Uplink Networked Sensing via Multiuser Correlation Exploitation

1 Uplink Netw orked Sensing via Multiuser Correlati on Ex p loitation Jingying Bao, J. An drew Zhang, Se nior Member , IEEE , Kai W u, Member , IEEE , Christos Masouros, F ellow , IEEE , and Y . J ay Guo, F ellow , IEEE Abstract —In this cor respondence, we in vestigate network ed sensing in perceptiv e mobile networks und er a bistatic multi- transmitter single-receiv er upli n k topolog y , where multiple user equipments (UEs) transmit signals ov er orthogonal fr equency- division multi ple access (OFDMA ) resources and a single base station performs joint sensing. Uplink clock asynchronism in- troduces offsets that destroy inter-pack et coherence and hinder high-resolution sensing, whi le multi-u ser observations exhib i t exploitable cross-user correla tion. W e theref ore formulate an asynchronous multi-user uplink OF DMA sensing model and exploit common delay-cluster sparsity across UEs. A line-of- sight (LoS)-referenced calibration first sup presses the offsets, after which a shar ed-private delay-domain sparse Bayesian learning (SBL) model is used f or delay support reco very and user grouping. Doppler and angle of arriv al are then estimated from temporal and spatial p hase differences. Simulation results show that the proposed scheme outperforms per -user processing, particularly under limited subcarrier b udgets and in low signal- to-noise ratio (S NR) regimes. Index T erms —Netw orked sensing, perceptive mobile network (PMN), multi-user OFDMA, sparse B ayesia n learning (SBL), clock asynchronism. I . I N T RO D U C T I O N Networked sensing is a key capability of perceptive m o bile networks (PMNs) for in tegrated sensing and commun ication (ISA C), wh ere distrib uted connected n o des co operate to ac- quire environmental inf ormation at scale [1]. By leveraging multi-persp ectiv e observations across geo graphica lly separated transceivers (e.g., multi-site/coope r ativ e architectu res), net- worked sensing can im prove sen sin g coverage, r obustness, a nd estimation accuracy over isolated sen sing [2 ], [3] . Howe ver , realizing such gains typica lly re q uires effectiv e coord ination and multi-n ode f usion under practical network constraints, posing implem entation ch a llenges in PMN/ISA C [ 4]. Existing studies o n n etworked sensing in PMNs/ISA C mainly focus on multi-site cooperation or multi-receiver ar- chitectures, where sensing gains a r e achie ved by combining observations f rom distributed sites/base station s (BSs) [5], [6]. For example, Behdad et al. in [5] investigated m ulti-static target detection in cell-fr e e massive multiple-inp ut multiple- output via coo peration among distributed access points, while Li et al. in [6] stud ied cellular multistatic ISA C f or seamless J. Y . Bao, J. A. Z hang, K. W u and Y . J. Guo are with the Global Big Data T echnologies Centr e, Uni versity of T echnology Sydney , Syd- ney , NSW 2007, Australia (e-mail: Jingying .Bao@student .uts.edu.au; an- dre w .zhang@uts.edu.au; kai .wu@uts.edu.au; ja y .guo@uts.edu.au). C. Masouros is with the Department of Electroni c and Electri cal En- gineeri ng, Uni ve rsity Coll ege London, London WC1E 7JE , U. K. (e-mail: c.masouros@ucl.a c.uk) sensing coverage via coope ration among spatially separ ated nodes. Howe ver , realizing such cooperation in practice ty pi- cally requires tight in te r-site coordin ation, includ ing accurate synchro n ization an d coor d inated interference man agement, thereby in curring considerab le sy stem c o mplexity and signal- ing overhead. T o this en d , we consider a novel bistatic multi- tr ansmitter single-receiver (multi-Tx/single- Rx) topolog y for u p link net- worked sensing, where multiple user equip ments (UEs) trans- mit to one BS, co nsistent with most practica l commu nication networks, includ in g cellular and Wi-Fi uplinks. Compar ed with multi-site coo perative sensing , this parad igm-shiftin g topolog y avoids inter-site coordination while still benefiting from m ulti-user diversity , where UEs in similar prop a gation en vironm ents often exhibit cor related m ultipath sign als re- flected/defrac ted from common targets [ 7 ]. One o f th e ma jo r challenges her e is how to explo it suc h c orrelation as c on ven- tional sen sin g algorithms s uch as multiple signal classification (MUSIC) lacks su ch a capability . Another challeng e is bistatic asynchro nism, which introdu c es UE-depend ent and tim e- varying timing offset ( TO), carrier-frequency of fset (CFO), and phase offset (PO), there b y d egrading in ter-packet coheren ce and hinde ring high -resolution sensing [8]. T o ad dress these issues, we d ev elop an asynch r onous up link networked sen sing framework by introdu cing com pressiv e sensing techn iques to explo r e m ultiuser cor r elation, in com - bination with offsets cancellation tech niques due to bistatic asynchro nism. Th e framework is demo nstrated v ia exploiting common delay cluster only , but it can be n aturally extended to other sensing param eters, such as Dopp ler and ang les, and cor- relation patterns. Spec ifically , we first p erform LoS-r eferenced calibration to sup p ress th e dom inant TO and align packets to a c ommon delay refe r ence. Then , we propose a shared- priv ate delay-do main sp a rse Bayesian learning ( SBL) f o rmulation that jointly recovers delay suppo rts while automatica lly exploiting correlation . Fina lly , based on the r e covered delay taps, we compen sate CFO/PO and then estimate Dop pler and angle of arriv al (AoA). Simulation results show that explo iting common delay- cluster sparsity ou tperform s the individual- SBL baseline , especially at low signal-to -noise ratio (SNR) and with small per-UE subcarr ie r budgets. I I . S Y S T E M M O D E L A. S ignal Model W e con sider an up link ISAC scenario in a PMN as shown in Fig. 1, where K single-an te n na UEs transmit to a BS 2 Pream ble Data Data Ċ ... Pream ble Data Data Ċ ... A T OFDM Packet OFDM Symbols BS Target UE k UE1 Target Fig. 1. Illustrat ion of the syste m model for uplink sensing. equippe d with an M -element un iform linear array ( U L A) in an uplink orthog onal fr equency-d ivision multiple access (OFDMA) system. UE k occupies a dedicated su bcarrier set N k ⊂ { 1 , . . . , N } with N k , |N k | , where N is the total number of subcarrier s, eac h UE is assigned a contig u ous block of N k subcarriers and different UEs are allocated n on- overlapping block s (i.e. , N k ∩ N k ′ = ∅ for k 6 = k ′ ). The BS collects T OFDM p ackets indexed by t ∈ { 1 , . . . , T } . In particular, we a ssume th at the BS knows the distanc e from each fixed UE, hence the Lo S prop a gation delay τ geom k, 0 is kn own, and the Lo S path power is significan tly stronger than tha t of th e NL o S paths. This provid es a delay reference for uplink sensing an d sy nchron ization, while the remaining non-lin e -of-sight (NLoS) co m ponen ts are to be estimated. These assumption s are rea sonable and commo nly adopted in PMN/ISA C uplink sensing [8], [ 9]. As dep icte d in Fig. 1, each uplink co mmunica tio n packet consists of o ne referen ce-bearin g preamb le and a sequen ce of data symbols with OFDM mo d ulation, a nd two consecuti ve packets ar e spaced by T A . F or si mplicity , each UE is assumed to use o nly one pre a mble OFDM symbol fo r sensing in each packet. In ad d ition, due to the asynchron ous transceivers, the received signal may be corrup te d by the packet-d ependen t TO/CFO an d PO, denoted by δ τ k [ t ] , δ ν k [ t ] , and β k [ t ] , respec- ti vely . As a result, f or UE k at packet t , after pilot ma tc h ed filtering, the channel state inform ation (CSI) estimation can be written in the delay-do m ain form as Y k [ t ] = L k X ℓ =1 ψ k ( ˜ τ k,ℓ [ t ]) s ⊤ k,ℓ [ t ] + E k [ t ] , (1) where L k denotes the n umber o f paths fo r UE k , ψ k ( ˜ τ ) ,  e − j 2 π f n 1 ˜ τ , . . . , e − j 2 π f n N k ˜ τ  ⊤ ∈ C N k is the delay s teering vector over N k and ˜ τ k,ℓ [ t ] , τ k,ℓ + δ τ k [ t ] ; s k,ℓ [ t ] , α k,ℓ e j β k [ t ] e j 2 π tT A ( ν k,ℓ + δν k [ t ]) a ( θ k,ℓ ) is the effecti ve spatial c oefficient; { α k,ℓ , τ k,ℓ , ν k,ℓ , θ k,ℓ } denote the sensing par a meters; a ( θ k,ℓ ) =  1 , e j π sin( θ k,ℓ ) , . . . , e j π ( M − 1) sin( θ k,ℓ )  ⊤ ∈ C M is the re c eiv e array respo nse vector of the ULA, and E k [ t ] ∈ C N k × M denotes the add itive white Gau ssian no ise matrix. B. P r opo sed Sparsity Mo del Capturing Delay Correlation T o facilitate delay estimation, we d iscretize the effective delay dom a in into a comm on uniform grid { ¯ τ 1 , . . . , ¯ τ G } ⊂ [0 , τ max ] , where G is the num ber o f grid points, an d construct Ψ k ,  ψ k ( ¯ τ 1 ) , . . . , ψ k ( ¯ τ G )  ∈ C N k × G . (2) According ly , by ap p roximatin g ˜ τ k,ℓ [ t ] to its near est grid point in { ¯ τ g } G g =1 , (1 ) can be simplified as the following multiple-me a su rement-vector (MMV) model Y k [ t ] = Ψ k W k [ t ] + E k [ t ] , (3) where W k [ t ] = [ w k, 1 [ t ] , w k, 2 [ t ] , . . . , w k,G [ t ]] ⊤ is a r ow- sparse coefficient matrix on th e delay grid, w ⊤ k,g [ t ] , P ℓ : g k,ℓ = g s ⊤ k,ℓ [ t ] and w ⊤ k,g [ t ] = 0 if n o path is assign e d to ¯ τ g . Hence, estimating delay r educes to recovering the row suppo rt of W k [ t ] . 1 Howe ver , (3) igno r es stru c tured sparsity across UEs. I n clustered m ulti-user I SAC/OFDMA channe ls, after exclu ding the TO-indu ced δ τ k [ t ] , UE s may exhib it cr o ss-user commo n- ality in the sensing parameters un der similar pro pagation condition s. Moti vated by this o bservation, we form clusters based on delay similarity , which leads to par tially comm on delay suppor ts among UEs in the same clu ster . T o this end , we intro duce the following shared–private decomp osition [7]: W nTO k [ t ] = W sh k [ t ] + W pr k [ t ] , (4) where W nTO k [ t ] den otes th e sparse delay -domain co efficient matrix af te r TO removal, W sh k [ t ] collec ts the coefficients on the delay tap s comm o nly sha r ed by UEs in the same cluster , wh ile W pr k [ t ] captures user-specific componen ts. As a simple example, if two UEs in o ne cluster have active delay sup