Applets:Physical Signal & Equivalent Lowpass Signal - Revision history

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	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{Example 1:}$		$\text{Example 1:}$
	The constellation given here results, for example, in the [[Modulation_Methods/~~Zweiseitenband~~-~~Amplitudenmodulation~~#~~AM-Signale_und_-Spektren_bei_harmonischem_Eingangssignal~~\|"double-sideband amplitude modulation"]] of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$. This is discussed frequently in the exercises.		The constellation given here results, for example, in the [[Modulation_Methods/Double-Sideband_Amplitude_Modulation#AM_signals_and_spectra_with_a_harmonic_input_signal\|"double-sideband amplitude modulation"]] of the message signal $x_{\rm N}(t) = A_{\rm N}\cdot \cos\left(2\pi f_{\rm N}\cdot t- \varphi_{\rm N}\right)$ with the carrier signal $x_{\rm T}(t) = A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t - \varphi_{\rm T}\right)$. This is discussed frequently in the exercises.

	[[File:Ortskurve_5.png\|center\|frame\|Spectrum $X_{\rm TP}(f)$ of the equivalent low–pass signal for different phase constellations \|class=fit]]		[[File:Ortskurve_5.png\|center\|frame\|Spectrum $X_{\rm TP}(f)$ of the equivalent low–pass signal for different phase constellations \|class=fit]]
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	==About the Authors==		==About the Authors==
	This interactive calculation was designed and realized at the  [https://www.ei.tum.de/en/lnt/home//startseite Institute for Communications Engineering]  of the  [https://www.tum.de/ Technical University of Munich] .		This interactive calculation was designed and realized at the  [https://www.ei.tum.de/en/lnt/home//startseite Institute for Communications Engineering]  of the  [https://www.tum.de/ Technical University of Munich] .
	*The original version was created in 2005 by  [[Biographies_and_Bibliographies/~~An_LNTwww_beteiligte_Studierende~~#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29\|Ji Li]]  as part of her Diploma thesis using "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 original version was created in 2005 by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29\|Ji Li]]  as part of her Diploma thesis using "FlashMX–Actionscript"  (Supervisor:  [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._Günter_Söder_(at_LNT_from_1974-2024)\|Günter Söder]]).
	*In 2018 this Applet was redesigned and updated to "HTML5" by  [[Biographies_and_Bibliographies/~~An_LNTwww_beteiligte_Studierende~~#Xiaohan_Liu_.28Bachelorarbeit_2018.29\|Xiaohan Liu]]  as part of her Bachelor's 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]]).		*In 2018 this Applet was redesigned and updated to "HTML5" by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Xiaohan_Liu_.28Bachelorarbeit_2018.29\|Xiaohan Liu]]  as part of her Bachelor's 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]]).


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	{{LntAppletLink\|physAnSignal_en}}         [https://lntwww.lnt.ei.tum.de/Applets:Physikalisches_Signal_%26_%C3%84quivalentes_TP-Signal '''English Applet with German WIKI description''']		{{LntAppletLink\|physAnSignal_en}}         [https://lntwww.lnt.ei.tum.de/Applets:Physikalisches_Signal_%26_%C3%84quivalentes_TP-Signal '''English Applet with German WIKI description''']
	[[de:Applets:~~Physical_Signal_~~&~~_Equivalent_Low-pass_Signal~~]]		[[de:Applets:Physical Signal & Equivalent Lowpass Signal]]

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	The associated equivalent low-pass signal is $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:		The associated equivalent low-pass signal is $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

	:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}		:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$
	A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

	[[File:Ortskurve_1.png\|right\|frame\|Equivalent low-pass signal currently $t=0$ for cosinusoidal carrier   ⇒   $\varphi_{\rm T} = 0$]]		[[File:Ortskurve_1.png\|right\|frame\|Equivalent low-pass signal currently $t=0$ for cosinusoidal carrier   ⇒   $\varphi_{\rm T} = 0$]]
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	The '''analytic signal''' $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:		The '''analytic signal''' $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:
	[[File:EN_Sig_T_4_2_S1c.png\|right\|frame\|spectral functions $X(f)$, $X_+(f)$ and $X_{\rm TP}(f)$\|class=fit]]		[[File:EN_Sig_T_4_2_S1c.png\|right\|frame\|spectral functions $X(f)$, $X_+(f)$ and $X_{\rm TP}(f)$\|class=fit]]
	:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \~~cdot~~		:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdotX(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$
	X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$

