Digital Signal Transmission/Burst Error Channels - Revision history

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	:$$p_{\rm G} = 0.01,$$		:$$p_{\rm G} = 0.01,$$
	:$$p_{\rm B} = 0.4,$$		:$$p_{\rm B} = 0.4,$$
	:$${\rm Pr}(\rm G\hspace{0.05cm}\vert\hspace{0.05cm} B) =		:$${\rm Pr}(\rm G\hspace{0.05cm}\vert\hspace{0.05cm} B) =0.1, $$
	0.1, $$		:$$ {\rm Pr}(\rm B\hspace{0.05cm}\vert\hspace{0.05cm}G) = 0.01\hspace{0.05cm}.$$
	:$$ {\rm Pr}(\rm B\hspace{0.05cm}\vert\hspace{0.05cm}
	G) = 0.01\hspace{0.05cm}.$$
	The underlying model is shown at the end of this example with the parameters given here.		The underlying model is shown at the end of this example with the parameters given here.

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	The upper graph shows the error distance distribution  $\rm (EDD)$  of the Gilbert-Elliott model  (red curve)  in linear and logarithmic representation for the parameters		The upper graph shows the error distance distribution  $\rm (EDD)$  of the Gilbert-Elliott model  (red curve)  in linear and logarithmic representation for the parameters
	:$${\rm Pr}(\rm		:$${\rm Pr}(\rmG\hspace{0.05cm}\|\hspace{0.05cm} B ) = 0.1 \hspace{0.05cm},\hspace{0.5cm}{\rm Pr}(\rmB\hspace{0.05cm}\|\hspace{0.05cm} G ) = 0.01 \hspace{0.05cm},\hspace{0.5cm}p_{\rm G} = 0.001, \hspace{0.5cm}p_{\rm B} = 0.4.$$
	G\hspace{0.05cm}\|\hspace{0.05cm} B ) = 0.1 \hspace{0.05cm},\hspace{0.5cm}{\rm Pr}(\rm
	B\hspace{0.05cm}\|\hspace{0.05cm} G ) = 0.01 \hspace{0.05cm},\hspace{0.5cm}p_{\rm G} = 0.001, \hspace{0.5cm}p_{\rm B} = 0.4.$$

	For comparison,  the corresponding  $V_a(k)$  curve for the BSC model with the same mean error probability  $p_{\rm M} = 4.5\%$  is also plotted as blue curve.<br>		For comparison,  the corresponding  $V_a(k)$  curve for the BSC model with the same mean error probability  $p_{\rm M} = 4.5\%$  is also plotted as blue curve.<br>
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	For the Gilbert-Elliott model, for  "renewing models" $\varphi_{e}(k)$  must always be calculated according to this equation.  The iterative calculation algorithm for  "renewing models",		For the Gilbert-Elliott model, for  "renewing models" $\varphi_{e}(k)$  must always be calculated according to this equation.  The iterative calculation algorithm for  "renewing models",
	:$$\varphi_{e}(k) = \sum_{\kappa = 1}^{k} {\rm Pr}(a = \kappa) \cdot		:$$\varphi_{e}(k) = \sum_{\kappa = 1}^{k} {\rm Pr}(a = \kappa) \cdot\varphi_{e}(k - \kappa), $$
	\varphi_{e}(k - \kappa), $$

	cannot be applied here, since the GE model is not renewing   ⇒   here,  the error distances are not statistically independent of each other.		cannot be applied here, since the GE model is not renewing   ⇒   here,  the error distances are not statistically independent of each other.
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	:$$p_{\rm G} = 0.01,$$		:$$p_{\rm G} = 0.01,$$
	:$$p_{\rm B} = 0.4,$$		:$$p_{\rm B} = 0.4,$$
	:$${\rm Pr}(\rm G\hspace{0.05cm}\vert\hspace{0.05cm} B) =		:$${\rm Pr}(\rm G\hspace{0.05cm}\vert\hspace{0.05cm} B) =0.1, $$
	0.1, $$		:$$ {\rm Pr}(\rm B\hspace{0.05cm}\vert\hspace{0.05cm}G) = 0.01\hspace{0.05cm}.$$
	:$$ {\rm Pr}(\rm B\hspace{0.05cm}\vert\hspace{0.05cm}
	G) = 0.01\hspace{0.05cm}.$$

	An error sequence of the equivalent McCullough model is drawn below.  The relations between the two models can be summarized as follows:		An error sequence of the equivalent McCullough model is drawn below.  The relations between the two models can be summarized as follows:
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	The parameters of the equivalent MC model can be calculated from the GE parameters as follows:		The parameters of the equivalent MC model can be calculated from the GE parameters as follows:

	:$$q_{\rm G} =1-\beta_{\rm		:$$q_{\rm G} =1-\beta_{\rmG}\hspace{0.05cm},$$
	G}\hspace{0.05cm},$$		:$$ q_{\rmB} = 1-\beta_{\rm B}\hspace{0.05cm}, $$
	:$$ q_{\rm		:$$q(\rm B\hspace{0.05cm}\|\hspace{0.05cm} G ) =\frac{\alpha_{\rm B} \cdot[{\rm Pr}(\rmB\hspace{0.05cm}\|\hspace{0.05cm} G ) + {\rm Pr}(\rmG\hspace{0.05cm}\|\hspace{0.05cm} B )]}{\alpha_{\rm G} \cdot q_{\rmB} + \alpha_{\rm B} \cdot q_{\rm G}} \hspace{0.05cm},$$
	B} = 1-\beta_{\rm B}\hspace{0.05cm}, $$		:$$q(\rm G\hspace{0.05cm}\|\hspace{0.05cm} B ) =\frac{\alpha_{\rm G}}{\alpha_{\rm B}} \cdot q(\rmB\hspace{0.05cm}\|\hspace{0.05cm} G )\hspace{0.05cm}.$$
	:$$q(\rm B\hspace{0.05cm}\|\hspace{0.05cm} G ) =\frac{\alpha_{\rm B} \cdot[{\rm Pr}(\rm
	B\hspace{0.05cm}\|\hspace{0.05cm} G ) + {\rm Pr}(\rm
	G\hspace{0.05cm}\|\hspace{0.05cm} B )]}{\alpha_{\rm G} \cdot q_{\rm
	B} + \alpha_{\rm B} \cdot q_{\rm G}} \hspace{0.05cm},$$
	:$$q(\rm G\hspace{0.05cm}\|\hspace{0.05cm} B ) =
	\frac{\alpha_{\rm G}}{\alpha_{\rm B}} \cdot q(\rm
	B\hspace{0.05cm}\|\hspace{0.05cm} G )\hspace{0.05cm}.$$

	*Here again the following auxiliary quantities are used:		*Here again the following auxiliary quantities are used:

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	{{Display}}		{{Display}}
			[[de:Digitalsignalübertragung/Bündelfehlerkanäle]]

Guenter at 16:10, 24 October 2022

2022-10-24T16:10:08Z

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	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{Example 7:}$  We now consider a channel with burst error characteristics.		$\text{Example 7:}$  We now consider a channel with burst error characteristics.
	[[File:EN_Dig_T_5_3_S5h_ganz_neu.png\|right\|frame\|Results of the Wilhelm-L model with  $\alpha = 0.7$ and  $p_{\rm S} = 0.2$<br><br> ~~'''Korrektur''' distanz~~\|class=fit]]		[[File:EN_Dig_T_5_3_S5h_ganz_neu.png\|right\|frame\|Results of the Wilhelm-L model with  $\alpha = 0.7$ and  $p_{\rm S} = 0.2$<br><br> \|class=fit]]

	#The graph shows as green circles the results for the Wilhelm–L model with  $\alpha = 0.7$.		#The graph shows as green circles the results for the Wilhelm–L model with  $\alpha = 0.7$.
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	$\text{Example 9:}$		$\text{Example 9:}$
	As in  [[Digital_Signal_Transmission/Burst_Error_Channels#Error_distance_consideration_according_to_the_Wilhelm_A_model\| $\text{Example 8}$]]:  $p_{\rm S} = 0.01$.  The error correlation functions shown here again represent		As in  [[Digital_Signal_Transmission/Burst_Error_Channels#Error_distance_consideration_according_to_the_Wilhelm_A_model\| $\text{Example 8}$]]:  $p_{\rm S} = 0.01$.  The error correlation functions shown here again represent
	[[File:~~EN_Dig_T_5_3_S5korr_version2~~.png\|right\|frame\|\|ECF results of the Wilhelm model ~~'''Korrektur''' Strich fehlt~~]]		[[File:EN_Dig_T_5_3_S5korr_version3.png\|right\|frame\|\|ECF results of the Wilhelm model]]

	*Red:        $\alpha = 1.0$   ⇒   BSC model,<br>		*Red:        $\alpha = 1.0$   ⇒   BSC model,<br>

Guenter at 13:40, 24 October 2022

2022-10-24T13:40:57Z

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	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{Example 7:}$  We now consider a channel with burst error characteristics.		$\text{Example 7:}$  We now consider a channel with burst error characteristics.
	[[File:~~EN_Dig_T_5_3_S5h~~.png\|right\|frame\|Results of the Wilhelm-L model with  $\alpha = 0.7$ and  $p_{\rm S} = 0.2$<br><br> '''Korrektur''' distanz\|class=fit]]		[[File:EN_Dig_T_5_3_S5h_ganz_neu.png\|right\|frame\|Results of the Wilhelm-L model with  $\alpha = 0.7$ and  $p_{\rm S} = 0.2$<br><br> '''Korrektur''' distanz\|class=fit]]

	#The graph shows as green circles the results for the Wilhelm–L model with  $\alpha = 0.7$.		#The graph shows as green circles the results for the Wilhelm–L model with  $\alpha = 0.7$.

Guenter at 12:21, 19 October 2022

2022-10-19T12:21:05Z

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	G) = 0.01\hspace{0.05cm}.$$		G) = 0.01\hspace{0.05cm}.$$

	~~Below is drawn error sequence of the equivalent McCullough model. '''KORREKTUR:~~ An error sequence of the equivalent McCullough model is drawn below.~~'''~~  The relations between the two models can be summarized as follows:		An error sequence of the equivalent McCullough model is drawn below.  The relations between the two models can be summarized as follows:
	#In the GE error sequence,  a change from state  "GOOD"  (white background)  to state  "BAD"  (gray background)  and vice versa is possible at any time  $\nu$,  even when  $e_\nu = 0$.		#In the GE error sequence,  a change from state  "GOOD"  (white background)  to state  "BAD"  (gray background)  and vice versa is possible at any time  $\nu$,  even when  $e_\nu = 0$.
	#In contrast,  in the ML error sequence,  a change of state at time  $\nu$  is only possible at  $e_\nu = 1$.  The last error value before a gray background is always  "$1$".<br>		#In contrast,  in the ML error sequence,  a change of state at time  $\nu$  is only possible at  $e_\nu = 1$.  The last error value before a gray background is always  "$1$".<br>
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	The conversion of the GE parameters to the MC parameters is illustrated in  [[Aufgaben:Aufgabe_5.7:_McCullough-Parameter_aus_Gilbert-Elliott-Parameter\|"Exercise 5.7"]]  using a simple example.   [[Aufgaben:Aufgabe_5.7Z:_Nochmals_McCullough-Modell\|"Exercise 5.7Z"]]  further shows how ~~'''KORREKTUR:~~ they~~'''~~ can be determined directly from the  $q$ parameters:		The conversion of the GE parameters to the MC parameters is illustrated in  [[Aufgaben:Aufgabe_5.7:_McCullough-Parameter_aus_Gilbert-Elliott-Parameter\|"Exercise 5.7"]]  using a simple example.   [[Aufgaben:Aufgabe_5.7Z:_Nochmals_McCullough-Modell\|"Exercise 5.7Z"]]  further shows how they can be determined directly from the  $q$ parameters:
	#the mean error probability,		#the mean error probability,
	#the error distance distribution,		#the error distance distribution,
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	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{Example 6:}$  We start with the  [[Digital_Signal_Transmission/Binary_Symmetric_Channel_(BSC)#Binary_Symmetric_Channel_.E2.80.93_Model_and_Error_Correlation_Function\|"BSC model"]].		$\text{Example 6:}$  We start with the  [[Digital_Signal_Transmission/Binary_Symmetric_Channel_(BSC)#Binary_Symmetric_Channel_.E2.80.93_Model_and_Error_Correlation_Function\|"BSC model"]].
	[[File:EN_Dig_T_5_3_S5h_ganz_neu.png\|right\|frame\|~~'''KORREKTUR:~~ BSC model, and~~''' Parameters of the BSC model~~ for  $p_{\rm S} = 0.2$\|class=fit]]		[[File:EN_Dig_T_5_3_S5h_ganz_neu.png\|right\|frame\|BSC model and parameters for  $p_{\rm S} = 0.2$\|class=fit]]

	#For presentation reasons,  we set the falsification probability very high to  $p_{\rm S} = 0.2$.		#For presentation reasons,  we set the falsification probability very high to  $p_{\rm S} = 0.2$.
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	With  $\alpha = 1$  one obtains from the equation in the last section:		With  $\alpha = 1$  one obtains from the equation in the last section:

	::<math>V_a\hspace{0.05cm}(k) = \big [k^{\alpha}-(k-1)^{\alpha}\big ] \cdot		::<math>V_a\hspace{0.05cm}(k) \hspace{-0.05cm}=\hspace{-0.05cm} \big [k^{\alpha}-(k-1)^{\alpha}\big ] \hspace{-0.05cm} \cdot \hspace{-0.05cm}
	\big [ 1 - {p_{\rm S}^{1/\alpha} }\big ]^{k-1} = (1 - p_{\rm S} )^{k-1}		\big [ 1 - {p_{\rm S}^{1/\alpha} }\big ]^{k-1} \hspace{-0.1cm} = \hspace{-0.1cm} (1 - p_{\rm S} )^{k-1}
	\hspace{0.05cm}.</math>		\hspace{0.05cm}.</math>

Guenter at 16:08, 17 October 2022

2022-10-17T16:08:55Z

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	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{Example 6:}$  We start with the  [[Digital_Signal_Transmission/Binary_Symmetric_Channel_(BSC)#Binary_Symmetric_Channel_.E2.80.93_Model_and_Error_Correlation_Function\|"BSC model"]].		$\text{Example 6:}$  We start with the  [[Digital_Signal_Transmission/Binary_Symmetric_Channel_(BSC)#Binary_Symmetric_Channel_.E2.80.93_Model_and_Error_Correlation_Function\|"BSC model"]].
	[[File:~~P ID2827 Dig T 5 3 S5d ganz neu~~.png\|right\|frame\|'''KORREKTUR: BSC model, and''' Parameters of the BSC model for  $p_{\rm S} = 0.2$\|class=fit]]		[[File:EN_Dig_T_5_3_S5h_ganz_neu.png\|right\|frame\|'''KORREKTUR: BSC model, and''' Parameters of the BSC model for  $p_{\rm S} = 0.2$\|class=fit]]

	#For presentation reasons,  we set the falsification probability very high to  $p_{\rm S} = 0.2$.		#For presentation reasons,  we set the falsification probability very high to  $p_{\rm S} = 0.2$.

Hwang at 11:28, 15 October 2022

2022-10-15T11:28:53Z

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	== Analysis of error structures with the Wilhelm A model==		== Analysis of error structures with the Wilhelm A model==
	<br>		<br>
	Wilhelm developed his channel model mainly in order to be able to draw conclusions about the errors occurring from measured error sequences.  From the multitude of analyses in  [Wil11]<ref name='Wil11'>Wilhelm, C.:  A-Model and L-Model, New Channel Models with Formulas for Probabilities of Error Structures.   [http://www.channels-networks.net/ Internet Publication to Channels-Networks,] 2011ff.</ref>.  only a few are to be quoted here,  whereby always the symbol error probability  $p_{\rm S} = 10^{-3}$  is the basis.		Wilhelm developed his channel model mainly in order to be able to draw conclusions about the errors occurring from measured error sequences.  From the multitude of analyses in  [Wil11]<ref name='Wil11'>Wilhelm, C.:  A-Model and L-Model, New Channel Models with Formulas for Probabilities of Error Structures.   [http://www.channels-networks.net/ Internet Publication to Channels-Networks,] 2011ff.</ref>  only a few are to be quoted here,  whereby always the symbol error probability  $p_{\rm S} = 10^{-3}$  is the basis.
	*In the diagrams,  the red curve applies in each case to statistically independent errors  $($BSC or  $\alpha = 1)$,		*In the diagrams,  the red curve applies in each case to statistically independent errors  $($BSC or  $\alpha = 1)$,

Hwang at 11:21, 15 October 2022

2022-10-15T11:21:40Z

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	The following statements can be generalized to a large extent,  see also  [[Digital_Signal_Transmission/Binary_Symmetric_Channel_(BSC)#Binary_Symmetric_Channel_.E2.80.93_Model_and_Error_Correlation_Function\|"Gilbert-Elliott model"]]:		The following statements can be generalized to a large extent,  see also  [[Digital_Signal_Transmission/Binary_Symmetric_Channel_(BSC)#Binary_Symmetric_Channel_.E2.80.93_Model_and_Error_Correlation_Function\|"Gilbert-Elliott model"]]:
	*The ECF value ~~att~~  $k = 0$  is equal to  $p_{\rm S} = 10^{-2}$  for all channels  $($marked by the circle with gray filling$)$ and the limit value for  $k \to \infty$  is always  $p_{\rm S}^2 = 10^{-4}$.<br>		*The ECF value at  $k = 0$  is equal to  $p_{\rm S} = 10^{-2}$  for all channels  $($marked by the circle with gray filling$)$ and the limit value for  $k \to \infty$  is always  $p_{\rm S}^2 = 10^{-4}$.<br>

	*In the BSC model,  this final value is already reached at  $k = 1$  $($marked by a red filled circle$)$.  Therefore, the ECF can only assume here the two values  $p_{\rm S}$  and  $p_{\rm S}^2$.  <br>		*In the BSC model,  this final value is already reached at  $k = 1$  $($marked by a red filled circle$)$.  Therefore, the ECF can only assume here the two values  $p_{\rm S}$  and  $p_{\rm S}^2$.  <br>

Hwang at 10:54, 15 October 2022

2022-10-15T10:54:20Z

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	The Wilhelm model with  $p_{\rm S} = 0.2$  and  $\alpha = 1$  has exactly the same error distance ~~distributionas~~ the corresponding  [[Digital_Signal_Transmission/Binary_Symmetric_Channel_(BSC)#Binary_Symmetric_Channel_.E2.80.93_Error_Distance_Distribution\| "BSC model"]].  This is also shown by the calculation.		The Wilhelm model with  $p_{\rm S} = 0.2$  and  $\alpha = 1$  has exactly the same error distance distribution as the corresponding  [[Digital_Signal_Transmission/Binary_Symmetric_Channel_(BSC)#Binary_Symmetric_Channel_.E2.80.93_Error_Distance_Distribution\| "BSC model"]].  This is also shown by the calculation.

