Examples of Communication Systems/xDSL as Transmission Technology - Revision history

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2026-03-16T13:46:54Z

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	*With the basic transmission pulse  $g_s(t)$  is then valid in the two branches of the block diagram:		*With the basic transmission pulse  $g_s(t)$  is then valid in the two branches of the block diagram:
	:$$ s_{\rm I}(t) = \sum_{n = - \infty}^{+\infty}a_{\rm		:$$ s_{\rm I}(t) = \sum_{n = - \infty}^{+\infty}a_{\rm I\hspace{0.03cm}\it n} \cdot g_s (t - n \cdot T)\hspace{0.05cm},\hspace{0.5cm}s_{\rm Q}(t) = \sum_{n = - \infty}^{+\infty}a_{\rm Q\hspace{0.03cm}\it n} \cdot g_s (t - n \cdotT)\hspace{0.05cm}.$$
	I\hspace{0.03cm}\it n} \cdot g_s (t - n \cdot T)\hspace{0.05cm},\hspace{0.5cm}
	s_{\rm Q}(t) = \sum_{n = - \infty}^{+\infty}a_{\rm
	Q\hspace{0.03cm}\it n} \cdot g_s (t - n \~~cdot~~
	T)\hspace{0.05cm}.$$
	:<u>Note:</u>  Because of the redundancy-free conversion to a higher-level code,  the symbol duration  $T$  of these signals is greater by a factor  $b$  than the bit duration  $T_{\rm B}$  of the binary input sequence.  In the drawn example  $($16-QAM$)$,  $T = 4 ⋅ T_{\rm B}$.		:<u>Note:</u>  Because of the redundancy-free conversion to a higher-level code,  the symbol duration  $T$  of these signals is greater by a factor  $b$  than the bit duration  $T_{\rm B}$  of the binary input sequence.  In the drawn example  $($16-QAM$)$,  $T = 4 ⋅ T_{\rm B}$.
	*The  »'''QAM transmitted signal'''«  $s(t)$  is then the sum of the two partial signals  $ s_{\rm I}(t)$  and  $ s_{\rm Q}(t)$,  multiplied by  "cosine"  resp.  "minus-sine"  $($possibly followed by a band limit to prevent interference to adjacent bands, as indicated in the graph below$)$:		*The  »'''QAM transmitted signal'''«  $s(t)$  is then the sum of the two partial signals  $ s_{\rm I}(t)$  and  $ s_{\rm Q}(t)$,  multiplied by  "cosine"  resp.  "minus-sine"  $($possibly followed by a band limit to prevent interference to adjacent bands, as indicated in the graph below$)$:
	:$$s(t) = s_{\rm I}(t) \cdot \cos (2 \pi f_{\rm T}\,t) - s_{\rm Q}(t) \cdot \sin (2 \pi f_{\rm T}\,t)		:$$s(t) = s_{\rm I}(t) \cdot \cos (2 \pi f_{\rm T}\,t) - s_{\rm Q}(t) \cdot \sin (2 \pi f_{\rm T}\,t)\hspace{0.05cm}. $$
	\hspace{0.05cm}. $$
	*The two branches  $(\rm I$  and  $\rm Q)$  can be considered as two completely separate  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#M.E2.80.93level_amplitude_shift_keying_.28M.E2.80.93ASK.29 \|"M–level ASK systems"]]  which do not interfere with each other as long as all components are optimally designed.  This means at the same time:		*The two branches  $(\rm I$  and  $\rm Q)$  can be considered as two completely separate  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#M.E2.80.93level_amplitude_shift_keying_.28M.E2.80.93ASK.29 \|"M–level ASK systems"]]  which do not interfere with each other as long as all components are optimally designed.  This means at the same time:

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	*in OFDM,  the above drawn spectrum  $S(f)$  in reality describes an  "equivalent low-pass spectrum"  $S_{\rm TP}(f)$  and still the shift around a carrier frequency  $f_{\rm T}$  has to be considered:		*in OFDM,  the above drawn spectrum  $S(f)$  in reality describes an  "equivalent low-pass spectrum"  $S_{\rm TP}(f)$  and still the shift around a carrier frequency  $f_{\rm T}$  has to be considered:

