The GNSS integer ambiguities: estimation and validation

matrix D. From equation (3.21) the n intervals that are used for the search follow as:
(ˆz1 − z1) 2 ≤ σ 2 1χ 2
(ˆz 2|1 − z2) 2 ≤ σ 2 2|1 (χ2 − (ˆz1 − z1) 2
) (3.22)
.
σ 2 1
Besides the shape of the search space, also the size is important in order for the search
to be efficient. That means that it should not contain too many integer grid points. The
choice of χ2 should therefore be such that the search space is small but at the same time
such that it is guaranteed that it contains at least one integer, or two integers in the
case the solution will be validated, see section 3.5. Since the bootstrapped solution gives
a good approximation of the ILS estimator and is easy to compute, and by definition
ˆz − ˇzLS2 Qˆz ≤ ˆz − ˇzB 2 , the size of the search space can be chosen as:
Qˆz
χ 2 = (ˆz − ˇzB) T Q −1
ˆz (ˆz − ˇzB) (3.23)
If more than one candidate must be determined, bootstrapping can still be used to set
χ2 . The conditional estimates ˆz i|I are then not only rounded to their nearest integer, but
also to the second nearest integer. In this way n + 1 conditionally rounded integers ˇzB,j
are obtained, for which also the squared norms Rj = ˆz − ˇzB,j2 are determined. They
Qˆz
are collected in a vector (R1 · · · Rn+1) with the ordering Ri ≤ Ri+1, ∀i = 1, . . . , n. If
the requested number of candidates is l ≤ n+1, the size of the search space is chosen as
χ2 = Rl since it is then guaranteed that the search space contains at least l candidates.
Note that this approach to set the size of the search space is different from the one
described in De Jonge and Tiberius (1996), which is based on the relation between
the volume of the ellipsoid and the number of candidates it contains. This may result
in much larger search spaces. Note, however, that if more than n + 1 candidates are
requested, the approximation based on the volume of the search space must be used.
In Teunissen et al. (1996) it was shown that the volume, Vn, is a good indicator of the
number of integers, Nz, contained in an ellipsoidal region if this number is larger than
a few:
with
Nz ≈ [Vn] (3.24)
Vn = λ n
Un |Qˆz| (3.25)
and Un the volume of the unit sphere in Rn which is given as Un = π n
2 /Γ( n
2 +1), where
Γ(x) is the gamma function, see equation (A.11).
Once the size of the search space is set, the actual search can be carried out. In De Jonge
and Tiberius (1996), it is described how to do that in an efficient way.
Integer estimation 35