Signal Processing - September 2017 - 106

innovation, is then scaled by a gain matrix to obtain a posteriori
estimation of the state vector. If the noise processes associated
with the state function and the measurement function are both
wide-sense stable, the Kalman gain matrix will converge to a
value that optimizes to the noise level. If the measurement error
increases, the innovation will contain less reliable information.
The Kalman filter will decrease the gain matrix to discount the
contribution from innovation. Equation (8) is the updated equation of a basic KFP at a particular time epoch k [56]:
Tzt k
Tzr k
t
> ~ k H = > ~r k H + L^ y k - yr k h,
at k
ar k

(8)

where the state vector components z, ~, and a are the carrier
phase, frequency, and acceleration of the signal, respectively,
and y is the measurement function. The "^" and "-" indicate
a posteriori and a priori state estimations, respectively. L is
the Kalman filter gain matrix and is related to the loop bandwidth. Lowering the gain matrix is equivalent to narrowing
the loop bandwidth.
The performance of a fixed bandwidth KFP was evaluated
using simulated scintillation data in [55] and using real scintillation in [52]. In both studies, the KFP performance was compared with a conventional PLL and an FAP, both with a fixed
bandwidth. Both studies show that all three methods have similar tracking loop errors under all scintillation levels. However,
the fixed gain KFP method incurred the least number of cycle
slips and loss-of-lock occurrences. In essence, the fixed gain
KFP works very much like a conventional PLL, although it
carries some advantages associated with a Kalman filter such
as having less noise in its estimated outputs.
Humphreys et al. [51] implemented a KFP with a variable
bandwidth. The KFP itself is based on [56]. The bandwidth
is optimized in the sense that the phase error is minimized
under the assumption of additive white Gaussian noise. The
KFP's bandwidth is an implicit function of C/N 0 and the
determination of the loop bandwidth requires the KFP to
continuously estimate C/N 0 . Performance of this KFP was
evaluated using both simulated and real scintillation data.
The simulation results showed that the KFP has drastic

C/N0

Weighting Factor q

cnrU
cnrL

cnrTH0 cnrTH1
(a)

C/N0

BKFP_L BKFP_U

BKFP

(b)

FIGURE 6. Mapping relationships between (a) C/N0 and weighting factor

q and (b) KFP bandwidth BKFP and C/N0 for an adaptive gain KFP (Figure
used with permission from [57]).

106

improvement in terms of loss of lock and lock threshold
over the best constant bandwidth PLL. However, processing of real strong scintillation data ^S 4 2 0.9h collected in
Brazil show that the KFP has similar performance as a conventional PLL with a narrow constant bandwidth of ~5 Hz.

Mitigation techniques for strong equatorial scintillation
The previously described techniques demonstrated improved
performance for weak to moderate scintillation signals. However, they do not perform well for very strong scintillation.
The adaptive gain KFP and vector processing approaches discussed in this section are designed to handle these extremely
violent scintillations.

Adaptive gain KFP
During strong equatorial scintillation, the amplitude fading
and rapidly changing signal phase push the noise limit and
dynamic limit of GPS signal tracking loops closer toward each
other. The noise limit and dynamic limit defines the tracking
loop filter functional bandwidth [57]. The dynamic limit is the
smallest bandwidth at which the carrier tracking loop can still
maintain lock during strong phase scintillation, while the noise
limit is the maximum bandwidth that enables the tracking loop
to maintain the lock of amplitude scintillation signal. The loop
bandwidth must be located within the functional range for the
tracking loop to maintain lock on a particular signal.
Under normal conditions, the noise limit is higher than the
dynamic limit. In extremely challenging scenarios, the noise
limit may be reduced to below the dynamic limit of the scintillation signals and the tracking loop bandwidth functional
range does not exist anymore. A fixed gain KFP will not work
regardless of the bandwidth value used. Zhang and Morton
[57] proposed a variable gain KFP with the following state
update equation for these scenarios:
Tzt k
Tzr k
^cnrk - cnrTH0 h q
t
$ y k - yr kE . (9)
> ~ k H = > ~r k H + L^cnrk h $ ; cnrTH
1 - cnrTH0
t
r
ak
ak
This update equation incorporates weighted carrier-phase
measurements through a weighting factor q and an adaptive
gain matrix L , which depends on cnrk, the real-time estimated
C/N 0 . Figure 6 shows the dependency of the weighting factor
q and the KFP bandwidth B KEP on C/N 0 . cnrk is estimated in
real time and compared to threshold values cnrTH1 and cnrTH0 .
If cnrk 2 cnrTH1, the KFP will continue in its normal mode
of operation. If cnrTH0 1 cnrk 1 cnrTH1, the phase measurements are considered less reliable and the weighting factor q is
applied to reduce the contribution of the phase error measurements in the update equation. If cnrk 1 cnrTH0, the phase error
measurement should be completely discarded.
The KFP gain L depends on the real-time C/N 0 estimate
through its dependency on the KFP bandwidth. The bandwidth
is determined based on two heuristically selected threshold
values for C/N 0 : cnrU and cnrL . For cnrL 1 C/N 0 1 cnrU,
the loop bandwidth will be linearly dependent on C/N 0 .

IEEE SIGNAL PROCESSING MAGAZINE

September 2017

Table of Contents for the Digital Edition of Signal Processing - September 2017