The GNSS integer ambiguities: estimation and validation

5.3.3 Difference test is an IA estimator
The difference test (3.86) as proposed by (Tiberius and De Jonge 1995) leads to acceptance
of ǎ if:
â − ǎ2 2 Qâ − â − ǎ2 Qâ
The acceptance region of this test is given as:
≥ µ (5.17)
ΩD = {x ∈ R n | x − ˇx 2 Qâ ≤ x − ˇx2 2 Qâ − µ} (5.18)
Let Ωz,D = ΩD ∩ Sz. Then:
⎧
⎪⎨
Ω0,D = {x ∈ R
⎪⎩
n | x2 Qâ ≤ x − z2Qâ − µ, ∀z ∈ Zn \ {0}}
Ωz,D = Ω0,D + z, ∀z ∈ Zn ΩD =
z∈Zn Ωz,D
(5.19)
The proof given in Teunissen (2003e) and Verhagen and Teunissen (2004a) is as follows:
Ωz,D = ΩD ∩ Sz
= {x ∈ Sz | x − ˇx 2 Qâ ≤ x − ˇx2 2 Qâ − µ}
= {x ∈ Sz | x − ˇx 2 Qâ ≤ x − u2 Qâ − µ, ∀u ∈ Zn \ {ˇx}}
= {x ∈ Sz | x − z 2 Qâ ≤ x − u2 Qâ − µ, ∀u ∈ Zn \ {z}}
= {x ∈ R n | x − z 2 Qâ ≤ x − u2 Qâ − µ, ∀u ∈ Zn \ {z}}
= {x ∈ R n | y 2 Qâ ≤ y − v2 Qâ − µ, ∀v ∈ Zn \ {0}, x = y + z}
= ΩD ∩ S0 + z
= Ω0,D + z
The first two equalities follow from the definition of the difference test, and the third
from ˆx − ˇx2 Qˆx ≤ ˆx − ˇx2 2 Qâ ≤ ˆx − u2Qâ , ∀u ∈ Zn \ {ˇx}. The fourth equality follows
since ˇx = z is equivalent to x ∈ Sz, and the fifth equality from the fact that µ ≥ 0. The
last equalities follow from a change of variables and the definition of Ω0,D = ΩD ∩ S0.
Finally note that
Ωz,D =
(ΩD ∩ Sz) = ΩD ∩ (
Sz) = ΩD
z∈Z n
z∈Z n
This ends the proof of (5.19).
z∈Z n
It has been shown that also in the case of the difference test the acceptance region
consists of an infinite number of integer translated copies of a subset of the ILS pull-in
region S0. And thus, the difference test is an IA estimator with aperture parameter µ.
It will be referred to as the DTIA estimator.
In a similar way as for the ratio test aperture pull-in region, it is possible to show how the
aperture pull-in region of the difference test is constructed, see (Verhagen and Teunissen
Ratio test, difference test and projector test 99