massive mimo transmission for leo satellite communications

IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 38, NO. 8, AUGUST 2020 1851

Massive MIMO Transmission for LEOSatellite Communications

Li You , Member, IEEE, Ke-Xin Li , Graduate Student Member, IEEE, Jiaheng Wang , Senior Member, IEEE,

Xiqi Gao , Fellow, IEEE, Xiang-Gen Xia, Fellow, IEEE, and Björn Ottersten , Fellow, IEEE

Abstract— Low earth orbit (LEO) satellite communicationsare expected to be incorporated in future wireless networks,in particular 5G and beyond networks, to provide global wire-less access with enhanced data rates. Massive multiple-inputmultiple-output (MIMO) techniques, though widely used interrestrial communication systems, have not been applied to LEOsatellite communication systems. In this paper, we propose amassive MIMO transmission scheme with full frequency reuse(FFR) for LEO satellite communication systems and exploitstatistical channel state information (sCSI) to address the dif-ficulty of obtaining instantaneous CSI (iCSI) at the transmit-ter. We first establish the massive MIMO channel model forLEO satellite communications and simplify the transmissiondesigns via performing Doppler and delay compensations atuser terminals (UTs). Then, we develop the low-complexity sCSIbased downlink (DL) precoder and uplink (UL) receiver inclosed-form, aiming to maximize the average signal-to-leakage-plus-noise ratio (ASLNR) and the average signal-to-interference-plus-noise ratio (ASINR), respectively. It is shown that the DLASLNRs and UL ASINRs of all UTs reach their upper boundsunder some channel condition. Motivated by this, we propose aspace angle based user grouping (SAUG) algorithm to schedulethe served UTs into different groups, where each group ofUTs use the same time and frequency resource. The proposedalgorithm is asymptotically optimal in the sense that the lower

Manuscript received October 1, 2019; revised January 15, 2020;accepted February 17, 2020. Date of publication June 8, 2020; date ofcurrent version August 20, 2020. This work was supported in part bythe National Key Research and Development Program of China underGrant 2018YFB1801103, in part by the Jiangsu Province Basic ResearchProject under Grant BK20192002, in part by the National Natural Sci-ence Foundation of China under Grant 61801114, Grant 61761136016, andGrant 61631018, in part by the Natural Science Foundation of JiangsuProvince under Grant BK20170688, and in part by the Huawei Cooper-ation Project. The work of Jiaheng Wang was supported in part by theNational Natural Science Foundation of China under Grant 61971130 andGrant 61720106003 and in part by the Natural Science Foundation of JiangsuProvince under Grant BK20160069. The work of Björn Ottersten wassupported in part by the Luxembourg National Research Fund (FNR) projects,titled Dynamic Beam Forming and In-band Signalling for Next GenerationSatellite Systems (DISBUS) and in part by the Energy and CompLexityEffiCienT mIllimeter-wave Large-Array Communications (ECLECTIC). Thisarticle was presented in part at the IEEE International Conference on Commu-nications, Dublin, Ireland, June 2020 [1]. (Corresponding author: Xiqi Gao.)

Li You, Ke-Xin Li, Jiaheng Wang, and Xiqi Gao are with the NationalMobile Communications Research Laboratory, Southeast University, Nan-jing 210096, China, and also with the Purple Mountain Laboratories,Nanjing 211100, China (e-mail: [email protected]; [email protected];[email protected]; [email protected]).

Xiang-Gen Xia is with the Department of Electrical and ComputerEngineering, University of Delaware, Newark, DE 19716 USA (e-mail:[email protected]).

Björn Ottersten is with the Interdisciplinary Centre for Security, Reliabilityand Trust (SnT), University of Luxembourg, 2721 Luxembourg, Luxembourg(e-mail: [email protected]).

Color versions of one or more of the figures in this article are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JSAC.2020.3000803

and upper bounds of the achievable rate coincide when thenumber of satellite antennas or UT groups is sufficiently large.Numerical results demonstrate that the proposed massive MIMOtransmission scheme with FFR significantly enhances the datarate of LEO satellite communication systems. Notably, the pro-posed sCSI based precoder and receiver achieve the similarperformance with the iCSI based ones that are often infeasiblein practice.

Index Terms— LEO satellite, massive MIMO, multibeam satel-lite, full frequency reuse, statistical CSI, user grouping.

I. INTRODUCTION

SATELLITE communication systems can provide seamlesswireless coverage so as to complement and extend terres-

trial communication networks and, as in recent standardizationendeavors [2], are expected to be incorporated in futurewireless networks, in particular 5G and beyond networks.Low earth orbit (LEO) satellite communications, with orbitsat altitudes of less than 2000 km, have recently gainedbroad research interests due to the potential in providingglobal wireless access with enhanced data rates. Comparedwith the geostationary earth orbit (GEO) counterpart, LEOsatellite communication systems impose much less stringentrequirements on, e.g., power consumption and transmissionsignal delays. Recently, several projects, e.g., OneWeb andSpaceX, on LEO satellite communication systems have beenlaunched [3].

In satellite communication systems, multibeam transmissiontechniques have been widely adopted to increase transmissiondata rates. As a well-know multibeam solution, a four-colorfrequency reuse (FR4) scheme where adjacent beams areallocated with non-overlapping frequency spectrum (or dif-ferent polarizations) is adopted to mitigate the co-channelinter-beam interference [4], [5]. To further enhance the spectralefficiency of satellite communications, the more aggressivefull frequency reuse (FFR) schemes [4]–[8], where frequencyresources are reused across neighboring beams, have beenconsidered to increase the total available bandwidth in eachbeam as that has been done in terrestrial cellular systems.Yet, in FFR the inter-beam interference becomes a criticalissue, which has to be properly handled. In general, inter-beaminterference management can be performed at either the trans-mitter via precoding or at the receiver via multi-user detection,similar as in terrestrial cellular communication systems [9].Compared with non-linear dirty paper coding (DPC) precodingand multi-user detection, in practice linear precoding anddetection are more preferred in multibeam satellite commu-nication systems due to their low computational complexityand near-optimal performance [10].

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1852 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 38, NO. 8, AUGUST 2020

It is worth noting that most of the existing works ondownlink (DL) precoding in multibeam satellite communica-tions, e.g., [4], [5], rely on precise instantaneous channel stateinformation (iCSI). However, obtaining iCSI at the transmittersides of satellite communication systems is usually difficultand even infeasible due to a number of practical factors,especially the long propagation delay between a satelliteand user terminals (UTs) as well as the mobility of UTsand satellites. In particular, for time-division duplex (TDD)systems, the coherence time of the channel is shorter thanthe transmission delay, which makes obtaining accurate iCSIvia the UL-DL reciprocity a mission impossible. On the otherhand, in more common frequency-division duplex (FDD) sys-tems, obtaining iCSI at the satellite side requires UL feedbackfrom UTs, which inevitably introduces a great amount oftraining and feedback overhead due to mobility of UTs andmore importantly could become outdated as a result of thelong propagation delay.

In recent years, massive multiple-input multiple-output(MIMO) transmission, where a large number of antennas areequipped at a base station to serve many UTs, has been appliedin terrestrial cellular wireless networks, e.g., 5G [11], [12],as an enabling technology. Massive MIMO can substan-tially increase available degrees of freedom, enhance spectralefficiency, and achieve high data rates. Motivated by this,we propose to exploit massive MIMO along with FFR forLEO satellite communication systems, where a large numberof antennas are equipped at the LEO satellite side. Our focusis particularized on the physical layer transmission designfor massive MIMO LEO satellite communication systems.We note that it is not necessary to perform predefined multiplebeamforming in fully digital-implemented FFR satellite com-munication systems. Exploiting massive MIMO for satellitecommunications with FFR can be seen as a technique withoutpredefined beamforming.

Albeit the existence of a large body of literature on massiveMIMO in terrestrial cellular communication systems [12],so far massive MIMO has not been applied to satellitecommunication systems. The performance of massive MIMOsystems relies substantially on the available CSI [13]–[15].As mentioned above, obtaining accurate DL iCSI at the LEOsatellite side is generally difficult and even infeasible due to thelong propagation delay and the mobility of satellites and UTs,which makes inapplicable of the existing terrestrial massiveMIMO transmission approaches relying on iCSI. Meanwhile,the implementation complexity accompanied with massiveMIMO becomes a critical concern in satellite communica-tion systems considering the payload limitation on satellites.Consequently, incorporating massive MIMO into LEO satellitecommunication systems is still an open and challenging task.

For massive MIMO, obtaining iCSI at the transmitter hasbeen a difficult problem even in terrestrial communicationsystems, especially in high-mobility scenarios. Compared withiCSI, statistical CSI (sCSI) varies much slower and thus canbe relatively easily obtained at both the satellite and theUTs with sufficiently high accuracy. Hence, sCSI based DLprecoding has been proposed in terrestrial massive MIMOsystems [14]–[16]. For massive MIMO satellite communica-tion systems, it is more practical to use sCSI, which can

overcome the difficulty of acquiring iCSI and significantlyreduce the computational overhead of satellite payloads via themuch less frequent update of transmission strategies including,e.g., DL precoding, UL receiving, and user grouping.

In this paper, we investigate massive MIMO transmissionfor LEO satellite communication systems using FFR basedon sCSI. In particular, we focus on devising DL precod-ing, UL receiving, and user grouping utilizing sCSI. Whilethis paper focuses on the LEO satellite communications,the proposed massive MIMO transmission schemes can alsobe extended to other non-terrestrial communication systems,e.g., GEO satellite communication systems, and high-altitudeplatform (HAP) communication systems. The major contribu-tions of the current work are summarized as follows:

• We introduce massive MIMO into LEO satellite com-munication systems using FFR and investigate low-complexity and low-overhead transmission strategiesbased on sCSI.

• We establish the massive MIMO channel model forLEO satellite communications by incorporating the LEOsatellite signal propagation properties, and simplify theUL/DL transmission designs via performing Doppler anddelay compensations at UTs.

• We develop the sCSI based DL precoder and UL receiverin closed-form, aiming to maximize the average signal-to-leakage-plus-noise ratio (ASLNR) and the averagesignal-to-interference-plus-noise ratio (ASINR), respec-tively, and theoretically prove that the proposed sCSIbased scheme asymptotically approaches the iCSI basedone.

• We propose a space angle based user grouping (SAUG)algorithm using only the channel space angle information,and show that the proposed algorithm is asymptoticallyoptimal in the sense that the lower and upper bounds ofthe achievable rate coincide when the number of satelliteantennas or UT groups is sufficiently large.

• Simulation results demonstrate that the proposed mas-sive MIMO transmission scheme with FFR significantlyenhances the data rate of LEO satellite communicationsystems. Notably, the proposed sCSI based precoder andreceiver achieve the similar performance with the iCSIbased ones that are often infeasible in practice.

The rest of the paper is organized as follows. In Section II,we investigate the channel model and the correspondingtransmission signal model for LEO satellite communicationsystems. Based on the system model, we then investigate theoptimal DL and UL transmission strategies for LEO satellitecommunications in Section III. In Section IV, we furtherinvestigate user grouping. We present the numerical resultsin Section V and conclude the paper in Section VI. The majorvariables adopted in the paper is listed in Table I for ease ofreference.

II. SYSTEM MODEL

A. System Setup

Consider a LEO satellite communication system wherea satellite provides services to a number of single-antennaUTs simultaneously. The satellite is equipped with a uniformplanar array (UPA) composed of M = MxMy antennas


YOU et al.: MASSIVE MIMO TRANSMISSION FOR LEO SATELLITE COMMUNICATIONS 1853

TABLE I

VARIABLE LIST

where Mx and My are the numbers of antennas on thex- and y-axes, respectively. Assume without loss of generalitythat the antennas are separated by one-half wavelength in boththe x- and y-axes, and both Mx and My are even. The systemsetup is illustrated in Fig. 1.

B. DL Channel Model

As different UTs are usually spatially separated by a fewwavelengths, it is reasonable to assume that the channel real-izations between the satellite and different UTs are uncorre-lated [17]. We focus on investigating the DL channel betweenthe satellite and UT k. Using a ray-tracing based channelmodeling approach, the complex baseband DL space domainchannel response between the LEO satellite and UT k atinstant t and frequency f can be represented by [18]–[20]

gdlk (t, f) =

Pk−1∑p=0

gdlk,p · exp {j2π [tνk,p − fτk,p]}

·vk,p ∈ CM×1, (1)

where CM×N denotes the M×N dimensional complex-valuedvector space, j =

√−1, Pk denotes the number of channel

Fig. 1. Illustration of the LEO satellite communication system setup.

propagation paths of UT k, and gdlk,p, νk,p, τk,p, and vk,p ∈

CM×1 are the complex-valued gain, the Doppler shift,the propagation delay, and the DL array response vectorassociated with path p of UT k, respectively. Note that thechannel model adopted in (1) is applicable over the timeintervals of interest where the relative positions of the LEOsatellite and UT k do not change significantly, and thus thephysical channel parameters, Pk, gdl

k,p, νk,p, τk,p, and vk,p, areassumed to be invariant. When the LEO satellite and/or theUT move over large distances, the above channel parameterswill vary and should be updated accordingly [19]. It is worthmentioning that the ray-tracing based channel model in (1)can be applied to different propagation scenarios, and furtheranalysis of the channel model will depend on the parameterproperties in the specific scenario. Hereafter, we detail somepropagation characteristics of the LEO satellite channels andtheir impact on the modeling of the channel parameters in (1).

1) Doppler: For LEO satellite communications, assumingthat the scatterers are stationary in the considered interval ofinterest, then the Doppler shift νk,p associated with propaga-tion path p of UT k is mainly composed of two independentDoppler shifts, νsat

k,p and νutk,p, that are caused by the motions

of the LEO satellite and the UT, respectively [21], [22].It is worth noting that due to the relatively high altitude of

the LEO satellite, the Doppler shifts νsatk,p caused by the motion

of the LEO satellite can be assumed to be identical for differentpropagation paths p of the same UT k [21], [22], and differentfor different UTs. Thus, for notation simplicity, we omit thepath index of the Doppler shift νsat

k,p due to the motion of theLEO satellite and rewrite the Doppler shifts as νsat

k,p = νsatk .

On the other hand, the Doppler shifts νutk,p due to the motion

of the UT are typically different for different propagationpaths, which contribute the Doppler spread of the LEO satellitechannels [21], [22]. As the scattering characteristics aroundthe UTs mainly determine the Doppler shifts caused by themovement of the UTs, the modeling of the Doppler spreadin LEO satellite communications can be similar to that in thetraditional terrestrial cellular communications [22].



2) Delay: Due to the relatively large distance between theLEO satellite and the UTs, the propagation delay τk,p associ-ated with path p of UT k exhibits a much larger value than thatin terrestrial wireless channels. Denote by τmin

k = minp {τk,p}and τmax

k = maxp {τk,p} the minimum and maximum valuesof the propagation delays of UT k, respectively. The delayspread of the LEO satellite channels τmax

k − τmink might be

much smaller than that of the terrestrial wireless channelsas observed in measurement results [22]–[24]. For notationalbrevity, we define τut

k,p � τk,p − τmink . Note that due to, e.g.,

the long propagation delays in LEO satellite communications,acquiring reliable iCSI at the transmitter sides is usuallyinfeasible, especially when the UTs are in high mobility. Thus,it is more practical to investigate transmission design with,e.g., sCSI, in LEO satellite communications.

3) Angle: The UPA response vector vk,p in (1) can berepresented by [25], [26]

vk,p � vxk,p ⊗ vy

k,p

= vx

(ϑx

k,p

)⊗ vy

(ϑy

k,p

)∈ C

M×1, (2)

where ⊗ denotes the Kronecker product, and vdk,p for d ∈ D �

{x, y} is the array response vector of the angle with respectto the x- or y-axis given by

vdk,p � vd

(ϑd

k,p

)=

1√Md

[1 exp

{−jπϑdk,p

}. . .

exp{−jπ(Md − 1)ϑd

k,p

}]T ∈ CMd×1, (3)

with the superscript (·)T denoting the transpose operation.In (3), the parameters ϑx

k,p and ϑyk,p are related to the

physical angles as ϑxk,p = sin

(θy

k,p

)cos(θx

k,p

)and ϑy

k,p =

cos(θy

k,p

)where θx

k,p and θyk,p are the angles with respect

to the x- and y-axes associated with the pth propagation pathof UT k, respectively. For satellite communication chan-nels, the angles of all propagation paths associated with thesame UT, can be assumed to be identical due to the relativelyhigh altitude of the satellite compared with that of the scatter-ers located in the vicinity of the UTs [27], i.e., ϑd

k,p = ϑdk. Note

that the parameters ϑdk can reflect the propagation properties of

the LEO satellite channels in the space domain, and we referto ϑd

k as the space angle parameters. Then, the array responsevector can be rewritten as

vk,p = vk = vxk ⊗ vy

k

= vx (ϑxk) ⊗ vy (ϑy

k) ∈ CM×1, (4)

which will be referred to as the DL channel direction vectorof UT k that is associated with the space angles ϑx

k and ϑyk.

Note that when the number of antennas Md for d ∈ Dtends to infinity, we can know from (3) and (4) that thechannel direction vectors of different UTs are asymptoticallyorthogonal, i.e.,

limMd→∞

(vd

k

)Hvd

k′ = δ (k − k′) , (5)

where (·)H denotes the conjugate-transpose operation.

Based on the above modeling of the propagation propertiesof LEO satellite communications, we can rewrite the channelresponse in (1) as follows

gdlk (t, f) = exp

{j2π[tνsat

k − fτmink

]} · gdlk (t, f) · vk, (6)

where gdlk (t, f) is the DL channel gain of UT k given by

gdlk (t, f) �

Pk−1∑p=0

gdlk,p · exp

{j2π[t(νk,p − νsat

k

)−f(τk,p − τmin

k

)]}=

Pk−1∑p=0

gdlk,p · exp

{j2π[tνut

k,p − fτutk,p

]}, (7)

which will be convenient for derivation of the transmissionsignal model later.

4) Gain: Note that the statistical properties of the fluctua-tions of the channel gain gdl

k (t, f) in LEO satellite commu-nications mainly depend on the propagation environment inwhich the UT is located. In addition, LEO satellite communi-cation systems are usually operated under line-of-sight (LOS)propagations and Rician channel model is widely accepted inLOS satellite communication systems. In this work, we focuson the case where both non-shadowed LOS and non-LOSpaths of the LEO satellite channels exist [6]. Then, the channelgain gdl

k (t, f) exhibits the Rician fading distribution with

the Rician factor κk and power E{∣∣gdl

k (t, f)∣∣2} = γk.

In other words, the real and imaginary parts of gdlk (t, f) are

independently and identically real-valued Gaussian distributedwith mean

√κkγk

2(κk+1) and variance γk

2(κk+1) , respectively.

C. UL Channel Model

Using the DL channel modeling approach presented inthe above subsections, we briefly investigate the UL channelmodel for LEO satellite communications in this subsection.Note that the UL channel response is the transpose of theDL channel response in TDD systems, and similar channelmodel can be obtained. Meanwhile, for FDD systems wherethe relative carrier frequency difference is small, the physicalchannel parameters, Pk, νu,p, τk,p, ϑx

k, and ϑyk are almost

identical between the UL and DL [28]–[30]. Thus, the majordifference between the UL and DL channels lies in the fastfading path gain terms. Similarly as (6), the UL space domainchannel response between UT k and the LEO satellite at timet and frequency f can be modeled as

gulk (t, f) = exp

{j2π[tνsat

k − fτmink

]} · gulk (t, f)

·uk ∈ CM×1, (8)

where gulk (t, f) is the UL channel gain of UT k given by

gulk (t, f) �

Pk−1∑p=0

gulk,p · exp

{j2π[tνut

k,p − fτutk,p

]}, (9)

which exhibits the same statistical properties as the DL chan-nel gain gdl

k (t, f), i.e., the real and imaginary parts of gulk (t, f)



are independently and identically real-valued Gaussian distrib-uted with mean

√κkγk

2(κk+1) and variance γk

2(κk+1) , respectively,

and uk is the UL channel direction vector given by

uk = uxk ⊗ uy

k

= ux (ϑxk) ⊗ uy (ϑy

k) ∈ CM×1, (10)

which exhibits a similar structure as the DL channel directionvector vk in (4) but with a center frequency offset for FDDsystems, and can be well approximated by vk when thefrequency separation between the UL and the DL is notsignificant [31]. Similarly as the DL case, the LEO satelliteUL channel also exhibits the asymptotic orthogonality as

limMd→∞

(ud

k

)Hud

k′ = δ (k − k′) . (11)

Note that the channel models in (6) and (8) are general inthe sense that they take into account the LEO satellite chan-nel propagation properties in the space, time, and frequencydomains.

D. DL/UL Transmission Signal Model

Consider a wideband massive MIMO LEO satellite com-munication system employing orthogonal frequency divisionmultiplexing (OFDM) modulation [21] with the number ofsubcarriers, Nus, and the cyclic prefix (CP), Ncp samples.Denote by Ts the system sampling interval. Then, the OFDMsymbol length and the CP length are given by Tus = NusTs

and Tcp = NcpTs, respectively. Note that with the delay andDoppler properties of the LEO satellite channels taken intoaccount, it is not difficult to select proper OFDM parameterssuch that the effects of the intersymbol and intercarrier inter-ference can be almost neglected [32].

Let{xdl

�,n

}Nus−1

n=0be the DL transmit symbols during sym-

bol �. Then, the transmitted signal xdl� (t) ∈ CM×1 can be

written as [33]

xdl� (t) =

Nus−1∑n=0

xdl�,n · exp

{j2π

n

Tust

}, (12)

where −Tcp ≤ t−� (Tcp + Tus) < Tus, and the correspondingreceived signal at UT k is given by (where the noise is omittedfor brevity)

ydlk,� (t) =

∞∫−∞

[gdl

k (t, τ)]T · xdl

� (t − τ) dτ, (13)

where gdlk (t, τ) is the inverse Fourier transform of gdl

k (t, f)in (6) in terms of τ .

Utilizing the Doppler and delay properties of the LEO satel-lite propagation channels addressed previously, we proceedto perform time and frequency synchronization. In particular,with delay compensation τ syn

k = τmink and Doppler compen-

sation νsynk = νsat

k,p applied to the received signal at UT k,the resultant signal can be represented by

ydl,synk,� (t) = ydl

k,� (t + τ synk ) · exp {−j2π (t + τ syn

k ) νsynk } .

(14)

Then, the corresponding signal dispersion in the delay andDoppler domains can be significantly reduced, and it is notdifficult to select proper OFDM parameters to mitigate theintersymbol and intercarrier interference [33]. Consequently,the demodulated DL received signal at UT k over subcarriern of OFDM symbol � can be represented by

ydlk,�,n =

(gdl

k,�,n

)Txdl

�,n, (15)

where gdlk,�,n is the DL channel of UT k over symbol � and

subcarrier n given by [33]

gdlk,�,n = vk · gdl

k,�,n ∈ CM×1, (16)

where gdlk,�,n = gdl

k (� (Tus + Tcp) , n/Tus).Besides, consider UL transmission employing OFDM mod-

ulation with similar parameters as DL transmission. Then, withproper delay and Doppler compensations performed at the UTside, the demodulated UL received signal at the satellite oversymbol � and subcarrier n can be represented as

yul�,n =

∑k

gulk,�,nxul

k,�,n ∈ CM×1, (17)

where xulk,�,n is the complex-valued symbols transmitted by

UT k, and gulk,�,n is the UL channel of UT k over subcarrier

n of OFDM symbol � given by

gulk,�,n = uk · gul

k,�,n ∈ CM×1, (18)

where gulk,�,n = gul

k (� (Tus + Tcp) , n/Tus). Note that thetransmission signal models in (15) and (17) are applicableprovided that delay and Doppler compensations are properlyperformed via exploiting the delay and Doppler properties ofthe LEO satellite channels described previously.

III. STATISTICAL CSI BASED DL/UL TRANSMISSIONS

In this section, we investigate DL precoder and UL receiverdesign for LEO satellite communications based on the channeland signal models established in the above section. Notethat the conventional designs of DL precoding vectors andUL receiving vectors in MIMO transmission usually requireknowledge of iCSI. However, it is in general infeasible toobtain precise iCSI at the satellite sides for DL of LEOsatellite communications. In addition, frequent update of theDL precoding vectors and UL receiving vectors using iCSIwill be challenging for implementation on payload of practicalsatellite communications. Hereafter, we focus on the design ofDL precoder and UL receiver utilizing slowly-varying sCSI forsatellite communications.

A. DL Precoder

We first consider DL transmission where K single antennaUTs are simultaneously served in the same time-frequencyblocks, and the served UT set is denoted by K ={0, 1, . . . , K − 1}. For DL linear precoding performed at thesatellite, the signal received by UT k ∈ K in (15) can berewritten as

ydlk =

(gdl

k

)T ∑i∈K

√qdli bis

dli + zdl

k , (19)



where the subcarrier and symbol indices are omitted forbrevity, qdl

k is the transmit power allocated to UT k, bk ∈CM×1 is the normalized transmit precoding vector satisfying

the ‖bk‖ =√

bHk bk = 1, sdl

k is the signal for UT k with mean

0 and variance 1, and zdlk is the additive circular symmetric

complex-valued Gaussian noise with mean 0 and variance σdlk ,

i.e., zdlk ∼ CN (0, σdl

k

).

Note that SLNR is a convenient and efficient design metricwidely adopted in DL multiuser MIMO transmission, and wefirst review the SLNR maximization criterion based precodingapproach. In particular, the SLNR of UT k in the DL is givenby [34], [35]

SLNRk =

∣∣∣(gdlk

)Tbk

∣∣∣2 qdlk∑

i�=k

∣∣∣(gdli

)Tbk

∣∣∣2 qdlk + σdl

k

=

∣∣∣(gdlk

)Tbk

∣∣∣2∑i�=k

∣∣∣(gdli

)Tbk

∣∣∣2 + 1ρdl

k

, (20)

where ρdlk � qdl

k /σdlk is the DL signal-to-noise ratio (SNR)

of UT k. Then the precoder of UT k that maximizes SLNRk

in (20) can be obtained as

bslnrk =

1ηslnr

k

⎡⎣(∑

i

gdli

(gdl

i

)H+

1ρdl

k

IM

)−1

gdlk

⎤⎦∗

, (21)

where (·)∗ denotes the conjugate operation and ηslnrk is

the power normalization coefficient that is set to satisfy∥∥bslnrk

∥∥ = 1. We mention that the SLNR maximization DLprecoder in (21) requires knowledge of iCSI gdl

k for all k.However, it is in general difficult to obtain precise DL iCSIfor transmitter at the satellite side.

In the following, we investigate DL precoding for satel-lite communications using long-term sCSI at the transmitter,including the channel direction vector vk and the statistics ofthe channel gain gdl

k,�,n. We consider the ASLNR performancemetric as follows [36]

ASLNRk �E

{∣∣∣(gdlk

)Tbk

∣∣∣2}

E

{∑i�=k

∣∣∣(gdli

)Tbk

∣∣∣2 + 1ρdl

k

}

=γk

∣∣∣(vk)T bk

∣∣∣2∑i�=k γi

∣∣∣(vi)T bk

∣∣∣2 + 1ρdl

k

, (22)

where the numerator and the denominator account for the aver-age power of the signal and leakage plus noise, respectively.The sCSI based precoder that maximizes ASLNRk is presentedin the following proposition.

Proposition 1: The precoding vector that maximizesASLNRk in (22) is given by

baslnrk =

1ηaslnr

k

⎡⎣(∑

i

γivivHi +

1ρdl

k

IM

)−1

vk

⎤⎦∗

, (23)

where ηaslnrk is the power normalization coefficient that is set

to satisfy∥∥baslnr

k

∥∥ = 1, and the corresponding maximumASLNR value is given by

ASLNRmaxk =

1

1 − γkvHk

(∑i γivivH

i + 1ρdl

k

IM

)−1

vk

− 1.

(24)

Proof: The proof is similar to the iCSI case in [35], andis omitted for brevity.

Proposition 1 provides a sCSI based DL precoder that maxi-mizes the ASLNR in closed-form. Note that the sCSI requiredin the proposed approach are the channel direction vectorsand the average power of all UTs’ channels, i.e., vk and γk,∀k. In addition, from the definition of the channel directionvector in (4), only the space angles, ϑx

k and ϑyk, are needed for

estimating vk. Thus, the number of parameters in statisticalCSI to estimate can be significantly reduced. As the proposedsCSI based DL precoding design is independent of subcarriersand OFDM symbols in transmission interval where the channelstatistics do not change significantly and thus is convenientfor practical implementation of the satellite payloads. Then,the computational overhead for DL precoding design can bereduced compared with the iCSI based approach.

B. UL Receiver

In this subsection, we investigate UL receiver design. TheUL received signal by the satellite in (17) can be rewritten as

yul =∑

k

gulk

√qulsul

k + zul, (25)

where the subcarrier and symbol indices are omitted forbrevity, qul is the transmit power of one UT, sul

k is the signalsent by UT k with mean 0 and variance 1, and zul is theadditive Gaussian noise distributed as CN (0, σulIM

). With a

linear receiver at the satellite, the recovered signal of UT kcan be expressed by

sulk = wT

k yul = wTk

∑i

guli

√qulsul

i + wTk zul, (26)

where wk is the linear receiving vector of UT k. Then theSINR of UT k is given by

SINRk =

∣∣wTk gul

k

∣∣2 qul∑i�=k

∣∣wTk gul

i

∣∣2 qul + σul ‖wk‖2

=

∣∣wTk gul

k

∣∣2∑i�=k

∣∣wTk gul

i

∣∣2 + 1ρul ‖wk‖2

, (27)

where ρul � qul/σul is the UL SNR. It is not difficult to obtainthe receiver of UT k that maximizes SINRk in (27) as

wsinrk =

⎡⎣(∑

i

guli

(gul

i

)H+

1ρul

IM

)−1

gulk

⎤⎦∗

. (28)

Note that the SINR maximization UL receiving vectors in(28) are in general difficult to be computed in practicalsatellite communications systems where the payload resource



is limited, as they are needed to be updated more frequentlyin time and frequency.

Similarly as the DL case, we investigate UL receiver designfor LEO satellite communications exploiting sCSI and con-sider the ASINR performance metric given by

ASINRk �E{∣∣wT

k gulk

∣∣2}E{∑

i�=k

∣∣wTk gul

i

∣∣2 + 1ρul ‖wk‖2

}

=γk

∣∣∣(uk)T wk

∣∣∣2∑i�=k γi

∣∣∣(ui)T wk

∣∣∣2 + 1ρul ‖wk‖2

. (29)

Using a similar proof procedure as in Proposition 1, we canobtain that the sCSI based UL receiver that maximizesASINRk in (29) is given by

wasinrk =

⎡⎣(∑

i

γiuiuHi +

1ρul

IM

)−1

uk

⎤⎦∗

, (30)

with the corresponding maximum ASINR of UT k being

ASINRmaxk =

1

1 − γkuHk

(∑i γiuiuH

i + 1ρul IM

)−1

uk

− 1.

(31)

Note that the sCSI based UL receiver in (30) is presented inclosed-form, and is based on sCSI, i.e., the channel directionvector uk and the statistics of the channel gain gdl

k,�,n, whichcan thus mitigate the payload complexity and cost in practicalsatellite communications.

C. DL-UL Duality

From (23) and (30), we can obtain the DL-UL dualitybetween the proposed sCSI based DL precoder and ULreceiver. Specifically, in the considered transmission intervalwhere the channel statistics do not change significantly, if theDL data transmission SNR ρdl

k equals the UL data transmissionSNR ρul, then the sCSI based DL precoding vectors in (23)are equal to the sCSI based UL receiving vectors in (30) withproper power normalization provided that the DL directionvector vk equals the UL direction vector uk, and the trans-mission complexity can be further reduced. Note that differentfrom the UL-DL duality results based on the perfect iCSIassumption in, e.g., [37] and [38], our result is establishedusing the sCSI at the satellite side.

D. Upper Bound of ASLNR/ASINR

In this subsection, we investigate the conditions under whichthe DL ASLNR and UL ASINR metrics considered above canbe upper bounded.

Proposition 2: The maximum DL ASLNR valueASLNRmax

k in (24) is upper bounded by

ASLNRmaxk ≤ ρdl

k γk, (32)

and the upper bound can be achieved under the condition that

(vxk)H vx

i = 0 or (vyk)H vy

i = 0, ∀k = i. (33)

Besides, the maximum UL ASINR value ASINRmaxk in (31)

is upper bounded by

ASINRmaxk ≤ ρulγk, (34)

and the upper bound can be achieved under the condition that

(uxk)H ux

i = 0 or (uyk)H uy

i = 0, ∀k = i. (35)

Proof: Please refer to Appendix A.Proposition 2 shows that the DL ASLNRs and UL ASINRs

of all served UTs with the proposed sCSI based precoderand receiver can reach their upper bounds provided that thecorresponding channel direction vectors of different UTs aremutually orthogonal. The result in Proposition 2 is physi-cally intuitive as the DL channel leakage power and the ULinter-user interference can be eliminated provided that theconditions in (33) and (35) are satisfied.

From (5) and (11), we can observe that the optimal condi-tions obtained in Proposition 2 can be asymptotically satisfiedwhen the number of antennas M tends to infinity. Thiscorroborates the rationality and potential of exploiting massiveMIMO in enhancing the transmission performance of satellitecommunications.

Remark 1: When the channel direction vectors of the UTsscheduled over the same time-frequency resource blocks sat-isfy the conditions in (33) and (35) or the number of antennasat the satellite side is sufficiently large, we can obtain fromthe matrix inversion lemma that the proposed sCSI basedprecoder/receiver in (23) and (30) will reduce to

baslnrk = v∗

k, wasinrk = u∗

k. (36)

Notably, the sCSI based DL precoder and UL receiver pre-sented in (36) approach the ones using iCSI as the number ofantennas tends to infinity [11], which demonstrates the asymp-totic optimality of the proposed precoder/receiver exploitingsCSI.

Remark 2: Note that for the case with a sufficiently largenumber of antennas at the satellite side, the precoder/receiverin (36) will asymptotically tend to the discrete Fourier trans-form (DFT) based fixed precoder/receiver as follows

bk =[vx

(ϑ

x

k

)⊗ vy

(ϑ

y

k

)]∗,

wasinrk = wk =

[ux

(ϑ

x

k

)⊗ uy

(ϑ

y

k

)]∗, (37)

where ϑd

k is the nearest point of ϑdk in the DFT grid satisfying

ϑd

k = −1 + 2ndk/Md with nd

k ∈ [0, Md − 1] being integers

and∣∣∣ϑd

k − ϑdk

∣∣∣ < 2/Md for d ∈ D. In this case, the pre-coding/receiving vectors for the simultaneously served UTsin the same user group are orthogonal, and can be efficientlyimplemented with fast Fourier transform (FFT).

IV. USER GROUPING

From the results in the above section, we can observe thatthe performance of the proposed sCSI based precoder andreceiver in massive MIMO LEO satellite communications willlargely depend on the channel statistics of the simultaneouslyserved UTs. As the number of the UTs to be served is usually



much larger than that of antennas equipped at the satellites,user grouping is of practical importance. Compared with theterrestrial counterpart, user grouping is of greater interest asthe satellite service provider generally aims at serving all UTsin satellite communications. In this section, we investigate usergrouping for massive MIMO LEO satellite communications.

A. Space Angle Based User Grouping

Although the conditions in Proposition 2 are desirablefor optimizing the performance of DL ASLNRs and ULASINRs in satellite communications, it is in general difficult toschedule the UTs that rigorously satisfy this condition, and theoptimal user grouping pattern can be found through exhaustivesearch. However, due to the large number of existing UTs insatellite communications, it is usually infeasible to perform anexhaustive search in practical systems.

The optimal user grouping condition presented in Propo-sition 2 indicates that the channel direction vectors of UTsin the same group should be as orthogonal as possible. Fromthe definitions in (4) and (10), the channel direction vectorsare directly related to the channel propagation propertiesin the space domain, i.e., the channel space angles. Then,the conditions for achieving the upper bounds of ASLNRand ASINR presented in (33) and (35) can be reduced to thecondition that the channel space angles should satisfy

ϑxk − ϑx

i =2

Mxnx

k,i or ϑyk − ϑy

i =2

Myny

k,i, ∀k = i, (38)

where both nxk,i and ny

k,i are non-zero integers. Motivated bythe condition in (38), we propose a space angle based usergrouping (SAUG) approach as follows. Specifically, we uni-formly divide the space angle range [−1, 1) into MxGx andMyGy equal sectors in the x- and y-axes, respectively, whereGx and Gy are both integers and their physical meaning willbe clear later. Then, the space angle intervals after divisioncan be represented by

A(m,n)(g,r) =

{(φx, φy)|φx ∈

[φx

g,m − Δx

2, φx

g,m +Δx

2

),

φy ∈[φy

r,n − Δy

2, φy

r,n +Δy

2

)}, (39)

where φda,b for d ∈ D is the center space angle of the interval

in the x-/y-axis given by

φda,b = −1 +

Δd

2+ (a + bGd)Δd, (40)

where 0 ≤ a ≤ Gd−1, 0 ≤ b ≤ Md−1, and Δd = 2/ (MdGd)is the length of the space angle interval in the x-/y-axis.

With the above definition of the space angle interval divi-sion, the UTs can be grouped as follows. A given UT k isscheduled into the (g, r)th group if there exist 0 ≤ m ≤ Mx−1and 0 ≤ n ≤ My−1 such that the corresponding channel spaceangles satisfy

(ϑxk, ϑy

k) ∈ A(m,n)(g,r) . (41)

Denote by K(m,n)(g,r) =

{k : (ϑx

k, ϑyk) ∈ A(m,n)

(g,r)

}the set of UTs

whose space angles lie in the interval A(m,n)(g,r) . In the proposed

SAUG approach, we always require∣∣∣K(m,n)

(g,r)

∣∣∣ ≤ 1 to avoidintra-beam interference. Note that other UTs located in thesame space angle interval can be scheduled over differenttime-frequency resources in a round-robin manner to preservefairness. Based on the above user grouping procedure, the UTsare scheduled into at most GxGy groups, where the (g, r)thUT group is defined as

K(g,r) �⋃

0≤m≤Mx−10≤n≤My−1

K(m,n)(g,r) . (42)

Note that the UTs scheduled in the same group will per-form transmission over the same time-frequency resources,while UTs in different groups will be allocated with differenttime-frequency transmission resources.

B. Achievable Rate Performance

In this subsection, we investigate the achievable rate per-formance of the proposed SAUG approach. We focus on theDL transmission case, and the UL results can be similarlyobtained.

From the DL signal model in (19) and the proposed SAUGapproach in the above subsection, the DL achievable ergodicsum rate is given by

Rdl =1

GxGy

Gx−1∑g=0

Gy−1∑r=0

∑k∈K(g,r)

E

⎧⎨⎩log2

⎧⎨⎩1 +

∣∣gdlk

∣∣2 ∣∣vTk baslnr

k

∣∣2 qdlk∑i�=k

i∈K(g,r)

∣∣gdlk


i

∣∣2 qdli + σdl

k

⎫⎬⎭⎫⎬⎭ ,

(43)

where baslnrk is the sCSI based precoder of UT k presented

in (23), and K(g,r) is the UT group defined in (42). Theergodic rate expression in (43) is in general difficult to handle.Therefore, we resort to investigate the bounds of the achievableergodic rate for further analysis. In the following proposition,we first present an upper bound of the DL achievable ergodicsum rate.

Proposition 3: With linear precoder utilizing only sCSI,the DL achievable ergodic sum rate Rdl in (43) is upperbounded by

Rdl ≤ Rubdl

� 1GxGy

Gx−1∑g=0

Gy−1∑r=0

∑k∈K(g,r)

E{log2

{1 + ρdl

k

∣∣gdlk

∣∣2}} ,

(44)

where the corresponding upper bound can be achieved pro-vided that the channel direction vectors of the UTs served inthe same group satisfy

(vxk)H vx

i = 0or (vyk)H vy

i = 0, ∀k, i ∈ K(g,r), k = i,

(45a)

baslnrk = (vx

k ⊗ vyk)∗ , ∀k. (45b)

Proof: Please refer to Appendix B.



Proposition 3 provides some insights for optimal DL pre-coding design with sCSI at the transmitter. In particular,the channel direction vectors of the UTs scheduled to beserved over the same time-frequency resources should be asorthogonal as possible. Meanwhile, the beamforming vector ofa given UT should be aligned with the corresponding channeldirection vector. Note that the previously proposed SAUGapproach attempts to schedule the UTs to satisfy the conditionin (45a), and the proposed ASLNR based precoding strives toreduce the inter-user interference to approach the conditionin (45b) as remarked in (36).

In the following, we further investigate the asymptoticperformance of the proposed approach. Before proceeding,we first provide an upper bound of the inner product

∣∣vHk vj

∣∣for UTs k, j(∀k = j) that are scheduled over the sametime-frequency transmission resources via the proposed SAUGapproach. From (3), we have

∣∣vHk vj

∣∣ =∣∣∣∣∣ sin πϕxMx

2

Mx sin πϕx2

∣∣∣∣∣∣∣∣∣∣ sin πϕyMy

2

My sin πϕy2

∣∣∣∣∣ , ∀k = j. (46)

where ϕd = ϑdk − ϑd

j for d ∈ D. With the proposed SAUGapproach, it is not difficult to show that

2Md

m − Δd ≤ ϕd ≤ 2Md

m + Δd, (47)

where 1 ≤ m ≤ Md − 1, d ∈ D. Then, we can further upperbound the inner product

∣∣vHk vj

∣∣ as

∣∣vHk vj

∣∣ ≤∣∣∣∣∣∣

sin π(1 − 1

Gx

)Mx sin π

Mx

(1 − 1

Gx

)∣∣∣∣∣∣∣∣∣∣∣∣

sin π(1 − 1

Gy

)My sin π

My

(1 − 1

Gy

)∣∣∣∣∣∣ ,(48)

for k = j. Thus, with sufficiently large numbers of groupsGx and Gy, the inner product

∣∣vHk vj

∣∣ can be sufficientlysmall. Motivated by this, we present the asymptotic optimal-ity of the proposed SAUG approach combined with sCSIbased DL precoder in each UT group in the followingproposition.

Proposition 4: The DL achievable ergodic sum rate Rdl

with the proposed sCSI based ASLNR maximization DLprecoder and the SAUG approach is lower bounded by

Rdl ≥ Rlbdl � 1

GxGy

Gx−1∑g=0

Gy−1∑r=0

∑k∈K(g,r)

E

⎧⎨⎩log2

⎧⎨⎩1 +

∣∣gdlk

∣∣2 (1 − δdl (ε))qdlk∑i�=k

i∈K(g,r)

∣∣gdlk

∣∣2 βdlk,i(ε)

ξdl(ε) qdli + σdl

k

⎫⎬⎭⎫⎬⎭ ,

(49)

where δdl (ε), βdlk,i(ε), and ξdl(ε) are given by

δdl (ε) =

(ρdlmaxγmax

)2 (Kmax − 1)2ε2

χdl(ε), (50a)

χdl(ε) =1

1ρdlminγmin

+ 1 + (Kmax − 1)ε, (50b)

βdlk,i (ε) =

(ρdlmaxγmax

)4(ρdl

k γiγk

)2 (Kmax − 1)2ε2, (50c)

ξdl(ε) =1(

1/ρdlmin + γmax + γmax(Kmax − 1)ε

)2 ,

(50d)

respectively, with Kmax = maxg,r

∣∣K(g,r)

∣∣, ρdlmax = max

g,rmax

k∈K(g,r)

ρdlk , ρdl

min = ming,r

mink∈K(g,r)

ρdlk , γmax = max

g,rmax

k∈K(g,r)

γk, and

γmin = ming,r

mink∈K(g,r)

γk, provided that the inner product of

the channel direction vectors of the UTs scheduled over thesame time-frequency resource blocks satisfies

∣∣vHk vj

∣∣ ≤ ε for∀k = j and k, j ∈ K(g,r). Moreover, when ε → 0, the lowedbound of the DL achievable ergodic rate in (49) asymptoticallytends to be equal to the upper bound of the DL achievableergodic rate in (44), i.e.,

limε→0

Rlbdl =

1GxGy

Gx−1∑g=0

Gy−1∑r=0

∑k∈K(g,r)

E{log2

{1 + ρdl

k

∣∣gdlk

∣∣2}} = Rubdl . (51)

Proof: Please refer to Appendix C.Proposition 4 shows that the proposed approach with SAUG

and sCSI based ASLNR maximization DL precoder per-formed in each UT group is asymptotically optimal when∣∣vH

k vj

∣∣→ 0. Note that this condition coincides with theupper bound achieving condition presented in Proposition 2.Therefore, when the number of satellite antennas M and/or thenumber of scheduled UT groups is sufficiently large, the pro-posed approach is asymptotically optimal, which indicates thepotential of adopting massive MIMO to serve a large numberof UTs in LEO satellite communications. In addition, whenthe previously derived conditions are not rigorously satisfied(which is the usual case in practice), the proposed sCSI basedprecoder and receiver can mitigate the inter-user interferenceand further enhance the transmission performance for satellitecommunications.

V. SIMULATION RESULTS

In this section, we provide simulation results to evaluatethe performance of the proposed massive MIMO transmissionapproach for LEO satellite communications. The major sim-ulation setup parameters are listed as follows. The numbersof antennas equipped at the satellite side are set to be Mx =My = 16 with half-wavelength antenna spacing in both thex- and y-axes. The channel Rician factor is set to be κk = κ =10 dB, and the channel power is normalized as γk = MxMy

for all UT k. In addition, the channel space angles ϑxk,p and

ϑyk,p are independently and uniformly distributed in the interval

[−1, 1) for all UTs. The numbers of UT groups in the proposedSAUG approach are set to be equal for both x- and y-axes,i.e., Gx = Gy = G. The number of UTs to be grouped is setas G2M .

Note that massive MIMO has not been applied to LEOsatellite communications, and we consider and compare thefollowing DL precoding and UL receiving approaches in thesimulations:



Fig. 2. Sum rate performance comparison between the proposed sCSI (using the true and the estimated sCSI that are obtained via averaging over 50 samples,respectively) and iCSI based precoding/receiving approaches.

Fig. 3. Sum rate performance of SAUG with different transmission approaches versus the number of scheduled UT groups for different SNRs when FFR isadopted.

• IntF: An ideal interference-free (IntF) case where theinterference from other scheduled UTs over the same timeand frequency resource is “genie-aided” eliminated willbe considered as the performance upper bound.

• iCSI: Relying on the iCSI, the SLNR maximization DLprecoder in (21) and the SINR maximization UL receiverin (28) are adopted, with the assumption that the iCSI canbe “genie-aided” obtained.

• sCSI: The proposed sCSI based ASLNR maximizationDL precoder and ASINR maximization UL receiverin (23) and (30) are adopted, respectively.

• Fixed: DFT based fixed DL precoding and UL receivingvectors in (37) are adopted.

In Fig. 2, we evaluate the performance of the proposedsCSI based precoding/receiving approaches, and comparethem with the iCSI based ones where UTs are groupedusing the proposed SAUG with G = 1. We considerboth cases that utilize the true and estimated sCSI that isobtained via averaging 50 samples. We can observe that inboth UL and DL transmissions, the proposed sCSI basedprecoder and receivers exhibit almost identical performance

as the iCSI based ones, while having significantly reducedcomputational overhead. In addition, the sum rate perfor-mance loss utilizing the estimated sCSI can be almostneglected.

In Fig. 3, we evaluate the performance of the proposedSAUG approach with different precoding/receiving approachesversus the number of scheduled groups G when FFR isadopted across neighboring beams. We can observe that theperformance of the proposed sCSI based precoder/receiver canapproach that of the interference-free scenario, especially inthe case with a large number of scheduled groups, whichdemonstrates the asymptotic optimality of the proposed trans-mission approach. In addition, the performance gap betweenthe approach with fixed precoding/receiving vectors and theproposed sCSI based ones becomes smaller as the number ofscheduled groups increases, especially in the low SNR regime,which indicates the near-optimality of the approach with fixedprecoding/receiving vectors in the case where interference isnot dominated.

In Fig. 4, the performance between the proposed transmis-sion approach with FFR and the conventional FR4 approach



Fig. 4. Sum rate performance comparison between the proposed approach with FFR and the conventional FR4 approach under different Rician factors.

is compared for different SNRs and channel Rician factors.Similarly as FFR, only one UT is scheduled per beam overthe same time and frequency resource in FR4. Note thatin the case of FR4, the UTs with the same color are agroup of UTs performing transmission over the same timeand frequency resource. For FR4, we consider two trans-mission approaches where “FR4, Conventional” denotes thefixed precoder/receiver in (37) and “FR4, sCSI” denotes theproposed sCSI based precoder/receiver in (23)/(30) applied tothe group of UTs over the same time and frequency resourcefor interference mitigation, respectively. We can observethat the proposed sCSI based precoder/receiver applied toFR4 show sum rate performance gains over the conventionalFR4 approach. Moreover, with FFR across neighboring beams,the proposed sCSI based precoder/receiver combined withSAUG can provide significant sum rate performance gains overthe conventional FR4 approach, especially in the cases withhigh SNRs and large Rician factors. Notably, for both ULand DL with an SNR of 20 dB and κ = 10 dB, the pro-posed transmission approach with G = 4 can provide abouteight-folded sum rate performance gain over the conventionalFR4 approach.

VI. CONCLUSION

In this paper, we have investigated massive MIMO trans-mission for LEO satellite communications exploiting sCSIwith FFR. We first established the massive MIMO chan-nel model for LEO satellite communications by taking intoaccount the LEO satellite signal propagation properties andsimplified the UL/DL transmission designs via performingDoppler and delay compensations at UTs. Then, we developedthe sCSI based DL precoder and UL receiver in closed-form, under the criteria of maximizing the ASLNR and theASINR, respectively, and revealed the duality between them.We further showed that the DL ASLNRs and UL ASINRs canreach their upper bounds provided that the channel directionvectors of the simultaneously served UTs are orthogonal,and proposed a space angle based user grouping (SAUG)

approach motivated by this condition. Besides, we showedthe asymptotic optimality of the proposed massive MIMOtransmission approach exploiting sCSI. Simulation resultsshowed that the proposed massive MIMO transmission schemewith FFR significantly enhances the data rate of LEO satel-lite communication systems. Notably, the proposed sCSIbased precoder and receiver achieved the similar perfor-mance with the iCSI based ones that are often infeasiblein practice. Future work includes detailed investigation onlow complexity sCSI estimation, transmission designs for thecases with UTs using multiple antenna or directive antennas,low peak-to-average power ratio transmission signal design,and extension to the multiple LEO satellite communicationsystems, etc.

APPENDIX APROOF OF PROPOSITION 2

We focus on the proof of the DL case and the proof of theUL case can be similarly obtained. We first show the upperbound of ASLNRmax

k in (32). From (24), we can obtain thatASLNRmax

k with the proposed sCSI based precoder can beupper bounded by

ASLNRmaxk =

1

1 − γkvHk

(∑i γivivH

i + 1ρdl

k

IM

)−1

vk

−1

(a)

≤ 1

1 − γkvHk

(γkvkvH

k + 1ρdl

k

IM

)−1

vk

−1

=(b) ρdlk γk, (52)

where (a) follows from that γivivHi is positive semidefinite

for ∀i, and (b) follows from the Sherman-Morrison formula[39, Eq. (15.2b)].

We then show the achievability of the upper bound. Fromthe definition of vk in (4), the condition given in (33)is equivalent to vH

k vi = 0 for ∀k = i. Thus, we can obtain



that (∑i

γivivHi +

1ρdl

k

IM

)vkvH

k

=(

γkvkvHk +

1ρdl

k

IM

)vkvH

k

= vkvHk

(γkvkvH

k +1

ρdlk

IM

), (53)

which yields(∑i

γivivHi +

1ρdl

k

IM

)−1

vkvHk

= vkvHk

(γkvkvH

k +1

ρdlk

IM

)−1

. (54)

Taking the traces of both sides of (54), we can further obtain

vHk

(∑i

γivivHi +

1ρdl

k

IM

)−1

vk

vHk

(γkvkvH

k +1

ρdlk

IM

)−1

vk. (55)

Thus, the inequality in (a) of (52) can be obtained when thecondition in (33) is satisfied. This concludes the proof.

APPENDIX BPROOF OF PROPOSITION 3

The achievable ergodic rate in (43) can be upper boundedby

Rdl

(a)

≤ 1GxGy

Gx−1∑g=0

Gy−1∑r=0

∑k∈K(g,r)

E{log2

{1 + ρdl

k

∣∣gdlk


k

∣∣2}}(b)

≤ 1GxGy

Gx−1∑g=0

Gy−1∑r=0

∑k∈K(g,r)

E{log2

{1 + ρdl

k

∣∣gdlk

∣∣2}} ,

(56)

where (a) follows from |·|2 ≥ 0, and (b) follows from theCauchy-Schwarz inequality.

We then examine the condition under which the upperbound in (44) can be achieved. The inequality (a) in (56)becomes tight when baslnr

i is orthogonal to (vxk ⊗ vy

k)∗ fori = k. In addition, as the equality in (b) can be achievedwhen baslnr

k = (vxk ⊗ vy

k)∗ [40], [41], we can obtain thatfor ∀k = i ∈ K(g,r), the channel direction vectors shouldsatisfy (vx

k ⊗ vyk)H (vx

i ⊗ vyi ) = (vx

k)H vxi (vy

k)H vyi = 0,

i.e., (vxk)H vx

i = 0 or (vyk)H vy

i = 0. This concludes the proof.

APPENDIX CPROOF OF PROPOSITION 4

We first define some auxiliary variables for clarity of furtherproof. For notational brevity, we focus on a specific UT group,namely, the (g, r)th UT group K(g,r), and omit the groupindex as the UTs in different groups are scheduled over dif-ferent time-frequency transmission resources. For a given UT

k ∈ K(g,r), we define baslnrk �

(∑i γivivH

i + 1ρdl

k

IM

)−1

vk.

From (23), we can have baslnrk =

(baslnr

k /∥∥∥baslnr

k

∥∥∥)∗. Then,the DL sum rate in (43) can be rewritten as

Rdl

=1

GxGy

Gx−1∑g=0

Gy−1∑r=0

∑k∈K(g,r)

E

⎧⎪⎪⎨⎪⎪⎩log2

⎧⎪⎪⎨⎪⎪⎩1 +

∣∣gdlk

∣∣2 |vHk baslnr

k |2‖baslnr

k ‖2 qdlk

∑i�=ki∈K(g,r)

∣∣gdlk

∣∣2 |vHk baslnr

i |2‖baslnr

i ‖2 qdli + σdl

k

⎫⎪⎪⎬⎪⎪⎭

⎫⎪⎪⎬⎪⎪⎭ .

(57)

In order to obtain a lower bound of Rdl in (57), we provide

an upper bound of∣∣∣vH

k baslnri

∣∣∣2 for ∀k = i, a lower bound

of∥∥∥baslnr

k

∥∥∥2, and a lower bound of∣∣∣vH

k baslnrk

∣∣∣2 /∥∥∥baslnr

k

∥∥∥2,respectively, in the following.

A. Upper Bound of∣∣∣vH

k baslnri

∣∣∣2 for ∀k = i

Denote by K �∣∣K(g,r)

∣∣, V � [v1, . . . ,vK ], and Γ �diag

{[γ1, . . . , γK ]T

}. Then, vH

k baslnri can be expressed by

the (k, i)th element of the following matrix

A � VH

(VΓVH +

1ρdl

k

IM

)−1

V

=(a) Γ−1 − 1ρdl

k

Γ−1

(1

ρdlk

Γ−1 + VHV)−1

Γ−1,

(58)

where (a) follows from the matrix inversion lemma. LetB � 1

ρdlk

Γ−1 + VHV. Then, vHk baslnr

i can be further writtenas

vHk baslnr

i = [A]k,i =

{− 1ρdl

k γiγk

[B−1

]k,i

, if i = k

1γk

− 1ρdl

k γ2k

[B−1

]k,k

, if i = k.

(59)

Denote by bk,j the (k, j)th element of B, and D′k (B) �∑

j �=k |bk,j | =∑

j �=k

∣∣vHk vj

∣∣. Then, according to Geršgorindisc theorem [42, Theorem 6.1.1], for an arbitrary eigen-value λ of B, there exists an integer 1 ≤ p ≤ K suchthat

|λ − bp,p| ≤ D′p (B)

(a)

≤ (K − 1) ε(b)

≤ (Kmax − 1) ε, (60)

where (a) follows from∣∣vH

p vj

∣∣ ≤ ε for all j = p, and(b) follows from K ≤ Kmax. Denote by λmax and λmin thelargest and smallest eigenvalues of B, respectively. For ∀i = k,we can have the following inequality∣∣∣[B−1

]k,i

∣∣∣ =∣∣eT

k B−1ei

∣∣(a)

≤ 1/λmin − 1/λmax

1/λmin + 1/λmax

√eT

k B−1ek

√eT

i B−1ei

=λmax − λmin

λmax + λmin

√[B−1]k,k

√[B−1]i,i



(b)

≤ 2 (Kmax − 1) ε

λmax + λmin

√[B−1]k,k

√[B−1]i,i

(c)

≤ ρdlmaxγmax (Kmax − 1) ε

√[B−1]k,k

√[B−1]i,i

(d)

≤ (ρdlmaxγmax

)2(Kmax − 1) ε, (61)

where ek is the kth column of identity matrix, (a) followsfrom Wielandt’s inequality [42, Eq. (7.4.12.2)], (b) followsfrom (60), (c) follows from Weyl’s inequality [42, Corol-lary 4.3.15], and (d) follows from Rayleigh quotient theorem[42, Theorem 4.2.2(c)]. From (59) and the inequality in (61),we can obtain∣∣∣vH

k baslnri

∣∣∣2 ≤(ρdlmaxγmax

)4(ρdl

k γiγk

)2 (Kmax − 1)2 ε2 � βdlk,i(ε),

(62)

for i = k.

B. Lower Bound of∥∥∥baslnr

k

∥∥∥2Defining ζmax(X) as the maximum eigenvalue of matrix X.

Then, a lower bound of∥∥∥baslnr

k

∥∥∥2 can be obtained as

∥∥∥baslnrk

∥∥∥2

= vHk

(VΓVH +

1ρdl

k

IM

)−2

vk

(a)

≥ 1

ζ2max

(VΓVH + 1

ρdlk

IM

)(b)

≥ 1(1/ρdl

k + ζmax (VΓVH))2

(c)

≥ 1(1/ρdl

min + γmax + γmax(Kmax − 1)ε)2

� ξdl (ε) , (63)

where (a) follows from Rayleigh quotient theorem [42, Theo-rem 4.2.2(c)], (b) follows from Weyl’s inequality [42, Corol-lary 4.3.15], and (c) holds by applying Geršgorin disc theorem[42, Theorem 6.1.1] to matrix VΓVH .

C. Lower Bound of∣∣∣vH

k baslnrk


k

∥∥∥2

Before proceeding, we first present some preliminary

results. An upper bound of∥∥∥baslnr

k

∥∥∥2 can be obtained as

∥∥∥baslnrk

∥∥∥2 =

∥∥∥∥∥(VΓVH +

1ρdl

k

IM

)−1

Vek

∥∥∥∥∥2

=(a)∥∥VB−1Γ−1ek

∥∥2= eT

k Γ−1B−1VHVB−1Γ−1ek

=(b) 1γ2

k

[B−1

(B− 1

ρdlk

Γ−1

)B−1

]k,k

=1γ2

k

([B−1

]k,k

− 1ρdl

k

[B−1Γ−1B−1

]k,k

)

=1γ2

k

⎛⎝[B−1

]k,k

−K∑

j=1

1ρdl

k γj

∣∣∣[B−1]k,j

∣∣∣2⎞⎠

≤ 1γ2

k

[B−1

]k,k

(1 − 1

ρdlk γk

[B−1

]k,k

)� ηdl

k ,

(64)

where (a) follows from the matrix inversion lemma, and(b) follows from the definition of Γ and B. In addition,[B−1

]k,k

can be lower bounded by[B−1

]k,k

(a)

≥ 1λmax

(b)

≥ 1bp,p + (Kmax − 1)ε

≥ 11

ρdlminγmin

+ 1 + (Kmax − 1)ε� χdl (ε) ,

(65)

where (a) follows from Rayleigh quotient theorem [42, The-orem 4.2.2(c)], (b) follows from Geršgorin disc theorem [42,Theorem 6.1.1]. Moreover, the following inequality can beobtained

1 =∣∣∣[BB−1

]k,k

∣∣∣ =∣∣∣∣∣∣

K∑j=1

bk,j

[B−1

]j,k

∣∣∣∣∣∣(a)

≥ bk,k

[B−1

]k,k

−∑j �=k

|bk,j |∣∣∣[B−1

]j,k

∣∣∣(b)

≥ bk,k

[B−1

]k,k

−∑j �=k

(ρdlmaxγmax

)2(Kmax − 1)ε2

=(

1ρdl

k γk+1)[

B−1]k,k

−(ρdlmaxγmax

)2(Kmax − 1)2ε2,

(66)

where (a) follows from the triangle inequality, and (b) followsfrom |bk,j | =

∣∣vHk vj

∣∣ ≤ ε for ∀j = k and the inequalityin (61). Then, we can obtain

1[B−1]k,k

− 1ρdl

k γk

(a)

≥ 1 − 1[B−1]k,k

(ρdlmaxγmax

)2(Kmax − 1)2ε2

(b)

≥ 1 −(ρdlmaxγmax

)2 (Kmax − 1)2ε2

χdl(ε)� 1 − δdl (ε) ,

(67)

where (a) follows from (66), and (b) follows from (65).

Consequently,∣∣∣vH

k baslnrk


k

∥∥∥2 can be lowerbounded by∣∣∣vH

k baslnrk

∣∣∣2∥∥∥baslnrk

∥∥∥2(a)

≥

∣∣∣vHk baslnr

k

∣∣∣2ηdl

k

(b)=

(1 − 1

ρdlk γk

[B−1

]k,k

)2

γ2kηdl

k

=(c) 1[B−1]k,k

− 1ρdl

k γk

(d)

≥ 1 − δdl (ε) , (68)

where (a) follows from the inequality in (64), (b) followsfrom (59), (c) follows from the definition of ηdl

k in (64),and (d) follows from (67).



Combining (57), (62), (63), and (68), we can obtain a lowerbound of Rdl as

Rdl

≥ 1GxGy

Gx−1∑g=0

Gy−1∑r=0

∑k∈K(g,r)

E

⎧⎨⎩log2

⎧⎨⎩1 +

∣∣gdlk

∣∣2 (1 − δdl (ε))qdlk∑i�=k

i∈K(g,r)

∣∣gdlk

∣∣2 βdlk,i(ε)

ξdl(ε)qdli + σdl

k

⎫⎬⎭⎫⎬⎭� Rlb

dl.

(69)

Note that limε→0 δdl (ε) = 0, limε→0 βdlk,i(ε) = 0, and

ξdl(ε) > 0, thus we can obtain limε→0 Rlbdl = Rub

dl . Thisconcludes the proof.

ACKNOWLEDGMENT

The authors would like to thank the anonymous reviewersfor their constructive comments and suggestions.

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Li You (Member, IEEE) received the B.E. and M.E.degrees from the Nanjing University of Aeronau-tics and Astronautics, Nanjing, China, in 2009 and2012, respectively, and the Ph.D. degree from South-east University, Nanjing, in 2016, all in electricalengineering.

From 2014 to 2015, he conducted visiting researchwith the Center for Pervasive Communications andComputing, University of California at Irvine, Irvine,CA, USA. Since 2016, he has been with the Facultyof the National Mobile Communications Research

Laboratory, Southeast University. His research interests lie in the general areasof communications, signal processing, and information theory with the currentemphasis on massive MIMO communications.

Ke-Xin Li (Graduate Student Member, IEEE)received the B.E. degree in communication engi-neering from the University of Electronic Scienceand Technology of China, Chengdu, China, in 2016.He is currently pursuing the Ph.D. degree with theNational Mobile Communications Research Labo-ratory, Southeast University, Nanjing, China. Hisresearch interests are mainly on communications,signal processing, and information theory, withemphasis on satellite communications.

Jiaheng Wang (Senior Member, IEEE) received theB.E. and M.S. degrees from Southeast University,Nanjing, China, in 2001 and 2006, respectively,and the Ph.D. degree in electronic and computerengineering from The Hong Kong University ofScience and Technology, Hong Kong, in 2010.

From 2010 to 2011, he was with the SignalProcessing Laboratory, KTH Royal Institute of Tech-nology, Stockholm, Sweden. He also held visit-ing positions at the Friedrich Alexander UniversityErlangen-Nürnberg, Nürnberg, Germany, and the

University of Macau, Macau. He is currently a Full Professor with the NationalMobile Communications Research Laboratory (NCRL), Southeast University,Nanjing, China. He has published more than 100 articles on internationaljournals and conferences. His research interests include optimization in signalprocessing and wireless communications.

Dr. Wang was a recipient of the Humboldt Fellowship for ExperiencedResearchers and the best paper awards of the IEEE GLOBECOM 2019,ADHOCNETS 2019, and WCSP 2014. He served as an Associate Editorfor the IEEE SIGNAL PROCESSING LETTERS from 2014 to 2018. Since2018, he has been serving as a Senior Area Editor for the IEEE SIGNAL

PROCESSING LETTERS.

Xiqi Gao (Fellow, IEEE) received the Ph.D. degreein electrical engineering from Southeast University,Nanjing, China, in 1997.

In April 1992, he joined the Department ofRadio Engineering, Southeast University. FromSeptember 1999 to August 2000, he was a VisitingScholar with the Massachusetts Institute of Tech-nology, Cambridge, MA, USA, and Boston Uni-versity, Boston, MA, USA. Since May 2001,he has been a Professor of information systems andcommunications with Southeast University. From

August 2007 to July 2008, he visited the Darmstadt University of Tech-nology, Darmstadt, Germany, as a Humboldt Scholar. His current researchinterests include broadband multicarrier communications, MIMO wirelesscommunications, channel estimation and turbo equalization, and multiratesignal processing for wireless communications.

Dr. Gao received the Science and Technology Awards of the State EducationMinistry of China in 1998, 2006, and 2009, the National Technological Inven-tion Award of China in 2011, and the 2011 IEEE Communications SocietyStephen O. Rice Prize Paper Award in the Field of communications theory.From 2007 to 2012, he served as an Editor for the IEEE TRANSACTIONS ON

WIRELESS COMMUNICATIONS. From 2009 to 2013, he served as an Editorfor the IEEE TRANSACTIONS ON SIGNAL PROCESSING. From 2015 to 2017,he served as an Editor for the IEEE TRANSACTIONS ON COMMUNICATIONS.

Xiang-Gen Xia (Fellow, IEEE) received the B.S.degree in mathematics from Nanjing Normal Uni-versity, Nanjing, China, the M.S. degree in mathe-matics from Nankai University, Tianjin, China, andthe Ph.D. degree in electrical engineering from theUniversity of Southern California, Los Angeles, CA,USA, in 1983, 1986, and 1992, respectively.

He was a Senior/Research Staff Member at HughesResearch Laboratories, Malibu, CA, USA, from1995 to 1996. In September 1996, he joined theDepartment of Electrical and Computer Engineering,

University of Delaware, Newark, DE, USA, where he is currently the CharlesBlack Evans Professor. He has authored the book Modulated Coding forIntersymbol Interference Channels (New York, Marcel Dekker, 2000). Hiscurrent research interests include space-time coding, MIMO and OFDMsystems, digital signal processing, and SAR and ISAR imaging.

Dr. Xia received the National Science Foundation (NSF) Faculty EarlyCareer Development (CAREER) Program Award in 1997, the Office ofNaval Research (ONR) Young Investigator Award in 1998, and the Outstand-ing Overseas Young Investigator Award from the National Nature ScienceFoundation of China in 2001. He received the 2019 Information TheoryOutstanding Overseas Chinese Scientist Award, the Information TheorySociety of Chinese Institute of Electronics. He is currently serving and hasserved as an Associate Editor for numerous international journals includingthe IEEE WIRELESS COMMUNICATIONS LETTERS, the IEEE TRANSAC-TIONS ON SIGNAL PROCESSING, the IEEE TRANSACTIONS ON WIRELESS

COMMUNICATIONS, the IEEE TRANSACTIONS ON MOBILE COMPUTING,and the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY. He is aTechnical Program Chair of the Signal Processing Symposium, GLOBECOM2007, Washington, DC, USA, and the General Co-Chair of ICASSP 2005,Philadelphia, PA, USA.

Björn Ottersten (Fellow, IEEE) was born in Stock-holm, Sweden, in 1961. He received the M.S. degreein electrical engineering and applied physics fromLinköping University, Linköping, Sweden, in 1986,and the Ph.D. degree in electrical engineering fromStanford University, Stanford, CA, USA, in 1990.He has held research positions with the Depart-ment of Electrical Engineering, Linköping Univer-sity, the Information Systems Laboratory, StanfordUniversity, the Katholieke Universiteit Leuven, Leu-ven, Belgium, and the University of Luxembourg,

Luxembourg. From 1996 to 1997, he was the Director of research withArrayComm, Inc., a start-up in San Jose, CA, USA, based on his patentedtechnology. In 1991, he was appointed a Professor of signal processingwith the Royal Institute of Technology (KTH), Stockholm, Sweden. From1992 to 2004, he was the Head of the Department for Signals, Sensors,and Systems, KTH. From 2004 to 2008, he was the Dean of the School ofElectrical Engineering, KTH. He is currently the Director for the Interdisci-plinary Centre for Security, Reliability and Trust, University of Luxembourg.As Digital Champion of Luxembourg, he acts as an Adviser to the EuropeanCommission.

He is a fellow of EURASIP. He was a recipient of the IEEE SignalProcessing Society Technical Achievement Award in 2011 and the EuropeanResearch Council advanced research grant twice, from 2009 to 2013 andfrom 2017 to 2022. He has coauthored journal articles that received the IEEESignal Processing Society Best Paper Award in 1993, 2001, 2006, and 2013,and seven other IEEE conference papers best paper awards. He has served asEditor-in-Chief of EURASIP Signal Processing Journal, an Associate Editorfor the IEEE TRANSACTIONS ON SIGNAL PROCESSING and the EditorialBoard of the IEEE Signal Processing Magazine. He is currently a member ofthe editorial boards of EURASIP Journal of Advances Signal Processing andFoundations and Trends in Signal Processing.


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