Applets:Linear Distortions of Periodic Signals - Revision history

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

2026-03-16T13:49:02Z

Join split $$ formula lines for correct rendering

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	<br>		<br>
	The generally complex valued frequency response can be represented as follows:		The generally complex valued frequency response can be represented as follows:
	:$$H(f) = \|H(f)\| \cdot {\rm e}^{-{\rm j} \hspace{0.05cm} \cdot		:$$H(f) = \|H(f)\| \cdot {\rm e}^{-{\rm j} \hspace{0.05cm} \cdot\hspace{0.05cm} b(f)} = {\rm e}^{-a(f)}\cdot {\rm e}^{-{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm} b(f)}.$$
	\hspace{0.05cm} b(f)} = {\rm e}^{-a(f)}\cdot {\rm e}^{-{\rm j}
	\hspace{0.05cm} \cdot \hspace{0.05cm} b(f)}.$$

	This results in the following description variables:		This results in the following description variables:
	*The absolute value $\|H(f)\|$ is called '''amplitude response''' and in logarithmic form '''attenuation function''':		*The absolute value $\|H(f)\|$ is called '''amplitude response''' and in logarithmic form '''attenuation function''':
	:$$a(f) = - \ln \|H(f)\|\hspace{0.2cm}{\rm in \hspace{0.1cm}Neper		:$$a(f) = - \ln \|H(f)\|\hspace{0.2cm}{\rm in \hspace{0.1cm}Neper\hspace{0.1cm}(Np) } = - 20 \cdot \lg \|H(f)\|\hspace{0.2cm}{\rm in\hspace{0.1cm}Decibel \hspace{0.1cm}(dB) }.$$
	\hspace{0.1cm}(Np) } = - 20 \cdot \lg \|H(f)\|\hspace{0.2cm}{\rm in
	\hspace{0.1cm}Decibel \hspace{0.1cm}(dB) }.$$
	*The '''phase function''' $b(f)$ indicates the negative frequency–dependent angle of $H(f)$ in the complex plane based on the real axis:		*The '''phase function''' $b(f)$ indicates the negative frequency–dependent angle of $H(f)$ in the complex plane based on the real axis:
	:$$b(f) = - {\rm arc} \hspace{0.1cm}H(f) \hspace{0.2cm}{\rm in		:$$b(f) = - {\rm arc} \hspace{0.1cm}H(f) \hspace{0.2cm}{\rm in\hspace{0.1cm}Radians \hspace{0.1cm}(rad)}.$$
	\hspace{0.1cm}Radians \hspace{0.1cm}(rad)}.$$


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	==About the Authors==		==About the Authors==
	This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technische Universität München].		This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technische Universität München].
	*The original version was created in 2005 by [[Biographies_and_Bibliographies/~~An_LNTwww_beteiligte_Studierende~~#Bettina_Hirner_.28Diplomarbeit_LB_2005.29\|Bettina Hirner]] 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#Bettina_Hirner_.28Diplomarbeit_LB_2005.29\|Bettina Hirner]] 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~~#Jimmy_He_.28Bachelorarbeit_2018.29\|Jimmy He]] as part of his 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#Jimmy_He_.28Bachelorarbeit_2018.29\|Jimmy He]] as part of his 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==

Maintenance script: Add German interlanguage link

2026-03-16T12:39:13Z

Add German interlanguage link

← Older revision		Revision as of 14:39, 16 March 2026
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	{{LntAppletLinkEn\|linDistortions_en}}         [https://lntwww.lnt.ei.tum.de/Applets:Lineare_Verzerrungen_periodischer_Signale '''English Applet with German WIKI description''']		{{LntAppletLinkEn\|linDistortions_en}}         [https://lntwww.lnt.ei.tum.de/Applets:Lineare_Verzerrungen_periodischer_Signale '''English Applet with German WIKI description''']
			[[de:Applets:Linear_Distortions_of_Periodic_Signals]]

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

2026-03-16T11:09:50Z

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

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

	{{LntAppletLinkEn\|linDistortions_en}}         [https://~~www~~.~~lntwww~~.de/Applets:Lineare_Verzerrungen_periodischer_Signale '''English Applet with German WIKI description''']		{{LntAppletLinkEn\|linDistortions_en}}         [https://lntwww.lnt.ei.tum.de/Applets:Lineare_Verzerrungen_periodischer_Signale '''English Applet with German WIKI description''']

Guenter at 15:16, 13 April 2023

2023-04-13T15:16:48Z

← Older revision		Revision as of 17:16, 13 April 2023
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	*'''Nonlinear distortions''' occur, if at all times $t$ the nonlinear correlation $y = g(x) \ne {\rm const.} \cdot x$ exists between the signal values $x = x(t)$ at the input and $y = y(t)$ at the output, whereby $y = g(x)$ is defined as the system's nonlinear characteristic. By creating a cosine signal at the input with frequency $f_0$ the output signal includes $f_0$, as well as multiple harmonic waves. We conclude that new frequencies arise through nonlinear distortion.		*'''Nonlinear distortions''' occur, if at all times $t$ the nonlinear correlation $y = g(x) \ne {\rm const.} \cdot x$ exists between the signal values $x = x(t)$ at the input and $y = y(t)$ at the output, whereby $y = g(x)$ is defined as the system's nonlinear characteristic. By creating a cosine signal at the input with frequency $f_0$ the output signal includes $f_0$, as well as multiple harmonic waves. We conclude that new frequencies arise through nonlinear distortion.

