Global Positioning System Reference
In-Depth Information
TABLE 7.1. Various types of Costas phase lock loop discriminators
Discriminator
Description
I
k
Q
k
D
=
sign
(
)
The output of the discriminator is proportional to sin
(ϕ)
.
I
k
Q
k
D
=
The discriminator output is proportional to sin
(
2
ϕ)
.
tan
−
1
Q
k
I
k
D
=
The discriminator output is the phase error.
Figure 7.8 shows the responses corresponding to the different discriminators.
The phase discriminator outputs in this figure are computed using expressions in
Table 7.1 for all possible phase errors. In the same figure it can be seen that the
discriminator outputs are zero when the real phase error is 0 and
180
◦
.Thisis
why the Costas loop is insensitive to the 180
◦
phase shifts in case of a navigation
bit transition.
The behavior of the Costas loop when a 180
◦
phase shift occurs is more clearly
illustrated in Figure 7.9. In this figure the vector sum of
I
k
and
Q
k
is shown as
the vector in the coordinate system. If the local carrier wave were in phase with
the input signal, the vector would be aligned to the
I
-axis, but in the present case
a small phase error is illustrated. When the signal is tracked correctly the vector
sum of
I
k
and
Q
k
tends to remain aligned with the
I
-axis. This property ensures
that if a navigation bit transition occurs, the vector on the phasor diagram will
flip 180
◦
(showed by the dashed vector in the figure). If a navigation bit transition
occurs, the Costas loop will still track the signal and nothing will happen. This
property does make Costas loop the commonly chosen phase lock loop in GPS
receivers; see Kaplan & Hegarty (2006), pages 166-170.
The output of the phase discriminator is filtered to predict and estimate any
relative motion of the satellite and to estimate the Doppler frequency.
±
100
arctan(
Q
/
I
)
sign(
I
)*
Q
I
*
Q
50
0
−50
−100
−150
−100
−50
0
50
100
150
True phase error [°]
FIGURE 7.8. Comparison between the common Costas phase lock loop discriminator re-
sponses.
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