Global Positioning System Reference
In-Depth Information
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ϕ
2
=
cos [
(
ω + ∆ω
) t
−
(k
+ ∆
k) x
]
(6.71)
These two harmonic waves can be superimposed by addition,
cos
t
k
x
2 cos
∆ω
t
− ∆
kx
ω +
∆
2
+
∆
k
2
ϕ
s
=
ϕ
1
+
ϕ
2
=
−
(6.72)
2
Th
is resultant wave is displayed in Figure 6.13. The combined signal shows two
co
mponent waves of significantly different frequency. The slowly varying amplitude
m
odulation represented by the envelope wave is
2 cos
2
(
Ψ =
∆ω
t
− ∆
kx)
(6.73)
ha
ving a propagation velocity wave of
∆ω
/
∆
k
. At the limit,
∆ω →
0 and
∆
k
→
0
[21
w
e obtain
d
dk
Lin
—
6.5
——
Nor
PgE
c
g
=
(6.74)
The quantity
c
g
is the velocity of the modulation and called the group velocity. In
the context of GPS signals
c
g
is the velocity of the P-code or C/A-codes. The second
wave component in (6.72) can be viewed as representing the carrier.
[21
Figure 6.13 Concept of group and phase propagation.
A point on the envelope travels
with group velocity
c
g
, whereas the waveform within the envelope travels with phase veloc-
ity
c
ϕ
.
