Biomedical Engineering Reference
In-Depth Information
multiple inputs with multiple output systems or single input with multiple outputs sys-
tems [1]. The relationships between the input transfer function of the system and the
output are analyzed by several techniques [2].
This chapter will deal primarily with coherence function of single input, single
output systems only. The coherence function is a function of frequency. Its maximum
values occur at the frequencies where the greatest transfer of energy within a system takes
place.
18.2 DESCRIPTION OF THE COHERENCE FUNCTION
The simplest explanation of the coherence function would be that it is similar to the
correlation coefficient,
r
[3]. The correlation coefficient,
r
, is a measure of the linear
relationship between two variables, with one being the independent variable and the
other being the dependent variable. Mathematically, the correlation coefficient,
r
,is
shown in terms of variance as given by (18.1) through (18.5), where (18.1) through
(18.3) are the mathematical definitions of terms.
n
1
n
2
X
σ
=
Xi Xi
=
Variance of
X
(18.1)
i
=
1
n
YiYi
1
2
y
σ
=
=
Variance of
Y
(18.2)
n
YiXi
1
yx
=
=
Covariance of
X
and
Y
(18.3)
t
−1
(
x
t
s
xy
s
x
s
y
=
−
x
)(
y
t
−
y
)
r
xy
=
ρ
ˆ
=
(18.4)
t
−1
(
x
t
y
)
2
1
/
2
xy
x
)
2
t
−1
(
y
t
−
−
or
t
−1
x
t
y
t
−
N x y
=
(18.5)
t
−1
x
t
N x
2
t
−1
y
t
N y
2
1
/
2
−
−
1. The correlation
coefficient squared term
r
2
is the coefficient of determination, which is often referred to as
The correlation coefficient,
r
, value has a range from
−
1to
+
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