Galileo Signal - A Software-Defined GPS and Galileo Receiver: A Single-Frequency Approach - page 45

X k Y k Z k The geocentric coordinates of satellite k (the superscript k denotes

satellite k , and not the power k ) at time t j are given as
⎡
⎣

( , , )

⎤
⎦ =

⎡
⎣

⎤
⎦ .

X k

r j cos f j

( t j )

Y k

k

i j )

k

r j sin f j

0

( t j ) R 3 ( −

j )

R 1 ( − R 3 ( − ω

j )

(3.15)

Z k ( t j )

For a definition of the parameters , i , ω ,and f , see Figure 8.5. The quantities

r j

= r ( t j ) , a , e ,and E are evaluated for t according to the following procedure:

Time elapsed since t oe t j = t − t oe

GM

n t j Mean anomaly at time t j µ j = µ 0 + / a 3 +

10 14 m 3 s 2 GM = 3 . 986 005 · /

Iterative solution for E j E j = µ j + e sin E j

arctan √ 1

e 2 sin E j
cos E j −

−

True anomaly f j =

e

= 0 + ( ˙

Longitude for ascending node j − ω e ) t j − ω e t oe

10 − 5 rad

ω e = 7 . 292 115 147 · / s

ω j = ω +

f j +

(ω +

f j ) +

(ω +

f j ) Argument of perigee C ω c cos 2 C ω s sin 2

Radial distance r j = a ( 1 − e cos E j ) + C rc cos 2 (ω + f j )

+ C rs sin 2 (ω + f j )

i 0 + it j + Inclination i j = C ic cos 2 (ω + f j ) + C is sin 2 (ω + f j ).

As usual the mean Earth rotation rate is denoted ω e . This algorithm is similar to

the one for GPS and is coded as the M -file satpos . The function computes the
position of any Galileo satellite at any time. It is fundamental to every position
calculation.

3.4.1 Time and Clock Correction Parameters
As for GPS, Galileo has its own system time, called Galileo System Time (GST).
Its starting epoch still has to be determined. GST consists of two parts: week
number, WN, and time of week, TOW. The WN covers 4096 weeks and is then
reset to zero. A week has 604,800 s and is reset at midnight between Saturday and
Sunday. Hence GST is described as a 32-bit binary number split into the two parts
just mentioned. Table 3.3 shows these parameters.
Let a signal be transmitted at time t k , Gal from satellite k , and let the same signal
be received at time t Gal
i

at receiver i . Then the travel time is

k
i

t Gal
i

t k , Gal τ = − . (3.16)