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Biomedical Signal and Image Processing
first-order variations of the signal, while signal complexity deals with the secondorder variations. Consider a biomedical signal x i , i = 1,…, N. Also, let signal d j ,
j = 1,…, N − 1, represent the vector of the first-order variations in x, i.e.,
d j = x j +1 − x j
(6.1)
Moreover, define the signal g k , k = 1,…, N − 2, as the vector of the second-order
variations in x, i.e.,
g = d − d
(
k
k +1
k
6.2)
Then, using the concepts x, d, and g, we define the following fundamental first- and
second-order factors:
x i
S =
∑
N
2
i=1
0
,
(6.3)
N
∑
N −1
d
2
j
S 1 =
j =2
,
(6.4)
N 1
−
∑
N −2
g
2
k
S 2 =
k =3
,
(6.5)
N − 2
Now, we can define signal complexity and signal mobility as follows:
S
2
l complexity =
2
S
2
Signa
1
(6.6)
S
2
−
S
2
1
0
and
S
Signal mobility =
1
(6.7)
S 0
These two measures are heavily used in biomedical signal processing specially in
processing of EEG, ECG, and electromyogram (EMG) signals as described in Part  II
of this book.
6.2.2 FRACTAL DIMENSION
Fractal dimension, which is frequently used in analysis of biomedical signals such
as EEG and ECG, is a nonlocal measure that describes the complexity of the fundamental patterns hidden in a signal. Fractal dimension can also be considered as a