Biomedical Engineering Reference
In-Depth Information
9.2 PROPERTIES OF THE AUTOCORRELATION
FUNCTION
One of the properties of an autocorrelation function is that although it retains all har-
monics of the given function, it discards all their phase angles. In other words, all periodic
functions having the same harmonic amplitudes but differing in their initial phase angles
have the same autocorrelation function or that the power spectrum of a periodic function
is
independent
of the phase angles of the harmonics.
Another important property is that the autocorrelation function is an
Even Func-
tion
of
τ
. An even periodic function satisfies the expression given by (9.20).
ϕ
11
(
−
τ
)
=
ϕ
11
(
τ
)
(9.20)
The proof is as follows: writing (9.16) as a function of a negative delay,
−
τ
, gives
(9.21).
T
1
/
2
1
T
1
ϕ
11
(
−
τ
)
=
f
1
(
t
)
f
1
(
t
−
τ
)
dt
(9.21)
−
T
1
/
2
Using the change of variable, since
x
=
(
t
−
τ
)or
x
=
(
t
+
τ
), then the limits be-
come
T
1
/
2
−
τ
;
−
T
1
/
2
−
τ
, and
f
1
(
t
−
τ
) becomes
f
1
(
x
+
τ
−
τ
)
=
f
1
(
x
). Likewise,
f
1
(
t
) becomes
f
1
(
t
+
τ
). Also since
ϕ
11
(
τ
) is a periodic function of period
T
1
, integra-
tion over the interval,
−
T
1
/
2
−
τ
to
T
1
/
2
−
τ
, is the same as
T
1
/
2
to
−
T
1
/
2
. The result
is 9.22:
T
1
/
2
1
T
1
ϕ
11
(
−
τ
)
=
f
1
(
t
)
f
1
(
t
+
τ
)
dt
or that
ϕ
11
(
−
τ
)
=
ϕ
11
(
τ
)
(9.22)
−
T
1
/
2
9.3 STEPS IN THE AUTOCORRELATION PROCESS
The autocorrelation process requires the five steps listed as follows:
1.
Change of variable from time (
t
) to delay (
τ
)
2.
Negative translation along the horizontal
x
-axis (
N
is the total number of
delays)
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