On Channel Estimation for Group-Connected Beyond Diagonal RIS Assisted Multi-User MIMO Communication

1 On Channel Estimation for Group-Conn ected Be yond Diagonal RIS Ass isted Multi- User MIMO Communication Rui W ang, Junyua n Gao, Shuowen Zhang, Bruno Clerckx, and Lia n g Liu Abstract —Bey ond diagonal reco nﬁgurabl e intelligent surface (BD-RIS) architectures offer superior beamform ing gain over con ventional diagonal RISs. Howe ver , the channel estimation ov erhead is the main h urdle for reaping the above gain in practice. This letter addre sses this issue f or group-connected BD- RIS a ided uplink communication from mul t i ple multi-antenn a users to one multi-antenna base station (BS). W e ﬁrst re veal that within each BD-RIS group, the cascaded channel associated with one user antenna and one BD-RIS element is a scaled version of that associated with any other user antenna and BD-RIS element due to the common RIS-B S channel. This i nsight drastically reduces the dimensionality of th e channel estimation p roblem. Buildin g on this p roperty , we propose an efﬁcient two-phase channel estimation protocol. In the ﬁrst ph ase, the refer ence cascaded ch annels for all groups are estimated in parallel based on common re ceived signals while determining the sca ling coef- ﬁcients f or a single refer ence antenna. In the se cond phase, the scaling coefﬁcients for all remaining user an t en nas are estimated. Numerical results demonstrate that our proposed framework achiev es substantiall y lo wer estimation error wit h fewer p ilot signals compared to state-of-the-art benchmark schemes. Index T erms —Bey ond diagonal reconﬁgurable in telligent sur - face (BD-RIS), group-connected architecture, channel estimation, low-o verhead communication. I . I N T R O D U C T I O N Reconﬁgura ble intellig e n t surface (RIS) is a key te c hnology for the sixth-gener ation (6G) networks. Beyond diagon al RIS (BD-RIS) architectures, wh ich use inter-element c o nnections to create n on-diago nal scatter ing ma trices, o ffer more ver - satile wa ve manipu la tio n, en h anced beamforming gains, and expanded network coverage over con ventiona l diago n al RISs [1]–[4]. Howev er, chan n el estimation is a main hur dle to re ap the above gains, because the interco nnected nature of BD-RIS elements le a ds to a d ramatic increase in the n umber of chann el coefﬁcients n eeded to be estimated compared to con ventional diagona l RIS systems [5 ]– [ 7]. Several recen t stu d ies have ex- plored cha n nel estimation for BD-RIS aid ed comm unications. A least squares (LS)- based method was introduc e d in [5] to obtain a closed-for m estimation of the cascaded ch annel. The work in [6] developed two distinct ten so r decom p osition- based algorithm s, utilizing Kha tr i-Rao factorization (KRF) and Rui W ang , Junyuan Gao, Shuo wen Zhang, and Liang Liu are with the Departmen t of Elec trical and Electroni c Engineering, The Hong Ko ng Polytechnic Univ ersity , Hong Kong SAR, China (e- mails: rui-eie.wa ng@connect .polyu.hk, { junyuan.gao, shuowen.zha ng, l iang- eie.liu } @polyu.edu.hk). Bruno Clerckx is with the D epartment of E lect rical and Elec- tronic Engineeri ng, Imperial College London, London, U.K. (e-mail: b .clerckx@i mperial.ac.uk). alternating least squar es (AL S) to redu c e e stima tio n overhead. Building on this tensor paradigm, a semi-blind scheme was propo sed in [8], while the ALS framew ork was extended to multi-user scen a rios in [9]. A limitation of these metho ds is that their chann el estimation overhead is much hig her th an that in the conv ention al RIS aided systems. Although the prope r ty that ca scad ed channels of different users share a common RIS-base station ( BS) compo nent has been utilized in conventional RIS systems [10], [11], this letter focuses on the unique co r relation among the cascaded channels associate d with intercon nected elements in a BD-RIS. Recently , o ur work in [7] unv eiled a crucial channel p roperty in fully-c onnected BD-RIS aided systems, i.e., the cascaded channel a ssoc ia ted with any user an tenna and BD-RIS elemen t is a scaled version of that associated with anoth er user anten na and BD-RIS element, stemming fr om the com mon RIS-BS channel. By exploiting this property , [7] p roposed a novel two-phase estimation scheme that ﬁrst estimate s a reference cascaded cha n nel and then only estimates the scaling coefﬁ- cients of the oth er cascad ed ch annels, showing that the cha nnel estimation overhead in BD-RIS aided systems is of the same order as that in c on ventional RIS aided systems [10]. In this letter, we aim to extend this metho d to th e g roup- connected BD-RIS aided co mmunicatio n systems. In a g roup- connected BD-RIS, the reﬂecting elements are partioned into se veral disjoint groups and th e elements in each g roup are inter-connected. On e heuristic sch eme is to tur n on one group of elements durin g each time interval, then the results in [7] can be applied to e stima te each group’ s channels. Howe ver , this sch eme is no t optimal because the channels in each g roup are estimated individually based on separate pilot signals. Our contribution in this letter is th e joint estimation acro ss all group s, wh ich utilizes common pilot signals to determine the reference channels in all grou ps. This method achie ves a further reduction in channel estimation overhead, as veriﬁed by n umerical re su lts. I I . S Y S T E M M O D E L W e co nsider an uplink multi-u ser MIMO commun ication system. The system , illustrated in Fig. 1, in corpor ates a BS equippe d with N an tennas, a B D-RIS compr ising M passive reﬂecting eleme n ts, and K users, each employing V antennas. This letter co nsiders a g roup-c o nnected BD-RIS architecture. The set of M BD-RIS elem ents is p artitioned into U d isjoint group s a n d the BD-RIS elements in e ach group are inter- connected . Let M u denote the number of elem ents in group u , 2 Fig. 1: A BD-RIS assisted MU-MI MO uplink co mmunicatio n system. u = 1 , · · · , U , such tha t P U u =1 M u = M . At time instant t , the scattering matrix of the BD-RIS, den oted by Φ t ∈ C M × M , is a block-d iagonal ma trix Φ t = blkdiag ( Φ 1 ,t , Φ 2 ,t , . . . , Φ U,t ) , ∀ t, (1) where Φ u,t = [ φ u,t, 1 , · · · , φ u,t,M u ] , with φ u,t,m denoting the reﬂecting co e fﬁ cients o f the m -th element in group u at time instant t , ∀ u , m = 1 , · · · , M u . Accordin g to [4], it fo llows that Φ H u,t Φ u,t = Φ u,t Φ H u,t = I M u , ∀ u, t. (2) A quasi-static narrowband blo c k fading channel mode l is considered , un d er which the channels ar e assumed to remain approx imately constan t within each coherence block. In this letter , we assume that the dir ect links from users to the BS are blocked. The b aseband equiv alent chan n el fr om the v - th antenna of user k to the m -th BD-RIS reﬂecting elem e nt in gro up u is deno ted by r k,v ,u,m ∈ C , while th e ch annel from the m -th BD-RIS element in group u to the BS is giv en by g u,m ∈ C N × 1 , k = 1 , . . . , K , v = 1 , . . . , V , u = 1 , · · · , U , and m = 1 , . . . , M u . The overall channel from u ser k to the group u of BD-RIS elements is deﬁned as R k,u ∈ C M u × V , ∀ k, u , with the ( m, v ) -th element bein g r k,v ,u,m , m = 1 · · · , M u , v = 1 , · · · , V , a n d the overall channel f r om the group u of BD-RIS elem ents to the BS is G u = [ g u, 1 , . . . , g u,M u ] ∈ C N × M u . Then , th e overall channel f r om user k to the BD-RIS is denoted by R k = [ R T k, 1 , · · · , R T k,U ] T ∈ C M × V , and the overall channel f r om the BD-RIS to th e BS is den oted b y G = [ G 1 , · · · , G U ] ∈ C N × M . Accordingly , the uplink user -RIS-BS cascaded chan - nel from user k to the BS is H k,t = G Φ t R k = X U u =1 G u Φ u,t R k,u , ∀ k , t. (3) It can be sho wn that at a ny time instant t , we ha ve G u Φ u,t R k,u = unv ec( J k,u φ u,t ) , u = 1 , · · · , U, (4) where un vec( · ) deno tes the inv erse operation of vectoriza tio n, φ u,t = vec( Φ u,t ) with vec( · ) deno ting the vector ization operation , an d J k,u = R T k,u ⊗ G u , (5) denotes the cascaded ch annel associated with user k and th e u - th group of BD-RIS elemen ts, with ⊗ de noting the Kronecker produ ct. Then, the signa l received at th e BS at time instant t is g i ven as y t = X K k =1 H k,t √ p a k,t + z t = X K k =1 X U u =1 un vec( J k,u φ u,t ) √ p a k,t + z t , ( a ) = X K k =1 X U u =1 √ p ( a T k,t ⊗ I N ) J k,u φ u,t + z t , t = 1 , · · · , τ , (6) where p is the ide ntical transmit p ower for all users, a k,t = [ a k, 1 ,t , · · · , a k,V ,t ] T ∈ C V × 1 is the unit-power pilot signal from user k at time instant t , z t ∼ C N ( 0 , σ 2 I N ) is the additive white Gaussian n oise (A WGN) at the BS at time instant t , and τ d enotes the length of th e pilot sequ ence in each coh erence block, a nd ( a ) is because vec( AX C ) = ( C T ⊗ A )vec( X ) gi ven any matrix A , X and C . In this letter , we m ainly focus on th e estimation o f cascaded channe ls J k,u ’ s based on (6), which is needed for BD-RIS scattering matrix design [5], [6], and pro p ose an innovati ve scheme with low training overhead by exploiting a novel chann el prop e rty . I I I . C H A N N E L P R O P E RT Y A N D P RO B L E M S TA T E M E N T T o estimate J k,u ’ s based on (6 ), e xisting app roaches pro- posed in [5], [6] have treate d all the K V N P U u =1 M 2 u entries in J k,u ’ s as indepe ndent un known variables. Howe ver, th e cas- caded channels a ssoc iate d with the BD-RIS elements with in each group are hig hly c orrelated. Speciﬁcally , accor ding to (5), we ha ve J k,u =    Q k, 1 ,u, 1 · · · Q k, 1 ,u,M u . . . . . . . . . Q k,V ,u, 1 · · · Q k,V ,u,M u    ∈ C V N × M 2 u , (7) where Q k,v ,u,m = r k,v ,u,m G u , ∀ k , v , u, m = 1 , · · · , M u , (8) denotes the ( v , m ) -th sub - block of J k,u . Note that Q k,v ,u,m can be vie wed as the cascaded ch annel correspo nding to the path from th e antenna v of user k to the m -th elem ent in group u via wireless channe l, then to all the elem e nts in this gro up via inter-conne c ted circuit, and last to the BS via wireless channel. A key observation according to (7) and (8) is th at different cascade d cha n nels Q k,v ,u,m ’ s shar e a commo n RIS-BS ch annel comp onent G u , ∀ u . Due to this comm on compon ent, according to (8), we hav e Q k,v ,u,m = β k,v ,u,m Q 1 , 1 ,u, 1 , ∀ u, ( k , v , m ) 6 = (1 , 1 , 1) , (9) where β k,v ,u,m = r k,v ,u,m r 1 , 1 ,u, 1 . (10) This structural insight r eveals that t he estimation task can be substantially simpliﬁed . W ith eac h grou p u , we d e ﬁne Q 1 , 1 ,u, 1 associated with its ﬁrst element and the ﬁrst anten na of user 1 as th e reference chan nel. After the reference channel of each gr oup u is determin ed, any other cascaded chann e ls of this group , i.e., Q k,v ,u,m , ∀ ( k , v , m ) 6 = (1 , 1 , 1) , can be reconstru cted by merely estimating the scalar β k,v ,u,m . Consequently , the set of ind ependen t u nknowns comprises the 3 U reference cascaded channels, i.e., Q 1 , 1 ,u, 1 ’ s, u = 1 , · · · , U , and K V M − U associated scaling coefﬁcients, i.e., β k,v ,u,m ’ s, ∀ ( k , v , m ) 6 = (1 , 1 , 1) an d ∀ u . The total number of inde- penden t unknowns is thus N M + K V M − U . This nu mber is signiﬁcantly smaller than the K V N P U u =1 M 2 u unknowns required by existing schemes such as [5] an d [6], which do not exploit the speciﬁc chann el p roperty in (9) and (1 0). In the fo llowing, we propose a efﬁcient scheme to estimate the above reduced number o f channel coefﬁcients. I V . P RO P O S E D T W O - P H A S E C H A N N E L E S T I M AT I O N P RO T O C O L Deﬁne B k,u = [ β k, 1 ,u , · · · , β k,V ,u ] ∈ C M u × V , where β k,v ,u = [ β k,v ,u, 1 , · · · , β k,v ,u,M u ] T , and β 1 , 1 ,u, 1 = 1 , ∀ u . I t can b e sh own that the received signals gi ven in (6) red u ce to y t = X K k =1 X U u =1 √ p Q 1 , 1 ,u, 1 Φ u,t B k,u a k,t + z t . (11) In this section, we propose a tw o- p hase channel estima tio n protoco l to estimate all cascaded channels b ased on (11), which is organized as follows. Phase I ( τ 1 time instants): The BS estimates the cascaded channels associated with the 1 st antenna of user 1 , includin g the referen ce chann els Q 1 , 1 ,u, 1 for all grou ps u = 1 , · · · , U , alongside th e scaling coefﬁcients β 1 , 1 ,u,m ’ s for the elements within those groups. Phase II ( τ 2 = τ − τ 1 time instants): Th e BS estimates th e remaining scaling coefﬁcients β k,v ,u,m ’ s f or all o ther a n tennas of u ser 1 and a ll antennas of users k = 2 , · · · , K . Finally , the ov erall channels Q k,v ,u,m are r e covered based on (9). The detailed implementatio n o f e ach ph ase is presented below . A. Phase I: Estimation of the Cascad ed Cha nnels Associated with th e 1st Antenna of User 1 In this p h ase, we aim to estimate the refere n ce cascad e d channels Q 1 , 1 ,u, 1 ’ s for all group s u = 1 , . . . , U , an d the scaling co efﬁcients associated with the ﬁrst anten na of u ser 1. T o implement this, we let only the 1 st antenna of u ser 1 transmit non-ze ro p ilot signals wh ile all other K V − 1 anten nas remain silent, i.e., a k,v ,t = a t , if k = 1 , v = 1 , a n d a k,v ,t = 0 otherwise, t = 1 , · · · , τ 1 . Under this conditio n , the r eceiv ed signal ( 1 1) simp liﬁes to: y t = U X u =1 f u,t, 1 + M u X m =2 β 1 , 1 ,u,m f u,t,m ! , t = 1 , · · · , τ 1 . (12) where f u,t,m = √ pa t Q 1 , 1 ,u, 1 φ u,t,m . The pr im ary challeng e in estimating Q 1 , 1 ,u, 1 ’ s and the coefﬁcients β 1 , 1 ,u,m ’ s stems from the fact that the rece i ved signals y t in (12) are no n-linear function s of th em. Recen tly , our work [7] propo sed an efﬁcient channel estimation a p proach in the spe c ial case of a fully- connected BD-RIS, i.e., U = 1 . A straightforward method for the c ase of grou p-conn ected BD-RIS is thus as follows. W e divide the o verall time into U blocks, while at th e u -th block, we sh ut do wn g roups 1 , · · · , u − 1 , u + 1 , · · · , U , and apply the app roach in [7] to estimate the channels associated with grou p u , i.e., Q 1 , 1 ,u, 1 and β 1 , 1 ,u,m ’ s, u = 1 , · · · , U . Howe ver , this approa c h is sub-optimal becau se it r equires a base received signal and its variant f o r ev ery group to estimate the associated reference channel, i.e., Q 1 , 1 ,u, 1 , ∀ u . A unique base recei ved sign al is assigned for group u to build a linear function of Q 1 , 1 ,u, 1 to estimate it, ∀ u . In practice, this base received signal c a n be repeatedly used for all group s. Thu s, we propo se to utilize a common base sign al shared by all gr oups and ass ign a v ariant based o n it for each gr o up, red ucing the overhead to estimate Q 1 , 1 ,u, 1 ’ s. Speciﬁcally , we d i vide the overall τ 1 time in stants in Phase I into U + 1 parts to facilitate the dif ferential estimation. The ﬁrst part pr ovid es the base sign al, inc lu ding τ 1 , 1 time in stants and th e (1 + i ) -th par t consisting of δ i ≤ τ 1 , 1 time instants, i = 1 , · · · , U , provid es the variants for group i . The B S-RIS scattering matrice s and the user pilot signals in these U + 1 parts ar e giv en a s follows: At time in stants t = 1 , · · · , τ 1 , 1 in the ﬁrst p a rt, the user pilot signals a t ’ s are set as rando m non-ze r o scalar s, and the BD-RIS scattering matrices Φ u,t = [ φ u,t, 1 , · · · , φ u,t,M u ] are set as random un itary matrices. Le t t

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