18 files changed, 816 insertions, 815 deletions

diff --git a/buch/papers/ellfilter/einleitung.tex b/buch/papers/ellfilter/einleitung.tex
index ae7127f..05061d1 100644
--- a/buch/papers/ellfilter/einleitung.tex
+++ b/buch/papers/ellfilter/einleitung.tex
@@ -1,17 +1,17 @@
 \section{Einleitung}
 
-Filter sind womöglich eines der wichtigsten Elementen in der Signalverarbeitung und finden Anwendungen in der digitalen und analogen Elektrotechnik.
+Filter sind womöglich eines der wichtigsten Elemente 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.
+Elektronische Schaltungen mit linearen Bauelementen wie Kondensatoren, Spulen und Widerständen führen immer zu linearen zeitinvarianten Systemen (LTI-System von englisch \textit{time-invariant system}).
+Durch die Linearität werden beim Filtern keine neuen Frequenzanteile erzeugt, was es erlaubt, einen Frequenzanteil eines Signals verzerrungsfrei herauszufiltern.
+Diese Eigenschaft macht es sinnvoll, lineare Filter im Frequenzbereich zu beschreiben.
+Die Übertragungsfunktion $H(\Omega)$ eines linearen Filters im Frequenzbereich 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.
+Die Polynome haben dabei immer reelle oder komplexkonjugierte Nullstellen.
 
-Ein breit angewendeter Filtertyp ist das Tiefpassfilter, welches beabsichtigt alle Frequenzen eines Signals oberhalb der Grenzfrequenz $\Omega_p$ auszulöschen.
+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.
+Aus dem Tiefpassfilter können dann durch Transformationen auch Hochpassfilter, Bandpassfilter und Bandsperren realisiert werden.
 Ein solches Filter hat idealerweise die Frequenzantwort
 \begin{equation}
     H(\Omega) =
@@ -32,7 +32,7 @@ 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}
+\begin{equation} \label{ellfilter:eq:quadratic_transfer}
     | 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.
@@ -40,8 +40,9 @@ Des weiteren müssen alle Nullstellen und Pole von $F_N$ auf der linken Halbeben
 $w$ 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.
+Je höher $N$ gewählt wird, desto steiler ist der Übergang im Sperrbereich.
+Grössere $N$ 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}
@@ -62,12 +63,15 @@ Eine Reihe von rationalen Funktionen können für $F_N$ eingesetzt werden, um Ti
 \end{align}
 Mit der Ausnahme vom Butterworth-Filter sind alle Filter nach speziellen Funktionen benannt.
 Alle diese Filter sind optimal hinsichtlich einer Eigenschaft.
+Es scheint so, als sind gewisse Eigenschaften dieser speziellen Funktionen verantwortlich für die Optimalität dieser Filter.
 Das Butterworth-Filter, zum Beispiel, ist maximal flach im Durchlassbereich.
-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.
+In vielen Anwendung sind Filter mit einem steilen Übergang gewünscht.
+Da es technisch nicht möglich ist, mit einer rationalen Funktion mit begrenzter Anzahl Pole eine steile Flanke zu erreichen, während der Durchlass- und Sperrbereich flach und monoton sind, gibt es Filtertypen, die absichtlich Welligkeiten in der Frequenzantwort aufweisen.
+Besonders effizient sind Filter mit Equiripple-Verhalten, wessen Welligkeit optimal definiert wird für eine maximal steile Flanke, während die maximale Abweichung zum idealen Filter begrenzt ist.
+Die Welligkeit beansprucht dabei einen begrenzen Verstärkungsintervall und nützt diesen vollständig aus, indem sie periodisch die Grenzen des Intervalls berührt.
+Das Tschebyscheff-1 Filter, zum Beispiel, hat Equiripple-Verhalten im Durchlassbereich, währendem es im Sperrbereich monoton abfallend ist.
+Beim Tschebyscheff-2 Filter ist es umgekehrt.
 
 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.
+Es hat Equiripple-Verhalten im Durchlass und Sperrbereich und hat dadurch 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 67bcca0..651d6bc 100644
--- a/buch/papers/ellfilter/elliptic.tex
+++ b/buch/papers/ellfilter/elliptic.tex
@@ -1,14 +1,17 @@
-\section{Elliptische rationale Funktionen}
+\section{Rationale elliptische Funktionen}
 
-Kommen wir nun zum eigentlichen Teil dieses Papers, den elliptischen rationalen Funktionen \cite{ellfilter:bib:orfanidis}
+Kommen wir nun zum eigentlichen Teil dieses Papers, den rationalen elliptischen 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) \label{ellfilter:eq:elliptic}\\
+    R_N(w, \xi) &= \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}
 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.
+Wie bei den Tschebyscheff-Polynomen ist die Formel mit speziellen Funktionen geschrieben.
+Es kann jedoch gezeigt werden, dass es sich tatsächlich um rationale Funktionen handelt, wie es für ein lineares Filter vorausgesetzt wird.
+Die elliptischen Funktionen werden also genau so eingesetzt, dass die resultierenden Nullstellen und Pole eine rationale Funktion ergeben.
+Anstelle des Kosinus bei den Tschebyscheff-Polynomen kommt hier die $\cd$-Funktion zum Einsatz.
+Die Ordnungszahl $N$ kommt auch als Faktor vor.
 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.
 
@@ -24,12 +27,12 @@ Die $\cd^{-1}(w, k)$-Funktion ist um $K$ verschoben zur $\sn^{-1}(w, k)$-Funktio
     \label{ellfilter:fig:cd}
 \end{figure}
 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.
+Die Idee des elliptischen Filter ist es, diese zwei Equiripple-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/cd2.tikz.tex}
     \caption{
-        $z_1=N\frac{K_1}{K}\cd^{-1}(w, k)$-Ebene der elliptischen rationalen Funktionen.
+        $z_1=N\frac{K_1}{K}\cd^{-1}(w, k)$-Ebene der rationalen elliptischen Funktionen.
         Je grösser die Ordnung $N$ gewählt wird, desto mehr Nullstellen werden passiert.
         Als Vereinfachung ist die Funktion nur für $w>0$ dargestellt.
     }
@@ -37,13 +40,10 @@ Die Idee des elliptischen Filter ist es, diese zwei Equirippel-Zonen abzufahren,
 \end{figure}
 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.
+Statt dass $z_1$ für alle $w>1$ in die imaginäre Richtung geht, bewegen wir uns im Sperrbereich wieder in reeller Richtung, wo Pole und Punkte mit $\pm 1/k$ 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.
+% Falls es möglich ist diese Werte abzufahren im Stil der Tschebyscheff-Polynome, kann ein Filter gebaut werden, dass Equiripple-Verhalten im Durchlass- und Sperrbereich aufweist.
+Abbildung \ref{ellfilter:fig:elliptic_freq} zeigt eine rationale elliptische Funktion und die Frequenzantwort des daraus resultierenden Filters.
 \begin{figure}
     \centering
     \input{papers/ellfilter/python/elliptic.pgf}
@@ -51,15 +51,22 @@ Abbildung \ref{ellfilter:fig:elliptic_freq} zeigt eine elliptisch rationale Funk
     \label{ellfilter:fig:elliptic_freq}
 \end{figure}
 
+Da sich die Funktion im Übergangsbereich nur zur nächsten Reihe von Polstellen bewegt, ist der Übergangsbereich monoton steigend.
+Theoretisch könnte eine gleiches Durchlass- und Sperrbereichsverhalten erreicht werden, wenn die Funktion auf eine andere Reihe ansteigen würde, z.B. $\mathrm{Im(z) = 3K^\prime}$.
+Dies würde jedoch zu Oszillationen zwischen $1$ und $1/k$ im Übergangsbereich führen.
+
 \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.
+In der reellen Richtung müssen sich die Periodizitäten $K$ und $K_1$ um den Faktor $N$ unterscheiden, während die imaginäre 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/k.pgf}
-    \caption{Die Periodizitäten in realer und imaginärer Richtung in Abhängigkeit vom elliptischen Modul $k$.}
+    \caption{
+        Die Periodizitäten in realer und imaginärer Richtung in Abhängigkeit vom elliptischen Modul $k$.
+        In der rechten Grafik sind $K$ und $K^\prime$ gegenübergestellt, wobei alle möglichen Kombinationen auf der eingezeichneten Ortskurve liegen.
+    }
     \label{ellfilter:fig:kprime}
 \end{figure}
 $K$ 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.
@@ -75,26 +82,48 @@ Algebraisch kann so die Gradgleichung
     N \frac{K^\prime}{K} = \frac{K^\prime_1}{K_1}
 \end{equation}
 aufgestellt werden, dessen Lösung ist gegeben durch
-\begin{equation} %TODO check
+\begin{equation}\label{ellfilter:eq:degeqsol}
 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.
+Die Herleitung ist sehr umfangreich 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}
+\subsection{Berechnung der rationalen Funktion}
 
-\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 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.
+$k_1$ muss jedoch gar nicht berechnet werden, um $R_N$ in der Form einer rationale Funktion erhalten.
+Die Ordnung $N$ und der Parameter $k$ können frei gewählt werden.
+% $k_1$ muss dann mit \eqref{ellfilter:eq:degeqsol} oder mit numerischen Methoden berechnet werden.
+Je kleiner $k$ gewählt wird, desto grösser wird die Dämpfung des Filters im Sperrbereich im Verhältnis zum Durchlassbereich.
+Allerdings verliert das Filter dabei auch an Steilheit.
+Wenn $k$ und $N$ bekannt sind, können die Position der Pol- und Nullstellen $p_i$ und $n_i$ in einem Raster konstruiert werden, wie dargestellt in Abbildung \ref{ellfilter:fig:pn}.
+\begin{figure}
+    \centering
+    \input{papers/ellfilter/tikz/pn.tikz.tex}
+    \caption{
+        Pole und Nullstellen in der $z = \cd^{-1}(w, k)$-Ebene für die Rücktransformation zur einer rationalen Funktion.
+    }
+    \label{ellfilter:fig:pn}
+\end{figure}
+Dabei muss aufgepasst werden, dass insgesamt nur $N$ Nullstellen und $N$ Pole gesetzt werden, da bei der Transformation mit dem $\cd$ mehrere Werte auf einen abgebildet werden und mehrfache Pole und Nullstellen nicht erwünscht sind.
+Wegen der Periodizität sind diese in der komplexen $z$-Ebene linear angeordnet:
+\begin{align}
+    n_i(k) &= K\frac{2i+1}{N} \\
+    p_i(k) &= n_i + jK^\prime.
+\end{align}
+Durch das Rücktransformieren mit der $\cd$-Funktion gelangt man schlussendlich zu der rationalen Funktion
+\begin{equation}
+    R_N(w, k) = r_0 \prod_{i=1}^N \frac{w - \cd \big(n_i(k), k \big)}{w - \cd \big(p_i(k), k \big)},
+\end{equation}
+wobei $r_0$ so gewählt werden muss, dass $R_N(w, k) = 1$.
 
+\section{Elliptisches Filter}
 
+Um ein elliptisches Filter auszulegen, werden aber nicht die Pol- und Nullstellen der rationalen Funktion gebraucht, sondern diejenigen der Übertragungsfunktion $H(s)$ der komplexen Frequenz $s = j\Omega + \sigma$.
+Der Bezug zum quadratischen Amplitudengang \eqref{ellfilter:eq:quadratic_transfer} ist dabei
+\begin{equation}
+    |H(\Omega)|^2 = H(s) H(s^*),
+\end{equation}
+wobei $*$ die komplexe Konjugation kennzeichnet.
+Die genaue Berechnung geht einiges tiefer in die Filtertheorie, und verlässt das Gebiet der speziellen Funktionen.
+Der interessierte Leser wird auf \cite[Kapitel~5]{ellfilter:bib:orfanidis} verwiesen.
diff --git a/buch/papers/ellfilter/jacobi.tex b/buch/papers/ellfilter/jacobi.tex
index 567bbcc..06548a5 100644
--- a/buch/papers/ellfilter/jacobi.tex
+++ b/buch/papers/ellfilter/jacobi.tex
@@ -1,15 +1,17 @@
 \section{Jacobische elliptische Funktionen}
 
-Für das elliptische Filter werden, wie es der Name bereits deutet, elliptische Funktionen gebraucht.
+Für das elliptische Filter werden, wie es der Name bereits andeutet, 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.
+Es ist daher naheliegend, dass der Kosinus des Tschebyscheff-Filters gegen ein elliptisches Pendant ausgetauscht werden könnte.
+Der Begriff elliptische Funktion wird für sehr viele Funktionen gebraucht, daher ist es hier wichtig zu erwähnen, dass es hier ausschliesslich um die Jacobischen elliptischen Funktionen geht.
+
+\subsection{Grundlegende Eigenschaften}
 
 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.
-Den \textit{elliptische Modul} $k$, der die Exzentrizität der Ellipse parametrisiert und das Winkelargument $z$.
+Im Gegensatz zum den trigonometrischen Funktionen haben die elliptischen Funktionen zwei Parameter.
+Den elliptischen 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 Kreisbogenlänge nicht linear zum Winkel verläuft.
 Darum kann hier nicht der gewohnte Winkel verwendet werden.
@@ -30,7 +32,7 @@ Das Winkelargument $z$ kann durch das elliptische Integral erster Art
 \end{equation}
 mit dem Winkel $\phi$ in Verbindung gebracht werden.
 
-Dabei wird das vollständige und unvollständige Elliptische integral unterschieden.
+Dabei wird das vollständige und unvollständige elliptische Integral unterschieden.
 Beim vollständigen Integral
 \begin{equation}
     K(k)
@@ -44,11 +46,11 @@ 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 Viertelellipsenbogen 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 elliptischen Funktionen sind somit $4K$-periodisch.
 
