Merge pull request #50 from ntobler/master

Ellfilter

19 files changed, 1953 insertions, 1567 deletions

diff --git a/buch/papers/ellfilter/einleitung.tex b/buch/papers/ellfilter/einleitung.tex
index 37fd89f..ae7127f 100644
--- a/buch/papers/ellfilter/einleitung.tex
+++ b/buch/papers/ellfilter/einleitung.tex
@@ -1,56 +1,73 @@
 \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}
+Filter sind womöglich eines der wichtigsten Elementen in der Signalverarbeitung und finden Anwendungen in der digitalen und analogen Elektrotechnik.
+Besonders hilfreich ist die Untergruppe der linearen Filter.
+Elektronische Schaltungen mit linearen Bauelementen wie Kondensatoren, Spulen und Widerständen führen immer zu linearen zeitinvarianten Systemen (LTI-System von englich \textit{time-invariant system}).
+Durch die Linearität werden beim das Filtern keine neuen Frequenzanteile erzeugt, was es erlaubt, einen Frequenzanteil eines Signals verzerrungsfrei herauszufiltern. %TODO review sentence
+Diese Eigenschaft macht es Sinnvoll, lineare Filter im Frequenzbereich zu beschreiben.
+Die Übertragungsfunktion eines linearen Filters im Frequenzbereich $H(\Omega)$ ist dabei immer eine rationale Funktion, also ein Quotient von zwei Polynomen.
+Dabei ist $\Omega = 2 \pi f$ die Frequenzeinheit.
+Die Polynome haben dabei immer reelle oder komplex-konjugierte Nullstellen.
+
+Ein breit angewendeter Filtertyp ist das Tiefpassfilter, welches beabsichtigt alle Frequenzen eines Signals oberhalb der Grenzfrequenz $\Omega_p$ auszulöschen.
+Der Rest soll dabei unverändert passieren.
+Aus dem Tiefpassifilter können dann durch Transformationen auch Hochpassfilter, Bandpassfilter und Bandsperren realisiert werden.
+Ein solches Filter hat idealerweise die Frequenzantwort
+\begin{equation}
+    H(\Omega) =
+    \begin{cases}
+        1  & \Omega < \Omega_p \\
+        0  & \Omega < \Omega_p
+    \end{cases},
 \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 $F_N(w) = w^N$.
+wie dargestellt in Abbildung \ref{ellfilter:fig:lp}
+\begin{figure}
+    \centering
+    \input{papers/ellfilter/tikz/filter.tikz.tex}
+    \caption{Frequenzantwort eines Tiefpassfilters.}
+    \label{ellfilter:fig:lp}
+\end{figure}
+Leider ist eine solche Funktion nicht als rationale Funktion darstellbar.
+Aus diesem Grund sind realisierbare Approximationen gesucht.
+Jede Approximation wird einen kontinuierlichen Übergang zwischen Durchlassbereich und Sperrbereich aufweisen.
+Oft wird dabei der Faktor $1/\sqrt{2}$ als Schwelle zwischen den beiden Bereichen gewählt.
+Somit lassen sich lineare Tiefpassfilter mit folgender Funktion zusammenfassen:
+\begin{equation}
+    | H(\Omega)|^2 = \frac{1}{1 + \varepsilon_p^2 F_N^2(w)}, \quad w=\frac{\Omega}{\Omega_p},
+\end{equation}
+wobei $F_N(w)$ eine rationale Funktion ist, $|F_N(w)| \leq 1 ~\forall~ |w| \leq 1$ erfüllt und für $|w| \geq 1$ möglichst schnell divergiert.
+Des weiteren müssen alle Nullstellen und Pole von $F_N$ auf der linken Halbebene liegen, damit das Filter implementierbar und stabil ist.
+$w$ ist die normalisierte Frequenz, die es erlaubt ein Filter unabhängig von der Grenzfrequenz zu beschrieben.
+Bei $w=1$ hat das Filter eine Dämpfung von $1/(1+\varepsilon^2)$.
+$N \in \mathbb{N} $ gibt die Ordnung des Filters vor, also die maximale Anzahl Pole oder Nullstellen.
+Je hoher $N$ gewählt wird, desto steiler ist der Übergang in denn Sperrbereich.
+Grössere $N$ sind erfordern jedoch aufwendigere Implementierungen und haben mehr Phasenverschiebung.
+Eine einfache Funktion, die für $F_N$ eingesetzt werden kann, ist das Polynom $w^N$.
 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.}
+    \caption{$F_N$ für Butterworth filter. Der grüne und gelbe 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?
-
+Eine Reihe von rationalen Funktionen können für $F_N$ eingesetzt werden, um Tiefpassfilter\-approximationen mit unterschiedlichen Eigenschaften zu erhalten:
 \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)}    \\
+        R_N(w, \xi)                    & \text{Elliptisch}    \\
     \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.
+Mit der Ausnahme vom Butterworth-Filter sind alle Filter nach speziellen Funktionen benannt.
+Alle diese Filter sind optimal hinsichtlich einer Eigenschaft.
 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.
+Das Tschebyscheff-1 Filter ist 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.
+
+Dieses Paper betrachtet die Theorie hinter dem elliptischen Filter, dem wohl exotischsten dieser Auswahl.
+Es weist sich aus durch den steilsten Übergangsbereich für eine gegebene Filterdesignspezifikation.
+Des weiteren kann es als Verallgemeinerung des Tschebyscheff-Filters angesehen werden.
+
+% wenn $F_N(w)$ eine rationale Funktion ist, ist auch $H(\Omega)$ eine rationale Funktion und daher ein lineares Filter. %proof?
diff --git a/buch/papers/ellfilter/elliptic.tex b/buch/papers/ellfilter/elliptic.tex
index 88bfbfe..67bcca0 100644
--- a/buch/papers/ellfilter/elliptic.tex
+++ b/buch/papers/ellfilter/elliptic.tex
@@ -1,92 +1,100 @@
 \section{Elliptische rationale Funktionen}
 
-Kommen wir nun zum eigentlichen Teil dieses Papers, den elliptischen rationalen Funktionen
+Kommen wir nun zum eigentlichen Teil dieses Papers, den elliptischen rationalen Funktionen \cite{ellfilter:bib:orfanidis}
 \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) \\
+    R_N(\xi, w) &= \cd \left(N~f_1(\xi)~\cd^{-1}(w, 1/\xi), f_2(\xi)\right) \label{ellfilter:eq:elliptic}\\
+                &= \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}
+Beim Betrachten dieser Definition, fällt die Ähnlichkeit zur trigonometrische Darstellung der Tsche\-byschef-Polynome \eqref{ellfilter:eq:chebychef_polynomials} auf.
 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.
+Zusätzlich werden noch zwei verschiedene elliptische Moduli $k$ und $k_1$ gebraucht.
+Bei $k = k_1 = 0$ wird der $\cd$ zum Kosinus und wir erhalten in diesem Spezialfall die Tschebyschef-Polynome.
 
+Durch das Konzept vom fundamentalen Rechteck, siehe Abbildung \ref{buch:elliptisch:fig:ellall} können für alle inversen Jacobi elliptischen Funktionen die Positionen der Null- und Polstellen anhand eines Diagramms ermittelt 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 $z = \sn^{-1}(w, k)$.
+        $z$-Ebene der Funktion $z = \cd^{-1}(w, k)$.
         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.
+Auffallend an der $w = \cd(z, k)$-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 Idee des elliptischen Filter ist es, diese zwei Equirippel-Zonen abzufahren, wie ersichtlich in Abbildung \ref{ellfilter:fig:cd2}, welche Analog zu Abbildung \ref{ellfilter:fig:arccos2} gesehen werden kann.
 \begin{figure}
     \centering
-    \input{papers/ellfilter/tikz/fundamental_rectangle.tikz.tex}
+    \input{papers/ellfilter/tikz/cd2.tikz.tex}
     \caption{
-        Fundamentales Rechteck der inversen Jaccobi elliptischen Funktionen.
+        $z_1=N\frac{K_1}{K}\cd^{-1}(w, k)$-Ebene der elliptischen rationalen Funktionen.
+        Je grösser die Ordnung $N$ gewählt wird, desto mehr Nullstellen werden passiert.
+        Als Vereinfachung ist die Funktion nur für $w>0$ dargestellt.
     }
-    \label{ellfilter:fig:fundamental_rectangle}
+    \label{ellfilter:fig:cd2}
 \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.
+Das elliptische Filter hat im Gegensatz zum Tschebyscheff-Filter drei Zonen.
+Im Durchlassbereich werden wie beim Tschebyscheff-Filter die Nullstellen durchlaufen.
+Statt dass $z_1$ für alle $w>1$ in die imaginäre Richtung geht, bewegen wir uns im Sperrbereich wieder in reeller Richtung, wo Pole durchlaufen werden.
+Aus dieser Sicht kann der Sperrbereich vom Tschebyscheff-Filter als unendlich langer Übergangsbereich angesehen werden.
+% Falls es möglich ist diese Werte abzufahren im Stil der Tschebyscheff-Polynome, kann ein Filter gebaut werden, dass Equirippel-Verhalten im Durchlass- und Sperrbereich aufweist.
+Da sich die Funktion im Übergangsbereich nur zur nächsten Reihe bewegt, ist der Übergangsbereich monoton steigend.
+Theoretisch könnte eine gleiches Durchlass- und Sperrbereichverhalten erreicht werden, wenn die Funktion auf eine andere Reihe ansteigen würde.
+Dies würde jedoch zu Oszillationen zwischen $1$ und $1/k$ im Übergangsbereich führen.
+Abbildung \ref{ellfilter:fig:elliptic_freq} zeigt eine elliptisch rationale Funktion und die Frequenzantwort des daraus resultierenden Filters.
 \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}
+    \input{papers/ellfilter/python/elliptic.pgf}
+    \caption{$F_N$ und die resultierende Frequenzantwort eines elliptischen Filters.}
+    \label{ellfilter:fig:elliptic_freq}
 \end{figure}
-% Da die $\cd^{-1}$-Funktion 
-
 
