72
M. Congedo
We end up this section with some considerations about TFA analysis. The Hilbert
transform can be obtained by the FFT algorithm [64]. The use of this algorithm
requires the choice of a tapering window in the time domain to counteract spectral
leakage due to finite window size (see Harris [39]). As illustrated in Fig. 4.4, the
analytic signal does not necessarily represent adequately the phase of the original
signal. The study of Chavez et al. [13] has stressed that this is the case in general only
if the original signal is a simple oscillator with a narrow-band frequency support.
These authors have provided useful measures to check empirically the goodness of
the analytic signal representation. Because of this limitation, for a signal displaying
multiple spectral power peaks or broad-band behavior, which is the case in general
of EEG and ERP, the application of a filter bank to extract narrow-band behavior
is necessary. When applying the filter bank, one has to make sure not to distort the
phase of the signal. In general, a finite impulse response filter with linear phase
response is adopted (see Widmann et al. [98], for a review). The choice of the filters
band width and frequency resolution is usually a matter of trials and errors; the band
width should be large enough to capture the oscillating behavior and small enough
to avoid capturing several oscillators in adjacent frequencies. Also, the use of filter
banks engenders edge effects, that is, severe distortions of the analytic signal at the
left and right extremities of the time window under analysis [67]. This latter problem
is easily solved defining a larger time window centered at the window of interest and
successively trimming an adequate number of samples at both sizes, as we have done
in the example of Fig. 4.7. The estimation of instantaneous phase for sweeps, time
sample and frequencies featuring a low SNR are meaningless; the phase being an
angle, it is defined for vectors of any length, even if the length (i.e., the amplitude) is
negligible. However, phase measures can be interpreted only where the amplitude is
high [7]. The effect is exacerbated if we apply the non-linear normalization, since in
this case very small coefficients are weighted as the others in the average, whereas
they should better be ignored.
4.5 Spatial Domain Analysis
Scalp topography and tomography (source localization) of ERPs are the basic tools
to perform analysis in the spatial domain of the electrical activity generating ERPs.
This is fundamental for linking experimental results to brain anatomy and physiology. It also represents an important dimension for studying ERP dynamics per se,
complementing the information provided in time and/or frequency dimensions [52].
The spatial pattern of ERP scalp potential or of an ERP source component provides
useful information to recognize and categorize ERP features, as well as to identify
artifacts and background EEG. Early ERP research was carried out using only a
few electrodes. Current research typically uses several tens and even hundreds of
electrodes covering the whole scalp surface. More and more high-density EEG studies involve realistic head models for increasing the precision of source localization
M. Congedo
We end up this section with some considerations about TFA analysis. The Hilbert
transform can be obtained by the FFT algorithm [64]. The use of this algorithm
requires the choice of a tapering window in the time domain to counteract spectral
leakage due to finite window size (see Harris [39]). As illustrated in Fig. 4.4, the
analytic signal does not necessarily represent adequately the phase of the original
signal. The study of Chavez et al. [13] has stressed that this is the case in general only
if the original signal is a simple oscillator with a narrow-band frequency support.
These authors have provided useful measures to check empirically the goodness of
the analytic signal representation. Because of this limitation, for a signal displaying
multiple spectral power peaks or broad-band behavior, which is the case in general
of EEG and ERP, the application of a filter bank to extract narrow-band behavior
is necessary. When applying the filter bank, one has to make sure not to distort the
phase of the signal. In general, a finite impulse response filter with linear phase
response is adopted (see Widmann et al. [98], for a review). The choice of the filters
band width and frequency resolution is usually a matter of trials and errors; the band
width should be large enough to capture the oscillating behavior and small enough
to avoid capturing several oscillators in adjacent frequencies. Also, the use of filter
banks engenders edge effects, that is, severe distortions of the analytic signal at the
left and right extremities of the time window under analysis [67]. This latter problem
is easily solved defining a larger time window centered at the window of interest and
successively trimming an adequate number of samples at both sizes, as we have done
in the example of Fig. 4.7. The estimation of instantaneous phase for sweeps, time
sample and frequencies featuring a low SNR are meaningless; the phase being an
angle, it is defined for vectors of any length, even if the length (i.e., the amplitude) is
negligible. However, phase measures can be interpreted only where the amplitude is
high [7]. The effect is exacerbated if we apply the non-linear normalization, since in
this case very small coefficients are weighted as the others in the average, whereas
they should better be ignored.
4.5 Spatial Domain Analysis
Scalp topography and tomography (source localization) of ERPs are the basic tools
to perform analysis in the spatial domain of the electrical activity generating ERPs.
This is fundamental for linking experimental results to brain anatomy and physiology. It also represents an important dimension for studying ERP dynamics per se,
complementing the information provided in time and/or frequency dimensions [52].
The spatial pattern of ERP scalp potential or of an ERP source component provides
useful information to recognize and categorize ERP features, as well as to identify
artifacts and background EEG. Early ERP research was carried out using only a
few electrodes. Current research typically uses several tens and even hundreds of
electrodes covering the whole scalp surface. More and more high-density EEG studies involve realistic head models for increasing the precision of source localization