-Neben dem $\sn$ gibt es zwei weitere basis-elliptische Funktionen $\cn$ und $\dn$.
+Neben dem $\sn$ gibt es zwei weitere elliptische Basisfunktionen $\cn$ und $\dn$.
 Dazu kommen noch weitere abgeleitete Funktionen, die durch Quotienten und Kehrwerte dieser Funktionen zustande kommen.
 Insgesamt sind es die zwölf Funktionen
 \begin{equation*}
@@ -66,7 +68,7 @@ Insgesamt sind es die zwölf Funktionen
     \dc.
 \end{equation*}
 
-Die Jacobischen elliptischen Funktionen können mit der inversen Funktion des kompletten elliptischen Integrals erster Art
+Die Jacobischen elliptischen Funktionen können mit der inversen Funktion des vollständigen elliptischen Integrals erster Art
 \begin{equation}
     \phi = F^{-1}(z, k)
 \end{equation}
@@ -76,7 +78,7 @@ definiert werden. Dabei ist zu beachten dass nur das $z$ Argument der Funktion i
     \Leftrightarrow
     \phi = F^{-1}(z, k).
 \end{equation}
-Mithilfe von $F^{-1}$ kann zum Beispiel $sn^{-1}$ mit dem Elliptischen integral dargestellt werden:
+Mithilfe von $F^{-1}$ kann zum Beispiel $sn^{-1}$ mit dem elliptischen Integral dargestellt werden:
 \begin{equation}
     \sin(\phi)
     =
@@ -87,36 +89,10 @@ Mithilfe von $F^{-1}$ kann zum Beispiel $sn^{-1}$ mit dem Elliptischen integral
     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}
+\subsection{Die Funktion $\sn^{-1}$}
 
