Biomedical Engineering Reference
In-Depth Information
standard programs do this without your knowledge, but if you are writing
your own program, you must do so or a major error will result. Consider
x (t )
=
s
1
(t )
+
m
1
and
y(t )
=
s
2
(t )
+
m
2
, where
m
1
and
m
2
are the means of
s
1
and
s
2
, respectively.
T
R
xy
(τ )
=
(s
1
(t )
+
m
1
)(s
2
(t
+
τ)
+
m
2
) dt
0
T
T
=
s
1
(t )s
2
(t
+
τ)dt
+
m
1
s
2
(t
+
τ)dt
0
0
T
T
+
m
2
s
1
(t )dt
+
m
1
m
2
dt
0
0
Since the signals and
m
1 and
m
2 are uncorrelated, the 2
nd
and 3
rd
terms
will
=
0.
T
T
∴
R
xy
(τ )
=
s
1
(t )s
2
(t
+
τ)dt
+
m
1
m
2
dt
0
0
The 1
st
term is the desired cross-correlation, but a major bias will added
by the 2
nd
term, and the peak of
R
xy
(τ )
may be grossly exaggerated.
2.1.6 Digital Implementation of Auto- and Cross-Correlation
Functions
Since data are now routinely collected and stored in a computer, the
implementation of the auto- and cross-correlation is the digital equivalent of
Equations (2.2) and (2.3), shown below in Equations (2.8) and (2.9)
N
1
N
[
(x (n)
−
x )(x (n
+
τ)
−
x )
]
n
=
1
R
xx
(τ )
=
(2.8)
n
=
1
N
1
N
(x (n)
−
x )
2
N
(x (n)
y)
1
N
−
x )(y(n
+
τ)
−
n
=
1
R
xy
(τ )
=
(2.9)
N
1
N
(x (n)
−
x )(y(n)
−
y)
=
n
1
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