Applets:Graphical Convolution - Revision history

Guenter at 15:30, 10 April 2023

2023-04-10T15:30:35Z

← Older revision		Revision as of 17:30, 10 April 2023
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	[[File:~~P_ID579__Sig_T_3_4_S1_neu~~.png\|right\|frame\|For the calculation of signal and spectrum at the LTI output]]		[[File:EN_Sig_T_3_4_S1.png\|right\|frame\|For the calculation of signal and spectrum at the LTI output]]
	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{Example 1:}$  Any linear time-invariant (LTI) system can be described both by the frequency response  $H(f)$  and by the impulse response  $h(t)$,  the relationship between these two system quantities also being given by the Fourier transform.		$\text{Example 1:}$  Any linear time-invariant (LTI) system can be described both by the frequency response  $H(f)$  and by the impulse response  $h(t)$,  the relationship between these two system quantities also being given by the Fourier transform.
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	This result can be proved similarly to the  [[Applets:Graphical_Convolution#Convolution_in_the_time_domain\|"convolution theorem in the time domain"]].  However, the integration variable  $\nu$  now has the dimension of a frequency.		This result can be proved similarly to the  [[Applets:Graphical_Convolution#Convolution_in_the_time_domain\|"convolution theorem in the time domain"]].  However, the integration variable  $\nu$  now has the dimension of a frequency.

	[[File:~~P_ID580__Sig_T_3_4_S2_neu~~.png\|right\|frame\|Convolution in the frequency domain]]		[[File:EN_Sig_T_3_4_S2.png\|right\|frame\|Convolution in the frequency domain]]
	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{Example 2:}$  The  [[Modulation_Methods/Double-Sideband_Amplitude_Modulation#Description_in_the_time_domain\|"double-sideband amplitude modulation"]]  (DSB-AM) without carrier is described by the sketched model.		$\text{Example 2:}$  The  [[Modulation_Methods/Double-Sideband_Amplitude_Modulation#Description_in_the_time_domain\|"double-sideband amplitude modulation"]]  (DSB-AM) without carrier is described by the sketched model.
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	:$$Y( f ) = X_1 ( f ) \cdot X_2 ( f ) = X_1 ( f ) \cdot \alpha \cdot {\rm{e}}^{ - {\rm{j}}\hspace{0.03cm}2\hspace{0.03cm}{\rm{\pi }}\hspace{0.01cm}f\hspace{0.01cm}T} .$$		:$$Y( f ) = X_1 ( f ) \cdot X_2 ( f ) = X_1 ( f ) \cdot \alpha \cdot {\rm{e}}^{ - {\rm{j}}\hspace{0.03cm}2\hspace{0.03cm}{\rm{\pi }}\hspace{0.01cm}f\hspace{0.01cm}T} .$$

	The complex exponential function leads to a shift by  $T$   ⇒   [[Signal_Representation/~~Fourier_Transform_Theorems~~#Shifting_Theorem\|"shifting theorem"]], the factor  $\alpha$  to an attenuation  $(\alpha < 1)$  or an amplification  $(\alpha > 1)$. It follows that:		The complex exponential function leads to a shift by  $T$   ⇒   [[Signal_Representation/The_Fourier_Transform_Theorems#Shifting_Theorem\|"shifting theorem"]], the factor  $\alpha$  to an attenuation  $(\alpha < 1)$  or an amplification  $(\alpha > 1)$. It follows that:

	:$$x_1 (t) * x_2 (t) = \alpha \cdot x_1 ( {t - T} ).$$		:$$x_1 (t) * x_2 (t) = \alpha \cdot x_1 ( {t - T} ).$$
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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 2006 by [[Biographies_and_Bibliographies/~~An_LNTwww_beteiligte_Studierende~~#Markus_Elsberger_.28Diplomarbeit_LB_2006.29\|Markus Elsberger]]  as part of his bachelor 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 2006 by [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Markus_Elsberger_.28Diplomarbeit_LB_2006.29\|Markus Elsberger]]  as part of his bachelor 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]]).

	*Last revision and English version 2020/2021 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 2020/2021 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.

Noah at 21:41, 26 March 2023

2023-03-26T21:41:04Z

Hwang at 18:06, 21 March 2023

2023-03-21T18:06:08Z

← Older revision		Revision as of 20:06, 21 March 2023
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	*The first version was created in 2006 by [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Markus_Elsberger_.28Diplomarbeit_LB_2006.29\|Markus Elsberger]]  as part of his bachelor 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 2006 by [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Markus_Elsberger_.28Diplomarbeit_LB_2006.29\|Markus Elsberger]]  as part of his bachelor 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]]).

	*Last revision and English version 2020/2021 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.  ~~Translation using DEEPL.com.~~		*Last revision and English version 2020/2021 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.

Guenter at 17:02, 10 January 2023

2023-01-10T17:02:02Z

Hwang at 08:03, 21 October 2022

2022-10-21T08:03:36Z

← Older revision		Revision as of 10:03, 21 October 2022
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	#  '''Mirror''' second '''function''':   $h(\tau) \to h(-\tau)$.		#  '''Mirror''' second '''function''':   $h(\tau) \to h(-\tau)$.
	#  '''Shift''' mirrored '''function''' by  $t$:   $h(-\tau) \to h(t-\tau)$.		#  '''Shift''' mirrored '''function''' by  $t$:   $h(-\tau) \to h(t-\tau)$.
	#  '''~~Multiplikation~~''' ~~der beiden Funktionen~~  $x(\tau)$  ~~und~~  $h(t-\tau)$.		#  '''Multiplication''' of the two functions  $x(\tau)$  and  $h(t-\tau)$.
	#  '''Integration'''  ~~über das Produkt bezüglich~~  $\tau$  in ~~den Grenzen von~~  $-\infty$  ~~bis~~  $+\infty$.		#  '''Integration'''  over the product with respect to  $\tau$  in the limits from  $-\infty$  to  $+\infty$.


	~~Da die Faltung kommutativ ist~~, ~~kann anstelle von~~  $h(\tau)$  ~~auch~~  $x(\tau)$  ~~gespiegelt werden.~~		Since the convolution is commutative,  $h(\tau)$  can be mirrored instead of  $x(\tau)$.

	<br><br>		<br><br>
	~~Nebenstehende Grafik zeigt einen Bildschirmabzug einer älteren Version des vorliegenden Applets~~.		The accompanying graphic shows a screen shot of an older version of this applet.
	<br><br>		<br><br>


	[[File:P_ID582__Sig_T_3_4_S4_neu.png\|right\|frame\|~~Beispiel einer Faltungsoperation~~: <br>~~Sprungfunktion gefaltet mit Exponentialfunktion~~]]		[[File:P_ID582__Sig_T_3_4_S4_neu.png\|right\|frame\|Example of a convolution operation: <br>Jump function convolved with exponential function]]
	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{~~Beispiel~~ 4:}$		$\text{Example 4:}$
	~~Die Vorgehensweise bei der grafischen Faltung wird nun anhand eines ausführlichen Beispiels erklärt~~:		The procedure of graphical convolution is now explained with a detailed example:
	*~~Am Eingang eines Filters liege eine Sprungfunktion~~  $x(t) = \gamma(t)$  an.		*Let a jump function  $x(t) = \gamma(t)$  be applied to the input of a filter.
	*~~Die Impulsantwort des~~ RC-~~Tiefpasses sei~~  $h( t ) = {1}/{T} \cdot {\rm{e} }^{ - t/T}.$		*Let the impulse response of the RC low-pass filter be  $h( t ) = {1}/{T} \cdot {\rm{e} }^{ - t/T}.$


	~~Die Grafik zeigt rot das Eingangssignal~~  $x(\tau)$, ~~blau die Impulsantwort~~  $h(\tau)$ ~~und grau das Ausgangssignal~~  $y(\tau)$.		The graph shows the input signal  $x(\tau)$ in red, the impulse response  $h(\tau)$ in blue, and the output signal  $y(\tau)$ in gray.
	~~Die Zeitachse ist bereits in~~  $\tau$  ~~umbenannt.~~		The time axis has already been renamed  $\tau$.

	~~Das Ausgangssignal kann zum Beispiel nach folgender Gleichung berechnet werden~~:		The output signal can be calculated, for example, according to the following equation:

	:$$y(t) = h(t) * x(t) = \int_{ - \infty }^{ + \infty } {h( \tau )} \cdot x( {t - \tau } )\hspace{0.1cm}{\rm d}\tau.$$		:$$y(t) = h(t) * x(t) = \int_{ - \infty }^{ + \infty } {h( \tau )} \cdot x( {t - \tau } )\hspace{0.1cm}{\rm d}\tau.$$

	~~Noch einige Anmerkungen zur grafischen Faltung~~:		A few more remarks on the graphical convolution:
	*~~Der Ausgangswert bei~~  $t = 0$  ~~ergibt sich, indem man das Eingangssignal~~  $x(\tau)$  ~~spiegelt, dieses gespiegelte Signal~~  $x(-\tau)$  ~~mit der Impulsantwort~~  $h(\tau)$  ~~multipliziert und darüber integriert~~.		*The output value at  $t = 0$  is obtained by mirroring the input signal  $x(\tau)$,  multiplying this mirrored signal  $x(-\tau)$  by the impulse response  $h(\tau)$,  and integrating over it.
	*~~Da es hier kein Zeitintervall gibt, bei dem sowohl die blaue Kurve~~  $h(\tau)$  ~~und gleichzeitig auch die rot gestrichelte Spiegelung~~  $x(-\tau)$  ~~ungleich Null ist~~, ~~folgt daraus~~  $y(t=0)=0$.		*Since there is no time interval here where both the blue curve  $h(\tau)$  and at the same time the red dashed mirroring  $x(-\tau)$  is not equal to zero, it follows that  $y(t=0)=0$.
	*~~Für jeden anderen Zeitpunkt~~  $t$  ~~muss das Eingangssignal verschoben werden~~   ⇒   $x(t-\tau)$, ~~beispielsweise entsprechend der grün gestrichelten Kurve für~~  $t=T$.		*For any other time  $t$,  the input signal must be shifted   ⇒   $x(t-\tau)$, for example corresponding to the green dashed curve for  $t=T$.
	*Da in ~~diesem Beispiel auch~~  $x(t-\tau)$  ~~nur die Werte~~  $0$  ~~oder~~  $1$  ~~annehmen kann, wird die Integration~~  $($~~allgemein von~~  $\tau_1$  ~~bis~~  $\tau_2)$  ~~einfach und man erhält mit~~  $\tau_1 = 0$  ~~und~~  $\tau_2 = t$ :		*Since in this example also  $x(t-\tau)$  can only take the values  $0$  or  $1$,  the integration  $($generally from  $\tau_1$  to  $\tau_2)$  becomes simple and we obtain with  $\tau_1 = 0$  and  $\tau_2 = t$ :
	:$$y( t) = \int_0^{\hspace{0.05cm} t} {h( \tau)}\hspace{0.1cm} {\rm d}\tau = \frac{1}{T}\cdot\int_0^{\hspace{0.05cm} t} {{\rm{e}}^{ - \tau /T } }\hspace{0.1cm} {\rm d}\tau = 1 - {{\rm{e}}^{ - t /T } }.$$		:$$y( t) = \int_0^{\hspace{0.05cm} t} {h( \tau)}\hspace{0.1cm} {\rm d}\tau = \frac{1}{T}\cdot\int_0^{\hspace{0.05cm} t} {{\rm{e}}^{ - \tau /T } }\hspace{0.1cm} {\rm d}\tau = 1 - {{\rm{e}}^{ - t /T } }.$$

	~~Die Skizze gilt für~~  $t=T$  ~~und führt zum Ausgangswert~~  $y(t=T) = 1 – 1/\text{e} \approx 0.632$.}}		The sketch is valid for  $t=T$  and leads to the initial value  $y(t=T) = 1 – 1/\text{e} \approx 0.632$.}}

Hwang at 07:37, 21 October 2022

2022-10-21T07:37:46Z

← Older revision		Revision as of 09:37, 21 October 2022
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	:$$Y( f ) = X_1 ( f ) \cdot X_2 ( f ) = X_1 ( f ) \cdot \alpha \cdot {\rm{e}}^{ - {\rm{j}}\hspace{0.03cm}2\hspace{0.03cm}{\rm{\pi }}\hspace{0.01cm}f\hspace{0.01cm}T} .$$		:$$Y( f ) = X_1 ( f ) \cdot X_2 ( f ) = X_1 ( f ) \cdot \alpha \cdot {\rm{e}}^{ - {\rm{j}}\hspace{0.03cm}2\hspace{0.03cm}{\rm{\pi }}\hspace{0.01cm}f\hspace{0.01cm}T} .$$

	The complex exponential function leads to a shift by  $T$   ⇒   [[Signal_Representation/~~Fourier_Transform_Laws~~#~~Verschiebungssatz~~\|~~Verschiebungssatz~~]], ~~der Faktor~~  $\alpha$  ~~zu einer Dämpfung~~  $(\alpha < 1)$  ~~bzw. einer Verstärkung~~  $(\alpha > 1)$. ~~Daraus folgt~~:		The complex exponential function leads to a shift by  $T$   ⇒   [[Signal_Representation/Fourier_Transform_Theorems#Shifting_Theorem\|"shifting theorem"]], the factor  $\alpha$  to an attenuation  $(\alpha < 1)$  or an amplification  $(\alpha > 1)$. It follows that:

	:$$x_1 (t) * x_2 (t) = \alpha \cdot x_1 ( {t - T} ).$$		:$$x_1 (t) * x_2 (t) = \alpha \cdot x_1 ( {t - T} ).$$

	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{In ~~Worten~~: }$  ~~Die Faltung einer beliebigen Funktion mit einer Diracfunktion bei~~  $t = T$  ~~ergibt die um~~  $T$  ~~nach rechts verschobene Funktion, wobei noch die Gewichtung der Diracfunktion durch den Faktor~~  $\alpha$  ~~zu berücksichtigen ist~~.}}		$\text{In words: }$  The convolution of an arbitrary function with a Dirac function at  $t = T$  results in the function shifted to the right by  $T$,  whereby the weighting of the Dirac function by the factor  $\alpha$  must still be taken into account.}}


	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{~~Beispiel~~ 3:}$  ~~Ein Rechtecksignal~~  $x(t)$  ~~wird durch ein LZI-System um eine Laufzeit~~  $\tau = 3\,\text{ ms}$  ~~verzögert und um den Faktor~~  $\alpha = 0.5$  ~~gedämpft~~.		$\text{Example 3:}$  A rectangular signal  $x(t)$  is delayed by a running time  $\tau = 3\,\text{ ms}$  and attenuated by a factor  $\alpha = 0.5$  by an LTI system.

	[[File:P_ID522__Sig_T_3_4_S3_neu.png\|center\|frame\|~~Faltung eines Rechtecks mit einer Diracfunktion~~]]		[[File:P_ID522__Sig_T_3_4_S3_neu.png\|center\|frame\|Convolution of a rectangle with a Dirac function]]

	~~Verschiebung und Dämpfung erkennt man sowohl am Ausgangssignal~~  $y(t)$  ~~als auch an der Impulsantwort~~  $h(t)$.}}		Shift and attenuation can be seen in the output signal  $y(t)$  as well as in the impulse response  $h(t)$.}}


	===Graphical Convolution===		===Graphical Convolution===

	~~In diesem Applet wird von folgender Faltungsoperation ausgegangen~~:		This applet assumes the following convolution operation:
	[[File:P_ID2723__Sig_T_3_4_programm.png\|right\|frame\|~~Bildschirmabzug des Programms „Grafische Faltung”~~ (~~frühere Version~~)]]		[[File:P_ID2723__Sig_T_3_4_programm.png\|right\|frame\|Screen capture of the program "Graphical Convolution" (former version)]]
	:$$y(t) = x (t) * h (t) = \int_{ - \infty }^{ + \infty } {x ( \tau )} \cdot h ( {t - \tau } )\hspace{0.1cm}{\rm d}\tau.$$		:$$y(t) = x (t) * h (t) = \int_{ - \infty }^{ + \infty } {x ( \tau )} \cdot h ( {t - \tau } )\hspace{0.1cm}{\rm d}\tau.$$

	~~Die Lösung des Faltungsintegrals soll auf grafischem Wege erfolgen~~. ~~Es wird vorausgesetzt, dass~~  $x(t)$  ~~und~~  $h(t)$  ~~zeitkontinuierliche Signale sind~~.		The solution of the convolution integral is to be done graphically. It is assumed that  $x(t)$  and  $h(t)$  are continuous-time signals.


	~~Dann sind die folgenden Schritte erforderlich~~:		Then the following steps are required:
	#  ~~Die~~  '''~~Zeitvariablen~~'''~~  der beiden Funktionen~~  '''~~ändern~~''':   <br>    $x(t) \to x(\tau)$,   $h(t) \to h(\tau)$.		#    '''Change''' the  '''time variables'''  of the two functions:   <br>    $x(t) \to x(\tau)$,   $h(t) \to h(\tau)$.
	#  ~~Zweite~~ '''~~Funktion spiegeln~~''':   $h(\tau) \to h(-\tau)$.		#  '''Mirror''' second '''function''':   $h(\tau) \to h(-\tau)$.
	#  ~~Gespiegelte~~ '''~~Funktion~~''' um  $t$~~  '''verschieben'''~~:   $h(-\tau) \to h(t-\tau)$.		#  '''Shift''' mirrored '''function''' by  $t$:   $h(-\tau) \to h(t-\tau)$.
	#  '''Multiplikation''' der beiden Funktionen  $x(\tau)$  und  $h(t-\tau)$.		#  '''Multiplikation''' der beiden Funktionen  $x(\tau)$  und  $h(t-\tau)$.
	#  '''Integration'''  über das Produkt bezüglich  $\tau$  in den Grenzen von  $-\infty$  bis  $+\infty$.		#  '''Integration'''  über das Produkt bezüglich  $\tau$  in den Grenzen von  $-\infty$  bis  $+\infty$.

Hwang at 10:51, 19 October 2022

2022-10-19T10:51:07Z

← Older revision		Revision as of 12:51, 19 October 2022
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	:$$x_1 ( t ) \cdot x_2 ( t )\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,X_1 (f) * X_2 (f) = \int_{ - \infty }^{ + \infty } {X_1 ( \nu )} \cdot X_2 ( {f - \nu })\hspace{0.1cm}{\rm d}\nu.$$		:$$x_1 ( t ) \cdot x_2 ( t )\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,X_1 (f) * X_2 (f) = \int_{ - \infty }^{ + \infty } {X_1 ( \nu )} \cdot X_2 ( {f - \nu })\hspace{0.1cm}{\rm d}\nu.$$

	This result can be proved similarly to the  [[Applets:~~Zur_Verdeutlichung_der_grafischen_Faltung~~#~~Faltung_im_Zeitbereich~~\|"convolution theorem in the time domain"]].  However, the integration variable  $\nu$  now has the dimension of a frequency.		This result can be proved similarly to the  [[Applets:Graphical_Convolution#Convolution_in_the_time_domain\|"convolution theorem in the time domain"]].  However, the integration variable  $\nu$  now has the dimension of a frequency.

	[[File:P_ID580__Sig_T_3_4_S2_neu.png\|right\|frame\|~~Faltung im Frequenzbereich~~]]		[[File:P_ID580__Sig_T_3_4_S2_neu.png\|right\|frame\|Convolution in the frequency domain]]
	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{~~Beispiel~~ 2:}$  ~~Die~~  [[Modulation_Methods/~~Zweiseitenband~~-~~Amplitudenmodulation~~#~~Beschreibung_im_Zeitbereich~~\|~~Zweiseitenband~~-~~Amplitudenmodulation~~]]  (~~ZSB~~-AM) ~~ohne Träger wird durch das skizzierte Modell beschrieben~~.		$\text{Example 2:}$  The  [[Modulation_Methods/Double-Sideband_Amplitude_Modulation#Description_in_the_time_domain\|"double-sideband amplitude modulation"]]  (DSB-AM) without carrier is described by the sketched model.
	*~~Bei der Zeitbereichsdarstellung~~ (~~blau~~) ~~ergibt sich das modulierte Signal~~  $s(t)$  ~~als das Produkt aus dem Nachrichtensignal~~  $q(t)$  ~~und dem~~ (~~normierten~~) ~~Trägersignal~~  $z(t)$.		*In the time domain representation (blue), the modulated signal  $s(t)$  is the product of the message signal  $q(t)$  and the (normalized) carrier signal  $z(t)$.
	*~~Nach dem Faltungssatz folgt daraus für den Frequenzbereich~~ (~~rot~~)~~, dass das Ausgangsspektrum~~  $S(f)$  ~~gleich dem Faltungsprodukt aus~~  $Q(f)$  ~~und~~  $Z(f)$  ~~ist.~~}}		*According to the convolution theorem, it follows for the frequency domain (red) that the output spectrum  $S(f)$  is equal to the convolution product of  $Q(f)$  and  $Z(f)$.  }}


	===~~Faltung einer Funktion mit einer Diracfunktion~~===		===Convolution of a function with a Dirac function===

	~~Sehr einfach wird die Faltungsoperation, wenn einer der beiden Operanden eine~~  [[Signal_Representation/~~Gleichsignal_~~-~~_Grenzfall_eines_periodischen_Signals~~#~~Diracfunktion_im_Frequenzbereich~~\|~~Diracfunktion~~]]  ~~ist. Dies gilt für die Faltung im Zeit– und im Frequenzbereich gleichermaßen~~.		The convolution operation becomes very simple if one of the two operands is a  [[Signal_Representation/Direct_Current_Signal_-_Limit_Case_of_a_Periodic_Signal#Dirac_.28delta.29_function_in_frequency_domain\|"Dirac function"]].  This is equally true for convolution in the time and frequency domain.

	~~Wir betrachten beispielhaft die Faltung einer Funktion~~  $x_1(t)$  ~~mit der Funktion~~		As an example, we consider the convolution of a function  $x_1(t)$  with the function

	:$$x_2 ( t ) = \alpha \cdot \delta ( {t - T} ) \quad \circ\,\!\!\!-\!\!\!-\!\!\!-\!\!\bullet \quad X_2 ( f )= \alpha \cdot {\rm{e}}^{ - {\rm{j}}\hspace{0.03cm}2\hspace{0.03cm}{\rm{\pi }}\hspace{0.01cm}f\hspace{0.01cm}T}.$$		:$$x_2 ( t ) = \alpha \cdot \delta ( {t - T} ) \quad \circ\,\!\!\!-\!\!\!-\!\!\!-\!\!\bullet \quad X_2 ( f )= \alpha \cdot {\rm{e}}^{ - {\rm{j}}\hspace{0.03cm}2\hspace{0.03cm}{\rm{\pi }}\hspace{0.01cm}f\hspace{0.01cm}T}.$$

	~~Für die Spektralfunktion des Signals~~  $y(t) = x_1(t) \ast x_2(t)$  ~~gilt dann~~:		Then for the spectral function of the signal  $y(t) = x_1(t) \ast x_2(t)$  holds:

