INTRODUCTION
Before introducing the wavelet transform we review some of the concepts re-garding transforms presented in Chapter 2. A transform can be thought of as a remapping of a signal that provides more information than the original. The Fourier transform fits this definition quite well because the frequency informa-tion it provides often leads to new insights about the original signal. However, the inability of the Fourier transform to describe both time and frequency char-acteristics of the waveform led to a number of different approaches described in the last chapter. None of these approaches was able to completely solve the time–frequency problem. The wavelet transform can be used as yet another way to describe the properties of a waveform that changes over time, but in this case the waveform is divided not into sections of time, but segments of scale.
In the Fourier transform, the waveform was compared to a sine func-tion—in fact, a whole family of sine functions at harmonically related frequen-cies. This comparison was carried out by multiplying the waveform with the sinusoidal functions, then averaging (using either integration in the continuous domain, or summation in the discrete domain):
X(ωm)=
∫
∞
−∞
x(t)e−jωm tdt (1)
Eq. (1) is the continuous form of Eq. (6) in Chapter 3 used to define the discrete Fourier transform. As discussed in Chapter 2, almost any family of func-177
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tions could be used to probe the characteristics of a waveform, but sinusoidal func-tions are particularly popular because of their unique frequency characteristics: they contain energy at only one specific frequency. Naturally, this feature makes them ideal for probing the frequency makeup of a waveform, i.e., its frequency spectrum.
Other probing functions can be used, functions chosen to evaluate some particular behavior or characteristic of the waveform. If the probing function is of finite duration, it would be appropriate to translate, or slide, the function over the waveform, x(t), as is done in convolution and the short-term Fourier trans-form (STFT), Chapter 6’s Eq. (1), repeated here:
STFT(t,f ) =
∫
∞
−∞
x(τ)(w(t − τ)e−2jπfτ)dτ (2)
where f, the frequency, also serves as an indication of family member, and w(t− τ) is some sliding window function where t acts to translate the window over x. More generally, a translated probing function can be written as:
X(t,m)=
∫
∞
−∞
x(τ)f(t − τ)mdτ (3)
where f(t)mis some family of functions, with m specifying the family number.
This equation was presented in discrete form in Eq. (10), Chapter 2.
If the family of functions, f(t)m, is sufficiently large, then it should be able to represent all aspects the waveform x(t). This would then allow x(t) to be reconstructed from X(t,m) making this transform bilateral as defined in Chapter 2. Often the family of basis functions is so large that X(t,m) forms a redundant set of descriptions, more than sufficient to recover x(t). This redundancy can sometimes be useful, serving to reduce noise or acting as a control, but may be simply unnecessary. Note that while the Fourier transform is not redundant, most transforms represented by Eq. (3) (including the STFT and all the distribu-tions in Chapter 6) would be, since they map a variable of one dimension (t) into a variable of two dimensions (t,m).
THE CONTINUOUS WAVELET TRANSFORM
The wavelet transform introduces an intriguing twist to the basic concept de-fined by Eq. (3). In wavelet analysis, a variety of different probing functions may be used, but the family always consists of enlarged or compressed versions of the basic function, as well as translations. This concept leads to the defining equation for the continuous wavelet transform (CWT):
W(a,b)=
∫
Wavelet Analysis 179
FIGURE7.1 A mother wavelet (a= 1) with two dilations (a = 2 and 4) and one contraction (a= 0.5).
where b acts to translate the function across x(t) just as t does in the equations above, and the variable a acts to vary the time scale of the probing function,ψ.
If a is greater than one, the wavelet function,ψ, is stretched along the time axis, and if it is less than one (but still positive) it contacts the function. Negative values of a simply flip the probing function on the time axis. While the probing functionψ could be any of a number of different functions, it always takes on an oscillatory form, hence the term “wavelet.” The * indicates the operation of complex conjugation, and the normalizing factor l/
√
a ensures that the energy is the same for all values of a (all values of b as well, since translations do not alter wavelet energy). If b= 0, and a = 1, then the wavelet is in its natural form, which is termed the mother wavelet;* that is, ψ1,o(t)≡ ψ(t). A mother wavelet is shown in Figure 7.1 along with some of its family members produced by dilation and contraction. The wavelet shown is the popular Morlet wavelet, named after a pioneer of wavelet analysis, and is defined by the equation:ψ(t) = e−t2cos(π
√
ln 22 t) (5)*Individual members of the wavelet family are specified by the subscripts a and b; i.e.,ψa,b. The mother wavelet,ψ1,0, should not to be confused with the mother of all Wavelets which has yet to be discovered.
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The wavelet coefficients, W(a,b), describe the correlation between the waveform and the wavelet at various translations and scales: the similarity be-tween the waveform and the wavelet at a given combination of scale and posi-tion, a,b. Stated another way, the coefficients provide the amplitudes of a series of wavelets, over a range of scales and translations, that would need to be added together to reconstruct the original signal. From this perspective, wavelet analy-sis can be thought of as a search over the waveform of interest for activity that most clearly approximates the shape of the wavelet. This search is carried out over a range of wavelet sizes: the time span of the wavelet varies although its shape remains the same. Since the net area of a wavelet is always zero by design, a waveform that is constant over the length of the wavelet would give rise to zero coefficients. Wavelet coefficients respond to changes in the wave-form, more strongly to changes on the same scale as the wavelet, and most strongly, to changes that resemble the wavelet. Although a redundant transfor-mation, it is often easier to analyze or recognize patterns using the CWT. An example of the application of the CWT to analyze a waveform is given in the section on MATLAB implementation.
If the wavelet function, ψ(t), is appropriately chosen, then it is possible to reconstruct the original waveform from the wavelet coefficients just as in the Fourier transform. Since the CWT decomposes the waveform into coefficients of two variables, a and b, a double summation (or integration) is required to recover the original signal from the coefficients:
x(t)= 1
and 0 < C < ∞ (the so-called admissibility condition) for recovery using Eq.
(6).
In fact, reconstruction of the original waveform is rarely performed using the CWT coefficients because of the redundancy in the transform. When recov-ery of the original waveform is desired, the more parsimonious discrete wavelet transform is used, as described later in this chapter.
Wavelet Time–Frequency Characteristics
Wavelets such as that shown in Figure 7.1 do not exist at a specific time or a specific frequency. In fact, wavelets provide a compromise in the battle between time and frequency localization: they are well localized in both time and
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Wavelet Analysis 181
quency, but not precisely localized in either. A measure of the time range of a specific wavelet,∆tψ, can be specified by the square root of the second moment of a given wavelet about its time center (i.e., its first moment) (Akansu &
Haddad, 1992):
∆tψ=冪−∞∞
∫
(t−∞− t∞∫
*ψ(t/a)*0)2*ψ(t/a)*2dt 2dt (7)where t0is the center time, or first moment of the wavelet, and is given by:
t0=
∫
Similarly the frequency range,∆ωψ, is given by:
∆ωtψ=冪−∞∞
∫
(ω − ω−∞∞∫
*Ψ(ω)*0)2*Ψ(ω)*2dω 2dω (9)where Ψ(ω) is the frequency domain representation (i.e., Fourier transform) of ψ(t/a), and ω0 is the center frequency ofΨ(ω). The center frequency is given by an equation similar to Eq. (8):
ω0=
The time and frequency ranges of a given family can be obtained from the mother wavelet using Eqs. (7) and (9). Dilation by the variable a changes the time range simply by multiplying∆tψby a. Accordingly, the time range of ψa,0 is defined as ∆tψ(a)= *a*∆tψ. The inverse relationship between time and frequency is shown in Figure 7.2, which was obtained by applying Eqs. (7–10) to the Mexican hat wavelet. (The code for this is given in Example 7.2.) The Mexican hat wavelet is given by the equation:
ψ(t) = (1 − 2t2)e−t2 (11)
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