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studium@daniel-thiem.de\\Version 0.7 - 20.09.2012} +\subsubtitle{Autor: Daniel Thiem - studium@daniel-thiem.de\\Version 0.8 - 21.09.2012} \maketitle \newpage @@ -223,7 +224,7 @@ X_s(j\Omega)&=\frac{1}{T_s}\sum\limits_{k=-\infty}^\infty X_c(j(\Omega-\frac{2\p \section{Transformation} \subsection{Rcktransformation} \begin{align} -x[n]&=\int\limits_{-\pi}^\pi X\freqe^{j\omega n}d\omega +x[n]&=\int\limits_{-\pi}^\pi X\freq e^{j\omega n}d\omega \end{align} \subsection{Zusammenhang $\Omega$ und $n$} \label{omegaToDescrete} ACHTUNG: Dieser zusammenhang ist in SSS etwas anders im gegensatz zu dem Hilfsblatt von DSS @@ -439,7 +440,7 @@ S_{XY}\freq=\mathcal{F}\{r_{XY}(\kappa)\}=\sum\limits_{k=-\infty}^\infty r_{XY}( \section{Kohrenz (coherence)} \label{coherence} \begin{equation} - \rm{Coh}_{XY}\freq=\frac{\left|S_{XY}\freq\right|^2}{S_{XX}\freqS_{YY}\freq} + \rm{Coh}_{XY}\freq=\frac{\left|S_{XY}\freq\right|^2}{S_{XX}\freq S_{YY}\freq} \end{equation} \subsection{Eigenschaften der Kohrenz} Die Kohrenz zwischen den Zufallsprozessen $X(n)$ und $Y(n)$ besagt, wie gut $X$ zu $Y$ bei einer gegebenen Frequenz $\omega$ korrespondiert. @@ -531,8 +532,8 @@ und $Y(n)$ ist \emph{station \begin{subequations} \begin{align} c_{YX}&=\sum\limits_{k=-\infty}^\infty h(k)c_{XX}(\kappa -k) \\ -C_{YX}\freq&=H\freqC_{XX}\freq\\ -c_{YX}&=\int\limits_{-\pi}^\pi H\freqC_{XX}\freqe^{-j\omega\kappa}\frac{d\omega}{2\pi} +C_{YX}\freq &=H\freq C_{XX}\freq\\ +c_{YX}&=\int\limits_{-\pi}^\pi H\freq C_{XX}\freq e^{-j\omega\kappa}\frac{d\omega}{2\pi} \end{align} \end{subequations} @@ -541,7 +542,7 @@ c_{YX}&=\int\limits_{-\pi}^\pi H\freqC_{XX}\freqe^{-j\omega\kappa}\frac{d\omega} \begin{align} c_{Y_1Y_2}(\kappa)&=\sum\limits_{k=-\infty}^\infty \sum\limits_{l=-\infty}^\infty h_1(k)h_2(l)\cdot c_{X_1X_2}(\kappa -k+l) \\ c_{Y_1Y_2}(\kappa)&=h_1(\kappa)\star h_2(\kappa)^*\star c_{X_1X_2}(\kappa)\\ -C_{Y_1Y_2}\freq&=H_1\freqH_2\freq^*C_{X_1X_2}\freq +C_{Y_1Y_2}\freq &=H_1\freq H_2\freq^*C_{X_1X_2}\freq \end{align} \end{subequations} @@ -555,7 +556,7 @@ C_{YX}\freq&=C_{XX}\freq\prod\limits_{i=1}^L H_i\freq \end{subequations} \section{Matched Filter} -\subsection{Annahmen} +\subsection{Annahmen des Matched Filters} \begin{itemize} \item Das eingehende Signal $X(n)$ besteht entweder aus einem Signal mit Rauschen oder nur Rauschen:\begin{equation} X(n)=\begin{cases} @@ -567,9 +568,72 @@ C_{YX}\freq&=C_{XX}\freq\prod\limits_{i=1}^L H_i\freq \item $E[V(n)]=0$ und $C_{VV}\freq$ bekannt \end{itemize} +\subsection{Ziel des Matched Filters} +Maximierung des Signal-Rausch-Verhltnis: +\begin{equation} +\left(\frac{S}{N}\right)=\max\frac{|s_0(n_0)|^2}{\E[V_0(n_0)^2]} +\end{equation} + +\subsection{bertragungsfunktion des Matched Filters} +Sei $S\freq = \mathcal{F}\{s(n)\}$, $C_{VV}$ das Spektrum des Rauschens, + $n_0$ die Abtastunszeit, bei welcher $(S/N)$ berechnet wird, und $k$ eine reele Konstante +\begin{equation} +H\freq = k \frac{S\freq^*}{C_{VV}\freq}e^{-j\omega n_0} +\end{equation} +Dabei