From 003cf9b62103a648a35892bcfedd29b09c85a996 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Jo=C3=ABl=20Leirer?= Date: Tue, 25 Oct 2022 11:13:58 +0200 Subject: [PATCH] Update --- RegT1E_HS22.tex | 80 +++++++------- RegT1E_HS22.xdv | Bin 60868 -> 13928 bytes img/DT-Glied.png | Bin 4703 -> 5066 bytes .../Integraltransformationen.tex | 54 ++++++++++ .../Kenngr\303\266ssen von Signalen.tex" | 97 +++++++++++++++++ .../img/Signalklassen.png" | Bin 0 -> 264813 bytes include/LTI-Systeme/LTI-Systeme.tex | 102 ++++++++++++++++++ include/Signalklassen/Signalklassen.tex | 20 ++++ include/Signalklassen/img/Signalklassen.png | Bin 0 -> 264813 bytes .../Wichtige Funktionen.tex | 58 ++++++++++ .../Wichtige Funktionen/img/Dreieckimpuls.png | Bin 0 -> 7075 bytes .../img/Eigenschaften _delta.png | Bin 0 -> 131196 bytes .../img/Impulsfunktion.png | Bin 0 -> 9942 bytes .../img/Rampenfunktion.png | Bin 0 -> 19434 bytes .../img/Rechteckimpuls.png | Bin 0 -> 8673 bytes .../img/Signumfunktion.png | Bin 0 -> 10096 bytes .../Wichtige Funktionen/img/SincFunktion.png | Bin 0 -> 47957 bytes .../img/Sprungfunktion.png | Bin 0 -> 11080 bytes .../Wichtige Werte & Vereinfachungen.tex | 7 ++ 19 files changed, 377 insertions(+), 41 deletions(-) create mode 100644 include/Integraltransformationen/Integraltransformationen.tex create mode 100644 "include/Kenngr\303\266ssen von Signalen/Kenngr\303\266ssen von Signalen.tex" create mode 100644 "include/Kenngr\303\266ssen von Signalen/img/Signalklassen.png" create mode 100644 include/LTI-Systeme/LTI-Systeme.tex create mode 100644 include/Signalklassen/Signalklassen.tex create mode 100644 include/Signalklassen/img/Signalklassen.png create mode 100644 include/Wichtige Funktionen/Wichtige Funktionen.tex create mode 100644 include/Wichtige Funktionen/img/Dreieckimpuls.png create mode 100644 include/Wichtige Funktionen/img/Eigenschaften _delta.png create mode 100644 include/Wichtige Funktionen/img/Impulsfunktion.png create mode 100644 include/Wichtige Funktionen/img/Rampenfunktion.png create mode 100644 include/Wichtige Funktionen/img/Rechteckimpuls.png create mode 100644 include/Wichtige Funktionen/img/Signumfunktion.png create mode 100644 include/Wichtige Funktionen/img/SincFunktion.png create mode 100644 include/Wichtige Funktionen/img/Sprungfunktion.png create mode 100644 include/Wichtige Werte & Vereinfachungen/Wichtige Werte & Vereinfachungen.tex diff --git a/RegT1E_HS22.tex b/RegT1E_HS22.tex index a90d31f..fcdc74e 100644 --- a/RegT1E_HS22.tex +++ b/RegT1E_HS22.tex @@ -7,12 +7,15 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Packages - +\usepackage{mutlicol} +\usepackage[export]{adjustbox} +\usepackage{bm} %% TODO: publish to CTAN \usepackage{tex/hsrstud} %% Language configuration \usepackage{polyglossia} + \setdefaultlanguage[variant=swiss]{german} %% License configuration @@ -23,6 +26,7 @@ lang={german}, ]{doclicense} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % Metadata @@ -69,28 +73,7 @@ \section*{Lizenz} \setcounter{page}{1} \pagenumbering{arabic} -\section{Wichtige Funktionen} -\label{func} -\begin{tabular}{p{5cm} p{10cm}} - \includegraphics[width = 2.5cm]{img/Sprungfunktion.png} & - Sprungfunktion: \newline - Normierter Einschaltvorgang \newline - $ t<0 : u(t) = 0, \ newline - t \geqslant 0: u(t) = 1 $ \\ - - \includegraphics[width = 2.5cm]{img/Impulsfunktion.png} & - Impulsfunktion = Deltafunktion = \newline - Diracimpuls = Delta-Distribution \newline - Unendlicher kurzer normierter Impuls mit unendlicher Amplitude - - Mathematische Eigenschaften: - $\int\limits _{-\infty} ^{+\infty} f(t) \cdot \delta (t-t_0) dt = f(t_0)$ \newline - $\int\limits _{-\infty} ^{+\infty} f(t) \cdot \delta (t) dt = f(0)$\newline - $\int\limits _{-\infty} ^{+\infty} \delta (t) dt = 1$ \\ - %TODO: Find Graphic - & - $\varepsilon(t) = \begin{cases} 0$ für $ t<0 \\ 1 $für$ t \geq 0 \end{cases}$ \\ -\end{tabular} +%\input{include/Wichtige Funktionen/Wichtige Funktionen.tex} \section{Begriffe} \begin{itemize} \item Prozess @@ -149,6 +132,7 @@ \section{Begriffe} \end{itemize} + \subsection{Grundglieder} \label{Grundglieder} \begin{itemize} @@ -176,32 +160,46 @@ \subsection{Grundglieder} \item PT\textsubscript{2}-Glied \begin{itemize} - \item %TODO: + \item Übertragungsglied mit Verhalten 2. Ordnung. + \item $T^2 \cdot \ddot{y}(t) + 2 \zeta T \cdot\dot{y}(t) + y(t) = K \cdot u(t)$ \end{itemize} \item D-Glied(idealer Differenzierer) \begin{itemize} - \item %TODO: + \item Bildet die Zeitliche Ableitung des Signals, mit einer Verstärkung K. + \item $y(t) = K \cdot \frac{du(t)}{dt} = K\cdot \dot{u}(t)$ \end{itemize} \item DT$_1$-Glied \begin{itemize} - \item %TODO: + \item Realiserbar, im Gegensatz zum theoretischen D-Glied. + \item $T \cdot \dot{y}(t) + y(t) = K \cdot \dot{u}(t)$ \end{itemize} \item PI-, PD- PID-Glied (PDT$_1$ und PIDT$_1$) \begin{itemize} - \item %TODO: + \item Werden häufig als Regler eingesetzt. + \item Name setzt sich aus den Aktiven Pfaden zusammen. + \begin{itemize} + \item P: Proportional + \item I: integrierend + \item D: Differenzierend + \item P, I und D in Kombination sind Parallel + \item T\textsubscript{1,2,t} bezieht sich auf die vorangehende Komponente. + \end{itemize} + \item Berechnet sich aus Superposition der Teilantworten + \item $y(t) = A \cdot [K_P + K_I \cdot t + \frac{K_D}{T_C} \cdot e^{\frac{-t}{T_C}}]$ \end{itemize} \item Lead-Glied, Lag-Glied, Lead-Lag-Glied \begin{itemize} - \item %TODO: + \item Varianten von PDT\textsubscript{1}-Gliedern. + \item Unterscheiden sich im Vorzeichen von $K_P$ und $K_D$. \end{itemize} \end{itemize} \end{itemize} +\input{/LTI-Systeme/LTI-Systeme.tex} - -\section{Tabllen} +\section{Tabellen} \subsection*{Blockschaltbilder-DIN} \begin{tabular}{|c|c|c|c|c|} \hline @@ -232,22 +230,22 @@ \subsection*{Blockschaltbilder-DIN} \subsection*{Blockschaltbilder MatLab} \begin{tabular}{|c|c|c|c|c|} \hline - \textbf{Summierer} & - \textbf{Differenzbilder} & - \textbf{Konstante} & - \textbf{Proportionalglied} & - \textbf{Integrierer} \\ + \textbf{Summierer} & + \textbf{Differenzbilder} & + \textbf{Konstante} & + \textbf{Proportionalglied} & + \textbf{Integrierer} \\ \includegraphics[]{img/matlab/sum_block_icon.png} & \includegraphics[]{img/matlab/difference_block_icon.png} & \includegraphics[]{img/matlab/constant_block_icon.png} & \includegraphics[]{img/matlab/gain_block_icon.png} & - \includegraphics[]{img/matlab/integrator_block_icon.png} \\ + \includegraphics[]{img/matlab/integrator_block_icon.png} \\ \hline - \textbf{Totzeitglied} & - \textbf{Differenzierer} & - \textbf{} & - \textbf{} & - \textbf{} \\ + \textbf{Totzeitglied} & + \textbf{Differenzierer} & + \textbf{} & + \textbf{} & + \textbf{} \\ \includegraphics[]{img/matlab/transport_delay_block_icon.png} & \includegraphics[]{img/matlab/derivative_block_icon.png} & \\ diff --git a/RegT1E_HS22.xdv b/RegT1E_HS22.xdv index 0a58eaa5d50fd894ddd931f640b1a7f8fcc52721..11f8746e2efa497462e22171e1bac7096c3bcae2 100644 GIT binary patch literal 13928 zcmdU032@X^nt!%k zXyQ2@jU}U_PGLr8x73++tf5@g(Xi1t=%PD=YClRv!wlI43S!y0>gM~&UkX>yuxH7RzP+g)7>UA11RQ|qhrc9){dWbW*8DXJ=T zN_9nLRfXN<9{vEIX}1~8)(%@;ovPLB>NXW^F4tGWPrq5MRWx_SVCBThfR$ymqQ|Y+>=?aD_fbWU(bdjk)w-Ux?IxqEs93ecVKTUEj-r(Y zp$5) 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zg5#99Rtf@Eh|rNZg_Dj;CWm2Pp19i=Vd~$$tqkkwNC}lP)+Uu}0G1jN7p~aLCVFrN zip9hkNq{6*R#qx>tiXHTjA`B!7pEN_9$vG#l0nwnLP{}+CUP}4 z=-)%hm4>bXt?vym`5_)ZVZabVi2xPY06BuPvMlm`iwyk;3=P_UsiQJ~lg$_?U?rtL z$$UVV7ITbP#-}6$@PpNh)cLNfjcsjg3(@Ihx*!J|Qq@*=okGp}i(uJr)K|p+?s-gd zBS6f>+?&AZB3L&Vs7fU31!$-cizI|tMM2bAQk)=mwfpz)Cq|aJSmU@n@_r@N zXkml_(#d}x(*PPSX=Dfnj@JO67WTjO|Cb7lAd<^K7Y6e{e#fMy5TLJZr1eh2A@08b DO|jkf diff --git a/include/Integraltransformationen/Integraltransformationen.tex b/include/Integraltransformationen/Integraltransformationen.tex new file mode 100644 index 0000000..bacb074 --- /dev/null +++ b/include/Integraltransformationen/Integraltransformationen.tex @@ -0,0 +1,54 @@ +\section{Integraltransformationen} + +\subsubsection*{Faltung} + +$$y(t) = (x_1 * x_2)(t) = \int \limits _{-infty} ^{infty} x_1(\tau) \cdot x_2(t-\tau) d\tau$$ +\textbf{Eigenschaften:} +\begin{itemize} + \item Kommutativ ($f * g = g * f$) + \item Assoziatiov ($(f*(g*h)) = ((f*g)*h)$) + \item Distributiv ($f*(g+h)= f*g + f*h$) +\end{itemize} + +\subsection{Fourier-Reihe und Transformation} +\subsubsection*{Fourier-Reihe} +\begin{tabular}{p{4.5cm}p{14.5cm}} + Trigonometrische Form & + $x(t) = \frac{u_0}{2} + \sum \limits _{n = 1} ^{\infty} u_n \cdot cos(2\pi n f_0 \cdot t) + v_n \cdot sin(2\pi n f_0 \cdot t) $ + \newline $u_n = \frac{2}{T} \int \limits _{T} x(t) \cdot cos(2\pi n f_0 \cdot t)dt$ + \newline $v_n = \frac{2}{T} \int \limits _{T} x(t) \cdot sin(2\pi n f_0 \cdot t)dt$ + \\[20pt] + Harmonische Form & + $x(t) = r_0 + \sum \limits _{n = 1} ^{\infty} r_n \cdot cos(2\pi n f_0 \cdot t + \varpi_n)$ + \newline $r_0 = \frac{u_0}{2} = \frac{1}{T} \int \limits _{T} x(t) dt $ + $r_n > 0 = \sqrt{u_n^2 + v_n^2}$ + $\varphi = arg(u_n - j \cdot v_n) $ + \\ + Komplexe Form & + $x(t) = \sum \limits _{n= -\infty} ^{\infty} c_n \cdot e^{jn2\pi f_0 \cdot t}$ + \newline $c_n =\overline{c_{-n}} = \frac{1}{T} \int \limits _{0} ^{T} x(t) \cdot e^{-j n 2 \pi f_0 \cdot t} dt$ + \newline $ 2\pi f_0$ wird auch als \textbf{Kreisfrequenz} $\omega$ bezeichnet, $f_0$ = Frequenz des Grundsignals $x(t)$. + \\[20pt] + Umrechnung Koeffizienten & + $c_n =\overline{c_{-n}} = \frac{a_n - jb_n}{2} (n = 0,1,2,3,..., b_0 =0)$ + \newline $a_n = 2 \cdot Re(c_n); \; b_n = -2 \cdot Im(c_n) (n = 0,1,2,3,..., b_0 =0)$\\ +\end{tabular} + + +\subsubsection*{Fouriertransformation $\mathcal{F}(\omega)$} +$$ X(\omega) = \mathcal{F}[x(t)] = \int \limits _{-\infty} ^{+\infty} x(t) \cdot e^{-j \omega t} dt $$ +$$ x(t) = \mathcal{F}^{-1}[X(\omega)] = \frac{1}{2 \pi} \int \limits _{- \infty} ^{+ \infty} X(\omega) \cdot e^{j \omega t} d\omega$$ + +\subsubsection*{Komplex sin/cos} + +$$cos \varphi = \frac{e^{j\varphi}+ e^{-j\varphi}}{2}, \; sin \varphi = \frac{e^{j\varphi} - e^{-j\varphi}}{2j} $$ + +\subsection{Laplace-Transformation} +TODO: %LAPLAAAAAAAAACEEEE + +\subsection{Hilbert-Transformation} +Hilbert Transformation ist die Anwendung eines Quadraturfilters. \\ +Definition im \textbf{Zeitbereich:} +$$\hat{x}(t) = x(t) * \frac{1}{\pi t} = \frac{1}{\pi} \int \limits _{-\infty} ^{\infty} \frac{x(\tau)}{t-\tau} d\tau$$ +Im \textbf{Frequenzbereich}: +$$\hat{X}(\omega) = X(\omega) \cdot H(\omega) = -j \cdot sgn(\omega) \cdot X(\omega)$$ diff --git "a/include/Kenngr\303\266ssen von Signalen/Kenngr\303\266ssen von Signalen.tex" "b/include/Kenngr\303\266ssen von Signalen/Kenngr\303\266ssen von Signalen.tex" new file mode 100644 index 0000000..c4d69fc --- /dev/null +++ "b/include/Kenngr\303\266ssen von Signalen/Kenngr\303\266ssen von Signalen.tex" @@ -0,0 +1,97 @@ +\section{Kenngrössen von Signalen} +\begin{tabular}{p{6cm}p{12cm}} + Energie & + $W_n = \lim_{T \to \infty} \int \limits _{-T/2} ^{T/2} |f(t)|^2 dt$ \\ + Leistung & + $P_n = lim_{T \to \infty} \frac{1}{T} \int \limits _{-T/2} ^{T/2} |f(t)|^2 dt$ \\ + *Linearer Mittelwert + \newline \tiny(auch: $ \bar{x}, x_m$) & + $X_0 = \frac{1}{T} \int \limits _{-T/2}^{T/2} x(t) dt $ \\ + *Quadratischer Mittelwert & + $X^2 = \frac{1}{T} \int \limits _{-T/2}^{T/2} x^2(t) dt$ \\ + *Effektivwert \newline \tiny{("Quadratischer Mittelwert", RMS)} & + $X^2 = \frac{1}{T} \int \limits _{-T/2}^{T/2} \sqrt{x^2(t)} dt $ \\ + Mittelwert n. Ordnung \newline \tiny(nur Signale $\in \mathbb{R}$) & + $X^n = \frac{1}{T} \int \limits _{-T/2} ^{T/2} x^n(t)dt$ \\ + Varianz & + $Var(x) = \sigma^2 = \frac{1}{T} \int \limits _{-T/2} ^{T/2} (x(t) - X_0)^2 dt$ \\ + Standardabweichung & + $\sigma = \sqrt{Var(x)} = \sqrt{X^2 - (X_0)^2}$ \\ +\end{tabular} +\textbf{\tiny *Hinweis: Formeln sind für Klasse 2a angegeben. \newline + Für Klasse 2b mit: $\lim_{t \to \infty}\;$ für Klasse 1: ohne $\frac{1}{T}$ } + +\subsubsection*{Die Autokorrelationsfunktion (AKF)} +Für Klasse 2a +$$ \varphi_{xx}(\pm \tau) = \frac{1}{T} \int \limits _{-T/2} ^{T/2} x(t) \cdot x(t- \tau) dt + = \frac{1}{T} \int \limits _{-T/2} ^{T/2} x(t + \tau) \cdot x(t)dt$$ +\textbf{Eigenschaften:} +\begin{itemize} + \item $\varphi_{xx}(0) = X^2 = (X_0)^2 + \sigma^2$ \tiny Quadratischer Mittelwert \normalsize + \item $\varphi_{xx}(\tau) = \varphi_{xx}(\tau \pm n \cdot T)$, mit $n \in \mathbb{N}$, + AKF hat gleiche Periode $T$ wie $x(t)$ + \item $\varphi_{xx}(\tau) = \varphi_{xx}(-\tau)$, AKF ist gerade Funktion + \item $\varphi_{xx}(0) \geq \left|\varphi_{xx}(\tau)\right|$ + \item $\varphi_{xx}(\tau) \geq (X_0)^2 - \sigma^2$ + \item \textbf{Klasse 2b:} $\lim_{t \to \infty}$ voran; \textbf{Klasse 1:} $\lim_{t \to \infty}$ anstelle $\frac{1}{T}$ +\end{itemize} + +\subsubsection*{Die Kreuzkorrelationsfunktion (KKF)} +Für Klasse 2a: +$$ (x \star y)(\tau) = \varphi_{x_1x_2}(\tau) = \frac{1}{T} \int \limits _{-T/2} ^{T/2} x_1(t) \cdot