Digital Signal Processing Reference
In-Depth Information
P
(
j
ω
)
baseband signal
(a)
P
-
P
+
ω
−σ
σ
0
(b)
DSB signal
P
-
P
+
P
-
P
+
ω
−ω
c
ω
c
0
SSB signal
(type 1)
(c)
P
+
P
-
ω
ω
c
−ω
c
0
SSB signal
(type 2)
(d)
P
-
P
+
ω
ω
c
−ω
c
0
Figure 2.18
. Fourier transforms of (a) the baseband signal
p
(
t
)
,
(b) the double side
band (DSB) modulated version, and (c), (d) single side band (SSB) modulated versions.
Intricacies.
If
s
(
n
)
is a wide sense stationary random process with power spec-
trum
S
ss
(
e
jω
)
,
then
x
(
t
)
in Eq.
(2.33) is a cyclo WSS process (Sec.
E.3 in
Appendix E) with average power spectrum
S
xx
(
jω
)=
1
T
S
ss
(
e
jωT
)
|
2
.
|
P
(
jω
)
(2
.
35)
Since
S
ss
(
e
jωT
)
is periodic with period
2
π/T,
the bandwidth of
x
(
t
)
is determined
essentially by
P
(
jω
)
.
In the following discussion, which involves frequency bands,
we shall therefore consider the modulation of the pulse
p
(
t
)
even though, in actual
practice, it is
x
(
t
)
that is modulated.
The cosine-modulated pulse waveform has the form
p
DSB
(
t
)=
p
(
t
)cos
ω
c
t.
(2
.
36)
Thus, assuming
p
(
t
) has an approximately bandlimited Fourier transform
P
(
jω
)
as demonstrated in Fig. 2.18(a), the modulated version
p
DSB
(
t
) has the Fourier
transform shown in Fig. 2.18(b). This occupies twice as much bandwidth as
the baseband signal. A more economic version of the modulated signal can be
Search WWH ::
Custom Search