The GNSS integer ambiguities: estimation and validation

in order to guarantee a low fail rate the critical value should somehow depend on the
distance to all integer vectors in the neighborhood of â, where the distance is measured
in the metric of Qâ.
2-D examples
By comparing 2-D aperture pull-in regions of the RTIA, DTIA and OIA estimators, it is
possible to get a better idea of the differences in performance of the three estimators.
Therefore three aperture pull-in regions for are plotted for different vc-matrices in figure
5.25. The following vc-matrices are considered:
2
3 Qˆz 02 01, Qˆz 02 01, 4Qˆz 02 01, Qˆz 02 01 +
The regions correspond to the following three situations:
0.8 0
0 0
a : µ chosen such that for each estimator the fail rate is obtained for which the percentage
of RTIA / DTIA solutions identical to the OIA solution, Pid, is minimum;
b : Ratio Test with fixed critical value of µ = 0.5,
µ for DTIA and OIA chosen such that fail rates of all estimators are equal
c : all estimators with fixed fail rate of Pf = 0.025
It follows that the first situation, smallest Pid for the same fail rate, in general occurs
when the OIA pull-in region just touches the ILS pull-in region, and thus the aperture
parameter is large. The reason is that for a large aperture parameter the shape of the
RTIA and DTIA pull-in regions start to resemble the shape of the ILS pull-in region
and is larger in the direction of the corner points of the ILS pull-in regions where little
probability mass is located. The OIA pull-in region, on the other hand, touches the
ILS pull-in regions there where more probability mass is located in Sa, since the PDF
of â is elliptically shaped. The smallest Pid is not obtained for a large aperture pull-in
region when the precision is so high that the PDF of â is peaked since then there is little
probability mass close to the boundaries of the ILS pull-in region.
If the aperture pull-in regions are chosen even larger than shown in figure 5.25, the OIA
pull-in region is cut off by the ILS pull-in region. So, if µ is increased the OIA pull-in
region cannot expand in all directions, and therefore the fail rate will not change much.
The aperture pull-in regions of the RTIA and DTIA estimators are especially different
from those of the OIA estimator in the direction of the vertices of the ILS pull-in region.
That is because in those directions there are two integers, ǎ2 and ǎ3 with the same
distance to â, but only â − ǎ22 is considered by the ratio test and difference test,
Qâ
whereas with OIA estimation the likelihood of all integers is considered.
Using a fixed critical value for the ratio test (b) results in very different and sometimes
large fail rates, depending on the vc-matrix. The reason is that the size of the aperture
pull-in regions is not adjusted if the precision changes. This can be seen for the first
128 Integer Aperture estimation