	The ''Signum function'' is for positive values of $f$ equal to $+1$ and for negative $f$ values equal to $-1$.		The ''Signum function'' is for positive values of $f$ equal to $+1$ and for negative $f$ values equal to $-1$.
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	The associated equivalent low-pass signal is with $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:		The associated equivalent low-pass signal is with $f_{\rm O}\hspace{0.01cm}' = f_{\rm O}- f_{\rm T} > 0$,   $f_{\rm U}\hspace{0.01cm}' = f_{\rm U}- f_{\rm T} < 0$  and  $f_{\rm T}\hspace{0.01cm}' = 0$:

	:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}		:$$x_{\rm TP}(t) = x_\text{TP, T}(t) + x_\text{TP, O}(t) + x_\text{TP, U}(t) = A_{\rm T}\cdot {\rm e}^{-{\rm j} \varphi_{\rm T} } \hspace{0.1cm}+ \hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{-{\rm j} \varphi_{\rm O} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm}+ \hspace{0.1cm}A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$
	A_{\rm U}\cdot {\rm e}^{-{\rm j} \varphi_{\rm U} } \cdot {\rm e}^{ {\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.01cm}'\hspace{0.05cm}\cdot \hspace{0.05cm}t} . $$

	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=

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	{{LntAppletLink\|physAnSignal_en}}         [https://lntwww.lnt.ei.tum.de/Applets:Physikalisches_Signal_%26_%C3%84quivalentes_TP-Signal '''English Applet with German WIKI description''']		{{LntAppletLink\|physAnSignal_en}}         [https://lntwww.lnt.ei.tum.de/Applets:Physikalisches_Signal_%26_%C3%84quivalentes_TP-Signal '''English Applet with German WIKI description''']
			[[de:Applets:Physical_Signal_&_Equivalent_Low-pass_Signal]]

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	{{LntAppletLink\|physAnSignal_en}}         [https://~~www~~.~~lntwww~~.de/Applets:Physikalisches_Signal_%26_%C3%84quivalentes_TP-Signal '''English Applet with German WIKI description''']		{{LntAppletLink\|physAnSignal_en}}         [https://lntwww.lnt.ei.tum.de/Applets:Physikalisches_Signal_%26_%C3%84quivalentes_TP-Signal '''English Applet with German WIKI description''']
	==Applet Description==		==Applet Description==
	<br>		<br>
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	==Once again: Open Applet in new Tab==		==Once again: Open Applet in new Tab==

	{{LntAppletLink\|physAnSignal_en}}         [https://~~www~~.~~lntwww~~.de/Applets:Physikalisches_Signal_%26_%C3%84quivalentes_TP-Signal '''English Applet with German WIKI description''']		{{LntAppletLink\|physAnSignal_en}}         [https://lntwww.lnt.ei.tum.de/Applets:Physikalisches_Signal_%26_%C3%84quivalentes_TP-Signal '''English Applet with German WIKI description''']

Guenter at 15:04, 13 April 2023

2023-04-13T15:04:56Z

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	The '''analytic signal''' $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:		The '''analytic signal''' $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:
	[[File:~~EN_Sig_T_4_2_S1b~~.png\|right\|frame\|spectral functions $X(f)$, $X_+(f)$ and $X_{\rm TP}(f)$\|class=fit]]		[[File:EN_Sig_T_4_2_S1c.png\|right\|frame\|spectral functions $X(f)$, $X_+(f)$ and $X_{\rm TP}(f)$\|class=fit]]
	:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot		:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot
	X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$		X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$

Guenter at 15:03, 13 April 2023

2023-04-13T15:03:16Z

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	The graph is intended to illustrate this relationship, assuming $A_{\rm U} > A_{\rm O}$   ⇒   the green pointer (for the lower sideband) is longer than the blue pointer (upper sideband). This is a snapshot at time $t_0$:		The graph is intended to illustrate this relationship, assuming $A_{\rm U} > A_{\rm O}$   ⇒   the green pointer (for the lower sideband) is longer than the blue pointer (upper sideband). This is a snapshot at time $t_0$:

