Digital Signal Transmission/Carrier Frequency Systems with Coherent Demodulation - Revision history

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2026-03-16T13:46:46Z

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	*By energy normalization one gets from the basic transmission pulse  $g_s(t)$  to the basis function		*By energy normalization one gets from the basic transmission pulse  $g_s(t)$  to the basis function

	:$$\varphi_1(t) = { g_s(t)}/{\sqrt{E_{gs}}} \hspace{0.4cm} {\rm with} \hspace{0.4cm} E_{gs} =		:$$\varphi_1(t) = { g_s(t)}/{\sqrt{E_{gs}}} \hspace{0.4cm} {\rm with} \hspace{0.4cm} E_{gs} =\int_{0}^{T} g_s(t)^2 \,{\rm d} t$$
	\int_{0}^{T} g_s(t)^2 \,{\rm d} t$$		:$$ \hspace{0.3cm}\Rightarrow \hspace{0.3cm} s_{\rm TP}(t) \big {\|}_{m\hspace{0.05cm} =\hspace{0.05cm} m_i} = s_{{\rm I}i} \cdot \varphi_1(t) + s_{{\rm Q}i} \cdot {\rm j} \cdot \varphi_1(t)\hspace{0.05cm}.$$
	:$$ \hspace{0.3cm}
	\Rightarrow \hspace{0.3cm} s_{\rm TP}(t) \big {\|}_{m\hspace{0.05cm} =\hspace{0.05cm} m_i} = s_{{\rm I}i} \cdot \varphi_1(t) + s_{{\rm Q}i} \cdot {\rm j} \cdot \varphi_1(t)
	\hspace{0.05cm}.$$

	*While the coefficients   $a_{{\rm I}i}$   and   $a_{{\rm Q}i}$   are dimensionless,  the new coefficients   $s_{{\rm I}i}$   and   $s_{{\rm Q}i}$   have the unit  "root of energy"   ⇒   see section  [[Digital_Signal_Transmission/Signals,_Basis_Functions_and_Vector_Spaces#Nomenclature_in_the_fourth_chapter\|"Nomenclature in the fourth main chapter"]]:		*While the coefficients   $a_{{\rm I}i}$   and   $a_{{\rm Q}i}$   are dimensionless,  the new coefficients   $s_{{\rm I}i}$   and   $s_{{\rm Q}i}$   have the unit  "root of energy"   ⇒   see section  [[Digital_Signal_Transmission/Signals,_Basis_Functions_and_Vector_Spaces#Nomenclature_in_the_fourth_chapter\|"Nomenclature in the fourth main chapter"]]:
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	*Since each of the  $M$  symbols represents  $b = \log_2 (M)$  bits,  the  "average energy per bit"  is:		*Since each of the  $M$  symbols represents  $b = \log_2 (M)$  bits,  the  "average energy per bit"  is:
	:$$E_{\rm B} = \frac{E_{\rm S}}{b} = \frac{E_{\rm S}}{{\rm log_2}\, (M)} =\frac{M^2 -1}{3 \cdot {\rm log_2}\, (M)} \cdot E$$		:$$E_{\rm B} = \frac{E_{\rm S}}{b} = \frac{E_{\rm S}}{{\rm log_2}\, (M)} =\frac{M^2 -1}{3 \cdot {\rm log_2}\, (M)} \cdot E$$
	:$$\Rightarrow\hspace{0.3cm}M= 8\hspace{-0.1cm}: \hspace{0.2cm} E_{\rm S}/E = 21		:$$\Rightarrow\hspace{0.3cm}M= 8\hspace{-0.1cm}: \hspace{0.2cm} E_{\rm S}/E = 21\hspace{0.05cm}, \hspace{0.2cm}E_{\rm B}/E = 7\hspace{0.05cm}.$$
	\hspace{0.05cm}, \hspace{0.2cm}E_{\rm B}/E = 7\hspace{0.05cm}.$$

	*The probability that one of the two outer symbols is falsified due to AWGN noise is therefore the same:		*The probability that one of the two outer symbols is falsified due to AWGN noise is therefore the same:
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	*For  $M = 2$  (BPSK)  and  $M = 4$  (identity between 4–PSK and 4–QAM),  the bit error probability can be specified exactly:		*For  $M = 2$  (BPSK)  and  $M = 4$  (identity between 4–PSK and 4–QAM),  the bit error probability can be specified exactly:
	:$$p_{\rm B} = {\rm Q} \left ( \sqrt{ { {2E_{\rm B} } }/{ N_0} }\right )		:$$p_{\rm B} = {\rm Q} \left ( \sqrt{ { {2E_{\rm B} } }/{ N_0} }\right )\hspace{0.05cm}.$$}}
	\hspace{0.05cm}.$$}}

	== Binary frequency shift keying (2–FSK) ==		== Binary frequency shift keying (2–FSK) ==

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2026-03-16T12:26:51Z

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	{{Display}}		{{Display}}
			[[de:Digitalsignalübertragung/Trägerfrequenzsysteme_mit_kohärenter_Demodulation]]

