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-rw-r--r--buch/papers/ellfilter/einleitung.tex2
-rw-r--r--buch/papers/ellfilter/elliptic.tex30
-rw-r--r--buch/papers/ellfilter/jacobi.tex6
-rw-r--r--buch/papers/ellfilter/python/elliptic.pgf977
-rw-r--r--buch/papers/ellfilter/python/elliptic2.py12
-rw-r--r--buch/papers/ellfilter/tschebyscheff.tex9
6 files changed, 428 insertions, 608 deletions
diff --git a/buch/papers/ellfilter/einleitung.tex b/buch/papers/ellfilter/einleitung.tex
index a2fc981..0c51ae0 100644
--- a/buch/papers/ellfilter/einleitung.tex
+++ b/buch/papers/ellfilter/einleitung.tex
@@ -66,7 +66,7 @@ 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.
In vielen Anwendung sind Filter mit einem steilen Übergang gewünscht.
-Da es technisch nicht möglich ist, mit einer rationalen Funktion mit begrezter 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.
+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.
diff --git a/buch/papers/ellfilter/elliptic.tex b/buch/papers/ellfilter/elliptic.tex
index 39f9b8d..fc9d5b6 100644
--- a/buch/papers/ellfilter/elliptic.tex
+++ b/buch/papers/ellfilter/elliptic.tex
@@ -40,7 +40,7 @@ Die Idee des elliptischen Filter ist es, diese zwei Equiripple-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 Equiripple-Verhalten im Durchlass- und Sperrbereich aufweist.
Abbildung \ref{ellfilter:fig:elliptic_freq} zeigt eine elliptisch rationale Funktion und die Frequenzantwort des daraus resultierenden Filters.
@@ -52,7 +52,7 @@ Abbildung \ref{ellfilter:fig:elliptic_freq} zeigt eine elliptisch rationale Funk
\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 Sperrbereichverhalten erreicht werden, wenn die Funktion auf eine andere Reihe ansteigen würde.
+Theoretisch könnte eine gleiches Durchlass- und Sperrbereichsverhalten erreicht werden, wenn die Funktion auf eine andere Reihe ansteigen würde.
Dies würde jedoch zu Oszillationen zwischen $1$ und $1/k$ im Übergangsbereich führen.
\subsection{Gradgleichung}
@@ -85,22 +85,16 @@ k_1 = k^N \prod_{i=1}^L \sn^4 \Bigg( \frac{2i - 1}{N} K, k \Bigg),
N = 2L+r.
\end{equation}
Die Herleitung ist sehr umfassend und wird in \cite{ellfilter:bib:orfanidis} im Detail angeschaut.
-Für das Auslegen von elliptischen Filtern müssen $k$ und $k_1$ mit \eqref{ellfilter:eq:degeqsol} oder mit numerischen Methoden berechnet werden.
-Die Position der Pol- und Nullstellen können dann konstruiert werden, wie dargestellt in Abbildung \ref{ellfilter:fig:cd2} und mit der $\cd$-Funktion zu der elliptischen rationalen Funktion transformiert werden.
-
-% \begin{figure}
-% \centering
-% \input{papers/ellfilter/tikz/elliptic_transform1.tikz}
-% \caption{Die Gradgleichung als geometrisches Problem.}
-% \end{figure}
-
-\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.
-
+Um ein elliptisches Filter auszulegen, kann die Ordnung $N$ und der Parameter $k$ 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 $k_1$ 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:cd2}.
+Durch das Rücktransformieren mit der $\cd$-Funktion gelangt man schlussendlich zu der elliptischen rationalen Funktion. %TODO check
+% \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.
diff --git a/buch/papers/ellfilter/jacobi.tex b/buch/papers/ellfilter/jacobi.tex
index eed9a12..841cd7d 100644
--- a/buch/papers/ellfilter/jacobi.tex
+++ b/buch/papers/ellfilter/jacobi.tex
@@ -3,7 +3,7 @@
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 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 ausschliesslich um die Jacobischen elliptischen Funktionen geht.
+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}
@@ -11,7 +11,7 @@ Die Jacobi elliptischen Funktionen werden ausführlich im Kapitel \ref{buch:elli
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 $z$.
+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.
@@ -173,7 +173,7 @@ 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
@@ -51,16 +51,16 @@
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@@ -438,7 +267,7 @@
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+\pgftext[x=0.730012in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.00}\)}%
\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%
@@ -932,8 +756,8 @@
\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}%
\pgfsetstrokecolor{currentstroke}%
\pgfsetdash{}{0pt}%
-\pgfpathmoveto{\pgfqpoint{1.232715in}{0.370218in}}%
-\pgfpathlineto{\pgfqpoint{1.232715in}{1.473611in}}%
+\pgfpathmoveto{\pgfqpoint{1.227693in}{0.548769in}}%
+\pgfpathlineto{\pgfqpoint{1.227693in}{1.481479in}}%
\pgfusepath{stroke}%
\end{pgfscope}%
\begin{pgfscope}%
@@ -951,7 +775,7 @@
\pgfusepath{stroke,fill}%
}%
\begin{pgfscope}%
-\pgfsys@transformshift{1.232715in}{0.370218in}%
+\pgfsys@transformshift{1.227693in}{0.548769in}%
\pgfsys@useobject{currentmarker}{}%
\end{pgfscope}%
\end{pgfscope}%
@@ -959,10 +783,10 @@
\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
\pgfsetstrokecolor{textcolor}%
\pgfsetfillcolor{textcolor}%
-\pgftext[x=1.232715in,y=0.272996in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.25}\)}%
+\pgftext[x=1.227693in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.25}\)}%
\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%
@@ -970,8 +794,8 @@
\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}%
\pgfsetstrokecolor{currentstroke}%
\pgfsetdash{}{0pt}%
-\pgfpathmoveto{\pgfqpoint{1.731899in}{0.370218in}}%
-\pgfpathlineto{\pgfqpoint{1.731899in}{1.473611in}}%
+\pgfpathmoveto{\pgfqpoint{1.725373in}{0.548769in}}%
+\pgfpathlineto{\pgfqpoint{1.725373in}{1.481479in}}%
\pgfusepath{stroke}%
\end{pgfscope}%
\begin{pgfscope}%
@@ -989,7 +813,7 @@
\pgfusepath{stroke,fill}%
}%
\begin{pgfscope}%
-\pgfsys@transformshift{1.731899in}{0.370218in}%
+\pgfsys@transformshift{1.725373in}{0.548769in}%
\pgfsys@useobject{currentmarker}{}%
\end{pgfscope}%
\end{pgfscope}%
@@ -997,10 +821,10 @@
\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
\pgfsetstrokecolor{textcolor}%
\pgfsetfillcolor{textcolor}%
-\pgftext[x=1.731899in,y=0.272996in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.50}\)}%
+\pgftext[x=1.725373in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.50}\)}%
\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%
@@ -1008,8 +832,8 @@
\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}%
\pgfsetstrokecolor{currentstroke}%
\pgfsetdash{}{0pt}%
-\pgfpathmoveto{\pgfqpoint{2.231083in}{0.370218in}}%
-\pgfpathlineto{\pgfqpoint{2.231083in}{1.473611in}}%
+\pgfpathmoveto{\pgfqpoint{2.223054in}{0.548769in}}%
+\pgfpathlineto{\pgfqpoint{2.223054in}{1.481479in}}%
\pgfusepath{stroke}%
\end{pgfscope}%
\begin{pgfscope}%
@@ -1027,7 +851,7 @@
\pgfusepath{stroke,fill}%
}%
\begin{pgfscope}%
-\pgfsys@transformshift{2.231083in}{0.370218in}%
+\pgfsys@transformshift{2.223054in}{0.548769in}%
\pgfsys@useobject{currentmarker}{}%
\end{pgfscope}%
\end{pgfscope}%
@@ -1035,10 +859,10 @@
\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
\pgfsetstrokecolor{textcolor}%
\pgfsetfillcolor{textcolor}%
-\pgftext[x=2.231083in,y=0.272996in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.75}\)}%
+\pgftext[x=2.223054in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.75}\)}%
\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%
@@ -1046,8 +870,8 @@
\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}%
\pgfsetstrokecolor{currentstroke}%
\pgfsetdash{}{0pt}%
-\pgfpathmoveto{\pgfqpoint{2.730268in}{0.370218in}}%
-\pgfpathlineto{\pgfqpoint{2.730268in}{1.473611in}}%
+\pgfpathmoveto{\pgfqpoint{2.720735in}{0.548769in}}%
+\pgfpathlineto{\pgfqpoint{2.720735in}{1.481479in}}%
\pgfusepath{stroke}%
\end{pgfscope}%
\begin{pgfscope}%
@@ -1065,7 +889,7 @@
\pgfusepath{stroke,fill}%
}%
\begin{pgfscope}%
-\pgfsys@transformshift{2.730268in}{0.370218in}%
+\pgfsys@transformshift{2.720735in}{0.548769in}%
\pgfsys@useobject{currentmarker}{}%
\end{pgfscope}%
\end{pgfscope}%
@@ -1073,10 +897,10 @@
\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
\pgfsetstrokecolor{textcolor}%
\pgfsetfillcolor{textcolor}%
-\pgftext[x=2.730268in,y=0.272996in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {1.00}\)}%
+\pgftext[x=2.720735in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {1.00}\)}%
\end{pgfscope}%
\begin{pgfscope}%
-\pgfpathrectangle{\pgfqpoint{0.733531in}{0.370218in}}{\pgfqpoint{3.993473in}{1.103393in}}%
+\pgfpathrectangle{\pgfqpoint{0.730012in}{0.548769in}}{\pgfqpoint{3.981446in}{0.932710in}}%
\pgfusepath{clip}%
\pgfsetrectcap%
\pgfsetroundjoin%
@@ -1084,8 +908,8 @@
\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}%
\pgfsetstrokecolor{currentstroke}%
\pgfsetdash{}{0pt}%
-\pgfpathmoveto{\pgfqpoint{3.229452in}{0.370218in}}%
-\pgfpathlineto{\pgfqpoint{3.229452in}{1.473611in}}%
+\pgfpathmoveto{\pgfqpoint{3.218416in}{0.548769in}}%
+\pgfpathlineto{\pgfqpoint{3.218416in}{1.481479in}}%
\pgfusepath{stroke}%
\end{pgfscope}%
\begin{pgfscope}%
@@ -1103,7 +927,7 @@
\pgfusepath{stroke,fill}%
}%
\begin{pgfscope}%
-\pgfsys@transformshift{3.229452in}{0.370218in}%
+\pgfsys@transformshift{3.218416in}{0.548769in}%
\pgfsys@useobject{currentmarker}{}%
\end{pgfscope}%
\end{pgfscope}%
@@ -1111,10 +935,10 @@
\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
\pgfsetstrokecolor{textcolor}%
\pgfsetfillcolor{textcolor}%
-\pgftext[x=3.229452in,y=0.272996in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {1.25}\)}%
+\pgftext[x=3.218416in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {1.25}\)}%
\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%
@@ -1122,8 +946,8 @@
\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}%
\pgfsetstrokecolor{currentstroke}%
\pgfsetdash{}{0pt}%
-\pgfpathmoveto{\pgfqpoint{3.728636in}{0.370218in}}%
-\pgfpathlineto{\pgfqpoint{3.728636in}{1.473611in}}%
+\pgfpathmoveto{\pgfqpoint{3.716097in}{0.548769in}}%
+\pgfpathlineto{\pgfqpoint{3.716097in}{1.481479in}}%
\pgfusepath{stroke}%
\end{pgfscope}%
\begin{pgfscope}%
@@ -1141,7 +965,7 @@
\pgfusepath{stroke,fill}%
}%
\begin{pgfscope}%
-\pgfsys@transformshift{3.728636in}{0.370218in}%
+\pgfsys@transformshift{3.716097in}{0.548769in}%
\pgfsys@useobject{currentmarker}{}%
\end{pgfscope}%
\end{pgfscope}%
@@ -1149,10 +973,10 @@
\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
\pgfsetstrokecolor{textcolor}%
\pgfsetfillcolor{textcolor}%
-\pgftext[x=3.728636in,y=0.272996in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {1.50}\)}%
+\pgftext[x=3.716097in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {1.50}\)}%
\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%
@@ -1160,8 +984,8 @@
\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}%
\pgfsetstrokecolor{currentstroke}%
\pgfsetdash{}{0pt}%
-\pgfpathmoveto{\pgfqpoint{4.227820in}{0.370218in}}%
-\pgfpathlineto{\pgfqpoint{4.227820in}{1.473611in}}%
+\pgfpathmoveto{\pgfqpoint{4.213778in}{0.548769in}}%
+\pgfpathlineto{\pgfqpoint{4.213778in}{1.481479in}}%
\pgfusepath{stroke}%
\end{pgfscope}%
\begin{pgfscope}%
@@ -1179,7 +1003,7 @@
\pgfusepath{stroke,fill}%
}%
\begin{pgfscope}%
-\pgfsys@transformshift{4.227820in}{0.370218in}%
+\pgfsys@transformshift{4.213778in}{0.548769in}%
\pgfsys@useobject{currentmarker}{}%
\end{pgfscope}%
\end{pgfscope}%
@@ -1187,10 +1011,10 @@
\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
\pgfsetstrokecolor{textcolor}%
\pgfsetfillcolor{textcolor}%
-\pgftext[x=4.227820in,y=0.272996in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {1.75}\)}%
+\pgftext[x=4.213778in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {1.75}\)}%
\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%
@@ -1198,8 +1022,8 @@
\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}%
\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}%
@@ -1217,7 +1041,7 @@
\pgfusepath{stroke,fill}%
}%
\begin{pgfscope}%
-\pgfsys@transformshift{4.727004in}{0.370218in}%
+\pgfsys@transformshift{4.711458in}{0.548769in}%
\pgfsys@useobject{currentmarker}{}%
\end{pgfscope}%
\end{pgfscope}%
@@ -1225,10 +1049,16 @@
\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
\pgfsetstrokecolor{textcolor}%
\pgfsetfillcolor{textcolor}%
-\pgftext[x=4.727004in,y=0.272996in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {2.00}\)}%
+\pgftext[x=4.711458in,y=0.451547in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {2.00}\)}%
+\end{pgfscope}%
+\begin{pgfscope}%
+\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
+\pgfsetstrokecolor{textcolor}%
+\pgfsetfillcolor{textcolor}%
+\pgftext[x=2.720735in,y=0.272534in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle w\)}%
\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%
@@ -1236,8 +1066,8 @@
\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}%
\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}%
@@ -1255,7 +1085,7 @@
\pgfusepath{stroke,fill}%
}%
\begin{pgfscope}%
-\pgfsys@transformshift{0.733531in}{0.370218in}%
+\pgfsys@transformshift{0.730012in}{0.548769in}%
\pgfsys@useobject{currentmarker}{}%
\end{pgfscope}%
\end{pgfscope}%
@@ -1263,10 +1093,10 @@
\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
\pgfsetstrokecolor{textcolor}%
\pgfsetfillcolor{textcolor}%
-\pgftext[x=0.458839in, y=0.321992in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.0}\)}%
+\pgftext[x=0.455320in, y=0.500544in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.0}\)}%
\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%
@@ -1274,8 +1104,8 @@
\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}%
\pgfsetstrokecolor{currentstroke}%
\pgfsetdash{}{0pt}%
-\pgfpathmoveto{\pgfqpoint{0.733531in}{0.921914in}}%
-\pgfpathlineto{\pgfqpoint{4.727004in}{0.921914in}}%
+\pgfpathmoveto{\pgfqpoint{0.730012in}{1.015124in}}%
+\pgfpathlineto{\pgfqpoint{4.711458in}{1.015124in}}%
\pgfusepath{stroke}%
\end{pgfscope}%
\begin{pgfscope}%
@@ -1293,7 +1123,7 @@
\pgfusepath{stroke,fill}%
}%
\begin{pgfscope}%
-\pgfsys@transformshift{0.733531in}{0.921914in}%
+\pgfsys@transformshift{0.730012in}{1.015124in}%
\pgfsys@useobject{currentmarker}{}%
\end{pgfscope}%
\end{pgfscope}%
@@ -1301,10 +1131,10 @@
\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
\pgfsetstrokecolor{textcolor}%
\pgfsetfillcolor{textcolor}%
-\pgftext[x=0.458839in, y=0.873689in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.5}\)}%
+\pgftext[x=0.455320in, y=0.966899in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {0.5}\)}%
\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%
@@ -1312,8 +1142,8 @@
\definecolor{currentstroke}{rgb}{0.690196,0.690196,0.690196}%
\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}%
\begin{pgfscope}%
@@ -1331,7 +1161,7 @@
\pgfusepath{stroke,fill}%
}%
\begin{pgfscope}%
-\pgfsys@transformshift{0.733531in}{1.473611in}%
+\pgfsys@transformshift{0.730012in}{1.481479in}%
\pgfsys@useobject{currentmarker}{}%
\end{pgfscope}%
\end{pgfscope}%
@@ -1339,16 +1169,16 @@
\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
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\pgfsetfillcolor{textcolor}%
-\pgftext[x=0.458839in, y=1.425386in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {1.0}\)}%
+\pgftext[x=0.455320in, y=1.433254in, left, base]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle {1.0}\)}%
\end{pgfscope}%
\begin{pgfscope}%
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-\pgftext[x=0.403284in,y=0.921914in,,bottom,rotate=90.000000]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle |H(w)|\)}%
+\pgftext[x=0.399764in,y=1.015124in,,bottom,rotate=90.000000]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle |H(w)|\)}%
\end{pgfscope}%
\begin{pgfscope}%
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+\pgfpathrectangle{\pgfqpoint{0.730012in}{0.548769in}}{\pgfqpoint{3.981446in}{0.932710in}}%
\pgfusepath{clip}%
\pgfsetrectcap%
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@@ -1356,69 +1186,65 @@
\definecolor{currentstroke}{rgb}{0.000000,0.501961,0.000000}%
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-\pgfpathlineto{\pgfqpoint{0.778480in}{1.472092in}}%
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-\pgfpathrectangle{\pgfqpoint{0.733531in}{0.370218in}}{\pgfqpoint{3.993473in}{1.103393in}}%
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@@ -1426,15 +1252,15 @@
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-\pgfpathrectangle{\pgfqpoint{0.733531in}{0.370218in}}{\pgfqpoint{3.993473in}{1.103393in}}%
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@@ -1442,37 +1268,34 @@
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diff --git a/buch/papers/ellfilter/python/elliptic2.py b/buch/papers/ellfilter/python/elliptic2.py
index 20a7428..6f03ecf 100644
--- a/buch/papers/ellfilter/python/elliptic2.py
+++ b/buch/papers/ellfilter/python/elliptic2.py
@@ -80,12 +80,12 @@ 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 +93,11 @@ 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 +105,7 @@ 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"]):
@@ -135,11 +137,13 @@ 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()
diff --git a/buch/papers/ellfilter/tschebyscheff.tex b/buch/papers/ellfilter/tschebyscheff.tex
index 5024bec..0a48949 100644
--- a/buch/papers/ellfilter/tschebyscheff.tex
+++ b/buch/papers/ellfilter/tschebyscheff.tex
@@ -17,7 +17,7 @@ Bemerkenswert ist, dass die Polynome im Intervall $[-1, 1]$ mit der trigonometri
\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}
@@ -37,7 +37,6 @@ Wenn wir die Tschebyscheff-Polynome quadrieren, passen sie perfekt in die Forder
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.
-
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}
@@ -92,7 +91,7 @@ Das Einzeichnen von Pol- und Nullstellen ist hilfreich für die Betrachtung der
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}.
+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}
@@ -105,5 +104,5 @@ Somit passiert $\cos( N~\cos^{-1}(w))$ im Intervall $[-1, 1]$ $N$ Nullstellen, w
}
\label{ellfilter:fig:arccos2}
\end{figure}
-Durch die spezielle Anordnung der Nullstellen hat die Funktion auf der reellen Achse Equiripple-Verhalten und ist dennoch ein Polynom, was sich perfekt für linear Filter eignet.
-Equiripple 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 $|w| <= 1$ ist die Funktion begrenzt zwischen $-1$ und $1$.
generated by cgit v1.2.1 (git 2.18.0) at 2024-12-23 13:29:09 +0100