A Tractable and Accurate Cross-Layer Model for Multi-Hop MIMO Networks

Transcription

1 Ths fu text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for pubcaton n the IEEE INFOCOM 2010 proceedngs Ths paper was presented as part of the man Technca Program at IEEE INFOCOM A Tractabe and Accurate Cross-Layer Mode for Mut-Hop MIMO Networks Ja Lu Y Sh Y. Thomas Hou Bradey Department of Eectrca and Computer Engneerng Vrgna Poytechnc Insttute and State Unversty, Backsburg, VA Abstract MIMO-based communcatons have great potenta to mprove network capacty for mut-hop wreess networks. Athough there has been sgnfcant progress on MIMO at the physca ayer or snge-hop communcaton, advances n the theory of MIMO for mut-hop wreess networks reman mted. Ths stagnaton s many due to the ack of an accurate and more mportant, anaytcay tractabe mode that can be used by networkng researchers. In ths paper, we propose such a mode to enabe the networkng communty to carry out crossayer research for mut-hop MIMO networks. In partcuar, at the physca ayer, we deveop a smpe mode for MIMO channe capacty computaton that captures the essence of spata mutpexng and transmt power mt wthout nvovng compex matrx operatons and the water-fng agorthm. We show that the approxmaton gap n ths mode s neggbe. At the nk ayer, we devse a space-tme schedung scheme caed OBIC that sgnfcanty advances the exstng zero-forcng beamformng (ZFBF) to hande nterference n a mut-hop network settng. The proposed OBIC scheme empoys smpe agebrac computaton on matrx dmensons to smpfy ZFBF n a mut-hop network. As a resut, we can characterze nk ayer schedung behavor wthout entangng wth beamformng detas. Fnay, we appy both the new physca and nk ayer modes n cross-ayer performance optmzaton for a mut-hop MIMO network. I. INTRODUCTION Snce ts ncepton [1], [2], MIMO has been wdey accepted as a key technoogy to ncrease wreess capacty. Researchers have shown that by empoyng mutpe antennas on the transmttng and recevng nodes, wreess channe capacty can scae amost neary wth the number of antennas. Such capabty s the drvng force for the wde depoyment of MIMO n wreess LAN (802.11n), WMAX access networks (802.16), and 4G ceuar networks (LTE). Athough there have been extensve studes on MIMO at the physca ayer for pont-to-pont and ceuar communcatons (see, e.g., [3] for an overvew), fundamenta understandng and resuts on MIMO n mut-hop networks reman mted, partcuary from a cross-ayer perspectve. Ths stagnaton s many due to the ack of an accurate and more mportanty, tractabe mode that s amenabe for anayss by networkng researchers. Tradtona sgna processng and channe modes for MIMO n communcatons research are cogged wth compex matrx representatons and operatons, renderng enormous chaenges for mut-hop network optmzatons. Due to these chaenges, most efforts on mut-hop MIMO networks to date [4] [12] fa nto the foowng two approaches. The frst approach s to formuate the probems by fathfuy ncorporatng the MIMO channe and sgna modes wthout any oss of accuracy. However, the probem formuatons under ths approach soon become ntractabe due to the heavy burden from the underyng modes. For exampe, Km et a. studed a maxmn optmzaton probem n [4] for mut-hop MIMO backhau networks where they formuated a nonnear optmzaton probem to maxmze the far throughput of the access ponts n the network under the routng, MAC, and physca ayer constrants. The physca ayer n [4] s based on mnmum mean square error (MMSE) beamformng. In [5], Chu and Wang aso studed cross-ayer agorthms for MIMO ad hoc networks where MMSE sequenta nterference canceaton technque (MMSE-SIC) was empoyed at the physca ayer to maxmze sgna to nterference and nose rato (SINR). Due to the compex MMSE mechancs, the cross-ayer optmzaton probems n [4] and [5] are ntractabe and the authors had to resort to heurstc agorthms. The second approach s to smpfy MIMO physca ayer behavor so that tractabe anayss can be deveoped for networkng research. Athough such approach s attractve, the probem wth exstng modes under ths approach suffer from over smpfcaton. That s, exstng smpe modes gnore some mportant characterstcs of MIMO and thus ead to resuts far from MIMO s achevabe performance. In [6], [7], a smpfed MIMO cross-ayer mode was empoyed to study dfferent throughput optmzaton probems. By usng ths mode, the network throughput performance can be characterzed smpy by countng the number of degrees of freedom (DoF) n