convergence}{1}{}\protected@file@percent }
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces The signal value of $x[n]=a^n u[n]$ and its absolute sum\relax }}{1}{}\protected@file@percent }
|
||||
\providecommand*\caption@xref[2]{\@setref\relax\@undefined{#1}}
|
||||
\newlabel{fig:abs_sum_anun}{{1}{1}}
|
||||
\@writefile{toc}{\contentsline {section}{\numberline {3}Conclusions}{1}{}\protected@file@percent }
|
||||
\gdef \@abspage@last{2}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces The DTFT of $x[n] = a^n u[n]$\relax }}{2}{}\protected@file@percent }
|
||||
\newlabel{fig:dtft_anun}{{2}{2}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces The truncated DTFT of $x[n] = a^n u[n]$ for various values of $K$\relax }}{2}{}\protected@file@percent }
|
||||
\newlabel{fig:truncated_DTFTs_anun}{{3}{2}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {4}{\ignorespaces The maximum error of the truncated DTFT of $x[n] = a^n u[n]$ for various values of $K$\relax }}{3}{}\protected@file@percent }
|
||||
\newlabel{fig:max_error_anun}{{4}{3}}
|
||||
\@writefile{toc}{\contentsline {subsection}{\numberline {1.2}More Complex Example of Uniform Convergence}{3}{}\protected@file@percent }
|
||||
\newlabel{eqn:n_geometric_series}{{4}{3}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {5}{\ignorespaces The signal value of $x[n] = n a^n u[n-1]$ and its absolute sum\relax }}{3}{}\protected@file@percent }
|
||||
\newlabel{fig:abs_sum_nanun-1}{{5}{3}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces Varying levels of approximation of the DTFT of $x[n] = n a^n u[n-1]$\relax }}{4}{}\protected@file@percent }
|
||||
\newlabel{fig:truncated_dtfts_nanun-1}{{6}{4}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces The maximum error of the truncated DTFT of $x[n] = n a^n u[n-1]$ for increasing values of $K$\relax }}{5}{}\protected@file@percent }
|
||||
\newlabel{fig:max_error_nanun-1}{{7}{5}}
|
||||
\@writefile{toc}{\contentsline {subsection}{\numberline {1.3}The Inverse DTFT}{5}{}\protected@file@percent }
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {8}{\ignorespaces The frequency response of an ideal lowpass filter\relax }}{6}{}\protected@file@percent }
|
||||
\newlabel{fig:ideal_lowpass_fresponse}{{8}{6}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {9}{\ignorespaces The inverse DTFT of an ideal lowpass filter\relax }}{7}{}\protected@file@percent }
|
||||
\newlabel{fig:recovered_lowpass}{{9}{7}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {10}{\ignorespaces Finite sum approximations of an ideal lowpass filter with varying bounds of summation\relax }}{7}{}\protected@file@percent }
|
||||
\newlabel{fig:truncated_DTFTs_lowpass}{{10}{7}}
|
||||
\@writefile{toc}{\contentsline {subsection}{\numberline {1.4}Convolutions: Pulse Response}{7}{}\protected@file@percent }
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {11}{\ignorespaces The response of $x[n] = u[n] - u[n-2]$ to an LTI system\relax }}{8}{}\protected@file@percent }
|
||||
\newlabel{fig:number4}{{11}{8}}
|
||||
\@writefile{toc}{\contentsline {subsection}{\numberline {1.5}Convolutions: Exponential Decay Response}{8}{}\protected@file@percent }
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {12}{\ignorespaces Convolution of $x[n] = 0.95^n u[n]$ with $h[n]$\relax }}{9}{}\protected@file@percent }
|
||||
\newlabel{fig:number5a}{{12}{9}}
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {13}{\ignorespaces Step response of an LTI system\relax }}{10}{}\protected@file@percent }
|
||||
\newlabel{fig:number5b}{{13}{10}}
|
||||
\@writefile{toc}{\contentsline {subsection}{\numberline {1.6}Correlation Video Summaries}{10}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {subsection}{\numberline {1.7}Crosscorrelation}{11}{}\protected@file@percent }
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {14}{\ignorespaces Cross correlation of $x[n]$ and $h[n]$\relax }}{12}{}\protected@file@percent }
|
||||
\newlabel{fig:number7}{{14}{12}}
|
||||
\@writefile{toc}{\contentsline {subsection}{\numberline {1.8}Autocorrelation}{12}{}\protected@file@percent }
|
||||
\@writefile{lof}{\contentsline {figure}{\numberline {15}{\ignorespaces Autocorrelation of $x[n] = [1,3,-2,4]$\relax }}{13}{}\protected@file@percent }
|
||||
\newlabel{fig:number8}{{15}{13}}
|
||||
\@writefile{toc}{\contentsline {subsection}{\numberline {1.9}Crosscorrelation and Autocorrelation}{13}{}\protected@file@percent }
|
||||
\@writefile{toc}{\contentsline {section}{\numberline {2}Conclusions}{13}{}\protected@file@percent }
|
||||
\gdef \@abspage@last{15}
|
||||
|
||||
@ -1,5 +1,5 @@
|
||||
# Fdb version 4
|
||||
["pdflatex"] 1710771429.52682 "dsp-lab-03.tex" "dsp-lab-03.pdf" "dsp-lab-03" 1710771430.37106 0
|
||||
["pdflatex"] 1711165609.71406 "dsp-lab-03.tex" "dsp-lab-03.pdf" "dsp-lab-03" 1711165610.4111 0
|
||||
"/usr/share/texlive/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 ""
|
||||
"/usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex7.tfm" 1246382020 1004 54797486969f23fa377b128694d548df ""
|
||||
"/usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/cmextra/cmex8.tfm" 1246382020 988 bdf658c3bfc2d96d3c8b02cfc1c94c20 ""
|
||||
@ -10,7 +10,6 @@
|
||||
"/usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm5.tfm" 1246382020 940 75ac932a52f80982a9f8ea75d03a34cf ""
|
||||
"/usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm7.tfm" 1246382020 940 228d6584342e91276bf566bcf9716b83 ""
|
||||
"/usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmbx12.tfm" 1136768653 1324 c910af8c371558dc20f2d7822f66fe64 ""
|
||||
"/usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmbx9.tfm" 1136768653 1328 5442e22a7072966dbaf88ca900acf3f0 ""
|
||||
"/usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmex10.tfm" 1136768653 992 662f679a0b3d2d53c1b94050fdaa3f50 ""
|
||||
"/usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmmi12.tfm" 1136768653 1524 4414a8315f39513458b80dfc63bff03a ""
|
||||
"/usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmmi6.tfm" 1136768653 1512 f21f83efb36853c0b70002322c1ab3ad ""
|
||||
@ -19,17 +18,17 @@
|
||||
"/usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmr17.tfm" 1136768653 1292 296a67155bdbfc32aa9c636f21e91433 ""
|
||||
"/usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmr6.tfm" 1136768653 1300 b62933e007d01cfd073f79b963c01526 ""
|
||||
"/usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmr8.tfm" 1136768653 1292 21c1c5bfeaebccffdb478fd231a0997d ""
|
||||
"/usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmr9.tfm" 1136768653 1292 6b21b9c2c7bebb38aa2273f7ca0fb3af ""
|
||||
"/usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmsy10.tfm" 1136768653 1124 6c73e740cf17375f03eec0ee63599741 ""
|
||||
"/usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmsy6.tfm" 1136768653 1116 933a60c408fc0a863a92debe84b2d294 ""
|
||||
"/usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmsy8.tfm" 1136768653 1120 8b7d695260f3cff42e636090a8002094 ""
|
||||
"/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx12.pfb" 1248133631 32080 340ef9bf63678554ee606688e7b5339d ""
|
||||
"/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx9.pfb" 1248133631 32298 c6d25bb16d1eac01ebdc6d7084126a1e ""
|
||||
"/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb" 1248133631 30251 6afa5cb1d0204815a708a080681d4674 ""
|
||||
"/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb" 1248133631 36299 5f9df58c2139e7edcf37c8fca4bd384d ""
|
||||
"/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.pfb" 1248133631 37912 77d683123f92148345f3fc36a38d9ab1 ""
|
||||
"/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb" 1248133631 36281 c355509802a035cadc5f15869451dcee ""
|
||||
"/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb" 1248133631 35752 024fb6c41858982481f6968b5fc26508 ""
|
||||
"/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr12.pfb" 1248133631 32722 d7379af29a190c3f453aba36302ff5a9 ""
|
||||
"/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr17.pfb" 1248133631 32362 179c33bbf43f19adbb3825bb4e36e57a ""
|
||||
"/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb" 1248133631 32762 224316ccc9ad3ca0423a14971cfa7fc1 ""
|
||||
"/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb" 1248133631 32569 5e5ddc8df908dea60932f3c484a54c0d ""
|
||||
"/usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb" 1248133631 32716 08e384dc442464e7285e891af9f45947 ""
|
||||
@ -67,9 +66,23 @@
|
||||
"/usr/share/texlive/texmf-dist/web2c/texmf.cnf" 1684972800 40326 aba987258e6d6b6da5dec3a727ea174b ""
|
||||
"/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map" 1695741510.34335 4547801 12750a1f41d88f5207b57129561a9960 ""
|
||||
"/var/lib/texmf/web2c/pdftex/pdflatex.fmt" 1710008668 7862952 9b0773ccc0e23d0927f5089e19dad72c ""
|
||||
"abs_sum_anun.png" 1710771266.33299 37956 60ebbb3c6d1ad8738c8d690f36c9fb74 ""
|
||||
"dsp-lab-03.aux" 1710771430.28467 718 5c33022ef6ba7c7aef855e50dc2a6508 "pdflatex"
|
||||
"dsp-lab-03.tex" 1710771426.85566 1933 fa3010c7f6ea1a21f805136c3562c49f ""
|
||||
"abs_sum_anun.png" 1711126439.38711 20154 753aaa0a4e80f2b9af930c92f00c8c82 ""
|
||||
"abs_sum_nanun-1.png" 1711126497.81334 25229 689877851eb9866fea81a91a2affb998 ""
|
||||
"dsp-lab-03.aux" 1711165610.35571 4529 baca7a985437a6c0170d98ca2446d701 "pdflatex"
|
||||
"dsp-lab-03.tex" 1711165608.51071 14906 462e6456bce911a027bd88e231b97d72 ""
|
||||
"dtft_anun.png" 1711126454.79217 19025 0e48fe92a04ea50dc9a6dce7470a3c83 ""
|
||||
"ideal_lowpass_fresponse.png" 1711126575.88965 15024 654c4eede7194b62075c04a6a02430d5 ""
|
||||
"max_error_anun.png" 1711126468.34922 16679 edd03290a09e08385b6c8ae9057fb2f3 ""
|
||||
"max_error_nanun-1.png" 1711127851.51682 17724 5b043ae1bece942d7b1ae0059c6a80c3 ""
|
||||
"num4.png" 1711155764.25805 11264 11e6a652ed7e2d096a11fad7aa9bccf5 ""
|
||||
"num5a.png" 1711155859.68335 23945 1a9c7256aaa5706454faec9583cc548b ""
|
||||
"num5b.png" 1711155881.71842 35300 9dba3c67f88ff694ee7a919375dc8202 ""
|
||||
"num7.png" 1711155787.38012 26817 e3954576558130401034391f390dd59f ""
|
||||
"num8real.png" 1711155843.8023 27648 cfbd73982c009413e761d3f8aef0bfff ""
|
||||
"recovered_lowpass_signal.png" 1711126566.13661 15886 211ae06161932eafbb9330f27d7551ab ""
|
||||
"truncated_DTFTs_anun.png" 1711126462.1062 46985 f9c04f8f91f10b6c057c1ae5806fcd7e ""
|
||||
"truncated_DTFTs_lowpass.png" 1711126582.65168 54154 b5180ed7dda429b34e185937a0c09730 ""
|
||||
"truncated_DTFTs_nanun.png" 1711126512.4204 46020 ac444e92d104c163824dece5a7295ded ""
|
||||
(generated)
|
||||
"dsp-lab-03.aux"
|
||||
"dsp-lab-03.log"
|
||||
|
||||
@ -305,8 +305,6 @@ INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm10.tfm
|
||||
INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/amsfonts/symbols/msbm7.tfm
|
||||
OUTPUT dsp-lab-03.pdf
|
||||
INPUT /var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map
|
||||
INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmr9.tfm
|
||||
INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmbx9.tfm
|
||||
INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmr12.tfm
|
||||
INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmbx12.tfm
|
||||
INPUT /usr/share/texlive/texmf-dist/fonts/tfm/public/cm/cmbx12.tfm
|
||||
@ -323,14 +321,85 @@ INPUT ./abs_sum_anun.png
|
||||
INPUT abs_sum_anun.png
|
||||
INPUT ./abs_sum_anun.png
|
||||
INPUT ./abs_sum_anun.png
|
||||
INPUT ./dtft_anun.png
|
||||
INPUT ./dtft_anun.png
|
||||
INPUT dtft_anun.png
|
||||
INPUT ./dtft_anun.png
|
||||
INPUT ./dtft_anun.png
|
||||
INPUT ./truncated_DTFTs_anun.png
|
||||
INPUT ./truncated_DTFTs_anun.png
|
||||
INPUT truncated_DTFTs_anun.png
|
||||
INPUT ./truncated_DTFTs_anun.png
|
||||
INPUT ./truncated_DTFTs_anun.png
|
||||
INPUT ./max_error_anun.png
|
||||
INPUT ./max_error_anun.png
|
||||
INPUT max_error_anun.png
|
||||
INPUT ./max_error_anun.png
|
||||
INPUT ./max_error_anun.png
|
||||
INPUT ./abs_sum_nanun-1.png
|
||||
INPUT ./abs_sum_nanun-1.png
|
||||
INPUT abs_sum_nanun-1.png
|
||||
INPUT ./abs_sum_nanun-1.png
|
||||
INPUT ./abs_sum_nanun-1.png
|
||||
INPUT ./truncated_DTFTs_nanun.png
|
||||
INPUT ./truncated_DTFTs_nanun.png
|
||||
INPUT truncated_DTFTs_nanun.png
|
||||
INPUT ./truncated_DTFTs_nanun.png
|
||||
INPUT ./truncated_DTFTs_nanun.png
|
||||
INPUT ./max_error_nanun-1.png
|
||||
INPUT ./max_error_nanun-1.png
|
||||
INPUT max_error_nanun-1.png
|
||||
INPUT ./max_error_nanun-1.png
|
||||
INPUT ./max_error_nanun-1.png
|
||||
INPUT ./ideal_lowpass_fresponse.png
|
||||
INPUT ./ideal_lowpass_fresponse.png
|
||||
INPUT ideal_lowpass_fresponse.png
|
||||
INPUT ./ideal_lowpass_fresponse.png
|
||||
INPUT ./ideal_lowpass_fresponse.png
|
||||
INPUT ./recovered_lowpass_signal.png
|
||||
INPUT ./recovered_lowpass_signal.png
|
||||
INPUT recovered_lowpass_signal.png
|
||||
INPUT ./recovered_lowpass_signal.png
|
||||
INPUT ./recovered_lowpass_signal.png
|
||||
INPUT ./truncated_DTFTs_lowpass.png
|
||||
INPUT ./truncated_DTFTs_lowpass.png
|
||||
INPUT truncated_DTFTs_lowpass.png
|
||||
INPUT ./truncated_DTFTs_lowpass.png
|
||||
INPUT ./truncated_DTFTs_lowpass.png
|
||||
INPUT ./num4.png
|
||||
INPUT ./num4.png
|
||||
INPUT num4.png
|
||||
INPUT ./num4.png
|
||||
INPUT ./num4.png
|
||||
INPUT ./num5a.png
|
||||
INPUT ./num5a.png
|
||||
INPUT num5a.png
|
||||
INPUT ./num5a.png
|
||||
INPUT ./num5a.png
|
||||
INPUT ./num5b.png
|
||||
INPUT ./num5b.png
|
||||
INPUT num5b.png
|
||||
INPUT ./num5b.png
|
||||
INPUT ./num5b.png
|
||||
INPUT ./num7.png
|
||||
INPUT ./num7.png
|
||||
INPUT num7.png
|
||||
INPUT ./num7.png
|
||||
INPUT ./num7.png
|
||||
INPUT ./num8real.png
|
||||
INPUT ./num8real.png
|
||||
INPUT num8real.png
|
||||
INPUT ./num8real.png
|
||||
INPUT ./num8real.png
|
||||
INPUT dsp-lab-03.aux
|
||||
INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx12.pfb
|
||||
INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx9.pfb
|
||||
INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb
|
||||
INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb
|
||||
INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.pfb
|
||||
INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb
|
||||
INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb
|
||||
INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr12.pfb
|
||||
INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr17.pfb
|
||||
INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb
|
||||
INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb
|
||||
INPUT /usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb
|
||||
|
||||
@ -1,4 +1,4 @@
|
||||
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022/CVE-2023-32700 patched) (preloaded format=pdflatex 2024.3.9) 18 MAR 2024 10:17
|
||||
This is pdfTeX, Version 3.141592653-2.6-1.40.24 (TeX Live 2022/CVE-2023-32700 patched) (preloaded format=pdflatex 2024.3.9) 22 MAR 2024 23:46
|
||||
entering extended mode
|
||||
restricted \write18 enabled.
|
||||
file:line:error style messages enabled.
|
||||
@ -269,39 +269,147 @@ File: umsb.fd 2013/01/14 v3.01 AMS symbols B
|
||||
) [1
|
||||
|
||||
{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}]
|
||||
Underfull \hbox (badness 10000) in paragraph at lines 32--39
|
||||
Underfull \hbox (badness 10000) in paragraph at lines 27--34
|
||||
|
||||
[]
|
||||
|
||||
|
||||
Underfull \hbox (badness 10000) in paragraph at lines 32--39
|
||||
Underfull \hbox (badness 10000) in paragraph at lines 27--34
|
||||
|
||||
[]
|
||||
|
||||
|
||||
Package amsmath Warning: Foreign command \over;
|
||||
(amsmath) \frac or \genfrac should be used instead
|
||||
(amsmath) on input line 40.
|
||||
(amsmath) on input line 35.
|
||||
|
||||
<abs_sum_anun.png, id=8, 1373.13pt x 708.9687pt>
|
||||
<abs_sum_anun.png, id=9, 682.2288pt x 331.7193pt>
|
||||
File: abs_sum_anun.png Graphic file (type png)
|
||||
<use abs_sum_anun.png>
|
||||
Package pdftex.def Info: abs_sum_anun.png used on input line 46.
|
||||
(pdftex.def) Requested size: 469.75502pt x 242.53905pt.
|
||||
[1 <./abs_sum_anun.png>] (./dsp-lab-03.aux) )
|
||||
Here is how much of TeX's memory you used:
|
||||
5560 strings out of 476182
|
||||
90295 string characters out of 5796582
|
||||
1855793 words of memory out of 6000000
|
||||
25834 multiletter control sequences out of 15000+600000
|
||||
520449 words of font info for 64 fonts, out of 8000000 for 9000
|
||||
1137 hyphenation exceptions out of 8191
|
||||
55i,6n,63p,483b,336s stack positions out of 10000i,1000n,20000p,200000b,200000s
|
||||
</usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx12.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx9.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr12.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/symbols/msbm10.pfb>
|
||||
Output written on dsp-lab-03.pdf (2 pages, 150885 bytes).
|
||||
PDF statistics:
|
||||
69 PDF objects out of 1000 (max. 8388607)
|
||||
40 compressed objects within 1 object stream
|
||||
0 named destinations out of 1000 (max. 500000)
|
||||
6 words of extra memory for PDF output out of 10000 (max. 10000000)
|
||||
Package pdftex.def Info: abs_sum_anun.png used on input line 41.
|
||||
(pdftex.def) Requested size: 469.75502pt x 228.41075pt.
|
||||
<dtft_anun.png, id=11, 682.2288pt x 331.7193pt>
|
||||
File: dtft_anun.png Graphic file (type png)
|
||||
<use dtft_anun.png>
|
||||
Package pdftex.def Info: dtft_anun.png used on input line 54.
|
||||
(pdftex.def) Requested size: 469.75502pt x 228.41075pt.
|
||||
[1 <./abs_sum_anun.png>]
|
||||
<truncated_DTFTs_anun.png, id=24, 682.2288pt x 331.7193pt>
|
||||
File: truncated_DTFTs_anun.png Graphic file (type png)
|
||||
<use truncated_DTFTs_anun.png>
|
||||
Package pdftex.def Info: truncated_DTFTs_anun.png used on input line 65.
|
||||
(pdftex.def) Requested size: 469.75502pt x 228.41075pt.
|
||||
<max_error_anun.png, id=25, 682.2288pt x 331.7193pt>
|
||||
File: max_error_anun.png Graphic file (type png)
|
||||
<use max_error_anun.png>
|
||||
Package pdftex.def Info: max_error_anun.png used on input line 70.
|
||||
(pdftex.def) Requested size: 469.75502pt x 228.41075pt.
|
||||
[2 <./dtft_anun.png> <./truncated_DTFTs_anun.png>]
|
||||
<abs_sum_nanun-1.png, id=31, 682.2288pt x 331.7193pt>
|
||||
File: abs_sum_nanun-1.png Graphic file (type png)
|
||||
<use abs_sum_nanun-1.png>
|
||||
Package pdftex.def Info: abs_sum_nanun-1.png used on input line 84.
|
||||
(pdftex.def) Requested size: 469.75502pt x 228.41075pt.
|
||||
|
||||
Underfull \hbox (badness 10000) in paragraph at lines 89--94
|
||||
|
||||
[]
|
||||
|
||||
[3 <./max_error_anun.png> <./abs_sum_nanun-1.png>]
|
||||
Underfull \hbox (badness 10000) in paragraph at lines 96--100
|
||||
|
||||
[]
|
||||
|
||||
<truncated_DTFTs_nanun.png, id=37, 682.2288pt x 331.7193pt>
|
||||
File: truncated_DTFTs_nanun.png Graphic file (type png)
|
||||
<use truncated_DTFTs_nanun.png>
|
||||
Package pdftex.def Info: truncated_DTFTs_nanun.png used on input line 105.
|
||||
(pdftex.def) Requested size: 469.75502pt x 228.41075pt.
|
||||
<max_error_nanun-1.png, id=38, 682.2288pt x 331.7193pt>
|
||||
File: max_error_nanun-1.png Graphic file (type png)
|
||||
<use max_error_nanun-1.png>
|
||||
Package pdftex.def Info: max_error_nanun-1.png used on input line 111.
|
||||
(pdftex.def) Requested size: 469.75502pt x 228.41075pt.
|
||||
[4 <./truncated_DTFTs_nanun.png>]
|
||||
<ideal_lowpass_fresponse.png, id=43, 682.2288pt x 331.7193pt>
|
||||
File: ideal_lowpass_fresponse.png Graphic file (type png)
|
||||
<use ideal_lowpass_fresponse.png>
|
||||
Package pdftex.def Info: ideal_lowpass_fresponse.png used on input line 123.
|
||||
(pdftex.def) Requested size: 469.75502pt x 228.41075pt.
|
||||
[5 <./max_error_nanun-1.png>]
|
||||
<recovered_lowpass_signal.png, id=48, 682.2288pt x 331.7193pt>
|
||||
File: recovered_lowpass_signal.png Graphic file (type png)
|
||||
<use recovered_lowpass_signal.png>
|
||||
Package pdftex.def Info: recovered_lowpass_signal.png used on input line 145.
|
||||
(pdftex.def) Requested size: 469.75502pt x 228.41075pt.
|
||||
[6 <./ideal_lowpass_fresponse.png>]
|
||||
<truncated_DTFTs_lowpass.png, id=55, 682.2288pt x 331.7193pt>
|
||||
File: truncated_DTFTs_lowpass.png Graphic file (type png)
|
||||
<use truncated_DTFTs_lowpass.png>
|
||||
Package pdftex.def Info: truncated_DTFTs_lowpass.png used on input line 152.
|
||||
(pdftex.def) Requested size: 469.75502pt x 228.41075pt.
|
||||
[7 <./recovered_lowpass_signal.png> <./truncated_DTFTs_lowpass.png>]
|
||||
<num4.png, id=61, 526.96875pt x 395.22656pt>
|
||||
File: num4.png Graphic file (type png)
|
||||
<use num4.png>
|
||||
Package pdftex.def Info: num4.png used on input line 171.
|
||||
(pdftex.def) Requested size: 469.75502pt x 352.3355pt.
|
||||
<num5a.png, id=62, 526.96875pt x 395.22656pt>
|
||||
File: num5a.png Graphic file (type png)
|
||||
<use num5a.png>
|
||||
Package pdftex.def Info: num5a.png used on input line 181.
|
||||
(pdftex.def) Requested size: 469.75502pt x 352.3355pt.
|
||||
[8 <./num4.png (PNG copy)>]
|
||||
<num5b.png, id=66, 352.1958pt x 277.035pt>
|
||||
File: num5b.png Graphic file (type png)
|
||||
<use num5b.png>
|
||||
Package pdftex.def Info: num5b.png used on input line 188.
|
||||
(pdftex.def) Requested size: 469.75502pt x 369.51294pt.
|
||||
[9 <./num5a.png (PNG copy)>]
|
||||
Underfull \hbox (badness 10000) in paragraph at lines 195--202
|
||||
|
||||
[]
|
||||
|
||||
|
||||
Underfull \hbox (badness 10000) in paragraph at lines 195--202
|
||||
|
||||
[]
|
||||
|
||||
[10 <./num5b.png (PNG copy)>]
|
||||
Underfull \hbox (badness 10000) in paragraph at lines 207--211
|
||||
|
||||
[]
|
||||
|
||||
<num7.png, id=73, 344.487pt x 288.1164pt>
|
||||
File: num7.png Graphic file (type png)
|
||||
<use num7.png>
|
||||
Package pdftex.def Info: num7.png used on input line 216.
|
||||
(pdftex.def) Requested size: 469.75502pt x 392.90579pt.
|
||||
[11]
|
||||
<num8real.png, id=77, 344.487pt x 288.1164pt>
|
||||
File: num8real.png Graphic file (type png)
|
||||
<use num8real.png>
|
||||
Package pdftex.def Info: num8real.png used on input line 229.
|
||||
(pdftex.def) Requested size: 469.75502pt x 392.90579pt.
|
||||
[12 <./num7.png (PNG copy)>]
|
||||
Underfull \hbox (badness 10000) in paragraph at lines 234--238
|
||||
|
||||
[]
|
||||
|
||||
[13 <./num8real.png (PNG copy)>] [14] (./dsp-lab-03.aux) )
|
||||
Here is how much of TeX's memory you used:
|
||||
5670 strings out of 476182
|
||||
92745 string characters out of 5796582
|
||||
1856793 words of memory out of 6000000
|
||||
25932 multiletter control sequences out of 15000+600000
|
||||
519842 words of font info for 62 fonts, out of 8000000 for 9000
|
||||
1137 hyphenation exceptions out of 8191
|
||||
55i,8n,63p,965b,332s stack positions out of 10000i,1000n,20000p,200000b,200000s
|
||||
</usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmbx12.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmex10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi5.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmmi7.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr12.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr17.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmr7.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy10.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/cm/cmsy7.pfb></usr/share/texlive/texmf-dist/fonts/type1/public/amsfonts/symbols/msbm10.pfb>
|
||||
Output written on dsp-lab-03.pdf (15 pages, 506420 bytes).
|
||||
PDF statistics:
|
||||
139 PDF objects out of 1000 (max. 8388607)
|
||||
72 compressed objects within 1 object stream
|
||||
0 named destinations out of 1000 (max. 500000)
|
||||
76 words of extra memory for PDF output out of 10000 (max. 10000000)
|
||||
|
||||
|
||||
Binary file not shown.
Binary file not shown.
@ -10,7 +10,7 @@
|
||||
\usepackage[margin=1in]{geometry}
|
||||
|
||||
|
||||
\title{}
|
||||
\title{Uniform Convergence, Convolutions, and Correlation of Signals}
|
||||
|
||||
\author{Aidan Sharpe \& Elise Heim}
|
||||
|
||||
@ -21,15 +21,10 @@
|
||||
\maketitle
|
||||
\end{titlepage}
|
||||
|
||||
\begin{abstract}
|
||||
\end{abstract}
|
||||
|
||||
\section{Introduction}
|
||||
|
||||
\section{Results \& Discussion}
|
||||
|
||||
\subsection{Uniform Convergence}
|
||||
Consider the discrete time signal, $x[n] = a^n u[n]$, where $u[n]$ is the unit-step and $a=0.9$. Let $X(e^{j\omega})$ be the discrete time Fourier transform (DTFT) of $x[n]$. For the DTFT of a signal to exist, it must be absolutely summable over all $n \in \mathbb{Z}$. If the signal is absolutely summable, then it must also be bounded, meaning $|x[n]| < \infty, \forall n \in \mathbb{Z}$. In other words, $x[n]$ must have a finite maximum value.
|
||||
\subsection{Simple Example of Uniform Convergence}
|
||||
Consider the discrete time signal, $x[n] = a^n u[n]$, where $u[n]$ is the unit-step and $a=0.9$. Let $X(e^{j\omega})$ be the discrete time Fourier transform (DTFT) of $x[n]$. For the DTFT of a signal to exist, it must be absolutely summable over all $n \in \mathbb{Z}$. If the signal is absolutely summable, then it must also be bounded, meaning there exists some non-negative real number $B$, such that $|x[n]| \le B, \forall n \in \mathbb{Z}$. At a bare minimum, $x[n]$ must have a finite maximum value.
|
||||
\\
|
||||
\\
|
||||
In this case, $x[n]$ is bounded because $a^n$ grows with smaller values of $n$, and $u[n]$ is zero when $n<0$. Therefore, the maximum value of $x[n] = a^n u[n]$ is $x[0] = a$.
|
||||
@ -41,15 +36,211 @@ Additionally, $x[n]$ takes the form of a geometric series, so its sum is given b
|
||||
\end{equation}
|
||||
In this case, $m=0$, because the signal has no value for $n<0$. Since $a=0.9$, the sum evaluates to is ${1 \over 1-0.9} = 10$. Considering that $x[n]$ is always a positive real number, each term is its own absolute value, so the sum and the absolute sum are equivalent. By taking the absolute sum of the first 200 terms of $x[n]$, it becomes clear that it approaches 10 in the limit, seen in figure \ref{fig:abs_sum_anun}.
|
||||
|
||||
\begin{figure}[h]
|
||||
\begin{figure}[H]
|
||||
\center
|
||||
\includegraphics[width=\textwidth]{abs_sum_anun.png}
|
||||
\caption{The signal value of $x[n]=a^n u[n]$ and its absolute sum}
|
||||
\label{fig:abs_sum_anun}
|
||||
\end{figure}
|
||||
\noindent
|
||||
Since the signal is absolutely summable, the DTFT, $X(e^{j\omega})$ exists, and can be found by evaluating the sum:
|
||||
\begin{equation}
|
||||
X(e^{j\omega}) = \sum_{n=-\infty}^{\infty} x[n] e^{-j\omega n}.
|
||||
\end{equation}
|
||||
Evaluating this sum reveals that
|
||||
$$X(e^{j\omega}) = {1\over 1 - a e^{-j\omega}},$$
|
||||
where $a = 0.9$. The plot for $X(e^{j\omega})$ is seen in figure \ref{fig:dtft_anun}.
|
||||
\begin{figure}[H]
|
||||
\includegraphics[width=\textwidth]{dtft_anun.png}
|
||||
\caption{The DTFT of $x[n] = a^n u[n]$}
|
||||
\label{fig:dtft_anun}
|
||||
\end{figure}
|
||||
\noindent
|
||||
In this case, the DTFT can be easily evalueated because the sum is a geometric series. However, for more complicated signals, the sum can only be approximated. These non-infinite sums are called truncated DTFTs and take the form:
|
||||
\begin{equation}
|
||||
X(e^{j\omega}) = \sum_{-K}^K x[n] e^{-j\omega n}.
|
||||
\end{equation}
|
||||
Seen in figure \ref{fig:truncated_DTFTs_anun}, the approximation seems to be quite poor for small values of $K$, but the accuracy seems to quickly increases for slightly larger values of $K$. The actual performance, quantified by the maximum error for a given value of $K$, lines up with these observations, as seen in figure \ref{fig:max_error_anun}. It also becomes apparent through figure \ref{fig:max_error_anun} that large values of $K$ have diminishing returns.
|
||||
\begin{figure}[H]
|
||||
\includegraphics[width=\textwidth]{truncated_DTFTs_anun.png}
|
||||
\caption{The truncated DTFT of $x[n] = a^n u[n]$ for various values of $K$}
|
||||
\label{fig:truncated_DTFTs_anun}
|
||||
\end{figure}
|
||||
\begin{figure}[H]
|
||||
\includegraphics[width=\textwidth]{max_error_anun.png}
|
||||
\caption{The maximum error of the truncated DTFT of $x[n] = a^n u[n]$ for various values of $K$}
|
||||
\label{fig:max_error_anun}
|
||||
\end{figure}
|
||||
|
||||
\subsection{More Complex Example of Uniform Convergence}
|
||||
Consider the signal $x[n] = n a^n u[n-1]$. Again, to determine if the DTFT of the signal exists, it must be absolutely summable. For $a<1$, its sum is given by:
|
||||
\begin{equation}
|
||||
\sum_{n=1}^\infty n a^n = {a \over (1-a)^2}.
|
||||
\label{eqn:n_geometric_series}
|
||||
\end{equation}
|
||||
For $a=0.9$, the infinite series evaluates to 90, and this result is verified by the sum of the first 200 samples of $x[n]$ seen in figure \ref{fig:abs_sum_nanun-1}.
|
||||
|
||||
\begin{figure}[H]
|
||||
\includegraphics[width=\textwidth]{abs_sum_nanun-1.png}
|
||||
\caption{The signal value of $x[n] = n a^n u[n-1]$ and its absolute sum}
|
||||
\label{fig:abs_sum_nanun-1}
|
||||
\end{figure}
|
||||
|
||||
\noindent
|
||||
The signal, $x[n]$ has a maximum amplitude of 3.4868 at $n=9$ and $n=10$. Therefore, the signal is bounded by 3.4868. It is also clear that the signal must be bounded because it is absolutely summable. If the signal was unbounded, then its absolute sum would also be unbounded.
|
||||
\\
|
||||
\\
|
||||
The DTFT of $x[n]$ can be found using equation \ref{eqn:n_geometric_series}. In this case, the value of $a$ in the equation is $a e^{-j\omega}$. Therefore, the DTFT of $x[n]$ is
|
||||
\begin{equation}
|
||||
X(e^{j\omega}) = \sum_{-\infty}^\infty n a^n u[n-1] e^{-j\omega n} = \sum_1^\infty n\left(a e^{-j\omega} \right)^n = {a e^{-j\omega} \over \left(1 - a e^{-j\omega} \right)^2}.
|
||||
\end{equation}
|
||||
\\
|
||||
\\
|
||||
The truncated DTFT is in this case is a variation on the geometric series of the form $k r^k$. The sum of the first $n$ terms is given by:
|
||||
\begin{equation}
|
||||
\sum_{k=1}^n k r^k = {r - r^{n+2} \over (1-r)^2} - {(n+1) r^{n+1} \over 1 - r}.
|
||||
\end{equation}
|
||||
Applying this formula allows for varying degrees of approximation of the DTFT, as seen in figure \ref{fig:truncated_dtfts_nanun-1}.
|
||||
\begin{figure}[H]
|
||||
\includegraphics[width=\textwidth]{truncated_DTFTs_nanun.png}
|
||||
\caption{Varying levels of approximation of the DTFT of $x[n] = n a^n u[n-1]$}
|
||||
\label{fig:truncated_dtfts_nanun-1}
|
||||
\end{figure}
|
||||
It can clearly be seen that as $K$ gets larger, the approximation of the DTFT becomes increasingly better. By plotting the maximum error for increasing values of $K$, as seen in figure \ref{fig:max_error_nanun-1}, it becomes clear that, once again, there are diminishing returns for larger values.
|
||||
\begin{figure}[H]
|
||||
\includegraphics[width=\textwidth]{max_error_nanun-1.png}
|
||||
\caption{The maximum error of the truncated DTFT of $x[n] = n a^n u[n-1]$ for increasing values of $K$}
|
||||
\label{fig:max_error_nanun-1}
|
||||
\end{figure}
|
||||
|
||||
\subsection{The Inverse DTFT}
|
||||
An ideal lowpass filter, whose frequency response is seen in figure \ref{fig:ideal_lowpass_fresponse}, has unity gain for all frequencies less than the cutoff frequency, $\omega_c$, and zero gain for all frequencies greater than the cutoff frequency. Therefore, the DTFT of a lowpass signal would be a pulse of amplitude one from $-\omega_c$ to $\omega_c$. To recover the corresponding signal, the inverse DTFT should be applied. This operation is defined as:
|
||||
\begin{equation}
|
||||
x[n] = {1\over 2\pi} \int\limits_{-\pi}^{\pi} X(e^{j\omega}) e^{j\omega n} d\omega
|
||||
\end{equation}
|
||||
where $X(e^{j\omega})$ is the DTFT of the signal $x[n]$.
|
||||
\begin{figure}[H]
|
||||
\includegraphics[width=\textwidth]{ideal_lowpass_fresponse.png}
|
||||
\caption{The frequency response of an ideal lowpass filter}
|
||||
\label{fig:ideal_lowpass_fresponse}
|
||||
\end{figure}
|
||||
Considering that $X(e^{j\omega})$ has a value of one between $-\omega_c$ and $\omega_c$ and 0 elsewhere, the integral can be rewritten as:
|
||||
\begin{equation}
|
||||
x[n] = {1\over 2\pi} \int\limits_{-\omega_c}^{\omega_c} e^{j\omega n} d\omega.
|
||||
\end{equation}
|
||||
Evaluating the integral gives:
|
||||
\begin{equation}
|
||||
x[n] = {\omega_c \over \pi} \sinc(\omega_c n),
|
||||
\end{equation}
|
||||
where $\sinc(x)$ is the continuous normalized $\sinc$ function defined as:
|
||||
\begin{equation}
|
||||
\sinc(x) = \begin{cases}
|
||||
1 & x = 0 \\
|
||||
{\sin(\pi x) \over \pi x} & x \ne 0
|
||||
\end{cases}
|
||||
\end{equation}
|
||||
\noindent
|
||||
The plot of $x[n]$, where $\omega_c = 0.4\pi$ is seen in figure \ref{fig:recovered_lowpass}.
|
||||
\begin{figure}[H]
|
||||
\includegraphics[width=\textwidth]{recovered_lowpass_signal.png}
|
||||
\caption{The inverse DTFT of an ideal lowpass filter}
|
||||
\label{fig:recovered_lowpass}
|
||||
\end{figure}
|
||||
\noindent
|
||||
Given this signal, finite sum approximations of the DTFT yield oscillating frequency responses that resemble the ideal case, as seen in figure \ref{fig:truncated_DTFTs_lowpass}. Similar to other approximations of DTFTs, the more terms in the summation, the better the accuracy of the approximation.
|
||||
\begin{figure}[H]
|
||||
\includegraphics[width=\textwidth]{truncated_DTFTs_lowpass.png}
|
||||
\caption{Finite sum approximations of an ideal lowpass filter with varying bounds of summation}
|
||||
\label{fig:truncated_DTFTs_lowpass}
|
||||
\end{figure}
|
||||
|
||||
\subsection{Convolutions: Pulse Response}
|
||||
Consider a linear, time-invariant signal with the impulse response $h[n] = [1, 2, 1]$. If the input of the system is given as $x[n] = u[n] - u[n-2]$ where $u[n]$ is the unit step, the output of the system can be calculated analytically.
|
||||
\begin{equation}
|
||||
y[n] = x[0]h[n] + x[1]h[n-1] = h[n] + h[n-1]
|
||||
\end{equation}
|
||||
$$y[0] = 1 + 0 = 1$$
|
||||
$$y[1] = 2 + 1 = 3$$
|
||||
$$y[2] = 1 + 2 = 3$$
|
||||
$$y[3] = 0 + 1 = 1$$
|
||||
|
||||
\noindent
|
||||
With this in mind, one can use MATLAB to check and confirm the answer, as shown below in figure \ref{fig:number4}.
|
||||
|
||||
\begin{figure}[H]
|
||||
\includegraphics[width=\textwidth]{num4.png}
|
||||
\caption{The response of $x[n] = u[n] - u[n-2]$ to an LTI system}
|
||||
\label{fig:number4}
|
||||
\end{figure}
|
||||
|
||||
\subsection{Convolutions: Exponential Decay Response}
|
||||
Consider a linear, time-invariant system with the impulse response $h[n] = u[n]$, where $u[n]$ is the unit step. If the input to the system is an expression for $y[n]$ can be calculated, as shown below.
|
||||
$$y[n] = \sum_{k=0}^{99} x[k] h[n-k]$$
|
||||
The plot for $y[n]$ is seen in figure \ref{fig:number5a}.
|
||||
\begin{figure}[H]
|
||||
\includegraphics[width=\textwidth]{num5a.png}
|
||||
\caption{Convolution of $x[n] = 0.95^n u[n]$ with $h[n]$}
|
||||
\label{fig:number5a}
|
||||
\end{figure}
|
||||
|
||||
The unit step response occurs when $x[n] = u[n]$, and it is seen in figure \ref{fig:number5b}.
|
||||
\begin{figure}[H]
|
||||
\includegraphics[width=\textwidth]{num5b.png}
|
||||
\caption{Step response of an LTI system}
|
||||
\label{fig:number5b}
|
||||
\end{figure}
|
||||
This system is BIBO stable because its impulse response is a bounded signal.
|
||||
|
||||
\subsection{Correlation Video Summaries}
|
||||
The first video discusses correlation. As mentioned by the video, “correlation is a measure of how similar signals are.” The video provides examples of three different signals: x, y, and z. It also displays a formula for calculating a correlation measurement. A greater correlation measurement between signals with similar energy means that they are correlated, however the measurement tells no information on how similar or different two signals are when the signals have significantly different energy. As demonstrated by altering the signals, the correlation measurement on its own does not provide an accurate representation of correlation. This is why it is necessary to use normalized correlation.
|
||||
\\
|
||||
\\
|
||||
In the second video, a new formula is presented to determine the normalized correlation. This takes our original equation and divides it by the square of the sum of the energies of the two signals being compared. The denominator is a scaling factor comprised of the energy of the signals being compared. Instead of taking the form of an integer, like a regular correlation, the resulting normalized correlation now takes the form of a value between -1 and 1. The more similar the signals, the greater the value. With this in mind, it might seem as though calculating correlation without normalizing would be trivial. However, through a demonstration with a MATLAB script, the value of non-normalized correlation is displayed. By calculating the correlation of multiple signals, it is easy to determine that some signals are twice as strongly present as other signals. However, the same information cannot be gleaned from the normalized correlation.
|
||||
\\
|
||||
\\
|
||||
The third and final video explains crosscorrelation, which is a measure of similarity between signals at different time lag positions. It introduces two signals, which are displayed such that there is zero lag. This means that the first, second, third, etc. samples are aligned vertically with each other. In order to calculate the correlation, it is necessary to multiply the vertically aligned samples and sum them all. At this zero lag position, a correlation value is calculated. If sample number zero of the first signal is aligned vertically with sample number one of the second signal, a lag of one is introduced. In this way, the vertically aligned samples can be multiplied then summed, and another correlation value can be determined. For each lag position, including negative lags, a new correlation value can be calculated to create a correlation sequence. Such a sequence can be generated and plotted in MATLAB.
|
||||
|
||||
\subsection{Crosscorrelation}
|
||||
Crosscorrelation between two real signals $x[n]$ and $h[n]$ at different lags $n$ is defined as:
|
||||
\begin{equation}
|
||||
c_{xh}[n] = \sum_k x[k] h[k-n]
|
||||
\end{equation}
|
||||
\\
|
||||
\\
|
||||
If $x[n] = [1 3 -2 4]$ and $h[n] = [2, 3, -1, 3]$, the crosscorrelation can be computed for $c_{xh}[0]$, $c_{xh}[1]$, and $c_{xh}[-1]$, as shown below.
|
||||
$$c_{xh}[0] = 25$$
|
||||
$$c_{xh}[1] = -4$$
|
||||
$$c_{xh}[-1] = -6$$
|
||||
The relationship of crosscorrelation versus lag can be plotted, as demonstrated by figure \ref{fig:number7}.
|
||||
\begin{figure}[H]
|
||||
\includegraphics[width=\textwidth]{num7.png}
|
||||
\caption{Cross correlation of $x[n]$ and $h[n]$}
|
||||
\label{fig:number7}
|
||||
\end{figure}
|
||||
|
||||
|
||||
\subsection{Autocorrelation}
|
||||
The autocorrelation of a real signal $x[n]$ is defined as:
|
||||
\begin{equation}
|
||||
c_{xx}[n] = \sum_k x[k] x[k-n]
|
||||
\end{equation}
|
||||
which is the crosscorrelation of a signal with itself. For example, the autocorrelation of $x[n] = [1, 3, -2, 4]$ is seen in figure \ref{fig:number8}.
|
||||
\begin{figure}[H]
|
||||
\includegraphics[width=\textwidth]{num8real.png}
|
||||
\caption{Autocorrelation of $x[n] = [1,3,-2,4]$}
|
||||
\label{fig:number8}
|
||||
\end{figure}
|
||||
\subsection{Crosscorrelation and Autocorrelation}
|
||||
Given the signal $x[n] = \begin{bmatrix}1&1&1&1&1&1&1&1\end{bmatrix}$, its autocorrelation at zero lag is 1.5.
|
||||
\\
|
||||
\\
|
||||
Consider the signals $v[n] = 2x[n]$, $w[n] = -x[n]$, and $y[n] = \begin{bmatrix}1&1&-1&-1&1&1&-1&-1\end{bmatrix}$.
|
||||
$$c_{xv}[0] = 3$$
|
||||
$$c_{xw}[0] = -1.5$$
|
||||
$$c_{xy}[0] = 1$$
|
||||
Since $w[n] = -x[n]$, it makes sense that the crosscorrelation of $x[n]$ and $w[n]$ is the negation of the autocorrelation of $x[n]$.
|
||||
|
||||
\section{Conclusions}
|
||||
Multiple concepts were reinforced through this lab. Discrete-Time Fourier Transformations (DTFT), convolutions, as well as correlation were explored through a variety of exercises. This lab showed how the inverse DTFT can be utilized to model a filter, as well as how the DTFT is very easy to approximate with non-infinite sums. Additionally, a variety of different signals were represented in MATLAB to gain a visual understanding of the signals. Through some helpful sources that were included within the lab, one can learn a great deal about correlation values and how they can be used to compare signals.
|
||||
|
||||
\end{document}
|
||||
|
||||
BIN
6th-Semester-Spring-2024/DSP/Labs/Lab-03/dtft_anun.png
Normal file
BIN
6th-Semester-Spring-2024/DSP/Labs/Lab-03/dtft_anun.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 19 KiB |
BIN
6th-Semester-Spring-2024/DSP/Labs/Lab-03/dtft_nanun-1.png
Normal file
BIN
6th-Semester-Spring-2024/DSP/Labs/Lab-03/dtft_nanun-1.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 18 KiB |
Binary file not shown.
|
After Width: | Height: | Size: 15 KiB |
@ -1,11 +1,10 @@
|
||||
import numpy as np
|
||||
import matplotlib.pyplot as plt
|
||||
|
||||
import scipy.signal
|
||||
n = np.arange(0,200)
|
||||
|
||||
a = 0.9
|
||||
|
||||
# x[n] = a^n u[n]
|
||||
# $x[n] = a^n u[n]$
|
||||
x = a**n * np.heaviside(n, 1)
|
||||
|
||||
asums = np.zeros(len(n))
|
||||
@ -25,3 +24,42 @@ plt.plot(n, asums)
|
||||
plt.xlabel("$n$")
|
||||
plt.ylabel("$\sum_n x[n]$", rotation="horizontal")
|
||||
plt.show()
|
||||
|
||||
# Compute DTFT of $x[n]$
|
||||
X = scipy.signal.freqz(1, (1, -a))
|
||||
omega, h = X
|
||||
|
||||
# Plot DTFT of $x[n]$
|
||||
plt.plot(omega, np.abs(h))
|
||||
plt.ylabel("Amplitude")
|
||||
plt.xlabel("Frequency [rad/sample]")
|
||||
plt.show()
|
||||
|
||||
# Set up plot for DTFT of $x[n]$ for comparrison with truncated DTFTs
|
||||
plt.plot(omega, np.abs(h), label="Actual DTFT")
|
||||
plt.ylabel("Amplitude")
|
||||
plt.xlabel("Frequency [rad/sample]")
|
||||
|
||||
# Calculate the truncated DTFTs of $x[n]$ as a function of $K$, $\sum_{n=-K}^K x[n] e^{-j\omega n}$
|
||||
for K in (3, 10, 20):
|
||||
# Finite geometric series formula
|
||||
X_K = (1 - (a**(K+1) * np.exp(-1j * omega * (K+1)))) / (1 - a*np.exp(-1j*omega))
|
||||
plt.plot(omega, np.abs(X_K), label=f"Truncated DTFT ($K={K}$)")
|
||||
|
||||
plt.legend()
|
||||
plt.show()
|
||||
|
||||
# Frequency of maximum difference between actual and truncated DTFT
|
||||
K_range = np.arange(1,200+1)
|
||||
max_diffs = np.zeros(K_range.shape)
|
||||
for K in K_range:
|
||||
X_K = (1 - a**(K+1) * np.exp(-1j * omega * (K+1))) / (1 - a*np.exp(-1j*omega))
|
||||
abs_diff = np.abs(X_K - h)
|
||||
max_diffs[K-1] = np.max(abs_diff)
|
||||
|
||||
plt.plot(K_range, max_diffs)
|
||||
plt.xlabel("K")
|
||||
plt.ylabel("Maximum Error")
|
||||
plt.show()
|
||||
|
||||
|
||||
|
||||
70
6th-Semester-Spring-2024/DSP/Labs/Lab-03/lab-03-2.py
Normal file
70
6th-Semester-Spring-2024/DSP/Labs/Lab-03/lab-03-2.py
Normal file
@ -0,0 +1,70 @@
|
||||
import numpy as np
|
||||
import matplotlib.pyplot as plt
|
||||
import scipy.signal
|
||||
|
||||
n = np.arange(0,200)
|
||||
|
||||
a = 0.9
|
||||
|
||||
# $x[n] = a^n u[n]$
|
||||
x = n*a**n * np.heaviside(n-1, 1)
|
||||
|
||||
asums = np.zeros(len(n))
|
||||
|
||||
for i in range(len(n)):
|
||||
asums[i] = np.sum(x[0:i])
|
||||
|
||||
