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Wavelet Matrix

A Matrix composed of Haar Functions which is used in the Wavelet Transform. The fourth-order wavelet matrix is given by


$\displaystyle {W}_4$ $\textstyle =$ $\displaystyle \left[\begin{array}{cccc}1 & 1 & 1 & 0\\  1 & 1 & -1 & 0\\  1 & -1 & 0 & 1\\  1 & -1 & 0 & -1\end{array}\right]$  
  $\textstyle =$ $\displaystyle \left[\begin{array}{cccc}1 & 1 & & \\  1 & -1 & & \\  & & 1 & 1\\...
...array}{cccc}1 & 1 & & \\  1 & -1 & & \\  & & 1 & \\  & & & 1\end{array}\right].$  

A wavelet matrix can be computed in ${\mathcal O}(n)$ steps, compared to ${\mathcal O}(n\lg n)$ for the Fourier Matrix, where $\lg x=\log_2 x$ is the base-2 Logarithm.

See also Fourier Matrix, Wavelet, Wavelet Transform




© 1996-9 Eric W. Weisstein
1999-05-26