Applets:Pulses and Spectra - Revision history

Maintenance script: Join split $$ formula lines for correct rendering

2026-03-16T13:49:05Z

Join split $$ formula lines for correct rendering

Maintenance script: Add German interlanguage link

2026-03-16T12:39:07Z

Add German interlanguage link

← Older revision		Revision as of 14:39, 16 March 2026
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	{{LntAppletLinkEnDe\|pulsesAndSpectra_en\|pulsesAndSpectra}}		{{LntAppletLinkEnDe\|pulsesAndSpectra_en\|pulsesAndSpectra}}
	<br><br>		<br><br>
			[[de:Applets:Impulse_und_Spektren]]

Guenter at 12:10, 21 April 2023

2023-04-21T12:10:40Z

← Older revision		Revision as of 14:10, 21 April 2023
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	==Applet Manual==		==Applet Manual==
	~~<br>~~
	[[File:EN_Impulse_Man_neu.png\|right\|frame\|Screenshot]]		[[File:EN_Impulse_Man_neu.png\|right\|frame\|Screenshot]]


	'''(A)'''     Theme (changeable graphical user interface design)		'''(A)'''     Theme (changeable graphical user interface design)
	:* Dark:   dark background  (recommended by the authors)		:* Dark:   dark background  (recommended by the authors)

Guenter at 12:09, 21 April 2023

2023-04-21T12:09:01Z

← Older revision		Revision as of 14:09, 21 April 2023
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	==Applet Manual==		==Applet Manual==
	<br>		<br>
	[[File:EN_Impulse_Man_neu.png\|right\|frame\|Screenshot ~~'''Korrektur''' Pulses~~]]		[[File:EN_Impulse_Man_neu.png\|right\|frame\|Screenshot]]
	'''(A)'''     Theme (changeable graphical user interface design)		'''(A)'''     Theme (changeable graphical user interface design)
	:* Dark:   dark background  (recommended by the authors)		:* Dark:   dark background  (recommended by the authors)
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	:* Protanopia:   for users with pronounced red visual impairment		:* Protanopia:   for users with pronounced red visual impairment

	'''(B)'''     Preselection for pulse shape  $x_1(t)$  (red curve)		'''(B)'''     Preselection for pulse shape  $x_1(t)$   ⇒   red curve

	'''(C)'''     Parameter definition for  $x_1(t)$		'''(C)'''     Parameter definition for  $x_1(t)$
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	'''(D)'''     Numeric output for  $x_1(t_)$  and  $X_1(f_)$		'''(D)'''     Numeric output for  $x_1(t_)$  and  $X_1(f_)$

	'''(E)'''     Preselection for pulse shape  $x_2(t)$  (blue curve)		'''(E)'''     Preselection for pulse shape  $x_2(t)$  ⇒   blue curve

	'''(F)'''     Parameter definition for  $x_2(t)$		'''(F)'''     Parameter definition for  $x_2(t)$
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	'''(N)'''     Show and hide sample solution		'''(N)'''     Show and hide sample solution
			<br clear=all>

	==About the Authors==		==About the Authors==

Guenter at 16:32, 20 April 2023

2023-04-20T16:32:15Z

← Older revision		Revision as of 18:32, 20 April 2023
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	==Applet Manual==		==Applet Manual==
	<br>		<br>
	[[File:~~Exercise_impuls_v2~~.png\|right\|frame\|Screenshot '''Korrektur''' Pulses]]		[[File:EN_Impulse_Man_neu.png\|right\|frame\|Screenshot '''Korrektur''' Pulses]]
	'''(A)'''     Theme (changeable graphical user interface design)		'''(A)'''     Theme (changeable graphical user interface design)
	:* Dark:   dark background  (recommended by the authors)		:* Dark:   dark background  (recommended by the authors)

Guenter at 15:52, 20 April 2023

2023-04-20T15:52:50Z

← Older revision		Revision as of 17:52, 20 April 2023
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	==Applet Manual==		==Applet Manual==
	<br>		<br>
	[[File:~~Exercise_impuls~~.png \|right\|frame\|Screenshot '''Korrektur''' Pulses]]		[[File:Exercise_impuls_v2.png\|right\|frame\|Screenshot '''Korrektur''' Pulses]]
	'''(A)'''     Theme (changeable graphical user interface design)		'''(A)'''     Theme (changeable graphical user interface design)
	:* Dark:   dark background  (recommended by the authors)		:* Dark:   dark background  (recommended by the authors)

Guenter at 12:36, 14 April 2023

2023-04-14T12:36:41Z

← Older revision		Revision as of 14:36, 14 April 2023
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	==Applet Manual==		==Applet Manual==
	<br>		<br>
	[[File:Exercise_impuls.png \|right\|frame\|Screenshot]]		[[File:Exercise_impuls.png \|right\|frame\|Screenshot '''Korrektur''' Pulses]]
	'''(A)'''     Theme (changeable graphical user interface design)		'''(A)'''     Theme (changeable graphical user interface design)
	:* Dark:   dark background  (recommended by the authors)		:* Dark:   dark background  (recommended by the authors)

Noah at 18:20, 25 March 2023

2023-03-25T18:20:32Z

← Older revision		Revision as of 20:20, 25 March 2023
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	*From this follows:  $X(f)$  contains instead of the  ${\rm si}$-function the  ${\rm si}^2$-function.		*From this follows:  $X(f)$  contains instead of the  ${\rm si}$-function the  ${\rm si}^2$-function.
	*$X(f)$  thus also has zeros at equidistant intervals  $1/\rm f$ .		*$X(f)$  thus also has zeros at equidistant intervals  $1/\rm f$ .
	*The asymptotic decay of  $X(f)$  occurs here with  $1/f^2$, while for comparison the rectangular ~~momentum~~ decays with  $1/f$ .		*The asymptotic decay of  $X(f)$  occurs here with  $1/f^2$, while for comparison the rectangular pulse decays with  $1/f$ .

Noah at 17:44, 25 March 2023

2023-03-25T17:44:44Z

Show changes

Noah at 16:48, 25 March 2023

2023-03-25T16:48:54Z

← Older revision		Revision as of 18:48, 25 March 2023
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	*The relationship between the time function  $x(t)$  and the spectrum  $X(f)$  is given by the  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_first_Fourier_integral\|"first Fourier integral"]] :		*The relationship between the time function  $x(t)$  and the spectrum  $X(f)$  is given by the  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_first_Fourier_integral\|"first Fourier integral"]] :
	:$$X(f)={\rm FT} [x(t)] = \int_{-\infty}^{+\infty}x(t)\cdot {\rm e}^{-{\rm j}2\pi f t}\hspace{0.15cm} {\rm d}t\hspace{1cm}		:$$X(f)={\rm FT} [x(t)] = \int_{-\infty}^{+\infty}x(t)\cdot {\rm e}^{-{\rm j}2\pi f t}\hspace{0.15cm} {\rm d}t\hspace{1cm}
	\rm FT\hspace{-0.1cm}: \ ~~Fouriertransformation~~.$$		\rm FT\hspace{-0.1cm}: \ Fourier \ transform.$$

	*~~Um aus der Spektralfunktion~~  $X(f)$  ~~die Zeitfunktion~~  $x(t)$  ~~berechnen zu können, benötigt man das~~  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_second_Fourier_integral\|"~~zweite Fourierintegral~~"]]:		*In order to calculate the time function  $x(f)$  from the spectral function  $x(t)$  one needs the  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_second_Fourier_integral\|"second Fourier integral"]]:
	:$$x(t)={\rm IFT} [X(f)] = \int_{-\infty}^{+\infty}X(f)\cdot {\rm e}^{+{\rm j}2\pi f t}\hspace{0.15cm} {\rm d}f\hspace{1cm}		:$$x(t)={\rm IFT} [X(f)] = \int_{-\infty}^{+\infty}X(f)\cdot {\rm e}^{+{\rm j}2\pi f t}\hspace{0.15cm} {\rm d}f\hspace{1cm}
	{\rm IFT}\hspace{-0.1cm}: \rm Inverse \ ~~Fouriertransformation~~.$$		{\rm IFT}\hspace{-0.1cm}: \rm Inverse \ Fourier \ transform.$$