 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.
+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)
@@ -153,7 +129,7 @@ Dazu betrachten wir wieder den Integranden
         }
     }.
 \end{equation}
-Beim $\cos^{-1}(x)$ haben wir gesehen, dass die analytische Fortsetzung bei $x < -1$ und $x > 1$ rechtwinklig in die Komplexen zahlen wandert.
+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 $k \leq 1$.
@@ -169,7 +145,7 @@ Abbildung \ref{ellfilter:fig:sn} zeigt den Verlauf der Funktion in der komplexen
     }
     \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:
+In der reellen Richtung ist sie $4K(k)$-periodisch und in der imaginären Richtung $4K^\prime$-periodisch, wobei $K^\prime$ das komplementäre vollständige Elliptische Integral ist:
 \begin{equation}
     K^\prime(k)
     =
diff --git a/buch/papers/ellfilter/python/elliptic.pgf b/buch/papers/ellfilter/python/elliptic.pgf
index 32485c1..a8d06d1 100644
--- a/buch/papers/ellfilter/python/elliptic.pgf
+++ b/buch/papers/ellfilter/python/elliptic.pgf
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@@ -720,21 +542,19 @@
 \pgfusepath{stroke,fill}%
 }%
 \begin{pgfscope}%
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 \begin{pgfscope}%
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 \begin{pgfscope}%
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@@ -752,11 +572,15 @@
 \pgfusepath{stroke,fill}%
 }%
 \begin{pgfscope}%
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@@ -913,7 +737,7 @@
 \pgfusepath{stroke,fill}%
 }%
 \begin{pgfscope}%
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@@ -951,7 +775,7 @@
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 \begin{pgfscope}%
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@@ -989,7 +813,7 @@
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 }%
 \begin{pgfscope}%
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 \begin{pgfscope}%
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@@ -1065,7 +889,7 @@
 \pgfusepath{stroke,fill}%
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 \begin{pgfscope}%
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@@ -1073,10 +897,10 @@
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+\pgfpathlineto{\pgfqpoint{2.696834in}{1.341847in}}%
+\pgfpathlineto{\pgfqpoint{2.710776in}{1.481369in}}%
+\pgfpathlineto{\pgfqpoint{2.712768in}{1.469582in}}%
+\pgfpathlineto{\pgfqpoint{2.715756in}{1.408523in}}%
+\pgfpathlineto{\pgfqpoint{2.720735in}{1.209077in}}%
+\pgfpathlineto{\pgfqpoint{2.720735in}{1.209077in}}%
 \pgfusepath{stroke}%
 \end{pgfscope}%
 \begin{pgfscope}%
-\pgfpathrectangle{\pgfqpoint{0.733531in}{0.370218in}}{\pgfqpoint{3.993473in}{1.103393in}}%
+\pgfpathrectangle{\pgfqpoint{0.730012in}{0.548769in}}{\pgfqpoint{3.981446in}{0.932710in}}%
 \pgfusepath{clip}%
 \pgfsetrectcap%
 \pgfsetroundjoin%
@@ -1426,15 +1252,15 @@
 \definecolor{currentstroke}{rgb}{1.000000,0.647059,0.000000}%
 \pgfsetstrokecolor{currentstroke}%
 \pgfsetdash{}{0pt}%
-\pgfpathmoveto{\pgfqpoint{2.730268in}{1.151360in}}%
-\pgfpathlineto{\pgfqpoint{2.739978in}{0.709346in}}%
-\pgfpathlineto{\pgfqpoint{2.746805in}{0.536003in}}%
-\pgfpathlineto{\pgfqpoint{2.750075in}{0.480388in}}%
-\pgfpathlineto{\pgfqpoint{2.750075in}{0.480388in}}%
+\pgfpathmoveto{\pgfqpoint{2.720735in}{1.209077in}}%
+\pgfpathlineto{\pgfqpoint{2.730426in}{0.835162in}}%
+\pgfpathlineto{\pgfqpoint{2.737243in}{0.688588in}}%
+\pgfpathlineto{\pgfqpoint{2.740483in}{0.641897in}}%
+\pgfpathlineto{\pgfqpoint{2.740483in}{0.641897in}}%
 \pgfusepath{stroke}%
 \end{pgfscope}%
 \begin{pgfscope}%
-\pgfpathrectangle{\pgfqpoint{0.733531in}{0.370218in}}{\pgfqpoint{3.993473in}{1.103393in}}%
+\pgfpathrectangle{\pgfqpoint{0.730012in}{0.548769in}}{\pgfqpoint{3.981446in}{0.932710in}}%
 \pgfusepath{clip}%
 \pgfsetrectcap%
 \pgfsetroundjoin%
@@ -1442,37 +1268,34 @@
 \definecolor{currentstroke}{rgb}{1.000000,0.000000,0.000000}%
 \pgfsetstrokecolor{currentstroke}%
 \pgfsetdash{}{0pt}%
-\pgfpathmoveto{\pgfqpoint{2.750075in}{0.480388in}}%
-\pgfpathlineto{\pgfqpoint{2.756998in}{0.398490in}}%
-\pgfpathlineto{\pgfqpoint{2.759965in}{0.373708in}}%
-\pgfpathlineto{\pgfqpoint{2.760954in}{0.373969in}}%
-\pgfpathlineto{\pgfqpoint{2.768865in}{0.418088in}}%
-\pgfpathlineto{\pgfqpoint{2.776777in}{0.445199in}}%
-\pgfpathlineto{\pgfqpoint{2.784689in}{0.461928in}}%
-\pgfpathlineto{\pgfqpoint{2.792600in}{0.471986in}}%
-\pgfpathlineto{\pgfqpoint{2.800512in}{0.477584in}}%
-\pgfpathlineto{\pgfqpoint{2.809413in}{0.480262in}}%
-\pgfpathlineto{\pgfqpoint{2.820291in}{0.480059in}}%
-\pgfpathlineto{\pgfqpoint{2.834137in}{0.476323in}}%
-\pgfpathlineto{\pgfqpoint{2.852927in}{0.467773in}}%
-\pgfpathlineto{\pgfqpoint{2.886552in}{0.448435in}}%
-\pgfpathlineto{\pgfqpoint{2.959734in}{0.406057in}}%
-\pgfpathlineto{\pgfqpoint{3.006216in}{0.383080in}}%
-\pgfpathlineto{\pgfqpoint{3.036873in}{0.370620in}}%
-\pgfpathlineto{\pgfqpoint{3.085332in}{0.388800in}}%
-\pgfpathlineto{\pgfqpoint{3.137747in}{0.405140in}}%
-\pgfpathlineto{\pgfqpoint{3.195107in}{0.419767in}}%
-\pgfpathlineto{\pgfqpoint{3.259389in}{0.432926in}}%
-\pgfpathlineto{\pgfqpoint{3.332572in}{0.444663in}}%
-\pgfpathlineto{\pgfqpoint{3.415645in}{0.454798in}}%
-\pgfpathlineto{\pgfqpoint{3.512563in}{0.463443in}}%
-\pgfpathlineto{\pgfqpoint{3.627282in}{0.470496in}}%
-\pgfpathlineto{\pgfqpoint{3.766725in}{0.475859in}}%
-\pgfpathlineto{\pgfqpoint{3.940782in}{0.479318in}}%
-\pgfpathlineto{\pgfqpoint{4.167253in}{0.480557in}}%
-\pgfpathlineto{\pgfqpoint{4.484709in}{0.478977in}}%
-\pgfpathlineto{\pgfqpoint{4.727004in}{0.476474in}}%
-\pgfpathlineto{\pgfqpoint{4.727004in}{0.476474in}}%
+\pgfpathmoveto{\pgfqpoint{2.740483in}{0.641897in}}%
+\pgfpathlineto{\pgfqpoint{2.747385in}{0.572668in}}%
+\pgfpathlineto{\pgfqpoint{2.750343in}{0.551719in}}%
+\pgfpathlineto{\pgfqpoint{2.751329in}{0.551940in}}%
+\pgfpathlineto{\pgfqpoint{2.759217in}{0.589234in}}%
+\pgfpathlineto{\pgfqpoint{2.767105in}{0.612151in}}%
+\pgfpathlineto{\pgfqpoint{2.774992in}{0.626293in}}%
+\pgfpathlineto{\pgfqpoint{2.782880in}{0.634794in}}%
+\pgfpathlineto{\pgfqpoint{2.791754in}{0.639917in}}%
+\pgfpathlineto{\pgfqpoint{2.801614in}{0.641963in}}%
+\pgfpathlineto{\pgfqpoint{2.813446in}{0.641168in}}%
+\pgfpathlineto{\pgfqpoint{2.829221in}{0.636806in}}%
+\pgfpathlineto{\pgfqpoint{2.852885in}{0.626689in}}%
+\pgfpathlineto{\pgfqpoint{2.998810in}{0.558502in}}%
+\pgfpathlineto{\pgfqpoint{3.026417in}{0.549109in}}%
+\pgfpathlineto{\pgfqpoint{3.078674in}{0.565615in}}%
+\pgfpathlineto{\pgfqpoint{3.135861in}{0.580381in}}%
+\pgfpathlineto{\pgfqpoint{3.198964in}{0.593453in}}%
+\pgfpathlineto{\pgfqpoint{3.270941in}{0.605140in}}%
+\pgfpathlineto{\pgfqpoint{3.353763in}{0.615363in}}%
+\pgfpathlineto{\pgfqpoint{3.450389in}{0.624079in}}%
+\pgfpathlineto{\pgfqpoint{3.565749in}{0.631263in}}%
+\pgfpathlineto{\pgfqpoint{3.705758in}{0.636764in}}%
+\pgfpathlineto{\pgfqpoint{3.881263in}{0.640439in}}%
+\pgfpathlineto{\pgfqpoint{4.111982in}{0.642013in}}%
+\pgfpathlineto{\pgfqpoint{4.439328in}{0.640922in}}%
+\pgfpathlineto{\pgfqpoint{4.711458in}{0.638589in}}%
+\pgfpathlineto{\pgfqpoint{4.711458in}{0.638589in}}%
 \pgfusepath{stroke}%
 \end{pgfscope}%
 \begin{pgfscope}%
@@ -1482,8 +1305,8 @@
 \definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}%
 \pgfsetstrokecolor{currentstroke}%
 \pgfsetdash{}{0pt}%
-\pgfpathmoveto{\pgfqpoint{0.733531in}{0.370218in}}%
-\pgfpathlineto{\pgfqpoint{0.733531in}{1.473611in}}%
+\pgfpathmoveto{\pgfqpoint{0.730012in}{0.548769in}}%
+\pgfpathlineto{\pgfqpoint{0.730012in}{1.481479in}}%
 \pgfusepath{stroke}%
 \end{pgfscope}%
 \begin{pgfscope}%
@@ -1493,8 +1316,8 @@
 \definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}%
 \pgfsetstrokecolor{currentstroke}%
 \pgfsetdash{}{0pt}%
-\pgfpathmoveto{\pgfqpoint{4.727004in}{0.370218in}}%
-\pgfpathlineto{\pgfqpoint{4.727004in}{1.473611in}}%
+\pgfpathmoveto{\pgfqpoint{4.711458in}{0.548769in}}%
+\pgfpathlineto{\pgfqpoint{4.711458in}{1.481479in}}%
 \pgfusepath{stroke}%
 \end{pgfscope}%
 \begin{pgfscope}%
@@ -1504,8 +1327,8 @@
 \definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}%
 \pgfsetstrokecolor{currentstroke}%
 \pgfsetdash{}{0pt}%
-\pgfpathmoveto{\pgfqpoint{0.733531in}{0.370218in}}%
-\pgfpathlineto{\pgfqpoint{4.727004in}{0.370218in}}%
+\pgfpathmoveto{\pgfqpoint{0.730012in}{0.548769in}}%
+\pgfpathlineto{\pgfqpoint{4.711458in}{0.548769in}}%
 \pgfusepath{stroke}%
 \end{pgfscope}%
 \begin{pgfscope}%
@@ -1515,8 +1338,8 @@
 \definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}%
 \pgfsetstrokecolor{currentstroke}%
 \pgfsetdash{}{0pt}%
-\pgfpathmoveto{\pgfqpoint{0.733531in}{1.473611in}}%
-\pgfpathlineto{\pgfqpoint{4.727004in}{1.473611in}}%
+\pgfpathmoveto{\pgfqpoint{0.730012in}{1.481479in}}%
+\pgfpathlineto{\pgfqpoint{4.711458in}{1.481479in}}%
 \pgfusepath{stroke}%
 \end{pgfscope}%
 \end{pgfpicture}%
diff --git a/buch/papers/ellfilter/python/elliptic2.py b/buch/papers/ellfilter/python/elliptic2.py
index 20a7428..3d9065d 100644
--- a/buch/papers/ellfilter/python/elliptic2.py
+++ b/buch/papers/ellfilter/python/elliptic2.py
@@ -29,6 +29,9 @@ def ellip_filter(N, mode=-1):
         fs=None
     )
 
+    print("poles", a)
+    print("zeros", b)
+
     if mode == 0:
         w = np.linspace(0*omega_c,omega_c, 2000)
     elif mode == 1:
@@ -80,12 +83,12 @@ axs[0].add_patch(Rectangle(
 ))
 
 zeros = [0,0.87,0.995]
-poles = [1.01,1.155]
+poles = [1.01,1.155, 2.05]
 
 import matplotlib.transforms
 axs[0].plot( # mark errors as vertical bars
     zeros,
-    np.zeros_like(zeros),
+    np.zeros_like(zeros)-0.075,
     "o",
     mfc='none',
     color='black',
@@ -93,10 +96,11 @@ axs[0].plot( # mark errors as vertical bars
         axs[0].transData,
         axs[0].transAxes,
     ),
+    clip_on=False,
 )
 axs[0].plot( # mark errors as vertical bars
     poles,
-    np.ones_like(poles),
+    np.ones_like(poles)+0.075,
     "x",
     mfc='none',
     color='black',
@@ -104,6 +108,7 @@ axs[0].plot( # mark errors as vertical bars
         axs[0].transData,
         axs[0].transAxes,
     ),
+    clip_on=False,
 )
 
 for mode, c in enumerate(["green", "orange", "red"]):
@@ -135,17 +140,19 @@ axs[1].add_patch(Rectangle(
 
 axs[0].set_xlim([0,2])
 axs[0].set_ylim([1e-4,1e6])
+axs[0].tick_params(bottom = False)
 axs[0].grid()
-axs[0].set_ylabel("$F^2_N(w)$")
+axs[0].set_ylabel("$|F_N(w)|^2$")
 axs[1].grid()
 axs[1].set_ylim([0,1])
 axs[1].set_ylabel("$|H(w)|$")
+axs[1].set_xlabel("$w$")
 plt.tight_layout()
 plt.savefig("elliptic.pgf")
 plt.show()
 
-print("zeros", a)
-print("poles", b)
+print("poles", a)
+print("zeros", b)
 
 
 
diff --git a/buch/papers/ellfilter/python/elliptic3.py b/buch/papers/ellfilter/python/elliptic3.py
new file mode 100644
index 0000000..10accbb
--- /dev/null
+++ b/buch/papers/ellfilter/python/elliptic3.py
@@ -0,0 +1,101 @@
+# %%
+
+import matplotlib.pyplot as plt
+import scipy.signal
+import numpy as np
+import matplotlib
+from matplotlib.patches import Rectangle
+import scipy.special
+import scipyx as spx
+
+# import plot_params
+
+def last_color():
+    return plt.gca().lines[-1].get_color()
+
+# define elliptic functions
+
+def ell_int(k):
+    """ Calculate K(k) """
+    m = k**2
+    return scipy.special.ellipk(m)
+
+def sn(z, k):
+    return spx.ellipj(z, k**2)[0]
+
+def cn(z, k):
+    return spx.ellipj(z, k**2)[1]
+
+def dn(z, k):
+    return spx.ellipj(z, k**2)[2]
+
+def cd(z, k):
+    sn, cn, dn, ph = spx.ellipj(z, k**2)
+    return cn / dn
+
+N = 6
+L = (N//2) * 2
+r = N - L
+
+k = 0.9143
+
+i = np.arange(1, L+1)
+ui = (2*i - 1) / N
+k1 = k**N * np.prod(sn(ui*ell_int(k), k)**4)
+k1 = 0.0165
+k1 = 0.0058
+
+
+kp = np.sqrt(1-k**2)
+k1p = np.sqrt(1-k1**2)
+
+K = ell_int(k)
+Kp = ell_int(kp)
+K1 = ell_int(k1)
+K1p = ell_int(k1p)
+
+# assert np.allclose(Kp*K1*N/K, K1p, rtol=0.001)
+
+zeros = K/N * (np.arange(N)*2 + 1)
+poles = zeros + (1j * Kp)
+# if len(poles) % 2 == 0:
+#     poles = np.delete(poles, len(poles)//2)
+
+
+plt.plot(np.real(zeros), np.imag(zeros), "o")
+plt.plot(np.real(poles), np.imag(poles), "x")
+# plt.plot([0,K1], [0,K1p])
+# plt.plot([0,K], [0,Kp])
+plt.show()
+
+zeros = cd(zeros, k)
+poles = cd(poles, k)
+
+plt.plot(np.real(zeros), np.imag(zeros), "o")
+plt.plot(np.real(poles), np.imag(poles), "x")
+plt.ylim([-0.1,0.1])
+plt.xlim([-2.5,2.5])
+plt.show()
+
+w = np.linspace(0,2, 2000)
+
+def make_RN(w):
+    y = np.prod(w[:, None] - zeros[None], axis=-1) / np.prod(w[:, None] - poles[None], axis=-1)
+    y /= np.prod(1 - zeros) / np.prod(1 - poles)
+    return y
+
+
+RN = make_RN(w)
+
+plt.semilogy(w, np.abs(RN))
+plt.ylim([0.1,1000])
+
+plt.plot(w, np.ones_like(w) / k1)
+
+plt.show()
+
+H = 1 / (1 + RN**2)
+
+plt.semilogy(w, np.abs(H))
+plt.ylim([0.00001,1])
+plt.show()
diff --git a/buch/papers/ellfilter/tikz/arccos.tikz.tex b/buch/papers/ellfilter/tikz/arccos.tikz.tex
index b11c25d..538ac35 100644
--- a/buch/papers/ellfilter/tikz/arccos.tikz.tex
+++ b/buch/papers/ellfilter/tikz/arccos.tikz.tex
@@ -1,4 +1,4 @@
-\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2]
+\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2, thick]
 
     \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}}
@@ -22,6 +22,19 @@
 
         \clip(-7.5,-2) rectangle (7.5,2);
 
