Antti Lehtinen Doppler Positioning with GPS - Matematiikan laitos

Dic
α = arccos . As a special case of equation 4.6 when the Doppler shift
L1vi
Di equals zero the receiver must be located on a plane that is orthogonal to vi
and passes through ri.
ru
vi ✛
α
ri
vi·cos α
Figure 4.1: Doppler effect in satellite positioning
If we had three ideal Doppler measurements, we could solve for the three unknowns,
namely the user position vector ru =[xu yu zu] T . The user position
would be found at the intersection of the three different cones defined by the
equation 4.3. However, things are unfortunately not that simple. It is extremely
difficult to measure the Doppler shift with the precision required for the positioning.
4.3.2 The Clock Drift Error
In order to accurately measure the frequency of a signal, one must have a very
stable clock. The same applies also to accurately generating a desired frequency.
Satellite frequency generation is based on a highly accurate free running atomic
standard. The GPS control segment makes sure that the frequencies transmitted
by the satellites are very close to the nominal frequency L1 and adds correction
terms to the navigation messages when necessary [Kaplan, p. 50]. Thus, the
error in the transmitted frequency causes little harm for the receiver.
By contrast, the receiver very seldom has sufficiently good an oscillator to accurately
estimate the Doppler shift of the received signal. Let us denote the time
derivative of the receiver clock bias, the receiver clock drift, by˙tu. If˙tu is positive,
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