	[[File:~~EN_LZI_T_2_2_S3~~.png\|center\|frame\|For clarification of nonlinear distortions \|class=fit]]		[[File:EN_LZI_T_2_2_S3_v2.png\|center\|frame\|For clarification of nonlinear distortions \|class=fit]]

	[[File:P_ID899__LZI_T_2_3_S1_neu.png\|right \|frame\| Description of a linear system\|class=fit]]		[[File:P_ID899__LZI_T_2_3_S1_neu.png\|right \|frame\| Description of a linear system\|class=fit]]

Guenter at 15:36, 10 April 2023

2023-04-10T15:36:22Z

← Older revision		Revision as of 17:36, 10 April 2023
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	*'''Nonlinear distortions''' occur, if at all times $t$ the nonlinear correlation $y = g(x) \ne {\rm const.} \cdot x$ exists between the signal values $x = x(t)$ at the input and $y = y(t)$ at the output, whereby $y = g(x)$ is defined as the system's nonlinear characteristic. By creating a cosine signal at the input with frequency $f_0$ the output signal includes $f_0$, as well as multiple harmonic waves. We conclude that new frequencies arise through nonlinear distortion.		*'''Nonlinear distortions''' occur, if at all times $t$ the nonlinear correlation $y = g(x) \ne {\rm const.} \cdot x$ exists between the signal values $x = x(t)$ at the input and $y = y(t)$ at the output, whereby $y = g(x)$ is defined as the system's nonlinear characteristic. By creating a cosine signal at the input with frequency $f_0$ the output signal includes $f_0$, as well as multiple harmonic waves. We conclude that new frequencies arise through nonlinear distortion.

	[[File:~~LZI_T_2_2_S3_vers2~~.png\|center\|frame\|For clarification of nonlinear distortions \|class=fit]]		[[File:EN_LZI_T_2_2_S3.png\|center\|frame\|For clarification of nonlinear distortions \|class=fit]]

	[[File:P_ID899__LZI_T_2_3_S1_neu.png\|right \|frame\| Description of a linear system\|class=fit]]		[[File:P_ID899__LZI_T_2_3_S1_neu.png\|right \|frame\| Description of a linear system\|class=fit]]

Noah at 22:12, 26 March 2023

2023-03-26T22:12:21Z

← Older revision		Revision as of 00:12, 27 March 2023
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	*The '''phase function''' $b(f)$ indicates the negative frequency–dependent angle of $H(f)$ in the complex plane based on the real axis:		*The '''phase function''' $b(f)$ indicates the negative frequency–dependent angle of $H(f)$ in the complex plane based on the real axis:
	:$$b(f) = - {\rm arc} \hspace{0.1cm}H(f) \hspace{0.2cm}{\rm in		:$$b(f) = - {\rm arc} \hspace{0.1cm}H(f) \hspace{0.2cm}{\rm in
	\hspace{0.1cm}~~Radian~~ \hspace{0.1cm}(rad)}.$$		\hspace{0.1cm}Radians \hspace{0.1cm}(rad)}.$$

Noah at 22:09, 26 March 2023

2023-03-26T22:09:15Z

← Older revision		Revision as of 00:09, 27 March 2023
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	The frequency response of a realizable high–pass (HP) of order $N$ is:		The frequency response of a realizable high–pass (HP) of order $N$ is:
	:$$H(f) = \left [\frac{ {\rm j}\cdot f/f_0 }{1 + {\rm j}\cdot f/f_0 }\right ]^N\hspace{0.05cm}.$$		:$$H(f) = \left [\frac{ {\rm j}\cdot f/f_0 }{1 + {\rm j}\cdot f/f_0 }\right ]^N\hspace{0.05cm}.$$
	For example the LC high pass is a first order high pass. Consequently we can obtain		For example the LC high-pass is a first order high-pass. Consequently we can obtain
	*the attenuation function:		*the attenuation function:
	:$$a(f) =N/2 \cdot \ln [1+( f_0/f)^2] \hspace{0.05cm},$$		:$$a(f) =N/2 \cdot \ln [1+( f_0/f)^2] \hspace{0.05cm},$$

Noah at 22:08, 26 March 2023

2023-03-26T22:08:45Z

← Older revision		Revision as of 00:08, 27 March 2023
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	The next block in the model above is ''Matching'': The output signal $y(t)$ is adjusted in amplitude and phase with equal ~~quantities~~ $k_{\rm M}$ and $\tau_{\rm M}$ for all frequencies which means that this is not a frequency-dependent equalization. Using the signal $z(t)$, one can differentiate between:		The next block in the model above is ''Matching'': The output signal $y(t)$ is adjusted in amplitude and phase with equal variables $k_{\rm M}$ and $\tau_{\rm M}$ for all frequencies which means that this is not a frequency-dependent equalization. Using the signal $z(t)$, one can differentiate between:
	*attenuation distortion and frequency–~~independant~~ attenuation, as well as		*attenuation distortion and frequency–independent attenuation, as well as
	*phase distortion and frequency–~~independant~~ delay.		*phase distortion and frequency–independent delay.


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	{{GraueBox\|TEXT=		{{GraueBox\|TEXT=
	$\text{Example:}$		$\text{Example:}$
	This graphic shows the phase function $b(f)$ with the ~~cut–off~~ frequency $f_0 = 1\ \rm kHz$ and order $N=1$		This graphic shows the phase function $b(f)$ with the cutoff frequency $f_0 = 1\ \rm kHz$ and order $N=1$
	* of a low–pass (green curve),		* of a low–pass (green curve),
	* of a high–pass (violet curve).		* of a high–pass (violet curve).
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	''Remark:'' The shown signals were created using the interactive applet [[Applets:Kausale_Systeme_-_Laplacetransformation\|Causal systems – Laplace transform]]. }}		''Remark:'' The shown signals were created using the interactive applet [[Applets:Kausale_Systeme_-_Laplacetransformation\|"Causal systems – Laplace transform"]]. }}

	=== Attenuation and Phase Distortions ===		=== Attenuation and Phase Distortions ===
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	'''(C)'''     Selction of channel parameters per slider: Dämpfungsfaktoren und Phasenlaufzeiten		'''(C)'''     Selction of channel parameters per slider: Dämpfungsfaktoren und Phasenlaufzeiten

	'''(D)'''     Selection of channel parameters for High and Low pass: Order$n$, ~~cut-off-~~frequency $f_0$		'''(D)'''     Selection of channel parameters for High and Low pass: Order$n$, cutoff frequency $f_0$

	'''(E)'''     Selection of matching parameters $k_{\rm M}$ and $\varphi_{\rm M}$		'''(E)'''     Selection of matching parameters $k_{\rm M}$ and $\varphi_{\rm M}$

Guenter at 14:30, 13 January 2023

2023-01-13T14:30:12Z

← Older revision		Revision as of 16:30, 13 January 2023
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	{{~~LntAppletLinkEn~~\|~~linDistortions_en~~}}         [https://~~www~~.lntwww.de/Applets:~~Lineare_Verzerrungen_periodischer_Signale~~ '''English Applet with ~~German~~ WIKI description''']		{{LntAppletLink\|physAnSignal_en}}         [https://en.lntwww.de/Applets:Physical_Signal_%26_Analytic_Signal '''English Applet with English WIKI description''']

	==Applet Description==		==Applet Description==

Javier: Text replacement - "„" to """

2021-05-28T14:45:47Z

Text replacement - "„" to """

← Older revision		Revision as of 16:45, 28 May 2021
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	*An exercise description is displayed.		*An exercise description is displayed.
	*Parameter values are adjusted to the respective exercises.		*Parameter values are adjusted to the respective exercises.
	*Click ~~&bdquo;~~Hide solution" to display the solution.		*Click "Hide solution" to display the solution.


	Number ~~&bdquo;~~0" is a ~~&bdquo;~~Reset" button:		Number "0" is a "Reset" button:
	*Sets parameters to initial values (when loading the page).		*Sets parameters to initial values (when loading the page).
	*Displays a ~~&bdquo;~~Reset text" to describe the applet further.		*Displays a "Reset text" to describe the applet further.


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	'''(M)'''     Variation possibilities for the graphic display		'''(M)'''     Variation possibilities for the graphic display

	$\hspace{1.5cm}$Zoom–functions ~~&bdquo;~~$+$" (scale up), ~~&bdquo;~~$-$" (scale down) und $\rm o$ (reset)		$\hspace{1.5cm}$Zoom–functions "$+$" (scale up), "$-$" (scale down) und $\rm o$ (reset)

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

	$\hspace{1.5cm}$'''Other options'':		$\hspace{1.5cm}$'''Other options'':
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	==About the Authors==		==About the Authors==
	This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technische Universität München].		This interactive calculation was designed and realized at the [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik] of the [https://www.tum.de/ Technische Universität München].
	*The original version was created in 2005 by [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Bettina_Hirner_.28Diplomarbeit_LB_2005.29\|Bettina Hirner]] 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#Bettina_Hirner_.28Diplomarbeit_LB_2005.29\|Bettina Hirner]] 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#Jimmy_He_.28Bachelorarbeit_2018.29\|Jimmy He]] as part of his 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#Jimmy_He_.28Bachelorarbeit_2018.29\|Jimmy He]] as part of his 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\|linDistortions_en}}         [https://www.lntwww.de/Applets:Lineare_Verzerrungen_periodischer_Signale '''English Applet with German WIKI description''']		{{LntAppletLinkEn\|linDistortions_en}}         [https://www.lntwww.de/Applets:Lineare_Verzerrungen_periodischer_Signale '''English Applet with German WIKI description''']