+\subsection{Gradgleichung}
 
+Damit die Pol- und Nullstellen genau in dieser Konstellation durchfahren werden, müssen die elliptischen Moduli des inneren und äusseren $\cd$ aufeinander abgestimmt werden.
+In der reellen Richtung müssen sich die Periodizitäten $K$ und $K_1$ um den Faktor $N$ unterscheiden, während die imagiäre Periodizitäten $K^\prime$ und $K^\prime_1$ gleich bleiben müssen.
+Zur Erinnerung, $K$ und $K^\prime$ sind durch elliptische Integrale definiert und vom Modul $k$ abhängig wie ersichtlich in Abbildung \ref{ellfilter:fig:kprime}.
 \begin{figure}
     \centering
-    \input{papers/ellfilter/python/F_N_elliptic.pgf}
-    \caption{$F_N$ für ein elliptischs filter.}
-    \label{ellfilter:fig:elliptic}
+    \input{papers/ellfilter/python/k.pgf}
+    \caption{Die Periodizitäten in realer und imaginärer Richtung in Abhängigkeit vom elliptischen Modul $k$.}
+    \label{ellfilter:fig:kprime}
 \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.
-
+$K$ und $K^\prime$ sind durch die Ortskurve $K + jK^\prime$ aneinander gebunden und benötigen den Zusatzfaktor $K_1/K$ in \eqref{ellfilter:eq:elliptic}, um die genanten Forderungen einzuhalten.
+Abbildung \ref{ellfilter:fig:degree_eq} zeigt das Problem geometrisch auf, wobei zwei Punkte $K+jK^\prime$ und $K_1+jK_1^\prime$ auf der Ortskurve gesucht sind.
+\begin{figure}
+    \centering
+    \input{papers/ellfilter/tikz/elliptic_transform2.tikz}
+    \caption{Die Gradgleichung als geometrisches Problem ($N=3$).}
+    \label{ellfilter:fig:degree_eq}
+\end{figure}
+Algebraisch kann so die Gradgleichung
 \begin{equation}
     N \frac{K^\prime}{K} = \frac{K^\prime_1}{K_1}
 \end{equation}
+aufgestellt werden, dessen Lösung ist gegeben durch
+\begin{equation} %TODO check
+k_1 = k^N \prod_{i=1}^L \sn^4 \Bigg( \frac{2i - 1}{N} K, k \Bigg),
+\quad \text{wobei} \quad
+N = 2L+r.
+\end{equation}
+Die Herleitung ist sehr umfassend und wird in \cite{ellfilter:bib:orfanidis} im Detail angeschaut.
 
+% \begin{figure}
+%     \centering
+%     \input{papers/ellfilter/tikz/elliptic_transform1.tikz}
+%     \caption{Die Gradgleichung als geometrisches Problem.}
+% \end{figure}
 
-Leider ist das lösen dieser Gleichung nicht trivial.
-Die Rechnung wird in \ref{ellfilter:bib:orfanidis} im Detail angeschaut.
-
-
-\subsection{Polynome?}
+\subsection{Schlussfolgerung}
 
+Die elliptischen Filter können als direkte Erweiterung der Tschebyscheff-Filter verstanden werden.
 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.
+Im elliptischen Fall entstehen so rationale Funktionen mit Nullstellen und 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.
+% 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
index 6a208fa..567bbcc 100644
--- a/buch/papers/ellfilter/jacobi.tex
+++ b/buch/papers/ellfilter/jacobi.tex
@@ -1,17 +1,17 @@
 \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.
+Für das elliptische Filter werden, wie es der Name bereits deutet, elliptische Funktionen gebraucht.
+Wie die trigonometrischen Funktionen Zusammenhänge eines Kreises darlegen, beschreiben die elliptischen Funktionen Ellipsen.
+Es ist daher naheliegend, dass Kosinus des Tschebyscheff-Filters mit einem elliptischen Pendant ausgetauscht werden könnte.
 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.
 
+Die Jacobi elliptischen Funktionen werden ausführlich im Kapitel \ref{buch:elliptisch:section:jacobi} behandelt.
 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 $z$.
+Den \textit{elliptische Modul} $k$, der die Exzentrizität der Ellipse parametrisiert und 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.
+Dies hat zur Folge, dass bei einer Ellipse die Kreisbogenlänge 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}
@@ -27,17 +27,8 @@ Das Winkelargument $z$ kann durch das elliptische Integral erster Art
             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.
+mit dem Winkel $\phi$ in Verbindung gebracht werden.
 
 Dabei wird das vollständige und unvollständige Elliptische integral unterschieden.
 Beim vollständigen Integral
@@ -53,9 +44,9 @@ Beim vollständigen Integral
         }
     }
 \end{equation}
-wird über ein viertel Ellipsenbogen integriert also bis $\phi=\pi/2$ und liefert das Winkelargument für eine Vierteldrehung.
+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.
+Alle elliptischen 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.
@@ -93,37 +84,40 @@ Mithilfe von $F^{-1}$ kann zum Beispiel $sn^{-1}$ mit dem Elliptischen integral
     =
     \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)
+    w.
 \end{equation}
 
+% \begin{equation} %TODO remove unnecessary equations
+%     \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}
+
+Beim Tschebyscheff-Filter konnten wir mit Betrachten des Arcuscosinus die Funktionalität erklären.
+Für das Elliptische Filter machen wir die gleiche Betrachtung mit der $\sn^{-1}$-Funktion.
+Der $\sn^{-1}$ ist durch das elliptische Integral
 \begin{align}
     \sn^{-1}(w, k)
         & =
@@ -148,11 +142,8 @@ Mithilfe von $F^{-1}$ kann zum Beispiel $sn^{-1}$ mit dem Elliptischen integral
         }
     }
 \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 $k \leq 1$.
+beschrieben.
+Dazu betrachten wir wieder den Integranden
 \begin{equation}
     \frac{
         1
@@ -160,24 +151,15 @@ Der erste Term unter der Wurzel wird dann negativ, während der zweite noch posi
         \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
-
+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 $k \leq 1$.
+Ab diesem Punkt knickt die Funktion in die imaginäre Richtung ab.
+Bei $t = 1/k$ ist auch der zweite Term negativ und die Funktion verläuft in die negative reelle Richtung.
+Abbildung \ref{ellfilter:fig:sn} zeigt den Verlauf der Funktion in der komplexen Ebene.
 \begin{figure}
     \centering
     \input{papers/ellfilter/tikz/sn.tikz.tex}
@@ -185,5 +167,20 @@ In der reellen Richtung ist sie $4K(k)$-periodisch und in der imaginären Richtu
         $z$-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}
+    \label{ellfilter:fig:sn}
 \end{figure}
+In der reellen Richtung ist sie $4K(k)$-periodisch und in der imaginären Richtung $4K^\prime(k)$-periodisch, wobei $K^\prime$ das komplementäre vollständige Elliptische Integral ist:
+\begin{equation}
+    K^\prime(k)
+    =
+    \int_{0}^{\pi / 2}
+    \frac{
+        d\theta
+    }{
+        \sqrt{
+            1-{k^\prime}^2 \sin^2 \theta
+        }
+    },
+    \quad
+    k^\prime = \sqrt{1-k^2}.
+\end{equation}
diff --git a/buch/papers/ellfilter/presentation/presentation.tex b/buch/papers/ellfilter/presentation/presentation.tex
index 7fdb864..96bdfd3 100644
--- a/buch/papers/ellfilter/presentation/presentation.tex
+++ b/buch/papers/ellfilter/presentation/presentation.tex
@@ -76,9 +76,9 @@
 
 %Title Page
 \title{Elliptische Filter}
-\subtitle{Eine Anwendung der Jaccobi elliptischen Funktionen}
+\subtitle{Eine Anwendung der Jacobi elliptischen Funktionen}
 \author{Nicolas Tobler}
-% \institute{OST Ostschweizer Fachhochschule}
+\institute{Mathematisches Seminar 2022 | Spezielle Funktionen}
 % \institute{\includegraphics[scale=0.3]{../img/ost_logo.png}}
 \date{\today}
 
@@ -113,25 +113,38 @@
 	\end{frame}
 
 	\begin{frame}
-		\frametitle{Content}
+		\frametitle{Inhalt}
 		\tableofcontents
 	\end{frame}
 
-	\section{Linear Filter}
+	\section{Lineare Filter}
 
 	\begin{frame}
 		\frametitle{Lineare Filter}
 
+		\begin{center}
+			\scalebox{0.75}{
+				\input{../tikz/filter.tikz.tex}
+			}
+		\end{center}
 
-		\begin{equation}
+
+		\begin{equation*}
 			| H(\Omega)|^2 = \frac{1}{1 + \varepsilon_p^2 F_N^2(w)}, \quad w=\frac{\Omega}{\Omega_p}
-		\end{equation}
+		\end{equation*}
 
 		\pause
 
-		\begin{equation}
+		\begin{align*}
+			|F_N(w)| &< 1 \quad \forall \quad |w| < 1 \\
+			|F_N(w)| &= 1 \quad \forall \quad |w| = 1 \\
+			|F_N(w)| &> 1 \quad \forall \quad |w| > 1
+		\end{align*}
+
+
+		\begin{equation*}
 			F_N(w) = w^N
-		\end{equation}
+		\end{equation*}
 
 	\end{frame}
 
@@ -218,10 +231,36 @@
 
 		Darstellung mit trigonometrischen Funktionen:
 
-		\begin{align} \label{ellfilter:eq:chebychef_polynomials}
+		\begin{align*}
 			T_N(w) &= \cos \left( N \cos^{-1}(w) \right) \\
 				&= \cos \left(N~z \right), \quad w= \cos(z)
-		\end{align}
+		\end{align*}
+
+		\pause
+
+		\begin{align*}
+			\cos^{-1}(x)
+			&=
+			\int_{x}^{1}
+			\frac{
+				dz
+			}{
+				\sqrt{
+					1-z^2
+				}
+			}\\
+			&=
+			\int_{0}^{x}
+			\frac{
+				-1
+			}{
+				\sqrt{
+					1-z^2
+				}
+			}
+			~dz
+			+ \frac{\pi}{2}
+		\end{align*}
 
 
 	\end{frame}
@@ -229,15 +268,41 @@
 	\begin{frame}
 		\frametitle{Tschebyscheff-Filter}
 
-		\begin{equation*}
-			z = \cos^{-1}(w)
-		\end{equation*}
+		\begin{columns}
+
+			\begin{column}{0.2\textwidth}
+
+				\begin{equation*}
+					z = \cos^{-1}(w)
+				\end{equation*}
+
+				\vspace{0.5cm}
+
+				Integrand:
+				\begin{equation*}
+					\frac{
+						-1
+					}{
+						\sqrt{
+							1-z^2
+						}
+					}
+				\end{equation*}
+
+			\end{column}
+			\begin{column}{0.8\textwidth}
+
+
+				\begin{center}
+					\scalebox{0.7}{
+						\input{../tikz/arccos.tikz.tex}
+					}
+				\end{center}
+
+			\end{column}
+		\end{columns}
+
 
