13 files changed, 1183 insertions, 1366 deletions

diff --git a/buch/papers/ellfilter/einleitung.tex b/buch/papers/ellfilter/einleitung.tex
index 18913fb..5bc2ead 100644
--- a/buch/papers/ellfilter/einleitung.tex
+++ b/buch/papers/ellfilter/einleitung.tex
@@ -56,7 +56,7 @@ Das Tschebyscheff-1 Filter ist maximal steil für eine definierte Welligkeit im
 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.
+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 861600b..8c60e46 100644
--- a/buch/papers/ellfilter/elliptic.tex
+++ b/buch/papers/ellfilter/elliptic.tex
@@ -6,47 +6,26 @@ Kommen wir nun zum eigentlichen Teil dieses Papers, den elliptischen rationalen
                 &= \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 Tschebyschef-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.
+Bei $k = k_1 = 0$ wird der $\cd$ zum Kosinus und wir erhalten in diesem Spezialfall die Tschebyschef-Polynome.
 
-
-
-Sinus entspricht $\sn$
-
-Damit die Nullstellen an ähnlichen Positionen zu liegen kommen wie bei den Tschebyscheff-Polynomen, muss die $\cd$-Funktion gewählt werden.
-
+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 Jacobi elliptischen Funktionen die Positionen der Null- und Polstellen anhand eines Diagramms ermittelt werden.
-Der erste Buchstabe bestimmt die Position der Nullstelle und der zweite Buchstabe die Polstelle.
-\begin{figure}
-    \centering
-    \input{papers/ellfilter/tikz/fundamental_rectangle.tikz.tex}
-    \caption{
-        Fundamentales Rechteck der inversen Jacobi elliptischen Funktionen.
-    }
-    \label{ellfilter:fig:fundamental_rectangle}
-\end{figure}
-
-Auffallend an der $w = \sn(z, k)$-Funktion ist, dass sich $w$ auf der reellen Achse wie der Kosinus immer zwischen $-1$ und $1$ bewegt, während bei $\mathrm{Im(z) = K^\prime}$ die Werte zwischen $\pm 1/k$ und $\pm \infty$ verlaufen.
-Die Funktion hat also Equirippel-Verhalten um $w=0$ und um $w=\pm \infty$.
-Falls es möglich ist diese Werte abzufahren im Sti der Tschebyscheff-Polynome, kann ein Filter gebaut werden, dass Equirippel-Verhalten im Durchlass- und Sperrbereich aufweist.
-
-
+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 Funktion hat also Equirippel-Verhalten um $w=0$ und um $w=\pm \infty$. %TODO Check
+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.
 
 Analog zu Abbildung \ref{ellfilter:fig:arccos2} können wir auch bei den elliptisch rationalen Funktionen die komplexe $z$-Ebene betrachten, wie ersichtlich in Abbildung \ref{ellfilter:fig:cd2}, um die besser zu verstehen.
 \begin{figure}
@@ -60,20 +39,10 @@ Analog zu Abbildung \ref{ellfilter:fig:arccos2} können wir auch bei den ellipti
 \end{figure}
 % Da die $\cd^{-1}$-Funktion 
 
-
-
-\begin{figure}
-    \centering
-    \input{papers/ellfilter/python/F_N_elliptic.pgf}
-    \caption{$F_N$ für ein elliptischs filter.}
-    \label{ellfilter:fig:elliptic}
-\end{figure}
-
-
 \begin{figure}
     \centering
     \input{papers/ellfilter/python/elliptic.pgf}
-    \caption{Die resultierende frequenzantwort eines elliptischs filter.}
+    \caption{$F_N$ und die resultierende Frequenzantwort eines elliptischen Filters.}
     \label{ellfilter:fig:elliptic_freq}
 \end{figure}
 
@@ -90,6 +59,10 @@ Dies trifft ein wenn die Gradengleichung erfüllt ist.
 Leider ist das lösen dieser Gleichung nicht trivial.
 Die Rechnung wird in \ref{ellfilter:bib:orfanidis} im Detail angeschaut.
 
