Global Positioning System Reference
In-Depth Information
θ
i
K
d
F
(
z
)
+
−
N
(
z
)
θ
0
FIGURE 7.2. Linearized digital second-order PLL model.
The above description of the demodulation is only for a signal with one satel-
lite. This is done to reduce the complexity of the equations and to give a simpler
idea of the demodulation scheme. In the real signal there is a signal contribution
from each visible satellite resulting in larger noise terms in the equations; see
Haykin (2000), page 95.
In the demodulation scheme seen in Figure 7.1, two local signal replicas are
required. To produce the exact replica some kind of feedback is needed. The feed-
back loop to produce the carrier replica is referred to as the carrier tracking loop,
and the feedback loop to produce the exact code replica is referred to as the code
tracking loop.
7.3
Second-Order PLL
Both the carrier tracking (Costas loop) and code tracking [delay lock loop (DLL)]
have an analytic linear phase lock loop model that can be used to predict perfor-
mance. This linear model has been derived by Ziemer & Peterson (1985) and is an
extremely powerful tool to predict the performance of the tracking loop. Another
excellent reference, once the fundamental models for Costas and DLL have been
derived, for linear phase lock loop and its parameters and performance is by Best
(2003).
Extending the linear PLL model has been derived by Chung et al. (1993). This
approach will be followed by the implementation of both the Costas loop and
DLL, yet the linear model references earlier can still be the basis of performance
prediction and analysis.
The second-order PLL system contains a first-order filter and a voltage con-
trolled oscillator (VCO). Note that the transfer function of an analog loop filter
and a VCO are
1
s
τ
2
s
+
1
F
(
s
)
=
,
(7.7)
τ
1
K
o
s
,
(
)
=
N
s
(7.8)
where
F
are the transfer functions of the filter and NCO, respectively.
K
o
is the NCO gain. The transfer function of a linearized analog PLL is [refer to
(
s
)
and
N
(
s
)
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