Applets:Frequency & Impulse Responses - Revision history

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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"

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"

Guenter at 14:17, 30 June 2023

2023-06-30T14:17:25Z

Guenter at 14:06, 13 March 2023

2023-03-13T14:06:49Z

Guenter at 13:54, 13 March 2023

2023-03-13T13:54:12Z

Guenter at 13:47, 13 March 2023

2023-03-13T13:47:31Z

Guenter at 13:46, 13 March 2023

2023-03-13T13:46:38Z

Guenter at 13:35, 13 March 2023

2023-03-13T13:35:02Z

Hwang at 19:27, 22 February 2023

2023-02-22T19:27:28Z

Hwang at 19:24, 22 February 2023

2023-02-22T19:24:05Z

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 This interactive calculation tool was designed and implemented at the&nbsp; [https://www.ei.tum.de/en/lnt/home/ $\text{Institute for Communications Engineering}$]&nbsp; at the&nbsp; [https://www.tum.de/en $\text{Technical University of Munich}$].
-*The first version was created in 2005 by&nbsp; [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|&raquo;Ji Li&laquo;]]&nbsp; as part of her diploma thesis with&nbsp; “FlashMX – Actionscript”&nbsp; (Supervisor:&nbsp; [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29| &raquo;Günter Söder&laquo;]]).
+*The first version was created in 2005 by&nbsp; [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|&raquo;Ji Li&laquo;]]&nbsp; as part of her diploma thesis with&nbsp; “FlashMX – Actionscript”&nbsp; (Supervisor:&nbsp; [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29| &raquo;Günter Söder&laquo;]]).
 *In 2017 the program was redesigned by&nbsp; [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#David_Jobst_.28Ingenieurspraxis_Math_2017.29|&raquo;David Jobst&laquo;]]&nbsp; (Ingenieurspraxis_Math,&nbsp; Supervisor:&nbsp; [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|&raquo;Tasnád Kernetzky&laquo;]] ) via "HTML5".

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 *The first version was created in 2005 by&nbsp; [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|&raquo;Ji Li&laquo;]]&nbsp; as part of her diploma thesis with&nbsp; “FlashMX – Actionscript”&nbsp; (Supervisor:&nbsp; [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29| &raquo;Günter Söder&laquo;]]).
-*In 2017 the program was redesigned by&nbsp; [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#David_Jobst_.28Ingenieurspraxis_Math_2017.29|&raquo;David Jobst&laquo;]]&nbsp; (Ingenieurspraxis_Math,&nbsp; Supervisor:&nbsp; [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28at_L.C3.9CT_since_2014.29|&raquo;Tasnád Kernetzky&laquo;]] ) via "HTML5".
+*In 2017 the program was redesigned by&nbsp; [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#David_Jobst_.28Ingenieurspraxis_Math_2017.29|&raquo;David Jobst&laquo;]]&nbsp; (Ingenieurspraxis_Math,&nbsp; Supervisor:&nbsp; [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28at_L.C3.9CT_from_2014-2022.29|&raquo;Tasnád Kernetzky&laquo;]] ) via "HTML5".
 *Last revision and English version 2020 by&nbsp; [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|&raquo;Carolin Mirschina&laquo;]]&nbsp; in the context of a working student activity.&nbsp;

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 <br>
-Real and symmetric low-passes&nbsp; $H(f)$&nbsp; and the corresponding impulse responses&nbsp; $h(t)$&nbsp; are shown,&nbsp; namely
+Real and symmetric low-pass frquency responses&nbsp; $H(f)$&nbsp; and the corresponding impulse responses&nbsp; $h(t)$&nbsp; are shown,&nbsp; namely
 #Gaussian low-pass,
 #rectangular low-pass,
 #triangular low-pass,
 #trapezoidal low-pass,
-#cosine rolloff low-pass &nbsp; &rArr; &nbsp; "raised-cosine low-pass",
+#cosine rolloff low-pass &nbsp; &rArr; &nbsp; &raquo;raised-cosine low-pass&laquo;,
-#cosine rolloff squared low-pass &nbsp; &rArr; &nbsp; "cosine-square low-pass".
+#cosine rolloff squared low-pass &nbsp; &rArr; &nbsp; &raquo;cosine-square low-pass&laquo;.

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 This interactive calculation tool was designed and implemented at the&nbsp; [https://www.ei.tum.de/en/lnt/home/ $\text{Institute for Communications Engineering}$]&nbsp; at the&nbsp; [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&nbsp; [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Ji_Li_.28Bachelorarbeit_EI_2003.2C_Diplomarbeit_EI_2005.29|&raquo;Ji Li&laquo;]]&nbsp; as part of her diploma thesis with&nbsp; “FlashMX – Actionscript”&nbsp; (Supervisor:&nbsp; [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29| &raquo;Günter Söder&laquo;]]).
 *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&nbsp; [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|$Carolin Mirschina]]&nbsp; in the context of a working student activity.&nbsp;
+*In 2017 the program was redesigned by&nbsp; [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#David_Jobst_.28Ingenieurspraxis_Math_2017.29|&raquo;David Jobst&laquo;]]&nbsp; (Ingenieurspraxis_Math,&nbsp; Supervisor:&nbsp; [[Biographies_and_Bibliographies/LNTwww_members_from_LÜT#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28at_L.C3.9CT_since_2014.29|&raquo;Tasnád Kernetzky&laquo;]] ) via "HTML5".
 ==Once again: Open Applet in new Tab==
 {{LntAppletLinkEnDe|frequImpResp_en|frequImpResp}}

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 {{BlaueBox|TEXT=
 '''(2)''' &nbsp; Leave the settings unchanged.&nbsp; Which low-pass&nbsp; $H(f)$&nbsp; fulfills the first or the second Nyquist criterion?<br>&nbsp; &nbsp; &nbsp; &nbsp; Here&nbsp; $H(f)$&nbsp; denotes the total frequency response of transmitter,
-channel and '''Korrektur'''receiver''' filter.}}
+channel and recettion filter.}}
 * First Nyquist criterion:&nbsp; The impulse response&nbsp; $h(t)$&nbsp; must have equidistant zero crossings at the (normalized) times&nbsp; $t = 1,\ 2$, ...  <br>
-* The impulse response&nbsp; $h(t) = {\rm si}(\pi \cdot \delta f \cdot t)$&nbsp; '''KORREKTUR: sinc?''' of the rectangular low-pass filter fulfils this criterion  with&nbsp; $\Delta f = 1$. <br>
+* The impulse response&nbsp; $h(t) = {\rm sinc}(\delta f \cdot t)$&nbsp; of the rectangular low-pass filter fulfils this criterion  with&nbsp; $\Delta f = 1$. <br>
 * In contrast, the first Nyquist criterion is never fulfilled for the Gaussian low-pass and there is always impulse interference.<br>
 * The second Nyquist criterion is met by neither the rectangular low-pass nor the Gaussian low-pass.
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   {{BlaueBox|TEXT=
-'''(3)''' &nbsp; Compare the&nbsp; &raquo;<b>red rectangular low-pass</b>&laquo;&nbsp; $(K_1 = 0.5, \Delta f_1 = 2)$&nbsp; to the&nbsp; &raquo;<b>blue rectangular low-pass</b>&laquo;&nbsp; $(K_2 = 1, \Delta f_2 = 1)$.&nbsp; Then vary&nbsp; $\Delta f_1$&nbsp; between&nbsp; $2$&nbsp; and&nbsp; $0.5$.}}
+'''(3)''' &nbsp; Compare the&nbsp; &raquo;<b>red rectangular low-pass</b>&laquo;&nbsp; $(K_1 = 0.5, \Delta f_1 = 2)$&nbsp; to the&nbsp; &raquo;<b>blue rectangular low-pass</b>&laquo;&nbsp; $(K_2 = 1, \Delta f_2 = 1)$.&nbsp; Then vary&nbsp; $\Delta f_1$&nbsp; between&nbsp; $2$&nbsp; ...&nbsp; $0.5$.}}
 * With&nbsp; $\Delta f_1 = 2$&nbsp; the zeros of&nbsp; $h_1(t)$&nbsp; are multiples&nbsp; of&nbsp; $0.5$ &nbsp; &rArr; &nbsp; $h_1(t)$ &nbsp; will decay twice as fast as&nbsp; $h_2(t)$. <br>
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   {{BlaueBox|TEXT=
-'''(4)''' &nbsp; Compare the&nbsp; &raquo;<b>red trapezoidal low-pass</b>&laquo;&nbsp; $(K_1 = 1, \  \Delta f_1 = 1, \ r_1 = 0.5)$&nbsp; with the&nbsp; &raquo;<b>blue rectangular low-pass</b>&laquo; &nbsp; $(K_2 = 1, \  \Delta f_2 = 1)$.&nbsp; $r_1$&nbsp; varies between&nbsp; $0$&nbsp; and&nbsp; $1$. }}
+'''(4)''' &nbsp; Compare the&nbsp; &raquo;<b>red trapezoidal low-pass</b>&laquo;&nbsp; $(K_1 = 1, \  \Delta f_1 = 1, \ r_1 = 0.5)$&nbsp; with&nbsp; &raquo;<b>blue rectangular low-pass</b>&laquo; &nbsp; $(K_2 = 1, \  \Delta f_2 = 1)$.&nbsp; $r_1$&nbsp; varies between&nbsp; $0$&nbsp; ...&nbsp; $1$. }}
 * With&nbsp; $r_1 = 0.5$&nbsp; the&nbsp; followers/precursors of&nbsp; $h_1(t)$&nbsp; 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&nbsp; [https://www.ei.tum.de/en/lnt/home/ $\text{Institute for Communications Engineering}$]&nbsp; at the&nbsp; [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&nbsp; [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|$\text{Carolin Mirschina}$]]&nbsp; in the context of a working student activity.&nbsp; Translation using DEEPL.com.
+*Last revision and English version 2020 by&nbsp; [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|$Carolin Mirschina]]&nbsp; in the context of a working student activity.&nbsp;
 ==Once again: Open Applet in new Tab==
 {{LntAppletLinkEnDe|frequImpResp_en|frequImpResp}}

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 :$$H(f) = \left\{ \begin{array}{l} \hspace{0.25cm}K  \\  K\cdot \cos^2\Big(\frac{|f|-f_1}{f_2-f_1}\cdot {\pi}/{2}\Big) \\ \hspace{0.25cm} 0 \\  \end{array} \right.\quad \quad \begin{array}{*{20}c}   {\rm{for}}  \\   {\rm{for}}  \\   {\rm{for}}  \\ \end{array}\begin{array}{*{20}c}   {\left| \hspace{0.05cm} f\hspace{0.05cm} \right| \le f_1,}  \\   {f_1\le \left| \hspace{0.05cm}f\hspace{0.05cm} \right| \le f_2,}  \\   {\left|\hspace{0.05cm} f \hspace{0.05cm} \right| \ge f_2.}  \\ \end{array}$$
 *For the equivalent bandwidth&nbsp; $($equal-area rectangle$)$&nbsp; the following applies:&nbsp; $\Delta f = f_1+f_2$.
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 *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&nbsp; ${\rm si}$ '''KORREKTUR: sinc?''' function,&nbsp; $h(t)=0$&nbsp; for all multiples of&nbsp; $T=1/\Delta f$ &nbsp; &rArr; &nbsp;  The equidistant zero crossings of the cosine rolloff low-pass are preserved.
 *Due to the expression in parentheses,&nbsp; $h(t)$ &nbsp;now shows further zero crossings at&nbsp; $t=\pm1.5 T$,&nbsp; $\pm2.5 T$,&nbsp; $\pm3.5 T$, ...&nbsp;
 *For&nbsp; $t=\pm T/2$&nbsp; the impulse response has the value&nbsp; $K\cdot \Delta f/2$.

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 :$$H(f) = \left\{ \begin{array}{l} \hspace{0.25cm}K\cdot \Big(1-\frac{|f|}{\Delta f}\Big)  \\ \hspace{0.25cm} 0 \\  \end{array} \right.\quad \quad \begin{array}{*{20}c}   {\rm{for}}  \\    {\rm{for}}  \\ \end{array}\begin{array}{*{20}c}   {\left| \hspace{0.05cm} f\hspace{0.05cm} \right| < \Delta f,}  \\   {\left| \hspace{0.05cm}f\hspace{0.05cm} \right| \ge \Delta f.}   \\ \end{array}$$
 *The absolute physical bandwidth&nbsp; $B$ &nbsp; &rArr; &nbsp;[positive frequencies only] &nbsp; is also equal&nbsp; $\Delta f$,&nbsp; thus is as large as for the rectangular low-pass.
 *For the impulse response&nbsp; $h(t)$&nbsp; one obtains according to the second Fourier transform:
-:$$h(t)=K\cdot \Delta f \cdot {\rm si}^2(\pi\cdot \Delta f \cdot t) \quad \text{with} \quad {\rm si}(x)={\sin(x)}/{x}.$$
+:$$h(t)=K\cdot \Delta f \cdot {\rm si}^2(\pi\cdot \Delta f \cdot t) \quad \text{with} \quad {\rm si}(x)={\sin(x)}/{x}={\rm sinc}(x/\pi).$$
 *$H(f)$&nbsp; can be represented as a convolution of two rectangular functions&nbsp; $($each with width&nbsp; $\Delta f)$.
-*It follows:&nbsp; $h(t)$&nbsp; contains instead of the&nbsp; ${\rm si}$&ndash;function the&nbsp; ${\rm si}^2$&ndash;function. '''KORREKTUR: sinc?'''
+*It follows:&nbsp; $h(t)$&nbsp; contains instead of the&nbsp; ${\rm si}$&ndash;function the&nbsp; ${\rm si}^2$&ndash;function.
 *$h(t)$&nbsp; thus also exhibits zeros at equidistant intervals&nbsp; $1/\Delta f$.
@@ Line 117: / Line 116: @@
 *The trapezoidal low-pass with height&nbsp; $K$&nbsp; and the two corner frequencies&nbsp; $f_1$&nbsp; and&nbsp; $f_2$&nbsp; reads:
 :$$H(f) = \left\{ \begin{array}{l} \hspace{0.25cm}K  \\  K\cdot \frac{f_2-|f|}{f_2-f_1} \\ \hspace{0.25cm} 0 \\  \end{array} \right.\quad \quad \begin{array}{*{20}c}   {\rm{for}}  \\   {\rm{for}}  \\   {\rm{for}}  \\ \end{array}\begin{array}{*{20}c}   {\left| \hspace{0.05cm} f\hspace{0.05cm} \right| \le f_1,}  \\   {f_1\le \left| \hspace{0.05cm}f\hspace{0.05cm} \right| \le f_2,}  \\   {\left|\hspace{0.05cm} f \hspace{0.05cm} \right| \ge f_2.}  \\ \end{array}$$
 *For the equivalent bandwidth&nbsp; $($equal-area rectangle$)$&nbsp; the following applies:&nbsp; $\Delta f = f_1+f_2$.