Merge pull request #31 from JODBaer/master

Kapitel umschreibungen

38 files changed, 54 insertions, 187 deletions

diff --git a/buch/papers/fm/01_AM-FM.tex b/buch/papers/fm/01_AM-FM.tex
index ef55d55..163c792 100644
--- a/buch/papers/fm/01_AM-FM.tex
+++ b/buch/papers/fm/01_AM-FM.tex
@@ -7,30 +7,41 @@
 \rhead{AM- FM}
 
 Das sinusförmige Trägersignal hat die übliche Form: 
-\(x_c(t) = A_c \cdot cos(\omega_c(t)+\varphi)\).
+\(x_c(t) = A_c \cdot \cos(\omega_c(t)+\varphi)\).
 Wobei die konstanten Amplitude \(A_c\) und Phase \(\varphi\) vom Nachrichtensignal \(m(t)\) verändert wird.
 Der Parameter \(\omega_c\), die Trägerkreisfrequenz bzw. die Trägerfrequenz \(f_c = \frac{\omega_c}{2\pi}\),
 steht nicht für die modulation zur verfügung, statt dessen kann durch ihn die Frequenzachse frei gewählt werden.
 \newblockpunct
-Jedoch ist das für die Vilfalt der Modulationsarten keine Einschrenkung.
+Jedoch ist das für die Vielfalt der Modulationsarten keine Einschrenkung.
 Ein Nachrichtensignal kann auch über die Momentanfrequenz (instantenous frequency) \(\omega_i\) eines trägers verändert werden.
 Mathematisch wird dann daraus
 \[
     \omega_i = \omega_c + \frac{d \varphi(t)}{dt}
 \]
-mit der Ableitung der Phase.
+mit der Ableitung der Phase\cite{fm:NAT}.
+Mit diesen drei parameter ergeben sich auch drei modulationsarten, die Amplitudenmodulation welche \(A_c\) benutzt, 
+die Phasenmodulation \(\varphi\) und dann noch die Momentankreisfrequenz \(\omega_i\):
 \newline
 \newline
-TODO:
-Hier beschrieib ich was AmplitudenModulation ist und mache dan den link zu Frequenzmodulation inkl Formel \[cos( cos x)\]
-
-
-
-%Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam
-%nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam
-erat, sed diam voluptua \cite{fm:bibtex}.
-%At vero eos et accusam et justo duo dolores et ea rebum.
-%Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum
-%dolor sit amet.
-
+To do: Bilder jeder Modulationsart
 
+\subsection{AM - Amplitudenmodulation}
+Das Ziel ist FM zu verstehen doch dazu wird zuerst AM erklärt welches einwenig einfacher zu verstehen ist und erst dann übertragen wir die Ideeen in FM.
+Nun zur Amplitudenmodulation verwenden wir das bevorzugte Trägersignal
+\[
+    x_c(t) = A_c \cdot \cos(\omega_ct).
+\]
+Dies bringt den grossen Vorteil das, dass modulierend Signal sämtliche Anteile im Frequenzspektrum inanspruch nimmt 
+und das Trägersignal nur  zwei komplexe Schwingungen besitzt. 
+Dies sieht man besonders in der Eulerischen Formel
+\[
+    x_c(t) = \frac{A_c}{2} \cdot e^{j\omega_ct}\;+\;\frac{A_c}{2} \cdot e^{-j\omega_ct}.
+\]
+Dabei ist die negative Frequenz der zweiten komplexen Schwingung zwingend erforderlich, damit in der Summe immer ein reelwertiges Trägersignal ergibt.
+Nun wird der parameter \(A_c\) durch das  Moduierende Signal \(m(t)\) ersetzt, wobei so \(m(t) \leqslant |1|\) normiert wurde.
+\newline
+\newline
+TODO:
+Hier beschrieib ich was AmplitudenModulation ist und mache dan den link zu Frequenzmodulation inkl Formel \[\cos( \cos x)\]
+so wird beschrieben das daraus eigentlich \(x_c(t) = A_c \cdot \cos(\omega_i)\) wird und somit \(x_c(t) = A_c \cdot \cos(\omega_c + \frac{d \varphi(t)}{dt})\).