	:$$S_{\rm TP}(f ) = \sum_{k = 1}^{255} D_k \cdot \delta (f - k \cdot f_0)\hspace{0.3cm}\Rightarrow \hspace{0.3cm}		:$$S_{\rm TP}(f ) = \sum_{k = 1}^{255} D_k \cdot \delta (f - k \cdot f_0)\hspace{0.3cm}\Rightarrow \hspace{0.3cm}S(f) = \frac{1}{2} \big [ S_{\rm TP}(f - f_{\rm T}) + S^*_{\rm TP}(-(f + f_{\rm T}))\big ]\hspace{0.05cm};$$
	S(f) = \frac{1}{2} \big [ S_{\rm TP}(f - f_{\rm T}) + S^*_{\rm TP}(-(f + f_{\rm T}))\big ]
	\hspace{0.05cm};$$

	*in DMT,  on the other hand,  the components at negative frequencies must still be taken into account,  which are to be weighted with the conjugate-complex spectral coefficients:		*in DMT,  on the other hand,  the components at negative frequencies must still be taken into account,  which are to be weighted with the conjugate-complex spectral coefficients:

	:$$S(f ) = \sum_{k = 1}^{255} \big [ D_k \cdot \delta (f - k \cdot f_0) + D^*_k \cdot \delta (f + k \cdot f_0) \big ]		:$$S(f ) = \sum_{k = 1}^{255} \big [ D_k \cdot \delta (f - k \cdot f_0) + D^*_k \cdot \delta (f + k \cdot f_0) \big ]\hspace{0.05cm}.$$
	\hspace{0.05cm}.$$

	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=

Maintenance script: Add German interlanguage link

2026-03-16T12:26:15Z

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	{{Display}}		{{Display}}
			[[de:Beispiele_von_Nachrichtensystemen/xDSL_als_Übertragungstechnik]]

Guenter at 15:05, 24 April 2023

2023-04-24T15:05:38Z

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	[[File:EN_Bei_T_2_3_S9d_ganz_neuV8.png\|right\|frame\|Discrete multitone transmission system]]		[[File:EN_Bei_T_2_3_S9d_ganz_neuV8.png\|right\|frame\|Discrete multitone transmission system]]
	[[File:~~P_ID1951__Bei_T_2_3_S9b_v3~~.png\|right\|frame\|DMT transmitter and receiver ~~'''Korrektur''' das Bild muss ich noch machen~~]]		[[File:EN_Bei_T_2_3_S9d_60Prozent.png\|right\|frame\|DMT transmitter and receiver ]]


	*To perform a DMT modulation,  the transmitter accumulates a block of input bits in a data buffer to be transmitted as one frame.		*To perform a DMT modulation,  the transmitter accumulates a block of input bits in a data buffer to be transmitted as one frame.

Guenter at 17:26, 25 March 2023

2023-03-25T17:26:44Z

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	#The symbol error probability $p$  depends on the Euclidean distance of the two points and the AWGN noise power density  $N_0$.		#The symbol error probability $p$  depends on the Euclidean distance of the two points and the AWGN noise power density  $N_0$.
	#A distortion to the more distant blue point instead of one of the two neighboring brown points is rather unlikely with Gaussian noise.		#A distortion to the more distant blue point instead of one of the two neighboring brown points is rather unlikely with Gaussian noise.
	[[File:~~EN Bei T 2 3 S6b neu~~.png\|right\|frame\|Signal space constellation:  "$\rm 16\hspace{0.05cm}–\hspace{-0.02cm}QAM$";  left: ITU propsal,  right:  Gray encoding~~. '''KORREKTUR: Symbolfehler, Bitfehler'''~~]]		[[File:EN_Bei_T_2_3_S6b_neu_v2.png\|right\|frame\|Signal space constellation:  "$\rm 16\hspace{0.05cm}–\hspace{-0.02cm}QAM$";  left: ITU propsal,  right:  Gray encoding]]

	An analysis shows:		An analysis shows:

Guenter at 09:21, 22 March 2023

2023-03-22T09:21:52Z

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	[[File:EN_Bei_T_2_3_S7d_neu.png\|right\|frame\|Reference model for  ADSL–CAP]]		[[File:EN_Bei_T_2_3_S7d_neu.png\|right\|frame\|Reference model for  ADSL–CAP]]

