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Satellite Navigation 155
receiver is important. Most of these requirements are provided by a mes-
sage contained in a separate signal transmitted by the satellite (the navi-
gation message), which reports the satellite ephemeris (position and orbit
information) and a timing signal. The frequency of transmission is known
by all users and must be provided by a very stable source aboard the satel-
lite. The following sections show how this information is combined to
allow an observer to compute a position on the earth.
Doppler Ranging
The frequency received from the satellite (fR) consists of the transmit-
ted frequency (fT) plus a Doppler frequency (+,) due to the relative
motion between the satellite and receiver. The receiver also has a stable
oscillator which produces a reference signal at approximately the same
frequency as transmitted by the satellite (fG). Subtracting this frequency
from the received frequency, the receiver determines the shift in frequen-
cy in terms of a Doppler count. The Doppler count represents a counting
of the number of frequency cycles occurring between timing signals trans-
mitted by the satellite. In equation form:
where tl + Rl/c represents the time the signal was received after it was trans-
mitted from the satellite at time tl and traveled to the receiver over a slant
range of R1 at the speed of light c. Expanding equation 7-1 into two parts:
t2 +R~lc t2 +R2 IC
NI=j f,dt-! tl+Rl/c fR dt (7 - 2)
ti +Rl/c
The first part is simply the integral of a constant since the receiver pro-
duces a stable frequency (fc), but the second part contains the changing
received frequency fW However, the second integral also represents the
number of cycles received between the two timing signals (sent by the
satellite) which must be equal to the number of cycles sent by the satel-
lite between those same two times. This allows us to rewrite the second
part in terms of simply the transmitted frequency and the transmitter tim-
ing signals:
