On Uplink-Downlink Sum-MSE Duality of Multi-hop MIMO Relay Channel

Transcription

1 On Upn-Downn Sum-MSE Duat of Mut-hop MIMO Rea Channe A Cagata Cr, Muhammad R. A. handaer, Yue Rong and Yngbo ua Department of Eectrca Engneerng, Unverst of Caforna Rversde, Rversde, CA, 95 Centre for Wreess Communcatons, Unverst of Ouu, Ouu, nand Department of Eectronc & Eectrca Engneerng, Unverst Coege ondon, ondon, WCE 7JE, U Department of Eectrca and Computer Engneerng, Curtn Unverst, Bente, WA 60, Austraa Ema: {acr, hua}@ee.ucr.edu, m.handaer@uc.ac.u,.rong@curtn.edu.au Abstract The upn and downn sum mean-squared error MSE duat for mut-hop ampf-and-forward A mutpenput mutpe-output MIMO rea channes s estabshed, whch s a generazaton of severa sum-mse duat resuts. Une the prevous resuts that prove the duat b computng the MSEs for each stream drect, we ntroduce an nterestng perspectve to the reaton of the upn-downn duat based on the arush-uhn-tucer T condtons assocated wth upn and downn transcever desgn optmzaton probems. Index Terms Ampf-and-forward, duat, MIMO rea. I. INTRODUCTION One of the e technques to sove the downn optmzaton probems s to transform the downn probem nto an upn probem va upn-downn duat reatonshp, and sove t n the upn doman snce the upn channe has a smper mathematca structure, and ess coupng of varabes. The MSE duat for a snge-hop was estabshed under a sum-power constrant when perfect channe state nformaton CSI s avaabe at a the nodes n the sstem n []-[], and for mperfect CSI n [3]-[5]. It has been shown that an MSE pont achevabe n the upn can aso be acheved n the downn under the sum-power constrant. Recent, the upn-downn sum-mse duat for snge-hop sstems []- [5] has been extended to two-hop and mut-hop A MIMO rea sstems n [6] and [7], respectve. Due to the mut-hop topoog, MSE s a compcated functon of the source, rea and recever matrces, whch maes both the proof of duat and the optmzaton probems assocated wth mut-hop MIMO rea networs much more chaengng than the exstng wors wth smper networ topoog. As a drect appcaton of the duat resuts, the compcated downn MIMO mut-hop transcever source precodng, rea ampfng and recever matrces desgn probems can be carred out effcent b focusng on an equvaent upn MIMO mut-hop rea sstem [7], [8]. A. Contrbutons of Ths Wor MSE duat n []-[4] and [7] s estabshed b cacuatng the MSE of each stream of a users drect. ere, we estabsh the upn-downn duat based on the Note that sgna-to-nterference-nose rato SINR duat for mut-hop A MIMO rea sstems has been estabshed n [8]. s s s B G B B G G g.. v x v 3 x 3 x v Upn mut-hop A MIMO rea sstem. T condtons of the upn and downn transcever optmzaton probems, whch s an nterestng perspectve to the reaton of the upn-downn duat. The duat resut estabshed n ths paper generazes the resuts n [5] and [6], whch aso use T condtons to prove the sum-mse duat for snge-hop and twohop MIMO channes, respectve. 3 The sum-mse duat for mut-hop A MIMO rea sstems n [7] s estabshed under the assumpton that recevers empo near mnmum MSE MMSE recevers, the sum-mse duat resut n ths paper s appcabe to an nd of near recever. The notatons used n ths paper are as foows. T and denote transpose and conugate transpose, respectve. E[ ], I N and tr denote the statstca expectaton, N N dentt matrx and trace, respectve. or matrces A, A A...A. or exampe, 3 A A A A 3 and 3 A A 3 A A. A A...A for and s equa to dentt matrx for >. II. SYSTEM MODE Smar to the sstem mode n [7]-[8], we consder a wreess communcaton sstem wth users, haf-dupex A rea nodes, and one base staton BS node, where each node s equpped wth mutpe antennas. The number of antennas at the th rea node of the upn sstem s N,,..., and the BS s equpped wth N antennas. Due to the path-oss n the wreess channes, we assume that the sgna transmtted b the th node can on be receved b the th node, so the sgna transmtted from the source node trave through hops to reach to ts destnaton. The th user s equpped wth M antennas, and transmts receves M ndependent data streams. W W s ˆ s ˆ

