improved Einleitung

9 files changed, 222 insertions, 90 deletions

diff --git a/buch/papers/ellfilter/tikz/arccos.tikz.tex b/buch/papers/ellfilter/tikz/arccos.tikz.tex
index 987f885..4211053 100644
--- a/buch/papers/ellfilter/tikz/arccos.tikz.tex
+++ b/buch/papers/ellfilter/tikz/arccos.tikz.tex
@@ -8,29 +8,6 @@
 
     \begin{scope}[xscale=0.6]
 
-        \clip(-7.5,-2) rectangle (7.5,2);
-
-        \draw[ultra thick, ->, darkgreen] (0, 0) -- (0,1.5);
-        \draw[ultra thick, ->, orange] (1, 0) -- (0,0);
-        \draw[ultra thick, ->, red] (2, 0) -- (1,0);
-        \draw[ultra thick, ->, blue] (2,1.5) -- (2, 0);
-
-        \foreach \i in {-2,...,1} {
-            \begin{scope}[opacity=0.5, xshift=\i*4cm]
-                \draw[->, orange] (-1, 0) -- (0,0);
-                \draw[->, darkgreen] (0, 0) -- (0,1.5);
-                \draw[->, darkgreen] (0, 0) -- (0,-1.5);
-                \draw[->, orange] (1, 0) -- (0,0);
-                \draw[->, red] (2, 0) -- (1,0);
-                \draw[->, blue] (2,1.5) -- (2, 0);
-                \draw[->, blue] (2,-1.5) -- (2, 0);
-                \draw[->, red] (2, 0) -- (3,0);
-
-                \node[zero] at (1,0) {};
-                \node[zero] at (3,0) {};
-            \end{scope}
-        }
-
         \node[gray, anchor=north] at (-6,0) {$-3\pi$};
         \node[gray, anchor=north] at (-4,0) {$-2\pi$};
         \node[gray, anchor=north] at (-2,0) {$-\pi$};
@@ -43,8 +20,40 @@
         % \node[gray, anchor=south east] at (0, 0) {$0$};
         \node[gray, anchor=east] at (0, 1.5) {$\infty$};
 
+        \clip(-7.5,-2) rectangle (7.5,2);
+
+        % \pause
+        \draw[ultra thick, ->, orange] (1, 0) -- (0,0);
+        % \pause
+        \draw[ultra thick, ->, darkgreen] (0, 0) -- (0,1.5);
+        % \pause
+        \draw[ultra thick, ->, red] (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);
+                    \draw[->, blue] (2,1.5) -- (2, 0);
+                    \draw[->, blue] (2,-1.5) -- (2, 0);
+                    \draw[->, red] (2, 0) -- (3,0);
+                \end{scope}
+                \node[zero] at (1,0) {};
+                \node[zero] at (3,0) {};
+            \end{scope}
+        }
+
     \end{scope}
 
+    \node[zero] at (4,2) (n) {};
+    \node[anchor=west] at (n.east) {Zero};
+
     \begin{scope}[yshift=-3cm]
 
         \draw[gray, ->] (-5,0) -- (5,0) node[anchor=west]{$w$};
diff --git a/buch/papers/ellfilter/tikz/arccos2.tikz.tex b/buch/papers/ellfilter/tikz/arccos2.tikz.tex
index 3fc3cc6..755e8a0 100644
--- a/buch/papers/ellfilter/tikz/arccos2.tikz.tex
+++ b/buch/papers/ellfilter/tikz/arccos2.tikz.tex
@@ -14,7 +14,6 @@
             \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]{$\infty$};
 
-
             \node[zero] at (-7,0) {};
             \node[zero] at (-5,0) {};
             \node[zero] at (-3,0) {};
@@ -24,7 +23,6 @@
             \node[zero] at (5,0) {};
             \node[zero] at (7,0) {};
 
