The GNSS integer ambiguities: estimation and validation

5.8.2 OIA estimation versus RTIA and DTIA
The results in the preceding section have shown that the RTIA and the DTIA estimator
both perform close to optimal for most examples. Therefore, in this section the relation
between these estimators and the OIA estimator is investigated.
Recall that with the ratio test the fixed solution is accepted if and only if:
R1
≤ µR
R2
With the difference test the acceptance criterion is:
R2 − R1 ≥ µD
With OIA estimation the fixed solution is used if and only if:
with
fˇɛ(â − ǎ)
≤ µO
fâ(â − ǎ)
fˇɛ(â − ǎ)
fâ(â − ǎ) =
∞
i=1
exp{− 1
2 Ri}
exp{− 1
2
1 exp{− 2 = 1 +
R1} R2}
exp{− 1 +
R1}
2
∞
i=3
exp{− 1
2 Ri}
exp{− 1
2 R1}
(5.63)
where the i refers to the solution that is best (1), second-best (2), etcetera. This
expression follows from equations (3.26) and (3.48).
An important difference with the ratio test statistic and the difference test statistic is
that in this expression all Ri, i = 1, . . . , ∞ are taken into account and not just those
that correspond to the best and second-best integer solution. From equation (5.63)
follows that:
fˇɛ(â − ǎ)
1
≥ 1 + exp{−
fâ(â − ǎ) 2 (R2 − R1)} (5.64)
So, if Ω ′ 0 is the aperture pull-in region corresponding to:
1 + exp{− 1
2 (R2 − R1)} ≤ µO
if follows that Ω0,O ⊂ Ω ′ 0. Equation (5.65) can be rewritten as:
(5.65)
R2 − R1 ≥ −2 ln(µO − 1) (5.66)
Equation (5.66) is identical to the difference test. Hence, it follows that Ω0,O ⊂ Ω0,D if
µD = −2 ln(µO − 1), and the fail rate and success rate of the DTIA estimator will be
higher than with OIA estimation.
If the precision as described by the vc-matrix Qâ is high the squared norms Ri, i > 1 will
be large and the last term on the right-hand side of equation (5.63) will become small.
Comparison of the different IA estimators 125