-		\begin{center}
-			\scalebox{0.85}{
-				\input{../tikz/arccos.tikz.tex}
-			}
-		\end{center}
 
 	\end{frame}
 
@@ -245,7 +310,7 @@
 		\frametitle{Tschebyscheff-Filter}
 
 		\begin{equation*}
-			z_1 = N~\cos^{-1}(w)
+			T_N(w) = \cos \left(z_1 \right), \quad z_1 = N~\cos^{-1}(w)
 		\end{equation*}
 
 		\begin{center}
@@ -257,15 +322,14 @@
 	\end{frame}
 
 
-	\section{Jaccobi elliptische Funktionen}
+	\section{Jacobi elliptische Funktionen}
 
 	\begin{frame}
-		\frametitle{Jaccobi elliptische Funktionen}
+		\frametitle{Jacobi elliptische Funktionen}
 
+		Elliptisches Integral erster Art
 
-		\begin{equation}
-			z
-			=
+		\begin{equation*}
 			F(\phi, k)
 			=
 			\int_{0}^{\phi}
@@ -276,18 +340,18 @@
 					1-k^2 \sin^2 \theta
 				}
 			}
-			=
-			\int_{0}^{\phi}
-			\frac{
-				dt
-			}{
-				\sqrt{
-					(1-t^2)(1-k^2 t^2)
-				}
-			}
-		\end{equation}
+			% =
+			% \int_{0}^{\phi}
+			% \frac{
+			% 	dt
+			% }{
+			% 	\sqrt{
+			% 		(1-t^2)(1-k^2 t^2)
+			% 	}
+			% }
+		\end{equation*}
 
-		\begin{equation}
+		\begin{equation*}
 			K(k)
 			=
 			\int_{0}^{\pi / 2}
@@ -298,24 +362,88 @@
 					1-k^2 \sin^2 \theta
 				}
 			}
-		\end{equation}
+		\end{equation*}
 
 
 
 	\end{frame}
 
+
+
+
+
 	\begin{frame}
-		\frametitle{Jaccobi elliptische Funktionen}
+		\frametitle{Jacobi elliptische Funktionen}
+
+			\begin{equation*}
+				\sn^{-1}(w, k)
+				=
+				F(\phi, k),
+				\quad
+				\phi = \sin^{-1}(w)
+			\end{equation*}
+
+			\begin{align*}
+				\sn^{-1}(w, k)
+					& =
+				\int_{0}^{\phi}
+				\frac{
+					d\theta
+				}{
+					\sqrt{
+						1-k^2 \sin^2 \theta
+					}
+				},
+				\quad
+				\phi = \sin^{-1}(w)
+				\\
+					& =
+				\int_{0}^{w}
+				\frac{
+					dt
+				}{
+					\sqrt{
+						(1-t^2)(1-k^2 t^2)
+					}
+				}
+			\end{align*}
 
-		\begin{equation*}
-			z = \sn^{-1}(w, k)
-		\end{equation*}
 
-		\begin{center}
-			\scalebox{0.7}{
-				\input{../tikz/sn.tikz.tex}
-			}
-		\end{center}
+
+		\end{frame}
+
+	\begin{frame}
+		\frametitle{Jacobi elliptische Funktionen}
+		\begin{columns}
+			\begin{column}{0.2\textwidth}
+
+				\begin{equation*}
+					z = \sn^{-1}(w, k)
+				\end{equation*}
+
+				\vspace{0.5cm}
+
+				Integrand:
+				\begin{equation*}
+					\frac{
+						1
+					}{
+						\sqrt{
+							(1-t^2)(1-k^2 t^2)
+						}
+					}
+				\end{equation*}
+
+			\end{column}
+			\begin{column}{0.8\textwidth}
+				\begin{center}
+					\scalebox{0.75}{
+						\input{../tikz/sn.tikz.tex}
+					}
+				\end{center}
+			\end{column}
+		\end{columns}
+
 
 	\end{frame}
 
@@ -334,7 +462,7 @@
 
 
 	\begin{frame}
-		\frametitle{Jaccobi elliptische Funktionen}
+		\frametitle{Jacobi elliptische Funktionen}
 
 		\begin{equation*}
 			z = \cd^{-1}(w, k)
@@ -349,6 +477,23 @@
 
 	\end{frame}
 
+	\section{Elliptisches Filter}
+
+	\begin{frame}
+		\frametitle{Elliptisches Filter}
+
+		% \begin{equation*}
+		% 	z_1 = N~\frac{K_1}{K}~\cd^{-1}(w, k)
+		% \end{equation*}
+
+		\begin{center}
+			\scalebox{0.75}{
+				\input{../tikz/cd3.tikz.tex}
+			}
+		\end{center}
+
+	\end{frame}
+
 	\begin{frame}
 		\frametitle{Periodizität in realer und imaginärer Richtung}
 
@@ -360,20 +505,42 @@
 	\end{frame}
 
 	\begin{frame}
+		\frametitle{Gradgleichung}
+
+		\begin{center}
+			\scalebox{0.95}{
+				\input{../tikz/elliptic_transform2.tikz}
+			}
+		\end{center}
+
+		\onslide<5->{
+		\begin{equation*}
+			N \frac{K^\prime}{K} = \frac{K^\prime_1}{K_1}
+		\end{equation*}
+		}
+
+	\end{frame}
+
+	\begin{frame}
 		\frametitle{Elliptisches Filter}
 
 		\begin{equation*}
-			z_1 = N~\frac{K_1}{K}~\cd^{-1}(w, k)
+			R_N = \cd(z_1, k_1),
+			\quad
+			z_1 = N~\frac{K_1}{K}~\cd^{-1}(w, k),
+			\quad
+			N \frac{K^\prime}{K} = \frac{K^\prime_1}{K_1}
 		\end{equation*}
 
 		\begin{center}
-			\scalebox{0.8}{
+			\scalebox{0.75}{
 				\input{../tikz/cd2.tikz.tex}
 			}
 		\end{center}
 
