ow to Scale Up the Spectral Effcency of Mult-way Massve MIMO Relayng? Chung Duc o, en Quoc Ngo, Mchal Matthaou, and Trung Q. Duong School of Electroncs, Electrcal Engneerng and Computer Scence, Queen s Unversty Belfast, BT7 NN, Belfast, U.K. Department of Electrcal Engneerng (ISY), Lnöpng Unversty, 58 83 Lnöpng, Sweden Emal:{choduc0, m.matthaou, trung.q.duong}@qub.ac.u, hen.ngo@lu.se arxv:703.0697v [cs.it] 30 Mar 07 Abstract Ths paper consders a decode-and-forward (DF) mult-way massve multple-nput multple-output (MIMO) relay system where many users exchange ther data wth the ad of a relay staton equpped wth a massve antenna array. We propose a new transmsson protocol whch leverages successve cancelaton decodng and zero-forcng (ZF) at the users. By usng propertes of massve MIMO, a tght analytcal approxmaton of the spectral effcency s derved. We show that our proposed scheme uses only half of the tme-slots requred n the conventonal scheme (n whch the number of tme-slots s equal to the number of users []), to exchange data across dfferent users. As a result, the sum spectral effcency of our proposed scheme s nearly double the one of the conventonal scheme, thereby boostng the performance of mult-way massve MIMO to unprecedented levels. Index Terms Amplfy-and-forward, decode-and-forward, maxmum-rato processng, mult-way relay massve MIMO. I. INTRODUCTION In the past few years, massve MIMO technology has attracted sgnfcant research attenton for ts ablty to mprove the spectral and energy effcency [], [3]. In massve MIMO systems, many users can be served by a base staton equpped wth very large antenna arrays. Wth very large antenna arrays at the base staton, the channels between dfferent users become parwse orthogonal, and hence, the nose and nteruser nterference reduce notceably wthout mprovng the complexty of the system [3]. Furthermore, by usng tme dvson duplex (TDD) mode, the channel estmaton overhead depends only on the number of actve users regardless of number of base staton antennas [4]. Ths maes massve MIMO scalable and become one of the ey canddates for future wreless communcaton systems. On a parallel avenue, mult-way relayng networs have also been nvestgated to enhance the robustness aganst the channel varatons n dstngushed areas, where the drect channels among users are unavalable due to large obstacle and/or heavy path loss n the propagaton envronment [5]. Wth the help of the relay staton, users that are geographcally separated can communcate or exchange ther data-bearng symbols much easer. Moreover, a sgnfcant number of papers demonstrate that mult-way relayng networs provde much hgher spectral effcency and communcaton relablty compared to one-way or two-way relayng systems [6], [7]. The combnaton of mult-way relayng and massve MIMO s very promsng snce t reaps all benefts of both technologes. Recently, some papers have evaluated the performance of mult-way relayng networs wth massve arrays at the relay [8], [9]. In these wors, the authors showed that multway massve MIMO relay systems can offer huge spectral and energy effcency. In addton, by usng smple lnear processng (e.g. ZF and maxmum rato processng) and employng a large number of antennas at the relay staton, the transmt power of each user can be scaled down proportonally to the number of relay antennas, whle mantanng a gven qualty of servce. owever, all of aforementoned studes consdered a conventonal transmsson protocol whch requres K tmeslots to exchange data among K users. Dfferent wth prevous wors, n ths paper we propose a novel transmsson protocol for mult-way massve MIMO relay networs whch requres only K tme-slots for the nformaton exchange among the K users. We consder the DF operaton at the relay, and assume that the relay and the users have perfect nowledge of the channel state nformaton (CSI). We derve an approxmate closed-form expresson for the spectral effcency. The approxmaton s shown to be very tght, especally when the number of relay antennas s large. Notatons: Matrces and vectors are expressed as upper and lower case boldface letter, respectvely. The superscrpts ( ) and Tr( ) stand for ermtan transpose and the trace, respectvely. We denote by a the -th column of matrx A. The symbol ndcates the norm of a vector. The notaton E{ } s the expectaton operator. The notaton [A] mn or a mn denotes the (m, n)-th element of matrx A, and I K s the K K dentty matrx. II. SYSTEM MODEL We consder a DF mult-way relay networs wth a very large antenna array at the relay staton. The system ncludes one relay staton equpped wth M antennas and K sngleantenna users. The bearng-messages from K users are exchanged wth the help of the relay staton. Each user wants to detect the sgnals transmtted from K other users. We assume that the users and the relay staton operate n halfduplex mode and now perfectly CSI. Furthermore, we assume that the drect lns (user-to-user lns) are unavalable due to large path loss and/or severe shadowng.
The channel matrx between the K users and M antennas at the relay s denoted by G C M K and s modeled as G D /, () where C M K models small-scale fadng wth ndependent CN (0, ) components, and D C K K s the dagonal matrx of large-scale fadng (path loss and log-normal attenuaton). Let g m and h m be the (m, )-th element of G and, respectvely. Then g m β h m, () where β s the -th dagonal element of D. In general, the transmsson protocol s dvded nto two phases: multpleaccess phase and broadcast phase. In the multple-access phase, all K users transmt sgnals to the relay staton. In the broadcast phase, the relay staton broadcast sgnals (whch are decoded n the mult-access phase) to the users. In the next sectons, we wll frst present the conventonal transmsson protocol, followed by the proposed transmsson scheme. III. CONVENTIONAL TRANSMISSION PROTOCOL In ths secton, we frst summarze a conventonal transmsson protocol talored to mult-way massve DF relayng networs. The upln and downln spectral effcences are then provded n closed-form. A. Multple-Access Phase Ths phase requres only one tme-slot. All the K users transmt ther data to the relay n the same tme-frequency resource. The M receved vector at the relay s y R P u Gx n R, (3) where x [x, x..., x K ] T s the sgnal vector transmtted from the K users, wth E { xx } I K, n R s the nose vector wth..d. CN (0, ) elements, and P u s the normalzed transmt power of each user. After recevng the transmtted sgnals from the K users, the relay employs maxmum rato combnng scheme by multplyng y R wth G as follows: r G y R P u G Gx G n R. (4) Then, the -th element of r, denoted by r, s used to decode the sgnal transmtted from user. From (4), r s gven by r P u g x P u K g g x g n R, (5) where g s the -th column of G. Therefore, the upln spectral effcency of the system n (5) (measured n bt/s/z) s gven by R ul E log P u g 4. (6) P u g g By usng Jensen s nequalty, a closed-form expresson lower bound of the spectral effcency (6) s gven by [3, Eq. (6)] R ul R ul log P u(m )β K. (7) P u β B. Broadcast Phase In ths phase, the relay staton transmts all sgnals decoded n the multple-access phase to all users n K tme slots. In the t tme-slot, the relay ams to transmt x j(,t) to user,,..., K, where { ( t) modulo K, f ( t) K j(, t) (8) K, otherwse. More precsely, n the t-th tme-slot, the relay staton transmts s (t) β g x j(,t), (9) where s the normalzed transmt power at the relay. Then, the receved sgnal at the -th user s y (t) g s (t) n (t) β g g x j(,t) n (t), (0) respectvely. The -th user nows ts own transmtted sgnal x (or x j( t,t) ), so t can remove the self-nterference pror to decodng. The receved sgnal after self-nterference cancelaton s ỹ (t) β g x j(,t) β j(,t) j(,t),j( t,t) g g x j(,t) n (t). () The correspondng downln spectral effcency for the t-th tme-slot s E log β g 4 β. () g j(,t) j(,t),j( t,t) Proposton : The spectral effcency gven by () can be lower bounded by log (M )(M )β (M )β β M K j(,t) j(,t),j( t,t) β. (3)
Proof: See Appendx VII-A. IV. MULTI-WAY TRANSMISSION WIT SUCCESSIVE CANCELATION DECODING In ths secton, we propose a novel transmsson scheme whch requres only K tme-slots for the nformaton exchange among the K users. A. Multple-Access Phase The multple-access phase s the same as the one of conventonal transmsson scheme. See Secton III-A. B. Broadcast Phase ere, we need only K tme-slots to transmt all K symbols to all users. The man dea s that: at a gven tme-slot, the -th user subtracts all symbols decoded n prevous tmeslots pror to decodng the desred symbol. Furthermore, after K K tme-slots, user receves sgnals, and each sgnal s a lnear combnaton of K K symbols. So t can detect all K K symbols wthout any nter-user nterference through the zero-forcng technque. A detaled presentaton of the proposed scheme s now provded. ) Frst tme-slot: The relay ntends to send x j(,) to the - th user, for,..., K. The sgnal vector transmtted from the relay s s () β g x j(,). (4) Thus, the receved sgnal at the -th user s y () g s () n () β g g x j(,) n (), (5) where n () CN (0, ) s the addtve nose at the -th user n the frst tme-slot. Snce user nows ts transmtted sgnal x (or x j(,) ), t can subtract the self-nterference before detectng sgnal x j(,). Therefore, the receved sgnal at user after self-nterference cancelaton s where ỹ () β g x j(,) β g j(,) / V, g x j(,) n (), (6) V,t {j( t, t), j( t, t),..., j(, t)}. (7) Then, the correspondng spectral effcency s gven by R dl,() E log β g 4 β. (8) g j(,) / V, ) Second tme-slot: The relay ntends to send x j(,) to the -th user, for,..., K. The sgnal vector transmtted from the relay s s () β g x j(,), (9) and hence, the sgnal receved at the -th user s y () g s () n () β g g x j(,) n (). (0) The -th user nows ts own transmtted symbol x as well as the symbol detected n the frst tme-slot x j(,), so t can subtract these symbols before detectng the desred sgnal x j(,). The receved sgnal at the -th user after subtractng the above symbols s ỹ () β g x j(,) β g j(,) / V, g x j(,) n (). () Then, the spectral effcency of user at the second tmeslot s R dl,() E log β g 4 β. () g j(,) / V, 3) t-th tme-slot: At the t-tme-slot, the relay ntends to send x j(,t) to the -th user, for,..., K. The sgnal vector transmtted from the relay s s (t) Then, the -th user sees y (t) β g s (t) n (t) β g x j(,t). (3) g g x j(,t) n (t). (4)
The -th users now ts own transmtted symbols x. Furthermore, t also nows ts detected symbols n prevous tmeslots. So t nows {x j(,), x j(,), x j(,),..., x j(,t ) }, and, hence, t can remove these symbols to obtan ỹ (t) β g x j(,t) β g j(,t) / V,t g x j(,t) n (t). (5) Then, the spectral effcency of the -th user at the t-th tme-slot s E log β g 4 β. (6) g g j(,t) / V,t Proposton : The spectral effcency gven by (6) can be lower bounded by log (M )(M )β (M )β β M K j(,t) / V,t β. (7) Proof: Followng a smlar methodology as the proof of Proposton. 4) After t K tme-slots, the -th user has receved t sgnals (the t-th receved sgnal s gven by (4)). Furthermore, t has decoded t symbols. So t can subtract all t detected symbols from each receved sgnal to obtan the followng results:, Pr g β g j(,)x j(,t ) n (t ),,., Pr,t β Pr β j(,t ) / V,t j(,t ) / V,t j(,t ) / V,t g g j(,)x j(,t ) n (t ),, g g j(,t )x j(,t ) n (t ),t. (8) We can see that we have t equatons, each equaton has (K t ) unnown varables {x j(,t )}. Snce t K, the number of equatons s greater than or equal to the number of unnown varables. Therefore, the -th user can detect all remanng (K t ) symbols {x j(,t )} va the ZF scheme as follows. Denote by, n (t ), n (t ),,., n(t )., (9),t n (t ),t g g j(,) g g j(,)... g g j(,k t ) g A g j(,) g g j(,3)... g g j(,k t )..., g g j(,t ) g g j(,t )... g g j(,k ) (30) and x [ x j(,t ) x j(,t )... x j(,k ) ] T. (3) Then, (8) can be rewrtten n matrx-vector form as β A x n (t ). (3) The -th user apples the ZF scheme to decode the remanng symbols as follows: r (t ) Z T β Z T A x Z T n (t ), (33) where Z T ( A A ) A. (34) The n-th