Global Positioning System Reference
In-Depth Information
|
S
(
f
)
|
|
S
(
f
c
)
|
f
−
f
c
f
c
2
W
2
W
|
S
(
f
)
|
2
|
S
(
f
c
)
|
f
−
W
0
W
FIGURE 1.12. Magnitude spectrum of bandpass signal
s
(
t
)
and complex envelope
s
(
t
)
.
1.5
Representation of Bandpass Signals
Signals can also be classified into lowpass, bandpass, and highpass categories de-
pending on their spectra. We define a bandpass signal as a signal with frequency
content concentrated in a band of frequencies above zero frequency. Bandpass sig-
nals arise in the GPS and Galileo systems where the information-bearing signals
are transmitted over bandpass channels from the satellite to the receiver.
Let us now consider an analog signal
s
with frequency content limited to
a narrow band of total extent 2
W
and centered about some frequency
(
t
)
±
f
c
;see
Figure 1.12. We term such a signal a
bandpass signal
and represent it as
cos
2
)
,
s
(
t
)
=
a
(
t
)
π
f
c
t
+
ϕ(
t
(1.23)
(
)
ϕ(
)
where
a
is called
the phase of the signal. The frequency
f
c
is called the carrier frequency. Equa-
tion (1.23) represents a hybrid form of amplitude modulation and angle modu-
lation and it includes amplitude modulation, frequency modulation, and phase
modulation as special cases. For more details on this topic we refer to Haykin
(2000).
By expanding the cosine function in (1.23) we obtain an alternative representa-
tion of the bandpass signal:
t
is called the
amplitude
or
envelope of the signal
and
t
cos
ϕ(
)
cos
sin
ϕ(
)
sin
s
(
t
)
=
a
(
t
)
t
(
2
π
f
c
t
)
−
a
(
t
)
t
(
2
π
f
c
t
)
=
s
I
(
t
)
cos
(
2
π
f
c
t
)
−
s
Q
(
t
)
sin
(
2
π
f
c
t
),
(1.24)
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