 	\end{frame}
 
+
 	\begin{frame}
 		\frametitle{Elliptisches Filter}
 
@@ -401,13 +568,4 @@
 
 	\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_elliptic.pgf b/buch/papers/ellfilter/python/F_N_elliptic.pgf
deleted file mode 100644
index 03084c6..0000000
--- a/buch/papers/ellfilter/python/F_N_elliptic.pgf
+++ /dev/null
@@ -1,847 +0,0 @@
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diff --git a/buch/papers/ellfilter/python/elliptic.pgf b/buch/papers/ellfilter/python/elliptic.pgf
index 31b77d4..32485c1 100644
--- a/buch/papers/ellfilter/python/elliptic.pgf
+++ b/buch/papers/ellfilter/python/elliptic.pgf
@@ -23,7 +23,7 @@
 \begingroup%
 \makeatletter%
 \begin{pgfpicture}%
-\pgfpathrectangle{\pgfpointorigin}{\pgfqpoint{4.000000in}{2.500000in}}%
+\pgfpathrectangle{\pgfpointorigin}{\pgfqpoint{5.000000in}{3.000000in}}%
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 \begin{pgfscope}%
 \pgfsetbuttcap%
@@ -34,9 +34,9 @@
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+\pgfpathlineto{\pgfqpoint{5.000000in}{0.000000in}}%
+\pgfpathlineto{\pgfqpoint{5.000000in}{3.000000in}}%
+\pgfpathlineto{\pgfqpoint{0.000000in}{3.000000in}}%
 \pgfpathlineto{\pgfqpoint{0.000000in}{0.000000in}}%
 \pgfpathclose%
 \pgfusepath{}%
@@ -51,16 +51,16 @@
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-\pgfpathlineto{\pgfqpoint{0.617954in}{0.548769in}}%
+\pgfpathmoveto{\pgfqpoint{0.733531in}{1.746607in}}%
+\pgfpathlineto{\pgfqpoint{4.727004in}{1.746607in}}%
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+\pgfpathlineto{\pgfqpoint{0.733531in}{2.850000in}}%
+\pgfpathlineto{\pgfqpoint{0.733531in}{1.746607in}}%
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 \begin{pgfscope}%
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+\pgfpathrectangle{\pgfqpoint{0.733531in}{1.746607in}}{\pgfqpoint{3.993473in}{1.103393in}}%
 \pgfusepath{clip}%
 \pgfsetbuttcap%
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@@ -72,16 +72,16 @@
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-\pgfpathlineto{\pgfqpoint{0.617954in}{1.788459in}}%
+\pgfpathmoveto{\pgfqpoint{0.733531in}{-108.151374in}}%
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+\pgfpathlineto{\pgfqpoint{0.733531in}{2.187964in}}%
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 \pgfpathclose%
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 \end{pgfscope}%
 \begin{pgfscope}%
-\pgfpathrectangle{\pgfqpoint{0.617954in}{0.548769in}}{\pgfqpoint{3.143642in}{1.753186in}}%
+\pgfpathrectangle{\pgfqpoint{0.733531in}{1.746607in}}{\pgfqpoint{3.993473in}{1.103393in}}%
 \pgfusepath{clip}%
 \pgfsetbuttcap%
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@@ -93,16 +93,16 @@
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-\pgfpathlineto{\pgfqpoint{2.189776in}{0.724087in}}%
+\pgfpathmoveto{\pgfqpoint{2.730268in}{2.187964in}}%
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+\pgfpathlineto{\pgfqpoint{2.750075in}{2.408643in}}%
+\pgfpathlineto{\pgfqpoint{2.730268in}{2.408643in}}%
+\pgfpathlineto{\pgfqpoint{2.730268in}{2.187964in}}%
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 \pgfusepath{clip}%
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@@ -114,16 +114,16 @@
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@@ -131,8 +131,8 @@
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 \pgfusepath{stroke}%
 \end{pgfscope}%
 \begin{pgfscope}%
@@ -150,7 +150,295 @@
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 }%
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+\pgfsys@transformshift{0.733531in}{1.746607in}%
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+\end{pgfscope}%
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@@ -188,7 +951,7 @@
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 }%
 \begin{pgfscope}%
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@@ -196,10 +959,10 @@
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@@ -207,8 +970,8 @@
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 \begin{pgfscope}%
@@ -226,7 +989,7 @@
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 }%
 \begin{pgfscope}%
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@@ -234,10 +997,10 @@
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@@ -245,8 +1008,8 @@
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 \begin{pgfscope}%
@@ -264,7 +1027,7 @@
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-\pgftext[x=2.975686in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {1.5}\)}%
+\pgftext[x=2.231083in,y=0.272996in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.75}\)}%
 \end{pgfscope}%
 \begin{pgfscope}%
-\pgfpathrectangle{\pgfqpoint{0.617954in}{0.548769in}}{\pgfqpoint{3.143642in}{1.753186in}}%
+\pgfpathrectangle{\pgfqpoint{0.733531in}{0.370218in}}{\pgfqpoint{3.993473in}{1.103393in}}%
 \pgfusepath{clip}%
 \pgfsetrectcap%
 \pgfsetroundjoin%
@@ -283,8 +1046,8 @@
 \definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}%
 \pgfsetstrokecolor{currentstroke}%
 \pgfsetdash{}{0pt}%
-\pgfpathmoveto{\pgfqpoint{3.761597in}{0.548769in}}%
-\pgfpathlineto{\pgfqpoint{3.761597in}{2.301955in}}%
+\pgfpathmoveto{\pgfqpoint{2.730268in}{0.370218in}}%
+\pgfpathlineto{\pgfqpoint{2.730268in}{1.473611in}}%
 \pgfusepath{stroke}%
 \end{pgfscope}%
 \begin{pgfscope}%
@@ -302,7 +1065,7 @@
 \pgfusepath{stroke,fill}%
 }%
 \begin{pgfscope}%
-\pgfsys@transformshift{3.761597in}{0.548769in}%
+\pgfsys@transformshift{2.730268in}{0.370218in}%
 \pgfsys@useobject{currentmarker}{}%
 \end{pgfscope}%
 \end{pgfscope}%
@@ -310,16 +1073,48 @@
 \definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
 \pgfsetstrokecolor{textcolor}%
 \pgfsetfillcolor{textcolor}%
-\pgftext[x=3.761597in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {2.0}\)}%
+\pgftext[x=2.730268in,y=0.272996in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {1.00}\)}%
+\end{pgfscope}%
+\begin{pgfscope}%
+\pgfpathrectangle{\pgfqpoint{0.733531in}{0.370218in}}{\pgfqpoint{3.993473in}{1.103393in}}%
+\pgfusepath{clip}%
+\pgfsetrectcap%
+\pgfsetroundjoin%
+\pgfsetlinewidth{0.803000pt}%
+\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}%
+\pgfsetstrokecolor{currentstroke}%
+\pgfsetdash{}{0pt}%
+\pgfpathmoveto{\pgfqpoint{3.229452in}{0.370218in}}%
+\pgfpathlineto{\pgfqpoint{3.229452in}{1.473611in}}%
+\pgfusepath{stroke}%
+\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}}{%
+\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}%
+\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}%
+\pgfusepath{stroke,fill}%
+}%
+\begin{pgfscope}%
+\pgfsys@transformshift{3.229452in}{0.370218in}%
+\pgfsys@useobject{currentmarker}{}%
+\end{pgfscope}%
 \end{pgfscope}%
 \begin{pgfscope}%
 \definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
 \pgfsetstrokecolor{textcolor}%
 \pgfsetfillcolor{textcolor}%
-\pgftext[x=2.189776in,y=0.272534in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle w\)}%
+\pgftext[x=3.229452in,y=0.272996in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {1.25}\)}%
 \end{pgfscope}%
 \begin{pgfscope}%
-\pgfpathrectangle{\pgfqpoint{0.617954in}{0.548769in}}{\pgfqpoint{3.143642in}{1.753186in}}%
+\pgfpathrectangle{\pgfqpoint{0.733531in}{0.370218in}}{\pgfqpoint{3.993473in}{1.103393in}}%
 \pgfusepath{clip}%
 \pgfsetrectcap%
 \pgfsetroundjoin%
@@ -327,8 +1122,8 @@
 \definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}%
 \pgfsetstrokecolor{currentstroke}%
 \pgfsetdash{}{0pt}%
-\pgfpathmoveto{\pgfqpoint{0.617954in}{0.548769in}}%
-\pgfpathlineto{\pgfqpoint{3.761597in}{0.548769in}}%
+\pgfpathmoveto{\pgfqpoint{3.728636in}{0.370218in}}%
+\pgfpathlineto{\pgfqpoint{3.728636in}{1.473611in}}%
 \pgfusepath{stroke}%
 \end{pgfscope}%
 \begin{pgfscope}%
@@ -340,13 +1135,13 @@
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-\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}%
-\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}%
+\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{%
+\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}%
+\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}%
 \pgfusepath{stroke,fill}%
 }%
 \begin{pgfscope}%
-\pgfsys@transformshift{0.617954in}{0.548769in}%
+\pgfsys@transformshift{3.728636in}{0.370218in}%
 \pgfsys@useobject{currentmarker}{}%
 \end{pgfscope}%
 \end{pgfscope}%
@@ -354,10 +1149,10 @@
 \definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
 \pgfsetstrokecolor{textcolor}%
 \pgfsetfillcolor{textcolor}%
-\pgftext[x=0.343262in, y=0.500544in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.0}\)}%
+\pgftext[x=3.728636in,y=0.272996in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {1.50}\)}%
 \end{pgfscope}%
 \begin{pgfscope}%
-\pgfpathrectangle{\pgfqpoint{0.617954in}{0.548769in}}{\pgfqpoint{3.143642in}{1.753186in}}%
+\pgfpathrectangle{\pgfqpoint{0.733531in}{0.370218in}}{\pgfqpoint{3.993473in}{1.103393in}}%
 \pgfusepath{clip}%
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@@ -365,8 +1160,8 @@
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-\pgfpathmoveto{\pgfqpoint{0.617954in}{0.899406in}}%
-\pgfpathlineto{\pgfqpoint{3.761597in}{0.899406in}}%
+\pgfpathmoveto{\pgfqpoint{4.227820in}{0.370218in}}%
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 \end{pgfscope}%
 \begin{pgfscope}%
@@ -378,13 +1173,13 @@
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-\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}%
-\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}%
+\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{%
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 \pgfusepath{stroke,fill}%
 }%
 \begin{pgfscope}%
-\pgfsys@transformshift{0.617954in}{0.899406in}%
+\pgfsys@transformshift{4.227820in}{0.370218in}%
 \pgfsys@useobject{currentmarker}{}%
 \end{pgfscope}%
 \end{pgfscope}%
@@ -392,10 +1187,10 @@
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-\pgftext[x=0.343262in, y=0.851181in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.2}\)}%
+\pgftext[x=4.227820in,y=0.272996in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {1.75}\)}%
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 \begin{pgfscope}%
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+\pgfpathrectangle{\pgfqpoint{0.733531in}{0.370218in}}{\pgfqpoint{3.993473in}{1.103393in}}%
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@@ -403,8 +1198,8 @@
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-\pgfpathmoveto{\pgfqpoint{0.617954in}{1.250043in}}%
-\pgfpathlineto{\pgfqpoint{3.761597in}{1.250043in}}%
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 \begin{pgfscope}%
@@ -416,13 +1211,13 @@
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-\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}%
+\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{%
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 \pgfusepath{stroke,fill}%
 }%
 \begin{pgfscope}%
-\pgfsys@transformshift{0.617954in}{1.250043in}%
+\pgfsys@transformshift{4.727004in}{0.370218in}%
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@@ -430,10 +1225,10 @@
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-\pgftext[x=0.343262in, y=1.201818in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.4}\)}%
+\pgftext[x=4.727004in,y=0.272996in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {2.00}\)}%
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 \begin{pgfscope}%
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@@ -441,8 +1236,8 @@
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 \begin{pgfscope}%
@@ -460,7 +1255,7 @@
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 \begin{pgfscope}%
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@@ -468,10 +1263,10 @@
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@@ -479,8 +1274,8 @@
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@@ -498,7 +1293,7 @@
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 \begin{pgfscope}%
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@@ -517,8 +1312,8 @@
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 \begin{pgfscope}%
@@ -667,8 +1482,8 @@
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-\pgfpathmoveto{\pgfqpoint{0.617954in}{0.548769in}}%
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 \pgfusepath{stroke}%
 \end{pgfscope}%
 \begin{pgfscope}%
@@ -678,8 +1493,8 @@
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 \pgfusepath{stroke}%
 \end{pgfscope}%
 \begin{pgfscope}%
@@ -689,8 +1504,8 @@
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diff --git a/buch/papers/ellfilter/python/elliptic.py b/buch/papers/ellfilter/python/elliptic.py
index b3336a1..6e0fd12 100644
--- a/buch/papers/ellfilter/python/elliptic.py
+++ b/buch/papers/ellfilter/python/elliptic.py
@@ -324,9 +324,9 @@ 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].plot(k, K, linewidth=1)
 axs[0].text(k[30], K[30]+0.1, f"$K$")
-axs[0].plot(k, K_prime, linewidth=0.1)
+axs[0].plot(k, K_prime, linewidth=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])
@@ -342,9 +342,9 @@ 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].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])
diff --git a/buch/papers/ellfilter/python/elliptic2.py b/buch/papers/ellfilter/python/elliptic2.py
index 29c6f47..20a7428 100644
--- a/buch/papers/ellfilter/python/elliptic2.py
+++ b/buch/papers/ellfilter/python/elliptic2.py
@@ -1,5 +1,6 @@
 # %%
 
+import enum
 import matplotlib.pyplot as plt
 import scipy.signal
 import numpy as np
@@ -8,7 +9,9 @@ from matplotlib.patches import Rectangle
 
 import plot_params
 
-def ellip_filter(N):
+N=5
+
+def ellip_filter(N, mode=-1):
 
     order = N
     passband_ripple_db = 3
@@ -26,7 +29,16 @@ def ellip_filter(N):
         fs=None
     )
 
