Continuous Wavelet Transform

The transform starts by selecting a wavelet function, called mother wavelet, that can be a function of space and time, hence a function of time, Ψ(𝑡), is used here since the signal to be analyzed is a temporal signal. However, the mother wavelet must satisfy the square-integrability and the admissibility condition for finite energy and inverse function, respectively (Rao and Bopardikar, 1998;

Mallat, 1999). The square-integrability is

𝐸 = ∫ |Ψ(𝑡)|_−∞^{∞ 2}𝑑𝑡< ∞ (2.17)

where 𝐸 is the energy of a function and the vertical brackets, |∙|, represent the absolute operator.

The admissibility condition is

𝐶_𝑎 = ∫₀^{∞|FT[Ψ(𝑡)]|}_𝑓 ²𝑑𝑓 < ∞ (2.18)

where the FT[∙] denotes FT and 𝐶_𝑎 is the admissibility constant. For complex wavelet functions, an additional criterion must be held that the FT must be both real and vanished for negative frequencies (Addison, 2017). Figure 2.5 shows some mother wavelets commonly used in practice.

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All derivatives of the Gaussian function can be employed as a mother wavelet. For example, the first derivative of the Gaussian function is called Gaussian wavelet and the second derivative is called Mexican hat wavelet. These two are often used in most of applications; higher-order derivatives are less commonplace. The Haar wavelet is the simplest example of wavelet functions and is also commonly used as a mother wavelet. The Morlet wavelet (or referred to as Gabor wavelet) is a wavelet function composed of a complex exponential function as a carrier and a Gaussian window as an envelope.

To perform a good wavelet analysis, two basic manipulations can be implemented to make the wavelet functions more flexible: stretching/squeezing and forward/backward moving. The stretching creates a dilation and the squeezing creates a contraction, and the first manipulation is named as dilating or scaling. The second manipulation, forward and backward moving along the time axis, is named as translating or shifting. According to these two manipulations, countless wavelet functions with different time and frequency localization can be generated using one selected mother wavelet.

By using the selected mother wavelet and the manipulations, the continuous wavelet transform (CWT) of a time series, 𝑥(𝑡), is defined as: wavelet, the multiplication of ¹

√𝑎 accounts for energy conservation, and 𝑎 and 𝑏 are the dilation and translation parameters, respectively. Both two parameters are real numbers and 𝑎 must be positive. It is noted that various wavelet functions can be generated by scaling (or dilating) and shifting (or translating) the mother wavelet using these two parameters; the dilation and contraction are governed by the dilation parameter and the movement along the time axis is governed by the translation parameter. To be elaborately described, the translation parameter, 𝑏, indicates the location of wavelet function in CWT and changing this parameter to shift the wavelet function along the time axis implies examining the time series in the neighborhood of current location. Therefore, the information in the time domain can be retained, in contrast with FT where the information in time domain becomes invisible after the integration over the whole time axis. The dilation parameter, 𝑎,

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indicates the width of wavelet function and replacing this parameter by a smaller value to contract the wavelet implies a higher-resolution filter in the frequency domain, i.e., the time series is examined through a narrower wavelet function, and vice versa.

Thus, the wavelet coefficients can be calculated for various locations and various scales using Equations (2.19) and (2.20). Eventually, the transform is done in a continuous and smooth fashion, and the time-frequency plane can be filled up with the wavelet coefficients. A visual representation of energy in the frequency plane is often used to interpret the information hiding in the time-frequency domain. This visualization is called wavelet scalogram and Figure 2.6 shows the wavelet spectrogram of the time series in Equation (2.15).

As a counterpart of inverse FT, there is an inverse CWT which is defined as:

𝑥(𝑡) =_𝐶¹

𝑎∫ ∫_−∞^∞ ₀^∞_𝑎¹₂𝑊_𝑎,𝑏Ψ_𝑎,𝑏(𝑡)𝑑𝑎𝑑𝑏 (2.21) This equation allows the original time series to be reconstructed from its wavelet coefficients by integrating over all the dilation and translation parameters. It can be observed in Equation (2.18) that the reconstruction can be done only if the selected mother wavelet satisfies the admissibility condition.

In CWT, the dilation parameters are inversely proportional to its characteristic frequencies (center frequency). The following relationship can be used to calculate the pseudo-frequencies, 𝐹_𝑎, corresponding to the dilation parameters

𝐹_𝑎 = ^𝑓𝑠_𝑎^𝐹𝑐 (2.22)

where 𝑓_𝑠 is the sampling rate (the reciprocal for sampling interval), 𝐹_𝑐 is the characteristic frequency of the wavelet function in Hertz, and 𝐹_𝑎 is also presented in Hertz. The time-frequency plane expressed by the pseudo-frequencies rather than scales is known as wavelet spectrogram, as shown in Figure 2.7.

According to the uncertainty principle, the resolution in the time and frequency domains cannot be arbitrarily small because its product is lower bounded. An important characteristic of CWT is that the ratio of the pseudo-frequency to the width of the frequency band is independent of the dilation and translation parameters. It provides the so-called zoom-in and zoom-out capability that the time window narrows in the high frequency range and widens in the low frequency range. Therefore, in wavelet analysis, the time resolution becomes arbitrarily good at high frequencies, while the frequency resolution becomes arbitrarily good at low frequencies. This is exactly the most desirable

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characteristic in time-frequency analysis and it overcomes the limitation of FT and STFT (Chui, 1992).