Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 36

A Tutorial on Kalman Filter-Based Techniques
PLL is straightforward considering the carrier tracking state-space
formulation example in Section III.B. Again, a difference between
both gains must be commented: if the standard PLL implementation has a set of three coefficients equal to {α1, α2, α3}, the strictly
equivalent KF must consider the following constant gains due to
the effect of the transition matrix (i.e., phase expressed in cycles),
K = [α1α 2 / Ts + 2α 3 / Ts2 ].
This subtle adjustment leads to the following equation (with α
= α1 + α2/Ts + 2α3/Ts2):

ˆ

k 1|k

ˆ
Ts2 fk|k 1
ˆ
ˆ

 k|k 1  Ts f k|k 1 
  k ,
2

where the different gains in the Doppler frequency and Doppler
ˆ
frequency rate terms Ts fˆk | k −1 and Ts2 fk | k −1/2, naturally appear in the
standard PLL predicted phase expression when considering the
corresponding loop gains K and the PLL state-space formulation
as in (29).
From an architectural point of view, there is a clear parallelism
between the well-known PLL and the standard KF formulation.
The main difference is that the loop filter gain is somehow heuristically adjusted in the PLL but optimally computed in the KF. If the
system is time invariant and the PLL bandwidth is set according to
the expected actual working conditions, this heuristic adjustment
may not be an inconvenience. In this case, the Kalman gain tends
rapidly to its steady-state value K∞. However, the flexibility of the
KF optimal gain plays an important role in real-life time-varying
scenarios, in which the optimal gain does not tend to a steady-state
value but evolves with time. Therefore, the PLL is a simplified
particular case of the general KF.
The following one-sigma equation is typically used to determine the desired ("optimal") static PLL loop bandwidth,

σ noise ( Bw ) +



measurement noise

θ e ( Bw )
3



≤ threshold → Bopt ,

(30)

COOPERATIVE LOOPS VERSUS KF JOINT ESTIMATION
The noise reduction versus dynamic range trade-off introduced in
Section II, which is the main problematic of standard constantbandwidth stand-alone PLLs, is clear from (30): if the noise affecting the system (σnoise(Bw)) increases, to maintain the jitter below
the threshold, one must lower the loop bandwidth for an optimal
behavior. But if the system dynamics (θe(Bw)) increase, one must
raise the loop bandwidth, which is inversely related to the dynamic
stress.
In practice, the loop bandwidth is set to the minimum value
that is able to cope with the maximum expected dynamics, which is
suboptimal most of the times. A well-established solution to cope
with this trade-off under non-nominal propagation conditions is
the use of cooperative loops. A popular approach is the F-PLL [47],
which uses a FLL to permanently assist a PLL, thus providing a
frequency aiding. The key idea directly arises from the bandwidth
determination using (30). If one is capable to reduce as much as
possible the dynamic stress of the loop, then a much lower loop
bandwidth may be used to cope with low SNℜ scenarios. Under
these circumstances, the main filter only copes with the residual
frequency errors and focuses on noise reduction. The classic second-order FLL-assisted third-order PLL architectures are sketched
in Fig. 6 [47], where the FLL loop structure is preserved to give
a clear picture of the corresponding frequency aiding interaction
with the PLL. Note that both the FLL and PLL bandwidths are
heuristically adjusted, relying on the correct operation of the frequency aiding provided by the FLL.
In the literature, a comparison between the F-PLL and the KF
at a theoretical level or looking for the architectural equivalence, as
done for the standard PLL, does not exist as far as authors' knowledge. Using the state-space formulation for the PLL introduced in
(29), and considering that the FLL tracks [ f k fk ], the interaction
between both filters is expressed as
ˆ
ρ kFLL = fkFLL
| k −1 + ( β1 + β 2 ) e f , k ,

(32)

dynamics

where the predefined threshold is usually known as loss-of-lock
threshold. This threshold is typically set to 1/12 of the pull-in range
of the discriminator, that is, 30° (coherent) and 15° (noncoherent).
Considering an arctangent Costas discriminator, the thermal noise
jitter contribution is

with β1 and β2 the FLL gains and ef,k the frequency discriminator
output. Using an equivalent third-order PLL state-space model and
the frequency aiding, the output of the PLL loop filter is
ˆ
ρ kPLL = fˆk |k −1 + fk |k −1 + ρ kFLL + (α1 + α 2 + α 3 )eθ , k ,


(33)

Frequency aiding

σ noise =


Bw 
1
1 +
 (rad),
C/N 0 
2C/N 0Ts 

(31)

and the dynamic stress θe is related to the maximum line-of-sight
(LOS) expected phase dynamics [3]. For instance, in a second-order PLL the phase model considers a constant Doppler shift; thus,
the dynamic stress is the maximum LOS acceleration. In a thirdorder PLL, the filter tracks a Doppler shift and Doppler frequency
rate, then the dynamic stress is the maximum LOS jerk.
36

where the frequency aiding of the FLL is done via the estimate

of the frequency rate. Optimally, fkFLL
| k −1 → f k ; thus, the frequency
rate that the PLL has to track fk | k −1 → 0. This frequency aiding is
made explicit by considering the architecture in Fig. 6 but is hidden
in the overall expression if considering the conventional compact
implementation,
ˆ
ρ kPLL = fˆk |k −1 + fk |k −1 + α eθ , k + β e f , k ,

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(34)

JULY 2017, Part II of II

Table of Contents for the Digital Edition of Aerospace and Electronic Systems Magazine July 2017 Tutorial XI