Modulation Methods/Linear Digital Modulation - Revision history

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2026-03-16T13:47:07Z

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← Older revision		Revision as of 15:47, 16 March 2026
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	*The received signal  $r(t) = s(t)$  is first multiplied by the receiver-side carrier  $z_{\rm E}(t) = 2 · \sin(2π\hspace{0.05cm}f_{\rm T}\hspace{0.05cm}t)$   ⇒   twice the amplitude compared to  $z(t)$,  no frequency and phase offset.  This gives:		*The received signal  $r(t) = s(t)$  is first multiplied by the receiver-side carrier  $z_{\rm E}(t) = 2 · \sin(2π\hspace{0.05cm}f_{\rm T}\hspace{0.05cm}t)$   ⇒   twice the amplitude compared to  $z(t)$,  no frequency and phase offset.  This gives:
	:$$b(t) = 2 \cdot z(t)\cdot r(t)= 2 \cdot z^2(t)\cdot q(t) $$		:$$b(t) = 2 \cdot z(t)\cdot r(t)= 2 \cdot z^2(t)\cdot q(t) $$
	:$$\Rightarrow \hspace{0.35cm}b(t) = q(t) \cdot \big [ 1 - \cos(4\pi\hspace{0.05cm} f_{\rm T}\hspace{0.05cm} t)\big]		:$$\Rightarrow \hspace{0.35cm}b(t) = q(t) \cdot \big [ 1 - \cos(4\pi\hspace{0.05cm} f_{\rm T}\hspace{0.05cm} t)\big]\hspace{0.05cm}.$$
	\hspace{0.05cm}.$$
	*The low-pass filter with frequency response  $H_{\rm E}(f) = {\rm sinc}(f_{\rm T}\hspace{0.05cm} T)$  and corresponding rectangular impulse response  $h_{\rm E}(t)$  forms from the signal  $b(t)$  the detection signal  $d(t) = b(t) \star h_{\rm E}(t)$  .		*The low-pass filter with frequency response  $H_{\rm E}(f) = {\rm sinc}(f_{\rm T}\hspace{0.05cm} T)$  and corresponding rectangular impulse response  $h_{\rm E}(t)$  forms from the signal  $b(t)$  the detection signal  $d(t) = b(t) \star h_{\rm E}(t)$  .
	* $h_{\rm E}(t)$  is matched to the rectangular basic pulse  $g_q(t)$.  This is called the   [[Theory_of_Stochastic_Signals/Matched_Filter\|$\text{matched filter configuration}$]]   ⇒   best possible compromise between equalization and noise power limitation.		* $h_{\rm E}(t)$  is matched to the rectangular basic pulse  $g_q(t)$.  This is called the   [[Theory_of_Stochastic_Signals/Matched_Filter\|$\text{matched filter configuration}$]]   ⇒   best possible compromise between equalization and noise power limitation.
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	:$$d_1(ν · T) = 0,\hspace{0.5cm} d_2(ν · T) = 0.$$		:$$d_1(ν · T) = 0,\hspace{0.5cm} d_2(ν · T) = 0.$$
	*Otherwise  $(a_ν = 1)$  applies:		*Otherwise  $(a_ν = 1)$  applies:
	:$$d_1(\nu \cdot T) = s_{\rm 0} \cdot \cos( \phi_{\rm T}),		:$$d_1(\nu \cdot T) = s_{\rm 0} \cdot \cos( \phi_{\rm T}),\hspace{0.5cm}d_2(\nu \cdot T) = -s_{\rm 0} \cdot \sin( \phi_{\rm T}) \hspace{0.05cm}.$$
	\hspace{0.5cm}d_2(\nu \cdot T) = -s_{\rm 0} \cdot \sin( \phi_{\rm
	T}) \hspace{0.05cm}.$$
	*After squaring the two partial signals we get for the sum signal:		*After squaring the two partial signals we get for the sum signal:
	:$$d(\nu \cdot T) = \left\{ \begin{array}{c} 0 \\ {s^2_0} \end{array} \right.\quad		:$$d(\nu \cdot T) = \left\{ \begin{array}{c} 0 \\ {s^2_0} \end{array} \right.\quad\begin{array}{*{1}c} {\rm if}\hspace{0.15cm}a_\nu = 0,\\ {\rm if}\hspace{0.15cm}a_\nu = 1. \\ \end{array}$$
	\begin{array}{*{1}c} {\rm if}\hspace{0.15cm}a_\nu = 0,
	\\ {\rm if}\hspace{0.15cm}a_\nu = 1. \\ \end{array}$$
	*By threshold decision – sensibly with the decision threshold  $E = {s_0}^2/2$ – the amplitude coefficients  $a_ν$  can be decided.  <br>However in the noisy case,  this results in a somewhat larger  [[Modulation_Methods/Linear_Digital_Modulation#Error_probabilities_-_a_brief_overview\|$\text{bit error probability}$]]  than with coherent demodulation.		*By threshold decision – sensibly with the decision threshold  $E = {s_0}^2/2$ – the amplitude coefficients  $a_ν$  can be decided.  <br>However in the noisy case,  this results in a somewhat larger  [[Modulation_Methods/Linear_Digital_Modulation#Error_probabilities_-_a_brief_overview\|$\text{bit error probability}$]]  than with coherent demodulation.

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	we obtain in the  $ν$–th time interval:		we obtain in the  $ν$–th time interval:
	:$$s_{\rm BPSK}(t) = \left\{ \begin{array}{c} s_0 \cdot \cos(2 \pi f_{\rm T} t +		:$$s_{\rm BPSK}(t) = \left\{ \begin{array}{c} s_0 \cdot \cos(2 \pi f_{\rm T} t +\pi+ \pi/2) \\ s_0 \cdot \cos(2 \pi f_{\rm T} t + \pi- \pi/2) \end{array} \right.\quad \begin{array}{*{1}c} {\rm if}\hspace{0.15cm}a_\nu = +1, \\ {\rm if}\hspace{0.15cm}a_\nu = -1. \\ \end{array}$$
	\pi+ \pi/2) \\ s_0 \cdot \cos(2 \pi f_{\rm T} t + \pi- \pi/2) \end{array} \right.\quad \begin{array}{*{1}c} {\rm if}\hspace{0.15cm}a_\nu = +1, \\ {\rm if}\hspace{0.15cm}a_\nu = -1. \\ \end{array}$$

	This equation for  »'''Binary Phase Shift Keying'''«  $($ $\rm BPSK)$  can be transformed as follows:		This equation for  »'''Binary Phase Shift Keying'''«  $($ $\rm BPSK)$  can be transformed as follows:
	:$$s_{\rm BPSK}(t) =		:$$s_{\rm BPSK}(t) =a_\nu \cdot s_0 \cdot \sin(2 \pi f_{\rm T} t ) $$
	a_\nu \cdot s_0 \cdot \sin(2 \pi f_{\rm T} t ) $$		:$$\Rightarrow \hspace{0.3cm}s_{\rm BPSK}(t) = \left\{ \begin{array}{c} s_0 \cdot \sin(2 \pi f_{\rm T} t ) \\-s_0 \cdot \sin(2 \pi f_{\rm T} t ) \end{array} \right.\quad \begin{array}{*{1}c} {\rm if}\hspace{0.15cm}a_\nu = +1, \\ {\rm if}\hspace{0.15cm}a_\nu = -1. \\ \end{array}$$
	:$$\Rightarrow \hspace{0.3cm}s_{\rm BPSK}(t) = \left\{ \begin{array}{c} s_0 \cdot \sin(2 \pi f_{\rm T} t ) \\
	-s_0 \cdot \sin(2 \pi f_{\rm T} t ) \end{array} \right.\quad \begin{array}{*{1}c} {\rm if}\hspace{0.15cm}a_\nu = +1, \\ {\rm if}\hspace{0.15cm}a_\nu = -1. \\ \end{array}$$
	<br clear=all>		<br clear=all>
	In the diagram the signals and the corresponding power-spectral densities are sketched.  It can be seen:		In the diagram the signals and the corresponding power-spectral densities are sketched.  It can be seen:
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	*The power-spectral densities with BPSK differ from those with ASK only by the missing Dirac delta functions  $($since now  $q(t)$  does not contain a DC component$)$  and by the factor  $4$  with respect to the continuous PSD components.		*The power-spectral densities with BPSK differ from those with ASK only by the missing Dirac delta functions  $($since now  $q(t)$  does not contain a DC component$)$  and by the factor  $4$  with respect to the continuous PSD components.
	*It further follows that binary phase modulation  $\rm (BPSK)$  can be counted as a linear modulation method.  In general, (analog) phase modulation  $\rm (PM)$  is,  with a few exceptions,  non-linear with respect to the source signal.		*It further follows that binary phase modulation  $\rm (BPSK)$  can be counted as a linear modulation method.  In general, (analog) phase modulation  $\rm (PM)$  is,  with a few exceptions,  non-linear with respect to the source signal.
	*For the diagrams,  different carrier phases were chosen for display reasons in the section  [[Modulation_Methods/~~Lineare_digitale_Modulation~~#~~ASK_.E2.80.93_Amplitude_Shift_Keying~~\|"ASK"]]   ⇒   "sine"  and here for  "BPSK"  ⇒   "minus cosine".  However,  this arbitrary determination is not a restriction.  Both methods work in the same way for other carrier phases.		*For the diagrams,  different carrier phases were chosen for display reasons in the section  [[Modulation_Methods/Linear_Digital_Modulation#ASK_–_Amplitude_Shift_Keying\|"ASK"]]   ⇒   "sine"  and here for  "BPSK"  ⇒   "minus cosine".  However,  this arbitrary determination is not a restriction.  Both methods work in the same way for other carrier phases.

