A Wavelet Transform (WT) of a function is a decomposition of that function into a weighted sum of a particular family of functions generated from a mother wavelet and forming a basis forL2(R). Wavelets are functions that satisfy certain mathematical demands in multiresolution analysis. In brief, a wavelet is an oscillation that decays quickly (Sifuzzaman et al.[210]. The name wavelet comes from the requirement that : (1) the average value of the function should be zero (2) the function has to be well localized (Muller
&Vidakovic [207]). A wavelet is a mathematical function used to divide a given function or continuous- time signal into different scale components. Usually one can assign a frequency range to each scale component. Each scale component can then be studied with a resolution that matches its scale.
The wavelets are scaled and translated copies (known as ”daughter wavelets”) of a finite-length or fast- decaying oscillating waveform (known as the ”mother wavelet”). The fundamental idea behind wavelets is to analyze according to scale. Wavelet transform can be continuous or discrete. The CWT is performed in a smooth continuous fashion and represents the energy content of a signal that contains features on different scales at any instant in time (Seena&Sung [92]). As explained inFarge [32], for data analysis it is better to use a complex wavelet, since the modulus of the complex-valued wavelet coefficients gives the energy density, in a fashion similar to Fourier analysis.
It is important to notice the significant differences between Fourier analysis and wavelet analysis (Strang[211]).
Fourier basis functions are localized in frequency but not in time. Small frequency changes in the Fourier domain will produce changes everywhere in the time domain. Wavelets are, however, local in both fre- quency (scale) and time. This localization is a major advantage of the WT. Another important feature is that a large class of functions can be represented by wavelets in a compact mode. For example, functions with discontinuities or with sharp transitions usually take substantially fewer wavelet basis functions than sine-cosine basis functions to obtain a comparable approximation. Furthermore, large data sets can be easily and quickly transformed by the WT. From the theoretical viewpoint, a brief discussion about the fundamentals of wavelet analysis is presented inTorrence and Compo [82].
The continuous wavelet transform (CWT) is simply the correlation of an input functionf(x)∈ L2(R) with a family (in particular, an orthogonal family) of wavelet functionsψa,b(x) (witha, b∈Randa6= 0) generated by scaling (dilating or compressing) and shifting a single mother waveletψ(x)∈ L2(R), and is expressed mathematically (Seena &Sung [92]) as :
Wf(s, x) = 1
√s Z ∞
−∞
f(τ)ψ∗ x−τ
s
dτ (9.3)
where
ψs,τ(x) = 1
√sψ x−τ
s
(9.4) represents a chosen wavelet family andsis the scale factor,τ is the translation factor,√
s is for energy normalization across the different scales, and ∗ denotes the complex conjugate. The function ψ(x) is the mother wavelet, which is dilated or contracted by s and translated by τ to generate a family of wavelets, ψs,τ. Large values of the scaling parameter s mean large scale and they correspond to small frequency ranges, while small values of parameterscorrespond to high frequencies and very fine scales.
The variation of s has a dilatation effect (when s > 1 ) and a contraction effect (when s < 1) of the mother-wavelet function. By changing parameterτ, the wavelet function can be translated along the time axis : Eachψs,τ(t) is localized aroundt=τ. Therefore, it is possible to analyze the long and short
period features of the signal or the low and high frequency aspects of the signal. Asbvaries, the function f is locally analyzed in the vicinities of this point.
The mother wavelet adopted in this analysis is the complexMorlet wavelet which has been found to be a common choice in fluid mechanics (Farge [32]). However it has been proven that the physical results obtained do not depend on the choice of the wavelet type (Farge [32] andCamussi &Felice [90]). The Morlet wavelet is a plane wave modulated by a Gaussian envelope defined as :
ψ(x) =π−14eiωoxe−12x2 (9.5) where the parameterωois the wavenumber associated with theMorlet wavelet, roughly corresponding to the number of oscillations of the wavelet. The coefficientωois usually taken equal to 6 in order to minimize errors related to the non-zero mean (Torrence &Compo [82]). The advantage of the Morlet wavelet over other candidates such as the Mexican hat wavelet resides in its good definition in the spectral space. For ωo= 6 (used here), the Morlet wavelet scale is almost identical to the corresponding Fourier period of the complex exponential, and the terms scale and period may conveniently be used synonymously (Torrence
&Compo[82];Torrence &Webster [212]). Figure 9.2 shows the real part of the complexMorlet wavelet adopted in the analysis reported here. Since this wavelet is a complex function it is possible to analyze the phase and the modulus of the decomposed signal (Seena &Sung [92]).
Figure 9.2: ComplexMorletmother-wavelet withωo= 6, (a) real component (b) imaginary component
In practice, Eqn. (9.3) is implemented using digital techniques. The continuous wavelet transformWn
of a discrete sequence of observationsxn is defined as the correlation ofxn with a scaled and translated waveletψ(η) that depends on a non-dimensional time parameterη,
Wn(s) = 1
√s
N−1
X
n′=0
x′nψ∗
n′−n s
δt (9.6)
wherenis the localized time index,sis the wavelet scale,δtis the sampling period,N is the number of points in the time series, and the asterik,∗indicates the complex conjugate.
By varying the wavelet scale, s and translating along the localized time index, n, one can construct a picture showing both amplitude of any features versus the scale and how this amplitude varies in time.
Such a transform is called the continuous wavelet transform, because the scale and localization parameters assume continuous values. By choosingN points, the correlation described by Eqn. (9.3) allows us to do allN convolutions simultaneously in Fourier space using discrete Fourier transform (DFT). The DFT of xn is :
ˆ xk = 1
N
N−1
X
n=0
xne−2πikn/N (9.7)
wherek= 0· · ·N−1 is the frequency index. In the continuous limit, the Fourier transform of a function ψ(t/s) is given by ˆψ(sω). By the correlation theorem, the wavelet transform is the inverse Fourier transform of the product :
Wn(s) =
N−1
X
k=0
ˆ
xkψˆ∗(sωk)eiωknδt (9.8)
To ensure that the wavelet transforms at each scalesare directly comparable to each other and to the transforms of other time series, the wavelet function at each scale isnormalized to have unity energy as follows :
ψ(sωˆ k) = 2πs
δt 12
ψˆo(sωk) (9.9)
The result of these correlations are referred to aswavelet coefficients. When two signals are correlated with each other, a measure of similarity is obtained between them. Thus, when the WT is computed at a scale corresponding to the compressed wavelet, a measure of similarity between the signal and the high-frequency wavelet is obtained. Likewise, when the wavelet function is dilated, a measure of how similar the input signal is to the low-frequency wavelet is obtained. In other words, the WT can be interpreted as frequency decomposition with corresponding coefficients which provide information about the frequency contributions of the original signal, as well as their spatial position. This kind of analysis is also referred to asmulti-resolution analysis (Azimifar [213]). As the wavelet function ψ is complex, the wavelet transform is also complex. The transform can then be divided into the real partℜ[Wn(s)], and the imaginary part,ℑ[Wn(s)]. The wavelet amplitude and wavelet energy are then calculated from the coefficients as :
WA(s) =ℜ[Wn(s)] (9.10)
and
WE(s) = (ℜ[Wn(s)])2 (9.11)
respectively, while the wavelet phase is given by :
WP(s) =tan−1(ℑ[Wn(s)]/ℜ[Wn(s)]) (9.12)
whereℜand ℑrepresent the real andimaginary parts of the wavelet coefficients.
The wavelet power spectrum is also complex and is defined as :
WP(s) =|Wn(s)|2 (9.13)
The wavelet power spectrum is a convenient description of the fluctuation of the variance at different frequencies (Torrence&Compo[82] ,Ancti&Coulibaly[214] ). Further, when normalized by the variance σ2, it gives a measure of the power relative to white noise, since the expectation value for a white-noise process isσ2at all nand s(Anctil &Coulibaly [214]).
The total energy in a signal is conserved under the wavelet transform, and the equivalent ofParseval’s theorem for wavelet analysis is given byTorrence &Compo [82] as the variance of the signal, expressed as :
WE =σ2= δjδt CδN
N−1
X
n=0 J
X
j=0
|Wn(sj)|2
sj (9.14)
whereCδ is a scale independent constant, which for the Morlet wavelet is 0.776.
The scales are fractional powers of two given by :
sj=so2jδj, j = 0,1,· · ·, J (9.15) and
J =δj−1log2(N δt/so) (9.16)
wheresois the smallest resolvable scale andJ is the largest scale.
The distribution of energy in space at any prescribed scale can be obtained using Eqn. (9.14). The summation of energies at all spatial scales gives the total energy,ET which can be expressed as :
ET =Es1 +Es2+Es3 +· · · (9.17)
whereEs1, Es2, Es3,· · · are energies at each scale, respectively. As mentioned previously and to recap, the Fourier Transform of a signal is a mapping from a function of timex(t) to a function of frequency, F(ω). The functionF(ω) gives information on the extent to which a signal component with frequencyω is present in the analyzed signal. It does not however indicate how that signal component evolves with time t. For that a transform is needed that returns a bivariate function of the form F(t, ω). Wavelet transform provides a time-frequency description of function f as did the windowed Fourier transform.
A difference is in the shapes of the analyzing functions g andψ. They are both shifted along the time axis but all the functionsg have the same time-width while functions ψ have widths adapted to their frequency. At high frequenciesψs,τ, are very narrow and at low frequencies they are much broader. This is why the wavelet transform gives a more flexible approach than windowed Fourier transform. With the
aid of a wavelet transform, it is possible to better zoom in on very short-lived high frequency phenomena, like transients in signals or singularities in functions. Wavelets are mathematical microscopes that are able to magnify a given part of a function with a certain factor represented by a value of scale parameter.
This magnification is associated with the extraction of the information at a certain scale hidden in a local area of the analyzed function.