element of r (t ) wll be used to detect x j(,t n). From (33) and the fact that Z T A I K (t ), the n-th element of r (t ) s gven by r (t ),n β x j(,t n) z T ) n n(t. (35) Thus, the correspondng spectral effcency of the system n (35) s E log E log R dl,(t n) β β z n [ (A A ) ] nn. (36) Snce (36) has a complcated form that nvolves a matrx nverse, we cannot obtan an exact closed-form. owever, thans to the trace lemma and law of large numbers (as M goes to nfnty) [0], we can obtan the followng approxmatng result. Proposton 3: As M, the spectral effcency R dl,(t n) gven by (36) converges to ( R dl,(t n) log P t rβ β ) j(,n ) K β. (37)
Proof: See Appendx VII-B. V. NUMERICAL RESULTS In ths secton, we provde numercal results to evaluate the performance of our proposed scheme. We consder the sum spectral effcency, defned as SE sum K ( ) t mn R ul,, (38) t where t K s the t -th tme-slot of the transmsson protocol n broadcast phase. Frst, we examne the tghtness of our analytcal results. Fgure shows the sum spectral effcency of our proposed scheme versus the number of relay antennas wth dfferent K for the smple case β,. The analyss curves represent our analytcal results obtaned by usng the lower bounds (7), (7), and the asymptotc result (37). The smulaton curves are generated from the outputs of a Monte- Carlo smulator usng (6), (6), and (36). We can see that the proposed approxmaton s very tght, even wth small number of antennas. Furthermore, as expected, the sum spectral effcency ncreases sgnfcantly when the number of relay antennas ncreases. We next compare the performance of our proposed scheme wth the one of the conventonal DF scheme (Secton III-B) and the conventonal AF scheme n [] (see Fg. ). We can see that our proposed scheme sgnfcantly outperforms other schemes. The sum spectral effcency of our proposed scheme mproves by factors of nearly and 3 compared wth the conventonal DF scheme and the conventonal AF scheme, respectvely. Ths s due to the fact that wth the conventonal DF scheme, we need n total K tme-slots to exchange the nformaton among the K users, whle wth our proposed scheme, we need only K. Fnally, we consder a more practcal scenaro where the large-scale fadng β changes dependng on the locatons of users and the shadow fadng. To generate the large-scale fadng, we use the same model as n []. Fgure 3 llustrates the cumulatve dstrbuton of the sum spectral effcency of our proposed scheme for K 5, 7, and 0. As expected, the sum spectral effcency ncreases when K ncreases. The 95%-lely sum spectral effcency wth K 0 s about 4.5 bt/s/z whch s nearly 4 tmes and tmes hgher than that wth K 5 and K 7, respectvely. VI. CONCLUSION We proposed a novel and useful transmsson scheme for mult-way massve MIMO relay systems wth decode-andforward protocol at the relay. Whle the conventonal scheme needs K tme-slots to exchange all data among K users, our proposed scheme, whch s based on successve cancelaton decodng, needs only K tme-slots. Thus, the sum spectral effcency of our proposed scheme s nearly double the sum spectral effcency of the conventonal scheme. Fg.. The sum spectral effcency of the system model wth dfferent K versus the number of relay antennas. We set P u 0 db, 0 db, β. Fg.. The comparson of the sum spectral effcency wth dfferent schemes versus the number of relay antennas. We choose P u 0 db, 0 db, K 0, β. A. Proof of Proposton VII. APPENDICES By usng Jensen s nequalty, we obtan where X (t) ( ( { log E β j(,t) j(,t),j( t,t) X (t) g }) ), (39) β g 4. (40)
Fg. 3. Cumulatve dstrbuton of the sum spectral effcency for dfferent K. We choose P u 0 db, 0 db, M 00. By dvdng the numerator and the denomnator of the rghthand sde of (40) by g, we get g { } E X (t) j(,t) j(,t),j( t,t) E g, (4) g 4 β where g g g g. Condtoned on g, g s Gaussan dstrbuted wth zero mean and varance β. Snce the varance of g does not depend on g, g s a CN (0, β ) random varable and s ndependent of g. Therefore, { E X (t) } j(,t) j(,t),j( t,t) { E g } { } E g E g 4. (4) β { } By usng [, Lemma.0], we obtan E g { } M M and E g 4 (M ) 3 (M ), and hence, we arrve at (3). B. Proof of Proposton 3 From (30), the (m, n)-th element of A A s gven by [ A A ] t mn g g j(, m) gj(, n) g. (43) Usng the trace lemma [0, Lemmas 4, 5], we have M g g j(,) gj(,) g β ( ) M Tr g j(,) gj(,) a.s. 0, M (44) where a.s. stands for almost sure convergence. ( Snce Tr g j(,) gj(,) large numbers, we get ( M Tr g j(,) gj(,) The substtuton of (45) nto (44) yelds ) g j(,), and from the law of ) a.s. M β j(,). (45) M g g j(,) gj(,) g a.s. β β j(,). (46) M Smlarly, we obtan M g j(,) g g a.s. g j(,) 0. (47) M From (43), (46), and (47), we have [ (A ) ] M A nn a.s. β t β j(,n ) Substtutng (48) nto (36), we obtan (37). ACKNOWLEDGMENT. (48) Ths wor was supported by project no. 38/QD-UBND, Bnh Duong government, Vetnam. The wor of. Q. Ngo was supported by the Swedsh Research Councl (VR) and ELLIIT. The wor of M. Matthaou was supported n part by the EPSRC under grant EP/P000673/. The wor of T. Q. Duong was supported by the U.K. Royal Academy of Engneerng Research Fellowshp under Grant RF45\4\, and by the EPSRC under Grant EP/P09374/. REFERENCES [] C. D. o,. Q. Ngo, M. Matthaou, and T. Q. Duong, On the performance of zero-forcng processng n mult-way massve MIMO relay networs, to appear IEEE Commun. Letters, 07. [] E. G. Larsson, O. Edfors, F. Tufvesson, and T. L. Marzetta, Massve MIMO for next generaton wreless systems, IEEE Commun. Mag., vol. 5, no., pp. 86 95, Feb. 04. [3]. Q. Ngo, E. G. Larsson, and T. L. Marzetta, Energy and spectral effcency of very large multuser MIMO systems, IEEE Trans. Commun., vol. 6, no. 4, pp. 436 449, Apr. 03. [4] T. L. Marzetta, E. G. Larsson,. Yang, and. Q. Ngo, Fundamentals of Massve MIMO. Cambrdge Unversty Press, 06. [5] D. Gündüz, A. Yener, A. Goldsmth, and. V. Poor, The multway relay channel, IEEE Trans. Inf. Theory, vol. 59, no., pp. 5 63, Jan. 03. [6] Y. Tan and A. Yener, Degrees of freedom for the MIMO mult-way relay channel, IEEE Trans. Inf. Theory, vol. 60, no. 5, pp. 495 5, May 04. [7] A. Amah and A. Klen, Non-regeneratve mult-way relayng wth lnear beamformng. n Proc. IEEE PIMRC, Sep. 009, pp. 843 847. [8] G. Amarasurya, E. G. Larsson, and. V. Poor, Wreless nformaton and power transfer n mult-way massve MIMO relay networs, IEEE Trans. Wreless Commun., vol. 5, no. 6, pp. 3837 3855, June 05. [9] G. Amarasurya and. V. Poor, Mult-way amplfy-and-forward relay networs wth massve MIMO, n Proc. IEEE PIMRC, Sep. 04, pp. 595 600. [0] S. Wagner, R. Coullet, M. Debbah, and D. T. Sloc, Large system analyss of lnear precodng n MISO broadcast channels wth lmted feedbac, IEEE Trans. Inf. Theory, vol. 58, no. 7, pp. 4509 4537, July 0. [] C. o,. Q. Ngo, M. Matthaou, and T. Q. Duong, Mult-way massve MIMO relay networs wth maxmum-rato processng, n Proc. IEEE SgTelCom, Jan. 07, pp. 4 8. [] A. M. Tulno and S. Verdú, Random matrx theory and wreless communcatons, Foundatons and Trends n Commun. and Inf. Theory, vol., no., pp. 8, Jun. 004.