Global Positioning System Reference
In-Depth Information
The noise bandwidth can be found as
(
1
)
2
ζ
2
ω
ω
n
∞
∞
1
+
ω
n
2
π
2
df
B
n
=
|
H(ω)
|
=
2
dω
1
−
2
2
ω
ω
n
2
ζ
0
0
ω
ω
n
+
1
+
4
ζ
2
ω
ω
n
2
∞
ζ
+
ω
n
2
π
ω
n
2
1
4
ζ
=
dω
=
(8.26)
−
1
)
ω
ω
n
2
ω
ω
4
0
+
2
(
2
ζ
2
+
1
This integration can be found in the appendix at the end of this chapter.
The error transfer function can be obtained from Equation (8.10) as
s
2
H
e
(s)
=
1
−
H(s)
=
(
8
.
27
)
s
2
+
2
ζω
n
s
+
ω
n
When the input is
θ
i
(s)
=
1
/s
, the error function is
s
(s)
=
(
8
.
28
)
s
2
+
2
ζω
n
s
+
ω
n
The steady-state error is
=
lim
s
=
0
lim
(t)
0
s(s)
(
8
.
29
)
t
→∞
→
When the input is
θ
i
(s)
=
1
/s
2
, the error function is
1
(s)
=
(
8
.
30
)
s
2
ω
n
+
2
ζω
n
s
+
The steady-state error is
lim
(t)
=
lim
s
0
s(s)
=
0
(
8
.
31
)
t
→∞
→
In contrast to the first-order loop, the steady-state error is zero for the frequency-
modulated signal. This means the second-order loop tracks a frequency-modulated
signal and returns the phase comparator characteristic to the null point. The con-
ventional phase-locked loop in a GPS receiver is usually a second-order one.
8.5 TRANSFORM FROM CONTINUOUS TO DISCRETE SYSTEMS
(
5,6
)
In the previous sections, the discussion is based on continuous systems. In order
to build a phase-locked loop in software for digitized data, the continuous system
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