	[[File:~~Ortskurve_3_neu~~.png\|center\|frame\|band-pass spectrum $X(f)$ ~~'''Korrektur'''~~\|class=fit]]		[[File:EN_Sig_T_4_2_ortskurve.png\|center\|frame\|band-pass spectrum $X(f)$\|class=fit]]

	*For these system parameters, the top of the pointer cluster $x_{\rm TP}(t)$ – that is, the geometric sum of red, blue and green pointers – on an ellipse.		*For these system parameters, the top of the pointer cluster $x_{\rm TP}(t)$ – that is, the geometric sum of red, blue and green pointers – on an ellipse.

Guenter at 14:11, 13 April 2023

2023-04-13T14:11:38Z

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	The '''analytic signal''' $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:		The '''analytic signal''' $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:
	[[File:~~Ortskurve_2~~.png\|right\|frame\|spectral functions $X(f)$, $X_+(f)$ and $X_{\rm TP}(f)$ ~~'''Korrektur'''~~\|class=fit]]		[[File:EN_Sig_T_4_2_S1b.png\|right\|frame\|spectral functions $X(f)$, $X_+(f)$ and $X_{\rm TP}(f)$\|class=fit]]
	:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot		:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot
	X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$		X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$

Guenter at 15:44, 10 April 2023

2023-04-10T15:44:41Z

← Older revision		Revision as of 17:44, 10 April 2023
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	The '''analytic signal''' $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:		The '''analytic signal''' $x_+(t)$ belonging to the physical signal $x(t)$ is the time function whose spectrum fulfills the following property:
	[[File:Ortskurve_2.png\|right\|frame\|spectral functions $X(f)$, $X_+(f)$ and $X_{\rm TP}(f)$ \|class=fit]]		[[File:Ortskurve_2.png\|right\|frame\|spectral functions $X(f)$, $X_+(f)$ and $X_{\rm TP}(f)$ '''Korrektur'''\|class=fit]]
	:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot		:$$X_+(f)=\big[1+{\rm sign}(f)\big] \cdot X(f) = \left\{ {2 \cdot
	X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$		X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$
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	The graph is intended to illustrate this relationship, assuming $A_{\rm U} > A_{\rm O}$   ⇒   the green pointer (for the lower sideband) is longer than the blue pointer (upper sideband). This is a snapshot at time $t_0$:		The graph is intended to illustrate this relationship, assuming $A_{\rm U} > A_{\rm O}$   ⇒   the green pointer (for the lower sideband) is longer than the blue pointer (upper sideband). This is a snapshot at time $t_0$:

	[[File:Ortskurve_3_neu.png\|center\|frame\|band-pass spectrum $X(f)$ \|class=fit]]		[[File:Ortskurve_3_neu.png\|center\|frame\|band-pass spectrum $X(f)$ '''Korrektur'''\|class=fit]]

	*For these system parameters, the top of the pointer cluster $x_{\rm TP}(t)$ – that is, the geometric sum of red, blue and green pointers – on an ellipse.		*For these system parameters, the top of the pointer cluster $x_{\rm TP}(t)$ – that is, the geometric sum of red, blue and green pointers – on an ellipse.

Noah at 22:35, 26 March 2023

2023-03-26T22:35:45Z

Show changes

Hwang at 16:07, 23 January 2023

2023-01-23T16:07:52Z

← Older revision		Revision as of 18:07, 23 January 2023
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	Beside the physical signal $x(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)$, one can also use the following descriptions of bandpass signals:		Beside the physical signal $x(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)$, one can also use the following descriptions of bandpass signals:
	*the analytic signal $x_+(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_+(f)$, see applet [[Applets:Physical_Signal_%26_Analytic_Signal\|Physical Signal & Analytic Signal]],		*the analytic signal $x_+(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_+(f)$, see applet [[Applets:Physical_Signal_%26_Analytic_Signal\|Physical Signal & Analytic Signal]],
	*the equivalent lowpass signal $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, see next ~~page~~		*the equivalent lowpass signal $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, see next section
	<br><br>		<br><br>