+$K$ und $K^\prime$ sind voneinender abhängig.
+
+Das Problem lässt sich grafisch darstellen.
+
 \begin{figure}
     \centering
     \input{papers/ellfilter/python/k.pgf}
@@ -108,8 +81,6 @@ Die Rechnung wird in \ref{ellfilter:bib:orfanidis} im Detail angeschaut.
     \caption{Die Gradgleichung als geometrisches Problem.}
 \end{figure}
 
-
-
 \subsection{Polynome?}
 
 Bei den Tschebyscheff-Polynomen haben wir gesehen, dass die Trigonometrische Formel zu einfachen Polynomen umgewandelt werden kann.
diff --git a/buch/papers/ellfilter/jacobi.tex b/buch/papers/ellfilter/jacobi.tex
index 6a208fa..3940171 100644
--- a/buch/papers/ellfilter/jacobi.tex
+++ b/buch/papers/ellfilter/jacobi.tex
@@ -2,14 +2,16 @@
 
 %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.
 Darum kann hier nicht der gewohnte Winkel verwendet werden.
@@ -27,17 +29,17 @@ 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?
+    % =
+    % \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 +55,9 @@ 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.
@@ -96,7 +98,7 @@ Mithilfe von $F^{-1}$ kann zum Beispiel $sn^{-1}$ mit dem Elliptischen integral
     w
 \end{equation}
 
