LoS MIMO Transmission for LEO Satellite Communication Systems

Lingxuan Li,Tingting Chen,Wenjin Wang,*,Xiaohang Song,Li You,Xiqi Gao

1 National Mobile Communications Research Laboratory,Southeast University,Nanjing 210096,China

2 Vodafone Chair Mobile Communications Systems,Technische Universit¨at Dresden,01062 Dresden,Germany

*The corresponding author,email:wangwj@seu.edu.cn

Abstract:To provide global service with low latency,the broadband low earth orbits(LEO)satellite constellation based communication systems have become one of the focuses in academic and industry.To allow for wideband access for user links,the feeder link of LEO satellite is correspondingly required to support high throughput data communications.To this end,we propose to apply line-of-sight(LoS)multiple-input multiple-output(MIMO)transmission for the feeder link to achieve spatial multiplexing by optimizing the antenna arrangement.Unlike the LoS MIMO applications for static scenarios,the movement of LEO satellites make it impractical to adjust the optimal antenna separation for all possible satellite positions.To address this issue,we propose to design the antenna placement to maximize the ergodic channel capacity during the visible region of the ground station.We frst derive the closed-form probability distribution of the satellite trajectory in visible region.Based on which the ergodic channel capacity can be then calculated numerically.The antenna placement can be further optimized to maximize the ergodic channel capacity.Numerical results verify the derived probability distribution of the satellite trajectory,and show that the proposed LoS MIMO scheme can signifcantly increase the ergodic channel capacity compared with the existing SISO one.

Keywords:LoS MIMO;LEO satellite;ergodic channel capacity;Beyond 5G

I.INTRODUCTION

Taking the advantages of moderate propagation delay,low launch cost and high data rate,the low earth orbit(LEO)satellites with an orbit altitude between 200 km and 2000 km have currently become one of the major focuses in academic and industry.In particular,the LEO satellite constellation based communication systems are considered as a complement and extension of the land mobile communications system and can also be a signifcant part of 5G and beyond networks[1–7].In addition to the LEO satellite constellation system that operates quite maturely like the Iridium system,several companies have proposed to provide wideband access from space using large constellations of LEO satellites,such as OneWeb,Telesat,and SpaceX[8].For scenarios such as backhaul communication at millimeter wave(mmWave)frequencies and satellite communication,the line-of-sight(LoS)path has the absolute dominance and other paths can be much weaker.Based on this,some studies were conducted on the theoretical and engineering applications of LoS multiple-output(MIMO)transmission in the above-mentioned scenarios.The core principle of LoS MIMO transmission is to place the antenna elements suffciently far apart to obtain spatial multiplexing in LoS environment[9–14].The mmWave LoS MIMO architecture for short-range indoor applications was investigated in[15–17],which maximized the spatial degrees of freedom by adjusting the antenna spacing without a rich scattering environment.In[18],a 60GHz mmWave LoS 2×2 MIMO demonstrator was presented for practical data transmission tests,which verifed the effectiveness of mmWave LoS MIMO technique.Additionally,[18]showed the robustness of LoS spatial multiplexing theoretically,while[19]showed that via numerical evaluation.For the feeder link of geostationary earth orbit(GEO)satellite,a theoretical derivation demonstrated that the MIMO orthogonal channel matrix for the satellite link can be setup to achieve optimum MIMO capacity via optimizing the antenna element spacing[20].This work has been extended to satellite broadcasting systems with regenerative payloads and transparent payloads,respectively[21,22].Moreover,a 2×2 LoS MIMO measurement over satellite was also performed to prove the effectiveness of exploiting the spatial multiplexing in LoS MIMO architecture[23].

Although the application of LoS MIMO in GEO system has been fully verifed,there is no related research on applying LoS MIMO to LEO feeder links to the authors’best knowledge.It is worth noting that in static LoS MIMO scenarios such as the short-range indoor mmWave transmission and the GEO communication system,the variation of the relative position between the transmitter and the receiver can be ignored.The optimal antenna placement can then be optimized to form a near full-rank channel matrix.However,since the optimal antenna placement depends on the locations of satellite and the LEO satellite is constantly moving relative to earth,the LoS MIMO design for these static scenarios cannot be directly applied in the LEO satellite systems.Authors of[18]and[19]performed a LoS MIMO communication sensitivity study,which revealed high robustness of the spatial multiplexing gain for linear arrays over a wide angle range centered on the array normal direction.Since this angle range corresponds to the elevation angle range for LEO satellite communication system,large spatial multiplexing gains can be expected even if the satellite-ground antenna array geometry is timevarying.Based on this theory,we investigate the LoS MIMO time-average channel capacity(ergodic channel capacity)during the LEO satellite visible region.

In this paper,we propose to apply LoS MIMO transmission for the feeder link of LEO satellite to maximize the ergodic channel capacity by optimizing the antenna array geometry.The major contributions of this paper are summarized as follows:

·Starting from the motion characteristics of the LEO satellite,we simplify the feeder link ergodic channel capacity analysis of the satellite constellation into one of the satellites by investigating the parameters related to the capacity calculation.

·In order to obtain the ergodic channel capacity during the visible region,we derive the closedform expression of the satellite trajectory probability distribution during the ground station visible region.

·Based on the closed-form probability distribution obtained above,the LoS MIMO ergodic channel capacity is numerically calculated and the satellite and ground antenna placements which can maximize the ergodic channel capacity are derived.

Outline:The rest of this paper is organized as follows.Section II introduces the LEO satellite constellation system model and the issues involving the ergodic channel capacity calculation.Section III gives the calculation process of the ergodic channel capacity and the antenna arrangement that can maximize the corresponding capacity.Simulations are contained in Section IV.Finally,the conclusion is drawn in Section V.

Notations:The following notation throughout the paper is adopted:We useto denote the imaginary unit.Upper case boldface letters denote matrices.We adopt INto denote theN×Ndimensional identity matrix.The superscripts(·)Hand(·)Tstand for the conjugate-transpose and transpose of a matrix,respectively.gcd(a,b)denotes the greatest common divisor ofaandb.denotes the smallest integer that is not less thanx.The notation‖.‖represents the Euclidean vector norm and|.|gives the absolute value of a scalar.

II.SYSTEM MODEL

In this paper we investigate the LoS MIMO channel capacity of the feeder link between a given ground station and LEO satellite constellation based communication system,as shown in Figure 1.

Figure 1.LEO satellite constellation system.

Figure 2.MIMO feeder link architecture.

Assuming that the LEO satellite constellation is comprised ofNorbitsatellite orbits,where all orbits are circular with height beingD,and the inclination angle between each orbital plane and the earth’s equatorial plane isμ.Since the longitudes of the ascending node(where the moving satellite passes from the southern hemisphere into the northern hemisphere)are different for different satellite orbits,we defne the ascending node longitude of thei-th satellite orbit asθi,1≤i≤Norbit.Supposing that there are a total ofNsatLEO satellites operating in the orbits defned above,and the period of the satellite’s rotation around the earth isTS.For thej-th,1≤j≤Nsatsatellite operating in one of the satellite orbits,we representtjas the instant when the satellite passes through the ascending node of the orbit.

Since a LEO satellite constellation consists of multiple LEO satellites distributed in different orbital planes,we start from studying the channel capacity between a single LEO satellite and the ground station of multiple antennas(see Figure 2),and then extend it to the entire constellation.

Figure 3.ECEF coordinate system.

2.1 Channel Capacity of LEO Satellite Communication on a Given Position

In order to describe the geographic locations of the satellite and the ground antennas,an Earth-centered Earth-fxed(ECEF)coordinate system[24]is established as illustrated in Figure 3.The origin of the coordinate system is located at the center of the earth,the equatorial plane is the XOY plane,the OX-axis intersects with the meridian of zero degrees,and the line connecting the center of the earth and the pole is the Z-axis.Assuming there arePantennas on the satellite andQantennas on the ground station,and the antennas are arranged as uniform linear arrays(ULAs).For the sake of clarity,only two orange circles G1and Gpare drawn to represent the frst and thep-th ground antenna element of the ULA array,respectively.φGandθGare used to indicate the latitude and the longitude of the ground station,respectively.

To further describe the relative position of the two ground antennas,we defnedGpas the distance between thep-th antenna and the frst one,andδGas the angle formed by the connection between the two antennas and the east direction.According to the geometric relationships,the coordinates of the two ground antennas G1(xG1,yG1,zG1)and Gp(xGp,yGp,zGp)in the ECEF coordinate system can be respectively ex-pressed as

whereRE=6378.1 km is the mean earth radius.Similar to the ground station,the coordinates of the satellite antennas S1(xS1,yS1,zS1)and Sq(xSq,ySq,zSq)can be expressed as

respectively,whereDis the height of the satellite’s orbit,φSandθSrepresent the latitude and longitude of the satellite,respectively.dSqis the distance between theq-th satellite antenna and the frst one.δSis the angle between the two satellite antennas connection and the east direction.

In contrast to the land mobile communications scenario where multipath components play a main part,the signal propagation of the satellite-ground link is dominated by the direct path signal component.Hence the channel matrix H∈CQ×Pcan be described using a deterministic LoS model based on the free space wave propagation

whereHqpis the element at the positionq,pof H.rqpandaqp=c0eˉ?ζ0/(4πfcrqp)denote the transmission distance and signal amplitude attenuation between thep-th ground antenna and theq-th satellite antenna,respectively.c0is the speed of light in free space,ζ0is the channel phase andfcis the carrier frequency.Considering the channel capacity is independent from the phase angleζ0,henceζ0=0 is assumed.In addition,since the distance between the transmitting antenna and the receiving antenna is far enough,the approximation ofaqp≈a=c0/(4πfcr11)is applied.1

For a time invariant MIMO channel without the channel information at the transmitter,the channel capacity is given by[25]:

whereσcorresponds to the system Signal-to-Noise Ratio(SNR).It should be mentioned that,although Eq.(6)holds for anyP＞1,Q＞1,due to the small size of the LEO satellite and the limited area of the ground station,the 2×2 LoS MIMO system is analyzed in the following part of this paper.Applying Eq.(5)to Eq.(6)withP=Q=2 results in

where

Here we note that the capacity analysis of the GEO LoS MIMO channel capacity in[21]is a special case of the analysis above withφS=0.According to the geometric relationship of the coordinate system,we haverqp=‖Sq-Gp‖.Since the ground station position(θG,φG)and satellite orbit heightDare fxed parameters,the channel capacity in Eq.(7)can be expressed as a function of the parameterdG,δG,dSandδS.In other words,with respect to a fxed ground station and a specifc SNR,the channel capacity of the MIMO feeder link is determined by two factors:

1.the relative position of the satellite and the ground station;

2.the placement of the satellite and ground antenna.

2.2 Ergodic Channel Capacity of LEO Satellite Constellation System

Based on the analysis of Section 2.1,in the following we investigate the channel capacity of LEO satellite LoS MIMO feeder link.Unlike the GEO system where the satellite is fxed relative to earth,a LEO satellite is always moving at a high speed relative to a fxed ground station.Taking the Iridium system as an example[26],the LEO satellite orbits around the earth everyTS=100.13 min.It can be known from Eq.(7)that the MIMO channel capacity is decided by the relative position of the antennas,therefore the channel capacity for LEO MIMO satellite system varies over time.

LetC(t)be the channel capacity obtained at instanttduring the satellite passing overhead,andTis the total period that the satellite passing overhead,then the average channel capacity amongTis given by

whenT→∞the ergodic channel capacityCcan be obtained as

Since only when the elevation angle between the satellite and ground station is greater than the minimal elevation angle required by the satellite,the communication process can work properly.From this,the concept ofground station visible regioncan be defned.

As shown in Figure 4,the blue area is the ground station visible region corresponding to the ground station G.ε0denotes the minimal elevation angle of the satellite,andβ0is the corresponding maximum satellite geocentric angle.In order to accurately describe each point on the ground station visible region,a ground level coordinate system is established with the ground station G as the origin,which is illustrated in Figure 5.The x′Gy′plane is tangential to the surface of the earth at point G,and the reference axes Gx′,Gy′and Gz′are pointing in the east,north and zenith direction,respectively.The azimuth angleηis the angle between the projection of the satellite-ground vector on the horizontal plane and the north direction,andβ

Figure 4.Ground station visible region.

Figure 5.Ground level coordinate system.

is the geocentric angle between the satellite and the ground station.It is clear that

then any point on the ground station visible region can be uniquely determined by the parameter pair(β,η).

It is worth mentioning that,the mathematical relationship between the satellite rotation periodTSand the earth rotation periodTEdetermines the characteristic of the satellite motion trajectory within the visible region.This mathematical relationship can be divided into two categories:

1.gcd(TS,TE)/=1,

2.gcd(TS,TE)=1.

Here we recall that the function gcd(a,b)returns the greatest common divisor ofaandb.The condition gcd(TS,TE)/=1 means that there is an exact multiple relationship between the earth rotation period and the satellite rotation period.In other words,this condition requires that the rotation period of the satellite should be accurate to the millisecond level.However,such requirements can hardly be precisely realized in practice,so we do not consider this case in this paper.When condition 2 is satisfed andT→∞,the satellite can traverse all the locations of the visible region,and the number of times that the satellite passes through each(β,η)is different.Here we defneρ(β,η)to represent the probability distribution of satellite appearing on the visible region for a given ground station.Thus,for ground stations with different latitudeφG,the ergodic channel capacity can be equivalently expressed as a weighted integral of the channel capacityC(β,η)obtained at different(β,η)locations:

Accordingly,in order to solve Eq.(12),the problem is transformed into solvingρ(β,η)of the location(β,η)on the visible region under condition 2.Moreover,another important point is that condition 2 also shows the ergodic channel capacity for a given LEO satellite has nothing to do with the satellite ascending node longitudeθiand the time passing the ascending nodetj.That is to say,in a LEO satellite constellation based communication system such as Iridium,the ergodic channel capacity provided by any one of the satellites is equal.Meanwhile,due to the fact that the ground station only communicates with one LEO satellite at each moment,the ergodic channel capacity of the entire constellation system is equal to that of a single satellite,so the following statement is made.

Proposition 1.Given a LEO satellite constellation based communication system,when the satellite orbit inclination angles are all equal andgcd(TS,TE)=1,the MIMO ergodic channel capacity Ccst of the entireconstellation system satisfies:

where Cj denotes the ergodic channel capacity between the ground station and the j-th LEO satellite.

III.LOS MIMO ERGODIC CHANNEL CAPACITY

3.1 Satellite Trajectory Probability Distribution on Ground Station Visible Region

According to Proposition 1,the problem of solving the channel capacity is reduced from the entire system to any one of the LEO satellites.To understand the characteristic of the satellite trajectory distribution during the visible region of a given ground station,it is necessary to comprehend the motion of LEO satellites relative to the earth.From[27],the latitudeφand longitudeθof the ground track(the intersection of the satellite orbital plane with the surface of the earth)can be obtained as follows:

where

θ0denotes the longitude of the ascending node,tANis the satellite’s passage time of the ascending node and the time origin is writtent=tAN.n=2π/TSandωE=2π/TEare the angle speed of satellite rotation and earth rotation,respectively.

Since the earth rotates uniformly and the LEO satellite rotates around the center of the earth at a constant speed,the satellite trajectory distribution on the earth’s surface is only related to the latitudeφ.In other words,the satellite trajectory distribution at different longitudes under the same latitude is the same.For that reason,when the recording timeT→∞,the satellite trajectory distributionρφat a point on the satellite’s orbit with latitudeφcan be expressed as

wheremφis the amount of satellite trajectory distribution at a given latitudeφ,andCφis the circumference of the latitude circle atφ.

In order to solvemφ,the satellite trajectory distribution in the range of[φ,φ+Δφ]is frst required,which is equivalent to solve the time lengthΔtof the satellites fying through the latitude range[φ,φ+Δφ].Since the instant when the satellite passes the ascending node does not affect the property of the ground track,tANin Eq.(14)can be set to 0,which leads to

From Eq.(19)we know that

then we have

AsΔφ→0,mφcan be obtained

Substituting Eq.(22)into Eq.(18),we have

In particular,whenμ=90°,i.e.polar orbit satellite,there is;whenμ=0°,i.e.the satellite orbit is an equatorial orbit,because the satellite’s ground track is fxed,there is no need to calculateρφ.According to Eq.(23),sincenandμare fxed parameters,ρφis only related to the latitudeφof the point.From the analysis in Section II,it is known that for a ground station located at(θG,φG),each point on the visible region is determined by the parameter pair(β,η).Therefore,it is necessary to establish the transformation relationship between the parameter pair(β,η)and the latitudeφ.

Proposition 2.The mathematical relationship between the latitude φ and parameter pair(β,η)of the point within the ground station visible region is as follows.When η=0,π

when η=π/2

Proof.The proof is given in Appendix.

Put Eq.(24)-Eq.(26)into Eq.(23),we can getρφwhenη∈[0,π].Because the satellite visible region is symmetrical,we obtain

3.2 Optimization of the MIMO Ergodic Channel Capacity

According to Section II and Proposition 2,it is known that there are four parametersdG,δG,dSandδSwhich will affect the value of the ergodic channel capacityC,thereforeCcan be expressed asC(dG,δG,dS,δS).SincedSis usually fxed due to the limitation of satellite antenna size,the parametersdG,δGandδSare set as independent variables and can be optimized to maximize the ergodic channel capacity.

3.2.1 Optimization of the Antenna Spacing

First,we consider the relationship between the ground station antenna spacingdGand the ergodic channel capacityC,assuming thatδSandδGare both equal to zero.For the purpose of achieving the optimal ergodic channel capacity by the ground antenna spacing arrangement,we have

It can be seen that the minimumdGsatisfyingis the optimized valuedG-opt.Here we note that although the analytical expression of the satellite trajectory probability distributionρ(β,η)is derived in Section 3.1,the analytical expression ofCcannot be obtained by substitutingρ(β,η)into Eq.(12).Based on this,the optimal ground antenna spacingdG-optto maximize the ergodic channel capacity is obtained through numerical computer simulations in Section 4.2.

3.2.2 Optimization of the Antenna Placement Angle

Next,the impact of the ground station antenna placement angleδGon the ergodic channel capacityCis considered.Assuming that the attitude of the satellite can be controlled and the antenna placement angleδSof the satellite antenna can be adjusted in real-time.According to Eq.(7),the conditions that the channel capacity obtains its optimum value is given by

After some mathematical derivations,the optimum value ofδSwhen the satellite is in different positions can be calculated

where

Therefore,we assume that the satellite can calculateδS-optcorresponding to the optimal ergodic channel capacity according to the relative position between itself and the ground station,then perform adaptive attitude adjustment.

IV.NUMERICAL RESULTS

4.1 Satellite Trajectory Probability Distribution

In order to verify the correctness of the calculatedρ(β,η),the following simulation were performed.Considering a ground station with coordinates(θG=18°,φG=30°),the operating parameters of the LEO satellite areε0=8.2°,μ=86.4°,θ0=-138.4°,tAN=0,TS=100.13 min,the numerical evaluation period is set to one year,i.e.T=365×24×60×60 s.First we record the sampling satellite position(θ,φ)within the ground station’s visible region duringTand then convert the(θ,φ)into the azimuth coordinates(β,η)according to(42)and(44).Next we discretize the data according to the following criteria.Set the number of segments of the geocentric angleβand the azimuth angleηasNβ=45 andNη=180,respectively.Since(β,η)is in the range

Figure 6.Theoretical and statistical simulation results of satellite trajectory distribution(μ=90°).

wherep=0,1,...,Nβ-1 andq=0,1,...,Nη-1.Calculate the indexes(Bp,Eq)corresponding to the sampling points(β,η)according to Eq.(36),and count the cumulative value of the same index(Bp,Eq)to form the simulated distribution matrixMp,q.Since the theoretical result in Eq.(23)is the calculated satellite trajectory probability distributionρ(β,η),and the simulation result is a statistical value obtained by dividing(β,η)within the value range.Therefore the double integration ofρ(β,η)needs to be performed according to the equal division scheme,so as to compare with the actual simulation result.Considering the visible region range atβ∈[βp,βp+1),η∈[ηq,ηq+1),the theoretical number of visible satellites should be:

Figure 7.Theoretical and statistical simulation results of satellite trajectory distribution(μ=60°).

For a more visual representation of the relationship between the theoretical analysis and practical simulation results,we perform data comparison from theβandηdimensions,respectively(see Figure 6 and 7 for different orbital inclinationμ).The theoretical distribution curve of geocentric angle(the blue curve)is obtained bywhile the red curve denotes the azimuth angle’s theoretical distribution calculated byCorrespondingly,the distribution histogram ofβis obtained by statistics onand similarlyis forη.Using the maximum value obtained from the simulated result as a standardto scale the theoretical data,it can be seen that the theoretical model is basically consistent with the numerical simulation.The reason for the slight errors in the fgure is that the statistical simulation timeTis a fnite value,but the theoretical result is an ideal value whenTtends to infnity.

Table 1.Parameters used in the simulation.

4.2 Ergodic Channel Capacity

Based on the above analysis,the Matlab simulations of the ergodic channel capacity between a given LEO satellite constellation based communication system and a given ground station are performed in this section.This paper takes the orbital parameters of Iridium satellite constellation as an example.Since the channel capacity provided by the entire constellation is equivalent to a single satellite,one of them is selected for capacity simulation analysis with parameters summarized in Table 1.In addition,in order to compare with the proposed LoS MIMO scheme,the ergodic channel capacity based on the existing SISO structure is also simulated.For the sake of fairness,the total transmission power of the SISO system should be the same as that of the MIMO system,so the channel capacity of the SISO system can be calculated

whereis the channel transfer coeffcient between the TX antenna and RX antenna.Correspondingly,the ergodic channel capacity based on the SISO structure can be achieved

Figure 8.Uplink ergodic channel capacity as a function of dG.

It is worth mentioning that since the ergodic channel capacity of MIMO and SISO systems(i.e.,Eq.(12)and Eq.(39))is challenging to be expressed in closedform,computer simulations are carried out here by means of numerical integration.

4.2.1 Optimization of the Antenna Spacing

Corresponding to Section 3.2.1,the relationship between the ground station antenna spacingdGand the uplink ergodic channel capacityC(dG)is simulated here,where the ground station coordinates areφG=59°,θG=18°.

As shown in Figure 8,the three curves in the fgure correspond to three parameter settings,which are determined by differently allowed array aperture at the satellitedS.The straight line shows the SISO ergodic channel capacity under the condition of equal transmission power.Taking the green curve as an example,the pointPcorresponds to the conditionand the optimized ground antenna spacingdG-opt=3.4 km.The black dashed lines show that under the current parameter conditions when the ground antenna spacingdG≥dG-opt,the ergodic channel capacityCwill fuctuate periodically betweenCminandCmaxasdGincreases.The relationship between extreme values isCmax/Cmin≈1.04.In particular,whendG≥dG-opt,the ergodic channel capacity of the proposed MIMO system is nearly twice that of the SISO system under the same energy consumption condition,which refects the advantages of the MIMO scheme and also verifes the robustness of the dynamic LoS MIMO array structure[19].The larger thedS,the shorter the period of the channel capacity fuctuation withdG.

Figure 9.Ergodic channel capacity comparison of uplink and downlink as a function of dG.

The ergodic channel capacity comparison of uplink and downlink is shown in Figure 9,where solid lines and dashed lines represent uplink and downlink respectively.As can be seen,higher carrier frequency used by the uplink results in greater signal attenuation,which leads to lower channel capacity.In addition,the optimal ground antenna spacingdG-optdecreases with the increase of carrier frequency.Considering the limitation of ground station specifcations,the current system can set the ground antenna spacing to the optimal uplink antenna spacing=3.4 km,which can achieve good performance although it does not make the downlink capacity optimal.

Figure 10.Uplink ergodic channel capacity as a function of δG.

4.2.2 Optimization of the Antenna Placement Angle

Next,we simulate the impact of the ground station antenna placement angleδGon the uplink ergodic channel capacityC.The three curves in the Figure 10 correspond to three different satellite antenna spacingdSconditions,where the ground station antenna spacing is set todG=3 km;the straight line shows the SISO ergodic channel capacity as a comparison.It can be known from the simulation result that when the satellite can adaptively adjustδSaccording to Eq.(32),the ergodic channel capacity can reach a nearly optimal state regardless of the ground antenna placement angleδG.Moreover,under the condition that the ground antenna spacingdGremains unchanged,the ergodic channel capacity increases with the rise of the satellite antenna spacingdS.

V.CONCLUSION

In this paper,we investigated the LoS MIMO ergodic channel capacity of the feeder link for a given LEO satellite constellation based communication system.Starting from the motion characteristics of the LEO satellite,the capacity analysis of the entire constellation system was simplifed to the analysis of a single satellite in the system.Next,the closed-form probability distribution of the satellite trajectory during the visible region was derived based on the evaluation of satellite ground track.Furthermore,the ergodic channel capacity was numerically calculated and the optimal antenna placement that can maximize the capacity was obtained.Finally,the numerical simulation results verifed the closed-form probability distribution of satellite trajectory,and showed that the proposed LoS MIMO scheme can signifcantly improve the ergodic channel capacity under the same energy consumption condition without increasing the system power consumption.

ACKNOWLEDGEMENT

This work was supported by the National Key R&D Program of China under Grant 2019YFB1803102.

NOTES

1The following example is used to numerically verify the reasonability of this approximation:Assuming that the ground station is located at the sub-satellite point of the satellite with an orbital altitude of 780 km,the antenna placementdG=10,km,dS=2 m,δG=δS=0 and the carrier frequency isfc=30 GHz.Then under the above conditions,(aqp)dB≈(a11)dB±7×10-4dB can be calculated,where(·)dBrepresents the attenuation value measured in dB,therefore the relative error can be ignored.

APPENDIX

Set the rectangular coordinates of the ground station in the ECF coordinate system as G=(xG,yG,zG)T,the longitude and latitude coordinates as(θG,φG),the satellite rectangular coordinates as S=(xS,yS,zS)T,and the longitude and latitude coordinates as(θS,φS).In order to use(β,η)to indicateφS,we need to fnd out the transformation relationship between the ECF coordinate system and the local tangent coordinate system.The unit vectors pointing in the east,north,and zenith directions at point G are as follows:

then the orthogonal transformation matrix E=(eE,eN,eZ)Tcan be defned:

Therefore,the satellite coordinates SE=(xES,yES,zES)Tin the local tangent coordinate system can be expressed as

Since E is a rotation matrix,E is also an orthogonal matrix,which has property E-1=ET,then

Meanwhile,the azimuth relationship of the local tangent coordinate system can be written as follows:

whered(β)denotes the distance between satellite and the ground station.After some mathematical derivation,we haveφ(β,η)as shown in Proposition 2.