Digital Signal Processing Reference
In-Depth Information
and from (5.11),
cos
(v
0
t)
Î
f
"
p[d(v - v
0
) + d(v + v
0
)].
Then, applying the convolution equation (5.17), we have
q
T
2
L
F(v) =
[d(v - l - v
0
) + d(v - l + v
0
)]sinc (lT/2)dl.
-
q
Because of the sifting property of the impulse function, (2.41), the integrand has a
nonzero value only when and Therefore, the convolution
integral is easily evaluated, and the transform pair is found to be
l = v
-
v
0
l = v + v
0
.
sinc
(v
-
v
0
)T
2
+ sinc
(v
+
v
0
)T
2
T
2
rect(t/T)cos(v
0
t)
Î
f
"
B
R
.
(5.33)
The frequency spectrum of the signal is shown in Figure 5.16(b). The sinc wave-
forms generated by the rectangular pulse are shifted in frequency so that one sinc pulse
is centered at and another at Each of the sinc pulses has one-half the magni-
tude of the single sinc function that represents the Fourier transform of rect(
t/T
). Since
each of the sinc waveforms has nonzero frequency components over an infinite range
of frequencies, there will be some overlap of frequency components from the two sinc
waveforms in However, if the effect of the overlap is usually negli-
gible in practical applications. Notice that the bandwidth of the sinc waveform is in-
versely proportional to the time duration of the time-domain rectangular pulse.
v
0
-v
0
.
F(v).
v
0
W 2p/T,
f(t) = e
-at
u(t), a 7 0,
The signal is shown in Figure 5.17(a). The Fourier transform
of this signal will be derived directly from the defining Equation (5.1):
q
q
1
a + jv
.
e
-at
u(t)e
-jvt
dt =
L
e
-(a+ jv)t
dt =
F(v) =
L
-
q
0
The frequency spectra of this signal are shown in Figures 5.17(b) and (c).
It can be shown that this derivation applies also for
a
complex, with
5
6
Re
a
7 0.
Therefore, the transform pair can be written as
1
a + jv
.
7 0
Î
f
"
e
-at
u(t),
5
6
Re
a
(5.34)
In Chapter 4, we determined that a periodic function of time could be represented
by its Fourier series,
