Applets:Principle of Pseudo-Ternary Coding - Revision history

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	*The actual encoding from binary  $(M_q = 2)$  to ternary  $(M = M_c = 3)$  is done by the  "linear coding network"  by the conventional subtraction		*The actual encoding from binary  $(M_q = 2)$  to ternary  $(M = M_c = 3)$  is done by the  "linear coding network"  by the conventional subtraction
	:$$c(t) ={1}/{2} \cdot \big [b(t) - K_{\rm C} \cdot b(t- N_{\~~rmC~~}\cdot T)\big] \in \{-1, \ 0, +1\}\hspace{0.05cm},$$		:$$c(t) ={1}/{2} \cdot \big [b(t) - K_{\rm C} \cdot b(t- N_{\rm C}\cdot T)\big] \in \{-1, \ 0, +1\}\hspace{0.05cm},$$

	:which can be described by the following  [[Linear_and_Time_Invariant_Systems/System_Description_in_Time_Domain#Impulse_response\|"impulse response"]]  resp.  [[Linear_and_Time_Invariant_Systems/System_Description_in_Frequency_Domain#Frequency_response_.E2.80.93_Transfer_function\|"transfer function"]]  with respect to the input signal  $b(t)$  and the output signal  $c(t)$:		:which can be described by the following  [[Linear_and_Time_Invariant_Systems/System_Description_in_Time_Domain#Impulse_response\|"impulse response"]]  resp.  [[Linear_and_Time_Invariant_Systems/System_Description_in_Frequency_Domain#Frequency_response_.E2.80.93_Transfer_function\|"transfer function"]]  with respect to the input signal  $b(t)$  and the output signal  $c(t)$:
	:$$h_{\rm C}(t) = {1}/{2} \cdot \big [\delta(t) - K_{\rm C} \cdot \delta(t- N_{\~~rmC~~}\cdot T)\big] \ \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\ \ \ H_{\rm C}(f) ={1}/{2} \cdot \left [1 - K_{\rm C} \cdot {\rm e}^{- {\rm j}\hspace{0.05cm}\cdot\hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}N_{\rm C}\hspace{0.05cm}\cdot \hspace{0.05cm}T}\right]\hspace{0.05cm}. $$		:$$h_{\rm C}(t) = {1}/{2} \cdot \big [\delta(t) - K_{\rm C} \cdot \delta(t- N_{\rm C}\cdot T)\big] \ \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\ \ \ H_{\rm C}(f) ={1}/{2} \cdot \left [1 - K_{\rm C} \cdot {\rm e}^{- {\rm j}\hspace{0.05cm}\cdot\hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}N_{\rm C}\hspace{0.05cm}\cdot \hspace{0.05cm}T}\right]\hspace{0.05cm}. $$

	*The relative redundancy is the same for all pseudo-ternary codes.  Substituting  $M_q=2$,  $M_c=3$  and  $T_c =T_q$  into the  [[Digital_Signal_Transmission/Redundancy-Free_Coding#Symbolwise_coding_vs._blockwise_coding\|"general definition equation"]],  we obtain		*The relative redundancy is the same for all pseudo-ternary codes.  Substituting  $M_q=2$,  $M_c=3$  and  $T_c =T_q$  into the  [[Digital_Signal_Transmission/Redundancy-Free_Coding#Symbolwise_coding_vs._blockwise_coding\|"general definition equation"]],  we obtain
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	:$${\it \Phi}_s(f) = {\it \Phi}_a(f) \cdot {1}/{T} \cdot\|G_s(f)\|^2 \hspace{0.05cm}.$$		:$${\it \Phi}_s(f) = {\it \Phi}_a(f) \cdot {1}/{T} \cdot\|G_s(f)\|^2 \hspace{0.05cm}.$$
	*The first term  ${\it \Phi}_a(f)$  is dimensionless and describes the spectral shaping of the transmitted signal by the statistical relations of the source:<br>		*The first term  ${\it \Phi}_a(f)$  is dimensionless and describes the spectral shaping of the transmitted signal by the statistical relations of the source:<br>
	:$$\varphi_a(\lambda) \hspace{0.4cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet \hspace{0.4cm}{\it \Phi}_a(f) = \sum_{\lambda =-\infty}^{+\infty}\varphi_a(\lambda)\cdot {\rm e}^{- {\~~rmj~~}\hspace{0.05cm} 2 \pi f \hspace{0.02cm} \lambda \hspace{0.02cm}T} =\varphi_a(0) + 2 \cdot \sum_{\lambda =1}^{\infty}\varphi_a(\lambda)\cdot\cos ( 2 \pi f\lambda T) \hspace{0.05cm}.$$		:$$\varphi_a(\lambda) \hspace{0.4cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet \hspace{0.4cm}{\it \Phi}_a(f) = \sum_{\lambda =-\infty}^{+\infty}\varphi_a(\lambda)\cdot {\rm e}^{- {\rm j}\hspace{0.05cm} 2 \pi f \hspace{0.02cm} \lambda \hspace{0.02cm}T} =\varphi_a(0) + 2 \cdot \sum_{\lambda =1}^{\infty}\varphi_a(\lambda)\cdot\cos ( 2 \pi f\lambda T) \hspace{0.05cm}.$$
	*${\it \Phi^{^{\hspace{0.08cm}\bullet}}_{g}}(f)$  takes into account the spectral shaping by  $g(t)$. The narrower this is, the wider  $\vert G(f) \vert^2$  and thus the larger the bandwidth requirement:		*${\it \Phi^{^{\hspace{0.08cm}\bullet}}_{g}}(f)$  takes into account the spectral shaping by  $g(t)$. The narrower this is, the wider  $\vert G(f) \vert^2$  and thus the larger the bandwidth requirement:
	:$$\varphi^{^{\hspace{0.05cm}\bullet}}_{g}(\tau) \hspace{0.4cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet \hspace{0.4cm}{\it \Phi}^{^{\hspace{0.08cm}\bullet}}_{g}(f) = \|G(f)\|^2\hspace{0.05cm}.$$		:$$\varphi^{^{\hspace{0.05cm}\bullet}}_{g}(\tau) \hspace{0.4cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet \hspace{0.4cm}{\it \Phi}^{^{\hspace{0.08cm}\bullet}}_{g}(f) = \|G(f)\|^2\hspace{0.05cm}.$$
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	<br>		<br>
	The frequency response of the linear code network of a pseudo-ternary code is generally:		The frequency response of the linear code network of a pseudo-ternary code is generally:
	:$$H_{\rm C}(f) = {1}/{2} \cdot \big [1 - K_{\rm C} \cdot {\rm		:$$H_{\rm C}(f) = {1}/{2} \cdot \big [1 - K_{\rm C} \cdot {\rm e}^{-{\rm j}\hspace{0.03cm}\cdot \hspace{0.03cm}2\pi\hspace{0.03cm}\cdot \hspace{0.03cm}f \hspace{0.03cm}\cdot\hspace{0.03cm} N_{\rm C}\hspace{0.03cm}\cdot \hspace{0.03cm}T}\big] ={1}/{2} \cdot \big [1 - K \cdot {\rm e}^{-{\rm j}\hspace{0.03cm}\cdot \hspace{0.03cm}\alpha}\big ]\hspace{0.05cm}.$$This gives the power-spectral density  $\rm (PSD)$  of the amplitude coefficients  $(K$  and  $\alpha$  are abbreviations according to the above equation$)$::$$ {\it \Phi}_a(f) = \| H_{\rm C}(f)\|^2 = \frac{\big [1 - K \cos(\alpha) + {\rm j}\cdot K \sin (\alpha) \big ] \big [1 - K \cos(\alpha) - {\rm j}\cdot K \sin (\alpha) \big ] }{4} = \text{...} = {1}/{4} \cdot \big [2 - 2 \cdot K \cdot \cos(\alpha) \big ] $$[[File:P_ID1347__Dig_T_2_4_S2b_v2.png\|right\|frame\|Power-spectral density of the AMI code\|class=fit]]:$$ \Rightarrow \hspace{0.3cm}{\it \Phi}_a(f) = \| H_{\rm C}(f)\|^2 = {1}/{2} \cdot \big [1 - K_{\rm C} \cdot \cos(2\pi f N_{\rm C} T)\big ]\hspace{0.4cm}\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ \hspace{0.4cm}\varphi_a(\lambda \cdot T)\hspace{0.05cm}.$$In particular,  for the power-spectral density of the AMI code  $(N_{\rm C} = K_{\rm C} = 1)$,  we obtain::$${\it \Phi}_a(f) = {1}/{2} \cdot \big [1 - \cos(2\pi f T)\big ] = \sin^2(\pi f T)\hspace{0.05cm}.$$
	e}^{-{\rm j}\hspace{0.03cm}\cdot \hspace{0.03cm}
	2\pi\hspace{0.03cm}\cdot \hspace{0.03cm}f \hspace{0.03cm}\cdot
	\hspace{0.03cm} N_{\rm C}\hspace{0.03cm}\cdot \hspace{0.03cm}T}
	\big] ={1}/{2} \cdot \big [1 - K \cdot {\rm
	e}^{-{\rm j}\hspace{0.03cm}\cdot \hspace{0.03cm}
	\alpha}
	\big ]\hspace{0.05cm}.$$This gives the power-spectral density  $\rm (PSD)$  of the amplitude coefficients  $(K$  and  $\alpha$  are abbreviations according to the above equation$)$::$$ {\it \Phi}_a(f) = \| H_{\rm C}(f)\|^2 = \frac{\big [1 - K \cos
	(\alpha) + {\rm j}\cdot K \sin (\alpha) \big ] \big [1 - K \cos
	(\alpha) - {\rm j}\cdot K \sin (\alpha) \big ] }{4} = \text{...} = {1}/{4} \cdot \big [2 - 2 \cdot K \cdot \cos
	(\alpha) \big ] $$[[File:P_ID1347__Dig_T_2_4_S2b_v2.png\|right\|frame\|Power-spectral density of the AMI code\|class=fit]]:$$ \Rightarrow \hspace{0.3cm}{\it \Phi}_a(f) = \| H_{\rm C}(f)\|^2 = {1}/{2} \cdot \big [1 - K_{\rm C} \cdot \cos
	(2\pi f N_{\rm C} T)\big ]
	\hspace{0.4cm}\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ \hspace{0.4cm}
	\varphi_a(\lambda \cdot T)\hspace{0.05cm}.$$In particular,  for the power-spectral density of the AMI code  $(N_{\rm C} = K_{\rm C} = 1)$,  we obtain::$${\it \Phi}_a(f) = {1}/{2} \cdot \big [1 - \cos
	(2\pi f T)\big ] = \sin^2
	(\pi f T)\hspace{0.05cm}.$$

	The graph shows		The graph shows
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	==About the Authors==		==About the Authors==
	This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ Institute for Communications Engineering]  at the  [https://www.tum.de/en Technical University of Munich].		This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ Institute for Communications Engineering]  at the  [https://www.tum.de/en Technical University of Munich].
	*The first version was created in 2010 by  [[~~Biografien_und_Bibliografien~~/~~An_LNTwww_beteiligte_Studierende~~#Stefan_M.C3.BCller_.28Diplomarbeit_LB_2010.29\|Stefan Müller]]    as part of his diploma thesis with “FlashMX – Actionscript” (Supervisor: [[Biographies_and_Bibliographies/~~An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten~~#Prof._Dr.-Ing._habil.~~_G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29~~\|Günter Söder]]).		*The first version was created in 2010 by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Stefan_M.C3.BCller_.28Diplomarbeit_LB_2010.29\|Stefan Müller]]    as part of his diploma thesis with “FlashMX – Actionscript” (Supervisor: [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._Günter_Söder_(at_LNT_from_1974-2024)\|Günter Söder]]).
	*In 2020 the program was redesigned by  [[Biographies_and_Bibliographies/~~An_LNTwww_beteiligte_Studierende~~#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29\|Carolin Mirschina]]  as part of her bachelor thesis  (Supervisor: [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29\|Tasnád Kernetzky]] ) via "HTML5".		*In 2020 the program was redesigned by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29\|Carolin Mirschina]]  as part of her bachelor thesis  (Supervisor: [[Biographies_and_Bibliographies/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29\|Tasnád Kernetzky]] ) via "HTML5".
	*Last revision and English version 2021 by  [[~~Biografien_und_Bibliografien~~/~~An_LNTwww_beteiligte_Studierende~~#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29\|Carolin Mirschina]]  in the context of a working student activity.		*Last revision and English version 2021 by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29\|Carolin Mirschina]]  in the context of a working student activity.

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2026-03-16T13:49:04Z

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← Older revision		Revision as of 15:49, 16 March 2026
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	*The  "non-linear pre-encoder"  obtains the precoded symbols  $b_\nu$,  which are also binary,  by a modulo–2 addition  ("antivalence")  between the symbols  $q_\nu$  and  $K_{\rm C} \cdot b_{\nu-N_{\rm C}} $:		*The  "non-linear pre-encoder"  obtains the precoded symbols  $b_\nu$,  which are also binary,  by a modulo–2 addition  ("antivalence")  between the symbols  $q_\nu$  and  $K_{\rm C} \cdot b_{\nu-N_{\rm C}} $:
	:$$q_\nu \in \{-1, +1\},\hspace{0.1cm} K_{\rm C} \in \{-1,		:$$q_\nu \in \{-1, +1\},\hspace{0.1cm} K_{\rm C} \in \{-1,+1\}\hspace{0.3cm}\Rightarrow \hspace{0.3cm}b_\nu \in \{-1,+1\}\hspace{0.05cm}.$$
	+1\}\hspace{0.3cm}\Rightarrow \hspace{0.3cm}b_\nu \in \{-1,
	+1\}\hspace{0.05cm}.$$
	*Like the source symbols  $q_\nu$,  the symbols  $b_\nu$  are statistically independent of each other.  Thus,  the pre-encoder does not add any redundancy.  However,  it allows a simpler realization of the decoder and prevents error propagation after a transmission error.<br>		*Like the source symbols  $q_\nu$,  the symbols  $b_\nu$  are statistically independent of each other.  Thus,  the pre-encoder does not add any redundancy.  However,  it allows a simpler realization of the decoder and prevents error propagation after a transmission error.<br>

	*The actual encoding from binary  $(M_q = 2)$  to ternary  $(M = M_c = 3)$  is done by the  "linear coding network"  by the conventional subtraction		*The actual encoding from binary  $(M_q = 2)$  to ternary  $(M = M_c = 3)$  is done by the  "linear coding network"  by the conventional subtraction
	:$$c(t) ={1}/{2} \cdot \big [b(t) - K_{\rm C} \cdot b(t- N_{\rm		:$$c(t) ={1}/{2} \cdot \big [b(t) - K_{\rm C} \cdot b(t- N_{\rmC}\cdot T)\big] \in \{-1, \ 0, +1\}\hspace{0.05cm},$$
	C}\cdot T)\big] \in \{-1, \ 0, +1\}\hspace{0.05cm},$$

	:which can be described by the following  [[Linear_and_Time_Invariant_Systems/System_Description_in_Time_Domain#Impulse_response\|"impulse response"]]  resp.  [[Linear_and_Time_Invariant_Systems/System_Description_in_Frequency_Domain#Frequency_response_.E2.80.93_Transfer_function\|"transfer function"]]  with respect to the input signal  $b(t)$  and the output signal  $c(t)$:		:which can be described by the following  [[Linear_and_Time_Invariant_Systems/System_Description_in_Time_Domain#Impulse_response\|"impulse response"]]  resp.  [[Linear_and_Time_Invariant_Systems/System_Description_in_Frequency_Domain#Frequency_response_.E2.80.93_Transfer_function\|"transfer function"]]  with respect to the input signal  $b(t)$  and the output signal  $c(t)$:
	:$$h_{\rm C}(t) = {1}/{2} \cdot \big [\delta(t) - K_{\rm C} \cdot \delta(t- N_{\rm		:$$h_{\rm C}(t) = {1}/{2} \cdot \big [\delta(t) - K_{\rm C} \cdot \delta(t- N_{\rmC}\cdot T)\big] \ \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\ \ \ H_{\rm C}(f) ={1}/{2} \cdot \left [1 - K_{\rm C} \cdot {\rm e}^{- {\rm j}\hspace{0.05cm}\cdot\hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}N_{\rm C}\hspace{0.05cm}\cdot \hspace{0.05cm}T}\right]\hspace{0.05cm}. $$
	C}\cdot T)\big] \ \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\ \ \ H_{\rm C}(f) ={1}/{2} \cdot \left [1 - K_{\rm C} \cdot {\rm e}^{- {\rm j}\hspace{0.05cm}\cdot\hspace{0.05cm} 2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}N_{\rm C}\hspace{0.05cm}\cdot \hspace{0.05cm}T}\right]\hspace{0.05cm}. $$

	*The relative redundancy is the same for all pseudo-ternary codes.  Substituting  $M_q=2$,  $M_c=3$  and  $T_c =T_q$  into the  [[Digital_Signal_Transmission/Redundancy-Free_Coding#Symbolwise_coding_vs._blockwise_coding\|"general definition equation"]],  we obtain		*The relative redundancy is the same for all pseudo-ternary codes.  Substituting  $M_q=2$,  $M_c=3$  and  $T_c =T_q$  into the  [[Digital_Signal_Transmission/Redundancy-Free_Coding#Symbolwise_coding_vs._blockwise_coding\|"general definition equation"]],  we obtain
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	*  If  $s(t)$  is the pattern function of a stationary and ergodic random process, then for the  [[Theory_of_Stochastic_Signals/Auto-Correlation_Function#Random_processes\|"Auto-Correlation Function"]]  $\rm (ACF)$:		*  If  $s(t)$  is the pattern function of a stationary and ergodic random process, then for the  [[Theory_of_Stochastic_Signals/Auto-Correlation_Function#Random_processes\|"Auto-Correlation Function"]]  $\rm (ACF)$:
	:$$\varphi_s(\tau) = {\rm E}\big [s(t) \cdot s(t + \tau)\big ] = \sum_{\lambda = -\infty}^{+\infty}{1}/{T}		:$$\varphi_s(\tau) = {\rm E}\big [s(t) \cdot s(t + \tau)\big ] = \sum_{\lambda = -\infty}^{+\infty}{1}/{T}\cdot \varphi_a(\lambda)\cdot\varphi^{^{\bullet} }_{gs}(\tau -\lambda \cdot T)\hspace{0.05cm}.$$
	\cdot \varphi_a(\lambda)\cdot\varphi^{^{\bullet} }_{gs}(\tau -
	\lambda \cdot T)\hspace{0.05cm}.$$

	*  This equation describes the convolution of the discrete ACF  $\varphi_a(\lambda) = {\rm E}\big [ a_\nu \cdot a_{\nu + \lambda}\big]$  of the amplitude coefficients with the energy–ACF of the base impulse:		*  This equation describes the convolution of the discrete ACF  $\varphi_a(\lambda) = {\rm E}\big [ a_\nu \cdot a_{\nu + \lambda}\big]$  of the amplitude coefficients with the energy–ACF of the base impulse:

	:$$\varphi^{^{\bullet} }_{g}(\tau) =		:$$\varphi^{^{\bullet} }_{g}(\tau) =\int_{-\infty}^{+\infty} g ( t ) \cdot g ( t +\tau)\,{\rm d} t \hspace{0.05cm}.$$
	\int_{-\infty}^{+\infty} g ( t ) \cdot g ( t +
	\tau)\,{\rm d} t \hspace{0.05cm}.$$

	*The point is to indicate that  $\varphi^{^{\bullet} }_{g}(\tau)$  has the unit of an energy, while  $\varphi_s(\tau)$  indicates a power and  $\varphi_a(\lambda)$  is dimensionless.		*The point is to indicate that  $\varphi^{^{\bullet} }_{g}(\tau)$  has the unit of an energy, while  $\varphi_s(\tau)$  indicates a power and  $\varphi_a(\lambda)$  is dimensionless.
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	=== Calculating the PSD of a digital signal ===		=== Calculating the PSD of a digital signal ===
	The corresponding quantity to the ACF in the frequency domain is the [[Theory_of_Stochastic_Signals/Power-Spectral_Density#Wiener-Khintchine_Theorem\|"Power Spectral Density"]]  $\rm (PSD)$  ${\it \Phi}_s(f)$, which is fixedly related to  $\varphi_s(\tau)$  via the Fourier integral:<br>		The corresponding quantity to the ACF in the frequency domain is the [[Theory_of_Stochastic_Signals/Power-Spectral_Density#Wiener-Khintchine_Theorem\|"Power Spectral Density"]]  $\rm (PSD)$  ${\it \Phi}_s(f)$, which is fixedly related to  $\varphi_s(\tau)$  via the Fourier integral:<br>
	:$$\varphi_s(\tau) \hspace{0.4cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet \hspace{0.4cm}		:$$\varphi_s(\tau) \hspace{0.4cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet \hspace{0.4cm}{\it \Phi}_s(f) = \int_{-\infty}^{+\infty} \varphi_s(\tau) \cdot{\rm e}^{- {\rm j}\hspace{0.05cm} 2 \pi f \hspace{0.02cm} \tau}\,{\rm d} \tau \hspace{0.05cm}.$$
	{\it \Phi}_s(f) = \int_{-\infty}^{+\infty} \varphi_s(\tau) \cdot
	{\rm e}^{- {\rm j}\hspace{0.05cm} 2 \pi f \hspace{0.02cm} \tau}
	\,{\rm d} \tau \hspace{0.05cm}.$$
	*The power spectral density  ${\it \Phi}_s(f)$  can be represented as a product of two functions, taking into account the dimensional adjustment  $(1/T)$ :		*The power spectral density  ${\it \Phi}_s(f)$  can be represented as a product of two functions, taking into account the dimensional adjustment  $(1/T)$ :
	:$${\it \Phi}_s(f) = {\it \Phi}_a(f) \cdot {1}/{T} \cdot		:$${\it \Phi}_s(f) = {\it \Phi}_a(f) \cdot {1}/{T} \cdot\|G_s(f)\|^2 \hspace{0.05cm}.$$
	\|G_s(f)\|^2 \hspace{0.05cm}.$$
	*The first term  ${\it \Phi}_a(f)$  is dimensionless and describes the spectral shaping of the transmitted signal by the statistical relations of the source:<br>		*The first term  ${\it \Phi}_a(f)$  is dimensionless and describes the spectral shaping of the transmitted signal by the statistical relations of the source:<br>
	:$$\varphi_a(\lambda) \hspace{0.4cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet \hspace{0.4cm}{\it \Phi}_a(f) = \sum_{\lambda =		:$$\varphi_a(\lambda) \hspace{0.4cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet \hspace{0.4cm}{\it \Phi}_a(f) = \sum_{\lambda =-\infty}^{+\infty}\varphi_a(\lambda)\cdot {\rm e}^{- {\rmj}\hspace{0.05cm} 2 \pi f \hspace{0.02cm} \lambda \hspace{0.02cm}T} =\varphi_a(0) + 2 \cdot \sum_{\lambda =1}^{\infty}\varphi_a(\lambda)\cdot\cos ( 2 \pi f\lambda T) \hspace{0.05cm}.$$
	-\infty}^{+\infty}\varphi_a(\lambda)\cdot {\rm e}^{- {\rm
	j}\hspace{0.05cm} 2 \pi f \hspace{0.02cm} \lambda \hspace{0.02cm}T} =
	\varphi_a(0) + 2 \cdot \sum_{\lambda =
	1}^{\infty}\varphi_a(\lambda)\cdot\cos ( 2 \pi f
	\lambda T) \hspace{0.05cm}.$$
	*${\it \Phi^{^{\hspace{0.08cm}\bullet}}_{g}}(f)$  takes into account the spectral shaping by  $g(t)$. The narrower this is, the wider  $\vert G(f) \vert^2$  and thus the larger the bandwidth requirement:		*${\it \Phi^{^{\hspace{0.08cm}\bullet}}_{g}}(f)$  takes into account the spectral shaping by  $g(t)$. The narrower this is, the wider  $\vert G(f) \vert^2$  and thus the larger the bandwidth requirement:
	:$$\varphi^{^{\hspace{0.05cm}\bullet}}_{g}(\tau) \hspace{0.4cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet \hspace{0.4cm}		:$$\varphi^{^{\hspace{0.05cm}\bullet}}_{g}(\tau) \hspace{0.4cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet \hspace{0.4cm}{\it \Phi}^{^{\hspace{0.08cm}\bullet}}_{g}(f) = \|G(f)\|^2\hspace{0.05cm}.$$
	{\it \Phi}^{^{\hspace{0.08cm}\bullet}}_{g}(f) = \|G(f)\|^2
	\hspace{0.05cm}.$$
	*The energy spectral density ${\it \Phi^{^{\hspace{0.08cm}\bullet}}_{g}}(f)$  has the unit  $\rm Ws/Hz$  and the power spectral density  ${\it \Phi_{s}}(f)$  after division by the symbol spacing  $T$  has the unit  $\rm W/Hz$.		*The energy spectral density ${\it \Phi^{^{\hspace{0.08cm}\bullet}}_{g}}(f)$  has the unit  $\rm Ws/Hz$  and the power spectral density  ${\it \Phi_{s}}(f)$  after division by the symbol spacing  $T$  has the unit  $\rm W/Hz$.

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	e}^{-{\rm j}\hspace{0.03cm}\cdot \hspace{0.03cm}		e}^{-{\rm j}\hspace{0.03cm}\cdot \hspace{0.03cm}
	\alpha}		\alpha}
	\big ]\hspace{0.05cm}.$$		\big ]\hspace{0.05cm}.$$This gives the power-spectral density  $\rm (PSD)$  of the amplitude coefficients  $(K$  and  $\alpha$  are abbreviations according to the above equation$)$::$$ {\it \Phi}_a(f) = \| H_{\rm C}(f)\|^2 = \frac{\big [1 - K \cos

	This gives the power-spectral density  $\rm (PSD)$  of the amplitude coefficients  $(K$  and  $\alpha$  are abbreviations according to the above equation$)$:
	:$$ {\it \Phi}_a(f) = \| H_{\rm C}(f)\|^2 = \frac{\big [1 - K \cos
	(\alpha) + {\rm j}\cdot K \sin (\alpha) \big ] \big [1 - K \cos		(\alpha) + {\rm j}\cdot K \sin (\alpha) \big ] \big [1 - K \cos
	(\alpha) - {\rm j}\cdot K \sin (\alpha) \big ] }{4} = \text{...} = {1}/{4} \cdot \big [2 - 2 \cdot K \cdot \cos		(\alpha) - {\rm j}\cdot K \sin (\alpha) \big ] }{4} = \text{...} = {1}/{4} \cdot \big [2 - 2 \cdot K \cdot \cos
	(\alpha) \big ] $$		(\alpha) \big ] $$[[File:P_ID1347__Dig_T_2_4_S2b_v2.png\|right\|frame\|Power-spectral density of the AMI code\|class=fit]]:$$ \Rightarrow \hspace{0.3cm}{\it \Phi}_a(f) = \| H_{\rm C}(f)\|^2 = {1}/{2} \cdot \big [1 - K_{\rm C} \cdot \cos
	[[File:P_ID1347__Dig_T_2_4_S2b_v2.png\|right\|frame\|Power-spectral density of the AMI code\|class=fit]]
	:$$ \Rightarrow \hspace{0.3cm}{\it \Phi}_a(f) = \| H_{\rm C}(f)\|^2 = {1}/{2} \cdot \big [1 - K_{\rm C} \cdot \cos
	(2\pi f N_{\rm C} T)\big ]		(2\pi f N_{\rm C} T)\big ]
	\hspace{0.4cm}\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ \hspace{0.4cm}		\hspace{0.4cm}\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ \hspace{0.4cm}
	\varphi_a(\lambda \cdot T)\hspace{0.05cm}.$$		\varphi_a(\lambda \cdot T)\hspace{0.05cm}.$$In particular,  for the power-spectral density of the AMI code  $(N_{\rm C} = K_{\rm C} = 1)$,  we obtain::$${\it \Phi}_a(f) = {1}/{2} \cdot \big [1 - \cos

	In particular,  for the power-spectral density of the AMI code  $(N_{\rm C} = K_{\rm C} = 1)$,  we obtain:
	:$${\it \Phi}_a(f) = {1}/{2} \cdot \big [1 - \cos
	(2\pi f T)\big ] = \sin^2		(2\pi f T)\big ] = \sin^2
	(\pi f T)\hspace{0.05cm}.$$		(\pi f T)\hspace{0.05cm}.$$
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	[[File:P_ID1348__Dig_T_2_4_S3b_v1.png\|right\|frame\|Power-spectral density of the duobinary code\|right\|class=fit]]		[[File:P_ID1348__Dig_T_2_4_S3b_v1.png\|right\|frame\|Power-spectral density of the duobinary code\|right\|class=fit]]

	:$${\it \Phi}_a(f) ={1}/{2} \cdot \big [1 + \cos		:$${\it \Phi}_a(f) ={1}/{2} \cdot \big [1 + \cos(2\pi f T)\big ] = \cos^2(\pi f T)\hspace{0.05cm},$$
	(2\pi f T)\big ] = \cos^2		:$$ {\it \Phi}_s(f) = s_0^2 \cdot T \cdot \cos^2(\pi f T)\cdot {\rm si}^2(\pi f T)= s_0^2 \cdot T \cdot {\rm si}^2(2 \pi f T) \hspace{0.05cm}.$$
	(\pi f T)\hspace{0.05cm},$$
	:$$ {\it \Phi}_s(f) = s_0^2 \cdot T \cdot \cos^2
	(\pi f T)\cdot {\rm si}^2
	(\pi f T)= s_0^2 \cdot T \cdot {\rm si}^2
	(2 \pi f T) \hspace{0.05cm}.$$

	The graph shows the power-spectral density		The graph shows the power-spectral density

Noah at 17:26, 14 April 2023

2023-04-14T17:26:59Z

← Older revision		Revision as of 19:26, 14 April 2023
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	==Applet Manual==		==Applet Manual==
	[[File:BS_Pseudoternär.png\|right\|600px\|frame\|~~Bildschirmabzug~~ (~~deutsche Version~~, ~~heller Hintergrund~~)]]		[[File:BS_Pseudoternär.png\|right\|600px\|frame\|Screenshot (German version, light background)]]
	<br>		<br>
	'''(A)'''     Theme (changeable graphical user interface design)		'''(A)'''     Theme (changeable graphical user interface design)

Noah at 17:26, 14 April 2023

2023-04-14T17:26:01Z

← Older revision		Revision as of 19:26, 14 April 2023
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	:* Protanopia:   for users with pronounced red visual impairment		:* Protanopia:   for users with pronounced red visual impairment

	'''(B)'''     ~~Zugrundeliegendes Blockschaltbild~~		'''(B)'''     Underlying block diagram

	'''(C)'''     ~~Auswahl des Pseudoternörcodes~~:		'''(C)'''     Selection of the pseudoternary code:
	<br>~~ ~~                  AMI–~~Code~~, ~~Duobinärcode~~, ~~Bipolarcode 2~~. ~~Ordnung~~		<br>                 AMI–code, duobinary code, 2nd order bipolar code.

	'''(D)'''     ~~Auswahl des Grundimpulses~~  $g(t)$:		'''(D)'''     Selection of the base pulse  $g(t)$:
	<br>                 ~~  Rechteckimpuls~~, ~~Nyquistimpuls~~, ~~Wurzel~~–~~Nyquistimpuls~~		<br>                  Rectangular pulse, Nyquist pulse, Root–Nyquist pulse.

	'''(E)'''     Rolloff–~~Faktor~~ (~~Frequenzbereich~~) ~~für~~ "Nyquist" ~~und~~ "~~Wurzel~~–Nyquist"		'''(E)'''     Rolloff–factor (frequency range) for "Nyquist" and "Root–Nyquist"

	'''(F)'''     ~~Einstellung von~~  $3 \cdot 4 = 12$  Bit ~~der Quellensymbolfolge~~		'''(F)'''     Setting of  $3 \cdot 4 = 12$  Bit of source symbol sequence.

	'''(G)'''     ~~Auswahl einer drei voreingestellten Quellensymbolfolgen~~		'''(G)''''     Selection of one of three preset source symbol sequences.

	'''(H)'''     ~~Zufällige binäre Quellensymbolfolge~~		'''(H)'''     Random binary source symbol sequence.

	'''( I )'''     ~~Schrittweise Verdeutlichung der Pseudoternärcodierung~~		'''( I )'''     Stepwise elucidation of pseudoternary coding.

	'''(J)'''     ~~Ergebnis der Pseudoternärcodierung~~:  ~~Signale~~  $q(t)$,  $b(t)$,  $c(t)$,  $s(t)$		'''(J)'''     Result of pseudoternary coding:  Signals  $q(t)$,  $b(t)$,  $c(t)$,  $s(t)$

	'''(K)'''     ~~Löschen der Signalverläufe im Grafikbereich~~  $\rm M$		'''(K)'''     Deleting the signal waveforms in the graphics area  $\rm M$

	'''(L)'''     ~~Skizzen für Autokorrelationsfunktion~~ & ~~Leistungsdichtespektrum~~		'''(L)'''     Sketches for autocorrelation function & power spectral density.

	'''(M)'''     ~~Grafikbereich~~:  ~~Quellensignal~~  $q(t)$, ~~Signal~~  $b(t)$  ~~nach Vorcodierung~~, <br>                 ~~  Codersignal~~  $c(t)$  ~~mit Rechtecken~~, ~~Sendesignal~~  $s(t)$  ~~gemäß~~  $g(t)$		'''(M)'''     Graphics area:   Source signal  $q(t)$, signal  $b(t)$  after precoding, <br>                 encoder signal  $c(t)$  with rectangles, transmission signal  $s(t)$  according to  $g(t)$

	'''(N)'''     ~~Bereich für Übungen~~:  ~~Aufgabenauswahl~~, ~~Fragen~~, ~~Musterlösung~~		'''(N)'''     Area for exercises:  Exercise selection, questions, sample solution.
	<br clear=all>		<br clear=all>
	==About the Authors==		==About the Authors==

Noah at 17:19, 14 April 2023

2023-04-14T17:19:05Z

← Older revision		Revision as of 19:19, 14 April 2023
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	[[File:BS_Pseudoternär.png\|right\|600px\|frame\|Bildschirmabzug (deutsche Version, heller Hintergrund)]]		[[File:BS_Pseudoternär.png\|right\|600px\|frame\|Bildschirmabzug (deutsche Version, heller Hintergrund)]]
	<br>		<br>
	'''(A)'''     Theme (~~veränderbare grafische Oberflächengestaltung~~)		'''(A)'''     Theme (changeable graphical user interface design)
	:* Dark:   ~~schwarzer Hintergrund~~  (~~wird von den Autoren empfohlen~~)		:* Dark:   dark background  (recommended by the authors)
	:* Bright:   ~~weißer Hintergrund~~  (~~empfohlen für Beamer und Ausdrucke~~)		:* Bright:   white background  (recommended for beamers and printouts)
	:* Deuteranopia:   ~~für Nutzer mit ausgeprägter Grün–Sehschwäche~~		:* Deuteranopia:   for users with pronounced green visual impairment
	:* Protanopia:   ~~für Nutzer mit ausgeprägter Rot–Sehschwäche~~		:* Protanopia:   for users with pronounced red visual impairment

	'''(B)'''     Zugrundeliegendes Blockschaltbild		'''(B)'''     Zugrundeliegendes Blockschaltbild

Guenter at 12:35, 14 April 2023

2023-04-14T12:35:07Z

← Older revision		Revision as of 14:35, 14 April 2023
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	==Applet Manual==		==Applet Manual==
	[[File:~~Anleitung_4B3T~~.png\|right\|600px\|frame\|Bildschirmabzug (deutsche Version, heller Hintergrund)]]		[[File:BS_Pseudoternär.png\|right\|600px\|frame\|Bildschirmabzug (deutsche Version, heller Hintergrund)]]
	<br>		<br>
	'''(A)'''     Theme (veränderbare grafische Oberflächengestaltung)		'''(A)'''     Theme (veränderbare grafische Oberflächengestaltung)

Guenter at 12:30, 14 April 2023

2023-04-14T12:30:30Z

← Older revision		Revision as of 14:30, 14 April 2023
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	==Applet Manual==		==Applet Manual==
	[[File:~~BS_Pseudoternär~~.png\|right\|600px\|frame\|Bildschirmabzug (deutsche Version, heller Hintergrund)]]		[[File:Anleitung_4B3T.png\|right\|600px\|frame\|Bildschirmabzug (deutsche Version, heller Hintergrund)]]
	<br>		<br>
	'''(A)'''     Theme (veränderbare grafische Oberflächengestaltung)		'''(A)'''     Theme (veränderbare grafische Oberflächengestaltung)

Noah at 18:19, 25 March 2023

2023-03-25T18:19:59Z

← Older revision		Revision as of 20:19, 25 March 2023
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	*of the binary–precoded sequence  $\langle \hspace{0.05cm}b_\nu \hspace{0.05cm}\rangle \ \in \{+1, -1\}$,  represented by the also redundancy-free binary bipolar rectangular signal  $b(t)$,		*of the binary–precoded sequence  $\langle \hspace{0.05cm}b_\nu \hspace{0.05cm}\rangle \ \in \{+1, -1\}$,  represented by the also redundancy-free binary bipolar rectangular signal  $b(t)$,
	*the pseudo-ternary code sequence  $\langle \hspace{0.05cm}c_\nu \hspace{0.05cm}\rangle \ \in \{+1,\ 0, -1\}$,  represented by the redundant ternary bipolar rectangular signal  $c(t)$,		*the pseudo-ternary code sequence  $\langle \hspace{0.05cm}c_\nu \hspace{0.05cm}\rangle \ \in \{+1,\ 0, -1\}$,  represented by the redundant ternary bipolar rectangular signal  $c(t)$,
	*the equally redundant ternary transmitted signal  $s(t)$, characterized by the amplitude coefficients  $a_\nu $,  and the (transmitted–) ~~fundamental momentum~~  $g(t)$:		*the equally redundant ternary transmitted signal  $s(t)$, characterized by the amplitude coefficients  $a_\nu $,  and the (transmitted–) base impulse  $g(t)$:
	:$$s(t) = \sum_{\nu = -\infty}^{+\infty} a_\nu \cdot g( t - \nu \cdot T)\hspace{0.05cm}.$$		:$$s(t) = \sum_{\nu = -\infty}^{+\infty} a_\nu \cdot g( t - \nu \cdot T)\hspace{0.05cm}.$$

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	The applet also shows that the total power spectral density  ${\it \Phi}_s (f)$ can be split into the part  ${\it \Phi}_a (f)$ that takes into account the statistical relations of the amplitude coefficients  $a_\nu$   and the energy spectral density $ {\it \Phi}^{^{\hspace{0.08cm}\bullet}}_{g}(f) = \|G(f)\|^2 $, characterized by the ~~momentum~~ shape  $g(t)$.		The applet also shows that the total power spectral density  ${\it \Phi}_s (f)$ can be split into the part  ${\it \Phi}_a (f)$ that takes into account the statistical relations of the amplitude coefficients  $a_\nu$   and the energy spectral density $ {\it \Phi}^{^{\hspace{0.08cm}\bullet}}_{g}(f) = \|G(f)\|^2 $, characterized by the shape  $g(t)$.

	''Note''   In the applet, no distinction is made between the encoder symbols  $c_\nu \in \{+1,\ 0, -1\}$  and the amplitude coefficients  $a_\nu \in \{+1,\ 0, -1\}$  .   It should be remembered that the  $a_\nu$  are always numerical values, while for the encoder symbols also the notation  $c_\nu \in \{\text{plus},\ \text{zero},\ \text{minus}\}$  would be admissible.		''Note''   In the applet, no distinction is made between the encoder symbols  $c_\nu \in \{+1,\ 0, -1\}$  and the amplitude coefficients  $a_\nu \in \{+1,\ 0, -1\}$  .   It should be remembered that the  $a_\nu$  are always numerical values, while for the encoder symbols also the notation  $c_\nu \in \{\text{plus},\ \text{zero},\ \text{minus}\}$  would be admissible.

Noah at 17:16, 24 March 2023

2023-03-24T17:16:52Z

← Older revision		Revision as of 19:16, 24 March 2023
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	:$$g_{\rm Nyq}(t)={\rm const.} \cdot \frac{\cos(\pi \cdot r\cdot t/T)}{1-(2\cdot r\cdot t/T)^2} \cdot {\rm si}(\pi \cdot t/T) \ \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\ \ \ G_{\rm Nyq}(f),$$		:$$g_{\rm Nyq}(t)={\rm const.} \cdot \frac{\cos(\pi \cdot r\cdot t/T)}{1-(2\cdot r\cdot t/T)^2} \cdot {\rm si}(\pi \cdot t/T) \ \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\ \ \ G_{\rm Nyq}(f),$$

	:*the root Nyquist impulse   ⇒   impulse response of the root raised cosine ~~lowpass~~ with rolloff factor $r$:		:*the root Nyquist impulse   ⇒   impulse response of the root raised cosine low-pass with rolloff factor $r$:
	:$$g_{\sqrt{\rm Nyq} }(t)\ \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\ \ \ G_{\sqrt{\rm Nyq} }(f)={\rm const.} \cdot \sqrt{G_{\rm Nyq}(f)} .$$ }}		:$$g_{\sqrt{\rm Nyq} }(t)\ \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\ \ \ G_{\sqrt{\rm Nyq} }(f)={\rm const.} \cdot \sqrt{G_{\rm Nyq}(f)} .$$ }}
	<br><br>		<br><br>
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	*The decoder detects this code violation and replaces "$+\ 0\ 0\ +$" with "$0\ 0\ 0\ 0$" again.<br>		*The decoder detects this code violation and replaces "$+\ 0\ 0\ +$" with "$0\ 0\ 0\ 0$" again.<br>
	<br><br>		<br><br>
	=== ~~Zur AKF–Berechnung eines Digitalsignals~~ ===		=== Calculating the ACF of a digital signal ===
	In ~~der Versuchsdurchführung werden einige Größen und Zusamenhänge verwendet~~, ~~die hier kurz eräutert werden sollen~~:		In the execution of the experiment some quantities and correlations are used, which shall be briefly explained here:

	*  ~~Das~~ (~~zeitlich unbegrenzte~~) ~~Digitalsignal beinhaltet sowohl die Quellenstatistik~~ $($~~Amplitudenkoeffizienten~~  $a_\nu$)  ~~als auch die Sendeimpulsform~~  $g(t)$:		*  The (time-unlimited) digital signal includes both the source statistics $($amplitude coefficients  $a_\nu$)  and the transmitted pulse shape  $g(t)$:
	:$$s(t) = \sum_{\nu = -\infty}^{+\infty} a_\nu \cdot g ( t - \nu \cdot T)\hspace{0.05cm}.$$		:$$s(t) = \sum_{\nu = -\infty}^{+\infty} a_\nu \cdot g ( t - \nu \cdot T)\hspace{0.05cm}.$$

	*  ~~Ist~~  $s(t)$  ~~die Musterfunktion eines stationären und ergodischen Zufallsprozesses~~, ~~so gilt für die~~  [[~~Stochastische_Signaltheorie~~/~~Autokorrelationsfunktion_(AKF)~~#~~Zufallsprozesse_.281.29~~\|~~Autokorrelationsfunktion~~]]  $\rm (~~AKF~~)$:		*  If  $s(t)$  is the pattern function of a stationary and ergodic random process, then for the  [[Theory_of_Stochastic_Signals/Auto-Correlation_Function#Random_processes\|"Auto-Correlation Function"]]  $\rm (ACF)$:
	:$$\varphi_s(\tau) = {\rm E}\big [s(t) \cdot s(t + \tau)\big ] = \sum_{\lambda = -\infty}^{+\infty}{1}/{T}		:$$\varphi_s(\tau) = {\rm E}\big [s(t) \cdot s(t + \tau)\big ] = \sum_{\lambda = -\infty}^{+\infty}{1}/{T}
	\cdot \varphi_a(\lambda)\cdot\varphi^{^{\bullet} }_{gs}(\tau -		\cdot \varphi_a(\lambda)\cdot\varphi^{^{\bullet} }_{gs}(\tau -
	\lambda \cdot T)\hspace{0.05cm}.$$		\lambda \cdot T)\hspace{0.05cm}.$$

	*  ~~Diese Gleichung beschreibt die Faltung der diskreten AKF~~  $\varphi_a(\lambda) = {\rm E}\big [ a_\nu \cdot a_{\nu + \lambda}\big]$  ~~der Amplitudenkoeffizienten mit der Energie~~–~~AKF des Grundimpulses~~:		*  This equation describes the convolution of the discrete ACF  $\varphi_a(\lambda) = {\rm E}\big [ a_\nu \cdot a_{\nu + \lambda}\big]$  of the amplitude coefficients with the energy–ACF of the base impulse:

	:$$\varphi^{^{\bullet} }_{g}(\tau) =		:$$\varphi^{^{\bullet} }_{g}(\tau) =
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	\tau)\,{\rm d} t \hspace{0.05cm}.$$		\tau)\,{\rm d} t \hspace{0.05cm}.$$

	*~~Der Punkt soll darauf hinweisen, dass~~  $\varphi^{^{\bullet} }_{g}(\tau)$  ~~die Einheit einer Energie besitzt~~, ~~während~~  $\varphi_s(\tau)$  ~~eine Leistung angibt und~~  $\varphi_a(\lambda)$  ~~dimensionslos ist~~.		*The point is to indicate that  $\varphi^{^{\bullet} }_{g}(\tau)$  has the unit of an energy, while  $\varphi_s(\tau)$  indicates a power and  $\varphi_a(\lambda)$  is dimensionless.
	<br>		<br>
	=== ~~Zur LDS-Berechnung eines Digitalsignals~~ ===		=== Calculating the PSD of a digital signal ===
	~~Die Entsprechungsgröße zur AKF ist im Frequenzbereich das~~ [[~~Stochastische_Signaltheorie~~/~~Leistungsdichtespektrum_(LDS)~~#~~Theorem_von_Wiener~~-~~Chintchine~~\|~~Leistungsdichtespektrum~~]]  $\rm (~~LDS~~)$  ${\it \Phi}_s(f)$, ~~das mit~~  $\varphi_s(\tau)$  ~~über das Fourierintegral fest verknüpft ist~~:<br>		The corresponding quantity to the ACF in the frequency domain is the [[Theory_of_Stochastic_Signals/Power-Spectral_Density#Wiener-Khintchine_Theorem\|"Power Spectral Density"]]  $\rm (PSD)$  ${\it \Phi}_s(f)$, which is fixedly related to  $\varphi_s(\tau)$  via the Fourier integral:<br>
	:$$\varphi_s(\tau) \hspace{0.4cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet \hspace{0.4cm}		:$$\varphi_s(\tau) \hspace{0.4cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet \hspace{0.4cm}
	{\it \Phi}_s(f) = \int_{-\infty}^{+\infty} \varphi_s(\tau) \cdot		{\it \Phi}_s(f) = \int_{-\infty}^{+\infty} \varphi_s(\tau) \cdot
	{\rm e}^{- {\rm j}\hspace{0.05cm} 2 \pi f \hspace{0.02cm} \tau}		{\rm e}^{- {\rm j}\hspace{0.05cm} 2 \pi f \hspace{0.02cm} \tau}
	\,{\rm d} \tau \hspace{0.05cm}.$$		\,{\rm d} \tau \hspace{0.05cm}.$$
	*~~Das Leistungsdichtespektrum~~  ${\it \Phi}_s(f)$  ~~kann unter Berücksichtigung der Dimensionsbereinigung~~  $(1/T)$  ~~als Produkt zweier Funktionen dargestellt werden~~:		*The power spectral density  ${\it \Phi}_s(f)$  can be represented as a product of two functions, taking into account the dimensional adjustment  $(1/T)$ :
	:$${\it \Phi}_s(f) = {\it \Phi}_a(f) \cdot {1}/{T} \cdot		:$${\it \Phi}_s(f) = {\it \Phi}_a(f) \cdot {1}/{T} \cdot
	\|G_s(f)\|^2 \hspace{0.05cm}.$$		\|G_s(f)\|^2 \hspace{0.05cm}.$$
	*~~Der erste Term~~  ${\it \Phi}_a(f)$  ~~ist dimensionslos und beschreibt die spektrale Formung des Sendesignals durch die statistischen Bindungen der Quelle~~:<br>		*The first term  ${\it \Phi}_a(f)$  is dimensionless and describes the spectral shaping of the transmitted signal by the statistical relations of the source:<br>
	:$$\varphi_a(\lambda) \hspace{0.4cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet \hspace{0.4cm}{\it \Phi}_a(f) = \sum_{\lambda =		:$$\varphi_a(\lambda) \hspace{0.4cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet \hspace{0.4cm}{\it \Phi}_a(f) = \sum_{\lambda =
	-\infty}^{+\infty}\varphi_a(\lambda)\cdot {\rm e}^{- {\rm		-\infty}^{+\infty}\varphi_a(\lambda)\cdot {\rm e}^{- {\rm
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	1}^{\infty}\varphi_a(\lambda)\cdot\cos ( 2 \pi f		1}^{\infty}\varphi_a(\lambda)\cdot\cos ( 2 \pi f
	\lambda T) \hspace{0.05cm}.$$		\lambda T) \hspace{0.05cm}.$$
	*${\it \Phi^{^{\hspace{0.08cm}\bullet}}_{g}}(f)$  ~~berücksichtigt die spektrale Formung durch~~  $g(t)$. ~~Je schmaler dieser ist~~, ~~desto breiter ist~~  $\vert G(f) \vert^2$  ~~und um so größer ist damit der Bandbreitenbedarf~~:		*${\it \Phi^{^{\hspace{0.08cm}\bullet}}_{g}}(f)$  takes into account the spectral shaping by  $g(t)$. The narrower this is, the wider  $\vert G(f) \vert^2$  and thus the larger the bandwidth requirement:
	:$$\varphi^{^{\hspace{0.05cm}\bullet}}_{g}(\tau) \hspace{0.4cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet \hspace{0.4cm}		:$$\varphi^{^{\hspace{0.05cm}\bullet}}_{g}(\tau) \hspace{0.4cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet \hspace{0.4cm}
	{\it \Phi}^{^{\hspace{0.08cm}\bullet}}_{g}(f) = \|G(f)\|^2		{\it \Phi}^{^{\hspace{0.08cm}\bullet}}_{g}(f) = \|G(f)\|^2
	\hspace{0.05cm}.$$		\hspace{0.05cm}.$$
	*~~Das Energiedichtespektrum~~ ${\it \Phi^{^{\hspace{0.08cm}\bullet}}_{g}}(f)$  ~~hat die Einheit~~  $\rm Ws/Hz$  ~~und das Leistungsdichtespektrum~~  ${\it \Phi_{s}}(f)$  ~~nach der Division durch den Symbolabstand~~  $T$  ~~die Einheit~~  $\rm W/Hz$.		*The energy spectral density ${\it \Phi^{^{\hspace{0.08cm}\bullet}}_{g}}(f)$  has the unit  $\rm Ws/Hz$  and the power spectral density  ${\it \Phi_{s}}(f)$  after division by the symbol spacing  $T$  has the unit  $\rm W/Hz$.

Noah at 15:56, 24 March 2023

2023-03-24T15:56:25Z

← Older revision		Revision as of 17:56, 24 March 2023
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	==Applet Description==		==Applet Description==
	<br>		<br>
	~~Das Applet behandelt die Eigenschaften der bekanntesten Pseudoternärcodes~~, ~~nämlich~~:		The applet covers the properties of the best known pseudo-ternary codes, namely:
	#  ~~Bipolarcode erster Ordnung bzw.~~  $\rm AMI$~~–Code~~  (~~von~~: ''Alternate Mark Inversion''),  ~~gekennzeichnet durch die Parameter~~  $N_{\rm C} = 1, \ K_{\rm C} = +1$,		#  First-order bipolar code,  $\rm AMI$ code  (from: ''Alternate Mark Inversion''),  characterized by the parameters  $N_{\rm C} = 1, \ K_{\rm C} = +1$,
	#  ~~Duobinärcode~~,  $(\rm DUOB)$,  ~~Codeparameter~~:  $N_{\rm C} = 1, \ K_{\rm C} = -1$,		#  Duobinary code,  $(\rm DUOB)$,  code parameters:  $N_{\rm C} = 1, \ K_{\rm C} = -1$,
	#  ~~Bipolarcode zweiter Ordnung~~  $(\rm BIP2)$,  ~~Codeparameter~~:  $N_{\rm C} = 2, \ K_{\rm C} = +1$.		#  Second order bipolar code  $(\rm BIP2)$,  code parameters:  $N_{\rm C} = 2, \ K_{\rm C} = +1$.


	~~Am Eingang liegt die redundanzfreie binäre bipolare Quellensymbolfolge~~  $\langle \hspace{0.05cm}q_\nu \hspace{0.05cm}\rangle \ \in \{+1, -1\}$   ⇒   ~~Rechtecksignal~~  $q(t)$  an.  ~~Verdeutlicht wird die Generierung~~		At the input is the redundancy-free binary bipolar source symbol sequence  $\langle \hspace{0.05cm}q_\nu \hspace{0.05cm}\rangle \ \in \{+1, -1\}$   ⇒   rectangular signal  $q(t)$  an.  Illustrating the generation.
	*~~der binär~~–~~vorcodierten Folge~~  $\langle \hspace{0.05cm}b_\nu \hspace{0.05cm}\rangle \ \in \{+1, -1\}$,  ~~dargestellt durch das ebenfalls redundanzfreie binäre bipolare Rechtecksignal~~  $b(t)$,		*of the binary–precoded sequence  $\langle \hspace{0.05cm}b_\nu \hspace{0.05cm}\rangle \ \in \{+1, -1\}$,  represented by the also redundancy-free binary bipolar rectangular signal  $b(t)$,
	*~~der pseudoternären Codefolge~~  $\langle \hspace{0.05cm}c_\nu \hspace{0.05cm}\rangle \ \in \{+1,\ 0, -1\}$,  ~~dargestellt durch das redundante ternäre bipolare Rechtecksignal~~  $c(t)$,		*the pseudo-ternary code sequence  $\langle \hspace{0.05cm}c_\nu \hspace{0.05cm}\rangle \ \in \{+1,\ 0, -1\}$,  represented by the redundant ternary bipolar rectangular signal  $c(t)$,
	*~~das gleichermaßen redundante ternäre Sendesignal~~  $s(t)$, ~~gekennzeichnet durch die Amplitudenkoeffizienten~~  $a_\nu $,  ~~und den~~ (~~Sende~~–) ~~Grundimpuls~~  $g(t)$:		*the equally redundant ternary transmitted signal  $s(t)$, characterized by the amplitude coefficients  $a_\nu $,  and the (transmitted–) fundamental momentum  $g(t)$:
	:$$s(t) = \sum_{\nu = -\infty}^{+\infty} a_\nu \cdot g( t - \nu \cdot T)\hspace{0.05cm}.$$		:$$s(t) = \sum_{\nu = -\infty}^{+\infty} a_\nu \cdot g( t - \nu \cdot T)\hspace{0.05cm}.$$

	~~Der Grundimpuls~~  $g(t)$  –  ~~im Applet~~  "~~Rechteck~~",  "Nyquist" ~~und~~  "~~Wurzel~~–Nyquist"  –  ~~bestimmt nicht nur die Form des Sendesignals~~, ~~sondern auch den Verlauf~~		The base impulse  $g(t)$  –  in the applet  "Rectangle",  "Nyquist" and  "Root–Nyquist"  –  determines not only the shape of the transmitted signal, but also the course of
	* ~~der Autokorrelationsfunktion~~  $\rm (~~AKF~~)$  $\varphi_s (\tau)$  ~~und~~		* of the auto-correlation function  $\rm (ACF)$  $\varphi_s (\tau)$  and
	* ~~des zugehörigen Leistungsdichtespektrums~~  $\rm (~~LDS~~)$  ${\it \Phi}_s (f)$.		* of the associated power spectral density  $\rm (PSD)$  ${\it \Phi}_s (f)$.


	~~Das Applet zeigt auch, dass das gesamte Leistungsdichtespektrum~~  ${\it \Phi}_s (f)$ ~~aufgeteilt werden kann in den Anteil~~  ${\it \Phi}_a (f)$~~, der die statistischen Bindungen der Amplitudenkoeffizienten~~  $a_\nu$   ~~berücksichtigt, und das Energiedichtespektrum~~ $ {\it \Phi}^{^{\hspace{0.08cm}\bullet}}_{g}(f) = \|G(f)\|^2 $, ~~gekennzeichnet durch die Impulsform~~  $g(t)$.		The applet also shows that the total power spectral density  ${\it \Phi}_s (f)$ can be split into the part  ${\it \Phi}_a (f)$ that takes into account the statistical relations of the amplitude coefficients  $a_\nu$   and the energy spectral density $ {\it \Phi}^{^{\hspace{0.08cm}\bullet}}_{g}(f) = \|G(f)\|^2 $, characterized by the momentum shape  $g(t)$.

	''~~Anmerkung~~'':   ~~Im Applet wird kein Unterschied zwischen den Codersymbolen~~  $c_\nu \in \{+1,\ 0, -1\}$  ~~und den Amplitudenkoeffizienten~~  $a_\nu \in \{+1,\ 0, -1\}$  ~~gemacht~~.  ~~Dabei sollte nicht vergessen werden, dass die~~  $a_\nu$  ~~stets Zahlenwerte sind~~, ~~während für die Codersymbole auch die Notation~~  $c_\nu \in \{\text{~~Plus~~},\ \text{~~Null~~},\ \text{~~Minus~~}\}$  ~~zulässig wäre~~.		''Note''   In the applet, no distinction is made between the encoder symbols  $c_\nu \in \{+1,\ 0, -1\}$  and the amplitude coefficients  $a_\nu \in \{+1,\ 0, -1\}$  .   It should be remembered that the  $a_\nu$  are always numerical values, while for the encoder symbols also the notation  $c_\nu \in \{\text{plus},\ \text{zero},\ \text{minus}\}$  would be admissible.

	==Theoretical Background==		==Theoretical Background==