Applets:Frequency & Impulse Responses - Revision history

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

2026-03-16T13:49:00Z

Join split $$ formula lines for correct rendering

Höfler: Text replacement - "Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29" to "Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29"

2025-02-24T13:39:51Z

Text replacement - "Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29" to "Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29"

← Older revision		Revision as of 15:39, 24 February 2025
Line 270:		Line 270:

	This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ $\text{Institute for Communications Engineering}$]  at the  [https://www.tum.de/en $\text{Technical University of Munich}$].		This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ $\text{Institute for Communications Engineering}$]  at the  [https://www.tum.de/en $\text{Technical University of Munich}$].
	*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._G.C3.BCnter_S.C3.B6der_.~~28at_LNT_since_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._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29\| »Günter Söder«]]).

	*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/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29\|»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/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29\|»Tasnád Kernetzky«]] ) via "HTML5".

Höfler: Text replacement - "Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28at_L.C3.9CT_since_2014.29" to "Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29"

2023-10-26T10:17:44Z

Text replacement - "Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28at_L.C3.9CT_since_2014.29" to "Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29"

← Older revision		Revision as of 12:17, 26 October 2023
Line 272:		Line 272:
	*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._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_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._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29\| »Günter Söder«]]).

	*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/LNTwww_members_from_LÜT#~~Tasn~~.C3.~~A1d_Kernetzky~~.~~2C_M.Sc~~._.28at_L.C3.~~9CT_since_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/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29\|»Tasnád Kernetzky«]] ) via "HTML5".

	*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.		*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.

Guenter at 14:17, 30 June 2023

2023-06-30T14:17:25Z

← Older revision		Revision as of 16:17, 30 June 2023
Line 5:		Line 5:
	<br>		<br>

	Real and symmetric low-~~passes~~  $H(f)$  and the corresponding impulse responses  $h(t)$  are shown,  namely		Real and symmetric low-pass frquency responses  $H(f)$  and the corresponding impulse responses  $h(t)$  are shown,  namely
	#Gaussian low-pass,		#Gaussian low-pass,
	#rectangular low-pass,		#rectangular low-pass,
	#triangular low-pass,		#triangular low-pass,
	#trapezoidal low-pass,		#trapezoidal low-pass,
	#cosine rolloff low-pass   ⇒   "raised-cosine low-pass",		#cosine rolloff low-pass   ⇒   »raised-cosine low-pass«,
	#cosine rolloff squared low-pass   ⇒   "cosine-square low-pass".		#cosine rolloff squared low-pass   ⇒   »cosine-square low-pass«.

Guenter at 14:06, 13 March 2023

2023-03-13T14:06:49Z

← Older revision		Revision as of 16:06, 13 March 2023
Line 270:		Line 270:

	This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ $\text{Institute for Communications Engineering}$]  at the  [https://www.tum.de/en $\text{Technical University of Munich}$].		This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ $\text{Institute for Communications Engineering}$]  at the  [https://www.tum.de/en $\text{Technical University of Munich}$].
	*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._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_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._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29\| »Günter Söder«]]).
	*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/LNTwww_members_from_LÜT#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28at_L.C3.9CT_since_2014.29\|Tasnád Kernetzky]] ) via "HTML5".
	*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.		*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/LNTwww_members_from_LÜT#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28at_L.C3.9CT_since_2014.29\|»Tasnád Kernetzky«]] ) via "HTML5".

			*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.

