Merge pull request #17 from ntobler/master

ellfilter draft

29 files changed, 7785 insertions, 216 deletions

diff --git a/buch/papers/ellfilter/Makefile.inc b/buch/papers/ellfilter/Makefile.inc
index 8f20278..97e4089 100644
--- a/buch/papers/ellfilter/Makefile.inc
+++ b/buch/papers/ellfilter/Makefile.inc
@@ -3,12 +3,11 @@
 #
 # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
 #
-dependencies-ellfilter =						\
-	papers/ellfilter/packages.tex					\
-	papers/ellfilter/main.tex					\
-	papers/ellfilter/references.bib				\
-	papers/ellfilter/teil0.tex					\
-	papers/ellfilter/teil1.tex					\
-	papers/ellfilter/teil2.tex					\
-	papers/ellfilter/teil3.tex
-
+dependencies-ellfilter =                        \
+    papers/ellfilter/packages.tex               \
+    papers/ellfilter/main.tex                   \
+    papers/ellfilter/references.bib             \
+    papers/ellfilter/einleitung.tex             \
+    papers/ellfilter/tschebyscheff.tex          \
+    papers/ellfilter/jacobi.tex                 \
+    papers/ellfilter/elliptic.tex
diff --git a/buch/papers/ellfilter/einleitung.tex b/buch/papers/ellfilter/einleitung.tex
new file mode 100644
index 0000000..37fd89f
--- /dev/null
+++ b/buch/papers/ellfilter/einleitung.tex
@@ -0,0 +1,56 @@
+\section{Einleitung}
+
+% Lineare filter
+
+% Filter, Signalverarbeitung
+
+
+Der womöglich wichtigste Filtertyp ist das Tiefpassfilter.
+Dieses soll im Durchlassbereich unter der Grenzfrequenz $\Omega_p$ unverstärkt durchlassen und alle anderen Frequenzen vollständig auslöschen.
+
+% Bei der Implementierung von Filtern
+
+In der Elektrotechnik führen Schaltungen mit linearen Bauelementen wie Kondensatoren, Spulen und Widerständen immer zu linearen zeitinvarianten Systemen (LTI-System von englich \textit{time-invariant system}).
+Die Übertragungsfunktion im Frequenzbereich $|H(\Omega)|$ eines solchen Systems ist dabei immer eine rationale Funktion, also eine Division von zwei Polynomen.
+Die Polynome habe dabei immer reelle oder komplex-konjugierte Nullstellen.
+
+
+\begin{equation} \label{ellfilter:eq:h_omega}
+    | H(\Omega)|^2 = \frac{1}{1 + \varepsilon_p^2 F_N^2(w)}, \quad w=\frac{\Omega}{\Omega_p}
+\end{equation}
+
+$\Omega = 2 \pi f$ ist die analoge Frequenz
+
+
+% Linear filter
+Damit das Filter implementierbar und stabil ist, muss $H(\Omega)^2$ eine rationale Funktion sein, deren Nullstellen und Pole auf der linken Halbebene liegen.
+
+$N \in \mathbb{N} $ gibt dabei die Ordnung des Filters vor, also die maximale Anzahl Pole oder Nullstellen.
+
+Damit ein Filter die Passband Kondition erfüllt muss $|F_N(w)| \leq 1 \forall |w| \leq 1$ und für $|w| \geq 1$ sollte die Funktion möglichst schnell divergieren.
+Eine einfaches Polynom, dass das erfüllt, erhalten wir wenn $F_N(w) = w^N$.
+Tatsächlich erhalten wir damit das Butterworth Filter, wie in Abbildung \ref{ellfilter:fig:butterworth} ersichtlich.
+\begin{figure}
+    \centering
+    \input{papers/ellfilter/python/F_N_butterworth.pgf}
+    \caption{$F_N$ für Butterworth filter. Der grüne Bereich definiert die erlaubten Werte für alle $F_N$-Funktionen.}
+    \label{ellfilter:fig:butterworth}
+\end{figure}
+
+wenn $F_N(w)$ eine rationale Funktion ist, ist auch $H(\Omega)$ eine rationale Funktion und daher ein lineares Filter. %proof?
+
+\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}
+
+Mit der Ausnahme vom Butterworth filter sind alle Filter nach speziellen Funktionen benannt.
+Alle diese Filter sind optimal für unterschiedliche Anwendungsgebiete.
+Das Butterworth-Filter, zum Beispiel, ist maximal flach im Durchlassbereich.
+Das Tschebyscheff-1 Filter sind maximal steil für eine definierte Welligkeit im Durchlassbereich, währendem es im Sperrbereich monoton abfallend ist.
+Es scheint so als sind gewisse Eigenschaften dieser speziellen Funktionen verantwortlich für die Optimalität dieser Filter.
diff --git a/buch/papers/ellfilter/elliptic.tex b/buch/papers/ellfilter/elliptic.tex
new file mode 100644
index 0000000..88bfbfe
--- /dev/null
+++ b/buch/papers/ellfilter/elliptic.tex
@@ -0,0 +1,92 @@
+\section{Elliptische rationale Funktionen}
+
+Kommen wir nun zum eigentlichen Teil dieses Papers, den elliptischen rationalen Funktionen
+\begin{align}
+    R_N(\xi, w) &= \cd \left(N~f_1(\xi)~\cd^{-1}(w, 1/\xi), f_2(\xi)\right) \\
+                &= \cd \left(N~\frac{K_1}{K}~\cd^{-1}(w, k), k_1)\right) , \quad k= 1/\xi, k_1 = 1/f(\xi) \\
+                &= \cd \left(N~K_1~z , k_1 \right), \quad w= \cd(z K, k)
+\end{align}
+
+
+sieht ähnlich aus wie die trigonometrische Darstellung der Tschebyschef-Polynome \eqref{ellfilter:eq:chebychef_polynomials}
+Anstelle vom Kosinus kommt hier die $\cd$-Funktion zum Einsatz.
+Die Ordnungszahl $N$ kommt auch als Faktor for.
+Zusätzlich werden noch zwei verschiedene elliptische Module $k$ und $k_1$ gebraucht.
+
+
+
+Sinus entspricht $\sn$
+
+Damit die Nullstellen an ähnlichen Positionen zu liegen kommen wie bei den Tschebyscheff-Polynomen, muss die $\cd$-Funktion gewählt werden.
+
+Die $\cd^{-1}(w, k)$-Funktion ist um $K$ verschoben zur $\sn^{-1}(w, k)$-Funktion, wie ersichtlich in Abbildung \ref{ellfilter:fig:cd}.
+\begin{figure}
+    \centering
+    \input{papers/ellfilter/tikz/cd.tikz.tex}
+    \caption{
+        $z$-Ebene der Funktion $z = \sn^{-1}(w, k)$.
+        Die Funktion ist in der realen Achse $4K$-periodisch und in der imaginären Achse $2jK^\prime$-periodisch.
+    }
+    \label{ellfilter:fig:cd}
+\end{figure}
+Auffallend ist, dass sich alle Nullstellen und Polstellen um $K$ verschoben haben.
+
+Durch das Konzept vom fundamentalen Rechteck, siehe Abbildung \ref{ellfilter:fig:fundamental_rectangle} können für alle inversen Jaccobi elliptischen Funktionen die Positionen der Null- und Polstellen anhand eines Diagramms ermittelt werden.
+Der erste Buchstabe bestimmt die Position der Nullstelle und der zweite Buchstabe die Polstelle.
+\begin{figure}
+    \centering
+    \input{papers/ellfilter/tikz/fundamental_rectangle.tikz.tex}
+    \caption{
+        Fundamentales Rechteck der inversen Jaccobi elliptischen Funktionen.
+    }
+    \label{ellfilter:fig:fundamental_rectangle}
+\end{figure}
+
+Auffallend an der $w = \sn(z, k)$-Funktion ist, dass sich $w$ auf der reellen Achse wie der Kosinus immer zwischen $-1$ und $1$ bewegt, während bei $\mathrm{Im(z) = K^\prime}$ die Werte zwischen $\pm 1/k$ und $\pm \infty$ verlaufen.
+Die Funktion hat also Equirippel-Verhalten um $w=0$ und um $w=\pm \infty$.
+Falls es möglich ist diese Werte abzufahren im Sti der Tschebyscheff-Polynome, kann ein Filter gebaut werden, dass Equirippel-Verhalten im Durchlass- und Sperrbereich aufweist.
+
+
+
+Analog zu Abbildung \ref{ellfilter:fig:arccos2} können wir auch bei den elliptisch rationalen Funktionen die komplexe $z$-Ebene betrachten, wie ersichtlich in Abbildung \ref{ellfilter:fig:cd2}, um die besser zu verstehen.
+\begin{figure}
+    \centering
+    \input{papers/ellfilter/tikz/cd2.tikz.tex}
+    \caption{
+        $z_1$-Ebene der elliptischen rationalen Funktionen.
+        Je grösser die Ordnung $N$ gewählt wird, desto mehr Nullstellen passiert.
+    }
+    \label{ellfilter:fig:cd2}
+\end{figure}
+% Da die $\cd^{-1}$-Funktion 
+
+
+
+\begin{figure}
+    \centering
+    \input{papers/ellfilter/python/F_N_elliptic.pgf}
+    \caption{$F_N$ für ein elliptischs filter.}
+    \label{ellfilter:fig:elliptic}
+\end{figure}
+
+\subsection{Degree Equation}
+
+Der $\cd^{-1}$ Term muss so verzogen werden, dass die umgebene $\cd$-Funktion die Nullstellen und Pole trifft.
+Dies trifft ein wenn die Degree Equation erfüllt ist.
+
+\begin{equation}
+    N \frac{K^\prime}{K} = \frac{K^\prime_1}{K_1}
+\end{equation}
+
+
+Leider ist das lösen dieser Gleichung nicht trivial.
+Die Rechnung wird in \ref{ellfilter:bib:orfanidis} im Detail angeschaut.
+
+
+\subsection{Polynome?}
+
+Bei den Tschebyscheff-Polynomen haben wir gesehen, dass die Trigonometrische Formel zu einfachen Polynomen umgewandelt werden kann.
+Im gegensatz zum $\cos^{-1}$ hat der $\cd^{-1}$ nicht nur Nullstellen sondern auch Pole.
+Somit entstehen bei den elliptischen rationalen Funktionen, wie es der name auch deutet, rationale Funktionen, also ein Bruch von zwei Polynomen.
+
+Da Transformationen einer rationalen Funktionen mit Grundrechenarten, wie es in \eqref{ellfilter:eq:h_omega} der Fall ist, immer noch rationale Funktionen ergeben, stellt dies kein Problem für die Implementierung dar.
diff --git a/buch/papers/ellfilter/jacobi.tex b/buch/papers/ellfilter/jacobi.tex
new file mode 100644
index 0000000..6a208fa
--- /dev/null
+++ b/buch/papers/ellfilter/jacobi.tex
@@ -0,0 +1,189 @@
+\section{Jacobische elliptische Funktionen}
+
+%TODO $z$ or $u$ for parameter?
+
+Für das elliptische Filter wird statt der, für das Tschebyscheff-Filter benutzen Kreis-Trigonometrie die elliptischen Funktionen gebraucht.
+Der Begriff elliptische Funktion wird für sehr viele Funktionen gebraucht, daher ist es hier wichtig zu erwähnen, dass es ausschliesslich um die Jacobischen elliptischen Funktionen geht.
+
+Im Wesentlichen erweitern die Jacobi elliptischen Funktionen die trigonometrische Funktionen für Ellipsen.
+Zum Beispiel gibt es analog zum Sinus den elliptischen $\sn(z, k)$.
+Im Gegensatz zum den trigonometrischen Funktionen haben die elliptischen Funktionen zwei parameter.
+Zum einen gibt es den \textit{elliptische Modul} $k$, der die Exzentrizität der Ellipse parametrisiert.
+Zum andern das Winkelargument $z$.
+Im Kreis ist der Radius für alle Winkel konstant, bei Ellipsen ändert sich das.
+Dies hat zur Folge, dass bei einer Ellipse die Kreisbodenstrecke nicht linear zum Winkel verläuft.
+Darum kann hier nicht der gewohnte Winkel verwendet werden.
+Das Winkelargument $z$ kann durch das elliptische Integral erster Art
+\begin{equation}
+    z
+    =
+    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)
+        }
+    } %TODO which is right? are both functions from phi?
+\end{equation}
+mit dem Winkel $\phi$ in Verbindung liegt.
+
+Dabei wird das vollständige und unvollständige Elliptische integral unterschieden.
+Beim vollständigen Integral
+\begin{equation}
+    K(k)
+    =
+    \int_{0}^{\pi / 2}
+    \frac{
+        d\theta
+    }{
+        \sqrt{
+            1-k^2 \sin^2 \theta
+        }
+    }
+\end{equation}
+wird über ein viertel Ellipsenbogen integriert also bis $\phi=\pi/2$ und liefert das Winkelargument für eine Vierteldrehung.
+Die Zahl wird oft auch abgekürzt mit $K = K(k)$ und ist für das elliptische Filter sehr relevant.
+Alle elliptishen Funktionen sind somit $4K$-periodisch.
+
+Neben dem $\sn$ gibt es zwei weitere basis-elliptische Funktionen $\cn$ und $\dn$.
+Dazu kommen noch weitere abgeleitete Funktionen, die durch Quotienten und Kehrwerte dieser Funktionen zustande kommen.
+Insgesamt sind es die zwölf Funktionen
+\begin{equation*}
+    \sn \quad
+    \ns \quad
+    \scelliptic \quad
+    \sd \quad
+    \cn \quad
+    \nc \quad
+    \cs \quad
+    \cd \quad
+    \dn \quad
+    \nd \quad
+    \ds \quad
+    \dc.
+\end{equation*}
+
+Die Jacobischen elliptischen Funktionen können mit der inversen Funktion des kompletten elliptischen Integrals erster Art
+\begin{equation}
+    \phi = F^{-1}(z, k)
+\end{equation}
+definiert werden. Dabei ist zu beachten dass nur das $z$ Argument der Funktion invertiert wird, also
+\begin{equation}
+    z = F(\phi, k)
+    \Leftrightarrow
+    \phi = F^{-1}(z, k).
+\end{equation}
+Mithilfe von $F^{-1}$ kann zum Beispiel $sn^{-1}$ mit dem Elliptischen integral dargestellt werden:
+\begin{equation}
+    \sin(\phi)
+    =
+    \sin \left( F^{-1}(z, k) \right)
+    =
+    \sn(z, k)
+    =
+    w
+\end{equation}
+
+\begin{equation}
+    \phi
+    =
+     F^{-1}(z, k)
+     =
+     \sin^{-1} \big( \sn (z, k ) \big)
+     =
+    \sin^{-1} ( w )
+\end{equation}
+
+\begin{equation}
+    F(\phi, k)
+    =
+    z
+    =
+    F( \sin^{-1} \big( \sn (z, k ) \big) , k)
+    =
+    F( \sin^{-1} ( w ), k)
+\end{equation}
+
+\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}
+
+Beim $\cos^{-1}(x)$ haben wir gesehen, dass die analytische Fortsetzung bei $x < -1$ und $x > 1$ rechtwinklig in die Komplexen zahlen wandert.
+Wenn man das gleiche mit $\sn^{-1}(w, k)$ macht, erkennt man zwei interessante Stellen.
+Die erste ist die gleiche wie beim $\cos^{-1}(x)$ nämlich bei $t = \pm 1$.
+Der erste Term unter der Wurzel wird dann negativ, während der zweite noch positiv ist, da $k \leq 1$.
+\begin{equation}
+    \frac{
+        1
+    }{
+        \sqrt{
+            (1-t^2)(1-k^2 t^2)
+        }
+    }
+    \in \mathbb{R}
+    \quad \forall \quad
+    -1 \leq t \leq 1
+\end{equation}
+Die zweite stelle passiert wenn beide Faktoren unter der Wurzel negativ werden, was bei $t = 1/k$ der Fall ist.
+
+
+
+
+Funktion in relle und komplexe Richtung periodisch
+
+In der reellen Richtung ist sie $4K(k)$-periodisch und in der imaginären Richtung $4K^\prime(k)$-periodisch.
+
+
+
+%TODO sn^{-1} grafik
+
+\begin{figure}
+    \centering
+    \input{papers/ellfilter/tikz/sn.tikz.tex}
+    \caption{
+        $z$-Ebene der Funktion $z = \sn^{-1}(w, k)$.
+        Die Funktion ist in der realen Achse $4K$-periodisch und in der imaginären Achse $2jK^\prime$-periodisch.
+    }
+    % \label{ellfilter:fig:cd2}
+\end{figure}
diff --git a/buch/papers/ellfilter/main.tex b/buch/papers/ellfilter/main.tex
index 26aaec1..c58dfa7 100644
--- a/buch/papers/ellfilter/main.tex
+++ b/buch/papers/ellfilter/main.tex
@@ -8,29 +8,10 @@
 \begin{refsection}
 \chapterauthor{Nicolas Tobler}
 
-Ein paar Hinweise für die korrekte Formatierung des Textes
-\begin{itemize}
-\item
-Absätze werden gebildet, indem man eine Leerzeile einfügt.
-Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet.
-\item
-Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende
-Optionen werden gelöscht.
-Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen.
-\item
-Beginnen Sie jeden Satz auf einer neuen Zeile.
-Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen
-in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt
-anzuwenden.
-\item
-Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren
-Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern.
-\end{itemize}
-
-\input{papers/ellfilter/teil0.tex}
-\input{papers/ellfilter/teil1.tex}
-\input{papers/ellfilter/teil2.tex}
-\input{papers/ellfilter/teil3.tex}
+\input{papers/ellfilter/einleitung.tex}
+\input{papers/ellfilter/tschebyscheff.tex}
+\input{papers/ellfilter/jacobi.tex}
+\input{papers/ellfilter/elliptic.tex}
 
 \printbibliography[heading=subbibliography]
 \end{refsection}
diff --git a/buch/papers/ellfilter/packages.tex b/buch/papers/ellfilter/packages.tex
index c94db34..9a550e2 100644
--- a/buch/papers/ellfilter/packages.tex
+++ b/buch/papers/ellfilter/packages.tex
@@ -8,3 +8,20 @@
 % following example
 %\usepackage{packagename}
 
+% \usepackage[dvipsnames]{xcolor}
+
+\usetikzlibrary{trees,shapes,decorations}
+
+\DeclareMathOperator{\sn}{\mathrm{sn}}
+\DeclareMathOperator{\ns}{\mathrm{ns}}
+\DeclareMathOperator{\scelliptic}{\mathrm{sc}}
+\DeclareMathOperator{\sd}{\mathrm{sd}}
+\DeclareMathOperator{\cn}{\mathrm{cn}}
+\DeclareMathOperator{\nc}{\mathrm{nc}}
+\DeclareMathOperator{\cs}{\mathrm{cs}}
+\DeclareMathOperator{\cd}{\mathrm{cd}}
+\DeclareMathOperator{\dn}{\mathrm{dn}}
+\DeclareMathOperator{\nd}{\mathrm{nd}}
+\DeclareMathOperator{\ds}{\mathrm{ds}}
+\DeclareMathOperator{\dc}{\mathrm{dc}}
+
diff --git a/buch/papers/ellfilter/presentation/presentation.tex b/buch/papers/ellfilter/presentation/presentation.tex
new file mode 100644
index 0000000..7fdb864
--- /dev/null
+++ b/buch/papers/ellfilter/presentation/presentation.tex
@@ -0,0 +1,413 @@
+\documentclass[ngerman, aspectratio=169, xcolor={rgb}]{beamer}
+
+% style
+\mode<presentation>{
+	\usetheme{Frankfurt}
+}
+%packages
+\usepackage[utf8]{inputenc}\DeclareUnicodeCharacter{2212}{-}
+\usepackage[english]{babel}
+\usepackage{graphicx}
+\usepackage{array}
+
+\newcolumntype{L}[1]{>{\raggedright\let\newline\\\arraybackslash\hspace{0pt}}m{#1}}
+\usepackage{ragged2e}
+
+\usepackage{bm} % bold math
+\usepackage{amsfonts}
