The GNSS integer ambiguities: estimation and validation

1
0.5
0
−0.5
−1
−1 −0.5 0 0.5 1
Figure 5.8: Construction of DTIA aperture pull-in region (grey) for Q ˆz 02 01.
2004a). From the definition of Ω0,D follows that:
Ω0,D : x 2 Qâ ≤ x − z2Qâ − µ, ∀z ∈ Zn \ {0}, µ ≥ 0
⇐⇒ x T Q −1 1
â z ≤
2 (z2Qâ − µ)
⇐⇒ zT Q −1
â x
≤
zQâ
z2 − µ Qâ
,
2zQâ
∀z ∈ Z n \ {0} (5.20)
On the left-hand side of equation (5.20) the orthogonal projection of x onto the direction
z can be recognized. This shows that the aperture pull-in region Ω0,D is constructed as
intersecting half-spaces which are bounded by the planes orthogonal to z and passing
through the points 1
2 )z.
(1 − µ
z 2 Q â
The construction of Ω0,D is thus very similar to that of the ILS pull-in region S0, see
equation (3.17), which is also constructed of half-spaces bounded by the planes orthogonal
to z. The difference is that these planes pass through the midpoint 1
2z, whereas
in the case of the difference test this point depends on the distance z2 and on µ.
Qâ
This implies that the difference test aperture pull-in region is a down-scaled version of
the ILS pull-in region, but that the scaling is different in the various directions.
2-D example
Figure 5.8 shows a 2-D example of the construction of the aperture pull-in region. The
black lines are the planes orthogonal to c and passing through the point 1 µ
2 (1 − c2 )c,
Qâ where the c are the six adjacent integers.
Figure 5.9 shows in black all float samples for which ā = ǎ for two different fail rates.
100 Integer Aperture estimation