Channel Coding/Error Probability and Application Areas - Revision history

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

2026-03-16T13:46:43Z

Join split $$ formula lines for correct rendering

← Older revision		Revision as of 15:46, 16 March 2026
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	*the parameter  $\varepsilon_{\rm S}$  of the  "8–BSC"  model   ⇒   "symbol error probability"  or  "block error probability":		*the parameter  $\varepsilon_{\rm S}$  of the  "8–BSC"  model   ⇒   "symbol error probability"  or  "block error probability":

	:$$\varepsilon_{\rm S} = 1 - (1 - 1.1 \cdot 10^{-3})^8 = 1 - 0.9989^8 = 1 - 0.9912 \approx 0.88 \%		:$$\varepsilon_{\rm S} = 1 - (1 - 1.1 \cdot 10^{-3})^8 = 1 - 0.9989^8 = 1 - 0.9912 \approx 0.88 \%\hspace{0.2cm}(\approx 8 \cdot \varepsilon)\hspace{0.05cm}.$$
	\hspace{0.2cm}(\approx 8 \cdot \varepsilon)\hspace{0.05cm}.$$

	⇒   So every single symbol is transmitted with more than  $99$  percent probability without errors.}}<br>		⇒   So every single symbol is transmitted with more than  $99$  percent probability without errors.}}<br>
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	Dominant here is the first term  $($for  $f=9)$, which already accounts for  $\approx 80\%$ :		Dominant here is the first term  $($for  $f=9)$, which already accounts for  $\approx 80\%$ :

	:$${255 \choose 9} \approx 1.1 \cdot 10^{16}\hspace{0.05cm},\hspace{0.15cm}		:$${255 \choose 9} \approx 1.1 \cdot 10^{16}\hspace{0.05cm},\hspace{0.15cm}\varepsilon_{\rm S}^9 \approx 4 \cdot 10^{-20}\hspace{0.05cm},\hspace{0.15cm}(1 -\varepsilon_{\rm S})^{246} \approx 0.18$$
	\varepsilon_{\rm S}^9 \approx 4 \cdot 10^{-20}\hspace{0.05cm},\hspace{0.15cm}		::$$ \Rightarrow \hspace{0.15cm} {\rm Pr}(f = 9) \approx 8 \cdot 10^{-5}\hspace{0.05cm}.$$
	(1 -\varepsilon_{\rm S})^{246} \approx 0.18$$
	::$$ \Rightarrow \hspace{0.15cm} {\rm Pr}(f = 9) \approx 8 \cdot 10^{-5}
	\hspace{0.05cm}.$$

	This should be taken as proof that one may stop the summation already after a few terms.<br>		This should be taken as proof that one may stop the summation already after a few terms.<br>

Maintenance script: Add German interlanguage link

2026-03-16T12:27:37Z

Add German interlanguage link

← Older revision		Revision as of 14:27, 16 March 2026
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	{{Display}}		{{Display}}
			[[de:Kanalcodierung/Fehlerwahrscheinlichkeit_und_Anwendungsgebiete]]

Guenter at 13:51, 13 December 2022

2022-12-13T13:51:36Z

← Older revision		Revision as of 15:51, 13 December 2022
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	<br>		<br>
	The prerequisites for the following calculation of block error probability of a Reed–Solomon coding system and conversion to binary symbols are summarized in the diagram:		The prerequisites for the following calculation of block error probability of a Reed–Solomon coding system and conversion to binary symbols are summarized in the diagram:
	[[File:EN_KC_T_2_6_S2a_neu_v3.png\|right\|frame\|Application of Reed-Solomon coding for binary channels '''~~Korrektur~~'''\|class=fit]]		[[File:EN_KC_T_2_6_S2a_neu_v3.png\|right\|frame\|Application of Reed-Solomon coding for binary channels.  at point<br>'''(1)'''  $k$–bit symbols;   '''(2)'''  $n$–bit symbols;   '''(3)'''  $n$–bit symbols;   '''(4)'''  $(n \cdot m)$–bit blocks;\|class=fit]]

	# Assume  $(n,\ k)$ Reed–Solomon coding with symbols of  $c_i ∈ {\rm GF}(2^m)$.  The smaller the code rate  $R=k/n$,  the less information can be transmitted at a fixed data rate.<br>		# Assume  $(n,\ k)$ Reed–Solomon coding with symbols of  $c_i ∈ {\rm GF}(2^m)$.  The smaller the code rate  $R=k/n$,  the less information can be transmitted at a fixed data rate.<br>

