Saeid Sanei - EEG Signal Processing and Machine Learning

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Explore cutting edge techniques at the forefront of electroencephalogram research and artificial intelligence from leading voices in the field

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(4.61) where ω lis the nearest frequency to the original point ω a b ω is the - фото 226

where ω lis 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 - фото 227,∆ 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 lof 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 lmust 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 463 As an example if - фото 228

This function has its maximum value at the origin as

(4.63) EEG Signal Processing and Machine Learning - изображение 229

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

EEG Signal Processing and Machine Learning - изображение 230

(4.64) The ambiguity function A x τν will be in the form of 465 This concept - фото 231

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

(4.65) This concept is very important in separation of signals using the TF domain - фото 232

This concept is very important in separation of signals using the TF domain. This will be addressed in the context of blind source separation (BSS) later in this chapter. Figure 4.8demonstrates this concept.

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) which changes to the dual form of the ambiguity function as 467 A - фото 233

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

(4.67) A quadratic form for the TF representation with the WignerVille distribution - фото 234

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) The WignerVille distribution is real and has very good resolution in both the - фото 235

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 | > t 0, then X WV( t , ω ) = 0 for | t | > t 0, and if X ( ω ) = 0 for | ω | > ω 0, then X WV( t , ω ) = 0 for | ω | > ω 0. It has also both time‐marginal and frequency‐marginal conditions of the form:

(4.69) and 470 If x t is the sum of two signals x 1 t and x 2 t ie x - фото 236

and

(4.70) If x t is the sum of two signals x 1 t and x 2 t ie x t x 1 - фото 237

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) where Re denotes the real part of a complex value and 472 It is seen - фото 238

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

(4.72) It is seen that the distribution is related to the spectra of both auto and - фото 239

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.:

(4.73) Figure 48 a A segment of a signal consisting of two modulated components - фото 240

Figure 48 a A segment of a signal consisting of two modulated components - фото 241

Figure 4.8 (a) A segment of a signal consisting of two modulated components, (b) ambiguity function for x 1( t ) only, and (c) the ambiguity function for x ( t ) = x 1( t ) + x 2( t ).

In order to suppress the undesired cross‐terms the two‐dimensional Wigner–Ville (WV) distribution may be convolved with a TF‐domain window. The window is a two‐dimensional lowpass filter, which satisfies the time and frequency‐marginal (uncertainty) conditions, as described earlier. This can be performed as:

(4.74) where 475 and φ τ ν is often selected from a set of well known - фото 242

where

(4.75) and φ τ ν is often selected from a set of well known waveforms called - фото 243

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:

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