Hwang at 10:04, 17 November 2022

2022-11-17T10:04:53Z

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	\hspace{0.05cm}.$$		\hspace{0.05cm}.$$

	*While the coefficients   $a_{{\rm I}i}$   and   $a_{{\rm Q}i}$   are dimensionless,  the new coefficients   $s_{{\rm I}i}$   and   $s_{{\rm Q}i}$   have the unit  "root of energy"   ⇒   see ~~page~~  [[Digital_Signal_Transmission/Signals,_Basis_Functions_and_Vector_Spaces#Nomenclature_in_the_fourth_chapter\|"Nomenclature in the fourth main chapter"]]:		*While the coefficients   $a_{{\rm I}i}$   and   $a_{{\rm Q}i}$   are dimensionless,  the new coefficients   $s_{{\rm I}i}$   and   $s_{{\rm Q}i}$   have the unit  "root of energy"   ⇒   see section  [[Digital_Signal_Transmission/Signals,_Basis_Functions_and_Vector_Spaces#Nomenclature_in_the_fourth_chapter\|"Nomenclature in the fourth main chapter"]]:

	:$$s_{{\rm I}i} = {\sqrt{E_{gs}}} \cdot a_{{\rm I}i}\hspace{0.05cm}, $$		:$$s_{{\rm I}i} = {\sqrt{E_{gs}}} \cdot a_{{\rm I}i}\hspace{0.05cm}, $$

Guenter at 15:17, 6 September 2022

2022-09-06T15:17:20Z

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	<br>		<br>
	In the following,  we always assume the equivalent low-pass signal unless explicitly stated otherwise.  In particular,  the signals		In the following,  we always assume the equivalent low-pass signal unless explicitly stated otherwise.  In particular,  the signals
	[[File:~~EN_Dig_T_4_4_S2~~.png\|right\|frame\|AWGN channel model for complex signals ~~'''KORREKTUR: optimal '''~~\|class=fit]]		[[File:EN_Dig_T_4_4_S2_v2.png\|right\|frame\|AWGN channel model for complex signals\|class=fit]]

	*$s(t) = s_{\rm TP}(t)$  and		*$s(t) = s_{\rm TP}(t)$  and

Guenter at 12:05, 3 September 2022

2022-09-03T12:05:17Z

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	in the graph are  "low-pass signals"  and thus generally complex.  The suffix  "TP"  is omitted in the remainder of this paper ~~'''KORREKTUR: in the following'''~~.<br>		in the graph are  "low-pass signals"  and thus generally complex.  The suffix  "TP"  is omitted in the remainder of this paper .<br>

	To this figure is to be noted:		To this figure is to be noted:

Hwang at 18:19, 1 September 2022

2022-09-01T18:19:35Z

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	::<math>\Rightarrow\hspace{0.3cm} M = 4\hspace{-0.1cm}:\hspace{0.1cm}d_{\rm min}/E^{1/2} = \sqrt{2} \approx 1.414 \hspace{0.05cm}, \hspace{0.8cm} M = 8\hspace{-0.1cm}:\hspace{0.1cm}d_{\rm min}/E^{1/2} \approx 0.765 \hspace{0.05cm},\hspace{0.8cm} M = 16\hspace{-0.1cm}:\hspace{0.1cm}d_{\rm min}/E^{1/2} \approx 0.390 \hspace{0.05cm}.</math>		::<math>\Rightarrow\hspace{0.3cm} M = 4\hspace{-0.1cm}:\hspace{0.1cm}d_{\rm min}/E^{1/2} = \sqrt{2} \approx 1.414 \hspace{0.05cm}, \hspace{0.8cm} M = 8\hspace{-0.1cm}:\hspace{0.1cm}d_{\rm min}/E^{1/2} \approx 0.765 \hspace{0.05cm},\hspace{0.8cm} M = 16\hspace{-0.1cm}:\hspace{0.1cm}d_{\rm min}/E^{1/2} \approx 0.390 \hspace{0.05cm}.</math>

	*The upper bound  $p_{\rm UB}$  for the AWGN symbol error probability  $p_{\rm S}$  after the  [[~~Digitalsignal%C3%BCbertragung~~/~~Approximation_der_Fehlerwahrscheinlichkeit~~#Union_Bound_-~~_Obere_Schranke_f.C3.BCr_die_Fehlerwahrscheinlichkeit~~\|"Union Bound"]]  yields:		*The upper bound  $p_{\rm UB}$  for the AWGN symbol error probability  $p_{\rm S}$  after the  [[Digital_Signal_Transmission/Approximation_of_the_Error_Probability#Union_Bound_-_Upper_bound_for_the_error_probability\|"Union Bound"]]  yields:

	::<math>p_{\rm UB} = 2 \cdot {\rm Q} \left ( \sin ({ \pi}/{ M}) \cdot \sqrt{ { {2E_{\rm S}}}/{ N_0} }\right ) \ge p_{\rm S}		::<math>p_{\rm UB} = 2 \cdot {\rm Q} \left ( \sin ({ \pi}/{ M}) \cdot \sqrt{ { {2E_{\rm S}}}/{ N_0} }\right ) \ge p_{\rm S}

