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include/Integraltransformationen/Integraltransformationen.tex
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| \section{Integraltransformationen} | ||
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| \subsubsection*{Faltung} | ||
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| $$y(t) = (x_1 * x_2)(t) = \int \limits _{-infty} ^{infty} x_1(\tau) \cdot x_2(t-\tau) d\tau$$ | ||
| \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} | ||
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| \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} | ||
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| \subsubsection*{Fouriertransformation $\mathcal{F}(\omega)$} | ||
| $$ X(\omega) = \mathcal{F}[x(t)] = \int \limits _{-\infty} ^{+\infty} x(t) \cdot e^{-j \omega t} dt $$ | ||
| $$ x(t) = \mathcal{F}^{-1}[X(\omega)] = \frac{1}{2 \pi} \int \limits _{- \infty} ^{+ \infty} X(\omega) \cdot e^{j \omega t} d\omega$$ | ||
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| \subsubsection*{Komplex sin/cos} | ||
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| $$cos \varphi = \frac{e^{j\varphi}+ e^{-j\varphi}}{2}, \; sin \varphi = \frac{e^{j\varphi} - e^{-j\varphi}}{2j} $$ | ||
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| \subsection{Laplace-Transformation} | ||
| TODO: %LAPLAAAAAAAAACEEEE | ||
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| \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}: | ||
| $$\hat{X}(\omega) = X(\omega) \cdot H(\omega) = -j \cdot sgn(\omega) \cdot X(\omega)$$ |
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include/Kenngrössen von Signalen/Kenngrössen von Signalen.tex
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| \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 $\frac{1}{T}$ } | ||
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| \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} | ||
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| \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} | ||
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| \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. | ||
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| $$10 \cdot \log_{10} (\frac{P_y}{P_x}) = | ||
| 10 \cdot \log_{10}(\frac{(y_{rms})^2}{(x_{rms})^2}) = | ||
| 20 \cdot log_{10}(\frac{y_{rms}}{x_{rms}}) = k[\textrm{dB}]$$ | ||
| daraus folgt: | ||
| $$P_y = P_x \cdot 10^{\frac{k}{10}} \; P_x = \frac{P_y}{10^{\frac{k}{10}}}$$ | ||
| bzw: | ||
| $$y_{rms} = x_{rms} \cdot 10^{\frac{k}{20}} \; x_{rms} = \frac{y_{rms}}{10^{\frac{k}{20}}}$$ | ||
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| \subsection{Rauschen} | ||
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| 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\\ | ||
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| 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: $A_F = 10 \cdot log_{10}(F) = a_{r_{in}} - a_{r_{out}}$ | ||
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| \subsection{Amplitudenanalyse von Signalen} | ||
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| \textbf{Amplitudendichte (WSK-Dichte)} $p(a) = lim_{da \to 0} \frac{\sum t (a-\frac{da}{2} < x(t) \leq a + \frac{da}{2})}{T \cdot da}= \frac{1}{T} \cdot \frac{dt}{da}$ | ||
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| \subsubsection*{Mittelwerte} | ||
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| \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 | ||
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| \subsubsection*{Faltung zweier Amplitudendichten} | ||
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| $p(a) = \int \limits _{-\infty} ^{\infty} p_2(x) \cdot p_1(a-x)dx$ | ||
| Note: werden 2 Normalverteilungen gefaltet, entsteht eine Normalverteilung. | ||
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| \subsubsection*{Weiter Funktionen} | ||
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| Q-Funktion: $Q(\xi) = \frac{1}{\sqrt{2\pi}}\int \limits _{\xi} ^{\infty} e^{-\frac{y^2}{2}}d\xi$ | ||
| Fehlerfunktion $erf(\xi) = \frac{2}{\sqrt{\pi}} \int \limits _{0} ^{\xi} e^{-y^2} dy$ | ||
| komplementäre Fehlerfunktion $erfc(\xi) = \frac{2}{\sqrt{\pi}} \int \limits _{\xi} ^{\infty} e^{-y^2} dy $ |
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