Satellite Constellation - Fundamentals of Global Positioning System Receivers: A Software Approach - page 36

Substituting these results into Equation (3.3), one obtains

v s r e cos θ
AS

v s r e cos θ
r e + v d = = ( 3 . 6 )

r s − 2 r e r s sin θ

This velocity can be plotted as a function θ and is shown in Figure 3.4.
As expected, when θ π/ 2, the Doppler velocity is zero. The maximum

Doppler velocity can be found by taking the derivative of v d with respect to θ
and setting the result equal to zero. The result is

=

vr e [ r e r s sin 2 θ dv d

dθ

− (r e + r s ) sin θ + r e r s ]

= = 0 ( 3 . 7 )

(r e + r s − 2 r e r s sin θ) 3 / 2

Thus sin θ can be solved as

= sin − 1 r e
r s

r e
r s

sin θ = or θ ≈ 0 . 242 rad ( 3 . 8 )

At this angle θ the satellite is at the horizontal position referenced to the user.
Intuitively, one expects that the maximum Doppler velocity occurs when the
satellite is at the horizon position and this calculation confirms it. From the orbit

FIGURE 3.4 Velocity component toward the user versus angle θ .