Digital Signal Transmission/Error Probability with Intersymbol Interference - Revision history

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			[[de:Digitalsignalübertragung/Fehlerwahrscheinlichkeit_unter_Berücksichtigung_von_Impulsinterferenzen]]

Guenter at 11:15, 10 October 2022

2022-10-10T11:15:23Z

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	$\text{Example 2:}$  The same preconditions apply as for the last example.  The graph shows the detection signal  $d(t)$  after the Gaussian low-pass  $($before the decision$)$  for two different cutoff frequencies,  namely  $f_{\rm G} \cdot T = 0.8$  and  $f_{\rm G} \cdot T = 0.4$.  We want to analyze these images in terms of intersymbol interference.		$\text{Example 2:}$  The same preconditions apply as for the last example.  The graph shows the detection signal  $d(t)$  after the Gaussian low-pass  $($before the decision$)$  for two different cutoff frequencies,  namely  $f_{\rm G} \cdot T = 0.8$  and  $f_{\rm G} \cdot T = 0.4$.  We want to analyze these images in terms of intersymbol interference.

	[[File:~~P ID1384 Dig T 3 2 S1c version2~~.png\|right\|frame\|Detection signal with Gaussian receiver filter ~~'''Benedikt'''~~]]		[[File:Dig_T_3_2_S1c_version2_ret.png\|right\|frame\|Detection signal with Gaussian receiver filter]]
	In both diagrams are shown:		In both diagrams are shown:
	*the component  $d_{\rm S}(\nu \cdot T)$  of the detection signal without considering the noise  $($blue circles at the detection times$)$,<br>		*the component  $d_{\rm S}(\nu \cdot T)$  of the detection signal without considering the noise  $($blue circles at the detection times$)$,<br>

Guenter at 09:14, 6 July 2022

2022-07-06T09:14:26Z

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	== Worst-case error probability==		== Worst-case error probability==
	<br>		<br>
	In the past,  a variety of approximations for the average error probability have been given, among others: ~~'''Bitte dieses Kapitel sehr kritisch checken'''~~		In the past,  a variety of approximations for the average error probability have been given, among others:

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Guenter at 14:13, 15 June 2022

2022-06-15T14:13:53Z

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	$\text{Definition:}$  As a very simple approximation for the actual error probability  $p_{\rm S}$,		$\text{Definition:}$  As a very simple approximation for the actual error probability  $p_{\rm S}$,
	the  '''worst-case error probability'''  (German:  "ungünstigste Fehlerwahrscheinlichkeit"   ⇒   subscript:  "U")  is often used:		the  '''worst-case error probability'''  (German:  "ungünstigste Fehlerwahrscheinlichkeit"   ⇒   subscript:  "U")  is often used:
	[[File:EN_Dig_T_3_2_S4_neu.png\|right\|frame\|"Mean symbol error probability"  $p_{\rm S}$  vs.  "worst-case symbol error probability"  $p_{\rm U}$ ~~'''KORREKTUR: actual PDF'''~~\|class=fit]]		[[File:EN_Dig_T_3_2_S4_neu.png\|right\|frame\|"Mean symbol error probability"  $p_{\rm S}$  vs.  "worst-case symbol error probability"  $p_{\rm U}$ \|class=fit]]
	:$$p_{\rm U} = {\rm Q} \left( \frac{\ddot{o}(T_{\rm D})/2}{ \sigma_d}		:$$p_{\rm U} = {\rm Q} \left( \frac{\ddot{o}(T_{\rm D})/2}{ \sigma_d}
	\right) \hspace{0.05cm}.$$		\right) \hspace{0.05cm}.$$
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	The left graph is for the  (normalized)  cutoff frequency  $f_{\rm G} \cdot T = 0.4$,  the right one for a broader band receiver filter with  $f_{\rm G} \cdot T = 0.8$.		The left graph is for the  (normalized)  cutoff frequency  $f_{\rm G} \cdot T = 0.4$,  the right one for a broader band receiver filter with  $f_{\rm G} \cdot T = 0.8$.

	[[File:EN_Dig_T_3_2_S4_b_neu.png\|right\|frame\|Mean error probability  $p_{\rm S}$  and  worst-case error probability  $p_{\rm U}$  as a function of  $E_{\rm B}/N_0$ ~~'''KORREKTUR: error probability'''~~\|class=fit]]		[[File:EN_Dig_T_3_2_S4_b_neu.png\|right\|frame\|Mean error probability  $p_{\rm S}$  and  worst-case error probability  $p_{\rm U}$  as a function of  $E_{\rm B}/N_0$\|class=fit]]
	The results can be interpreted as follows:		The results can be interpreted as follows:
	*$p_{\rm U}$  is always an upper bound for the actual symbol error probability  $p_{\rm S}$.  The smaller the influence of the intersymbol interference  (large cutoff frequency),  the closer  $p_{\rm S}$  and  $p_{\rm U}$  are to each other.  For the optimal receiver  $p_{\rm S} = p_{\rm U}.$		*$p_{\rm U}$  is always an upper bound for the actual symbol error probability  $p_{\rm S}$.  The smaller the influence of the intersymbol interference  (large cutoff frequency),  the closer  $p_{\rm S}$  and  $p_{\rm U}$  are to each other.  For the optimal receiver  $p_{\rm S} = p_{\rm U}.$

Guenter at 16:08, 14 June 2022

2022-06-14T16:08:43Z

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	$\text{Definition:}$  As a very simple approximation for the actual error probability  $p_{\rm S}$,		$\text{Definition:}$  As a very simple approximation for the actual error probability  $p_{\rm S}$,
	the  '''worst-case error probability'''  (German:  "ungünstigste Fehlerwahrscheinlichkeit"   ⇒   subscript:  "U")  is often used:		the  '''worst-case error probability'''  (German:  "ungünstigste Fehlerwahrscheinlichkeit"   ⇒   subscript:  "U")  is often used:
	[[File:~~EN_Dig_T_3_2_S4~~.png\|right\|frame\|"Mean symbol error probability"  $p_{\rm S}$  vs.  "worst-case symbol error probability"  $p_{\rm U}$ '''KORREKTUR: actual PDF'''\|class=fit]]		[[File:EN_Dig_T_3_2_S4_neu.png\|right\|frame\|"Mean symbol error probability"  $p_{\rm S}$  vs.  "worst-case symbol error probability"  $p_{\rm U}$ '''KORREKTUR: actual PDF'''\|class=fit]]
	:$$p_{\rm U} = {\rm Q} \left( \frac{\ddot{o}(T_{\rm D})/2}{ \sigma_d}		:$$p_{\rm U} = {\rm Q} \left( \frac{\ddot{o}(T_{\rm D})/2}{ \sigma_d}
	\right) \hspace{0.05cm}.$$		\right) \hspace{0.05cm}.$$
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	The left graph is for the  (normalized)  cutoff frequency  $f_{\rm G} \cdot T = 0.4$,  the right one for a broader band receiver filter with  $f_{\rm G} \cdot T = 0.8$.		The left graph is for the  (normalized)  cutoff frequency  $f_{\rm G} \cdot T = 0.4$,  the right one for a broader band receiver filter with  $f_{\rm G} \cdot T = 0.8$.

