Satellite Constellation - Fundamentals of Global Positioning System Receivers: A Software Approach - page 33

From the distances in Figure 3.1 one can see that the time delays from the
satellites are in the range of 67 ms (20,192 km/ c ) to 86 ms (25,785 km/ c ), where
c is the speed of light. If the user is on the surface of the earth, the maxi-
mum differential delay time from two different satellites should be within 19
( 86 67 ) ms. In this figure, the angle α is approximately 76.13 degrees and the

angle β is approximately 13.87 degrees. Therefore, the transmitting antenna on
the satellite need only have a solid angle of 13.87 degrees to cover the earth.
However, the antenna for the L 1 band is 21.3 degrees and the antenna for the
L 2 band is 23.4 degrees. Both are wider than the minimum required angle. The
solid angle of 21.3 degrees will be used in Section 3.13 to estimate the power to
the receiver. The antenna pattern will be further discussed in Section 5.2.

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3.4 MAXIMUM DIFFERENTIAL POWER LEVEL FROM DIFFERENT
SATELLITES

From Figure 3.1 one can calculate the relative power level of the received sig-
nals on the surface of the earth. The transmitting antenna pattern is designed to
directly aim at the earth underneath it. However, the distances from the receiver
to various satellites are different. The shortest distance to a satellite is at zenith
and the farthest distance to a satellite is at horizon. Suppose the receiver has an
omnidirectional antenna pattern. Since the power level is inversely proportional
to the distance square, the difference in power level can be found as

10 log 25785 2

20192 2

p = ≈ 2 . 1dB ( 3 . 1 )

It is desirable to receive signals from different satellites with similar strength.
In order to achieve this goal, the transmitting antenna pattern must be prop-
erly designed. The beam is slightly weaker at the center to compensate for the
power difference.

3.5 SIDEREAL DAY ( 10,11 )

Table 3.1 indicates that the satellite rotates around the earth twice in a sidereal
day. The sidereal day is slightly different from an apparent solar day. The apparent
day has 24 hours and it is the time used daily. The apparent solar day is measured
by the time between two successive transits of the sun across our local meridian,
because we use the sun as our reference. A sidereal day is the time for the earth
to turn one revolution. Figure 3.2 shows the difference between the apparent
solar day and a sidereal day. In this figure, the effect is exaggerated and it is
obvious that a sidereal day is slightly shorter than a solar day. The difference
should be approximately equal to one day in one year which corresponds to about
4min ( 24 × 60 / 365 ) per day. The mean sidereal day is 23 hrs, 56 min, 4.09 sec.