Differential GPS - Understanding GPS: Principles and Applications - page 408

Equation (8.23) represents a set of scalar complementary Kalman filter equa-
tions that can operate on each of the requisite DD measurement pairs (code and car-
rier-phase) in turn. Alternatively, these equations can be set up in matrix form and
accomplish the same end once all DD measurements for a given epoch are calculated
and collected together in respective arrays. Either approach is satisfactory, but, for
ease of programming, the scalar formulation is used here.
Figure 8.15 shows actual carrier phase (top line) and code (bottom line) DD
measurements collected over a period of 20 minutes during a flight test [25]. The off-
set between the two plots is arbitrary but can be thought of in terms of some
unknown ambiguity included in the carrier-phase DD measurements. It is apparent
that the two sets of data are quite similar with the exception of apparent noise on the
code DDs.
Figure 8.16 shows the output of the complementary Kalman filter (i.e., the
smoothed-code DDs, bottom, and the original carrier phase DDs, top, over the same
20-minute interval). With the exception of the first few epochs (nominally about
10), the smoothed-code DD virtually mirrors the carrier-phase DD. It has the added
advantage that it is centered about the original code DD measurements and is thus
unambiguous.
Depending upon the multipath in the local environment, the smoothed-code
DD, once the complementary Kalman filter is initialized, is generally within ± 1-2m.

In terms of carrier wavelengths, this represents approximately at L1.

Figure 8.17 shows the difference between the carrier-phase and smooth-code
DDs of Figure 8.16 with the nominal offset removed from the former. For this par-
ticular set of data, the difference is well within

± 5-10 λ

1m and is indicative of low
multipath at both the ground and airborne antennas. Again, the behavior prior to
completing the initialization of the complementary Kalman filter is clearly evi-
dent during the first few epochs, but, once initialized, the difference is very well
behaved.

±

30

20

10

0

− 10

− 20

− 30

0 5 15 20

10

Run time (minutes)

Figure 8.15 (Top line) Carrier-phase and (bottom line) raw-code DDs.