presentation FM-Bessel

28 files changed, 211 insertions, 124 deletions

diff --git a/buch/papers/fm/RS presentation/FM_presentation.pdf b/buch/papers/fm/RS presentation/FM_presentation.pdf
new file mode 100644
index 0000000..496e35e
--- /dev/null
+++ b/buch/papers/fm/RS presentation/FM_presentation.pdf
diff --git a/buch/papers/fm/RS presentation/FM_presentation.tex b/buch/papers/fm/RS presentation/FM_presentation.tex
new file mode 100644
index 0000000..92cb501
--- /dev/null
+++ b/buch/papers/fm/RS presentation/FM_presentation.tex
@@ -0,0 +1,125 @@
+%% !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}
+	%\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(\beta) + 2\sum_{m=1}^\infty J_{2m}(\beta) \cos(2m\varphi)
+			\\
+			\sin(\beta\sin\varphi)
+			&=
+			J_0(\beta) + 2\sum_{m=1}^\infty J_{2m}(\beta) \cos(2m\varphi)
+			\\
+			J_{-n}(\beta) &= (-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\omega_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\omega_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/RS presentation/Frequency modulation (FM) and Bessel functions.pdf b/buch/papers/fm/RS presentation/Frequency modulation (FM) and Bessel functions.pdf
new file mode 100644
index 0000000..a6e701c
--- /dev/null
+++ b/buch/papers/fm/RS presentation/Frequency modulation (FM) and Bessel functions.pdf
diff --git a/buch/papers/fm/RS presentation/README.txt b/buch/papers/fm/RS presentation/README.txt
new file mode 100644
index 0000000..4d0620f
--- /dev/null
+++ b/buch/papers/fm/RS presentation/README.txt
@@ -0,0 +1 @@
+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
index 8e3de17..8a67619 100644
--- a/buch/papers/fm/RS presentation/RS.tex
+++ b/buch/papers/fm/RS presentation/RS.tex
@@ -1,3 +1,5 @@
+%% !TeX root = RS.tex
+
 \documentclass[11pt,aspectratio=169]{beamer}
 \usepackage[utf8]{inputenc}
 \usepackage[T1]{fontenc}
@@ -13,7 +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}{}
@@ -24,139 +26,98 @@
 \section{Einführung}
 	\begin{frame}
 		\frametitle{Frequenzmodulation}
-		\begin{itemize}
-		\visible<1->{\item Für Übertragung von Daten}
-		\visible<2->{\item Amplituden unabhängig}
-		\end{itemize}
+		
+		\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}
 %-------------------------------------------------------------------------------	
-	\begin{frame}
-	\frametitle{Parameter}
-	\begin{center}
-		\begin{tabular}{ c c c } 
-			\hline
-			Nutzlas & Fehler & Versenden \\
-			\hline 
-			3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ 
-			4 & 2 & 8 Werte eines Polynoms vom Grad 3 \\
-\visible<1->{3}& 
-\visible<1->{3}& 
-\visible<1->{9 Werte eines Polynoms vom Grad 2} \\ 
-			&&\\
-\visible<1->{$k$} & 
-\visible<1->{$t$} & 
-\visible<1->{$k+2t$ Werte eines Polynoms vom Grad $k-1$} \\ 
-			\hline
-			&&\\
-			&&\\
-			\multicolumn{3}{l} {
-				\visible<1>{Ausserdem können bis zu $2t$ Fehler erkannt werden!}
-			}
-		\end{tabular}
-	\end{center}	
-	\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}
 
-\section{Diskrete Fourier Transformation}
-	\begin{frame}
-		\frametitle{Idee}
-		\begin{itemize}
-			\item Fourier-transformieren
-			\item Übertragung
-			\item Rücktransformieren
-		\end{itemize}
+%-------------------------------------------------------------------------------	
+\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[width=0.9\linewidth]{images/fig1.pdf}
-			}
-			\only<2>{
-			\includegraphics[width=0.9\linewidth]{images/fig2.pdf}
-			}
-			\only<3>{
-			\includegraphics[width=0.9\linewidth]{images/fig3.pdf}
-			}
-			\only<4>{
-			\includegraphics[width=0.9\linewidth]{images/fig4.pdf}
