Digital Signal Processing Reference
In-Depth Information
F
( )
f
(
t
)
1
T
1
T
1
2
0
T
1
2
t
4
2
2
4
T
1
T
1
T
1
T
1
(a)
G
( )
g
(
t
)
1
T
1
2
T
1
4
T
1
4
0
t
4
4
T
1
T
1
(b)
Figure 5.6
Rectangular pulses and their frequency spectra.
From the result of Example 5.1,
V
rect
(t/T)
Î
f
"
TV sinc
(T
v/2),
[eq(5.4)]
it is seen that is simply the particular case where
and as shown in Figure 5.6(a). Applying the time-scaling property to the transform
pair obtained in (5.4) results in
g(t)
T = T
1
, the time-scaling factor a = 2,
V = 1,
g(t) = f(2t),
where
f(t) = rect
(t/T
1
).
Therefore, from (5.12),
T
1
2
vT
1
4
1
2
F(v/2) =
¢
≤
G(v) =
sinc
.
This result is illustrated in Figure 5.6(b).
The following MATLAB program illustrates the duration-bandwidth relationship:
%
This program displays the relationship of the magnitude frequency spectra of two
%
rectangular pulses.
%