Hwang at 12:54, 1 September 2022

2022-09-01T12:54:39Z

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	in the graph are  "low-pass signals"  and thus generally complex.  The suffix  "TP"  is omitted in the remainder of this paper.<br>		in the graph are  "low-pass signals"  and thus generally complex.  The suffix  "TP"  is omitted in the remainder of this paper '''KORREKTUR: in the following'''.<br>

	To this figure is to be noted:		To this figure is to be noted:

Hwang at 12:53, 1 September 2022

2022-09-01T12:53:21Z

← Older revision		Revision as of 14:53, 1 September 2022
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	<br>		<br>
	In the following,  we always assume the equivalent low-pass signal unless explicitly stated otherwise.  In particular,  the signals		In the following,  we always assume the equivalent low-pass signal unless explicitly stated otherwise.  In particular,  the signals
	[[File:EN_Dig_T_4_4_S2.png\|right\|frame\|AWGN channel model for complex signals\|class=fit]]		[[File:EN_Dig_T_4_4_S2.png\|right\|frame\|AWGN channel model for complex signals '''KORREKTUR: optimal '''\|class=fit]]

	*$s(t) = s_{\rm TP}(t)$  and		*$s(t) = s_{\rm TP}(t)$  and

Guenter at 14:49, 24 August 2022

2022-08-24T14:49:47Z

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	In the first three chapters of this  [[Digital_Signal_Transmission\|fourth main chapter:]]   "Generalized Description of Digital Modulation Methods",  the structure of the optimal receiver and the signal representation by means of basis functions were treated by the example of baseband transmission.		In the first three chapters of this  [[Digital_Signal_Transmission\|fourth main chapter:]]   "Generalized Description of Digital Modulation Methods",  the structure of the optimal receiver and the signal representation by means of basis functions were treated by the example of baseband transmission.

	[[File:~~EN_Dig_T_4_4_S1~~.png\|right\|frame\|Equivalent low-pass model of carrier-modulated transmission methods ~~'''Korrektur''' hinten etwas verschieben~~\|class=fit]]		[[File:EN_Dig_T_4_4_S1_v2.png\|right\|frame\|Equivalent low-pass model of carrier-modulated transmission methods\|class=fit]]

	With the same systematics and the same uniformity,  band–pass systems will now also be considered which have already been described in earlier books or chapters,  namely		With the same systematics and the same uniformity,  band–pass systems will now also be considered which have already been described in earlier books or chapters,  namely
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	*with sinusoidal carrier:   $s_{{\rm I}i} \equiv 0$  and  $s_{{\rm Q}i} = \{\pm E^{1/2}\}$.		*with sinusoidal carrier:   $s_{{\rm I}i} \equiv 0$  and  $s_{{\rm Q}i} = \{\pm E^{1/2}\}$.

	[[File:~~EN_Dig_T_4_4_S3~~.png\|right\|frame\|Signal space constellations of the BPSK ~~'''Korrektur''' BPSK statt OOK~~\|class=fit]]		[[File:EN_Dig_T_4_4_S4_v2.png\|right\|frame\|Signal space constellations of the BPSK\|class=fit]]

	The improvements compared to on–off keying can be seen from the equations given in the graphic  $($in the field with green background$)$:		The improvements compared to on–off keying can be seen from the equations given in the graphic  $($in the field with green background$)$:
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	*the two innermost signal space points  $(\pm 1)$  do not differ in terms of amplitude  ("envelope"),		*the two innermost signal space points  $(\pm 1)$  do not differ in terms of amplitude  ("envelope"),
	*but only in terms of phase  $(0^\circ$ or $180^\circ)$.		*but only in terms of phase  $(0^\circ$ or $180^\circ)$.
	[[File:EN_Dig_T_4_4_S5.png\|right\|frame\|Signal room constellation of the 8-ASK~~'''Korrektur'''~~\|class=fit]]		[[File:EN_Dig_T_4_4_S5.png\|right\|frame\|Signal room constellation of the 8-ASK\|class=fit]]

Guenter at 08:05, 23 August 2022

2022-08-23T08:05:24Z

← Older revision		Revision as of 10:05, 23 August 2022
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	*with sinusoidal carrier:   $s_{{\rm I}i} \equiv 0$  and  $s_{{\rm Q}i} = \{\pm E^{1/2}\}$.		*with sinusoidal carrier:   $s_{{\rm I}i} \equiv 0$  and  $s_{{\rm Q}i} = \{\pm E^{1/2}\}$.

	[[File:EN_Dig_T_4_4_S3.png\|right\|frame\|Signal space constellations of the BPSK '''Korrektur'''\|class=fit]]		[[File:EN_Dig_T_4_4_S3.png\|right\|frame\|Signal space constellations of the BPSK '''Korrektur''' BPSK statt OOK\|class=fit]]

	The improvements compared to on–off keying can be seen from the equations given in the graphic  $($in the field with green background$)$:		The improvements compared to on–off keying can be seen from the equations given in the graphic  $($in the field with green background$)$: