Biomedical Engineering Reference
In-Depth Information
T
T
T
x (t )
2
dt
−
τ)
2
dt
+
x (t
−
2
x (t )x (t
−
τ)dt
≥
0
0
0
0
For these integrations,
τ
is constant; thus, the second term is equal to the
first term, and the denominator for the autocorrelation is the same for all
terms and is not shown. Thus:
R
xx
(
0
)
+
R
xx
(
0
)
−
2
R
xx
(τ )
≥
0
R
xx
(
0
)
−
R
xx
(τ )
≥
0
(2.4)
Property #2.
An autocorrelation function is an even function, which means
that the function for a
−
ve
phase shift is a mirror image of the function for
a
+
ve
phase shift. This can be easily derived as follows; for simplicity, we
will only derive the numerator of the equation:
T
1
T
R
xx
(τ )
=
x (t )x (t
+
τ)dt
0
dt = dt
:
(t
−
Substituting
t
=
τ)
and taking the derivative, we have
T
1
T
x (t
−
τ)x (t
)dt
=
R
xx
(τ )
=
R
xx
(
−
τ)
(2.5)
0
Therefore, we have to calculate only the function for
+
ve
phase shifts
−
ve
phase shifts.
because the function is a mirror image for
Property #3.
The autocorrelation function for a periodic signal is also peri-
odic, but the phase of the function is lost. Consider the autocorrelation of a
sine wave; again we derive only the numerator of the equation.
x (t )
=
E
sin
(ωt )
T
1
T
R
xx
(τ )
=
E
sin
(ωt )E
sin
(ω(t
−
τ))dt
0
Using the common trig identity: sin
(a)
sin
(b)
=
1
/
2
(
cos
(a
−
b)
−
cos
(a
+
b))
, we get:
t
cos
(ωt )
−
2
ω
sin
(
2
ωt
+
ωτ)
T
0
E
2
2
T
1
R
xx
(τ )
=
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