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z^%55mB{L!-QpBtqfiyS~Lk*kr!LJRIn0tZEz=c6#(W%}vI$uEFZWquD49S{)C|HLF z2jZPRzHSVX8t3Z~>vB$bJxFi2^X9PaOC#Do57LKrot}CDs0ZshuvV~tr?lmmxg#8w zdnT|ZWBTX8YSQalL_|bH4zv<5UkpZ~GTXwDtzHgCD*GZ3b&W|PC^Jum^?wtLe#MxG z1LCiT?S6^9zoIy&grj_AoVx5B7u0k9i@jf*IVXU5C|t&a5pCZKW_?DoGX%`9z&sa@ zPPMSm}HooSVaCE)MHf0*8aS5X^QkJHcE89FfuXCE?J| zmt}0{)L?zO`y$Z37EGER567G-YBV$F1TfzKdwIAW>%e?7Sf5~JgYC9wg~K+t@ADbK zD0Tqm%y9an;cH-2IK1@?Fk8BgT&B^?M5HiS>H-mwqB01a8@ew}{Sv6pns_icS4VtA zJPygah(;!XnFUkYH%7PTzYXd8 zgJjJ*VjqVCzk}o(os$__!TXusQX(QEA|f3LOEWV9xINJhi4m`j6@w5fLRbA|g_pDl`5U@~O@>z@@-B!BcYy@F)Xx zV;vC@Q8FVUA_tUt7|iPg_lfxjF!kXD`Gsgzki1y+;M@-WU`YQ?`&dLoL_|aiMGY{b z|C)0?2WAtPC&Ap6`^~Yh4VLY^rE6wHq!4H{GZ7Jy{4oe(lf%btkX)AI<<~-Nc> +#+end_src + +#+NAME: fig:coh_geophones +#+CAPTION: Cohererence between the signals of the two geophones +#+RESULTS: fig:coh_geophones +[[file:figs/coh_geophones.png]] + +** Huddle Test +#+NAME: fig:huddle_test +#+HEADER: :headers '("\\usepackage{tikz}" "\\usepackage{import}" "\\import{$HOME/MEGA/These/LaTeX/}{config.tex}") +#+HEADER: :imagemagick t :fit yes :iminoptions -scale 100% -density 150 :imoutoptions -quality 100 +#+HEADER: :results raw replace :buffer no :eval no-export :exports both :mkdirp yes +#+HEADER: :output-dir figs +#+begin_src latex :file huddle-test.pdf :post pdf2svg(file=*this*, ext="png") :exports results + \begin{tikzpicture} + \coordinate[] (U) at (0, 0) {}; + \node[block, above right=0.5 and 2 of U] (S1) {$S_1$}; + \node[block, below right=0.5 and 2 of U] (S2) {$S_2$}; + \node[addb={+}{}{}{}{}, right=0.5 of S1] (add1) {}; + \node[addb={+}{}{}{}{}, right=0.5 of S2] (add2) {}; + + \draw[] (U) node[above right]{$U$} -- ++(1, 0) node[]{$\bullet$}; + \draw[->] ($(U)+(1, 0)$) |- (S1.west); + \draw[->] ($(U)+(1, 0)$) |- (S2.west); + + \draw[->] (S1.east) -- (add1.west); + \draw[->] (S2.east) -- (add2.west); + + \draw[->] (add1.east) -- ++(1, 0) node[above]{$X_1$}; + \draw[->] (add2.east) -- ++(1, 0) node[above]{$X_2$}; + + \draw[<-] (add1.north) -- ++(0, 0.8)node[right]{$N_1$}; + \draw[<-] (add2.north) -- ++(0, 0.8)node[right]{$N_2$}; + \end{tikzpicture} +#+end_src + +#+NAME: fig:huddle_test +#+CAPTION: Huddle test block diagram +#+RESULTS: fig:huddle_test +[[file:figs/huddle-test.png]] + +We are measuring $X_1$ and $X_2$. +The goal is to determine $N$. + +\begin{align*} + X_1(\omega) &= S_1(\omega) U(\omega) + N_1(\omega)\\ + X_2(\omega) &= S_2(\omega) U(\omega) + N_2(\omega) +\end{align*} + +Then +\[ X_2(\omega) = \frac{S_2(\omega)}{S_1(\omega)} X_1(\omega) + N_2(\omega) - \frac{S_2(\omega)}{S_1(\omega)}N_1(\omega) \] + +We suppose $N_1 = N_2 = N$ +\[ N_2(\omega) - \frac{S_2(\omega)}{S_1(\omega)}N_1(\omega) = \left( 1 - \frac{S_2(\omega)}{S_1(\omega)}\right) N(\omega) \] +and +\[ N(\omega) = \frac{S_2(\omega)}{S_1(\omega)} X_2(\omega) - N_2(\omega) - \frac{S_2(\omega)}{S_1(\omega)}N_1(\omega) \] + + + +#+begin_src matlab :results none + S = abs(T12.*pxx1); + + N = pxx2 - (T12.^2).*pxx1; + N = abs(N)/2; +#+end_src + +#+begin_src matlab :results none + figure; + hold on; + plot(f, pxx1, '-'); + plot(f, pxx2, '-'); + plot(f, N, 'k:', 'linewidth', 1); + hold off; + set(gca, 'xscale', 'log'); set(gca, 'yscale', 'log'); + xlim([1, 500]); + legend('$\Phi_{ss} (f)$','$\Phi_{nn} (f)$') +#+end_src + +#+NAME: fig:huddle_test_results +#+HEADER: :tangle no :exports results :results value raw replace :noweb yes +#+begin_src matlab :var filepath="figs/huddle_test_results.pdf" :var figsize="wide-tall" :post pdf2svg(file=*this*, ext="png") + <> +#+end_src + +#+NAME: fig:huddle_test_results +#+CAPTION: Results of the Huddle test +#+RESULTS: fig:huddle_test_results +[[file:figs/huddle_test_results.png]]