ports { τ 2 , τ 5 , τ 9 , τ 14 } an d { τ 2 , τ 5 , τ 9 , τ 20 } , respec- ti vely , then the com mon taps { τ 2 , τ 5 , τ 9 } are rep resented by { W sh 1 [ t ] , W sh 2 [ t ] } , wher eas τ 14 and τ 20 are mod eled by W pr 1 [ t ] an d W pr 2 [ t ] , respectively . This d e composition allows common sup ports to be reinf orced jointly across UEs while preserving individual flexibility , thereby improvin g support recovery . Therefo re, after rem oving δ τ k [ t ] , th e resulting equivalent model can be written as ¯ Y k [ t ] = Ψ τ ,k ¯ W k [ t ] + E k [ t ] , (5) where Ψ τ ,k , [ Ψ k Ψ k ] and ¯ W k [ t ] ,  ( W sh k [ t ]) ⊤ ( W pr k [ t ]) ⊤  ⊤ . The c o rrespon ding TO calibr ation will be detailed in the next section . Giv en the estimated CSI { Y k [ t ] } T t =1 , we aim to r ecover the sparse coefficient matrix { ¯ W k [ t ] } T t =1 after TO rem oval, whose nonzer o supp ort identifies the active delay-grid indices, wh ile the a ssociated coefficients absorb th e Do ppler/CFO/PO a nd AoA infor mation. Hence, under (5), the problem red u ces to support recovery an d co efficient estimatio n , from which the remaining sensing par ameters can b e infer red. I I I . S E N S I N G P A R A M E T E R E S T I M A T I O N S C H E M E In this section, we d ev elop an e fficient estimation scheme for OFDMA multi-u ser uplink sensing with delay- cluster common ality under clock async hronism. As detailed below , 1 Since Ψ is paramet erized by delay alone, Doppler/CFO/PO and AoA are absorbed into w ⊤ k,g [ t ] and only a ffe ct the v alues of t he nonzero co ef ficients, without changing t he acti ve de lay i ndices. 3 we estimate and co mpensate th e TO using the known g e - ometric Lo S delay r eference. Th en, lev eraging the delay- clustered multi-u ser structur e, w e perform d e lay-clustering SBL to recover the active delay sup port. Finally , based on the recovered delay taps, CFO/PO are canceled using the LoS referenc e tap. Doppler is then estimated f rom adjace n t-packet phase ev olution, while AoA is obtained from a d jacent-anten na phase differences and the path ga in is recovered from the correspo n ding averaged coefficient cor relation. A. De lay-Clustering SBL T o recover { ¯ W k [ t ] } T t =1 in (5), we first need to sup p ress TO fr om { Y k [ t ] } T t =1 . Since the LoS path is assumed to be dominan t, the observed LoS delay can be identified fr om the dominan t peak of the delay -doma in perio dogram : fo r packet t , we fo rm p k ( τ g ; t ) , 1 M k d H k ( τ g ) Y k [ t ] k 2 2 , with d k ( τ g ) , [ e − j 2 π f n τ g ] n ∈N k . The stro ngest grid peak yields a coarse LoS delay estimate, which is fur th er refined by the parab olic interpolatio n around the dominant peak to obtain ˜ τ obs k, 0 [ t ] [10 ]. W e then estimate the TO as c δ τ k [ t ] = ˜ τ obs k, 0 [ t ] − τ geom k, 0 , and compen sate it by ¯ Y k [ t ] = diag ( φ k [ t ]) Y k [ t ] , whe re φ k [ t ] , [ e j 2 π f n c δτ k [ t ] ] n ∈N k . Above calibration mitigates th e domin ant TO and aligns th e observations to a commo n de la y r eference for the subsequ ent spar se suppor t recovery . Then, based o n (4), to mod el user clustering, we consider C can didate clusters and define a laten t cluster indic a tor z k , [ z k, 1 , . . . , z k,C ] ⊤ for UE k : if UE k belongs to cluster c ∈ { 1 , . . . , C } , th e n z k,c = 1 and z k,c ′ = 0 fo r all c ′ 6 = c , so that P C c =1 z k,c = 1 . L e t π = [ π 1 , . . . , π C ] ⊤ denote the clu ster- propo rtion vector , and assign it a Dirichlet pr io r π ∼ D ir( α 0 ) with concentratio n α 0 = [ α 0 , 1 , . . . , α 0 ,C ] ⊤ . Given π , z k follows a categor ical distribution: p ( z k | π ) = C Y c =1 π z k,c c , k = 1 , . . . , K . (6) T o pro ceed, following the classical sp a rse Bayesian mod el, we let w sh k,g [ t ] a n d w pr k,g [ t ] d enote the g -th ro ws of W sh k [ t ] a n d W pr k [ t ] respectively , which admit the following Gaussian p rior distributions: p  w sh k,g [ t ] | z k , γ g  = C Y c =1 C N  0 , γ − 1 g,c I M  z k,c = p sh k,g [ t ] , ( 7) p  w pr k,g [ t ] | η k,g  = C N  0 , η − 1 k,g I M  = p pr k,g [ t ] . (8) where γ g , [ γ g, 1 , . . . , γ g,C ] ⊤ , { γ g,c } are shared ro w pr eci- sions governing th e co mmon support with in clu ster c , an d { η k,g } are user k -specific r ow precisions captur ing individual compon ents. Gi ven { γ g,c } and { η k,g } , all rows ar e assumed to be inde penden t, y ielding p  ¯ W k [ t ] | z k , { γ g } G g =1 , { η k,g } G g =1  = G Y g =1 p sh k,g [ t ] p pr k,g [ t ] . ( 9 ) T o en able conjugate variational up dates, we impose Gamma hyperp riors to the p recisions as p ( γ g,c ) = Γ( γ g,c | a 0 , b 0 ) , p ( η k,g ) = Γ( η k,g | a 0 , b 0 ) , (10) where Γ( x | a 0 , b 0 ) d enotes the Gamma density with shape a 0 and rate b 0 . In this work, we set a 0 = b 0 = 0 . 0 1 . Moreover , the add itive noise is mo d eled as i.i.d. circularly symmetr ic complex Gaussian: vec( E k [ t ]) ∼ C N ( 0 , β − 1 I ) with p ( β ) = Γ( β | a 0 , b 0 ) . Based on ¯ Y k [ t ] , the condition al likelihood is giv en by th e circularly symmetric com plex Gaussian distribution: p ( ¯ Y k [ t ] | ¯ W k [ t ] , β ) ∝ exp  − β   ¯ Y k [ t ] − Ψ τ ,k ¯ W k [ t ]   2 F  . (11) Furthermo re, to enhance the ro bustness of delay-do main clustering, we leverage m u ltiple packets and form a stacked mu ltiple m e asurement vector ( MMV) observation. Namely , the CSI estimation c a n be stacked as ˜ Y k , [ ¯ Y k [ t 1 ] , . . . , ¯ Y k [ t T ]] ∈ C N k × M T and the stacked co efficient matrix can be den oted a s ˜ W k , [ ¯ W k [ t 1 ] , . . . , ¯ W k [ t T ]] . Let us define all th e p arameters to be estimated as Ω , n ˜ W k } K k =1 , { γ g,c } G,C g =1 ,c =1 , { η k,g } K,G k =1 ,g =1 , { z k } K k =1 , π , β o . Nev ertheless, the posterior p (Ω | { ˜ Y k } ) is an alytically in- tractable due to { z k } and { γ g,c } . T o address this, we ad opt a mean-field variational inferen c e (VI) approach following [7], which can approx imate p (Ω | { ˜ Y k } ) by the factorized distribution q (Ω) : q (Ω) = K Y k =1 q ( ˜ W k ) Y g,c q ( γ g,c ) Y k,g q ( η k,g ) K Y k =1 q ( z k ) q ( π ) q ( β ) . (12) According to [7], the variational factors in (1 2) c a n b e iterativ ely u p dated b y coordina te ascen t as log q ( x ) ∝ E q (Ω \ x ) [log p ( { ˜ Y k } , Ω)] , which yields clo sed-form up dates as summarized below . a) U pdate of q ( ˜ W k ) : The variational posterior of ˜ W k is a cir c ularly symmetr ic comp lex Gaussian matrix distribution with independen t colu m ns s haring a com mon ro w cov ariance, i.e., q ( ˜ W k ) = C N  ˜ W k ; U k , Σ row k  , where Σ row k and U k denote the po sterior r ow-covariance and mean, respectively Σ row k =  E [ β ] G k + diag ( λ k )  − 1 , U k = E [ β ] Σ row k B k , (13) where G k , Ψ H τ ,k Ψ τ ,k , B k , Ψ H τ ,k ˜ Y k and λ k =  ¯ γ sh k ; E [ η k ]  , ¯ γ sh k ,  ¯ γ sh k, 1 , . . . , ¯ γ sh k,G  ⊤ and ¯ γ sh k,g , P C c =1 r k,c E [ γ g,c ] ; η k , [ η k, 1 , . . . , η k,G ] ⊤ . Moreover , the required row-wise secon d mome n ts are E h k w sh k,g k 2 2 i = k U k ( g , :) k 2 2 + M T [ Σ row k ] g,g and E h k w pr k,g k 2 2 i = k U k ( G + g , :) k 2 2 + M T [ Σ row k ] G + g, G + g . b) Update of q ( γ g,c ) an d q ( η k,g ) : The posterior s remain Gamma distributed: q ( γ g,c ) = Γ  a γ g,c , b γ g,c  , q ( η k,g ) = Γ  a η k,g , b η k,g  . ( 14) where a γ g,c = a 0 + M T P K k =1 r k,c , b γ g,c = b 0 + P K k =1 r k,c E h k w sh k,g k 2 2 i , and a η k,g = a 0 + M T , b η k,g = b 0 + E h k w pr k,g k 2 2 i . Thu s E [ γ g,c ] = a γ g,c /b γ g,c and E [ η k,g ] = a η k,g /b η k,g . 4 c) Update of q ( z k ) and q ( π ) : q ( z k ) is categorical with responsibilities r k,c , q ( z k,c = 1) , updated as r k,c = exp( ξ k,c ) / C X c ′ =1 exp( ξ k,c ′ ) , (15) where ξ k,c , E [log π c ] + M T P G g =1 E [log γ g,c ] − P G g =1 E [ γ g,c ] E h k w sh k,g k 2 2 i , E [log π c ] = ψ ( α c ) − ψ ( P c ′ α c ′ ) , E [log γ g,c ] = ψ ( a γ g,c ) − log b γ g,c , where ψ ( · ) den otes the digamma functio n. Giv en a Dir ichlet prior p ( π ) = Dir( α 0 ) , we have q ( π ) = Dir( α ) with α c = α 0 ,c + P K k =1 r k,c . d) Up date of q ( β ) : The variational posterior is q ( β ) = Γ( a p ost β , b p ost β ) , where a p ost β = a 0 + M T P K k =1 N k , b p ost β = b 0 + P K k =1    ˜ Y k − Ψ τ ,k U k   2 F + M T tr  Σ row k G k   and E [ β ] = a p ost β /b p ost β . See Algorithm 1 fo r a summar y of above propo sed proce- dure, where b I k denotes the estimated acti ve d e la y in dex set for UE k , with b L k = | b I k | . B. Do ppler , A oA and Chan nel Ga ins Estimation Giv en the estimated active delay index set b I k in Section III- A, we n ext reco nstruct the per-tap coefficients and extract Doppler/Ao A for UE k . Before the fixed-support reconstru ction, we mitigate the off- grid erro r of the co a r se d elay grid via a co arse-to-fine refine- ment: for each detected coar se- grid tap ˆ τ k,ℓ , w e construct a fine grid T k,ℓ ,  ˆ τ k,ℓ + i ∆ τ F   i = − I , . . . , I  , wh ere ∆ τ = τ max G − 1 is the coarse-grid spacing , F = 4 denotes the r efinement factor , and I = 8 spe cifies the neigh borho od span. For each candidate τ ∈ T k,ℓ , we co mpute th e de la y steering vector d k ( τ ) = exp( − j 2 π f N k τ ) ∈ C N k , an d refine the de la y by maximizing a loc al de lay -doma in p eriodog ram over ˜ Y k , i.e., ¯ τ k,ℓ , arg ma x τ ∈T k,ℓ    d H k ( τ ) ˜ Y k    2 2 . Define ¯ τ k , [ ¯ τ k, 1 , . . . , ¯ τ k, b L k ] ⊤ . Accor d ingly , the re- fined reduced delay dictionary is giv en b y B k , [ d k ( ¯ τ k, 1 ) , . . . , d k ( ¯ τ k, b L k )] ∈ C N k × b L k and th u s the p er-packet CSI estimation admits the fixed-su p port mo del ¯ Y k [ t ] ≈ B k b X k [ t ] + E k [ t ] , ( 16) where b X k [ t ] ∈ C b L k × M collects the per-tap co mplex coeffi- cients across the M a ntennas. Based on (16), we next extract b X k [ t ] for each p acket t via a regularized least-squares (LS) pro jection as b X k [ t ] =  B H k B k + λ I  − 1 B H k ¯ Y k [ t ] , (17) where λ = 10 − 3 , and th e ( ℓ, m ) - th entry of b X k [ t ] ap- proxim a tely follows the slow-time phase model ˆ x k,ℓ,m [ t ] ≈ α k,ℓ [ a ( θ k,ℓ )] m exp  j β k [ t ]  exp  j 2 πtT A  ν k,ℓ + δ ν k [ t ]   . Next, in o rder to estimate the Doppler frequ ency , we construct the adjace n t-packet conjugate prod uct ˆ x ∗ k,ℓ,m [ t ] ˆ x k,ℓ,m [ t +1] ≈ | α k,ℓ | 2 | a ( θ k,ℓ ) m | 2 e j ( β k [ t +1] − β k [ t ]) e j 2 π T A ( ν k,ℓ +( t +1) δν k [ t +1] − tδ ν k [ t ]) . Ex ploiting the known LoS referenc e tap ( ν k,ℓ ref = 0 ), we multiply the above produ ct by Algorithm 1 M ean-field VI for clu ster delay- SBL Require: { Y k [ t ] } t ∈ T , τ geom k, 0 , Ψ τ ,k . 1: Prepare: Ob tain TO-compensated { ¯ Y k [ t ] } t ∈ T and stack T packets to fo rm ˜ Y k ∈ C N k × M T . 2: Initialize { r k,c } , { E [ γ g,c ] } , { E [ η k,g ] } , and E [ β ] . 3: repeat 4: Update q ( ˜ W k ) via a) . 5: Update q ( γ g,c ) and q ( η k,g ) via b) . 6: Update q ( π ) and q ( z k ) via c) . 7: Update q ( β ) via d) . 8: until con vergence 9: O utput: { b I k } K k =1 , wher e the delay support ˆ I k is ob tained by selectin g the do minant peak s of k U k ( g , :) k 2 2 + k U k ( G + g , :) k 2 2 , yielding { ˆ τ k,ℓ } b L k ℓ =1 . its conjug ated LoS counter part so that the co m mon CFO/PO terms are canceled and we can ob tain the Doppler estimate as b ν k,ℓ = 1 2 π T A ∠ T − 1 X t =1  M X m =1 ˆ x k,ℓ ref ,m [ t ] ˆ x ∗ k,ℓ ref ,m [ t +1]  ×  M X m =1 ˆ x ∗ k,ℓ,m [ t ] ˆ x k,ℓ,m [ t +1]  ! . (18) Finally , since the adjac e n t-antenn a conju gate product of the recovered ℓ -th tap satisfies ˆ x k,ℓ,m [ t ] ˆ x ∗ k,ℓ,m +1 [ t ] ≈ | α k,ℓ | 2 e j π sin ( θ k,ℓ ) , we can d erive the AoA and ch annel g ain of the ℓ -th path by a veraging the cross-correlations between any two adjacent anten n a elements averaged over t and m as sin( b θ k,ℓ ) ≈ 1 π ∠ 1 T ( M − 1) T X t =1 M − 1 X m =1 ˆ x k,ℓ,m [ t ] ˆ x ∗ k,ℓ,m +1 [ t ] ! , (19) | b α k,ℓ | 2 ≈      1 T ( M − 1) T X t =1 M − 1 X m =1 ˆ x k,ℓ,m [ t ] ˆ x ∗ k,ℓ,m +1 [ t ]      . (20) I V . S IM U L A T I O N R E S U LT S In this section, we evaluate the propo sed estimation pipeline for OFDMA up link sen sing under asynchr onous tran sceiv ers. Unless otherwise specified , th e system opera tes at a carrier frequen cy of f c = 3 . 5 GHz, with a signal bandwidth of B = 140 MHz . The total numb e r of sub carriers is N = 2048 , the subca rrier spacing is ∆ f = 60 kHz, an d the packet interval is T A ≈ 0 . 25 ms . W e con sider a BS equip ped with a M = 8 - antenna ULA with h alf-wav elength spacing, serv ing K = 8 uplink UEs over T = 16 packets. The K UEs are p artitioned into S = 3 delay clusters, w h ere UEs within th e same cluster share a set of commo n delay taps. Specifically , each UE contains L sh = 3 commonly shar ed pa th s and L pr = 1 priv ate path (set as the LoS tap), L k = L sh + L pr . Th e maximu m delay is τ max = 2 . 5 µ s and the grid n umber is G = 256 . The Dop p ler freq uencies are a ssum ed to lie in [ − 0 . 35 , 0 . 35] kHz, an d the AoAs in [ − 90 ◦ , 90 ◦ ] . T he TOs and CFOs ar e u niformly drawn fro m [0 s , 20 /B ] and [0 Hz , 150 Hz ] , respectively . Before pr esenting th e simulation results, we define the perfor mance m etrics averaged over all K UEs: for x ∈ { τ , ν } , 5 -5 0 5 10 15 SNR [dB] 10 -3 10 -2 10 -1 10 0 NMSE of Delay / Doppler 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 RMSE of AoA (rad) Delay: Proposed method Doppler: Proposed method AoA: Proposed method Delay: Individual-SBL Doppler: Individual-SBL AoA: Individual-SBL Fig. 2. Delay and Doppl