Digital Signal Processing Reference
In-Depth Information
Energy spectral density of a rectangular pulse
EXAMPLE 5.19
For the rectangular waveform shown in Figure 5.33(a), we have previously found the fre-
quency spectrum to be described by the sinc function shown in Figure 5.33(b). We now find
the energy spectral density. The magnitude of this curve is squared and divided by to form
the frequency spectrum of the energy from (5.48); the result is shown in Figure 5.33(c). Next,
we fold the energy frequency spectrum about the axis and add the frequency compo-
nents as they overlap. This result is shown in Figure 5.33(d), which is a plot of the energy
spectral density, of the rectangular waveform.
We find the energy contained in some band of frequencies of particular interest by
finding the area under the energy spectral density curve over that band of frequencies. For
example, in Figure 5.33(d), the amount of energy contained in the band of frequencies
2p
v = 0
f
(v),
F
( )
AT
f
(
t
)
A
T
2
T
2
t
4
2
2
4
T
T
T
T
(a)
(b)
1
2
2
F
( )
A
2
T
2
2
0
1
2
T
2
T
4
2
1
(c)
1
2
F
( )
e
f
(
)
A
2
T
2
0
1
2
2
T
4
T
(d)
Figure 5.33
A rectangular voltage pulse and its energy spectrum.
