The GNSS integer ambiguities: estimation and validation

Figure 5.9: 2-D example for DTIA estimation. Float samples for which ā = ǎ are shown
as black dots. Left: Pf = 0.001; Right: Pf = 0.025. Also shown is the IA least-squares
pull-in region for the same fail rates (in grey).
5.3.4 Projector test is an IA estimator
The tests defined in Wang et al. (1998) and in Han (1997) are both based on testing:
H0 : E{â} = ǎ versus Ha : E{â} = ǎ + c∇, with c = ǎ2 − ǎ
ǎ2 − ǎQâ
ǎ2 is the second-best integer candidate. Ha can be tested against H0 using the w-test
of equation (2.82). The fixed solution is accepted if:
(ǎ2
− ǎ)
T Q −1
â (â − ǎ)
ǎ2 − ≤ µ (5.21)
ǎQâ
This test is referred to as the projector test since the term on the left-hand side of
equation (5.21) equals a projector which projects â − ǎ orthogonally on the direction of
ǎ2 − ǎ, in the metric of Qâ. Furthermore note that from equation (3.90) it follows that:
(ǎ2
− ǎ)
T Q −1
â (â − ǎ)
ǎ2 − ǎQâ
≤ 1
2 ǎ2 − ǎQâ
This implies that the fixed solution is always accepted if µ ≥ 1
2 ǎ2 − ǎQâ .
The acceptance region of the projector test is given as:
(5.22)
ΩP = {x ∈ R n | (ˇx − ˇx2) T Q −1
â (x − ˇx) ≤ µˇx − ˇx2Qâ } (5.23)
In a similar way as for the difference test it can be proven that
⎧
Ω0,P ⎪⎨
= {x ∈ S0 | c
⎪⎩
T Q −1
â x ≤ µcQâ , c = arg min
z∈Zn \{0} x − z2Qâ }
Ωz,P = Ω0,P + z, ∀z ∈ Zn ΩP =
Ωz,P
z∈Z n
(5.24)
Ratio test, difference test and projector test 101