Digital Signal Transmission/Binary Symmetric Channel - Revision history

Guenter at 15:23, 6 September 2022

2022-09-06T15:23:58Z

Guenter at 14:56, 5 September 2022

2022-09-05T14:56:05Z

Guenter at 12:25, 5 September 2022

2022-09-05T12:25:46Z

Guenter at 12:23, 5 September 2022

2022-09-05T12:23:41Z

Guenter at 11:20, 5 September 2022

2022-09-05T11:20:28Z

Guenter at 11:07, 5 September 2022

2022-09-05T11:07:46Z

Guenter at 14:31, 4 September 2022

2022-09-04T14:31:18Z

Guenter: Guenter moved page Digital Signal Transmission/Binary Symmetric Channel (BSC) to Digital Signal Transmission/Binary Symmetric Channel

2022-09-02T13:00:04Z

Guenter moved page Digital Signal Transmission/Binary Symmetric Channel (BSC) to Digital Signal Transmission/Binary Symmetric Channel

Guenter at 15:28, 1 September 2022

2022-09-01T15:28:06Z

Hwang at 11:52, 26 August 2022

2022-08-26T11:52:23Z

Show changes

@@ Line 71: / Line 71: @@
 <br>
 The BSC model is the&nbsp; "digital equivalent"&nbsp; of the simplest analog model &nbsp; &#8658; &nbsp; [[Digital_Signal_Transmission/Structure_of_the_Optimal_Receiver#Some_properties_of_the_AWGN_channel|"AWGN"]]&nbsp; for a time-invariant digital system corresponding to the following graph.&nbsp; It holds
-[[File:P ID1830 Dig T 5 2 S3 version1.png|right|frame|On the use of the BSC model|class=fit]]
+[[File:EN_Dig_T_5_2_S2b.png|right|frame|On the use of the BSC model|class=fit]]
 #The only degradation is noise.
 #There is no distortion.<br>

@@ Line 48: / Line 48: @@
 *This fact is somewhat surprising at first glance:&nbsp;
-:With&nbsp; $p = 0.01$,&nbsp; for example,&nbsp; the mean error distance is&nbsp; ${\rm E}\big[a\big] = 100$.&nbsp; Nevertheless,&nbsp; two consecutive errors&nbsp; $(a = 1)$&nbsp; are more probable by a factor of&nbsp; $0.99^{99} \approx 2.7$&nbsp; than the error distance&nbsp; $a = 100$.<br><br>
+:With&nbsp; $p = 0.01$,&nbsp; for example,&nbsp; the mean error distance is&nbsp; ${\rm E}\big[a\big] = 100$.&nbsp; Nevertheless,&nbsp; two consecutive errors&nbsp; $(a = 1)$&nbsp; are more probable by a factor of&nbsp; $0.99^{99} \approx 2.7$&nbsp; than the error distance&nbsp; $a = 100$.<br>
 *The error distance distribution is obtained by summation according to the&nbsp; [[Digital_Signal_Transmission/Parameters_of_Digital_Channel_Models#Error_distance_distribution|"general definition"]]:&nbsp;
@@ Line 85: / Line 85: @@
 *threshold decision with symmetric decision threshold,<br>
-*no extraneous influences such as crosstalk, dial pulses, electromagnetic fields, ...<br>
+*no extraneous interference influences such as: <br>crosstalk, dial pulses, electromagnetic fields, ...<br>

← Older revision		Revision as of 12:25, 5 September 2022
Line 91:		Line 91:
	*the application of the BSC model is often justified,		*the application of the BSC model is often justified,

−	*but not if fading influences   $($ ~~ ~~[[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Non-Coherent_Demodulation#Rayleigh_and_Rice_Distribution\|"Rayleigh or Rice"]]$)$  play a role or if echoes may occur   <br>⇒  [[Mobile_Communications/Multi-Path_Reception_in_Mobile_Communications\|"multi-path reception"]].<br>	+	*but not if fading influences   $($[[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Non-Coherent_Demodulation#Rayleigh_and_Rice_Distribution\|"Rayleigh or Rice"]]$)$  play a role or if echoes may occur   <br>⇒  [[Mobile_Communications/Multi-Path_Reception_in_Mobile_Communications\|"multi-path reception"]].<br>