er NMSE, AoA RMSE versus SNR. N k = 128 . T ABLE I M U L T I - U S E R C L US T E RI N G AC CU R AC Y V E R S U S S N R. SNR (dB) -5 0 5 10 15 Clusteri ng accur acy 0.502 0.689 0.812 0.868 0.899 NMSE( x ) , 1 K P K k =1 P ℓ 6 = ℓ LoS  b x k,ℓ − x k,ℓ  2 P ℓ 6 = ℓ LoS x 2 k,ℓ , and for AoA θ we have RMSE( θ ) , 1 K P K k =1 q 1 L k P L k ℓ =1  b θ k,ℓ − θ k,ℓ  2 . In Fig. 2, we e valuate the estimatio n performan c e versus SNR for two sch emes: our pro posed mu lti-user SBL a n d the Individu al-SBL baseline (standard per-UE SBL [11] without exploiting cross-user structure ). From Fig. 2, it ca n be o b- served that the NMSE/RMSE metrics decrease mono tonically with SNR. Mo re impor tantly , the p roposed scheme ac h iev es unifor m ly lower delay/Do ppler NMSE and AoA RMSE than the Individual-SBL baseline acro ss th e SNR range, with a more significant gain in the low-SNR regime. T o gain more insights, we record the mu lti- user clustering accu racy versus SNR in T ab le I. As we c an see, the clusterin g accura cy in- creases steadily with SNR, indica tin g that the prop osed method can identify the u nderlyin g delay clusters more reliably as th e observation quality imp roves. Fig. 3 depicts th e delay -estimation NMSE versus th e number o f allo cated subc a rriers N k under ( L sh , L pr ) ∈ { (3 , 1) , (2 , 2 ) } . As N k increases, the delay NMSE of all schemes decreases due to richer frequency-d o main observa- tions. No tably , the I ndividual-SBL baseline is largely insensi- ti ve to the share d /priv ate composition, since it processes each UE in depend ently . In contrast, the pro p osed m ulti-user scheme benefits more fro m stron ger shared sparsity , an d thu s perform s better in th e (3 , 1) case at large N k . Furtherm o re, th e p roposed multi-user scheme achieves a clear NMSE re duction over the Individual-SBL baseline, with a m ore significant g ain wh en N k is small, indicating e xploiting cro ss-user share d sparsity is most ben eficial in th e N k -limited regime. Finally , Fig. 4 rep orts the delay NMSE, Do p pler NM SE, a nd AoA RMSE versus the number of packets T . As T increases, all three metrics d ecrease, as mo re packets provide r icher temporal o b servations fo r delay suppo rt recovery , enh ance slow-time div ersity for Dopp ler e stimation, an d enab le more effecti ve coh erent averaging for Ao A estimation. Mean while, the pr oposed scheme consistently outperf orms the Individual- SBL baselin e ac r oss all T . This gain mainly stem s fro m the improved d e lay supp ort recovery enabled b y explo iting com- mon delay - cluster sparsity acr o ss users, which y ields lower delay NMSE an d su bsequently lead s to better Do ppler and AoA estimation comp ared to Individual-SBL. 64 96 128 192 256 The number of subcarriers for each User N k 10 -4 10 -3 10 -2 10 -1 NMSE of Delay Individual-SBL (L sh =3,L pr =1) Individual-SBL (L sh =2,L pr =2) Proposed method (L sh =3,L pr =1) Proposed method (L sh =2,L pr =2) Fig. 3. Delay NMSE versus N k . SNR = 5 dB. 8 16 32 64 128 The number of packets T 10 -3 10 -2 10 -1 NMSE of Delay/ Doppler 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 RMSE of AoA (rad) Delay NMSE: Proposed method Doppler NMSE: Proposed method AoA RMSE: Proposed method Delay NMSE: Individual-SBL Doppler NMSE: Individual-SBL AoA RMSE: Individual-SBL Fig. 4. 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