Modulation Methods/Non-Linear Digital Modulation - Revision history

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← Older revision		Revision as of 16:15, 16 March 2026
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	*While in conventional Offset-QPSK implementation any basic transmission pulse  $g_s(t)$ is possible, e.g. a rectangular or a root-Nyquist pulse,  there is only one suitable basic pulse for MSK implementation.  This spans two symbol durations:		*While in conventional Offset-QPSK implementation any basic transmission pulse  $g_s(t)$ is possible, e.g. a rectangular or a root-Nyquist pulse,  there is only one suitable basic pulse for MSK implementation.  This spans two symbol durations:
	:$$g_{\rm MSK}(t) = \left\{ \begin{array}{l} s_0 \cdot \cos \big ({\pi/2 \cdot t}/T \big ) \\		:$$g_{\rm MSK}(t) = \left\{ \begin{array}{l} s_0 \cdot \cos \big ({\pi/2 \cdot t}/T \big ) \\0 \\ \end{array} \right.\quad\begin{array}{{5}c} {\rm{for}}\\ \\ \end{array}\begin{array}{{10}c}-T \le t \le +T \hspace{0.05cm}, \\{\rm otherwise}\hspace{0.05cm}. \\\end{array}$$
	0 \\ \end{array} \right.\quad
	\begin{array}{*{5}c} {\rm{for}}
	\\ \\ \end{array}\begin{array}{*{10}c}
	-T \le t \le +T \hspace{0.05cm}, \\
	{\rm otherwise}\hspace{0.05cm}. \\
	\end{array}$$
	<br clear=all>		<br clear=all>
	{{GraueBox\|TEXT=$\text{Example 2:}$  The graph shows		{{GraueBox\|TEXT=$\text{Example 2:}$  The graph shows
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	The recoding  $a_k = (–1)^{k+1} · a_{k–1} · q_k$  is already taken into account,  as well as the MSK basic pulse:		The recoding  $a_k = (–1)^{k+1} · a_{k–1} · q_k$  is already taken into account,  as well as the MSK basic pulse:
	:$$g_{\rm MSK}(t) = \left\{ \begin{array} {l} s_0 \cdot \cos ({\pi \cdot t}/{2 \cdot T}) \\		:$$g_{\rm MSK}(t) = \left\{ \begin{array} {l} s_0 \cdot \cos ({\pi \cdot t}/{2 \cdot T}) \\0 \\ \end{array} \right.\quad\begin{array}{{5}c}{\rm{for} }\\ \\ \end{array}\begin{array}{{10}c}-T \le t \le +T \hspace{0.05cm}, \\{\rm otherwise}\hspace{0.05cm}. \\\end{array}$$
	0 \\ \end{array} \right.\quad
	\begin{array}{*{5}c}{\rm{for} }
	\\ \\ \end{array}\begin{array}{*{10}c}
	-T \le t \le +T \hspace{0.05cm}, \\
	{\rm otherwise}\hspace{0.05cm}. \\
	\end{array}$$

	From the comparison of the top and the bottom diagrams,  one can see:		From the comparison of the top and the bottom diagrams,  one can see:
	#The MSK phase response  $ϕ(t)$  is step-wise linear and increases or decreases by  $90^\circ \ (π/2)$ within each symbol duration,  depending on whether  $q_k = +1$  or  $q_k = -1$  currently holds.		#The MSK phase response  $ϕ(t)$  is step-wise linear and increases or decreases by  $90^\circ \ (π/2)$ within each symbol duration,  depending on whether  $q_k = +1$  or  $q_k = -1$  currently holds.
	#The associated transmitted signal  $s(t)$  contains the two frequencies  $f_{\rm T} ± 1/(4T)$ in sections. It has basically the same shape as the signal  $s_{\rm D}(t)$  in the section [[Modulation_Methods/~~Nonlinear_Digital_Modulation~~#Binary_FSK_with_Continuous_Phase_Matching\|"Binary FSK with CPM"]].		#The associated transmitted signal  $s(t)$  contains the two frequencies  $f_{\rm T} ± 1/(4T)$ in sections. It has basically the same shape as the signal  $s_{\rm D}(t)$  in the section [[Modulation_Methods/Non-Linear_Digital_Modulation#Binary_FSK_with_Continuous_Phase_Matching\|"Binary FSK with CPM"]].


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	:$$\int_{-\infty}^{+\infty} g (t)\hspace{0.1cm} {\rm d}t = 1 \hspace{0.05cm}.$$		:$$\int_{-\infty}^{+\infty} g (t)\hspace{0.1cm} {\rm d}t = 1 \hspace{0.05cm}.$$
	*But the following relationship also holds with the   »'''phase pulse'''«   $g_ϕ(t)$:		*But the following relationship also holds with the   »'''phase pulse'''«   $g_ϕ(t)$:
	:$$\phi(t) = {\pi}\cdot h \cdot\sum_{\nu = - \infty}^{+\infty}a_{ \nu} \cdot g_\phi (t - \nu\cdot T),\hspace{0.2cm}{\rm where}\hspace{0.2cm}g_\phi(t) =\int_{-\infty}^{t} g (\tau )\hspace{0.1cm} {\~~rmd~~}\tau\hspace{0.05cm}.$$		:$$\phi(t) = {\pi}\cdot h \cdot\sum_{\nu = - \infty}^{+\infty}a_{ \nu} \cdot g_\phi (t - \nu\cdot T),\hspace{0.2cm}{\rm where}\hspace{0.2cm}g_\phi(t) =\int_{-\infty}^{t} g (\tau )\hspace{0.1cm} {\rm d}\tau\hspace{0.05cm}.$$

	Many CPM variants can be realized by appropriately choosing the pulses  $g(t)$  and  $g_ϕ(t)$ . Some of them are shown on the right.		Many CPM variants can be realized by appropriately choosing the pulses  $g(t)$  and  $g_ϕ(t)$ . Some of them are shown on the right.
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	*This results in a smoother phase response at point   '''(3)'''  than in  [[Modulation_Methods/~~Nonlinear_Digital_Modulation~~#~~MSK_.E2.80.93_Minimum_Shift_Keying~~\|$\text{MSK}$]],  where  $ϕ(t)$  rises / falls linearly within each symbol.		*This results in a smoother phase response at point   '''(3)'''  than in  [[Modulation_Methods/Non-Linear_Digital_Modulation#MSK_–_Minimum_Shift_Keying\|$\text{MSK}$]],  where  $ϕ(t)$  rises / falls linearly within each symbol.

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2026-03-16T13:47:08Z

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← Older revision		Revision as of 15:47, 16 March 2026
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	:$$s(t) = s_0 \cdot \cos\hspace{-0.05cm}\big [\psi(t)\big ] \hspace{0.5cm} {\rm with} \hspace{0.5cm} \psi(t) = 2\pi f_{\rm T} \hspace{0.05cm}t + K_{\rm FM} \cdot \int q(t)\hspace{0.1cm} {\rm d}t,$$		:$$s(t) = s_0 \cdot \cos\hspace{-0.05cm}\big [\psi(t)\big ] \hspace{0.5cm} {\rm with} \hspace{0.5cm} \psi(t) = 2\pi f_{\rm T} \hspace{0.05cm}t + K_{\rm FM} \cdot \int q(t)\hspace{0.1cm} {\rm d}t,$$
	*the rectangular binary source signal with  $a_ν ∈ \{+1, –1\}$   ⇒   bipolar signaling:		*the rectangular binary source signal with  $a_ν ∈ \{+1, –1\}$   ⇒   bipolar signaling:
	:$$q(t) = \sum_{\nu = - \infty}^{+\infty}a_{ \nu} \cdot g_s (t - \nu		:$$q(t) = \sum_{\nu = - \infty}^{+\infty}a_{ \nu} \cdot g_s (t - \nu\cdot T) \hspace{0.5cm} {\rm with} \hspace{0.5cm} g_s(t) = \left\{ \begin{array}{l} A \\ 0 \\ \end{array} \right.\quad \begin{array}{{5}c}{\rm{for} } \\{\rm{ } } \\ \end{array}\begin{array}{{10}c}\hspace{0.05cm}0 < t < T\hspace{0.05cm}, \\ {\rm otherwise} \hspace{0.05cm}. \\ \end{array}$$}}
	\cdot T) \hspace{0.5cm} {\rm with} \hspace{0.5cm} g_s(t) = \left\{ \begin{array}{l} A \\ 0 \\ \end{array} \right.\quad \begin{array}{{5}c}{\rm{for} } \\{\rm{ } } \\ \end{array}\begin{array}{{10}c}
	\hspace{0.05cm}0 < t < T\hspace{0.05cm}, \\ {\rm otherwise} \hspace{0.05cm}. \\ \end{array}$$}}


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	::$$s(t) = s_0 \cdot \cos\hspace{-0.05cm}\big [2 \pi \cdot t \cdot (f_{\rm T}+a_{ \nu} \cdot {\rm \Delta}f_{\rm A} ) \big ]\hspace{0.05cm}.$$		::$$s(t) = s_0 \cdot \cos\hspace{-0.05cm}\big [2 \pi \cdot t \cdot (f_{\rm T}+a_{ \nu} \cdot {\rm \Delta}f_{\rm A} ) \big ]\hspace{0.05cm}.$$
	*This can be calculated with the two possible signal frequencies		*This can be calculated with the two possible signal frequencies
	::$$f_{\rm +1} = f_{\rm T} +{\rm \Delta}f_{\rm A} \hspace{0.05cm},		::$$f_{\rm +1} = f_{\rm T} +{\rm \Delta}f_{\rm A} \hspace{0.05cm},\hspace{0.2cm}f_{\rm -1} = f_{\rm T} -{\rm \Delta}f_{\rm A}$$
	\hspace{0.2cm}f_{\rm -1} = f_{\rm T} -{\rm \Delta}f_{\rm A}$$
	:and written as:		:and written as:
	::$$s(t) = \left\{ \begin{array}{l} s_0 \cdot \cos (2 \pi \cdot f_{\rm +1} \cdot t ) \\ s_0 \cdot \cos (2 \pi \cdot f_{\rm -1} \cdot t ) \\ \end{array} \right.\quad \begin{array}{{5}c}{\rm{for}} \\{\rm{for}} \\ \end{array}\begin{array}{{10}c} \hspace{0.05cm}a_{ \nu} = +1 \hspace{0.05cm}, \\ \hspace{0.05cm}a_{ \nu} = -1\hspace{0.05cm}. \\ \end{array}$$		::$$s(t) = \left\{ \begin{array}{l} s_0 \cdot \cos (2 \pi \cdot f_{\rm +1} \cdot t ) \\ s_0 \cdot \cos (2 \pi \cdot f_{\rm -1} \cdot t ) \\ \end{array} \right.\quad \begin{array}{{5}c}{\rm{for}} \\{\rm{for}} \\ \end{array}\begin{array}{{10}c} \hspace{0.05cm}a_{ \nu} = +1 \hspace{0.05cm}, \\ \hspace{0.05cm}a_{ \nu} = -1\hspace{0.05cm}. \\ \end{array}$$
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	*is defined in the same way as for analog frequency modulation, namely		*is defined in the same way as for analog frequency modulation, namely
	*as the maximum deviation of the instantaneous frequency  $f_{\rm A}(t)$  from the carrier frequency  $f_{\rm T}$:		*as the maximum deviation of the instantaneous frequency  $f_{\rm A}(t)$  from the carrier frequency  $f_{\rm T}$:
	::$$ {\rm \Delta}f_{\rm A} =f_{\rm +1} - f_{\rm T}\hspace{0.05cm},		::$$ {\rm \Delta}f_{\rm A} =f_{\rm +1} - f_{\rm T}\hspace{0.05cm},\hspace{0.2cm}f_{\rm -1} = f_{\rm T} -f_{\rm -1}.$$
	\hspace{0.2cm}f_{\rm -1} = f_{\rm T} -f_{\rm -1}.$$

	Sometimes,  the  »'''total frequency deviation'''«  $ 2 \cdot {\rm \Delta}f_{\rm A} =f_{\rm +1} - f_{\rm -1}$  is used in the literature. }}		Sometimes,  the  »'''total frequency deviation'''«  $ 2 \cdot {\rm \Delta}f_{\rm A} =f_{\rm +1} - f_{\rm -1}$  is used in the literature. }}
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	#This means:  $a_ν = +1$  was probably sent if the following condition is satisfied:		#This means:  $a_ν = +1$  was probably sent if the following condition is satisfied:
	::$$d_{\rm +1}(\nu \cdot T) > d_{\rm -1}(\nu \cdot T) $$		::$$d_{\rm +1}(\nu \cdot T) > d_{\rm -1}(\nu \cdot T) $$
	::$$\Rightarrow \hspace{0.3cm}		::$$\Rightarrow \hspace{0.3cm}d(\nu \cdot T) = d_{\rm +1}(\nu \cdot T) - d_{\rm -1}(\nu \cdotT) > 0\hspace{0.05cm}.$$
	d(\nu \cdot T) = d_{\rm +1}(\nu \cdot T) - d_{\rm -1}(\nu \~~cdot~~
	T) > 0\hspace{0.05cm}.$$

	The upper block diagram has been drawn according to this description for better understanding.		The upper block diagram has been drawn according to this description for better understanding.
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	then the bit error probability is given by:		then the bit error probability is given by:
	:$$p_{\rm B} = {\rm Q}\left ( \sqrt{{E_{\rm B}}/{N_0 }} \hspace{0.1cm}\right		:$$p_{\rm B} = {\rm Q}\left ( \sqrt{{E_{\rm B}}/{N_0 }} \hspace{0.1cm}\right) = {1}/{2}\cdot {\rm erfc}\left ( \sqrt{{E_{\rm B}}/(2 N_0 }) \hspace{0.1cm}\right ).$$
	) = {1}/{2}\cdot {\rm erfc}\left ( \sqrt{{E_{\rm B}}/(2 N_0 }) \hspace{0.1cm}\right ).$$

	This corresponds to a degradation of  $3 \ \rm dB$  compared to [[Modulation_Methods/Linear_Digital_Modulation#Error_probabilities_-_a_brief_overview\|$\text{BSPK}$]]  because		This corresponds to a degradation of  $3 \ \rm dB$  compared to [[Modulation_Methods/Linear_Digital_Modulation#Error_probabilities_-_a_brief_overview\|$\text{BSPK}$]]  because
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	*the digital source signal at point  '''(1)''',  a sequence of Dirac delta pulses spaced by  $T$,  weighted by the coefficients  $a_ν ∈ \{–1, +1\}$:		*the digital source signal at point  '''(1)''',  a sequence of Dirac delta pulses spaced by  $T$,  weighted by the coefficients  $a_ν ∈ \{–1, +1\}$:
	:$$q_\delta(t) = \sum_{\nu = - \infty}^{+\infty}a_{ \nu} \cdot \delta (t - \nu		:$$q_\delta(t) = \sum_{\nu = - \infty}^{+\infty}a_{ \nu} \cdot \delta (t - \nu\cdot T)\hspace{0.05cm};$$
	\cdot T)\hspace{0.05cm};$$
	*the rectangular signal  $q_{\rm R}(t)$  at point  '''(2)'''  after convolution with the rectangular pulse $g(t)$  of duration  $T$  and height  $1/T$:		*the rectangular signal  $q_{\rm R}(t)$  at point  '''(2)'''  after convolution with the rectangular pulse $g(t)$  of duration  $T$  and height  $1/T$:
	:$$q_{\rm R}(t) = \sum_{\nu = - \infty}^{+\infty}a_{ \nu} \cdot g (t - \nu		:$$q_{\rm R}(t) = \sum_{\nu = - \infty}^{+\infty}a_{ \nu} \cdot g (t - \nu\cdot T)\hspace{0.05cm},$$
	\cdot T)\hspace{0.05cm},$$
	*the  [[Modulation_Methods/Frequency_Modulation_(FM)#Signal_characteristics_with_frequency_modulation\|$\text{frequency modulator}$]]  (integrator and following phase modulator).  For the signal at point '''(3)''':		*the  [[Modulation_Methods/Frequency_Modulation_(FM)#Signal_characteristics_with_frequency_modulation\|$\text{frequency modulator}$]]  (integrator and following phase modulator).  For the signal at point '''(3)''':
	:$$\phi(t) = {\pi}/{2}\cdot \int_{0}^{t}		:$$\phi(t) = {\pi}/{2}\cdot \int_{0}^{t}q_{\rm R}(\tau)\hspace{0.1cm} {\rm d}\tau \hspace{0.05cm}.$$
	q_{\rm R}(\tau)\hspace{0.1cm} {\rm d}\tau \hspace{0.05cm}.$$
	:The phase values at multiples of the symbol duration  $T$  are multiples of  $π/2$, when taking into account the modulation index  $h = 0.5$  for the MSK method. The phase response is linear.		:The phase values at multiples of the symbol duration  $T$  are multiples of  $π/2$, when taking into account the modulation index  $h = 0.5$  for the MSK method. The phase response is linear.
	*From this,  the MSK signal at point  '''(4)'''  of the diagram is given by:		*From this,  the MSK signal at point  '''(4)'''  of the diagram is given by:
	:$$s(t) = s_0 \cdot \cos (2 \pi f_{\rm T} \hspace{0.05cm}t +		:$$s(t) = s_0 \cdot \cos (2 \pi f_{\rm T} \hspace{0.05cm}t +\phi(t)) $$
	\phi(t)) $$
	:$$\Rightarrow \hspace{0.3cm} s(t) = s_0 \cdot \cos (2 \pi \cdot t \cdot (f_{\rm T}+a_{ \nu} \cdot {\rm \Delta}f_{\rm A} )) \hspace{0.05cm}.$$		:$$\Rightarrow \hspace{0.3cm} s(t) = s_0 \cdot \cos (2 \pi \cdot t \cdot (f_{\rm T}+a_{ \nu} \cdot {\rm \Delta}f_{\rm A} )) \hspace{0.05cm}.$$

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	The  »'''phase function'''«  $ϕ(t)$  in  "Continuous Phase Modulation"  $\rm (CPM)$  can be generally represented in the following form   $(h$  denotes the modulation index$)$:		The  »'''phase function'''«  $ϕ(t)$  in  "Continuous Phase Modulation"  $\rm (CPM)$  can be generally represented in the following form   $(h$  denotes the modulation index$)$:
	:$$\phi(t) = {\pi}\cdot h \cdot\int_{-\infty}^{t}		:$$\phi(t) = {\pi}\cdot h \cdot\int_{-\infty}^{t}\sum_{\nu = - \infty}^{+\infty}a_{ \nu} \cdot g (\tau - \nu \cdotT)\hspace{0.1cm} {\rm d}\tau \hspace{0.05cm}.$$
	\sum_{\nu = - \infty}^{+\infty}a_{ \nu} \cdot g (\tau - \nu \~~cdot~~
	T)\hspace{0.1cm} {\rm d}\tau \hspace{0.05cm}.$$
	*In this representation,  $g(t)$  denotes the  »'''frequency pulse'''«,  which must satisfy the following condition:		*In this representation,  $g(t)$  denotes the  »'''frequency pulse'''«,  which must satisfy the following condition:
	:$$\int_{-\infty}^{+\infty} g (t)\hspace{0.1cm} {\rm d}t = 1 \hspace{0.05cm}.$$		:$$\int_{-\infty}^{+\infty} g (t)\hspace{0.1cm} {\rm d}t = 1 \hspace{0.05cm}.$$
	*But the following relationship also holds with the   »'''phase pulse'''«   $g_ϕ(t)$:		*But the following relationship also holds with the   »'''phase pulse'''«   $g_ϕ(t)$:
	:$$\phi(t) = {\pi}\cdot h \cdot		:$$\phi(t) = {\pi}\cdot h \cdot\sum_{\nu = - \infty}^{+\infty}a_{ \nu} \cdot g_\phi (t - \nu\cdot T),\hspace{0.2cm}{\rm where}\hspace{0.2cm}g_\phi(t) =\int_{-\infty}^{t} g (\tau )\hspace{0.1cm} {\rmd}\tau\hspace{0.05cm}.$$
	\sum_{\nu = - \infty}^{+\infty}a_{ \nu} \cdot g_\phi (t - \nu
	\cdot T),\hspace{0.2cm}{\rm where}\hspace{0.2cm}g_\phi(t) =
	\int_{-\infty}^{t} g (\tau )\hspace{0.1cm} {\rm
	d}\tau\hspace{0.05cm}.$$

	Many CPM variants can be realized by appropriately choosing the pulses  $g(t)$  and  $g_ϕ(t)$ . Some of them are shown on the right.		Many CPM variants can be realized by appropriately choosing the pulses  $g(t)$  and  $g_ϕ(t)$ . Some of them are shown on the right.
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	*One achieves the smoother GMSK phase transitions by using a  [[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory#Gaussian_low-pass_filter\|$\text{Gaussian low-pass}$]].  With the system-theoretic cutoff frequency  $f_{\rm G}$,  its frequency response and impulse response are as follows:		*One achieves the smoother GMSK phase transitions by using a  [[Linear_and_Time_Invariant_Systems/Some_Low-Pass_Functions_in_Systems_Theory#Gaussian_low-pass_filter\|$\text{Gaussian low-pass}$]].  With the system-theoretic cutoff frequency  $f_{\rm G}$,  its frequency response and impulse response are as follows:
	:$$H_{\rm G}(f) = {\rm e}^{-\pi\cdot (\frac{f}{2 \cdot f_{\rm G}})^2} \hspace{0.2cm}\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\, \hspace{0.2cm}		:$$H_{\rm G}(f) = {\rm e}^{-\pi\cdot (\frac{f}{2 \cdot f_{\rm G}})^2} \hspace{0.2cm}\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\, \hspace{0.2cm}h_{\rm G}(t) = 2 f_{\rm G} \cdot {\rm e}^{-\pi\cdot (2 \cdot f_{\rm G}\cdot t)^2}\hspace{0.05cm}.$$
	h_{\rm G}(t) = 2 f_{\rm G} \cdot {\rm e}^{-\pi\cdot (2 \cdot f_{\rm G}\cdot t)^2}\hspace{0.05cm}.$$

	*The resulting frequency pulse  $g(t)$  at point  '''(2)'''  is obtained by convolution of  $g_{\rm R}(t)$   and   $h_{\rm G}(t)$.		*The resulting frequency pulse  $g(t)$  at point  '''(2)'''  is obtained by convolution of  $g_{\rm R}(t)$   and   $h_{\rm G}(t)$.
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	*between the  "system-theoretic cutoff frequency"  $f_\text{G}$		*between the  "system-theoretic cutoff frequency"  $f_\text{G}$
	*and the  "3dB cutoff frequency"  $f_\text{3dB}$:		*and the  "3dB cutoff frequency"  $f_\text{3dB}$:
	:$$H_{\rm G}(f= f_{\rm 3 \hspace{0.03cm}dB}) = {\rm e}^{-\pi\cdot ({f_{\rm 3 \hspace{0.03cm}{dB} } }/{2 f_{\rm G} })^2} =		:$$H_{\rm G}(f= f_{\rm 3 \hspace{0.03cm}dB}) = {\rm e}^{-\pi\cdot ({f_{\rm 3 \hspace{0.03cm}{dB} } }/{2 f_{\rm G} })^2} ={1}/{\sqrt{2} }\hspace{0.3cm} \Rightarrow\hspace{0.3cm}f_{\rm 3 dB} = f_{\rm G} \cdot \sqrt { {4}/{\pi}\cdot {\rm ln}\sqrt{2} }\approx {2}/{3}\cdot f_{\rm G}\hspace{0.05cm}.$$
	{1}/{\sqrt{2} }\hspace{0.3cm} \Rightarrow\hspace{0.3cm}
	f_{\rm 3 dB} = f_{\rm G} \cdot \sqrt { {4}/{\pi}\cdot {\rm ln
	}\sqrt{2} }\approx {2}/{3}\cdot f_{\rm G}
	\hspace{0.05cm}.$$
	From  $f_{\rm 3dB} = 0.3/T$   ⇒  $f_{\rm G} ≈ 0.45/T$. }}		From  $f_{\rm 3dB} = 0.3/T$   ⇒  $f_{\rm G} ≈ 0.45/T$. }}

Maintenance script: Add German interlanguage link

2026-03-16T12:28:14Z

Add German interlanguage link

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

Guenter at 14:21, 17 January 2023

2023-01-17T14:21:03Z

← Older revision		Revision as of 16:21, 17 January 2023
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	*This type of modulation is used,  for example,  in the [[Examples_of_Communication_Systems/General_Description_of_GSM\|$\text{GSM}$]]  mobile communications standard.		*This type of modulation is used,  for example,  in the [[Examples_of_Communication_Systems/General_Description_of_GSM\|$\text{GSM}$]]  mobile communications standard.

	[[File:~~EN_Mod_T_4_4_S9~~.png \|right\|frame\| Signal waveforms in Gaussian Minimum Shift Keying  $\rm (GMSK)$ ~~'''KORREKTUR: Phasenmodulator'''~~]]		[[File:EN_Mod_T_4_4_S9_v3.png \|right\|frame\| Signal waveforms in Gaussian Minimum Shift Keying  $\rm (GMSK)$]]

Guenter at 16:55, 16 January 2023

2023-01-16T16:55:26Z

← Older revision		Revision as of 18:55, 16 January 2023
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	*This type of modulation is used,  for example,  in the [[Examples_of_Communication_Systems/General_Description_of_GSM\|$\text{GSM}$]]  mobile communications standard.		*This type of modulation is used,  for example,  in the [[Examples_of_Communication_Systems/General_Description_of_GSM\|$\text{GSM}$]]  mobile communications standard.

	[[File:~~EN_Mod_T_4_4_S9_v2~~.png \|right\|frame\| Signal waveforms in Gaussian Minimum Shift Keying  $\rm (GMSK)$]]		[[File:EN_Mod_T_4_4_S9.png \|right\|frame\| Signal waveforms in Gaussian Minimum Shift Keying  $\rm (GMSK)$ '''KORREKTUR: Phasenmodulator''']]

Guenter at 16:45, 16 January 2023

2023-01-16T16:45:06Z

← Older revision		Revision as of 18:45, 16 January 2023
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	*This type of modulation is used,  for example,  in the [[Examples_of_Communication_Systems/General_Description_of_GSM\|$\text{GSM}$]]  mobile communications standard.		*This type of modulation is used,  for example,  in the [[Examples_of_Communication_Systems/General_Description_of_GSM\|$\text{GSM}$]]  mobile communications standard.

	[[File:~~EN_Mod_T_4_4_S9~~.png \|right\|frame\| Signal waveforms in Gaussian Minimum Shift Keying  $\rm (GMSK)$ ~~'''KORREKTUR: Phasenmodulator'''~~]]		[[File:EN_Mod_T_4_4_S9_v2.png \|right\|frame\| Signal waveforms in Gaussian Minimum Shift Keying  $\rm (GMSK)$]]

Guenter at 13:54, 13 January 2023

2023-01-13T13:54:02Z

Hwang at 15:23, 11 January 2023

2023-01-11T15:23:04Z

← Older revision		Revision as of 17:23, 11 January 2023
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	[[File:EN_Mod_T_4_4_S6.png \| right\|frame\| Block diagram for generating a MSK signal]]		[[File:EN_Mod_T_4_4_S6.png \| right\|frame\| Block diagram for generating a MSK signal]]

	*the digital source signal at point  '''(1)''',  a sequence of Dirac impulses '''KORREKTUR: delta pulses''' spaced by  $T$,  weighted by the coefficients  $a_ν ∈ \{–1, +1\}$:		*the digital source signal at point  '''(1)''',  a sequence of Dirac impulses '''KORREKTUR: Dirac delta pulses''' spaced by  $T$,  weighted by the coefficients  $a_ν ∈ \{–1, +1\}$:
	:$$q_\delta(t) = \sum_{\nu = - \infty}^{+\infty}a_{ \nu} \cdot \delta (t - \nu		:$$q_\delta(t) = \sum_{\nu = - \infty}^{+\infty}a_{ \nu} \cdot \delta (t - \nu
	\cdot T)\hspace{0.05cm};$$		\cdot T)\hspace{0.05cm};$$

Hwang at 13:39, 11 January 2023

2023-01-11T13:39:19Z

← Older revision		Revision as of 15:39, 11 January 2023
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	*This type of modulation is used,  for example,  in the [[Examples_of_Communication_Systems/General_Description_of_GSM\|$\text{GSM}$]]  mobile communications standard.		*This type of modulation is used,  for example,  in the [[Examples_of_Communication_Systems/General_Description_of_GSM\|$\text{GSM}$]]  mobile communications standard.

	[[File:EN_Mod_T_4_4_S9.png \|right\|frame\| Signal waveforms in Gaussian Minimum Shift Keying  $\rm (GMSK)$]]		[[File:EN_Mod_T_4_4_S9.png \|right\|frame\| Signal waveforms in Gaussian Minimum Shift Keying  $\rm (GMSK)$ '''KORREKTUR: Phasenmodulator''']]

Hwang at 13:38, 11 January 2023

2023-01-11T13:38:13Z

← Older revision		Revision as of 15:38, 11 January 2023
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	The spectral calculation of CPM procedures is complicated in general.  Only the special case  "MSK"  leads to equations which are easy to handle,  as shown in  [[Aufgaben:Exercise_4.14:_Phase_Progression_of_the_MSK\|"Exercise 4.14"]].		The spectral calculation of CPM procedures '''KORREKTUR: methods''' is complicated in general.  Only the special case  "MSK"  leads to equations which are easy to handle,  as shown in  [[Aufgaben:Exercise_4.14:_Phase_Progression_of_the_MSK\|"Exercise 4.14"]].

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← Older revision		Revision as of 15:54, 13 January 2023
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	This demodulator operates according to following principle  (see upper arrangement):		This demodulator operates according to following principle  (see upper arrangement):
	#We are dealing with a [[Digital_Signal_Transmission/Optimal_Receiver_Strategies#Maximum-a-posteriori_and_maximum.E2.80.93likelihood_decision_rule\|$\text{maximum–likelihood receiver}$]]  $\rm (ML)$  with  [[Theory_of_Stochastic_Signals/Matched_Filter\|$\text{~~Matched-Filter~~}$]]  ~~'''KORREKTUR: matched filter'''~~ realization.  This filter with frequency response $H_{\rm MF}(f)$  can also be realized as an integrator with rectangular basic transmission pulse  $g_s(t)$.		#We are dealing with a [[Digital_Signal_Transmission/Optimal_Receiver_Strategies#Maximum-a-posteriori_and_maximum.E2.80.93likelihood_decision_rule\|$\text{maximum–likelihood receiver}$]]  $\rm (ML)$  with  [[Theory_of_Stochastic_Signals/Matched_Filter\|$\text{matched filter}$]]  realization.  This filter with frequency response $H_{\rm MF}(f)$  can also be realized as an integrator with rectangular basic transmission pulse  $g_s(t)$.
	#Both signals  $b_{+1}(t)$  and  $b_{–1}(t)$  before their corresponding ~~matched–filters '''KORREKTUR:~~ matched filters~~'''~~ are obtained by phase-appropriate multiplication with the oscillations of the frequencies  $f_{+1}$  and  $f_{–1}$,  respectively.		#Both signals  $b_{+1}(t)$  and  $b_{–1}(t)$  before their corresponding matched filters are obtained by phase-appropriate multiplication with the oscillations of the frequencies  $f_{+1}$  and  $f_{–1}$,  respectively.
	#The maximum-likelihood receiver is known to decide on the branch  (for the symbol)  with the larger  "metric",  taking into account the following matched filter.		#The maximum-likelihood receiver is known to decide on the branch  (for the symbol)  with the larger  "metric",  taking into account the following matched filter.
	#This means:  $a_ν = +1$  was probably sent if the following condition is satisfied:		#This means:  $a_ν = +1$  was probably sent if the following condition is satisfied:
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	[[File:EN_Mod_T_4_4_S6.png \| right\|frame\| Block diagram for generating a MSK signal]]		[[File:EN_Mod_T_4_4_S6.png \| right\|frame\| Block diagram for generating a MSK signal]]

	*the digital source signal at point  '''(1)''',  a sequence of ~~Dirac impulses '''KORREKTUR:~~ Dirac delta pulses~~'''~~ spaced by  $T$,  weighted by the coefficients  $a_ν ∈ \{–1, +1\}$:		*the digital source signal at point  '''(1)''',  a sequence of Dirac delta pulses spaced by  $T$,  weighted by the coefficients  $a_ν ∈ \{–1, +1\}$:
	:$$q_\delta(t) = \sum_{\nu = - \infty}^{+\infty}a_{ \nu} \cdot \delta (t - \nu		:$$q_\delta(t) = \sum_{\nu = - \infty}^{+\infty}a_{ \nu} \cdot \delta (t - \nu
	\cdot T)\hspace{0.05cm};$$		\cdot T)\hspace{0.05cm};$$
Line 295:		Line 295:


	The spectral calculation of CPM ~~procedures '''KORREKTUR:~~ methods~~'''~~ is complicated in general.  Only the special case  "MSK"  leads to equations which are easy to handle,  as shown in  [[Aufgaben:Exercise_4.14:_Phase_Progression_of_the_MSK\|"Exercise 4.14"]].		The spectral calculation of CPM methods is complicated in general.  Only the special case  "MSK"  leads to equations which are easy to handle,  as shown in  [[Aufgaben:Exercise_4.14:_Phase_Progression_of_the_MSK\|"Exercise 4.14"]].

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