The GNSS integer ambiguities: estimation and validation

0.6
0.4
0.2
0
−0.2
−0.4
−0.6
y
−0.6 −0.4 −0.2 0 0.2 0.4 0.6
Figure 5.17: Aperture parameter for OIA estimation approximated using IAB estimation
with Pf,IAB = 0.025. y is an element of the boundary of both the IAB pull-in region and
the approximated OIA pull-in region. The dashed OIA pull-in region is the one for which
Pf,OIA = 0.025.
Since it is quite easy to determine the aperture parameter µ for a fixed fail rate in the case
of IAB or EIA estimation, it would be interesting to investigate whether this parameter
can be used to approximate the aperture parameter for e.g. Optimal IA estimation such
that Pf,OIA(µ ′ ) ≈ Pf,IAB(µ) with µ ′ the approximated aperture parameter.
One approach would be to determine the aperture parameter for IAB estimation using
a root finding method. Then an ambiguity vector on the boundary of the bootstrapped
aperture pull-in region Ω0,B is determined. From the definition of the bootstrapped pullin
region in equation (3.12) it can be seen that S0,B is constructed as the intersection
of half-spaces which always pass through the points ± 1
2 ci, with ci the canonical vector
with a one as its ith entry. From equation (3.10) follows then that â = 1
2 cn will lie on
the boundary of S0,B and Scn,B since
ǎj,B =
ǎn,B =
=⇒
j−1
âj − σâjâ σ i|I
i=1
−2
â (âi|I − ǎi)
i|I
⎡
n−1
⎢
⎣ân −
i=1
ǎB = 0 or
ǎB = cn
Now y is chosen as:
σânâ i|I σ −2
⎤
⎥
â (âi|I − ǎi) ⎦ = [ân] =
i|I
=0 ∀i
= 0, ∀j = 1, . . . , n − 1
1
2
y = µ · 1
2 cn, (5.57)
since this will be an element of the boundary of the scaled bootstrapped pull-in region,
i.e. the IAB pull-in region. It must then be checked if y ∈ S0,LS, which will generally
116 Integer Aperture estimation