-\begin{equation}
+\begin{equation} %TODO remove unnecessary equations
     \phi
     =
      F^{-1}(z, k)
@@ -153,31 +155,9 @@ Beim $\cos^{-1}(x)$ haben wir gesehen, dass die analytische Fortsetzung bei $x <
 Wenn man das gleiche mit $\sn^{-1}(w, k)$ macht, erkennt man zwei interessante Stellen.
 Die erste ist die gleiche wie beim $\cos^{-1}(x)$ nämlich bei $t = \pm 1$.
 Der erste Term unter der Wurzel wird dann negativ, während der zweite noch positiv ist, da $k \leq 1$.
-\begin{equation}
-    \frac{
-        1
-    }{
-        \sqrt{
-            (1-t^2)(1-k^2 t^2)
-        }
-    }
-    \in \mathbb{R}
-    \quad \forall \quad
-    -1 \leq t \leq 1
-\end{equation}
-Die zweite stelle passiert wenn beide Faktoren unter der Wurzel negativ werden, was bei $t = 1/k$ der Fall ist.
-
-
-
-
-Funktion in relle und komplexe Richtung periodisch
-
-In der reellen Richtung ist sie $4K(k)$-periodisch und in der imaginären Richtung $4K^\prime(k)$-periodisch.
-
-
-
-%TODO sn^{-1} grafik
-
+Ab diesem Punkt verläuft 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 \label{ellfilter:fig:sn} zeigt den Verlauf der Funktion in der komplexen Ebene.
 \begin{figure}
     \centering
     \input{papers/ellfilter/tikz/sn.tikz.tex}
@@ -185,5 +165,20 @@ 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 komplemenäre 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/python/F_N_elliptic.pgf b/buch/papers/ellfilter/python/F_N_elliptic.pgf
deleted file mode 100644
index 50faaaa..0000000
--- a/buch/papers/ellfilter/python/F_N_elliptic.pgf
+++ /dev/null
@@ -1,834 +0,0 @@
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diff --git a/buch/papers/ellfilter/python/elliptic.pgf b/buch/papers/ellfilter/python/elliptic.pgf
index 89ffb60..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}}%
 \pgfusepath{use as bounding box, clip}%
 \begin{pgfscope}%
 \pgfsetbuttcap%
@@ -34,9 +34,9 @@
 \pgfsetstrokeopacity{0.000000}%
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-\pgfpathlineto{\pgfqpoint{4.000000in}{2.500000in}}%
-\pgfpathlineto{\pgfqpoint{0.000000in}{2.500000in}}%
+\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 @@
 \pgfsetstrokecolor{currentstroke}%
 \pgfsetstrokeopacity{0.000000}%
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-\pgfpathmoveto{\pgfqpoint{0.617954in}{0.548769in}}%
-\pgfpathlineto{\pgfqpoint{3.761597in}{0.548769in}}%
-\pgfpathlineto{\pgfqpoint{3.761597in}{2.301955in}}%
-\pgfpathlineto{\pgfqpoint{0.617954in}{2.301955in}}%
-\pgfpathlineto{\pgfqpoint{0.617954in}{0.548769in}}%
+\pgfpathmoveto{\pgfqpoint{0.733531in}{1.746607in}}%
+\pgfpathlineto{\pgfqpoint{4.727004in}{1.746607in}}%
+\pgfpathlineto{\pgfqpoint{4.727004in}{2.850000in}}%
+\pgfpathlineto{\pgfqpoint{0.733531in}{2.850000in}}%
+\pgfpathlineto{\pgfqpoint{0.733531in}{1.746607in}}%
 \pgfpathclose%
 \pgfusepath{fill}%
 \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%
 \pgfsetmiterjoin%
@@ -72,16 +72,16 @@
 \pgfsetstrokecolor{currentstroke}%
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 \pgfsetdash{}{0pt}%
-\pgfpathmoveto{\pgfqpoint{0.617954in}{1.788459in}}%
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-\pgfpathlineto{\pgfqpoint{0.617954in}{3.541645in}}%
-\pgfpathlineto{\pgfqpoint{0.617954in}{1.788459in}}%
+\pgfpathmoveto{\pgfqpoint{0.733531in}{-108.151374in}}%
+\pgfpathlineto{\pgfqpoint{2.730268in}{-108.151374in}}%
+\pgfpathlineto{\pgfqpoint{2.730268in}{2.187964in}}%
+\pgfpathlineto{\pgfqpoint{0.733531in}{2.187964in}}%
+\pgfpathlineto{\pgfqpoint{0.733531in}{-108.151374in}}%
 \pgfpathclose%
 \pgfusepath{fill}%
 \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%
 \pgfsetmiterjoin%
@@ -93,16 +93,16 @@
 \pgfsetstrokecolor{currentstroke}%
 \pgfsetstrokeopacity{0.200000}%
 \pgfsetdash{}{0pt}%
-\pgfpathmoveto{\pgfqpoint{2.189776in}{0.724087in}}%
-\pgfpathlineto{\pgfqpoint{2.205368in}{0.724087in}}%
-\pgfpathlineto{\pgfqpoint{2.205368in}{1.788459in}}%
-\pgfpathlineto{\pgfqpoint{2.189776in}{1.788459in}}%
-\pgfpathlineto{\pgfqpoint{2.189776in}{0.724087in}}%
+\pgfpathmoveto{\pgfqpoint{2.730268in}{2.187964in}}%
+\pgfpathlineto{\pgfqpoint{2.750075in}{2.187964in}}%
+\pgfpathlineto{\pgfqpoint{2.750075in}{2.408643in}}%
+\pgfpathlineto{\pgfqpoint{2.730268in}{2.408643in}}%
+\pgfpathlineto{\pgfqpoint{2.730268in}{2.187964in}}%
 \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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@@ -114,16 +114,16 @@
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-\pgfpathlineto{\pgfqpoint{2.205368in}{0.548769in}}%
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+\pgfpathlineto{\pgfqpoint{4.746812in}{2.850005in}}%
+\pgfpathlineto{\pgfqpoint{2.750075in}{2.850005in}}%
+\pgfpathlineto{\pgfqpoint{2.750075in}{2.408643in}}%
 \pgfpathclose%
 \pgfusepath{fill}%
 \end{pgfscope}%
 \begin{pgfscope}%
-\pgfpathrectangle{\pgfqpoint{0.617954in}{0.548769in}}{\pgfqpoint{3.143642in}{1.753186in}}%
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@@ -131,8 +131,8 @@
 \definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}%
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+\pgfpathlineto{\pgfqpoint{0.733531in}{2.850000in}}%