+\usepackage{amssymb}
+\usepackage{mathtools}
+\usepackage{amsmath}
+\usepackage{multirow} % multi row in tables
+\usepackage{booktabs} %toprule midrule bottomrue in tables
+\usepackage{scrextend}
+\usepackage{textgreek}
+\usepackage[rgb]{xcolor}
+
+\usepackage{ marvosym } % \Lightning
+
+\usepackage{multimedia} % embedded videos
+
+\usepackage{tikz}
+\usepackage{pgf}
+\usepackage{pgfplots}
+
+\usepackage{algorithmic}
+
+%citations
+\usepackage[style=verbose,backend=biber]{biblatex}
+\addbibresource{references.bib}
+
+
+%math font
+\usefonttheme[onlymath]{serif}
+
+%Beamer Template modifications
+%\definecolor{mainColor}{HTML}{0065A3} % HSR blue
+\definecolor{mainColor}{HTML}{D72864} % OST pink
+\definecolor{invColor}{HTML}{28d79b} % OST pink
+\definecolor{dgreen}{HTML}{38ad36} % Dark green
+
+%\definecolor{mainColor}{HTML}{000000} % HSR blue
+\setbeamercolor{palette primary}{bg=white,fg=mainColor}
+\setbeamercolor{palette secondary}{bg=orange,fg=mainColor}
+\setbeamercolor{palette tertiary}{bg=yellow,fg=red}
+\setbeamercolor{palette quaternary}{bg=mainColor,fg=white} %bg = Top bar, fg = active top bar topic
+\setbeamercolor{structure}{fg=black} % itemize, enumerate, etc (bullet points)
+\setbeamercolor{section in toc}{fg=black} % TOC sections
+\setbeamertemplate{section in toc}[sections numbered]
+\setbeamertemplate{subsection in toc}{%
+	\hspace{1.2em}{$\bullet$}~\inserttocsubsection\par}
+
+\setbeamertemplate{itemize items}[circle]
+\setbeamertemplate{description item}[circle]
+\setbeamertemplate{title page}[default][colsep=-4bp,rounded=true]
+\beamertemplatenavigationsymbolsempty
+
+\setbeamercolor{footline}{fg=gray}
+\setbeamertemplate{footline}{%
+	\hfill\usebeamertemplate***{navigation symbols}
+	\hspace{0.5cm}
+	\insertframenumber{}\hspace{0.2cm}\vspace{0.2cm}
+}
+
+\usepackage{caption}
+\captionsetup{labelformat=empty}
+
+%Title Page
+\title{Elliptische Filter}
+\subtitle{Eine Anwendung der Jaccobi elliptischen Funktionen}
+\author{Nicolas Tobler}
+% \institute{OST Ostschweizer Fachhochschule}
+% \institute{\includegraphics[scale=0.3]{../img/ost_logo.png}}
+\date{\today}
+
+\input{../packages.tex}
+
+\newcommand*{\QED}{\hfill\ensuremath{\blacksquare}}%
+
+\newcommand*{\HL}{\textcolor{mainColor}}
+\newcommand*{\RD}{\textcolor{red}}
+\newcommand*{\BL}{\textcolor{blue}}
+\newcommand*{\GN}{\textcolor{dgreen}}
+
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+
+
+\makeatletter
+\newcount\my@repeat@count
+\newcommand{\myrepeat}[2]{%
+	\begingroup
+	\my@repeat@count=\z@
+	\@whilenum\my@repeat@count<#1\do{#2\advance\my@repeat@count\@ne}%
+	\endgroup
+}
+\makeatother
+
+\usetikzlibrary{automata,arrows,positioning,calc,shapes.geometric, fadings}
+
+\begin{document}
+
+	\begin{frame}
+		\titlepage
+	\end{frame}
+
+	\begin{frame}
+		\frametitle{Content}
+		\tableofcontents
+	\end{frame}
+
+	\section{Linear Filter}
+
+	\begin{frame}
+		\frametitle{Lineare Filter}
+
+
+		\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{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} \label{ellfilter:eq:chebychef_polynomials}
+			T_N(w) &= \cos \left( N \cos^{-1}(w) \right) \\
+				&= \cos \left(N~z \right), \quad w= \cos(z)
+		\end{align}
+
+
+	\end{frame}
+
+	\begin{frame}
+		\frametitle{Tschebyscheff-Filter}
+
+		\begin{equation*}
+			z = \cos^{-1}(w)
+		\end{equation*}
+
+		\begin{center}
+			\scalebox{0.85}{
+				\input{../tikz/arccos.tikz.tex}
+			}
+		\end{center}
+
+	\end{frame}
+
+	\begin{frame}
+		\frametitle{Tschebyscheff-Filter}
+
+		\begin{equation*}
+			z_1 = N~\cos^{-1}(w)
+		\end{equation*}
+
+		\begin{center}
+			\scalebox{0.85}{
+				\input{../tikz/arccos2.tikz.tex}
+			}
+		\end{center}
+
+	\end{frame}
+
+
+	\section{Jaccobi elliptische Funktionen}
+
+	\begin{frame}
+		\frametitle{Jaccobi elliptische Funktionen}
+
+
+		\begin{equation}
+			z
+			=
+			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{Jaccobi elliptische Funktionen}
+
+		\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{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{Jaccobi 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}
+
+	\begin{frame}
+		\frametitle{Periodizität in realer und imaginärer Richtung}
+
+		\begin{center}
+			\input{../python/k.pgf}
+		\end{center}
+
+
+	\end{frame}
+
+	\begin{frame}
+		\frametitle{Elliptisches Filter}
+
+		\begin{equation*}
+			z_1 = N~\frac{K_1}{K}~\cd^{-1}(w, k)
+		\end{equation*}
+
+		\begin{center}
+			\scalebox{0.8}{
+				\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}
+
+	\begin{frame}
+		\frametitle{Gradgleichung}
+
+		\begin{equation}
+			N \frac{K^\prime}{K} = \frac{K^\prime_1}{K_1}
+		\end{equation}
+
+	\end{frame}
+
+	\end{document}
diff --git a/buch/papers/ellfilter/python/F_N_butterworth.pgf b/buch/papers/ellfilter/python/F_N_butterworth.pgf
new file mode 100644
index 0000000..857e363
--- /dev/null
+++ b/buch/papers/ellfilter/python/F_N_butterworth.pgf
@@ -0,0 +1,1083 @@
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diff --git a/buch/papers/ellfilter/python/F_N_chebychev.pgf b/buch/papers/ellfilter/python/F_N_chebychev.pgf
new file mode 100644
index 0000000..72d5834
--- /dev/null
+++ b/buch/papers/ellfilter/python/F_N_chebychev.pgf
@@ -0,0 +1,1066 @@
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diff --git a/buch/papers/ellfilter/python/F_N_chebychev2.pgf b/buch/papers/ellfilter/python/F_N_chebychev2.pgf
new file mode 100644
index 0000000..43ebb91
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diff --git a/buch/papers/ellfilter/python/F_N_elliptic.pgf b/buch/papers/ellfilter/python/F_N_elliptic.pgf
new file mode 100644
index 0000000..03084c6
--- /dev/null
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diff --git a/buch/papers/ellfilter/python/chebychef.py b/buch/papers/ellfilter/python/chebychef.py
new file mode 100644
index 0000000..254ad4b
--- /dev/null
+++ b/buch/papers/ellfilter/python/chebychef.py
@@ -0,0 +1,66 @@
+# %%
+
+import matplotlib.pyplot as plt
+import scipy.signal
+import numpy as np
+
+
+order = 5
+passband_ripple_db = 1
+omega_c = 1000
+
+a, b = scipy.signal.cheby1(
+    order,
+    passband_ripple_db,
+    omega_c,
+    btype='low',
+    analog=True,
+    output='ba',
+    fs=None,
+)
+
+w, mag, phase = scipy.signal.bode((a, b), w=np.linspace(0,2000,256))
+f, axs = plt.subplots(2,1, sharex=True)
+axs[0].plot(w, 10**(mag/20))
+axs[0].set_ylabel("$|H(\omega)| /$ db")
+axs[0].grid(True, "both")
+axs[1].plot(w, phase)
+axs[1].set_ylabel(r"$arg H (\omega) / $ deg")
+axs[1].grid(True, "both")
+axs[1].set_xlim([0, 2000])
+axs[1].set_xlabel("$\omega$")
+plt.show()
+
+
+# %% Cheychev filter F_N plot
+
+w = np.linspace(-1.1,1.1, 1000)
+plt.figure(figsize=(5.5,2.5))
+for N in [3,6,11]:
+    # F_N = np.cos(N * np.arccos(w))
+    F_N = scipy.special.eval_chebyt(N, w)
+    plt.plot(w, F_N, label=f"$N={N}$")
+plt.xlim([-1.2,1.2])
+plt.ylim([-2,2])
+plt.grid()
+plt.xlabel("$w$")
+plt.ylabel("$T_N(w)$")
+plt.legend()
+plt.tight_layout()
+plt.savefig("F_N_chebychev2.pgf")
+plt.show()
+
+# %% Build Chebychev polynomials
+
+N = 11
+
+zeros = (np.arange(N)+0.5) * np.pi
+zeros = np.cos(zeros/N)
+
+x = np.linspace(-1.2,1.2,1000)
+y = np.prod(x[:, None] - zeros[None, :], axis=-1)*2**(N-1)
+
+plt.plot(x, y)
+plt.ylim([-1,1])
+plt.grid()
+plt.show()
diff --git a/buch/papers/ellfilter/python/elliptic.pgf b/buch/papers/ellfilter/python/elliptic.pgf
new file mode 100644
index 0000000..31b77d4
--- /dev/null
+++ b/buch/papers/ellfilter/python/elliptic.pgf
@@ -0,0 +1,709 @@
+%% Creator: Matplotlib, PGF backend
+%%
+%% To include the figure in your LaTeX document, write
+%%   \input{<filename>.pgf}
+%%
+%% Make sure the required packages are loaded in your preamble
+%%   \usepackage{pgf}
+%%
+%% Also ensure that all the required font packages are loaded; for instance,
+%% the lmodern package is sometimes necessary when using math font.
