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%Title Page
\title{Elliptische Filter}
\subtitle{Eine Anwendung der Jacobi elliptischen Funktionen}
\author{Nicolas Tobler}
\institute{Mathematisches Seminar 2022 | Spezielle Funktionen}
% \institute{\includegraphics[scale=0.3]{../img/ost_logo.png}}
\date{\today}

\input{../packages.tex}

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\makeatletter
\newcount\my@repeat@count
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\usetikzlibrary{automata,arrows,positioning,calc,shapes.geometric, fadings}

\begin{document}

	\begin{frame}
		\titlepage
	\end{frame}

	\begin{frame}
		\frametitle{Inhalt}
		\tableofcontents
	\end{frame}

	\section{Lineare Filter}

	\begin{frame}
		\frametitle{Lineare Filter}

		\begin{center}
			\scalebox{0.75}{
				\input{../tikz/filter.tikz.tex}
			}
		\end{center}


		\begin{equation*}
			| H(\Omega)|^2 = \frac{1}{1 + \varepsilon_p^2 F_N^2(w)}, \quad w=\frac{\Omega}{\Omega_p}
		\end{equation*}

		\pause

		\begin{align*}
			|F_N(w)| &< 1 \quad \forall \quad |w| < 1 \\
			|F_N(w)| &= 1 \quad \forall \quad |w| = 1 \\
			|F_N(w)| &> 1 \quad \forall \quad |w| > 1
		\end{align*}


		\begin{equation*}
			F_N(w) = w^N
		\end{equation*}

	\end{frame}

	\begin{frame}
		\frametitle{Beispiel: Butterworth Filter}

		\begin{equation}
			F_N(w) = w^N
		\end{equation}

		\begin{center}
			\input{../python/F_N_butterworth.pgf}
		\end{center}

	\end{frame}


	\begin{frame}
		\frametitle{Arten von linearen filtern}

		\begin{align*}
			F_N(w) & =
			\begin{cases}
				w^N                            & \text{Butterworth} \\
				T_N(w)                         & \text{Tschebyscheff, Typ 1}  \\
				[k_1 T_N (k^{-1} w^{-1})]^{-1} & \text{Tschebyscheff, Typ 2}  \\
				R_N(w,\xi)                      & \text{Elliptisch (Cauer)}    \\
			\end{cases}
		\end{align*}

	\end{frame}

	\section{Tschebycheff Filter}

	\begin{frame}
		\frametitle{Tschebyscheff-Polynome}


		\begin{columns}
			\begin{column}[T]{0.35\textwidth}

				\begin{align*}
					T_{0}(x)&=1\\
					T_{1}(x)&=x\\
					T_{2}(x)&=2x^{2}-1\\
					T_{3}(x)&=4x^{3}-3x\\
					T_{n+1}(x)&=2x~T_{n}(x)-T_{n-1}(x)
				\end{align*}

			\end{column}
			\begin{column}[T]{0.65\textwidth}

				\begin{center}
					\resizebox{\textwidth}{!}{
					\input{../python/F_N_chebychev2.pgf}
					}
				\end{center}

			\end{column}
		\end{columns}



	\end{frame}

	\begin{frame}
		\frametitle{Tschebyscheff-Filter}

		\begin{equation*}
			| H(\Omega)|^2 = \frac{1}{1 + \varepsilon_p^2 T_N^2(w)}, \quad w=\frac{\Omega}{\Omega_p}
		\end{equation*}

		\begin{center}
			\scalebox{0.9}{
				\input{../python/F_N_chebychev.pgf}
			}
		\end{center}

	\end{frame}


	\begin{frame}
		\frametitle{Tschebyscheff-Filter}

		Darstellung mit trigonometrischen Funktionen:

		\begin{align*}
			T_N(w) &= \cos \left( N \cos^{-1}(w) \right) \\
				&= \cos \left(N~z \right), \quad w= \cos(z)
		\end{align*}

		\pause

		\begin{align*}
			\cos^{-1}(x)
			&=
			\int_{x}^{1}
			\frac{
				dz
			}{
				\sqrt{
					1-z^2
				}
			}\\
			&=
			\int_{0}^{x}
			\frac{
				-1
			}{
				\sqrt{
					1-z^2
				}
			}
			~dz
			+ \frac{\pi}{2}
		\end{align*}


	\end{frame}

	\begin{frame}
		\frametitle{Tschebyscheff-Filter}

		\begin{columns}

			\begin{column}{0.2\textwidth}

				\begin{equation*}
					z = \cos^{-1}(w)
				\end{equation*}

				\vspace{0.5cm}

				Integrand:
				\begin{equation*}
					\frac{
						-1
					}{
						\sqrt{
							1-z^2
						}
					}
				\end{equation*}

			\end{column}
			\begin{column}{0.8\textwidth}


				\begin{center}
					\scalebox{0.7}{
						\input{../tikz/arccos.tikz.tex}
					}
				\end{center}

			\end{column}
		\end{columns}



	\end{frame}

	\begin{frame}
		\frametitle{Tschebyscheff-Filter}

		\begin{equation*}
			T_N(w) = \cos \left(z_1 \right), \quad z_1 = N~\cos^{-1}(w)
		\end{equation*}

		\begin{center}
			\scalebox{0.85}{
				\input{../tikz/arccos2.tikz.tex}
			}
		\end{center}