-			}
-			\only<5>{
-			\includegraphics[width=0.9\linewidth]{images/fig5.pdf}
-			}
-			\only<6>{
-			\includegraphics[width=0.9\linewidth]{images/fig6.pdf}
-			}
-			\only<7>{
-			\includegraphics[width=0.9\linewidth]{images/fig7.pdf}
-			}
+	\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{Diskrete Fourier Transformation}
-	\begin{itemize}
-	\item Diskrete Fourier-Transformation gegeben durch:
-	\visible<1->{
-	\[
-	\label{ft_discrete}
-	\hat{c}_{k} 
-	= \frac{1}{N} \sum_{n=0}^{N-1}
-	{f}_n \cdot e^{-\frac{2\pi j}{N} \cdot kn}
-	\]}
-	\visible<2->{
-	\item Ersetzte
-	\[
-	w = e^{-\frac{2\pi j}{N} k}
-	\]}
-	\visible<3->{
-	\item Wenn $N$ konstant:
-	\[
-	\hat{c}_{k}=\frac{1}{N}( {f}_0 w^0 + {f}_1 w^1 + {f}_2 w^2 + \dots + {f}_{N-1} w^N)
-	\]}
-	\end{itemize}
-	\end{frame}	
-
-%-------------------------------------------------------------------------------	
-
-%-------------------------------------------------------------------------------
-	\begin{frame}
-		\frametitle{Ein Beispiel}
-		
-		\begin{itemize}
-			
-			\onslide<1->{\item endlicher Körper $q = 11$}
-			
-			\onslide<2->{ist eine Primzahl}
-
-			\onslide<3->{beinhaltet die Zahlen $\mathbb{F}_{11} = \{0,1,2,3,4,5,6,7,8,9,10\}$}
-			
-			\vspace{10pt}
-			
-			\onslide<4->{\item Nachrichtenblock $=$ Nutzlast $+$ Fehlerkorrekturstellen}
-			
-			\onslide<5->{$n = q - 1 = 10$ Zahlen}
-			
-			\vspace{10pt}
-			
-			\onslide<6->{\item Max.~Fehler $t = 2$}
-			
-			\onslide<7->{maximale Anzahl von Fehler, die wir noch korrigieren können}
-			
-			\vspace{10pt}
-			
-			\onslide<8->{\item Nutzlast $k = n -2t = 6$ Zahlen}
-			
-			\onslide<9->{Fehlerkorrkturstellen $2t = 4$ Zahlen}
-			
-			\onslide<10->{Nachricht $m = [0,0,0,0,4,7,2,5,8,1]$}
-			
-			\onslide<11->{als Polynom $m(X) = 4X^5 + 7X^4 + 2X^3 + 5X^2 + 8X + 1$}
-			
-		\end{itemize}
-		
+	\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/RS presentation/images/100HZ.png b/buch/papers/fm/RS presentation/images/100HZ.png
new file mode 100644
index 0000000..371b9bf
--- /dev/null
+++ b/buch/papers/fm/RS presentation/images/100HZ.png
diff --git a/buch/papers/fm/RS presentation/images/200HZ.png b/buch/papers/fm/RS presentation/images/200HZ.png
new file mode 100644
index 0000000..f6836bd
--- /dev/null
+++ b/buch/papers/fm/RS presentation/images/200HZ.png
diff --git a/buch/papers/fm/RS presentation/images/300HZ.png b/buch/papers/fm/RS presentation/images/300HZ.png
new file mode 100644
index 0000000..6762c1a
--- /dev/null
+++ b/buch/papers/fm/RS presentation/images/300HZ.png
diff --git a/buch/papers/fm/RS presentation/images/400HZ.png b/buch/papers/fm/RS presentation/images/400HZ.png
new file mode 100644
index 0000000..236c428
--- /dev/null
+++ b/buch/papers/fm/RS presentation/images/400HZ.png
diff --git a/buch/papers/fm/RS presentation/images/bessel.png b/buch/papers/fm/RS presentation/images/bessel.png
new file mode 100644
index 0000000..f4c83ea
--- /dev/null
+++ b/buch/papers/fm/RS presentation/images/bessel.png
diff --git a/buch/papers/fm/RS presentation/images/bessel2.png b/buch/papers/fm/RS presentation/images/bessel2.png
new file mode 100644
index 0000000..ccda3f9
--- /dev/null
+++ b/buch/papers/fm/RS presentation/images/bessel2.png
diff --git a/buch/papers/fm/RS presentation/images/bessel_beta1.png b/buch/papers/fm/RS presentation/images/bessel_beta1.png
new file mode 100644
index 0000000..1f5c47e
--- /dev/null
+++ b/buch/papers/fm/RS presentation/images/bessel_beta1.png
diff --git a/buch/papers/fm/RS presentation/images/bessel_frequenz.png b/buch/papers/fm/RS presentation/images/bessel_frequenz.png