2 A. Upn MIMO Rea Sstem The upn MIMO mut-hop rea sstem s shown n g.. The data streamss C M s near precoded b the th user wth the source precodng matrx B C MM and the th user transmts the precoded sgna vector u B s to the frst rea node. We assume compex, zero mean, ndependent [ and dentca dstrbuted..d. data streams ] wth E s s I M. The receved sgna at the frst rea node s gven b G B s v where G C NM,,...,, s the channe between the frst rea node and the th user and v s the N..d. addtve whte Gaussan nose AWGN vector at the frst rea. The th rea node,,...,, appes C N N to ampf and forward the receved sgnas, whch s gven b x,,..., where C N s the sgna that th rea node receves,,...,. rom and, the receved sgna vector at the rea nodes,,...,, and the receved sgna vector at the BS can be wrtten as A G B s v,,..., 3 where A s the equvaent channe matrx between the frst rea node and the th rea node, and v s the equvaent nose vector gven b { A,,..., 4 I N,, v { v v,,..., 5 v,. ere C N N,,...,, s the channe matrx at the th hop, and v s the..d. AWGN at the -th node of the upn sstem,,...,. We assume that a noses are compex sgnas wth zero mean and unt varance. rom 5, the covarance matrx of v can be wrtten as, C E [ v v ] I N,,...,, I N,. 6 To estmate the data streams transmtted, the BS appes a near recever,.e., ŝ W, whch s gven b [ ] ŝ W A G B s v,,..., 7 where W s the weght matrx of the near recever of sze M N. rom 3 and 7, the MSE matrx of the th user, s ˆ s ˆ E T T Z x x 3 Z 3 Z G n g.. [ s E be wrtten as E n n G G n n Downn mut-hop A MIMO rea sstem. ŝ s D D D s ˆ s ˆ ˆ s ] ŝ,,..., can I M W A G B B G A W W [ A A A C ] W 8 where A G B B G. The transmsson power consumed at the th rea node s [ tr E x x ] tr A A A C 9 The upn transcever optmzaton probem s formuated as: tr E 0 mn,b,w s.t. tr B B, tr A A A C, where and are the tota transmt power at the users and transmsson power constrants at each rea node, respectve, and P,,...,, are the power mt. B. Downn MIMO Rea Sstem The downn communcaton sstem s shown n g.. The BS near precodes the data streams of user, s C M wth the matrx T C NM and transmts the N precoded sgna vector T s. We assume compex [ data streams wth zero mean,..d data streams wth ] E s s I M. The sgna vector receved of sze N at the frst rea node of the downn sstem can be wrtten as T s n 3. where n C N s the AWGN vector at the frst rea. The th rea node n the downn sstem,,..., appes Z C N N to ampf and forward the receved sgnas,.e., x Z,,...,, where C N,,...,, s the receved sgna vector at the th rea node n the downn channe and s expressed as T s n,,...,.4

3 ere s the equvaent channe matrx between the frst rea node and the th rea node n the downn channe and n s the equvaent nose vector gven b { m m Z m,,..., 5 I N,, m m Z m n n n,,..., 6 n, where n s the..d. AWGN vector at the th rea node,,...,. The receved sgna vector at the th user,..., can be expressed as G Z n G Z T s n 7 where n G Z n n s the equvaent nose vector at the th user. rom 6, the covarance matrx of n, C, at the th rea node,,..., and the covarance matrx of n, C, at the th user can be wrtten as m Z m Z m m C m m I N,,...,, 8 C G Z C Z G I M. 9 To estmate the data streams s, th user appes a near recever matrx D C MM,.e., ŝ D,,...,, whch s wrtten as ŝ D G Z T s D n. 0 The MSE matrx of the th user,,...,,.e., E E[ s ] ŝ s ŝ can be wrtten as E I M D G Z T T Z G D [ ] D G Z A Z G C D where A T T. The transmsson power consumed at the th rea node [ tr E x x ]tr Z A C Z. The downn transcever optmzaton probem s formuated: tr E mn Z,T,D s.t. tr T T, 3 tr Z A C Z, 4 where 3 and 4 are the tota transmt power at the users and transmsson power constrants at each rea node, respectve, and P,,...,, are the power mt. III. UPIN-DOWNIN DUAITY The optmzaton probems 0- and -4 are both non-convex, but the obectve functons and constrants of them are contnuous dfferentabe. Thus the upn-downn duat can be estabshed based on ther T condtons [5]. A. The T Condtons of the Upn Probem The agrangan functon of 0- can be wrtten as tr E λ tr B B P 5 λ tr A A A C P whereλ andλ,,...,, are the agrange mutpers of the power constrants n and. The gradent functon of 5 wth respect to B,,W s gven b G A W λ I M λ G A A G G A W W A G B, 6 m m m m m λ m m W B G m m m m m W W A A m m m m W W λ A A A C m m m m m m A A m m m m m, 7 B G A W A A A C, 8 where we have used the denttes from [9] that traz RZ trbz A, RZ B T, traz IZ A, trbz IZ B T and

4 [ ] dfz dz fz Rz fz Iz. ere. The other T condtons assocated wth 0- are gven beow λ tr B B P 0 9 λ tr A A A C P 0 30 λ 0, tr B B 3 λ 0, tr A A A C.3 emma. [Reaton between the agrange mutpers, and the rea ampfng and receve matrces.] or an soutons satsfng the T condtons 6-3, the agrange mutpers are λ tr λ λ tr W W P m tr P 33 W W λ I N m m P m m m λ m 34 W W m m m m λ I N,,...,.35 m Proof: See Appendx A n [0]. B. The T Condtons of the Downn Probem The agrangan functon of -4 can be wrtten as tr E α tr T T P 36 α tr Z A C Z P whereα andα,,...,, are the agrange mutpers of the power constrants n 3 and 4. The gradent functon of 36 wth respect to T,Z,D s gven b Z G D α I M α Z Z Z G D D G Z T,37 X G D T Y Y c X G D D G Z A Y X G D D G Z α Z α X Z n nm T Z G D c c m A Y C A Y m Z my n m Z m, 38 G Z A Z G C.39 The other T condtons assocated wth the probem - 4 for,..., are gven beow α tr T T P 0 40 α tr Z Z A Z C Z P 0 4 α 0, tr T T 4 α 0, tr Z Z A Z C Z. 43 In 38, X c and Y c are defned as X c Y c { c m m Z m, otherwse 44 I N, c { m c Z m m, otherwse I N, c. 45 emma. or an soutons satsfng the T condtons 37-43, the agrange mutpers are α tr D D P 46 tr Z G D D G α I M Z α 47 α P tr Z P Z G D D G Z α Z Z α I N Z, 3,...,. 48

5 Proof: Smar to the proof of emma, emma can aso be proved eas. 0 0 Upn Downn C. Sum-MSE Upn-Downn Duat Theorem. Assume that the upn transcever matrces, { },{B },{W } satsf the upn T condtons 6-3. et T /λ W, D λ B, Z λ /λ,,...,. Then, when the power constrant of the th node of the downn channe s swapped wth the power constrant of the - th node of the upn channe,.e., P P,,...,, sum-mse acheved b { },{B },{W } can aso be acheved b the downn transcever matrces, {Z },{T },{D }, whch satsf the downn T condtons Converse, assume that the downn transcever matrces {Z },{T },{D } satsf the T condtons et B /α D, W α T and α /α Z,,...,. Then, when the power constrant of the th node of the upn channe s swapped wth the power constrant of the -th node of the downn channe,.e., P,,...,, the sum-mse acheved b P {Z },{T },{D } can aso be acheved b the upn transcever matrces { },{B },{W }, whch satsf the upn T condtons 6-3. Proof: See Appendx B n [0]. Theorem shows that sum-mse acheved b a transcever desgn that satsfes the T condtons of an upn optmzaton probem, can aso be acheved b a transcever desgn that satsfes the T condtons of a downn optmzaton probem, and vce versa. Therefore, the downn transcever optmzaton probems can be soved through sovng an equvaent upn probem, and vce versa. Snce the upn and downn optmzaton probems are non-convex, the T condtons are on necessar for oca mnmums n both channes. And b Theorem, ever possbe oca mnmum satsfng the T condtons of the upn sum-mse corresponds to a same oca mnmum n the downn. In other words, f the upn transcever matrces acheve a oca optmum of the upn sstem, the are aso oca optma for the downn. IV. NUMERICA EXAMPES In ths secton, we smuate fve-hop mutuser MIMO rea sstems. or smpct, we assume a users have the same number of antennas.e., M M,,, and a rea nodes and the destnaton node n the upn have the same number of antennas.e., N N,,,. We set P P 0dB and assume that P P P,,,. A smuaton resuts are averaged over 000 channe reazatons. We use the teratve agorthm n [] to desgn the optma upn transcevers { },{B },{W } and use the proposed duat resut to obtan the optma downn transcevers {Z },{T },{D }. g. 3 shows the MSE performance of the upn and downn sstems versus MSE P db g. 3. MSE versus P. 3, M, N 0, P P P P,,,. P 0dB, P wth 3, M, and N 0. It can be seen from gs. 3 that the curves overap, ndcatng that both the upn and downn sstems acheve the same sum-mse. V. CONCUSION We have estabshed the upn-downn sum-mse duat n a mut-hop A MIMO rea sstem, whch s a generazaton of severa sum-mse duat resuts. B anazng the T condtons of the upn and downn mnmum sum- MSE transcever optmzaton probems, t s shown that both the upn and the downn sstems share the same achevabe sum-mse regon. REERENCES [] S. Sh and M. Schubert, MMSE transmt optmzaton for mut-user mut-antenna sstems, n Proc. IEEE Int. Conf. Acoustcs, Speech, and Sgna Process. ICASSP, Phadepha, PA, Mar [] A. hachan, A. J. Tenenbaum, and R. Adve, near processng for the downn n mutuser MIMO sstems wth mutpe data streams, n Proc. IEEE Int. Conf. Commun., vo. 9, pp , June 006. [3] M. Dng and S. D. Bosten, Upn-downn duat n normazed MSE or SINR under mperfect channe nowedge, n Proc. IEEE GOBECOM 07, pp , Nov [4] M. B. Shenouda and T. Davdson, On the desgn of near transcevers for mutuser sstems wth channe uncertant, IEEE J. Se. Areas Commun., vo. 6, no. 6, pp , Aug [5] M. Dng and S. D. Bosten, Reaton between ont optmzatons for mutuser MIMO upn and downn wth mperfect CSI, IEEE ICASSP, pp , Apr [6] J. u and Z. Qu, Sum MSE upn-downn duat of mutuser ampf-and-forward MIMO rea sstems, n Proc. IEEE Vehcuar Techn. Conf. VTC a, pp.-5, Sept. 0. [7] A. C. Cr, Y. Rong, Y. Ma, and Y. ua, On MAC-BC duat of muthop MIMO rea channe wth mperfect channe nowedge, to appear n IEEE Trans. Wreess Commun, Apr 04. [8] Y. Rong and M. R. A. handaer, On upn-downn duat of muthop MIMO rea channe, IEEE Trans. Wreess Commun., vo. 0, pp , Jun. 0. [9] A. orungnes and D. Gesbert, Compex-vaued matrx dfferentatons: Technques and e resuts, IEEE Trans. Sgna Process., vo. 55, no. 6, pp , Jun [0] A. C. Cr, Muhammad R. A. handaer, Y. Rong and Y. ua, On upn-downn sum-mse duat of mut-hop MIMO rea channe, unpubshed. [] M. R. A. handaer and Y. Rong, Jont transcever optmzaton for mutuser MIMO rea communcaton sstems, IEEE Trans. Sgna Process., vo. 60, pp , Nov. 0.