Digital Signal Transmission/Burst Error Channels - Revision history

Guenter at 16:10, 24 October 2022

2022-10-24T16:10:08Z

Guenter at 13:40, 24 October 2022

2022-10-24T13:40:57Z

Guenter at 12:21, 19 October 2022

2022-10-19T12:21:05Z

Guenter at 16:08, 17 October 2022

2022-10-17T16:08:55Z

Hwang at 11:28, 15 October 2022

2022-10-15T11:28:53Z

Hwang at 11:21, 15 October 2022

2022-10-15T11:21:40Z

Hwang at 10:54, 15 October 2022

2022-10-15T10:54:20Z

Hwang at 10:52, 15 October 2022

2022-10-15T10:52:52Z

Hwang at 10:30, 15 October 2022

2022-10-15T10:30:48Z

Hwang at 10:25, 15 October 2022

2022-10-15T10:25:40Z

@@ Line 482: / Line 482: @@
 {{GraueBox|TEXT=
 $\text{Example 7:}$&nbsp;  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&nbsp; $\alpha = 0.7$ and&nbsp;  $p_{\rm S} = 0.2$<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&nbsp; $\alpha = 0.7$ and&nbsp;  $p_{\rm S} = 0.2$<br><br> |class=fit]]
 #The graph shows as green circles the results for the Wilhelm&ndash;L model with&nbsp; $\alpha = 0.7$.
@@ Line 568: / Line 568: @@
 $\text{Example 9:}$&nbsp;
 As in&nbsp; [[Digital_Signal_Transmission/Burst_Error_Channels#Error_distance_consideration_according_to_the_Wilhelm_A_model| $\text{Example 8}$]]:&nbsp;  $p_{\rm S} = 0.01$.&nbsp; 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:&nbsp; &nbsp;&nbsp; &nbsp; &nbsp;$\alpha = 1.0$ &nbsp; &#8658; &nbsp; BSC model,<br>

@@ Line 482: / Line 482: @@
 {{GraueBox|TEXT=
 $\text{Example 7:}$&nbsp;  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&nbsp; $\alpha = 0.7$ and&nbsp;  $p_{\rm S} = 0.2$<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&nbsp; $\alpha = 0.7$ and&nbsp;  $p_{\rm S} = 0.2$<br><br> '''Korrektur''' distanz|class=fit]]
 #The graph shows as green circles the results for the Wilhelm&ndash;L model with&nbsp; $\alpha = 0.7$.

@@ Line 227: / Line 227: @@
 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.'''&nbsp;  The relations between the two models can be summarized as follows:
+An error sequence of the equivalent McCullough model is drawn below.&nbsp;  The relations between the two models can be summarized as follows:
 #In the GE error sequence,&nbsp; a change from state&nbsp;  "GOOD"&nbsp; (white background)&nbsp; to state&nbsp; "BAD"&nbsp; (gray background)&nbsp; and vice versa is possible at any time&nbsp; $\nu$,&nbsp; even when&nbsp; $e_\nu = 0$.
 #In contrast,&nbsp; in the ML error sequence,&nbsp; a change of state at time&nbsp; $\nu$&nbsp; is only possible at&nbsp; $e_\nu = 1$.&nbsp; The last error value before a gray background is always&nbsp; "$1$".<br>
@@ Line 290: / Line 290: @@
-The conversion of the GE parameters to the MC parameters is illustrated in&nbsp; [[Aufgaben:Aufgabe_5.7:_McCullough-Parameter_aus_Gilbert-Elliott-Parameter|"Exercise 5.7"]]&nbsp; using a simple example. &nbsp; [[Aufgaben:Aufgabe_5.7Z:_Nochmals_McCullough-Modell|"Exercise 5.7Z"]]&nbsp; further shows how '''KORREKTUR: they''' can be determined directly from the &nbsp;$q$ parameters:
+The conversion of the GE parameters to the MC parameters is illustrated in&nbsp; [[Aufgaben:Aufgabe_5.7:_McCullough-Parameter_aus_Gilbert-Elliott-Parameter|"Exercise 5.7"]]&nbsp; using a simple example. &nbsp; [[Aufgaben:Aufgabe_5.7Z:_Nochmals_McCullough-Modell|"Exercise 5.7Z"]]&nbsp; further shows how they can be determined directly from the &nbsp;$q$ parameters:
 #the mean error probability,
 #the error distance distribution,
@@ Line 446: / Line 446: @@
 {{GraueBox|TEXT=
 $\text{Example 6:}$&nbsp; We start with the&nbsp; [[Digital_Signal_Transmission/Binary_Symmetric_Channel_(BSC)#Binary_Symmetric_Channel_.E2.80.93_Model_and_Error_Correlation_Function|"BSC model"]].&nbsp;
-[[File:EN_Dig_T_5_3_S5h_ganz_neu.png|right|frame|'''KORREKTUR: BSC model, and''' Parameters of the BSC model for&nbsp; $p_{\rm S} = 0.2$|class=fit]]
+[[File:EN_Dig_T_5_3_S5h_ganz_neu.png|right|frame|BSC model and parameters for&nbsp; $p_{\rm S} = 0.2$|class=fit]]
 #For presentation reasons,&nbsp; we set the falsification probability very high to&nbsp; $p_{\rm S} = 0.2$.
@@ Line 456: / Line 456: @@
 With&nbsp; $\alpha = 1$&nbsp; 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>

@@ Line 446: / Line 446: @@
 {{GraueBox|TEXT=
 $\text{Example 6:}$&nbsp; We start with the&nbsp; [[Digital_Signal_Transmission/Binary_Symmetric_Channel_(BSC)#Binary_Symmetric_Channel_.E2.80.93_Model_and_Error_Correlation_Function|"BSC model"]].&nbsp;
-[[File:P ID2827 Dig T 5 3 S5d ganz neu.png|right|frame|'''KORREKTUR: BSC model, and''' Parameters of the BSC model for&nbsp; $p_{\rm S} = 0.2$|class=fit]]
+[[File:EN_Dig_T_5_3_S5h_ganz_neu.png|right|frame|'''KORREKTUR: BSC model, and''' Parameters of the BSC model for&nbsp; $p_{\rm S} = 0.2$|class=fit]]
 #For presentation reasons,&nbsp; we set the falsification probability very high to&nbsp; $p_{\rm S} = 0.2$.

@@ Line 583: / Line 583: @@
 == Analysis of error structures with the Wilhelm A model==
 <br>
-Wilhelm developed his channel model mainly in order to be able to draw conclusions about the errors occurring from measured error sequences.&nbsp; From the multitude of analyses in&nbsp; [Wil11]<ref name='Wil11'>Wilhelm, C.:&nbsp; A-Model and L-Model, New Channel Models with Formulas for Probabilities of Error Structures. &nbsp; [http://www.channels-networks.net/ Internet Publication to Channels-Networks,] 2011ff.</ref>.&nbsp; only a few are to be quoted here,&nbsp; whereby always the symbol error probability&nbsp; $p_{\rm S} = 10^{-3}$&nbsp; 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.&nbsp; From the multitude of analyses in&nbsp; [Wil11]<ref name='Wil11'>Wilhelm, C.:&nbsp; A-Model and L-Model, New Channel Models with Formulas for Probabilities of Error Structures. &nbsp; [http://www.channels-networks.net/ Internet Publication to Channels-Networks,] 2011ff.</ref>&nbsp; only a few are to be quoted here,&nbsp; whereby always the symbol error probability&nbsp; $p_{\rm S} = 10^{-3}$&nbsp; is the basis.
 *In the diagrams,&nbsp; the red curve applies in each case to statistically independent errors&nbsp; $($BSC or&nbsp; $\alpha = 1)$,

@@ Line 575: / Line 575: @@
 The following statements can be generalized to a large extent,&nbsp;   see also&nbsp; [[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&nbsp; $k = 0$&nbsp; is equal to&nbsp; $p_{\rm S} = 10^{-2}$&nbsp; for all channels&nbsp; $($marked by the circle with gray filling$)$ and the limit value for&nbsp; $k \to \infty$&nbsp; is always&nbsp; $p_{\rm S}^2 = 10^{-4}$.<br>
+*The ECF value at&nbsp; $k = 0$&nbsp; is equal to&nbsp; $p_{\rm S} = 10^{-2}$&nbsp; for all channels&nbsp; $($marked by the circle with gray filling$)$ and the limit value for&nbsp; $k \to \infty$&nbsp; is always&nbsp; $p_{\rm S}^2 = 10^{-4}$.<br>
 *In the BSC model,&nbsp; this final value is already reached at&nbsp; $k = 1$&nbsp; $($marked by a red filled circle$)$.&nbsp; Therefore, the ECF can only assume here the two values&nbsp; $p_{\rm S}$&nbsp;  and&nbsp; $p_{\rm S}^2$.&nbsp; <br>

← Older revision		Revision as of 10:54, 15 October 2022
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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:

@@ Line 312: / Line 312: @@
 In a large number of measurements, the fact that
-&rArr; &nbsp; the course&nbsp; $p_{\rm B}(n)$&nbsp; in double-logarithmic representation show linear increases in the lower range
+&rArr; &nbsp; the course&nbsp; $p_{\rm B}(n)$&nbsp; in double-logarithmic representation shows linear increases in the lower range
 has been confirmed again and again&nbsp; $($see graph$)$.&nbsp; Thus,&nbsp; it holds for&nbsp; $n \le n^\star$: