Biomedical Engineering Reference
In-Depth Information
ϕ
(1,0,t)
ϕ
(2,
−
100,t)
1
1
0
0
1
1
−
−
0
200
400
600
800
0
200
400
600
800
ϕ
(2,0,t)
ϕ
(2,
−
200,t)
1
1
0
0
−
1
−
1
0
200
400
600
800
0
200
400
600
800
(4,0,t)
(2,
400,t)
ϕ
ϕ
−
1
1
0
0
−
1
−
1
0
200
400
600
800
0
200
400
600
800
TIME (ms) TIME (ms)
FIGURE 11.33
Illustrations of the db2 wavelet at several scales and positions. The upper left-hand corner illus-
trates the basic waveform
j
(t). The notation for the illustrations is given in the form
j
(scale, delay, t). Thus,
j
(t)
¼ j
(1,0,t),
j
(2t-100)
¼ j
(2,-100,t), and so on.
ðð
x
ð
t
Þ¼
K
'
C
ð
a
,
s
Þ'ð
a
,
s
,
t
Þ
dtds
ð
11
:
58
Þ
can be used to recover the original signal,
x
(
t
), from the wavelet coefficients,
C
(
a, s
).
K
'
is a
function of the wavelet used.
In practice, wavelet analysis is performed on digitized signals using a subset of scales
and positions (see MATLAB's Wavelet Toolbox). One computational process is to recur-
sively break the signal into low-frequency (“high-scale” or “approximation”) and high-
frequency (“low-scale” or “detail”) components using digital low-pass and high-pass filters
that are functions of the mother wavelet. The output of each filter will have the same num-
ber of points as the input. In order to keep the total number of data points the same at each
level, every other data point of the output sequences is discarded. This process is known as
“downsampling.” Using “upsampling” and a second set of digital filters, called reconstruc-
tion filters, the process can be reversed, and the original data set is reconstructed. Remark-
ably, the inverse discrete wavelet transform does exist!
While this process will rapidly yield wavelet transform coefficients, the power of discrete
wavelet analysis lies in its ability to examine waveform shapes at different resolutions
and to selectively reconstruct waveforms using only the levels of approximation and detail
that are desired. Applications include detecting discontinuities and breakdown points,
detecting long-term evolution, detecting self-similarity (e.g., fractal trees), identifying pure
