The GNSS integer ambiguities: estimation and validation

4.3 Comparison of the float, fixed, and BIE estimators
Although it is known that the BIE estimator outperforms its float and fixed counterparts
in terms of precision, it is not clear how large this difference will be under varying
measurement scenarios.
It follows that in the limits of the precision the following is true:
lim
σ→∞
ã = â and lim ã = ǎ (4.17)
σ→0
with the vc-matrix factored as Qâ = σ2Gâ. The first limit corresponds to an integer
grid size which is very small in relation to the size and extend of the PDF of the float
ambiguities. This implies that the summation over all integers will be approximately
equal to the integration over the space of reals, and then:
R
ã =
n
x exp{− 1
2â − x2Qâ }dx
exp{− 1
2â − x2 n
â(2π) 2 |Qâ|
=
}dx Qâ (2π) n = â (4.18)
2 |Qâ|
R n
This shows that if the precision is low the BIE estimator will approximate the float
estimator. If, on the other hand, the PDF of the float ambiguities becomes very peaked
the BIE estimator will approximate the fixed estimator. This can be shown as follows.
Let the ILS pull-in region of z ∈ Z n be given as Sz = {x ∈ R n | x − z 2 Qâ ≤
x − u 2 Qâ , ∀u ∈ Zn }. Then Sz is independent of σ, and the weight wz(x) is equal to
(see equation (4.8)):
wz(x) =
1 exp{− 2x − z2Qâ }
exp{− 1
2x − u2 Qâ }
u∈Z n
=
1 +
u∈Z n \{z}
1
exp{− 1
2σ2 (x − u2 Gâ − x − z2 Gâ )}
(4.19)
It follows thus that lim wz(x) = 1 if x ∈ Sz, and since the sum of all weights must
σ→0
equal one lim wu(x) = 0, ∀u = z if x ∈ Sz. So, wu(x) reduces to the indicator function
σ→0
of the ILS pull-in region in the limit σ → 0.
Obviously, the limiting cases in equation (4.17) imply that also the following is true:
lim
σ→∞ â − ã2Qâ = 0 and lim
σ→0 â − ã2Qâ = â − ǎ2Qâ (4.20)
It is now interesting to know how the BIE estimator performs in the intermediate cases
compared to the float and fixed estimators. In this section the three estimators are
therefore compared numerically. This also provides the possibility to compare their
distributional properties. The results were presented in Verhagen and Teunissen (2003).
72 Best Integer Equivariant estimation