Biomedical Engineering Reference
In-Depth Information
Since Limit
T
→∞
,
ω
0
→
0,
ω
0
can be denoted by
ω
(see (12.18))
T
/
2
2
π
ω
π
ω
2
1
T
f
T
(
t
)
e
−
jn
ω
0
t
Then
T
=
0
=
and
TF
n
=
dt
(12.18)
−
T
/
2
Since
TF
n
(
jn
ω
)
, the expression may be rewritten as
TF
n
=
F
(
jn
ω
)
.
∞
F
n
e
jn
ω
0
t
Then (12.17),
f
T
(
t
)
=
,
becomes (12.19).
n
=−∞
∞
F
(
jn
ω
)
e
(
jn
ω
0
t
)
t
f
T
(
t
)
=
T
n
=−∞
F
(
jn
∞
ω
)
e
(
jn
ω
0
t
)
t
f
T
(
t
)
=
ω
(12.19)
2
π
n
=−∞
Rewriting (12.19) in the “Limit”: as
T
→∞
,
ω
→
0,
f
T
(
t
)
→
f
(
t
), the equation takes
0
the form of (12.20), thus proving the limit process.
Lim
ω
→
F
(
jn
1
2
)
e
jn
ω
t
f
(
t
)
=
Lim
T
f
T
(
t
)
=
ω
ω
(12.20)
π
→∞
0
12.8 INVERSE FOURIER TRANSFORM
To return to the “Time Domain” after transforming into the “Frequency Domain,” the
inverse Fourier transform must be obtained. By definition, the Inverse Fourier Transform
is written as (12.21). The process is the same as taking the forward transform, except that
in lieu of multiplying the time domain function,
f
(
t
), by the Fourier Basis Function, all
the Fourier Coefficients form the Frequency domain function,
F
n
)
ω
2
,
are multiplied by the inverse Fourier basis function as in (12.19). Particular emphasis
is made to point out that the sign of the basis function exponential frequency,
j
=
F
(
jn
ω
π
ω
t
,is
positive for the inverse transformation.
∞
1
2
)
e
j
ω
t
f
(
t
)
=
ω
ω
(12.21)
F
(
j
d
π
−∞
where
n
ω
becomes
ω
(a continuous variable) and the function
F
(
j
ω
=
Lim
ω
→
)
0
F
(
jn
ω
)
.
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