Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 35
Vilà-Valls et al.
Figure 5.
Standard second-order PLL versus standard two-states (xk = [θk fk]T) KF.
where the initial phase error variance can be set to σ θ0 = π2/3
(squared radian), if the initial phase is uniform in [-π, π], or equal
to σ θ20 = 1/12 (squared cycles), if it is considered in [-1/2, 1/2].
The initial frequency and frequency rate error variances depend
on the acquisition stage. The maximum expected acquisition error
or acquisition resolution determines the maximum expected initial
tracking frequency error.
2
(
)
xˆ k +1|k = Fk xˆ k | k −1 + Fk K k y k − yˆ k | k −1 .
(26)
Considering the phase contribution (i.e., the first element of xk) and
the linear second-order loop form, this equation can be rewritten as
θˆk +1|k = θˆk |k −1 + Ts fˆk |k −1 + α1, k +
α 2, k
k ,
Ts
Output of the loop
PLL VERSUS KF ARCHITECTURE COMPARISON
(27)
Equivalent to NCO
In the previous sections, the basics of both PLL and KF-based carrier tracking architectures have been introduced in a separate manner. ℜegarding the problem at hand, some comparisons between
PLLs and KFs are found in the literature, but only taking into account basic architectures. This section provides a comparison of
those two approaches not only for conventional architectures but
also for most advanced PLL-based solutions.
with Kk = α1, kα 2, k and εk representing the KF discriminator
output, and where the main architectural contributions have been
identified to construct the parallelism with the standard PLL architecture.
The predicted phase in a standard second-order PLL is
k −1
ˆ PLL + (α + α ) + α .
θˆkPLL
2i
+1 = θ k
1
2
k
CONVENTIONAL PLL VERSUS STANDARD KF
i =1
The fact that a second-order PLL is equivalent to a second-order
KF in steady-state conditions (i.e., for a time-invariant system with
an a priori fixed Kalman gain) is well-known [66].
ℜecently, the equivalence between both techniques in
steady-state conditions has been shown for the third-order case
[67]. A block diagram comparison is sketched in Fig. 5, reusing the previously introduced standard architectures, where
it is easy to identify the block-by-block equivalence. In the
KF approach, the innovations' sequence goes through the discriminator to obtain the residual phase error to be used in the
linear KF implementation, as in the PLL. Then, the estimated
phase is constructed from the weighted residual error plus the
predicted value, being directly the implementation of the KF
equations. Notice that in both cases, the input to the carrier
generator block is the predicted phase; therefore, the equivalence is made more evident if the KF formulation is expressed
in the predictor form,
JULY 2017, Part II of II
(28)
It is straightforward from this expression (see Fig. 5) and the NCO
expression in (7) to see that the second-order joint PLL loop filter
plus NCO operation can be written by using a state-space formulation as [16]:
α1
1 Ts PLL 1 Ts
ˆx PLL
ˆ
xk +
α 2 k ,
k +1 =
0 1
0 1 T
s
(29)
= [θˆk fˆk ] ,
with εk the PLL discriminator output in cycles and xˆ PLL
k
which is strictly equivalent to the KF considering constant gains
(i.e., notice the effect of the state prediction matrix Fk, which implies a modification of the original gain α2). By considering frequency estimates at the output of the loop filter and a first-order
NCO, the standard loop filter and NCO block structure is recovered
from (29), as shown in (27). The equivalence for the third-order
IEEE A&E SYSTEMS MAGAZINE
35
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