Guenter at 16:20, 12 December 2022

2022-12-12T16:20:31Z

← Older revision		Revision as of 18:20, 12 December 2022
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	<br>		<br>
	The prerequisites for the following calculation of block error probability of a Reed–Solomon coding system and conversion to binary symbols are summarized in the diagram:		The prerequisites for the following calculation of block error probability of a Reed–Solomon coding system and conversion to binary symbols are summarized in the diagram:
	[[File:~~EN_KC_T_2_6_S2b_neu_v3~~.png\|right\|frame\|Application of Reed-Solomon coding for binary channels '''Korrektur'''\|class=fit]]		[[File:EN_KC_T_2_6_S2a_neu_v3.png\|right\|frame\|Application of Reed-Solomon coding for binary channels '''Korrektur'''\|class=fit]]

	# Assume  $(n,\ k)$ Reed–Solomon coding with symbols of  $c_i ∈ {\rm GF}(2^m)$.  The smaller the code rate  $R=k/n$,  the less information can be transmitted at a fixed data rate.<br>		# Assume  $(n,\ k)$ Reed–Solomon coding with symbols of  $c_i ∈ {\rm GF}(2^m)$.  The smaller the code rate  $R=k/n$,  the less information can be transmitted at a fixed data rate.<br>

Guenter at 16:46, 7 December 2022

2022-12-07T16:46:24Z

← Older revision		Revision as of 18:46, 7 December 2022
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	<br>		<br>
	The prerequisites for the following calculation of block error probability of a Reed–Solomon coding system and conversion to binary symbols are summarized in the diagram:		The prerequisites for the following calculation of block error probability of a Reed–Solomon coding system and conversion to binary symbols are summarized in the diagram:
	[[File:~~EN_KC_T_2_6_S2b_neu~~.png\|right\|frame\|Application of Reed-Solomon coding for binary channels '''Korrektur'''\|class=fit]]		[[File:EN_KC_T_2_6_S2b_neu_v3.png\|right\|frame\|Application of Reed-Solomon coding for binary channels '''Korrektur'''\|class=fit]]

	# Assume  $(n,\ k)$ Reed–Solomon coding with symbols of  $c_i ∈ {\rm GF}(2^m)$.  The smaller the code rate  $R=k/n$,  the less information can be transmitted at a fixed data rate.<br>		# Assume  $(n,\ k)$ Reed–Solomon coding with symbols of  $c_i ∈ {\rm GF}(2^m)$.  The smaller the code rate  $R=k/n$,  the less information can be transmitted at a fixed data rate.<br>

Guenter at 15:58, 7 December 2022

2022-12-07T15:58:46Z

← Older revision		Revision as of 17:58, 7 December 2022
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	<br>		<br>
	The prerequisites for the following calculation of block error probability of a Reed–Solomon coding system and conversion to binary symbols are summarized in the diagram:		The prerequisites for the following calculation of block error probability of a Reed–Solomon coding system and conversion to binary symbols are summarized in the diagram:
	[[File:~~P ID2565 KC T 2 6 S2a v2~~.png\|right\|frame\|Application of Reed-Solomon coding for binary channels '''Korrektur'''\|class=fit]]		[[File:EN_KC_T_2_6_S2b_neu.png\|right\|frame\|Application of Reed-Solomon coding for binary channels '''Korrektur'''\|class=fit]]

	# Assume  $(n,\ k)$ Reed–Solomon coding with symbols of  $c_i ∈ {\rm GF}(2^m)$.  The smaller the code rate  $R=k/n$,  the less information can be transmitted at a fixed data rate.<br>		# Assume  $(n,\ k)$ Reed–Solomon coding with symbols of  $c_i ∈ {\rm GF}(2^m)$.  The smaller the code rate  $R=k/n$,  the less information can be transmitted at a fixed data rate.<br>

Guenter at 17:32, 24 November 2022

2022-11-24T17:32:16Z

← Older revision		Revision as of 19:32, 24 November 2022
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	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Conclusion:}$		$\text{Conclusion:}$
	*Reed–Solomon codes are not very good at the memoryless AWGN channel.  Both codes are more than  $4 \, \rm dB$  away from the information-theoretical  [[Channel_Coding/Information_Theoretical_Limits_of_Channel_Coding#AWGN_channel_capacity_for_binary_input_signals\| $\text{Shannon ~~limit~~ }$ ~~'''KORREKTUR: bound'''~~]].<br>		*Reed–Solomon codes are not very good at the memoryless AWGN channel.  Both codes are more than  $4 \, \rm dB$  away from the information-theoretical  [[Channel_Coding/Information_Theoretical_Limits_of_Channel_Coding#AWGN_channel_capacity_for_binary_input_signals\| $\text{Shannon bound }$ ]].<br>

	*But Reed–Solomon codes are very effective with so-called  [[Digital_Signal_Transmission/Burst_Error_Channels\| $\text{Burst Error Channels}$]].  They are therefore mainly used for fading channels,  memory systems,  etc.<br>		*But Reed–Solomon codes are very effective with so-called  [[Digital_Signal_Transmission/Burst_Error_Channels\| $\text{Burst Error Channels}$]].  They are therefore mainly used for fading channels,  memory systems,  etc.<br>

Hwang at 12:12, 24 November 2022

2022-11-24T12:12:28Z

← Older revision		Revision as of 14:12, 24 November 2022
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	*The inner code is almost always a binary code and in satellite– and mobile communications often a convolutional code.		*The inner code is almost always a binary code and in satellite– and mobile communications often a convolutional code.

	*If one wishes to study only Reed–Solomon ~~coding~~/decoding, one replaces the grayed components in a simulation with a single block called a  "super channel".<br>		*If one wishes to study only Reed–Solomon encoding/decoding, one replaces the grayed components in a simulation with a single block called a  "super channel".<br>

	*Concatenated coding systems are very efficient if an interleaver is connected between the two encoders to further relax burst errors.<br>		*Concatenated coding systems are very efficient if an interleaver is connected between the two encoders to further relax burst errors.<br>

Hwang at 12:11, 24 November 2022

2022-11-24T12:11:30Z

← Older revision		Revision as of 14:11, 24 November 2022
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	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Conclusion:}$		$\text{Conclusion:}$
	*Reed–Solomon codes are not very good at the memoryless AWGN channel.  Both codes are more than  $4 \, \rm dB$  away from the information-theoretical  [[Channel_Coding/Information_Theoretical_Limits_of_Channel_Coding#AWGN_channel_capacity_for_binary_input_signals\| $\text{Shannon limit '''KORREKTUR: bound'''}$]].<br>		*Reed–Solomon codes are not very good at the memoryless AWGN channel.  Both codes are more than  $4 \, \rm dB$  away from the information-theoretical  [[Channel_Coding/Information_Theoretical_Limits_of_Channel_Coding#AWGN_channel_capacity_for_binary_input_signals\| $\text{Shannon limit }$ '''KORREKTUR: bound''']].<br>

	*But Reed–Solomon codes are very effective with so-called  [[Digital_Signal_Transmission/Burst_Error_Channels\| $\text{Burst Error Channels}$]].  They are therefore mainly used for fading channels,  memory systems,  etc.<br>		*But Reed–Solomon codes are very effective with so-called  [[Digital_Signal_Transmission/Burst_Error_Channels\| $\text{Burst Error Channels}$]].  They are therefore mainly used for fading channels,  memory systems,  etc.<br>

Hwang at 12:11, 24 November 2022

2022-11-24T12:11:07Z

← Older revision		Revision as of 14:11, 24 November 2022
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	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Conclusion:}$		$\text{Conclusion:}$
	*Reed–Solomon codes are not very good at the memoryless AWGN channel.  Both codes are more than  $4 \, \rm dB$  away from the information-theoretical  [[Channel_Coding/Information_Theoretical_Limits_of_Channel_Coding#AWGN_channel_capacity_for_binary_input_signals\| $\text{Shannon limit}$]].<br>		*Reed–Solomon codes are not very good at the memoryless AWGN channel.  Both codes are more than  $4 \, \rm dB$  away from the information-theoretical  [[Channel_Coding/Information_Theoretical_Limits_of_Channel_Coding#AWGN_channel_capacity_for_binary_input_signals\| $\text{Shannon limit '''KORREKTUR: bound'''}$]].<br>

	*But Reed–Solomon codes are very effective with so-called  [[Digital_Signal_Transmission/Burst_Error_Channels\| $\text{Burst Error Channels}$]].  They are therefore mainly used for fading channels,  memory systems,  etc.<br>		*But Reed–Solomon codes are very effective with so-called  [[Digital_Signal_Transmission/Burst_Error_Channels\| $\text{Burst Error Channels}$]].  They are therefore mainly used for fading channels,  memory systems,  etc.<br>