Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 39

Vilà-Valls et al.
the community. The second case proposes a KF-based architecture
in a deep space communications system, being an extremely challenging synchronization scenario.

AUGMENTED STATE AND MULTIPLE MODEL
FORMULATIONS
One of the most important features of the KF-based solution, apart
from its optimal approach, is the flexibility it provides to deal with
different problems, while strictly considering the same architecture, which from a practical point of view may be of capital importance in some applications. For instance, in the carrier tracking
problem, to extend a second-order loop to a third-order, one only
needs to add the frequency rate into the state evolution formulation
and to extend the corresponding covariance matrices to take it into
account. This can be used to include any prior knowledge of the
system into the state-space formulation. In some cases, this can
be the only way to solve the problem and provide a robust carrier
tracking solution. For instance, if some specific propagation conditions are a priori known and effectively modeled using a dynamic
state-space model, they can be merged together with the CP of interest into a single state-space formulation. In this case, the KF is
aware of those specific propagation conditions and may be able to
mitigate undesired effects. An illustrative example [28], [29] of a
real application of these ideas is presented in Section VI.
All the methods introduced in this article, from the standard
architectures to the most advanced adaptive and augmented state
KF-based approaches, rely on a specific dynamic model, which
defines the evolution of the parameters of interest (e.g., the CP
in this case). The KF is optimal when the state-space formulation
perfectly matches the real system. If a mild modeling mismatch
or slightly time-varying scenario is considered (i.e., a weak uncertainty about the state evolution Qk or the measurement noise
Rk) the natural solution is given by the AKF, but this approach
does not provide a robust solution to strongly time-varying scenarios. High variability in the sense that the system uncertainty
is not only on the system noise but on the state-space formulation
itself. To overcome this model-based uncertainty, the best solution
is to consider model matching or selection strategies. Among the
different solutions available in the literature, the most promising is
the so-called interactive multiple model (IMM) approach, which
has been thoroughly used in target tracking, navigation, and high
dynamics applications [56], [72]. The main idea behind the IMM
is to overcome the main problem of stand-alone KFs following a
divide and conquer strategy, dealing with changing scenarios by
using several more easily fixed operation KFs. In other words, it is
a bank of interacting KFs running in parallel. Each KF is designed
for a specific scenario, and the filter is in charge to construct a
final estimate using models' likelihood. Notice that a key point to
obtain a good estimate is the interconnection among individual filters. This concept has already been successfully applied to carrier
synchronization [27].

COMPUTER SIMULATIONS
To support the discussion on the PLL versus KF dilemma, two
illustrative examples of interest to the aerospace community are
given in the sequel. The first case deals with the GNSS carrier
tracking under harsh propagation conditions, namely considering
ionospheric scintillation disturbances, which is a popular topic in
JULY 2017, Part II of II

CASE 1: ROBUST GNSS CP TRACKING
Ionospheric scintillation is the name given to the disturbance
caused by electron density irregularities along the propagation
path through the ionosphere. These irregularities affect the GNSS
signals with amplitude fades and phase variations. An important
feature is the existing correlation between deep amplitude fades
and phase variations in a simultaneous random manner, the socalled canonical fades [73]. This is certainly the most challenging scenario in GNSS carrier tracking problems. This particular
example is used to support the fact that prior knowledge on the
propagation conditions can be introduced into the system by state
augmentation, providing an extra capability to the filter, a fact that
is impossible to take into account using PLL-based architectures
(see Section V.B).
The scintillation can be modeled as a multiplicative channel
[74] ξs(t) = ρs(t)e jθ s ( t ) and synthesized by using the Cornell scintillation model (CSM)5 [75], where ρs(t) and θs(t) are the corresponding
envelope and phase components.
The scintillation phase is a correlated stochastic process,
which, in turn, can be fairly modeled as an AR(1) process: θs,k =
βθs,k−1 + ηk, with ηk ∼  (0,σ η2 ), which can be included in the KF
state-space formulation to jointly track the desired phase θd,k Doppler frequency fd,k frequency rate fd , k and possible scintillation effect θs,k. This idea was first introduced in [27] within a multiple
model approach and further extended in [28], [29]. The simplified
model for the samples at the input of the carrier tracking stage is

(

yk = α k e jθk + nk ; nk ~  0,σ n2, k

)

α k = Ak ρ s , k ; θ k = θ d , k + θ s , k ,

(40a)

(40b)

where ρs,k refers to the scintillation amplitude effects. The state
evolution [27] is given by
 1 Ts Ts2 / 2 0 


0 1
Ts
0
xk = 
x k −1 + v k ,
0 0
1
0


0
β
0 0

(41)

where the process noise, vk ∼  (0,Q), stands for possible uncertainties or errors on the state transition model.
In this example, the signal of interest is corrupted by moderate
and severe scintillation, and the following parameters are used: Ts
5

The CSM has been embedded in the so-called Cornell scintillation simulation MATLAB toolkit, which is available at http://
gps.ece.cornell.edu/tools.php. This software will be used in the
computer simulations to generate the desired scintillation effect.

IEEE A&E SYSTEMS MAGAZINE

http://gps.ece.cornell.edu/tools.php http://gps.ece.cornell.edu/tools.php

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