1、 Rec. ITU-R S.1257-3 1 RECOMMENDATION ITU-R S.1257-3 Analytical method to calculate short-term visibility and interference statistics for non-geostationary satellite orbit satellites as seen from a point on the Earths surface (Questions ITU-R 206/4 and ITU-R 231/4) (1997-2000-2001-2002) The ITU Radi
2、ocommunication Assembly, considering a) that there may be a need to calculate the probabilities of exceeding a given interference level in interference or sharing studies between non-GSO satellite orbit stations and other stations; b) that there are computer simulation programmes to derive the requi
3、red statistics but such programmes may not be commonly available; c) that computer simulations can be used to provide a large amount of statistics with good accuracy, but they may need expertise in their use and may require considerable time to run; d) that in some cases analytical methods may be ti
4、me saving and need less sophisticated means for calculation of limited amount of statistics, recommends 1 that the analytical methodology given in Annex 1 can be used to obtain short-term visibility statistics for interference and sharing studies between non-GSO networks and other GSO FSS networks (
5、The term “probability” is not used in the strict mathematical sense. It signifies “percentage of time”.); 2 that the methodology given in Annex 2 can then be used to obtain short-term interference statistics for interference and sharing studies between non-GSO networks and other GSO FSS networks; 3
6、that the information given in Annex 3 may be used to extend the methodology in Annex 1 to enable information on the frequencies and durations of short-term interference events to be derived; 4 that the following Notes should be regarded as part of this Recommendation: NOTE 1 The methodology can be u
7、sed for interference calculations between non-GSO feeder links and GSO stations. They can also be applied to any case where one antenna is pointing to a fixed direction and the other is either on board a non-GSO satellite, or is tracking the non-GSO satellite. NOTE 2 The methodology gives an average
8、 value for a non-synchronous satellite constellation. It is applicable also for a synchronous orbit constellation (satellite tracks are repeating after a certain period of time) if there are several satellite tracks through the area of interest. NOTE 3 The limitations of the methodology described in
9、 Annex 1 have to be taken into account. 2 Rec. ITU-R S.1257-3 ANNEX 1 Appendix 1 gives a method to define an area for which the probability that a satellite falls within the area to be calculated. Appendix 2 gives the method to calculate the probability. Appendix 3 gives the derivation of the formul
10、ae used in Appendix 2. APPENDIX 1 TO ANNEX 1 Calculation of the needed discrimination angle for a given interference level The case considered here is non-GSO/GSO sharing. The area is defined for interference: a) from non-GSO earth station to GSO satellite; b) from non-GSO satellite to GSO earth sta
11、tion; c) from GSO earth station to non-GSO satellite; d) from GSO satellite to non-GSO earth station. In the following equations: : elevation angle of a satellite from a station r : earth radius = 6 378 km H : altitude of the GEO satellite (km) h : altitude of the non-GSO satellite (km) dI: distance
12、 from the interference source to the interfered receiver (km) dN: distance to the non-GSO satellite from a station (km) dG: distance to the GSO satellite from a station (km) f : frequency (GHz) E : e.i.r.p. spectral density (dB(W/Hz) N0: noise spectral density (dB(W/Hz) G() : antenna gain to the dir
13、ection (degrees) from the main beam axis (dB) G : antenna on-axis gain (dB) Suffix G refers to the GSO network Suffix N refers to the non-GSO network Suffix E refers to an earth station Suffix S refers to a satellite Suffix T refers to transmitting parameter Suffix R refers to receiving parameter PR
14、 : protection ratio Rec. ITU-R S.1257-3 3 Distances to a non-GSO and GSO satellite from a station for the in-line case are: += sin2sin222rhhrrdN(1) += sin2sin222rHHrrdG(2) C0 /I0method NEGElineinEEIC )/(00=for case a) (3) NGNSGSlineinddEEIC log20log20)/(00+= for case b) (4) GENElineinEEIC )/(00=for
15、case c) (5) GNGSNSlineinddEEIC log20log20=)/(00+ for case d) (6) The GSO and non-GSO earth stations should be either co-located or, in cases a) and d), the GSO satellite antenna discrimination has to be taken into account, and in cases b) and c) the non-GSO satellite antenna discrimination has to be
16、 taken into account. The required antenna discrimination for a non-GSO earth station, in cases a) and d), and for a GSO earth station in cases b) and c), is: RPICGGlinein=)/()(00(7) To calculate the required avoidance angle , the appropriate antenna gain patterns has to be used. For GSO earth statio
17、ns the side lobe pattern used in equation (8) is given in Recommendation ITU-R S.580. Avoidance angle at side lobe area is: 25/)(29(10=G(8) For the main beam area, the discrimination angle may be calculated by: 12)(0=GG(9) In case b) the distance from the non-GSO satellite to the GSO earth station c
18、hanges as the satellite moves on its orbit. For small values of and for high elevation angles it need not be taken into account but in other cases the distance dN has to be calculated separately for the highest elevation ( + ) and for the lowest elevation ( ) and the new values have to be used in eq
19、uation (2). For calculations in Appendices 2 and 3 the angle is the radius of the area for which the probability is to be calculated. For the calculation of statistics 1= /2 and 2= + /2. I0 /N0method To calculate the required avoidance angle for the case based on I0 /N0requirement, the following equ
20、ations are used: requiredlineinNINIGG )/()/()(0000=(10) I0 /N0for the in-line case can be calculated by: 5.92log20log20)/(000=fdNENIIlinein(11) 4 Rec. ITU-R S.1257-3 where: E : e.i.r.p. of the interfering transmitter N0: noise spectral density of the wanted receiving system. APPENDIX 2 TO ANNEX 1 Ca
21、lculation of in-area statistics for non-GSO satellites 1 Introduction The method given in this Appendix can be used in calculating the probability to find a satellite of a constellation in a circular or rectangular area (azimuth/elevation or latitude/longitude). A circular area may be for example a
22、radio-relay link or satellite earth station antenna main beam or side lobe area. If the area is based on an offset angle where certain interference level is achieved, the result is the probability that a given interference level is exceeded. The method can be used to calculate the interference betwe
23、en non-GSO and GSO networks. The method can also be used in calculating the probability of interference between a non-GSO earth station and a receiving fixed satellite station or FSS earth station. The calculation uses the discrimination angle to define an area where the permissible interference lev
24、el is exceeded. The method can be used for any observation point (e.g. earth station) latitude and for any satellite altitude, inclination, azimuth and elevation, but only in the case that the satellite may be visible in the defined area. (See 5.) 2 Symbols used (in Appendix 2 and Appendix 3) Ts : S
25、atellite orbital period (min) Te : Earth rotation time (min) L0 : Latitude of the observation point (rad) L: Latitude of the area (rad) lm : Medium length of the tracks through a circle b: Length of the area in longitudinal direction (rad) i: Inclination of satellite orbit (rad) rc : Radius of the c
26、ircular area (rad) : Angle between ground track and latitude line (rad) A: Area on a spherical surface (sterad) Ac : Area of a circle on a spherical surface (sterad) Ph : Probability to hit the area (one satellite calculation) Pi : Probability to be inside the area if hit during the revolution (one
27、satellite calculation) Rec. ITU-R S.1257-3 5 P: Probability for a satellite to be inside the defined area (one satellite calculation) Pc : Probability that any one of the satellites in a constellation is inside the area N: Number of satellites in the constellation k= r / (r + h) : Azimuth of the cen
28、tre of the area (rad) : Elevation of the centre of the area (rad) r : Radius of the Earth h : Altitude of the satellite : Nadir angle from subsatellite point (see Fig. 3) (rad) : Geocentric angle in elevation direction corresponding to (rad) :2 Geocentric angle for the highest point of the area in e
29、levation direction (rad) :1 Geocentric angle for the lowest point of the area in elevation direction (rad) : Geocentric angle difference in direction perpendicular to (rad) : Width of the area in azimuthal direction (e.g. antenna beamwidth in azimuthal direction) 1, 2: The highest and lowest elevati
30、on of the area (rad) (1, 2is e.g. the antenna beamwidth in the elevation direction) 3 Calculation of probability The following formulae are a collection of those in Appendix 3 and given here are only those which are needed for the probability calculation. The numbering of formulae is the same as in
31、Appendix 3, explanation is given for the calculated parameters. hrrk+= (21) 111)cos(arccos)2/(1=+=k (22) 222)cos(arccos)2/(2=+=k (22) 12= (23) 221+= (24) =cossin)2/(tanarctan2 (25) L = arcsin (cos sin L0+ sin cos L0 cos ) (27) Licoscosarccos= (15) 6 Rec. ITU-R S.1257-3 LAPcos1sin122= (19a) PNPc= For
32、 a circular area: A = Ac=4cA For a rectangular area: A = (2 1) (26) APPENDIX 3*TO ANNEX 1 Derivation of the formulae given in Appendix 2 1 Probability that a satellite is in a given area Figure 1 shows a non-GSO satellite orbit around the Earth. A simple case is the orbit over the poles. If the sate
33、llite is projected on the Earth surface, ground tracks are created. If the Earth would be stationary, there would be only one track over the poles. However, as the Earth rotates during the time it takes the satellite to make one revolution, the next ground track will be shifted by a longitude differ
34、ence equal to: Ts/ Te2 (12) Figure 1 shows several tracks over a longer period of time. The rectangular area is an area for which the probability is to be calculated. The probability that the satellite hits the area is the portion of length 2 b from the whole length of the shadowed band around the E
35、arth (see Fig. 1). The multiple 2 takes into account the fact that the satellite crosses the band twice during one revolution around the Earth. The length of the band is 2 cos L and the probability to hit a rectangular area as in Fig. 2 is: LbPhcos1= (13) In this equation, the value of b is in radia
36、ns and it is the actual length expressed as a geocentric angle. If the longitude difference is used, then: b = (Longitude difference) cos L (13a) It should be noted that the area used in calculations is the area at the satellite orbit shell. For a circular antenna beam that area is an ellipsoid whos
37、e major axis is in elevation direction. The latitude L used in calculations is not the station latitude but the latitude of a point from the orbit shell projected to the Earth surface. In this case, the projected point is the centre of the area. _ *The symbols used in this Appendix are given in 2 of
38、 Appendix 2. Rec. ITU-R S.1257-3 7 For a common case, when the satellite orbit inclination is different than 90, the probability of a satellite to be inside the defined area is the probability to hit the area multiplied by the medium length of the tracks inside the area and divided by the length of
39、one revolution. The probability to hit the area is dependent on the length of its projection on a latitude line crossing the middle of the area. The projection is made parallel to the ground tracks. Calculations are here presented for a circular area case, because it is more illustrative. 1257-01b2r
40、cbSatellite orbitGround tracksCircular area for calculationLatitude lineTrack directionFIGURE 2Length b of a circulararea projection on latitude lineFIGURE 1Tracks of polar orbit satellite(Rectangular area for calculation)According to Fig. 2, the length of b is: =sin2crb (14) Licoscosarccos= (15) An
41、gle is the angle between the satellite track and latitude line. The rotation speed of the Earth does not need to be taken into account in the probability calculation, but it should be taken into account if the real angle, in relation to the rotating Earth, is needed. The average length of a large nu
42、mber of equally spaced paths traversing a circle is: 2cmrl= (16) If the satellite hits the area during one revolution, the probability that it is inside the area is the length of the track inside the area divided by the length of one revolution. 42cmirlP = (17) 8 Rec. ITU-R S.1257-3 The total probab
43、ility is then: 2cos1sin1ccihrLrPPP= (18) 1257-03CGC2GC1rOEALL0hGC2L0O12GC2GC3L0OEquatorOrbit shell ofnon-GSO satelliteEarthEquatorial planeof orbit shellProjection of GC1on horizontal planE: Earths centreO: observation pointC: centre of the areaGC1: great circle line between observation point and su
44、b-satellite point of the centre of the areaGC2: projection of vertical line on orbit shellGC3: projection of horizontal line on orbit shellFIGURE 3aProjection of circular beam on orbit shellFIGURE 3cArea size in horizontal directionFIGURE 3bArea size in vertical directionRec. ITU-R S.1257-3 9 From t
45、hat follows: LAPccos1sin122= (19a) LiAPc222sinsin12= (19b) Part of the area may be under the horizon as in the case of the radio-relay link antenna. In that case, only the area above the horizon is used. The last part of equation (19b) may be taken to be a conversion factor: Lic22sinsin1= (19c) Valu
46、es of this factor are given in Fig. 4. 1257-0440 5060708085 9014121086420102030405060708090FIGURE 4Conversion factors for different area latitudes and inclinationsConversionfactor,kArea latitude (degrees)Satellite inclination = Other calculation or simulation results may be converted to other latitu
47、des and inclinations for the same satellite altitude by using conversion factor: 122222121221sinsinsinsinccLiLic = (19d) 10 Rec. ITU-R S.1257-3 The rotation speed of the Earth does not need to be taken into account for the probability calculation but if the time the satellite is inside an area is to
48、 be calculated, the vector sum of the satellite speed and the local Earth speed should be used. It can be shown that the result is independent of the shape of the area A, which in the case of a circle is r2. For a rectangular area (azimuth, elevation) the area is: A = (2 1) 2 Calculation of the area
49、 In the following, the area which has been defined by elevation and azimuth values is projected onto the spherical orbit shell of the satellite. From the triangle EOC in Fig. 3a: ()rhr=+ sin2/sin(20) hrrk+= (21) = (/2 + ) = arccos (k cos ) (22) The angle is calculated separately for the highest point of the area, ,2and for th