# Plot settings for $x[n]$
|
||||
plt.subplot(121)
|
||||
plt.plot(n, x)
|
||||
plt.xlabel("$n$")
|
||||
plt.ylabel("$x[n]$", rotation="horizontal")
|
||||
|
||||
# Plot settings for sum of $x[n]$
|
||||
plt.subplot(122)
|
||||
plt.plot(n, asums)
|
||||
plt.xlabel("$n$")
|
||||
plt.ylabel("$\sum_n x[n]$", rotation="horizontal")
|
||||
plt.show()
|
||||
|
||||
# Compute DTFT of $x[n]$
|
||||
X = scipy.signal.freqz((0, a), (1, -2*a, a*a))
|
||||
omega, h = X
|
||||
|
||||
# Set up plot for DTFT of $x[n]$
|
||||
plt.plot(omega, np.abs(h))
|
||||
plt.ylabel("Amplitude")
|
||||
plt.xlabel("Frequency [rad/sample]")
|
||||
plt.show()
|
||||
|
||||
# Set up plot of actual DTFT of $x[n]$ for comparisson with truncated DTFTs
|
||||
plt.plot(omega, np.abs(h), label="Actual DTFT")
|
||||
plt.ylabel("Amplitude")
|
||||
plt.xlabel("Frequency [rad/sample]")
|
||||
|
||||
# Calculate the truncated DTFTs of $x[n]$ as a function of $K$, $\sum_{n=-K}^K x[n] e^{-j\omega n}$
|
||||
for K in (3, 10, 20, 40):
|
||||
# Finite geometric series formula
|
||||
#X_K = ((a*np.exp(-1j*omega)) - (a*np.exp(-1j*omega))**(K+3)) / (1 - a*np.exp(-1j*omega))**2 - (K+2)*(a*np.exp(-1j*omega))**(K+2) / (1 - a*np.exp(-1j*omega))
|
||||
n_K = np.arange(-K,K+1)
|
||||
X_K = np.zeros(omega.shape, np.complex128)
|
||||
for n in n_K:
|
||||
x_n = n* a**n * np.heaviside(n-1, 1)
|
||||
X_K += x_n * np.exp(-1j * n * omega)
|
||||
plt.plot(omega, np.abs(X_K), label=f"Truncated DTFT ($K={K}$)")
|
||||
|
||||
plt.legend()
|
||||
plt.show()
|
||||
|
||||
# Frequency of maximum difference between actual and truncated DTFT
|
||||
K_range = np.arange(1,200+1)
|
||||
max_diffs = np.empty(K_range.shape)
|
||||
for K in K_range:
|
||||
X_K = ((a*np.exp(-1j*omega)) - (a*np.exp(-1j*omega))**(K+3)) / (1 - a*np.exp(-1j*omega))**2 - (K+2)*(a*np.exp(-1j*omega))**(K+2) / (1 - a*np.exp(-1j*omega))
|
||||
abs_diff = np.abs(X_K - h)
|
||||
max_diffs[K-1] = np.max(abs_diff)
|
||||
|
||||
plt.plot(K_range, max_diffs)
|
||||
plt.xlabel("K")
|
||||
plt.ylabel("Maximum Error")
|
||||
plt.show()
|
||||
39
6th-Semester-Spring-2024/DSP/Labs/Lab-03/lab-03-3.py
Normal file
39
6th-Semester-Spring-2024/DSP/Labs/Lab-03/lab-03-3.py
Normal file
@ -0,0 +1,39 @@
|
||||
import numpy as np
|
||||
import matplotlib.pyplot as plt
|
||||
|
||||
omega_c = 0.4*np.pi
|
||||
omega = np.linspace(-np.pi, np.pi, 300)
|
||||
|
||||
# Inverse DTFT of Ideal Lowpass: $x[n] = {\omega_c \over \pi} \sinc(\omega_c n)$
|
||||
|
||||
# Plot $x[n]$ for values of $n$ between -50 and 50
|
||||
n = np.arange(-50,51)
|
||||
x = np.sinc(omega_c*n/np.pi)*omega_c/np.pi
|
||||
plt.stem(n,x)
|
||||
plt.ylabel("$x[n]$")
|
||||
plt.xlabel("$n$")
|
||||
plt.show()
|
||||
|
||||
# Plot the frequency response of $X(e^{j\omega})$
|
||||
f_response = np.compress(np.where(omega >= 0, True, False), omega)
|
||||
X_f_response = np.where(np.abs(f_response) <= omega_c, 1, 0)
|
||||
plt.plot(f_response, X_f_response)
|
||||
plt.ylabel("Amplitude")
|
||||
plt.xlabel("Frequency [rad/sample]")
|
||||
plt.show()
|
||||
|
||||
# Calculate truncated DTFTs of $x[n]$
|
||||
plt.plot(f_response, X_f_response, label="Actual DTFT of $x[n]$")
|
||||
for K in (10, 20, 30):
|
||||
n_K = np.arange(-K, K+1)
|
||||
X_K = np.zeros(f_response.shape, np.complex128)
|
||||
for n in n_K:
|
||||
x_n = np.sinc(omega_c*n/np.pi)*omega_c/np.pi
|
||||
X_K += x_n * np.exp(-1j*f_response*n)
|
||||
plt.plot(f_response, X_K, label=f"Truncated DTFT (K = {K})")
|
||||
|
||||
# Plot truncated DTFTs of $x[n]$
|
||||
plt.ylabel("Amplitude")
|
||||
plt.xlabel("Frequency [rad/sample]")
|
||||
plt.legend()
|
||||
plt.show()
|
||||
Binary file not shown.
|
After Width: | Height: | Size: 19 KiB |
BIN
6th-Semester-Spring-2024/DSP/Labs/Lab-03/max_error_anun.png
Normal file
BIN
6th-Semester-Spring-2024/DSP/Labs/Lab-03/max_error_anun.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 16 KiB |
BIN
6th-Semester-Spring-2024/DSP/Labs/Lab-03/max_error_nanun-1.png
Normal file
BIN
6th-Semester-Spring-2024/DSP/Labs/Lab-03/max_error_nanun-1.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 17 KiB |
BIN
6th-Semester-Spring-2024/DSP/Labs/Lab-03/num4.png
Normal file
BIN
6th-Semester-Spring-2024/DSP/Labs/Lab-03/num4.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 11 KiB |
BIN
6th-Semester-Spring-2024/DSP/Labs/Lab-03/num5a.png
Normal file
BIN
6th-Semester-Spring-2024/DSP/Labs/Lab-03/num5a.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 23 KiB |
BIN
6th-Semester-Spring-2024/DSP/Labs/Lab-03/num5b.png
Normal file
BIN
6th-Semester-Spring-2024/DSP/Labs/Lab-03/num5b.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 34 KiB |
BIN
6th-Semester-Spring-2024/DSP/Labs/Lab-03/num7.png
Normal file
BIN
6th-Semester-Spring-2024/DSP/Labs/Lab-03/num7.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 26 KiB |
BIN
6th-Semester-Spring-2024/DSP/Labs/Lab-03/num8real.png
Normal file
BIN
6th-Semester-Spring-2024/DSP/Labs/Lab-03/num8real.png
Normal file
Binary file not shown.
|
After Width: | Height: | Size: 27 KiB |
Binary file not shown.
|
After Width: | Height: | Size: 16 KiB |
Binary file not shown.
|
After Width: | Height: | Size: 46 KiB |
Binary file not shown.
|
After Width: | Height: | Size: 53 KiB |
Binary file not shown.
|
After Width: | Height: | Size: 45 KiB |
Reference in New Issue
Block a user