	*Digital filtering is used instead of multiplication by cosine and minus sine.  $g_{\rm I}(t)$  and  $g_{\rm Q}(t)$  are impulse responses of two transversal ~~bandpass '''KORREKTUR:~~ band-pass~~'''~~ filters with same amplitude characteristic,  phase-shifted by  $π/2$.		*Digital filtering is used instead of multiplication by cosine and minus sine.  $g_{\rm I}(t)$  and  $g_{\rm Q}(t)$  are impulse responses of two transversal band-pass filters with same amplitude characteristic,  phase-shifted by  $π/2$.

	*Both are orthogonal to each other,  which means that the integral of the product  $g_{\rm I}(t) · g_{\rm Q}(t)$  over a symbol duration gives zero.		*Both are orthogonal to each other,  which means that the integral of the product  $g_{\rm I}(t) · g_{\rm Q}(t)$  over a symbol duration gives zero.

Guenter at 09:20, 22 March 2023

2023-03-22T09:20:28Z

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	$\text{Example 1:}$		$\text{Example 1:}$
	An important QAM parameter is the bit number  $b$  processed to the amplitude coefficient pair  $(a_{\rm I}, a_{\rm Q})$.  Here  $b$  is always an even number.		An important QAM parameter is the bit number  $b$  processed to the amplitude coefficient pair  $(a_{\rm I}, a_{\rm Q})$.  Here  $b$  is always an even number.
	[[File:EN_Bei_T_2_3_S6a_v2.png\|right\|frame\|Signal space constellations  "$\rm 4\hspace{0.05cm}–\hspace{-0.02cm}QAM$"  $($left$)$  and  "$\rm 16\hspace{0.05cm}–\hspace{-0.02cm}QAM$"  $($right$)$		[[File:EN_Bei_T_2_3_S6a_v2.png\|right\|frame\|Signal space constellations  "$\rm 4\hspace{0.05cm}–\hspace{-0.02cm}QAM$"  $($left$)$  and  "$\rm 16\hspace{0.05cm}–\hspace{-0.02cm}QAM$"  $($right$)$]]
	'''Korrektur''': Jiwoo, ich habe die gelben Punkte durch braune Punkte ersetzt, weil meine schlechten Auge diese nicht erkennen konnten. Könntest u bitte den Text kontrollieren und mit der deutschen Version vergleichen, ob jetzt wirklich alle gelben Punkte braun sind.<br><br> Das gleiche gilt für Example 3.]]

	*If  $b = 2$,  both  $a_{\rm I}$  and  $a_{\rm Q}$  can only take the values  $±1$  resulting in the  $\rm 4\hspace{0.05cm}–\hspace{-0.02cm}QAM$  corresponding to the left constellation.		*If  $b = 2$,  both  $a_{\rm I}$  and  $a_{\rm Q}$  can only take the values  $±1$  resulting in the  $\rm 4\hspace{0.05cm}–\hspace{-0.02cm}QAM$  corresponding to the left constellation.

Guenter at 21:03, 21 March 2023

2023-03-21T21:03:05Z

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	*QAM is a  "single carrier modulation method"  around the carrier frequency  $f_{\rm T}$.  First,  a blockwise serial/parallel conversion of the bit stream and signal space assignment is performed.		*QAM is a  "single carrier modulation method"  around the carrier frequency  $f_{\rm T}$.  First,  a blockwise serial/parallel conversion of the bit stream and signal space assignment is performed.

	*From each  $b$  binary symbols,  two ~~multilevel '''KORREKTUR:~~ multi-level~~'''~~ amplitude coefficients  $a_{{\rm I}n}$  and  $a_{{\rm Q}n}$  are derived  $($"in-phase component"  and  "quadrature component"$)$.  Both coefficients can each take one of  $M = 2^{b/2}$  possible amplitude values.		*From each  $b$  binary symbols,  two multi-level amplitude coefficients  $a_{{\rm I}n}$  and  $a_{{\rm Q}n}$  are derived  $($"in-phase component"  and  "quadrature component"$)$.  Both coefficients can each take one of  $M = 2^{b/2}$  possible amplitude values.

