Biomedical Engineering Reference
In-Depth Information
A 2 0 x
(
)=
(
)
n
x
n
h
g
A 2 1 x
D 2 1 x
(
n
)
(
n
)
h
g
A 2 2 x
D 2 2 x
(
n
)
(
n
)
h
g
...
...
FIGURE 2.17: Cascade of filters producing the discrete wavelet transform coef-
ficients. In this algorithm the signal is passed through the cascade, where at each
step the approximation of signal from the previous step is low-pass and high-pass
filtered, by convolving with h and g , respectively. Each of the filtered sequences is
downsampled by factor 2 (on the scheme it is marked as
). Next the approximation
is passed to the following step of the cascade.
with the mirror filter g
(
n
)=
g
(
n
)
of the filter g
(
n
)
defined by:
g
(
n
)=
Ψ 2 1
(
u
) ,
φ
(
u
n
)
(2.107)
The filter g
(
n
)
can also be computed from filter h
(
n
)
with the formula:
1
n h
g
(
n
)=(
1
)
(
1
n
)
(2.108)
is a high-pass filter. The A 2 j and D 2 j can be obtained from A 2 j + 1 by
convolving with h
(
)
Filter g
n
(
)
(
)
, respectively, and downsampling by factor 2. Com-
putation of discrete approximation by repeating the process for j
n
and g
n
<
0 is called the
pyramidal transform. It is illustrated in Figure 2.17.
For a continuous signal the cascade could be infinite. In practical cases of sampled
signals, the process of consecutive downsamplings has the natural stop. The coef-
ficients of wavelet transform are the outputs of all the detail branches and the last
approximation branch.
For correctly designed filters (quadrature mirror filters) the process of decompo-
sition can be reversed yielding the inverse discrete wavelet transform. A technical
discussion of how to design these filters is beyond the scope of this topic and is
available, e.g., in [Strang and Nguyen, 1996]. If we implement the wavelet transform
as an iterated filter bank we do not have to specify the wavelet explicitly.
There is one important consequence of discretization of the wavelet transform.
The transform is not shift invariant, which means that the wavelet transform of a
signal and of the wavelet transform of the same time-shifted signal are not simply
shifted versions of each other. An illustration of the structure of the time-frequency
space related to the discrete wavelet representation is shown in Figure 2.18.
 
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