+        \foreach \i in {-2,...,1} {
+            \begin{scope}[xshift=\i*4cm]
+                \draw[->, thick, darkgreen!50] (-1, 0) -- (0,0);
+                \draw[->, thick, orange!50] (0, 0) -- (0,1.5);
+                \draw[->, thick, orange!50] (0, 0) -- (0,-1.5);
+                \draw[->, thick, darkgreen!50] (1, 0) -- (0,0);
+                \draw[->, thick, cyan!50] (2, 0) -- (1,0);
+                \draw[->, thick, blue!50] (2,1.5) -- (2, 0);
+                \draw[->, thick, blue!50] (2,-1.5) -- (2, 0);
+                \draw[->, thick, cyan!50] (2, 0) -- (3,0);
+            \end{scope}
+        }
+
         % \pause
         \draw[ultra thick, ->, darkgreen] (1, 0) -- (0,0);
         % \pause
@@ -34,16 +47,6 @@
 
         \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}
diff --git a/buch/papers/ellfilter/tikz/arccos2.tikz.tex b/buch/papers/ellfilter/tikz/arccos2.tikz.tex
index 2cec75f..d0e0430 100644
--- a/buch/papers/ellfilter/tikz/arccos2.tikz.tex
+++ b/buch/papers/ellfilter/tikz/arccos2.tikz.tex
@@ -1,4 +1,4 @@
-\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2]
+\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2, thick]
 
     \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}}
diff --git a/buch/papers/ellfilter/tikz/cd.tikz.tex b/buch/papers/ellfilter/tikz/cd.tikz.tex
index 0cf2417..ffe3b9c 100644
--- a/buch/papers/ellfilter/tikz/cd.tikz.tex
+++ b/buch/papers/ellfilter/tikz/cd.tikz.tex
@@ -1,4 +1,4 @@
-\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2]
+\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2, thick]
 
     \tikzstyle{zero} = [draw, circle, inner sep =0, minimum height=0.15cm]
 
@@ -9,63 +9,59 @@
         \draw[gray, ->] (0,-1.5) -- (0,1.5) node[anchor=south]{$\mathrm{Im}~z$};
         \draw[gray, ->] (-5,0) -- (5,0) node[anchor=west]{$\mathrm{Re}~z$};
 
-        \draw[gray] ( 1,0) +(0,0.1) -- +(0, -0.1) node[inner sep=0, anchor=north] {\small $K$};
 
-        \draw[gray]  (0, 0.5) +(0.1, 0) -- +(-0.1, 0) node[inner sep=0, anchor=east]{\small $jK^\prime$};
-
-
-        \begin{scope}
-
-            \begin{scope}[xshift=0cm]
-
-                \clip(-4.5,-1.25) rectangle (4.5,1.25);
-
-                \fill[yellow!30] (0,0) rectangle (1, 0.5);
-
-                \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}
-                    }
+        \begin{scope}[xshift=0cm]
+
+            \clip(-4.5,-1.25) rectangle (4.5,1.25);
+
+            \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[->, thick, orange!50] (0, 0) -- (0,0.5);
+                        \draw[->, thick, darkgreen!50] (1, 0) -- (0,0);
+                        \draw[->, thick, cyan!50] (2, 0) -- (1,0);
+                        \draw[->, thick, blue!50] (2,0.5) -- (2, 0);
+                        \draw[->, thick, purple!50] (1, 0.5) -- (2,0.5);
+                        \draw[->, thick, red!50] (0, 0.5) -- (1,0.5);
+                        \draw[->, thick, orange!50] (0,1) -- (0,0.5);
+                        \draw[->, thick, blue!50] (2,0.5) -- (2, 1);
+                        \draw[->, thick, purple!50] (3, 0.5) -- (2,0.5);
+                        \draw[->, thick, red!50] (4, 0.5) -- (3,0.5);
+                        \draw[->, thick, cyan!50] (2, 0) -- (3,0);
+                        \draw[->, thick, darkgreen!50] (3, 0) -- (4,0);
+                    \end{scope}
                 }
-
-                \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]
-                            \node[zero] at ( 1, 0) {};
-                            \node[zero] at ( 3, 0) {};
-                            \node[pole] at ( 1,0.5) {};
-                            \node[pole] at ( 3,0.5) {};
-
-                        \end{scope}
-                    }
+            }
+
+            \draw[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]
+                        \node[zero] at ( 1, 0) {};
+                        \node[zero] at ( 3, 0) {};
+                        \node[pole] at ( 1,0.5) {};
+                        \node[pole] at ( 3,0.5) {};
+
+                    \end{scope}
                 }
-
-            \end{scope}
+            }
 
         \end{scope}
 
+        \draw[gray] ( 1,0) +(0,0.05) -- +(0, -0.05) node[inner sep=0, anchor=north west] {\small $K$};
+        \draw[gray]  (0, 0.5) +(0.1, 0) -- +(-0.1, 0) node[inner sep=0, anchor=south east]{\small $jK^\prime$};
+
     \end{scope}
 
+
     \node[zero] at (4,3) (n) {};
     \node[anchor=west] at (n.east) {Nullstelle};
     \node[pole, below=0.25cm of n] (n) {};
diff --git a/buch/papers/ellfilter/tikz/cd2.tikz.tex b/buch/papers/ellfilter/tikz/cd2.tikz.tex
index d4187c4..47efa53 100644
--- a/buch/papers/ellfilter/tikz/cd2.tikz.tex
+++ b/buch/papers/ellfilter/tikz/cd2.tikz.tex
@@ -1,4 +1,4 @@
-\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2]
+\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2, thick]
 
     \tikzstyle{zero} = [draw, circle, inner sep =0, minimum height=0.15cm]
     \tikzstyle{dot} = [fill, circle, inner sep =0, minimum height=0.1cm]
@@ -7,11 +7,15 @@
 
     \begin{scope}[xscale=1.25, yscale=3.5]
 
+        \fill[orange!30] (0,0) rectangle (5, 0.5);
+        % \fill[yellow!30] (0,0) rectangle (1, 0.1);
+        \node[] at (2.5, 0.25) {\small $N=5$};
+
         \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 $K_1$};
-        \draw[gray] ( 5,0) +(0,0.05) -- +(0, -0.05) node[inner sep=0, anchor=north] {\small $5K_1$};
+        \draw[gray] ( 1,0) +(0,0.035) -- +(0, -0.035) node[inner sep=0, anchor=north] {\small $K_1$};
+        \draw[gray] ( 5,0) +(0,0.035) -- +(0, -0.035) 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}
@@ -34,9 +38,6 @@
             % \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, 0.1);
-            \node[] at (2.5, 0.25) {\small $N=5$};
 
 
             \draw[decorate,decoration={brace,amplitude=3pt,mirror}, yshift=0.05cm]
diff --git a/buch/papers/ellfilter/tikz/cd3.tikz.tex b/buch/papers/ellfilter/tikz/cd3.tikz.tex
index ae18519..158a5ec 100644
--- a/buch/papers/ellfilter/tikz/cd3.tikz.tex
+++ b/buch/papers/ellfilter/tikz/cd3.tikz.tex
@@ -1,4 +1,4 @@
-\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2]
+\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2, thick]
 
