Added content, presentation

20 files changed, 7056 insertions, 239 deletions

diff --git a/buch/papers/ellfilter/main.tex b/buch/papers/ellfilter/main.tex
index 29ebf7a..e9d6aba 100644
--- a/buch/papers/ellfilter/main.tex
+++ b/buch/papers/ellfilter/main.tex
@@ -11,16 +11,15 @@
 
 \section{Einleitung}
 
-Lineare filter
+% Lineare filter
 
-Filter, Signalverarbeitung
+% 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
-
+% 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.
@@ -39,15 +38,12 @@ Damit das Filter implementierbar und stabil ist, muss $H(\Omega)^2$ eine rationa
 
 $N \in \mathbb{N} $ gibt dabei die Ordnung des Filters vor, also die maximale Anzahl Pole oder Nullstellen.
 
-% In \eqref{ellfilter:eq:h_omega} wird $F_N(w)$ so verzogen, dass $F_N(w) \forall |w| < 1$
-
-
 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
-    \includegraphics[scale=1]{papers/ellfilter/python/F_N_butterworth.pdf}
+    \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}
@@ -60,7 +56,7 @@ wenn $F_N(w)$ eine rationale Funktion ist, ist auch $H(\Omega)$ eine rationale F
         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)                         & \text{Elliptisch (Cauer)}    \\
+        R_N(w, \xi)                    & \text{Elliptisch (Cauer)}    \\
     \end{cases}
 \end{align}
 
@@ -73,9 +69,9 @@ Es scheint so als sind gewisse Eigenschaften dieser speziellen Funktionen verant
 \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 Fitler ein Spezialfall davon.
+Genauer ausgedrückt sind die Tschebyscheff-1 und -2 Filter Spezialfälle davon.
 
-Der Name des Filters deutet schon an, dass die Tschebyschff-Polynome $T_N$ relevant sind für das Filter:
+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\\
@@ -83,15 +79,17 @@ Der Name des Filters deutet schon an, dass die Tschebyschff-Polynome $T_N$ relev
     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{equation} \label{ellfilter:eq:chebychef_polynomials}
-    T_N(w) = \cos \left( N \cos^{-1}(w) \right)
-\end{equation}
+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
-    \includegraphics[scale=1]{papers/ellfilter/python/F_N_chebychev2.pdf}
+    \input{papers/ellfilter/python/F_N_chebychev2.pgf}
     \caption{Die Tschebyscheff-Polynome $C_N$.}
     \label{ellfilter:fig:chebychef_polynomials}
 \end{figure}
@@ -101,103 +99,123 @@ 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
-    \includegraphics[scale=1]{papers/ellfilter/python/F_N_chebychev.pdf}
+    \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 zu verstehen.
+Die genauere Betrachtung wird uns dann helfen die elliptischen Filter besser zu verstehen.
 
-\begin{equation}
+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_{0}^{x}
+    &=
+    \int_{x}^{1}
     \frac{
         dz
     }{
         \sqrt{
             1-z^2
         }
-    }
-\end{equation} %TOdO is it minus dz?
-
-\begin{equation}
+    }\\
+    &=
+    \int_{0}^{x}
     \frac{
-        1
+        -1
     }{
         \sqrt{
             1-z^2
         }
     }
-    \in \mathbb{R}
-    \quad
-    \forall
-    \quad
-    -1  \leq z \leq 1
-\end{equation}
-Wenn $|z|$ über 1 hinausgeht, wird der Term unter der Wurzel negativ.
-Durch die Quadratwurzel entstehen zwei Reinkomplexe Lösungen
+    ~dz
+    + \frac{\pi}{2}
+\end{align}
+Der Integrand oder auch die Ableitung
 \begin{equation}
     \frac{
-        1
+        -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}
-
+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{
-        $z$-Ebene der Tschebyscheff-Funktion.
-        Je grösser die Ordnung $N$ gewählt wird, desto mehr Nullstellen werden hat das Tschebyscheff-Polynom.
+        $z_1=N \cos^{-1}(w)$-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:arccos}
+    \label{ellfilter:fig:arccos2}
 \end{figure}
-
-
-
-
-
-% Analytische Fortsetzung
-
-
+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.
 
