The GNSS integer ambiguities: estimation and validation

can be distinguished:
â ∈ Ωa success: correct integer estimation
â ∈ Ω \ Ωa failure: incorrect integer estimation
â /∈ Ω undecided: ambiguity not fixed to an integer
The corresponding probabilities of success (s), failure (f) and undecidedness (u) are given
by:
Ps = P (ā = a) = fâ(x)dx
Pf =
Ω\Ωa
Ωa
fâ(x)dx =
Ω0
fˇɛ(x)dx −
Pu = 1 − Ps − Pf = 1 − fˇɛ(x)dx
Ω0
Ωa
fâ(x)dx (5.4)
These probabilities are referred to as success rate, fail rate, and undecided rate respectively.
5.1.2 Distribution functions
From the definition of the new estimator in (5.2) it follows that the distribution function
will take a hybrid form, which means that it is a combination of a probability mass
function for all values within Ω, and a probability density function otherwise:
⎧
⎪⎨
fâ(x) if x ∈ R
fā(x) =
⎪⎩
n \{Ω}
fâ(y)dy = P (ā = z) if x = z, z ∈ Z
Ωz
n
0 otherwise
= fâ(x)¯ω(x) +
fâ(y)dyδ(x − z) (5.5)
z∈Zn Ωz
where δ(x) is the impulse function, ¯ω(x) = (1 −
z∈Zn ωz(x)) is the indicator function
of Rn \Ω. Note that P (ā = z) ≤ P (ǎ = z).
Also the distribution function of the corresponding baseline estimator ¯b is different compared
to the PDF of ˇb: f¯b(x) =
fˆb|â (x|z)P (ā = z) + fˆb|â (x|y)fâ(y)dy (5.6)
z∈Z n
R n \{Ω}
In the 1-D case the aperture space is built up of intervals around the integers. As an
example, it is assumed that a = 0 and Ωa = [−0.2, 0.2] and thus Ωz = [−0.2+z, 0.2+z].
The top panel of figure 5.1 shows the PDF of â in black, the PMF of ǎ in dark grey,
Integer Aperture estimation 89