The GNSS integer ambiguities: estimation and validation
The GNSS integer ambiguities: estimation and validation
The GNSS integer ambiguities: estimation and validation
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<strong>and</strong> the hybrid distribution of ā in light grey. <strong>The</strong> bars indicate probability masses. <strong>The</strong><br />
bottom panel in the figure shows the corresponding distributions of the three baseline<br />
estimators. It can be seen that the PDF of ā always falls in between the PDFs of ˆ b <strong>and</strong><br />
ˇ b.<br />
5.1.3 IA <strong>estimation</strong><br />
It is now possible to design different IA estimators by defining the size <strong>and</strong> shape of<br />
the aperture pull-in regions Ωz. In classical hypothesis testing theory the size of an<br />
acceptance region is determined by the choice of the testing parameters: the false alarm<br />
rate <strong>and</strong> the detection power. However, in the case of <strong>integer</strong> ambiguity resolution it is<br />
not obvious how to choose these parameters. It is especially important that the probability<br />
of incorrect ambiguity fixing is small. <strong>The</strong>refore the concept of Integer Aperture<br />
<strong>estimation</strong> with a fixed fail rate is introduced. This means that the size of the aperture<br />
space is determined by the condition that the fail rate is equal to or lower than a fixed<br />
value. At the same time the shape of the aperture pull-in regions should preferably be<br />
chosen such that the success rate is still as high as possible. In the following sections<br />
several examples of IA estimators are presented.<br />
Note that if all probability mass of â is located within the pull-in region Sa the fail rate<br />
is automatically equal to zero, <strong>and</strong> thus the IA solution will always be equal to the fixed<br />
solution.<br />
It is very important to note that Integer Aperture <strong>estimation</strong> with a fixed fail rate is an<br />
overall approach of <strong>integer</strong> <strong>estimation</strong> <strong>and</strong> <strong>validation</strong>, <strong>and</strong> allows for an exact <strong>and</strong> overall<br />
probabilistic evaluation of the solution. With the traditional approaches, e.g. the ratio<br />
test applied with a fixed critical value, this is not possible. Two important probabilistic<br />
measures are the fail rate, which will never exceed the user-defined threshold, <strong>and</strong> the<br />
probability that ǎ = a if ā = z:<br />
P s|ā=z = P (ǎ = a|ā = z)<br />
= P (ǎ = a, ā = z)<br />
=<br />
P (ā = z)<br />
Ps<br />
Ps + Pf<br />
(5.7)<br />
<strong>The</strong> success <strong>and</strong> fail rate, Ps <strong>and</strong> Pf , from equation (5.4) are used. Note that if<br />
Pf