Page 40 - Satellite Communications, Fourth Edition

20 Chapter One

relative motion the wavelength is lengthened and therefore the received
frequency decreased. It should be kept in mind that the radio-beacon
emits a constant frequency, and the electromagnetic wave travels at con-
stant velocity, that of light. Denoting the constant emitted frequency by
f , the relative velocity between satellite and beacon, measured along the
0
line of sight as v, and the velocity of light as c, then to a close approxi-
mation the received frequency is given by (assuming v c):
v
f Q1 (1.1)
c Rf 0
The relative velocity v is positive when the line of sight distance is
decreasing, (satellite and beacon moving closer together) and negative
when it is increasing (satellite and beacon moving apart). The relative
velocity v is a function of the satellite motion and of the earth’s rotation.
The frequency difference resulting from the relative motion is
v (1.2)
f
f f f 0 0
c
The fractional change is
f v (1.3)

f 0 c
When v is zero, the received frequency is the same as the transmitted
frequency. When the beacon and satellite are approaching each other, v
is positive, which results in a positive value of Δf. When the beacon and
satellite are receding, v is negative, resulting in a negative value of Δf.
The time at which Δf is zero is known as the time of closest approach.
Figure 1.9 shows how the beacon frequency, as received at the satel-
lite, varies for different passes. In all cases, the received frequency goes
from being higher to being lower than the transmitted value as the
satellite approaches and then recedes from the beacon. The longest
record and the greatest change in frequency are obtained if the satellite
passes over the site, as shown for pass no. 2. This is so because the satel-
lite is visible for the longest period during this pass. Knowing the orbital
parameters for the satellite, the beacon frequency, and the Doppler shift
for any one pass, the distance of the beacon relative to the projection of
the orbit on the earth can be determined. However, whether the beacon
is east or west of the orbit cannot be determined easily from a single
pass. For two successive passes, the effect of the earth’s rotation on the
Doppler shift can be estimated more accurately, and from this it can
be determined whether the orbital path is moving closer to, or moving
away from the beacon. In this way, the ambiguity in east-west posi-
tioning is resolved. The satellite must of course get the information
back to an earth station so that the search and rescue operation can be
completed, successfully one hopes. The SARSAT communicates on a