Quelltext der Seite Applets:Generation of Walsh functions

{{LntAppletLinkEn|walsh}}

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

:$$
\mathbf{H}_2 =
\left[ \begin{array}{rr}
+1 & +1\\
+1 & -1 \\
\end{array}\right]
$$
the following relationship applies to the generation of further Hadamard matrices:
:$$
 \mathbf{H}_{2N} =
\left[ \begin{array}{rr}
+\mathbf{H}_N & +\mathbf{H}_N\\
+\mathbf{H}_N & -\mathbf{H}_N \\
\end{array}\right]
$$
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{{GraueBox|TEXT=
$\text{Example:}$&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;]] 
*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 submatrix &nbsp;$\mathbf H_4$&nbsp; is highlighted in yellow.}}}
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==How to use the applet==

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[[Datei:Bildschirm_Walsh_EN_3.png|right|600px]]

&nbsp; &nbsp; '''(A)''' &nbsp; &nbsp; Selection of&nbsp; $G$ &nbsp; &rArr; &nbsp; Band spread factor:&nbsp; $J= 2^G$

&nbsp; &nbsp; '''(B)''' &nbsp; &nbsp; Selection of the Walsh function&nbsp; $w_j$&nbsp; to be marked&nbsp; 
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== About the authors==

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]]).
*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]] ).
*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==
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{{LntAppletLinkEn|walsh}}