File:MATH 414 Filter-Based Wavelet Decomposition.svg

Flowchart of wavelet decomposition using filters, downsampling, and upsampling.

For example, here are the following definitions in terms of the Haar wavelet.

• ${\displaystyle \ell =\left(\dots ,0,0,0,\underbrace {\frac {1}{2}} _{k=-1},\underbrace {\frac {1}{2}} _{k=0},0,0,0,\dots \right)}$ is a low-pass filter
• ${\displaystyle {\tilde {\ell }}=\left(\dots ,0,0,0,\underbrace {1} _{k=0},\underbrace {1} _{k=1},0,0,0,\dots \right)}$ is the "inverse" low-pass filter
• ${\displaystyle h=\left(\dots ,0,0,0,\underbrace {-{\frac {1}{2}}} _{k=-1},\underbrace {\frac {1}{2}} _{k=0},0,0,0\dots \right)}$ is a high-pass filter
• ${\displaystyle {\tilde {h}}=\left(\dots ,0,0,0,\underbrace {1} _{k=0},\underbrace {-1} _{k=1},0,0,0,\dots \right)}$ is the "inverse" high-pass filter
• ${\displaystyle D}$ is a downsampling that accepts only the even terms
• ${\displaystyle U}$ is an upsampling that inserts zeroes for odd terms

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