-
         \end{scope}
 
         \node[gray, anchor=north] at (-8,0) {$-4\pi$};
@@ -42,4 +40,7 @@
 
     \end{scope}
 
+    \node[zero] at (4,2) (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 7a2767b..b2b0090 100644
--- a/buch/papers/ellfilter/tikz/cd.tikz.tex
+++ b/buch/papers/ellfilter/tikz/cd.tikz.tex
@@ -63,6 +63,11 @@
 
     \end{scope}
 
+    \node[zero] at (4,3) (n) {};
+    \node[anchor=west] at (n.east) {Zero};
+    \node[pole, below=0.25cm of n] (n) {};
+    \node[anchor=west] at (n.east) {Pole};
+
     \begin{scope}[yshift=-4cm, xscale=0.75]
 
         \draw[gray, ->] (-6,0) -- (6,0) node[anchor=west]{$w$};
diff --git a/buch/papers/ellfilter/tikz/cd2.tikz.tex b/buch/papers/ellfilter/tikz/cd2.tikz.tex
index 425db95..bba5789 100644
--- a/buch/papers/ellfilter/tikz/cd2.tikz.tex
+++ b/buch/papers/ellfilter/tikz/cd2.tikz.tex
@@ -47,21 +47,6 @@
                 (5,0) node(t_k_unten){} -- node[right, xshift=0.1cm]{$K^\prime \frac{K_1N}{K} = K^\prime_1$}
                 (5,0.5) node(t_k_opt_unten){};
 
-            \foreach \i in {-2,...,1} {
-                \foreach \j in {-2,...,1} {
-                    \begin{scope}[xshift=\i*4cm, yshift=\j*1cm]
-
-                        \node[zero] at ( 1, 0) {};
-                        \node[zero] at ( 3, 0) {};
-                        \node[pole] at ( 1,0.5) {};
-                        \node[pole] at ( 3,0.5) {};
-
-                    \end{scope}
-                }
-            }
-
-
-
 
             \draw[ultra thick, ->, darkgreen] (5, 0) -- node[yshift=-0.5cm]{Durchlassbereich} (0,0);
             \draw[ultra thick, ->, orange] (-0, 0) --  node[align=center]{Übergangs-\\berech} (0,0.5);
@@ -74,10 +59,20 @@
             \draw (2,0.5)  node[dot]{} node[anchor=north]      {\small $-1/k$};
             \draw (4,0.5)  node[dot]{} node[anchor=north]      {\small $1/k$};
 
+            \foreach \i in {-2,...,1} {
+                \foreach \j in {-2,...,1} {
+                    \begin{scope}[xshift=\i*4cm, yshift=\j*1cm]
 
+                        \node[zero] at ( 1, 0) {};
+                        \node[zero] at ( 3, 0) {};
+                        \node[pole] at ( 1,0.5) {};
+                        \node[pole] at ( 3,0.5) {};
 
-        \end{scope}
+                    \end{scope}
+                }
+            }
 
+        \end{scope}
 
     \end{scope}
 
diff --git a/buch/papers/ellfilter/tikz/cd3.tikz.tex b/buch/papers/ellfilter/tikz/cd3.tikz.tex
index fa9cc08..ae18519 100644
--- a/buch/papers/ellfilter/tikz/cd3.tikz.tex
+++ b/buch/papers/ellfilter/tikz/cd3.tikz.tex
@@ -36,7 +36,6 @@
 
             % \fill[orange!30] (0,0) rectangle (5, 0.5);
             \fill[yellow!30] (0,0) rectangle (1, 0.5);
-            \node[] at (2.5, 0.25) {\small $N=5$};
 
 
             % \draw[decorate,decoration={brace,amplitude=3pt,mirror}, yshift=0.05cm]
@@ -62,11 +61,13 @@
 
 
 
-
+            \onslide<2->{
             \draw[ultra thick, ->, darkgreen] (5, 0) -- node[yshift=-0.4cm]{Durchlassbereich} (0,0);
             \draw[ultra thick, ->, orange] (-0, 0) --  node[align=center]{Übergangs-\\berech} (0,0.5);
             \draw[ultra thick, ->, red] (0,0.5) -- node[align=center, yshift=0.4cm]{Sperrbereich} (5, 0.5);
-
+            \node[] at (2.5, 0.25) {\small $N=5$};
+            }
+            \onslide<1->{
             \draw (4,0  )  node[dot]{} node[anchor=south]      {\small $1$};
             \draw (2,0  )  node[dot]{} node[anchor=south]      {\small $-1$};
             \draw (0,0  )  node[dot]{} node[anchor=south west] {\small $1$};
@@ -74,6 +75,7 @@
             \draw (2,0.5)  node[dot]{} node[anchor=north]      {\small $-1/k$};
             \draw (4,0.5)  node[dot]{} node[anchor=north]      {\small $1/k$};
 
+            }
 
 
         \end{scope}
diff --git a/buch/papers/ellfilter/tikz/elliptic_transform1.tikz.tex b/buch/papers/ellfilter/tikz/elliptic_transform1.tikz.tex
new file mode 100644
index 0000000..2a36ee0
--- /dev/null
+++ b/buch/papers/ellfilter/tikz/elliptic_transform1.tikz.tex
@@ -0,0 +1,76 @@
+\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2]
+
+    \tikzstyle{zero} = [draw, circle, inner sep =0, minimum height=0.15cm]
+
+    \tikzset{pole/.style={cross out, draw, minimum size=(0.15cm-\pgflinewidth), inner sep=0pt, outer sep=0pt}}
+
+    \begin{scope}[xscale=1, yscale=1.5]
+
+    \begin{scope}[]
+
+        \fill[orange!25] (0,0) rectangle (1.5, 0.75);
+        \fill[yellow!50] (0,0) rectangle (0.5, 0.25);
+
+        \draw[gray, ->] (0,-0.75) -- (0,1.25) node[anchor=south]{$\mathrm{Im}~z$};
+        \draw[gray, ->] (-1.75,0) -- (1.75,0) node[anchor=west]{$\mathrm{Re}~z$};
+
+        \draw[gray] ( 0.5,0) +(0,0.05) -- +(0, -0.05) node[inner sep=0, anchor=north] {\small $K$};
+        \draw[gray]  (0, 0.25) +(0.05, 0) -- +(-0.05, 0) node[inner sep=0, anchor=east]{\small $jK^\prime$};
+
+        % \draw[gray] ( 1.5,0) +(0,0.05) -- +(0, -0.05) node[inner sep=0, anchor=north] {\small $K_1$};
+        % \draw[gray]  (0, 0.75) +(0.05, 0) -- +(-0.05, 0) node[inner sep=0, anchor=east]{\small $jK_1^\prime$};
+
+        \clip(-1.6,-0.6) rectangle (1.6,1.6);
+        \begin{scope}[xscale=0.5, yscale=0.25, blue]
+            \foreach \i in {-1,...,1} {
+                \foreach \j in {-1,...,2} {
+                    \begin{scope}[xshift=\i*2cm, yshift=\j*2cm]
+                        \node[zero] at ( 1, 0) {};
+                        \node[zero] at ( -1, 0) {};
+                        \node[pole] at ( 1,1) {};
+                        \node[pole] at ( -1,1) {};
+                    \end{scope}
+                }
+            }
+        \end{scope}
+
+        \node at (0,2) {$\cd \left(N~K_1~z , k_1 \right)$};
+        \node at (0,2) {$w= \cd(z K, k)$};
+
+        \draw[scale=0.2, domain=0.02:5,  variable=\x, red] plot ({\x1+3}, {1/\x+2});
+
+    \end{scope}
+
+    \begin{scope}[xshift=5cm]
+
+        \fill[orange!50] (0,0) rectangle (1.5, 0.75);
+        \fill[yellow!25] (0,0) rectangle (0.5, 0.25);
+
+        \draw[gray, ->] (0,-0.75) -- (0,1.25) node[anchor=south]{$\mathrm{Im}~z$};
+        \draw[gray, ->] (-1.75,0) -- (1.75,0) node[anchor=west]{$\mathrm{Re}~z$};
+
+        % \draw[gray] ( 0.5,0) +(0,0.05) -- +(0, -0.05) node[inner sep=0, anchor=north] {\small $K$};
+        % \draw[gray]  (0, 0.25) +(0.05, 0) -- +(-0.05, 0) node[inner sep=0, anchor=east]{\small $jK^\prime$};
+
+        \draw[gray] ( 0.5,0) +(0,0.05) -- +(0, -0.05) node[inner sep=0, anchor=north] {\small $K_1$};
+        \draw[gray]  (0, 0.75) +(0.05, 0) -- +(-0.05, 0) node[inner sep=0, anchor=east]{\small $jK_1^\prime$};
+
+        \clip(-1.6,-0.6) rectangle (1.6,1.6);
+        \begin{scope}[xscale=0.5, yscale=0.75, red]
+            \foreach \i in {-1,...,1} {
+                \foreach \j in {-1,...,0} {
+                    \begin{scope}[xshift=\i*2cm, yshift=\j*2cm]
+                        \node[zero] at ( 1, 0) {};
+                        \node[zero] at ( -1, 0) {};
+                        \node[pole] at ( 1,1) {};
+                        \node[pole] at ( -1,1) {};
+                    \end{scope}
+                }
+            }
+        \end{scope}
+
+    \end{scope}
+
+\end{scope}
+
+\end{tikzpicture}
diff --git a/buch/papers/ellfilter/tikz/elliptic_transform.tikz.tex b/buch/papers/ellfilter/tikz/elliptic_transform2.tikz.tex
index c91ecf1..20c2d82 100644
--- a/buch/papers/ellfilter/tikz/elliptic_transform.tikz.tex
+++ b/buch/papers/ellfilter/tikz/elliptic_transform2.tikz.tex
@@ -9,55 +9,66 @@
     \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}}
+    \tikzset{pole/.style={cross out, draw, minimum size=(0.15cm-\pgflinewidth), inner sep=0pt, outer sep=0pt}}
 
     \begin{scope}[xscale=3, yscale=3]
 
     \begin{scope}[]
-
+        % \onslide<4->{
         \fill[orange!30, scale=1.735] (0,0) rectangle (\d*\a+0.5, \d/\a+0.5);
+        % }
+        % \onslide<2->{
         \fill[yellow!30] (0,0) rectangle (\d*\a+0.5, \d/\a+0.5);
+        % }
 
-        \begin{scope}[scale=1.735, red]
-            \draw (0,0) rectangle (\d*\a/\n+0.5/\n, \d/\a+0.5);
-            \draw[gray] (0,0) -- (\d*\a/\n+0.5/\n, \d/\a+0.5);
-
-            \node[zero] at ( \d*\a/\n+0.5/\n, \d/\a+0.5) {};
-            \node[pole, color=red] at ( \d*\a/\n+0.5/\n, 0) {};
-
-
-            \draw[] ( \d*\a/\n+0.5/\n,0)  node[anchor=north] {\small $K_1$};
-            \draw[]  (0, \d/\a+0.5)  node[anchor=east]{\small $jK_1^\prime$};
-
+        \begin{scope}[]
+            \clip(0,0) rectangle (2,1.25);
+            \draw[thick, scale=1, domain=0.1:10,  variable=\x, smooth, samples=200] plot ({\d*\x1+0.5}, {\d/\x+0.5});
+            \node at(1.25,0.7) {$K + jK^\prime$ Ortskurve};
         \end{scope}
 