 \pgfusepath{stroke}%
 \end{pgfscope}%
 \begin{pgfscope}%
@@ -150,7 +150,295 @@
 \pgfusepath{stroke,fill}%
 }%
 \begin{pgfscope}%
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+\pgfsys@transformshift{0.733531in}{1.746607in}%
+\pgfsys@useobject{currentmarker}{}%
+\end{pgfscope}%
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+\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}%
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+\pgfusepath{stroke}%
+\end{pgfscope}%
+\begin{pgfscope}%
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+\pgfusepath{stroke,fill}%
+}%
+\begin{pgfscope}%
+\pgfsys@transformshift{1.232715in}{1.746607in}%
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+\end{pgfscope}%
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+}%
+\begin{pgfscope}%
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+\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}%
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+}%
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 \end{pgfscope}%
 \begin{pgfscope}%
@@ -188,7 +951,7 @@
 \pgfusepath{stroke,fill}%
 }%
 \begin{pgfscope}%
-\pgfsys@transformshift{1.403865in}{0.548769in}%
+\pgfsys@transformshift{1.232715in}{0.370218in}%
 \pgfsys@useobject{currentmarker}{}%
 \end{pgfscope}%
 \end{pgfscope}%
@@ -196,10 +959,10 @@
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-\pgftext[x=1.403865in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.5}\)}%
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 \pgfsetrectcap%
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@@ -207,8 +970,8 @@
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 \pgfusepath{stroke}%
 \end{pgfscope}%
 \begin{pgfscope}%
@@ -226,7 +989,7 @@
 \pgfusepath{stroke,fill}%
 }%
 \begin{pgfscope}%
-\pgfsys@transformshift{2.189776in}{0.548769in}%
+\pgfsys@transformshift{1.731899in}{0.370218in}%
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 \end{pgfscope}%
@@ -234,10 +997,10 @@
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-\pgftext[x=2.189776in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {1.0}\)}%
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@@ -245,8 +1008,8 @@
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 \end{pgfscope}%
 \begin{pgfscope}%
@@ -264,7 +1027,7 @@
 \pgfusepath{stroke,fill}%
 }%
 \begin{pgfscope}%
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+\pgfsys@transformshift{2.231083in}{0.370218in}%
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 \end{pgfscope}%
 \end{pgfscope}%
@@ -272,10 +1035,10 @@
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@@ -283,8 +1046,8 @@
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 \end{pgfscope}%
 \begin{pgfscope}%
@@ -302,7 +1065,7 @@
 \pgfusepath{stroke,fill}%
 }%
 \begin{pgfscope}%
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@@ -310,16 +1073,48 @@
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+\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}%
+\pgfusepath{stroke,fill}%
+}%
+\begin{pgfscope}%
+\pgfsys@transformshift{3.229452in}{0.370218in}%
+\pgfsys@useobject{currentmarker}{}%
+\end{pgfscope}%
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@@ -327,8 +1122,8 @@
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 \begin{pgfscope}%
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 \begin{pgfscope}%
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 \begin{pgfscope}%
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+\pgfpathlineto{\pgfqpoint{2.460573in}{1.467921in}}%
+\pgfpathlineto{\pgfqpoint{2.473559in}{1.472628in}}%
+\pgfpathlineto{\pgfqpoint{2.484546in}{1.473549in}}%
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+\pgfpathlineto{\pgfqpoint{2.505522in}{1.466094in}}%
+\pgfpathlineto{\pgfqpoint{2.516510in}{1.456841in}}%
+\pgfpathlineto{\pgfqpoint{2.528496in}{1.442365in}}%
+\pgfpathlineto{\pgfqpoint{2.541482in}{1.421636in}}%
+\pgfpathlineto{\pgfqpoint{2.556465in}{1.391771in}}%
+\pgfpathlineto{\pgfqpoint{2.575443in}{1.346664in}}%
+\pgfpathlineto{\pgfqpoint{2.642367in}{1.179908in}}%
+\pgfpathlineto{\pgfqpoint{2.654354in}{1.161187in}}%
+\pgfpathlineto{\pgfqpoint{2.662345in}{1.153583in}}%
+\pgfpathlineto{\pgfqpoint{2.668338in}{1.151382in}}%
+\pgfpathlineto{\pgfqpoint{2.673332in}{1.152493in}}%
+\pgfpathlineto{\pgfqpoint{2.678326in}{1.156976in}}%
+\pgfpathlineto{\pgfqpoint{2.683321in}{1.165718in}}%
+\pgfpathlineto{\pgfqpoint{2.689314in}{1.183573in}}%
+\pgfpathlineto{\pgfqpoint{2.695307in}{1.212352in}}%
+\pgfpathlineto{\pgfqpoint{2.701300in}{1.256368in}}%
+\pgfpathlineto{\pgfqpoint{2.708292in}{1.333653in}}%
+\pgfpathlineto{\pgfqpoint{2.719280in}{1.472196in}}%
+\pgfpathlineto{\pgfqpoint{2.720279in}{1.473481in}}%
+\pgfpathlineto{\pgfqpoint{2.722277in}{1.459537in}}%
+\pgfpathlineto{\pgfqpoint{2.725273in}{1.387304in}}%
+\pgfpathlineto{\pgfqpoint{2.730268in}{1.151360in}}%
+\pgfpathlineto{\pgfqpoint{2.730268in}{1.151360in}}%
+\pgfusepath{stroke}%
+\end{pgfscope}%
+\begin{pgfscope}%
+\pgfpathrectangle{\pgfqpoint{0.733531in}{0.370218in}}{\pgfqpoint{3.993473in}{1.103393in}}%
 \pgfusepath{clip}%
 \pgfsetrectcap%
 \pgfsetroundjoin%
@@ -636,15 +1426,15 @@
 \definecolor{currentstroke}{rgb}{1.000000,0.647059,0.000000}%
 \pgfsetstrokecolor{currentstroke}%
 \pgfsetdash{}{0pt}%
-\pgfpathmoveto{\pgfqpoint{2.189776in}{1.789929in}}%
-\pgfpathlineto{\pgfqpoint{2.198207in}{1.037413in}}%
-\pgfpathlineto{\pgfqpoint{2.204331in}{0.756839in}}%
-\pgfpathlineto{\pgfqpoint{2.205368in}{0.723819in}}%
-\pgfpathlineto{\pgfqpoint{2.205368in}{0.723819in}}%
+\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}}%
 \pgfusepath{stroke}%
 \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%
@@ -652,40 +1442,37 @@
 \definecolor{currentstroke}{rgb}{1.000000,0.000000,0.000000}%
 \pgfsetstrokecolor{currentstroke}%
 \pgfsetdash{}{0pt}%
-\pgfpathmoveto{\pgfqpoint{2.205368in}{0.723819in}}%
-\pgfpathlineto{\pgfqpoint{2.211596in}{0.579693in}}%
-\pgfpathlineto{\pgfqpoint{2.213153in}{0.554315in}}%
-\pgfpathlineto{\pgfqpoint{2.213932in}{0.554729in}}%
-\pgfpathlineto{\pgfqpoint{2.220938in}{0.631412in}}%
-\pgfpathlineto{\pgfqpoint{2.227945in}{0.675812in}}%
-\pgfpathlineto{\pgfqpoint{2.234951in}{0.701496in}}%
-\pgfpathlineto{\pgfqpoint{2.241958in}{0.715641in}}%
-\pgfpathlineto{\pgfqpoint{2.248186in}{0.721878in}}%
-\pgfpathlineto{\pgfqpoint{2.254414in}{0.724034in}}%
-\pgfpathlineto{\pgfqpoint{2.261420in}{0.723044in}}%
-\pgfpathlineto{\pgfqpoint{2.269984in}{0.718482in}}%
-\pgfpathlineto{\pgfqpoint{2.281661in}{0.708528in}}%
-\pgfpathlineto{\pgfqpoint{2.300345in}{0.688017in}}%
-\pgfpathlineto{\pgfqpoint{2.384424in}{0.591070in}}%
-\pgfpathlineto{\pgfqpoint{2.417121in}{0.560087in}}%
-\pgfpathlineto{\pgfqpoint{2.431134in}{0.549408in}}%
-\pgfpathlineto{\pgfqpoint{2.464610in}{0.575028in}}%
-\pgfpathlineto{\pgfqpoint{2.499642in}{0.597878in}}%
-\pgfpathlineto{\pgfqpoint{2.536232in}{0.618072in}}%
-\pgfpathlineto{\pgfqpoint{2.575936in}{0.636444in}}%
-\pgfpathlineto{\pgfqpoint{2.618753in}{0.652857in}}%
-\pgfpathlineto{\pgfqpoint{2.666242in}{0.667715in}}%
-\pgfpathlineto{\pgfqpoint{2.719180in}{0.680961in}}%
-\pgfpathlineto{\pgfqpoint{2.778347in}{0.692514in}}%
-\pgfpathlineto{\pgfqpoint{2.845298in}{0.702393in}}%
-\pgfpathlineto{\pgfqpoint{2.922370in}{0.710591in}}%
-\pgfpathlineto{\pgfqpoint{3.012676in}{0.717023in}}%
-\pgfpathlineto{\pgfqpoint{3.120888in}{0.721532in}}%
-\pgfpathlineto{\pgfqpoint{3.254012in}{0.723855in}}%
-\pgfpathlineto{\pgfqpoint{3.424505in}{0.723575in}}%
-\pgfpathlineto{\pgfqpoint{3.658834in}{0.719887in}}%
-\pgfpathlineto{\pgfqpoint{3.761597in}{0.717600in}}%
-\pgfpathlineto{\pgfqpoint{3.761597in}{0.717600in}}%
+\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}}%
 \pgfusepath{stroke}%
 \end{pgfscope}%
 \begin{pgfscope}%
@@ -695,8 +1482,8 @@
 \definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}%
 \pgfsetstrokecolor{currentstroke}%
 \pgfsetdash{}{0pt}%
-\pgfpathmoveto{\pgfqpoint{0.617954in}{0.548769in}}%
-\pgfpathlineto{\pgfqpoint{0.617954in}{2.301955in}}%
+\pgfpathmoveto{\pgfqpoint{0.733531in}{0.370218in}}%
+\pgfpathlineto{\pgfqpoint{0.733531in}{1.473611in}}%
 \pgfusepath{stroke}%
 \end{pgfscope}%
 \begin{pgfscope}%
@@ -706,8 +1493,8 @@
 \definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}%
 \pgfsetstrokecolor{currentstroke}%
 \pgfsetdash{}{0pt}%
-\pgfpathmoveto{\pgfqpoint{3.761597in}{0.548769in}}%
-\pgfpathlineto{\pgfqpoint{3.761597in}{2.301955in}}%
+\pgfpathmoveto{\pgfqpoint{4.727004in}{0.370218in}}%
+\pgfpathlineto{\pgfqpoint{4.727004in}{1.473611in}}%
 \pgfusepath{stroke}%
 \end{pgfscope}%
 \begin{pgfscope}%
@@ -717,8 +1504,8 @@
 \definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}%
 \pgfsetstrokecolor{currentstroke}%
 \pgfsetdash{}{0pt}%
-\pgfpathmoveto{\pgfqpoint{0.617954in}{0.548769in}}%
-\pgfpathlineto{\pgfqpoint{3.761597in}{0.548769in}}%
+\pgfpathmoveto{\pgfqpoint{0.733531in}{0.370218in}}%
+\pgfpathlineto{\pgfqpoint{4.727004in}{0.370218in}}%
 \pgfusepath{stroke}%
 \end{pgfscope}%
 \begin{pgfscope}%
@@ -728,8 +1515,8 @@
 \definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}%
 \pgfsetstrokecolor{currentstroke}%
 \pgfsetdash{}{0pt}%
-\pgfpathmoveto{\pgfqpoint{0.617954in}{2.301955in}}%
-\pgfpathlineto{\pgfqpoint{3.761597in}{2.301955in}}%
+\pgfpathmoveto{\pgfqpoint{0.733531in}{1.473611in}}%
+\pgfpathlineto{\pgfqpoint{4.727004in}{1.473611in}}%
 \pgfusepath{stroke}%
 \end{pgfscope}%
 \end{pgfpicture}%
diff --git a/buch/papers/ellfilter/python/elliptic.py b/buch/papers/ellfilter/python/elliptic.py
index c9cf5bd..6e0fd12 100644
--- a/buch/papers/ellfilter/python/elliptic.py
+++ b/buch/papers/ellfilter/python/elliptic.py
@@ -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 cfa16ea..20a7428 100644
--- a/buch/papers/ellfilter/python/elliptic2.py
+++ b/buch/papers/ellfilter/python/elliptic2.py
@@ -50,20 +50,20 @@ def ellip_filter(N, mode=-1):
     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 mode, c in enumerate(["green", "orange", "red"]):
     w, FN2, mag, a, b = ellip_filter(N, mode=mode)
-    plt.semilogy(w, FN2, label=f"$N={N}, k=0.1$", linewidth=1, color=c)
+    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),
     0.00992, 1e2-1,
     fc ='orange',
@@ -71,7 +71,7 @@ plt.gca().add_patch(Rectangle(
     lw = 10,
 ))
 
-plt.gca().add_patch(Rectangle(
+axs[0].add_patch(Rectangle(
     (1.00992, 100),
     1, 1e6,
     fc ='red',
@@ -83,54 +83,41 @@ 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))
 for mode, c in enumerate(["green", "orange", "red"]):
     w, FN2, mag, a, b = ellip_filter(N, mode=mode)
-    plt.plot(w, mag, linewidth=1, color=c)
+    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),
     0.00992, np.sqrt(2)/2 - 0.1,
     fc ='orange',
@@ -138,7 +125,7 @@ plt.gca().add_patch(Rectangle(
     lw = 10,
 ))
 
-plt.gca().add_patch(Rectangle(
+axs[1].add_patch(Rectangle(
     (1.00992, 0),
     1, 0.1,
     fc ='red',
@@ -146,11 +133,13 @@ plt.gca().add_patch(Rectangle(
     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 52dd705..bbb823a 100644
--- a/buch/papers/ellfilter/python/k.pgf
+++ b/buch/papers/ellfilter/python/k.pgf
@@ -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 4211053..a139fc4 100644
--- a/buch/papers/ellfilter/tikz/arccos.tikz.tex
+++ b/buch/papers/ellfilter/tikz/arccos.tikz.tex
@@ -23,26 +23,26 @@
         \clip(-7.5,-2) rectangle (7.5,2);
 
         % \pause
-        \draw[ultra thick, ->, orange] (1, 0) -- (0,0);
+        \draw[ultra thick, ->, darkgreen] (1, 0) -- (0,0);
         % \pause
-        \draw[ultra thick, ->, darkgreen] (0, 0) -- (0,1.5);
+        \draw[ultra thick, ->, orange] (0, 0) -- (0,1.5);
         % \pause
-        \draw[ultra thick, ->, red] (2, 0) -- (1,0);
+        \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}[opacity=0.5]
-                    \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);
+                \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[->, red] (2, 0) -- (3,0);
+                    \draw[->, cyan] (2, 0) -- (3,0);
                 \end{scope}
                 \node[zero] at (1,0) {};
                 \node[zero] at (3,0) {};
@@ -58,10 +58,10 @@
 
         \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 755e8a0..c3f11bb 100644
--- a/buch/papers/ellfilter/tikz/arccos2.tikz.tex
+++ b/buch/papers/ellfilter/tikz/arccos2.tikz.tex
@@ -2,21 +2,34 @@
 
     \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) {};
@@ -25,9 +38,9 @@
 
         \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) {$2\pi$};
@@ -35,12 +48,12 @@
         \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 (4,2) (n) {};
+    \node[zero] at (6.5,2) (n) {};
     \node[anchor=west] at (n.east) {Zero};
 
 \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 b2b0090..cc5852c 100644
--- a/buch/papers/ellfilter/tikz/cd.tikz.tex
+++ b/buch/papers/ellfilter/tikz/cd.tikz.tex
@@ -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[ultra thick, ->, darkgreen] (0, 0) -- (0,0.5);
-                \draw[ultra thick, ->, orange] (1, 0) -- (0,0);
-                \draw[ultra thick, ->, red] (2, 0) -- (1,0);
+                \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, ->, cyan] (0, 0.5) -- (1,0.5);
-
-
+                \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) {};
@@ -72,12 +75,12 @@
 
         \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/sn.tikz.tex b/buch/papers/ellfilter/tikz/sn.tikz.tex
index 8e4d223..c3df8d1 100644
--- a/buch/papers/ellfilter/tikz/sn.tikz.tex
+++ b/buch/papers/ellfilter/tikz/sn.tikz.tex
@@ -17,39 +17,45 @@
 
             \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, <-, orange] (2, 0) -- (1,0);
+                \draw[ultra thick, <-, darkgreen] (2, 0) -- (1,0);
                 % \pause
-                \draw[ultra thick, <-, darkgreen] (2,0.5) -- (2, 0);
+                \draw[ultra thick, <-, orange] (2,0.5) -- (2, 0);
                 % \pause
-                \draw[ultra thick, <-, cyan] (1, 0.5) -- (2,0.5);
+                \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, <-, red] (1, 0) -- (0,0);
+                \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, <-, blue] (0, 0) -- (0,0.5);
-                            \draw[opacity=0.5, <-, red] (1, 0) -- (0,0);
-                            \draw[opacity=0.5, <-, orange] (2, 0) -- (1,0);
-                            \draw[opacity=0.5, <-, darkgreen] (2,0.5) -- (2, 0);
-                            \draw[opacity=0.5, <-, cyan] (1, 0.5) -- (2,0.5);
-                            \draw[opacity=0.5, <-, purple] (0, 0.5) -- (1,0.5);
-                            \draw[opacity=0.5, <-, blue] (0,1) -- (0,0.5);
-                            \draw[opacity=0.5, <-, darkgreen] (2,0.5) -- (2, 1);
-                            \draw[opacity=0.5, <-, cyan] (3, 0.5) -- (2,0.5);
-                            \draw[opacity=0.5, <-, purple] (4, 0.5) -- (3,0.5);
-                            \draw[opacity=0.5, <-, orange] (2, 0) -- (3,0);
-                            \draw[opacity=0.5, <-, red] (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}
                     }
                 }
@@ -72,12 +78,12 @@
 
         \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..8a82c5f 100644
--- a/buch/papers/ellfilter/tschebyscheff.tex
+++ b/buch/papers/ellfilter/tschebyscheff.tex
@@ -1,8 +1,7 @@
 \section{Tschebyscheff-Filter}
 
-Als Einstieg betrachent Wir das Tschebyscheff-Filter, welches sehr verwand ist mit dem elliptischen Filter.
+Als Einstieg betrachten wir das Tschebyscheff-Filter, welches sehr verwand ist mit dem elliptischen Filter.
 Genauer ausgedrückt sind die Tschebyscheff-1 und -2 Filter Spezialfälle davon.
-
 Der Name des Filters deutet schon an, dass die Tschebyscheff-Polynome $T_N$ für das Filter relevant sind:
 \begin{align}
     T_{0}(x)&=1\\
@@ -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}
@@ -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)
     &=
@@ -88,46 +86,21 @@ Abbildung \ref{ellfilter:fig:arccos} zeigt den $\arccos$ 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 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 $N$.
+        Die eingefärbten Pfade sind Verläufe von $w~\forall~[-\infty, \infty]$ für $N = 4$.
         Je grösser die Ordnung $N$ gewählt wird, desto mehr Nullstellen werden passiert.
     }
     \label{ellfilter:fig:arccos2}
 \end{figure}
-Somit passert $\cos( N~\cos^{-1}(w))$ im Intervall $[-1, 1]$ $N$ Nullstellen.
 Durch die spezielle Anordnung der Nullstellen hat die Funktion Equirippel-Verhalten und ist dennoch ein Polynom, was sich perfekt für linear Filter eignet.