Difference between revisions of "Applets:Complementary Gaussian Error Functions"

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    ''Hinweis:''   Das Applet ist für den '''CHROME'''–Browser optimiert. Bei anderen Browsern kommt es teilweise zu Darstellungsproblemen.
 
  
  
  
==Programmbeschreibung==
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==Applet Description==
 
<br>
 
<br>
Dieses Applet ermöglicht die Berechnung und graphische Darstellung der Gaußschen Fehlerfunktionen&nbsp; ${\rm Q}(x)$&nbsp; und &nbsp;$1/2\cdot {\rm erfc}(x)$, die für die Fehlerwahrscheinlichkeitsberechnung von großer Bedeutung sind.  
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This applet allows the calculation and graphical representation of the (complementary) Gaussian error functions&nbsp; ${\rm Q}(x)$&nbsp; and &nbsp;$1/2\cdot {\rm erfc}(x)$, which are of great importance for error probability calculation.  
*Sowohl die Abszisse als auch der Funktionswert können entweder linear oder logarithmisch dargestellt werden.
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*Both the abscissa and the function value can be represented either linearly or logarithmically.
*Für beide Funktionen wird jeweils eine obere Schranke (englisch:&nbsp; ''Upper Bound '') und eine untere Schranke (englisch:&nbsp; ''Lower Bound'') angegeben.
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*For both functions an upper bound&nbsp; $\rm (UB)$&nbsp; and a lower bound&nbsp; $\rm (LB)$&nbsp; are given.
  
 
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==Theoretical Background==
==Theoretischer Hintergrund==
 
 
<br>
 
<br>
Bei der Untersuchung digitaler Übertragungssysteme muss oft die Wahrscheinlichkeit bestimmt werden, dass eine (mittelwertfreie) gaußverteilte Zufallsgröße&nbsp; $x$&nbsp; mit der Varianz&nbsp; $σ^2$&nbsp; einen vorgegebenen Wert&nbsp; $x_0$&nbsp; überschreitet. Für diese Wahrscheinlichkeit gilt:  
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In the study of digital transmission systems, it is often necessary to determine the probability that a (zero mean) Gaussian distributed random variable&nbsp; $x$&nbsp; with variance&nbsp; $σ^2$&nbsp; exceeds a given value&nbsp; $x_0$.&nbsp; For this probability holds:  
 
:$${\rm Pr}(x > x_0)={\rm Q}(\frac{x_0}{\sigma}) = 1/2 \cdot {\rm erfc}(\frac{x_0}{\sqrt{2} \cdot \sigma}).$$
 
:$${\rm Pr}(x > x_0)={\rm Q}(\frac{x_0}{\sigma}) = 1/2 \cdot {\rm erfc}(\frac{x_0}{\sqrt{2} \cdot \sigma}).$$
<br>
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===The function ${\rm Q}(x )$===
===Die Funktion ${\rm Q}(x )$===
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<br>
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The function&nbsp; ${\rm Q}(x)$&nbsp; is called the&nbsp; '''complementary Gaussian error integral'''.&nbsp; The following calculation rule applies:  
Die Funktion&nbsp; ${\rm Q}(x)$&nbsp; bezeichnet man als das ''Komplementäre Gaußsche Fehlerintegral''. Es gilt folgende Berechnungsvorschrift:  
 
 
:$${\rm Q}(x ) = \frac{1}{\sqrt{2\pi}}\int_{x}^{ +\infty}\hspace{-0.2cm}{\rm e}^{-u^{2}/\hspace{0.05cm} 2}\,{\rm d} u .$$
 
:$${\rm Q}(x ) = \frac{1}{\sqrt{2\pi}}\int_{x}^{ +\infty}\hspace{-0.2cm}{\rm e}^{-u^{2}/\hspace{0.05cm} 2}\,{\rm d} u .$$
*Dieses Integral ist nicht analytisch lösbar und muss &ndash; wenn man dieses Applet nicht zur Verfügung hat &ndash; aus Tabellen entnommen werden.  
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*This integral cannot be solved analytically and must be taken from tables if one does not have this applet available.
*Speziell für größere&nbsp; $x$–Werte (also für kleine Fehlerwahrscheinlichkeiten) liefern die nachfolgend angegebenen Schranken eine brauchbare Abschätzung für das Komplementäre Gaußsche Fehlerintegral, die auch ohne Tabellen berechnet werden können.  
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*Specially for larger&nbsp; $x$&nbsp; values&nbsp; (i.e., for small error probabilities), the bounds given below provide a useful estimate for&nbsp; ${\rm Q}(x)$, which can also be calculated without tables.  
*Eine obere Schranke (englisch:&nbsp; ''Upper Bound '') des Komplementären Gaußschen Fehlerintegrals lautet:  
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*An upper bound&nbsp; $\rm  (UB)$&nbsp; of this function is:  
:$${\rm Q}_{\rm UB}(x )=\text{Upper Bound }\big [{\rm Q}(x ) \big ] = \frac{ 1}{\sqrt{2\pi}\cdot x}\cdot {\rm e}^{- x^{2}/\hspace{0.05cm}2} > {\rm Q}(x).$$
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:$${\rm Q}_{\rm UB}(x )=\text{Upper Bound }\big [{\rm Q}(x ) \big ] = \frac{ 1}{\sqrt{2\pi}\cdot x}\cdot {\rm e}^{- x^{2}/\hspace{0.05cm}2} > {\rm Q}(x).$$
*Entsprechend gilt für die untere Schranke (englisch:&nbsp; ''Lower Bound ''):  
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*Correspondingly, for the lower bound&nbsp; $\rm  (LB)$:  
:$${\rm Q}_{\rm LB}(x )=\text{Lower Bound }\big [{\rm Q}(x ) \big ] =\frac{1-1/x^2}{\sqrt{2\pi}\cdot x}\cdot {\rm e}^{-x^ 2/\hspace{0.05cm}2} ={\rm Q}_{\rm UB}(x ) \cdot (1-1/x^2)< {\rm Q}(x).$$
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:$${\rm Q}_{\rm LB}(x )=\text{Lower Bound }\big [{\rm Q}(x ) \big ] =\frac{1-1/x^2}{\sqrt{2\pi}\cdot x}\cdot {\rm e}^{-x^ 2/\hspace{0.05cm}2} ={\rm Q}_{\rm UB}(x ) \cdot (1-1/x^2)< {\rm Q}(x).$$
  
In vielen Programmbibliotheken findet man allerdings  die Funktion&nbsp; ${\rm Q}(x )$&nbsp; nicht.
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However, in many program libraries, the function&nbsp; ${\rm Q}(x )$&nbsp; cannot be found.
 
<br>
 
<br>
  
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===The function $1/2 \cdot {\rm erfc}(x )$===
  
===Die Funktion $1/2 \cdot {\rm erfc}(x )$===
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On the other hand, in almost all program libraries, you can find the&nbsp; '''complementary Gaussian error function''':
<br>
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:$${\rm erfc}(x) = \frac{2}{\sqrt{\pi}}\int_{x}^{ +\infty}\hspace{-0.2cm}{\rm e}^{-u^{2}}\,{\rm d} u ,$$
In fast allen Programmbibliotheken findet man dagegen die ''Komplementäre Gaußsche Fehlerfunktion'' (englisch:&nbsp; ''Complementary Gaussian Error Function'')
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which is related to&nbsp; ${\rm Q}(x)$&nbsp; as follows: &nbsp; ${\rm Q}(x)=1/2\cdot {\rm erfc}(x/{\sqrt{2}}).$  
:$${\rm erfc}(x) = \frac{2}{\sqrt{\pi}}\int_{x}^{ +\infty}\hspace{-0.2cm}{\rm e}^{-u^{2}}\,{\rm d} u .$$
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*Since in almost all applications this function is used with the factor&nbsp; $1/2$, in this applet exactly this function was realized:
die mit&nbsp; ${\rm Q}(x)$&nbsp; wie folgt zusammenhängt: &nbsp; ${\rm Q}(x)=1/2\cdot {\rm erfc}(x/{\sqrt{2}}).$ Da bei fast allen Anwendungen diese Funktion mit dem Faktor&nbsp; $1/2$&nbsp; verwendet wird, wurde in diesem Applet genau diese Funktion realisiert:
 
 
:$$1/2 \cdot{\rm erfc}(x) = \frac{1}{\sqrt{\pi}}\int_{x}^{ +\infty}\hspace{-0.2cm}{\rm e}^{-u^{2}}\,{\rm d} u .$$
 
:$$1/2 \cdot{\rm erfc}(x) = \frac{1}{\sqrt{\pi}}\int_{x}^{ +\infty}\hspace{-0.2cm}{\rm e}^{-u^{2}}\,{\rm d} u .$$
  
*Auch für diese Funktion kann wieder eine obere und eine untere Schranke angegeben werden:  
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*Once again, an upper and lower bound can be specified for this function:  
:$$\text{Upper Bound }\big [1/2 \cdot{\rm erfc}(x) \big ] = \frac{ 1}{\sqrt{\pi}\cdot 2x}\cdot {\rm e}^{- x^{2}} ,$$
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:$$\text{Upper Bound }\big [1/2 \cdot{\rm erfc}(x) \big ] = \frac{ 1}{\sqrt{\pi}\cdot 2x}\cdot {\rm e}^{- x^{2}} ,$$
:$$\text{Lower Bound }\big [1/2 \cdot{\rm erfc}(x) \big ] = \frac{ {1-1/(2x^2)}}{\sqrt{\pi}\cdot 2x}\cdot {\rm e}^{- x^{2}} .$$
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:$$\text{Lower Bound }\big [1/2 \cdot{\rm erfc}(x) \big ] = \frac{ {1-1/(2x^2)}}{\sqrt{\pi}\cdot 2x}\cdot {\rm e}^{- x^{2}} .$$
<br>
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===When which function offers advantages?===
  
===Wann bietet welche Funktion Vorteile?===
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{{GreyBox|TEXT=   
<br>
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$\text{Example 1:}$&nbsp; We consider binary baseband transmission. Here, the bit error probability&nbsp; $p_{\rm B} = {\rm Q}({s_0}/{\sigma_d})$, where the useful signal can take the values&nbsp; $\pm s_0$&nbsp; and the noise root-mean-square value&nbsp; $\sigma_d$&nbsp;.
{{GraueBox|TEXT=   
 
$\text{Beispiel 1:}$&nbsp; Wir betrachten die binäre Basisbandübertragung. Hier lautet die Bitfehlerwahrscheinlichkeit&nbsp; $p_{\rm B} = {\rm Q}({s_0}/{\sigma_d})$, wobei das Nutzsignal die Werte&nbsp; $\pm s_0$&nbsp; annehmen kann und der Rauscheffektivwert&nbsp; $\sigma_d$&nbsp; ist.
 
  
Es wird vorausgesetzt, dass Tabellen zur Verfügung stehen, in denen das Argument der Gaußschen Fehlerfunktionen im Abstand&nbsp; $0.1$&nbsp; aufgelistet sind. Mit&nbsp; $s_0/\sigma_d = 4$&nbsp; erhält man für die Bitfehlerwahrscheinlichkeit gemäß der Q&ndash;Funktion:
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It is assumed that tables are available listing the argument of the two Gaussian error functions at distance&nbsp; $0.1$.&nbsp; With&nbsp; $s_0/\sigma_d = 4$&nbsp; one obtains for the bit error probability according to the function&nbsp; ${\rm Q}(x )$:
 
:$$p_{\rm B} = {\rm Q} (4) \approx 0.317 \cdot 10^{-4}\hspace{0.05cm}.$$
 
:$$p_{\rm B} = {\rm Q} (4) \approx 0.317 \cdot 10^{-4}\hspace{0.05cm}.$$
Nach der zweiten Gleichung ergibt sich:
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According to the second equation, we get:
 
:$$p_{\rm B} = {1}/{2} \cdot {\rm erfc} ( {4}/{\sqrt{2} })= {1}/{2} \cdot {\rm erfc} ( 2.828)\approx {1}/{2} \cdot {\rm erfc} ( 2.8)= 0.375 \cdot 10^{-4}\hspace{0.05cm}.$$
 
:$$p_{\rm B} = {1}/{2} \cdot {\rm erfc} ( {4}/{\sqrt{2} })= {1}/{2} \cdot {\rm erfc} ( 2.828)\approx {1}/{2} \cdot {\rm erfc} ( 2.8)= 0.375 \cdot 10^{-4}\hspace{0.05cm}.$$
*Richtiger ist der erste Wert. Bei der zweiten Berechnungsart muss man runden oder &ndash; noch besser &ndash; interpolieren, was aufgrund der starken Nichtlinearität dieser Funktion sehr schwierig ist.<br>
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*The first value is more correct.&nbsp; In the second method of calculation, one must round or &ndash; even better &ndash; interpolate, which is very difficult due to the strong nonlinearity of this function.<br>
*Bei den gegebenen Zahlenwerten ist demnach  Q&ndash;Funktion besser geeignet. Außerhalb von Übungsbeispielen wird allerdings&nbsp; $s_0/\sigma_d$&nbsp; in der Regel einen &bdquo;krummen&rdquo; Wert besitzen. In diesem Fall bietet&nbsp; ${\rm Q}(x)$&nbsp; natürlich keinen Vorteil gegenüber&nbsp; $1/2 \cdot{\rm erfc}(x)$. }}
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*Accordingly, with the given numerical values, ${\rm Q}(x )$&nbsp; is more suitable.&nbsp; However, outside of exercise examples&nbsp; $s_0/\sigma_d$&nbsp; will usually have a "curvilinear" value.&nbsp; In this case, of course,&nbsp; ${\rm Q}(x)$&nbsp; offers no advantage over&nbsp; $1/2 \cdot{\rm erfc}(x)$. }}
  
  
{{GraueBox|TEXT=   
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{{GreyBox|TEXT=   
$\text{Beispiel 2:}$&nbsp;  
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$\text{Example 2:}$&nbsp;  
Mit der Energie pro Bit&nbsp; $(E_{\rm B})$&nbsp; und der Rauschleistungsdichte&nbsp; $(N_0)$&nbsp; gilt für die Bitfehlerwahrscheinlichkeit von ''Binary Phase Shift Keying''&nbsp; (BPSK):
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With the energy per bit&nbsp; $(E_{\rm B})$&nbsp; and the noise power density&nbsp; $(N_0)$&nbsp; the bit error probability of ''Binary Phase Shift Keying''&nbsp; (BPSK) is:
:$$p_{\rm B} = {\rm Q} \left ( \sqrt{ {2 E_{\rm B} }/{N_0} }\right ) = {1}/{2} \cdot {\rm erfc} \left ( \sqrt{ {E_{\rm B} }/{N_0} }\right ) \hspace{0.05cm}.$$
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:$$p_{\rm B} = {\rm Q} \left ( \sqrt{ {2 E_{\rm B} }/{N_0} }\right ) = {1}/{2} \cdot { \rm erfc} \left ( \sqrt{ {E_{\rm B} }/{N_0} }\right ) \hspace{0.05cm}.$$
Für die Zahlenwerte&nbsp; $E_{\rm B} = 16 \ \rm mWs$ und $N_0 = 16 \ \rm mW/Hz$&nbsp; erhält man:
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For the numerical values&nbsp; $E_{\rm B} = 16 \rm mWs$&nbsp; and&nbsp; $N_0 = 1 \rm mW/Hz$&nbsp; we obtain:
:$$p_{\rm B} = {\rm Q} \left (4 \cdot \sqrt{ 2} \right ) = {1}/{2} \cdot {\rm erfc} \left ( 4\right ) \hspace{0.05cm}.$$
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:$$p_{\rm B} = {\rm Q} \left (4 \cdot \sqrt{ 2} \right ) = {1}/{2} \cdot {\rm erfc} \left ( 4\right ) \hspace{0.05cm}.$$
*Der erste Weg führt zum Ergebnis&nbsp; $p_{\rm B} = {\rm Q} (5.657) \approx {\rm Q} (5.7) = 0.6 \cdot 10^{-8}\hspace{0.05cm}$, während&nbsp; $1/2 \cdot{\rm erfc}(x)$&nbsp; hier den richtigeren Wert&nbsp; $p_{\rm B} \approx 0.771 \cdot 10^{-8}$&nbsp; liefert.  
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*The first way leads to the result&nbsp; $p_{\rm B} = {\rm Q} (5.657) \approx {\rm Q} (5.7) = 0.6 \cdot 10^{-8}\hspace{0.01cm}$, while &nbsp; $1/2 \cdot{\rm erfc}(x)$&nbsp; yields the more correct value&nbsp; $p_{\rm B} \approx 0.771 \cdot 10^{-8}$&nbsp; here.  
*Wie im ersten Beispiel erkennt man aber auch hier: &nbsp; Die Funktionen&nbsp; ${\rm Q}(x)$&nbsp; und&nbsp; $1/2 \cdot{\rm erfc}(x)$&nbsp; sind grundsätzlich gleich gut geeignet. Vor&ndash; oder Nachteile der einen oder anderen Funktion ergeben sich nur bei konkreten Zahlenwerten.}}
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*As in the first example, however, you can see: &nbsp; The functions&nbsp; ${\rm Q}(x)$&nbsp; and&nbsp; $1/2 \cdot{\rm erfc}(x)$&nbsp; are basically equally well suited.  
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*Advantages or disadvantages of one or the other function arise only for concrete numerical values.}}
 
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<br>
  
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==Exercises==
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* First select the number&nbsp; $(1, 2, \text{...})$&nbsp; of the exercise.&nbsp; The number&nbsp; $0$&nbsp; corresponds to a "Reset":&nbsp; Same setting as at program start.
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*A task description is displayed.&nbsp; The parameter values ​​are adjusted.&nbsp; Solution after pressing "Show solution". <br>
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{{BlueBox|TEXT= 
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'''(1)''' &nbsp; Find the values of the function&nbsp; ${\rm Q}(x)$&nbsp; for&nbsp; $x=1$,&nbsp; $x=2$,&nbsp; $x=4$&nbsp; and&nbsp; $x=6$.&nbsp; Interpret the graphs for linear and logarithmic ordinates.}}
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*The applet returns the values&nbsp; ${\rm Q}(1)=1.5866 \cdot 10^{-1}$,&nbsp; ${\rm Q}(2)=2. 275 \cdot 10^{-2}$,&nbsp; ${\rm Q}(4)=3.1671 \cdot 10^{-5}$&nbsp; and&nbsp; ${\rm Q}(6)=9.8659 \cdot 10^{-10}$.
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*With linear ordinate, the values for&nbsp; $x>3$&nbsp; are indistinguishable from the zero line.&nbsp; More interesting is the plot with logarithmic ordinate.
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{{BlueBox|TEXT= 
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'''(2)''' &nbsp; Evaluate the two bounds&nbsp; ${\rm UB}(x )=\text{Upper Bound }\big [{\rm Q}(x ) \big ]$&nbsp; and&nbsp; ${\rm LB}(x )=\text{Lower Bound }\big [{\rm Q}(x ) \big ]$&nbsp; for the&nbsp; ${\rm Q}$&nbsp; function. }}
  
==Zur Handhabung des Applets==
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*For&nbsp; $x \ge 2$&nbsp; the upper bound is only slightly above&nbsp; ${\rm Q}(x)$&nbsp; and the lower bound is only slightly below&nbsp; ${\rm Q}(x)$.&nbsp;
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*For example:&nbsp; ${\rm Q}(x=4)=3.1671 \cdot 10^{-5}$ &nbsp; &rArr; &nbsp; ${\rm LB}(x=4)=3.1366 \cdot 10^{-5}$, &nbsp; ${\rm UB}(x=4)=3.3458 \cdot 10^{-5}$.
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*The upper bound has greater significance for assessing a communications system than "LB",&nbsp; since this corresponds to a "worst case" consideration.
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{{BlueBox|TEXT= 
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'''(3)''' &nbsp; Try to use the app to determine&nbsp; ${\rm Q}(x=2 \cdot \sqrt{2} \approx 2.828)$&nbsp; as accurately as possible despite the quantization of the input parameter. }}
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*The program returns for&nbsp; $x=2.8$&nbsp; the too large result&nbsp; $2.5551 \cdot 10^{-3}$&nbsp; and for&nbsp; $x=2.85$&nbsp; the result&nbsp; $2.186 \cdot 10^{-3}$.&nbsp; The exact value lies in between.
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*But it also holds:&nbsp; ${\rm Q}(x=2 \cdot \sqrt{2})=0.5 \cdot {\rm erfc}(x=2)$.&nbsp; This gives the exact value&nbsp; ${\rm Q}(x=2 \cdot \sqrt{2})=2.3389 \cdot 10^{-3}$.
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{{BlueBox|TEXT= 
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'''(4)''' &nbsp; Find the values of the function&nbsp; $0.5 \cdot {\rm erfc}(x)$&nbsp; for&nbsp; $x=1$,&nbsp; $x=2$,&nbsp; $x=3$&nbsp; and&nbsp; $x=4$.&nbsp; Interpret the exact results and the bounds.}}
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*The applet returns:&nbsp; $0.5 \cdot {\rm erfc}(1)=7.865 \cdot 10^{-2}$,&nbsp; $0.5 \cdot {\rm erfc}(2)=2. 3389 \cdot 10^{-3}$,&nbsp; $0.5 \cdot {\rm erfc}(3)=1.1045 \cdot 10^{-5}$&nbsp; and&nbsp; $0.5 \cdot {\rm erfc}(4)=7.7086 \cdot 10^{-9}$.
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*All the above statements about&nbsp; ${\rm Q}(x)$&nbsp; with respect to suitable representation type and upper and lower bounds also apply to the function&nbsp; $0.5 \cdot {\rm erfc}(x)$.
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{{BlueBox|TEXT= 
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'''(5)''' &nbsp; The results of&nbsp; '''(4)'''&nbsp; are now to be converted for the case of a logarithmic abscissa.&nbsp; The conversion is done according to&nbsp; $\rho\big[{\rm dB}\big ] = 20 \cdot \lg(x)$. }}
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* The linear abscissa value&nbsp; $x=1$&nbsp; leads to the logarithmic abscissa value&nbsp; $\rho=0\ \rm dB$ &nbsp; &rArr; &nbsp; $0. 5 \cdot {\rm erfc}(\rho=0\ {\rm dB})={0.5 \cdot \rm erfc}(x=1)=7.865 \cdot 10^{-2}$.
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*Similarly&nbsp; $0.5 \cdot {\rm erfc}(\rho=6.021\ {\rm dB}) =0.5 \cdot {\rm erfc}(x=2)=2. 3389 \cdot 10^{-3}$, &nbsp; &nbsp; $0.5 \cdot {\rm erfc}(\rho=9.542\ {\rm dB})=0.5 \cdot {\rm erfc}(3)=1.1045 \cdot 10^{-5}$,&nbsp; &nbsp;
 +
*$0.5 \cdot {\rm erfc}(\rho=12.041\ {\rm dB})= 0.5 \cdot {\rm erfc}(4)=7.7086 \cdot 10^{-9}$.
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*As per right diagram:&nbsp; $0.5 \cdot {\rm erfc}(\rho=6\ {\rm dB}) =2.3883 \cdot 10^{-3}$, &nbsp; &nbsp; $0.5 \cdot {\rm erfc}(\rho=9. 5\ {\rm dB}) =1.2109 \cdot 10^{-5}$, &nbsp; &nbsp; $0.5 \cdot {\rm erfc}(\rho=12\ {\rm dB}) =9.006 \cdot 10^{-9}$.
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{{BlueBox|TEXT=
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'''(6)''' &nbsp; Find&nbsp; ${\rm Q}(\rho=0\ {\rm dB})$,&nbsp; ${\rm Q}(\rho=5\ {\rm dB})$&nbsp; and&nbsp; ${\rm Q}(\rho=10\ {\rm dB})$,&nbsp; and establish the relationship between linear and logarithmic abscissa.}}
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*The program returns for logarithmic abscissa&nbsp; ${\rm Q}(\rho=0\ {\rm dB})=1. 5866 \cdot 10^{-1}$,&nbsp; ${\rm Q}(\rho=5\ {\rm dB})=3.7679 \cdot 10^{-2}$,&nbsp; ${\rm Q}(\rho=10\ {\rm dB})=7.827 \cdot 10^{-4}$.
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*The conversion is done according to the equation&nbsp; $x=10^{\hspace{0.05cm}0.05\hspace{0.05cm} \cdot\hspace{0.05cm} \rho[{\rm dB}]}$.&nbsp; For&nbsp; $\rho=0\ {\rm dB}$&nbsp; we get&nbsp; $x=1$ &nbsp; &rArr; &nbsp; ${\rm Q}(\rho=0\ {\rm dB})={\rm Q}(x=1) =1.5866 \cdot 10^{-1}$.
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*For&nbsp; $\rho=5\ {\rm dB}$&nbsp; we get&nbsp; $x=1.1778$ &nbsp; &rArr; &nbsp; ${\rm Q}(\rho=5\ {\rm dB})={\rm Q}(x=1. 778) =3.7679 \cdot 10^{-2}$.&nbsp; From the left diagram:&nbsp; ${\rm Q}(x=1.8) =3.593 \cdot 10^{-2}$.
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*For&nbsp; $\rho=10\ {\rm dB}$&nbsp; we get&nbsp; $x=3.162$ &nbsp; &rArr; &nbsp; ${\rm Q}(\rho=10\ {\rm dB})={\rm Q}(x=3. 162) =7.827 \cdot 10^{-4}$.&nbsp; After "quantization":&nbsp; ${\rm Q}(x=3.15) =8.1635 \cdot 10^{-4}$.
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==Applet Manual==
 
<br>
 
<br>
  
 
[[File:Qfunction bedienung.png|right|550px]]
 
[[File:Qfunction bedienung.png|right|550px]]
&nbsp; &nbsp; '''(A)''' &nbsp; &nbsp; Verwendete Gleichungen am Beispiel &nbsp;${\rm Q}(x)$
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&nbsp; &nbsp; '''(A)''' &nbsp; &nbsp; Equations used in the example &nbsp;${\rm Q}(x)$
  
&nbsp; &nbsp; '''(B)''' &nbsp; &nbsp; Auswahloption für &nbsp;${\rm Q}(x)$&nbsp; oder &nbsp;${\rm 0.5 \cdot erfc}(x)$
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&nbsp; &nbsp; '''(B)''' &nbsp; &nbsp; Selection option for &nbsp;${\rm Q}(x)$&nbsp; or &nbsp;${\rm 0.5 \cdot erfc}(x)$
  
&nbsp; &nbsp; '''(C)''' &nbsp; &nbsp; Schranken &nbsp;${\rm LB}$&nbsp; und &nbsp;${\rm UB}$&nbsp; werden gezeichnet
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&nbsp; &nbsp; '''(C)''' &nbsp; &nbsp; Bounds &nbsp;${\rm LB}$&nbsp; and &nbsp;${\rm UB}$&nbsp; are drawn
  
&nbsp; &nbsp; '''(D)''' &nbsp; &nbsp; Auswahl, ob Abszisse linear &nbsp;$\rm (lin)$&nbsp; oder logarithmisch &nbsp;$\rm (log)$&nbsp;
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&nbsp; &nbsp; '''(D)''' &nbsp; &nbsp; Selection whether abscissa is linear &nbsp;$\rm (lin)$&nbsp; or logarithmic &nbsp;$\rm (log)$&nbsp;
  
&nbsp; &nbsp; '''(E)''' &nbsp; &nbsp; Auswahl, ob Ordinate linear &nbsp;$\rm (lin)$&nbsp; oder logarithmisch &nbsp;$\rm (log)$&nbsp;
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&nbsp; &nbsp; '''(E)''' &nbsp; &nbsp; Select whether ordinate is linear &nbsp;$\rm (lin)$&nbsp; or logarithmic &nbsp;$\rm (log)$&nbsp;
  
&nbsp; &nbsp; '''(F)''' &nbsp; &nbsp; Numerikausgabe am Beispiel &nbsp;${\rm Q}(x)$&nbsp; bei linearer Abszisse
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&nbsp; &nbsp; '''(F)''' &nbsp; &nbsp; Numerical output using the example &nbsp;${\rm Q}(x)$&nbsp; with linear abscissa
  
&nbsp; &nbsp; '''(G)''' &nbsp; &nbsp; Slidereingabe des Abszissenwertes &nbsp;$x$&nbsp; für lineare Abszisse
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&nbsp; &nbsp; '''(G)''' &nbsp; &nbsp; Slider input of abscissa value &nbsp;$x$&nbsp; for linear abscissa
  
&nbsp; &nbsp; '''(H)''' &nbsp; &nbsp; Slidereingabe des Abszissenwertes &nbsp;$\rho \ \rm [dB]$&nbsp; für logarithmische Abszisse
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&nbsp; &nbsp; '''(H)''' &nbsp; &nbsp; Slider input of abscissa value &nbsp;$\rho \ \rm [dB]$&nbsp; for logarithmic abscissa
  
&nbsp; &nbsp; '''(I)''' &nbsp; &nbsp; Grafikausgabe der Funktion  &nbsp;${\rm Q}(x)$&nbsp; &ndash; hier:&nbsp; lineare Abszisse
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&nbsp; &nbsp; '''(I)''' &nbsp; &nbsp; Graphical output of function &nbsp;${\rm Q}(x)$&nbsp; &ndash; here:&nbsp; linear abscissa
  
&nbsp; &nbsp; '''(J)''' &nbsp; &nbsp; Grafikausgabe der Funktion  &nbsp;${\rm 0.5 \cdot erfc}(x)$&nbsp; &ndash; hier:&nbsp; lineare Abszisse
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&nbsp; &nbsp; '''(J)''' &nbsp; &nbsp; Graph output of function &nbsp;${\rm 0.5 \cdot erfc}(x)$&nbsp; &ndash; here:&nbsp; linear abscissa
  
&nbsp; &nbsp; '''(K)''' &nbsp; &nbsp; Variationsmöglichkeit für die graphischen Darstellungen
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&nbsp; &nbsp; '''(K)''' &nbsp; &nbsp; Variation possibility for the graphical representations
  
$\hspace{1.5cm}$&bdquo;$+$&rdquo; (Vergrößern),  
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$\hspace{1.5cm}$"$+$" (zoom in),  
  
$\hspace{1.5cm}$ &bdquo;$-$&rdquo; (Verkleinern)
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$\hspace{1.5cm}$"$-$" (zoom out)
  
$\hspace{1.5cm}$ &bdquo;$\rm o$&rdquo; (Zurücksetzen)
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$\hspace{1.5cm}$ "$\rm o$" (Reset)
  
$\hspace{1.5cm}$ &bdquo;$\leftarrow$&rdquo; (Verschieben nach links), usw.
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$\hspace{1.5cm}$ "$\leftarrow$" (Move left), etc.
 
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==Über die Autoren==
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==About the Authors==
Dieses interaktive Berechnungstool  wurde am&nbsp; [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik]&nbsp; der&nbsp; [https://www.tum.de/ Technischen Universität München]&nbsp; konzipiert und realisiert.  
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*Die erste Version wurde 2007 von&nbsp; [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Thomas_Gro.C3.9Fer_.28Diplomarbeit_LB_2006.2C_danach_freie_Mitarbeit_bis_2010.29|Thomas Großer]]&nbsp; im Rahmen seiner Diplomarbeit mit &bdquo;FlashMX&ndash;Actionscript&rdquo; erstellt (Betreuer:&nbsp; [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).  
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This interactive calculation tool was designed and implemented at the&nbsp; [https://www.ei.tum.de/en/lnt/home/ Institute for Communications Engineering]&nbsp; at the&nbsp; [https://www.tum.de/en Technical University of Munich].  
*2018/2019 wurde das Programm  von&nbsp; ''Marwen Ben Ammar''&nbsp;  und&nbsp; [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]]&nbsp; (Bachelorarbeit, Betreuer:&nbsp; [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]] ) auf  &bdquo;HTML5&rdquo; umgesetzt und neu gestaltet.
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*The first version was created in 2007 by&nbsp; [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Thomas_Gro.C3.9Fer_.28Diplomarbeit_LB_2006.2C_danach_freie_Mitarbeit_bis_2010.29|Thomas Großer]]&nbsp; as part of his diploma thesis with “FlashMX – Actionscript” (Supervisor: [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).  
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*In 2018 the program was redesigned by&nbsp; [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]]&nbsp; as part of her bachelor thesis&nbsp; (Supervisor: [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]] ) via "HTML5".
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*Last revision and English version 2021 by&nbsp; [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|Carolin Mirschina]]&nbsp; in the context of a working student activity.&nbsp;
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The conversion of this applet to HTML 5 was financially supported by&nbsp; [https://www.ei.tum.de/studium/studienzuschuesse/ "Studienzuschüsse"]&nbsp; (Faculty EI of the TU Munich).&nbsp; We thank.
  
Die Umsetzung dieses Applets auf HTML 5 wurde durch&nbsp; [https://www.ei.tum.de/studium/studienzuschuesse/ Studienzuschüsse]&nbsp; der Fakultät EI der TU München finanziell unterstützt. Wir bedanken uns.
 
  
  
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{{LntAppletLink|qfunction}}
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{{LntAppletLinkEnDe|qfunction_en|qfunction}}
&nbsp; &nbsp; ''Hinweis:'' &nbsp; Das Applet ist für '''CHROME'''&ndash;Browser optimiert. Bei anderen Browsern kommt es teilweise zu Darstellungsproblemen.
 

Latest revision as of 18:49, 16 April 2023

Open Applet in new Tab   Deutsche Version Öffnen


Applet Description


This applet allows the calculation and graphical representation of the (complementary) Gaussian error functions  ${\rm Q}(x)$  and  $1/2\cdot {\rm erfc}(x)$, which are of great importance for error probability calculation.

  • Both the abscissa and the function value can be represented either linearly or logarithmically.
  • For both functions an upper bound  $\rm (UB)$  and a lower bound  $\rm (LB)$  are given.

Theoretical Background


In the study of digital transmission systems, it is often necessary to determine the probability that a (zero mean) Gaussian distributed random variable  $x$  with variance  $σ^2$  exceeds a given value  $x_0$.  For this probability holds:

$${\rm Pr}(x > x_0)={\rm Q}(\frac{x_0}{\sigma}) = 1/2 \cdot {\rm erfc}(\frac{x_0}{\sqrt{2} \cdot \sigma}).$$

The function ${\rm Q}(x )$

The function  ${\rm Q}(x)$  is called the  complementary Gaussian error integral.  The following calculation rule applies:

$${\rm Q}(x ) = \frac{1}{\sqrt{2\pi}}\int_{x}^{ +\infty}\hspace{-0.2cm}{\rm e}^{-u^{2}/\hspace{0.05cm} 2}\,{\rm d} u .$$
  • This integral cannot be solved analytically and must be taken from tables if one does not have this applet available.
  • Specially for larger  $x$  values  (i.e., for small error probabilities), the bounds given below provide a useful estimate for  ${\rm Q}(x)$, which can also be calculated without tables.
  • An upper bound  $\rm (UB)$  of this function is:
$${\rm Q}_{\rm UB}(x )=\text{Upper Bound }\big [{\rm Q}(x ) \big ] = \frac{ 1}{\sqrt{2\pi}\cdot x}\cdot {\rm e}^{- x^{2}/\hspace{0.05cm}2} > {\rm Q}(x).$$
  • Correspondingly, for the lower bound  $\rm (LB)$:
$${\rm Q}_{\rm LB}(x )=\text{Lower Bound }\big [{\rm Q}(x ) \big ] =\frac{1-1/x^2}{\sqrt{2\pi}\cdot x}\cdot {\rm e}^{-x^ 2/\hspace{0.05cm}2} ={\rm Q}_{\rm UB}(x ) \cdot (1-1/x^2)< {\rm Q}(x).$$

However, in many program libraries, the function  ${\rm Q}(x )$  cannot be found.

The function $1/2 \cdot {\rm erfc}(x )$

On the other hand, in almost all program libraries, you can find the  complementary Gaussian error function:

$${\rm erfc}(x) = \frac{2}{\sqrt{\pi}}\int_{x}^{ +\infty}\hspace{-0.2cm}{\rm e}^{-u^{2}}\,{\rm d} u ,$$

which is related to  ${\rm Q}(x)$  as follows:   ${\rm Q}(x)=1/2\cdot {\rm erfc}(x/{\sqrt{2}}).$

  • Since in almost all applications this function is used with the factor  $1/2$, in this applet exactly this function was realized:
$$1/2 \cdot{\rm erfc}(x) = \frac{1}{\sqrt{\pi}}\int_{x}^{ +\infty}\hspace{-0.2cm}{\rm e}^{-u^{2}}\,{\rm d} u .$$
  • Once again, an upper and lower bound can be specified for this function:
$$\text{Upper Bound }\big [1/2 \cdot{\rm erfc}(x) \big ] = \frac{ 1}{\sqrt{\pi}\cdot 2x}\cdot {\rm e}^{- x^{2}} ,$$
$$\text{Lower Bound }\big [1/2 \cdot{\rm erfc}(x) \big ] = \frac{ {1-1/(2x^2)}}{\sqrt{\pi}\cdot 2x}\cdot {\rm e}^{- x^{2}} .$$

When which function offers advantages?

$\text{Example 1:}$  We consider binary baseband transmission. Here, the bit error probability  $p_{\rm B} = {\rm Q}({s_0}/{\sigma_d})$, where the useful signal can take the values  $\pm s_0$  and the noise root-mean-square value  $\sigma_d$ .

It is assumed that tables are available listing the argument of the two Gaussian error functions at distance  $0.1$.  With  $s_0/\sigma_d = 4$  one obtains for the bit error probability according to the function  ${\rm Q}(x )$:

$$p_{\rm B} = {\rm Q} (4) \approx 0.317 \cdot 10^{-4}\hspace{0.05cm}.$$

According to the second equation, we get:

$$p_{\rm B} = {1}/{2} \cdot {\rm erfc} ( {4}/{\sqrt{2} })= {1}/{2} \cdot {\rm erfc} ( 2.828)\approx {1}/{2} \cdot {\rm erfc} ( 2.8)= 0.375 \cdot 10^{-4}\hspace{0.05cm}.$$
  • The first value is more correct.  In the second method of calculation, one must round or – even better – interpolate, which is very difficult due to the strong nonlinearity of this function.
  • Accordingly, with the given numerical values, ${\rm Q}(x )$  is more suitable.  However, outside of exercise examples  $s_0/\sigma_d$  will usually have a "curvilinear" value.  In this case, of course,  ${\rm Q}(x)$  offers no advantage over  $1/2 \cdot{\rm erfc}(x)$.


$\text{Example 2:}$  With the energy per bit  $(E_{\rm B})$  and the noise power density  $(N_0)$  the bit error probability of Binary Phase Shift Keying  (BPSK) is:

$$p_{\rm B} = {\rm Q} \left ( \sqrt{ {2 E_{\rm B} }/{N_0} }\right ) = {1}/{2} \cdot { \rm erfc} \left ( \sqrt{ {E_{\rm B} }/{N_0} }\right ) \hspace{0.05cm}.$$

For the numerical values  $E_{\rm B} = 16 \rm mWs$  and  $N_0 = 1 \rm mW/Hz$  we obtain:

$$p_{\rm B} = {\rm Q} \left (4 \cdot \sqrt{ 2} \right ) = {1}/{2} \cdot {\rm erfc} \left ( 4\right ) \hspace{0.05cm}.$$
  • The first way leads to the result  $p_{\rm B} = {\rm Q} (5.657) \approx {\rm Q} (5.7) = 0.6 \cdot 10^{-8}\hspace{0.01cm}$, while   $1/2 \cdot{\rm erfc}(x)$  yields the more correct value  $p_{\rm B} \approx 0.771 \cdot 10^{-8}$  here.
  • As in the first example, however, you can see:   The functions  ${\rm Q}(x)$  and  $1/2 \cdot{\rm erfc}(x)$  are basically equally well suited.
  • Advantages or disadvantages of one or the other function arise only for concrete numerical values.


Exercises

  • First select the number  $(1, 2, \text{...})$  of the exercise.  The number  $0$  corresponds to a "Reset":  Same setting as at program start.
  • A task description is displayed.  The parameter values ​​are adjusted.  Solution after pressing "Show solution".


(1)   Find the values of the function  ${\rm Q}(x)$  for  $x=1$,  $x=2$,  $x=4$  and  $x=6$.  Interpret the graphs for linear and logarithmic ordinates.

  • The applet returns the values  ${\rm Q}(1)=1.5866 \cdot 10^{-1}$,  ${\rm Q}(2)=2. 275 \cdot 10^{-2}$,  ${\rm Q}(4)=3.1671 \cdot 10^{-5}$  and  ${\rm Q}(6)=9.8659 \cdot 10^{-10}$.
  • With linear ordinate, the values for  $x>3$  are indistinguishable from the zero line.  More interesting is the plot with logarithmic ordinate.


(2)   Evaluate the two bounds  ${\rm UB}(x )=\text{Upper Bound }\big [{\rm Q}(x ) \big ]$  and  ${\rm LB}(x )=\text{Lower Bound }\big [{\rm Q}(x ) \big ]$  for the  ${\rm Q}$  function.

  • For  $x \ge 2$  the upper bound is only slightly above  ${\rm Q}(x)$  and the lower bound is only slightly below  ${\rm Q}(x)$. 
  • For example:  ${\rm Q}(x=4)=3.1671 \cdot 10^{-5}$   ⇒   ${\rm LB}(x=4)=3.1366 \cdot 10^{-5}$,   ${\rm UB}(x=4)=3.3458 \cdot 10^{-5}$.
  • The upper bound has greater significance for assessing a communications system than "LB",  since this corresponds to a "worst case" consideration.


(3)   Try to use the app to determine  ${\rm Q}(x=2 \cdot \sqrt{2} \approx 2.828)$  as accurately as possible despite the quantization of the input parameter.

  • The program returns for  $x=2.8$  the too large result  $2.5551 \cdot 10^{-3}$  and for  $x=2.85$  the result  $2.186 \cdot 10^{-3}$.  The exact value lies in between.
  • But it also holds:  ${\rm Q}(x=2 \cdot \sqrt{2})=0.5 \cdot {\rm erfc}(x=2)$.  This gives the exact value  ${\rm Q}(x=2 \cdot \sqrt{2})=2.3389 \cdot 10^{-3}$.


(4)   Find the values of the function  $0.5 \cdot {\rm erfc}(x)$  for  $x=1$,  $x=2$,  $x=3$  and  $x=4$.  Interpret the exact results and the bounds.

  • The applet returns:  $0.5 \cdot {\rm erfc}(1)=7.865 \cdot 10^{-2}$,  $0.5 \cdot {\rm erfc}(2)=2. 3389 \cdot 10^{-3}$,  $0.5 \cdot {\rm erfc}(3)=1.1045 \cdot 10^{-5}$  and  $0.5 \cdot {\rm erfc}(4)=7.7086 \cdot 10^{-9}$.
  • All the above statements about  ${\rm Q}(x)$  with respect to suitable representation type and upper and lower bounds also apply to the function  $0.5 \cdot {\rm erfc}(x)$.


(5)   The results of  (4)  are now to be converted for the case of a logarithmic abscissa.  The conversion is done according to  $\rho\big[{\rm dB}\big ] = 20 \cdot \lg(x)$.

  • The linear abscissa value  $x=1$  leads to the logarithmic abscissa value  $\rho=0\ \rm dB$   ⇒   $0. 5 \cdot {\rm erfc}(\rho=0\ {\rm dB})={0.5 \cdot \rm erfc}(x=1)=7.865 \cdot 10^{-2}$.
  • Similarly  $0.5 \cdot {\rm erfc}(\rho=6.021\ {\rm dB}) =0.5 \cdot {\rm erfc}(x=2)=2. 3389 \cdot 10^{-3}$,     $0.5 \cdot {\rm erfc}(\rho=9.542\ {\rm dB})=0.5 \cdot {\rm erfc}(3)=1.1045 \cdot 10^{-5}$,   
  • $0.5 \cdot {\rm erfc}(\rho=12.041\ {\rm dB})= 0.5 \cdot {\rm erfc}(4)=7.7086 \cdot 10^{-9}$.
  • As per right diagram:  $0.5 \cdot {\rm erfc}(\rho=6\ {\rm dB}) =2.3883 \cdot 10^{-3}$,     $0.5 \cdot {\rm erfc}(\rho=9. 5\ {\rm dB}) =1.2109 \cdot 10^{-5}$,     $0.5 \cdot {\rm erfc}(\rho=12\ {\rm dB}) =9.006 \cdot 10^{-9}$.


(6)   Find  ${\rm Q}(\rho=0\ {\rm dB})$,  ${\rm Q}(\rho=5\ {\rm dB})$  and  ${\rm Q}(\rho=10\ {\rm dB})$,  and establish the relationship between linear and logarithmic abscissa.

  • The program returns for logarithmic abscissa  ${\rm Q}(\rho=0\ {\rm dB})=1. 5866 \cdot 10^{-1}$,  ${\rm Q}(\rho=5\ {\rm dB})=3.7679 \cdot 10^{-2}$,  ${\rm Q}(\rho=10\ {\rm dB})=7.827 \cdot 10^{-4}$.
  • The conversion is done according to the equation  $x=10^{\hspace{0.05cm}0.05\hspace{0.05cm} \cdot\hspace{0.05cm} \rho[{\rm dB}]}$.  For  $\rho=0\ {\rm dB}$  we get  $x=1$   ⇒   ${\rm Q}(\rho=0\ {\rm dB})={\rm Q}(x=1) =1.5866 \cdot 10^{-1}$.
  • For  $\rho=5\ {\rm dB}$  we get  $x=1.1778$   ⇒   ${\rm Q}(\rho=5\ {\rm dB})={\rm Q}(x=1. 778) =3.7679 \cdot 10^{-2}$.  From the left diagram:  ${\rm Q}(x=1.8) =3.593 \cdot 10^{-2}$.
  • For  $\rho=10\ {\rm dB}$  we get  $x=3.162$   ⇒   ${\rm Q}(\rho=10\ {\rm dB})={\rm Q}(x=3. 162) =7.827 \cdot 10^{-4}$.  After "quantization":  ${\rm Q}(x=3.15) =8.1635 \cdot 10^{-4}$.



Applet Manual


Qfunction bedienung.png

    (A)     Equations used in the example  ${\rm Q}(x)$

    (B)     Selection option for  ${\rm Q}(x)$  or  ${\rm 0.5 \cdot erfc}(x)$

    (C)     Bounds  ${\rm LB}$  and  ${\rm UB}$  are drawn

    (D)     Selection whether abscissa is linear  $\rm (lin)$  or logarithmic  $\rm (log)$ 

    (E)     Select whether ordinate is linear  $\rm (lin)$  or logarithmic  $\rm (log)$ 

    (F)     Numerical output using the example  ${\rm Q}(x)$  with linear abscissa

    (G)     Slider input of abscissa value  $x$  for linear abscissa

    (H)     Slider input of abscissa value  $\rho \ \rm [dB]$  for logarithmic abscissa

    (I)     Graphical output of function  ${\rm Q}(x)$  – here:  linear abscissa

    (J)     Graph output of function  ${\rm 0.5 \cdot erfc}(x)$  – here:  linear abscissa

    (K)     Variation possibility for the graphical representations

$\hspace{1.5cm}$"$+$" (zoom in),

$\hspace{1.5cm}$"$-$" (zoom out)

$\hspace{1.5cm}$ "$\rm o$" (Reset)

$\hspace{1.5cm}$ "$\leftarrow$" (Move left), etc.

About the Authors


This interactive calculation tool was designed and implemented at the  Institute for Communications Engineering  at the  Technical University of Munich.

  • The first version was created in 2007 by  Thomas Großer  as part of his diploma thesis with “FlashMX – Actionscript” (Supervisor: Günter Söder).
  • In 2018 the program was redesigned by  Xiaohan Liu  as part of her bachelor thesis  (Supervisor: Tasnád Kernetzky ) via "HTML5".
  • Last revision and English version 2021 by  Carolin Mirschina  in the context of a working student activity. 


The conversion of this applet to HTML 5 was financially supported by  "Studienzuschüsse"  (Faculty EI of the TU Munich).  We thank.


Once again: Open Applet in new Tab


Open Applet in new Tab   Deutsche Version Öffnen