Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 33
Vilà-Valls et al.
1 Ts Ts2 / 2
Fk −1 = 0 1
Ts ,
0 0
1
where the phase is expressed in cycles (radian/2π). It is straightforward to extend this state formulation to higher-order frequency
terms if needed.
Figure 3.
Linear KF-based carrier tracking architecture with noisy carrier observations.
Carrier Tracking Observation Equation
functions and directly applies the KF equations. The key point is
to use linearized transition matrices (Jacobian matrices), such as3
k −1 = ∇f ( x )
F
k −1
k −1 ˆ
xk −1|k −1
k = ∇h ( x )
; H
k
k ˆ
x k |k −1
CARRIER TRACKING STATE-SPACE FORMULATION
Carrier Tracking Process Equation
(19)
where the additive noise includes any possible modeling mismatch.
The state to be tracked includes the CP and the Doppler frequency
terms (i.e., using a Taylor series expansion of the CP and truncating at the order of interest), and the so-called transition matrix Fk-1
defines the phase evolution due to receiver dynamics. For instance,
assuming that tracking phase θk (radian), Doppler shift fk (hertz),
and Doppler frequency rate fk (hertz per second) are enough for a
given application (i.e., assuming a third-order Taylor approximation of the phase), the phase is
θ k = θ 0 + 2π f k kTs +
1 2 2
f k k Ts ,
2
(20)
where k refers to the discrete-time instants and Ts is the sampling
period. In this scenario, the state to be tracked is xk θ k f k fk ,
and the transition matrix is given by
The vector differential operator is defined as ∇ = [∂/∂x1,...,∂/∂xn].
JULY 2017, Part II of II
C
Linear observation equation: the inputs to the carrier tracking block are linearly related to carrier observables,
lin
y lin
k = H k x k + nθ , k → yk = θ k + nθ , k ,
(22)
with Hk the measurement transition matrix and nk the measurement noise, including thermal and phase noise contributions, as well as other propagation disturbances.
C
It is usually assumed that the phase variations of the signal of interest are due either to the relative movement between the transmitter
and the receiver or to synchronization mismatches, and on top of it,
there is a random behavior due to the noises affecting the system.
The state-space formulation of this problem is defined via both
process and measurement equations, as shown hereafter.
x k = Fk −1x k −1 + ν k ,
Two cases may be considered: i) the measurements are noisy CP
observables (linear equivalent model); and ii) the observations are
directly the received signal baseband complex samples.
,
and plug them into Steps 3, 5, and 7 of Algorithm 1. Notice that
in Steps 2 and 4, we can use nonlinear functions. It is beyond the
scope of this article to provide a detailed discussion on KF theory;
for further details on the topic, refer to [51], [60].
3
(21)
Nonlinear observation equation: the inputs to the carrier
tracking block are the complex baseband signal samples,
which are nonlinearly related to the carrier observables,
y k = h k ( x k ) + n k → yk = γ k e jθk + nk ,
(23)
with hk(·) the nonlinear measurement function, and γk refers
to the time-varying envelope of the received signal, which
may be affected by different propagation disturbances such
as fading, multipath, or scintillation.
Early approaches to this problem, such as the α-β and the α-β-γ
filters [61], do not require a detailed system model, trading computational load by a degradation in performance with respect to the
KF [62] due to their static, heuristically chosen gains.
KF-BASED CARRIER SYNCHRONIZATION ARCHITECTURES
In this section, different architectures to implement the KF-based
carrier tracking solution are provided, coupling the general formulation given in Section III.A with the specific state-space model of
Section III.B [16], [25].
Linear Observation Architectures
The inputs to the tracking block are directly phase observables
and thus use the state-space model defined by (19) and (22). Note
that this architecture is called standard linear KF throughout the
article, but in the literature, it is also referred to as direct-state
KF [16]. The closed-loop block diagram is sketched in Fig. 3. An
alternative linear architecture named error-state KF or rate-only
feedback loop, typically used in GNSS [63], [64], is presented and
IEEE A&E SYSTEMS MAGAZINE
33
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