Channel Coding/Information Theoretical Limits of Channel Coding - Revision history

Maintenance script: Add German interlanguage link

2026-03-16T12:27:40Z

Add German interlanguage link

← Older revision		Revision as of 14:27, 16 March 2026
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	#For all binary codes holds per se  $0 < R ≤ 1$. Only with non-binary codes  ⇒   rates  $R > 1$  are possible.  For example,  the maximum possible code rate of a quaternary code:  $R = \log_2 \, M_y = \log_2 \, 4 = 2$.<br>		#For all binary codes holds per se  $0 < R ≤ 1$. Only with non-binary codes  ⇒   rates  $R > 1$  are possible.  For example,  the maximum possible code rate of a quaternary code:  $R = \log_2 \, M_y = \log_2 \, 4 = 2$.<br>
	#All one-dimensional modulation types – i.e., those methods that use only the in-phase– or only the quadrature component such as  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#On.E2.80.93off_keying_.282.E2.80.93ASK.29\|$\text{2–ASK}$]],  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#Binary_phase_shift_keying_.28BPSK.29 \|$\text{BPSK}$]]  and  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Non-Coherent_Demodulation#Non-coherent_demodulation_of_binary_FSK_.282.E2.80.93FSK.29\|$\text{2–FSK}$]]  must be in the blue area of the present graphic.<br>		#All one-dimensional modulation types – i.e., those methods that use only the in-phase– or only the quadrature component such as  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#On.E2.80.93off_keying_.282.E2.80.93ASK.29\|$\text{2–ASK}$]],  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#Binary_phase_shift_keying_.28BPSK.29 \|$\text{BPSK}$]]  and  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Non-Coherent_Demodulation#Non-coherent_demodulation_of_binary_FSK_.282.E2.80.93FSK.29\|$\text{2–FSK}$]]  must be in the blue area of the present graphic.<br>
	#As shown in the chapter  [[Information_Theory/~~AWGN_Channel_Capacity_for_Discrete_Input~~#Maximum_code_rate_for_QAM_structures\|"Maximum code rate for QAM structures"]],  there is a  "friendlier"  limit curve for two-dimensional modulation types such as the  [[Modulation_Methods/Quadrature_Amplitude_Modulation\|"Quadrature Amplitude Modulation"]].		#As shown in the chapter  [[Information_Theory/AWGN_Channel_Capacity_for_Discrete-Valued_Input#Maximum_code_rate_for_QAM_structures\|"Maximum code rate for QAM structures"]],  there is a  "friendlier"  limit curve for two-dimensional modulation types such as the  [[Modulation_Methods/Quadrature_Amplitude_Modulation\|"Quadrature Amplitude Modulation"]].

	== AWGN channel capacity for binary input signals ==		== AWGN channel capacity for binary input signals ==
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	$\text{Conclusion:}$  It should be noted that Shannon recognized and proved as early as 1948 that no one-dimensional modulation method can lie to the left of the AWGN limit curve "Gaussian (real)" drawn throughout.		$\text{Conclusion:}$  It should be noted that Shannon recognized and proved as early as 1948 that no one-dimensional modulation method can lie to the left of the AWGN limit curve "Gaussian (real)" drawn throughout.
	*For two-dimensional methods such as QAM and multilevel PSK, on the other hand, the "Gaussian (complex)" limit curve applies, which is drawn here as a dashed line and always lies to the left of the solid curve.		*For two-dimensional methods such as QAM and multilevel PSK, on the other hand, the "Gaussian (complex)" limit curve applies, which is drawn here as a dashed line and always lies to the left of the solid curve.
	*For more details, see the  [[Information_Theory/~~AWGN_Channel_Capacity_for_Discrete_Input~~#Maximum_code_rate_for_QAM_structures\|$\text{Maximum code rate for QAM structures}$]]  section of the "Information Theory" book.<br>		*For more details, see the  [[Information_Theory/AWGN_Channel_Capacity_for_Discrete-Valued_Input#Maximum_code_rate_for_QAM_structures\|$\text{Maximum code rate for QAM structures}$]]  section of the "Information Theory" book.<br>
	*Also, this limit curve has now been nearly reached with QAM methods and very long channel codes, without ever being exceeded.}}<br>		*Also, this limit curve has now been nearly reached with QAM methods and very long channel codes, without ever being exceeded.}}<br>

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	{{Display}}		{{Display}}
			[[de:Kanalcodierung/Informationstheoretische_Grenzen_der_Kanalcodierung]]

Guenter at 10:12, 22 November 2022

2022-11-22T10:12:08Z

← Older revision		Revision as of 12:12, 22 November 2022
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	#For all binary codes holds per se  $0 < R ≤ 1$. Only with non-binary codes  ⇒   rates  $R > 1$  are possible.  For example,  the maximum possible code rate of a quaternary code:  $R = \log_2 \, M_y = \log_2 \, 4 = 2$.<br>		#For all binary codes holds per se  $0 < R ≤ 1$. Only with non-binary codes  ⇒   rates  $R > 1$  are possible.  For example,  the maximum possible code rate of a quaternary code:  $R = \log_2 \, M_y = \log_2 \, 4 = 2$.<br>
	#All one-dimensional modulation types – i.e., those methods that use only the in-phase– or only the quadrature component such as  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#On.E2.80.93off_keying_.282.E2.80.93ASK.29\|$\text{2–ASK}$]],  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#Binary_phase_shift_keying_.28BPSK.29 \|$\text{BPSK}$]]  and  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Non-Coherent_Demodulation#Non-coherent_demodulation_of_binary_FSK_.282.E2.80.93FSK.29\|$\text{2–FSK}$]]  must be in the blue area of the present graphic.<br>		#All one-dimensional modulation types – i.e., those methods that use only the in-phase– or only the quadrature component such as  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#On.E2.80.93off_keying_.282.E2.80.93ASK.29\|$\text{2–ASK}$]],  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#Binary_phase_shift_keying_.28BPSK.29 \|$\text{BPSK}$]]  and  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Non-Coherent_Demodulation#Non-coherent_demodulation_of_binary_FSK_.282.E2.80.93FSK.29\|$\text{2–FSK}$]]  must be in the blue area of the present graphic.<br>
	#As shown in the chapter  [[Information_Theory/AWGN_Channel_Capacity_for_Discrete_Input#Maximum_code_rate_for_QAM_structures\|~~$\text{~~Maximum code rate for QAM structures}$]],  there is a  "friendlier"  limit curve for two-dimensional modulation types such as the  [[Modulation_Methods/Quadrature_Amplitude_Modulation\|~~$\text{~~Quadrature Amplitude Modulation}$]].		#As shown in the chapter  [[Information_Theory/AWGN_Channel_Capacity_for_Discrete_Input#Maximum_code_rate_for_QAM_structures\|"Maximum code rate for QAM structures"]],  there is a  "friendlier"  limit curve for two-dimensional modulation types such as the  [[Modulation_Methods/Quadrature_Amplitude_Modulation\|"Quadrature Amplitude Modulation"]].

