Digital Signal Processing Reference
In-Depth Information
e
-jTv/2
- e
+jTv/2
-jv
e
jTv/2
- e
-jTv/2
j2
TV
vT/2
B
R
B
R
= V
=
sin(Tv/2)
Tv/2
B
R
= TV
= TV sinc(Tv/2),
and we have derived our first Fourier transform:
f
5
V[u(t + T/2) - u(t - T/2)]
6
= TV sinc(Tv/2).
Note that the Fourier transform of the nonperiodic rectangular pulse has the
same form as the envelope of the Fourier series representation of the periodic rec-
tangular pulse train derived in Example 4.6 and Table 4.3.
The waveforms of Example 5.1, the rectangular pulse and the sinc function,
play important roles in signal representation and analysis. Many important wave-
forms, such as a digital “1” or a radar pulse, can be approximated by a rectangular
pulse similar to the one used in the example. Because of the frequent use of the rec-
tangular pulse in the study of communication signals, it is often defined with a spe-
cial function name, such as
rect(t/T) = [u(t + T/2) - u(t - T/2)].
Therefore, in our table of transform pairs we will list
rect(t/T)
Î
f
"
T sin
c(Tv/2)
(5.4)
as representing the transform pair shown in Figure 5.3.
The transform pair (5.4) is valid even though we have not yet taken into con-
sideration the fact that some waveforms do not have Fourier transforms.
Sufficient conditions for the existence of the Fourier transform are similar to
those given earlier for the Fourier series. They are the
Dirichlet conditions
:
1.
On any finite interval,
a.
f
(
t
) is bounded;
b.
f
(
t
) has a finite number of maxima and minima; and
c.
f
(
t
) has a finite number of discontinuities.
2.
f
(
t
) is absolutely integrable; that is,
q
-
q
ƒ f(t) ƒ
dt 6
q
.
L
Note that these are
sufficient
conditions and not
necessary
conditions. Use of
the Fourier transform for the analysis of many useful signals would be impossible if
these were necessary conditions.
