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-rw-r--r--buch/papers/fm/01_AM-FM.tex41
-rw-r--r--buch/papers/fm/02_frequenzyspectrum.tex2
-rw-r--r--buch/papers/fm/03_bessel.tex24
-rw-r--r--buch/papers/fm/04_fazit.tex32
-rw-r--r--buch/papers/fm/FM presentation/FM_presentation.pdf (renamed from buch/papers/fm/RS presentation/FM_presentation.pdf)bin357597 -> 357597 bytes
-rw-r--r--buch/papers/fm/FM presentation/FM_presentation.tex (renamed from buch/papers/fm/RS presentation/FM_presentation.tex)4
-rw-r--r--buch/papers/fm/FM presentation/README.txt1
-rw-r--r--buch/papers/fm/FM presentation/images/100HZ.png (renamed from buch/papers/fm/RS presentation/images/100HZ.png)bin8601 -> 8601 bytes
-rw-r--r--buch/papers/fm/FM presentation/images/200HZ.png (renamed from buch/papers/fm/RS presentation/images/200HZ.png)bin8502 -> 8502 bytes
-rw-r--r--buch/papers/fm/FM presentation/images/300HZ.png (renamed from buch/papers/fm/RS presentation/images/300HZ.png)bin9059 -> 9059 bytes
-rw-r--r--buch/papers/fm/FM presentation/images/400HZ.png (renamed from buch/papers/fm/RS presentation/images/400HZ.png)bin9949 -> 9949 bytes
-rw-r--r--buch/papers/fm/FM presentation/images/bessel.png (renamed from buch/papers/fm/RS presentation/images/bessel.png)bin40393 -> 40393 bytes
-rw-r--r--buch/papers/fm/FM presentation/images/bessel2.png (renamed from buch/papers/fm/RS presentation/images/bessel2.png)bin102494 -> 102494 bytes
-rw-r--r--buch/papers/fm/FM presentation/images/bessel_beta1.png (renamed from buch/papers/fm/RS presentation/images/bessel_beta1.png)bin40696 -> 40696 bytes
-rw-r--r--buch/papers/fm/FM presentation/images/bessel_frequenz.png (renamed from buch/papers/fm/RS presentation/images/bessel_frequenz.png)bin11264 -> 11264 bytes
-rw-r--r--buch/papers/fm/FM presentation/images/beta_0.001.png (renamed from buch/papers/fm/RS presentation/images/beta_0.001.png)bin6233 -> 6233 bytes
-rw-r--r--buch/papers/fm/FM presentation/images/beta_0.1.png (renamed from buch/papers/fm/RS presentation/images/beta_0.1.png)bin6630 -> 6630 bytes
-rw-r--r--buch/papers/fm/FM presentation/images/beta_0.5.png (renamed from buch/papers/fm/RS presentation/images/beta_0.5.png)bin8167 -> 8167 bytes
-rw-r--r--buch/papers/fm/FM presentation/images/beta_1.png (renamed from buch/papers/fm/RS presentation/images/beta_1.png)bin11303 -> 11303 bytes
-rw-r--r--buch/papers/fm/FM presentation/images/beta_2.png (renamed from buch/papers/fm/RS presentation/images/beta_2.png)bin14703 -> 14703 bytes
-rw-r--r--buch/papers/fm/FM presentation/images/beta_3.png (renamed from buch/papers/fm/RS presentation/images/beta_3.png)bin20377 -> 20377 bytes
-rw-r--r--buch/papers/fm/FM presentation/images/fm_10Hz.png (renamed from buch/papers/fm/RS presentation/images/fm_10Hz.png)bin6781 -> 6781 bytes
-rw-r--r--buch/papers/fm/FM presentation/images/fm_20hz.png (renamed from buch/papers/fm/RS presentation/images/fm_20hz.png)bin7834 -> 7834 bytes
-rw-r--r--buch/papers/fm/FM presentation/images/fm_30Hz.png (renamed from buch/papers/fm/RS presentation/images/fm_30Hz.png)bin8601 -> 8601 bytes
-rw-r--r--buch/papers/fm/FM presentation/images/fm_3Hz.png (renamed from buch/papers/fm/RS presentation/images/fm_3Hz.png)bin6558 -> 6558 bytes
-rw-r--r--buch/papers/fm/FM presentation/images/fm_40Hz.png (renamed from buch/papers/fm/RS presentation/images/fm_40Hz.png)bin8795 -> 8795 bytes
-rw-r--r--buch/papers/fm/FM presentation/images/fm_5Hz.png (renamed from buch/papers/fm/RS presentation/images/fm_5Hz.png)bin5766 -> 5766 bytes
-rw-r--r--buch/papers/fm/FM presentation/images/fm_7Hz.png (renamed from buch/papers/fm/RS presentation/images/fm_7Hz.png)bin6337 -> 6337 bytes
-rw-r--r--buch/papers/fm/FM presentation/images/fm_frequenz.png (renamed from buch/papers/fm/RS presentation/images/fm_frequenz.png)bin11042 -> 11042 bytes
-rw-r--r--buch/papers/fm/FM presentation/images/fm_in_time.png (renamed from buch/papers/fm/RS presentation/images/fm_in_time.png)bin27400 -> 27400 bytes
-rw-r--r--buch/papers/fm/Quellen/A2-14.pdfbin0 -> 259673 bytes
-rw-r--r--buch/papers/fm/Quellen/FM_presentation.pdfbin0 -> 357597 bytes
-rw-r--r--buch/papers/fm/Quellen/Frequency modulation (FM) and Bessel functions.pdf (renamed from buch/papers/fm/RS presentation/Frequency modulation (FM) and Bessel functions.pdf)bin159598 -> 159598 bytes
-rw-r--r--buch/papers/fm/Quellen/Seydel2022_Book_HöhereMathematikImAlltag.pdfbin0 -> 4118379 bytes
-rw-r--r--buch/papers/fm/RS presentation/README.txt1
-rw-r--r--buch/papers/fm/RS presentation/RS.tex123
-rw-r--r--buch/papers/fm/main.tex2
-rw-r--r--buch/papers/fm/references.bib11
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
Binary files differ
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
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index f6836bd..f6836bd 100644
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index 6762c1a..6762c1a 100644
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index ccda3f9..ccda3f9 100644
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index 1f5c47e..1f5c47e 100644
--- a/buch/papers/fm/RS presentation/images/bessel_beta1.png
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index 4f228b9..4f228b9 100644
--- a/buch/papers/fm/RS presentation/images/bessel_frequenz.png
+++ b/buch/papers/fm/FM presentation/images/bessel_frequenz.png
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index 7e4e276..7e4e276 100644
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index e7722b3..e7722b3 100644
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index 6d3535c..6d3535c 100644
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index 6930eae..6930eae 100644
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index c6df82c..c6df82c 100644
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index 126ecf3..126ecf3 100644
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index 371b9bf..371b9bf 100644
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index d4098af..d4098af 100644
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index b3dd7e3..b3dd7e3 100644
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+++ b/buch/papers/fm/FM presentation/images/fm_7Hz.png
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index 26bfd86..26bfd86 100644
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+++ b/buch/papers/fm/FM presentation/images/fm_frequenz.png
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index 068eafc..068eafc 100644
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new file mode 100644
index 0000000..7348cca
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new file mode 100644
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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
Binary files differ
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
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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 },