Digital Signal Processing Reference
In-Depth Information
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(a)
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Keep one sample out of two
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Convolution with the filter X
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Insert one zero between each two samples
Figure 2.14. 1-D wavelet packet transform: (a) forward transform, (b) recon-
struction.
As in the case of the DWT, recursion (2.39) gives us a fast pyramidal algorithm to
compute the whole set of wavelet packet coefficients. This is displayed in the upper
part of Fig. 2.14. The reconstruction or synthesis corresponding to (2.40) is depicted
in the lower part of Fig. 2.14. For a discrete signal of
N
equally spaced samples,
the wavelet packet forward transform and the reconstruction require
O
(
N
log
N
)
operations.
2.8.4 Best Wavelet Packet Basis
B
The wavelet packet implies a large library of bases
; that is, the wavelet packet
dictionary is a union of orthobases. Within this family, the best basis for a particular
signal
f
is the one that will be best adapted to its time-frequency content. Coifman
and Wickerhauser (1992) proposed to cast this best basis search as an optimization
problem that will minimize some additive information cost function. Although
wavelet packet bases form a large library, they are organized in a full binary tree,
where each subtree
corresponds to a specific dyadic covering of the frequency
axis. Thus the optimization problem over bases amounts to an optimization over
T
