Applets:Bessel Functions of the First Kind - Revision history

Noah at 20:20, 26 March 2023

2023-03-26T20:20:48Z

Hwang at 16:07, 23 January 2023

2023-01-23T16:07:42Z

Hwang at 16:10, 18 January 2023

2023-01-18T16:10:29Z

Guenter at 17:23, 11 January 2023

2023-01-11T17:23:58Z

Javier: Text replacement - "”" to """

2021-05-28T14:47:33Z

Text replacement - "”" to """

Javier: Text replacement - "„" to """

2021-05-28T14:41:34Z

Text replacement - "„" to """

Guenter at 10:22, 25 February 2021

2021-02-25T10:22:01Z

Guenter at 13:50, 23 January 2021

2021-01-23T13:50:43Z

Guenter at 13:49, 23 January 2021

2021-01-23T13:49:34Z

Guenter at 13:48, 23 January 2021

2021-01-23T13:48:15Z

@@ Line 21: / Line 21: @@
 \cdot {\rm J}_n (x)= 0. $$
-This is an ordinary second-order linear differential equation.&nbsp; The parameter&nbsp; $ n $&nbsp; is usually integer, also in this program.&nbsp; These mathematical functions, introduced by&nbsp; [https://en.wikipedia.org/wiki/Friedrich_Bessel Friedrich Wilhelm Bessel]&nbsp; in 1844, can also be represented in closed form as integrals:
+This is an ordinary second-order linear differential equation.&nbsp; The parameter&nbsp; $ n $&nbsp; is usually integer, also in this program.&nbsp; These mathematical functions, introduced by&nbsp; [https://en.wikipedia.org/wiki/Friedrich_Bessel "Friedrich Wilhelm Bessel"]&nbsp; in 1844, can also be represented in closed form as integrals:
 :$${\rm J}_n (x) = \frac{1}{2\pi}\cdot \int_{-\pi}^{+\pi} {{\rm e}^{\hspace{0.05cm}{\rm j}\hspace{0.05cm}\cdot \hspace{0.05cm}[\hspace{0.05cm}x \hspace{0.05cm}\cdot \hspace{0.05cm}\sin(\alpha) -\hspace{0.05cm} n \hspace{0.05cm}\cdot \hspace{0.05cm}\alpha \hspace{0.05cm}]}}\hspace{0.1cm}{\rm d}\alpha
@@ Line 54: / Line 54: @@
 {{GraueBox|TEXT=
-$\text{Example (B):}$&nbsp; &nbsp;For the spectrum of the analytic signal, the following applies to the case&nbsp; "Phase Modulation of a sinusoidal signal":
+$\text{Example (B):}$&nbsp; &nbsp;For the spectrum of the analytic signal, the following applies to the case&nbsp; "phase modulation of a sinusoidal signal":
 [[File:Mod_T_3_1_S4_version2.png|right|frame|Phase Modulation: &nbsp; Spectrum of the Analytic Signal]]
 :$$S_{\rm +}(f) = A_{\rm T} \cdot \sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta) \cdot \delta \big[f - (f_{\rm T}+ n \cdot f_{\rm N})\big]\hspace{0.05cm}.$$
@@ Line 99: / Line 99: @@
-The task of the&nbsp; [[Signal_Representation/Spectrum_Analysis|Spectral Analysis]]&nbsp; is to limit the influence of the spectral leakage effect by providing suitable window functions.
+The task of the&nbsp; [[Signal_Representation/Spectrum_Analysis|"Spectrum Analysis"]]&nbsp; is to limit the influence of the spectral leakage effect by providing suitable window functions.
-Such a window function provides, for example, the Kaiser&ndash;Bessel window &nbsp; &rArr; &nbsp; see section&nbsp; [[Signal_Representation/Spectrum_Analysis#Spezielle_Fensterfunktionen|Special Window Functions]].&nbsp; Its time-discrete window function with the Bessel function zero order &nbsp; &rArr; &nbsp; ${\rm J}_0(x)$, the parameter&nbsp; $\alpha=3.5$&nbsp; and the window length&nbsp; $N$:
+Such a window function provides, for example, the Kaiser&ndash;Bessel window &nbsp; &rArr; &nbsp; see section&nbsp; [[Signal_Representation/Spectrum_Analysis#Special_window_functions|"Special Window Functions"]].&nbsp; Its time-discrete window function with the Bessel function zero order &nbsp; &rArr; &nbsp; ${\rm J}_0(x)$, the parameter&nbsp; $\alpha=3.5$&nbsp; and the window length&nbsp; $N$:
