The GNSS integer ambiguities: estimation and validation

Figure 5.13: 2-D example for IALS estimation. Float samples for which ā = ǎ are shown
as black dots. Left: Pf = 0.001; Right: Pf = 0.025.
This shows that the IALS success rate can be computed in the same way as the ILS
success rate by replacing Qâ with the up-scaled version 1
µ 2 Qâ. As explained in section
3.2.2 exact evaluation of the ILS success rate is too complex, but the up-scaled version
of the vc-matrix can also be used to compute the lower and upper bounds of the IALS
success rate as given in section 3.2.2.
Lower and upper bounds for the IALS fail rate can be obtained by bounding the integration
region with ellipsoidal regions. The lower bound can be obtained identically to
the one for EIA estimation as described in section 5.2. The upper bound follows in a
similar way by using µS0 ⊂ C ɛ 0 with C ɛ 0 an ellipsoidal region centered at 0 and with size
governed by ɛ = µmax
x∈S0
x Qâ .
Summarizing, the following lower and upper bounds for the IALS success rate and fail
rate can be used:
Pf ≥
z∈Z n \{0}
Pf ≤
Ps ≥
Ps ≤ P
z∈Z n \{0}
n
[2Φ( µ
i=1
P χ 2 (n, λz) ≤ 1
4 µ2 min
z∈Zn \{0} z 2
Qâ
P χ 2 (n, λz) ≤ µ 2 max x
x∈S0
2
Qâ
2σ i|I
(5.33)
(5.34)
) − 1] (5.35)
χ 2 (n, 0) ≤ µ2cn ADOP 2
(5.36)
with λz = z T Q −1
â z, and cn and ADOP as given in section 3.2.2. Note that the lower
bound of the success rate is equal to the IAB success rate.
If IALS estimation with a fixed fail rate is applied, especially the upper bound of the fail
rate is interesting. Unfortunately, equation (5.34) is not useful then, since max x
x∈S0
2 Qâ
must be known.
106 Integer Aperture estimation