Biomedical Engineering Reference
In-Depth Information
The fact that the autocorrelation is composed of cosines implies that it must be an
even function of
0, all of the cosine functions reinforce each other to give
a peak positive value for the autocorrelation. Whether this peak value is ever attained
for other shifts of
τ
. When
τ
=
depends on whether the components of the signal are harmonically
related or not. The peak value of the autocorrelation function at
τ
0 is simply the mean
square value, or average power of the signal, and is given by (17.3) [1].
τ
=
T
0
/
)
τ
=
2
lim
[
f
(
t
)]
2
dt
R
xx
(
τ
=
R
xx
(0)
=
(17.3)
0
T
0
→∞
−
T
0
/
2
17.4 THE CROSS-CORRELATION FUNCTION
The cross-correlation function is essentially a time-averaged measure of shared signal
properties and is defined as (17.4).
T
0
/
2
lim
1
T
0
R
xy
(
τ
)
=
f
1
(
t
)
f
2
(
t
+
τ
)
dt
(17.4)
T
0
→∞
−
T
0
/
2
where
is a time shift on one of the signals. Since signals to be compared must be of
finite duration, the function is modified as shown in (17.5).
τ
∞
R
xy
(
τ
)
=
f
1
(
t
)
f
2
(
t
+
τ
)
dt
(17.5)
−∞
The cross-correlation function reflects the product of the amplitudes of
f
1
(
t
) and
f
2
(
t
), their common frequency,
, and their relative phase angle. When the two signals
being cross-correlated share a number of common frequencies, each gives a corresponding
contribution to the cross-correlation function as shown in Fig. 17.3. Since it retains
information about the relative phases of common frequency components in two signals,
the cross-correlation function, unlike the autocorrelation function, is not normally an
even function of
ω
(4).
From the previous discussions, it can be seen that the autocorrelation and cross-
correlation functions provide all analytical means, which may evaluate the magnitude
τ
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