+%%   \usepackage{lmodern}
+%%
+%% Figures using additional raster images can only be included by \input if
+%% they are in the same directory as the main LaTeX file. For loading figures
+%% from other directories you can use the `import` package
+%%   \usepackage{import}
+%%
+%% and then include the figures with
+%%   \import{<path to file>}{<filename>.pgf}
+%%
+%% Matplotlib used the following preamble
+%%
+\begingroup%
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diff --git a/buch/papers/ellfilter/python/elliptic.py b/buch/papers/ellfilter/python/elliptic.py
new file mode 100644
index 0000000..b3336a1
--- /dev/null
+++ b/buch/papers/ellfilter/python/elliptic.py
@@ -0,0 +1,356 @@
+
+# %%
+
+import scipy.special
+import scipyx as spx
+import numpy as np
+import matplotlib.pyplot as plt
+from matplotlib.patches import Rectangle
+
+import plot_params
+
+def last_color():
+    return plt.gca().lines[-1].get_color()
+
+# define elliptic functions
+
+def ell_int(k):
+    """ Calculate K(k) """
+    m = k**2
+    return scipy.special.ellipk(m)
+
+def sn(z, k):
+    return spx.ellipj(z, k**2)[0]
+
+def cn(z, k):
+    return spx.ellipj(z, k**2)[1]
+
+def dn(z, k):
+    return spx.ellipj(z, k**2)[2]
+
+def cd(z, k):
+    sn, cn, dn, ph = spx.ellipj(z, k**2)
+    return cn / dn
+
+# https://mathworld.wolfram.com/JacobiEllipticFunctions.html eq 3-8
+
+def sn_inv(z, k):
+    m = k**2
+    return scipy.special.ellipkinc(np.arcsin(z), m)
+
+def cn_inv(z, k):
+    m = k**2
+    return scipy.special.ellipkinc(np.arccos(z), m)
+
+def dn_inv(z, k):
+    m = k**2
+    x = np.sqrt((1-z**2) / k**2)
+    return scipy.special.ellipkinc(np.arcsin(x), m)
+
+def cd_inv(z, k):
+    m = k**2
+    x = np.sqrt(((m - 1) * z**2) / (m*z**2 - 1))
+    return scipy.special.ellipkinc(np.arccos(x), m)
+
+
+k = 0.8
+z = 0.5
+
+assert np.allclose(sn_inv(sn(z ,k), k), z)
+assert np.allclose(cn_inv(cn(z ,k), k), z)
+assert np.allclose(dn_inv(dn(z ,k), k), z)
+assert np.allclose(cd_inv(cd(z ,k), k), z)
+
+
+# %% Buttwerworth filter F_N plot
+
+w = np.linspace(0,1.5, 100)
+plt.figure(figsize=(4,2.5))
+
+for N in range(1,5):
+    F_N = w**N
+    plt.plot(w, F_N**2, label=f"$N={N}$")
+plt.gca().add_patch(Rectangle(
+    (0, 0),
+    1, 1,
+    fc ='green',
+    alpha=0.2,
+    lw = 10,
+))
+plt.gca().add_patch(Rectangle(
+    (1, 1),
+    0.5, 1,
+    fc ='orange',
+    alpha=0.2,
+    lw = 10,
+))
+plt.xlim([0,1.5])
+plt.ylim([0,2])
+plt.grid()
+plt.xlabel("$w$")
+plt.ylabel("$F^2_N(w)$")
+plt.legend()
+plt.tight_layout()
+plt.savefig("F_N_butterworth.pgf")
+plt.show()
+
+# %% Cheychev filter F_N plot
+
+w = np.linspace(0,1.5, 100)
+
+plt.figure(figsize=(4,2.5))
+for N in range(1,5):
+    # F_N = np.cos(N * np.arccos(w))
+    F_N = scipy.special.eval_chebyt(N, w)
+    plt.plot(w, F_N**2, label=f"$N={N}$")
+plt.gca().add_patch(Rectangle(
+    (0, 0),
+    1, 1,
+    fc ='green',
+    alpha=0.2,
+    lw = 10,
+))
+plt.gca().add_patch(Rectangle(
+    (1, 1),
+    0.5, 1,
+    fc ='orange',
+    alpha=0.2,
+    lw = 10,
+))
+plt.xlim([0,1.5])
+plt.ylim([0,2])
+plt.grid()
+plt.xlabel("$w$")
+plt.ylabel("$F^2_N(w)$")
+plt.legend()
+plt.tight_layout()
+plt.savefig("F_N_chebychev.pgf")
+plt.show()
+
+
+# %% plot arcsin
+
+def lattice(a1, b1, c1, a2, b2, c2):
+    r1 = np.logspace(a1, b1, c1)
+    x1 = np.concatenate((-np.flip(r1), [0], r1), axis=0)
+    x1 = x1.astype(np.complex128)
+    r2 = np.logspace(a2, b2, c2)
+    x2 = np.concatenate((-np.flip(r2), [0], r2), axis=0)
+    x2 = x2.astype(np.complex128)
+    x = (x1[:, None] + (x2[None, :] * 1j))
+    return x
+
+plt.figure(figsize=(12,12))
+y = np.arcsin(lattice(-1,6,1000, -1,5,10))
+plt.plot(np.real(y), np.imag(y), "-", color="red", lw=0.5)
+y = np.arcsin(lattice(-1,6,10, -1,5,100)).T
+plt.plot(np.real(y), np.imag(y), "-", color="red", lw=0.5)
+y = np.arcsin(lattice(-1,6,10, -1,5,10))
+plt.plot(np.real(y), np.imag(y), ".", color="red", lw=0.5)
+plt.show()
+
+# %% plot cd^-1 TODO complex cd^-1 missing
+
+
+r = np.logspace(-1,8, 50)
+
+
+
+x = np.concatenate((-np.flip(r), [0], r), axis=0)
+y = cd_inv(x, 0.99)
+
+plt.figure(figsize=(12,12))
+plt.plot(np.real(y), np.imag(y), "-")
+plt.show()
+
+# %%plot cd
+plt.figure(figsize=(10,6))
+z = np.linspace(-4,4, 500)
+for k in [0, 0.9, 0.99, 0.999, 0.99999]:
+    w = cd(z*ell_int(k), k)
+    plt.plot(z, w, label=f"$k={k}$")
+plt.grid()
+plt.legend()
+# plt.xlim([-4,4])
+plt.xlabel("$u$")
+plt.ylabel("$cd(uK, k)$")
+plt.show()
+
+# %% Test ????
+
+N = 5
+k = 0.9
+k1 = k**N
+
+assert np.allclose(k**(-N), k1**(-1))
+
+K = ell_int(k)
+Kp = ell_int(np.sqrt(1-k**2))
+
+K1 = ell_int(k1)
+Kp1 = ell_int(np.sqrt(1-k1**2))
+
+print(Kp * (K1 / K) * N, Kp1)
+
+
+# %%
+
+
+k = 0.9
+k_prim = np.sqrt(1 - k**2)
+K = ell_int(k)
+Kp = ell_int(k_prim)
+
+print(K, Kp)
+
+zs = [
+    0 + (K + 0j) * np.linspace(0,1,25),
+    K +  (Kp*1j) * np.linspace(0,1,25),
+    (K + Kp*1j) + (-K) * np.linspace(0,1,25),
+]
+
+
+for z in zs:
+    plt.plot(np.real(z),  np.imag(z))
+plt.show()
+
+
+
+for z in zs:
+    w = cd(z, k)
+    plt.plot(np.real(w),  np.imag(w))
+plt.show()
+
+
+
+
+
+# %%
+
+for i in range(10):
+    x = np.linspace(i*1,i*1+1,10, dtype=np.complex64)
+    w = np.arccos(x)
+
+    x2 = np.cos(w)
+    x4 = np.cos(w+ 2*np.pi)
+    x3 = np.cos(np.conj(w))
+
+    assert np.allclose(x2, x4, rtol=0.001, atol=1e-5)
+
+    assert np.allclose(x2, x3)
+    assert np.allclose(x2, x, rtol=0.001, atol=1e-5)
+
+    plt.plot(np.real(w), np.imag(w), ".-")
+
+for i in range(10):
+    x = -np.linspace(i*1,i*1+1,100, dtype=np.complex64)
+    w = np.arccos(x)
+    plt.plot(np.real(w), np.imag(w), ".-")
+
+plt.grid()
+plt.show()
+
+
+
+
+# %%
+
+plt.plot(omega, np.abs(G))
+plt.show()
+
+
+def cd_inv(u, m):
+    return K(1/2) - F(np.arcsin())
+
+def K(m):
+    return scipy.special.ellipk(m)
+
+def L(n, xi):
+    return 1 #TODO
+
+def R(n, xi, x):
+    cn(n*K(1/L(n, xi))/K(1/xi) * cd_inv(x, 1/xi, 1/L(n, xi)))
+
+epsilon = 0.1
+n = 3
+omega = np.linspace(0, np.pi, 1000)
+omega_0 = 1
+xi = 1.1
+
+G = 1 / np.sqrt(1 + epsilon**2 * R(n, xi, omega/omega_0)**2)
+
+
+plt.plot(omega, np.abs(G))
+plt.show()
+
+
+
+# %% Chebychef
+
+epsilon = 0.5
+omega = np.linspace(0, np.pi, 1000)
+omega_0 = 1
+n = 4
+
+def chebychef_poly(n, x):
+    x = x.astype(np.complex64)
+    y = np.cos(n* np.arccos(x))
+    return np.real(y)
+
+F_omega = chebychef_poly
+
+for n in (1,2,3,4):
+    plt.plot(omega, F_omega(n, omega/omega_0)**2)
+plt.ylim([0,5])
+plt.xlim([0,1.5])
+plt.grid()
+plt.show()
+
+for n in (1,2,3,4):
+    G = 1 / np.sqrt(1 + epsilon**2 * F_omega(n, omega/omega_0)**2)
+    plt.plot(omega, np.abs(G))
+plt.grid()
+plt.show()
+
+
+
+
+# %%
+
+
+k = np.concatenate(([0.00001,0.0001,0.001], np.linspace(0,1,101)[1:-1], [0.999,0.9999, 0.99999]), axis=0)
+K = ell_int(k)
+K_prime = ell_int(np.sqrt(1-k**2))
+
+
+f, axs = plt.subplots(1,2, figsize=(5,2.5))
+axs[0].plot(k, K, linewidth=0.1)
+axs[0].text(k[30], K[30]+0.1, f"$K$")
+axs[0].plot(k, K_prime, linewidth=0.1)
+axs[0].text(k[30], K_prime[30]+0.1, f"$K^\prime$")
+axs[0].set_xlim([0,1])
+axs[0].set_ylim([0,4])
+axs[0].set_xlabel("$k$")
+
+axs[1].axvline(x=np.pi/2, color="gray", linewidth=0.5)
+axs[1].axhline(y=np.pi/2, color="gray", linewidth=0.5)
+axs[1].text(0.1, np.pi/2 + 0.1, "$\pi/2$")
+axs[1].text(np.pi/2+0.1, 0.1, "$\pi/2$")
+axs[1].plot(K, K_prime, linewidth=1)
+
+k = np.array([0.1,0.2,0.4,0.6,0.9,0.99])
+K = ell_int(k)
+K_prime = ell_int(np.sqrt(1-k**2))
+
+axs[1].plot(K, K_prime, '.', color=last_color(), markersize=2)
+for x, y, n in zip(K, K_prime, k):
+    axs[1].text(x+0.1, y+0.1, f"$k={n:.2f}$", rotation_mode="anchor")
+axs[1].set_ylabel("$K^\prime$")
+axs[1].set_xlabel("$K$")
+axs[1].set_xlim([0,6])
+axs[1].set_ylim([0,5])
+plt.tight_layout()
+plt.savefig("k.pgf")
+plt.show()
+
+print(K[0], K[-1])
diff --git a/buch/papers/ellfilter/python/elliptic2.py b/buch/papers/ellfilter/python/elliptic2.py
new file mode 100644
index 0000000..29c6f47
--- /dev/null
