Applets:Generation of Walsh functions - Versionsgeschichte

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Textersetzung - „Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29“ durch „Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_von_1974-2024.29“

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Guenter: Die Seite wurde neu angelegt: „== program description==
This applet allows to display the Hadamard matrices $\mathbf{H}_J$ for the construction of the Walsh functions …“

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Die Seite wurde neu angelegt: „== program description== This applet allows to display the Hadamard matrices $\mathbf{H}_J$ for the construction of the Walsh functions …“

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== program description==
 
This applet allows to display the Hadamard matrices  $\mathbf{H}_J$  for the construction of the Walsh functions  $w_j$. The factor  $J$  of the band spreading as well as the selection of the individual Walsh functions (by blue bordering the rows of the matrix) can be changed.

==Theoretical background==
 
===Application===
 
The  '''Walsh functions'''  are a group of periodic orthogonal functions. Their application in digital signal processing is mainly in the use for band spreading in CDMA systems, for example the mobile radio standard UMTS.
*Due to their orthogonal properties and the favourable PKKF conditions (periodic KKF), the Walsh functions represent optimal spreading sequences for a distortion-free channel and a synchronous CDMA system. If you take any two lines and form the correlation (averaging over the products), the PKKF value is always zero.
*In asynchronous operation (example:   uplink of a mobile radio system) or de-orthogonalization due to multipath propagation, Walsh functions alone are not necessarily suitable for band spreading - see   [Tasks:5.4_Walsh Functions_(PKKF,_PAKF)|Task 5.4]].
*In terms of PAKF (periodic AKF) these sequences are less good:   Each individual Walsh function has a different PAKF and each individual PAKF is less good than a comparable PN sequence. That means:   The synchronization is more difficult with Walsh functions than with PN sequences.

=== Construction==
 
The construction of Walsh functions can be done recursively using the '''Hadamard matrices''. A Hadamard matrix $\mathbf{H}_J$ of order $J$ is a $J\times J$ matrix, which contains line by line the $\pm 1$ weights of the Walsh sequences. The orders of the Hadamard matrices are fixed to powers of two, i.e. $J = 2^G$ applies to a natural number $G$. Starting from $\mathbf{H}_1 = [+1]$ and

\begin{equation}
\mathbf{H}_2 =
\left[ \begin{array}{rr}
+1 & +1\\
+1 & -1 \\
\end [array}\right]
\end{equation}
the following relationship applies to the generation of further Hadamard matrices:
\begin{equation}
\mathbf{H}_{2N} =
\left[ \begin{array}{rr}
+\mathbf{H}_N & +\mathbf{H}_N\\\\
+\mathbf{H}_N & -\mathbf{H}_N \\\\
\end [array}\right]
\end{equation}
 
{{{{GrayBox|TEXT=
$\text{example:}$  The graphic shows the Hadamard matrix  $\mathbf H_8$  (right) and the spreading sequences which can be constructed with it  $J -1$ .
[[File:P_ID1882__Mod_T_5_3_S7_new.png|right|frame| Walsh spreading sequences  $(J = 8)$  and Hadamard matrix  $\mathbf H_8$ ]]
*$J - 1$ because the unspread sequence  $w_0(t)$  is usually not used.
*Please note the color assignment between the lines of the Hadamard matrix and the spreading sequences  $w_j(t)$.
*The matrix  $\mathbf H_4$  is highlighted in yellow.}}}
 

==To use the applet==

 
[[File:Walsh Handling.png|right|550px]]

    '''(A)'''     Selection of the factor for band spreading as power of two of $G$

    '''(B)'''     Selection of the respective Walsh function $w_j$
 

== About the authors==
This interactive calculation tool was desi

Translated with www.DeepL.com/Translator (free version)

@@ Zeile 60: / Zeile 60: @@
 This interactive calculation tool was designed and realized at the&nbsp; [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik]&nbsp; $\rm (LNT)$&nbsp; of the&nbsp; [https://www.tum.de/ Technical University of Munich]&nbsp; $\rm (TUM)$.
-*The first German version was created in 2007 by&nbsp;  [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Thomas_Gro.C3.9Fer_.28Diplomarbeit_LB_2006.2C_danach_freie_Mitarbeit_bis_2010.29|Thomas Großer]]&nbsp;&nbsp; in the context of his diploma thesis with "FlashMX&ndash;Actionscript"&nbsp;  (Supervisor:&nbsp; [[Biografien_und_Bibliografien/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 German version was created in 2007 by&nbsp;  [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Thomas_Gro.C3.9Fer_.28Diplomarbeit_LB_2006.2C_danach_freie_Mitarbeit_bis_2010.29|Thomas Großer]]&nbsp;&nbsp; in the context of his diploma thesis with "FlashMX&ndash;Actionscript"&nbsp;  (Supervisor:&nbsp; [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_von_1974-2024.29|Günter Söder]]).
 *2018/2019 the applet was converted on "HTML5" and redesigned by&nbsp;  [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|Carolin Mirschina]]&nbsp; (Engineering practice, supervisor:&nbsp; [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_LÜT-Angehörige#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28bei_L.C3.9CT_von_2014-2022.29|Tasnád Kernetzky]] ).
-*2020 this English version was made by&nbsp; [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|Carolin Mirschina]]&nbsp; (working student) and&nbsp; [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]].&nbsp; Translation using "www.DeepL.com/Translator" (free version).
+*2020 this English version was made by&nbsp; [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|Carolin Mirschina]]&nbsp; (working student) and&nbsp; [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_von_1974-2024.29|Günter Söder]].&nbsp; Translation using "www.DeepL.com/Translator" (free version).
 ==Call the applet again==
 <br>
 {{LntAppletLinkEn|walsh_en}}

@@ Zeile 61: / Zeile 61: @@
 *The first German version was created in 2007 by&nbsp;  [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Thomas_Gro.C3.9Fer_.28Diplomarbeit_LB_2006.2C_danach_freie_Mitarbeit_bis_2010.29|Thomas Großer]]&nbsp;&nbsp; in the context of his diploma thesis with "FlashMX&ndash;Actionscript"&nbsp;  (Supervisor:&nbsp; [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
-*2018/2019 the applet was converted on "HTML5" and redesigned by&nbsp;  [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|Carolin Mirschina]]&nbsp; (Engineering practice, supervisor:&nbsp; [[Biografien_und_Bibliografien/Beteiligte_der_Professur_Leitungsgebundene_%C3%9Cbertragungstechnik#Tasn.C3.A1d_Kernetzky.2C_M.Sc._.28bei_L.C3.9CT_seit_2014.29|Tasnád Kernetzky]] ).
+*2018/2019 the applet was converted on "HTML5" and redesigned by&nbsp;  [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|Carolin Mirschina]]&nbsp; (Engineering practice, supervisor:&nbsp; [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_LÜT-Angehörige#Dr.-Ing._Tasn.C3.A1d_Kernetzky_.28bei_L.C3.9CT_von_2014-2022.29|Tasnád Kernetzky]] ).
 *2020 this English version was made by&nbsp; [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|Carolin Mirschina]]&nbsp; (working student) and&nbsp; [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]].&nbsp; Translation using "www.DeepL.com/Translator" (free version).
 ==Call the applet again==
 <br>
 {{LntAppletLinkEn|walsh_en}}

@@ Zeile 1: / Zeile 1: @@
-{{LntAppletLinkEn|walsh}}
+{{LntAppletLinkEn|walsh_en}}
 == Program description==
@@ Zeile 65: / Zeile 65: @@
 ==Call the applet again==
 <br>
-{{LntAppletLinkEn|walsh}}
+{{LntAppletLinkEn|walsh_en}}

@@ Zeile 49: / Zeile 49: @@
 <br>
-[[Datei:Walsh Handhabung.png|right|550px]]
+[[Datei:Bildschirm_Walsh_EN_3.png|right|600px]]
 &nbsp; &nbsp; '''(A)''' &nbsp; &nbsp; Selection of&nbsp; $G$ &nbsp; &rArr; &nbsp; Band spread factor:&nbsp; $J= 2^G$