 :$$w_\nu = \frac{ {\rm J}_0\big(\pi \cdot \alpha \cdot \sqrt{1 - (2\nu/N)^2}\big)}{ {\rm J}_0\big(\pi \cdot \alpha \big)}.$$
-In the section&nbsp; [[Signal_Representation/Spectrum_Analysis#G.C3.BCtekriterien_von_Fensterfunktionen|Quality Criteria of Window Functions]]&nbsp; the parameters of the Kaiser-Bessel window are given.
+In the section&nbsp; [[Signal_Representation/Spectrum_Analysis#Quality_criteria_of_window_functions|"Quality Criteria of Window Functions"]]&nbsp; the parameters of the Kaiser-Bessel window are given.
 *The large "Minimum Distance between the main lobe and side lobes" and the desired small "maximum Scaling Error" are favourable.
 *Due to the very large "Equivalent Noise Width" the Kaiser-Bessel window cuts in the most important comparison criterion "Maximum Process Loss" but worse than the established Hamming and Hanning windows.}}
@@ Line 111: / Line 111: @@
 $\text{Example (D):} \hspace{0.5cm} \text{Rice Fading Channel Model}$
-The&nbsp; [[Mobile_Communications/Wahrscheinlichkeitsdichte_des_Rayleigh%E2%80%93Fadings#Allgemeine_Beschreibung_des_Mobilfunkkanals| Rayleigh distribution]]&nbsp; describes the mobile channel on the assumption that there is no direct path and thus the multiplicative factor&nbsp; $z(t) = x(t) + {\rm j} \cdot y(t)$&nbsp; is composed solely of diffusely scattered components.
+The&nbsp; [[Mobile_Communications/Probability_Density_of_Rayleigh_Fading#A_very_general_description_of_the_mobile_communication_channel| "Rayleigh distribution"]]&nbsp; describes the mobile channel on the assumption that there is no direct path and thus the multiplicative factor&nbsp; $z(t) = x(t) + {\rm j} \cdot y(t)$&nbsp; is composed solely of diffusely scattered components.
-In the case of a direct component&nbsp; (<i>Line of Sight</i>, LoS) one has to add  in the model DC components&nbsp; $x_0$&nbsp; and/or&nbsp; $y_0$&nbsp; for the mean-free Gaussian processes&nbsp; $x(t)$&nbsp; and&nbsp; $y(t)$&nbsp;:
+In the case of a direct component&nbsp; (<i>Line of Sight</i>, LoS) one has to add  in the model DC components&nbsp; $x_0$&nbsp; and/or&nbsp; $y_0$&nbsp; for the zero mean Gaussian processes&nbsp; $x(t)$&nbsp; and&nbsp; $y(t)$&nbsp;:
 [[File:EN_Mob_T_1_4_S1.png|right|frame|Rice fading channel model|class=fit]]
@@ Line 140: / Line 140: @@
 $\text{Example (B)}$&nbsp;  has already been shown that the angle modulation of a harmonic oscillation of the frequency $f_{\rm N}$&nbsp; leads to a line spectrum.&nbsp; The spectral lines are around the carrier frequency $f_{\rm T}$&nbsp; at&nbsp; $f_{\rm T} + n \cdot f_{\rm N}$&nbsp; with&nbsp; $n \in \{ \ \text{...}, -2, -1, \ 0, +1, +2, \text{...} \ \}$.&nbsp; The weights of the Dirac delta lines are&nbsp; ${\rm J }_n(\eta)$, depending on the modulation index&nbsp; $\eta$.
-The graph shows the magnitude spectrum&nbsp; $\vert S_{\rm +}(f) \vert$&nbsp; of the analytic signal in phase modulation (PM) and frequency modulation (FM), two different forms of angle modulation (WM).&nbsp; Bessel lines with values less than&nbsp; $0.03$&nbsp; are neglected in both cases.
+The graph shows the magnitude spectrum&nbsp; $\vert S_{\rm +}(f) \vert$&nbsp; of the analytic signal in phase modulation (PM) and frequency modulation (FM), two different forms of angle modulation.&nbsp; Bessel lines with values less than&nbsp; $0.03$&nbsp; are neglected in both cases.
 [[File:P_ID1095__Mod_T_3_2_S4_neu.png|right|frame|Discrete spectra with phase modulation (left) and frequency modulation (right)]]

@@ Line 103: / Line 103: @@
 Such a window function provides, for example, the Kaiser&ndash;Bessel window &nbsp; &rArr; &nbsp; see section&nbsp; [[Signal_Representation/Spectrum_Analysis#Spezielle_Fensterfunktionen|Special Window Functions]].&nbsp; Its time-discrete window function with the Bessel function zero order &nbsp; &rArr; &nbsp; ${\rm J}_0(x)$, the parameter&nbsp; $\alpha=3.5$&nbsp; and the window length&nbsp; $N$:
 :$$w_\nu = \frac{ {\rm J}_0\big(\pi \cdot \alpha \cdot \sqrt{1 - (2\nu/N)^2}\big)}{ {\rm J}_0\big(\pi \cdot \alpha \big)}.$$
-On the page&nbsp; [[Signal_Representation/Spectrum_Analysis#G.C3.BCtekriterien_von_Fensterfunktionen|Quality Criteria of Window Functions]]&nbsp; the parameters of the Kaiser-Bessel window are given.
+In the section&nbsp; [[Signal_Representation/Spectrum_Analysis#G.C3.BCtekriterien_von_Fensterfunktionen|Quality Criteria of Window Functions]]&nbsp; the parameters of the Kaiser-Bessel window are given.
 *The large "Minimum Distance between the main lobe and side lobes" and the desired small "maximum Scaling Error" are favourable.
 *Due to the very large "Equivalent Noise Width" the Kaiser-Bessel window cuts in the most important comparison criterion "Maximum Process Loss" but worse than the established Hamming and Hanning windows.}}

@@ Line 138: / Line 138: @@
 $\text{Example (E):} \hspace{0.5cm} \text{Analysis of the Frequency Spectrum of Frequency Modulated Signals}$
-$\text{Example (B)}$&nbsp;  has already been shown that the angle modulation of a harmonic oscillation of the frequency $f_{\rm N}$&nbsp; leads to a line spectrum.&nbsp; The spectral lines are around the carrier frequency $f_{\rm T}$&nbsp; at&nbsp; $f_{\rm T} + n \cdot f_{\rm N}$&nbsp; with&nbsp; $n \in \{ \ \text{...}, -2, -1, \ 0, +1, +2, \text{...} \ \}$.&nbsp; The weights of the Dirac lines are&nbsp; ${\rm J }_n(\eta)$, depending on the modulation index&nbsp; $\eta$.
+$\text{Example (B)}$&nbsp;  has already been shown that the angle modulation of a harmonic oscillation of the frequency $f_{\rm N}$&nbsp; leads to a line spectrum.&nbsp; The spectral lines are around the carrier frequency $f_{\rm T}$&nbsp; at&nbsp; $f_{\rm T} + n \cdot f_{\rm N}$&nbsp; with&nbsp; $n \in \{ \ \text{...}, -2, -1, \ 0, +1, +2, \text{...} \ \}$.&nbsp; The weights of the Dirac delta lines are&nbsp; ${\rm J }_n(\eta)$, depending on the modulation index&nbsp; $\eta$.
 The graph shows the magnitude spectrum&nbsp; $\vert S_{\rm +}(f) \vert$&nbsp; of the analytic signal in phase modulation (PM) and frequency modulation (FM), two different forms of angle modulation (WM).&nbsp; Bessel lines with values less than&nbsp; $0.03$&nbsp; are neglected in both cases.

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-{{LntAppletLinkEn|besselFuns_en}}
+{{LntAppletLinkEnDe|besselFuns_en|besselFuns}}
 ==Applet Description==
 <br>
@@ Line 186: / Line 186: @@
 ==Once again: Open Applet in new Tab==
-{{LntAppletLinkEn|besselFuns_en}}
+{{LntAppletLinkEnDe|besselFuns_en|besselFuns}}

@@ Line 105: / Line 105: @@
 On the page&nbsp; [[Signal_Representation/Spectrum_Analysis#G.C3.BCtekriterien_von_Fensterfunktionen|Quality Criteria of Window Functions]]&nbsp; the parameters of the Kaiser-Bessel window are given.
 *The large "Minimum Distance between the main lobe and side lobes" and the desired small "maximum Scaling Error" are favourable.
-*Due to the very large "Equivalent Noise Width&rdquo; the Kaiser-Bessel window cuts in the most important comparison criterion "Maximum Process Loss" but worse than the established Hamming and Hanning windows.}}
+*Due to the very large "Equivalent Noise Width" the Kaiser-Bessel window cuts in the most important comparison criterion "Maximum Process Loss" but worse than the established Hamming and Hanning windows.}}
@@ Line 161: / Line 161: @@
 &nbsp; &nbsp; '''(D)''' &nbsp; &nbsp; Variation of the graphic representation
-$\hspace{1.5cm}$"$+$&rdquo; (Enlarge),
+$\hspace{1.5cm}$"$+$" (Enlarge),
-$\hspace{1.5cm}$ "$-$&rdquo; (Decrease)
+$\hspace{1.5cm}$ "$-$" (Decrease)
-$\hspace{1.5cm}$ "$\rm o$&rdquo; (Reset to default)
+$\hspace{1.5cm}$ "$\rm o$" (Reset to default)
-$\hspace{1.5cm}$ "$\leftarrow$&rdquo; (pushed to the left),  usw.
+$\hspace{1.5cm}$ "$\leftarrow$" (pushed to the left),  usw.
 &nbsp; &nbsp; '''(E)''' &nbsp; &nbsp; Selection of the abscissa value&nbsp; $x_1$&nbsp; for the left numeric output
@@ Line 181: / Line 181: @@
 This interactive applet was designed and realized  at the&nbsp; [https://www.ei.tum.de/en/lnt/home/ Institute for Communications Engineering]&nbsp; at the&nbsp; [https://www.tum.de/en Technical University of Munich].
-*The original version was created in 2005 by&nbsp; [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Markus_Elsberger_.28Diplomarbeit_LB_2006.29|Markus Elsberger]]&nbsp; and&nbsp; [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Slim_Lamine_.28Studienarbeit_EI_2006.29|Slim Lamine]] as part of thesis using  "FlashMX&ndash;Actionscript&rdquo; (Supervisor:&nbsp; [[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&nbsp; [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Markus_Elsberger_.28Diplomarbeit_LB_2006.29|Markus Elsberger]]&nbsp; and&nbsp; [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Slim_Lamine_.28Studienarbeit_EI_2006.29|Slim Lamine]] as part of thesis using  "FlashMX&ndash;Actionscript" (Supervisor:&nbsp; [[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 "HTML5&rdquo; by&nbsp; [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]]&nbsp; as part of her Bachelor's thesis (Supervisor:&nbsp; [[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&nbsp; [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Studierende#Xiaohan_Liu_.28Bachelorarbeit_2018.29|Xiaohan Liu]]&nbsp; as part of her Bachelor's thesis (Supervisor:&nbsp; [[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==
 {{LntAppletLinkEn|besselFuns_en}}

← Older revision		Revision as of 13:50, 23 January 2021
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−	~~[https://www.lntwww.de/Applets:Besselfunktionen_erster_Art '''German description''']~~

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

← Older revision		Revision as of 13:48, 23 January 2021
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−	{{LntAppletLinkEn\|besselFuns_en}}	+	{{LntAppletLinkEn\|besselFuns_en}}         [https://en.lntwww.de/Applets:Bessel_Functions_of_the_First_Kind '''with German desciption''']

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