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Filters Used to Calculate the DWT and IDWT

For an orthogonal wavelet, in the multiresolution framework (see [Dau92] in Using Wavelet Packets), we start with the scaling function phi and the wavelet function psi. One of the fundamental relations is the twin-scale relation (dilation equation or refinement equation):

All the filters used in DWT and IDWT are intimately related to the sequence

Clearly if phi is compactly supported, the sequence (wn) is finite and can be viewed as a filter. The filter W, which is called the scaling filter (nonnormalized), is

For example, for the db3 scaling filter,

From filter W, we define four FIR filters, of length 2N and of norm 1, organized as follows.

Filters
Low-Pass
High-Pass
Decomposition
Lo_D
Hi_D
Reconstruction
Lo_R
Hi_R

The four filters are computed using the following scheme.

where qmf is such that Hi_R and Lo_R are quadrature mirror filters (i.e., Hi_R(k) = (-1) k Lo_R(2N + 1 - k)) for k = 1, 2, ..., 2N.

Note that wrev flips the filter coefficients. So Hi_D and Lo_D are also quadrature mirror filters. The computation of these filters is performed using orthfilt. Next, we illustrate these properties with the db6 wavelet. The plots associated with the following commands are shown in Figure 6-8.

Figure 6-8: The Four Wavelet Filters for db6


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