+++ b/buch/papers/ellfilter/python/elliptic2.py
@@ -0,0 +1,149 @@
+# %%
+
+import matplotlib.pyplot as plt
+import scipy.signal
+import numpy as np
+import matplotlib
+from matplotlib.patches import Rectangle
+
+import plot_params
+
+def ellip_filter(N):
+
+    order = N
+    passband_ripple_db = 3
+    stopband_attenuation_db = 20
+    omega_c = 1
+
+    a, b = scipy.signal.ellip(
+        order,
+        passband_ripple_db,
+        stopband_attenuation_db,
+        omega_c,
+        btype='low',
+        analog=True,
+        output='ba',
+        fs=None
+    )
+
+    w, mag_db, phase = scipy.signal.bode((a, b), w=np.linspace(0*omega_c,2*omega_c, 4000))
+
+    mag = 10**(mag_db/20)
+
+    passband_ripple = 10**(-passband_ripple_db/20)
+    epsilon2 = (1/passband_ripple)**2 - 1
+
+    FN2 = ((1/mag**2) - 1)
+
+    return w/omega_c, FN2 / epsilon2, mag, a, b
+
+
+plt.figure(figsize=(4,2.5))
+
+for N in [5]:
+    w, FN2, mag, a, b = ellip_filter(N)
+    plt.semilogy(w, FN2, label=f"$N={N}, k=0.1$", linewidth=1)
+
+plt.gca().add_patch(Rectangle(
+    (0, 0),
+    1, 1,
+    fc ='green',
+    alpha=0.2,
+    lw = 10,
+))
+plt.gca().add_patch(Rectangle(
+    (1, 1),
+    0.01, 1e2-1,
+    fc ='orange',
+    alpha=0.2,
+    lw = 10,
+))
+
+plt.gca().add_patch(Rectangle(
+    (1.01, 100),
+    1, 1e6,
+    fc ='red',
+    alpha=0.2,
+    lw = 10,
+))
+
+zeros = [0,0.87,1]
+poles = [1.01,1.155]
+
+import matplotlib.transforms
+plt.plot( # mark errors as vertical bars
+    zeros,
+    np.zeros_like(zeros),
+    "o",
+    mfc='none',
+    color='black',
+    transform=matplotlib.transforms.blended_transform_factory(
+        plt.gca().transData,
+        plt.gca().transAxes,
+    ),
+)
+plt.plot( # mark errors as vertical bars
+    poles,
+    np.ones_like(poles),
+    "x",
+    mfc='none',
+    color='black',
+    transform=matplotlib.transforms.blended_transform_factory(
+        plt.gca().transData,
+        plt.gca().transAxes,
+    ),
+)
+
+plt.xlim([0,2])
+plt.ylim([1e-4,1e6])
+plt.grid()
+plt.xlabel("$w$")
+plt.ylabel("$F^2_N(w)$")
+plt.legend()
+plt.tight_layout()
+plt.savefig("F_N_elliptic.pgf")
+plt.show()
+
+
+
+plt.figure(figsize=(4,2.5))
+plt.plot(w, mag, linewidth=1)
+
+plt.gca().add_patch(Rectangle(
+    (0, np.sqrt(2)/2),
+    1, 1,
+    fc ='green',
+    alpha=0.2,
+    lw = 10,
+))
+plt.gca().add_patch(Rectangle(
+    (1, 0.1),
+    0.01, np.sqrt(2)/2 - 0.1,
+    fc ='orange',
+    alpha=0.2,
+    lw = 10,
+))
+
+plt.gca().add_patch(Rectangle(
+    (1.01, 0),
+    1, 0.1,
+    fc ='red',
+    alpha=0.2,
+    lw = 10,
+))
+
+plt.grid()
+plt.xlim([0,2])
+plt.ylim([0,1])
+plt.xlabel("$w$")
+plt.ylabel("$|H(w)|$")
+plt.tight_layout()
+plt.savefig("elliptic.pgf")
+plt.show()
+
+print("zeros", a)
+print("poles", b)
+
+
+
+
diff --git a/buch/papers/ellfilter/python/k.pgf b/buch/papers/ellfilter/python/k.pgf
new file mode 100644
index 0000000..95d61d4
--- /dev/null
+++ b/buch/papers/ellfilter/python/k.pgf
@@ -0,0 +1,1157 @@
+%% Creator: Matplotlib, PGF backend
+%%
+%% To include the figure in your LaTeX document, write
+%%   \input{<filename>.pgf}
+%%
+%% Make sure the required packages are loaded in your preamble
+%%   \usepackage{pgf}
+%%
+%% Also ensure that all the required font packages are loaded; for instance,
+%% the lmodern package is sometimes necessary when using math font.
+%%   \usepackage{lmodern}
+%%
+%% Figures using additional raster images can only be included by \input if
+%% they are in the same directory as the main LaTeX file. For loading figures
+%% from other directories you can use the `import` package
+%%   \usepackage{import}
+%%
+%% and then include the figures with
+%%   \import{<path to file>}{<filename>.pgf}
+%%
+%% Matplotlib used the following preamble
+%%
+\begingroup%
+\makeatletter%
+\begin{pgfpicture}%
+\pgfpathrectangle{\pgfpointorigin}{\pgfqpoint{5.000000in}{2.500000in}}%
+\pgfusepath{use as bounding box, clip}%
+\begin{pgfscope}%
+\pgfsetbuttcap%
+\pgfsetmiterjoin%
+\pgfsetlinewidth{0.000000pt}%
+\definecolor{currentstroke}{rgb}{1.000000,1.000000,1.000000}%
+\pgfsetstrokecolor{currentstroke}%
+\pgfsetstrokeopacity{0.000000}%
+\pgfsetdash{}{0pt}%
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+\pgfpathlineto{\pgfqpoint{0.000000in}{2.500000in}}%
+\pgfpathlineto{\pgfqpoint{0.000000in}{0.000000in}}%
+\pgfpathclose%
+\pgfusepath{}%
+\end{pgfscope}%
+\begin{pgfscope}%
+\pgfsetbuttcap%
+\pgfsetmiterjoin%
+\definecolor{currentfill}{rgb}{1.000000,1.000000,1.000000}%
+\pgfsetfillcolor{currentfill}%
+\pgfsetlinewidth{0.000000pt}%
+\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}%
+\pgfsetstrokecolor{currentstroke}%
+\pgfsetstrokeopacity{0.000000}%
+\pgfsetdash{}{0pt}%
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+\pgfpathlineto{\pgfqpoint{0.316407in}{2.301955in}}%
+\pgfpathlineto{\pgfqpoint{0.316407in}{0.548769in}}%
+\pgfpathclose%
+\pgfusepath{fill}%
+\end{pgfscope}%
+\begin{pgfscope}%
+\pgfsetbuttcap%
+\pgfsetroundjoin%
+\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}%
+\pgfsetfillcolor{currentfill}%
+\pgfsetlinewidth{0.803000pt}%
+\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}%
+\pgfsetstrokecolor{currentstroke}%
+\pgfsetdash{}{0pt}%
+\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{%
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diff --git a/buch/papers/ellfilter/python/plot_params.py b/buch/papers/ellfilter/python/plot_params.py
new file mode 100644
index 0000000..4ddd1d8
--- /dev/null
+++ b/buch/papers/ellfilter/python/plot_params.py
@@ -0,0 +1,9 @@
+import matplotlib
+
+matplotlib.rcParams.update({
+    "pgf.texsystem": "pdflatex",
+    'font.family': 'serif',
+    'font.size': 9,
+    'text.usetex': True,
+    'pgf.rcfonts': False,
+})
diff --git a/buch/papers/ellfilter/references.bib b/buch/papers/ellfilter/references.bib
index 81b3577..8f21971 100644
--- a/buch/papers/ellfilter/references.bib
+++ b/buch/papers/ellfilter/references.bib
@@ -4,32 +4,19 @@
 % (c) 2020 Autor, Hochschule Rapperswil
 %
 
-@online{ellfilter:bibtex,
-	title = {BibTeX},
-	url = {https://de.wikipedia.org/wiki/BibTeX},
-	date = {2020-02-06},
-	year = {2020},
-	month = {2},
-	day = {6}
-}
-
-@book{ellfilter:numerical-analysis,
-	title = {Numerical Analysis},
-	author = {David Kincaid and Ward Cheney},
-	publisher = {American Mathematical Society},
-	year = {2002},
-	isbn = {978-8-8218-4788-6},
-	inseries = {Pure and applied undegraduate texts},
-	volume = {2}
-}
-
-@article{ellfilter:mendezmueller,
-        author = { Tabea Méndez and Andreas Müller },
-        title = { Noncommutative harmonic analysis and image registration },
-        journal = { Appl. Comput. Harmon. Anal.},
-        year = 2019,
-        volume = 47,
-        pages = {607--627},
-        url = {https://doi.org/10.1016/j.acha.2017.11.004}
+@online{ellfilter:bib:orfanidis,
+        author = { Sophocles J. Orfanidis},
+        title = { LECTURE NOTES ON ELLIPTIC FILTER DESIGN },
+        year = 2006,
+        url = {https://www.ece.rutgers.edu/~orfanidi/ece521/notes.pdf}
 }
 
+% Schwalm
+% https://en.wikipedia.org/wiki/Elliptic_rational_functions
+% https://en.wikipedia.org/wiki/Rational_function
+% https://en.wikipedia.org/wiki/Jacobi_elliptic_functions
+% https://de.wikipedia.org/wiki/Elliptisches_Integral
+% https://de.wikipedia.org/wiki/Tschebyschow-Polynom
+% https://en.wikipedia.org/wiki/Chebyshev_filter
+% https://mathworld.wolfram.com/JacobiEllipticFunctions.html
+% https://mathworld.wolfram.com/EllipticIntegraloftheFirstKind.html
diff --git a/buch/papers/ellfilter/teil0.tex b/buch/papers/ellfilter/teil0.tex
deleted file mode 100644
index fd04ba9..0000000
--- a/buch/papers/ellfilter/teil0.tex
+++ /dev/null
@@ -1,22 +0,0 @@
-%
-% einleitung.tex -- Beispiel-File für die Einleitung
-%
-% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\section{Teil 0\label{ellfilter:section:teil0}}
-\rhead{Teil 0}
-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{ellfilter: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.
