Applets:Eye Pattern and Worst-Case Error Probability - Revision history

Maintenance script: Join split $$ formula lines for correct rendering

2026-03-16T13:49:00Z

Join split $$ formula lines for correct rendering

← Older revision		Revision as of 15:49, 16 March 2026
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	*The rolloff of  $H_{\rm CRO}(f)$  is point symmetric about the Nyquist frequency  $1/(2T)$. The larger the rolloff factor  $r_{ \hspace {-0.05cm}f}$,  ithe flatter the Nyquist slope.		*The rolloff of  $H_{\rm CRO}(f)$  is point symmetric about the Nyquist frequency  $1/(2T)$. The larger the rolloff factor  $r_{ \hspace {-0.05cm}f}$,  ithe flatter the Nyquist slope.
	*The basic detection pulse  $g_d(t) = s_0 \cdot T \cdot {\mathcal F}^{-1}\big[H_{\rm CRO}(f)\big]$  has zeros at times  $\nu \cdot T$  independent of  $r_{ \hspace {-0.05cm}f}$.  There are further zero crossings depending on  $r_{ \hspace {-0.05cm}f}$.  For the pulse holds:		*The basic detection pulse  $g_d(t) = s_0 \cdot T \cdot {\mathcal F}^{-1}\big[H_{\rm CRO}(f)\big]$  has zeros at times  $\nu \cdot T$  independent of  $r_{ \hspace {-0.05cm}f}$.  There are further zero crossings depending on  $r_{ \hspace {-0.05cm}f}$.  For the pulse holds:
	:$$g_d(t) = s_0 \hspace{-0.05cm}\cdot\hspace{-0.05cm} {\rm sinc}( t/T )\hspace{-0.05cm}\cdot\hspace{-0.05cm}\frac {\cos(\pi \cdot r_{\hspace{-0.05cm}f} \cdot t/T )}{1 - (2 \~~cdot~~		:$$g_d(t) = s_0 \hspace{-0.05cm}\cdot\hspace{-0.05cm} {\rm sinc}( t/T )\hspace{-0.05cm}\cdot\hspace{-0.05cm}\frac {\cos(\pi \cdot r_{\hspace{-0.05cm}f} \cdot t/T )}{1 - (2 \cdotr_{\hspace{-0.05cm}f} \cdot t/T)^2}.$$
	r_{\hspace{-0.05cm}f} \cdot t/T)^2}.$$
	*It follows:  As with the matched filter receiver, the eye is maximally open   ⇒   $ö_{\rm norm} =1$.		*It follows:  As with the matched filter receiver, the eye is maximally open   ⇒   $ö_{\rm norm} =1$.

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	:$$H_{\rm S}(f) = {\rm sinc}(f T).$$		:$$H_{\rm S}(f) = {\rm sinc}(f T).$$
	*Gaussian receiver filter with cutoff frequency  $f_{\rm G}$:		*Gaussian receiver filter with cutoff frequency  $f_{\rm G}$:
	:$$H_{\rm E}(f) = H_{\rm G}(f) = {\rm e}^{- \pi \hspace{0.05cm}\cdot \hspace{0.03cm} f^2/(2\hspace{0.05cm}\cdot \hspace{0.03cm}f_{\rm G})^2 } \hspace{0.2cm} \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ		:$$H_{\rm E}(f) = H_{\rm G}(f) = {\rm e}^{- \pi \hspace{0.05cm}\cdot \hspace{0.03cm} f^2/(2\hspace{0.05cm}\cdot \hspace{0.03cm}f_{\rm G})^2 } \hspace{0.2cm} \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\hspace{0.2cm}h_{\rm E}(t) = h_{\rm G}(t) = {\rm e}^{- \pi \cdot (2\hspace{0.05cm}\cdot \hspace{0.03cm}f_{\rm G}\hspace{0.05cm}\cdot \hspace{0.02cm} t)^2}\hspace{0.05cm}.$$
	\hspace{0.2cm}h_{\rm E}(t) = h_{\rm G}(t) = {\rm e}^{- \pi \cdot (2\hspace{0.05cm}\cdot \hspace{0.03cm}
	f_{\rm G}\hspace{0.05cm}\cdot \hspace{0.02cm} t)^2}
	\hspace{0.05cm}.$$

	Based on the assumptions made here, the following applies to the basic detection pulse:		Based on the assumptions made here, the following applies to the basic detection pulse:

	[[File:Auge_5_neu.png\|right\|frame\|Frequency response and impulse response of the receiver filter]]		[[File:Auge_5_neu.png\|right\|frame\|Frequency response and impulse response of the receiver filter]]
	:$$g_d(t) = s_0 \cdot T \cdot \big [h_{\rm S}(t) \star h_{\rm G}(t)\big ] = 2 f_{\rm G} \cdot s_0 \cdot \int_{t-T/2}^{t+T/2}		:$$g_d(t) = s_0 \cdot T \cdot \big [h_{\rm S}(t) \star h_{\rm G}(t)\big ] = 2 f_{\rm G} \cdot s_0 \cdot \int_{t-T/2}^{t+T/2}{\rm e}^{- \pi \hspace{0.05cm}\cdot\hspace{0.05cm} (2 \hspace{0.05cm}\cdot\hspace{0.02cm}f_{\rm G}\hspace{0.05cm}\cdot\hspace{0.02cm} \tau )^2} \,{\rm d} \tau \hspace{0.05cm}.$$
	{\rm e}^{- \pi \hspace{0.05cm}\cdot\hspace{0.05cm} (2 \hspace{0.05cm}\cdot\hspace{0.02cm}
	f_{\rm G}\hspace{0.05cm}\cdot\hspace{0.02cm} \tau )^2} \,{\rm d} \tau \hspace{0.05cm}.$$

	The integration leads to the result:		The integration leads to the result:

	:$$g_d(t) = s_0 \cdot \big [ {\rm Q} \left ( 2 \cdot \sqrt {2 \pi}		:$$g_d(t) = s_0 \cdot \big [ {\rm Q} \left ( 2 \cdot \sqrt {2 \pi}\cdot f_{\rm G}\cdot ( t - {T}/{2})\right )- {\rm Q} \left (2 \cdot \sqrt {2 \pi} \cdot f_{\rm G}\cdot ( t + {T}/{2})\right ) \big ],$$
	\cdot f_{\rm G}\cdot ( t - {T}/{2})\right )- {\rm Q} \left (
	2 \cdot \sqrt {2 \pi} \cdot f_{\rm G}\cdot ( t + {T}/{2}
	)\right ) \big ],$$

	using the complementary Gaussian error function		using the complementary Gaussian error function

	:$${\rm Q} (x) = \frac{\rm 1}{\sqrt{\rm 2\pi}}\int_{\it		:$${\rm Q} (x) = \frac{\rm 1}{\sqrt{\rm 2\pi}}\int_{\it x}^{+\infty}\rm e^{\it -u^{\rm 2}/\rm 2}\,d {\it u}\hspace{0.05cm}.$$
	x}^{+\infty}\rm e^{\it -u^{\rm 2}/\rm 2}\,d {\it u}
	\hspace{0.05cm}.$$

	The module  [[Applets:Komplementäre_Gaußsche_Fehlerfunktionen\|"Complementary Gaussian Error Functions"]]  provides the numerical values of  ${\rm Q} (x)$.<br>		The module  [[Applets:Komplementäre_Gaußsche_Fehlerfunktionen\|"Complementary Gaussian Error Functions"]]  provides the numerical values of  ${\rm Q} (x)$.<br>

Guenter at 14:06, 17 January 2024

2024-01-17T14:06:27Z

← Older revision		Revision as of 16:06, 17 January 2024
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	This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ Institute for Communications Engineering]  at the  [https://www.tum.de/en Technical University of Munich].		This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ Institute for Communications Engineering]  at the  [https://www.tum.de/en Technical University of Munich].
	*The first version was created in 2008 by [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Thomas_Gro.C3.9Fer_.28Diplomarbeit_LB_2006.2C_danach_freie_Mitarbeit_bis_2010.29\|Thomas Großer]]  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]]).		*The first version was created in 2008 by [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Thomas_Gro.C3.9Fer_.28Diplomarbeit_LB_2006.2C_danach_freie_Mitarbeit_bis_2010.29\|Thomas Großer]]  as part of his diploma thesis with “FlashMX – Actionscript” (Supervisor: [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29\|Günter Söder]]).

	*Last revision and English version 2020/2021 by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29\|Carolin Mirschina]]  in the context of a working student activity.		*Last revision and English version 2020/2021 by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29\|Carolin Mirschina]]  in the context of a working student activity.