	:$$Y( f ) = X_1 ( f ) \cdot X_2 ( f ) = X_1 ( f ) \cdot \alpha \cdot {\rm{e}}^{ - {\rm{j}}\hspace{0.03cm}2\hspace{0.03cm}{\rm{\pi }}\hspace{0.01cm}f\hspace{0.01cm}T} .$$		:$$Y( f ) = X_1 ( f ) \cdot X_2 ( f ) = X_1 ( f ) \cdot \alpha \cdot {\rm{e}}^{ - {\rm{j}}\hspace{0.03cm}2\hspace{0.03cm}{\rm{\pi }}\hspace{0.01cm}f\hspace{0.01cm}T} .$$

	~~Die komplexe Exponentialfunktion führt zur Verschiebung um~~  $T$   ⇒   [[Signal_Representation/Fourier_Transform_Laws#Verschiebungssatz\|Verschiebungssatz]], der Faktor  $\alpha$  zu einer Dämpfung  $(\alpha < 1)$  bzw. einer Verstärkung  $(\alpha > 1)$. Daraus folgt:		The complex exponential function leads to a shift by  $T$   ⇒   [[Signal_Representation/Fourier_Transform_Laws#Verschiebungssatz\|Verschiebungssatz]], der Faktor  $\alpha$  zu einer Dämpfung  $(\alpha < 1)$  bzw. einer Verstärkung  $(\alpha > 1)$. Daraus folgt:

	:$$x_1 (t) * x_2 (t) = \alpha \cdot x_1 ( {t - T} ).$$		:$$x_1 (t) * x_2 (t) = \alpha \cdot x_1 ( {t - T} ).$$

Hwang at 10:29, 19 October 2022

2022-10-19T10:29:09Z

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	''~~Anmerkung~~'':   ~~Die Faltung ist~~  '''~~kommutativ~~'''   ⇒   ~~Die Reihenfolge der Operanden ist vertauschbar~~:   ${ {x}}_{\rm{1}} ( t ) * { {x}}_{\rm{2}} (t ) ={ {x}}_{\rm{2}} ( t ) * { {x}}_{\rm{1}} (t ) $.		''Note'':   The convolution is  '''commutative'''   ⇒   The order of the operands is interchangeable:   ${ {x}}_{\rm{1}} ( t ) * { {x}}_{\rm{2}} (t ) ={ {x}}_{\rm{2}} ( t ) * { {x}}_{\rm{1}} (t ) $.


	[[File:P_ID579__Sig_T_3_4_S1_neu.png\|right\|frame\|~~Zur Berechnung von Signal und Spektrum am LZI–Ausgang~~]]		[[File:P_ID579__Sig_T_3_4_S1_neu.png\|right\|frame\|For the calculation of signal and spectrum at the LTI output]]
	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{~~Beispiel~~ 1:}$  ~~Ein jedes lineare zeitinvariante~~ (~~LZI-~~) ~~System kann sowohl durch den Frequenzgang~~  $H(f)$  ~~als auch durch die Impulsantwort~~  $h(t)$  ~~beschrieben werden, wobei der Zusammenhang zwischen diesen beiden Systemgrößen ebenfalls durch die Fouriertransformation gegeben ist~~.		$\text{Example 1:}$  Any linear time-invariant (LTI) system can be described both by the frequency response  $H(f)$  and by the impulse response  $h(t)$,  the relationship between these two system quantities also being given by the Fourier transform.

	~~Legt man an den Eingang ein Signal~~  $x(t)$  ~~mit dem Spektrum~~  $X(f)$  an, ~~so gilt für das Spektrum des Ausgangssignals~~:		If a signal  $x(t)$  with the spectrum  $X(f)$  is applied to the input, the following applies to the spectrum of the output signal:

	:$$Y(f) = X(f) \cdot H(f)\hspace{0.05cm}.$$		:$$Y(f) = X(f) \cdot H(f)\hspace{0.05cm}.$$

	~~Mit dem Faltungssatz ist es nun möglich, das Ausgangssignal auch direkt im Zeitbereich zu berechnen~~:		With the convolution theorem it is now possible to calculate the output signal also directly in the time domain:

	:$$y( t ) = x(t) * h( t ) = \int_{ - \infty }^{ + \infty } \hspace{-0.15cm}{x( \tau )} \cdot h( {t - \tau } )\hspace{0.1cm}{\rm d}\tau = \int_{ - \infty }^{ + \infty } \hspace{-0.15cm} {h( \tau )} \cdot x( {t - \tau } )\hspace{0.1cm}{\rm d}\tau = h(t) * x( t ).$$		:$$y( t ) = x(t) * h( t ) = \int_{ - \infty }^{ + \infty } \hspace{-0.15cm}{x( \tau )} \cdot h( {t - \tau } )\hspace{0.1cm}{\rm d}\tau = \int_{ - \infty }^{ + \infty } \hspace{-0.15cm} {h( \tau )} \cdot x( {t - \tau } )\hspace{0.1cm}{\rm d}\tau = h(t) * x( t ).$$

	~~Aus dieser Gleichung geht nochmals hervor~~, ~~dass die Faltungsoperation~~  ''~~kommutativ~~''  ~~ist.~~}}		From this equation, it is again clear that the convolution operation is  ''commutative''.  }}


	===~~Faltung im Frequenzbereich~~===		===Convolution in the frequency domain===

	~~Die Dualität zwischen Zeit– und Frequenzbereich erlaubt auch Aussagen hinsichtlich des Spektrums eines Produktsignals~~:		The duality between the time and frequency domains also allows us to make statements regarding the spectrum of a product signal:

	:$$x_1 ( t ) \cdot x_2 ( t )\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,X_1 (f) * X_2 (f) = \int_{ - \infty }^{ + \infty } {X_1 ( \nu )} \cdot X_2 ( {f - \nu })\hspace{0.1cm}{\rm d}\nu.$$		:$$x_1 ( t ) \cdot x_2 ( t )\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\,X_1 (f) * X_2 (f) = \int_{ - \infty }^{ + \infty } {X_1 ( \nu )} \cdot X_2 ( {f - \nu })\hspace{0.1cm}{\rm d}\nu.$$

	~~Dieses Resultat lässt sich ähnlich wie der~~  [[Applets:Zur_Verdeutlichung_der_grafischen_Faltung#Faltung_im_Zeitbereich\|~~Faltungssatz im Zeitbereich~~]]  ~~beweisen. Die Integrationsvariable~~  $\nu$  ~~hat aber nun die Dimension einer Frequenz~~.		This result can be proved similarly to the  [[Applets:Zur_Verdeutlichung_der_grafischen_Faltung#Faltung_im_Zeitbereich\|"convolution theorem in the time domain"]].  However, the integration variable  $\nu$  now has the dimension of a frequency.

	[[File:P_ID580__Sig_T_3_4_S2_neu.png\|right\|frame\|Faltung im Frequenzbereich]]		[[File:P_ID580__Sig_T_3_4_S2_neu.png\|right\|frame\|Faltung im Frequenzbereich]]

Hwang at 10:06, 19 October 2022

2022-10-19T10:06:02Z

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	For the output signal  $y(t)$  corresponding to the block diagram in  $\text{Example 1}$,  then, as stated in the chapter  [[Applets:~~Zur_Verdeutlichung_der_grafischen_Faltung~~#~~Grafische_Faltung~~\|~~Grafische Faltung~~]]  ~~dargelegt:~~		For the output signal  $y(t)$  corresponding to the block diagram in  $\text{Example 1}$,  then, as stated in the chapter  [[Applets:Graphical_Convolution#Graphical_Convolution\|"Graphical Convolution"]]:
	:$$y( t ) = x(t) * h( t ) = \int_{ - \infty }^{ + \infty } \hspace{-0.15cm}{x( \tau )} \cdot h( {t - \tau } )\hspace{0.1cm}{\rm d}\tau .$$		:$$y( t ) = x(t) * h( t ) = \int_{ - \infty }^{ + \infty } \hspace{-0.15cm}{x( \tau )} \cdot h( {t - \tau } )\hspace{0.1cm}{\rm d}\tau .$$

	~~Bei kausalen Systemen~~   ⇒    $h(t) \equiv 0$  ~~für~~  $t < 0$  (~~Beispiele~~: ~~Spalt~~–~~Tiefpass sowie Tiefpass erster und zweiter Ordnung~~)   ~~kann hierfür auch geschrieben werden~~:		For causal systems   ⇒    $h(t) \equiv 0$  for  $t < 0$  (examples: slit low–pass as well as first and second order low–pass)   can be written for this also:

	:$$y( t ) = \int_{ - \infty }^{ t } \hspace{-0.15cm}{x( \tau )} \cdot h( {t - \tau } )\hspace{0.1cm}{\rm d}\tau .$$		:$$y( t ) = \int_{ - \infty }^{ t } \hspace{-0.15cm}{x( \tau )} \cdot h( {t - \tau } )\hspace{0.1cm}{\rm d}\tau .$$

	~~Bitte beachten Sie~~:		Please note:
	*~~Alle Größen~~ – ~~auch die Zeit~~ $t$ – ~~sind normiert~~ (~~dimensionslos~~) ~~zu verstehen~~.		*All quantities – also the time $t$ – are to be understood normalized (dimensionless).
	*~~Die Zeitfunktionen~~  $x(t)$,  $h(t)$  ~~und~~  $y(t)$  ~~können im Programm keine negativen Signalwerte annehmen~~.		*The time functions  $x(t)$,  $h(t)$  and  $y(t)$  cannot assume negative signal values in the program.
	*~~Die~~ ''absolute ~~Dauer~~''  ~~eines Impulses~~  $y(t)$  ~~ist der~~ (~~zusammenhängende~~) ~~Zeitbereich, für den~~  $y(t) > 0$  ~~gilt.~~		*The ''absolute duration''  of a pulse  $y(t)$  is the (continuous) time range for which  $y(t) > 0$.
	*~~Die~~ ''~~äquivalente Dauer~~''  ~~eines Impulses ist über das flächengleiche Rechteck berechenbar~~.		*The ''equivalent duration''  of a pulse can be calculated by the rectangle of equal area.

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

	===~~Faltung im Zeitbereich~~===		===Convolution in the time domain===

	~~Der~~  [[Signal_Representation/The_Convolution_Theorem_and_Operation\|~~Faltungssatz~~]]  ~~ist mit das wichtigste Gesetz der Fouriertransformation~~. ~~Wir betrachten zunächst den Faltungssatz im Zeitbereich und setzen voraus, dass die Spektren zweier Zeitfunktionen~~  $x_1(t)$  ~~und~~  $x_2(t)$  ~~bekannt sind~~:		The  [[Signal_Representation/The_Convolution_Theorem_and_Operation\|"convolution theorem"]]  is one of the most important laws of the Fourier transform. We first consider the convolution theorem in the time domain and assume that the spectra of two time functions  $x_1(t)$  and  $x_2(t)$  are known:

	:$$X_1 ( f )\hspace{0.15cm}\bullet\!\!\!-\!\!\!-\!\!\!-\!\!\circ\hspace{0.15cm}x_1( t ),\quad X_2 ( f )\hspace{0.1cm}\bullet\!\!\!-\!\!\!-\!\!\!-\!\!\circ\hspace{0.1cm}x_2 ( t ).$$		:$$X_1 ( f )\hspace{0.15cm}\bullet\!\!\!-\!\!\!-\!\!\!-\!\!\circ\hspace{0.15cm}x_1( t ),\quad X_2 ( f )\hspace{0.1cm}\bullet\!\!\!-\!\!\!-\!\!\!-\!\!\circ\hspace{0.1cm}x_2 ( t ).$$

	~~Dann gilt für die Zeitfunktion des Produktes~~  $X_1(f) \cdot X_2(f)$:		Then for the time function of the product  $X_1(f) \cdot X_2(f)$ holds:

	:$$X_1 ( f ) \cdot X_2 ( f )\hspace{0.15cm}\bullet\!\!\!-\!\!\!-\!\!\!-\!\!\circ\hspace{0.15cm}\int_{ - \infty }^{ + \infty } {x_1 ( \tau )} \cdot x_2 ( {t - \tau } )\hspace{0.1cm}{\rm d}\tau.$$		:$$X_1 ( f ) \cdot X_2 ( f )\hspace{0.15cm}\bullet\!\!\!-\!\!\!-\!\!\!-\!\!\circ\hspace{0.15cm}\int_{ - \infty }^{ + \infty } {x_1 ( \tau )} \cdot x_2 ( {t - \tau } )\hspace{0.1cm}{\rm d}\tau.$$

	~~Hierbei ist~~  $\tau$  ~~eine formale Integrationsvariable mit der Dimension einer Zeit~~.		Here  $\tau$  is a formal integration variable with the dimension of a time.

	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{Definition:}$  ~~Die obige Verknüpfung der Zeitfunktion~~  $x_1(t)$  ~~und~~  $x_2(t)$  ~~bezeichnet man als~~  '''~~Faltung~~'''  ~~und stellt diesen Funktionalzusammenhang mit einem Stern dar~~:		$\text{Definition:}$  The above connection of the time function  $x_1(t)$  and  $x_2(t)$  is called  '''convolution'''  and represents this functional connection with a star:

	:$$x_{\rm{1} } (t) * x_{\rm{2} } (t) = \int_{ - \infty }^{ + \infty } {x_1 ( \tau ) } \cdot x_2 ( {t - \tau } ) \hspace{0.1cm}{\rm d}\tau.$$		:$$x_{\rm{1} } (t) * x_{\rm{2} } (t) = \int_{ - \infty }^{ + \infty } {x_1 ( \tau ) } \cdot x_2 ( {t - \tau } ) \hspace{0.1cm}{\rm d}\tau.$$

	~~Damit lässt sich obige Fourierkorrespondenz auch wie folgt schreiben~~:		Thus, above Fourier correspondence can also be written as follows:

	:$$X_1 ( f ) \cdot X_2 ( f )\hspace{0.15cm}\bullet\!\!\!-\!\!\!-\!\!\!-\!\!\circ\hspace{0.15cm}{ {x} }_{\rm{1} } ( t ) * { {x} }_{\rm{2} } (t ).$$		:$$X_1 ( f ) \cdot X_2 ( f )\hspace{0.15cm}\bullet\!\!\!-\!\!\!-\!\!\!-\!\!\circ\hspace{0.15cm}{ {x} }_{\rm{1} } ( t ) * { {x} }_{\rm{2} } (t ).$$

	[[Signal_Representation/The_Convolution_Theorem_and_Operation#~~Beweis_des_Faltungssatzes~~\|$\text{~~Beweis~~}$]]}}		[[Signal_Representation/The_Convolution_Theorem_and_Operation#Proof_of_the_Convolution_Theorem\|$\text{Proof}$]]}}

Hwang at 09:35, 19 October 2022

2022-10-19T09:35:56Z

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	== Applet Description==		== Applet Description==
	<br>		<br>
	~~Dieses Applet verdeutlicht die Faltungsoperation im Zeitbereich~~		This applet illustrates the convolution operation in the time domain
	*~~zwischen einem Eingangsimpuls~~  $x(t)$   ⇒   ~~Rechteck~~, ~~Dreieck~~, ~~Gauß~~, ~~Exponentialfunktion~~		*between an input pulse  $x(t)$   ⇒   rectangle, triangle, Gaussian, exponential function
	*~~und der Impulsantwort~~  $h(t)$  ~~eines LZI~~–~~Systems mit Tiefpass–Charakter~~  ⇒   ~~Spalt~~–~~Tiefpass~~, ~~Tiefpass erster bzw. zweiter Ordnung~~, ~~Gauß~~–~~Tiefpass~~.		*and the impulse response  $h(t)$  of an LTI system with low–pass character  ⇒   slit low–pass, first or second order low–pass, Gaussian low–pass.


	~~Für das Ausgangssignal~~  $y(t)$  ~~entsprechend dem Blockschaltbild im~~  $\text{~~Beispiel~~ 1}$  ~~gilt dann~~, ~~wie im Kapitel~~  [[Applets:Zur_Verdeutlichung_der_grafischen_Faltung#Grafische_Faltung\|Grafische Faltung]]  dargelegt:		For the output signal  $y(t)$  corresponding to the block diagram in  $\text{Example 1}$,  then, as stated in the chapter  [[Applets:Zur_Verdeutlichung_der_grafischen_Faltung#Grafische_Faltung\|Grafische Faltung]]  dargelegt:
	:$$y( t ) = x(t) * h( t ) = \int_{ - \infty }^{ + \infty } \hspace{-0.15cm}{x( \tau )} \cdot h( {t - \tau } )\hspace{0.1cm}{\rm d}\tau .$$		:$$y( t ) = x(t) * h( t ) = \int_{ - \infty }^{ + \infty } \hspace{-0.15cm}{x( \tau )} \cdot h( {t - \tau } )\hspace{0.1cm}{\rm d}\tau .$$

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	===~~Grafische Faltung~~===		===Graphical Convolution===

	In diesem Applet wird von folgender Faltungsoperation ausgegangen:		In diesem Applet wird von folgender Faltungsoperation ausgegangen:

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	:$$X_1 ( f ) \cdot X_2 ( f )\hspace{0.15cm}\bullet\!\!\!-\!\!\!-\!\!\!-\!\!\circ\hspace{0.15cm}{ {x} }_{\rm{1} } ( t ) * { {x} }_{\rm{2} } (t ).$$		:$$X_1 ( f ) \cdot X_2 ( f )\hspace{0.15cm}\bullet\!\!\!-\!\!\!-\!\!\!-\!\!\circ\hspace{0.15cm}{ {x} }_{\rm{1} } ( t ) * { {x} }_{\rm{2} } (t ).$$

	[[Signal_Representation/The_Convolution_Theorem_and_Operation#Proof_of_the_Convolution_Theorem\|$\text{Proof}$]]}}		[[Signal_Representation/The_Convolution_Theorem_and_Operation#Proof_of_the_Convolution_Theorem\|$\text{"Proof"}$]]}}


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	===Convolution of a function with a Dirac function===		===Convolution of a function with a Dirac delta function===

	The convolution operation becomes very simple if one of the two operands is a  [[Signal_Representation/Direct_Current_Signal_-_Limit_Case_of_a_Periodic_Signal#Dirac_.28delta.29_function_in_frequency_domain\|"Dirac function"]].  This is equally true for convolution in the time and frequency domain.		The convolution operation becomes very simple if one of the two operands is a  [[Signal_Representation/Direct_Current_Signal_-_Limit_Case_of_a_Periodic_Signal#Dirac_.28delta.29_function_in_frequency_domain\|"Dirac delta function"]].  This is equally true for convolution in the time and frequency domain.

	As an example, we consider the convolution of a function  $x_1(t)$  with the function		As an example, we consider the convolution of a function  $x_1(t)$  with the function
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	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	$\text{In words: }$  The convolution of an arbitrary function with a Dirac function at  $t = T$  results in the function shifted to the right by  $T$,  whereby the weighting of the Dirac function by the factor  $\alpha$  must still be taken into account.}}		$\text{In words: }$  The convolution of an arbitrary function with a Dirac delta function at  $t = T$  results in the function shifted to the right by  $T$,  whereby the weighting of the Dirac delta function by the factor  $\alpha$  must still be taken into account.}}


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	$\text{Example 3:}$  A rectangular signal  $x(t)$  is delayed by a running time  $\tau = 3\,\text{ ms}$  and attenuated by a factor  $\alpha = 0.5$  by an LTI system.		$\text{Example 3:}$  A rectangular signal  $x(t)$  is delayed by a running time  $\tau = 3\,\text{ ms}$  and attenuated by a factor  $\alpha = 0.5$  by an LTI system.

	[[File:P_ID522__Sig_T_3_4_S3_neu.png\|center\|frame\|Convolution of a rectangle with a Dirac function]]		[[File:P_ID522__Sig_T_3_4_S3_neu.png\|center\|frame\|Convolution of a rectangle with a Dirac delta function]]

	Shift and attenuation can be seen in the output signal  $y(t)$  as well as in the impulse response  $h(t)$.}}		Shift and attenuation can be seen in the output signal  $y(t)$  as well as in the impulse response  $h(t)$.}}

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	==Exercises==		==Exercises==

	*First, select the number  $(1,\ 2, \text{...} \ )$  of the ~~task '''KORREKTUR:~~ exercise~~'''~~ to be processed.  The number  $0$  corresponds to a "Reset":  Same setting as at program start.		*First, select the number  $(1,\ 2, \text{...} \ )$  of the exercise to be processed.  The number  $0$  corresponds to a "Reset":  Same setting as at program start.
	*~~A task '''KORREKTUR:~~ An exercise~~'''~~ description is displayed.  The parameter values are adjusted.  Solution after pressing "Show Solution".		* An exercise description is displayed.  The parameter values are adjusted.  Solution after pressing "Show Solution".
	*Both the input signal  $x(t)$  and the impulse response  $h(t)$  of the filter are are normalized, dimensionless and energy-limited ("time-limited pulses").		*Both the input signal  $x(t)$  and the impulse response  $h(t)$  of the filter are are normalized, dimensionless and energy-limited ("time-limited pulses").

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	<br>       Analyze and interpret this  "system"  in terms of causality and the resulting distortions for the rectangular pulse. }}		<br>       Analyze and interpret this  "system"  in terms of causality and the resulting distortions for the rectangular pulse. }}

	* The low-pass is not causal (and therefore non–realizable):  For  $t < 0$~~  does  $h(t) \equiv 0$  not hold. '''KORREKTUR~~:  $h(t) \equiv 0$  does not hold.~~'''~~ Suitable model if infinite delay is ignored.		* The low-pass is not causal (and therefore non–realizable):  For  $t < 0$:  $h(t) \equiv 0$  does not hold. Suitable model if infinite delay is ignored.
	* The larger  $\Delta t_h$  is, the wider the output pulse and the stronger the degradation of a digital system due to intersymbol interference.		* The larger  $\Delta t_h$  is, the wider the output pulse and the stronger the degradation of a digital system due to intersymbol interference.
	* The frequency response  $H(f)$  is the Fourier transform of  $h(t)$. The larger  $\Delta t_h$  is, the smaller  $\Delta f_h = 1/\Delta t_h$   ⇒   The system is more narrowband.		* The frequency response  $H(f)$  is the Fourier transform of  $h(t)$. The larger  $\Delta t_h$  is, the smaller  $\Delta f_h = 1/\Delta t_h$   ⇒   The system is more narrowband.
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	'''(K)'''      Successive representation of the output signal  $y(t)$  ⇒   brown curve		'''(K)'''      Successive representation of the output signal  $y(t)$  ⇒   brown curve

	'''(L)'''      Area for exercise execution:   ~~Task '''KORREKTUR:~~ Exercise~~'''~~ selection.		'''(L)'''      Area for exercise execution:   Exercise selection.

	'''(M)'''      Exercise execution:   Area for ~~task '''KORREKTUR:~~ exercise~~'''~~ description		'''(M)'''      Exercise execution:   Area for exercise description

	'''(N)'''      Exercise execution:   Area for the sample solution		'''(N)'''      Exercise execution:   Area for the sample solution
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	The conversion of this applet to HTML 5 was financially supported by  [https://www.ei.tum.de/studium/studienzuschuesse/ Studienzuschüsse]  ("study grants")  of the TUM Faculty EI.  ~~We thank. '''KORREKTUR:~~ Many thanks.~~'''~~		The conversion of this applet to HTML 5 was financially supported by  [https://www.ei.tum.de/studium/studienzuschuesse/ Studienzuschüsse]  ("study grants")  of the TUM Faculty EI.  Many thanks.