x_2(t- \tau) dt + = \frac{1}{T} \int \limits _{-T/2} ^{T/2} x_1(t + \tau) \cdot x_2(t)dt$$ +\textbf{Eigenschaften:} +\begin{itemize} + \item $\varphi_{x_1x_2}(\tau) = \varphi_{x_1x_2}(-\tau)$ + \item ist \textbf{nicht} Kommutativ($(x \star y)(\tau) \neq (y \star x)(\tau)$) + \item \textbf{Klasse 2b:} $\lim_{t \to \infty}$ voran; \textbf{Klasse 1:} $\lim_{t \to \infty}$ anstelle $\frac{1}{T}$ +\end{itemize} + + +\subsection{Vergleich Signalleistung / physikalische Leistung} +Leistungsverhältnisse zweier Leistungen wird oft in dB, Dezibel angegeben. +Bel steht für das Verhältnis zweier Werte im Zehnerlogarithmus. +Aufgrund des d (=dezi) muss ein Faktor 10 verwendet werden. +Werden anstelle Leistungen Effektivwerte genommen wird ein Faktor 20 benötigt. +Bei Referenzwerten $P_0$ ist dieser für $P_x$ resp $x_{rms}$ einzusetzen. + +$$10 \cdot \log_{10} (\frac{P_y}{P_x}) = + 10 \cdot \log_{10}(\frac{(y_{rms})^2}{(x_{rms})^2}) = + 20 \cdot log_{10}(\frac{y_{rms}}{x_{rms}}) = k[\textrm{dB}]$$ +daraus folgt: +$$P_y = P_x \cdot 10^{\frac{k}{10}} \; P_x = \frac{P_y}{10^{\frac{k}{10}}}$$ +bzw: +$$y_{rms} = x_{rms} \cdot 10^{\frac{k}{20}} \; x_{rms} = \frac{y_{rms}}{10^{\frac{k}{20}}}$$ + +\subsection{Rauschen} + +effektive thermische Rauschleistung: $P_r = k \cdot T \cdot \Delta f $ \\ +daraus folgt: $U_r = \sqrt{4\cdot k \cdot T \cdot \Delta f \cdot R} $\\ +wobei $k = 1.380662\cdot 10^{-23} \frac{J}{K}$ = Boltzmann-Konstante\\ + +Signal-Rausch-Verhältnis: $a_r = 10 \cdot log_{10}(\frac{P_s}{P_r}) = 20 \cdot log_{10}(\frac{U_s}{U_r})$ +Rauschzahl: $F = \frac{P_{s_{in}}}{P_{r_{in}}}\cdot \frac{P_{r_{in}}}{P_{s_{out}}} $ +logarithmisch: $A_F = 10 \cdot log_{10}(F) = a_{r_{in}} - a_{r_{out}}$ + +\subsection{Amplitudenanalyse von Signalen} + +\textbf{Amplitudendichte (WSK-Dichte)} $p(a) = lim_{da \to 0} \frac{\sum t (a-\frac{da}{2} < x(t) \leq a + \frac{da}{2})}{T \cdot da}= \frac{1}{T} \cdot \frac{dt}{da}$ + +\subsubsection*{Mittelwerte} + +\textbf{Linear:}$X_0 = \int \limits _{-\infty} ^{\infty} a \cdot p(a) da$ +\textbf{N-te Ordnung:}$X^n = \int \limits _{-\infty} ^{\infty} a^n \cdot p(a) da$ +Gauss verteilung \textbf{für Stochastische Signale} $p(a) = N(\mu, \sigma) = \frac{1}{\sigma \cdot \sqrt{2\pi}}\cdot e^{\frac{-(a-\mu)^2}{2\sigma^2}} $ +wobei: $\mu$ = Linearer Mittelwert ($X_0$) und $\sigma$ = Varianz + +\subsubsection*{Faltung zweier Amplitudendichten} + +$p(a) = \int \limits _{-\infty} ^{\infty} p_2(x) \cdot p_1(a-x)dx$ +Note: werden 2 Normalverteilungen gefaltet, entsteht eine 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