-    w, mag_db, phase = scipy.signal.bode((a, b), w=np.linspace(0*omega_c,2*omega_c, 4000))
+    if mode == 0:
+        w = np.linspace(0*omega_c,omega_c, 2000)
+    elif mode == 1:
+        w = np.linspace(omega_c,1.00992*omega_c, 2000)
+    elif mode == 2:
+        w = np.linspace(1.00992*omega_c,2*omega_c, 2000)
+    else:
+        w = np.linspace(0*omega_c,2*omega_c, 4000)
+
+    w, mag_db, phase = scipy.signal.bode((a, b), w=w)
 
     mag = 10**(mag_db/20)
 
@@ -38,105 +50,96 @@ def ellip_filter(N):
     return w/omega_c, FN2 / epsilon2, mag, a, b
 
 
-plt.figure(figsize=(4,2.5))
+f, axs = plt.subplots(2, 1, figsize=(5,3), sharex=True)
 
-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)
+for mode, c in enumerate(["green", "orange", "red"]):
+    w, FN2, mag, a, b = ellip_filter(N, mode=mode)
+    axs[0].semilogy(w, FN2, label=f"$N={N}, k=0.1$", linewidth=1, color=c)
 
-plt.gca().add_patch(Rectangle(
+axs[0].add_patch(Rectangle(
     (0, 0),
     1, 1,
     fc ='green',
     alpha=0.2,
     lw = 10,
 ))
-plt.gca().add_patch(Rectangle(
+axs[0].add_patch(Rectangle(
     (1, 1),
-    0.01, 1e2-1,
+    0.00992, 1e2-1,
     fc ='orange',
     alpha=0.2,
     lw = 10,
 ))
 
-plt.gca().add_patch(Rectangle(
-    (1.01, 100),
+axs[0].add_patch(Rectangle(
+    (1.00992, 100),
     1, 1e6,
     fc ='red',
     alpha=0.2,
     lw = 10,
 ))
 
-zeros = [0,0.87,1]
+zeros = [0,0.87,0.995]
 poles = [1.01,1.155]
 
 import matplotlib.transforms
-plt.plot( # mark errors as vertical bars
+axs[0].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,
+        axs[0].transData,
+        axs[0].transAxes,
     ),
 )
-plt.plot( # mark errors as vertical bars
+axs[0].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,
+        axs[0].transData,
+        axs[0].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)
+for mode, c in enumerate(["green", "orange", "red"]):
+    w, FN2, mag, a, b = ellip_filter(N, mode=mode)
+    axs[1].plot(w, mag, linewidth=1, color=c)
 
-plt.gca().add_patch(Rectangle(
+axs[1].add_patch(Rectangle(
     (0, np.sqrt(2)/2),
     1, 1,
     fc ='green',
     alpha=0.2,
     lw = 10,
 ))
-plt.gca().add_patch(Rectangle(
+axs[1].add_patch(Rectangle(
     (1, 0.1),
-    0.01, np.sqrt(2)/2 - 0.1,
+    0.00992, np.sqrt(2)/2 - 0.1,
     fc ='orange',
     alpha=0.2,
     lw = 10,
 ))
 
-plt.gca().add_patch(Rectangle(
-    (1.01, 0),
+axs[1].add_patch(Rectangle(
+    (1.00992, 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("$|H(w)|$")
+axs[0].set_xlim([0,2])
+axs[0].set_ylim([1e-4,1e6])
+axs[0].grid()
+axs[0].set_ylabel("$F^2_N(w)$")
+axs[1].grid()
+axs[1].set_ylim([0,1])
+axs[1].set_ylabel("$|H(w)|$")
 plt.tight_layout()
 plt.savefig("elliptic.pgf")
 plt.show()
diff --git a/buch/papers/ellfilter/python/k.pgf b/buch/papers/ellfilter/python/k.pgf
index 95d61d4..bbb823a 100644
--- a/buch/papers/ellfilter/python/k.pgf
+++ b/buch/papers/ellfilter/python/k.pgf
@@ -320,7 +320,7 @@
 \pgfusepath{clip}%
 \pgfsetrectcap%
 \pgfsetroundjoin%
-\pgfsetlinewidth{0.100375pt}%
+\pgfsetlinewidth{1.003750pt}%
 \definecolor{currentstroke}{rgb}{0.121569,0.466667,0.705882}%
 \pgfsetstrokecolor{currentstroke}%
 \pgfsetdash{}{0pt}%
@@ -434,7 +434,7 @@
 \pgfusepath{clip}%
 \pgfsetrectcap%
 \pgfsetroundjoin%
-\pgfsetlinewidth{0.100375pt}%
+\pgfsetlinewidth{1.003750pt}%
 \definecolor{currentstroke}{rgb}{1.000000,0.498039,0.054902}%
 \pgfsetstrokecolor{currentstroke}%
 \pgfsetdash{}{0pt}%
@@ -1011,56 +1011,6 @@
 \pgfusepath{stroke}%
 \end{pgfscope}%
 \begin{pgfscope}%
-\pgfpathrectangle{\pgfqpoint{2.874885in}{0.548769in}}{\pgfqpoint{1.940523in}{1.753186in}}%
-\pgfusepath{clip}%
-\pgfsetbuttcap%
-\pgfsetroundjoin%
-\definecolor{currentfill}{rgb}{0.121569,0.466667,0.705882}%
-\pgfsetfillcolor{currentfill}%
-\pgfsetlinewidth{1.003750pt}%
-\definecolor{currentstroke}{rgb}{0.121569,0.466667,0.705882}%
-\pgfsetstrokecolor{currentstroke}%
-\pgfsetdash{}{0pt}%
-\pgfsys@defobject{currentmarker}{\pgfqpoint{-0.006944in}{-0.006944in}}{\pgfqpoint{0.006944in}{0.006944in}}{%
-\pgfpathmoveto{\pgfqpoint{0.000000in}{-0.006944in}}%
-\pgfpathcurveto{\pgfqpoint{0.001842in}{-0.006944in}}{\pgfqpoint{0.003608in}{-0.006213in}}{\pgfqpoint{0.004910in}{-0.004910in}}%
-\pgfpathcurveto{\pgfqpoint{0.006213in}{-0.003608in}}{\pgfqpoint{0.006944in}{-0.001842in}}{\pgfqpoint{0.006944in}{0.000000in}}%
-\pgfpathcurveto{\pgfqpoint{0.006944in}{0.001842in}}{\pgfqpoint{0.006213in}{0.003608in}}{\pgfqpoint{0.004910in}{0.004910in}}%
-\pgfpathcurveto{\pgfqpoint{0.003608in}{0.006213in}}{\pgfqpoint{0.001842in}{0.006944in}}{\pgfqpoint{0.000000in}{0.006944in}}%
-\pgfpathcurveto{\pgfqpoint{-0.001842in}{0.006944in}}{\pgfqpoint{-0.003608in}{0.006213in}}{\pgfqpoint{-0.004910in}{0.004910in}}%
-\pgfpathcurveto{\pgfqpoint{-0.006213in}{0.003608in}}{\pgfqpoint{-0.006944in}{0.001842in}}{\pgfqpoint{-0.006944in}{0.000000in}}%
-\pgfpathcurveto{\pgfqpoint{-0.006944in}{-0.001842in}}{\pgfqpoint{-0.006213in}{-0.003608in}}{\pgfqpoint{-0.004910in}{-0.004910in}}%
-\pgfpathcurveto{\pgfqpoint{-0.003608in}{-0.006213in}}{\pgfqpoint{-0.001842in}{-0.006944in}}{\pgfqpoint{0.000000in}{-0.006944in}}%
-\pgfpathlineto{\pgfqpoint{0.000000in}{-0.006944in}}%
-\pgfpathclose%
-\pgfusepath{stroke,fill}%
-}%
-\begin{pgfscope}%
-\pgfsys@transformshift{3.384190in}{1.844597in}%
-\pgfsys@useobject{currentmarker}{}%
-\end{pgfscope}%
-\begin{pgfscope}%
-\pgfsys@transformshift{3.388110in}{1.606330in}%
-\pgfsys@useobject{currentmarker}{}%
-\end{pgfscope}%
-\begin{pgfscope}%
-\pgfsys@transformshift{3.405294in}{1.376014in}%
-\pgfsys@useobject{currentmarker}{}%
-\end{pgfscope}%
-\begin{pgfscope}%
-\pgfsys@transformshift{3.441114in}{1.248396in}%
-\pgfsys@useobject{currentmarker}{}%
-\end{pgfscope}%
-\begin{pgfscope}%
-\pgfsys@transformshift{3.612461in}{1.128939in}%
-\pgfsys@useobject{currentmarker}{}%
-\end{pgfscope}%
-\begin{pgfscope}%
-\pgfsys@transformshift{3.960478in}{1.102320in}%
-\pgfsys@useobject{currentmarker}{}%
-\end{pgfscope}%
-\end{pgfscope}%
-\begin{pgfscope}%
 \pgfsetrectcap%
 \pgfsetmiterjoin%
 \pgfsetlinewidth{0.803000pt}%
@@ -1116,42 +1066,6 @@
 \pgfsetfillcolor{textcolor}%
 \pgftext[x=3.415254in,y=0.583833in,left,base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle \pi/2\)}%
 \end{pgfscope}%
-\begin{pgfscope}%
-\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
-\pgfsetstrokecolor{textcolor}%
-\pgfsetfillcolor{textcolor}%
-\pgftext[x=3.416532in,y=1.879661in,left,base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle k=0.10\)}%
-\end{pgfscope}%
-\begin{pgfscope}%
-\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
-\pgfsetstrokecolor{textcolor}%
-\pgfsetfillcolor{textcolor}%
-\pgftext[x=3.420452in,y=1.641394in,left,base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle k=0.20\)}%
-\end{pgfscope}%
-\begin{pgfscope}%
-\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
-\pgfsetstrokecolor{textcolor}%
-\pgfsetfillcolor{textcolor}%
-\pgftext[x=3.437636in,y=1.411078in,left,base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle k=0.40\)}%
-\end{pgfscope}%
-\begin{pgfscope}%
-\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
-\pgfsetstrokecolor{textcolor}%
-\pgfsetfillcolor{textcolor}%
-\pgftext[x=3.473456in,y=1.283460in,left,base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle k=0.60\)}%
-\end{pgfscope}%
-\begin{pgfscope}%
-\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
-\pgfsetstrokecolor{textcolor}%
-\pgfsetfillcolor{textcolor}%
-\pgftext[x=3.644803in,y=1.164003in,left,base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle k=0.90\)}%
-\end{pgfscope}%
-\begin{pgfscope}%
-\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
-\pgfsetstrokecolor{textcolor}%
-\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/tikz/arccos.tikz.tex b/buch/papers/ellfilter/tikz/arccos.tikz.tex
index 2772620..b11c25d 100644
--- a/buch/papers/ellfilter/tikz/arccos.tikz.tex
+++ b/buch/papers/ellfilter/tikz/arccos.tikz.tex
@@ -8,29 +8,6 @@
 
     \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) {$-\pi$};
@@ -43,16 +20,53 @@
         % \node[gray, anchor=south east] at (0, 0) {$0$};
         \node[gray, anchor=east] at (0, 1.5) {$\infty$};
 
+        \clip(-7.5,-2) rectangle (7.5,2);
+
+        % \pause
+        \draw[ultra thick, ->, darkgreen] (1, 0) -- (0,0);
+        % \pause
+        \draw[ultra thick, ->, orange] (0, 0) -- (0,1.5);
+        % \pause
+        \draw[ultra thick, ->, cyan] (2, 0) -- (1,0);
+        \draw[ultra thick, ->, blue] (2,1.5) -- (2, 0);
+
+        % \pause
+
+        \foreach \i in {-2,...,1} {
+            \begin{scope}[xshift=\i*4cm]
+                \begin{scope}[]
+                    \draw[->, darkgreen] (-1, 0) -- (0,0);
+                    \draw[->, orange] (0, 0) -- (0,1.5);
+                    \draw[->, orange] (0, 0) -- (0,-1.5);
+                    \draw[->, darkgreen] (1, 0) -- (0,0);
+                    \draw[->, cyan] (2, 0) -- (1,0);
+                    \draw[->, blue] (2,1.5) -- (2, 0);
+                    \draw[->, blue] (2,-1.5) -- (2, 0);
+                    \draw[->, cyan] (2, 0) -- (3,0);
+                \end{scope}
+                \node[zero] at (1,0) {};
+                \node[zero] at (3,0) {};
+            \end{scope}
+        }
+
     \end{scope}
 
-    \begin{scope}[yshift=-2.5cm]
+    \node[zero] at (4,2) (n) {};
+    \node[anchor=west] at (n.east) {Nullstelle};
+
+    \begin{scope}[yshift=-3.25cm]
+
+        \draw[->, thick](0,0) -- node[anchor=center, fill=white]{$z = \cos^{-1}(w)$} (0,1);
+
+    \end{scope}
+    \begin{scope}[yshift=-4cm]
 
         \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);
+        \draw[ultra thick, ->, blue]      (-4, 0) -- (-2, 0);
+        \draw[ultra thick, ->, cyan]       (-2, 0) -- (0, 0);
+        \draw[ultra thick, ->, darkgreen]    (0, 0) -- (2, 0);
+        \draw[ultra thick, ->, orange] (2, 0) -- (4, 0);
 
         \node[anchor=south] at (-4,0) {$-\infty$};
         \node[anchor=south] at (-2,0) {$-1$};
diff --git a/buch/papers/ellfilter/tikz/arccos2.tikz.tex b/buch/papers/ellfilter/tikz/arccos2.tikz.tex
index 3fc3cc6..2cec75f 100644
--- a/buch/papers/ellfilter/tikz/arccos2.tikz.tex
+++ b/buch/papers/ellfilter/tikz/arccos2.tikz.tex
@@ -2,34 +2,45 @@
 
     \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}}
+    \tikzstyle{dot} = [fill, circle, inner sep =0, minimum height=0.1cm]
 
-    \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}[xscale=0.75]
+
+        \draw[gray, ->] (0,-1) -- (0,2) node[anchor=south]{$\mathrm{Im}~z_1$};
+        \draw[gray, ->] (-2,0) -- (9,0) 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]{$\infty$};
+            \draw[->, ultra thick, blue] (8, 1.5) --  node[align=center]{Sperrbereich} (8,0);
+            \draw[->, ultra thick, cyan] (8, 0) -- node[yshift=-0.5cm]{Durchlassbereich}(4,0);
+            \draw[->, ultra thick, darkgreen] (4, 0) -- node[yshift=-0.5cm]{Durchlassbereich} (0,0);
+            \draw[->, ultra thick, orange] (0, 0) --  node[align=center]{Sperrbereich} (0,1.5);
+
+            \node[anchor=north east] at (8, 1.5) {$-\infty$};
+            \draw (8, 0) node[dot]{} node[anchor=south east] {$1$};
+            \draw (6, 0) node[dot]{} node[anchor=south] {$-1$};
+            \draw (4, 0) node[dot]{} node[anchor=south] {$1$};
+            \draw (2, 0) node[dot]{} node[anchor=south] {$-1$};
+            \draw (0, 0) node[dot]{} node[anchor=south west] {$1$};
+            \node[anchor=north west] at (0, 1.5){$\infty$};
 
+            \node at(4,1) {$N = 4$};
 
-            \node[zero] at (-7,0) {};
-            \node[zero] at (-5,0) {};
-            \node[zero] at (-3,0) {};
+            % \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 (-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) {$2\pi$};
@@ -37,9 +48,29 @@
         \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) {$-\infty$};
         \node[gray, anchor=east] at (0, 1.5) {$\infty$};
 
     \end{scope}
 
+    \node[zero] at (6.5,2) (n) {};
+    \node[anchor=west] at (n.east) {Nullstelle};
+
+    \begin{scope}[xshift=2.75cm, yshift=-2cm]
+
+        \draw[gray, ->] (-5,0) -- (5,0) node[anchor=west]{$w$};
+
+        \draw[ultra thick, ->, blue]      (-4, 0) -- (-2, 0);
+        \draw[ultra thick, ->, cyan]       (-2, 0) -- (0, 0);
+        \draw[ultra thick, ->, darkgreen]    (0, 0) -- (2, 0);
+        \draw[ultra thick, ->, orange] (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/cd.tikz.tex b/buch/papers/ellfilter/tikz/cd.tikz.tex
index 7155a85..0cf2417 100644
--- a/buch/papers/ellfilter/tikz/cd.tikz.tex
+++ b/buch/papers/ellfilter/tikz/cd.tikz.tex
@@ -4,7 +4,7 @@
 
     \tikzset{pole/.style={cross out, draw=black, minimum size=(0.15cm-\pgflinewidth), inner sep=0pt, outer sep=0pt}}
 
-    \begin{scope}[xscale=1, yscale=2]
+    \begin{scope}[xscale=0.9, yscale=1.8]
 
         \draw[gray, ->] (0,-1.5) -- (0,1.5) node[anchor=south]{$\mathrm{Im}~z$};
         \draw[gray, ->] (-5,0) -- (5,0) node[anchor=west]{$\mathrm{Re}~z$};
@@ -22,32 +22,35 @@
 
                 \fill[yellow!30] (0,0) rectangle (1, 0.5);
 
+                \foreach \i in {-2,...,1} {
+                    \foreach \j in {-2,...,1} {
+                        \begin{scope}[xshift=\i*4cm, yshift=\j*1cm]
+                            \draw[->, orange!50] (0, 0) -- (0,0.5);
+                            \draw[->, darkgreen!50] (1, 0) -- (0,0);
+                            \draw[->, cyan!50] (2, 0) -- (1,0);
+                            \draw[->, blue!50] (2,0.5) -- (2, 0);
+                            \draw[->, purple!50] (1, 0.5) -- (2,0.5);
+                            \draw[->, red!50] (0, 0.5) -- (1,0.5);
+                            \draw[->, orange!50] (0,1) -- (0,0.5);
+                            \draw[->, blue!50] (2,0.5) -- (2, 1);
+                            \draw[->, purple!50] (3, 0.5) -- (2,0.5);
+                            \draw[->, red!50] (4, 0.5) -- (3,0.5);
+                            \draw[->, cyan!50] (2, 0) -- (3,0);
+                            \draw[->, darkgreen!50] (3, 0) -- (4,0);
+                        \end{scope}
+                    }
+                }
 
-                \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);
-
-
+                \draw[ultra thick, ->, orange] (0, 0) -- (0,0.5);
+                \draw[ultra thick, ->, darkgreen] (1, 0) -- (0,0);
+                \draw[ultra thick, ->, cyan] (2, 0) -- (1,0);
+                \draw[ultra thick, ->, blue] (2,0.5) -- (2, 0);
+                \draw[ultra thick, ->, purple] (1, 0.5) -- (2,0.5);
+                \draw[ultra thick, ->, red] (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) {};
@@ -63,16 +66,21 @@
 
     \end{scope}
 
-    \begin{scope}[yshift=-3.5cm, xscale=0.75]
+    \node[zero] at (4,3) (n) {};
+    \node[anchor=west] at (n.east) {Nullstelle};
+    \node[pole, below=0.25cm of n] (n) {};
+    \node[anchor=west] at (n.east) {Polstelle};
+
+    \begin{scope}[yshift=-4cm, 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);
+        \draw[ultra thick, ->, purple] (-5, 0) -- (-3, 0);
+        \draw[ultra thick, ->, blue]      (-3, 0) -- (-2, 0);
+        \draw[ultra thick, ->, cyan]       (-2, 0) -- (0, 0);
+        \draw[ultra thick, ->, darkgreen]    (0, 0) -- (2, 0);
+        \draw[ultra thick, ->, orange] (2, 0) -- (3, 0);
+        \draw[ultra thick, ->, red] (3, 0) -- (5, 0);
 
         \node[anchor=south] at (-5,0) {$-\infty$};
         \node[anchor=south] at (-3,0) {$-1/k$};
diff --git a/buch/papers/ellfilter/tikz/cd2.tikz.tex b/buch/papers/ellfilter/tikz/cd2.tikz.tex
index 0743f7d..d4187c4 100644
--- a/buch/papers/ellfilter/tikz/cd2.tikz.tex
+++ b/buch/papers/ellfilter/tikz/cd2.tikz.tex
@@ -5,9 +5,9 @@
 
     \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]
+    \begin{scope}[xscale=1.25, yscale=3.5]
 
-        \draw[gray, ->] (0,-0.75) -- (0,1.25) node[anchor=south]{$\mathrm{Im}~z_1$};
+        \draw[gray, ->] (0,-0.55) -- (0,1.05) 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 $K_1$};
@@ -35,18 +35,30 @@
             % \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);
+            % \fill[yellow!30] (0,0) rectangle (1, 0.1);
             \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]{$NK$}
+                (5,0.5) node(t_k_unten){} -- node[above, yshift=0.1cm]{$NK_1$}
                 (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){};
 