	[[File:~~EN_Dig_T_3_2_S4_b~~.png\|right\|frame\|Mean error probability  $p_{\rm S}$  and  worst-case error probability  $p_{\rm U}$  as a function of  $E_{\rm B}/N_0$ '''KORREKTUR: error probability'''\|class=fit]]		[[File:EN_Dig_T_3_2_S4_b_neu.png\|right\|frame\|Mean error probability  $p_{\rm S}$  and  worst-case error probability  $p_{\rm U}$  as a function of  $E_{\rm B}/N_0$ '''KORREKTUR: error probability'''\|class=fit]]
	The results can be interpreted as follows:		The results can be interpreted as follows:
	*$p_{\rm U}$  is always an upper bound for the actual symbol error probability  $p_{\rm S}$.  The smaller the influence of the intersymbol interference  (large cutoff frequency),  the closer  $p_{\rm S}$  and  $p_{\rm U}$  are to each other.  For the optimal receiver  $p_{\rm S} = p_{\rm U}.$		*$p_{\rm U}$  is always an upper bound for the actual symbol error probability  $p_{\rm S}$.  The smaller the influence of the intersymbol interference  (large cutoff frequency),  the closer  $p_{\rm S}$  and  $p_{\rm U}$  are to each other.  For the optimal receiver  $p_{\rm S} = p_{\rm U}.$

Hwang at 08:45, 6 June 2022

2022-06-06T08:45:13Z

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	[[Aufgaben:Exercise_3.2:_Eye_Pattern_according_to_Gaussian_Low-Pass\|Exercise 3.2: Eye Pattern according to Gaussian Low-Pass]]		[[Aufgaben:Exercise_3.2:_Eye_Pattern_according_to_Gaussian_Low-Pass\|Exercise 3.2: Eye Pattern according to Gaussian Low-Pass]]

	[[Aufgaben:Exercise_3.2Z:_Optimum_Cut-off_Frequency_for_Gaussian_Low-pass\|Exercise 3.2Z: Optimum ~~Cut-off~~ Frequency for Gaussian Low-pass]]		[[Aufgaben:Exercise_3.2Z:_Optimum_Cut-off_Frequency_for_Gaussian_Low-pass\|Exercise 3.2Z: Optimum Cutoff Frequency for Gaussian Low-pass]]


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Hwang at 08:11, 6 June 2022

2022-06-06T08:11:16Z

← Older revision		Revision as of 10:11, 6 June 2022
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	The left graph is for the  (normalized)  cutoff frequency  $f_{\rm G} \cdot T = 0.4$,  the right one for a broader band receiver filter with  $f_{\rm G} \cdot T = 0.8$.		The left graph is for the  (normalized)  cutoff frequency  $f_{\rm G} \cdot T = 0.4$,  the right one for a broader band receiver filter with  $f_{\rm G} \cdot T = 0.8$.

	[[File:EN_Dig_T_3_2_S4_b.png\|right\|frame\|Mean error probability  $p_{\rm S}$  and  worst-case error probability  $p_{\rm U}$  as a function of  $E_{\rm B}/N_0$\|class=fit]]		[[File:EN_Dig_T_3_2_S4_b.png\|right\|frame\|Mean error probability  $p_{\rm S}$  and  worst-case error probability  $p_{\rm U}$  as a function of  $E_{\rm B}/N_0$ '''KORREKTUR: error probability'''\|class=fit]]
	The results can be interpreted as follows:		The results can be interpreted as follows:
	*$p_{\rm U}$  is always an upper bound for the actual symbol error probability  $p_{\rm S}$.  The smaller the influence of the intersymbol interference  (large cutoff frequency),  the closer  $p_{\rm S}$  and  $p_{\rm U}$  are to each other.  For the optimal receiver  $p_{\rm S} = p_{\rm U}.$		*$p_{\rm U}$  is always an upper bound for the actual symbol error probability  $p_{\rm S}$.  The smaller the influence of the intersymbol interference  (large cutoff frequency),  the closer  $p_{\rm S}$  and  $p_{\rm U}$  are to each other.  For the optimal receiver  $p_{\rm S} = p_{\rm U}.$

Hwang at 08:09, 6 June 2022

2022-06-06T08:09:23Z

← Older revision		Revision as of 10:09, 6 June 2022
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	$\text{Definition:}$  As a very simple approximation for the actual error probability  $p_{\rm S}$,		$\text{Definition:}$  As a very simple approximation for the actual error probability  $p_{\rm S}$,
	the  '''worst-case error probability'''  (German:  "ungünstigste Fehlerwahrscheinlichkeit"   ⇒   subscript:  "U")  is often used:		the  '''worst-case error probability'''  (German:  "ungünstigste Fehlerwahrscheinlichkeit"   ⇒   subscript:  "U")  is often used:
	[[File:EN_Dig_T_3_2_S4.png\|right\|frame\|"Mean symbol error probability"  $p_{\rm S}$  vs.  "worst-case symbol error probability"  $p_{\rm U}$\|class=fit]]		[[File:EN_Dig_T_3_2_S4.png\|right\|frame\|"Mean symbol error probability"  $p_{\rm S}$  vs.  "worst-case symbol error probability"  $p_{\rm U}$ '''KORREKTUR: actual PDF'''\|class=fit]]
	:$$p_{\rm U} = {\rm Q} \left( \frac{\ddot{o}(T_{\rm D})/2}{ \sigma_d}		:$$p_{\rm U} = {\rm Q} \left( \frac{\ddot{o}(T_{\rm D})/2}{ \sigma_d}
	\right) \hspace{0.05cm}.$$		\right) \hspace{0.05cm}.$$

Guenter at 15:46, 2 June 2022

2022-06-02T15:46:18Z

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	erfc} \left ( 2 \cdot \sqrt {\pi} \cdot f_{\rm G}\cdot ( t +		erfc} \left ( 2 \cdot \sqrt {\pi} \cdot f_{\rm G}\cdot ( t +
	{T}/{2} )\right ) \big ]\hspace{0.05cm}.$$		{T}/{2} )\right ) \big ]\hspace{0.05cm}.$$
	[[File:~~Dig_T_3_2_S1_version2~~.png\|right\|frame\|Block diagram for the chapter  "Error Probability with Intersymbol Interference"]]		[[File:EN_Dig_T_3_2_S1.png\|right\|frame\|Block diagram for the chapter  "Error Probability with Intersymbol Interference"]]

	Here,  two variants of the complementary Gaussian error function are used,  viz.		Here,  two variants of the complementary Gaussian error function are used,  viz.
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	$\text{Example 3:}$  We assume a redundancy-free binary bipolar NRZ rectangular signal  $s(t)$  and the Gaussian low-pass filter with cutoff frequency  $f_{\rm G} \cdot T = 0.4$.		$\text{Example 3:}$  We assume a redundancy-free binary bipolar NRZ rectangular signal  $s(t)$  and the Gaussian low-pass filter with cutoff frequency  $f_{\rm G} \cdot T = 0.4$.
	[[File:~~P ID1375 Dig T 3 2 S2 version1~~.png\|right\|frame\|On the left:  Eye diagram with noise  ⇒   signal  $d(t)=d_{\rm S}(t) +d_{\rm N}(t)$, <br>on the right:  Eye diagram without noise  ⇒   signal  $d_{\rm S}(t)$\|class=fit]]		[[File:EN_Dig_T_3_2_S2.png\|right\|frame\|On the left:  Eye diagram with noise  ⇒   signal  $d(t)=d_{\rm S}(t) +d_{\rm N}(t)$, <br>on the right:  Eye diagram without noise  ⇒   signal  $d_{\rm S}(t)$\|class=fit]]
	In the graphic shown are the eye diagrams after the Gaussian low-pass,		In the graphic shown are the eye diagrams after the Gaussian low-pass,
	*left inclusive the noise component   ⇒   signal  $d(t)=d_{\rm S}(t) +d_{\rm N}(t)$,		*left inclusive the noise component   ⇒   signal  $d(t)=d_{\rm S}(t) +d_{\rm N}(t)$,
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	$\text{Definition:}$  As a very simple approximation for the actual error probability  $p_{\rm S}$,		$\text{Definition:}$  As a very simple approximation for the actual error probability  $p_{\rm S}$,
	the  '''worst-case error probability'''  (German:  "ungünstigste Fehlerwahrscheinlichkeit"   ⇒   subscript:  "U")  is often used:		the  '''worst-case error probability'''  (German:  "ungünstigste Fehlerwahrscheinlichkeit"   ⇒   subscript:  "U")  is often used:
	[[File:~~P ID1379 Dig T 3 2 S4 version1~~.png\|right\|frame\|"Mean symbol error probability"  $p_{\rm S}$  vs.  "worst-case symbol error probability"  $p_{\rm U}$\|class=fit]]		[[File:EN_Dig_T_3_2_S4.png\|right\|frame\|"Mean symbol error probability"  $p_{\rm S}$  vs.  "worst-case symbol error probability"  $p_{\rm U}$\|class=fit]]
	:$$p_{\rm U} = {\rm Q} \left( \frac{\ddot{o}(T_{\rm D})/2}{ \sigma_d}		:$$p_{\rm U} = {\rm Q} \left( \frac{\ddot{o}(T_{\rm D})/2}{ \sigma_d}
	\right) \hspace{0.05cm}.$$		\right) \hspace{0.05cm}.$$
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	The left graph is for the  (normalized)  cutoff frequency  $f_{\rm G} \cdot T = 0.4$,  the right one for a broader band receiver filter with  $f_{\rm G} \cdot T = 0.8$.		The left graph is for the  (normalized)  cutoff frequency  $f_{\rm G} \cdot T = 0.4$,  the right one for a broader band receiver filter with  $f_{\rm G} \cdot T = 0.8$.