+Da \(\sin \) abgeleitet \(\cos \) ergibt, so wird aus dem \(m(t)\) ein \( \frac{d \varphi(t)}{dt}\)  in der momentan frequenz. \[ \Rightarrow \cos( \cos x) \]
diff --git a/buch/papers/fm/02_frequenzyspectrum.tex b/buch/papers/fm/02_frequenzyspectrum.tex
index 1c6044d..80e1c65 100644
--- a/buch/papers/fm/02_frequenzyspectrum.tex
+++ b/buch/papers/fm/02_frequenzyspectrum.tex
@@ -7,7 +7,9 @@
 \label{fm:section:teil1}}
 \rhead{Problemstellung}
 
+TODO
 Hier Beschreiben ich das Frequenzspektrum und wie AM und FM aussehen und generiert werden.
+Somit auch die Herleitung des Frequenzspektrum.
 %Sed ut perspiciatis unde omnis iste natus error sit voluptatem
 %accusantium doloremque laudantium, totam rem aperiam, eaque ipsa
 %quae ab illo inventore veritatis et quasi architecto beatae vitae
diff --git a/buch/papers/fm/03_bessel.tex b/buch/papers/fm/03_bessel.tex
index fdaa0d1..aed084e 100644
--- a/buch/papers/fm/03_bessel.tex
+++ b/buch/papers/fm/03_bessel.tex
@@ -7,22 +7,16 @@
 \label{fm:section:teil2}}
 \rhead{Teil 2}
 
+
+TODO
 Hier wird beschrieben wie die Bessel Funktion der FM im Frequenzspektrum hilft, wieso diese gebrauch wird und ihre Vorteile.
-%Sed ut perspiciatis unde omnis iste natus error sit voluptatem
-%accusantium doloremque laudantium, totam rem aperiam, eaque ipsa
-%quae ab illo inventore veritatis et quasi architecto beatae vitae
-%dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit
-%aspernatur aut odit aut fugit, sed quia consequuntur magni dolores
-%eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam
-%est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci
-%velit, sed quia non numquam eius modi tempora incidunt ut labore
-%et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima
-%veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam,
-%nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure
-%reprehenderit qui in ea voluptate velit esse quam nihil molestiae
-%consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla
-%pariatur?
-%
+\begin{itemize}
+    \item Zuerest einmal die Herleitung von FM zu der Besselfunktion
+    \item Im Frequenzspektrum darstellen mit Farben, ersichtlich machen. 
+    \item Parameter tuing der Trägerfrequenz, Modulierende frequenz und Beta. 
+\end{itemize}
+
+
 %\subsection{De finibus bonorum et malorum
 %\label{fm:subsection:bonorum}}
 
diff --git a/buch/papers/fm/04_fazit.tex b/buch/papers/fm/04_fazit.tex
index 8c6c002..8d5eca4 100644
--- a/buch/papers/fm/04_fazit.tex
+++ b/buch/papers/fm/04_fazit.tex
@@ -6,35 +6,7 @@
 \section{Fazit
 \label{fm:section:fazit}}
 \rhead{Zusamenfassend}
-%Sed ut perspiciatis unde omnis iste natus error sit voluptatem
-%accusantium doloremque laudantium, totam rem aperiam, eaque ipsa
-%quae ab illo inventore veritatis et quasi architecto beatae vitae
-%dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit
-%aspernatur aut odit aut fugit, sed quia consequuntur magni dolores
-%eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam
-%est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci
-%velit, sed quia non numquam eius modi tempora incidunt ut labore
-%et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima
-%veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam,
-%nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure
-%reprehenderit qui in ea voluptate velit esse quam nihil molestiae
-%consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla
-%pariatur?
-%
-%\subsection{De finibus bonorum et malorum
-%\label{fm:subsection:malorum}}
-%At vero eos et accusamus et iusto odio dignissimos ducimus qui
-%blanditiis praesentium voluptatum deleniti atque corrupti quos
-%dolores et quas molestias excepturi sint occaecati cupiditate non
-%provident, similique sunt in culpa qui officia deserunt mollitia
-%animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis
-%est et expedita distinctio. Nam libero tempore, cum soluta nobis
-%est eligendi optio cumque nihil impedit quo minus id quod maxime
-%placeat facere possimus, omnis voluptas assumenda est, omnis dolor
-%repellendus. Temporibus autem quibusdam et aut officiis debitis aut
-%rerum necessitatibus saepe eveniet ut et voluptates repudiandae
-%sint et molestiae non recusandae. Itaque earum rerum hic tenetur a
-%sapiente delectus, ut aut reiciendis voluptatibus maiores alias
-%consequatur aut perferendis doloribus asperiores repellat.
+
+TODO  Anwendungen erklären und Sinn des Ganzen.
 
 
diff --git a/buch/papers/fm/RS presentation/FM_presentation.pdf b/buch/papers/fm/FM presentation/FM_presentation.pdf
index 496e35e..496e35e 100644
--- a/buch/papers/fm/RS presentation/FM_presentation.pdf
+++ b/buch/papers/fm/FM presentation/FM_presentation.pdf
diff --git a/buch/papers/fm/RS presentation/FM_presentation.tex b/buch/papers/fm/FM presentation/FM_presentation.tex
index 92cb501..2801e69 100644
--- a/buch/papers/fm/RS presentation/FM_presentation.tex
+++ b/buch/papers/fm/FM presentation/FM_presentation.tex
@@ -1,4 +1,4 @@
-%% !TeX root = RS.tex
+%% !TeX root = .tex
 
 \documentclass[11pt,aspectratio=169]{beamer}
 \usepackage[utf8]{inputenc}
@@ -15,7 +15,7 @@
 	\logo{}
 	\institute{OST Ostschweizer Fachhochschule}
 	\date{16.5.2022}
-	\subject{Mathematisches Seminar}
+	\subject{Mathematisches Seminar - Spezielle Funktionen}
 	%\setbeamercovered{transparent}
 	\setbeamercovered{invisible}
 	\setbeamertemplate{navigation symbols}{}
diff --git a/buch/papers/fm/FM presentation/README.txt b/buch/papers/fm/FM presentation/README.txt
new file mode 100644
index 0000000..65f390d
--- /dev/null
+++ b/buch/papers/fm/FM presentation/README.txt
@@ -0,0 +1 @@
+Dies ist die Presentation des FM - Bessel
\ No newline at end of file
diff --git a/buch/papers/fm/RS presentation/images/100HZ.png b/buch/papers/fm/FM presentation/images/100HZ.png
index 371b9bf..371b9bf 100644
--- a/buch/papers/fm/RS presentation/images/100HZ.png
+++ b/buch/papers/fm/FM presentation/images/100HZ.png
diff --git a/buch/papers/fm/RS presentation/images/200HZ.png b/buch/papers/fm/FM presentation/images/200HZ.png
index f6836bd..f6836bd 100644
--- a/buch/papers/fm/RS presentation/images/200HZ.png
+++ b/buch/papers/fm/FM presentation/images/200HZ.png
diff --git a/buch/papers/fm/RS presentation/images/300HZ.png b/buch/papers/fm/FM presentation/images/300HZ.png
index 6762c1a..6762c1a 100644
--- a/buch/papers/fm/RS presentation/images/300HZ.png
+++ b/buch/papers/fm/FM presentation/images/300HZ.png
diff --git a/buch/papers/fm/RS presentation/images/400HZ.png b/buch/papers/fm/FM presentation/images/400HZ.png
index 236c428..236c428 100644
--- a/buch/papers/fm/RS presentation/images/400HZ.png
+++ b/buch/papers/fm/FM presentation/images/400HZ.png
diff --git a/buch/papers/fm/RS presentation/images/bessel.png b/buch/papers/fm/FM presentation/images/bessel.png
index f4c83ea..f4c83ea 100644
--- a/buch/papers/fm/RS presentation/images/bessel.png
+++ b/buch/papers/fm/FM presentation/images/bessel.png
diff --git a/buch/papers/fm/RS presentation/images/bessel2.png b/buch/papers/fm/FM presentation/images/bessel2.png
index ccda3f9..ccda3f9 100644
--- a/buch/papers/fm/RS presentation/images/bessel2.png
+++ b/buch/papers/fm/FM presentation/images/bessel2.png
diff --git a/buch/papers/fm/RS presentation/images/bessel_beta1.png b/buch/papers/fm/FM presentation/images/bessel_beta1.png
index 1f5c47e..1f5c47e 100644
--- a/buch/papers/fm/RS presentation/images/bessel_beta1.png
+++ b/buch/papers/fm/FM presentation/images/bessel_beta1.png
diff --git a/buch/papers/fm/RS presentation/images/bessel_frequenz.png b/buch/papers/fm/FM presentation/images/bessel_frequenz.png
index 4f228b9..4f228b9 100644
--- a/buch/papers/fm/RS presentation/images/bessel_frequenz.png
+++ b/buch/papers/fm/FM presentation/images/bessel_frequenz.png
diff --git a/buch/papers/fm/RS presentation/images/beta_0.001.png b/buch/papers/fm/FM presentation/images/beta_0.001.png
index 7e4e276..7e4e276 100644
--- a/buch/papers/fm/RS presentation/images/beta_0.001.png
+++ b/buch/papers/fm/FM presentation/images/beta_0.001.png
diff --git a/buch/papers/fm/RS presentation/images/beta_0.1.png b/buch/papers/fm/FM presentation/images/beta_0.1.png
index e7722b3..e7722b3 100644
--- a/buch/papers/fm/RS presentation/images/beta_0.1.png
+++ b/buch/papers/fm/FM presentation/images/beta_0.1.png
diff --git a/buch/papers/fm/RS presentation/images/beta_0.5.png b/buch/papers/fm/FM presentation/images/beta_0.5.png
index 5261b43..5261b43 100644
--- a/buch/papers/fm/RS presentation/images/beta_0.5.png
+++ b/buch/papers/fm/FM presentation/images/beta_0.5.png
diff --git a/buch/papers/fm/RS presentation/images/beta_1.png b/buch/papers/fm/FM presentation/images/beta_1.png
index 6d3535c..6d3535c 100644
--- a/buch/papers/fm/RS presentation/images/beta_1.png
+++ b/buch/papers/fm/FM presentation/images/beta_1.png
diff --git a/buch/papers/fm/RS presentation/images/beta_2.png b/buch/papers/fm/FM presentation/images/beta_2.png
index 6930eae..6930eae 100644
--- a/buch/papers/fm/RS presentation/images/beta_2.png
+++ b/buch/papers/fm/FM presentation/images/beta_2.png
diff --git a/buch/papers/fm/RS presentation/images/beta_3.png b/buch/papers/fm/FM presentation/images/beta_3.png
index c6df82c..c6df82c 100644
--- a/buch/papers/fm/RS presentation/images/beta_3.png
+++ b/buch/papers/fm/FM presentation/images/beta_3.png
diff --git a/buch/papers/fm/RS presentation/images/fm_10Hz.png b/buch/papers/fm/FM presentation/images/fm_10Hz.png
index 51bddc7..51bddc7 100644
--- a/buch/papers/fm/RS presentation/images/fm_10Hz.png
+++ b/buch/papers/fm/FM presentation/images/fm_10Hz.png
diff --git a/buch/papers/fm/RS presentation/images/fm_20hz.png b/buch/papers/fm/FM presentation/images/fm_20hz.png
index 126ecf3..126ecf3 100644
--- a/buch/papers/fm/RS presentation/images/fm_20hz.png
+++ b/buch/papers/fm/FM presentation/images/fm_20hz.png
diff --git a/buch/papers/fm/RS presentation/images/fm_30Hz.png b/buch/papers/fm/FM presentation/images/fm_30Hz.png
index 371b9bf..371b9bf 100644
--- a/buch/papers/fm/RS presentation/images/fm_30Hz.png
+++ b/buch/papers/fm/FM presentation/images/fm_30Hz.png
diff --git a/buch/papers/fm/RS presentation/images/fm_3Hz.png b/buch/papers/fm/FM presentation/images/fm_3Hz.png
index d4098af..d4098af 100644
--- a/buch/papers/fm/RS presentation/images/fm_3Hz.png
+++ b/buch/papers/fm/FM presentation/images/fm_3Hz.png
diff --git a/buch/papers/fm/RS presentation/images/fm_40Hz.png b/buch/papers/fm/FM presentation/images/fm_40Hz.png
index 4cf11d4..4cf11d4 100644
--- a/buch/papers/fm/RS presentation/images/fm_40Hz.png
+++ b/buch/papers/fm/FM presentation/images/fm_40Hz.png
diff --git a/buch/papers/fm/RS presentation/images/fm_5Hz.png b/buch/papers/fm/FM presentation/images/fm_5Hz.png
index e495b5c..e495b5c 100644
--- a/buch/papers/fm/RS presentation/images/fm_5Hz.png
+++ b/buch/papers/fm/FM presentation/images/fm_5Hz.png
diff --git a/buch/papers/fm/RS presentation/images/fm_7Hz.png b/buch/papers/fm/FM presentation/images/fm_7Hz.png
index b3dd7e3..b3dd7e3 100644
--- a/buch/papers/fm/RS presentation/images/fm_7Hz.png
+++ b/buch/papers/fm/FM presentation/images/fm_7Hz.png
diff --git a/buch/papers/fm/RS presentation/images/fm_frequenz.png b/buch/papers/fm/FM presentation/images/fm_frequenz.png
index 26bfd86..26bfd86 100644
--- a/buch/papers/fm/RS presentation/images/fm_frequenz.png
+++ b/buch/papers/fm/FM presentation/images/fm_frequenz.png
diff --git a/buch/papers/fm/RS presentation/images/fm_in_time.png b/buch/papers/fm/FM presentation/images/fm_in_time.png
index 068eafc..068eafc 100644
--- a/buch/papers/fm/RS presentation/images/fm_in_time.png
+++ b/buch/papers/fm/FM presentation/images/fm_in_time.png
diff --git a/buch/papers/fm/Quellen/A2-14.pdf b/buch/papers/fm/Quellen/A2-14.pdf
new file mode 100644
index 0000000..7348cca
--- /dev/null
+++ b/buch/papers/fm/Quellen/A2-14.pdf
diff --git a/buch/papers/fm/Quellen/FM_presentation.pdf b/buch/papers/fm/Quellen/FM_presentation.pdf
new file mode 100644
index 0000000..496e35e
--- /dev/null
+++ b/buch/papers/fm/Quellen/FM_presentation.pdf
diff --git a/buch/papers/fm/RS presentation/Frequency modulation (FM) and Bessel functions.pdf b/buch/papers/fm/Quellen/Frequency modulation (FM) and Bessel functions.pdf
index a6e701c..a6e701c 100644
--- a/buch/papers/fm/RS presentation/Frequency modulation (FM) and Bessel functions.pdf
+++ b/buch/papers/fm/Quellen/Frequency modulation (FM) and Bessel functions.pdf
diff --git a/buch/papers/fm/Quellen/Seydel2022_Book_HöhereMathematikImAlltag.pdf b/buch/papers/fm/Quellen/Seydel2022_Book_HöhereMathematikImAlltag.pdf
new file mode 100644
index 0000000..2a0bddd
--- /dev/null
+++ b/buch/papers/fm/Quellen/Seydel2022_Book_HöhereMathematikImAlltag.pdf
diff --git a/buch/papers/fm/RS presentation/README.txt b/buch/papers/fm/RS presentation/README.txt
deleted file mode 100644
index 4d0620f..0000000
--- a/buch/papers/fm/RS presentation/README.txt
+++ /dev/null
@@ -1 +0,0 @@
-Dies ist die Presentation des Reed-Solomon-Code
\ No newline at end of file
diff --git a/buch/papers/fm/RS presentation/RS.tex b/buch/papers/fm/RS presentation/RS.tex
deleted file mode 100644
index 8a67619..0000000
--- a/buch/papers/fm/RS presentation/RS.tex
+++ /dev/null
@@ -1,123 +0,0 @@
-%% !TeX root = RS.tex
-
-\documentclass[11pt,aspectratio=169]{beamer}
-\usepackage[utf8]{inputenc}
-\usepackage[T1]{fontenc}
-\usepackage{lmodern}
-\usepackage[ngerman]{babel}
-\usepackage{tikz}
-\usetheme{Hannover}
-
-\begin{document}
-	\author{Joshua Bär}
-	\title{FM - Bessel}
-	\subtitle{}
-	\logo{}
-	\institute{OST Ostschweizer Fachhochschule}
-	\date{16.5.2022}
-	\subject{Mathematisches Seminar- Spezielle Funktionen}
-	%\setbeamercovered{transparent}
-	\setbeamercovered{invisible}
-	\setbeamertemplate{navigation symbols}{}
-	\begin{frame}[plain]
-		\maketitle
-	\end{frame}
-%-------------------------------------------------------------------------------	
-\section{Einführung}
-	\begin{frame}
-		\frametitle{Frequenzmodulation}
-		
-		\visible<1->{\begin{equation} \cos(\omega_c t+\beta\sin(\omega_mt))\end{equation}}
-		
-		\only<2>{\includegraphics[scale= 0.7]{images/fm_in_time.png}}
-		\only<3>{\includegraphics[scale= 0.7]{images/fm_frequenz.png}}
-		\only<4>{\includegraphics[scale= 0.7]{images/bessel_frequenz.png}}
-		
-		
-	\end{frame}
-%-------------------------------------------------------------------------------	
-\section{Proof}
-\begin{frame}
-	\frametitle{Bessel}
-
-	\visible<1->{\begin{align}
-			\cos(\beta\sin\varphi)
-			&=
-			J_0(\beat) + 2\sum_{m=1}^\infty J_{2m}(\beta) \cos(2m\varphi)
-			\\
-			\sin(\beta\sin\varphi)
-			&=
-			J_0(\beat) + 2\sum_{m=1}^\infty J_{2m}(\beta) \cos(2m\varphi)
-			\\
-			J_{-n}(\beat) &= (-1)^n J_n(\beta)
-		\end{align}}
-	\visible<2->{\begin{align}
-			\cos(A + B) 
-			&= 
-			\cos(A)\cos(B)-\sin(A)\sin(B)
-			\\
-			2\cos (A)\cos (B)
-			&=
-			\cos(A-B)+\cos(A+B)
-			\\
-			2\sin(A)\sin(B)
-			&=
-			\cos(A-B)-\cos(A+B)
-		\end{align}}
-\end{frame}
-
-%-------------------------------------------------------------------------------	
-\begin{frame}
-		\frametitle{Prof->Done}
-		\begin{align}
-			\cos(\omega_ct+\beta\sin(\omega_mt))
-			&=
-			\sum_{k= -\infty}^\infty J_{k}(\beta) \cos((\omega_c+k\omgea_m)t)
-		\end{align}
-	\end{frame}
-%-------------------------------------------------------------------------------	
-	\begin{frame}
-	\begin{figure}
-		\only<1>{\includegraphics[scale = 0.75]{images/fm_frequenz.png}}
-		\only<2>{\includegraphics[scale = 0.75]{images/bessel_frequenz.png}}
-	\end{figure}
-	\end{frame}
-%-------------------------------------------------------------------------------		
-\section{Input Parameter}
-	\begin{frame}
-	\frametitle{Träger-Frequenz Parameter}
-	\onslide<1->{\begin{equation}\cos(\omega_ct+\beta\sin(\omega_mt))\end{equation}}
-	\only<1>{\includegraphics[scale=0.75]{images/100HZ.png}}
-	\only<2>{\includegraphics[scale=0.75]{images/200HZ.png}}
-	\only<3>{\includegraphics[scale=0.75]{images/300HZ.png}}
-	\only<4>{\includegraphics[scale=0.75]{images/400HZ.png}}
-	\end{frame}
-%-------------------------------------------------------------------------------
-\begin{frame}
-\frametitle{Modulations-Frequenz Parameter}
-\onslide<1->{\begin{equation}\cos(\omega_ct+\beta\sin(\omega_mt))\end{equation}}
-\only<1>{\includegraphics[scale=0.75]{images/fm_3Hz.png}}
-\only<2>{\includegraphics[scale=0.75]{images/fm_5Hz.png}}
-\only<3>{\includegraphics[scale=0.75]{images/fm_7Hz.png}}
-\only<4>{\includegraphics[scale=0.75]{images/fm_10Hz.png}}
-\only<5>{\includegraphics[scale=0.75]{images/fm_20Hz.png}}
-\only<6>{\includegraphics[scale=0.75]{images/fm_30Hz.png}}
-\end{frame}	
-%-------------------------------------------------------------------------------
-\begin{frame}
-\frametitle{Beta Parameter}
-	\onslide<1->{\begin{equation}\sum_{k= -\infty}^\infty J_{k}(\beta) \cos((\omega_c+k\omgea_m)t)\end{equation}}
-	\only<1>{\includegraphics[scale=0.7]{images/beta_0.001.png}}
-	\only<2>{\includegraphics[scale=0.7]{images/beta_0.1.png}}
-	\only<3>{\includegraphics[scale=0.7]{images/beta_0.5.png}}
-	\only<4>{\includegraphics[scale=0.7]{images/beta_1.png}}
-	\only<5>{\includegraphics[scale=0.7]{images/beta_2.png}}
-	\only<6>{\includegraphics[scale=0.7]{images/beta_3.png}}
-	\only<7>{\includegraphics[scale=0.7]{images/bessel.png}}
-\end{frame}	
-%-------------------------------------------------------------------------------
-\begin{frame}
-	\includegraphics[scale=0.5]{images/beta_1.png}
-	\includegraphics[scale=0.5]{images/bessel.png}
-\end{frame}	
-\end{document}
diff --git a/buch/papers/fm/main.tex b/buch/papers/fm/main.tex
index fcf4d1a..6af3386 100644
--- a/buch/papers/fm/main.tex
+++ b/buch/papers/fm/main.tex
@@ -27,7 +27,7 @@ welches Digital einfach umzusetzten ist,
 genauso als Trägersignal genutzt werden kann.
 Zuerst wird erklärt was \textit{FM-AM} ist, danach wie sich diese im Frequenzspektrum verhalten.
 Erst dann erklär ich dir wie die Besselfunktion mit der Frequenzmodulation( acro?) zusammenhängt.
-Nun zur Modulation im nächsten Abschnitt.
+Nun zur Modulation im nächsten Abschnitt.\cite{fm:NAT}
 
 \input{papers/fm/01_AM-FM.tex}
 \input{papers/fm/02_frequenzyspectrum.tex}
diff --git a/buch/papers/fm/references.bib b/buch/papers/fm/references.bib
index 76eb265..21b910b 100644
--- a/buch/papers/fm/references.bib
+++ b/buch/papers/fm/references.bib
@@ -23,6 +23,17 @@
 	volume = {2}
 }
 
+@book{fm:NAT,
+	title = {Nachrichtentechnik 1 + 2},
+	author = {Thomas Kneubühler},
+	publisher = {None},
+	year = {2021},
+	isbn = {},
+	inseries = {Script for students},
+	volume = {}
+}
+
+
 @article{fm:mendezmueller,
         author = { Tabea Méndez and Andreas Müller },
         title = { Noncommutative harmonic analysis and image registration },