+
+            \draw[ultra thick, ->, darkgreen] (5, 0) -- node[yshift=-0.5cm]{Durchlassbereich} (0,0);
+            \draw[ultra thick, ->, orange] (-0, 0) --  node[align=center]{Übergangs-\\berech} (0,0.5);
+            \draw[ultra thick, ->, red] (0,0.5) -- node[align=center, yshift=0.7cm]{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 $1/k$};
+
             \foreach \i in {-2,...,1} {
                 \foreach \j in {-2,...,1} {
                     \begin{scope}[xshift=\i*4cm, yshift=\j*1cm]
@@ -60,24 +72,22 @@
                 }
             }
 
+        \end{scope}
 
+    \end{scope}
 
+    \begin{scope}[xshift=1cm , yshift=-3cm, xscale=0.75]
 
-            \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 $1/k$};
-
-
+        \draw[gray, ->] (-1,0) -- (6,0) node[anchor=west]{$w$};
 
-        \end{scope}
+        \draw[ultra thick, ->, darkgreen]    (0, 0) -- (2, 0);
+        \draw[ultra thick, ->, orange] (2, 0) -- (3, 0);
+        \draw[ultra thick, ->, red] (3, 0) -- (5, 0);
 
+        \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}
 
diff --git a/buch/papers/ellfilter/tikz/cd3.tikz.tex b/buch/papers/ellfilter/tikz/cd3.tikz.tex
new file mode 100644
index 0000000..ae18519
--- /dev/null
+++ b/buch/papers/ellfilter/tikz/cd3.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]
+    \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,-0.55) -- (0,1.05) node[anchor=south]{$\mathrm{Im}$};
+        \draw[gray, ->] (-1.5,0) -- (6,0) node[anchor=west]{$\mathrm{Re}$};
+
+        % \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);
+
+
+            % \draw[decorate,decoration={brace,amplitude=3pt,mirror}, yshift=0.05cm]
+            %     (5,0.5) node(t_k_unten){} -- node[above, yshift=0.1cm]{$NK_1$}
+            %     (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}
+                }
+            }
+
+
+
+            \onslide<2->{
+            \draw[ultra thick, ->, darkgreen] (5, 0) -- node[yshift=-0.4cm]{Durchlassbereich} (0,0);
+            \draw[ultra thick, ->, orange] (-0, 0) --  node[align=center]{Übergangs-\\berech} (0,0.5);
+            \draw[ultra thick, ->, red] (0,0.5) -- node[align=center, yshift=0.4cm]{Sperrbereich} (5, 0.5);
+            \node[] at (2.5, 0.25) {\small $N=5$};
+            }
+            \onslide<1->{
+            \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/elliptic_transform1.tikz.tex b/buch/papers/ellfilter/tikz/elliptic_transform1.tikz.tex
new file mode 100644
index 0000000..2a36ee0
--- /dev/null
+++ b/buch/papers/ellfilter/tikz/elliptic_transform1.tikz.tex
@@ -0,0 +1,76 @@
+\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, minimum size=(0.15cm-\pgflinewidth), inner sep=0pt, outer sep=0pt}}
+
+    \begin{scope}[xscale=1, yscale=1.5]
+
+    \begin{scope}[]
+
+        \fill[orange!25] (0,0) rectangle (1.5, 0.75);
+        \fill[yellow!50] (0,0) rectangle (0.5, 0.25);
+
+        \draw[gray, ->] (0,-0.75) -- (0,1.25) node[anchor=south]{$\mathrm{Im}~z$};
+        \draw[gray, ->] (-1.75,0) -- (1.75,0) node[anchor=west]{$\mathrm{Re}~z$};
+
+        \draw[gray] ( 0.5,0) +(0,0.05) -- +(0, -0.05) node[inner sep=0, anchor=north] {\small $K$};
+        \draw[gray]  (0, 0.25) +(0.05, 0) -- +(-0.05, 0) node[inner sep=0, anchor=east]{\small $jK^\prime$};
+
+        % \draw[gray] ( 1.5,0) +(0,0.05) -- +(0, -0.05) node[inner sep=0, anchor=north] {\small $K_1$};
+        % \draw[gray]  (0, 0.75) +(0.05, 0) -- +(-0.05, 0) node[inner sep=0, anchor=east]{\small $jK_1^\prime$};
+
+        \clip(-1.6,-0.6) rectangle (1.6,1.6);
+        \begin{scope}[xscale=0.5, yscale=0.25, blue]
+            \foreach \i in {-1,...,1} {
+                \foreach \j in {-1,...,2} {
+                    \begin{scope}[xshift=\i*2cm, yshift=\j*2cm]
+                        \node[zero] at ( 1, 0) {};
+                        \node[zero] at ( -1, 0) {};
+                        \node[pole] at ( 1,1) {};
+                        \node[pole] at ( -1,1) {};
+                    \end{scope}
+                }
+            }
+        \end{scope}
+
+        \node at (0,2) {$\cd \left(N~K_1~z , k_1 \right)$};
+        \node at (0,2) {$w= \cd(z K, k)$};
+
+        \draw[scale=0.2, domain=0.02:5,  variable=\x, red] plot ({\x1+3}, {1/\x+2});
+
+    \end{scope}
+
+    \begin{scope}[xshift=5cm]
+
+        \fill[orange!50] (0,0) rectangle (1.5, 0.75);
+        \fill[yellow!25] (0,0) rectangle (0.5, 0.25);
+
+        \draw[gray, ->] (0,-0.75) -- (0,1.25) node[anchor=south]{$\mathrm{Im}~z$};
+        \draw[gray, ->] (-1.75,0) -- (1.75,0) node[anchor=west]{$\mathrm{Re}~z$};
+
+        % \draw[gray] ( 0.5,0) +(0,0.05) -- +(0, -0.05) node[inner sep=0, anchor=north] {\small $K$};
+        % \draw[gray]  (0, 0.25) +(0.05, 0) -- +(-0.05, 0) node[inner sep=0, anchor=east]{\small $jK^\prime$};
+
+        \draw[gray] ( 0.5,0) +(0,0.05) -- +(0, -0.05) node[inner sep=0, anchor=north] {\small $K_1$};
+        \draw[gray]  (0, 0.75) +(0.05, 0) -- +(-0.05, 0) node[inner sep=0, anchor=east]{\small $jK_1^\prime$};
+
+        \clip(-1.6,-0.6) rectangle (1.6,1.6);
+        \begin{scope}[xscale=0.5, yscale=0.75, red]
+            \foreach \i in {-1,...,1} {
+                \foreach \j in {-1,...,0} {
+                    \begin{scope}[xshift=\i*2cm, yshift=\j*2cm]
+                        \node[zero] at ( 1, 0) {};
+                        \node[zero] at ( -1, 0) {};
+                        \node[pole] at ( 1,1) {};
+                        \node[pole] at ( -1,1) {};
+                    \end{scope}
+                }
+            }
+        \end{scope}
+
+    \end{scope}
+
+\end{scope}
+
+\end{tikzpicture}
diff --git a/buch/papers/ellfilter/tikz/elliptic_transform2.tikz.tex b/buch/papers/ellfilter/tikz/elliptic_transform2.tikz.tex
new file mode 100644
index 0000000..20c2d82
--- /dev/null
+++ b/buch/papers/ellfilter/tikz/elliptic_transform2.tikz.tex
@@ -0,0 +1,75 @@
+
+\def\d{0.2}
+\def\n{3}
+\def\nn{2}
+\def\a{2.5}
+
+\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, minimum size=(0.15cm-\pgflinewidth), inner sep=0pt, outer sep=0pt}}
+
+    \begin{scope}[xscale=3, yscale=3]
+
+    \begin{scope}[]
+        % \onslide<4->{
+        \fill[orange!30, scale=1.735] (0,0) rectangle (\d*\a+0.5, \d/\a+0.5);
+        % }
+        % \onslide<2->{
+        \fill[yellow!30] (0,0) rectangle (\d*\a+0.5, \d/\a+0.5);
+        % }
+
+        \begin{scope}[]
+            \clip(0,0) rectangle (2,1.25);
+            \draw[thick, scale=1, domain=0.1:10,  variable=\x, smooth, samples=200] plot ({\d*\x1+0.5}, {\d/\x+0.5});
+            \node at(1.25,0.7) {$K + jK^\prime$ Ortskurve};
+        \end{scope}
+
+        % \onslide<2->{
+        \begin{scope}[blue]
+            \draw[] (0,0) rectangle (\d*\a+0.5, \d/\a+0.5);
+
+
+            \node[pole] at ( \d*\a+0.5, \d/\a+0.5) {};
+            \node[zero] at ( \d*\a+0.5, 0) {};
+
+            \draw[] ( \d*\a+0.5,0)  node[anchor=north] {\small $K$};
+            \draw[]  (0, \d/\a+0.5)  node[anchor=east]{\small $jK^\prime$};
+
+            % \onslide<3->{
+
+            \foreach \i in {1,...,\nn} {
+                \draw[gray, dotted] (\i*\d*\a/\n+\i*0.5/\n, 0) -- (\i*\d*\a/\n+\i*0.5/\n, \d/\a+0.5);
+            }
+
+            \node[dot, gray] at (\d*\a/\n+0.5/\n, \d/\a+0.5) {};
+            \node[above] at (0.5*\d*\a/\n+0.5*0.5/\n, \d/\a+0.5) {\small $K/N$};
+            % }
+        \end{scope}
+        % }
+
+        % \onslide<4->{
+        \begin{scope}[scale=1.735, red]
+            \draw (0,0) rectangle (\d*\a/\n+0.5/\n, \d/\a+0.5);
+            \draw[gray] (0,0) -- (\d*\a/\n+0.5/\n, \d/\a+0.5);
+
+            \node[pole] at ( \d*\a/\n+0.5/\n, \d/\a+0.5) {};
+            \node[zero] at ( \d*\a/\n+0.5/\n, 0) {};
+
+
+            \draw[] ( \d*\a/\n+0.5/\n,0)  node[anchor=north] {\small $K_1$};
+            \draw[]  (0, \d/\a+0.5)  node[anchor=east]{\small $jK_1^\prime$};
+
+        \end{scope}
+        % }
+
+        \draw[gray, ->] (0,-0.25) -- (0,1.25) node[anchor=south]{$\mathrm{Im}$};
+        \draw[gray, ->] (-0.25,0) -- (2,0) node[anchor=west]{$\mathrm{Re}$};
+
+    \end{scope}
+
+\end{scope}
+
+\end{tikzpicture}
diff --git a/buch/papers/ellfilter/tikz/filter.tikz.tex b/buch/papers/ellfilter/tikz/filter.tikz.tex
new file mode 100644
index 0000000..769602a
--- /dev/null
+++ b/buch/papers/ellfilter/tikz/filter.tikz.tex
@@ -0,0 +1,32 @@
+\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=3, yscale=2.5]
+
+        \fill[darkgreen!15] (0,0) rectangle  (1,1);
+        \node[darkgreen] at (0.5,0.5) {Durchlassbereich};
+        \fill[orange!15] (1,0) rectangle  (2.5,1);
+        \node[orange] at (1.75,0.5) {Sperrbereich};
+
+        \draw[gray, ->] (0,0) -- (0,1.25) node[anchor=south]{$|H(\Omega)|$};
+        \draw[gray, ->] (0,0) -- (2.75,0) node[anchor=west]{$\Omega$};
+
+        \draw[dashed] (0,0.707) node[left] {$\sqrt{\frac{1}{1+\varepsilon^2}}$} -| (1,0) node[below] {$\Omega_p$};
+        \draw[dashed] (0,0.707) node[left] {$\sqrt{\frac{1}{1+\varepsilon^2}}$} -| (1,0) node[below] {$\Omega_p$};
+
+        \node[left] at(0,1) {$1$};
+
+        \draw[red, thick] (0,1) -- (1,1) -- (1,0) -- (2.5,0);
+
+        \node[anchor=north, red] at (0.5,1) {Ideal};
+
+        \draw[thick, domain=0:2.5,  variable=\x, smooth, samples=200] plot
+        ({\x}, {sqrt(abs(1/ (1 + \x^10)))});
+        \node[anchor=south] at (0.5,1) {Butterworth ($N=5$)};
+
+    \end{scope}
+
+\end{tikzpicture}
diff --git a/buch/papers/ellfilter/tikz/sn.tikz.tex b/buch/papers/ellfilter/tikz/sn.tikz.tex
index 87c63c0..0546fda 100644
--- a/buch/papers/ellfilter/tikz/sn.tikz.tex
+++ b/buch/papers/ellfilter/tikz/sn.tikz.tex
@@ -4,7 +4,7 @@
 
