Subjects signal processing

Morlet Wavelet E414A4

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1. The problem is to provide the Morlet mother wavelet and the continuous wavelet transform (CWT) equations in a format suitable for copying into Word. 2. The Morlet mother wavelet is defined as: $$\psi^{Morelet}(t) = \frac{1}{\pi^{1/4}} e^{i t \omega_0} e^{-t^2/2}$$ where $t$ is the normalized time and $\omega_0$ is the normalized frequency. 3. According to the literature, $\omega_0 = 6$ is chosen to balance time and frequency localization. 4. The continuous wavelet transform (CWT) of a time series $x$ is given by: $$W_x(\tau, s) = \int_{-\infty}^{\infty} x(t) \tilde{\psi}_{\tau,s}^*(t) dt, \quad s, \tau \in \mathbb{R}, s \neq 0$$ where $\tilde{\psi}_{\tau,s}^*(t)$ is the complex conjugate of the scaled and translated mother wavelet. These equations can be copied directly into Word using the equation editor with LaTeX support.