-
-Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam
-nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam
-erat, sed diam voluptua.
-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.
-
-
diff --git a/buch/papers/ellfilter/teil1.tex b/buch/papers/ellfilter/teil1.tex
deleted file mode 100644
index 7e62a2f..0000000
--- a/buch/papers/ellfilter/teil1.tex
+++ /dev/null
@@ -1,55 +0,0 @@
-%
-% teil1.tex -- Beispiel-File für das Paper
-%
-% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\section{Teil 1
-\label{ellfilter:section:teil1}}
-\rhead{Problemstellung}
-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
-\begin{equation}
-\int_a^b x^2\, dx
-=
-\left[ \frac13 x^3 \right]_a^b
-=
-\frac{b^3-a^3}3.
-\label{ellfilter:equation1}
-\end{equation}
-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{ellfilter:subsection:finibus}}
-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 \eqref{000tempmlate:equation1}.
-
-Et harum quidem rerum facilis est et expedita distinctio
-\ref{ellfilter:section:loesung}.
-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
-\ref{ellfilter:section:folgerung}.
-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.
-
-
diff --git a/buch/papers/ellfilter/teil2.tex b/buch/papers/ellfilter/teil2.tex
deleted file mode 100644
index 71fdc6d..0000000
--- a/buch/papers/ellfilter/teil2.tex
+++ /dev/null
@@ -1,40 +0,0 @@
-%
-% teil2.tex -- Beispiel-File für teil2 
-%
-% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\section{Teil 2 
-\label{ellfilter:section:teil2}}
-\rhead{Teil 2}
-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{ellfilter:subsection:bonorum}}
-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.
-
-
diff --git a/buch/papers/ellfilter/teil3.tex b/buch/papers/ellfilter/teil3.tex
deleted file mode 100644
index 79a5f3d..0000000
--- a/buch/papers/ellfilter/teil3.tex
+++ /dev/null
@@ -1,40 +0,0 @@
-%
-% teil3.tex -- Beispiel-File für Teil 3
-%
-% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\section{Teil 3
-\label{ellfilter:section:teil3}}
-\rhead{Teil 3}
-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{ellfilter: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.
-
-
diff --git a/buch/papers/ellfilter/tikz/arccos.tikz.tex b/buch/papers/ellfilter/tikz/arccos.tikz.tex
new file mode 100644
index 0000000..2772620
--- /dev/null
+++ b/buch/papers/ellfilter/tikz/arccos.tikz.tex
@@ -0,0 +1,66 @@
+\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2]
+
+    \tikzstyle{zero} = [draw, circle, inner sep =0, minimum height=0.15cm]
+    \tikzset{pole/.style={cross out, draw=black, minimum size=(0.15cm-\pgflinewidth), inner sep=0pt, outer sep=0pt}}
+
+    \draw[gray, ->] (0,-2) -- (0,2) node[anchor=south]{$\mathrm{Im}~z$};
+    \draw[gray, ->] (-5,0) -- (5,0) node[anchor=west]{$\mathrm{Re}~z$};
+
+    \begin{scope}[xscale=0.6]
+
+        \clip(-7.5,-2) rectangle (7.5,2);
+
+        \draw[thick, ->, darkgreen] (0, 0) -- (0,1.5);
+        \draw[thick, ->, orange] (1, 0) -- (0,0);
+        \draw[thick, ->, red] (2, 0) -- (1,0);
+        \draw[thick, ->, blue] (2,1.5) -- (2, 0);
+
+        \foreach \i in {-2,...,1} {
+            \begin{scope}[opacity=0.5, xshift=\i*4cm]
+                \draw[->, orange] (-1, 0) -- (0,0);
+                \draw[->, darkgreen] (0, 0) -- (0,1.5);
+                \draw[->, darkgreen] (0, 0) -- (0,-1.5);
+                \draw[->, orange] (1, 0) -- (0,0);
+                \draw[->, red] (2, 0) -- (1,0);
+                \draw[->, blue] (2,1.5) -- (2, 0);
+                \draw[->, blue] (2,-1.5) -- (2, 0);
+                \draw[->, red] (2, 0) -- (3,0);
+
+                \node[zero] at (1,0) {};
+                \node[zero] at (3,0) {};
+            \end{scope}
+        }
+
+        \node[gray, anchor=north] at (-6,0) {$-3\pi$};
+        \node[gray, anchor=north] at (-4,0) {$-2\pi$};
+        \node[gray, anchor=north] at (-2,0) {$-\pi$};
+        % \node[gray, anchor=north] at (0,0) {$0$};
+        \node[gray, anchor=north] at (2,0) {$\pi$};
+        \node[gray, anchor=north] at (4,0) {$2\pi$};
+        \node[gray, anchor=north] at (6,0) {$3\pi$};
+
+        \node[gray, anchor=east] at (0,-1.5) {$-\infty$};
+        % \node[gray, anchor=south east] at (0, 0) {$0$};
+        \node[gray, anchor=east] at (0, 1.5) {$\infty$};
+
+    \end{scope}
+
+    \begin{scope}[yshift=-2.5cm]
+
+        \draw[gray, ->] (-5,0) -- (5,0) node[anchor=west]{$w$};
+
+        \draw[thick, ->, blue]      (-4, 0) -- (-2, 0);
+        \draw[thick, ->, red]       (-2, 0) -- (0, 0);
+        \draw[thick, ->, orange]    (0, 0) -- (2, 0);
+        \draw[thick, ->, darkgreen] (2, 0) -- (4, 0);
+
+        \node[anchor=south] at (-4,0) {$-\infty$};
+        \node[anchor=south] at (-2,0) {$-1$};
+        \node[anchor=south] at (0,0) {$0$};
+        \node[anchor=south] at (2,0) {$1$};
+        \node[anchor=south] at (4,0) {$\infty$};
+
+    \end{scope}
+
+
+\end{tikzpicture}
\ No newline at end of file
diff --git a/buch/papers/ellfilter/tikz/arccos2.tikz.tex b/buch/papers/ellfilter/tikz/arccos2.tikz.tex
new file mode 100644
index 0000000..3fc3cc6
--- /dev/null
+++ b/buch/papers/ellfilter/tikz/arccos2.tikz.tex
@@ -0,0 +1,45 @@
+\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2]
+
+    \tikzstyle{zero} = [draw, circle, inner sep =0, minimum height=0.15cm]
+    \tikzset{pole/.style={cross out, draw=black, minimum size=(0.15cm-\pgflinewidth), inner sep=0pt, outer sep=0pt}}
+
+    \begin{scope}[xscale=0.5]
+
+        \draw[gray, ->] (0,-2) -- (0,2) node[anchor=south]{$\mathrm{Im}~z_1$};
+        \draw[gray, ->] (-10,0) -- (10,0) node[anchor=west]{$\mathrm{Re}~z_1$};
+
+        \begin{scope}
+
+            \draw[>->, line width=0.05, thick, blue]   (2, 1.5) -- (2,0.05)  -- node[anchor=south, pos=0.5]{$N=1$} (0.1,0.05) -- (0.1,1.5);
+            \draw[>->, line width=0.05, thick, orange] (4, 1.5) -- (4,0)     -- node[anchor=south, pos=0.25]{$N=2$} (0,0) -- (0,1.5);
+            \draw[>->, line width=0.05, thick, red]    (6, 1.5) node[anchor=north west]{$-\infty$} -- (6,-0.05) node[anchor=west]{$-1$} -- node[anchor=north]{$0$} node[anchor=south, pos=0.1666]{$N=3$} (-0.1,-0.05) node[anchor=east]{$1$}  -- (-0.1,1.5) node[anchor=north east]{$\infty$};
+
+
+            \node[zero] at (-7,0) {};
+            \node[zero] at (-5,0) {};
+            \node[zero] at (-3,0) {};
+            \node[zero] at (-1,0) {};
+            \node[zero] at (1,0) {};
+            \node[zero] at (3,0) {};
+            \node[zero] at (5,0) {};
+            \node[zero] at (7,0) {};
+
+
+        \end{scope}
+
+        \node[gray, anchor=north] at (-8,0) {$-4\pi$};
+        \node[gray, anchor=north] at (-6,0) {$-3\pi$};
+        \node[gray, anchor=north] at (-4,0) {$-2\pi$};
+        \node[gray, anchor=north] at (-2,0) {$-\pi$};
+        \node[gray, anchor=north] at (2,0) {$\pi$};
+        \node[gray, anchor=north] at (4,0) {$2\pi$};
+        \node[gray, anchor=north] at (6,0) {$3\pi$};
+        \node[gray, anchor=north] at (8,0) {$4\pi$};
+
+
+        \node[gray, anchor=east] at (0,-1.5) {$-\infty$};
+        \node[gray, anchor=east] at (0, 1.5) {$\infty$};
+
+    \end{scope}
+
+\end{tikzpicture}
\ No newline at end of file
diff --git a/buch/papers/ellfilter/tikz/cd.tikz.tex b/buch/papers/ellfilter/tikz/cd.tikz.tex
new file mode 100644
index 0000000..7155a85
--- /dev/null
+++ b/buch/papers/ellfilter/tikz/cd.tikz.tex
@@ -0,0 +1,87 @@
+\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2]
+
+    \tikzstyle{zero} = [draw, circle, inner sep =0, minimum height=0.15cm]
+
+    \tikzset{pole/.style={cross out, draw=black, minimum size=(0.15cm-\pgflinewidth), inner sep=0pt, outer sep=0pt}}
+
+    \begin{scope}[xscale=1, yscale=2]
+
+        \draw[gray, ->] (0,-1.5) -- (0,1.5) node[anchor=south]{$\mathrm{Im}~z$};
+        \draw[gray, ->] (-5,0) -- (5,0) node[anchor=west]{$\mathrm{Re}~z$};
+
+        \draw[gray] ( 1,0) +(0,0.1) -- +(0, -0.1) node[inner sep=0, anchor=north] {\small $K$};
+
+        \draw[gray]  (0, 0.5) +(0.1, 0) -- +(-0.1, 0) node[inner sep=0, anchor=east]{\small $jK^\prime$};
+
+
+        \begin{scope}
+
+            \begin{scope}[xshift=0cm]
+
+                \clip(-4.5,-1.25) rectangle (4.5,1.25);
+
+                \fill[yellow!30] (0,0) rectangle (1, 0.5);
+
+
+                \draw[thick, ->, darkgreen] (0, 0) -- (0,0.5);
+                \draw[thick, ->, orange] (1, 0) -- (0,0);
+                \draw[thick, ->, red] (2, 0) -- (1,0);
+                \draw[thick, ->, blue] (2,0.5) -- (2, 0);
+                \draw[thick, ->, purple] (1, 0.5) -- (2,0.5);
+                \draw[thick, ->, cyan] (0, 0.5) -- (1,0.5);
+
+
+
+                \foreach \i in {-2,...,1} {
+                    \foreach \j in {-2,...,1} {
+                        \begin{scope}[xshift=\i*4cm, yshift=\j*1cm]
+                            \draw[opacity=0.5, ->, darkgreen] (0, 0) -- (0,0.5);
+                            \draw[opacity=0.5, ->, orange] (1, 0) -- (0,0);
+                            \draw[opacity=0.5, ->, red] (2, 0) -- (1,0);
+                            \draw[opacity=0.5, ->, blue] (2,0.5) -- (2, 0);
+                            \draw[opacity=0.5, ->, purple] (1, 0.5) -- (2,0.5);
+                            \draw[opacity=0.5, ->, cyan] (0, 0.5) -- (1,0.5);
+                            \draw[opacity=0.5, ->, darkgreen] (0,1) -- (0,0.5);
+                            \draw[opacity=0.5, ->, blue] (2,0.5) -- (2, 1);
+                            \draw[opacity=0.5, ->, purple] (3, 0.5) -- (2,0.5);
+                            \draw[opacity=0.5, ->, cyan] (4, 0.5) -- (3,0.5);
+                            \draw[opacity=0.5, ->, red] (2, 0) -- (3,0);
+                            \draw[opacity=0.5, ->, orange] (3, 0) -- (4,0);
+
+                            \node[zero] at ( 1, 0) {};
+                            \node[zero] at ( 3, 0) {};
+                            \node[pole] at ( 1,0.5) {};
+                            \node[pole] at ( 3,0.5) {};
+
+                        \end{scope}
+                    }
+                }
+
+            \end{scope}
+
+        \end{scope}
+
+    \end{scope}
+
+    \begin{scope}[yshift=-3.5cm, xscale=0.75]
+
+        \draw[gray, ->] (-6,0) -- (6,0) node[anchor=west]{$w$};
+
+        \draw[thick, ->, purple] (-5, 0) -- (-3, 0);
+        \draw[thick, ->, blue]      (-3, 0) -- (-2, 0);
+        \draw[thick, ->, red]       (-2, 0) -- (0, 0);
+        \draw[thick, ->, orange]    (0, 0) -- (2, 0);
+        \draw[thick, ->, darkgreen] (2, 0) -- (3, 0);
+        \draw[thick, ->, cyan] (3, 0) -- (5, 0);
+
+        \node[anchor=south] at (-5,0) {$-\infty$};
+        \node[anchor=south] at (-3,0) {$-1/k$};
+        \node[anchor=south] at (-2,0) {$-1$};
+        \node[anchor=south] at (0,0) {$0$};
+        \node[anchor=south] at (2,0) {$1$};
+        \node[anchor=south] at (3,0) {$1/k$};
+        \node[anchor=south] at (5,0) {$\infty$};
+
+    \end{scope}
+
+\end{tikzpicture}
\ No newline at end of file
diff --git a/buch/papers/ellfilter/tikz/cd2.tikz.tex b/buch/papers/ellfilter/tikz/cd2.tikz.tex
new file mode 100644
index 0000000..0743f7d
--- /dev/null
+++ b/buch/papers/ellfilter/tikz/cd2.tikz.tex
@@ -0,0 +1,84 @@
+\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2]
+
+    \tikzstyle{zero} = [draw, circle, inner sep =0, minimum height=0.15cm]
+    \tikzstyle{dot} = [fill, circle, inner sep =0, minimum height=0.1cm]
+
+    \tikzset{pole/.style={cross out, draw=black, minimum size=(0.15cm-\pgflinewidth), inner sep=0pt, outer sep=0pt}}
+
+    \begin{scope}[xscale=1.25, yscale=2.5]
+
+        \draw[gray, ->] (0,-0.75) -- (0,1.25) node[anchor=south]{$\mathrm{Im}~z_1$};
+        \draw[gray, ->] (-1.5,0) -- (6,0) node[anchor=west]{$\mathrm{Re}~z_1$};
+
+        \draw[gray] ( 1,0) +(0,0.05) -- +(0, -0.05) node[inner sep=0, anchor=north] {\small $K_1$};
+        \draw[gray] ( 5,0) +(0,0.05) -- +(0, -0.05) node[inner sep=0, anchor=north] {\small $5K_1$};
+        \draw[gray]  (0, 0.5) +(0.1, 0) -- +(-0.1, 0) node[inner sep=0, anchor=east]{\small $jK^\prime_1$};
+
+        \begin{scope}
+
+            \clip(-1.5,-0.75) rectangle (6.8,1.25);
+
+            % \draw[>->, line width=0.05, thick, blue]   (1, 0.45) -- (2, 0.45) -- (2, 0.05) -- ( 0.1, 0.05) -- ( 0.1,0.45) -- (1, 0.45);
+            % \draw[>->, line width=0.05, thick, orange] (2, 0.5 ) -- (4, 0.5 ) -- (4, 0   ) -- ( 0  , 0   ) -- ( 0  ,0.5 ) -- (2, 0.5 );
+            % \draw[>->, line width=0.05, thick, red]    (3, 0.55) -- (6, 0.55) -- (6,-0.05) -- (-0.1,-0.05) -- (-0.1,0.55) -- (3, 0.55);
+            % \node[blue] at (1, 0.25) {$N=1$};
+            % \node[orange] at (3, 0.25) {$N=2$};
+            % \node[red] at (5, 0.25) {$N=3$};
+
+
+
+            % \draw[line width=0.1cm, fill, red!50] (0,0) rectangle (3, 0.5);
+            % \draw[line width=0.05cm, fill, orange!50] (0,0) rectangle (2, 0.5);
+            % \fill[yellow!50] (0,0) rectangle (1, 0.5);
+            % \node[] at (0.5, 0.25) {\small $N=1$};
+            % \node[] at (1.5, 0.25) {\small $N=2$};
+            % \node[] at (2.5, 0.25) {\small $N=3$};
+
+            \fill[orange!30] (0,0) rectangle (5, 0.5);
+            \fill[yellow!30] (0,0) rectangle (1, 0.5);
+            \node[] at (2.5, 0.25) {\small $N=5$};
+
+
+            \draw[decorate,decoration={brace,amplitude=3pt,mirror}, yshift=0.05cm]
+                (5,0.5) node(t_k_unten){} -- node[above, yshift=0.1cm]{$NK$}
+                (0,0.5) node(t_k_opt_unten){};
+
+            \draw[decorate,decoration={brace,amplitude=3pt,mirror}, xshift=0.1cm]
+                (5,0) node(t_k_unten){} -- node[right, xshift=0.1cm]{$K^\prime \frac{K_1N}{K} = K^\prime_1$}
+                (5,0.5) node(t_k_opt_unten){};
+
+            \foreach \i in {-2,...,1} {
+                \foreach \j in {-2,...,1} {
+                    \begin{scope}[xshift=\i*4cm, yshift=\j*1cm]
+
+                        \node[zero] at ( 1, 0) {};
+                        \node[zero] at ( 3, 0) {};
+                        \node[pole] at ( 1,0.5) {};
+                        \node[pole] at ( 3,0.5) {};
+
+                    \end{scope}
+                }
+            }
+
+
+
+
+            \draw[thick, ->, darkgreen] (5, 0) -- node[yshift=-0.5cm]{Durchlassbereich} (0,0);
+            \draw[thick, ->, orange] (-0, 0) --  node[align=center]{Übergangs-\\berech} (0,0.5);
+            \draw[thick, ->, red] (0,0.5) -- node[align=center, yshift=0.5cm]{Sperrbereich} (5, 0.5);
+
+            \draw (4,0  )  node[dot]{} node[anchor=south]      {\small $1$};
+            \draw (2,0  )  node[dot]{} node[anchor=south]      {\small $-1$};
+            \draw (0,0  )  node[dot]{} node[anchor=south west] {\small $1$};
+            \draw (0,0.5)  node[dot]{} node[anchor=north west] {\small $1/k$};
+            \draw (2,0.5)  node[dot]{} node[anchor=north]      {\small $-1/k$};
+            \draw (4,0.5)  node[dot]{} node[anchor=north]      {\small $1/k$};
+
+
+
+        \end{scope}
+
+
+    \end{scope}
+
+\end{tikzpicture}
\ No newline at end of file
diff --git a/buch/papers/ellfilter/tikz/fundamental_rectangle.tikz.tex b/buch/papers/ellfilter/tikz/fundamental_rectangle.tikz.tex
new file mode 100644
index 0000000..921dbfa
--- /dev/null
+++ b/buch/papers/ellfilter/tikz/fundamental_rectangle.tikz.tex
@@ -0,0 +1,26 @@
+\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2]
+
+    \tikzstyle{zero} = [draw, circle, inner sep =0, minimum height=0.15cm]
+
+    \tikzset{pole/.style={cross out, draw=black, minimum size=(0.15cm-\pgflinewidth), inner sep=0pt, outer sep=0pt}}
+
+    \begin{scope}[xscale=2, yscale=2]
+
+        \draw[gray, ->] (0,-0.25) -- (0,1.25) node[anchor=south]{$\mathrm{Im}~z$};
+        \draw[gray, ->] (-0.25,0) -- (1.5,0) node[anchor=west]{$\mathrm{Re}~z$};
+
+        \draw[gray] ( 1,0) +(0,0.05) -- +(0, -0.05) node[inner sep=0, anchor=north] {\small $K$};
+
+        \draw[gray]  (0, 1) +(0.05, 0) -- +(-0.05, 0) node[inner sep=0, anchor=east]{\small $jK^\prime$};
+
+        \fill[yellow!50] (0,0) rectangle (1, 1);
+
+        \node[anchor=south east] at ( 1,0) {$c$};
+        \node[anchor=north east] at ( 1,1) {$d$};
+        \node[anchor=north west] at ( 0,1) {$n$};
+        \node[anchor=south west] at ( 0,0) {$s$};
+
+    \end{scope}
+
+
+\end{tikzpicture}
\ No newline at end of file
diff --git a/buch/papers/ellfilter/tikz/sn.tikz.tex b/buch/papers/ellfilter/tikz/sn.tikz.tex
new file mode 100644
index 0000000..87c63c0
--- /dev/null
+++ b/buch/papers/ellfilter/tikz/sn.tikz.tex
@@ -0,0 +1,86 @@
+\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2]
+
+    \tikzstyle{zero} = [draw, circle, inner sep =0, minimum height=0.15cm]
+
+    \tikzset{pole/.style={cross out, draw=black, minimum size=(0.15cm-\pgflinewidth), inner sep=0pt, outer sep=0pt}}
+
+    \begin{scope}[xscale=1, yscale=2]
+
+        \draw[gray, ->] (0,-1.5) -- (0,1.5) node[anchor=south]{$\mathrm{Im}~z$};
+        \draw[gray, ->] (-5,0) -- (5,0) node[anchor=west]{$\mathrm{Re}~z$};
+
+        \begin{scope}
+
+            \clip(-4.5,-1.25) rectangle (4.5,1.25);
+
+            \fill[yellow!30] (0,0) rectangle (1, 0.5);
+
+            \begin{scope}[xshift=-1cm]
+
+                \draw[thick, ->, darkgreen] (0, 0) -- (0,0.5);
+                \draw[thick, ->, orange] (1, 0) -- (0,0);
+                \draw[thick, ->, red] (2, 0) -- (1,0);
+                \draw[thick, ->, blue] (2,0.5) -- (2, 0);
+                \draw[thick, ->, purple] (1, 0.5) -- (2,0.5);
+                \draw[thick, ->, cyan] (0, 0.5) -- (1,0.5);
+
+
+                \foreach \i in {-2,...,2} {
+                    \foreach \j in {-2,...,1} {
+                        \begin{scope}[xshift=\i*4cm, yshift=\j*1cm]
+                            \draw[opacity=0.5, ->, darkgreen] (0, 0) -- (0,0.5);
+                            \draw[opacity=0.5, ->, orange] (1, 0) -- (0,0);
+                            \draw[opacity=0.5, ->, red] (2, 0) -- (1,0);
+                            \draw[opacity=0.5, ->, blue] (2,0.5) -- (2, 0);
+                            \draw[opacity=0.5, ->, purple] (1, 0.5) -- (2,0.5);
+                            \draw[opacity=0.5, ->, cyan] (0, 0.5) -- (1,0.5);
+                            \draw[opacity=0.5, ->, darkgreen] (0,1) -- (0,0.5);
+                            \draw[opacity=0.5, ->, blue] (2,0.5) -- (2, 1);
+                            \draw[opacity=0.5, ->, purple] (3, 0.5) -- (2,0.5);
+                            \draw[opacity=0.5, ->, cyan] (4, 0.5) -- (3,0.5);
+                            \draw[opacity=0.5, ->, red] (2, 0) -- (3,0);
+                            \draw[opacity=0.5, ->, orange] (3, 0) -- (4,0);
+
+                            \node[zero] at ( 1, 0) {};
+                            \node[zero] at ( 3, 0) {};
+                            \node[pole] at ( 1,0.5) {};
+                            \node[pole] at ( 3,0.5) {};
+
+                        \end{scope}
+                    }
+                }
+
+            \end{scope}
+
+        \end{scope}
+
+        \draw[gray] ( 1,0) +(0,0.1) -- +(0, -0.1) node[inner sep=0, anchor=north] {\small $K$};
+        \draw[gray]  (0, 0.5) +(0.1, 0) -- +(-0.1, 0) node[inner sep=0, anchor=east]{\small $jK^\prime$};
+
+
+
+    \end{scope}
+
+    \begin{scope}[yshift=-3.5cm, xscale=0.75]
+
+        \draw[gray, ->] (-6,0) -- (6,0) node[anchor=west]{$w$};
+
+        \draw[thick, ->, purple] (-5, 0) -- (-3, 0);
+        \draw[thick, ->, blue]      (-3, 0) -- (-2, 0);
+        \draw[thick, ->, red]       (-2, 0) -- (0, 0);
+        \draw[thick, ->, orange]    (0, 0) -- (2, 0);
+        \draw[thick, ->, darkgreen] (2, 0) -- (3, 0);
+        \draw[thick, ->, cyan] (3, 0) -- (5, 0);
+
+        \node[anchor=south] at (-5,0) {$-\infty$};
+        \node[anchor=south] at (-3,0) {$-1/k$};
+        \node[anchor=south] at (-2,0) {$-1$};
+        \node[anchor=south] at (0,0) {$0$};
+        \node[anchor=south] at (2,0) {$1$};
+        \node[anchor=south] at (3,0) {$1/k$};
+        \node[anchor=south] at (5,0) {$\infty$};
+
+    \end{scope}
+
+
+\end{tikzpicture}
\ No newline at end of file
diff --git a/buch/papers/ellfilter/tschebyscheff.tex b/buch/papers/ellfilter/tschebyscheff.tex
new file mode 100644
index 0000000..7d426b6
--- /dev/null
+++ b/buch/papers/ellfilter/tschebyscheff.tex
@@ -0,0 +1,133 @@
+\section{Tschebyscheff-Filter}
+
+Als Einstieg betrachent Wir das Tschebyscheff-Filter, welches sehr verwand ist mit dem elliptischen Filter.
+Genauer ausgedrückt sind die Tschebyscheff-1 und -2 Filter Spezialfälle davon.
+
+Der Name des Filters deutet schon an, dass die Tschebyscheff-Polynome $T_N$ für das Filter relevant sind:
+\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}
+Bemerkenswert ist, dass die Polynome im Intervall $[-1, 1]$ mit der trigonometrischen Funktion
+\begin{align} \label{ellfilter:eq:chebychef_polynomials}
+    T_N(w) &= \cos \left( N \cos^{-1}(w) \right) \\
+           &= \cos \left(N~z \right), \quad w= \cos(z)
+\end{align}
+übereinstimmt.
+Der Zusammenhang lässt sich mit den Doppel- und Mehrfachwinkelfunktionen der trigonometrischen Funktionen erklären.
+Abbildung \ref{ellfilter:fig:chebychef_polynomials} zeigt einige Tschebyscheff-Polynome.
+\begin{figure}
+    \centering
+    \input{papers/ellfilter/python/F_N_chebychev2.pgf}
+    \caption{Die Tschebyscheff-Polynome $C_N$.}
+    \label{ellfilter:fig:chebychef_polynomials}
+\end{figure}
+Da der Kosinus begrenzt zwischen $-1$ und $1$ ist, sind auch die Tschebyscheff-Polynome begrenzt.
+Geht man aber über das Intervall $[-1, 1]$ hinaus, divergieren die Funktionen mit zunehmender Ordnung immer steiler gegen $\pm \infty$.
+Diese Eigenschaft ist sehr nützlich für ein Filter.
+Wenn wir die Tschebyscheff-Polynome quadrieren, passen sie perfekt in die Voraussetzungen für Filterfunktionen, wie es Abbildung \ref{ellfiter:fig:chebychef} demonstriert.
+\begin{figure}
+    \centering
+    \input{papers/ellfilter/python/F_N_chebychev.pgf}
+    \caption{Die Tschebyscheff-Polynome füllen den erlaubten Bereich besser, und erhalten dadurch eine steilere Flanke im Sperrbereich.}
+    \label{ellfiter:fig:chebychef}
+\end{figure}
+
+
+Die analytische Fortsetzung von \eqref{ellfilter:eq:chebychef_polynomials} über das Intervall $[-1,1]$ hinaus stimmt mit den Polynomen überein, wie es zu erwarten ist.
+Die genauere Betrachtung wird uns dann helfen die elliptischen Filter besser zu verstehen.
+
+Starten wir mit der Funktion, die als erstes auf $w$ angewendet wird, dem Arcuscosinus.
+Die invertierte Funktion des Kosinus kann als definites Integral dargestellt werden:
+\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}
+Der Integrand oder auch die Ableitung
+\begin{equation}
+    \frac{
+        -1
+    }{
+        \sqrt{
+            1-z^2
+        }
+    }
+\end{equation}
+bestimmt dabei die Richtung, in der die Funktion verläuft.
+Der reelle Arcuscosinus is bekanntlich nur für $|z| \leq 1$ definiert.
+Hier bleibt der Wert unter der Wurzel positiv und das Integral liefert reelle Werte.
+Doch wenn $|z|$ über 1 hinausgeht, wird der Term unter der Wurzel negativ.
+Durch die Quadratwurzel entstehen für den Integranden zwei rein komplexe Lösungen.
+Der Wert des Arcuscosinus verlässt also bei $z= \pm 1$ den reellen Zahlenstrahl und knickt in die komplexe Ebene ab.
+Abbildung \ref{ellfilter:fig:arccos} zeigt den $\arccos$ in der komplexen Ebene.
+\begin{figure}
+    \centering
+    \input{papers/ellfilter/tikz/arccos.tikz.tex}
+    \caption{Die Funktion $z = \cos^{-1}(w)$ dargestellt in der komplexen ebene.}
+    \label{ellfilter:fig:arccos}
+\end{figure}
+Wegen der Periodizität des Kosinus ist auch der Arcuscosinus $2\pi$-periodisch und es entstehen periodische Nullstellen.
+% \begin{equation}
+%     \frac{
+%         1
+%     }{
+%         \sqrt{
+%             1-z^2
+%         }
+%     }
+%     \in \mathbb{R}
+%     \quad
+%     \forall
+%     \quad
+%     -1  \leq z \leq 1
+% \end{equation}
+% \begin{equation}
+%     \frac{
+%         1
+%     }{
+%         \sqrt{
+%             1-z^2
+%         }
+%     }
+%     = i \xi \quad | \quad \xi \in \mathbb{R}
+%     \quad
+%     \forall
+%     \quad
+%     z \leq -1 \cup z \geq 1
+% \end{equation}
+
+Die Tschebyscheff-Polynome skalieren diese Nullstellen mit dem Ordnungsfaktor $N$, wie dargestellt in Abbildung \ref{ellfilter:fig:arccos2}.
+\begin{figure}
+    \centering
+    \input{papers/ellfilter/tikz/arccos2.tikz.tex}
+    \caption{
+        $z_1=N \cos^{-1}(w)$-Ebene der Tschebyscheff-Funktion.
+        Die eingefärbten Pfade sind Verläufe von $w~\forall~[-\infty, \infty]$ für verschiedene Ordnungen $N$.
+        Je grösser die Ordnung $N$ gewählt wird, desto mehr Nullstellen werden passiert.
+    }
+    \label{ellfilter:fig:arccos2}
+\end{figure}
+Somit passert $\cos( N~\cos^{-1}(w))$ im Intervall $[-1, 1]$ $N$ Nullstellen.
+Durch die spezielle Anordnung der Nullstellen hat die Funktion Equirippel-Verhalten und ist dennoch ein Polynom, was sich perfekt für linear Filter eignet.