@@ Zeile 3: / Zeile 3: @@
 == Program description==
 <br>
-This applet allows to display the Hadamard matrices&nbsp; $\mathbf{H}_J$&nbsp; for the construction of the Walsh functions&nbsp; $w_j$.&nbsp; The factor&nbsp; $J$&nbsp; of the band spreading as well as the selection of the individual Walsh functions&nbsp; (by blue bordering the rows of the matrix)&nbsp; can be changed.
+This applet allows to display the Hadamard matrices&nbsp; $\mathbf{H}_J$&nbsp; for the construction of the Walsh functions&nbsp; $w_j$.&nbsp; The factor&nbsp; $J$&nbsp; of the band spreading as well as the selection of the individual Walsh functions&nbsp; (by means of a blue border around rows of the matrix)&nbsp; can be changed.
 ==Theoretical background==
@@ Zeile 9: / Zeile 9: @@
 ===Application===
 <br>
-The&nbsp; '''Walsh functions'''&nbsp; are a group of periodic orthogonal functions.&nbsp; Their application in digital signal processing is mainly in the use for band spreading in CDMA systems, for example the mobile radio standard UMTS.
+The&nbsp; '''Walsh functions'''&nbsp; are a group of periodic orthogonal functions.&nbsp; Their application in digital signal processing mainly lies in the use for band spreading in CDMA systems, for example the mobile radio standard UMTS.
 *Due to their orthogonal properties and the favourable periodic cross-correlation function&nbsp; $\rm (PCCF)$, the Walsh functions represent optimal spreading sequences for a distortion-free channel and a synchronous CDMA system.&nbsp; If you take any two lines and form the correlation (averaging over the products), the PCCF value is always zero.
 *In asynchronous operation&nbsp; (example: &nbsp; uplink of a mobile radio system)&nbsp; or de-orthogonalization due to multipath propagation, Walsh functions alone are not necessarily suitable for band spreading.
-*In terms of&nbsp; $\rm (PACF)$&nbsp; (''periodic autocorrelation function'') these sequences are less good:&nbsp; Each individual Walsh function has a different PACF and each individual PACF is less good than a comparable pseudo noise&nbsp; $\rm (PN)$&nbsp; sequence. That means: &nbsp; The synchronization is more difficult with Walsh functions than with PN sequences.
+*In terms of&nbsp; $\rm (PACF)$&nbsp; (''periodic autocorrelation function'') these sequences are not as good:&nbsp; Each individual Walsh function has a different PACF and each individual PACF is less good than a comparable pseudo noise&nbsp; $\rm (PN)$&nbsp; sequence. That means: &nbsp; The synchronization is more difficult with Walsh functions than with PN sequences.
 <br>

@@ Zeile 12: / Zeile 12: @@
 *Due to their orthogonal properties and the favourable periodic cross-correlation function&nbsp; $\rm (PCCF)$, the Walsh functions represent optimal spreading sequences for a distortion-free channel and a synchronous CDMA system.&nbsp; If you take any two lines and form the correlation (averaging over the products), the PCCF value is always zero.
 *In asynchronous operation&nbsp; (example: &nbsp; uplink of a mobile radio system)&nbsp; or de-orthogonalization due to multipath propagation, Walsh functions alone are not necessarily suitable for band spreading.
-*In terms of&nbsp; $\rm (PACF)$&nbsp; (''periodic cross-correlation function'') these sequences are less good:&nbsp; Each individual Walsh function has a different PACF and each individual PACF is less good than a comparable pseudo noise&nbsp; $\rm (PN)$&nbsp; sequence. That means: &nbsp; The synchronization is more difficult with Walsh functions than with PN sequences.
+*In terms of&nbsp; $\rm (PACF)$&nbsp; (''periodic autocorrelation function'') these sequences are less good:&nbsp; Each individual Walsh function has a different PACF and each individual PACF is less good than a comparable pseudo noise&nbsp; $\rm (PN)$&nbsp; sequence. That means: &nbsp; The synchronization is more difficult with Walsh functions than with PN sequences.
 <br>
@@ Zeile 39: / Zeile 39: @@
 {{GraueBox|TEXT=
-$\text{Example:}$&nbsp; The graphic shows the Hadamard matrix &nbsp;$\mathbf H_8$&nbsp; (right) and the &nbsp;$J -1$&nbsp; spreading sequences which can be constructed with it.
+$\text{Example:}$&nbsp; The graphic shows the Hadamard matrix &nbsp;$\mathbf H_8$&nbsp; (right) and the &nbsp;$J\hspace{-0.09cm} -\hspace{-0.09cm}1$&nbsp; spreading sequences which can be constructed with it.
 [[Datei:P_ID1882__Mod_T_5_3_S7_neu.png|right|frame| Walsh spreading sequences &nbsp;$(J = 8)$&nbsp; and Hadamard matrix &nbsp;$\mathbf H_8$&nbsp;]]
-*$J - 1$ because the unspread sequence &nbsp;$w_0(t)$&nbsp; is usually not used.
+*Only&nbsp; $J\hspace{-0.09cm} -\hspace{-0.09cm}1$, because the unspreaded sequence &nbsp;$w_0(t)$&nbsp; is usually not used.
 *Please note the color assignment between the lines of the Hadamard matrix and the spreading sequences &nbsp;$w_j(t)$.
-*The matrix &nbsp;$\mathbf H_4$&nbsp; is highlighted in yellow.}}}
+*The submatrix &nbsp;$\mathbf H_4$&nbsp; is highlighted in yellow.}}}
 <br clear=all>
@@ Zeile 51: / Zeile 51: @@
 [[Datei:Walsh Handhabung.png|right|550px]]
-&nbsp; &nbsp; '''(A)''' &nbsp; &nbsp; Selection of the factor for band spreading as power of two of $G$
+&nbsp; &nbsp; '''(A)''' &nbsp; &nbsp; Selection of&nbsp; $G$ &nbsp; &rArr; &nbsp; Band spread factor:&nbsp; $J= 2^G$
-&nbsp; &nbsp; '''(B)''' &nbsp; &nbsp; Selection of the respective Walsh function $w_j$
+&nbsp; &nbsp; '''(B)''' &nbsp; &nbsp; Selection of the Walsh function&nbsp; $w_j$&nbsp; to be marked&nbsp;
 <br clear=all>
 == About the authors==
-Dieses interaktive Berechnungstool  wurde am&nbsp; [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik]&nbsp; der&nbsp; [https://www.tum.de/ Technischen Universität München]&nbsp; konzipiert und realisiert.
+This interactive calculation tool was designed and realized at the&nbsp; [http://www.lnt.ei.tum.de/startseite Lehrstuhl für Nachrichtentechnik]&nbsp; $\rm (LNT)$&nbsp; of the&nbsp; [https://www.tum.de/ Technical University of Munich]&nbsp; $\rm (TUM)$.
-==Nochmalige Aufrufmöglichkeit des Applets in neuem Fenster==
+*The first German version was created in 2007 by&nbsp;  [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Thomas_Gro.C3.9Fer_.28Diplomarbeit_LB_2006.2C_danach_freie_Mitarbeit_bis_2010.29|Thomas Großer]]&nbsp;&nbsp; in the context of his diploma thesis with "FlashMX&ndash;Actionscript"&nbsp;  (Supervisor:&nbsp; [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
 <br>
 {{LntAppletLinkEn|walsh}}

@@ Zeile 1: / Zeile 1: @@
-== program description==
+{{LntAppletLinkEn|walsh}}
 <br>
-This applet allows to display the Hadamard matrices&nbsp; $\mathbf{H}_J$&nbsp; for the construction of the Walsh functions&nbsp; $w_j$. The factor&nbsp; $J$&nbsp; of the band spreading as well as the selection of the individual Walsh functions (by blue bordering the rows of the matrix) can be changed.
+This applet allows to display the Hadamard matrices&nbsp; $\mathbf{H}_J$&nbsp; for the construction of the Walsh functions&nbsp; $w_j$.&nbsp; The factor&nbsp; $J$&nbsp; of the band spreading as well as the selection of the individual Walsh functions&nbsp; (by blue bordering the rows of the matrix)&nbsp; can be changed.
 ==Theoretical background==
@@ Zeile 7: / Zeile 9: @@
 ===Application===
 <br>
-The&nbsp; '''Walsh functions'''&nbsp; are a group of periodic orthogonal functions. Their application in digital signal processing is mainly in the use for band spreading in CDMA systems, for example the mobile radio standard UMTS.
+The&nbsp; '''Walsh functions'''&nbsp; are a group of periodic orthogonal functions.&nbsp; Their application in digital signal processing is mainly in the use for band spreading in CDMA systems, for example the mobile radio standard UMTS.
-*Due to their orthogonal properties and the favourable PKKF conditions (periodic KKF), the Walsh functions represent optimal spreading sequences for a distortion-free channel and a synchronous CDMA system. If you take any two lines and form the correlation (averaging over the products), the PKKF value is always zero.
+*Due to their orthogonal properties and the favourable periodic cross-correlation function&nbsp; $\rm (PCCF)$, the Walsh functions represent optimal spreading sequences for a distortion-free channel and a synchronous CDMA system.&nbsp; If you take any two lines and form the correlation (averaging over the products), the PCCF value is always zero.
-*In asynchronous operation (example: &nbsp; uplink of a mobile radio system) or de-orthogonalization due to multipath propagation, Walsh functions alone are not necessarily suitable for band spreading - see &nbsp; [Tasks:5.4_Walsh Functions_(PKKF,_PAKF)|Task 5.4]].
+*In asynchronous operation&nbsp; (example: &nbsp; uplink of a mobile radio system)&nbsp; or de-orthogonalization due to multipath propagation, Walsh functions alone are not necessarily suitable for band spreading.
-*In terms of PAKF (periodic AKF) these sequences are less good: &nbsp; Each individual Walsh function has a different PAKF and each individual PAKF is less good than a comparable PN sequence. That means: &nbsp; The synchronization is more difficult with Walsh functions than with PN sequences.
+*In terms of&nbsp; $\rm (PACF)$&nbsp; (''periodic cross-correlation function'') these sequences are less good:&nbsp; Each individual Walsh function has a different PACF and each individual PACF is less good than a comparable pseudo noise&nbsp; $\rm (PN)$&nbsp; sequence. That means: &nbsp; The synchronization is more difficult with Walsh functions than with PN sequences.
-=== Construction==
+=== Construction===
 <br>
-The construction of Walsh functions can be done recursively using the '''Hadamard matrices''. A Hadamard matrix $\mathbf{H}_J$ of order $J$ is a $J\times J$ matrix, which contains line by line the $\pm 1$ weights of the Walsh sequences. The orders of the Hadamard matrices are fixed to powers of two, i.e. $J = 2^G$ applies to a natural number $G$. Starting from $\mathbf{H}_1 = [+1]$ and
+The construction of Walsh functions can be done recursively using the&nbsp; '''Hadamard matrices'''.
-\begin{equation}
+:$$
 \mathbf{H}_2 =
 \left[ \begin{array}{rr}
 +1 & +1\\
 +1 & -1 \\
-\end [array}\right]
+\end{array}\right]
-\end{equation}
+$$
 the following relationship applies to the generation of further Hadamard matrices:
-\begin{equation}
+:$$
   \mathbf{H}_{2N} =
 \left[ \begin{array}{rr}
-+\mathbf{H}_N & +\mathbf{H}_N\\\\
++\mathbf{H}_N & +\mathbf{H}_N\\
-+\mathbf{H}_N & -\mathbf{H}_N \\\\
++\mathbf{H}_N & -\mathbf{H}_N \\
-\end [array}\right]
+\end{array}\right]
-\end{equation}
+$$
 <br>
-{{{{GrayBox|TEXT=
-$\text{example:}$&nbsp; The graphic shows the Hadamard matrix &nbsp;$\mathbf H_8$&nbsp; (right) and the spreading sequences which can be constructed with it &nbsp;$J -1$&nbsp;.
+{{GraueBox|TEXT=
-[[File:P_ID1882__Mod_T_5_3_S7_new.png|right|frame| Walsh spreading sequences &nbsp;$(J = 8)$&nbsp; and Hadamard matrix &nbsp;$\mathbf H_8$&nbsp;]]
+$\text{Example:}$&nbsp; The graphic shows the Hadamard matrix &nbsp;$\mathbf H_8$&nbsp; (right) and the &nbsp;$J -1$&nbsp; spreading sequences which can be constructed with it.
 *$J - 1$ because the unspread sequence &nbsp;$w_0(t)$&nbsp; is usually not used.
 *Please note the color assignment between the lines of the Hadamard matrix and the spreading sequences &nbsp;$w_j(t)$.
@@ Zeile 40: / Zeile 46: @@
 <br clear=all>
-==To use the applet==
+==How to use the applet==
 <br>
-[[File:Walsh Handling.png|right|550px]]
+[[Datei:Walsh Handhabung.png|right|550px]]
 &nbsp; &nbsp; '''(A)''' &nbsp; &nbsp; Selection of the factor for band spreading as power of two of $G$
@@ Zeile 51: / Zeile 57: @@
 == About the authors==
 Translated with www.DeepL.com/Translator (free version)