Graphics Programs Reference
In-Depth Information
cos
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
=
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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