	\end{frame}


	\section{Jacobi elliptische Funktionen}

	\begin{frame}
		\frametitle{Jacobi elliptische Funktionen}

		Elliptisches Integral erster Art

		\begin{equation*}
			F(\phi, k)
			=
			\int_{0}^{\phi}
			\frac{
				d\theta
			}{
				\sqrt{
					1-k^2 \sin^2 \theta
				}
			}
			% =
			% \int_{0}^{\phi}
			% \frac{
			% 	dt
			% }{
			% 	\sqrt{
			% 		(1-t^2)(1-k^2 t^2)
			% 	}
			% }
		\end{equation*}

		\begin{equation*}
			K(k)
			=
			\int_{0}^{\pi / 2}
			\frac{
				d\theta
			}{
				\sqrt{
					1-k^2 \sin^2 \theta
				}
			}
		\end{equation*}



	\end{frame}





	\begin{frame}
		\frametitle{Jacobi elliptische Funktionen}

			\begin{equation*}
				\sn^{-1}(w, k)
				=
				F(\phi, k),
				\quad
				\phi = \sin^{-1}(w)
			\end{equation*}

			\begin{align*}
				\sn^{-1}(w, k)
					& =
				\int_{0}^{\phi}
				\frac{
					d\theta
				}{
					\sqrt{
						1-k^2 \sin^2 \theta
					}
				},
				\quad
				\phi = \sin^{-1}(w)
				\\
					& =
				\int_{0}^{w}
				\frac{
					dt
				}{
					\sqrt{
						(1-t^2)(1-k^2 t^2)
					}
				}
			\end{align*}



		\end{frame}

	\begin{frame}
		\frametitle{Jacobi elliptische Funktionen}
		\begin{columns}
			\begin{column}{0.2\textwidth}

				\begin{equation*}
					z = \sn^{-1}(w, k)
				\end{equation*}

				\vspace{0.5cm}

				Integrand:
				\begin{equation*}
					\frac{
						1
					}{
						\sqrt{
							(1-t^2)(1-k^2 t^2)
						}
					}
				\end{equation*}

			\end{column}
			\begin{column}{0.8\textwidth}
				\begin{center}
					\scalebox{0.75}{
						\input{../tikz/sn.tikz.tex}
					}
				\end{center}
			\end{column}
		\end{columns}


	\end{frame}

	\begin{frame}
		\frametitle{Fundamentales Rechteck}

		Nullstelle beim ersten Buchstabe, Polstelle beim zweiten Buchstabe

		\begin{center}
			\scalebox{0.8}{
				\input{../tikz/fundamental_rectangle.tikz.tex}
			}
		\end{center}

	\end{frame}


	\begin{frame}
		\frametitle{Jacobi elliptische Funktionen}

		\begin{equation*}
			z = \cd^{-1}(w, k)
		\end{equation*}

		\begin{center}
			\scalebox{0.7}{
				\input{../tikz/cd.tikz.tex}

			}
		\end{center}

	\end{frame}

	\section{Elliptisches Filter}

	\begin{frame}
		\frametitle{Elliptisches Filter}

		% \begin{equation*}
		% 	z_1 = N~\frac{K_1}{K}~\cd^{-1}(w, k)
		% \end{equation*}

		\begin{center}
			\scalebox{0.75}{
				\input{../tikz/cd3.tikz.tex}
			}
		\end{center}

	\end{frame}

	\begin{frame}
		\frametitle{Periodizität in realer und imaginärer Richtung}

		\begin{center}
			\input{../python/k.pgf}
		\end{center}


	\end{frame}

	\begin{frame}
		\frametitle{Gradgleichung}

		\begin{center}
			\scalebox{0.95}{
				\input{../tikz/elliptic_transform2.tikz}
			}
		\end{center}

		\onslide<5->{
		\begin{equation*}
			N \frac{K^\prime}{K} = \frac{K^\prime_1}{K_1}
		\end{equation*}
		}

	\end{frame}

	\begin{frame}
		\frametitle{Elliptisches Filter}

		\begin{equation*}
			R_N = \cd(z_1, k_1),
			\quad
			z_1 = N~\frac{K_1}{K}~\cd^{-1}(w, k),
			\quad
			N \frac{K^\prime}{K} = \frac{K^\prime_1}{K_1}
		\end{equation*}

		\begin{center}
			\scalebox{0.75}{
				\input{../tikz/cd2.tikz.tex}
			}
		\end{center}

	\end{frame}


	\begin{frame}
		\frametitle{Elliptisches Filter}

		\begin{columns}

			\begin{column}[T]{0.5\textwidth}

				\begin{center}
					\resizebox{\textwidth}{!}{
					\input{../python/F_N_elliptic.pgf}
					}
				\end{center}

			\end{column}
			\begin{column}[T]{0.5\textwidth}

				\begin{center}
					\resizebox{\textwidth}{!}{
					\input{../python/elliptic.pgf}
					}
				\end{center}

			\end{column}
		\end{columns}

	\end{frame}

	\end{document}