Toward Multi-Satellite Cooperative Transmission: A Joint Framework for CSI Acquisition, Feedback, and Phase Synchronization

1 T o wa rd Multi-Sate llite Cooperati v e T ransmission: A Joint Frame work for CSI Acquisition, Feedback, and Phase Syn chronization Y iming Zhu, Graduate Student Member , IEEE , Y afei W ang , Graduate Stud ent Member , IEEE , Carla Amatetti, Member , IEEE , Ale ssand ro V anelli- Co r alli, Senior Memb er , IEEE , W enjin W ang, Memb er , IEEE , Rui Ding, Sym eon Chatzinotas, F ellow , IEEE , Bj ¨ orn Ottersten, F ell ow , IEEE Abstract —The stringent link budget, caused by long satellite- to-ground distances and payload constraints, poses a fu nda- mental bottleneck for sin gle-satellite transmission performance. Although the advent of low E arth orbit mega-constellations makes mu l ti-satellite cooperativ e transmission (MS CT), su ch as distributed precoding (DP), increasingly feasible, its promised cooperativ e gains critically rely on stringent time-frequency- phase synchronization (TFP-Sync), which is difﬁ cult to maintain under rapid chann el variation and su bstantial feedback latency . T o address this issue, this paper pro poses a joint channel state information (CSI) acqu isition, f eed b ack, and phase-level synchronization (J CAFPS) framew ork for MSCT . Speciﬁcally , to enable reliable and o verhead-efﬁcient CSI acquisition, we ﬁrst design a b eam-domain adjustable phase-shif t tracking reference signal transmission sc heme, along with design c riteria f or the TRS and CSI-feedb ack periods. Then, by exploiting d eterminis- tic orbital motion and dominant l ine-of-sight propagation, we establish a polynomial model for the temporal evo lution of delay and Doppler shift, and derive an orthogonal frequency division multiplexing-based multi-satellite signal mo del under non-ideal synchronization. The analysis rev eals that, unlike the regular single-satellite ca se, the composite multi-satellite channel exhibits nonlinear time-fr equ ency-vary ing phase beha vior , which necessitates symbol- and su bcarrier -wise p hase precompensation fo r coherent transmission. Based on these results, we de velop a practica l closed-loop realization that integrates sin gle-TRS- based chann el parameter estima tion, multi-TRS-b ased channel prediction, predictive CSI feedb ack, and user -sp eciﬁc TFP pre- compensation. Nu merical results demonstrate that th e proposed framewo rk achieves accurate CSI acquisition and precise TFP- Sync, enab ling DP-b ased dual-satellite cooperati ve transmission to approach the theoretical 6 dB power gain over single-satellite transmission, while remaining robust u nder extended p rediction durations and enlarged TRS p eriods. Index T e rms —Multi-satellite cooperation, CSI acquisi tion, phase synchronization, d i stributed precoding. I . I N T R O D U C T I O N As o n e key direction f or the sixth-gen e ration techn ology ev olu tio n, satellite comm u nication (SatCom) is anticipated to Manuscript rece i ved xxx (Correspon ding author: W enji n W ang.) Y iming Zhu, Y afei W ang, W enjin W ang are with the National Mobile Communicat ions Researc h Laboratory , So utheast Uni versity , Nan jing 21 0096, China, and also with Purple Mountain Laboratories, Nanjing 211100, China (e-mail: { ymzhu, wangyf, wangwj } @seu.edu.cn). Carla Amatett i and Alessandro V anelli-C oralli are with the Depart ment of Electric al, E lectro nic, and Information Engineering, Unive rsity of Bologna, Bologna 40136, Italy (e-mail: { carla.amatet ti2, alessandro.v anell i } @unibo.it). Rui Ding is with China Satell ite Network Group Company Ltd., Beijing 100029, Chi na (e-mail: great dn@qq.com) Symeon Chatzin otas and Bjorn Ottersten are with the Interdisciplin ary Centre for Security , Reliabil ity and Trust (SnT), Uni versity of L uxembou rg, (e-mail: { symeon.cha tzinot as, bjorn.otterst en } @uni.l u). integrate with ter r estrial networks (TNs) for ubiq uitous co n - nectivity [ 1 ]. Accordin gly , mo bile satellite Intern et, dedicated to delivering broad band access to handh e ld user eq uipments (UEs) anytime and anywhere, has attracted co nsiderable at- tention fro m b o th academia an d industry [ 2 ]–[ 4 ]. Howe ver, realizing such b roadb a n d access to hand held UEs is ch allenged by a conﬂu e n ce of link-budget-limiting factors: 1 ) large path loss in duced b y the long distance, 2 ) string ent size, weight, and power (SW aP) constrain ts on the p ayload, and 3) the compa c t form factor o f handheld terminals that restricts the achievable antenna gain and transmit power [ 5 ]. Consequen tly , the link budget of mobile satellite In te r net remains tight an d is often throug hput-limitin g , necessitating the development of innova- ti ve techn iques enab ling impr oved system p e rforman ce [ 2 ]. T o mitigate link- budge t limitations, pre c oding ser ves as an effecti ve enabler in SatCom [ 6 ], [ 7 ]. Altho ugh patter n-orien ted precod in g, inclu ding Earth-movin g and Earth - ﬁxed de signs, is prev alent in deployed systems due to comp utational ef ﬁciency and relaxed chann el state inf ormation (CSI) req uirements, it is hampered by limited ﬂexibility in interfere n ce mana gement, which results in low spectra l efﬁciency [ 7 ]. Conversely , with the ad vancing comp utational capab ilities of on-bo ard pro- cessors, user-speciﬁc precod in g schemes, su ch as max imum ratio tr ansmission, zero forcin g, and minimum mean squared error, constitute a more compe titi ve alternative [ 8 ]–[ 10 ]. These technique s exploit user CSI, e.g ., angle infor m ation, to d y- namically d ir ect high-g ain b eams toward spatially distributed UEs wh ile mitigating in terference , thus enhancin g both signal power and system capacity . Howe ver, the hig hly d y namic nature o f low Earth orbit (L E O) satellites exacerbates the chan- nel aging, th us comprom ising u ser-speciﬁc prec o ding perfo r- mance [ 8 ]. Fur thermor e , while the pro liferation of LEO mega- constellations establishes the found ation for multi-satellite co- operative transmission ( MSCT) parad igms to tran scen d single- satellite perform ance ceilings [ 11 ]–[ 14 ], the realization of such gains is contingen t up on stringent time-fre q uency-ph ase synchro n ization (TFP-Sync). Against this backdrop, this paper in vestigates a jo in t CSI ac quisition, feedback , a n d ph ase-lev el synchro n ization (JCAFPS) framework for MSCT , offering practical solutions for improvin g the u ser pea k ra te. A. Rela te d W orks and Motivations T o improve the CSI timeliness, existing studie s on single - satellite transm ission have investigated cha n nel prediction 2 methods that exploit the tempor al corr elation inherent in successiv e ch a nnel estimates [ 15 ], [ 16 ]. Nevertheless, a fun- damental bottlen eck remains: e ven with perfect CSI, th e precod in g power gain is physically boun ded b y sin gle-satellite antenna apertu re. Since en g ineering feasibility and cost consid- erations limit the scaling of array size, the attainable g ain m ay still be insufﬁcient to close the link-budge t deﬁcit f or mob ile satellite I n ternet. Enabled by recen t advances in LEO man ufac- turing and reusable lau nch vehicles, mega-co nstellations have rendere d M SCT increa sin g ly viable. Distributed precod ing (DP) has emerged as a compelling solutio n to overcome the single-satellite bo ttleneck. W ith da ta sha r ing via inter- satellite lin k s ( ISLs), coop erative satellites may simultan e o usly transmit identical data streams to UE s b y co herent p recodin g design. Crucially , coher ent superposition unlock s a theoretical received po wer gain th at scales q uadratically with the numb er of coo perative satellites u nder perfe c t synchro nization. Prior works in [ 11 ], [ 12 ] inves tigated d istributed multiple input multiple output (MIMO) -based satellite network d esign a n d provided spectral efﬁciency analyses to quantif y DP gains. Nev ertheless, the realiza tion o f su c h co operative gains hinges on rig orous TFP-Syn c among coop erative satellites. The rapid dyn amics and pr opagation latency of multi-satellite systems sign iﬁcantly co mplicate CSI acquisition, inevitably hamperin g accu rate synchro nization [ 17 ]. In [ 13 ], a rigoro u s analysis was pr ovided to character iz e how impe rfect syn c hro- nization degrad es the ach iev able rate of DP- based MSCT . T o a d dress this issue, the appr oach in [ 18 ] r elied on the assumption of perfect position and trajectory knowledge to execute geometric-ba sed pha se precom pensation and satellite- UE associa tio n to h arvest c ooperative gains. G iven that p erfect position and trajectory infor mation is typ ically u nav ailab le in practice, recen t efforts h ave focused on m ulti-satellite robust precod in g und e r imperfect synchro nization. T o mitigate timing impairmen ts, the authors in [ 19 ] proposed a delay es tima- tion scheme and d e riv ed a timing- robust co operative bea m - forming alg o rithm. Further more, co nsidering residu al delay - and Doppler-induced im pairments, the authors in [ 9 ], [ 10 ] in vestigated statistical CSI-based DP de sign and multi-stream beamspace tr ansmission f or o rthogo nal f requen cy division multiplexing (OFDM)- b ased multi-satellite systems. The achiev ab le gains of MSCT dep end o n accu rate TFP- Sync, and synchro nization imp erfections can substantially compro mise these gain s. For coh e rent transm ission, in ter- satellite phase offsets hin der coh erent co mbination and even yield d estructive interf erence. Desp ite the exploration o f robust precod in g designs un d er imper f ect sync h ronization , the piv otal challenge of TFP-Sync tailore d fo r MSCT has yet to b e ade - quately addressed. While an alternative par adigm is predicated on high-p recision real-time position to execute TFP-Sync, the associated physical con straints are substantial. Considering that a ca r rier frequency of 2 GHz co rrespond s to a w avelength of 0 . 15 m, the requisite c e ntimeter-le vel position accuracy entails pr o hibitive implementatio n costs [ 20 ], particu la r ly in the context of LEO mega-co n stellations. These o bservations highligh t a fu n damental an d timely research problem : How to design an efﬁcient JCAFPS framework to enab le TFP-S ync for MSCT? B. Main Contributions Motiv ated by the consid e r ations o utlined ab ove, this paper aims to design an efﬁcient JCAFPS f ramework tailor ed f o r MSCT . Our con tributions are su mmarized as follows: • W e propo se a compr ehensive JCAFPS fram ework for MSCT , in which CSI acquisition, pred ic tive feed back, and pre compensatio n a r e tigh tly integrated into a clo sed loop for en abling st ringen t TFP-Sync. T o suppor t reliable downlink CSI acquisition u n der severe p ropagatio n loss and mu lti- satellite asy nchron ization, we furth er d esign a dedicated tra cking ref e rence signal (TRS) transmission mechanism tailored to satellite-to- g round (S2G) ch an- nels. Speciﬁcally , by introducing beam position (BP )- speciﬁc beam- domain mappin g, TRS broadc a st area (TB A) -speciﬁc pr ecompen sation, and ad justable p h ase- shift design for TRS, the proposed d esign im proves CSI acq uisition reliab ility wh ile r educing p ilot overhead. Moreover , we analyz e the design criteria of th e TRS period and CSI feed back period , ther e by revealing the fundam ental trade-off amon g signaling overhead, predic- tion validity , an d p hase-unwr apping re liab ility . • W e estab lish a time- varying S2G channel mod el cha rac- terizing the dyn amic ev o lution of S2G chann el param- eters. Capitalizing on the determ inistic satellite mo tion and line-of - sight (Lo S) domin ance, we model the p ropa- gation delay and Do ppler sh ift via time p olynom ials and elucidate the qualitative relationship between polyn omial order and v alid time scale. Based on this characterizatio n, we fu rther d erive an OFDM-based m ulti-satellite signal model with imper fect synch ronization and obtain the cor- respond in g equiv alent TF-doma in chan n el repr e sentation. Moreover , we analy tica lly rev eal that, in co ntrast to th e regular sing le-satellite case, th e co m posite multi-satellite channel exhibits nonlinear TF-varying phase behavior , necessitating the ﬁne-grained sy m bol- an d su b carrier- wise phase precomp ensation for coheren t MSCT . • W e develop a complete alg orithmic realization of the pro- posed JCAFPS f ramework. For sing le-TRS processing, we devise an TRS-aided chan nel p a r ameter estimation scheme that combines initial channe l estimation , sup er- resolution delay-do m ain par ameter e xtraction, and coarse Doppler shift estimation. Buildin g on successi ve TRS ob- servations, we fur th er propose a multi-TRS-b ased chann el prediction method based o n Do ppler-assisted cross-TRS phase u nwrapp ing (PU) and polynom ial coefﬁcient esti- mation, thro ugh which the temporal evolution of channel parameters ca n b e compactly represen ted and predicted . On this basis, the UE f e eds back pre d ictiv e CSI to the cooper a tive satelli tes, enablin g UE-speciﬁc TFP p recom- pensation for coh erent transmission. Num e rical results further verify that the proposed framew o rk effecti vely supports th e gain re a liza tion o f MSCT . This pa p er is organized as follows: Section II p resents the framework overview . Section III develops the multi-satellite signal mod el. Sec tio n IV prop oses th e detailed d esign of the JCAFPS frame work . Simula tio n resu lts are provide d in Section V , and conclu sions are drawn in Section VI . SUBMITTED P APER 3 Recovered symbols Coherent transmission ? UE-specific TFP-Sync via CSI feedback Modulation S2G channel Single-TRS-based channel parameter estimation Multi-TRS-based channel prediction 1 m M ³ - Coherent: DMRS channel estimation, equalization… Non-coherent: DMRS channel estimation, equalization… Transmitted symbols TBA-specific TF-Sync via position data S2G channel CSI feedback: Polynomial model coefficients csi-fb period t T D ³ Y N Y N N Y Y Sync CSI feedback CSI acquisition T ransmitter Receiver ISL Distributed antenna array Master satellite Secondary satellite BP-specific Beam direction TBA-specific TF-sync for TRS UE-specific TFP-sync for CT BP TBA UE Coherent transmission ? Fig. 1. Flow ch art of JCAFPS framewo rk. Notations : L owercase, bold lowercase, a n d b old u ppercase represent scalars, colum n vectors, and matr ices, respectively . The superscripts ( · ) T , ( · ) ∗ , ( · ) H , ( · ) − 1 , an d ( · ) † denote trans- pose, conjuga te , conjug ate-transpo se, in verse, and pseu doin- verse. R , C , Z , and B denote the real, co mplex, integer, and binary sets. [ · ] i and [ · ] i,j denote the i -th vector elem ent and th e ( i, j ) -th m atrix element. For an index set S (a scen ding order ) , [ · ] S , [ · ] : , S , an d [ · ] S , : represent the selected elements, c o lumns, and rows. k · k 1 , k ·k 2 , and k · k F denote th e ℓ 1 , ℓ 2 , and Fr obenius norms. I M represents the M × M dimension al identity matrix. 1 N and 0 N denote the N dimen sional all-one and all-zero vectors. diag {·} , E {·} , and min {·} represen t the diago nal, expectation, and min im um op erators. CN ( x ; µ, τ ) d enotes the circularly sy m metric com p lex Gaussian distribution o f variable x with mea n µ a n d v arian ce τ . ⊗ , ⊙ , and ◦ d enotes the Kronecker, Hadamard , and Khatri- Rao pro ducts. ¯  = √ − 1 and δ ( · ) is the Dir ac d e lta function. I I . F R A M E W O R K O V E RV I E W This pap er considers a freq uency-division duplexing (FDD) OFDM-based mu lti- satellite commu nication system. As de- picted in Fig. 1 , m ultiple LEO satellites equipp e d with a unifor m planar ar ray cooper ativ ely serve m ultiple single- antenna UE s over shared frequency-d omain resources via space division multiple access with in a comm o n coverage area. Each arr ay consists of N t = N x N y antennas arrang e d in an N x × N y rectangu la r lattice with half-wavelength spacing. Owing to the non-u niform UE d istribution, the service ar ea is par titioned into ¯ B active pr edivided BPs. T o satisfy the coverage deman d while e ffectively m itigating in te r-beam in- terference u nder on b oard power an d hard ware constraints, th e satellites emp loy bea m hopp ing [ 21 ], such tha t ¯ B active BPs are served over N hop hops and each satellite simultaneo u sly illuminates B BPs. W ithout loss of generality , we focus on the tr a n smission under the p attern of the ﬁrst hop , a s the transceiver processing rema ins id entical ac r oss hop s. In particular, S satellites coopera ti vely serve ¯ U UEs within th e B illuminate d BPs, where on e satellite is designated as the “master” to coord inate co operative tra n smission [ 4 ] an d the remaining satellites act as “secondar y” ones. The OFDM system compr ises N subcarriers with spacing ∆ f and a cyclic preﬁx (CP) of leng th N cp , amon g which N sc subcarriers are occu pied for transmission. Acco rdingly , the sampling inter val, CP duratio n, symbo l duration s without and with CP , and slot duratio n are deﬁn e d by T s = 1 N ∆ f , T cp = N cp T s , T = N T s , T sym = T cp + T , and T slot , respectively . The car r ier freq uency is f c , co rrespon ding to the wa veleng th λ = v c /f c , where v c is the speed of ligh t. A. Overall F ramework Design T o enable MSCT , we d ev elop a comprehensive JCAFPS framework consisting of three coupled m odules, a s illustrated in Fig. 1 . The fram ew o rk fo llows a two-phase op eration. During TRS tr ansmission, coo perative satellites a p ply TBA- speciﬁc time-frequency (TF) precomp ensation to facilitate reliable CSI acquisition, with each TB A being a ﬁne-g r ained spatial partition within a BP . Based on the rec e ived TRSs, the UE estimates the downlink channel par ameters and pr o- gressiv ely establishes channe l p r ediction capability . During subsequen t coher ent transmission, the satellites exploit the fed-bac k p redictive CSI to p erform UE-sp eciﬁc TFP precom - pensation, ther eby enablin g the constru ctiv e super position of identical streams at the UE 1 . In this way , CSI ac q uisition, CSI feedback , a nd synch ronization fo r m a clo sed loo p that suppo rts the strin gent TFP-Sync requir e d b y MSCT . 1) CSI Acquisition Module: Cooperative satellites peri- odically transmit TB A-sp eciﬁc TRSs, f rom wh ich the UE estimates the chann el parame ters of different satellite-to - UE links, in cluding delay , Doppler shift, phase, a nd g ain. Owing to the deterministic orbital motion and the domin ant LoS pro pa- gation of S2 G channels [ 3 ], these parameters exhibit structu red temporal evolution. T h e UE therefo re f urther repr esents the channel dy namics u sing a co mpact set of coefﬁcients extracted from histor ical e stimates, th ereby establishin g the b asis for subsequen t channel prediction and synchr onization. The main cha llenge is to achieve ac c urate multi-satellite parameter acq uisition with affordable pilot overhead. T h is challenge stems from severe pro pagation loss, inter-satellite asynchro nization, a n d th e limited efﬁciency of conv entional single-por t TRS signalin g in the multi- satellite setting . These issues motiv ate the ded icated TRS tran sm ission an d ch annel estimation designs pre sen ted in Section II-B and Sectio n IV -A . 2) CSI F eedba ck Mod ule: After su fﬁcient TRS ob serva- tions have been accumulated , the UE periodically feeds bac k the estimated chan n el-ev o lution coefﬁcients to the master satellite, which then r e lays them to the seco n dary satellites throug h ISLs 2 . Based on this pred ictive feedbac k , the cooper- ativ e satellites can in fer the future chan nel ev olutio n and pre- pare the p recompe n sation req uired for coher ent tra n smission. According ly , the f eedback mod ule b ridges UE-side chann el prediction and tr a nsmitter-side preco mpensation . The key issue in this mod u le is CSI ag ing caused by the sub stantial ro und-tr ip latency . Accordin gly , the feedb ack 1 The proposed framewor k is applic able to MSCT with s ynchroni zation require ments, rather than bei ng limited to coherent transmission. 2 This work focuses on downli nk t ransmission, assuming that the reliabilit y of uplink feedbac k channel is guaranteed by robust transmission strategie s, such as conserv ativ e modulation and coding schemes. Detailed uplink opti- mizatio n remains beyond the current scope and is reserved for future researc h. 4 … … … … … … … … … … …… …… TRS 0 … … … … … … … … … … …… …… RB 0 trs offset T trs period T slot T sym T sc N f D N f D f D tc N f D s 3 tr p s 4 tr p TRS Resource Non-TRS Resource trs 1 p s 2 tr p Symbol index TRS 1 Fig. 2. TRS conﬁgurat ion with N trs sym = 4 , N trs slot = 2 , an d N tc = 4 [ 24 ]. should chara c terize c h annel evolution ra th er th an mere ly in - stantaneous CSI, wh ile keeping the signalin g overhead well controlled . This motiv ates th e m odel-based pr e dictiv e feed - back design developed in Section II-C and Section IV -B . 3) S y nchr onization Module: The sync hronizatio n modu le supports both TRS t ransmission and coh erent tr ansmission. In th e T RS ph ase, coo perative satellites p erform open-lo op TB A-sp e ciﬁc TF precompensation based o n ephemer is an d TB A-ce nter position infor m ation. S ince the UE shar es the same trajectory and position information [ 22 ], [ 23 ] and knows the ad opted pre c ompensation rule, it can reconstruct the transmitter-side c o mpensation and recover th e continuou s tem- poral tra je c to ries of the cha n nel para meters. In the cohe r ent transmission phase, the satellites further perf o rm closed- loop UE-speciﬁc TFP preco mpensation b a sed on th e pr edictive feedback so as to ensure coher ent comb ining at the UE. The f undame n tal challenge is that syn c hronizatio n errors ar e jointly co upled acro ss the time, frequ e ncy , and phase domain s, and e ven small mismatches can substan tially degrade co herent combinin g. This ob servation motiv a te s the sub seq uent phase- aware signal modeling and predictive syn chroniz a tio n design in Section III-C and Section IV -B . B. T racking Refer ence Signal T ransmission Design MSCT r equires a c curate downlink chan nel parame ter es- timation to enab le stringent TFP-Sync amo ng coo perative satellites. In FDD systems, downlink CSI is typic a lly acq uired at the UE based on CSI-refer ence signals (CSI-RSs) an d f e d back via th e uplink [ 25 ]. Howe ver, th e rapid time variation of S2G ch annels calls for con tinuous hig h-ﬁdelity CSI a c- quisition an d feedb ack, which is no t adequ a tely su pported by conv entional designs. W e the refore employ the TRS, deﬁned as a set of CSI-RS resources, to suppo rt the JCAFPS frame- work. As sho wn in Fig. 2 , one typical TRS conﬁguration in freque n cy range 1 co nsists of N trs sym single-por t p eriodic CSI-RS re sources sp anning N trs slot consecutive slots [ 24 ]. The conﬁgur ation is speciﬁed by the period T trs p erio d and offset T trs oﬀset , with symbol-in dex set within one perio d d enoted by N trs sym = { p trs i } N trs sym − 1 i =0 . In the f requency dom ain, each TRS symbol o ccupies N trs = N sc N tc subcarriers, deﬁn ed by the index set N trs sc and comb num ber N tc . The temp oral and spectral density of the TRS de te r mine the maximum trackable frequen cy and timin g o ffsets. T o enable efﬁcient CSI acq uisition for MSCT , we design the TRS tr ansmission tailore d to the S2 G cha n nel char acteristics. 1) BP -Speciﬁc Bea m-Domain TRS Br oa d casting: T o com- bat severe pro pagation loss inheren t in SatCom, th e TRS is mapped to th e beam dom ain r ather than the an tenna domain . T o a void the excessi ve pilot overhea d , each satellite steers its B simultaneous beam s to the cen ters of pred eﬁned BPs and broadcasts TRSs to th e UEs with in the illumin ated BPs, as shown in Fig. 1 . Note that inter -beam interference can be effectiv ely m itigated throu g h techniqu e s such as bea m hoppin g, eliminating the reliance on UE-speciﬁc CSI. 2) TBA-Speciﬁc TRS Precompensation: The spatial reso- lution o f do wn link b e a ms is physically co n strained by the satellite antenna aper ture. With limited array sizes, the r esult- ing BP radius m a y be sufﬁciently large to indu ce noticeab le geometric disparities in pro pagation delay and Dop pler shift between th e BP center an d edge. Direct precom p ensation targeting the BP center w ould th us expose ed ge users to mu lti- satellite asynchron o us interf erence. T o circumven t this, we propo se a ﬁne- grained app roach by subd ividing each BP in to multiple TBAs an d app ly ing distinct TF preco mpensation to the transmission of each TBA, a s depicted in Fig. 1 . Since the number o f active TB As within a BP is typically only a small subset o f the entire T BA set, the TRS transmissions associated with different active TB As within a hop a re orth ogonalize d in the time d omain. TF preco mpensation is then tailore d to the center of each active TBA, ensur in g that timing and frequency offsets ar e strictly con ﬁned within the CP and th e allowable tolerance 3 [ 26 ]–[ 28 ]. Given that the signal pr ocessing logic is g eneric across TBAs, the subseq uent analy sis focuses on the ﬁr st TB A within one hop with o ut lo ss o f gener a lity . Speciﬁcally , we con sider S satellites ser ving U U E s distributed across B TB As, where each TB A is associated with a distinct BP , an d all lin ks shar e the sam e TF resource s. Remark 1 . As satellite payloa ds evolve towar d extr emely lar ge-scale an tenna a rrays, the BP radius will pr ogr e ssively shrink. In this r egime , a BP may eventually coincide with a sing le TBA, thereby eliminating the need fo r time- division multiplexing of TRS r esou r ces a c r oss TBAs. 3) AP S-TRSs for Multi-S atellite CSI Acquisition: Since a TRS is conﬁned to a sin g le antenna port [ 24 ], m ulti- satellite CSI acquisition requ ires orthog onal TF resources across satellites and th us incu rs high p ilot overhead . W e therefor e prop ose adjustab le phase-shift TRSs (APS-TRSs) by lev eraging the sp arse de lay proﬁle of S2G ch annels [ 3 ], [ 2 9 ], [ 30 ]. Speciﬁcally , all co operative satellites transm it TRSs on identical TF resource s, while b eing disting u ished by unique frequen cy-domain phase sh ifts. As a fr equency-d omain linear phase shif t is equiv a le n t to a d elay-do main o ffset, the UE can resolve individual satelli te chan nels without additional resource overhead. The pro posed APS-TRS fro m the s -th satellite to the b - th TB A o n th e p -th symbo l of th e m -th TRS transmission is given by d trs , ( m ) sbp = d iag { f trs φ s } ¯ d trs , ( m ) bp , (1) 3 W ith the normalized freq uenc y of fset ǫ deﬁned rel ati ve to ∆ f , nume rical analysi s shows that | ǫ | = 0 . 05 yields a s ignal-to-inte rference ratio of ≈ 20 dB [ 26 ], [ 27 ]. In powe r-li mited SatCom scenarios, this implies that the impact of i nter- carrier interfere nce on link performance is negligi ble for | ǫ | ≤ 0 . 05 . SUBMITTED P APER 5 where ¯ d trs , ( m ) bp ∈ C N trs , satisfying ¯ d trs , ( m ) bp ( ¯ d trs , ( m ) bp ) H = I N trs , de notes the basic TRS sequ e nce shared by all co opera- ti ve satellites fo r th e b -th TB A. f trs φ s represents the satellite- speciﬁc phase-shift vector with [ f trs φ s ] k = e − ¯  2 π kφ s N trs and φ s = ( s + 0 . 5) N trs /S . This d esign co nﬁnes the equiv alent delay-do main chan nel af ter apply ing φ s to the interval T s = [ s/ ( S N tc ∆ f ) , ( s + 1) / ( S N tc ∆ f )) . C. TRS an d CSI F eedback P eriod Design T o enab le efﬁ cient TRS-aided downlink CSI ac q uisition and fee dback without incur ring excessi ve p ilot and feedback overhead, the TRS p eriod T trs p erio d and th e feedb ack p eriod T csi - fb p erio d should be d esigned in acco rdance with th e tempor a l characteristics of S2G chan n els. I n th e fo llowing, we analyze the design criteria for pilot and feed back period s. 1) Prediction Mod el V alid ity: Owing to the d e te r ministic orbital m otion of satellites, the temporal ev olu tio n of S2G channel parame te r s can be characterized by the prediction model over d ifferent time scales, as will be detailed in Sec- tion III-A . Le t T eﬀ denote th e validity in terval within which the approx imation error of the a dopted prediction model remain s below a prescribed threshold ǫ d . T o ensure reliable pr ediction, the CSI acquisition an d feedb ack mu st be co mpleted within this validity interval, implyin g: M T trs p erio d + T csi - fb p erio d < T eﬀ . (2) This cond ition r ev eals the fundamen ta l trade-off between overhead reduction and m odel ﬁdelity . 2) Ph ase Unwrappin g Effectiveness: Coherent tran smission hinges on p h ase synchr o nization across co operative satellites. Since phase o b servations are inhere n tly cyclic, reliable pha se prediction r e quires PU acr oss successiv e TRS transmissions. T o av oid unwrapping failure, the phase r otation over one inter - TRS inte r val caused b y max imum Doppler sh if t estimation error ∆ ν max must remain below π , i.e., ∆ ν max T trs p erio d < 0 . 5 . (3) This condition guarantees that the inter-TRS pha se ev o lution remains continu ous from the perspective o f PU. I I I . M U LT I - S AT E L L I T E S I G N A L M O D E L This section establishes a time-varying S2G chan nel model, constructing a p olynom ia l mode l to capture the temp oral ev o - lution of propagation dela y s and Dop pler shifts. Sub sequently , we derive an OFDM-based m ulti-satellite sign a l mod el that accounts for n on-idea l synchronization , followed by a detailed phase analysis of equivalent TF-d omain chann els. A. Chan nel Model The baseband chann el impulse response (CIR) between the s -th satellite and the u -th UE [ 2 2 ], [ 3 1 ] is d enoted as ˜ h su ( t, τ ) = L su − 1 X l =0 α sul ( t ) e ¯ ψ sul e − ¯  2 πf c τ sul ( t ) δ ( τ − τ sul ( t )) v ( θ sul ( t )) , (4) where L su denotes the n umber of multip ath co m ponen ts. ψ sul , α sul ( t ) , and τ sul ( t ) r epresent th e or iginal ph ase, r e al-valued gain, and propag ation d elay of the l - th path. The corr esponding 0 0.2 0.4 0.6 0.8 1 Time (s) 10 -8 10 -6 10 -4 10 -2 10 0 10 2 Error of Satellite-to-UE Distance Model (m) f c = 2 GHz, h sat = 350 km, θ incl = 53 ◦ , β max = 90 ◦ N ord = 1 N ord = 2 N ord = 3 η = 10 ◦ η = 30 ◦ η = 50 ◦ 0 . 1 λ 0 . 01 λ 0 . 001 λ 0.01 0.02 0.03 0.04 0.05 0.06 10 -4 10 -3 Fig. 3. Satell ite-to -UE distance modeling error based on T aylor series expa nsion with f c = 2 GHz, r sat = 6721 km, θ inc = 53 ◦ , β max = 90 ◦ . Doppler shift at f c is denoted as ν sul ( t ) = − d τ sul ( t ) d t f c . θ sul ( t ) = [ θ ele sul ( t ) , θ azi sul ( t )] T denotes the e le vation and azimu th angles of departur e (AoDs). The steering vector is given by v ( θ sul ( t )) = v N x ( θ x sul ( t )) ⊗ v N y ( θ y sul ( t )) , (5) with v N ( x ) = [1 , e − ¯ πx , · · · , e − ¯ π ( N − 1) x ] T , θ x sul ( t ) = sin ( θ ele sul ( t )) cos ( θ azi sul ( t )) , θ y sul ( t ) = sin ( θ ele sul ( t )) sin ( θ azi sul ( t )) . Since p ropag a tio n delay and Dopp ler shift are de termined by the S2G prop agation distance and its rate of change, we next cha r acterize the tempo ral ev o lution o f the S2G distance . For brevity , th e satellite and UE indices ( s, u ) are om itted . Since S2G chann els are d ominated b y the LoS com ponent [ 3 ], we focus on the tempo ral ev olu tio n of the LoS distance for stationary UEs. As illustrated in [ 32 ], th e relative angu lar velocity between a LE O satellite and a UE remains quasi- static over sho rt time scales. This allows f or the approximation ω se ≈ ω s − ω e cos θ inc , wh e re ω s and ω e denote the angu lar velocities of the satellite an d Earth in th e Ear th-centered inertial f rame. θ inc is th e in clination o f the satellite’ s orbit. W e deﬁne t = 0 to coincide with the UE’ s max im um elev ation a ngle β max and minimu m geocen tric angle γ min . The geo c e ntric ang le traversed by the satellite is η = ω se t , and the S2 G distance is d sat 0 ( t ) = p r sat + r ue − 2 r sat r ue cos γ min cos ( ω se t ) , (6) where r sat and r ue denote the geocentric distan ces of t he satellite a nd UE. T o chara c terize ch annel e volution over a local o bservation window , we app r oximate d sat 0 ( t ) b y an N ord -th-ord er T aylor series expansion, yieldin g d sat 0 ( t + ∆ t ) ≈ P N ord n =0 d sat , ( n ) 0 ( t )(∆ t ) n /n ! , (7) where d sat , ( n ) 0 ( t ) deno tes the n - th-orde r deriv ativ e of d sat 0 ( t ) . As shown in Fig. 3 , in creasing N ord extends the valid time scale of the approxim a tion. I n par ticu lar , f or errors below 10 − 3 λ , ﬁrst-, second -, and third-o rder mod els are suitable for millisecond-, sub-secon d-, and second- lev el inter vals. Remark 2. Under the N ord -th-order ap pr oxima tio n of the S2G distance, th e pr opagatio n delay τ sat 0 ( t ) and Doppler shift ν sat 0 ( t ) can b e repr esented by time po lynomials o f o rder N ord and ( N ord − 1) , r espectively . This polynomial characterization will serve as the basis for the subsequent signal modeling and channel pr ediction design. 6 B. Sig nal Model Based on the design in Section II-A , eith er TRS or coheren t transmission calls for TB A- or UE-speciﬁc pre c o mpensation , implying p er-stream OFDM mo dulation ra th er than joint mo d - ulation of ag gregated streams. Since the signal models for T RS and coherent transmission s mainly d iffer in the transmitted sig- nal model, we ﬁrst d ev elop the TRS transmitted signal mod el. Consider the u - th UE is located in the b u -th BP/TB A. Let { x sb u pk } N/ 2 − 1 k = − N / 2 denote the spatial- frequen cy-domain symbol vectors fro m the s - th s atellite to the b u -th TB A on the p - th symbol. Af ter OFDM mod ulation, preco mpensation , and frequen cy up conv ersion, the tra n smitted ban d pass sign a l is ˜ x cps sb u ( t ) = P − 1 X p =0 N/ 2 − 1 X k = − N / 2 x sb u pk e − ¯ ϕ cps sb u pk e ¯  2 π ( f c + k ∆ f )( t + τ cps sb u p − pT sym ) · e − ¯  2 πν cps sb u p ( t + τ cps sb u p ) U tx ( t + τ cps sb u p − pT sym ) , (8) where τ cps sb u p , ν cps sb u p , and ϕ cps sb u pk are th e TFP precom pensation terms calculated based on the ephem eris and TB A positions. For TRS tr ansmission, ϕ cps sb u pk = 0 . U tx ( t ) eq uals 1 when t ∈ [ − T cp , T ) , an d 0 otherwise, accountin g for CP insertion . The coheren t-transmission case ˜ x dcps su ( t ) is obtain ed by rep lacing these TBA-speciﬁc q uantities and tra nsmitted symbo ls with the UE-speciﬁc ones, i.e., { τ dcps sup , ν dcps sup , ϕ dcps supk } an d x d supk . The re c e i ved bandpass signal is obtaine d by conv o lving the transmitted signal ( 8 ) with the CIR ( 4 ), yielding ˜ y u ( t ) = P B − 1 b =0 P S − 1 s =0 R ˜ h T su ( t, τ ) ˜ x cps sb ( t − τ ) d τ . (9) After CP removal, the p -th-symbo l segmen t is ˜ y up ( t ) = ˜ y u ( t + pT sym ) U rx ( t ) , t ∈ [0 , T ) , (10) where U rx ( t ) equals 1 when t ∈ [0 , T ) , and 0 o therwise. The received signal co nsists of an effecti ve co m ponen t and an inter-beam interfe r ence comp onent. I n wha t f ollows, we ﬁrst analyze th e effecti ve co mpone n t. Beneﬁtting from the TB A-sp e ciﬁc TRS precom pensation in Sec tion II-B , timin g and freq uency offsets are strictly con ﬁned within the CP and the allowable tolerance [ 26 ]–[ 28 ]. Accor dingly , th e effecti ve received ba seb and signa l on the p - th sym bol is g iv en by ˜ y eﬀ up ( t ) = S − 1 X s =0 L su − 1 X l =0 N/ 2 − 1 X k = − N / 2 α supl ( t ) e ¯ ψ sul e − ¯  2 π f c ( τ supl ( t ) − τ cps sb u p ) · e ¯  2 π k ∆ f ( t − ( τ supl ( t ) − τ cps sb u p )) e − ¯  2 πν cps sb u p ( t + pT sym − ( τ supl ( t ) − τ cps sb u p )) · e − ¯ ϕ cps sb u pk v T ( θ supl ( t )) x sb u pk , t ∈ [0 , T ) , (11) where α supl ( t ) = α sul ( t + pT sym ) , τ supl ( t ) = τ sul ( t + pT sym ) , and θ supl ( t ) = θ sul ( t + pT sym ) . Follo wing the an alysis of S2G distance in Section III-A , we assume that the Doppler shift, Ao Ds, an d path g ain are q uasi- static within one symbo l [ 8 ], i.e., ν supl ( t ) = ν supl , θ supl ( t ) = θ supl , and α supl ( t ) = α supl . Thus, the pro pagation delay ov er one symbol is linearized as τ supl ( t ) = τ supl − ν supl t/f c , where τ supl is the de la y at th e symbol beginnin g. Since the satellite-to-UE distance is mu ch larger than the scatterer separation [ 8 ], the AoDs across paths are nearly identical, i.e., θ supl ≈ θ sup 0 , ∀ l . W e fu rther deﬁn e th e spatial-f r equency- domain sym b ol vector as x sb u pk = w sb u p d sb u pk , where w sb u p represents th e pr ecoder of the s -th satellite for the b u -th BP . W e assume that the satellites emp loy the angle-based precod e r usin g the ephemeris and BP center positions, yielding w sb u p = √ P tx v ∗ ( θ b sb u p ) , wh e re θ b sb u p is the AoD vector from the s -th satellite to the b u -th BP center on the p -th symbo l. Sampling th e effecti ve r eceiv ed signal ˜ y eﬀ up ( t ) yields ˜ y eﬀ up [ n ] = S − 1 X s =0 L su − 1 X l =0 N/ 2 − 1 X k = − N / 2 α res supl e ¯  ˜ ψ res supl e ¯  2 π n N ∆ f ˜ ν res supl · e − ¯  2 π k N T s ˜ τ res supl e ¯  2 π nk N (1+ ν supl f c ) e − ¯ ϕ cps sb u pk d sb u pk , n ∈ N , (1 2) where N = { n ∈ Z | 0 ≤ n ≤ N − 1 } . The residual phase, residual delay , residual Doppler shift, an d equiv alent real- valued path gain are deﬁned as follows: ˜ ψ res supl = ψ sul + ψ b sb u p − ϕ cps sb u pk (13a) − 2 π (( f c − ν cps sb u p ) ˜ τ res supl + pT sym ν cps sb u p ) , ˜ τ res supl = τ supl − τ cps sb u p , (13b) ˜ ν res supl = ν supl − ν cps sb u p , (13c) α res supl = α supl α b sb u p , (13d) where ψ b sb u p and α b sb u p are the phase and array gain due to precod in g, i.e., v T ( θ sup 0 ) w sb u p = α b sb u p e ψ b sb u p . The above analysis is consistent with the results in [ 9 ], [ 1 0 ]. After OFDM demod ulation, the effecti ve received signal vector of th e u - th UE on the p -th symbo l can b e exp r essed as y eﬀ up = S − 1 X s =0 L su − 1 X l =0 ΞH supl d sb u p , (14) where Ξ ∈ B N trs × N selects the su bcarriers in N trs , i.e. , the selected colu mns form I N trs . d sb u p ∈ C N denotes the symbol vector , with [ d sb u p ] k = d sb u pk . The equivalent TF-d omain channel matrix is giv en by H supl = α res supl e ¯  ˜ ψ res supl ˜ F N Λ ν ( ˜ ν res supl ) ˇ F H N ( ν supl ) diag { f τ ( ˜ τ res supl ) } , (15) where ˜ F N is the phase-shift discrete Four ier transform matrix of size N with [ ˜ F N ] k,n = 1 √ N e − ¯  2 π ( k − N/ 2) n N . The diagon al matrix Λ ν ( ˜ ν res supl ) captures the residual Doppler s hift, with [ Λ ν ( ˜ ν res supl )] n,n = e ¯  2 π n N ∆ f ˜ ν res supl . ˇ F N ( ν supl ) diag { f τ ( ˜ τ res supl ) } incorpo rates the Dop pler shift- induced phase rotatio n , with [ ˇ F N ( ν supl )] k,n = 1 √ N e − ¯  2 π ( k − N/ 2) n N (1+ ν supl f c ) and [ f τ ( ˜ τ res supl )] k = e − ¯  2 π k − N/ 2 N T s ˜ τ res supl . Th e effectiv e r eceiv ed signal model for coher ent tr ansmission is obtain e d an alogously by replacing the TBA-speciﬁc p r ecompen sation and symbol vec- tor with the UE-speciﬁc ones, i.e., { τ dcps sup , ν dcps sup , ϕ dcps sup } a n d d d sup , yielding y deﬀ up and H d supl . In ad dition to the effectiv e compo nent, the received sign al contains inter-beam interf erence, which is given b y z int up = P b 6 = b u P S − 1 s =0 P L su − 1 l =0 z int supl,b , (16) where z int supl,b ∈ C N trs is the interf erence of the u -th UE on the p -th symb o l cau sed by the tran sm ission from the s -th satellite to the b - th ( b 6 = b u ) TB A via th e l - th pa th. Owing to the substantial d ifferential p ropag a tion delays and Doppler shifts between satellites and UEs, z int up can be mode le d as Gau ssian distribution [ 9 ], [ 33 ] with SUBMITTED P APER 7 (a) (b) (c) Fig. 4. Charact eristic analysis of equiv alent T F-domain channe ls: (a) ICI po wer of single -satell ite channel; (b) phase and magnitude varia tions of single-satell ite channe l (left) and dual-satell ite composite channel (right); (c) equali zation results ov er single-sat ellit e ch annel (left) and dual-sat ellit e composite channel (right). Parame ters: f c = 2 GHz, N = 2048 , ∆ f = 15 kHz , ν 00 = 27 kHz, ν 10 = − 20 kHz, ˜ ν res 00 = 180 Hz, ˜ ν res 10 = − 260 Hz, α res 00 = 1 , and α res 10 = 0 . 92 . variance σ 2 int ,up = E [ P b 6 = b u P S − 1 s =0 P L su − 1 l =0 ( α res supl,b ) 2 ] , where α res supl,b = α supl √ P tx | v T ( θ sup 0 ) v ∗ ( θ b sbp ) | . The interference power is determined by th e re sid u al AoDs and can be effec- ti vely sup pressed by beam -hopp ing d e sign [ 8 ]. Based on the ab ove an alysis, the received signal of the u -th UE on the p -th symbol can be summ arized as y up = y eﬀ up + z up , (17) where z up = z nse up + z int up represents the complex Gaussian noise-plus-in terferenc e vector , with indep endent and identi- cally distributed (i.i.d. ) elements CN ([ z up ] k ; 0 , σ 2 nse + σ 2 int ,up ) . z nse up is the Gaussian noise vector with variance σ 2 nse . C. Ph ase An alysis of T ime-F requency-Domain Chann els The perfo rmance of multi-satellite coheren t tr a nsmission critically depends on th e phase alignment among signals from different satellites. Since the S2G chann els are typically dominated by th e LoS componen t, we focus o n the p hase characteristics o f the eq uiv alen t T F-domain ch annel associated with the Lo S pa th . For nota tio nal simplicity , the UE and path indices ( u, l ) ar e om itted. T o facilitate the analysis, we assume that time -freque n cy precompe nsation is perfo rmed a t th e 0 -th symbol and remain s ﬁxed over the sub sequent sym bols. W e ﬁrst analyze th e impact of the m atrix Φ ν sp = ˜ F N Λ ν ( ˜ ν res sp ) ˇ F H N ( ν sp ) on the ch annel ph ase. Th e off-diagonal and diago nal elements o f Φ ν sp characterize the ICI an d the phase shift, respectively . Since th e Dopple r shif t is small in the FFT -size-scaled car rier-normalized sense, i.e., | N ν sp 2 f c | ≪ 1 , and the T B A/UE-sp e ciﬁc pre compensatio n fur th er suppr esses the residual Doppler shift such that | ˜ ν res sp ∆ f | ≪ 1 , the magn itudes of the o ff-diagonal elements of Φ ν sp are negligible com p ared with those of the d iagonal ele m ents, as shown in Fig. 4(a) . Furthermo re, given the relati vely low signal-to- noise ratio (SNR) regime o f power-limited SatCom links, the resu ltin g ICI power un der the co nsidered system paramete r s is m uch lower th an the n oise ﬂoo r, an d can theref ore be neglected . According ly , Φ ν sp can b e appro ximated as a diagonal m a trix, and its diagona l elements can be further approximated by T a ylor expansion as [ Φ ν sp ] k,k ≈ e ¯ π ( k − N 2 ) N − 1 N ν sp f c · e ¯  π N − 1 N ˜ ν res sp ∆ f . (18) According ly , the equivalent single-satellite ch a nnel on the p -th OFDM symbol can be compa c tly expressed as H sp ≈ diag { h sp } = α res sp e ¯ ψ res sp diag { f τ ( τ res sp ) } . (1 9 ) Here, the eq uiv alen t residual delay and pha se are deﬁned as τ res sp = ˜ τ res s 0 − P p p ′ =1 ν sp ′ f c T sym − N − 1 2 N ∆ f ν sp f c , (20a) ψ res sp = ˜ ψ res s 0 + 2 π P p p ′ =1 ˜ ν res sp ′ T sym + π N − 1 N ˜ ν res sp ∆ f , (20b) which in dicate that the Dop pler shif t main ly ind uces a frequen cy-depend ent lin ear p hase te r m, whereas the residu al Doppler shift contributes a frequency-ind ependen t p hase term. Moreover , due to temp oral accumulatio n, b oth terms ev o lve with tim e, an d the former f urther enla rges the phase dif ference across su b carriers. T his expression shows that the sing le- satellite ch annel phase ev olves in a relatively regu lar ma n ner over time and frequ ency , as shown in Fig. 4(b) . Howe ver, f or multi-satellite tran smission, the phase of the composite channel on th e p - th symb o l and the k -th subcarr ier becomes ∠ ( P S − 1 s =0 α res sp e − ¯  2 π ( k − N 2 )∆ f τ res sp · e ¯  ψ res sp ) , (21 ) where ∠ ( · ) is the phase- a ngle extraction o p erator . Sin ce the signals fr om different satellites gen erally experience distinct path gain s an d Do ppler-induced ph a se ev o lutions, the inter- satellite r e lati ve p hase mismatch increases join tly over time and frequ ency . Consequently , the co mposite-chan nel phase and ma g nitude no lon ger exhibit the regular structu re of the single-satellite case, but in stead becom e nonlinear T F-varying quantities, as shown in Fig. 4(b) . Su ch p hase misalign ment leads to coheren t com bining loss and degrad es the e ffective signal-to-in terference- plus-noise ratio (SINR). This effect is fur ther illu stra ted by th e equalizatio n results in Fig. 4(c) . Assum e that the demo d ulation r eference signal (DMRS) is placed on the 0 -th OFDM symbol. T he UE obtains the chan n el estimate on the 0 - th symbol from the DMRS and then perf o rms o n e-tap equalization for th e r eceiv ed sign als on the subsequen t symb o ls u sing the n earest-neigh bor hold interpolatio n method. T o isolate the effect of the channel o n the equalization results, we assume no ise-free pilot-based chann e l estimation a n d eq ualization. In the single-satellite case, the equalized symbo ls main ly undergo a regular p hase rotation , which can still be co mpensated a t th e receiver . In contra st, 8 in th e mu lti-satellite case, th e super position of m ultiple asyn- chrono usly evolving phases causes irregular a m plitude and phase d istortions, which beco me more severe over tim e ∆ t and cannot be effectiv ely rem oved b y receiver - side postpro c ess- ing. Therefo re, accurate c o herent tr ansmission requires ﬁn e- grained phase preco mpensation at th e tra n smitter , perfo rmed on a symbol- and subcarr ier-wise basis. I V . D E TA I L E D D E S I G N O F T H E J C A F P S F R A M E W O R K This section presents the algorithmic rea lization of th e propo sed JCAFPS framework. The d esign consists of thr ee stages: sing le-TRS-based ch annel param eter estimation, multi- TRS-based channel pred ic tio n, and pred icti ve CSI feedb ack with UE-speciﬁc TFP preco mpensation . A. Sin gle-TRS-Ba sed Chan nel P arameter Estimation From on e TRS tran smission, the UE estimates th e de- lay , phase, gain, a n d Doppler shift of each s atellite-to-UE link with su fﬁcient accura cy to su p port subsequen t chan nel prediction . This is ach iev ed throu gh initial ch annel estima- tion, super-resolution delay - domain parameter extraction, and coarse Doppler shift estimation. 1) In itial Chan nel Estimatio n : Combining ( 14 ), ( 1 7 ),and ( 19 ), the d e m odulated frequen cy-domain received signal of the u -th UE o n th e p -th sym b ol of the m -th TRS tr a nsmission can be ap proxim ated as y ( m ) up ≈ P S − 1 s =0 diag { d trs , ( m ) sb u p } h trs , ( m ) sup + z trs , ( m ) up , (22) where d trs , ( m ) sb u p = Ξ d ( m ) sb u p and z trs , ( m ) up = Ξ z ( m ) up denote th e TRS vector and the no ise vector of the u -th UE on the p -th symbo l of th e m -th TRS transmission. Th e equiv a lent frequen cy-domain chann el resp onse vector is h trs , ( m ) sup = F trs τ ( τ res , ( m ) sup ) Λ ψ ( ψ res , ( m ) sup ) α res , ( m ) sup , (23) where α res , ( m ) sup ∈ R L su represents th e real- valued delay - domain chan n el gain vector . F trs τ ( τ res , ( m ) sup ) ∈ C N trs × L su is the FT matrix, w ith its l -th colu mn vector f trs τ ( τ res , ( m ) supl ) = Ξf τ ( τ res , ( m ) supl ) . W e fu rther deﬁne the equiv a len t CIR as c ( m ) sup = Λ ψ ( ψ res , ( m ) sup ) α res , ( m ) sup ∈ C L su . T o elimina te the impact of basic TRS seque nce, the UE ﬁrst perfor ms least-squares (LS) estimation on y ( m ) up as h ls , ( m ) up = d iag { ( ¯ d trs , ( m ) b u p ) ∗ } y ( m ) up = F trs τ ( τ ls , ( m ) up ) c ( m ) up + z ls , ( m ) up , (24) where τ ls , ( m ) up = τ res , ( m ) up + ˜ φ u . Here, τ res , ( m ) up = [( τ res , ( m ) 0 up ) T , . . . , ( τ res , ( m ) ( S − 1) up ) T ] T denotes the stacked residua l delay vector , and ˜ φ u = [ φ 0 N sc ∆ f 1 T L 0 u , . . . , φ S − 1 N sc ∆ f 1 T L ( S − 1) u ] T represents the delay -domain of fset vector introduced by the APS-TRS. c ( m ) up = Λ ψ ( ψ res , ( m ) up ) α res , ( m ) up represents the stacked CIR, where ψ res , ( m ) up = [( ψ res , ( m ) 0 up ) T , . . . , ( ψ res , ( m ) ( S − 1) up ) T ] T , and α res , ( m ) up = [( α res , ( m ) 0 up ) T , . . . , ( α res , ( m ) ( S − 1) up ) T ] T . The equiv alen t n o ise- plus-interf e rence vector z ls , ( m ) up = diag { ( ¯ d trs , ( m ) b u p ) ∗ } z trs , ( m ) up still fo llows the i.i.d. co mplex G a u ssian distribution with CN ([ z ls , ( m ) up ] k ; 0 , σ 2 nse + σ 2 int ,up ) . Remark 3. After r emoving the impa ct of basic TRS sequence, the s atellite-speciﬁc p hase shifts { φ s } S − 1 s =0 evenly distribute the equivalent delay-domain chann el r esponses of dif fer en t satellites, ther e b y p re venting inter-satellite pilot interfer ence. 2) Cha nnel P ar a meter Estimation: T o suppor t the sub se- quent channel prediction over multiple TRS transm ission s, the UE should ﬁrst estimate the channel parameter vec- tor ζ ( m ) up = [( τ res , ( m ) up ) T , ( ψ res , ( m ) up ) T , ( α res , ( m ) up ) T ] T based o n the initial ch annel estimation h ls , ( m ) up . Accord ingly , the channel parameter estimation prob le m is formu lated as min ζ ( m ) up   h ls , ( m ) up − F trs τ ( τ ls , ( m ) up ) c ( m ) up   2 2 . (25) The above optimization problem is a delay-domain p arametric estimation task [ 34 ], [ 3 5 ]. Since the facto r matrix F trs τ ( τ ls , ( m ) up ) has a V and ermond e matrix with gen erators {{ φ gen , ( m ) supl = e − ¯  2 π N tc ∆ f τ ls , ( m ) supl } L su − 1 l =0 } S − 1 s =0 and accurate delay extraction is required to sup press leakage-in duced amplitud e and p hase distortion, we ad opt the spatial-sm o othing estimation o f sig- nal p arameters v ia rota tio nal inv ariance techniq ues(ESPRIT) algorithm [ 36 ] algorith m for supe r-resolutio n CIR estimatio n. Considering the un iqueness co ndition of matrix decom po- sition, the sp atial-smoothin g param eters ( K ss , L ss ) for the spatial-smooth in g ESPRIT algorith m m ust satisfy [ 37 ]: 1 ) K ss + L ss = N trs + 1 an d 2) min { K ss − 1 , L ss } ≥ P S − 1 s =0 L su . W ith these par ameters, spatial smoothing [ 3 8 ] is applied to the initial channe l estimate h ls , ( m ) up for H ss , ( m ) up =  Ξ ss 0 h ls , ( m ) up , Ξ ss 1 h ls , ( m ) up , . . . , Ξ ss L ss − 1 h ls , ( m ) up  , (26 ) where Ξ ss l ss = [ 0 K ss × l ss , I K ss , 0 K ss × ( L ss − 1 − l ss ) ] denotes the selection matrix for the l ss -th c olumn vecto r . Then, we p erform the tr uncated singular value deco mposi- tion (SVD) o n the smoothed chan n el matrix H ss , ( m ) up , yieldin g H ss , ( m ) up = ˇ U ( m ) up ˇ Σ ( m ) up ( ˇ V ( m ) up ) H , (27) where ˇ U ( m ) up ∈ C K ss × ¯ L u , ˇ Σ ( m ) up ∈ R ¯ L u × ¯ L u , a nd ˇ V ( m ) up ∈ C L ss × ¯ L u denote th e tru ncated lef t singular vector matrix, singular value matr ix, and righ t sing ular vector matrix, re- spectiv ely . Note that th e n umber of singu lar values ¯ L u = P S − 1 s =0 ¯ L su is deter m ined by th e r a n k of the channel matr ix, which can be estimated by the minimum descrip tion leng th criterion [ 39 ]. Given the d ominant L oS pr opagation ch arac- teristic of S2 G chan nels [ 3 ], the estimatio n of ¯ L u typically equals the n umber of coopera tive satellites. Exploiting th e V andermo n de structure of F trs τ ( τ ls , ( m ) up ) , the stan d ard spatial-sm oothing ESPRIT p rinciple imp lies that the signa l subspace satisﬁes a shif t- in variance r elation. Ac- cording ly , there exists a un ique no nsingular diagon al m a trix ˇ Φ ( m ) up = d iag { ˇ φ ( m ) up } ∈ C ¯ L u × ¯ L u such that ˇ M ( m ) up ˇ Φ ( m ) up ( ˇ M ( m ) up ) − 1 =([ ˇ U ( m ) up ] 0: K ss − 2 , : ) † [ ˇ U ( m ) up ] 1: K ss − 1 , : . (28) Eigenv alue decompo sition of ( 28 ) yields the no rmalized eige n- values, i.e. , the estimation o f generato rs ˆ φ gen , ( m ) up , fro m which the d elays ar e estimated as ˆ τ ls , ( m ) supl = − 1 2 π N tc ∆ f ∠ ˆ φ gen , ( m ) supl . (29) Based on the estimated delays, the factor matr ix can be SUBMITTED P APER 9 reconstruc ted, and the CIR can be estimated via LS cr iterion as ˆ c ( m ) up =  F trs τ ( ˆ τ ls , ( m ) up )  † h ls , ( m ) up . (30) Since th e app lied preco mpensation may vary across sym- bols, abr upt ch a nges can b e introduc e d into the residu a l delays and Doppler shifts between a d jacent sy mbols. Therefor e, the contributions induced b y adjustme nt terms must be rem oved from the delay an d phase expressions in ( 29 ) and ( 30 ), such that the resu lting estimates exhibit temp oral d y namics. W e assume that both satellites and UEs h av e the k nowledge of the ephemer is, BP c enter p ositions, and TB A center p ositions [ 22 ], [ 23 ]. Accord ing to ( 13 ), the unadjusted delay , p hase, and real- valued gain can be expressed as ˆ τ ( m ) supl = ˆ τ ls , ( m ) supl − φ s / ( N sc ∆ f ) + τ cps , ( m ) sb u p , (31a) ˆ ψ w , ( m ) supl = u nwrap { ∠ ( ˆ c ( m ) supl e ¯ ψ adj , ( m ) sb u p ) } , (31b) ˆ α ( m ) supl = | ˆ c ( m ) supl | /α b , ( m ) sb u p , (31c) where ψ adj , ( m ) sb u p = − ψ b , ( m ) sb u p + 2 π  ( f c − ν cps , ( m ) sb u p ) τ cps , ( m ) sb u p + ( mT trs p erio d + pT sym + N − 1 2 N ∆ f ) ν cps , ( m ) sb u p  denotes the phase ad- justment. unwrap {·} represents the PU op e ration, wh ich guar- antees the p hase contin u ity acro ss N sym trs TRS symb o ls. Sinc e the recip r ocal o f the TRS symb ol spac in g in one TRS typically exceeds twice the f requen cy offset [ 24 ], the Nyq uist sampling condition is fulﬁlled, wh ich elimin a tes ph ase a m biguity . In other words, the PU in ( 31 b ) can be executed withou t a ny prior k nowledge of the freque n cy offset. 3) Do ppler S hift Coa rse Estimation: Based o n the phase estimates after intra- TRS PU, we n ext obtain a coar se Doppler estimate. Over the shor t duration of one T RS, the pr o pagation delay can be well approx imated as a lin ear function of time accordin g to the validity an alysis in Section III-A . Hence, the Doppler shift is trea ted as constant within one TRS and estimated from th e ph ase ro tation acro ss TRS sym b ols. Neglecting noise, the u nadjusted p hase is expr e ssed as ˆ ψ w , ( m ) sul = M ψ [ ψ ro , ( m ) sul , 2 π ν ro , ( m ) sul ] T , (32) where ν ro , ( m ) sul and ψ ro , ( m ) sul represent th e common Doppler shift and ph ase. The m easurement matrix M ψ ∈ C N trs sym × 2 is de- ﬁned with its ( i, j ) -th element (( p trs i − p trs 0 ) T sym ) j . Therefo re, the unad justed common Doppler shift is estimated by ˆ ν ro , ( m ) sul = [( M ψ ) † ( ˆ ψ w , ( m ) sul )] 1 / 2 π . (33) B. Multi-TRS -based Chan n el Prediction Based o n the c h annel para meters estimated fro m ind ividual TRS transmissions, the UE furthe r explo its their tempor al cor- relation across mu ltiple observations to establish a p redictive channel m odel. Th e objective is to convert d iscrete histor ical estimates in to a compa ct continuo us-time representatio n f o r subsequen t feedback and transmitter-side preco mpensation . 1) Cr oss-TRS Phase Unwrapping : Although the PU in ( 31b ) ensures pha se con tin uity within o ne TRS, it does n ot resolve the ambiguity acr oss adjace n t TRS transmissions, where the inter-TRS inter val is much longer . Therefore, cross- TRS PU is re q uired. Speciﬁcally , a Doppler estimatio n error ∆ ν causes an inter-TRS phase mismatch 2 π ∆ ν T trs p erio d . Hence reliable un wrapping requir es ∆ ν T trs p erio d < 0 . 5 , which is consistent with the design criterion in Section II-C . T o improve robustness, we ﬁrst smooth th e coarse Dop pler estimates over mu ltiple TRS transmission s throug h poly nomial ﬁtting and then perf orm cro ss-TRS PU. Since the cumulative observation interval lies in the sub- seco nd regime, a second - order delay model is sufﬁcient according to the v alid ity analysis in Section III-A . A c c ording ly , n eglecting n oise, th e Doppler shift vector ˆ ν ro sul ∈ R M follows an ( N ord − 1) -th- order polynomial with coefﬁ cient vector p ro ,ν sul ∈ R N ord , wh ich is estima ted by ˆ p ro ,ν sul = ( M ν ) † ˆ ν ro sul , (34) where M ν ∈ R M × N ord denotes th e cor respondin g measure- ment matrix, with its ( m, n ) -th elem ent ( mT trs p erio d ) n . Using th e above estimated po lynomial coefﬁcients, cross-TRS PU is perfor med on the wrap ped p hases {{ ˆ ψ w , ( m ) supl } p ∈N trs sym } M − 1 m =0 to recover their tempor al evolution {{ ˆ ψ uw , ( m ) supl } p ∈N trs sym } M − 1 m =0 . Their relationsh ip is given by ˆ ψ w , ( m +1) sup trs i l + 2 π k ( m +1) sup trs i l,p = ˆ ψ uw , ( m ) supl + Z ∆ t ( m +1) p trs i ∆ t ( m ) p t T N ord − 1 ˆ p ro ,ν sul d t, (3 5) where t N ord − 1 = [1 , t, . . . , t N ord − 1 ] T . ∆ t ( m ) p trs i = ( p trs i − p trs 0 ) T sym + mT trs p erio d represents the inter val b etween the p trs i - th sym b ol of the m -th TRS transmission and the p trs 0 -th symbo l of the 0 -th TRS tr ansmission. No te that the 0 -th TRS tr a n s- mission is used as the reference, such that ˆ ψ uw , (0) sup trs i l = ˆ ψ w , (0) sup trs i l . Due to noise and estimation erro r , the comp uted value of th e integer variable k ( m +1) sup trs i l,p trs j , acco unting f or pha se wr apping, generally deviates from an exact in teger . T o reﬁne the estima tio n, the unw r apped ph ases { ˆ ψ uw , ( m ) supl } p ∈N trs sym are employed to d eriv e the integers { k ( m +1) sup trs i l,p } p ∈N trs sym , which are ideally identical. Practically , we av erage these N trs sym indepen d ent estima tes and apply integer quantization to obtain the ﬁnal estimate: ˆ k ( m +1) sup trs i l = round  P p ∈N trs sym k ( m +1) sup trs i l,p / N trs sym  , (36 ) where round {·} den otes the nearest integer rounding operator . Therefo re, the unwr apped phase for the p trs i -th symb ol of the m -th TRS tr ansmission can be recovered as ˆ ψ uw , ( m ) sup trs i l = ˆ ψ w , ( m ) sup trs i l + 2 π ˆ k ( m ) sup trs i l . (37) 2) P olynomial Coefﬁcient Estimation : Based on the un- wrapped phase and the estimated chan nel p arameters, we next determine the p olynom ial coef ﬁcients that chara cterize the temporal evolution o f phase, delay , and path gain, thereby enabling co m pact pre d ictiv e CSI repre sentation. Based on ( 20b ), the un w r apped pha se can be char acterized by the Doppler shift as ˆ ψ uw , ( m ) sup trs i l = ψ ori sul + 2 π Z ∆ t ( m ) p trs i 0 ν sul ( t ) d t + π N − 1 N ∆ f ν sul (∆ t ( m ) p trs i ) , (38) where ψ ori sul denotes th e origin a l phase at the refer ence time. ν sul ( t ) is th e Dop pler shift at time t , which is modeled as an ( N ord − 1 )-th -order po lynomial, i.e., ν sul ( t ) = t T N ord − 1 p ν sul , (39) 10 where p ν sul ∈ R N ord denotes the correspon ding polyn o- mial co efﬁcient vector . By combining ( 38 ) an d ( 39 ), the unwrapp ed phase vector ˆ ψ uw sul ∈ R N trs sym M over M TRS transmissions is expressed by an N ord -th-ord er polyn omial, and the c orrespon ding poly nomial coefﬁcient vector p ψ sul = [ ψ ori sul , ( p ν sul ) T ] T ∈ R N ord +1 is estima ted by the LS criterion : ˆ p ψ sul = ( ˜ M ψ ) † ˆ ψ uw sul , (40) where ˜ M ψ = [ 1 N trs sym M , ¯ M ψ ] ∈ R N trs sym M × ( N ord +1) denotes the cor respondin g measur ement matrix. The ( i + j N trs sym , k ) - th element of ¯ M ψ is d eﬁned as (∆ t ( j ) p trs i ) k ( 2 π k +1 ∆ t ( j ) p trs i + π N − 1 N ∆ f ) . Follo wing the same poly nomial param eterization in ( 20a ), the d elay is also deter m ined by the Doppler ev olu tion, which is expr essed as ˆ τ ( m ) sup trs i l = τ ori sul − 1 f c Z ∆ t ( m ) p trs i 0 ν sul ( t ) d t − N − 1 2 f c T s ν sul (∆ t ( m ) p trs i ) , (41) where τ ori sul denotes the origin al delay at the re ference time instant. The delay vector ˆ τ sul ∈ R N trs sym M over M TRS trans- missions corr esponds to an N ord -th-ord er polynomial with coefﬁcients p τ sul = [ τ ori sul , ( p ν sul ) T ] T , whose LS estimate is ˆ p τ sul = ( ˜ M τ ) † ˆ τ sul , (42) where the m easuremen t matrix ˜ M τ = [ 1 N trs sym M , ¯ M τ ] ∈ R N trs sym M × ( N ord +1) . Th e ( i + j N trs sym , k ) - th element of ¯ M τ deﬁned as − 1 f c (∆ t ( j ) p trs i ) k ( 1 k +1 ∆ t ( j ) p trs i + N − 1 2 T s ) . Gi ven the reduced phase estimation erro r afforded by PU, ˆ p ψ sul typically exhibits markedly highe r accu racy than ˆ p τ sul . Thus, we reuse the high-p recision c oefﬁcients [ ˆ p ψ sul ] 1: N ord , and restrict the delay estimation to determining τ ori sul with lower co mplexity as ˆ τ ori sul =([ ˜ M τ ] : , 0 ) † ( ˆ τ sul − [ ˜ M τ ] : , 1: N ord [ ˆ p ψ sul ] 1: N ord ) . (43) Like wise, regard ing the tempor al evolution of path gain o n sub-second scales, we pred o minantly co n sider gain ﬂuctuation caused by the dynamic S2G distance, while treating remaining perturb ations as noise. Thus, th e recipro cal of path gain ˆ β ( m ) sup trs i l = 1 / ˆ α ( m ) sup trs i l is also characterized by Dop pler shift as ˆ β ( m ) sup trs i l = β ori sul + c β sul Z ∆ t ( m ) p trs i 0 ν sul ( t ) d t, (44) where the con stant c β sul comprises the effect of th e carrie r frequen cy , an te n na gain, and other f actors. β ori sul represents the orig inal reciprocal of path gain at the referen c e time. Therefo re, th e recipro cal of path gain ca n be m odeled as an N ord -th-ord er polyno mial. The ensemble of M TRS-derived gain ob ser vations facilitates the estimation o f the polyn omial coefﬁcient vector p β sul = [ β ori sul , ( c β sul p ν sul ) T ] T as ˆ p β sul = ( ˜ M β ) † ˆ β sul , (45) where ˜ M β ∈ R N trs sym M × ( N ord +1) , w ith the ( i + j N trs sym , k ) - th element given by (∆ t ( j ) p trs i ) k +1 / ( k + 1) . Sim ilar ly , we can restrict the path gain estimation to determine [ β ori sul , c β sul ] T by reusing [ ˆ p ψ sul ] 1: N ord , i.e., [ ˆ β ori sul , ˆ c β sul ] T =  ˜ M β h 1 0 0 N ord [ ˆ p ψ sul ] 1: N ord i † ˆ β sul , (46) Algorithm 1: ESPRIT -Po ly -PU Algorith m 1 Input: { { y ( m ) up } p ∈N trs sym } M − 1 m =0 . 2 O ut put: { { ˆ p ψ sul , ˆ p τ sul , ˆ p β sul } ¯ L su − 1 l =0 } S − 1 s =0 , { τ dcps , ( m ) sup , ν dcps , ( m ) sup , ϕ dcps , ( m ) sup } S − 1 s =0 , { ˆ h trs , ( m ) sup } S − 1 s =0 . 3 for m = 0 , · · · M − 1 do 4 % Single-TRS -Based Ch annel Parameter Esti mation: 5 Compute h ls , ( m ) up via ( 24 ), ∀ u , p 6 Compute H ss , ( m ) up via ( 26 ), ∀ u, p 7 Compute th e gen erator ˆ φ gen , ( m ) supl via ( 27 ), ( 28 ), ∀ s, u , p, l 8 Compute ˆ τ ( m ) supl , ˆ ψ w , ( m ) supl , ˆ α ( m ) supl via ( 31 ), ∀ s, u, p, l 9 Compute ˆ ν ro , ( m ) sul via ( 33 ), ∀ s, u, l 10 end 11 % Mul ti-TRS-Based Ch annel Prediction: 12 Com pute ˆ p ro ,ν sul via ( 34 ), ∀ s, u, l 13 Com pute ˆ k ( m ) supl , ˆ ψ uw , ( m ) supl via ( 36 ), ( 37 ), ∀ s, u, p, l , m 14 Com pute ˆ p ψ sul , ˆ p τ sul , ˆ p β sul via ( 40 ), ( 42 ), ( 4 5 ), ∀ s, u, l 15 % CSI Feedback and UE-Speciﬁ c TFP Precompensation: 16 Feed back ˆ p ψ sul , ˆ p τ sul , ˆ p β sul , ∀ s, u , l 17 Pred ict ˆ ψ uw , ( m ) supl , ˆ ν ( m ) supl , ˆ τ ( m ) supl , ˆ α ( m ) supl via ( 38 ), ( 39 ), ( 41 ), ( 44 ), ∀ s, u, p , l 18 Pred ict ˆ h trs , ( m ) sup via ( 23 ), ∀ s, u, p 19 TFP p recompen sation fo r co herent tr a nsmission: τ dcps , ( m ) sup , ν dcps , ( m ) sup , ϕ dcps , ( m ) supk via ( 47 ), ∀ s, u, p, k C. CSI F eedback and UE-S peciﬁc TFP Precompensation Based o n the polynomial coefﬁcients obtained from chann el prediction , the UE perio dically feeds back compac t p r edictive CSI { { ˆ p ψ sul , ˆ p τ sul , ˆ p β sul } ¯ L su − 1 l =0 } S − 1 s =0 to the coope r ativ e satel- lites. Using these coef ﬁcien ts, satellites c an reconstru ct the future ev olu tion of the ph ases ˆ ψ uw , ( m ) supl , Doppler shif ts ˆ ν ( m ) supl , delays ˆ τ ( m ) supl , path gains ˆ α ( m ) supl , and equiv alent freq uency- domain channel response vector ˆ h trs , ( m ) sup throug hout the va- lidity in terval via ( 38 ), ( 39 ), ( 41 ), ( 44 ), and ( 23 ). This predictive cap ability em powers the satellites to exe- cute UE-sp eciﬁc TFP prec ompensation for downlink coherent transmission as follows τ dcps , ( m ) sup = ˆ τ ( m ) sup 0 , ν dcps , ( m ) sup = ˆ ν ( m ) sup 0 , ϕ dcps , ( m ) supk = ∠ ˆ h trs , ( m ) supk , (47) such that the signals fr om different satellites remain aligne d for c o herent transmission. In this way , predicti ve feed back serves as the bridge between UE-side channel pred iction and transmitter-side p recompen sation. T o co nclude, the com - prehen sive JCAFPS framework term ed ESPRIT -Poly-PU is summarized in Alg orithm 1 . Remark 4. Repo rting full set {{ ˆ p ψ sul , ˆ p τ sul , ˆ p β sul } ¯ L su − 1 l =0 } S − 1 s =0 would incur a feedba ck load o f 3( N ord + 1) P S − 1 s =0 ¯ L su co- efﬁcients. Since these parameters’ evolution share the same Doppler-r elated p olynomial coe fﬁcients, the UE need o nly r epo rt compressed set { { ˆ p ψ sul , ˆ τ ori sul , ˆ β ori sul , ˆ c β sul } ¯ L su − 1 l =0 } S − 1 s =0 , r e- ducing the overhead to ( N ord + 4) P S − 1 s =0 ¯ L su coefﬁcients while pr eserving the informa tion needed for TFP-Sync in MS CT . SUBMITTED P APER 11 Per-Beam Transmit Power (dBW) (a) 4 8 12 16 NMSE (dB) -18 -15 -12 -9 -6 -3 0 ESPRIT-Poly-PU OMP-Poly-PU ESPRIT-AR-PU ESPRIT-Poly-woPU 4 8 12 16 Per-Beam Transmit Power (dBW) (b) 0 0.05 0.1 0.15 0.2 TEE (Sampling Interval) ESPRIT-Poly-PU OMP-Poly-PU ESPRIT-AR-PU ESPRIT-Poly-woPU Per-Beam Transmit Power (dBW) (c) 4 8 12 16 FEE (Hz) 0 10 20 30 40 ESPRIT-Poly-PU OMP-Poly-PU ESPRIT-AR-PU ESPRIT-Poly-woPU Per-Beam Transmit Power (dBW) (d) 4 8 12 16 PEE (deg) 0 10 20 30 40 50 60 70 80 90 ESPRIT-Poly-PU OMP-Poly-PU ESPRIT-AR-PU ESPRIT-Poly-woPU Per-Beam Transmit Power (dBW) (e) 4 8 12 16 SINR (dB) 0 3 6 9 12 15 18 ESPRIT-Poly-PU OMP-Poly-PU ESPRIT-AR-PU ESPRIT-Poly-woPU DualSat-woPhaseCps SingleSat Fig. 5. Per-beam po wer P b eam vs (a) NMSE, (b) TEE, (c) FEE , (d) PEE, and (e) SINR. T pred = 80 ms, M = 12 , T trs p eriod = 20 ms. V . S I M U L AT I O N R E S U LT S In th is section , we p r esent simula tio n re su lts to illustra te the perfor mance of the pro posed f ramework. The scenario and chan nel pa rameters are gener a ted using the Qu aDRiGa channel simulator, which implem ents a 3 -D g eometry- based stochastic chann el mo del [ 3 ], [ 40 ]–[ 4 2 ]. Recog n ized by 3rd generation partne rship pro ject (3GPP) as a feature- complete simulator, Qu aDRiGa’ s satellite-r elated modeling co mplies with 3GPP-NTN d ocumen ts and has been validated via various ﬁeld tests [ 40 ], [ 41 ]. Unless otherwise speciﬁed , the default system parameter s of simulation s are listed in T able I . In each Monte Carlo trial, a ref erence po int is ra n domly gene rated within the constellation’ s service area. The S satellites nearest to this re f erence point co nstitute the coo perative cluster . UEs are ran domly distributed within the overlapping cov erage area of these satellites. Regarding the b e a m hop ping strategy , the pattern is determined by selectin g B BPs subject to a steering vector q uasi-ortho gonality con straint [ 8 ]. Besi des, the BP radius is d eﬁned based on the 3 d B beam footprint at the nadir . A. P erformance Metric an d B aselines The per forman c e of the propo sed fram ework is examin ed across c h annel pred iction accuracy , TFP-Sync pr ecision, and downlink co herent transmission efﬁciency . T h e re le vant per- forman ce metrics are deﬁned as follows: • CSI acqu isition n ormalized mean squ a r e err or (NMSE) : NMSE ( m ) p = P U − 1 u =0 P S − 1 s =0 k h trs , ( m ) sup − ˆ h trs , ( m ) sup k 2 2 P U − 1 u =0 P S − 1 s =0 k h trs , ( m ) sup k 2 2 . (48) • Timing offset estimation err o r (TE E), f r equency offset estimation er r or (FEE), and phase estimation err o r (PEE): TEE ( m ) p = P S − 1 s =0 P U − 1 u =0 | τ ( m ) sup 0 − τ dcps , ( m ) sup | S U , (49a) FEE ( m ) p = P S − 1 s =0 P U − 1 u =0 | ν ( m ) sup 0 − ν dcps , ( m ) sup | S U , (49b) PEE ( m ) p = P S − 1 s =0 P U − 1 u =0 k ∠ ( h trs , ( m ) sup ⊙ ( ˆ h trs , ( m ) sup ) ∗ ) k 1 S U N . (49c) • SINR for coh erent tran smission: SINR ( m ) p = P U − 1 u =0 k y deﬀ , ( m ) up k 2 2 U ( k z int , ( m ) up k 2 2 + k z nse , ( m ) up k 2 2 ) . (50) T o demo nstrate the super iority of the p roposed sche me, we compare it against the following baselines: • OMP-Poly-PU : Utilizes the orthogo nal ma tch ing pur- suit ( OMP)-based alg o rithm with a unifor m dictionar y [ 43 ] for single- TRS-based channe l par ameter estimatio n, followed by cross-TRS PU and p olynom ial ﬁtting for channel prediction in Sec tio n IV - B . T ABLE I B A S I C S Y S T E M P A R A M E T E R S [ 3 ] , [ 9 ] , [ 2 2 ] , [ 4 1 ] System Parameters V alu e Carrier frequency f c 2 GHz FFT si ze N 2048 CP length N cp 144 Subcarrier spacing ∆ f 15 kHz Number of used subcarriers N sc 1632 Number of TRS transmissions M 12 Period of TRS T trs period 20 ms Prediction duration T pred 80 ms Number of symbols in one TRS N trs sym 4 Symbol indices of one TRS N trs sym { 4 , 9 , 18 , 23 } Number of slots in one TRS N trs slot 2 Number of transmission combs N tc 4 Number of cooperati ve satellites S 2 Per-antenna gain of satellites 0 dBi Number of beams per satellite B 10 Per-beam po wer of satelli tes P b eam 8 dBW Number of satellite antennas [ N x , N y ] [32 , 32] Gain-to-noise-temperature of UEs − 33 . 6 dB/K Radius of BPs 8 . 5 km Radius of TBAs 1 km Channel model NTN Urban LOS Other loss 4 dB Constellation T ype W alk er-Delta Orbital alt itude h sat 350 km Orbital P lanes 110 Satellites P er Plane 60 Inclination Angle 53 ◦ • ESPRIT -AR-P U : Utilizes the ESPRIT algo rithm for single-TRS-based channel pa r ameter estimation , fo llowed by cross-TRS PU in Section IV -B and au toregressiv e (AR) m o deling [ 44 ] fo r ch annel predictio n. • ESPRIT -Poly-woPU : Utilizes the ESPRIT algor ithm for single-TRS-based channel pa r ameter estimation , fo llowed by polynomial ﬁtting for channel prediction in Sec- tion IV -B , without cross-TRS PU. The baseline s are designed to isolate different compon ents of the p r oposed fram ew o rk. Speciﬁcally , OMP-Poly - PU ev aluates the impact of single-TRS-b a sed chan n el param eter extrac- tion, ESPRIT -AR-PU assesses the adopted tem poral ev olution model, and ESPRIT -Po ly-woPU highlights the role of cro ss- TRS PU. Th us, each b aseline differs fr om ESPRIT -Poly -PU in one key co m ponen t, enab ling a n interp r etable compar ison aligned with the algo rithmic stru cture in Sec tio n IV . I n terms of SINR, the evaluation extends beyon d th e aforeme ntioned baselines to in clude sing le-satellite transmission and d u al- satellite co operative transmission without phase preco mpen- sation, denoted as SingleSat and DualSat-woPhaseCps . 12 20 40 60 80 100 120 140 160 Prediction Duration (ms) (a) -20 -15 -10 -5 0 5 NMSE (dB) ESPRIT-Poly-PU OMP-Poly-PU ESPRIT-AR-PU ESPRIT-Poly-woPU 20 40 60 80 100 120 140 160 Prediction Duration (ms) (b) 0 0.05 0.1 0.15 0.2 TEE (Sampling Interval) ESPRIT-Poly-PU OMP-Poly-PU ESPRIT-AR-PU ESPRIT-Poly-woPU 20 40 60 80 100 120 140 160 Prediction Duration (ms) (c) 0 5 10 15 20 25 30 35 FEE (Hz) ESPRIT-Poly-PU OMP-Poly-PU ESPRIT-AR-PU ESPRIT-Poly-woPU 20 40 60 80 100 120 140 160 Prediction Duration (ms) (d) 0 10 20 30 40 50 60 70 80 90 PEE (deg) ESPRIT-Poly-PU OMP-Poly-PU ESPRIT-AR-PU ESPRIT-Poly-woPU 20 40 60 80 100 120 140 160 Prediction Duration (ms) (e) 3 4 5 6 7 8 9 10 SINR (dB) ESPRIT-Poly-PU OMP-Poly-PU ESPRIT-AR-PU ESPRIT-Poly-woPU DualSat-woPhaseCps SingleSat Fig. 6. Prediction duratio n T pred vs (a) NMSE, (b) TEE, (c) FEE , (d) PEE, (e) SINR. P b eam = 8 dBW , M = 12 , T trs p eriod = 20 ms. TRS Period (ms) (a) 10 20 30 40 NMSE (dB) -18 -15 -12 -9 -6 -3 0 ESPRIT-Poly-PU OMP-Poly-PU ESPRIT-AR-PU ESPRIT-Poly-woPU TRS Period (ms) (b) 10 20 30 40 TEE (Sampling Interval) 0 0.03 0.06 0.09 0.12 0.15 0.18 ESPRIT-Poly-PU OMP-Poly-PU ESPRIT-AR-PU ESPRIT-Poly-woPU TRS Period (ms) (c) 10 20 30 40 FEE (Hz) 0 10 20 30 40 ESPRIT-Poly-PU OMP-Poly-PU ESPRIT-AR-PU ESPRIT-Poly-woPU TRS Period (ms) (d) 10 20 30 40 PEE (deg) 0 10 20 30 40 50 60 70 80 90 ESPRIT-Poly-PU OMP-Poly-PU ESPRIT-AR-PU ESPRIT-Poly-woPU TRS Period (ms) (e) 10 20 30 40 SINR (dB) 3 4 5 6 7 8 9 10 ESPRIT-Poly-PU OMP-Poly-PU ESPRIT-AR-PU ESPRIT-Poly-woPU DualSat-woPhaseCps SingleSat Fig. 7. TRS period T trs p eriod vs (a) NMSE, (b) TEE, (c) FEE , (d) PEE , (e) SINR. T pred = 80 ms, P b eam = 8 dBW , M T trs p eriod = 240 ms. B. Effect o f P er -Beam P ower Fig. 5 pr esents the perfo rmance of the prop osed and baseline schemes as a fun c tio n of per -beam transmit power . Across the entire power regime, the pro posed sche me exhibits superio r perfor mance in CSI acquisition, TFP-Sync, and downlink coheren t transmission , co nsistently outper f orming the bench- marks. The gap to OMP-Poly-PU co n ﬁrms th at limited delay resolution cau ses leakag e-induce d estimation distortion, which propag ates to synchronization errors. Th e perform ance loss of ESPRIT -Poly - woPU fu rther shows that a c c urate single-T RS estimation alone is in sufﬁcient without cross-TRS p hase un- wrapping . In addition, th e co nsistent advantage over ESPRIT - AR-PU veriﬁes that th e polyn omial mo del better captures th e channel ev olution in the consider e d scenario. As a r esult, the propo sed sche m e enables d ual-satellite co herent transmission to approach the theoretical 6 dB and 3 dB SINR gains relativ e to Sing leSat a n d DualSat-woPhaseCps in Fig. 5(e) , respectively . C. Effect of Pr ediction Du ration Giv en the trade-off between the prediction duration and feedback overhead , we examine the ev olution of key perfor- mance metrics over the p r ediction d u ration in Fig. 6 . Ex cept for ESPRIT -Poly- woPU, all p redictive scheme s degrade mono - tonically as the p rediction duration T pred increases, whereas the propo sed metho d shows the slowest degradatio n by vir tue of the high ﬁde lity of the po lynomial model in captur ing channel temporal dynamics. In particular , at T pred = 1 6 0 ms, the pro p osed scheme ma intains robust per forman ce with NMSE b elow − 10 dB, T EE b elow 0 . 0 1 T s , FEE b elow 2 H z , and PEE below 25 ◦ , co rrespond ing to a marginal SINR loss of appro x imately 6% . OMP-Poly -PU follows a similar trend but re mains limited b y estimation leakage, while ESPRIT - AR-PU suffers from recursive error pro p agation and its SINR collapses to the DualSat-woPhaseCps ﬂoor once T pred reaches 120 ms, a s illustrated in Fig. 6(e) . These results conﬁrm th at the polyn o mial-based desig n is more suitable fo r long-hor izon predictive syn chroniza tion. D. Effect o f TRS P eriod Optimizing the trad e-off between perform a nce and pilot overhead co n stitutes a piv o tal aspect of system design. In Fig. 7 , we characterize the performa n ce of various sch e m es as a fun ction of the TRS period . It is obser ved th at schemes incor- porating PU exhibit a perf ormanc e decline as T trs p erio d extends, whereas ESPRIT -Poly- woPU displays a consistently poor per- forman ce ﬂoo r irrespectiv e of T trs p erio d , owing to its inability to recover p hase infor mation. This perfo rmance degradation can be attributed to two factors. First, enlarging the TRS p eriod reduces the tem poral density of chann el observations, thereb y increasing the estimation error of the polyno mial coefﬁcients. Second, reliable phase unwrapp ing req uires the TRS period to remain below a critical thr eshold, which is deter m ined b y the achiev ab le fr equency o ffset estimation acc uracy , as discussed in Section II-C . OMP-based baseline fails beyond 20 ms d u e to insufﬁcient freq uency offset estimation ac curacy fo r reliab le phase unwrapping, which causes severe SINR degradation. By compariso n , the prop osed scheme is marked ly mo r e robust and still ach iev es at T trs p erio d = 40 ms an SINR com parable to th at of th e best baseline at 2 0 ms. This indicates tha t the propo sed design can ef fectively h a lve the pilot ov erhead while preserving sup erior co herent tr ansmission perfor m ance. V I . C O N C L U S I O N This paper investigated th e stringen t TFP-Syn c challenge in MSCT and developed a closed-loo p JCAFPS framework that jointly integrates CSI acquisition, predictiv e f eedback, and UE-speciﬁc pr ecompen sation. By exploiting deterministic orbital mo tion a n d dom inant LoS propagation , we estab- lished a poly nomial channel evolution model and showed that the comp o site mu lti- satellite channe l exhibits nonline a r time- frequen cy-varying phase behavior , which fundamenta lly re- quires sym bol- an d subcar rier-wise pha se pr e compensatio n for coheren t transmission. Building on th is insight, the proposed framework enables p ractical pred ictiv e syn chroniza tion for MSCT with reliab le CSI ac q uisition and feed back. Num e r ical results veriﬁed that the pro posed fram ew o rk enables re liab le SUBMITTED P APER 13 CSI acquisition a n d strin gent TFP-Sy nc, thereby allowing dual-satellite DP to n early attain the theo retical 6 dB g a in over single- satellite tran smission, with strong robustness to long predictio n duration s a nd large TRS perio ds. R E F E R E N C E S [1] K. Ntontin, E . Lagunas, J. Querol, J. ur Rehman, J. Grotz, S. Chatz inotas, and B. Ottersten , “ A vision, survey , and roadmap towa rd spac e commu- nicat ions in the 6G a nd be yond era, ” Pr oc. IEEE , 2025. [2] W . W ang, Y . Zhu, Y . W ang, R. Ding, and S. Chatzinota s, “T owa rd mobile satell ite internet: T he fundamenta l limitation of wireless transmission and ena bling technol ogies, ” E nginee ring , 2025. [3] Study on New Radio (NR) to support non-terre s trial networks, V e rsion 15.4.0 . Standard 3GPP T . R. 38.811, Sep. 2020. [4] Solutions for NR to support non-terr estrial networks (NTN), V er sion 16.2.0 . Standard 3GPP T . R. 38.821, Sep. 2023. [5] J. Heo, S. Sung, H. Lee, I. Hwang, and D. Hong, “MIMO satellite communicat ion systems: A surve y from the PHY layer perspecti ve, ” IEEE Commun. Surveys T uts. , v ol. 25 , no. 3, pp. 1543–1570, 2023. [6] H. Al-Hraisha w i, H. Chougra ni, S. Kisselef f, E . Lagunas, and S. Chatzi notas, “ A survey on nongeostationa ry satellit e systems: The communicat ion perspecti ve, ” IEEE Commun. Survey s T uts. , vol. 25, no. 1, pp. 101–132, 2022. [7] Y . W ang, H. Hou, X. Y i, W . W ang, and S. Jin, “T owa rd uniﬁed AI m odels for MU-MIMO communications: A tensor equiv ariance frame work, ” IEEE T rans. W ire less Commun. , vol. 24, no. 12, pp. 10 517– 10 533, 2025. [8] L. Y ou, K.-X. Li, J. W ang, X. Gao, X. -G. Xia, and B. Otterste n, “Massi ve MIMO transmission for LEO satelli te communi cation s, ” IEEE J. Select. Area s Commun. , vol. 38, no. 8, pp. 1851 –1865, 2020. [9] Y . W ang, V . N. Ha, K. Ntontin, H. Y a n, W . W ang, S. Chatzinot as, and B. Ottersten, “Statistic al CSI-based distribu ted precoding design for OFDM-cooperat i ve m ulti-sa telli te systems, ” IEEE J. Sel. A reas Commun. , 2026 , Early Acce ss. [10] Y . W ang, Y . Zhu, V . N. Ha, W . W ang, R. Ding, S. Chatzinota s, and B. Ottersten, “Multi-sa telli te mult i-strea m beamspace m assi ve MIMO transmission, ” 2025. [Online]. A vail able: https:/ /arxi v .org/abs/2512.21998 [11] M. Y . Abdelsadek , G. K. Kurt, and H. Y a nikomer oglu, “Distribute d massi ve MIMO for LEO satellite networks, ” IEEE Open J. Commun. Soc. , vol . 3, pp. 2162–2177, 2022. [12] Z. Xu, G. Chen, R. Fernandez, Y . Gao, and R. T afazolli , “Enhanc ement of direct LEO satellite -to-smartpho ne communications by distribute d beamforming, ” IEEE T rans. V eh. T ec hnol. , vol. 73, no. 8, pp. 11 543– 11 555, 2024. [13] S. W u, Y . W a ng, G. Sun, W . W ang, J . W ang, and B . Ottersten, “Distrib uted beamforming for multiple LEO satelli tes with imperfect delay and Doppler compensations: Modeling and rate analysis, ” IEEE T rans. V eh. T e chno l. , vol. 74 , no. 9, pp. 14 978–14 984, 2025. [14] Y . W ang, X. Xu, Y . Z hu, W . W ang, R. Ding, S. Chatzinota s, and B. Ottersten, “Multi -LEO satellite cooperati ve transmission: A spatial- temporal -frequenc y perspec ti ve, ” IE EE W irel ess Commun. Mag . , 2026, Early Acc ess. [15] Y . Zhang, A. Liu, P . L i, and S. Jiang, “Deep learn ing (DL)-based channel predict ion and hybrid beamforming for LE O satellite massive MIMO system, ” IEE E Internet Things J , vol. 9, no. 23, pp. 23 705–23 715, 2022. [16] M. Y ing, X. Chen, Q. Qi, and W . Gerstacke r , “De ep learning -based joint channe l prediction and multibe am precoding for LEO satellite Interne t of T hings, ” IEEE T rans. W irel ess Commun. , vol. 23, no. 10, pp. 13 946– 13 960, 2024. [17] P . Y ue, J. Du, R. Zhang, H. Ding, S. W ang, and J. An, “Collabora ti ve LEO satellit es for secure and green Internet of remote things, ” IEE E Internet Things J , vo l. 10, no. 11, pp. 9283–92 94, 2022. [18] Q. Ouyang, Z. Qu, and Y . Gao, “ A nove l distribu ted beamforming scheme based on phase adjustmen t and dynamic tracki ng for LEO satell ite communication s, ” IEEE T rans. V eh. T ec hnol. , vol. 74, no. 6, pp. 9391 –9403, 2025. [19] X. Chen and Z . Luo, “ Asynchrono us interferenc e m itigat ion for LEO multi-sat ellit e cooperat i ve systems, ” IE EE T rans. W irel ess Commun. , vol. 23, no. 10, pp. 14 956–14 971, 2024. [20] F . Zangenehnej ad and Y . Gao, “GNSS smartphones positioni ng: Ad- v ances, challeng es, oppor tunitie s , and future perspecti ves, ” Sat ellit e navigat ion , vol. 2, no. 1, p. 24, 2021. [21] Z. Lin, Z. Ni, L. Kuang , C. Jiang, and Z. Huang, “Multi-satel lite beam hopping based on load bala ncing and interf erence av oidance for NGSO satell ite communicat ion systems, ” IEEE T rans. Commun. , vol. 71, no. 1, pp. 282– 295, 2022. [22] Z. Xia ng, X. Gao, K.-X. Li, and X.-G. Xia , “Massi ve MIMO do wnlink transmission for multiple LEO satelli te communicati on, ” IEEE T rans. Commun. , v ol. 72, no. 6, pp. 3352–3364, 2024. [23] S. Ch en, S. S un, and S. Kan g, “System inte gration of te rrestria l mobile communicat ion and satell ite communicat ion—the trends, challe nges and ke y tech nologie s in B5G and 6G, ” China Commun. , vol. 17, no. 12, pp. 156–171, 2021 . [24] NR, P hysical layer proce dur es for data, V er sion 15.8.0 . Stan dard 3GPP T .S. 38.214, Dec. 2019. [25] NR, P hysical channel s and modulation, V ersion 16.6.0 . Sta ndard 3GPP T .S. 38.211, Jun. 2021. [26] P . H. Moose, “ A techniq ue for orthogonal frequency divi sion multiplex- ing frequen cy of fset correc tion, ” IE EE T rans. Commun. , vol. 42, no. 10, pp. 2908 –2914, 2002. [27] J. Armstrong, “ Analysis of new and exist ing m ethods of reducing interc arrier interfe rence due to carrier frequenc y offse t in OF D M, ” IEE E T rans. Commun . , vol . 47, no. 3, pp. 365–369, 2002 . [28] NR, User Equipment (UE) radio transmission and re cepti on, P art 1: Range 1 Standal one, V er sion 16.3.0 . Standard 3GPP T .S. 38.101-1, Mar . 2020. [29] F . P . Fontan, M. V ´ azquez -Castro, C. E. Cabado, J. P . Garcia, and E. Kubista , “Statistica l m odeling of the LMS channel, ” IEEE T rans. V e h. T echnol. , v ol. 50, no . 6, pp. 1549–1567, 2001. [30] B. R. V ojcic, R. L. Pickholtz, and L. B. Milstein, “Performance of DS- CDMA with imperfec t power control operat ing ov er a lo w earth orbiting satell ite link, ” IEEE J . Sel ect. Ar eas Commun. , vol. 12, no. 4, pp. 560– 567, 200 2. [31] Y . Zhu, J. Zhuang, G. Sun, H. Hou, L. Y ou, and W . W ang, “Joint channe l estimati on and prediction for massive MIMO with frequency hopping sounding, ” IEEE T rans. Commun. , vol. 73, no. 7, pp. 5139–5154, 2025. [32] I. Ali, N. Al-Dhahir , and J. E. Hershey , “Doppler characteriz ation for LEO s atellites, ” IEEE T rans. Commun. , vol. 46, no. 3, pp. 309–313, 2002. [33] C. Geng, N. Naderial izade h, A. S. A vestimeh r , and S. A. J afa r , “On the optimality of treating interference as noise, ” IEEE T rans. Inform. Theory , v ol. 61, no. 4, pp. 17 53–1767, 2015. [34] Y . Z hu, G. Sun, W . W ang, L. Y ou, F . W ei, L . W ang, and Y . Chen, “OFDM-based massi ve grant-free transmission over frequenc y-selecti ve fadi ng channels, ” IEEE T rans. Commun. , vol. 70, no. 7, pp. 4543–4558, 2022. [35] C. Wu, X. Y i, Y . Zhu, W . W ang, L. Y ou, and X. Gao, “Channel prediction in high-mobility m assi ve MIMO: From s patio-t emporal autore gression to deep learning, ” IEEE J. Select. Areas Commun. , vol. 39, no. 7, pp. 1915–1930, Jul. 2021. [36] R. Roy and T . Kailat h, “ESPRIT-estimation of signal paramete rs via rotati onal in varian ce techniques, ” IE EE T rans. A coust., Speec h, Signal Pr ocessing , v ol. 37, no. 7, pp. 984–995, 1989. [37] J. B. Kruskal, “Three-way arrays: rank and uniqueness of triline ar decomposit ions, with applica tion to arithmet ic comple xity and statistics, ” Linear al gebr a and its ap plicat ions , vol. 18, no. 2, pp. 95–138, 1977. [38] J. Pan, M. Sun, Y . W ang, and X. Zhang, “ An enhance d s patial smoothing techni que with ESPRIT algorithm for direction of arri val estimation in coheren t scenarios, ” IEEE T rans. Signal Proc essing , vol. 68, pp. 3635– 3643, 2020. [39] P . D. Gr ¨ unw ald, I. J. Myung, and M. A. Pitt, Advances in minimum descripti on length: Theory and ap plicat ions . MIT press, 200 5. [40] F . Burkhardt, S. Jaeck el, E . Eberlein, and R. Prieto-Cerdei ra, “QuaDRiGa : A MIMO channel model for land mobile satellite, ” in P r oc. 8th Eur . Conf . Antennas Propa g. IEEE, 2014, pp. 1274–1278. [41] S. Jaeckel , L. Raschk o wski, and L. Thieley , “ A 5G-NR satell ite ex tension for the QuaDRiGa channel model, ” in Pro c. Jo int Eur . Conf. Netw . Commun. 6G Summit . IEEE , 20 22, pp. 142 –147. [42] Study on channel model for freq uencie s from 0.5 to 100 GHz, V ersi on 16.1.0 . document 3GPP T .R. 38.901, Dec . 2019. [43] J.-F . Determe, J. Louveaux, L. Jacques, and F . Horlin, “On the exac t reco very conditio n of simultaneous orthogonal matc hing pursuit, ” IEEE Signal Pr ocessing Lett . , vol. 23, no. 1, pp. 164–168, 2015. [44] C. Lv , J.-C. Lin, and Z . Y ang, “Channel predicti on for millimet er wa ve MIMO-OFDM communica tions in rapidly time-v arying frequenc y- select i ve fading chann els, ” IEE E Access , vol . 7, pp. 15 183–15 195, 2019.

Toward Multi-Satellite Cooperative Transmission: A Joint Framework for CSI Acquisition, Feedback, and Phase Synchronization

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