Biomedical Engineering Reference
In-Depth Information
by applying the definition of the Fourier Transform of the periodic function
f
(
t
)as
in (9.12),
T
1
/
2
1
T
1
f
(
t
)
e
−
jn
ω
1
t
dt
F
(
n
)
=
for
n
=
0
,
±
1
,
±
2
,
±
....
(9.12)
−
T
1
/
2
and substituting the convolution function for
f
(
t
) as in (9.13):
T
1
/
2
1
T
f
1
(
t
)
f
2
(
t
+
τ
)
dt
=
f
(
t
)
(9.13)
−
T
1
/
2
the complex spectrum
F
1
(
n
)
F
2
(
n
) becomes (9.14):
T
1
/
2
T
1
/
2
1
T
1
e
−
jn
ω
1
τ
d
F
1
(
n
)
F
2
(
n
)
=
τ
f
1
(
t
)
f
2
(
t
+
τ
)
dt
(9.14)
−
T
1
/
2
−
T
1
/
2
Again, the right integration is preformed first with respect to
t
and the resulting
function of
is then multiplied by
e
−
jn
ω
1
τ
. Then, the product is integrated with respect
τ
to
τ
in order to obtain a function of
n
.
Thus the equations in (9.15) are Fourier Transforms of each other.
T
1
/
2
1
T
1
f
1
(
t
)
f
2
(
t
+
τ
)
dt
and
F
1
(
n
)
F
2
(
n
)
(9.15)
−
T
1
/
2
This relationship is called the “Correlation Theorem” for periodic functions.
9.1 THE CORRELATION PROCESS
The correlation process involves three important operations:
(1) The periodic function
f
2
(
t
) is given as delays or time displacement,
τ
.
(2) The displacement function is multiplied by the other periodic function of the
same fundamental frequency.
(3) The product is averaged by integration over a complete period.
These steps are repeated for every value
) so that a function
is generated. In summary, the combination of the three operations—displacement,
τ
in the interval (
−∞
,
∞
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