	== AWGN channel capacity for binary input signals ==		== AWGN channel capacity for binary input signals ==
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	#These codes,  invented by  [http://perso.telecom-bretagne.eu/claudeberrou/ $\text{Claude Berrou}$]  around 1990,  can be decoded iteratively.  The  (red)  marks are each less than  $1 \, \rm dB$  from the Shannon bound.<br>		#These codes,  invented by  [http://perso.telecom-bretagne.eu/claudeberrou/ $\text{Claude Berrou}$]  around 1990,  can be decoded iteratively.  The  (red)  marks are each less than  $1 \, \rm dB$  from the Shannon bound.<br>
	#Similar behavior is shown by  [[Channel_Coding/The_Basics_of_Low-Density_Parity_Check_Codes\|$\text{LDPC codes}$]]  ("Low Density Parity–check Codes")  marked by white rectangles,  which have been used since in 2006 on  [https://en.wikipedia.org/wiki/DVB-S2 $\text{DVB–S(2)}$]  ("Digital Video Broadcast over Satellite").		#Similar behavior is shown by  [[Channel_Coding/The_Basics_of_Low-Density_Parity_Check_Codes\|$\text{LDPC codes}$]]  ("Low Density Parity–check Codes")  marked by white rectangles,  which have been used since in 2006 on  [https://en.wikipedia.org/wiki/DVB-S2 $\text{DVB–S(2)}$]  ("Digital Video Broadcast over Satellite").
	#These are well suited for iterative decoding using  "factor–graphs"    and   "exit charts"  due to the sparse occupancy of the parity-check matrix  $\boldsymbol {\rm H}$  (with "ones").  See [Hag02]<ref name='Hag02'>Hagenauer, J.: The Turbo Principle in Mobile Communications.  In: Int. Symp. on Information Theory and Its Applications, Oct.2010, [http://wwwmayr.in.tum.de/konferenzen/Jass05/courses/4/papers/prof_hagenauer.pdf PDF document.]</ref><br>		#These are well suited for iterative decoding using  "factor–graphs"    and   "exit charts"  due to the sparse occupancy of the parity-check matrix  $\boldsymbol {\rm H}$  (with "ones").  See [Hag02]<ref name='Hag02'>Hagenauer, J.: The Turbo Principle in Mobile Communications.  In: Int. Symp. on Information Theory and Its Applications, Oct.2010,  [http://wwwmayr.in.tum.de/konferenzen/Jass05/courses/4/papers/prof_hagenauer.pdf $\text{PDF document}$.]</ref><br>
	#The black dots mark the CCSDS specified  [[Channel_Coding/The_Basics_of_Low-Density_Parity_Check_Codes\|$\text{LDPC codes}$]],  all of which assume a constant number of information bits  $(k = 16384)$.		#The black dots mark the CCSDS specified  [[Channel_Coding/The_Basics_of_Low-Density_Parity_Check_Codes\|$\text{LDPC codes}$]],  all of which assume a constant number of information bits  $(k = 16384)$.
	#In contrast,  for all white rectangles,  the code length  $n = 64800$  is constant,  while the number  $k$  of information bits changes according to the rate  $R = k/n$.<br>		#In contrast,  for all white rectangles,  the code length  $n = 64800$  is constant,  while the number  $k$  of information bits changes according to the rate  $R = k/n$.<br>

Guenter at 10:03, 22 November 2022

2022-11-22T10:03:16Z

← Older revision		Revision as of 12:03, 22 November 2022
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	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Definition:}$  The  »'''channel capacity'''«  introduced by Shannon gives the maximum mutual information  $I(x; y)$  between the input variable  $x$  and ~~the~~ output variable  $y$:		$\text{Definition:}$  The  »'''channel capacity'''«  introduced by Shannon gives the maximum mutual information  $I(x; y)$  between the input variable  $x$  and output variable  $y$:
	::<math>C = \max_{{{\rm Pr}(x_i)}} \hspace{0.1cm} I(X; Y) \hspace{0.05cm}.</math>		::<math>C = \max_{{{\rm Pr}(x_i)}} \hspace{0.1cm} I(X; Y) \hspace{0.05cm}.</math>

	The pseudo–unit  "bit/channel use"  must be added.}}<br>		*The pseudo–unit  "bit/channel use"  must be added.}}<br>

	:Since the maximization of the mutual information  $I(x; y)$  must be done over all possible  (discrete)  input distributions  ${\rm Pr}(x_i)$,		:Since the maximization of the mutual information  $I(x; y)$  must be done over all possible  (discrete)  input distributions  ${\rm Pr}(x_i)$,

