From d3fce671dff9e1fd9c35d780eb736ba495782d41 Mon Sep 17 00:00:00 2001
From: Daniel Thiem <info@daniel-thiem.de>
Date: Tue, 25 Sep 2012 10:05:38 +0200
Subject: [PATCH] Wiener-Filter und Doppelte Faltung

---
 document.pdf | 4594 +++++++++++++++++++++++++-------------------------
 document.tex |   13 +-
 2 files changed, 2336 insertions(+), 2271 deletions(-)

diff --git a/document.pdf b/document.pdf
index 463a16d..cfad4c9 100644
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 /Limits [(equation.4.5) (equation.4.6d)]
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->> endobj
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->> endobj
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->> endobj
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->> endobj
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->> endobj
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->> endobj
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-/Limits [(equation.6.4d) (page.0)]
->> endobj
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-/Limits [(page.1) (page.14)]
->> endobj
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-/Limits [(page.15) (page.2)]
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 >> endobj
 963 0 obj <<
-/Names [(page.20) 759 0 R (page.21) 771 0 R (page.22) 780 0 R (page.23) 805 0 R (page.24) 812 0 R (page.25) 829 0 R]
-/Limits [(page.20) (page.25)]
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+/Limits [(equation.5.18a) (equation.5.4)]
 >> endobj
 964 0 obj <<
-/Names [(page.26) 841 0 R (page.27) 851 0 R (page.28) 859 0 R (page.29) 868 0 R (page.3) 516 0 R (page.30) 878 0 R]
-/Limits [(page.26) (page.30)]
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 >> endobj
 965 0 obj <<
-/Names [(page.4) 555 0 R (page.5) 559 0 R (page.6) 564 0 R (page.7) 578 0 R (page.8) 596 0 R (page.9) 607 0 R]
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-/Limits [(section*.3) (section.1.5)]
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-/Limits [(section.1.6) (section.3.2)]
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-/Names [(section.6.2) 398 0 R (subsection.1.1.1) 10 0 R (subsection.1.1.2) 14 0 R (subsection.1.2.1) 22 0 R (subsection.1.2.2) 26 0 R (subsection.1.3.1) 34 0 R]
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 971 0 obj <<
-/Names [(subsection.1.3.2) 38 0 R (subsection.1.3.3) 42 0 R (subsection.1.5.1) 54 0 R (subsection.1.5.2) 58 0 R (subsection.1.6.1) 66 0 R (subsection.1.6.2) 70 0 R]
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 972 0 obj <<
-/Names [(subsection.1.7.1) 78 0 R (subsection.1.7.2) 82 0 R (subsection.1.7.3) 86 0 R (subsection.1.7.4) 90 0 R (subsection.1.7.5) 94 0 R (subsection.2.1.1) 106 0 R]
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 973 0 obj <<
-/Names [(subsection.2.1.2) 110 0 R (subsection.2.2.1) 118 0 R (subsection.2.2.2) 122 0 R (subsection.2.2.3) 126 0 R (subsection.2.2.4) 130 0 R (subsection.3.2.1) 146 0 R]
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-/Names [(subsection.3.6.3) 214 0 R (subsection.3.6.4) 218 0 R (subsection.3.6.5) 222 0 R (subsection.3.6.6) 226 0 R (subsection.4.1.1) 238 0 R (subsection.4.1.2) 242 0 R]
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 977 0 obj <<
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-/Limits [(equation.1.5b) (equation.3.19)]
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 986 0 obj <<
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-/Limits [(equation.3.2) (equation.4.3)]
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 987 0 obj <<
-/Kids [950 0 R 951 0 R 952 0 R 953 0 R 954 0 R 955 0 R]
-/Limits [(equation.4.3a) (equation.5.5a)]
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 >> endobj
 988 0 obj <<
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-/Limits [(equation.5.5b) (page.14)]
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 >> endobj
 989 0 obj <<
-/Kids [962 0 R 963 0 R 964 0 R 965 0 R 966 0 R 967 0 R]
-/Limits [(page.15) (section.3.2)]
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 >> endobj
 990 0 obj <<
-/Kids [968 0 R 969 0 R 970 0 R 971 0 R 972 0 R 973 0 R]
-/Limits [(section.3.3) (subsection.3.2.1)]
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 >> endobj
 991 0 obj <<
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-/Limits [(subsection.3.2.2) (subsection.5.1.8)]
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 >> endobj
 992 0 obj <<
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-/Limits [(subsection.5.2.1) (subsubsection*.9)]
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 >> endobj
 993 0 obj <<
-/Kids [984 0 R 985 0 R 986 0 R 987 0 R 988 0 R 989 0 R]
-/Limits [(Doc-Start) (section.3.2)]
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 994 0 obj <<
-/Kids [990 0 R 991 0 R 992 0 R]
-/Limits [(section.3.3) (subsubsection*.9)]
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 >> endobj
 995 0 obj <<
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-/Limits [(Doc-Start) (subsubsection*.9)]
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 >> endobj
 996 0 obj <<
-/Dests 995 0 R
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 >> endobj
 997 0 obj <<
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-/Pages 930 0 R
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-/PageMode/UseOutlines/PageLabels<</Nums[0<</P(0)>>1<</S/D>>]>>
-/OpenAction 413 0 R
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 998 0 obj <<
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diff --git a/document.tex b/document.tex
index 0e8d8b6..68d962e 100644
--- a/document.tex
+++ b/document.tex
@@ -22,7 +22,7 @@
 
 \title{Stochastische Signale und Systeme}
 \subtitle{Zusammenfassung Formeln}
-\subsubtitle{Autor: Daniel Thiem - studium@daniel-thiem.de\\Version 0.9.5.1 - 24.09.2012}
+\subsubtitle{Autor: Daniel Thiem - studium@daniel-thiem.de\\Version 0.9.6 - 24.09.2012}
 
 \maketitle
 \newpage
@@ -665,6 +665,12 @@ H_{opt}\freq = \frac{C_{XY}\freq}{C_{YY}\freq}
 \end{equation}
 
 \subsection{Mean Square Error des Wiener Filters}
+Der Mean Square Error ist als der Erwartungswert des quadrates der Fehlerfunktion definiert
+\begin{align}
+q(h)&=\E[\epsilon_{X}^2(n)] \\
+h_{opt} &= \arg \min_h q(h) ,n\in \mathbb{Z}
+\end{align}
+Daraus folgt:
 \begin{subequations}
 \begin{align}
 q_{min}&=C_{XX}(0)-\sum\limits_{m=-\infty}^\infty h_{opt}(m)C_{XY}(m) \\
@@ -729,6 +735,9 @@ S_{WW}(e^{j \omega})&=\sigma_W^2
 |H\freq|^2=H\freq H(e^{-j \omega})
 \end{equation}
 
-
+\subsection{Doppelte Faltungssumme}
+\begin{equation}
+\sum\limits_{m=-\infty}^\infty \sum\limits_{k=-\infty}^\infty h(m)h(k)f(k-m) = h(n)\star f(0) \star h(-n)
+\end{equation}
 
 \end{document}
\ No newline at end of file