	[[File:~~P ID1385 Dig T 3 2 S4b version2~~.png\|right\|frame\|Mean error probability  $p_{\rm S}$  and  worst-case error probability  $p_{\rm U}$  as a function of  $E_{\rm B}/N_0$\|class=fit]]		[[File:EN_Dig_T_3_2_S4_b.png\|right\|frame\|Mean error probability  $p_{\rm S}$  and  worst-case error probability  $p_{\rm U}$  as a function of  $E_{\rm B}/N_0$\|class=fit]]
	The results can be interpreted as follows:		The results can be interpreted as follows:
	*$p_{\rm U}$  is always an upper bound for the actual symbol error probability  $p_{\rm S}$.  The smaller the influence of the intersymbol interference  (large cutoff frequency),  the closer  $p_{\rm S}$  and  $p_{\rm U}$  are to each other.  For the optimal receiver  $p_{\rm S} = p_{\rm U}.$		*$p_{\rm U}$  is always an upper bound for the actual symbol error probability  $p_{\rm S}$.  The smaller the influence of the intersymbol interference  (large cutoff frequency),  the closer  $p_{\rm S}$  and  $p_{\rm U}$  are to each other.  For the optimal receiver  $p_{\rm S} = p_{\rm U}.$
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	<br>		<br>
	For system optimization and system comparison, it turns out to be convenient,  instead of using the worst-case error probability  $p_{\rm U}$  to use the  "worst–case signal–to–noise power ratio"  $\text{(S/N ratio)}$:		For system optimization and system comparison, it turns out to be convenient,  instead of using the worst-case error probability  $p_{\rm U}$  to use the  "worst–case signal–to–noise power ratio"  $\text{(S/N ratio)}$:
	[[File:~~P ID1380 Dig T 3 2 S5 version1~~.png\|right\|frame\|SNR as a function of the cutoff frequency of a Gaussian low-pass filter\|class=fit]]		[[File:EN_Dig_T_3_2_S5.png\|right\|frame\|SNR as a function of the cutoff frequency of a Gaussian low-pass filter\|class=fit]]

	:$$\rho_{\rm U} = [\ddot{o}(T_{\rm D})]^2/ \sigma_d^2.$$		:$$\rho_{\rm U} = [\ddot{o}(T_{\rm D})]^2/ \sigma_d^2.$$