	*In ~~allen Beispielen verwenden wir reelle und gerade Funktionen~~.  ~~Somit gilt~~:		*In all examples we use real and even functions.  Thus:
	:$$x(t)=\int_{-\infty}^{+\infty}X(f)\cdot \cos(2\pi ft) \hspace{0.15cm} {\rm d}f \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\ \ \ X(f)=\int_{-\infty}^{+\infty}x(t)\cdot \cos(2\pi ft) \hspace{0.15cm} {\rm d}t .$$		:$$x(t)=\int_{-\infty}^{+\infty}X(f)\cdot \cos(2\pi ft) \hspace{0.15cm} {\rm d}f \ \ \circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\ \ \ X(f)=\int_{-\infty}^{+\infty}x(t)\cdot \cos(2\pi ft) \hspace{0.15cm} {\rm d}t .$$
	*$x(t)$  ~~und~~  $X(f)$  ~~haben unterschiedliche Einheiten~~, ~~beispielsweise~~  $x(t)$  in  $\rm V$,  $X(f)$  in  $\rm V/Hz$.		*$x(t)$  and  $X(f)$  have different units, for example  $x(t)$  in  $\rm V$,  $X(f)$  in  $\rm V/Hz$.
	*~~Der Zusammenhang zwischen diesem Modul und dem ähnlich aufgebauten Applet~~  [[Applets:Frequenzgang_und_Impulsantwort\|"~~Frequenzgang~~ & ~~Impulsantwort~~"]]  ~~basiert auf dem~~  [[Signal_Representation/The_Fourier_Transform_Theorems#Duality_Theorem\|"Duality Theorem"]].		*The relationship between this module and the similarly constructed applet  [[Applets:Frequenzgang_und_Impulsantwort\|"Frequency response & Impulse response"]]  is based on the  [[Signal_Representation/The_Fourier_Transform_Theorems#Duality_Theorem\|"Duality Theorem"]].
	*~~Alle Zeiten sind auf eine Zeit~~  $T$  ~~normiert und alle Frequenzen auf~~  $1/T$   ⇒   ~~die Spektralwerte~~  $X(f)$  ~~müssen noch mit der Normierungszeit~~  $T$  ~~multipliziert werden~~.		*All times are normalized to a time  $T$  and all frequencies are normalized to  $1/T$   ⇒   the spectral values  $X(f)$  still have to be multiplied by the normalization time  $T$ .


	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{~~Beispiel~~:}$   ~~Stellt man einen Rechteckimpuls mit Amplitude~~  $A_1 = 1$  ~~und äquivalenter Impulsdauer~~  $\Delta t_1 = 1$  ~~ein, so ist~~  $x_1(t)$  ~~im Bereich~~  $-0.5 < t < +0.5$  ~~gleich Eins und außerhalb dieses Bereichs gleich Null~~.  ~~Die Spektralfunktion~~  $X_1(f)$  ~~verläuft~~  $\rm si$–~~förmig mit~~  $X_1(f= 0) = 1$  ~~und der ersten Nullstelle bei~~  $f=1$.		$\text{Example:}$   If one sets a rectangular pulse with amplitude  $A_1 = 1$  and equivalent pulse duration  $\Delta t_1 = 1$  then  $x_1(t)$  in the range  $-0.5 < t < +0. 5$  equal to one and outside this range equal to zero.  The spectral function  $X_1(f)$  proceeds  $\rm si$–shaped with  $X_1(f= 0) = 1$  and the first zero at  $f=1$.

	*~~Soll mit dieser Einstellung ein Rechteckimpuls mit~~  $A = K = 3 \ \rm V$  ~~und~~  $\~~Delta~~ t = T = 2 \ \rm ms$  ~~nachgebildet werden~~, ~~dann sind alle Signalwerte mit~~  $K = 3 \ \rm V$  ~~und alle Spektralwerte mit~~  $K \cdot T = 0.006 \ \rm V/Hz$  ~~zu multiplizieren~~.		*If a rectangular pulse with  $A = K = 3 \ \rm V$  and  $\delta t = T = 2 \ \rm ms$  is to be simulated with this setting, then all signal values with  $K = 3 \ \rm V$  and all spectral values with  $K \cdot T = 0. 006 \ \rm V/Hz$  to be multiplied by.
	*~~Der maximale Spektralwert ist dann~~  $X(f= 0) = 0.006 \ \rm V/Hz$  ~~und die erste Nullstelle liegt bei~~  $f=1/T = 0.5 \ \rm kHz$.}}		*The maximum spectral value is then  $X(f= 0) = 0.006 \ \rm V/Hz$  and the first zero is at  $f=1/T = 0.5 \ \rm kHz$.}}

← Older revision		Revision as of 15:49, 16 March 2026
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	===Relationship $x(t)\Leftrightarrow X(f)$===		===Relationship $x(t)\Leftrightarrow X(f)$===
	*The relationship between the time function  $x(t)$  and the spectrum  $X(f)$  is given by the  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_first_Fourier_integral\|"first Fourier integral"]] :		*The relationship between the time function  $x(t)$  and the spectrum  $X(f)$  is given by the  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_first_Fourier_integral\|"first Fourier integral"]] :
	:$$X(f)={\rm FT} [x(t)] = \int_{-\infty}^{+\infty}x(t)\cdot {\rm e}^{-{\rm j}2\pi f t}\hspace{0.15cm} {\rm d}t\hspace{1cm}		:$$X(f)={\rm FT} [x(t)] = \int_{-\infty}^{+\infty}x(t)\cdot {\rm e}^{-{\rm j}2\pi f t}\hspace{0.15cm} {\rm d}t\hspace{1cm}\rm FT\hspace{-0.1cm}: \ Fourier \ transform.$$
	\rm FT\hspace{-0.1cm}: \ Fourier \ transform.$$

	*In order to calculate the time function  $x(f)$  from the spectral function  $x(t)$  one needs the  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_second_Fourier_integral\|"second Fourier integral"]]:		*In order to calculate the time function  $x(f)$  from the spectral function  $x(t)$  one needs the  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_second_Fourier_integral\|"second Fourier integral"]]:
	:$$x(t)={\rm IFT} [X(f)] = \int_{-\infty}^{+\infty}X(f)\cdot {\rm e}^{+{\rm j}2\pi f t}\hspace{0.15cm} {\rm d}f\hspace{1cm}		:$$x(t)={\rm IFT} [X(f)] = \int_{-\infty}^{+\infty}X(f)\cdot {\rm e}^{+{\rm j}2\pi f t}\hspace{0.15cm} {\rm d}f\hspace{1cm}{\rm IFT}\hspace{-0.1cm}: \rm Inverse \ Fourier \ transform.$$
	{\rm IFT}\hspace{-0.1cm}: \rm Inverse \ Fourier \ transform.$$

	*In all examples we use real and even functions.  Thus:		*In all examples we use real and even functions.  Thus:
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	<br>		<br>
	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 2005 by  [[~~Biografien_und_Bibliografien~~/~~An_LNTwww_beteiligte_Studierende~~#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29\|Ji Li]]  as part of her 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 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 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 2017 the program was redesigned by  [[~~Biografien_und_Bibliografien~~/~~An_LNTwww_beteiligte_Studierende~~#David_Jobst_.28Ingenieurspraxis_Math_2017.29\|David Jobst]]  (Ingenieurspraxis_Math, 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 2017 the program was redesigned by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#David_Jobst_.28Ingenieurspraxis_Math_2017.29\|David Jobst]]  (Ingenieurspraxis_Math, 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 2020 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 2020 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.