The GNSS integer ambiguities: estimation and validation

The optimization problem of (5.49) is thus equivalent to:
or to
max(Ps
+ Pf ) subject to: Pf = β (5.54)
Ω0
max
Ω
fâ(x)dx subject to:
Ω
fâ(x)(1−sa(x))dx = β, Ω = Ω+z, z ∈ Z n (5.55)
Ω
Lemma 5.6.1 can now be applied to obtain the following result:
Ω = {x ∈ R n |
fâ(x + z) ≥ λ
fâ(x + z)(1 − sa(x + z))}
Therefore
z∈Z n
z∈Z n
Ω0 = Ω ∩ S0
= {x ∈ R n |
fâ(x + z) ≥ λ(
fâ(x + z) − fâ(x + a)), x ∈ S0}
z∈Z n
z∈Z n
= {x ∈ R n |
fâ(x + z) ≤ λ
λ − 1 fâ(x + a), x ∈ S0}
z∈Z n
This ends the proof of (5.50).
The best choice for S0 is the ILS pull-in region. The reason is that Ps + Pf is independent
of S0, but Ps and Pf are not. Therefore, any choice of S0 which makes Ps
larger, automatically makes Pf smaller. So indeed the best choice for S0 follows from:
max
S0
Ω∩S0
fâ(x + a)dx subject to Ω = Ω + z, ∀z ∈ Z n
as the ILS pull-in region, see (Teunissen 1999a).
Recall that
fâ(x + z)s0(x) = fˇɛ(x). It follows thus that the aperture space is
z∈Z n
defined in the same way as for PIA estimation, only that µ does not depend on the
penalties anymore, but on the choice of Pf = β. This new approach will be referred to
as Optimal IA estimation (OIA estimation).
In practice it will be difficult to compute µ for a certain choice of the fail rate. Instead,
one could use the following upper bound:
β = Pf =
fâ(x + z)dx
z∈Zn \{a} Ω0
≤ (µ − 1)fâ(x + a)dx = (µ − 1) fâ(x)dx
Ω0
≤ (µ − 1) fâ(x)dx ≤ (µ − 1)Ps,LS
Sa
Ωa
(5.56)
Optimal Integer Aperture estimation 113