Graphics Programs Reference
In-Depth Information
cos
2π
f
0
t
∞
∞
frequency
0
Figure 3.2. Amplitude spectrum for a continuous sine wave.
f
0
f
0
f
1
()
A
=
cos
ω
0
t
(3.18)
The FT of
is
f
1
()
F
1
()
A
πδω ω
0
=
[
(
)
+
δ ω ω
0
(
+
)
]
(3.19)
where
δ⋅
(
)
is the Dirac delta function, and
ω
0
=
2π
f
0
. As indicated by
the amplitude spectrum shown in Fig. 3.2, the signal
f
1
()
has infinitesimal
bandwidth, located at
±
f
0
.
Next consider the time domain signal
f
2
()
given by
τ
---
τ
---
A
≤≤
t
t
-
f
2
()
ARect
=
=
(3.20)
0
otherwise
It follows that the FT is
F
2
()
A
τ
Sinc
ωτ
------
=
(3.21)
2
where
sin
π
x
π()
Sinc
()
=
-------------------
(3.22)
The amplitude spectrum of is shown in
Fig. 3.3
. In this case, the band-
width is infinite. Since infinite bandwidths cannot be physically implemented,
the signal bandwidth is approximated by
f
2
()
radians per second or
2πτ
⁄
1 τ
⁄
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