EEG Signal Processing and Machine Learning. Saeid Sanei

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Название EEG Signal Processing and Machine Learning
Автор произведения Saeid Sanei
Жанр Программы
Серия
Издательство Программы
Год выпуска 0
isbn 9781119386933



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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 ak ) is re‐allocated into Tf (ωl , b) as provided in the following equation (Eq. 4.61):

      4.5.3 Ambiguity Function and the Wigner–Ville Distribution

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

      (4.62)equation

      This function has its maximum value at the origin as

      (4.63)equation

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

equation

      (4.64)equation

      The ambiguity function Ax (τ,ν) will be in the form of:

      (4.65)equation

      The Wigner–Ville frequency distribution of a signal x(t) is then defined as the two‐dimensional Fourier transform of the ambiguity function:

      (4.66)equation

      which changes to the dual form of the ambiguity function as:

      (4.67)equation

      A quadratic form for the TF representation with the Wigner–Ville distribution can also be obtained using the signal in the frequency domain as:

      (4.68)equation

      The Wigner–Ville distribution is real and has very good resolution in both the time‐ and frequency‐domains. Also it has time and frequency support properties, i.e. if x(t) = 0 for | t | > t0 , then XWV (t, ω) = 0 for | t | > t0 , and if X(ω) = 0 for | ω | > ω 0, then XWV (t, ω) = 0 for | ω | > ω 0. It has also both time‐marginal and frequency‐marginal conditions of the form:

      (4.69)equation

      and

      (4.70)equation

      If x(t) is the sum of two signals x 1(t) and x 2(t), i.e. x(t) = x 1(t) + x 2(t), the Wigner–Ville distribution of x(t) with respect to the distributions of x 1(t) and x 2(t) will be:

      (4.71)equation

      where Re{.} denotes the real part of a complex value and

      It is seen that the distribution is related to the spectra of both auto‐ and cross‐correlations. A pseudo‐Wigner–Ville distribution (PWVD) is defined by applying a window function, w(τ), centred at τ = 0 to the time‐based correlations, i.e.:

      (4.74)equation

      where

      (4.75)equation

      and φ(τ, ν) is often selected from a set of well known waveforms called Cohen's class. The most popular member of the Cohen's class of functions is the bell‐shaped function defined as: