
Continuous Wavelet Transforms ◾ 79
and its inverse as
=
∞
ω
()
1
ˆ
()
ft
it
(3.14)
More detailed information on the Fourier transform and its properties
may be found in Brigham [6] and Mallat [7], but the essential properties
that we need are listed in Box3.3.
Example 3.2: Centers and Bandwidths of the Morlet Wavelet
As an illustrative example, we now nd the centers and bandwidths
of the Morlet wavelet given in Equation (3.8). For this wavelet, the
corresponding family is given by
ψ=
σπ
ω
−
−
−
σ
t
a
i
tb
a
tb
a
ab
()
11
expexp
1
2
.
,
2
4
0
2
(3.15)
BOX3.3 SELECTED PROPERTIES OF THE FOURIER TRANSFORM
• Convolution
∗⇔
ft ft ff()