The GNSS integer ambiguities: estimation and validation

1
0.5
0
−0.5
−1
−1 −0.5 0 0.5 1
Figure 5.10: Construction of projector test aperture pull-in region for Q ˆz,02 01, µ = 1.
And thus also the projector test is an IA estimator, and will be referred to as the PTIA
estimator.
From equation (5.24) follows that Ω0,P is bounded by the planes orthogonal to c and
passing through µc, and these planes themselves are bounded by the condition that
c = arg min
z∈Zn \{0} x − z2 for all x on the plane, see (Verhagen and Teunissen 2004a).
Qâ
2-D example
Figure 5.10 shows a 2-D example of the construction of the aperture pull-in region Ω0.
The black lines are the planes orthogonal to c and passing through µc. For the construction
of Ω0,P, these planes are bounded by the condition that c = arg min
z∈Z n \{0} x − z2 Qâ
for all x ∈ Ω0. Therefore, also sectors within the ILS pull-in region are shown as alternating
grey and white regions, with the sectors containing all x with a certain integer
c as second closest integer. The region Ω0,P follows as the region bounded by the intersection
of the black lines and the sectors, and by the boundaries of the sectors. This
results in the strange shape of the aperture pull-in regions in the direction of the vertices
of the ILS pull-in region.
Figure 5.11 shows in black all float samples for which ā = ǎ for two different fail rates.
5.4 Integer Aperture Bootstrapping and Least-Squares
A straightforward choice of the aperture pull-in regions would be to choose them as
down-scaled versions of the pull-in regions Sz corresponding to a certain admissible
integer estimator. This approach is applied to the bootstrapping and integer leastsquares
estimators.
102 Integer Aperture estimation