Biomedical Engineering Reference
In-Depth Information
CHAPTER
9
Correlation Functions
In the general theory of harmonic analysis, an expression of considerable importance and
interest is the correlation function given in (9.1).
T
1
/
2
1
T
f
1
(
t
)
f
2
(
t
+
τ
)
dt
(9.1)
−
T
1
/
2
For the case of periodic functions where
f
1
(
t
) and
f
2
(
t
) have the same fundamental
angular frequency
ω
1
and where
τ
is a continuous time of displacement in the range
(
), independent of
t
(the ongoing time).
One important property of the correlation expression is the fact that its Fourier Tra-
nsform is given by (9.2), where
f
1
(
t
) has the complex spectrum
F
1
(
n
), and
f
2
(
t
) has
F
2
(
n
).
−∞
,
∞
F
1
(
n
)
F
2
(
n
)
(9.2)
The bar (
F
1
(
n
)) indicates the complex conjugate of the quantity (function) over
which it is placed. To establish this fact, the Fourier expansion of
f
1
(
t
) and
f
2
(
t
) can be
expressed as (9.3) and (9.4).
∞
F
1
(
n
)
e
jn
ω
1
t
f
1
(
t
)
=
(9.3)
n
=−∞
∞
F
2
(
n
)
e
jn
ω
1
t
f
2
(
t
)
=
(9.4)
n
=−∞
where the complex spectrums are given by (9.5) and (9.6), respectively.
T
1
/
2
1
T
1
f
1
(
t
)
e
−
jn
ω
1
t
dt
F
1
(
n
)
=
(9.5)
−
T
1
/
2
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