Aufgaben:Exercise 3.11Z: Extremely Asymmetrical Channel - Revision history

Maintenance script: Fix interlanguage link: resolve redirect chain

2026-03-16T15:56:01Z

Fix interlanguage link: resolve redirect chain

← Older revision		Revision as of 17:56, 16 March 2026
Line 26:		Line 26:
	Hints:		Hints:
	*The exercise belongs to the chapter  [[Information_Theory/Anwendung_auf_die_Digitalsignalübertragung\|Application to digital signal transmission]].		*The exercise belongs to the chapter  [[Information_Theory/Anwendung_auf_die_Digitalsignalübertragung\|Application to digital signal transmission]].
	*Reference is made in particular to the page  [[Information_Theory/~~Anwendung_auf_die_Digitalsignalübertragung~~#Channel_capacity_of_a_binary_channel\|Channel capacity of a binary channel]].		*Reference is made in particular to the page  [[Information_Theory/Application_to_Digital_Signal_Transmission#Channel_capacity_of_a_binary_channel\|Channel capacity of a binary channel]].
	*In  [[Aufgaben:Exercise_3.14:_Channel_Coding_Theorem\|Exercise 3.14]]  the results found here are to be interpreted in comparison to the BSC channel.		*In  [[Aufgaben:Exercise_3.14:_Channel_Coding_Theorem\|Exercise 3.14]]  the results found here are to be interpreted in comparison to the BSC channel.

Line 128:		Line 128:

	[[Category:Information Theory: Exercises\|^3.3 Application to Digital Signal Transmission^]]		[[Category:Information Theory: Exercises\|^3.3 Application to Digital Signal Transmission^]]
	[[de:Aufgaben:3.~~10Z_Extrem_unsymmetrischer_Kanal~~]]		[[de:Aufgaben:Aufgabe 3.11Z: Extrem unsymmetrischer Kanal]]

Maintenance script: Join split $$ formula lines for correct rendering

2026-03-16T13:30:44Z

Join split $$ formula lines for correct rendering

← Older revision		Revision as of 15:30, 16 March 2026
Line 100:		Line 100:
	'''(5)'''  The channel capacity  $C$  is the mutual information  $I(X; Y)$  at best possible probabilities  $p_0$  and   $p_1$  of the source symbols.		'''(5)'''  The channel capacity  $C$  is the mutual information  $I(X; Y)$  at best possible probabilities  $p_0$  and   $p_1$  of the source symbols.
	*After differentiation, the determination equation is obtained:		*After differentiation, the determination equation is obtained:
	:$$\frac{\rm d}{{\rm d}p_0} \hspace{0.1cm} I(X;Y) =		:$$\frac{\rm d}{{\rm d}p_0} \hspace{0.1cm} I(X;Y) =\frac{\rm d}{{\rm d}p_0} \hspace{0.1cm} H_{\rm bin}(\frac{1+p_0}{2}) +1 \stackrel{!}{=} 0\hspace{0.05cm}.$$
	\frac{\rm d}{{\rm d}p_0} \hspace{0.1cm} H_{\rm bin}(\frac{1+p_0}{2}) +1 \stackrel{!}{=} 0
	\hspace{0.05cm}.$$
	*With the differential quotient of the binary entropy function		*With the differential quotient of the binary entropy function
	:$$ \frac{\rm d}{{\rm d}p} \hspace{0.1cm} H_{\rm bin}(p) = {\rm log}_2 \hspace{0.1cm} \frac{1-p}{ p} \hspace{0.05cm},$$		:$$ \frac{\rm d}{{\rm d}p} \hspace{0.1cm} H_{\rm bin}(p) = {\rm log}_2 \hspace{0.1cm} \frac{1-p}{ p} \hspace{0.05cm},$$

Maintenance script: Add German interlanguage link

2026-03-16T12:31:53Z

Add German interlanguage link

← Older revision		Revision as of 14:31, 16 March 2026
Line 130:		Line 130:

	[[Category:Information Theory: Exercises\|^3.3 Application to Digital Signal Transmission^]]		[[Category:Information Theory: Exercises\|^3.3 Application to Digital Signal Transmission^]]
			[[de:Aufgaben:3.10Z_Extrem_unsymmetrischer_Kanal]]

Guenter at 11:56, 24 September 2021

2021-09-24T11:56:37Z

← Older revision		Revision as of 13:56, 24 September 2021
Line 129:		Line 129:


	[[Category:Information Theory: Exercises\|^3.3 ~~Anwendung auf DSÜ-Kanäle~~^]]		[[Category:Information Theory: Exercises\|^3.3 Application to Digital Signal Transmission^]]

Noah at 08:24, 24 September 2021

2021-09-24T08:24:31Z

← Older revision		Revision as of 10:24, 24 September 2021
Line 1:		Line 1:

	{{quiz-Header\|Buchseite=~~Informationstheorie~~/~~Anwendung auf die Digitalsignalübertragung~~		{{quiz-Header\|Buchseite=Information_Theory/Application_to_Digital_Signal_Transmission
	}}		}}

Guenter at 12:14, 22 September 2021

2021-09-22T12:14:17Z

← Older revision		Revision as of 14:14, 22 September 2021
Line 107:		Line 107:
	:and corresponding post-differentialisation one obtains:		:and corresponding post-differentialisation one obtains:
	:$${1}/{2} \cdot {\rm log}_2 \hspace{0.1cm} \frac{(1-p_0)/2}{1- (1-p_0)/2} +1 \stackrel{!}{=} 0 \hspace{0.3cm} \Rightarrow \hspace{0.3cm} {1}/{2} \cdot {\rm log}_2 \hspace{0.1cm} \frac{(1-p_0)/2}{(1+p_0)/2} +1 \stackrel{!}{=} 0$$		:$${1}/{2} \cdot {\rm log}_2 \hspace{0.1cm} \frac{(1-p_0)/2}{1- (1-p_0)/2} +1 \stackrel{!}{=} 0 \hspace{0.3cm} \Rightarrow \hspace{0.3cm} {1}/{2} \cdot {\rm log}_2 \hspace{0.1cm} \frac{(1-p_0)/2}{(1+p_0)/2} +1 \stackrel{!}{=} 0$$
	:$$ \Rightarrow \hspace{0.3cm} {\rm log}_2 \hspace{0.1cm} \frac{1+p_0}{1-p_0} \stackrel{!}{=} 2 \hspace{0.3cm} \Rightarrow \hspace{0.3cm} \frac{1+p_0}{1-p_0} \stackrel{!}{=} 4 \hspace{0.3cm}\Rightarrow \hspace{0.3cm} p_0 \hspace{0.15cm} \underline {=0.6}\hspace{0.05cm}.$$		:$$ \Rightarrow \hspace{0.3cm} {\rm log}_2 \hspace{0.1cm} \frac{1+p_0}{1-p_0} \stackrel{!}{=} 2 \hspace{0.3cm} \Rightarrow \hspace{0.3cm} \frac{1+p_0}{1-p_0} \stackrel{!}{=} 4 \hspace{0.3cm}\Rightarrow \hspace{0.3cm} p_0 \hspace{0.15cm} \underline {=0.6}=p_0^{(*)}\hspace{0.05cm}.$$

Guenter at 12:09, 22 September 2021

2021-09-22T12:09:49Z

← Older revision		Revision as of 14:09, 22 September 2021
Line 27:		Line 27:
	*The exercise belongs to the chapter  [[Information_Theory/Anwendung_auf_die_Digitalsignalübertragung\|Application to digital signal transmission]].		*The exercise belongs to the chapter  [[Information_Theory/Anwendung_auf_die_Digitalsignalübertragung\|Application to digital signal transmission]].
	*Reference is made in particular to the page  [[Information_Theory/Anwendung_auf_die_Digitalsignalübertragung#Channel_capacity_of_a_binary_channel\|Channel capacity of a binary channel]].		*Reference is made in particular to the page  [[Information_Theory/Anwendung_auf_die_Digitalsignalübertragung#Channel_capacity_of_a_binary_channel\|Channel capacity of a binary channel]].
	*In  [[Aufgaben:3.~~14_Kanalcodierungstheorem~~\|Exercise 3.14]]  the results found here are to be interpreted in comparison to the BSC channel.		*In  [[Aufgaben:Exercise_3.14:_Channel_Coding_Theorem\|Exercise 3.14]]  the results found here are to be interpreted in comparison to the BSC channel.

Guenter at 12:08, 22 September 2021

2021-09-22T12:08:44Z

← Older revision		Revision as of 14:08, 22 September 2021
Line 5:		Line 5:
	[[File:P_ID2800__Inf_Z_3_10.png\|right\|frame\|One-sided distorting channel]]		[[File:P_ID2800__Inf_Z_3_10.png\|right\|frame\|One-sided distorting channel]]
	The channel shown opposite with the following properties is considered:		The channel shown opposite with the following properties is considered:
	* The symbol  $X = 0$  is always transmitted correctly and always ~~leads~~ to the result  $Y = 0$.		* The symbol  $X = 0$  is always transmitted correctly and leads always to the result  $Y = 0$.
	* The symbol  $X = 1$  is distorted to the maximum.		* The symbol  $X = 1$  is distorted to the maximum.

Noah at 11:33, 16 September 2021

2021-09-16T11:33:58Z

Show changes

Guenter: Guenter moved page Aufgaben:Aufgabe 3.11Z: Extrem unsymmetrischer Kanal to Aufgaben:Exercise 3.11Z: Extremely Asymmetrical Channel

2021-07-22T14:53:55Z

Guenter moved page Aufgaben:Aufgabe 3.11Z: Extrem unsymmetrischer Kanal to Aufgaben:Exercise 3.11Z: Extremely Asymmetrical Channel

← Older revision	Revision as of 16:53, 22 July 2021
(No difference)

← Older revision		Revision as of 14:09, 22 September 2021
Line 27:		Line 27:
	*The exercise belongs to the chapter  [[Information_Theory/Anwendung_auf_die_Digitalsignalübertragung\|Application to digital signal transmission]].		*The exercise belongs to the chapter  [[Information_Theory/Anwendung_auf_die_Digitalsignalübertragung\|Application to digital signal transmission]].
	*Reference is made in particular to the page  [[Information_Theory/Anwendung_auf_die_Digitalsignalübertragung#Channel_capacity_of_a_binary_channel\|Channel capacity of a binary channel]].		*Reference is made in particular to the page  [[Information_Theory/Anwendung_auf_die_Digitalsignalübertragung#Channel_capacity_of_a_binary_channel\|Channel capacity of a binary channel]].
	*In  [[Aufgaben:3.~~14_Kanalcodierungstheorem~~\|Exercise 3.14]]  the results found here are to be interpreted in comparison to the BSC channel.		*In  [[Aufgaben:Exercise_3.14:_Channel_Coding_Theorem\|Exercise 3.14]]  the results found here are to be interpreted in comparison to the BSC channel.