46
D.-W. Kim and C.-H. Im
once the signal recording is done. For instance, it is impossible to observe high
frequency oscillations (HFO, >100 Hz) once the recording is done with a sampling
rate of 200 Hz.
3.3.2 Analysis Window Size and Frequency Resolution
Suppose that we have a set of EEG data continuously recorded from a subject for
relatively long duration (e.g., 2 min). It is possible to estimate the power spectrum of
the signal by applying FFT to whole recording data; however, using so much data for
spectral estimation is not generally recommended because of the high computational
burden and possible artifacts that could be included during the long-term recordings
(e.g., gross motion artifacts or ocular artifacts). Long-term recordings may increase
the frequency resolution; however, because our main interest is usually the average
power in a certain frequency band (e.g., alpha band power), overly high frequency
resolution is generally regarded as excessive information. In addition, for an objective
comparison, it is important that the total data length used for data analysis should be
uniform among subjects. Therefore, a common procedure for EEG spectral analysis
is to divide the long-term recording into smaller pieces, called epochs, and take an
average of the spectral analysis results over artifact-free epochs [32].
The length of the epoch is a crucial factor, because it determines the frequency
resolution of the spectrum. The frequency resolution f c is determined by the following
equation:
f c
f s
N
,
(3.13)
where f s is the sampling frequency, and N is the number of samples in the epoch.
Hence, for a fixed sampling frequency, the number of samples, which is proportional
to time, determines the frequency resolution. Because the sampling frequency is fixed
before recording, researchers can change the frequency resolution of the spectrum
by adjusting the length of the analysis window.
Another issue regarding the frequency resolution is the so-called picket fence
effect [21, 31]. Since the Fourier transform is applied to a sampled (discrete) signal,
the spectrum has values only at discrete frequency samples that are multiples of the
fundamental frequency f c . Therefore, the spectral information is accurate only when
the frequency of the original signal matches nf c (n 0, 1, 2,…, N − 1); otherwise,
leaked frequency components can be observed in adjacent frequency samples. For
instance, suppose a signal has a convex spectrum peaking at f 1 , as shown in Fig. 3.7a.
With the frequency resolution f c , the main peak of the signal power spectrum can
be correctly represented at f 1 . However, if the peak of the spectrum is located at
f 2 , which is the middle point of f 1 and the next frequency sample (=(f 1 + f c )/2),
the spectral characteristics of the signal cannot be correctly estimated, as shown in
Fig. 3.7b.
D.-W. Kim and C.-H. Im
once the signal recording is done. For instance, it is impossible to observe high
frequency oscillations (HFO, >100 Hz) once the recording is done with a sampling
rate of 200 Hz.
3.3.2 Analysis Window Size and Frequency Resolution
Suppose that we have a set of EEG data continuously recorded from a subject for
relatively long duration (e.g., 2 min). It is possible to estimate the power spectrum of
the signal by applying FFT to whole recording data; however, using so much data for
spectral estimation is not generally recommended because of the high computational
burden and possible artifacts that could be included during the long-term recordings
(e.g., gross motion artifacts or ocular artifacts). Long-term recordings may increase
the frequency resolution; however, because our main interest is usually the average
power in a certain frequency band (e.g., alpha band power), overly high frequency
resolution is generally regarded as excessive information. In addition, for an objective
comparison, it is important that the total data length used for data analysis should be
uniform among subjects. Therefore, a common procedure for EEG spectral analysis
is to divide the long-term recording into smaller pieces, called epochs, and take an
average of the spectral analysis results over artifact-free epochs [32].
The length of the epoch is a crucial factor, because it determines the frequency
resolution of the spectrum. The frequency resolution f c is determined by the following
equation:
f c
f s
N
,
(3.13)
where f s is the sampling frequency, and N is the number of samples in the epoch.
Hence, for a fixed sampling frequency, the number of samples, which is proportional
to time, determines the frequency resolution. Because the sampling frequency is fixed
before recording, researchers can change the frequency resolution of the spectrum
by adjusting the length of the analysis window.
Another issue regarding the frequency resolution is the so-called picket fence
effect [21, 31]. Since the Fourier transform is applied to a sampled (discrete) signal,
the spectrum has values only at discrete frequency samples that are multiples of the
fundamental frequency f c . Therefore, the spectral information is accurate only when
the frequency of the original signal matches nf c (n 0, 1, 2,…, N − 1); otherwise,
leaked frequency components can be observed in adjacent frequency samples. For
instance, suppose a signal has a convex spectrum peaking at f 1 , as shown in Fig. 3.7a.
With the frequency resolution f c , the main peak of the signal power spectrum can
be correctly represented at f 1 . However, if the peak of the spectrum is located at
f 2 , which is the middle point of f 1 and the next frequency sample (=(f 1 + f c )/2),
the spectral characteristics of the signal cannot be correctly estimated, as shown in
Fig. 3.7b.