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Shi et al. Satell Navig (2021) 2:5 Page 4 of 13

received power can be described through the free-space Table 3 SISRE values in each system
transmission formula:
Constellation SISRE (m)
Pr = Pt + Gt(θ) − Ls − La + Gr(ϕ) (2) GPS 0.44
Te GNSS satellite amplifer output power Pt and the BDS 0.59
gain of transmitting antenna Gt(θ) are Equivalent to Iso- Galileo 0.35
tropically Radiated Power (EIRP), while EIRP is related GLONASS 1.56
to the θ GNSS considering Gt(θ) . Te free space path loss
( Ls ) is the main loss in transmission. In order to simplify
the experimental model, the atmospheric loss ( La ) and the main lobe signals with similar frequency bands in
the polarization loss of receiving antenna are assumed each GNSS are selected for analysis in the study (Teunis-
0.5 dB and 0 dB, respectively in this paper. In this study, sen and Montenbruck 2017). Considering the main lobe
the +Z direction antenna gain Gr(ϕ) is set to 10 dB at of GPS III satellite transmitting antenna is 47° (Ram-
0° and approximately − 0.75 dB at 40°. To simplify the akrishnan et al. 2013) and according to the satellites sta-
model, the gain for the receiving angle greater than 40° tus of other navigation systems, the frequency band of
is considered as − 1.8 dB (Lorga et al. 2010). Considering each constellation and the main lobe width in this simu-
that the EIRP of each GNSS satellite is diferent due to lation are listed in the Table 2.
diferent generations and various years in orbit, accord- Considering the external input noise, the received
ing to the references (Liu et al. 2016, 2017; Steigenberger power can be expressed by the Carrier to Noise ratio
et al. 2017; Toelert et al. 2019), the EIRP settings of ( C/N 0):
each GNSS in the paper are given in Fig. 3. In spite of
each GNSS satellite has diferent frequency bands, only C/N 0 = Pr − 10 log (k · T eff ) (3)
10
where parameter k is Boltzmann constant
k = 1.38 × 10 −23 J/K , and T eff is the efective tempera-
35
ture of the entire front end, whose value depends on the
30 front-end design of each GNSS. Te T eff is set to 290 K
25 in this paper based on the GPS typical value (Diggelen
2009). Ten according to Eq. (3) the C/N 0 and Pr difer
20 by approximately 204 dB, i.e. − 200 dB W is equal to 4 dB
EIRP (dB·W) 15 Hz.

10
5 Dilution of precision and position error
Te position error is mainly caused by the pseudorange
0 BDS IGO/GEO error between the navigation satellite and the receiver,
BDS MEO according Acharya (2014), which can be expressed as:
Galileo
-5 GPS/GLONASS

∂R ∂R ∂R ∂R T
-10 dR = ∂x ∂y ∂z ∂t ·[ dx dy dz dt ] = Q · dξ
0 5 10 15 20 25
Off-boresight angle (°) (4)
Fig. 3 EIRP settings of each constellation in this simulation (Liu et al. where Q is k × 4 matrix used to describe the 3D relative
2016, 2017; Steigenberger et al. 2017; Thoelert et al. 2019)
position between the receiver and k available naviga-
tion satellites at that moment. It is specifed by direction
Table 2 Signal band cosines:
Constellation Confguration
∗
X −X ∗ Y −Y ∗ Z −Z ∗ 

∗
∗
1 1 1 −1
Band Carrier Main lobe width (°)  R ∗ 1 R ∗ 1 R ∗ 1 
∗
∗
∗
frequency  X −X ∗ Y −Y ∗ Z −Z ∗ 
2
2
2
(MHz)  R ∗ 2 R ∗ 2 R ∗ 2 −1 
Q =  . . . 
 . . . . 
BDS B1 1575.42 50 (MEO), 38 (GEO/IGSO)  . . . . . 
∗
∗

X −X ∗ Y −Y ∗ Z −Z ∗ 
∗
GPS L1 1575.42 47 k k k −1
R ∗ k R ∗ k R ∗ k
GLONASS L1 1602 a 40
∗
∗
∗
Galileo E1 1575.42 41 where X , Y , Z are the receiver position parameter
∗
∗
∗
∗
a GLONASS frequency depends on channel number k, in this simulation the estimated at that moment, and X , Y , Z , R are the
k
k
k
k
value is simplifed to 1602