Biomedical Engineering Reference
In-Depth Information
and
T
1
/
2
1
T
1
f
2
(
t
)
e
−
jn
ω
1
t
dt
F
2
(
n
)
=
(9.6)
−
T
1
/
2
However, it should be noted from (9.1) that the original equation is not for
f
2
(
t
) but
rather for
f
2
(
t
). Thus, it was found that caution must be taken in the manner that
the correlation function is expressed:
+
τ
T
1
/
2
T
1
/
2
∞
1
T
1
T
F
2
(
n
)
e
jn
ω
1
(
t
+
τ
)
f
1
(
t
)
f
2
(
t
+
τ
)
dt
=
f
1
(
t
)
dt
(9.7)
n
=−∞
−
T
1
/
2
−
T
1
/
2
The manner in which the right-hand side of the equation is written indicates that
summation with respect to
n
comes first and is followed by integration with respect to
t
as shown in (9.7).
At this point, if the order of summation and integration is inversed, the result
would be given by (9.8):
,
T
1
/
2
T
1
/
2
∞
1
T
1
1
T
1
F
2
(
n
)
e
jn
ω
1
τ
f
1
(
t
)
e
jn
ω
1
t
dt
f
1
(
t
)
f
2
(
t
+
τ
)
dt
=
(9.8)
n
=−∞
−
T
1
/
2
−
T
1
/
2
Note that in (9.8), the exponent in the second term (the integral) has a positive,
+
j
,
j
, as shown in the integral (
dt
) of (9.5), where the last integral
is recognized as the complex conjugate of
F
1
(
n
). Hence, the result yields (9.10),
rather than a negative,
−
T
1
/
2
∞
1
T
1
[
F
1
(
n
)
F
2
(
n
)]
e
jn
ω
1
τ
f
1
(
t
)
f
2
(
t
+
τ
)
dt
=
(9.10)
n
=−∞
−
T
1
/
2
which results in the general form of the inverse Fourier Transform (9.11) for a periodic
function of fundamental angular frequency
ω
1
and complex spectrum
F
1
(
n
)
F
2
(
n
)of
(9.10).
∞
F
(
n
)
e
jn
1
t
f
(
t
)
=
(9.11)
n
=−∞
However, note that (9.10) is in terms of delays,
τ
, instead of time,
t
, as in (9.11). Then
Search WWH ::
Custom Search