Digital Signal Processing Reference
In-Depth Information
Figure 2.1. Morlet's wavelet: (left) real part and (right) imaginary part.
for some applications, and
is not necessarily a wavelet function. For example,
we will show in the next chapter (Section 3.6) how to reconstruct a signal from its
wavelet coefficients using nonnegative functions.
χ
2.3 EXAMPLES OF WAVELET FUNCTIONS
2.3.1 Morlet's Wavelet
The wavelet defined by Morlet (Coupinot et al. 1992; Goupillaud et al. 1985) is given
in the Fourier domain as
ˆ
2
(
ν
−
ν
0
)
2
e
−
2
π
ψ
(
ν
)
=
.
(2.5)
By taking the inverse Fourier transform, it is straightforward to obtain the complex
wavelet whose real and imaginary parts are
1
√
2
t
2
2
cos(2
e
−
ψ
=
πν
Re(
(
t
))
0
t
)
π
1
√
2
t
2
2
sin(2
e
−
Im(
ψ
(
t
))
=
πν
0
t
)
,
π
where
ν
0
is a constant. Morlet's transform is not admissible. For
ν
0
greater than ap-
proximately 0
8, the mean of the wavelet function is very small so that approximate
reconstruction is satisfactory. Figure 2.1 shows these two functions.
.
2.3.2 Mexican Hat
The Mexican hat used, for example, by Murenzi (1988) is in one dimension:
t
2
2
t
2
)
e
−
ψ
(
t
)
=
(1
−
.
(2.6)