 \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.
+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.
-
-%TODO $z$ or $u$ for parameter?
-
-neu zwei parameter
-$sn(z, k)$
-$z$ ist das winkelargument
+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.
-An deren stelle kommt der parameter $k$ zum Einsatz, welcher durch das elliptische Integral erster Art
+Das Winkelargument $z$ kann durch das elliptische Integral erster Art
 \begin{equation}
     z
     =
@@ -211,12 +229,18 @@ An deren stelle kommt der parameter $k$ zum Einsatz, welcher durch das elliptisc
             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}
@@ -231,25 +255,75 @@ Beim vollständigen Integral
         }
     }
 \end{equation}
-wird über ein viertel Ellipsenbogen integriert also bis $\phi=\pi/2$.
-
-Die Jacobischen elliptischen Funktionen können mit der inversen Funktion
+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
+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 $sn^{-1}$ mit dem Elliptischen integral dargestellt werden:
+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(u, k)
+    \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}
@@ -306,28 +380,90 @@ In der reellen Richtung ist sie $4K(k)$-periodisch und in der imaginären Richtu
 
 %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}
 
 \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}
+
 
-\begin{equation}
-    R_N(\xi, w) = \cd \left(N~f_1(\xi)~\cd^{-1}(w, 1/\xi), f_2(\xi)\right)
-\end{equation}
-\begin{equation}
-    R_N(\xi, w) = \cd (N~u K_1, k_1), \quad w= \cd(uK, k)
-\end{equation}
+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}
 
-sieht ähnlich aus wie die trigonometrische darstellung der Tschebyschef-Polynome
+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.
 
-der Ordnungszahl $N$ kommt auch als Faktor for 
 
-%TODO cd^{-1} grafik mit 
 
+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.
+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}
@@ -345,19 +481,7 @@ Bei den Tschebyscheff-Polynomen haben wir gesehen, dass die Trigonometrische For
 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.
 
-
-
-
-\begin{figure}
-    \centering
-    \includegraphics[scale=1]{papers/ellfilter/python/F_N_elliptic.pdf}
-    \caption{$F_N$ für ein elliptischs filter.}
-    \label{ellfilter:fig:elliptic}
-\end{figure}
-
-
-
-
+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.
 
 \input{papers/ellfilter/teil0.tex}
 \input{papers/ellfilter/teil1.tex}
diff --git a/buch/papers/ellfilter/packages.tex b/buch/papers/ellfilter/packages.tex
index 8045a1a..9a550e2 100644
--- a/buch/papers/ellfilter/packages.tex
+++ b/buch/papers/ellfilter/packages.tex
@@ -8,6 +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}}
 
-\DeclareMathOperator{\sn}{\text{sn}}
-\DeclareMathOperator{\cd}{\text{cd}}
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
--- /dev/null
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@@ -0,0 +1,1023 @@
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new file mode 100644
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diff --git a/buch/papers/ellfilter/python/chebychef.py b/buch/papers/ellfilter/python/chebychef.py
index a278989..254ad4b 100644
--- a/buch/papers/ellfilter/python/chebychef.py
+++ b/buch/papers/ellfilter/python/chebychef.py
@@ -35,7 +35,7 @@ plt.show()
 # %% Cheychev filter F_N plot
 
 w = np.linspace(-1.1,1.1, 1000)
-plt.figure(figsize=(5.5,2))
+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)
@@ -44,9 +44,10 @@ plt.xlim([-1.2,1.2])
 plt.ylim([-2,2])
 plt.grid()
 plt.xlabel("$w$")
-plt.ylabel("$C_N(w)$")
+plt.ylabel("$T_N(w)$")
 plt.legend()
-plt.savefig("F_N_chebychev2.pdf")
+plt.tight_layout()
+plt.savefig("F_N_chebychev2.pgf")
 plt.show()
 
 # %% Build Chebychev polynomials
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%
+\makeatletter%
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diff --git a/buch/papers/ellfilter/python/elliptic.py b/buch/papers/ellfilter/python/elliptic.py
index 9f209e9..b3336a1 100644
--- a/buch/papers/ellfilter/python/elliptic.py
+++ b/buch/papers/ellfilter/python/elliptic.py
@@ -5,19 +5,62 @@ import scipy.special
 import scipyx as spx
 import numpy as np
 import matplotlib.pyplot as plt
-import matplotlib
 from matplotlib.patches import Rectangle
 
-matplotlib.rcParams.update({
-    "pgf.texsystem": "pdflatex",
-    'font.family': 'serif',
-    'font.size': 9,
-    'text.usetex': True,
-    'pgf.rcfonts': False,
-})
+import plot_params
 
 def last_color():
-    plt.gca().lines[-1].get_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
 
@@ -37,7 +80,7 @@ plt.gca().add_patch(Rectangle(
 plt.gca().add_patch(Rectangle(
     (1, 1),
     0.5, 1,
-    fc ='green',
+    fc ='orange',
     alpha=0.2,
     lw = 10,
 ))
@@ -47,7 +90,8 @@ plt.grid()
 plt.xlabel("$w$")
 plt.ylabel("$F^2_N(w)$")
 plt.legend()
-plt.savefig("F_N_butterworth.pdf")
+plt.tight_layout()
+plt.savefig("F_N_butterworth.pgf")
 plt.show()
 
 # %% Cheychev filter F_N plot
@@ -69,7 +113,7 @@ plt.gca().add_patch(Rectangle(
 plt.gca().add_patch(Rectangle(
     (1, 1),
     0.5, 1,
-    fc ='green',
+    fc ='orange',
     alpha=0.2,
     lw = 10,
 ))
@@ -79,57 +123,10 @@ plt.grid()
 plt.xlabel("$w$")
 plt.ylabel("$F^2_N(w)$")
 plt.legend()
-plt.savefig("F_N_chebychev.pdf")
+plt.tight_layout()
+plt.savefig("F_N_chebychev.pgf")
 plt.show()
 
-# %% 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)
 
 # %% plot arcsin
 
@@ -314,3 +311,46 @@ for n in (1,2,3,4):
     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
index 92fefd9..29c6f47 100644
--- a/buch/papers/ellfilter/python/elliptic2.py
+++ b/buch/papers/ellfilter/python/elliptic2.py
@@ -6,13 +6,14 @@ 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 = 1000
+    omega_c = 1
 
     a, b = scipy.signal.ellip(
         order,
@@ -34,14 +35,14 @@ def ellip_filter(N):
 
     FN2 = ((1/mag**2) - 1)
 
-    return w/omega_c, FN2 / epsilon2
+    return w/omega_c, FN2 / epsilon2, mag, a, b
 
 
 plt.figure(figsize=(4,2.5))
 
 for N in [5]:
-    w, FN2 = ellip_filter(N)
-    plt.semilogy(w, FN2, label=f"$N={N}$")
+    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),
@@ -53,7 +54,7 @@ plt.gca().add_patch(Rectangle(
 plt.gca().add_patch(Rectangle(
     (1, 1),
     0.01, 1e2-1,
-    fc ='green',
+    fc ='orange',
     alpha=0.2,
     lw = 10,
 ))
@@ -61,18 +62,88 @@ plt.gca().add_patch(Rectangle(
 plt.gca().add_patch(Rectangle(
     (1.01, 100),
     1, 1e6,
-    fc ='green',
+    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.savefig("F_N_elliptic.pdf")
+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}%
+\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}%
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+\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}%
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+\pgfusepath{fill}%
+\end{pgfscope}%
+\begin{pgfscope}%
+\pgfsetbuttcap%
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+}%
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+}%
+\begin{pgfscope}%
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+\end{pgfscope}%
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+}%
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+\pgfpathrectangle{\pgfqpoint{2.874885in}{0.548769in}}{\pgfqpoint{1.940523in}{1.753186in}}%
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+\pgfusepath{stroke,fill}%
+}%
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+\end{pgfscope}%
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+\end{pgfscope}%
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+\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
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+\end{pgfscope}%
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+\end{pgfscope}%
+\begin{pgfscope}%
+\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
+\pgfsetstrokecolor{textcolor}%
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+\end{pgfscope}%
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+\end{pgfscope}%
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+\begin{pgfscope}%
+\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
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+\pgfsetfillcolor{textcolor}%
+\pgftext[x=3.992820in,y=1.137383in,left,base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle k=0.99\)}%
+\end{pgfscope}%
+\end{pgfpicture}%
+\makeatother%
+\endgroup%
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 2b873af..8f21971 100644
--- a/buch/papers/ellfilter/references.bib
+++ b/buch/papers/ellfilter/references.bib
@@ -11,3 +11,12 @@
         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/tikz/arccos.tikz.tex b/buch/papers/ellfilter/tikz/arccos.tikz.tex
index 2bdcc2d..2772620 100644
--- a/buch/papers/ellfilter/tikz/arccos.tikz.tex
+++ b/buch/papers/ellfilter/tikz/arccos.tikz.tex
@@ -1,81 +1,49 @@
 \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]{Im $z$};
-    \draw[gray, ->] (-5,0) -- (5,0) node[anchor=west]{Re $z$};
+    \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}
-        \draw[thick, ->, orange] (-1, 0) -- (0,0);
-        \draw[thick, ->, darkgreen] (0, 0) -- (0,1.5);
-        \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);
-        \draw[thick, ->, blue] (2,-1.5) -- (2, 0);
-        \draw[thick, ->, red] (2, 0) -- (3,0);
-
-        \node[anchor=south west] at (0,1.5) {$\infty$};
-        \node[anchor=south west] at (0,-1.5) {$\infty$};
-        \node[anchor=south west] at (0,0) {$1$};
-        \node[anchor=south] at (1,0) {$0$};
-        \node[anchor=south west] at (2,0) {$-1$};
-        \node[anchor=south west] at (2,1.5) {$-\infty$};
-        \node[anchor=south west] at (2,-1.5) {$-\infty$};
-        \node[anchor=south west] at (3,0) {$0$};
-    \end{scope}
+    \begin{scope}[xscale=0.6]
 