	==Demodulation and detection of BPSK signals==		==Demodulation and detection of BPSK signals==
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	$\text{Here without proof:}$		$\text{Here without proof:}$
	Let us first consider the  »'''bit error probability'''«  of  [[Modulation_Methods/Linear_Digital_Modulation#Demodulation_and_detection_of_BPSK_signals\|$\text{Binary Phase Shift Keying}$]]   $\rm (BPSK)$  assuming a coherent receiver:		Let us first consider the  »'''bit error probability'''«  of  [[Modulation_Methods/Linear_Digital_Modulation#Demodulation_and_detection_of_BPSK_signals\|$\text{Binary Phase Shift Keying}$]]   $\rm (BPSK)$  assuming a coherent receiver:
	::$$p_{\text{B, BPSK, coh} } = {\rm Q}\left ( \sqrt{ {2 \cdot E_{\rm B} }/{N_0 } } \hspace{0.1cm}\right		::$$p_{\text{B, BPSK, coh} } = {\rm Q}\left ( \sqrt{ {2 \cdot E_{\rm B} }/{N_0 } } \hspace{0.1cm}\right) = {1}/{2}\cdot {\rm erfc}\left ( \sqrt{ {E_{\rm B} }/{N_0 } } \hspace{0.1cm}\right ).$$
	) = {1}/{2}\cdot {\rm erfc}\left ( \sqrt{ {E_{\rm B} }/{N_0 } } \hspace{0.1cm}\right ).$$
	*[[Applets:Komplementäre_Gaußsche_Fehlerfunktionen\|"Two variants of the complementary Gaussian error function"]]  were used in the formulas:		*[[Applets:Komplementäre_Gaußsche_Fehlerfunktionen\|"Two variants of the complementary Gaussian error function"]]  were used in the formulas:
	::$${\rm Q} ({\it x}) = \frac{\rm 1}{\sqrt{\rm 2\pi} }\int_{\it		::$${\rm Q} ({\it x}) = \frac{\rm 1}{\sqrt{\rm 2\pi} }\int_{\it x}^{+\infty}{\rm e}^{ {\it -u}^{\rm 2}/\rm 2}\,{\rm d} {\it u}\hspace{0.05cm},\hspace{0.3cm} {\rm erfc} ( {\it x} ) = \frac{\rm2}{\sqrt{\rm \pi} }\int_{\it x}^{+\infty}{\rm e}^{ {\it -u}^{\rm2} }\,{\rm d} {\it u} \hspace{0.05cm}.$$
	x}^{+\infty}{\rm e}^{ {\it -u}^{\rm 2}/\rm 2}\,{\rm d} {\it u}
	\hspace{0.05cm},\hspace{0.3cm} {\rm erfc} ( {\it x} ) = \frac{\rm
	2}{\sqrt{\rm \pi} }\int_{\it x}^{+\infty}{\rm e}^{ {\it -u}^{\rm
	2} }\,{\rm d} {\it u} \hspace{0.05cm}.$$
	*In contrast, for  [[Modulation_Methods/Linear_Digital_Modulation#Coherent_demodulation_of_ASK_signals\|$\text{Amplitude Shift Keying}$]]  $\rm (ASK)$  with coherent demodulation:		*In contrast, for  [[Modulation_Methods/Linear_Digital_Modulation#Coherent_demodulation_of_ASK_signals\|$\text{Amplitude Shift Keying}$]]  $\rm (ASK)$  with coherent demodulation:
	::$$p_{\text{B, ASK, coh} } = {\rm Q}\left ( \sqrt{{E_{\rm B} }/{N_0 } } \hspace{0.1cm}\right		::$$p_{\text{B, ASK, coh} } = {\rm Q}\left ( \sqrt{{E_{\rm B} }/{N_0 } } \hspace{0.1cm}\right) ={1}/{2}\cdot {\rm erfc}\left ( \sqrt{ {E_{\rm B}}/{(2 \cdot N_0) } } \hspace{0.1cm}\right ).$$
	) ={1}/{2}\cdot {\rm erfc}\left ( \sqrt{ {E_{\rm B}}/{(2 \cdot N_0) } } \hspace{0.1cm}\right ).$$
	::If we plot the bit error probability over the quotient  $E_{\rm B}/N_0$  in double logarithmic scale,  the ASK curve is always  $3 \ \rm dB$  to the right of the BPSK curve.  This degradation is one of the reasons why ASK is rarely used in practice.		::If we plot the bit error probability over the quotient  $E_{\rm B}/N_0$  in double logarithmic scale,  the ASK curve is always  $3 \ \rm dB$  to the right of the BPSK curve.  This degradation is one of the reasons why ASK is rarely used in practice.
	*The decisive advantage of  [[Modulation_Methods/Linear_Digital_Modulation#Differential_coherent_demodulation_of_the_DPSK_signal\|$\text{Differential Phase Shift Keying}$]]  $\rm (DPSK)$  is that it  '''can be demodulated without knowledge of the carrier phase'''.  This simple realization is bought by an increased error probability compared to coherent BPSK:		*The decisive advantage of  [[Modulation_Methods/Linear_Digital_Modulation#Differential_coherent_demodulation_of_the_DPSK_signal\|$\text{Differential Phase Shift Keying}$]]  $\rm (DPSK)$  is that it  '''can be demodulated without knowledge of the carrier phase'''.  This simple realization is bought by an increased error probability compared to coherent BPSK:

Maintenance script: Add German interlanguage link

2026-03-16T12:28:13Z

Add German interlanguage link

← Older revision		Revision as of 14:28, 16 March 2026
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	{{Display}}		{{Display}}
			[[de:Modulationsverfahren/Lineare_digitale_Modulation]]

Guenter at 13:47, 13 January 2023

2023-01-13T13:47:38Z

← Older revision		Revision as of 15:47, 13 January 2023
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	<br>		<br>
	The graphic shows the block diagram of a transmission system with DPSK modulation  ("Differential Phase Shift Keying")  and differential coherent demodulation.  In bullet point form,  the given system can be described as follows:		The graphic shows the block diagram of a transmission system with DPSK modulation  ("Differential Phase Shift Keying")  and differential coherent demodulation.  In bullet point form,  the given system can be described as follows:
	[[File:EN_Mod_T_4_2_S7c_neu.png\|right\|frame\| DPSK modulation and differential coherent demodulation ~~'''KORREKTUR''': Takt~~]]		[[File:EN_Mod_T_4_2_S7c_neu.png\|right\|frame\| DPSK modulation and differential coherent demodulation]]
	*Ignoring the modulation with the carrier signals  $z(t)$  and  $2 · z(t)$  resp.,  the symbol  $m_ν = m_{ν-1} · q_ν$  is present in the interval  $ν$  at input  '''(1)'''  of the multiplier highlighted in yellow,  and the symbol  $m_{ν-1}$  is present at input   '''(2)'''.		*Ignoring the modulation with the carrier signals  $z(t)$  and  $2 · z(t)$  resp.,  the symbol  $m_ν = m_{ν-1} · q_ν$  is present in the interval  $ν$  at input  '''(1)'''  of the multiplier highlighted in yellow,  and the symbol  $m_{ν-1}$  is present at input   '''(2)'''.
	*The multiplication of  '''(1)'''  and  '''(2)'''  gives the desired result:		*The multiplication of  '''(1)'''  and  '''(2)'''  gives the desired result:

Guenter at 13:46, 13 January 2023

2023-01-13T13:46:40Z

← Older revision		Revision as of 15:46, 13 January 2023
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	<br>		<br>
	The diagram shows the block diagram of an ASK system including the receiver components.		The diagram shows the block diagram of an ASK system including the receiver components.
	[[File:EN_Mod_T_4_2_S3a_neu.png\|right\|frame\|Block diagram of an ASK system including receiver components ~~'''KORREKTUR''': Takt~~]]		[[File:EN_Mod_T_4_2_S3a_neu.png\|right\|frame\|Block diagram of an ASK system including receiver components]]
	*Let the source signal  $q(t)$  be NRZ-rectangular and unipolar:		*Let the source signal  $q(t)$  be NRZ-rectangular and unipolar:
	:$$a_ν ∈ \{0, \ 1\}.$$		:$$a_ν ∈ \{0, \ 1\}.$$

Hwang at 16:02, 9 January 2023

2023-01-09T16:02:46Z

Show changes

Guenter at 14:19, 11 April 2022

2022-04-11T14:19:23Z

← Older revision		Revision as of 16:19, 11 April 2022
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	<br>		<br>
	The diagram shows the block diagram of an ASK system including the receiver components.		The diagram shows the block diagram of an ASK system including the receiver components.
	[[File:~~EN_Mod_T_4_2_S3a~~.png\|right\|frame\|Block diagram of an ASK system including receiver components '''KORREKTUR''': Takt]]		[[File:EN_Mod_T_4_2_S3a_neu.png\|right\|frame\|Block diagram of an ASK system including receiver components '''KORREKTUR''': Takt]]
	*Let the source signal  $q(t)$  be NRZ-rectangular and unipolar:		*Let the source signal  $q(t)$  be NRZ-rectangular and unipolar:
	:$$a_ν ∈ \{0, \ 1\}.$$		:$$a_ν ∈ \{0, \ 1\}.$$
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	<br>		<br>
	We further assume ASK modulation as well as an ideal transmission channel, i.e.		We further assume ASK modulation as well as an ideal transmission channel, i.e.
	[[File:~~P_ID1691__Mod_T_4_2_S4a_neu~~.png\|right\|frame\|Block diagram of an incoherent ASK demodulator ~~'''KORREKTUR''': Takt~~]]		[[File:EN_Mod_T_4_2_S4a_neu.png\|right\|frame\|Block diagram of an incoherent ASK demodulator]]
	#distortion-free,		#distortion-free,
	#attenuation-free and		#attenuation-free and
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	<br>		<br>
	The graphic shows the block diagram of a transmission system with DPSK modulation  ("Differential Phase Shift Keying")  and differential coherent demodulation.  In bullet point form,  the given system can be described as follows:		The graphic shows the block diagram of a transmission system with DPSK modulation  ("Differential Phase Shift Keying")  and differential coherent demodulation.  In bullet point form,  the given system can be described as follows:
	[[File:~~EN_Mod_T_4_2_S7c~~.png \|right\|frame\| DPSK modulation and differential coherent demodulation '''KORREKTUR''': Takt]]		[[File:EN_Mod_T_4_2_S7c_neu.png\|right\|frame\| DPSK modulation and differential coherent demodulation '''KORREKTUR''': Takt]]
	*Ignoring the modulation with the carrier signals  $z(t)$  and  $2 · z(t)$  resp.,  the symbol  $m_ν = m_{ν-1} · q_ν$  is present in the interval  $ν$  at input  '''(1)'''  of the multiplier highlighted in yellow,  and the symbol  $m_{ν-1}$  is present at input   '''(2)'''.		*Ignoring the modulation with the carrier signals  $z(t)$  and  $2 · z(t)$  resp.,  the symbol  $m_ν = m_{ν-1} · q_ν$  is present in the interval  $ν$  at input  '''(1)'''  of the multiplier highlighted in yellow,  and the symbol  $m_{ν-1}$  is present at input   '''(2)'''.
	*The multiplication of  '''(1)'''  and  '''(2)'''  gives the desired result:		*The multiplication of  '''(1)'''  and  '''(2)'''  gives the desired result:

Reed at 13:57, 9 April 2022

2022-04-09T13:57:12Z

← Older revision		Revision as of 15:57, 9 April 2022
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	{{Header		{{Header
	\|Untermenü=Digital Modulation Methods		\|Untermenü=Digital Modulation Methods
	\|Vorherige Seite=~~Pulscodemodulation~~		\|Vorherige Seite=Pulse Code Modulation
	\|Nächste Seite=~~Quadratur–Amplitudenmodulation~~		\|Nächste Seite=Quadrature Amplitude Modulation
	}}		}}
	==Differences between analog and digital modulation methods==		==Differences between analog and digital modulation methods==

Guenter at 09:36, 8 April 2022

2022-04-08T09:36:09Z

← Older revision		Revision as of 11:36, 8 April 2022
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	<br>		<br>
	The diagram shows the block diagram of an ASK system including the receiver components.		The diagram shows the block diagram of an ASK system including the receiver components.
	[[File:EN_Mod_T_4_2_S3a.png\|right\|frame\|Block diagram of an ASK system including receiver components]]		[[File:EN_Mod_T_4_2_S3a.png\|right\|frame\|Block diagram of an ASK system including receiver components '''KORREKTUR''': Takt]]
	*Let the source signal  $q(t)$  be NRZ-rectangular and unipolar:		*Let the source signal  $q(t)$  be NRZ-rectangular and unipolar:
	:$$a_ν ∈ \{0, \ 1\}.$$		:$$a_ν ∈ \{0, \ 1\}.$$
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	<br>		<br>
	We further assume ASK modulation as well as an ideal transmission channel, i.e.		We further assume ASK modulation as well as an ideal transmission channel, i.e.
	[[File:P_ID1691__Mod_T_4_2_S4a_neu.png\|right\|frame\|Block diagram of an incoherent ASK demodulator]]		[[File:P_ID1691__Mod_T_4_2_S4a_neu.png\|right\|frame\|Block diagram of an incoherent ASK demodulator '''KORREKTUR''': Takt]]
	#distortion-free,		#distortion-free,
	#attenuation-free and		#attenuation-free and

Hwang at 11:13, 7 April 2022

2022-04-07T11:13:25Z

← Older revision		Revision as of 13:13, 7 April 2022
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	<br>		<br>
	The graphic shows the block diagram of a transmission system with DPSK modulation  ("Differential Phase Shift Keying")  and differential coherent demodulation.  In bullet point form,  the given system can be described as follows:		The graphic shows the block diagram of a transmission system with DPSK modulation  ("Differential Phase Shift Keying")  and differential coherent demodulation.  In bullet point form,  the given system can be described as follows:
	[[File:EN_Mod_T_4_2_S7c.png \|right\|frame\| DPSK modulation and differential coherent demodulation]]		[[File:EN_Mod_T_4_2_S7c.png \|right\|frame\| DPSK modulation and differential coherent demodulation '''KORREKTUR''': Takt]]
	*Ignoring the modulation with the carrier signals  $z(t)$  and  $2 · z(t)$  resp.,  the symbol  $m_ν = m_{ν-1} · q_ν$  is present in the interval  $ν$  at input  '''(1)'''  of the multiplier highlighted in yellow,  and the symbol  $m_{ν-1}$  is present at input   '''(2)'''.		*Ignoring the modulation with the carrier signals  $z(t)$  and  $2 · z(t)$  resp.,  the symbol  $m_ν = m_{ν-1} · q_ν$  is present in the interval  $ν$  at input  '''(1)'''  of the multiplier highlighted in yellow,  and the symbol  $m_{ν-1}$  is present at input   '''(2)'''.
	*The multiplication of  '''(1)'''  and  '''(2)'''  gives the desired result:		*The multiplication of  '''(1)'''  and  '''(2)'''  gives the desired result:

Guenter at 12:36, 4 April 2022

2022-04-04T12:36:11Z

← Older revision		Revision as of 14:36, 4 April 2022
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	Below the phase values  $ϕ_{\rm S}$  are given for the first six symbols:		Below the phase values  $ϕ_{\rm S}$  are given for the first six symbols:
	*Additional phase rotation on the channel,  e.g. by  $70.3π$, does change the absolute phase values to  $69.8π$,  $69.8π$,  $70.8π$,  $70.8π$,  $69.8π$  and  $70.8π$.		*Additional phase rotation on the channel,  e.g. by  $70.3π$,  does change the absolute phase values to  $69.8π$,  $69.8π$,  $70.8π$,  $70.8π$,  $69.8π$  and  $70.8π$.
	*However,  the phase difference of adjacent symbols is preserved, so the differential coherent demodulation still works.		*However,  the phase difference of adjacent symbols is preserved, so the differential coherent demodulation still works.
	*A corresponding demodulator is presented in the following section. }}		*A corresponding demodulator is presented in the following section. }}