Applets:Complementary Gaussian Error Functions - Revision history

Noah at 17:49, 16 April 2023

2023-04-16T17:49:53Z

Noah at 17:49, 16 April 2023

2023-04-16T17:49:23Z

Hwang at 18:06, 21 March 2023

2023-03-21T18:06:00Z

Javier: Text replacement - "„" to """

2021-05-28T14:44:54Z

Text replacement - "„" to """

Javier: Text replacement - "”" to """

2021-05-28T14:39:36Z

Text replacement - "”" to """

Guenter at 15:44, 1 March 2021

2021-03-01T15:44:34Z

Guenter at 10:17, 27 January 2021

2021-01-27T10:17:21Z

Guenter at 10:10, 27 January 2021

2021-01-27T10:10:10Z

Guenter at 10:06, 27 January 2021

2021-01-27T10:06:26Z

Guenter at 15:42, 26 January 2021

2021-01-26T15:42:14Z

@@ Line 125: / Line 125: @@
 &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; Bounds &nbsp;${\rm LB}$&nbsp; and &nbsp;${\rm UB}$&nbsp; are drawn
+&nbsp; &nbsp; '''(C)''' &nbsp; &nbsp; Bounds &nbsp;${\rm LB}$&nbsp; and &nbsp;${\rm UB}$&nbsp; are drawn
-&nbsp; &nbsp; '''(D)'''' &nbsp; &nbsp; Selection whether abscissa is linear &nbsp;$\rm (lin)$&nbsp; or logarithmic &nbsp;$\rm (log)$&nbsp;
+&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; Select whether ordinate is linear &nbsp;$\rm (lin)$&nbsp; or logarithmic &nbsp;$\rm (log)$&nbsp;
+&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; Numerical output using the example &nbsp;${\rm Q}(x)$&nbsp; with linear abscissa
+&nbsp; &nbsp; '''(F)''' &nbsp; &nbsp; Numerical output using the example &nbsp;${\rm Q}(x)$&nbsp; with linear abscissa
 &nbsp; &nbsp; '''(G)''' &nbsp; &nbsp; Slider input of abscissa value &nbsp;$x$&nbsp; for linear abscissa
@@ Line 137: / Line 137: @@
 &nbsp; &nbsp; '''(H)''' &nbsp; &nbsp; Slider input of abscissa value &nbsp;$\rho \ \rm [dB]$&nbsp; for logarithmic abscissa
-&nbsp; &nbsp; '''(I)'''' &nbsp; &nbsp; Graphical output of function &nbsp;${\rm Q}(x)$&nbsp; &ndash; here:&nbsp; linear abscissa
+&nbsp; &nbsp; '''(I)''' &nbsp; &nbsp; Graphical output of function &nbsp;${\rm Q}(x)$&nbsp; &ndash; here:&nbsp; linear abscissa
 &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; Variation possibility for the graphical representations
+&nbsp; &nbsp; '''(K)''' &nbsp; &nbsp; Variation possibility for the graphical representations
 $\hspace{1.5cm}$"$+$" (zoom in),

@@ Line 11: / Line 11: @@
 ==Theoretical Background==
 <br>
-In the study of digital transmission systems, it is often necessary to determine the probability that a (mean-free) 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:
+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:
 :$${\rm Pr}(x > x_0)={\rm Q}(\frac{x_0}{\sigma}) = 1/2 \cdot {\rm erfc}(\frac{x_0}{\sqrt{2} \cdot \sigma}).$$
@@ Line 30: / Line 30: @@
 ===The function $1/2 \cdot {\rm erfc}(x )$===
-On the other hand, in almost all program libraries, you can find the&nbsp; '''Complementary Gaussian Error Function''':
+On the other hand, in almost all program libraries, you can find the&nbsp; '''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&nbsp; ${\rm Q}(x)$&nbsp; as follows: &nbsp; ${\rm Q}(x)=1/2\cdot {\rm erfc}(x/{\sqrt{2}}).$
@@ Line 43: / Line 43: @@
 {{GreyBox|TEXT=
-$\text{Example 1:}$&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;.
+$\text{Example 1:}$&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;.
 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; ${\rm Q}(x )$:
@@ Line 59: / Line 59: @@
 For the numerical values&nbsp; $E_{\rm B} = 16 \rm mWs$&nbsp; and&nbsp; $N_0 = 1 \rm mW/Hz$&nbsp; we obtain:
 :$$p_{\rm B} = {\rm Q} \left (4 \cdot \sqrt{ 2} \right ) = {1}/{2} \cdot {\rm erfc} \left ( 4\right ) \hspace{0.05cm}.$$
-*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; here the more correct value&nbsp; $p_{\rm B} \approx 0.771 \cdot 10^{-8}$&nbsp; yields.
+*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.
 *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.
 *Advantages or disadvantages of one or the other function arise only for concrete numerical values.}}
@@ Line 121: / Line 121: @@
 [[File:Qfunction bedienung.png|right|550px]]
-&nbsp; &nbsp; '''(A)''' &nbsp; &nbsp; Verwendete Gleichungen am Beispiel &nbsp;${\rm Q}(x)$
+&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;${\rm 0.5 \cdot erfc}(x)$
+&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
+&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;
+&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;
+&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
+&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
+&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
+&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
+&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
+&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
+&nbsp; &nbsp; '''(K)'''' &nbsp; &nbsp; Variation possibility for the graphical representations
-$\hspace{1.5cm}$"$+$" (Vergrößern),
+$\hspace{1.5cm}$"$+$" (zoom in),
-$\hspace{1.5cm}$ "$-$" (Verkleinern)
+$\hspace{1.5cm}$"$-$" (zoom out)
-$\hspace{1.5cm}$ "$\rm o$" (Zurücksetzen)
+$\hspace{1.5cm}$ "$\rm o$" (Reset)
-$\hspace{1.5cm}$ "$\leftarrow$" (Verschieben nach links),  usw.
+$\hspace{1.5cm}$ "$\leftarrow$" (Move left), etc.
 <br clear=all>
 ==About the Authors==

@@ Line 156: / Line 156: @@
 *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]]).
 *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".
-*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; Translation using DEEPL.com.
+*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;

@@ Line 50: / Line 50: @@
 :$$p_{\rm B} = {1}/{2} \cdot {\rm erfc} ( {4}/{\sqrt{2} })= {1}/{2} \cdot {\rm erfc} ( 2.828)\approx {1}/{2} \cdot {\rm erfc} ( 2.8)= 0.375 \cdot 10^{-4}\hspace{0.05cm}.$$
 *The first value is more correct.&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>
-*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 &bdquo;curvilinear" value.&nbsp; In this case, of course,&nbsp; ${\rm Q}(x)$&nbsp; offers no advantage over&nbsp; $1/2 \cdot{\rm erfc}(x)$. }}
+*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; $1/2 \cdot{\rm erfc}(x)$. }}
@@ Line 112: / Line 112: @@
 *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; ${\rm Q}(\rho=0\ {\rm dB})={\rm Q}(x=1) =1.5866 \cdot 10^{-1}$.
 *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; ${\rm Q}(x=1.8) =3.593 \cdot 10^{-2}$.
-*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 &bdquo;quantization":&nbsp; ${\rm Q}(x=3.15) =8.1635 \cdot 10^{-4}$.
+*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; ${\rm Q}(x=3.15) =8.1635 \cdot 10^{-4}$.
@@ Line 143: / Line 143: @@
 &nbsp; &nbsp; '''(K)''' &nbsp; &nbsp; Variationsmöglichkeit für die graphischen Darstellungen
-$\hspace{1.5cm}$&bdquo;$+$" (Vergrößern),
+$\hspace{1.5cm}$"$+$" (Vergrößern),
-$\hspace{1.5cm}$ &bdquo;$-$" (Verkleinern)
+$\hspace{1.5cm}$ "$-$" (Verkleinern)
-$\hspace{1.5cm}$ &bdquo;$\rm o$" (Zurücksetzen)
+$\hspace{1.5cm}$ "$\rm o$" (Zurücksetzen)
-$\hspace{1.5cm}$ &bdquo;$\leftarrow$" (Verschieben nach links),  usw.
+$\hspace{1.5cm}$ "$\leftarrow$" (Verschieben nach links),  usw.
 <br clear=all>
 ==About the Authors==
@@ Line 155: / Line 155: @@
 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].
 *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]]).
-*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 &bdquo;HTML5".
+*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".
 *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; Translation using DEEPL.com.

@@ Line 50: / Line 50: @@
 :$$p_{\rm B} = {1}/{2} \cdot {\rm erfc} ( {4}/{\sqrt{2} })= {1}/{2} \cdot {\rm erfc} ( 2.828)\approx {1}/{2} \cdot {\rm erfc} ( 2.8)= 0.375 \cdot 10^{-4}\hspace{0.05cm}.$$
 *The first value is more correct.&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>
-*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 &bdquo;curvilinear&rdquo; value.&nbsp; In this case, of course,&nbsp; ${\rm Q}(x)$&nbsp; offers no advantage over&nbsp; $1/2 \cdot{\rm erfc}(x)$. }}
+*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 &bdquo;curvilinear" value.&nbsp; In this case, of course,&nbsp; ${\rm Q}(x)$&nbsp; offers no advantage over&nbsp; $1/2 \cdot{\rm erfc}(x)$. }}
@@ Line 112: / Line 112: @@
 *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; ${\rm Q}(\rho=0\ {\rm dB})={\rm Q}(x=1) =1.5866 \cdot 10^{-1}$.
 *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; ${\rm Q}(x=1.8) =3.593 \cdot 10^{-2}$.
-*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 &bdquo;quantization&rdquo;:&nbsp; ${\rm Q}(x=3.15) =8.1635 \cdot 10^{-4}$.
+*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 &bdquo;quantization":&nbsp; ${\rm Q}(x=3.15) =8.1635 \cdot 10^{-4}$.
@@ Line 143: / Line 143: @@
 &nbsp; &nbsp; '''(K)''' &nbsp; &nbsp; Variationsmöglichkeit für die graphischen Darstellungen
-$\hspace{1.5cm}$&bdquo;$+$&rdquo; (Vergrößern),
+$\hspace{1.5cm}$&bdquo;$+$" (Vergrößern),
-$\hspace{1.5cm}$ &bdquo;$-$&rdquo; (Verkleinern)
+$\hspace{1.5cm}$ &bdquo;$-$" (Verkleinern)
-$\hspace{1.5cm}$ &bdquo;$\rm o$&rdquo; (Zurücksetzen)
+$\hspace{1.5cm}$ &bdquo;$\rm o$" (Zurücksetzen)
-$\hspace{1.5cm}$ &bdquo;$\leftarrow$&rdquo; (Verschieben nach links),  usw.
+$\hspace{1.5cm}$ &bdquo;$\leftarrow$" (Verschieben nach links),  usw.
 <br clear=all>
 ==About the Authors==
@@ Line 155: / Line 155: @@
 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].
 *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]]).
-*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 &bdquo;HTML5&rdquo;.
+*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 &bdquo;HTML5".
 *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; Translation using DEEPL.com.

@@ Line 1: / Line 1: @@
-{{LntAppletLink|qfunction_en}} &nbsp; &nbsp; &nbsp; &nbsp; [https://www.lntwww.de/Applets:Komplementäre_Gaußsche_Fehlerfunktionen '''English Applet with German WIKI description''']
+{{LntAppletLinkEnDe|qfunction_en|qfunction}}
@@ Line 166: / Line 165: @@
 ==Once again: Open Applet in new Tab==
 <br>
-{{LntAppletLink|qfunction_en}} &nbsp; &nbsp; &nbsp; &nbsp; [https://www.lntwww.de/Applets:Komplementäre_Gaußsche_Fehlerfunktionen '''English Applet with German WIKI description''']
+{{LntAppletLinkEnDe|qfunction_en|qfunction}}

@@ Line 72: / Line 72: @@
 {{BlueBox|TEXT=
-'''(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 at linear and logarithmic ordinates.}}
+'''(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.}}
 *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; ${\rm Q}(6)=9.8659 \cdot 10^{-10}$.
@@ Line 100: / Line 100: @@
 {{BlueBox|TEXT=
-'''(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 accordingly&nbsp; $\rho\big[{\rm dB}\big ] = 20 \cdot \lg(x)$. }}
+'''(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; $\rho\big[{\rm dB}\big ] = 20 \cdot \lg(x)$. }}
 * The linear abscissa value&nbsp; $x=1$&nbsp; leads to the logarithmic abscissa value&nbsp; $\rho=0\ \rm dB$ &nbsp; &rArr; &nbsp; $0. 5 \cdot {\rm erfc}(\rho=0\ {\rm dB})={0.5 \cdot \rm erfc}(x=1)=7.865 \cdot 10^{-2}$.

← Older revision		Revision as of 10:06, 27 January 2021
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−	{{LntAppletLink\|qfunction_en}}	+	{{LntAppletLink\|qfunction_en}}         [https://lntwww.de/Applets:Komplementäre_Gaußsche_Fehlerfunktionen '''English Applet with German WIKI description''']
−	~~''Hinweis:''~~   ~~Das Applet ist für den~~ '''~~CHROME~~'''~~–Browser optimiert. Bei anderen Browsern kommt es teilweise zu Darstellungsproblemen.~~	+

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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:
 :$$p_{\rm B} = {\rm Q} \left ( \sqrt{ {2 E_{\rm B} }/{N_0} }\right ) = {1}/{2} \cdot { \rm erfc} \left ( \sqrt{ {E_{\rm B} }/{N_0} }\right ) \hspace{0.05cm}.$$
-For the numerical values&nbsp; $E_{\rm B} = 16 \rm mWs$&nbsp; and&nbsp; $N_0 = 16 \rm mW/Hz$&nbsp; we obtain:
+For the numerical values&nbsp; $E_{\rm B} = 16 \rm mWs$&nbsp; and&nbsp; $N_0 = 1 \rm mW/Hz$&nbsp; we obtain:
 :$$p_{\rm B} = {\rm Q} \left (4 \cdot \sqrt{ 2} \right ) = {1}/{2} \cdot {\rm erfc} \left ( 4\right ) \hspace{0.05cm}.$$
 *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; here the more correct value&nbsp; $p_{\rm B} \approx 0.771 \cdot 10^{-8}$&nbsp; yields.