Guenter at 10:00, 22 November 2022

2022-11-22T10:00:26Z

Hwang at 15:28, 18 November 2022

2022-11-18T15:28:08Z

← Older revision		Revision as of 17:28, 18 November 2022
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	#These codes,  invented by  [http://perso.telecom-bretagne.eu/claudeberrou/ $\text{Claude Berrou}$]  around 1990,  can be decoded iteratively.  The  (red)  marks are each less than  $1 \, \rm dB$  from the Shannon bound.<br>		#These codes,  invented by  [http://perso.telecom-bretagne.eu/claudeberrou/ $\text{Claude Berrou}$]  around 1990,  can be decoded iteratively.  The  (red)  marks are each less than  $1 \, \rm dB$  from the Shannon bound.<br>
	#Similar behavior is shown by  [[Channel_Coding/The_Basics_of_Low-Density_Parity_Check_Codes\|$\text{LDPC codes}$]]  ("Low Density Parity–check Codes")  marked by white rectangles,  which have been used since in 2006 on  [https://en.wikipedia.org/wiki/DVB-S2 $\text{DVB–S(2)}$]  ("Digital Video Broadcast over Satellite").		#Similar behavior is shown by  [[Channel_Coding/The_Basics_of_Low-Density_Parity_Check_Codes\|$\text{LDPC codes}$]]  ("Low Density Parity–check Codes")  marked by white rectangles,  which have been used since in 2006 on  [https://en.wikipedia.org/wiki/DVB-S2 $\text{DVB–S(2)}$]  ("Digital Video Broadcast over Satellite").
	#These are well suited for iterative decoding using  "factor–graphs"    and   "exit charts"  due to the sparse occupancy of the parity-check matrix  $\boldsymbol {\rm H}$  (with "ones").  See [Hag02]<ref name='Hag02'>Hagenauer, J.: The Turbo Principle in Mobile Communications.  In: Int. Symp. on Information Theory and Its Applications, Oct.2010, [http://wwwmayr.in.tum.de/konferenzen/Jass05/courses/4/papers/prof_hagenauer.pdf PDF ~~dokument~~.]</ref><br>		#These are well suited for iterative decoding using  "factor–graphs"    and   "exit charts"  due to the sparse occupancy of the parity-check matrix  $\boldsymbol {\rm H}$  (with "ones").  See [Hag02]<ref name='Hag02'>Hagenauer, J.: The Turbo Principle in Mobile Communications.  In: Int. Symp. on Information Theory and Its Applications, Oct.2010, [http://wwwmayr.in.tum.de/konferenzen/Jass05/courses/4/papers/prof_hagenauer.pdf PDF document.]</ref><br>
	#The black dots mark the CCSDS specified  [[Channel_Coding/The_Basics_of_Low-Density_Parity_Check_Codes\|$\text{LDPC codes}$]],  all of which assume a constant number of information bits  $(k = 16384)$.		#The black dots mark the CCSDS specified  [[Channel_Coding/The_Basics_of_Low-Density_Parity_Check_Codes\|$\text{LDPC codes}$]],  all of which assume a constant number of information bits  $(k = 16384)$.
	#In contrast,  for all white rectangles,  the code length  $n = 64800$  is constant,  while the number  $k$  of information bits changes according to the rate  $R = k/n$.<br>		#In contrast,  for all white rectangles,  the code length  $n = 64800$  is constant,  while the number  $k$  of information bits changes according to the rate  $R = k/n$.<br>

