Applets:Physical Signal & Analytic Signal - Revision history

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2026-03-16T13:49:03Z

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	The associated analytic signal is:		The associated analytic signal is:

	:$$x_+(t) = x_{\rm U+}(t) + x_{\rm T+}(t) + x_{\rm O+}(t) = A_{\rm U}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm U})}		:$$x_+(t) = x_{\rm U+}(t) + x_{\rm T+}(t) + x_{\rm O+}(t) = A_{\rm U}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm U})}\hspace{0.1cm}+ \hspace{0.1cm}A_{\rm T}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm T})}\hspace{0.1cm}+\hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm O})}. $$
	\hspace{0.1cm}+ \hspace{0.1cm}A_{\rm T}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm T})}
	\hspace{0.1cm}+\hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm O})}. $$

	[[File:Zeigerdiagramm_2a_version2.png\|right\|frame\|Analytic signal at the time $t=0$]]		[[File:Zeigerdiagramm_2a_version2.png\|right\|frame\|Analytic signal at the time $t=0$]]
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	[[File:EN_Sig_T_4_2_S1a.png\|right\|frame\|Construction of the spectral function $X_+(f)$\|class=fit]]		[[File:EN_Sig_T_4_2_S1a.png\|right\|frame\|Construction of the spectral function $X_+(f)$\|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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	For the '''Hilbert transform''' $ {\rm H}\left\{x(t)\right\}$ of a time function $x(t)$ we have:		For the '''Hilbert transform''' $ {\rm H}\left\{x(t)\right\}$ of a time function $x(t)$ we have:

	:$$y(t) = {\rm H}\left\{x(t)\right\} = \frac{1}{ {\rm \pi} } \cdot		:$$y(t) = {\rm H}\left\{x(t)\right\} = \frac{1}{ {\rm \pi} } \cdot\hspace{0.03cm}\int_{-\infty}^{+\infty}\frac{x(\tau)}{ {t -\tau} }\hspace{0.15cm} {\rm d}\tau.$$
	\hspace{0.03cm}\int_{-\infty}^{+\infty}\frac{x(\tau)}{ {t -
	\tau} }\hspace{0.15cm} {\rm d}\tau.$$

	This particular integral is not solvable in a simple, conventional way, but must be evaluated using the [https://en.wikipedia.org/wiki/Cauchy_principal_value "Cauchy principal value theorem"].		This particular integral is not solvable in a simple, conventional way, but must be evaluated using the [https://en.wikipedia.org/wiki/Cauchy_principal_value "Cauchy principal value theorem"].
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	Thus, the spectrum of the analytic signal (that is, without the Dirac delta function at the frequency $f =-f_{\rm T}$, but doubling at $f =+f_{\rm T}$):		Thus, the spectrum of the analytic signal (that is, without the Dirac delta function at the frequency $f =-f_{\rm T}$, but doubling at $f =+f_{\rm T}$):

	:$$X_+(f) = A \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\varphi}\cdot\delta (f - f_{\rm		:$$X_+(f) = A \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\varphi}\cdot\delta (f - f_{\rm T}) .$$
	T}) .$$

	The associated time function is obtained by applying the Displacement Law:		The associated time function is obtained by applying the Displacement Law:

	:$$x_+(t) = x_{\rm U+}(t) + x_{\rm T+}(t) + x_{\rm O+}(t) = A_{\rm U}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm U})}		:$$x_+(t) = x_{\rm U+}(t) + x_{\rm T+}(t) + x_{\rm O+}(t) = A_{\rm U}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm U})}\hspace{0.1cm}+ \hspace{0.1cm}A_{\rm T}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm T})}\hspace{0.1cm}+\hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm O})}. $$
	\hspace{0.1cm}+ \hspace{0.1cm}A_{\rm T}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm T})}
	\hspace{0.1cm}+\hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm O})}. $$

	This equation describes a pointer rotating at constant angular velocity $\omega_{\rm T} = 2\pi f_{\rm T}$.		This equation describes a pointer rotating at constant angular velocity $\omega_{\rm T} = 2\pi f_{\rm T}$.
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	The associated analytic signal is:		The associated analytic signal is:
	:$$x_+(t) = x_{\rm U+}(t) + x_{\rm T+}(t) + x_{\rm O+}(t) = A_{\rm U}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm U})}		:$$x_+(t) = x_{\rm U+}(t) + x_{\rm T+}(t) + x_{\rm O+}(t) = A_{\rm U}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm U}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm U})}\hspace{0.1cm}+ \hspace{0.1cm}A_{\rm T}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm T})}\hspace{0.1cm}+\hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm O})}. $$
	\hspace{0.1cm}+ \hspace{0.1cm}A_{\rm T}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm T}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm T})}
	\hspace{0.1cm}+\hspace{0.1cm} A_{\rm O}\cdot {\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}(2\pi \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm O}\hspace{0.05cm}\cdot \hspace{0.05cm}t- \varphi_{\rm O})}. $$

	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
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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]]).

