The GNSS integer ambiguities: estimation and validation

1
0.5
0
−0.5
−1
−1 −0.5 0 0.5 1
Figure 5.4: Construction of RTIA aperture pull-in region. For all x that fall in the grey
region, the second-closest integer z2 = [−1 0] T . The boundary of the aperture pull-in region
within this grey region is then equal to the ellipsoid with center − µ
z2 and size governed
1−µ
by √ µ
1−µ z2Q â . Examples are shown for µ is 0.3, 0.5, and 0.8, respectively. Ellipsoids are
grey; aperture pull-in regions black.
so that the integers z do not have any influence on the boundary of the aperture pull-in
region.
This means that the integer least-squares pull-in region S0 can be split up into sectors, all
having another integer c as second-closest, i.e. x2 Qâ ≤ x − c2Qˆx ≤ x − z2 , ∀z ∈
Qâ
Zn \ {0}. Within a sector, the aperture pull-in region is then equal to the intersection of
the sector with the ellipsoid with center − µ
1−µ c and size governed by √ µ
cQâ 1−µ . This
is illustrated for the 2-D case in figure 5.4.
It can be seen that especially for larger µ, the shape of the aperture pull-in region starts
to resemble that of the ILS pull-in region. The reason is that for larger µ the size of the
ellipsoid increases, and the center of the ellipsoid is further away, but in the direction of
[0, −c], and the ellipsoid has the same orientation as the ILS pull-in region.
2-D example
Figure 5.5 shows two 2-D examples of the construction of the aperture pull-in regions.
For a diagonal matrix the ILS pull-in region becomes square and there are only four
adjacent integers. It can be seen in the left panel that indeed only these four integers
determine the shape of the aperture pull-in region. For the vc-matrix Qˆz 02 01 there are
six adjacent integers that determine the shape.
Figure 5.6 shows in black all float samples for which ā = ǎ for two different fail rates.
96 Integer Aperture estimation