Aufgaben:Exercise 5.5: Error Sequence and Error Distance Sequence - Revision history

Hwang at 02:57, 18 September 2022

2022-09-18T02:57:28Z

Hwang at 02:56, 18 September 2022

2022-09-18T02:56:23Z

Guenter at 14:49, 5 September 2022

2022-09-05T14:49:49Z

Guenter at 14:41, 5 September 2022

2022-09-05T14:41:32Z

Guenter at 15:31, 1 September 2022

2022-09-01T15:31:50Z

Hwang at 18:46, 27 August 2022

2022-08-27T18:46:11Z

Guenter at 17:24, 23 August 2022

2022-08-23T17:24:27Z

Guenter: Guenter moved page Aufgaben:Aufgabe 5.5: Fehlerfolge und Fehlerabstandsfolge to Aufgaben:Exercise 5.5: Error Sequence and Error Distance Sequence

2022-08-23T16:36:34Z

Guenter moved page Aufgabe 5.5: Fehlerfolge und Fehlerabstandsfolge to Exercise 5.5: Error Sequence and Error Distance Sequence

Javier: Text replacement - "Category:Aufgaben zu Digitalsignalübertragung" to "Category:Digital Signal Transmission: Exercises"

2021-03-23T12:46:34Z

Text replacement - "Category:Aufgaben zu Digitalsignalübertragung" to "Category:Digital Signal Transmission: Exercises"

Javier: Text replacement - "[[Stochastische_Signaltheorie/" to "[[Theory_of_Stochastic_Signals/"

2020-07-09T09:52:42Z

Text replacement - "[[Stochastische_Signaltheorie/" to "[[Theory_of_Stochastic_Signals/"

← Older revision		Revision as of 02:57, 18 September 2022
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	'''(2)'''  With  $N = 10^6$  and  $p_{\rm M} = 10^{–3}$,  about thousand error distances are to be stored,  each one with  $4 \ \rm bytes$   ⇒   $G_a \ \underline {= 4 \rm kByte}$.		'''(2)'''  With  $N = 10^6$  and  $p_{\rm M} = 10^{–3}$,  about thousand error distances are to be stored,  each one with  $4 \ \rm bytes$   ⇒   $G_a \ \underline {= 4 \rm kByte}$.
−	*In contrast to the storage of the error sequence,  this value will vary slightly,  since in an error sequence of ~~the~~  (limited)  length  $N = 10^6$  not always exactly  $1000$  errors will occur.	+	*In contrast to the storage of the error sequence,  this value will vary slightly,  since in an error sequence of  (limited)  length  $N = 10^6$  not always exactly  $1000$  errors will occur.

← Older revision		Revision as of 02:56, 18 September 2022
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−	'''(2)'''  With  $N = 10^6$  and  $p_{\rm M} = 10^{–3}$,  about ~~thousend~~ error distances are to be stored,  each one with  $4 \ \rm bytes$   ⇒   $G_a \ \underline {= 4 \rm kByte}$.	+	'''(2)'''  With  $N = 10^6$  and  $p_{\rm M} = 10^{–3}$,  about thousand error distances are to be stored,  each one with  $4 \ \rm bytes$   ⇒   $G_a \ \underline {= 4 \rm kByte}$.
	*In contrast to the storage of the error sequence,  this value will vary slightly,  since in an error sequence of the  (limited)  length  $N = 10^6$  not always exactly  $1000$  errors will occur.		*In contrast to the storage of the error sequence,  this value will vary slightly,  since in an error sequence of the  (limited)  length  $N = 10^6$  not always exactly  $1000$  errors will occur.

@@ Line 50: / Line 50: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; For each element $e_{\nu}$ of the error sequence exactly one $\rm bit$ is needed.
+'''(1)'''&nbsp; For each element&nbsp; $e_{\nu}$&nbsp; of the error sequence exactly one&nbsp; $\rm bit$&nbsp; is needed.
-*Multiplication by $N$ results in $10^6 \ \rm bits$ corresponding to $G_e \ \underline {= 125  \ \rm kByte}$.
+*Multiplication by&nbsp; $N$&nbsp; results in&nbsp; $10^6 \ \rm bits$&nbsp; corresponding to&nbsp; $G_e \ \underline {= 125  \ \rm kByte}$.
-'''(2)'''&nbsp; With $N = 10^6$ and $p_{\rm M} = 10^{&ndash;3}$, about $1000$ error distances are to be stored, each one with $4 \ \rm bytes$ &nbsp; &#8658; &nbsp; $G_a \ \underline {= 4  \rm kByte}$.
+'''(2)'''&nbsp; With&nbsp; $N = 10^6$&nbsp; and&nbsp; $p_{\rm M} = 10^{&ndash;3}$,&nbsp; about thousend error distances are to be stored,&nbsp; each one with&nbsp; $4 \ \rm bytes$ &nbsp; &#8658; &nbsp; $G_a \ \underline {= 4  \rm kByte}$.
-*In contrast to the storage of the error sequence, this value will vary slightly, since in an error sequence of the (limited) length $N = 10^6$ not always exactly $1000$ errors will occur.
+*In contrast to the storage of the error sequence,&nbsp; this value will vary slightly,&nbsp; since in an error sequence of the&nbsp; (limited)&nbsp; length&nbsp; $N = 10^6$&nbsp; not always exactly&nbsp; $1000$&nbsp; errors will occur.
-'''(3)'''&nbsp; Now, on average, $0.5 \cdot 10^6$ errors will occur &nbsp; &#8658; &nbsp; $G_a \ \underline {= 2000 \ \rm KByte}$.
+'''(3)'''&nbsp; Now,&nbsp; on average,&nbsp; $0.5 \cdot 10^6$&nbsp; errors will occur &nbsp; &#8658; &nbsp; $G_a \ \underline {= 2000 \ \rm kByte}$.
-*From this it can be seen that storing the error distances only makes sense if the (average) error probability is not too large.
@@ Line 67: / Line 68: @@
 '''(4)'''&nbsp; From the explanations of the upper subtasks it follows:
 :$$N \cdot p_{\rm M} \cdot 4 < {N}/{8} \Rightarrow
-\hspace{0.3cm}p_{\rm M, \hspace{0.05cm}max} = {1}/{32} \hspace{0.15cm}\underline {=
+\hspace{0.3cm}p_{\rm M, \hspace{0.1cm}max} = {1}/{32} \hspace{0.15cm}\underline {=
 .125\%}\hspace{0.05cm}.$$
-*This result is independent of the sequence length $N$.
+*This result is independent of the sequence length&nbsp; $N$.
 {{ML-Fuß}}
 [[Category:Digital Signal Transmission: Exercises|^5.2 Binary Symmetric Channel^]]

@@ Line 2: / Line 2: @@
 [[File:EN_Dig_A_5_5.png|right|frame|Error sequence&nbsp; (blue), <br>error distance sequence&nbsp;  (red)]]
-Any error sequence&nbsp; $&#9001;e_{\nu}&#9002;$&nbsp; can also be specified as the sequence&nbsp; $&#9001;a_n&#9002;$&nbsp; of error distances. If the average error probability is not too large, then this results in a lower memory requirement than if the error sequence is stored. For the comparison in this exercise, the following assumptions are to be made:
+Any error sequence&nbsp; $&#9001;e_{\nu}&#9002;$&nbsp; can also be specified as the sequence&nbsp; $&#9001;a_n&#9002;$&nbsp; of error distances.&nbsp; If the average error probability is not too large,&nbsp; then this results in a lower memory requirement than if the error sequence is stored.
 * A error sequence of length&nbsp; $N = 10^6$&nbsp; elements is to be stored in each case.
 * The most memory-efficient method (one bit per error) is to be used for storing&nbsp; $&#9001;e_{\nu}&#9002;$.&nbsp;
-* Each error distance is represented by 4 bytes (32 bits).
+* The most memory-efficient method&nbsp; $($one bit per error$)$&nbsp; is to be used for storing&nbsp; $&#9001;e_{\nu}&#9002;$.&nbsp;
 If the underlying channel model is renewing, such as the BSC model, two different methods can be used to generate the error sequence&nbsp; $&#9001;e_{\nu}&#9002;$&nbsp; on a digital computer:
-* the symbol-wise generation of the errors, in the BSC model according to the probabilities&nbsp; $p$&nbsp; (error) and &nbsp;$1-p$&nbsp; (no error),
+# The symbol-wise generation of the errors,&nbsp; in the BSC model due to the probabilities&nbsp; $p$&nbsp; ("error")&nbsp; and &nbsp;$1-p$&nbsp; ("no error"),
-* the generation of the error distances, in the BSC model according to the&nbsp; [[Theory_of_Stochastic_Signals/Binomial_Distribution| "binomial distribution"]].
+# The generation of the error distances,&nbsp; in the BSC model according to the&nbsp; [[Theory_of_Stochastic_Signals/Binomial_Distribution| "binomial distribution"]].
+<u>Notes:</u>
+* The exercise belongs to  the chapter&nbsp; [[Digital_Signal_Transmission/Binary_Symmetric_Channel_(BSC)| "Binary Symmetric Channel"]].
-''Notes:''
-* The exercise belongs to the topic of the chapter&nbsp; [[Digital_Signal_Transmission/Binary_Symmetric_Channel_(BSC)| "Binary Symmetric Channel (BSC)"]].
+* In the following questions,&nbsp;
-* In the following questions,&nbsp; $G_e$&nbsp; indicates the required file size (in bytes) for storing the error sequence&nbsp; $&#9001;e_{\nu}&#9002;$&nbsp; and&nbsp; $G_a$&nbsp; indicates (also in bytes) the file size when storing the error distance sequence&nbsp; $&#9001;a_n&#9002;$.&nbsp;
+**$G_e$&nbsp; indicates the required file size&nbsp; (in bytes)&nbsp; for storing the error sequence&nbsp; $&#9001;e_{\nu}&#9002;$,&nbsp; and&nbsp;
@@ Line 26: / Line 31: @@
 ===Questions===
 <quiz display=simple>
-{How much storage space (in bytes) is required when saving an error sequence of length&nbsp;  $N = 10^6$&nbsp; directly?
+{How much storage space&nbsp; (in bytes)&nbsp; is required when saving an error sequence of length&nbsp;  $N = 10^6$&nbsp; directly?
 |type="{}"}
 $G_e \ = \ ${ 125 3% } $\  \rm kByte$
-{What is the approximate size of the file when storing the error distances? Let&nbsp; $p_{\rm M} = 10^{-3}$.
+{What is the approximate size of the file when storing the error distances?&nbsp; Let&nbsp; $p_{\rm M} = 10^{-3}$.
 |type="{}"}
 $G_a \ = \ ${ 4 3% } $\ \rm kByte$

@@ Line 1: / Line 1: @@
 {{quiz-Header|Buchseite=Digital_Signal_Transmission/Binary_Symmetric_Channel_(BSC)}}
-[[File:P_ID1834__Dig_A_5_5.png|right|frame|Error sequence (top, blue) and error distance sequence (bottom, red)]]
+[[File:EN_Dig_A_5_5.png|right|frame|Error sequence&nbsp; (blue), <br>error distance sequence&nbsp;  (red)]]
 Any error sequence&nbsp; $&#9001;e_{\nu}&#9002;$&nbsp; can also be specified as the sequence&nbsp; $&#9001;a_n&#9002;$&nbsp; of error distances. If the average error probability is not too large, then this results in a lower memory requirement than if the error sequence is stored. For the comparison in this exercise, the following assumptions are to be made:
 * A error sequence of length&nbsp; $N = 10^6$&nbsp; elements is to be stored in each case.

← Older revision		Revision as of 17:24, 23 August 2022
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−	[[Category:Digital Signal Transmission: Exercises\|^5.2 Binary Symmetric Channel ~~(BSC)~~^]]	+	[[Category:Digital Signal Transmission: Exercises\|^5.2 Binary Symmetric Channel^]]

← Older revision		Revision as of 12:46, 23 March 2021
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−	[[Category:~~Aufgaben zu Digitalsignalübertragung~~\|^5.2 Binary Symmetric Channel (BSC)^]]	+	[[Category:Digital Signal Transmission: Exercises\|^5.2 Binary Symmetric Channel (BSC)^]]

@@ Line 10: / Line 10: @@
 Ist das zugrundeliegende Kanalmodell erneuernd wie zum Beispiel das BSC&ndash;Modell, so können zur Generierung der Fehlerfolge&nbsp; $&#9001;e_{\nu}&#9002;$&nbsp; auf einem Digitalrechner zwei unterschiedliche Methoden angewandt werden:
 * die symbolweise Erzeugung der Fehler, beim BSC&ndash;Modell gemäß den Wahrscheinlichkeiten&nbsp; $p$&nbsp; (Fehler) und &nbsp;$1-p$&nbsp; (kein Fehler),
-* die Erzeugung der Fehlerabstände, beim BSC&ndash;Modell entsprechend der&nbsp; [[Stochastische_Signaltheorie/Binomialverteilung| Binomialverteilung]].
+* die Erzeugung der Fehlerabstände, beim BSC&ndash;Modell entsprechend der&nbsp; [[Theory_of_Stochastic_Signals/Binomialverteilung| Binomialverteilung]].