Definition Walshfunktion
Die kontinuierlichen Walshfunktionen (nach Josph L. Walsh (1895-1973) und Jacques Hadamard )
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{\displaystyle wal_{n}(x)}
definiert durch:
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{\displaystyle {\begin{matrix}wal_{0}(x)&=&\left\{{\begin{matrix}1&\quad -{\frac {1}{2}}\leq x\leq +{\frac {1}{2}}\\0&sonst\end{matrix}}\right.\\\\wal_{2n}(x)&=&(-1)^{\left\lfloor {\frac {n}{2}}\right\rfloor }\cdot \left(wal_{n}({\frac {4x+1}{2}})+(-1)^{n}wal_{n}({\frac {4x-1}{2}})\right)\\\\wal_{2n+1}(x)&=&(-1)^{\left\lfloor {\frac {n}{2}}\right\rfloor +1}\cdot \left(wal_{n}({\frac {4x+1}{2}})+(-1)^{n+1}wal_{n}({\frac {4x-1}{2}})\right).\end{matrix}}}
orthogonal
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{\displaystyle {\begin{matrix}\int _{-{\frac {1}{2}}}^{+{\frac {1}{2}}}&wal_{n}(x)\cdot wal_{m}(x)dx=\left\{{\begin{matrix}{}1\quad n=m\\0\quad n\neq m\end{matrix}}\right.\end{matrix}}}
Definition Hadamardmatritzen
Die Hadamardmatritzen sind definiert durch:
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{\displaystyle H_{2}=\left({\begin{matrix}{}1&1\\1&-1\end{matrix}}\right)}
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{\displaystyle H_{N}=H_{2}\otimes H_{\frac {N}{2}}=\left({\begin{matrix}{}H_{\frac {N}{2}}&H_{\frac {N}{2}}\\H_{\frac {N}{2}}&-H_{\frac {N}{2}}\end{matrix}}\right)}
Für
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{\displaystyle N=2^{p}}
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{\displaystyle {\begin{matrix}c=WHT(f)=H_{N}f\end{matrix}}}
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{\displaystyle f=WHT^{-1}(c)={\frac {1}{N}}H_{N}c}
Walsh-Hadamard-Transformation der Ordnung
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{\displaystyle N}
