Aerospace and Electronic Systems Magazine July 2017 Tutorial XI - 34
A Tutorial on Kalman Filter-Based Techniques
analyzed in [16], with respect to the standard (direct-state) KF. The
idea consists of using a state-space model, where the filter does not
track directly the CP but the phase error. As it does not provide any
advantage over the standard KF, the latter is preferred for practical
applications.
Nonlinear Observation Architectures
The previous linear architecture is of limited applicability in reallife implementations. At least, some extra information should be
added to explain how the phase observables are obtained. The
standard solution is to use a discriminator, as done in traditional
PLL architectures (Case 1), but the complex samples of the received baseband signal can be directly treated using a nonlinear
filter (Case 2).
C
C
Case 1, discriminator-based traditional approach:
Using a discriminator allows the use of a traditional KF,
avoids the derivation of suboptimal solutions, and is considered the reference KF-based architecture [43]. The main
differences with the linear standard KF are i) the carrier generator block, which uses the nonlinear observation equation
hk(·) and ii) the discriminator as phase detector. The block
diagram is shown in Fig. 4 (top).
Case 2, EKF solution:
Using a discriminator might break the Gaussianity assumption within the KF, making the filter not optimal anymore,
which may lead to poor filter performances or even divergence. Moreover, the discriminators may need to operate under saturation in low SNℜ scenarios. A solution is to directly
deal with the nonlinear Gaussian observation model [25],
[26]. The simplest solution is to use a linearization procedure (EKF-like solution) and then reuse the previous linear
standard KF approach, sketched in the bottom diagram of
Fig. 4.
In practice, there are some issues that are of capital importance
for the actual implementation, detailed hereafter.
Noise Statistics
It is common for the system model not to be perfectly known; thus,
the noise covariance matrices are set to some expected value. A
rule of thumb typically considered in the KF design is that noise
covariances must be equal or greater than the true ones to ensure
the filter convergence. Therefore, it is convenient to be rather conservative and not underestimate the noise impact into the system.
ˆ
If Q k > Qk,4 the steady-state filter performance may be worse, but
the filter tends to be more reactive and robust to model changes
so more suitable to rapidly time-varying scenarios. ℜegarding the
ˆ > R implies that K < K
measurement noise, R
; thus, the
k
k
k
optimal
filter relies more on the process transition model [19].
34
Standard (top) and extended (bottom) KF-based carrier tracking architectures. The standard KF uses a discriminator as a phase detector, while
the EKF directly operates with the input complex samples and computes
.
Py,k|k-1, Kk, and Px,k|k by using the linearized H
k
An expression for the approximated variance of the phase
noise, expressed in squared radians, at the output of the Costastype two quadrant arctangent discriminator is [65]
σ n2θ =
1
2C/N 0Ts
1
1 +
2C/N 0Ts
.
(24)
where C/N0 is the carrier-to-noise density ratio (independent of the
receiver bandwidth Bw), which is related to the SNℜ as
C/N 0 (dB-Hz) = SNR (dB) + 10log10 ( Bw (Hz) ) .
(25)
The process noise covariance matrix is fixed according to the expected dynamic working conditions. Considering the third-order
illustrative example, this covariance is related to the frequency
rate error variance (i.e., possible frequency rate modeling error or
higher-order expected dynamics).
ON THE IMPLEMENTATION ISSUES
4
Figure 4.
Filter Initialization
How to set the initial values is something completely application
dependent. In practice, the best option is to set these parameters according to some a priori information or physical meaning. Taking
into account the example at hand, where the state to be tracked is
xk θ k ; f k ; fk T, the initial values can be set to xˆ 0 = [0; 0; 0]T. The
initial error covariance is defined as
(
)
Px ,0|0 = diag σ θ20 ,σ 2f0 ,σ 2f ,
0
A > B means that A − B is non-negative definite [60].
IEEE A&E SYSTEMS MAGAZINE
JULY 2017, Part II of II
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