Classification of Normal and Epileptic Seizure EEG Signals Based on Empirical Mode Decomposition - Complex System Modelling and Control Through Intelligent Soft Computations - page 369

where, A ð t Þ is the amplitude envelope of z ð t Þ ,de ned as:

p

x 2 A ð t Þ ¼ ð t Þþ y 2 ð t Þ ð 8 Þ

and /ð t Þ is the instantaneous phase of z ð t Þ ,de ned as:

tan 1 y ð t Þ
x

/ð t Þ ¼ ð 9 Þ

ð t Þ

The instantaneous frequency of the analytic signal can be obtained by differ-
entiating ( 9 ) as:

d /ð t Þ

xð Þ ¼

t

dt

ð 10 Þ

dy ð t Þ
dt

dx ð t Þ
dt

ð Þ ð Þ

x t y t

¼

:

A 2 ð t Þ

is a measurement of
the rate of rotation in the complex plane. The Hilbert transform can be applied on
all IMFs obtained by EMD method. The IMFs are mono-component signals and
exhibit property of locally symmetry. Therefore, the instantaneous frequency is well
localized in the time-frequency domain and reveals a meaningful feature of the
signal (Huang et al. 1998 ).
The analytic signal can be obtained for all the IMFs using the Hilbert transform.
A complex signal can be represented as a sum of proper rotational components using
EMD method which makes it possible to compute the area in a complex plane
(Amoud et al. 2007 ). Since each IMF is a proper rotational component and has its own
rotation frequency, the plot of the analytic IMF follows circular geometry in complex
plane. The complex plane representation can be obtained by tracing the real part
against the imaginary part of the analytic signal. The analytic signal representations in
complex plane corresponding to the normal and epileptic seizure EEG signals and
their

The instantaneous frequency xð t Þ of the analytic signal z ð t Þ

first four intrinsic mode functions are depicted in Figs. 3 and 4 , respectively.
These figures present the traces of entire signals in the complex plane, as well as those
of each IMF for both signals. It can be observed that the shape of this trace is similar to
a rotating curve. The analytic signal representation of IMFs in complex plane possess
a proper structure of rotation with a unique center (Lai and Ye 2003 ).
Central tendency measure (CTM) provides a rapid way to summarize the visual
information related to a graph or plot (Cohen et al. 1996 ). The modi ed CTM can

be used to measure the degree of variability from analytic signal representation of
the signal. CTM can be used to determine the area of the complex plane repre-
sentation (Pachori and Bajaj 2011 ). The radius corresponding to 95 % modi ed

CTM can be used to compute the area of analytic signal representation of the IMF
in complex plane. The modi

ed CTM provides the ratio of points falling inside the

circular region of speci ed radius to the total number of points in analytic signal