EXTENSION OF SEDJOCO AND ITS USE IN A COMBINATION OF MULTICAST AND COORDINATED MULTI-POINT SYSTEMS

Transcription

1 EXTENSION OF SEDJOCO AND ITS USE IN A COMBINATION OF MULTICAST AND COORDINATED MULTI-POINT SYSTEMS Yao Cheng 1, Are Yeredor 2, and Martn Haardt 1 1 Communcatons Research Laboratory 2 School of Electrcal Engneerng Ilmenau Unversty of Technology Tel-Avv Unversty P.O. Box , D Ilmenau, Germany P.O. Box 9040, Tel-Avv 69978, Israel {y.cheng, martn.haardt}@tu-lmenau.de are@eng.tau.ac.l ABSTRACT Ths paper presents a new perspectve of beamformng desgns n Coordnated Mult-Pont (CoMP) downln systems that are combned wth multcast schemes. The beamformer computaton s expressed as a ont matrx transformaton that can be regarded as an extenson of the Sequentally Drlled Jont Congruence (SeDJoCo) decomposton. A soluton of the proposed ont matrx transformaton s devsed that taes nto account the elmnaton of the multuser nterference as well as the maxmzaton of the desred sgnal components. Therefore, t leads to a very effectve sem-algebrac soluton of beamformng desgns for the multcast CoMP downln, whch s evdent n the numercal smulatons. Index Terms ont matrx transformatons, beamformng, Coordnated Mult-Pont, multcast 1. INTRODUCTION Jont matrx transformatons are a crucal lnear algebrac tool n varous sgnal processng felds such as Blnd Source Separaton (BSS), Independent Vector Analyss (IVA), and beamformng for mult-user Multple-Input Multple-Output (MIMO) systems [1, [2, [, [4. Recently, a ont matrx transformaton termed Sequentally Drlled Jont Congruence (SeDJoCo) decomposton has been proposed [5. It fnds applcatons n the context of both BSS and Coordnated Beamformng (CBF) n the mult-user MIMO downln. The SeDJoCo transformaton s defned as follows: Gven a set of K (conugate-)symmetrc target matrces of sze K K denoted by Q for = 1,...,K, fnd a transformaton matrx B C K K such that the -th column (and the -th row) of the -th transformed matrx D = B Q B H C K K, = 1,...,K (1) s drlled,.e., all elements n the -th column and row are zeros, except for the (, )-th element. Ths drlled structure of SeDJoCo s depcted n Fg. 1. In the mult-user MIMO down- Fg. 1. Illustraton of the drlled structure n SeDJoCo for K = ln wth space dvson multple access, one base staton equpped The authors gratefully acnowledge the fnancal support by the German-Israel Foundaton (GIF), grant number I /2014. wth K antennas transmts to K users each havng K receve antennas. Assumng maxmum rato combnng s employed at each user termnal, the precodng vectors for the K users denoted by f C K ( = 1,...,K) are computed such that the mult-user nterference s mtgated. In ths context, the target matrces and the transformaton matrx are constructed as Q = H H H and B H = f 1 f K, respectvely. HereH C K K represents the channel matrx between the base staton and the -th user. Two solutons of SeDJoCo have been proposed n [5. In ths contrbuton, we focus on a more sophstcated Coordnated Mult-Pont (CoMP) downln system combned wth multcast schemes (cf. Fg. 2 for a three-cell example). A CBF scheme called Extended FlexCoBF has been developed for a smlar scenaro n [6, where the transmt and the receve beamformers are computed ontly and teratvely. In each teraton, Extended FlexCoBF requres the exchange of the precoded channels of the cell edge users (e.g., User n Fg. 2) among adacent cells, resultng n transmsson delays and a sgnalng overhead. Motvated by the ntrgung ln between SeDJoCo and CBF n the mult-user MIMO downln, we propose a new ont matrx transformaton that helps accomplsh the challengng CBF tas n multcast CoMP downln settngs. The soluton of the proposed ont matrx transformaton s nspred by the ont mult-user MIMO system (JMMS) [7 but s largely extended from t. Va numercal smulatons, we show that the proposed ont matrx transformaton, as an extenson of SeDJoCo, s a very effectve and effcent sem-algebrac tool for beamformng n multcast CoMP downln systems. It leads to a comparable sum rate performance as Extended FlexCoBF but requres a sgnfcantly smaller sgnalng overhead. 2. PROBLEM FORMULATION Consder a multcast CoMP downln settng where the users are dstngushed nto cell nteror users and cell edge users. An example of a three-cell scenaro s llustrated n Fg. 2. The base statons are each equpped wth M T transmt antennas. After user schedulng, each base staton serves three users. All user nodes have the same number of receve antennas denoted asm R, and a sngle data stream s sent to each user. Usng a cell-specfc multcast scheme, the same sgnal s transmtted to the user wth the same ndex n all the three cells,.e., denotng the sgnal for the -th user as x ( = 1,2,), User n Cell 1, Cell 2, and Cell all receve x from ther correspondng base statons, respectvely. User n all three cells belong to one multcast group [8, [9, [10, and, for nstance, they could be subscrbers of the same TV programs. Here User 1 and User 2 n each cell are treated as cell nteror users. Both the nter-cell nterference and the desred sgnals from the adacent cells are assumed to be neglgble due to the heavy path loss. Therefore, the cell nteror /16/$ IEEE 276 ICASSP 2016

2 To guarantee zero nterference for User, the followng has to be fulflled for l = 1, 2 f,n H H,n H (H,1 f l,1 +H,2 f l,2 +H, f l,) = 0. n=1,2, To formulate the coordnated beamformng tas descrbed above nto a ont matrx transformaton problem, we defne transformaton matrces B n (n = 1,2,) wth respect to Celln B H n = [ f 1,n f 2,n f,n C. (5) The target matrces can be dstngushed nto two sets,.e., Fg. 2. A three-cell multcast CoMP downln scenaro where two cell nteror users and one cell edge user are served n each cell users only receve sgnals from ther own base staton and suffer only from the ntra-cell nterference,.e., the mult-user nterference as n the sngle-cell mult-user MIMO downln scenaros. On the other hand, User s a cell edge user and s affected by both the ntra-cell nterference and the nter-cell nterference. The fact that each cell edge user receves the desred sgnals from the base statons n the adacent cells s exploted. Jont transmssons of the neghborng cells to the cell edge user are consdered to combat the nterference and to deal wth the greater path loss compared to the cell nteror users. To avod cumbersome notatons, we tae ths example to present the data model and the soluton n Secton. The extenson to a more general case s addressed at the end of Secton. The receved sgnal of User m as a cell nteror user (m = 1,2) n Cell n (n = 1,2,) s expressed as r m,n = H m,n f m,n x m +H m,n f l,n x l +n m,n, (2) l m whereh m,n C M R M T denotes the channel matrx from the base staton n Cellnto Userm,x m represents the sgnal for Userm, and f l,n C M T s the precodng vector computed at the base staton of Cellnfor Userl(l = 1,2,). In ths example,m T = M R = and s equal to the number of users served by each cell. It can be seen that for a cell nteror user, assumng that the nter-cell nterference s neglgble, the transmsson from the base staton n ts own cell s the same as n a sngle-cell mult-user MIMO downln system. On the other hand, the receved sgnal of User s wrtten as r = H,n f,n x + (H,n f l,n x l)+n. n=1,2, n=1,2, l=1,2 Based on the maxmum rato combnng crteron, we defne the receve combnng vectors as follows w m,n = H m,n f m,n, m = 1,2, n = 1,2, () w = H,n f,n. (4) n=1,2, For a cell nteror user, User m (m = 1,2) n Cell n (n = 1,2,), achevng zero nterference va the desgn of the beamformng vectors requres fm,n H Hm,n H H m,n f l,n= 0, where l m. For User, as a cell edge user, the desred component can be expressed as f,n H H,n H (H,1 f,1 +H,2 f,2 +H, f,) x. n=1,2, m = H H m,n H m,n, m = 1,2,, n = 1,2, that are all Hermtan matrces and Q (1,2) =H H,1 H,2, Q (1,) =H H,1 H,, Q (2,) =H H,2 H,. The resultng ont matrx transformaton s llustrated n Fg.. Fg.. Illustraton of the proposed ont matrx transformaton. SOLUTION OF THE PROPOSED MATRIX TRANSFORMATION The aforementoned ont matrx transformaton can be reformulated nto (D 1,D 2, and D are also llustrated n Fg. ) D [ 1= B 1 B 2 B D [ 2= B 1 B 2 B D [ = B 1 B 2 B Q (1,1) Q (2,2) Q (,) 1 Q (1,1) Q (2,2) Q (,) 2 Q (1,1) Q (1,2) Q (1,) Q (2,1) Q (2,2) Q (2,) Q (,1) Q (,2) Q (,) B H B H B H where the -th row and column of D ( = 1,2,) are drlled,.e., except for the (,)-th element, the rest of the elements are zeros., 277

3 Let us defne C ( = 1,2, <, n = 1,2,), such that (,) = pn, T(n) () = qn, (6) (,) = rn, T(n) (,) = sn, (7) whereas the rest of the dagonal elements of are ones, and the rest of the off-dagonal elements are zeros. Here for a matrx A we use A() to denote ts ()-th entry. Based on successve Jacob rotatons [11, the transformaton matrces and the target matrces are updated va B n H B n, n = 1,2, (8) H, = 1,2,, n = 1,2, (9) Q (n 1,n 2 ) T (n 1) H Q (n 1,n 2 ) T (n 2), (n 1,n 2)=(1,2),(1,),(2,). (10) Notce that the updated andq (n 1,n 2 ) resemble ther old versons before the updatng n (9) and (10) except for the -th and -th columns and rows. Then nstead of the ont matrx transformaton defned at the very begnnng of ths secton, the followng one wth a reduced dmenson s consdered D = p q r s where for = 1,2 wth = and Q = Q = [ Q (1,1) Q (1,1) Q (2,1) Q (,1) H Q p q r s Q(2,2) Q(,) C 2 2, (11) C 6 6 (12) (,) (, ) (,), n = 1,2,, (1) (, ) Q (1,2) Q (2,2) Q (,2) Q (1,) Q (2,) Q (,) C 6 6. Each two-bytwo element matrx of Q s defned smlarly as (1). In addton, D n (11) s lned tod ( = 1,2,) va D D(,) D = () C 2 2. (14) D (,) D (,) Note thatd (,) s the desred component, and the rest of the elements on the-th row as well as the-th column should be nulled, where = 1,2,...,K. Therefore, for a certan par of (), only D and D (.e., = or = ) are relevant. It s then desred to fnd,.e.,pn,qn,rn, andsn (n = 1,2,) to acheve the nullng of D () and D () as well as the maxmzaton of D (,) and D (,). Ths s equvalent to max p n,q n,r n,s n (n=1,2,) ( D (1,1) 2 + D (2,2) 2), (15) and n the meantme mnmzng the off-dagonal elements of D C 2 2 and D C 2 2. For = 1 and = 2 (both and correspond to the ndces of cell nteror users), (n = 1, 2, ) are calculated separately, each smlar to the case of the sngle-cell mult-user MIMO downln n [7. It s due to the fact that the cell nteror users are only affected by the ntra-cell nterference, and the transmsson from each base staton to ts correspondng cell nteror users resembles that n a sngle-cell mult-user MIMO downln settng. Ths also leads to the bloc dagonal structure of Q 1 and Q 2 n (12). The desred s obtaned by equvalently solvng the channel dagonalzaton problem of a two-user system (wth respect tod andd ) where the maxmzaton of the sgnal-to-nterference rato s acheved [7. Frst compute the egenvalue decomposton (EVD) of Q C 2 2 and then defne W = V Λ V H, (16) = V Λ 1 2 matrx. Further calculate the EVD ofw H as a whtenng Q W contanng the egenvectors that correspond C 2 2 and obtane to ts egenvalues sorted n a descendng order. Fnally, p n, q n, r n, s n are obtaned va pn q n r n s n = W E a1 0, (17) 0 a 2 where a 1 and a 2 are chosen to normalze the two columns of W E, respectvely. On the other hand, f = 1 or 2 and = ( and correspond to the ndces of a cell nteror user and a cell edge user, respectvely), we follow the same phlosophy descrbed above for the case where = 1 and = 2 and propose a way of obtanng (n = 1,2,) ontly. We frst compute the EVD of Q C 2 2 for n = 1,2,, respectvely, and obtan V C 2 2 that contans the egenvectors of Q. The egenvectors correspond to the egenvalues sorted n a descendng order. Defne W = V (1,1) V (2,2) V (,) C 6 2, (18) and compute Σ = W H Q W C 2 2. Here Σ s a dagonal matrx whose dagonal elements are actually the sum of the correspondng egenvalues of Q C 2 2 (n = 1,2,). The whtenng matrx W s obtaned as W = W Σ 1 2 C 6 2. Then, we further calculate the EVD of W H Q W C 2 2 and obtan E C 2 2 that contans the egenvectors (the correspondng egenvalues are also sorted n a descendng order). Fnally, the unnown elements n are computed as p q r s [ 0 a1 = W E a 2 0. (19) Note that the two columns of W E should be normalzed such that the two-norm of each column s, as the number of cells 278

4 consdered n ths example s three. In addton. the permutaton of the columns ofw E n (19) results from the roles ofq andq n the computaton of as compared to (17). The transformaton matrces are ntalzed asb n = I forn = 1, 2,, respectvely. At the end of each sweep, the transformaton matrces and the target matrces are updated va (8), (9), and (10). For the convergence, we can trac ether the resdual nterference or the followng term ǫ = n=1,2, p n q n 2 + r n 2 + s n 1 2. (20) When the convergence s acheved,ǫs close to zero, and (n = 1,2,) are close to dentty matrces. Remars: () The proposed ont matrx transformaton scheme s especally effectve for the case where there s a sngle cell edge user, as shown n Secton 4. In such scenaros, t provdes a performance comparable wth that of a CBF algorthm. Due to the lmted spatal capacty of a base staton, user schedulng s usually conducted before precodng. A sngle cell edge user can be scheduled for each transmsson. () When there are more than one cell edge user, for a certan par of ndces (), there exsts a thrd case where both and correspond to the ndces of cell edge users. To obtan (n = 1,...,N wth N denotng the number of cells), a whtenng matrx s computed asw = V Λ 1 2, where the columns ofv are the egenvectors ofq, andλ s a dagonal matrx whose dagonal elements [ are the correspondng egenvalues. Fnally, the stacng of pn q n forn = 1,...,N (smlar as the left-hand sde of (19) r n s n a1 0 forn = ) s obtaned asw E, wheree 0 a contans 2 the two egenvectors ofw H Q W that correspond to ts largest and smallest egenvalues. 4. SIMULATION RESULTS AND CONCLUSIONS In ths secton, we evaluate the performance of the proposed ont matrx transformaton. Two multcast CoMP downln settngs are consdered. The frst s a three-cell case smlar to the example scenaro depcted n Fg. 2, whle the number of cell nteror users n each cell s three, leadng to the fact that there are ten users n total. In addton, a sx-user two-cell scenaro s nvestgated, where each cell serves two cell nteror users, and there are two cell edge users. In both scenaros, M T = M R = 4. The path loss of the cell edge user s assumed to be 10 tmes larger than that for the cell nteror users [6. Here the SNR s defned as P T/σ 2 n, where P T represents the transmt power of each cell, and σ 2 n denotes the nose varance. The global channel state nformaton s avalable at all base statons. For each settng, both the Extended FlexCoBF algorthm [6 and the proposed extenson of SeDJoCo are employed to compute the beamformer. The Frobenus norm square ofb n (n = 1,...,N) s scaled to P T to fulfll the total transmt power constrant and to guarantee a far comparson of these two methods. The resultng comparson of the sum rate performance of these two schemes are presented n Fg. 4. It can be seen that for the scenaro wth a sngle cell edge user, the extenson of SeDJoCo acheves a smlar performance as Extended FlexCoBF, ndcatng that ths new ont matrx transformaton s a very promsng lnear algebrac tool for beamformng n the multcast CoMP downln system. On the other hand, when there are two cell edge users, Extended FlexCoBF slghtly outperforms the Sum Rate [bps/hz Extended FlexCoBF extenson of SeDJoCo three cell one cell edge user ten users n total 20 two cell two cell edge users sx users n total SNR [db Fg. 4. Performance comparson between Extended FlexCoBF and the proposed ont matrx transformaton (averaged over 500 trals) extenson of SeDJoCo. Ths observaton leads to the conecture that when both and n a certan sweep correspond to the ndces of cell edge users, there mght exst better ways of computng (n = 1,...,N) such that the performance of the proposed ont matrx transformaton s mproved. Fg. 5 shows the convergence behavor of the proposed extenson of SeDJoCo. For each multcast CoMP downln settng descrbed ε three cell, one cell edge user, ten users n total two cell, two cell edge users, sx users n total number of teratons Fg. 5. Convergence behavor wth respect toǫof the proposed ont matrx transformaton (10 ndependent trals) above, 10 ndependent trals have been smulated. We observe that especally n the three-cell case wth a sngle cell edge user, the convergence of the extenson of SeDJoCo s fast. To obtan the sum rate results shown n Fg. 4, t s enough to set a threshold of ǫ as 10 5,.e., once ǫ falls below 10 5, no further teratons are needed. Note that as a CBF scheme, Extended FlexCoBF computes the transmt and receve beamformers ontly and teratvely [6. For example, wth a threshold for the resdual nterference set to 10 5 [6 and the maxmum number of teratons set to 100, the mean number of teratons requred by Extended FlexCoBF n the three-cell case s around 47. Due to lmted space, detaled complexty comparson between these two schemes s not ncluded n ths paper. Stll, t s worth mentonng that for each sweep (each par of()), the extenson of SeDJoCo consders ont matrx transformatons wth a reduced dmenson (cf. (11)). Consequently, the ncrease of complexty caused by the ncreased number of cells or users s lmted. In addton, for each teraton of Extended FlexCoBF, to update the receve beamformng vector for each cell edge user (e.g., (4) for User n the example scenaro depcted n Fg. 2), the base statons of neghborng cells have to exchange the precoded channels. By contrast, such an nformaton exchange s not requred for the proposed extenson of SeDJoCo, leadng to a smaller sgnalng overhead. 279

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