Satellite Constellation - Fundamentals of Global Positioning System Receivers: A Software Approach - page 40

In this equation, only the magnitude is of interest, thus, the sign is neglected.
The corresponding rate of change of the Doppler frequency is

0 . 178 × 1575 . 42 × 10 6
3

dv d
dt

f r
c

δf dr | max = = = 0 . 936 Hz/s ( 3 . 16 )

× 10 8

This value is also very small. If the frequency accuracy measured through the
tracking program is assumed on the order of 1 Hz, the update rate is almost one
second, even at the maximum Doppler frequency changing rate.

3.9 RATE OF CHANGE OF THE DOPPLER FREQUENCY DUE TO USER
ACCELERATION

From the previous two sections, it is obvious that the rate of change of the
Doppler frequency caused by the satellite motion is rather low; therefore, it does
not affect the update rate of the tracking program significantly.
Now let us consider the motion of the user. If the user has an acceleration
of 1 g (gravitational acceleration with a value of 9.8 m/s 2 ) toward a satellite,
the corresponding rate of change of the Doppler frequency can be found from
Equation (3.15) by replacing dv d /dt by g . The corresponding result obtained
from Equation (3.16) is about 51.5 Hz/s. For a high-performance aircraft, the
acceleration can achieve several g values, such as 7 g. The corresponding rate of
change of the Doppler frequency will be close to 360 Hz/s. Comparing with the
rate of change of the Doppler frequency caused by the motions of the satellite
and the receiver, the acceleration of the receiver is the dominant factor.
In tracking the GPS signal in a software GPS receiver two factors are used to
update the tracking loop: the change of the carrier frequency and the alignment
of the input and the locally generated C/A codes. As discussed in Section 3.5,
the input data adjustment rate is about 20 ms due to the Doppler frequency on
the C/A code. If the carrier frequency of the tracking loop has a bandwidth of
the order of 1 Hz and the receiver accelerates at 7 g, the tracking loop must be
updated approximately every 2.8 ms (1/360) due to the carrier frequency change.
This might be a difficult problem because of the noise in the received signal.
The operation and performance of a receiver tracking loop greatly depends on
the acceleration of the receiver.

3.10 KEPLER'S LAWS ( 11,12 )

In the previous section, the position of a satellite is briefly described. This infor-
mation can be used to determine the differential power level and the Doppler
frequency on the input signal. However, this information is not sufficient to
calculate the position of a satellite. To find the position of a satellite, Kepler's
laws are needed. The discussion in this section provides the basic equations to
determine a satellite position.