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Wavelet Transform
CWT, this function is often referred to as the mother wavelet. In addition, now it
becomes clearer why this transformation is called “wavelet transform.” A closer
look at the definition of the mother wavelet tells that this function must be limited
in duration and therefore looks like a decaying small wave. All other basis functions are the shifted and scaled version of the mother wavelet. In contrast with
the STFT, instead of using sinusoidal functions in order to generate the basis
functions, a mother wavelet is continuously shifted and scaled to create all basis
functions in CWT.
Now that we have discussed the analysis equation or the CWT, it is the best time
to describe the synthesis equation that allows forming the time signal based on its
WT. The inverse continuous wavelet transform or ICWT is defined as
+∞ +∞
C
−1
⎛ t b
− ⎞
x t
( ) =
Ψ
−∞
∫
W Ψ
⎜
⎟ da
a
2
,X (a b
, )Ψ
db , a ≠ 0
(5.4)
⎝ a ⎠
−∞
∫
In this equation, C
−1
Ψ is a constant whose value depends on the exact choice of the
mother wavelet Ψ(t). As can be seen, while the CWT equation is a single integral,
the ICWT is a double integral based on two dummy variables a and b.
Every choice of the mother wavelet gives a particular CWT, and as a result, we
are dealing with infinite number of transformations under the same name CWT
(as opposed to only one transform for the continuous Fourier transform [CFT]).
Any choice of the mother wavelet gives certain unique properties that make the
resulting transformation a suitable choice for a particular task. A mother wavelet called Mexican hat is shown in Figure 5.8. The reason this mother wavelet
is called Mexican hat (or sombrero) is evident from its waveform. The waveform represented by this signal is the general shape of the most popular mother
wavelets.
Daubechies (dbX) wavelets are among the most popular mother wavelets that are
commonly used in signal and image processing. The index “X” in dbX identifies the
exact formulation of the function. These mother wavelets are simply the functions
FIGURE 5.8 Mexican hat mother wavelet.
Wavelet Transform
CWT, this function is often referred to as the mother wavelet. In addition, now it
becomes clearer why this transformation is called “wavelet transform.” A closer
look at the definition of the mother wavelet tells that this function must be limited
in duration and therefore looks like a decaying small wave. All other basis functions are the shifted and scaled version of the mother wavelet. In contrast with
the STFT, instead of using sinusoidal functions in order to generate the basis
functions, a mother wavelet is continuously shifted and scaled to create all basis
functions in CWT.
Now that we have discussed the analysis equation or the CWT, it is the best time
to describe the synthesis equation that allows forming the time signal based on its
WT. The inverse continuous wavelet transform or ICWT is defined as
+∞ +∞
C
−1
⎛ t b
− ⎞
x t
( ) =
Ψ
−∞
∫
W Ψ
⎜
⎟ da
a
2
,X (a b
, )Ψ
db , a ≠ 0
(5.4)
⎝ a ⎠
−∞
∫
In this equation, C
−1
Ψ is a constant whose value depends on the exact choice of the
mother wavelet Ψ(t). As can be seen, while the CWT equation is a single integral,
the ICWT is a double integral based on two dummy variables a and b.
Every choice of the mother wavelet gives a particular CWT, and as a result, we
are dealing with infinite number of transformations under the same name CWT
(as opposed to only one transform for the continuous Fourier transform [CFT]).
Any choice of the mother wavelet gives certain unique properties that make the
resulting transformation a suitable choice for a particular task. A mother wavelet called Mexican hat is shown in Figure 5.8. The reason this mother wavelet
is called Mexican hat (or sombrero) is evident from its waveform. The waveform represented by this signal is the general shape of the most popular mother
wavelets.
Daubechies (dbX) wavelets are among the most popular mother wavelets that are
commonly used in signal and image processing. The index “X” in dbX identifies the
exact formulation of the function. These mother wavelets are simply the functions
FIGURE 5.8 Mexican hat mother wavelet.