Biomedical Engineering Reference
In-Depth Information
Visual analysis of EEG signals in the time domain is an empirical science and
requires a considerable amount of clinical and neurological knowledge. Many brain
abnormalities are diagnosed by a doctor or an electroencephalographer after visual
inspection of brain rhythms in the EEG signals. However, long-term monitoring and
visual interpretation is very subjective and does not lend itself to statistical analysis
[27, 28]. Therefore, alternative methods have been used to
quantify
information
carried by an EEG signal. Among these are the Fourier transform, the wavelet trans-
form, chaos, entropy, and subband wavelet entropy methods [29-37].
The main goal of this chapter is to provide the reader with a broad perspective of
classical and modern spectral estimation techniques and their implementations. The
reader is assumed to have some fundamental knowledge of signals and systems that
covers continuous and discrete linear systems and transform theory. Practicing engi-
neers and neuroscientists working in neurological engineering will also find this
chapter useful in their research work in EEG signal processing. Because all of the
EEG spectral analysis techniques are performed using computers, the focus is
directed more toward discrete time EEG signals. Furthermore, because most practic-
ing scientists and researchers working with EEG signals use MATLAB, various
relevant MATLAB functions are also included.
The remainder of this section is organized into three major sections. In Section
3.1.1, classical spectral analysis is covered, including Fourier analysis, windowing,
correlation and estimation of the power spectrum, the periodogram, and Welch's
method. This is followed by an illustrative application. In Section 3.1.2, modern
spectral techniques using parametric modeling of EEG signals are covered. These
models include autoregressive moving average and autoregressive spectrum estima-
tion. In Section 3.1.3, time-frequency analysis techniques are detailed for analyzing
nonstationary EEG signals. The techniques included in this section are the
short-time Fourier transform and the wavelet transform.
3.1.1 Classical Spectral Analysis of EEGs
3.1.1.1 Fourier Analysis
The EEG signal can be represented in several ways including the time and frequency
domains. Fourier analysis is the process of decomposing a signal into its frequency
components. Fourier analysis is a very powerful method that can be used to reveal
information that cannot be easily seen in the time domain. The Fourier transform
uses sinusoidal functions or complex exponential signals as basis functions. The
Fourier transform of a continuous real-time aperiodic signal
x
(
t
) is defined as
follows [38-40]:
∞
(
{}
()
() ( )
∫
Fxt
=
X
ω
=
xt
exp
−
jtdt
ω
(3.1)
−∞
where
ω
2
π
f
is the angular frequency in radians/s, and
F
{
°
} is the Fourier operator.
The Fourier transform is complex for real signals.
The inverse Fourier transform is the operator that transforms a signal from the
frequency domain into the time domain. It represents the synthesis of signal
x
(
t
)asa
weighted combination of the complex exponential basis functions. It is defined as
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