Biomedical Engineering Reference
In-Depth Information
f
(
t
)
Nonperiodic
t
0
Periodic
f
T
(
t
)
0
FIGURE 12.7
:
A transient pulse. Top trace is a transient; bottom trace is periodic
It is an accepted fact that a periodic function,
f
T
(
t
), can be expressed as the sum of eternal
exponentials of frequencies 0
etc. Also, it is known that
the Fourier Transform,
F
n
, amplitude of the component of frequency,
jn
,
±
j
ω,
±
j
2
ω,
±
3
j
ω...
ω
, is expressed
)
ω
2
as
F
(
jn
ω
.
π
Proof of the Limiting Process:
Viewing Fig. 12.7, let us examine the proof of the limiting
process.
Let us begin with the general expression (12.16):
In the Limit as
T
→∞
: Lim
T
f
T
(
t
)
=
f
(
t
)
(12.16)
→∞
Then the exponential Fourier Series for
f
T
(
t
) is given by (12.17).
∞
F
n
e
jn
ω
0
t
f
T
(
t
)
=
(12.17)
n
=−∞
T
T
/
2
2
T
1
f
T
(
t
)
e
−
jn
ω
0
t
where
ω
=
and
F
n
=
dt
0
−
T
/
2
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