     \tikzstyle{zero} = [draw, circle, inner sep =0, minimum height=0.15cm]
     \tikzstyle{dot} = [fill, circle, inner sep =0, minimum height=0.1cm]
diff --git a/buch/papers/ellfilter/tikz/elliptic_transform1.tikz.tex b/buch/papers/ellfilter/tikz/elliptic_transform1.tikz.tex
index 2a36ee0..329cadd 100644
--- a/buch/papers/ellfilter/tikz/elliptic_transform1.tikz.tex
+++ b/buch/papers/ellfilter/tikz/elliptic_transform1.tikz.tex
@@ -1,4 +1,4 @@
-\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2]
+\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2, thick]
 
     \tikzstyle{zero} = [draw, circle, inner sep =0, minimum height=0.15cm]
 
diff --git a/buch/papers/ellfilter/tikz/elliptic_transform2.tikz.tex b/buch/papers/ellfilter/tikz/elliptic_transform2.tikz.tex
index 20c2d82..31c40dc 100644
--- a/buch/papers/ellfilter/tikz/elliptic_transform2.tikz.tex
+++ b/buch/papers/ellfilter/tikz/elliptic_transform2.tikz.tex
@@ -4,7 +4,7 @@
 \def\nn{2}
 \def\a{2.5}
 
-\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2]
+\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2, thick]
 
     \tikzstyle{zero} = [draw, circle, inner sep =0, minimum height=0.15cm]
     \tikzstyle{dot} = [fill, circle, inner sep =0, minimum height=0.1cm]
diff --git a/buch/papers/ellfilter/tikz/filter.tikz.tex b/buch/papers/ellfilter/tikz/filter.tikz.tex
index 769602a..1232bf0 100644
--- a/buch/papers/ellfilter/tikz/filter.tikz.tex
+++ b/buch/papers/ellfilter/tikz/filter.tikz.tex
@@ -1,4 +1,4 @@
-\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2]
+\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2, thick]
 
     \tikzstyle{zero} = [draw, circle, inner sep =0, minimum height=0.15cm]
 
@@ -14,8 +14,7 @@
         \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] {$\Omega_p$};
+        \draw[dashed] (0,0.707) node[left] {$\sqrt{\frac{1}{1+\varepsilon^2}}$} -- (1, 0.707) (1,0) node[below] {$\Omega_p$};
 
         \node[left] at(0,1) {$1$};
 
diff --git a/buch/papers/ellfilter/tikz/fundamental_rectangle.tikz.tex b/buch/papers/ellfilter/tikz/fundamental_rectangle.tikz.tex
index 921dbfa..1d1e45e 100644
--- a/buch/papers/ellfilter/tikz/fundamental_rectangle.tikz.tex
+++ b/buch/papers/ellfilter/tikz/fundamental_rectangle.tikz.tex
@@ -1,4 +1,4 @@
-\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2]
+\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2, thick]
 
     \tikzstyle{zero} = [draw, circle, inner sep =0, minimum height=0.15cm]
 
diff --git a/buch/papers/ellfilter/tikz/pn.tikz.tex b/buch/papers/ellfilter/tikz/pn.tikz.tex
new file mode 100644
index 0000000..5dd8fa4
--- /dev/null
+++ b/buch/papers/ellfilter/tikz/pn.tikz.tex
@@ -0,0 +1,63 @@
+\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2, thick]
+
+    \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=0.75, yscale=2.5]
+
+        \fill[orange!30] (0,0) rectangle (5, 0.5);
+        % \fill[yellow!30] (0,0) rectangle (1, 0.1);
+        \node[] at (5, 0.25) {\small $N=5$};
+
+        \draw[gray, ->] (0,-0.25) -- (0,0.75) node[anchor=south]{$\mathrm{Im}~z$};
+        \draw[gray, ->] (-1,0) -- (11,0) node[anchor=west]{$\mathrm{Re}~z$};
+
+        \draw[gray] ( 5,0) +(0,0.035) -- +(0, -0.035) node[inner sep=0, anchor=north] {\small $K$};
+        \draw[gray]  (0, 0.5) +(0.1, 0) -- +(-0.1, 0) node[inner sep=0, anchor=east]{\small $jK^\prime$};
+
+        \begin{scope}
+
+            \draw[ultra thick, ->, purple] (5, 0.5) -- (10,0.5);
+            \draw[ultra thick, ->, blue] (10, 0.5) --  (10,0);
+            \draw[ultra thick, ->, cyan] (10, 0) --  (5,0);
+            \draw[ultra thick, ->, darkgreen] (5, 0) --  (0,0);
+            \draw[ultra thick, ->, orange] (-0, 0) -- (0,0.5);
+            \draw[ultra thick, ->, red] (0,0.5) -- (5, 0.5);
+
+            \foreach \i in {1,...,5} {
+                \begin{scope}[xshift=(\i-1)*2cm]
+                    \node[zero] at ( 1, 0) {};
+                    \node[anchor=south west] at ( 1, 0) {$n_\i$};
+                    \node[pole] at ( 1,0.5) {};
+                    \node[anchor=south west] at ( 1, 0.5) {$p_\i$};
+                \end{scope}
+            }
+
+        \end{scope}
+
+    \end{scope}
+
+    % \begin{scope}[yshift=-1.5cm, xshift=3.75cm, xscale=0.75]
+
+    %     \draw[gray, ->] (-6,0) -- (6,0) node[anchor=west]{$w$};
+
+    %     \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) {$-1/k$};
+    %     \node[anchor=south] at (-2,0) {$-1$};
+    %     \node[anchor=south] at (0,0) {$0$};
+    %     \node[anchor=south] at (2,0) {$1$};
+    %     \node[anchor=south] at (3,0) {$1/k$};
+    %     \node[anchor=south] at (5,0) {$\infty$};
+
+    % \end{scope}
+
+\end{tikzpicture}
diff --git a/buch/papers/ellfilter/tikz/sn.tikz.tex b/buch/papers/ellfilter/tikz/sn.tikz.tex
index 0546fda..c34b619 100644
--- a/buch/papers/ellfilter/tikz/sn.tikz.tex
+++ b/buch/papers/ellfilter/tikz/sn.tikz.tex
@@ -1,4 +1,4 @@
-\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2]
+\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2, thick]
 
     \tikzstyle{zero} = [draw, circle, inner sep =0, minimum height=0.15cm]
 
@@ -6,66 +6,65 @@
 
     \begin{scope}[xscale=0.9, yscale=1.8]
 
+        \fill[yellow!30] (0,0) rectangle (1, 0.5);
+
         \draw[gray, ->] (0,-1.5) -- (0,1.5) node[anchor=south]{$\mathrm{Im}~z$};
         \draw[gray, ->] (-5,0) -- (5,0) node[anchor=west]{$\mathrm{Re}~z$};
 
-        \begin{scope}
-
-            \clip(-4.5,-1.25) rectangle (4.5,1.25);
-
-            \fill[yellow!30] (0,0) rectangle (1, 0.5);
-
-            \begin{scope}[xshift=-1cm]
-
-                \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]
-                            \node[zero] at ( 1, 0) {};
-                            \node[zero] at ( 3, 0) {};
-                            \node[pole] at ( 1,0.5) {};
-                            \node[pole] at ( 3,0.5) {};
-                        \end{scope}
-                    }
-                }
+        \clip(-4.5,-1.25) rectangle (4.5,1.25);
+
 
-            \end{scope}
+        \begin{scope}[xshift=-1cm]
+
+            \foreach \i in {-2,...,2} {
+                \foreach \j in {-2,...,1} {
+                    \begin{scope}[xshift=\i*4cm, yshift=\j*1cm]
+                        \draw[<-, thick, blue!50] (0, 0) -- (0,0.5);
+                        \draw[<-, thick, cyan!50] (1, 0) -- (0,0);
+                        \draw[<-, thick, darkgreen!50] (2, 0) -- (1,0);
+                        \draw[<-, thick, orange!50] (2,0.5) -- (2, 0);
+                        \draw[<-, thick, red!50] (1, 0.5) -- (2,0.5);
+                        \draw[<-, thick, purple!50] (0, 0.5) -- (1,0.5);
+                        \draw[<-, thick, blue!50] (0,1) -- (0,0.5);
+                        \draw[<-, thick, orange!50] (2,0.5) -- (2, 1);
+                        \draw[<-, thick, red!50] (3, 0.5) -- (2,0.5);
+                        \draw[<-, thick, purple!50] (4, 0.5) -- (3,0.5);
+                        \draw[<-, thick, darkgreen!50] (2, 0) -- (3,0);
+                        \draw[<-, thick, 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]
+                        \node[zero] at ( 1, 0) {};
+                        \node[zero] at ( 3, 0) {};
+                        \node[pole] at ( 1,0.5) {};
+                        \node[pole] at ( 3,0.5) {};
+                    \end{scope}
+                }
+            }
 
         \end{scope}
 
-        \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 $jK^\prime$};
+
+        \draw[gray] ( 1,0) +(0,0.05) -- +(0, -0.05) node[inner sep=0, anchor=north west] {\small $K$};
+        \draw[gray]  (0, 0.5) +(0.1, 0) -- +(-0.1, 0) node[inner sep=0, anchor=south east]{\small $jK^\prime$};
 
     \end{scope}
 
diff --git a/buch/papers/ellfilter/tschebyscheff.tex b/buch/papers/ellfilter/tschebyscheff.tex
index 639c87c..84095a7 100644
--- a/buch/papers/ellfilter/tschebyscheff.tex
+++ b/buch/papers/ellfilter/tschebyscheff.tex
@@ -2,22 +2,22 @@
 
 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}
+Der Name des Filters deutet schon an, dass die Tschebyscheff-Polynome $T_N$ (siehe auch Kapitel \ref{buch:polynome:section:tschebyscheff}) für das Filter relevant sind:
+\begin{align*}
     T_{0}(x)&=1\\
     T_{1}(x)&=x\\
     T_{2}(x)&=2x^{2}-1\\
     T_{3}(x)&=4x^{3}-3x\\
     T_{n+1}(x)&=2x~T_{n}(x)-T_{n-1}(x).
-\end{align}
+\end{align*}
 Bemerkenswert ist, dass die Polynome im Intervall $[-1, 1]$ mit der trigonometrischen Funktion
 \begin{align} \label{ellfilter:eq:chebychef_polynomials}
     T_N(w) &= \cos \left( N \cos^{-1}(w) \right) \\
-           &= \cos \left(N~z \right), \quad w= \cos(z)
+           &= \cos \left(N~z \right), \quad w= \cos(z) \label{ellfilter:eq:chebychef_polynomials2}
 \end{align}
 übereinstimmen.
 Der Zusammenhang lässt sich mit den Doppel- und Mehrfachwinkelfunktionen der trigonometrischen Funktionen erklären.
-Abbildung \ref{ellfilter:fig:chebychef_polynomials} zeigt einige Tschebyscheff-Polynome.
+Abbildung \ref{ellfilter:fig:chebychef_polynomials} zeigt einige Tschebyscheff-Polynome, wobei das Equiripple-Verhalten schon sichtbar ist.
 \begin{figure}
     \centering
     \input{papers/ellfilter/python/F_N_chebychev2.pgf}
@@ -36,8 +36,7 @@ Wenn wir die Tschebyscheff-Polynome quadrieren, passen sie perfekt in die Forder
 \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 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 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}
@@ -63,7 +62,7 @@ Die invertierte Funktion des Kosinus kann als bestimmtes Integral dargestellt we
     ~dz
     + \frac{\pi}{2}.
 \end{align}
-Der Integrand oder auch die Ableitung von $\cos^{-1}(x)$
+Der Integrand oder auch die Ableitung von $\cos^{-1}(x)$,
 \begin{equation}
     \frac{
         -1
@@ -71,10 +70,10 @@ Der Integrand oder auch die Ableitung von $\cos^{-1}(x)$
         \sqrt{
             1-z^2
         }
-    }
+    },
 \end{equation}
 bestimmt dabei die Richtung, in welche die Funktion verläuft.
-Der reelle Arcuscosinus is bekanntlich nur für $|z| \leq 1$ definiert.
+Der reelle Arcuscosinus ist 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.
@@ -83,27 +82,28 @@ Abbildung \ref{ellfilter:fig:arccos} zeigt den Arcuscosinus in der komplexen Ebe
 \begin{figure}
     \centering
     \input{papers/ellfilter/tikz/arccos.tikz.tex}
-    \caption{Die Funktion $z = \cos^{-1}(w)$ dargestellt in der komplexen ebene.}
+    \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.
+Wegen der Periodizität des Kosinus werden periodisch Werte in der $z$-Ebene auf den gleichen Wert in $w$ abgebildet.
+Das gleiche Muster kommt daher periodisch vor.
 Das Einzeichnen von Pol- und Nullstellen ist hilfreich für die Betrachtung der Funktion.
 
 
-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}.
+In \eqref{ellfilter:eq:chebychef_polynomials2} 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 $\cos \big( N~\cos^{-1}(w) \big)$ 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 $w\in(-\infty, \infty)$ für $N = 4$.
-        Je grösser die Ordnung $N$ gewählt wird, desto mehr Nullstellen werden passiert die zu Equirippel-Verhalten führen.
+        Je grösser die Ordnung $N$ gewählt wird, desto mehr Nullstellen werden passiert die zu Equiripple-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}
-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.
+Durch die spezielle Anordnung der Nullstellen hat die Funktion auf der reellen Achse Equiripple-Verhalten und ist dennoch ein Polynom, was sich perfekt für lineare Filter eignet.
+Für $|w| \le 1$ ist die Funktion begrenzt zwischen $-1$ und $1$.