EEG Signal Processing and Machine Learning. Saeid Sanei

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(4.59)

One assumption in the above equation (Eq. 4.59) is that the selected mother wavelet is concentrated within the positive energy range, which means , for ε < 0. If is concentrated around ε = ω ₀, then, W(a, b) is concentrated around a = ω ₀/ω. This is spread out over a region of the horizontal line (e.g. a = ω ₀/ω). In the situation where ω is almost but not exactly similar to the actual instantaneous frequency (IF) of the input signal, then, W(a, b) has non‐zero energy. By synchro‐squeezing, the idea is to move this energy away from ω. The proposed method in the SSWT aims to reassign the frequency locations which are closer to the actual IF to obtain an enhanced spectrum. To do that, first, the candidate IFs are calculated. For these IFs, W(a, b) ≠ 0:

      (4.60)

Considering the selected pure harmonic signal f(t) = Acos(ωt), it is simple to observe that ω(a, b) = ω. The candidate IFs are exploited to recover the actual frequencies. Therefore, a reallocation technique has been used to map the time domain into TF domain using (b, a) ⇒ (b, ω(a, b)). Based on this, each value of W(a, b) (computed at discrete values of a_k ) is re‐allocated into T_f (ω_l , b) as provided in the following equation (Eq. 4.61):

(4.61)

where ω_l is the nearest frequency to the original point ω(a, b), ∆ω is the width of the frequency bins ,∆ω = ω_l − ω_{l − 1}, and (∆a) _k = a_k − a_{k − 1}. T_f (ω_l , b) represents the synchro‐squeezed transform at the centres ω_l of consecutive frequency bins. For each fixed time point b, the reassigned frequencies should be estimated for all scales using Eq. (4.103). For each desired IF of ω_l , T_f (ω_l , b) is calculated using summation of all W(a_k , b) considering that the distance between the reassigned frequency ω(a_k , b) and ω_l must be within the specified frequency bin width (∆ω). It has been shown that the original signal can be reconstructed after the synchro‐squeezing process [23].

      4.5.3 Ambiguity Function and the Wigner–Ville Distribution

      The ambiguity function for a continuous time signal is defined as:

      (4.62)

      This function has its maximum value at the origin as

      (4.63)

      As an example, if we consider a continuous time signal consisting of two modulated signals with different carrier frequencies such as