	With  $\alpha = 1$  one obtains from the equation in the last section:		With  $\alpha = 1$  one obtains from the equation in the last section:

← Older revision		Revision as of 17:44, 16 March 2026
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	The upper graph shows the error distance distribution  $\rm (EDD)$  of the Gilbert-Elliott model  (red curve)  in linear and logarithmic representation for the parameters		The upper graph shows the error distance distribution  $\rm (EDD)$  of the Gilbert-Elliott model  (red curve)  in linear and logarithmic representation for the parameters
	:$${\rm Pr}(\~~rmG~~\hspace{0.05cm}\|\hspace{0.05cm} B ) = 0.1 \hspace{0.05cm},\hspace{0.5cm}{\rm Pr}(\~~rmB~~\hspace{0.05cm}\|\hspace{0.05cm} G ) = 0.01 \hspace{0.05cm},\hspace{0.5cm}p_{\rm G} = 0.001, \hspace{0.5cm}p_{\rm B} = 0.4.$$		:$${\rm Pr}(\rm G\hspace{0.05cm}\|\hspace{0.05cm} B ) = 0.1 \hspace{0.05cm},\hspace{0.5cm}{\rm Pr}(\rm B\hspace{0.05cm}\|\hspace{0.05cm} G ) = 0.01 \hspace{0.05cm},\hspace{0.5cm}p_{\rm G} = 0.001, \hspace{0.5cm}p_{\rm B} = 0.4.$$

	For comparison,  the corresponding  $V_a(k)$  curve for the BSC model with the same mean error probability  $p_{\rm M} = 4.5\%$  is also plotted as blue curve.<br>		For comparison,  the corresponding  $V_a(k)$  curve for the BSC model with the same mean error probability  $p_{\rm M} = 4.5\%$  is also plotted as blue curve.<br>
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	The parameters of the equivalent MC model can be calculated from the GE parameters as follows:		The parameters of the equivalent MC model can be calculated from the GE parameters as follows:

	:$$q_{\rm G} =1-\beta_{\~~rmG~~}\hspace{0.05cm},$$		:$$q_{\rm G} =1-\beta_{\rm G}\hspace{0.05cm},$$
	:$$ q_{\~~rmB~~} = 1-\beta_{\rm B}\hspace{0.05cm}, $$		:$$ q_{\rm B} = 1-\beta_{\rm B}\hspace{0.05cm}, $$
	:$$q(\rm B\hspace{0.05cm}\|\hspace{0.05cm} G ) =\frac{\alpha_{\rm B} \cdot[{\rm Pr}(\~~rmB~~\hspace{0.05cm}\|\hspace{0.05cm} G ) + {\rm Pr}(\~~rmG~~\hspace{0.05cm}\|\hspace{0.05cm} B )]}{\alpha_{\rm G} \cdot q_{\~~rmB~~} + \alpha_{\rm B} \cdot q_{\rm G}} \hspace{0.05cm},$$		:$$q(\rm B\hspace{0.05cm}\|\hspace{0.05cm} G ) =\frac{\alpha_{\rm B} \cdot[{\rm Pr}(\rm B\hspace{0.05cm}\|\hspace{0.05cm} G ) + {\rm Pr}(\rm G\hspace{0.05cm}\|\hspace{0.05cm} B )]}{\alpha_{\rm G} \cdot q_{\rm B} + \alpha_{\rm B} \cdot q_{\rm G}} \hspace{0.05cm},$$
	:$$q(\rm G\hspace{0.05cm}\|\hspace{0.05cm} B ) =\frac{\alpha_{\rm G}}{\alpha_{\rm B}} \cdot q(\~~rmB~~\hspace{0.05cm}\|\hspace{0.05cm} G )\hspace{0.05cm}.$$		:$$q(\rm G\hspace{0.05cm}\|\hspace{0.05cm} B ) =\frac{\alpha_{\rm G}}{\alpha_{\rm B}} \cdot q(\rm B\hspace{0.05cm}\|\hspace{0.05cm} G )\hspace{0.05cm}.$$

	*Here again the following auxiliary quantities are used:		*Here again the following auxiliary quantities are used:
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	:$$q_{\rm G} = 0.0186, \hspace{0.5cm} q_{\rm B} = 0.4613, \hspace{0.5cm} q(\rm B\hspace{0.05cm}\vert\hspace{0.05cm}G ) = 0.3602, \hspace{0.5cm} {\it q}(\rm G\hspace{0.05cm}\vert\hspace{0.05cm}B ) = 0.2240.$$		:$$q_{\rm G} = 0.0186, \hspace{0.5cm} q_{\rm B} = 0.4613, \hspace{0.5cm} q(\rm B\hspace{0.05cm}\vert\hspace{0.05cm}G ) = 0.3602, \hspace{0.5cm} {\it q}(\rm G\hspace{0.05cm}\vert\hspace{0.05cm}B ) = 0.2240.$$

	*If we compare in  $\text{Example 2}$  the red error sequence  $($GE,  change of state is always possible$)$  with the blue sequence  $($equivalent MC,  change of state only at  $e_\nu = 1$$)$,  we can see quite serious differences.		*If we compare in  $\text{Example 2}$  the red error sequence  $($GE,  change of state is always possible$)$  with the blue sequence  $($equivalent MC,  change of state only at  $e_\nu = 1$),  we can see quite serious differences.

	* But the blue error sequence of the equivalent McCullough model has exactly the same statistical properties as the red error sequence of the Gilbert-Elliott model.}}		* But the blue error sequence of the equivalent McCullough model has exactly the same statistical properties as the red error sequence of the Gilbert-Elliott model.}}
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	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{Example 6:}$  We start with the  [[Digital_Signal_Transmission/~~Binary_Symmetric_Channel_(BSC)~~#~~Binary_Symmetric_Channel_.E2.80.93_Model_and_Error_Correlation_Function~~\|"BSC model"]].		$\text{Example 6:}$  We start with the  [[Digital_Signal_Transmission/Binary_Symmetric_Channel#Binary_Symmetric_Channel_–_Model_and_Error_Correlation_Function\|"BSC model"]].
	[[File:EN_Dig_T_5_3_S5h_ganz_neu.png\|right\|frame\|BSC model and parameters for  $p_{\rm S} = 0.2$\|class=fit]]		[[File:EN_Dig_T_5_3_S5h_ganz_neu.png\|right\|frame\|BSC model and parameters for  $p_{\rm S} = 0.2$\|class=fit]]

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	The Wilhelm model with  $p_{\rm S} = 0.2$  and  $\alpha = 1$  has exactly the same error distance distribution as the corresponding  [[Digital_Signal_Transmission/~~Binary_Symmetric_Channel_(BSC)~~#~~Binary_Symmetric_Channel_.E2.80.93_Error_Distance_Distribution~~\| "BSC model"]].  This is also shown by the calculation.		The Wilhelm model with  $p_{\rm S} = 0.2$  and  $\alpha = 1$  has exactly the same error distance distribution as the corresponding  [[Digital_Signal_Transmission/Binary_Symmetric_Channel#Binary_Symmetric_Channel_–_Error_Distance_Distribution\| "BSC model"]].  This is also shown by the calculation.

	With  $\alpha = 1$  one obtains from the equation in the last section:		With  $\alpha = 1$  one obtains from the equation in the last section:
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	*Green:   $\alpha = 0.70$   ⇒   strong bursting.<br><br>		*Green:   $\alpha = 0.70$   ⇒   strong bursting.<br><br>

	The following statements can be generalized to a large extent,  see also  [[Digital_Signal_Transmission/~~Binary_Symmetric_Channel_(BSC)~~#~~Binary_Symmetric_Channel_.E2.80.93_Model_and_Error_Correlation_Function~~\|"Gilbert-Elliott model"]]:		The following statements can be generalized to a large extent,  see also  [[Digital_Signal_Transmission/Binary_Symmetric_Channel#Binary_Symmetric_Channel_–_Model_and_Error_Correlation_Function\|"Gilbert-Elliott model"]]:
	*The ECF value at  $k = 0$  is equal to  $p_{\rm S} = 10^{-2}$  for all channels  $($marked by the circle with gray filling$)$ and the limit value for  $k \to \infty$  is always  $p_{\rm S}^2 = 10^{-4}$.<br>		*The ECF value at  $k = 0$  is equal to  $p_{\rm S} = 10^{-2}$  for all channels  $($marked by the circle with gray filling$)$ and the limit value for  $k \to \infty$  is always  $p_{\rm S}^2 = 10^{-4}$.<br>