Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 38

A Tutorial on Kalman Filter-Based Techniques
computed and not set to constant values. One can conclude that this
KF-based approach provides an optimal solution to the suboptimal
F-PLL implementation.

ADAPTIVE PLLS VERSUS KF OPTIMAL APPROACH
As already stated in Section II, one of the main limitations of
standard PLLs is their constant bandwidth, which is a priori
fixed by the designer according to the expected working conditions [i.e., typically set using (30)] and of limited applicability
in time-varying scenarios. An alternative to counteract this lack
of adaptability or flexibility is to incorporate the capability to
estimate the actual working conditions (i.e., system noise and
dynamic stress), which leads to the A-PLL [48], [49]. The parameters used to set the loop bandwidth, σnoise and θe, can be
sequentially estimated from the input samples and the discriminator output.
The standard KF carrier tracking implementation is usually
said to inherently have an adaptive bandwidth, because the Kalman gain is optimally computed, taking into account both Qk and
Rk. However, this is only true if the noise statistics are completely
specified (i.e., known ∀ k). In practice, the measurement noise covariance is set according to the expected SNℜ and usually does not
take into account possible variations, and the process noise covariance is determined according to a single application-dependent
scenario. Considering that constant noise covariances lead to a
constant steady-state Kalman gain; therefore, the optimal adaptive
behavior of the KF is lost in the implementation for time-variant
systems. Analyzing (30), it is easy to see the analogy between the
first left-hand term σnoise and Rk and the relation between the dynamic stress and Qk. In the KF, the relation between those quantities and the filter bandwidth is computed in an optimal manner,
while in the A-PLL is suboptimally computed using predefined
thresholds. If the noise statistics are not fully determined a priori,
the solution is to estimate and sequentially adjust them into the
filter, which is usually known as AKF [19], [24] (see Section V.A
for details). From a practical architecture point of view, consider
the following:
C

Adaptive PLL versus optimal KF: the KF does not need
any additional processing to estimate the actual working
conditions and to optimally adjust the loop bandwidth,
which can be seen as an implementation of an optimized
A-PLL.
Adaptive PLL versus AKF: if the system is partially known,
an AKF solution is equivalent to the A-PLL. Both architecˆ and
tures make use of an estimate of the system noise (R
k
ˆ
ˆ
σˆ noise) and the dynamic stress (Q k and θ e), but the AKF optimally adjusts the loop bandwidth according to these estimates, while the A-PLL uses a somehow heuristic thresholddependent approach.

To conclude, with respect to the A-PLL, both KF-based adaptive approaches will always be superior in terms of performance,
optimality, and flexibility [11].

ADVANCED KF-BASED APPROACHES
In the previous section, the KF-based formulation of standard and
advanced PLL architectures has been detailed from theoretical, architectural, and conceptual points of view. The main idea was to
show that a standard PLL is a particular suboptimal implementation of the KF, that the cooperative loops can also be formulated
using KFs, and that the adaptive bandwidth loops are actually improved when using a KF approach. The goals of this section are
first to give a deeper insight on the AKF schemes and then to show
that the flexibility of the KF goes far beyond the implementation of
existing PLL-based architectures. The much more complex problems, which cannot be treated from a PLL point of view, can actually be solved by using powerful KF-based solutions.

AKF TRACKING SCHEMES
In standard KF-based tracking architectures, both the measurement noise variance σ n2, k and the process noise covariance matrix
Qk are assumed to be perfectly known, which is not realistic in
practical implementations and may lead to poor performances in
time-varying scenarios. The concept behind the AKF has already
been introduced in the previous section when compared with the
adaptive PLL approach. The main goal of the AKF is to sequentially adjust the noise statistics according to the actual working
conditions to obtain a robust and reliable tracking solution and to
provide the answer to the problem of interest here: variability. This
is equivalent to obtain a sequential optimal time-varying Kalman
gain adaptation (i.e., adaptive equivalent noise bandwidth). The
suboptimality introduced by the lack of precise knowledge of the
noise statistics within the KF framework may introduce several
estimation errors [68]. Therefore, a real-life robust system must
counteract the fact that accurate noise characteristics and dynamic
models are hardly available in practice. Two different approaches
based on the residuals (state estimate minus prediction) were presented in [20] for vehicle navigation; an adaptive two-stage KF
relying on the innovations' covariance is proposed in [21] for high
dynamics scenarios, and an ad hoc implementation, called variable gain AKF, was introduced in [22]. An interesting alternative
approach has been recently presented in [23], [24], where a C/N0
estimate (usually available at the receiver) is used to adjust the CP
error variance, which, in turn, is used to optimally compute the
Kalman gain in a time-varying manner.
From an optimal estimation point of view, the problem reduces
to the estimation of the covariance matrices of two Gaussian distributions. The carrier tracking problem using standard noise statistics estimation methods [69], [70] was studied in [68], but the
problem is not yet solved in a unified manner. In the literature,
we find a plethora of methods and different approaches to face
the noise statistics estimation problem. In the early 1970s, Mehra
[69] published a survey paper and classified the existing methods
into four categories: Bayesian, ML, covariance matching, and correlation methods. The most popular are the correlation methods
[69], and the more recent autocovariance least square [71] seems
to provide the best solution. A good analysis on the design of such
AKFs for carrier tracking is given in [19].

IEEE A&E SYSTEMS MAGAZINE

JULY 2017, Part II of II

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