	==Once again:  Open Applet in new Tab==		==Once again:  Open Applet in new Tab==

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	{{LntAppletLink\|physAnSignal_en}}         [https://lntwww.lnt.ei.tum.de/Applets:Physikalisches_Signal_%26_Analytisches_Signal '''English Applet with German WIKI description''']		{{LntAppletLink\|physAnSignal_en}}         [https://lntwww.lnt.ei.tum.de/Applets:Physikalisches_Signal_%26_Analytisches_Signal '''English Applet with German WIKI description''']
			[[de:Applets:Physical_Signal_&_Analytic_Signal]]

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Update domain links: en.lntwww.de → en.lntwww.lnt.ei.tum.de, www.lntwww.de → lntwww.lnt.ei.tum.de

← Older revision		Revision as of 13:09, 16 March 2026
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	{{LntAppletLink\|physAnSignal_en}}         [https://~~www~~.~~lntwww~~.de/Applets:Physikalisches_Signal_%26_Analytisches_Signal '''English Applet with German WIKI description''']		{{LntAppletLink\|physAnSignal_en}}         [https://lntwww.lnt.ei.tum.de/Applets:Physikalisches_Signal_%26_Analytisches_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_Analytisches_Signal '''English Applet with German WIKI description''']		{{LntAppletLink\|physAnSignal_en}}         [https://lntwww.lnt.ei.tum.de/Applets:Physikalisches_Signal_%26_Analytisches_Signal '''English Applet with German WIKI description''']

Guenter at 13:54, 13 April 2023

2023-04-13T13:54:40Z

← Older revision		Revision as of 15:54, 13 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:~~Zeigerdiagramm_3a~~.png\|right\|frame\|Construction of the spectral function $X_+(f)$ ~~'''Korrektur'''~~\|class=fit]]		[[File:EN_Sig_T_4_2_S1a.png\|right\|frame\|Construction of the spectral function $X_+(f)$\|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 15:41, 10 April 2023

2023-04-10T15:41:22Z

← Older revision		Revision as of 17:41, 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:Zeigerdiagramm_3a.png\|right\|frame\|Construction of the spectral function $X_+(f)$ \|class=fit]]		[[File:Zeigerdiagramm_3a.png\|right\|frame\|Construction of the spectral function $X_+(f)$ '''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.$$		X(f) \; \hspace{0.2cm}\rm for\hspace{0.2cm} {\it f} > 0, \atop {\,\,\,\, \rm 0 \; \hspace{0.9cm}\rm for\hspace{0.2cm} {\it f} < 0.} }\right.$$

Noah at 22:25, 26 March 2023

2023-03-26T22:25:25Z

Hwang at 16:08, 23 January 2023

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	Besides the physical signal $x(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)$, one can also use the following descriptions of bandpass signals:		Besides 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 next ~~page~~,		*the analytic signal $x_+(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_+(f)$, see next section,
	*the equivalent lowpass signal   (in German:   äquivalentes '''T'''ief '''P'''ass–Signal) $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, <br>see Applet [[Applets:Physical_Signal_%26_Equivalent_Low-pass_Signal\|Physical Signal & Equivalent Lowpass signal]].		*the equivalent lowpass signal   (in German:   äquivalentes '''T'''ief '''P'''ass–Signal) $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, <br>see Applet [[Applets:Physical_Signal_%26_Equivalent_Low-pass_Signal\|Physical Signal & Equivalent Lowpass signal]].
	<br><br>		<br><br>

Guenter at 14:33, 13 January 2023

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	{{~~LntAppletLinkEn~~\|physAnSignal_en}}		{{LntAppletLink\|physAnSignal_en}}         [https://www.lntwww.de/Applets:Physikalisches_Signal_%26_Analytisches_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==

	{{~~LntAppletLinkEn~~\|physAnSignal_en}}		{{LntAppletLink\|physAnSignal_en}}         [https://www.lntwww.de/Applets:Physikalisches_Signal_%26_Analytisches_Signal '''English Applet with German WIKI description''']

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← Older revision		Revision as of 00:25, 27 March 2023
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	==Applet Description==		==Applet Description==
	<br>		<br>
	This applet shows the relationship between the physical ~~bandpass~~ signal $x(t)$ and the associated analytic signal $x_+(t)$. It is assumed that the ~~bandpass~~ signal $x(t)$ has a frequency-discrete spectrum $X(f)$:		This applet shows the relationship between the physical band-pass signal $x(t)$ and the associated analytic signal $x_+(t)$. It is assumed that the band-pass signal $x(t)$ has a frequency-discrete spectrum $X(f)$:
	:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$		:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
	The physical signal $x(t)$ is thus composed of three harmonic oscillations, a constellation that can be found, for example, in the ''Double-sideband Amplitude Modulation''		The physical signal $x(t)$ is thus composed of three harmonic oscillations, a constellation that can be found, for example, in the ''Double-sideband Amplitude Modulation''
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	The time trace of $x_+(t)$ is also referred to below as ''Pointer Diagram''. The relationship between the physical ~~bandpass~~ signal $x(t)$ and the associated analytic signal $x_+(t)$ is:		The time trace of $x_+(t)$ is also referred to below as ''Pointer Diagram''. The relationship between the physical band-pass signal $x(t)$ and the associated analytic signal $x_+(t)$ is:

	:$$x(t) = {\rm Re}\big [x_+(t)\big ].$$		:$$x(t) = {\rm Re}\big [x_+(t)\big ].$$

	''Note:''   In the figure $\varphi_{\rm O} = +30^\circ$. This leads to the angle with respect to the coordinate system at $t=0$:   $\phi_{\rm O}=-\varphi_{\rm O}=-30^\circ$. Similarly, the ~~null~~ phase angle $\varphi_{\rm U}=-30^\circ$ of the lower sideband leads to the phase angle to be considered in the complex plane:   $\phi_{\rm U}=+30^\circ$.		''Note:''   In the figure $\varphi_{\rm O} = +30^\circ$. This leads to the angle with respect to the coordinate system at $t=0$:   $\phi_{\rm O}=-\varphi_{\rm O}=-30^\circ$. Similarly, the zero phase angle $\varphi_{\rm U}=-30^\circ$ of the lower sideband leads to the phase angle to be considered in the complex plane:   $\phi_{\rm U}=+30^\circ$.


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	==Theoretical Background==		==Theoretical Background==
	<br>		<br>
	===Description of ~~Bandpass~~ Signals===		===Description of Band-pass Signals===
	[[File:Zeigerdiagramm_1a.png\|right\|frame\|~~Bandpass~~ spectrum $X(f)$ \|class=fit]]		[[File:Zeigerdiagramm_1a.png\|right\|frame\|Band-pass spectrum $X(f)$ \|class=fit]]
	We consider '''~~bandpass~~ signals''' $x(t)$ with the property that their spectra $X(f)$ are not in the range around the frequency $f=0$, but around a carrier frequency $f_{\rm T}$. In most cases it can also be assumed that the bandwidth is $B \ll f_{\rm T}$.		We consider '''band-pass signals''' $x(t)$ with the property that their spectra $X(f)$ are not in the range around the frequency $f=0$, but around a carrier frequency $f_{\rm T}$. In most cases it can also be assumed that the bandwidth is $B \ll f_{\rm T}$.

	The figure shows such a ~~bandpass~~ spectrum $X(f)$. Assuming that the associated $x(t)$ is a physical signal and thus real, the spectral function $X(f)$ has a symmetry with respect to the frequency $f = 0$, if $x(t)$ is an even function   ⇒   $x(-t)=x(t)$, $X(f)$ is real and even.		The figure shows such a band-pass spectrum $X(f)$. Assuming that the associated $x(t)$ is a physical signal and thus real, the spectral function $X(f)$ has a symmetry with respect to the frequency $f = 0$, if $x(t)$ is an even function   ⇒   $x(-t)=x(t)$, $X(f)$ is real and even.


	Besides the physical signal $x(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)$, one can also use the following descriptions of ~~bandpass~~ signals:		Besides the physical signal $x(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X(f)$, one can also use the following descriptions of band-pass signals:
	*the analytic signal $x_+(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_+(f)$, see next section,		*the analytic signal $x_+(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_+(f)$, see next section,
	*the equivalent ~~lowpass~~ signal   (in German:   äquivalentes '''T'''ief '''P'''ass–Signal) $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, <br>see Applet [[Applets:Physical_Signal_%26_Equivalent_Low-pass_Signal\|Physical Signal & Equivalent ~~Lowpass~~ signal]].		*the equivalent low-pass signal   (in German:   äquivalentes '''T'''ief '''P'''ass–Signal) $x_{\rm TP}(t)\ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \ X_{\rm TP}(f)$, <br>see Applet [[Applets:Physical_Signal_%26_Equivalent_Low-pass_Signal\|"Physical Signal & Equivalent Low-pass signal"]].
	<br><br>		<br><br>

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	From the graph you can see the calculation rule for $X_+(f)$:		From the graph you can see the calculation rule for $X_+(f)$:

	The actual ~~bandpass~~ spectrum $X(f)$ becomes		The actual band-pass spectrum $X(f)$ becomes
	* doubled at the positive frequencies, and		* doubled at the positive frequencies, and
	* set to zero at the negative frequencies.		* set to zero at the negative frequencies.
Line 85:		Line 85:
	\tau} }\hspace{0.15cm} {\rm d}\tau.$$		\tau} }\hspace{0.15cm} {\rm d}\tau.$$

	This particular integral is not solvable in a simple, conventional way, but must be evaluated using the [https://en.wikipedia.org/wiki/Cauchy_principal_value Cauchy principal value theorem].		This particular integral is not solvable in a simple, conventional way, but must be evaluated using the [https://en.wikipedia.org/wiki/Cauchy_principal_value "Cauchy principal value theorem"].

	Accordingly, in the frequency domain:		Accordingly, in the frequency domain:
Line 92:		Line 92:

	The above result can be summarized with this definition as follows:		The above result can be summarized with this definition as follows:
	* The analytic signal $x_+(t)$ is obtained from the physical ~~bandpass~~ signal $x(t)$ by adding an imaginary part to $x(t)$ according to the Hilbert transform:		* The analytic signal $x_+(t)$ is obtained from the physical band-pass signal $x(t)$ by adding an imaginary part to $x(t)$ according to the Hilbert transform:

	:$$x_+(t) = x(t)+{\rm j} \cdot {\rm H}\left\{x(t)\right\} .$$		:$$x_+(t) = x(t)+{\rm j} \cdot {\rm H}\left\{x(t)\right\} .$$
Line 99:		Line 99:


	* From the analytic signal $x_+(t)$, the physical ~~bandpass~~ signal can be easily determined by the following operation:		* From the analytic signal $x_+(t)$, the physical band-pass signal can be easily determined by the following operation:
	:$$x(t) = {\rm Re}\big[x_+(t)\big] .$$		:$$x(t) = {\rm Re}\big[x_+(t)\big] .$$

Line 115:		Line 115:
	===Representation of the Harmonic Oscillation as an Analytic Signal===		===Representation of the Harmonic Oscillation as an Analytic Signal===

	The spectral function $X(f)$ of a harmonic oscillation $x(t) = A\cdot\text{cos}(2\pi f_{\rm T}\cdot t - \varphi)$ is known to consist of two Dirac functions at the frequencies		The spectral function $X(f)$ of a harmonic oscillation $x(t) = A\cdot\text{cos}(2\pi f_{\rm T}\cdot t - \varphi)$ is known to consist of two Dirac delta functions at the frequencies
	* $+f_{\rm T}$ with the complex weight $A/2 \cdot \text{e}^{-\text{j}\hspace{0.05cm}\varphi}$,		* $+f_{\rm T}$ with the complex weight $A/2 \cdot \text{e}^{-\text{j}\hspace{0.05cm}\varphi}$,
	* $-f_{\rm T}$ with the complex weight $A/2 \cdot \text{e}^{+\text{j}\hspace{0.05cm}\varphi}$.		* $-f_{\rm T}$ with the complex weight $A/2 \cdot \text{e}^{+\text{j}\hspace{0.05cm}\varphi}$.


