Geology Reference
In-Depth Information
T
G
(
f
)
-1/
T
1/
T
-3/
T
-5/
T
3/
T
5/
T
4/T
-4/
T
-2/
T
2/
T
f
0
Figure 2.6 The Fourier transform of the boxcar time function, a sinc function of
frequency. The sidelobes fall o
according to the reciprocal of the distance along
the frequency axis from the main lobe.
ff
Figure 2.6 shows a plot of
G
(
f
) for the band of frequencies
5/
T
<
f
< 5/
T
.
The height of the
n
th sidelobe of the sinc function (2.256) is approximately
−
1)
n
1)
n
2
T
(
−
(
−
1)/2
T
=
=
1)π
,
n
1,2,3,....
(2.257)
π(2
n
+
(2
n
+
The first sidelobe is negative, and 21.2% in magnitude, compared to the main lobe.
Next, we examine the Fourier transform of a di
erent boxcar, shown in
Figure 2.7. The area enclosed by this boxcar is unity. It is identical to the pre-
vious boxcar, except that it is scaled by 1/
T
. It then has the Fourier transform
sinc(π
fT
). This transform pair illustrates a general property of Fourier transforms.
A function of short duration in the time domain has a Fourier transform of long
duration in the frequency domain. Similarly, a function of long duration in the time
domain will have a Fourier transform of short duration in the frequency domain.
Now, consider the limit as
T
→
ff
0. The time domain function vanishes everywhere,
except at the origin where it goes to infinity. This infinity is such that the area under
the function remains unity. The frequency domain function becomes unity for all
frequencies.
In this limit of
T
0, the time domain function becomes a
Dirac delta
function
at the time origin. It is denoted by δ(
t
). Located at time
t
0
, a Dirac delta function is
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