From 50c73a0a29e0881584e4cb724470b8c5e46813d7 Mon Sep 17 00:00:00 2001 From: leo Date: Mon, 27 Feb 2023 03:56:47 +0100 Subject: [PATCH] tex: add batch 1, add related pics --- res/autocorrelation_u.jpg | Bin 19912 -> 0 bytes res/autocorrelation_u.png | Bin 0 -> 52998 bytes res/autocorrelation_y.jpg | Bin 19444 -> 0 bytes res/autocorrelation_y.png | Bin 0 -> 42178 bytes res/impulse-response.png | Bin 0 -> 53640 bytes res/impulse_func_estimate.png | Bin 0 -> 43977 bytes res/impulse_func_estimate_comparison.png | Bin 0 -> 45875 bytes res/mutual_correlation_uy.jpg | Bin 23679 -> 0 bytes res/mutual_correlation_uy.png | Bin 0 -> 46265 bytes res/mutual_correlation_yu.jpg | Bin 23598 -> 0 bytes res/mutual_correlation_yu.png | Bin 0 -> 48229 bytes res/signal_u.jpg | Bin 39221 -> 0 bytes res/signal_u.png | Bin 0 -> 116424 bytes res/signal_y.jpg | Bin 24608 -> 0 bytes res/signal_y.png | Bin 0 -> 106780 bytes solution.tex | 115 ++++++++++++++++++----- 16 files changed, 94 insertions(+), 21 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z4kvU8=^ZhkTkb5`iR^fgy-!@z10VW7EfmF>2Mf5RIBGwj*K1X@IN-KAur$U3-a_E> x0&DS~(m@3UZi}, následne sme si so zachytených vzoriek (jednorozmerné polia rovnakej dĺžky) vytvorili -\textit{maticu}. Túto maticu sme nakoniec zapísali príkazom +\textit{maticu}. Táto matica bola nakoniec zapísaná príkazom \texttt{writematrix} \footnote{V predošlom protokole sa vyskytovala chyba, keďže som poznamenal, že bol použitý príkaz \texttt{matrixwrite}} do želaného súboru. -Súbor bolo dodatočne potrebné manuálne upraviť - pridať mu CSV "hlavičku" s -názvami stĺpcov, aby ho bolo možné za pomoci knižnice -\href{https://pandas.pydata.org/}{\texttt{Pandas}} bez problémov spracovať. +Súbor bolo dodatočne potrebné manuálne upraviť - pridať mu CSV ``hlavičku'' +s~názvami stĺpcov, aby ho bolo možné za pomoci knižnice +\href{https://pandas.pydata.org/}{\texttt{Pandas}} ďalej bez problémov +spracovávať. +Go balíček \texttt{csv} zo štandardnej knižnice takýto nárok/problém nemá, +takže bolo možné po \textit{odstránení} hlavičky dáta zo súbory v Go programe +načítať a skonvertovať to natívnych typov (\texttt{[]float64}...) pomocou +funkcionality balíčka štandardnej knižnice \texttt{strconv}. Pre referenciu viď +súbor \texttt{p2/data.go}. \newpage \subsection{Získané signály} \begin{figure}[ht] \centering - \includegraphics[width=.77\textwidth]{res/signal_u} + \includegraphics[width=.57\textwidth]{res/signal_u} \caption{Náhodne vygenerovaný signál pred aplikáciou filtra - u} \label{fig:signalorig} \end{figure} \begin{figure}[ht] \centering - \includegraphics[width=.77\textwidth]{res/signal_y} + \includegraphics[width=.57\textwidth]{res/signal_y} \caption{Náhodne vygenerovaný signál po aplikácii filtra - y} \label{fig:signalfiltered} \end{figure} @@ -126,7 +132,7 @@ Vypočítali sme štatistické charakteristiky prvého a druhého rádu, ktorým \subsubsection{Autokorelačná funkcia u} \begin{figure}[ht] \centering - \includegraphics[width=.77\textwidth]{res/autocorrelation_u} + \includegraphics[width=.57\textwidth]{res/autocorrelation_u} \caption{Autokorelačná funkcia u} \label{fig:autocorrelation-u} \end{figure} @@ -134,7 +140,7 @@ Vypočítali sme štatistické charakteristiky prvého a druhého rádu, ktorým \subsubsection{Autokorelačná funkcia y} \begin{figure}[ht] \centering - \includegraphics[width=.77\textwidth]{res/autocorrelation_y} + \includegraphics[width=.57\textwidth]{res/autocorrelation_y} \caption{Autokorelačná funkcia y} \label{fig:autocorrelation-y} \end{figure} @@ -143,14 +149,14 @@ Vypočítali sme štatistické charakteristiky prvého a druhého rádu, ktorým \subsubsection{Vzájomne korelačná funkcia} \begin{figure}[ht] \centering - \includegraphics[width=.77\textwidth]{res/mutual_correlation_uy} + \includegraphics[width=.57\textwidth]{res/mutual_correlation_uy} \caption{Vzájomne korelačná funkcia uy} \label{fig:mutualcorrelation-uy} \end{figure} \begin{figure}[ht] \centering - \includegraphics[width=.77\textwidth]{res/mutual_correlation_yu} + \includegraphics[width=.57\textwidth]{res/mutual_correlation_yu} \caption{Vzájomne korelačná funkcia yu} \label{fig:mutualcorrelation-yu} \end{figure} @@ -164,8 +170,8 @@ Vypočítali sme štatistické charakteristiky prvého a druhého rádu, ktorým \begin{tabular}{r|cc} & \textbf{u} & \textbf{y} \\ \hline - \textbf{u} & 0.30774141 & -0.00436625 \\ - \textbf{y} & -0.00436625 & 0.01241115 \\ + \textbf{u} & 1.1583640130324268647 & 0.015851330298377188 \\ + \textbf{y} & 0.015851330298377188 & 0.0247634923750309523 \\ \end{tabular} \caption{Kovariančná matica} @@ -175,7 +181,8 @@ Vypočítali sme štatistické charakteristiky prvého a druhého rádu, ktorým Zo zaznamenaných dát signálov sme spočítali odhady stredných hodnôt a rozptylov. V~tabuľke~\ref{table:stats} môžeme vidieť výrazné posuny signálu po prechode -filtrom oproti pôvodnému signálu ako v strednej hodnote tak v rozptyle. +filtrom oproti pôvodnému signálu hlavne v rozptyle, ale zmeny zachytila aj +stredná hodnota. \begin{table}[!hbt] @@ -183,20 +190,86 @@ filtrom oproti pôvodnému signálu ako v strednej hodnote tak v rozptyle. \begin{tabular}{r|cc} & \textbf{u} & \textbf{y} \\ \hline - \textbf{$\hat{\mu}_{u,y}$ (stredná hodnota)} & -0.0016865221052993265 & 0.0030596815877553443 \\ - \textbf{$\hat{\sigma}^{2}_{u,y}$ (rozptyl)} & 0.3077414110318535 & 0.012411153945455982 \\ - \textbf{$\hat{r}(U,Y)$ (koeficient korelácie)} & \multicolumn{2}{c}{-0.07064959453763188} \\ + \textbf{$\hat{\mu}_{u,y}$ (stredná hodnota)} & -0.0731240350331255062 & -0.0109705557504332952 \\ + \textbf{$\hat{\sigma}^{2}_{u,y}$ (rozptyl)} & 1.1583640130324268647 & 0.0247634923750309523 \\ + \textbf{$\hat{r}(U,Y)$ (koeficient korelácie)} & \multicolumn{2}{c}{0.09359167949834103} \\ \end{tabular} \caption{Súhrn štatistických charakteristík} \label{table:stats} \end{table} -Hodnota koeficientu korelácie je mierne záporná, čo znamená jemný náznak -tvrdenia, že signály spolu nesúvisia, resp. vzhľadom na to ako blízko je -daná záporná hodnota 0 ide skôr o preukázanie neexistencie súvisu. +Hodnota koeficientu korelácie je mierne kladná, čo znamená jemný náznak +tvrdenia, že signály spolu súvisia. +\newpage +\subsection{Impulzná odozva} +Pomocou \textit{Control System Toolbox} (CST) a \textit{System Identification +Toolbox} (SIT) sme vypočítali zo zadaného modelu \textbf{impulznú odozvu}. +\begin{figure}[h] + \centering + \begin{varwidth}{\linewidth} + \begin{verbatim} + >> % define a system from the transfer function + >> sys = tf(1, [2 5 1]) + >> % get impulse response from CST + >> ir = impulse(sys) + >> % get impulse response from SIT + >> ircra = cra(out.u, out.y) + >> + >> % save CST impulse response to file + >> writematrix(ir, 'data/ir.csv') + >> % save SIT impulse response to file + >> writematrix(ircra, 'data/ircra.csv') + \end{verbatim} + \end{varwidth} + \caption{Výpočet impulznej odozvy pomocou CST a SIT} +\end{figure} + +\subsubsection{Výpočet impulznej odozvy} +Výpočet hodnôt impulznej odozvy pomocou korelačnej analýzy: hodnoty impulznej +funkcie vypočítame diskretizáciou Wiener-Hopfovej rovnice. + +Po dikretizácii Wiener-Hopfovej rovnice dostáváme: +\begin{equation*} + R_{uy} (\tau)=\sum_{i=1}^N R_{uu} (\tau - i\Delta t) g(i\Delta t) \Delta t +\end{equation*} + +Výslednú impulznú charakteristiku vykreslenú na základe odhadu impulznej +funkcie je možné vidieť na obrázku~\ref{fig:imp-res}. +\begin{figure}[ht] + \centering + \includegraphics[width=.57\textwidth]{res/impulse_func_estimate} + \caption{Impulzná charakteristika pomocou funkcie \texttt{gCustom}} + \label{fig:imp-res} +\end{figure} + +\newpage + +\subsubsection{Porovnanie odhadov impulznej funkcie} +Porovnanie odhadov impulznej funkcie \texttt{gCustom} s odhadom impulznej +funkcie zo SIT (Matlab príkaz \texttt{cra(out.u, out.y)}) je možné vidieť na +obrázku~\ref{fig:imp-res-comp}. +\begin{figure}[ht] + \centering + \includegraphics[width=.57\textwidth]{res/impulse_func_estimate_comparison} + \caption{Porovnanie odhadov impulznej funkcie pomocou \texttt{gCustom} a + \texttt{cra} zo SIT} + \label{fig:imp-res-comp} +\end{figure} + +% \newpage +Odhad impulznej funkcie podľa CST funkcie \texttt{impulse} je možné vidieť na +obrázku~\ref{fig:imp-res-cst}. +\begin{figure}[ht] + \centering + \includegraphics[width=.77\textwidth]{res/impulse-response} + \caption{Impulzná charakteristika podľa \texttt{impulse} z CST} + \label{fig:imp-res-cst} +\end{figure} + +Tento sa od odhadov pomocou \texttt{gCustom} a \texttt{gSIT} výrazne líši. \newpage \section{Záver}

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