From 8ced517966a5996ad659b155b7e0372107bbf116 Mon Sep 17 00:00:00 2001 From: Nicolas Tobler Date: Tue, 2 Aug 2022 23:54:02 +0200 Subject: improved Einleitung --- .../papers/ellfilter/presentation/presentation.tex | 239 ++++++++++++++++----- 1 file changed, 185 insertions(+), 54 deletions(-) (limited to 'buch/papers/ellfilter/presentation') diff --git a/buch/papers/ellfilter/presentation/presentation.tex b/buch/papers/ellfilter/presentation/presentation.tex index adbf925..96bdfd3 100644 --- a/buch/papers/ellfilter/presentation/presentation.tex +++ b/buch/papers/ellfilter/presentation/presentation.tex @@ -76,9 +76,9 @@ %Title Page \title{Elliptische Filter} -\subtitle{Eine Anwendung der Jaccobi elliptischen Funktionen} +\subtitle{Eine Anwendung der Jacobi elliptischen Funktionen} \author{Nicolas Tobler} -% \institute{OST Ostschweizer Fachhochschule} +\institute{Mathematisches Seminar 2022 | Spezielle Funktionen} % \institute{\includegraphics[scale=0.3]{../img/ost_logo.png}} \date{\today} @@ -113,7 +113,7 @@ \end{frame} \begin{frame} - \frametitle{Content} + \frametitle{Inhalt} \tableofcontents \end{frame} @@ -122,16 +122,29 @@ \begin{frame} \frametitle{Lineare Filter} + \begin{center} + \scalebox{0.75}{ + \input{../tikz/filter.tikz.tex} + } + \end{center} - \begin{equation} + + \begin{equation*} | H(\Omega)|^2 = \frac{1}{1 + \varepsilon_p^2 F_N^2(w)}, \quad w=\frac{\Omega}{\Omega_p} - \end{equation} + \end{equation*} \pause - \begin{equation} + \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{equation*} \end{frame} @@ -218,10 +231,36 @@ Darstellung mit trigonometrischen Funktionen: - \begin{align} \label{ellfilter:eq:chebychef_polynomials} + \begin{align*} T_N(w) &= \cos \left( N \cos^{-1}(w) \right) \\ &= \cos \left(N~z \right), \quad w= \cos(z) - \end{align} + \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} @@ -229,15 +268,41 @@ \begin{frame} \frametitle{Tschebyscheff-Filter} - \begin{equation*} - z = \cos^{-1}(w) - \end{equation*} + \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} + - \begin{center} - \scalebox{0.85}{ - \input{../tikz/arccos.tikz.tex} - } - \end{center} \end{frame} @@ -245,7 +310,7 @@ \frametitle{Tschebyscheff-Filter} \begin{equation*} - z_1 = N~\cos^{-1}(w) + T_N(w) = \cos \left(z_1 \right), \quad z_1 = N~\cos^{-1}(w) \end{equation*} \begin{center} @@ -257,15 +322,14 @@ \end{frame} - \section{Jaccobi elliptische Funktionen} + \section{Jacobi elliptische Funktionen} \begin{frame} - \frametitle{Jaccobi elliptische Funktionen} + \frametitle{Jacobi elliptische Funktionen} + Elliptisches Integral erster Art - \begin{equation} - z - = + \begin{equation*} F(\phi, k) = \int_{0}^{\phi} @@ -276,18 +340,18 @@ 1-k^2 \sin^2 \theta } } - = - \int_{0}^{\phi} - \frac{ - dt - }{ - \sqrt{ - (1-t^2)(1-k^2 t^2) - } - } - \end{equation} + % = + % \int_{0}^{\phi} + % \frac{ + % dt + % }{ + % \sqrt{ + % (1-t^2)(1-k^2 t^2) + % } + % } + \end{equation*} - \begin{equation} + \begin{equation*} K(k) = \int_{0}^{\pi / 2} @@ -298,24 +362,88 @@ 1-k^2 \sin^2 \theta } } - \end{equation} + \end{equation*} \end{frame} + + + + \begin{frame} - \frametitle{Jaccobi elliptische Funktionen} + \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*} - \begin{equation*} - z = \sn^{-1}(w, k) - \end{equation*} - \begin{center} - \scalebox{0.7}{ - \input{../tikz/sn.tikz.tex} - } - \end{center} + + \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} @@ -334,7 +462,7 @@ \begin{frame} - \frametitle{Jaccobi elliptische Funktionen} + \frametitle{Jacobi elliptische Funktionen} \begin{equation*} z = \cd^{-1}(w, k) @@ -354,9 +482,9 @@ \begin{frame} \frametitle{Elliptisches Filter} - \begin{equation*} - z_1 = N~\frac{K_1}{K}~\cd^{-1}(w, k) - \end{equation*} + % \begin{equation*} + % z_1 = N~\frac{K_1}{K}~\cd^{-1}(w, k) + % \end{equation*} \begin{center} \scalebox{0.75}{ @@ -379,16 +507,17 @@ \begin{frame} \frametitle{Gradgleichung} - \begin{equation} - N \frac{K^\prime}{K} = \frac{K^\prime_1}{K_1} - \end{equation} - \begin{center} \scalebox{0.95}{ - \input{../tikz/elliptic_transform.tikz} + \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} @@ -398,7 +527,9 @@ \begin{equation*} R_N = \cd(z_1, k_1), \quad - z_1 = N~\frac{K_1}{K}~\cd^{-1}(w, k) + 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} -- cgit v1.2.1