	*The example considered here applies to the  $\text{16-QAM}$  with  $b = M = 4$  and correspondingly sixteen  signal space points. For a   $\text{256-QAM}$  would hold:   $b = 8$  and  $M = 16$  $(2^b = M^2 = 256)$.		*The example considered here applies to the  $\text{16-QAM}$  with  $b = M = 4$  and correspondingly sixteen  signal space points. For a   $\text{256-QAM}$  would hold:   $b = 8$  and  $M = 16$  $(2^b = M^2 = 256)$.

Hwang at 19:53, 21 March 2023

2023-03-21T19:53:04Z

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	*After  $68$  data frames,  one synchronization frame is inserted each.		*After  $68$  data frames,  one synchronization frame is inserted each.

	*Due to the  "cyclic prefix"  $($see chapter  [[Examples_of_Communication_Systems/Methods_to_Reduce_the_Bit_Error_Rate_in_DSL#~~Insert.C3.BCgen_of_Guard.E2.80.93Interval_and_cyclic_Pr. C3.A4fix~~\|"Insertion of Guard–Interval and Cyclic Prefix"]]$)$  the symbol duration  $T = 1/f_0$  must still be shortened by the factor  $16/17$.		*Due to the  "cyclic prefix"  $($see chapter  [[Examples_of_Communication_Systems/Methods_to_Reduce_the_Bit_Error_Rate_in_DSL#Inserting_guard_interval_and_cyclic_prefix\|"Insertion of Guard–Interval and Cyclic Prefix"]]$)$  the symbol duration  $T = 1/f_0$  must still be shortened by the factor  $16/17$.

	*Thus,  the  »'''basic frequency'''«  of DMT is		*Thus,  the  »'''basic frequency'''«  of DMT is

Hwang at 19:44, 21 March 2023

2023-03-21T19:44:54Z

← Older revision		Revision as of 21:44, 21 March 2023
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	[[File:EN_Bei_T_2_3_S7d_neu.png\|right\|frame\|Reference model for  ADSL–CAP]]		[[File:EN_Bei_T_2_3_S7d_neu.png\|right\|frame\|Reference model for  ADSL–CAP]]

	*Digital filtering is used instead of multiplication by cosine and minus sine.  $g_{\rm I}(t)$  and  $g_{\rm Q}(t)$  are impulse responses of two transversal bandpass filters with same amplitude characteristic,  phase-shifted by  $π/2$.		*Digital filtering is used instead of multiplication by cosine and minus sine.  $g_{\rm I}(t)$  and  $g_{\rm Q}(t)$  are impulse responses of two transversal bandpass '''KORREKTUR: band-pass''' filters with same amplitude characteristic,  phase-shifted by  $π/2$.

	*Both are orthogonal to each other,  which means that the integral of the product  $g_{\rm I}(t) · g_{\rm Q}(t)$  over a symbol duration gives zero.		*Both are orthogonal to each other,  which means that the integral of the product  $g_{\rm I}(t) · g_{\rm Q}(t)$  over a symbol duration gives zero.

Hwang at 19:43, 21 March 2023

2023-03-21T19:43:56Z

← Older revision		Revision as of 21:43, 21 March 2023
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	#The symbol error probability $p$  depends on the Euclidean distance of the two points and the AWGN noise power density  $N_0$.		#The symbol error probability $p$  depends on the Euclidean distance of the two points and the AWGN noise power density  $N_0$.
	#A distortion to the more distant blue point instead of one of the two neighboring brown points is rather unlikely with Gaussian noise.		#A distortion to the more distant blue point instead of one of the two neighboring brown points is rather unlikely with Gaussian noise.
	[[File:EN Bei T 2 3 S6b neu.png\|right\|frame\|Signal space constellation:  "$\rm 16\hspace{0.05cm}–\hspace{-0.02cm}QAM$";  left: ITU propsal,  right:  Gray encoding. ]]		[[File:EN Bei T 2 3 S6b neu.png\|right\|frame\|Signal space constellation:  "$\rm 16\hspace{0.05cm}–\hspace{-0.02cm}QAM$";  left: ITU propsal,  right:  Gray encoding. '''KORREKTUR: Symbolfehler, Bitfehler''']]

	An analysis shows:		An analysis shows: