Precise Orbit Determination of Global Navigation Satellite System of ...

Chapter 2 Observations of Orbit Determination
2.2.2 Doppler
2.2.2.1 Basic Observation Equation
S(t3 )
S(t2 )
S(t 4 )
S 1
r(t 4 )
S 2 S 2
14
S 1
dS
Figure 2-5 Two-Way Doppler Shift
r(t 1 )
S(t 4 -dt)
dS
S(t 1 +dt)
S(t 1 )
One-way Doppler shift is expressed as
f 1 S&
r
1 = ( 1−
)
(2-69)
ft1
c
where
f received frequency (downlink) at transponder
r1
f = f
t1
t
transmitted frequency at satellite
S1 & change rate of distance between satellite and ground station for downlink
When the transponder receives the signal and changes it to the coherent frequency ft2 = Kfr1,
then the signal is
returned to the transmitter again. Doppler shift received by the satellite (the transponder) is written as
f 2 S&
r
2 = ( 1−
)
(2-70)
ft
2 c
where
f r2
= fr
the received frequency (uplink) at the satellite
f t2
the transmitted frequency at the transponder
S2 & change rate of distance between satellite and ground station for uplink
From Figure 2-5,
1
1
2
1
2
T
1
2
1
2
S = || r ( t ) − S ( t ) || = {[ r ( t ) − S ( t )] [ r ( t ) − S ( t )]}
(2-71)
T
1
2
S2
= || r ( t4)
− S ( t3)
|| = {[ r ( t4)
− S ( t3)]
[ r ( t4)
− S ( t3)]}
(2-72)
T
dS1
[ r(
t1)
S ( t2)]
[ r&
&
( t1)
S ( t2)]
S&
−
−
1 = =
dt
S1
(2-73)
T
dS2
[ r ( t4)
S ( t3)]
[ r&
&
( t4)
S ( t3)]
S&
−
−
2 = =
dt
S
(2-74)
2
Using Eq.(2-69), Eq.(2-70) becomes
S&
2
S&
2 S&
1 S&
2
fr 2 = fr
= ft
2(
1−
) = fr1K
( 1−
) = ftK
( 1−
)( 1−
)
(2-75)
c
c
c c
Eq.(2-75) can also be rewritten as