Global Positioning System Reference
In-Depth Information
Incoming
signal
c
2
z
−
1
c
1
FIGURE 7.3. Second-order phase lock loop filter.
Chung et al. (1993)]
(
)
(
)
K
d
F
s
N
s
H
(
s
)
=
)
,
(7.9)
+
(
)
(
1
K
d
F
s
N
s
where
K
d
is the gain of the phase discriminator. Substituting Equations (7.7) and
(7.8) into the transfer function (7.9) yields
2
n
2
ζω
n
s
+
ω
H
(
s
)
=
n
,
(7.10)
s
2
+
s
ζω
n
s
+
ω
√
(
where the natural frequency
ω
n
=
K
o
K
d
)/τ
1
, and the damping ratio
ζ
=
(τ
2
ω
n
)/
2. The above transfer functions are analog versions and to convert the
transfer functions to digital form, the bilinear transformation is used on (7.10).
This yields the following digital transfer functions for the PLL model:
4
2
+
+
(ω
n
T
ζω
n
T
z
−
2
2
z
−
1
2
ζω
n
T
+
(ω
n
T
)
2
(ω
n
T
)
)
−
4
2
z
−
2
.
(7.11)
The linearized digital second-order PLL model is shown in Figure 7.2, where
K
d
is the discriminator gain,
F
H
1
(
z
)
=
4
2
+
2
8
z
−
1
+
4
2
+
4
ζω
n
T
+
(ω
n
T
)
(ω
n
T
)
−
−
4
ζω
n
T
+
(ω
n
T
)
is the
transfer function of NCO. The transfer functions for the digital filter and NCO are
(
z
)
is the transfer function of the filter, and
N
(
z
)
C
1
z
−
1
)
=
(
C
1
+
C
2
)
−
F
(
z
,
(7.12)
−
z
−
1
1
K
o
z
−
1
N
(
z
)
=
1
,
(7.13)
1
−
z
−
where
F
(
z
)
is the transfer function of the filter and
N
(
z
)
is the transfer function
of the NCO. Figure 7.3 shows the phase lock filter
F
.
The goal is to find the coefficients
C
1
and
C
2
in the second-order PLL. This
is done by comparing the transfer function for the digital PLL and the transfer
function for the analog PLL. The transfer function for the digital version can be
found as
(
z
)
θ
o
(
z
)
K
d
F
(
z
)
N
(
z
)
H
(
z
)
=
=
)
.
(7.14)
θ
i
(
z
)
1
+
K
d
F
(
z
)
N
(
z
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