-    \begin{scope}[xshift=4cm]
-        \draw[thick, ->, orange] (-1, 0) -- (0,0);
-        \draw[thick, ->, darkgreen] (0, 0) -- (0,1.5);
-        \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);
-        % \draw[thick, ->, blue] (2,-1.5) -- (2, 0);
-        % \draw[thick, ->, red] (2, 0) -- (3,0);
-
-        \node[anchor=south west] at (0,1.5) {$\infty$};
-        \node[anchor=south west] at (0,-1.5) {$\infty$};
-        \node[anchor=south west] at (0,0) {$1$};
-        % \node[anchor=south] at (1,0) {$0$};
-        % \node[anchor=south west] at (2,0) {$-1$};
-        % \node[anchor=south west] at (2,1.5) {$-\infty$};
-        % \node[anchor=south west] at (2,-1.5) {$-\infty$};
-        % \node[anchor=south west] at (3,0) {$0$};
-    \end{scope}
+        \clip(-7.5,-2) rectangle (7.5,2);
 
-    \begin{scope}[xshift=-4cm]
-        % \draw[thick, ->, orange] (-1, 0) -- (0,0);
         \draw[thick, ->, darkgreen] (0, 0) -- (0,1.5);
-        \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);
-        \draw[thick, ->, blue] (2,-1.5) -- (2, 0);
-        \draw[thick, ->, red] (2, 0) -- (3,0);
-
-        \node[anchor=south west] at (0,1.5) {$\infty$};
-        \node[anchor=south west] at (0,-1.5) {$\infty$};
-        \node[anchor=south west] at (0,0) {$1$};
-        \node[anchor=south] at (1,0) {$0$};
-        \node[anchor=south west] at (2,0) {$-1$};
-        \node[anchor=south west] at (2,1.5) {$-\infty$};
-        \node[anchor=south west] at (2,-1.5) {$-\infty$};
-        \node[anchor=south west] at (3,0) {$0$};
-    \end{scope}
-
-    \node[gray, anchor=north west] at (-4,0) {$-2\pi$};
-    \node[gray, anchor=north west] at (-2,0) {$-\pi$};
-    \node[gray, anchor=north west] at (0,0) {$0$};
-    \node[gray, anchor=north west] at (2,0) {$\pi$};
-    \node[gray, anchor=north west] at (4,0) {$2\pi$};
-
-
-    \node[gray, anchor=south east] at (0,-1.5) {$-\infty$};
-    \node[gray, anchor=south east] at (0, 0) {$0$};
-    \node[gray, anchor=south east] at (0, 1.5) {$\infty$};
 
+        \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]
 
@@ -94,4 +62,5 @@
 
     \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
index dcf02fd..3fc3cc6 100644
--- a/buch/papers/ellfilter/tikz/arccos2.tikz.tex
+++ b/buch/papers/ellfilter/tikz/arccos2.tikz.tex
@@ -1,19 +1,18 @@
 \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]{Im $z$};
-        \draw[gray, ->] (-10,0) -- (10,0) node[anchor=west]{Re $z$};
+        \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) -- (6,-0.05) -- node[anchor=south, pos=0.1666]{$N=3$} (-0.1,-0.05) -- (-0.1,1.5);
+            \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) {};
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