	Thus, the spectrum of the analytic signal (that is, without the Dirac function at the frequency $f =-f_{\rm T}$, but doubling at $f =+f_{\rm T}$):		Thus, the spectrum of the analytic signal (that is, without the Dirac delta function at the frequency $f =-f_{\rm T}$, but doubling at $f =+f_{\rm T}$):

	:$$X_+(f) = A \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\varphi}\cdot\delta (f - f_{\rm		:$$X_+(f) = A \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\varphi}\cdot\delta (f - f_{\rm
Line 161:		Line 161:
	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{Example 3:}$		$\text{Example 3:}$
	Shown the constellation arises i.e. in the [https://en.wikipedia.org/wiki/Sideband Double-sideband Amplitude Modulation] (with carrier) 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.		Shown the constellation arises i.e. in the [https://en.wikipedia.org/wiki/Sideband "Double-sideband Amplitude Modulation"] (with carrier) 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.


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	'''(8)'''   Now  $\text{Red:} \hspace{0.05cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.05cm} A_{\rm U} = 0.4\ \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.05cm} A_{\rm O} = 0.2\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = +90^\circ$.		'''(8)'''   Now  $\text{Red:} \hspace{0.05cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.05cm} A_{\rm U} = 0.4\ \text{V}, \ f_{\rm U} = 80 \ \text{kHz}, \ \varphi_{\rm U} = -90^\circ$,   $\text{Blue:} \hspace{0.05cm} A_{\rm O} = 0.2\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = +90^\circ$.

	:Which constellation could be described here? Which shape results for the equivalent ~~lowpass~~ signal $x_{\rm TP}(t)$?}}		:Which constellation could be described here? Which shape results for the equivalent low-pass signal $x_{\rm TP}(t)$?}}

	::It could be a DSB–AM of a sinusoidal signal with cosinusoidal carrier and modulation degree $m=0.8$, in which the upper sideband is attenuated by a factor of 2. The equivalent ~~lowpass~~ signal $x_{\rm TP}(t)$ has an elliptical trace in the complex plane.		::It could be a DSB–AM of a sinusoidal signal with cosinusoidal carrier and modulation degree $m=0.8$, in which the upper sideband is attenuated by a factor of 2. The equivalent low-pass signal $x_{\rm TP}(t)$ has an elliptical trace in the complex plane.

	==Applet Manual==		==Applet Manual==

← Older revision		Revision as of 16:45, 28 May 2021
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	The nomenclature is also adapted to this case:		The nomenclature is also adapted to this case:
	* $x_{\rm O}(t)$ denotes the "upper sideband~~”~~   (in German:   '''O'''beres Seitenband) with the amplitude $A_{\rm O}= A_{\rm N}/2$, the frequency $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and the phase $\varphi_{\rm O} = \varphi_{\rm T} + \varphi_{\rm N}$.		* $x_{\rm O}(t)$ denotes the "upper sideband"   (in German:   '''O'''beres Seitenband) with the amplitude $A_{\rm O}= A_{\rm N}/2$, the frequency $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and the phase $\varphi_{\rm O} = \varphi_{\rm T} + \varphi_{\rm N}$.
	*Similarly, for the "lower sideband~~”~~   (in German:   '''U'''nteres Seitenband) $x_{\rm U}(t)$ with $f_{\rm U} = f_{\rm T} - f_{\rm N}$, $A_{\rm U}= A_{\rm O}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$.		*Similarly, for the "lower sideband"   (in German:   '''U'''nteres Seitenband) $x_{\rm U}(t)$ with $f_{\rm U} = f_{\rm T} - f_{\rm N}$, $A_{\rm U}= A_{\rm O}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$.


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	:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$		:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
	* Each of the three harmonic oscillations $x_{\rm T}(t)$, $x_{\rm U}(t)$ and $x_{\rm O}(t)$ is represented by an amplitude $(A)$, a frequency $(f)$ and a phase value $(\varphi)$.		* Each of the three harmonic oscillations $x_{\rm T}(t)$, $x_{\rm U}(t)$ and $x_{\rm O}(t)$ is represented by an amplitude $(A)$, a frequency $(f)$ and a phase value $(\varphi)$.
	*The indices are based on the ''Double-sideband Amplitude Modulation'' method. "T~~”~~ stands for "carrier~~”~~, "U~~”~~ for "lower sideband~~”~~ and "O~~”~~ for "upper Sideband~~”~~.		*The indices are based on the ''Double-sideband Amplitude Modulation'' method. "T" stands for "carrier", "U" for "lower sideband" and "O" for "upper Sideband".
	*Accordingly, $f_{\rm U} < f_{\rm T}$ and $f_{\rm O} > f_{\rm T}$. There are no restrictions for the amplitudes and phases.		*Accordingly, $f_{\rm U} < f_{\rm T}$ and $f_{\rm O} > f_{\rm T}$. There are no restrictions for the amplitudes and phases.

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	*A task description is displayed.		*A task description is displayed.
	*Parameter values are adjusted.		*Parameter values are adjusted.
	*Solution after pressing "Hide solition~~”~~.		*Solution after pressing "Hide solition".


	The number "0~~”~~ will reset the program and output a text with further explanation of the applet.		The number "0" will reset the program and output a text with further explanation of the applet.
	<br clear=all>		<br clear=all>
	In the following, $\rm Green$ denotes the lower sideband   ⇒   $\big (A_{\rm U}, f_{\rm U}, \varphi_{\rm U}\big )$,		In the following, $\rm Green$ denotes the lower sideband   ⇒   $\big (A_{\rm U}, f_{\rm U}, \varphi_{\rm U}\big )$,
Line 235:		Line 235:
	'''(7)'''     Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.8\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 90^\circ$.		'''(7)'''     Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.8\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 90^\circ$.

	:Which constellation is described here? Which figure is given for the equivalent low-pass signal $x_{\rm TP}(t)$?   ⇒   "locus~~”~~? <br>What changes with $A_{\rm U} = 0.8\ \text{V}$ and $A_{\rm O} = 0$?}}		:Which constellation is described here? Which figure is given for the equivalent low-pass signal $x_{\rm TP}(t)$?   ⇒   "locus"? <br>What changes with $A_{\rm U} = 0.8\ \text{V}$ and $A_{\rm O} = 0$?}}

	::In both cases, it is a [https://en.wikipedia.org/wiki/Single-sideband_modulation Single-sideband Amplitude Modulation] (SSB–AM) with the modulation degree $\mu = 0.8$ (in SSB we denote the degree of modulation with $\mu$ instead of $m$). The carrier signal is cosinusoidal and the message signal is sinusoidal. The equivalent low-pass signal $x_{\rm TP}(t)$ has a circular course in the complex plane.		::In both cases, it is a [https://en.wikipedia.org/wiki/Single-sideband_modulation Single-sideband Amplitude Modulation] (SSB–AM) with the modulation degree $\mu = 0.8$ (in SSB we denote the degree of modulation with $\mu$ instead of $m$). The carrier signal is cosinusoidal and the message signal is sinusoidal. The equivalent low-pass signal $x_{\rm TP}(t)$ has a circular course in the complex plane.
Line 270:		Line 270:
	'''(D)'''     Control elements:   Start – Step – Pause/Continue – Reset		'''(D)'''     Control elements:   Start – Step – Pause/Continue – Reset

	'''(E)'''     Speed of animation:   "Speed~~”~~   ⇒   Values: 1, 2, 3		'''(E)'''     Speed of animation:   "Speed"   ⇒   Values: 1, 2, 3

	'''(F)'''     "Trace~~”~~   ⇒   On or Off, trace of complex signal values $x_{\rm +}(t)$		'''(F)'''     "Trace"   ⇒   On or Off, trace of complex signal values $x_{\rm +}(t)$

	'''(G)'''     Numerical output of the time $t$ and the signal values  ${\rm Re}[x_{\rm +}(t)] = x(t)$  and  ${\rm Im}[x_{\rm +}(t)]$		'''(G)'''     Numerical output of the time $t$ and the signal values  ${\rm Re}[x_{\rm +}(t)] = x(t)$  and  ${\rm Im}[x_{\rm +}(t)]$
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	'''(H)'''     Variations for the graphical representation		'''(H)'''     Variations for the graphical representation

	$\hspace{1.5cm}$Zoom–Functions "$+$~~”~~ (Enlarge), "$-$~~”~~ (Decrease) and $\rm o$ (Reset to default)		$\hspace{1.5cm}$Zoom–Functions "$+$" (Enlarge), "$-$" (Decrease) and $\rm o$ (Reset to default)

	$\hspace{1.5cm}$Move with "$\leftarrow$~~”~~ (Section to the left, ordinate to the right), "$\uparrow$~~”~~ "$\downarrow$~~”~~ and "$\rightarrow$~~”~~		$\hspace{1.5cm}$Move with "$\leftarrow$" (Section to the left, ordinate to the right), "$\uparrow$" "$\downarrow$" and "$\rightarrow$"

	'''(I)'''     Experiment section:  Task selection and task		'''(I)'''     Experiment section:  Task selection and task
Line 295:		Line 295:
	==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/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]]).
	*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/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]]).

	==Once again:  Open Applet in new Tab==		==Once again:  Open Applet in new Tab==

	{{LntAppletLinkEn\|physAnSignal_en}}		{{LntAppletLinkEn\|physAnSignal_en}}

← Older revision		Revision as of 16:44, 28 May 2021
Line 11:		Line 11:

	The nomenclature is also adapted to this case:		The nomenclature is also adapted to this case:
	* $x_{\rm O}(t)$ denotes the ~~&bdquo;~~upper sideband”   (in German:   '''O'''beres Seitenband) with the amplitude $A_{\rm O}= A_{\rm N}/2$, the frequency $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and the phase $\varphi_{\rm O} = \varphi_{\rm T} + \varphi_{\rm N}$.		* $x_{\rm O}(t)$ denotes the "upper sideband”   (in German:   '''O'''beres Seitenband) with the amplitude $A_{\rm O}= A_{\rm N}/2$, the frequency $f_{\rm O} = f_{\rm T} + f_{\rm N}$ and the phase $\varphi_{\rm O} = \varphi_{\rm T} + \varphi_{\rm N}$.
	*Similarly, for the ~~&bdquo;~~lower sideband”   (in German:   '''U'''nteres Seitenband) $x_{\rm U}(t)$ with $f_{\rm U} = f_{\rm T} - f_{\rm N}$, $A_{\rm U}= A_{\rm O}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$.		*Similarly, for the "lower sideband”   (in German:   '''U'''nteres Seitenband) $x_{\rm U}(t)$ with $f_{\rm U} = f_{\rm T} - f_{\rm N}$, $A_{\rm U}= A_{\rm O}$ and $\varphi_{\rm U} = -\varphi_{\rm O}$.


Line 151:		Line 151:
	:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$		:$$x(t) = x_{\rm U}(t) + x_{\rm T}(t) + x_{\rm O}(t) = A_{\rm U}\cdot \cos\left(2\pi f_{\rm U}\cdot t- \varphi_{\rm U}\right)+A_{\rm T}\cdot \cos\left(2\pi f_{\rm T}\cdot t- \varphi_{\rm T}\right)+A_{\rm O}\cdot \cos\left(2\pi f_{\rm O}\cdot t- \varphi_{\rm O}\right). $$
	* Each of the three harmonic oscillations $x_{\rm T}(t)$, $x_{\rm U}(t)$ and $x_{\rm O}(t)$ is represented by an amplitude $(A)$, a frequency $(f)$ and a phase value $(\varphi)$.		* Each of the three harmonic oscillations $x_{\rm T}(t)$, $x_{\rm U}(t)$ and $x_{\rm O}(t)$ is represented by an amplitude $(A)$, a frequency $(f)$ and a phase value $(\varphi)$.
	*The indices are based on the ''Double-sideband Amplitude Modulation'' method. ~~&bdquo;~~T” stands for ~~&bdquo;~~carrier”, ~~&bdquo;~~U” for ~~&bdquo;~~lower sideband” and ~~&bdquo;~~O” for ~~&bdquo;~~upper Sideband”.		*The indices are based on the ''Double-sideband Amplitude Modulation'' method. "T” stands for "carrier”, "U” for "lower sideband” and "O” for "upper Sideband”.
	*Accordingly, $f_{\rm U} < f_{\rm T}$ and $f_{\rm O} > f_{\rm T}$. There are no restrictions for the amplitudes and phases.		*Accordingly, $f_{\rm U} < f_{\rm T}$ and $f_{\rm O} > f_{\rm T}$. There are no restrictions for the amplitudes and phases.

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	*A task description is displayed.		*A task description is displayed.
	*Parameter values are adjusted.		*Parameter values are adjusted.
	*Solution after pressing ~~&bdquo;~~Hide solition”.		*Solution after pressing "Hide solition”.


	The number ~~&bdquo;~~0” will reset the program and output a text with further explanation of the applet.		The number "0” will reset the program and output a text with further explanation of the applet.
	<br clear=all>		<br clear=all>
	In the following, $\rm Green$ denotes the lower sideband   ⇒   $\big (A_{\rm U}, f_{\rm U}, \varphi_{\rm U}\big )$,		In the following, $\rm Green$ denotes the lower sideband   ⇒   $\big (A_{\rm U}, f_{\rm U}, \varphi_{\rm U}\big )$,
Line 235:		Line 235:
	'''(7)'''     Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.8\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 90^\circ$.		'''(7)'''     Now let   $\text{Red:} \hspace{0.15cm} A_{\rm T} = 1\ \text{V}, \ f_{\rm T} = 100 \ \text{kHz}, \ \varphi_{\rm T} = 0^\circ$,   $\text{Green:} \hspace{0.15cm} A_{\rm U} = 0$,   $\text{Blue:} \hspace{0.15cm} A_{\rm O} = 0.8\ \text{V}, \ f_{\rm O} = 120 \ \text{kHz}, \ \varphi_{\rm O} = 90^\circ$.

	:Which constellation is described here? Which figure is given for the equivalent low-pass signal $x_{\rm TP}(t)$?   ⇒   ~~&bdquo;~~locus”? <br>What changes with $A_{\rm U} = 0.8\ \text{V}$ and $A_{\rm O} = 0$?}}		:Which constellation is described here? Which figure is given for the equivalent low-pass signal $x_{\rm TP}(t)$?   ⇒   "locus”? <br>What changes with $A_{\rm U} = 0.8\ \text{V}$ and $A_{\rm O} = 0$?}}

	::In both cases, it is a [https://en.wikipedia.org/wiki/Single-sideband_modulation Single-sideband Amplitude Modulation] (SSB–AM) with the modulation degree $\mu = 0.8$ (in SSB we denote the degree of modulation with $\mu$ instead of $m$). The carrier signal is cosinusoidal and the message signal is sinusoidal. The equivalent low-pass signal $x_{\rm TP}(t)$ has a circular course in the complex plane.		::In both cases, it is a [https://en.wikipedia.org/wiki/Single-sideband_modulation Single-sideband Amplitude Modulation] (SSB–AM) with the modulation degree $\mu = 0.8$ (in SSB we denote the degree of modulation with $\mu$ instead of $m$). The carrier signal is cosinusoidal and the message signal is sinusoidal. The equivalent low-pass signal $x_{\rm TP}(t)$ has a circular course in the complex plane.
Line 270:		Line 270:
	'''(D)'''     Control elements:   Start – Step – Pause/Continue – Reset		'''(D)'''     Control elements:   Start – Step – Pause/Continue – Reset

	'''(E)'''     Speed of animation:   ~~&bdquo;~~Speed”   ⇒   Values: 1, 2, 3		'''(E)'''     Speed of animation:   "Speed”   ⇒   Values: 1, 2, 3

	'''(F)'''     ~~&bdquo;~~Trace”   ⇒   On or Off, trace of complex signal values $x_{\rm +}(t)$		'''(F)'''     "Trace”   ⇒   On or Off, trace of complex signal values $x_{\rm +}(t)$

	'''(G)'''     Numerical output of the time $t$ and the signal values  ${\rm Re}[x_{\rm +}(t)] = x(t)$  and  ${\rm Im}[x_{\rm +}(t)]$		'''(G)'''     Numerical output of the time $t$ and the signal values  ${\rm Re}[x_{\rm +}(t)] = x(t)$  and  ${\rm Im}[x_{\rm +}(t)]$
Line 278:		Line 278:
	'''(H)'''     Variations for the graphical representation		'''(H)'''     Variations for the graphical representation

	$\hspace{1.5cm}$Zoom–Functions ~~&bdquo;~~$+$” (Enlarge), ~~&bdquo;~~$-$” (Decrease) and $\rm o$ (Reset to default)		$\hspace{1.5cm}$Zoom–Functions "$+$” (Enlarge), "$-$” (Decrease) and $\rm o$ (Reset to default)

	$\hspace{1.5cm}$Move with ~~&bdquo;~~$\leftarrow$” (Section to the left, ordinate to the right), ~~&bdquo;~~$\uparrow$” ~~&bdquo;~~$\downarrow$” and ~~&bdquo;~~$\rightarrow$”		$\hspace{1.5cm}$Move with "$\leftarrow$” (Section to the left, ordinate to the right), "$\uparrow$” "$\downarrow$” and "$\rightarrow$”

	'''(I)'''     Experiment section:  Task selection and task		'''(I)'''     Experiment section:  Task selection and task
Line 295:		Line 295:
	==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 ~~&bdquo;~~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/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]]).
	*In 2018 this Applet was redesigned and updated to ~~&bdquo;~~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/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]]).

	==Once again:  Open Applet in new Tab==		==Once again:  Open Applet in new Tab==

	{{LntAppletLinkEn\|physAnSignal_en}}		{{LntAppletLinkEn\|physAnSignal_en}}