Single-Channel EEG Analysis - Quantitative EEG Analysis Methods and Clinical Applications - page 56

Time domain Frequency domain

40

1

20
0

0.8

− 20

0.6

− 40

0.4

− 60

0.2

− 80

− 100

0

0 0.2 0.4 0.6 0.8

10 20 30 40 50 60

Samples Normalized frequency (

×π

rad/sample)

Figure 3.3 Hamming window of L = 65 and its Fourier transform magnitude in decibels.

assumed to be a random signal generated by a stochastic process [5-9]. The direct
application of the Fourier transform is not attractive for random processes such as
the EEG because the transform may not even exist. If we use power instead of volt-
age as a function of frequency, then such a spectral function will exist. The power
spectrum or the power spectral density (PSD) of a random signal x ( n ) is defined as
the Fourier transform of the autocorrelation function r xx ( m ). It is defined as follows:

N −

∑

1

( {} () () ( )

PSD xt = S ω = r m exp − j m ω (3.10)

xx
mN

( =− − 1

)

where

Nm −−

∑

1

1

() ()( )

rm = xnxn m + (3.11)

xx

N

n = 0

However, it can be shown that the PSD obtained in (3.10) is equivalent to that
obtained using the DFT in (3.3) [39-41]:

1

()

2

( {} ()

Xe j

ω

PSD xt = S ω = (3.12)

N

The PSD estimation using the DFT is known as the periodogram , which can eas-
ily be calculated using the FFT method. If we increase N , the mean value of the
periodogram will converge to the true PSD, but unfortunately, the variance does not
decrease to zero. Therefore, the periodogram is a biased estimator. To reduce the
variance of the periodogram, ensemble averaging is used. The resultant power spec-
trum is called the average periodogram. One of the most popular methods for com-
puting the average periodogram is the Welch method, in which windowed
overlapping segments are used [39, 40]. The procedure for computing the PSD of a
given sequence of N data points is as follows:

1. Divide the data sequence into K segments of M samples each.
2. Compute the periodogram of each windowed segment using the FFT
algorithm: