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%% !TeX root = .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(\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}
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