new file mode 100644
index 0000000..4f228b9
--- /dev/null
+++ b/buch/papers/fm/RS presentation/images/bessel_frequenz.png
diff --git a/buch/papers/fm/RS presentation/images/beta_0.001.png b/buch/papers/fm/RS presentation/images/beta_0.001.png
new file mode 100644
index 0000000..7e4e276
--- /dev/null
+++ b/buch/papers/fm/RS presentation/images/beta_0.001.png
diff --git a/buch/papers/fm/RS presentation/images/beta_0.1.png b/buch/papers/fm/RS presentation/images/beta_0.1.png
new file mode 100644
index 0000000..e7722b3
--- /dev/null
+++ b/buch/papers/fm/RS presentation/images/beta_0.1.png
diff --git a/buch/papers/fm/RS presentation/images/beta_0.5.png b/buch/papers/fm/RS presentation/images/beta_0.5.png
new file mode 100644
index 0000000..5261b43
--- /dev/null
+++ b/buch/papers/fm/RS presentation/images/beta_0.5.png
diff --git a/buch/papers/fm/RS presentation/images/beta_1.png b/buch/papers/fm/RS presentation/images/beta_1.png
new file mode 100644
index 0000000..6d3535c
--- /dev/null
+++ b/buch/papers/fm/RS presentation/images/beta_1.png
diff --git a/buch/papers/fm/RS presentation/images/beta_2.png b/buch/papers/fm/RS presentation/images/beta_2.png
new file mode 100644
index 0000000..6930eae
--- /dev/null
+++ b/buch/papers/fm/RS presentation/images/beta_2.png
diff --git a/buch/papers/fm/RS presentation/images/beta_3.png b/buch/papers/fm/RS presentation/images/beta_3.png
new file mode 100644
index 0000000..c6df82c
--- /dev/null
+++ b/buch/papers/fm/RS presentation/images/beta_3.png
diff --git a/buch/papers/fm/RS presentation/images/fm_10Hz.png b/buch/papers/fm/RS presentation/images/fm_10Hz.png
new file mode 100644
index 0000000..51bddc7
--- /dev/null
+++ b/buch/papers/fm/RS presentation/images/fm_10Hz.png
diff --git a/buch/papers/fm/RS presentation/images/fm_20hz.png b/buch/papers/fm/RS presentation/images/fm_20hz.png
new file mode 100644
index 0000000..126ecf3
--- /dev/null
+++ b/buch/papers/fm/RS presentation/images/fm_20hz.png
diff --git a/buch/papers/fm/RS presentation/images/fm_30Hz.png b/buch/papers/fm/RS presentation/images/fm_30Hz.png
new file mode 100644
index 0000000..371b9bf
--- /dev/null
+++ b/buch/papers/fm/RS presentation/images/fm_30Hz.png
diff --git a/buch/papers/fm/RS presentation/images/fm_3Hz.png b/buch/papers/fm/RS presentation/images/fm_3Hz.png
new file mode 100644
index 0000000..d4098af
--- /dev/null
+++ b/buch/papers/fm/RS presentation/images/fm_3Hz.png
diff --git a/buch/papers/fm/RS presentation/images/fm_40Hz.png b/buch/papers/fm/RS presentation/images/fm_40Hz.png
new file mode 100644
index 0000000..4cf11d4
--- /dev/null
+++ b/buch/papers/fm/RS presentation/images/fm_40Hz.png
diff --git a/buch/papers/fm/RS presentation/images/fm_5Hz.png b/buch/papers/fm/RS presentation/images/fm_5Hz.png
new file mode 100644
index 0000000..e495b5c
--- /dev/null
+++ b/buch/papers/fm/RS presentation/images/fm_5Hz.png
diff --git a/buch/papers/fm/RS presentation/images/fm_7Hz.png b/buch/papers/fm/RS presentation/images/fm_7Hz.png
new file mode 100644
index 0000000..b3dd7e3
--- /dev/null
+++ b/buch/papers/fm/RS presentation/images/fm_7Hz.png
diff --git a/buch/papers/fm/RS presentation/images/fm_frequenz.png b/buch/papers/fm/RS presentation/images/fm_frequenz.png
new file mode 100644
index 0000000..26bfd86
--- /dev/null
+++ b/buch/papers/fm/RS presentation/images/fm_frequenz.png
diff --git a/buch/papers/fm/RS presentation/images/fm_in_time.png b/buch/papers/fm/RS presentation/images/fm_in_time.png
new file mode 100644
index 0000000..068eafc
--- /dev/null
+++ b/buch/papers/fm/RS presentation/images/fm_in_time.png