	==Once again: Open Applet in new Tab==		==Once again: Open Applet in new Tab==
	{{LntAppletLinkEnDe\|frequImpResp_en\|frequImpResp}}		{{LntAppletLinkEnDe\|frequImpResp_en\|frequImpResp}}

Guenter at 13:54, 13 March 2023

2023-03-13T13:54:12Z

Guenter at 13:47, 13 March 2023

2023-03-13T13:47:31Z

← Older revision		Revision as of 15:47, 13 March 2023
Line 182:		Line 182:
	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	'''(2)'''   Leave the settings unchanged.  Which low-pass  $H(f)$  fulfills the first or the second Nyquist criterion?<br>        Here  $H(f)$  denotes the total frequency response of transmitter,		'''(2)'''   Leave the settings unchanged.  Which low-pass  $H(f)$  fulfills the first or the second Nyquist criterion?<br>        Here  $H(f)$  denotes the total frequency response of transmitter,
	channel and ~~reception~~ '''~~KORREKTUR:~~ receiver''' filter.}}		channel and '''Korrektur'''receiver''' filter.}}

	* First Nyquist criterion:  The impulse response  $h(t)$  must have equidistant zero crossings at the (normalized) times  $t = 1,\ 2$, ... <br>		* First Nyquist criterion:  The impulse response  $h(t)$  must have equidistant zero crossings at the (normalized) times  $t = 1,\ 2$, ... <br>

Guenter at 13:46, 13 March 2023

2023-03-13T13:46:38Z

← Older revision		Revision as of 15:46, 13 March 2023
Line 135:		Line 135:

	:$$H(f) = \left\{ \begin{array}{l} \hspace{0.25cm}K \\ K\cdot \cos^2\Big(\frac{\|f\|-f_1}{f_2-f_1}\cdot {\pi}/{2}\Big) \\ \hspace{0.25cm} 0 \\ \end{array} \right.\quad \quad \begin{array}{{20}c} {\rm{for}} \\ {\rm{for}} \\ {\rm{for}} \\ \end{array}\begin{array}{{20}c} {\left\| \hspace{0.05cm} f\hspace{0.05cm} \right\| \le f_1,} \\ {f_1\le \left\| \hspace{0.05cm}f\hspace{0.05cm} \right\| \le f_2,} \\ {\left\|\hspace{0.05cm} f \hspace{0.05cm} \right\| \ge f_2.} \\ \end{array}$$		:$$H(f) = \left\{ \begin{array}{l} \hspace{0.25cm}K \\ K\cdot \cos^2\Big(\frac{\|f\|-f_1}{f_2-f_1}\cdot {\pi}/{2}\Big) \\ \hspace{0.25cm} 0 \\ \end{array} \right.\quad \quad \begin{array}{{20}c} {\rm{for}} \\ {\rm{for}} \\ {\rm{for}} \\ \end{array}\begin{array}{{20}c} {\left\| \hspace{0.05cm} f\hspace{0.05cm} \right\| \le f_1,} \\ {f_1\le \left\| \hspace{0.05cm}f\hspace{0.05cm} \right\| \le f_2,} \\ {\left\|\hspace{0.05cm} f \hspace{0.05cm} \right\| \ge f_2.} \\ \end{array}$$

	~~::'''Korrektur:''' Warum sind in dieser Gleichung die Zeilenabstände rechts kleiner als~~


	*For the equivalent bandwidth  $($equal-area rectangle$)$  the following applies:  $\Delta f = f_1+f_2$.		*For the equivalent bandwidth  $($equal-area rectangle$)$  the following applies:  $\Delta f = f_1+f_2$.
Line 157:		Line 154:

	*For the impulse response one obtains according to the Fourier inverse transform:		*For the impulse response one obtains according to the Fourier inverse transform:
	:$$h(t)=K\cdot \Delta f \cdot {\pi}/{4}\cdot \big [{\rm si}~~(\pi~~(\Delta f\cdot t +0.5))+{\rm si}~~(\pi~~(\Delta f\cdot t -0.5))\big ]\cdot {\rm si}(~~\pi \cdot~~ \Delta f \cdot t).$$		:$$h(t)=K\cdot \Delta f \cdot {\pi}/{4}\cdot \big [{\rm sinc}(\Delta f\cdot t +0.5)+{\rm sinc}(\Delta f\cdot t -0.5)\big ]\cdot {\rm sinc}(\Delta f \cdot t)\quad \text{with} \quad {\rm sinc}(x)={\sin(x)}/{(\pi x})={\rm si}(\pi x).$$
	*Because of the last  ${\rm si}$ ~~'''KORREKTUR: sinc?'''~~ function,  $h(t)=0$  for all multiples of  $T=1/\Delta f$   ⇒   The equidistant zero crossings of the cosine rolloff low-pass are preserved.
			*Because of the last  ${\rm sinc}$–function,  $h(t)=0$  for all multiples of  $T=1/\Delta f$   ⇒   The equidistant zero crossings of the cosine rolloff low-pass are preserved.
	*Due to the expression in parentheses,  $h(t)$  now shows further zero crossings at  $t=\pm1.5 T$,  $\pm2.5 T$,  $\pm3.5 T$, ...		*Due to the expression in parentheses,  $h(t)$  now shows further zero crossings at  $t=\pm1.5 T$,  $\pm2.5 T$,  $\pm3.5 T$, ...
	*For  $t=\pm T/2$  the impulse response has the value  $K\cdot \Delta f/2$.		*For  $t=\pm T/2$  the impulse response has the value  $K\cdot \Delta f/2$.

Guenter at 13:35, 13 March 2023

2023-03-13T13:35:02Z

Hwang at 19:27, 22 February 2023

2023-02-22T19:27:28Z

← Older revision		Revision as of 21:27, 22 February 2023
Line 275:		Line 275:

	This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ $\text{Institute for Communications Engineering}$]  at the  [https://www.tum.de/en $\text{Technical University of Munich}$].		This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ $\text{Institute for Communications Engineering}$]  at the  [https://www.tum.de/en $\text{Technical University of Munich}$].
	*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\|$\text{Ji Li}$]] 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\|$\text{Günter Söder}$]]).		*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\|$\text{Ji Li}$]] as part of her diploma thesis with “FlashMX – Actionscript” (Supervisor: [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29\|$\text{Günter Söder}$]]).
	*In 2017 the program was redesigned by [[~~Biografien_und_Bibliografien~~/~~An_LNTwww_beteiligte_Studierende~~#David_Jobst_.28Ingenieurspraxis_Math_2017.29\|$\text{David Jobst}$]] (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\|$\text{Tasnád Kernetzky}$]] ) via "HTML5".		*In 2017 the program was redesigned by [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#David_Jobst_.28Ingenieurspraxis_Math_2017.29\|$\text{David Jobst}$]] (Ingenieurspraxis_Math, Supervisor: [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28at_L.C3.9CT_since_2014.29\|$\text{Tasnád Kernetzky}$]] ) 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\|$\text{Carolin Mirschina}$]]  in the context of a working student activity.  Translation using DEEPL.com.		*Last revision and English version 2020 by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29\|$\text{Carolin Mirschina}$]]  in the context of a working student activity.  Translation using DEEPL.com.

	==Once again: Open Applet in new Tab==		==Once again: Open Applet in new Tab==
	{{LntAppletLinkEnDe\|frequImpResp_en\|frequImpResp}}		{{LntAppletLinkEnDe\|frequImpResp_en\|frequImpResp}}

← Older revision		Revision as of 12:17, 26 October 2023
Line 272:		Line 272:
	*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._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_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._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29\| »Günter Söder«]]).

	*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/LNTwww_members_from_LÜT#~~Tasn~~.C3.~~A1d_Kernetzky~~.~~2C_M.Sc~~._.28at_L.C3.~~9CT_since_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/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29\|»Tasnád Kernetzky«]] ) via "HTML5".

	*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.		*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.

← Older revision		Revision as of 15:49, 16 March 2026
Line 34:		Line 34:

	*The properties of  $H(f)$  are expressed in the time domain by the  [[Linear_and_Time_Invariant_Systems/System_Description_in_Time_Domain#Impulse_response\|$\text{impulse response}$]]  $h(t)$.   According to the  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_second_Fourier_integral\|$\text{second Fourier integral}$]]  holds:		*The properties of  $H(f)$  are expressed in the time domain by the  [[Linear_and_Time_Invariant_Systems/System_Description_in_Time_Domain#Impulse_response\|$\text{impulse response}$]]  $h(t)$.   According to the  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_second_Fourier_integral\|$\text{second Fourier integral}$]]  holds:
	:$$h(t)={\rm IFT} [H(f)] = \int_{-\infty}^{+\infty}H(f)\cdot {\rm e}^{+{\rm j}2\pi f t}\hspace{0.15cm} {\rm d}f\hspace{1cm}		:$$h(t)={\rm IFT} [H(f)] = \int_{-\infty}^{+\infty}H(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.$$
	{\rm IFT}\hspace{-0.1cm}: \rm Inverse \ Fourier \ transform.$$
	*The inverse direction is described by the  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_first_Fourier_integral\|$\text{first Fourier integral}$]]:		*The inverse direction is described by the  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_first_Fourier_integral\|$\text{first Fourier integral}$]]:
	:$$H(f)={\rm FT} [h(t)] = \int_{-\infty}^{+\infty}h(t)\cdot {\rm e}^{-{\rm j}2\pi f t}\hspace{0.15cm} {\rm d}t\hspace{1cm}		:$$H(f)={\rm FT} [h(t)] = \int_{-\infty}^{+\infty}h(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 all examples we use real and even functions.  Thus:		*In all examples we use real and even functions.  Thus:
	:$$h(t)=\int_{-\infty}^{+\infty}H(f)\cdot \cos(2\pi ft) \hspace{0.15cm} {\rm d}f \ \circ\!\!-\!\!-\! \!\!-\!\!\bullet\ \ H(f)=\int_{-\infty}^{+\infty}h(t)\cdot \cos(2\pi ft) \hspace{0.15cm} {\rm d}t .$$		:$$h(t)=\int_{-\infty}^{+\infty}H(f)\cdot \cos(2\pi ft) \hspace{0.15cm} {\rm d}f \ \circ\!\!-\!\!-\! \!\!-\!\!\bullet\ \ H(f)=\int_{-\infty}^{+\infty}h(t)\cdot \cos(2\pi ft) \hspace{0.15cm} {\rm d}t .$$
Line 45:		Line 43:
	*The unit of impulse response is  $\rm 1/s$.  It is true that  $\rm 1/s = 1 \ Hz$,  but the unit  "Hertz"  is unusual in this context.		*The unit of impulse response is  $\rm 1/s$.  It is true that  $\rm 1/s = 1 \ Hz$,  but the unit  "Hertz"  is unusual in this context.

	*The relationship between this applet and the similarly constructed applet  [[Applets:Pulses_and_Spectra\|"Pulses and Spectra"]]  is based on the  [[Signal_Representation/~~Fourier_Transform_Theorems~~#Duality_Theorem\|$\text{Duality Theorem}$]].		*The relationship between this applet and the similarly constructed applet  [[Applets:Pulses_and_Spectra\|"Pulses and Spectra"]]  is based on the  [[Signal_Representation/The_Fourier_Transform_Theorems#Duality_Theorem\|$\text{Duality Theorem}$]].

	*All times are normalized to a normalization time  $T$  and all frequencies are normalized to  $1/T  \ \Rightarrow$  the numerical values of   $h(t)$  still have to be divided by  $T$.		*All times are normalized to a normalization time  $T$  and all frequencies are normalized to  $1/T  \ \Rightarrow$  the numerical values of   $h(t)$  still have to be divided by  $T$.
Line 72:		Line 70:
	:$$h(t)=K\cdot \Delta f \cdot {\rm e}^{-\pi(t\hspace{0.05cm}\cdot\hspace{0.05cm} \Delta f)^2} .$$		:$$h(t)=K\cdot \Delta f \cdot {\rm e}^{-\pi(t\hspace{0.05cm}\cdot\hspace{0.05cm} \Delta f)^2} .$$

	*The smaller  $\Delta f$,  the wider and lower is the impulse response   ⇒   [[Signal_Representation/~~Fourier_Transform_Theorems~~#Reciprocity_Theorem_of_time_duration_and_bandwidth\|$\text{Reciprocity theorem of bandwidth and impulse duration}$]].		*The smaller  $\Delta f$,  the wider and lower is the impulse response   ⇒   [[Signal_Representation/The_Fourier_Transform_Theorems#Reciprocity_Theorem_of_time_duration_and_bandwidth\|$\text{Reciprocity theorem of bandwidth and impulse duration}$]].

	*Both  $H(f)$  and  $h(t)$  are not exactly equal to zero at any value of  $f$  resp.  $t$.		*Both  $H(f)$  and  $h(t)$  are not exactly equal to zero at any value of  $f$  resp.  $t$.

← Older revision		Revision as of 15:54, 13 March 2023
Line 182:		Line 182:
	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	'''(2)'''   Leave the settings unchanged.  Which low-pass  $H(f)$  fulfills the first or the second Nyquist criterion?<br>        Here  $H(f)$  denotes the total frequency response of transmitter,		'''(2)'''   Leave the settings unchanged.  Which low-pass  $H(f)$  fulfills the first or the second Nyquist criterion?<br>        Here  $H(f)$  denotes the total frequency response of transmitter,
	channel and ~~'''Korrektur'''receiver'''~~ filter.}}		channel and recettion filter.}}

	* First Nyquist criterion:  The impulse response  $h(t)$  must have equidistant zero crossings at the (normalized) times  $t = 1,\ 2$, ... <br>		* First Nyquist criterion:  The impulse response  $h(t)$  must have equidistant zero crossings at the (normalized) times  $t = 1,\ 2$, ... <br>
	* The impulse response  $h(t) = {\rm si}(~~\pi \cdot~~ \delta f \cdot t)$  ~~'''KORREKTUR: sinc?'''~~ of the rectangular low-pass filter fulfils this criterion with  $\Delta f = 1$. <br>		* The impulse response  $h(t) = {\rm sinc}(\delta f \cdot t)$  of the rectangular low-pass filter fulfils this criterion with  $\Delta f = 1$. <br>
	* In contrast, the first Nyquist criterion is never fulfilled for the Gaussian low-pass and there is always impulse interference.<br>		* In contrast, the first Nyquist criterion is never fulfilled for the Gaussian low-pass and there is always impulse interference.<br>
	* The second Nyquist criterion is met by neither the rectangular low-pass nor the Gaussian low-pass.		* The second Nyquist criterion is met by neither the rectangular low-pass nor the Gaussian low-pass.
Line 191:		Line 191:

	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	'''(3)'''   Compare the  »<b>red rectangular low-pass</b>«  $(K_1 = 0.5, \Delta f_1 = 2)$  to the  »<b>blue rectangular low-pass</b>«  $(K_2 = 1, \Delta f_2 = 1)$.  Then vary  $\Delta f_1$  between  $2$  ~~and~~  $0.5$.}}		'''(3)'''   Compare the  »<b>red rectangular low-pass</b>«  $(K_1 = 0.5, \Delta f_1 = 2)$  to the  »<b>blue rectangular low-pass</b>«  $(K_2 = 1, \Delta f_2 = 1)$.  Then vary  $\Delta f_1$  between  $2$  ...  $0.5$.}}

	* With  $\Delta f_1 = 2$  the zeros of  $h_1(t)$  are multiples  of  $0.5$   ⇒   $h_1(t)$   will decay twice as fast as  $h_2(t)$. <br>		* With  $\Delta f_1 = 2$  the zeros of  $h_1(t)$  are multiples  of  $0.5$   ⇒   $h_1(t)$   will decay twice as fast as  $h_2(t)$. <br>
Line 199:		Line 199:

	{{BlaueBox\|TEXT=		{{BlaueBox\|TEXT=
	'''(4)'''   Compare the  »<b>red trapezoidal low-pass</b>«  $(K_1 = 1, \ \Delta f_1 = 1, \ r_1 = 0.5)$  with ~~the~~  »<b>blue rectangular low-pass</b>«   $(K_2 = 1, \ \Delta f_2 = 1)$.  $r_1$  varies between  $0$  ~~and~~  $1$. }}		'''(4)'''   Compare the  »<b>red trapezoidal low-pass</b>«  $(K_1 = 1, \ \Delta f_1 = 1, \ r_1 = 0.5)$  with  »<b>blue rectangular low-pass</b>«   $(K_2 = 1, \ \Delta f_2 = 1)$.  $r_1$  varies between  $0$  ...  $1$. }}

	* With  $r_1 = 0.5$  the  followers/precursors of  $h_1(t)$  for the "trapezoid" are less than for the "rectangle" due to the flatter edge drop .<br>		* With  $r_1 = 0.5$  the  followers/precursors of  $h_1(t)$  for the "trapezoid" are less than for the "rectangle" due to the flatter edge drop .<br>
Line 270:		Line 270:

	This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ $\text{Institute for Communications Engineering}$]  at the  [https://www.tum.de/en $\text{Technical University of Munich}$].		This interactive calculation tool was designed and implemented at the  [https://www.ei.tum.de/en/lnt/home/ $\text{Institute for Communications Engineering}$]  at the  [https://www.tum.de/en $\text{Technical University of Munich}$].
	*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\|~~$\text{~~Ji Li}$]] as part of her diploma thesis with “FlashMX – Actionscript” (Supervisor: [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29\|~~$\text{~~Günter Söder}$]]).		*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._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29\| Günter Söder}]).
	*In 2017 the program was redesigned by [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#David_Jobst_.28Ingenieurspraxis_Math_2017.29\|~~$\text{~~David Jobst}$]] (Ingenieurspraxis_Math, Supervisor: [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28at_L.C3.9CT_since_2014.29\|~~$\text{~~Tasnád Kernetzky}$]] ) 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/LNTwww_members_from_LÜT#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28at_L.C3.9CT_since_2014.29\|Tasnád Kernetzky]] ) via "HTML5".
	*Last revision and English version 2020 by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29\|$~~\text{~~Carolin Mirschina}$]]  in the context of a working student activity.  ~~Translation using DEEPL.com.~~		*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.

	==Once again: Open Applet in new Tab==		==Once again: Open Applet in new Tab==
	{{LntAppletLinkEnDe\|frequImpResp_en\|frequImpResp}}		{{LntAppletLinkEnDe\|frequImpResp_en\|frequImpResp}}

← Older revision		Revision as of 15:35, 13 March 2023
Line 99:		Line 99:
	:$$H(f) = \left\{ \begin{array}{l} \hspace{0.25cm}K\cdot \Big(1-\frac{\|f\|}{\Delta f}\Big) \\ \hspace{0.25cm} 0 \\ \end{array} \right.\quad \quad \begin{array}{{20}c} {\rm{for}} \\ {\rm{for}} \\ \end{array}\begin{array}{{20}c} {\left\| \hspace{0.05cm} f\hspace{0.05cm} \right\| < \Delta f,} \\ {\left\| \hspace{0.05cm}f\hspace{0.05cm} \right\| \ge \Delta f.} \\ \end{array}$$		:$$H(f) = \left\{ \begin{array}{l} \hspace{0.25cm}K\cdot \Big(1-\frac{\|f\|}{\Delta f}\Big) \\ \hspace{0.25cm} 0 \\ \end{array} \right.\quad \quad \begin{array}{{20}c} {\rm{for}} \\ {\rm{for}} \\ \end{array}\begin{array}{{20}c} {\left\| \hspace{0.05cm} f\hspace{0.05cm} \right\| < \Delta f,} \\ {\left\| \hspace{0.05cm}f\hspace{0.05cm} \right\| \ge \Delta f.} \\ \end{array}$$
	*The absolute physical bandwidth  $B$   ⇒  [positive frequencies only]   is also equal  $\Delta f$,  thus is as large as for the rectangular low-pass.		*The absolute physical bandwidth  $B$   ⇒  [positive frequencies only]   is also equal  $\Delta f$,  thus is as large as for the rectangular low-pass.
	~~::'''Korrektur:''' Warum sind in dieser Gleichung die Zeilenabstände rechts kleiner als lins~~

	*For the impulse response  $h(t)$  one obtains according to the second Fourier transform:		*For the impulse response  $h(t)$  one obtains according to the second Fourier transform:
	:$$h(t)=K\cdot \Delta f \cdot {\rm si}^2(\pi\cdot \Delta f \cdot t) \quad \text{with} \quad {\rm si}(x)={\sin(x)}/{x}.$$		:$$h(t)=K\cdot \Delta f \cdot {\rm si}^2(\pi\cdot \Delta f \cdot t) \quad \text{with} \quad {\rm si}(x)={\sin(x)}/{x}={\rm sinc}(x/\pi).$$
	*$H(f)$  can be represented as a convolution of two rectangular functions  $($each with width  $\Delta f)$.		*$H(f)$  can be represented as a convolution of two rectangular functions  $($each with width  $\Delta f)$.

	*It follows:  $h(t)$  contains instead of the  ${\rm si}$–function the  ${\rm si}^2$–function. ~~'''KORREKTUR: sinc?'''~~		*It follows:  $h(t)$  contains instead of the  ${\rm si}$–function the  ${\rm si}^2$–function.

	*$h(t)$  thus also exhibits zeros at equidistant intervals  $1/\Delta f$.		*$h(t)$  thus also exhibits zeros at equidistant intervals  $1/\Delta f$.
Line 117:		Line 116:
	*The trapezoidal low-pass with height  $K$  and the two corner frequencies  $f_1$  and  $f_2$  reads:		*The trapezoidal low-pass with height  $K$  and the two corner frequencies  $f_1$  and  $f_2$  reads:
	:$$H(f) = \left\{ \begin{array}{l} \hspace{0.25cm}K \\ K\cdot \frac{f_2-\|f\|}{f_2-f_1} \\ \hspace{0.25cm} 0 \\ \end{array} \right.\quad \quad \begin{array}{{20}c} {\rm{for}} \\ {\rm{for}} \\ {\rm{for}} \\ \end{array}\begin{array}{{20}c} {\left\| \hspace{0.05cm} f\hspace{0.05cm} \right\| \le f_1,} \\ {f_1\le \left\| \hspace{0.05cm}f\hspace{0.05cm} \right\| \le f_2,} \\ {\left\|\hspace{0.05cm} f \hspace{0.05cm} \right\| \ge f_2.} \\ \end{array}$$		:$$H(f) = \left\{ \begin{array}{l} \hspace{0.25cm}K \\ K\cdot \frac{f_2-\|f\|}{f_2-f_1} \\ \hspace{0.25cm} 0 \\ \end{array} \right.\quad \quad \begin{array}{{20}c} {\rm{for}} \\ {\rm{for}} \\ {\rm{for}} \\ \end{array}\begin{array}{{20}c} {\left\| \hspace{0.05cm} f\hspace{0.05cm} \right\| \le f_1,} \\ {f_1\le \left\| \hspace{0.05cm}f\hspace{0.05cm} \right\| \le f_2,} \\ {\left\|\hspace{0.05cm} f \hspace{0.05cm} \right\| \ge f_2.} \\ \end{array}$$

	~~::'''Korrektur:''' Warum sind in dieser Gleichung die Zeilenabstände rechts kleiner als lins~~

	*For the equivalent bandwidth  $($equal-area rectangle$)$  the following applies:  $\Delta f = f_1+f_2$.		*For the equivalent bandwidth  $($equal-area rectangle$)$  the following applies:  $\Delta f = f_1+f_2$.