+        % \onslide<2->{
         \begin{scope}[blue]
             \draw[] (0,0) rectangle (\d*\a+0.5, \d/\a+0.5);
-            \foreach \i in {1,...,\nn} {
-                \draw[gray, dotted] (\i*\d*\a/\n+\i*0.5/\n, 0) -- (\i*\d*\a/\n+\i*0.5/\n, \d/\a+0.5);
-            }
 
-            \node[zero] at ( \d*\a+0.5, \d/\a+0.5) {};
-            \node[pole, color=blue] at ( \d*\a+0.5, 0) {};
+
+            \node[pole] at ( \d*\a+0.5, \d/\a+0.5) {};
+            \node[zero] at ( \d*\a+0.5, 0) {};
 
             \draw[] ( \d*\a+0.5,0)  node[anchor=north] {\small $K$};
             \draw[]  (0, \d/\a+0.5)  node[anchor=east]{\small $jK^\prime$};
 
+            % \onslide<3->{
+
+            \foreach \i in {1,...,\nn} {
+                \draw[gray, dotted] (\i*\d*\a/\n+\i*0.5/\n, 0) -- (\i*\d*\a/\n+\i*0.5/\n, \d/\a+0.5);
+            }
+
             \node[dot, gray] at (\d*\a/\n+0.5/\n, \d/\a+0.5) {};
             \node[above] at (0.5*\d*\a/\n+0.5*0.5/\n, \d/\a+0.5) {\small $K/N$};
-
+            % }
         \end{scope}
+        % }
 
-        \draw[thick, gray, ->] (0,-0.25) -- (0,1.25) node[anchor=south]{$\mathrm{Im}$};
-        \draw[thick, gray, ->] (-0.25,0) -- (2,0) node[anchor=west]{$\mathrm{Re}$};
+        % \onslide<4->{
+        \begin{scope}[scale=1.735, red]
+            \draw (0,0) rectangle (\d*\a/\n+0.5/\n, \d/\a+0.5);
+            \draw[gray] (0,0) -- (\d*\a/\n+0.5/\n, \d/\a+0.5);
 
-        \begin{scope}[]
-            \clip(0,0) rectangle (2,1.25);
-            \draw[scale=1, domain=0.1:10,  variable=\x, smooth, samples=200] plot ({\d*\x1+0.5}, {\d/\x+0.5});
+            \node[pole] at ( \d*\a/\n+0.5/\n, \d/\a+0.5) {};
+            \node[zero] at ( \d*\a/\n+0.5/\n, 0) {};
+
+
+            \draw[] ( \d*\a/\n+0.5/\n,0)  node[anchor=north] {\small $K_1$};
+            \draw[]  (0, \d/\a+0.5)  node[anchor=east]{\small $jK_1^\prime$};
 
         \end{scope}
-    \end{scope}
+        % }
 
+        \draw[gray, ->] (0,-0.25) -- (0,1.25) node[anchor=south]{$\mathrm{Im}$};
+        \draw[gray, ->] (-0.25,0) -- (2,0) node[anchor=west]{$\mathrm{Re}$};
+
+    \end{scope}
 
 \end{scope}
 
diff --git a/buch/papers/ellfilter/tikz/filter.tikz.tex b/buch/papers/ellfilter/tikz/filter.tikz.tex
new file mode 100644
index 0000000..05b59b9
--- /dev/null
+++ b/buch/papers/ellfilter/tikz/filter.tikz.tex
@@ -0,0 +1,26 @@
+\begin{tikzpicture}[>=stealth', auto, node distance=2cm, scale=1.2]
+
+    \tikzstyle{zero} = [draw, circle, inner sep =0, minimum height=0.15cm]
+
+    \tikzset{pole/.style={cross out, draw=black, minimum size=(0.15cm-\pgflinewidth), inner sep=0pt, outer sep=0pt}}
+
+    \begin{scope}[xscale=2, yscale=2]
+
+        \fill[ gray!20] (0,0) rectangle  (1,0.707);
+
+        \draw[gray, ->] (0,-0.25) -- (0,1.25) node[anchor=south]{$|H(\Omega)|$};
+        \draw[gray, ->] (-0.25,0) -- (3,0) node[anchor=west]{$\Omega$};
+
+        \draw[fill = gray!20] (0,0.707) node[left] {$\sqrt{\frac{1}{1+\varepsilon^2}}$} -| (1,0) node[below] {$\Omega_p$};
+
+        \draw[fill = gray!20] (0,0.707) node[left] {$\sqrt{\frac{1}{1+\varepsilon^2}}$} -| (1,0) node[below] {$\Omega_p$};
+
+        \begin{scope}[]
+            \draw[thick, domain=0:2.5,  variable=\x, smooth, samples=200] plot
+                ({\x}, {sqrt(abs(1/ (1 + \x^10)))});
+
+        \end{scope}
+
+    \end{scope}
+
+\end{tikzpicture}
diff --git a/buch/papers/ellfilter/tikz/sn.tikz.tex b/buch/papers/ellfilter/tikz/sn.tikz.tex
index 6ced3c5..8e4d223 100644
--- a/buch/papers/ellfilter/tikz/sn.tikz.tex
+++ b/buch/papers/ellfilter/tikz/sn.tikz.tex
@@ -17,29 +17,33 @@
 
             \begin{scope}[xshift=-1cm]
 
-                \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, ->, 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);
-
+                % \pause
+                \draw[ultra thick, <-, orange] (2, 0) -- (1,0);
+                % \pause
+                \draw[ultra thick, <-, darkgreen] (2,0.5) -- (2, 0);
+                % \pause
+                \draw[ultra thick, <-, cyan] (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);
+                % \pause
 
                 \foreach \i in {-2,...,2} {
                     \foreach \j in {-2,...,1} {
                         \begin{scope}[xshift=\i*4cm, yshift=\j*1cm]
-                            \draw[opacity=0.5, ->, darkgreen] (0, 0) -- (0,0.5);
-                            \draw[opacity=0.5, ->, orange] (1, 0) -- (0,0);
-                            \draw[opacity=0.5, ->, red] (2, 0) -- (1,0);
-                            \draw[opacity=0.5, ->, blue] (2,0.5) -- (2, 0);
-                            \draw[opacity=0.5, ->, purple] (1, 0.5) -- (2,0.5);
-                            \draw[opacity=0.5, ->, cyan] (0, 0.5) -- (1,0.5);
-                            \draw[opacity=0.5, ->, darkgreen] (0,1) -- (0,0.5);
-                            \draw[opacity=0.5, ->, blue] (2,0.5) -- (2, 1);
-                            \draw[opacity=0.5, ->, purple] (3, 0.5) -- (2,0.5);
-                            \draw[opacity=0.5, ->, cyan] (4, 0.5) -- (3,0.5);
-                            \draw[opacity=0.5, ->, red] (2, 0) -- (3,0);
-                            \draw[opacity=0.5, ->, orange] (3, 0) -- (4,0);
+                            \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) {};
@@ -57,10 +61,13 @@
         \draw[gray] ( 1,0) +(0,0.1) -- +(0, -0.1) node[inner sep=0, anchor=north] {\small $K$};
         \draw[gray]  (0, 0.5) +(0.1, 0) -- +(-0.1, 0) node[inner sep=0, anchor=east]{\small $jK^\prime$};
 
-
-
     \end{scope}
 
+    \node[zero] at (4,3) (n) {};
+    \node[anchor=west] at (n.east) {Zero};
+    \node[pole, below=0.25cm of n] (n) {};
+    \node[anchor=west] at (n.east) {Pole};
+
     \begin{scope}[yshift=-4cm, xscale=0.75]
 
         \draw[gray, ->] (-6,0) -- (6,0) node[anchor=west]{$w$};