     \tikzset{pole/.style={cross out, draw=black, minimum size=(0.15cm-\pgflinewidth), inner sep=0pt, outer sep=0pt}}
 
-    \begin{scope}[xscale=1, yscale=2]
+    \begin{scope}[xscale=0.9, yscale=1.8]
 
         \draw[gray, ->] (0,-1.5) -- (0,1.5) node[anchor=south]{$\mathrm{Im}~z$};
         \draw[gray, ->] (-5,0) -- (5,0) node[anchor=west]{$\mathrm{Re}~z$};
@@ -17,35 +17,45 @@
 
             \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[<-, blue!50] (0, 0) -- (0,0.5);
+                            \draw[<-, cyan!50] (1, 0) -- (0,0);
+                            \draw[<-, darkgreen!50] (2, 0) -- (1,0);
+                            \draw[<-, orange!50] (2,0.5) -- (2, 0);
+                            \draw[<-, red!50] (1, 0.5) -- (2,0.5);
+                            \draw[<-, purple!50] (0, 0.5) -- (1,0.5);
+                            \draw[<-, blue!50] (0,1) -- (0,0.5);
+                            \draw[<-, orange!50] (2,0.5) -- (2, 1);
+                            \draw[<-, red!50] (3, 0.5) -- (2,0.5);
+                            \draw[<-, purple!50] (4, 0.5) -- (3,0.5);
+                            \draw[<-, darkgreen!50] (2, 0) -- (3,0);
+                            \draw[<-, cyan!50] (3, 0) -- (4,0);
+                        \end{scope}
+                    }
+                }
+
+                % \pause
+                \draw[ultra thick, <-, darkgreen] (2, 0) -- (1,0);
+                % \pause
+                \draw[ultra thick, <-, orange] (2,0.5) -- (2, 0);
+                % \pause
+                \draw[ultra thick, <-, red] (1, 0.5) -- (2,0.5);
+                % \pause
+                \draw[ultra thick, <-, blue] (0, 0) -- (0,0.5);
+                \draw[ultra thick, <-, purple] (0, 0.5) -- (1,0.5);
+                \draw[ultra thick, <-, cyan] (1, 0) -- (0,0);
+                % \pause
 
 
                 \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}
                     }
                 }
@@ -57,20 +67,23 @@
         \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]
+    \node[zero] at (4,3) (n) {};
+    \node[anchor=west] at (n.east) {Nullstelle};
+    \node[pole, below=0.25cm of n] (n) {};
+    \node[anchor=west] at (n.east) {Polstelle};
+
+    \begin{scope}[yshift=-4cm, 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);
+        \draw[ultra thick, ->, purple] (-5, 0) -- (-3, 0);
+        \draw[ultra thick, ->, blue]      (-3, 0) -- (-2, 0);
+        \draw[ultra thick, ->, cyan]       (-2, 0) -- (0, 0);
+        \draw[ultra thick, ->, darkgreen]    (0, 0) -- (2, 0);
+        \draw[ultra thick, ->, orange] (2, 0) -- (3, 0);
+        \draw[ultra thick, ->, red] (3, 0) -- (5, 0);
 
         \node[anchor=south] at (-5,0) {$-\infty$};
         \node[anchor=south] at (-3,0) {$-1/k$};
diff --git a/buch/papers/ellfilter/tschebyscheff.tex b/buch/papers/ellfilter/tschebyscheff.tex
index 7d426b6..639c87c 100644
--- a/buch/papers/ellfilter/tschebyscheff.tex
+++ b/buch/papers/ellfilter/tschebyscheff.tex
@@ -1,9 +1,8 @@
 \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:
+Als Einstieg betrachten wir das Tschebyscheff-Filter, welches sehr verwandt ist mit dem elliptischen Filter.
+Genauer ausgedrückt erhält man die Tschebyscheff-1 und -2 Filter bei Grenzwerten von Parametern beim elliptischen Filter.
+Der Name des Filters deutet schon an, dass die Tschebyscheff-Polynome $T_N$ (siehe auch Kapitel \label{buch:polynome:section:tschebyscheff}) für das Filter relevant sind:
 \begin{align}
     T_{0}(x)&=1\\
     T_{1}(x)&=x\\
@@ -16,7 +15,7 @@ Bemerkenswert ist, dass die Polynome im Intervall $[-1, 1]$ mit der trigonometri
     T_N(w) &= \cos \left( N \cos^{-1}(w) \right) \\
            &= \cos \left(N~z \right), \quad w= \cos(z)
 \end{align}
-übereinstimmt.
+übereinstimmen.
 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}
@@ -28,7 +27,7 @@ Abbildung \ref{ellfilter:fig:chebychef_polynomials} zeigt einige Tschebyscheff-P
 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.
+Wenn wir die Tschebyscheff-Polynome quadrieren, passen sie perfekt in die Forderungen für Filterfunktionen, wie es Abbildung \ref{ellfiter:fig:chebychef} demonstriert.
 \begin{figure}
     \centering
     \input{papers/ellfilter/python/F_N_chebychev.pgf}
@@ -36,12 +35,11 @@ Wenn wir die Tschebyscheff-Polynome quadrieren, passen sie perfekt in die Voraus
     \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.
+Die genauere Betrachtung wird uns 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:
+Starten wir mit der Funktion, die in \eqref{ellfilter:eq:chebychef_polynomials} als erstes auf $w$ angewendet wird, dem Arcuscosinus.
+Die invertierte Funktion des Kosinus kann als bestimmtes Integral dargestellt werden:
 \begin{align}
     \cos^{-1}(x)
     &=
@@ -63,9 +61,9 @@ Die invertierte Funktion des Kosinus kann als definites Integral dargestellt wer
         }
     }
     ~dz
-    + \frac{\pi}{2}
+    + \frac{\pi}{2}.
 \end{align}
-Der Integrand oder auch die Ableitung
+Der Integrand oder auch die Ableitung von $\cos^{-1}(x)$
 \begin{equation}
     \frac{
         -1
@@ -75,59 +73,37 @@ Der Integrand oder auch die Ableitung
         }
     }
 \end{equation}
-bestimmt dabei die Richtung, in der die Funktion verläuft.
+bestimmt dabei die Richtung, in welche 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.
+Abbildung \ref{ellfilter:fig:arccos} zeigt den Arcuscosinus 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}
+Wegen der Periodizität des Kosinus ist auch der Arcuscosinus $2\pi$-periodisch.
+Das Einzeichnen von Pol- und Nullstellen ist hilfreich für die Betrachtung der Funktion.
+
 
-Die Tschebyscheff-Polynome skalieren diese Nullstellen mit dem Ordnungsfaktor $N$, wie dargestellt in Abbildung \ref{ellfilter:fig:arccos2}.
+In \eqref{ellfilter:eq:chebychef_polynomials} wird $z$ mit dem Ordnungsfaktor $N$ multipliziert und durch die Kosinusfunktion zurück transformiert.
+Die Skalierung hat zur folge, dass bei der Rücktransformation durch den Kosinus mehrere Nullstellen durchlaufen werden.
+Somit passiert $\cos( N~\cos^{-1}(w))$ im Intervall $[-1, 1]$ $N$ Nullstellen, 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.
+        Die eingefärbten Pfade sind Verläufe von $w\in(-\infty, \infty)$ für $N = 4$.
+        Je grösser die Ordnung $N$ gewählt wird, desto mehr Nullstellen werden passiert die zu Equirippel-Verhalten führen.
+        Die vertikalen Segmente der Funktion sorgen für das Ansteigen der Funktion gegen $\infty$ nach der Grenzfrequenz.
+        Die eingezeichneten Nullstellen sind vom zurücktransformierenden Kosinus.
     }
     \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.
+Durch die spezielle Anordnung der Nullstellen hat die Funktion auf der reellen Achse Equirippel-Verhalten und ist dennoch ein Polynom, was sich perfekt für linear Filter eignet.
+Equirippel bedeutet, dass alle lokalen Maxima der Betragsfunktion gleich gross sind.