@@ Line 20: / Line 20: @@
 *The two symbols&nbsp; $($for example&nbsp; $\rm L$&nbsp; and &nbsp;$\rm H)$&nbsp; are each falsified with the same probability&nbsp; $p$,&nbsp; so that the mean error probability&nbsp; $p_{\rm M} = p$&nbsp; is also independent of the symbol probabilities&nbsp; $p_{\rm L}$&nbsp; and&nbsp;  $p_{\rm H}$.<br><br>
-The right graph shows the&nbsp; "error correlation function"&nbsp; $\rm (ECF)$&nbsp; of the BSC model:
+The right graph shows the&nbsp; [[Digital_Signal_Transmission/Parameters_of_Digital_Channel_Models#Error_correlation_function|"error correlation function"]]&nbsp; $\rm (ECF)$&nbsp; of the BSC model:
 ::<math>\varphi_{e}(k) =  {\rm E}\big[e_{\nu} \cdot e_{\nu + k}\big] =

@@ Line 81: / Line 81: @@
 *noise according to the AWGN model &nbsp; &#8658; &nbsp; additive,&nbsp; white,&nbsp; Gaussian distributed,<br>
-*no distortions due to the components of the transmitter and receiver,<br>
+*no&nbsp; $($linear & non-linear$)$&nbsp; distortions due to transmitter / receiver components,<br>
 *threshold decision with symmetric decision threshold,<br>
-*no extraneous influences such as crosstalk, dial pulses or electromagnetic interference fields.<br>
+*no extraneous influences such as crosstalk, dial pulses, electromagnetic fields, ...<br>
-For a&nbsp; ''radio system''&nbsp; with a direct line-of-sight between transmitter and receiver, the application of the BSC model is often justified, but not if fading influences &nbsp; &#8658; &nbsp;[[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Non-Coherent_Demodulation#Rayleigh_and_Rice_Distribution|"Rayleigh or Rice"]]&nbsp; play a role or if echoes may occur &nbsp; &#8658; &nbsp;[[Mobile_Communications/Multi-Path_Reception_in_Mobile_Communications|"multi-path reception"]].<br>
+For a&nbsp; "radio system"&nbsp; with a direct line-of-sight between transmitter and receiver,
-In contrast, according to network operators, statistically independent errors tend to be the exception in the case of&nbsp; ''wireline transmission''&nbsp; (for example,&nbsp; [[Examples_of_Communication_Systems/General_Description_of_DSL|"DSL"]], but also optical transmission). If errors occur during data transmission via the telephone network, they are usually clustered.
+*but not if fading influences &nbsp; $($ &nbsp;[[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Non-Coherent_Demodulation#Rayleigh_and_Rice_Distribution|"Rayleigh or Rice"]]$)$&nbsp; play a role or if echoes may occur &nbsp; <br>&#8658; &nbsp;[[Mobile_Communications/Multi-Path_Reception_in_Mobile_Communications|"multi-path reception"]].<br>
-In this case, we speak of so-called&nbsp; <i>burst errors</i>, which will be discussed in the next chapter.<br>
 ==Exercises for the chapter==

@@ Line 8: / Line 8: @@
 == Binary Symmetric Channel &ndash; Model and Error Correlation Function==
 <br>
-The left graph shows the BSC model, the simplest model of a digital transmission system.
+The left graph shows the BSC model,&nbsp; the simplest model of a digital transmission system.
-The name stands for&nbsp; '''Binary Symmetric Channel'''&nbsp; and states that this model can only be used for binary systems with symmetrical falsification properties. Further applies:
+The name stands for&nbsp; "'''Binary Symmetric Channel'''"&nbsp; and states that this model can only be used for binary systems with symmetrical falsification properties.
 [[File:EN_Dig_T_5_2_S1.png|right|frame|BSC model and associated error correlation function|class=fit]]
-*The BSC model is suitable for the study and generation of a sequence of errors with statistically independent errors. Such a channel is also called <i>memory-free</i> and, unlike the&nbsp; [[Digital_Signal_Transmission/Burst_Error_Channels|"burst error channel models"]],&nbsp; only a single channel state exists.<br>
+Further applies:
-*The two symbols $($for example&nbsp; $\rm L$&nbsp; and &nbsp;$\rm H)$&nbsp; are each falsified with the same probability&nbsp; $p$,&nbsp; so that the mean error probability&nbsp; $p_{\rm M} = p$&nbsp; is also independent of the symbol probabilities&nbsp; $p_{\rm L}$&nbsp; and&nbsp;  $p_{\rm H}$.<br><br>
+*The BSC model is suitable for the study and generation of an error sequence with statistically independent errors.&nbsp;
-The right graph shows the&nbsp; ''error correlation function''&nbsp; (ECF) of the BSC model:
+*Such a channel is also called&nbsp; "memory-free"&nbsp; and unlike the&nbsp; [[Digital_Signal_Transmission/Burst_Error_Channels|"burst error channel models"]]&nbsp; only a single channel state exists.<br>
 ::<math>\varphi_{e}(k) =  {\rm E}\big[e_{\nu} \cdot e_{\nu + k}\big] =
@@ Line 26: / Line 30: @@
 {{BlaueBox|TEXT=
 $\text{Conclusion:}$&nbsp;
-*In the BSC model, the final ECF value (square of the mean error probability), which in other models is valid only for&nbsp; $k \to \infty$,&nbsp; is reached exactly at&nbsp; $k = 1$&nbsp; and then remains constant.<br>
+*In the BSC model, the final ECF value&nbsp; $($square of the mean error probability$)$,&nbsp; which in other models is valid only for&nbsp; $k \to \infty$,&nbsp; is reached exactly at&nbsp; $k = 1$&nbsp; and then remains constant.<br>
-*The BSC model belongs to the class of ''renewal channel models''. In a renewal channel model, the error distances are statistically independent of each other and the error correlation function can be calculated iteratively in a simple way:
+*The BSC model belongs to the class of&nbsp; "renewal channel models".&nbsp; In a renewal channel model,&nbsp; the error distances are statistically independent of each other and the error correlation function can be calculated iteratively in a simple way:
 ::<math>\varphi_{e}(k) = \sum_{\kappa = 1}^{k} {\rm Pr}(a = \kappa) \cdot
@@ Line 35: / Line 39: @@
 == Binary Symmetric Channel &ndash; Error Distance Distribution==
 <br>
-We now consider the&nbsp; <i>error distance distribution</i>&nbsp; (EDD). The probability for the error distance&nbsp; $a=k$&nbsp; is obtained from the condition of&nbsp; $k-1$&nbsp; error-free symbols and one transmission error at time&nbsp; $\nu +k$, assuming that the last error occurred at time&nbsp; $\nu$.&nbsp; One obtains:
+We now consider the&nbsp; "error distance distribution"&nbsp; $\rm (EDD)$.&nbsp; The probability for the error distance&nbsp; $a=k$&nbsp; is obtained from the condition of&nbsp; $k-1$&nbsp; error-free symbols and one transmission error at time&nbsp; $\nu +k$,&nbsp; assuming that the last error occurred at time&nbsp; $\nu$.&nbsp; One obtains:
 ::<math>{\rm Pr}(a = k) = (1-p)^{k-1}\cdot p \hspace{0.05cm}.</math>
 It follows:
-*The error distance&nbsp; $a = 1$&nbsp; always occurs with the greatest probability in the BSC model, and this for any value of&nbsp; $p$.<br>
+*The error distance&nbsp; $a = 1$&nbsp; always occurs  in the BSC model with the greatest probability,&nbsp; and this for any value of&nbsp; $p$.<br>
-*This fact is somewhat surprising at first glance. With&nbsp; $p = 0.01$,&nbsp; for example, the mean error distance is&nbsp; ${\rm E}\big[a\big] = 100$.
-*Nevertheless, two consecutive errors&nbsp; $(a = 1)$&nbsp; are more probable by a factor of&nbsp; $0.99^{99} \approx 2.7$&nbsp; than the error distance&nbsp; $a = 100$.<br><br>
+*This fact is somewhat surprising at first glance:&nbsp;
-The error distance distribution is obtained by summation according to the&nbsp; [[Digital_Signal_Transmission/Parameters_of_Digital_Channel_Models#Error_distance_distribution|"general definition"]]:&nbsp;
+*The error distance distribution is obtained by summation according to the&nbsp; [[Digital_Signal_Transmission/Parameters_of_Digital_Channel_Models#Error_distance_distribution|"general definition"]]:&nbsp;
 ::<math>V_a(k) =  {\rm Pr}(a \ge k) = 1 - \sum_{\kappa = 1}^{k}  (1-p)^{\kappa-1}\cdot p = (1-p)^{k-1}\hspace{0.05cm}.</math>
@@ Line 56: / Line 62: @@
-The decrease is exponential with increasing&nbsp; $k$&nbsp; and is steeper the smaller&nbsp; $p$&nbsp; is.<br>
+&rArr; &nbsp; The decrease is exponential with increasing&nbsp; $k$&nbsp; and is steeper the smaller&nbsp; $p$&nbsp; is.<br>
-The right graph shows the logarithmic representation. Here the drop is linear according to
+&rArr; &nbsp; The right graph shows the logarithmic representation.&nbsp; Here the drop is linear according to
 ::<math>{\rm lg} \hspace{0.15cm}V_a(k) =  (k - 1) \cdot {\rm lg} \hspace{0.15cm}(1-p)\hspace{0.05cm}.</math>}}<br>
@@ Line 64: / Line 70: @@
 == Applications of the BSC model ==
 <br>
-The BSC model is the <i>digital equivalent</i>&nbsp; of the simplest analog model &nbsp; &#8658; &nbsp; [[Digital_Signal_Transmission/Structure_of_the_Optimal_Receiver#Some_properties_of_the_AWGN_channel|"AWGN"]]&nbsp; for a time-invariant digital system corresponding to the following graph. The only degradation is noise; there is no distortion.<br>
+The BSC model is the&nbsp; "digital equivalent"&nbsp; of the simplest analog model &nbsp; &#8658; &nbsp; [[Digital_Signal_Transmission/Structure_of_the_Optimal_Receiver#Some_properties_of_the_AWGN_channel|"AWGN"]]&nbsp; for a time-invariant digital system corresponding to the following graph.&nbsp; It holds
-[[File:P ID1830 Dig T 5 2 S3 version1.png|right|frame|On the use of the BSC model|class=fit]]
+*noise according to the AWGN model &nbsp; &#8658; &nbsp; additive,&nbsp; white,&nbsp; Gaussian distributed,<br>
 *threshold decision with symmetric decision threshold,<br>
 *no extraneous influences such as crosstalk, dial pulses or electromagnetic interference fields.<br>
-<br clear=all>
 For a&nbsp; ''radio system''&nbsp; with a direct line-of-sight between transmitter and receiver, the application of the BSC model is often justified, but not if fading influences &nbsp; &#8658; &nbsp;[[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Non-Coherent_Demodulation#Rayleigh_and_Rice_Distribution|"Rayleigh or Rice"]]&nbsp; play a role or if echoes may occur &nbsp; &#8658; &nbsp;[[Mobile_Communications/Multi-Path_Reception_in_Mobile_Communications|"multi-path reception"]].<br>

@@ Line 9: / Line 9: @@
 <br>
 The left graph shows the BSC model, the simplest model of a digital transmission system.
 The name stands for&nbsp; '''Binary Symmetric Channel'''&nbsp; and states that this model can only be used for binary systems with symmetrical falsification properties. Further applies:
 *The BSC model is suitable for the study and generation of a sequence of errors with statistically independent errors. Such a channel is also called <i>memory-free</i> and, unlike the&nbsp; [[Digital_Signal_Transmission/Burst_Error_Channels|"burst error channel models"]],&nbsp; only a single channel state exists.<br>
@@ Line 49: / Line 48: @@
 ::<math>V_a(k) =  {\rm Pr}(a \ge k) = 1 - \sum_{\kappa = 1}^{k}  (1-p)^{\kappa-1}\cdot p = (1-p)^{k-1}\hspace{0.05cm}.</math>
 {{GraueBox|TEXT=
 $\text{Example 1:}$&nbsp; The left graph shows&nbsp; $V_a(k)$&nbsp; in linear representation for
-*$p = 0.1$&nbsp; (blue curve), and&nbsp;
+[[File:EN_Dig_T_5_2_S2.png|right|frame||Error distance distribution for the BSC model in linear and logarithmic plots.|class=fit]]
 *$p = 0.02$&nbsp; (red curve).

← Older revision	Revision as of 13:00, 2 September 2022
(No difference)

@@ Line 10: / Line 10: @@
 The left graph shows the BSC model, the simplest model of a digital transmission system.
-[[File:P ID1828 Dig T 5 2 S1 version1.png|center|frame|BSC model and associated error correlation function|class=fit]]
+[[File:EN_Dig_T_5_2_S1.png|center|frame|BSC model and associated error correlation function|class=fit]]
 The name stands for&nbsp; '''Binary Symmetric Channel'''&nbsp; and states that this model can only be used for binary systems with symmetrical falsification properties. Further applies:
@@ Line 49: / Line 49: @@
 ::<math>V_a(k) =  {\rm Pr}(a \ge k) = 1 - \sum_{\kappa = 1}^{k}  (1-p)^{\kappa-1}\cdot p = (1-p)^{k-1}\hspace{0.05cm}.</math>
-[[File:P ID1829 Dig T 5 2 S2 version1.png|right|Error distance distribution for the BSC model in linear and logarithmic plots.|class=fit]]
+[[File:EN_Dig_T_5_2_S2.png|right|Error distance distribution for the BSC model in linear and logarithmic plots.|class=fit]]
 {{GraueBox|TEXT=
 $\text{Example 1:}$&nbsp; The left graph shows&nbsp; $V_a(k)$&nbsp; in linear representation for