Information Technology Reference
In-Depth Information
It can be seen that the real and the imaginary part of the wavelet differ in phase
by a quarter period. The
4
term is a normalization factor which ensures that the
wavelet has unit energy.
11.3.1.2
Spectral Decomposition and Heisenberg Boxes
The Fourier transform of the Morlet wavelet is given by
O
.f / D
4
p
2e
2
.2f 2f
0
/
2
(11.35)
which has the form of a Gaussian function displaced along the frequency axis by
f
0
. The energy spectrum (the squared magnitude of the Fourier transform) is given
by [
2
]
D 2
2
e
.2f 2f
0
/
2
j
O
.f /j
2
(11.36)
which is a Gaussian centered at f
0
. The integral of (
11.36
) gives the energy of the
Morlet wavelet. The energy spectrum of the Morlet wavelet depicted in diagram (c)
of Fig.
11.2
(i), for different values of the variance
2
is given in Fig.
11.2
b.
The central frequency f
0
is the frequency of the complex sinusoid and its value
determines the number of significant sinusoidal waveforms contained within the
envelope. The dilated and translated Morlet wavelet .
x
a
/ is given by
x
b
a
2
.
x
a
/
2
4
e
i2f
0
.
x
a
/
e
1
1
D
(11.37)
The Heisenberg boxes in the x-frequency plane for a wavelet at different scales
are shown in Fig.
11.3
a. To evaluate frequency composition a sample of a long
region of the signal is required. If instead, a small region of the signal is measured
with accuracy, then it becomes very difficult to determine the frequency content
of the signal in that region. That is, the more accurate the temporal measurement
(smaller
x
) is, the less accurate the spectral measurement (larger
f
) becomes, and
vice-versa [
119
].
The Morlet central frequency f
0
sets the location of the Heisenberg box in the
x-frequency plane. If the x-length of the wavelets remains the same, then no matter
the change of the central frequency f
0
the associated Heisenberg boxes will have
the same dimensions. This is depicted in Fig.
11.3
b.
Finally, in Fig.
11.4
a are shown the Heisenberg boxes in the x-frequency plane
for a number of wavelets with three different spectral frequencies (low, medium,
and high). The confining Gaussian windows have the same dimensions along the x
axis. Therefore, altering the central frequency of the wavelet shifts the Heisenberg
box up and down the x-frequency plane without altering its dimensions.