geht der Signalverlauf am Ende des Filters verloren und der Filter kann zur Signaldetektion genutzt werden +\subsection{Matched Filter fr Wei\ss es Rauschen} +Bei wei\ss em Rauschen wird die Impulsantwort des Filters zu +\begin{equation} +h(n) \equiv c \cdot s(n_0 - n) +\end{equation} +$\Rightarrow$ Die Impulsantwort des Filters ist das bekannte Signal ''rckwrts gespielt'' und um $n_0$ verschoben\\ +Der Signal zu Rausch Abstand ergibt sich dann zu: +\begin{equation} +\left(\frac{S}{N}\right)_{out}=\frac{E_s}{\sigma_V^2} +\end{equation} + + \section{Wiener Filter} +\subsection{Ziel des Wiener Filters} +Der Wiener Filter versucht die optimale Schtzung (nach (\ref{conv:mss})) eines Zufallsprozesses durch die Beobachtung eines anderen Prozesses +\subsection{Annahmen des Wiener Filters} +\includegraphics[width=\textwidth]{images/wienerfilter.JPG} +\begin{itemize} + \item $X(n)$ ist der zu schtzende Zufallsprozess + \item $Y(n)$ ist der betrachtete Zufallsprozess + \item $\epsilon(n)$ ist der Fehlerprozess + \item $X(n)$ und $Y(n)$ sind reelwertig, mittelwertfrei und \emph{gemeinsam stationr im weiteren Sinne} (\ref{jointstationary}) + \item Aufgrund der \emph{gemeinsamen Stationaritt im weiteren Sinne} (\ref{jointstationary}) der beiden Prozesse ist die Impulsantwort + $h(n)$ stabil und der Fehlerprozess $\epsilon(n)$ \emph{stationr im weiteren Sinne} (\ref{stationary}) +\end{itemize} +\subsection{Die bertragungsfunktion des Wiener Filters} +Enstehend aus den \emph{Wiener-Hopf-Gleichungen} +\begin{subequations} +\begin{alignat}{2} +c_{XY}(\kappa)&=h_{opt}(\kappa)\star C_{YY}(\kappa) \quad &\kappa \in \mathbb{Z} \\ +C_{XY}\freq &=G_{opt}\freq C_{YY}\freq \quad &\omega \in \mathbb{R} +\end{alignat} +\end{subequations} +erlangt man die optimale bertragungsfunktion: +\begin{equation} +H_{opt}\freq = \frac{C_{XY}\freq}{C_{YY}\freq} +\end{equation} + +\subsection{Mean Square Error des Wiener Filters} +\begin{subequations} +\begin{align} +q_{min}&=C_{XX}(0)-\sum\limits_{m=-\infty}^\infty h_{opt}(m)C_{XY}(m) \\ +q_{min}&=p(0) \quad \text{mit} \\ +p(\kappa) &= C_{XX}(\kappa)-h_{opt}(\kappa)\star c_{YX}(\kappa) \notag +\end{align} +\end{subequations} + +\subsection{Der Wiener Filter mit additivem Rasuchen} +\begin{equation} +H_{opt}\freq = \frac{C_{XX}\freq}{C_{XX}\freq + C_{VV}\freq} +\end{equation} \chapter{Sonstiges} \section{Spezielle Funktionen} \subsection{Gaussian white noise process} \label{whitenoise} @@ -589,4 +653,11 @@ S_{WW}(e^{j \omega})&=\sigma_W^2 0 &\kappa \neq 0 \end{cases} \end{equation} + + +\section{Mathematische ntzliche Formeln} +\subsection{Ungleichung von Schwarz} +\begin{equation} +\left|\int\limits_a^b \varphi_1(\omega)\varphi_2(\omega)d\omega \right|^2\leq \left( \int\limits_a^b |\varphi_1(\omega)|^2 d\omega \right)\cdot\left( \int\limits_a^b |\varphi_2(\omega)|^2 d\omega \right) +\end{equation} \end{document} diff --git a/images/wienerfilter.JPG b/images/wienerfilter.JPG new file mode 100644 index 0000000..effb0e8 Binary files /dev/null and b/images/wienerfilter.JPG differ