Single-Channel EEG Analysis - Quantitative EEG Analysis Methods and Clinical Applications - page 54

PY= angle(Y)

3.1.1.2 Windowing
EEG signals are often divided into finite time segments. Segmentation or truncation
in the time domain is equivalent to multiplication of the complete EEG signal with a
finite time rectangular window. Because multiplication in time is equivalent to con-
volution in frequency, the Fourier transform of the signal after windowing is more
complex and will leak or extend over a wider frequency range than the original sig-
nal [41]. The abrupt transition of the signal values in the case of a rectangular win-
dow results in the appearance of ripples in the discrete Fourier transform. These
ripples can be reduced using alternative window functions. Many window functions
are available in the literature [38-41]. The following examples represent four of the
most popular windowing functions:

1. Rectangular:

⎨
⎩

1
0

,
,

nN <

[]

Wn = (3.6)

R

otherwise

2. Bartlett:

Nn
N

−

⎧
⎪
⎪

, nN <

[]

Wn = (3.7)

B

0 otherwise

3. Hamming:

⎧
⎪
⎪

⎛ 2 π n ⎞

054 . − 046 . cos ⎜ ⎟ , nN <

[]

Wn = (3.8)

N − 1

⎝ ⎠

H

0 , otherwise

4. Hanning:

⎧
⎨

⎛ ⎛ 2 π n ⎞ ⎞

1
2

⎜ ⎟

⎪ 1 − cos ⎜ ⎟ , nN <

[]

Wn = (3.9)

N − 1

⎝ ⎠

⎝ ⎠

H

⎪

0 , otherwise

The selection of the most appropriate window is not a straightforward matter
and depends on the application at hand and may require some trial and error. The
window functions while reducing the ripples and tends to reduce sharp variations or
resolution of the discrete Fourier transform. For example, if we are interested in
resolving two narrowband, closely spaced spectral components, then a rectangular
window is appropriate because it has the narrowest mainlobe in the frequency
domain. If, on the other hand, we have two signals that are not closely spaced in the
frequency domain, then a window with rapidly decaying sidelobes is preferred.