Guenter at 14:04, 17 January 2024

2024-01-17T14:04:41Z

← Older revision		Revision as of 16:04, 17 January 2024
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	This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ Institute for Communications Engineering]  at the  [https://www.tum.de/en Technical University of Munich].		This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ Institute for Communications Engineering]  at the  [https://www.tum.de/en Technical University of Munich].
	*The first version was created in 2008 by [[~~Biografien_und_Bibliografien~~/~~An_LNTwww_beteiligte_Studierende~~#Thomas_Gro.C3.9Fer_.28Diplomarbeit_LB_2006.2C_danach_freie_Mitarbeit_bis_2010.29\|Thomas Großer]]  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]]).		*The first version was created in 2008 by [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Thomas_Gro.C3.9Fer_.28Diplomarbeit_LB_2006.2C_danach_freie_Mitarbeit_bis_2010.29\|Thomas Großer]]  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]]).

	*Last revision and English version 2020/2021 by  [[~~Biografien_und_Bibliografien~~/~~An_LNTwww_beteiligte_Studierende~~#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29\|Carolin Mirschina]]  in the context of a working student activity.		*Last revision and English version 2020/2021 by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29\|Carolin Mirschina]]  in the context of a working student activity.

Guenter at 15:29, 21 April 2023

2023-04-21T15:29:09Z

← Older revision		Revision as of 17:29, 21 April 2023
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	<br>		<br>
	[[File:Anleitung_Auge.png\|right\|600px\|frame\|Screenshot of the German version]]		[[File:Anleitung_Auge.png\|right\|600px\|frame\|Screenshot of the German version]]
	'''(A)'''     Selection:   ~~Coding~~ <br>                   (binary,  quaternary,  AMI code,  duobinary code)		'''(A)'''     Selection:   Encoding <br>                   (binary,  quaternary,  AMI code,  duobinary code)

	'''(B)'''     Selection:   Basic detection pulse<br>                    (according to Gauss–TP,  CRO–Nyquist,  according to gap–TP}		'''(B)'''     Selection:   Basic detection pulse<br>                    (according to Gauss–TP,  CRO–Nyquist,  according to gap–TP}

Guenter at 16:28, 20 April 2023

2023-04-20T16:28:58Z

← Older revision		Revision as of 18:28, 20 April 2023
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	===Intersymbol interference system with Gaussian receiver filter===		===Intersymbol interference system with Gaussian receiver filter===

	[[File:~~EN_Auge_4~~.png\|right\|frame\|System with Gaussian receiver filter]]		[[File:EN_Auge_4_neu.png\|right\|frame\|System with Gaussian receiver filter]]

	We start from the block diagram sketched on the right. Further it shall be valid:		We start from the block diagram sketched on the right. Further it shall be valid:

Guenter at 12:00, 17 April 2023

2023-04-17T12:00:18Z

← Older revision		Revision as of 14:00, 17 April 2023
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	===Intersymbol interference system with Gaussian receiver filter===		===Intersymbol interference system with Gaussian receiver filter===

	[[File:EN_Auge_4.png\|right\|frame\|System with Gaussian receiver filter ~~'''Korrektur'''~~]]		[[File:EN_Auge_4.png\|right\|frame\|System with Gaussian receiver filter]]

	We start from the block diagram sketched on the right. Further it shall be valid:		We start from the block diagram sketched on the right. Further it shall be valid:

Guenter at 15:17, 13 April 2023

2023-04-13T15:17:25Z

← Older revision		Revision as of 17:17, 13 April 2023
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	===Intersymbol interference system with Gaussian receiver filter===		===Intersymbol interference system with Gaussian receiver filter===

	[[File:~~EN_Auge_4_v2~~.png\|right\|frame\|System with Gaussian receiver filter]]		[[File:EN_Auge_4.png\|right\|frame\|System with Gaussian receiver filter '''Korrektur''']]

	We start from the block diagram sketched on the right. Further it shall be valid:		We start from the block diagram sketched on the right. Further it shall be valid:

Guenter at 14:13, 13 April 2023

2023-04-13T14:13:47Z

← Older revision		Revision as of 16:13, 13 April 2023
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	===Intersymbol interference system with Gaussian receiver filter===		===Intersymbol interference system with Gaussian receiver filter===

	[[File:~~EN_Auge_4~~.png\|right\|frame\|System with Gaussian receiver filter ~~'''Korrektur'''~~]]		[[File:EN_Auge_4_v2.png\|right\|frame\|System with Gaussian receiver filter]]

	We start from the block diagram sketched on the right. Further it shall be valid:		We start from the block diagram sketched on the right. Further it shall be valid:

Guenter at 13:37, 13 April 2023

2023-04-13T13:37:39Z

← Older revision		Revision as of 15:37, 13 April 2023
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	This ensures that there are no long  "+1"–  or  "–1" sequences in the AMI-encoded signal, which would be problematic for an DC signal-free channel.		This ensures that there are no long  "+1"–  or  "–1" sequences in the AMI-encoded signal, which would be problematic for an DC signal-free channel.
	<br>		<br>
	[[File:~~Auge_16a~~.png\|left\|frame\| ~~'''Korrektur'''~~\|class=fit]]		[[File:EN_Auge_16a.png\|left\|frame\| \|class=fit]]


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	* Also the duobinary encoded sequence consists to 50% of zeros. The enhancement factor due to the smaller  $E_{\rm B}/ N_0$  is equal to  $\sqrt{1/2} \approx 0.707$, as in the AMI code.		* Also the duobinary encoded sequence consists to 50% of zeros. The enhancement factor due to the smaller  $E_{\rm B}/ N_0$  is equal to  $\sqrt{1/2} \approx 0.707$, as in the AMI code.

	[[File:~~Auge_17a~~.png\|left\|frame\| ~~'''Korrektur'''~~\|class=fit]]		[[File:EN_Auge_17a.png\|left\|frame\| \|class=fit]]
	<br>		<br>
	The eye diagram is shown on the left.		The eye diagram is shown on the left.

Guenter at 13:35, 13 April 2023

2023-04-13T13:35:06Z

← Older revision		Revision as of 15:35, 13 April 2023
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	===Nyquist system with raised cosine overall frequency response===		===Nyquist system with raised cosine overall frequency response===

	[[File:~~Auge_2_neu~~.png\|right\|frame\|Raised cosine overall frequency response ~~'''Korrektur'''~~]]		[[File:EN_Auge_2.png\|right\|frame\|Raised cosine overall frequency response]]

	We assume that the overall frequency response between the Dirac-shaped source to the decision has the shape of a  [[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory#Raised-cosine_low-pass_filter\|"raised cosine low-pass"]]    ⇒   $H_{\rm S}(f)\cdot H_{\rm E}(f) = H_{\rm CRO}(f)$ .		We assume that the overall frequency response between the Dirac-shaped source to the decision has the shape of a  [[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory#Raised-cosine_low-pass_filter\|"raised cosine low-pass"]]    ⇒   $H_{\rm S}(f)\cdot H_{\rm E}(f) = H_{\rm CRO}(f)$ .
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	===Intersymbol interference system with Gaussian receiver filter===		===Intersymbol interference system with Gaussian receiver filter===

	[[File:~~Auge_4~~.png\|right\|frame\|System with Gaussian receiver filter '''Korrektur''']]		[[File:EN_Auge_4.png\|right\|frame\|System with Gaussian receiver filter '''Korrektur''']]

	We start from the block diagram sketched on the right. Further it shall be valid:		We start from the block diagram sketched on the right. Further it shall be valid:
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	In symbolwise coding, each incoming source symbol  $q_\nu$  generates an encoder symbol  $c_\nu$  that depends not only on the current input symbol  $q_\nu$  but also on the  $N_{\rm C}$  preceding symbols  $q_{\nu-1}$, ... , $q_{\nu-N_{\rm C}} $.   $N_{\rm C}$  is referred to as the ''order''  of the code.  It is typical for a symbolwise coding that		In symbolwise coding, each incoming source symbol  $q_\nu$  generates an encoder symbol  $c_\nu$  that depends not only on the current input symbol  $q_\nu$  but also on the  $N_{\rm C}$  preceding symbols  $q_{\nu-1}$, ... , $q_{\nu-N_{\rm C}} $.   $N_{\rm C}$  is referred to as the ''order''  of the code.  It is typical for a symbolwise coding that
	[[File:~~Dig_T_2_4_S1_v1_neu~~.png\|right\|frame\|Block diagram and equivalent circuit of a pseudo-ternary encoder ~~'''Korrektur'''~~\|class=fit]]		[[File:EN_Dig_T_2_4_S1_v26.png\|right\|frame\|Block diagram and equivalent circuit of a pseudo-ternary encoder\|class=fit]]

	*the symbol duration  $T$  of the encoded signal (and of the transmitted signal) coincides with the bit duration  $T_{\rm B}$  of the binary source signal, and		*the symbol duration  $T$  of the encoded signal (and of the transmitted signal) coincides with the bit duration  $T_{\rm B}$  of the binary source signal, and