Hwang at 15:27, 18 November 2022

2022-11-18T15:27:48Z

← Older revision		Revision as of 17:27, 18 November 2022
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	#These codes,  invented by  [http://perso.telecom-bretagne.eu/claudeberrou/ $\text{Claude Berrou}$]  around 1990,  can be decoded iteratively.  The  (red)  marks are each less than  $1 \, \rm dB$  from the Shannon bound.<br>		#These codes,  invented by  [http://perso.telecom-bretagne.eu/claudeberrou/ $\text{Claude Berrou}$]  around 1990,  can be decoded iteratively.  The  (red)  marks are each less than  $1 \, \rm dB$  from the Shannon bound.<br>
	#Similar behavior is shown by  [[Channel_Coding/The_Basics_of_Low-Density_Parity_Check_Codes\|$\text{LDPC codes}$]]  ("Low Density Parity–check Codes")  marked by white rectangles,  which have been used since in 2006 on  [https://en.wikipedia.org/wiki/DVB-S2 $\text{DVB–S(2)}$]  ("Digital Video Broadcast over Satellite").		#Similar behavior is shown by  [[Channel_Coding/The_Basics_of_Low-Density_Parity_Check_Codes\|$\text{LDPC codes}$]]  ("Low Density Parity–check Codes")  marked by white rectangles,  which have been used since in 2006 on  [https://en.wikipedia.org/wiki/DVB-S2 $\text{DVB–S(2)}$]  ("Digital Video Broadcast over Satellite").
	#These are well suited for iterative decoding using  "factor–graphs"    and   "exit charts"  due to the sparse occupancy of the parity-check matrix  $\boldsymbol {\rm H}$  (with "ones").  See [Hag02]<ref name='Hag02'>Hagenauer, J.: The Turbo Principle in Mobile Communications.  In: Int. Symp. on Information Theory and Its Applications, Oct.2010, [http://wwwmayr.in.tum.de/konferenzen/Jass05/courses/4/papers/prof_hagenauer.pdf PDF~~–Dokument~~.]</ref><br>		#These are well suited for iterative decoding using  "factor–graphs"    and   "exit charts"  due to the sparse occupancy of the parity-check matrix  $\boldsymbol {\rm H}$  (with "ones").  See [Hag02]<ref name='Hag02'>Hagenauer, J.: The Turbo Principle in Mobile Communications.  In: Int. Symp. on Information Theory and Its Applications, Oct.2010, [http://wwwmayr.in.tum.de/konferenzen/Jass05/courses/4/papers/prof_hagenauer.pdf PDF dokument.]</ref><br>
	#The black dots mark the CCSDS specified  [[Channel_Coding/The_Basics_of_Low-Density_Parity_Check_Codes\|$\text{LDPC codes}$]],  all of which assume a constant number of information bits  $(k = 16384)$.		#The black dots mark the CCSDS specified  [[Channel_Coding/The_Basics_of_Low-Density_Parity_Check_Codes\|$\text{LDPC codes}$]],  all of which assume a constant number of information bits  $(k = 16384)$.
	#In contrast,  for all white rectangles,  the code length  $n = 64800$  is constant,  while the number  $k$  of information bits changes according to the rate  $R = k/n$.<br>		#In contrast,  for all white rectangles,  the code length  $n = 64800$  is constant,  while the number  $k$  of information bits changes according to the rate  $R = k/n$.<br>

Hwang at 15:27, 18 November 2022

2022-11-18T15:27:19Z

← Older revision		Revision as of 17:27, 18 November 2022
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	*For two-dimensional methods such as QAM and multilevel PSK, on the other hand, the "Gaussian (complex)" limit curve applies, which is drawn here as a dashed line and always lies to the left of the solid curve.		*For two-dimensional methods such as QAM and multilevel PSK, on the other hand, the "Gaussian (complex)" limit curve applies, which is drawn here as a dashed line and always lies to the left of the solid curve.
	*For more details, see the  [[Information_Theory/AWGN_Channel_Capacity_for_Discrete_Input#Maximum_code_rate_for_QAM_structures\|$\text{Maximum code rate for QAM structures}$]]  section of the "Information Theory" book.<br>		*For more details, see the  [[Information_Theory/AWGN_Channel_Capacity_for_Discrete_Input#Maximum_code_rate_for_QAM_structures\|$\text{Maximum code rate for QAM structures}$]]  section of the "Information Theory" book.<br>
	*Also, this limit curve has now been nearly reached with QAM procedures and very long channel codes, without ever being exceeded.}}<br>		*Also, this limit curve has now been nearly reached with QAM procedures '''KORREKTUR: methods''' and very long channel codes, without ever being exceeded.}}<br>

	== Exercises for the chapter ==		== Exercises for the chapter ==

Hwang at 15:18, 18 November 2022

2022-11-18T15:18:34Z

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	#If  $10 \cdot \lg \, E_{\rm B}/N_0 < -1.59 \, \rm dB$,  error-free decoding is impossible in principle.  If the code rate  $R = 0.5$,  then  $10 \cdot \lg \, E_{\rm B}/N_0 > 0 \, \rm dB$  must be   ⇒  $E_{\rm B} > N_0$.<br>		#If  $10 \cdot \lg \, E_{\rm B}/N_0 < -1.59 \, \rm dB$,  error-free decoding is impossible in principle.  If the code rate  $R = 0.5$,  then  $10 \cdot \lg \, E_{\rm B}/N_0 > 0 \, \rm dB$  must be   ⇒  $E_{\rm B} > N_0$.<br>
	#For all binary codes holds per se  $0 < R ≤ 1$. Only with non-binary codes  ⇒   rates  $R > 1$  are possible.  For example,  the maximum possible code rate of a quaternary code:  $R = \log_2 \, M_y = \log_2 \, 4 = 2$.<br>		#For all binary codes holds per se  $0 < R ≤ 1$. Only with non-binary codes  ⇒   rates  $R > 1$  are possible.  For example,  the maximum possible code rate of a quaternary code:  $R = \log_2 \, M_y = \log_2 \, 4 = 2$.<br>
	#All one-dimensional modulation types – i.e., those methods that use only the in-phase– or only the quadrature component such as  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#On.E2.80.93off_keying_.282.E2.80.93ASK.29\|$\text{2–ASK}$]],  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#Binary_phase_shift_keying_.28BPSK.29 \|$\text{BPSK}$]]  and  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Non-Coherent_Demodulation#Non-~~Coh.C3.A4rente_Demodulation_of_bin.C3.A4rer_FSK_~~.282.E2.80.93FSK.29\|$\text{2–FSK}$]]  must be in the blue area of the present graphic.<br>		#All one-dimensional modulation types – i.e., those methods that use only the in-phase– or only the quadrature component such as  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#On.E2.80.93off_keying_.282.E2.80.93ASK.29\|$\text{2–ASK}$]],  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#Binary_phase_shift_keying_.28BPSK.29 \|$\text{BPSK}$]]  and  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Non-Coherent_Demodulation#Non-coherent_demodulation_of_binary_FSK_.282.E2.80.93FSK.29\|$\text{2–FSK}$]]  must be in the blue area of the present graphic.<br>
	#As shown in the chapter  [[Information_Theory/AWGN_Channel_Capacity_for_Discrete_Input#Maximum_code_rate_for_QAM_structures\|$\text{Maximum code rate for QAM structures}$]],  there is a  "friendlier"  limit curve for two-dimensional modulation types such as the  [[Modulation_Methods/Quadrature_Amplitude_Modulation\|$\text{Quadrature Amplitude Modulation}$]].		#As shown in the chapter  [[Information_Theory/AWGN_Channel_Capacity_for_Discrete_Input#Maximum_code_rate_for_QAM_structures\|$\text{Maximum code rate for QAM structures}$]],  there is a  "friendlier"  limit curve for two-dimensional modulation types such as the  [[Modulation_Methods/Quadrature_Amplitude_Modulation\|$\text{Quadrature Amplitude Modulation}$]].

Hwang at 15:16, 18 November 2022

2022-11-18T15:16:01Z

← Older revision		Revision as of 17:16, 18 November 2022
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	*However,  a comparison of different coding methods at constant  "energy per transmitted symbol"   ⇒   $E_{\rm S}$  is not fair.		*However,  a comparison of different coding '''KORREKTUR: encoding''' methods at constant  "energy per transmitted symbol"   ⇒   $E_{\rm S}$  is not fair.

	*Rather,  for this comparison,  the  "energy per source bit"   ⇒   $E_{\rm B}$  should be fixed.  The following relationships apply:		*Rather,  for this comparison,  the  "energy per source bit"   ⇒   $E_{\rm B}$  should be fixed.  The following relationships apply:

Hwang at 15:15, 18 November 2022

2022-11-18T15:15:02Z

← Older revision		Revision as of 17:15, 18 November 2022
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	::<math>P_X = \frac{E_{\rm S}}{T_{\rm S} } \hspace{0.05cm}, \hspace{0.4cm} P_N = \frac{N_0}{2T_{\rm S} }\hspace{0.05cm}. </math>		::<math>P_X = \frac{E_{\rm S}}{T_{\rm S} } \hspace{0.05cm}, \hspace{0.4cm} P_N = \frac{N_0}{2T_{\rm S} }\hspace{0.05cm}. </math>

	*Thus, the AWGN channel capacity can also be expressed by the  »'''transmit energy per symbol'''«  $(E_{\rm S})$  and the  »'''noise power density'''«  $(N_0)$:		*Thus, the AWGN channel capacity can also be expressed by the  »'''transmit '''KORREKTUR: transmitted?''' energy per symbol'''«  $(E_{\rm S})$  and the  »'''noise power density'''«  $(N_0)$:

	::<math>C = {1}/{2 } \cdot {\rm log_2 } \hspace{0.05cm}\left ( 1 + {2 E_{\rm S}}/{N_0 } \right )\hspace{0.05cm}, \hspace{1.9cm}		::<math>C = {1}/{2 } \cdot {\rm log_2 } \hspace{0.05cm}\left ( 1 + {2 E_{\rm S}}/{N_0 } \right )\hspace{0.05cm}, \hspace{1.9cm}

← Older revision		Revision as of 12:00, 22 November 2022
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	Notes:		Notes:
	*The statement  "The error probability approaches zero"  is not identical with the statement  "The transmission is error-free".  Example:   For an infinitely long sequence,  finitely many symbols are ~~corrupted '''KORREKTUR:~~ falsified~~'''~~.		*The statement  "The error probability approaches zero"  is not identical with the statement  "The transmission is error-free".  Example:   For an infinitely long sequence,  finitely many symbols are falsified.

	*For some channels,  even with  $R=C$  the error probability still goes towards zero  (but not for all).		*For some channels,  even with  $R=C$  the error probability still goes towards zero  (but not for all).
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	::<math>P_X = \frac{E_{\rm S}}{T_{\rm S} } \hspace{0.05cm}, \hspace{0.4cm} P_N = \frac{N_0}{2T_{\rm S} }\hspace{0.05cm}. </math>		::<math>P_X = \frac{E_{\rm S}}{T_{\rm S} } \hspace{0.05cm}, \hspace{0.4cm} P_N = \frac{N_0}{2T_{\rm S} }\hspace{0.05cm}. </math>

	*Thus, the AWGN channel capacity can also be expressed by the  »~~'''transmit '''KORREKTUR:~~ transmitted~~?'''~~ energy per symbol'''«  $(E_{\rm S})$  and the  »'''noise power density'''«  $(N_0)$:		*Thus, the AWGN channel capacity can also be expressed by the  »transmitted energy per symbol'''«  $(E_{\rm S})$  and the  »'''noise power density'''«  $(N_0)$:

	::<math>C = {1}/{2 } \cdot {\rm log_2 } \hspace{0.05cm}\left ( 1 + {2 E_{\rm S}}/{N_0 } \right )\hspace{0.05cm}, \hspace{1.9cm}		::<math>C = {1}/{2 } \cdot {\rm log_2 } \hspace{0.05cm}\left ( 1 + {2 E_{\rm S}}/{N_0 } \right )\hspace{0.05cm}, \hspace{1.9cm}
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	*However,  a comparison of different ~~coding '''KORREKTUR:~~ encoding~~'''~~ methods at constant  "energy per transmitted symbol"   ⇒   $E_{\rm S}$  is not fair.		*However,  a comparison of different encoding methods at constant  "energy per transmitted symbol"   ⇒   $E_{\rm S}$  is not fair.

	*Rather,  for this comparison,  the  "energy per source bit"   ⇒   $E_{\rm B}$  should be fixed.  The following relationships apply:		*Rather,  for this comparison,  the  "energy per source bit"   ⇒   $E_{\rm B}$  should be fixed.  The following relationships apply:
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	*For two-dimensional methods such as QAM and multilevel PSK, on the other hand, the "Gaussian (complex)" limit curve applies, which is drawn here as a dashed line and always lies to the left of the solid curve.		*For two-dimensional methods such as QAM and multilevel PSK, on the other hand, the "Gaussian (complex)" limit curve applies, which is drawn here as a dashed line and always lies to the left of the solid curve.
	*For more details, see the  [[Information_Theory/AWGN_Channel_Capacity_for_Discrete_Input#Maximum_code_rate_for_QAM_structures\|$\text{Maximum code rate for QAM structures}$]]  section of the "Information Theory" book.<br>		*For more details, see the  [[Information_Theory/AWGN_Channel_Capacity_for_Discrete_Input#Maximum_code_rate_for_QAM_structures\|$\text{Maximum code rate for QAM structures}$]]  section of the "Information Theory" book.<br>
	*Also, this limit curve has now been nearly reached with QAM ~~procedures '''KORREKTUR:~~ methods~~'''~~ and very long channel codes, without ever being exceeded.}}<br>		*Also, this limit curve has now been nearly reached with QAM methods and very long channel codes, without ever being exceeded.}}<br>

	== Exercises for the chapter ==		== Exercises for the chapter ==