in Figure 4.9. In this case the distribution is referred to as a Choi–Williams distribution.
The application of a discrete‐time form of the Wigner–Ville distribution to BSS will be discussed later in this chapter and its application to seizure detection has been explained in the chapter devoted to seizure. To improve the distribution a signal‐dependent kernel may also be used [24].
Figure 4.9 Illustration of
4.6 Empirical Mode Decomposition
Empirical mode decomposition (EMD) may be considered as a multiresolution signal decomposition technique. EMD is an adaptive time–space analysis method suitable for processing nonstationary and nonlinear time series. EMD partitions a series into ‘modes’ namely, intrinsic mode functions (IMFs) in the time domain. Like Fourier and WTs EMD does not follow any physical concept. However, the modes may provide insight into various signals contained within the data and have distinct frequency bands/components.
EMD is based on The Hilbert–Huang transform (HHT) which is a way to decompose a signal into IMFs along with a trend and obtain IF data. Its difference from other common transforms like the Fourier transform, is that the HHT is more like an algorithm (an empirical approach) that can be applied to a data set, rather than a theoretical tool.
IMF represents a simple oscillatory mode as a counterpart to the simple harmonic function, but instead of constant amplitude and frequency in a simple harmonic component, an IMF can have variable amplitude and frequency along the time axis.
The procedure of extracting an IMF is called sifting. The sifting process is as follows [25, 26]:
1 Identify all the local extrema in the test data.
2 Connect all the local maxima by a cubic spline line as the upper envelope.
3 Repeat the procedure for the local minima to produce the lower envelope.
The upper and lower envelopes should cover all the data between them. Their mean is m 1. The difference between the data and m 1 is the first component d 1:
(4.77)
Ideally, d 1 should satisfy the definition of an IMF, since the construction of d 1 described above should have made it symmetric and having all maxima positive and all minima negative. After the first round of sifting, a crest may become a local maximum. New extrema generated in this way actually reveal the proper modes lost in the initial examination. In the subsequent sifting process, d 1 can only be treated as a proto‐IMF. In the next step, d 1 is treated as data:
(4.78)
After repeated sifting up to k times, d 1 becomes an IMF, i.e.:
(4.79)
C 1 = d 1k is considered as the first IMF of the signal x(t).
The iteration above can be stopped in different ways such as when the power (standard deviation) of the difference (between current and previous iteration) signal becomes less than a predefined threshold, or when the number of iterations reaches a reasonable number [27].
For calculation of the other IMFs:
(4.80)
The residue r 1 is then treated as the new signal and the same processing is applied to that. Therefore
(4.81)
The sifting process finally stops when the residue, rn , becomes a monotonic function from which no more IMFs can be extracted. From the above equations, it is induced that:
(4.82)
This results in decomposition of the data into n‐empirical modes [25, 26].
Ensemble EMD (EEMD) is a noise assisted data analysis method. EEMD consists of ‘sifting’ an ensemble of white noise‐added signal. EEMD can separate scales naturally without any a priori subjective criterion selection as in the intermittence test for the original EMD algorithm. Complete ensemble EMD with adaptive noise (CEEMDAN) is a variation of the EEMD algorithm that provides an exact reconstruction of the original signal and a better spectral separation of the IMFs.
4.7 Coherency, Multivariate Autoregressive Modelling, and Directed Transfer Function
In some applications such as in detection and classification of finger movement, it is very useful to find out how the associated movement signals propagate within the neural network of the brain. As will be shown in Chapter 16, there is a consistent movement of the source signals from the occipital to temporal regions. It is also clear that during the mental tasks different regions within the brain communicate with each other. The interaction and cross‐talk among the EEG channels may be the only clue to understanding this process. This requires recognition of the transient periods of synchrony between various regions in the brain. These phenomena are not easy to observe by visual inspection of the EEGs. Therefore, some signal processing techniques have to be used in order to infer such causal relationships. One time series is said to be causal to another if the information contained in that time series enables the prediction of the other time series.
The spatial statistics of scalp EEG are usually presented as coherence in individual frequency bands, these coherences result both from correlations among neocortical sources and volume conduction through the tissues of the head, i.e. brain, cerebrospinal fluid, skull, and scalp. Therefore, spectral coherence [28] is a common method for determining the synchrony in EEG activity. Coherency is given as:
(4.83)
Figure 4.10 Cross‐spectral coherence for a set of three electrode EEGs, one second before the right‐finger movement. Each block refers to one electrode. By careful inspection of the figure, it is observed that the same waveform is transferred from Cz to C3.
where