the network. However, ths mode does not consder transmt power constrant and power aocaton at each node n the network. Aso, athough some deas of zero-forcng beamformng (ZFBF) were empoyed to hande nterference, the proposed nterference canceaton scheme at the nk ayer was not desgned effcenty, resutng n a sma DoF regon and nferor throughput performance. Aso, n [8] [12], varous studes on MAC desgns and routng schemes are gven based on very smpe MIMO modes that do not fuy expot MIMO physca capabtes. The goa of ths paper s to acheve the best of both approaches whe avodng ther ptfas. We want to construct a mode for MIMO that s both tractabe and accurate for crossayer optmzaton. Our man contrbutons are as foows. At the physca ayer, we devse a smpe mode for computng MIMO channe capacty. Ths mode captures the essence of both spata mutpexng and transmt power constrant. More mportanty, ths mode does not /10/$ IEEE

2 Ths fu text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for pubcaton n the IEEE INFOCOM 2010 proceedngs Ths paper was presented as part of the man Technca Program at IEEE INFOCOM requre compex matrces computaton and compcated water-fng process (whch does not admt a cose-form souton). We show that the gap between our proposed mode and the exact capacty mode s neggbe. At the nk ayer, we construct a mode that takes nto account the nterference nung/supresson by expotng ZFBF. More specfcay, we propose a space-tme schedung scheme caed OBIC (abbrevaton of orderbased nterference canceaton). The proposed OBIC empoys smpe agebrac computaton on matrx dmensons to mode ZFBF n a mut-hop network. Moreover, by carefuy arrangng the canceaton order among the nodes, OBIC does not waste unnecessary DoF resources on nterference mtgaton, thus offerng superor throughput performance than those n [6], [7]. As an appcaton, we use the proposed new modes to study a cross-ayer utty maxmzaton probem for mut-hop MIMO networks. We show that the resutng optmzaton probem no onger nvove compex matrx varabes and operatons. Further, the formuated probem shares a ot of smartes wth those cross-ayer optmzaton probems under snge-antenna ad hoc networks, whch have been actvey studed n recent years. Ths suggests that new soutons to mut-hop MIMO networks may be deveoped by drawng upon the experences ganed for snge-antenna ad hoc networks. The remander of ths paper s organzed as foows. Secton II presents a new channe capacty mode for MIMO at the physca ayer. Secton III presents a new nk ayer mode caed OBIC. In Secton IV, as an appcaton of our new modes, we study a cross-ayer optmzaton probem n a mut-hop MIMO network. Secton V concudes ths paper. II. A MODEL FOR PHYSICAL LAYER CAPACITY COMPUTATION From networkng research perspectve, the most mportant aspect of physca ayer modeng for MIMO s ts channe capacty computaton. In Secton II-A, we frst gve background on MIMO channe capacty computaton and anayze why t s dffcut to work wth for networkng research. Then, n Secton II-B, we propose a new mode for MIMO channe capacty that s both smpe and accurate. A. Why Exstng Physca Mode for MIMO s Dffcut to Use? The channe of a MIMO nk s characterzed by a matrx H, as shown n Fg. 1. Communcaton over such a MIMO channe wth n t transmt antennas and n r receve antennas can be descrbed by y = ρ α H x + n, (1) where x, y and n denote the vectors of transmtted sgna, receved sgna, and whte Gaussan nose wth unt varance, respectvey. In (1), ρ represents the receved SNR of the channe, α [0, 1] represents the fracton of the transmt power that s assgned to nk (n the case when the source of nk aso transmts on other nks). As we sha see ater x n t antennas Tx() Transmttng Node x Fg. 2. n t antennas Tx() Transmttng Node... Fg H A MIMO channe. Λ 1 2 d n r antennas... Rx() Recevng Node n r antennas... Rx() Recevng Node The equvaent parae scaar channes after transformaton. n Secton IV, α s usefu to mode the power aocaton at each node f mut-hop mut-path routng s empoyed n the network. For the snge nk case n Fg. 1, we have α =1. The channe gan matrx H s typcay assumed to be a compex random matrx wth each of ts entres beng..d. Gaussan dstrbuted [13] wth zero mean and unt varance. From basc near agebra, we know that by snguar vaue decomposton (SVD), the channe mode n (1) can be wrtten as y = ρ α U Λ V x + n, where U and V are untary matrces, Λ s a dagona matrx wth the snguar vaues of H on ts man dagona. By ettng x = V x, ỹ = U y, and ñ = U n, the channe mode can be re-wrtten as ỹ = ρ α Λ x + ñ, (2) whch s equvaent to a set of parae channes shown n Fg. 2. The number of non-zero snguar vaues (.e., non-zero dagona entres n Λ )sd mn{n t,n r },.e., the rank of H. The rank of H s aso caed the degrees of freedom (DoF), whch measures the number of ndependent sgnang dmensons that are avaabe n the channe. The capacty for the set of parae channes n (2) can be found by the water-fng power aocaton agorthm [1]: C (wf) = max W og 2 det(i + ρ α H Q H Q ) = d =1 W (og 2(ρ α μλ )) +, where W represents the bandwdth of the channe; Q = E{x x } s the nput covarance matrx representng the power aocaton of sgna x ; det( ) represents matrx determnant; I represents an n r n r dentty matrx; ( ) + represents max(0, ); λ denotes an egenvaue of matrx H H (havng the same number of non-zero snguar vaues n Λ and equa to the square of the snguar vaues of H ); and μ s the optma water-eve satsfyng d =1 (μ (ρ α λ ) 1 ) + =1. Further, snce H s a random matrx, the ergodc capacty y ỹ

3 Ths fu text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for pubcaton n the IEEE INFOCOM 2010 proceedngs Ths paper was presented as part of the man Technca Program at IEEE INFOCOM of such a fadng MIMO channe can be computed as [14]: C (wf),ergodc = E H [C (wf) ]=W d =1 E λ [ (og2 (ρ α μλ )) + ] = W d =1 (og2 (ρ α μλ )) + f λ (λ)dλ, (3) where E λ [ ] represents the expectaton taken over the dstrbuton of λ and f λ ( ) denotes the dstrbuton of λ. Athough (3) s the exact formua for computng MIMO channe capacty, there are some ssues that prevent (3) from beng easy adopted n cross-ayer optmzaton. 1) To determne the egenvaues λ of H H, one needs to sove the characterstc poynoma equaton. However, t s known that there s no formua for roots of poynomas of degree 5 or greater. Even for a cubc or quartc poynoma equaton (correspondng to 3 3 and 4 4 MIMO channes), the root formua s cumbersome to use and the equaton s often soved approxmatey by numerca methods nstead. Further, due to the compexty n computng λ,tseven harder to determne the dstrbuton of f λ ( ) from H H. 2) Even f we have soved λ s for a gven H, t remans to sove the optma water-eve μ for the optma power aocaton. However, due to the dscontnuty property of the water-fng souton, there s no cosed-form souton to determne μ. Instead, μ can ony be evauated numercay. 3) Snce t s dffcut to determne λ, f λ ( ), and μ, computng the ntegraton n (3) becomes a chaengng task. Instead of ntegratng (og 2 (ρ α μλ )) + f λ ( ), we can cacuate a sampe mean of (og 2 (ρ α μλ )) + as an approxmaton for the expectaton. However, ths cacuaton requres a arge number of random sampes of H (so as to obtan a good approxmaton). Due to the above dffcutes, Eq. (3) cannot be ready used to offer tractabe anayss n cross-ayer optmzaton. B. A Smpe and Accurate Mode for MIMO Channe Capacty To avod the dffcutes ncurred n usng (3), we propose a smpe and yet non-trva mode to approxmate the MIMO channe capacty computaton as foows: ( C (sm) = W d og 2 1+ ρ ) α. (4) d The constructon of (4) s based on the foowng ntuton. Frst, note that n (3), the capacty s determned by the averagng behavor of the egenvaues of H H. Athough these egenvaues are random, n practce they tend to be..d. faded. As a resut, when averaged over a arge number of channe reazatons, the mean channe gan for each parae spata channe (see Fg. 2) s roughy the same. Therefore, we approxmate the random matrx H by a determnstc dentty matrx (.e., we repace H by I d ), thus emnatng the expectaton computaton. Wth such a smpfcaton, t s easy to verfy that the optma water-fng scheme degenerates nto a trva equa power aocaton snce a spata channes have equa gan. It then foows that the channe capacty can be roughy approxmated by (4). The man beneft of (4) s TABLE I NORMALIZED GAP VERSUS THE NUMBER OF ANTENNAS. Number of Normazed gap antennas SNR = 20 db SNR = 30 db % 0.82% % 1.87% % 2.39% % 2.73% % 2.93% % 3.08% % 3.23% that we no onger need to expcty compute the egenvaues of H H, the p.d.f. of λ, the optma water eve μ, and the expectaton functon. Note that when d =1, (4) s reduced to Shannon formua for snge-antenna case. We now formay examne the accuracy of (4). Frst, we quantfy the gap between (3) and (4) for one channe reazaton. We have the foowng emma and ts proof s gven n [15]. Lemma 1. For a MIMO nk wth nstantaneous channe gan H of rank d, ΔC C (wf) C (sm) W d =1 og 2 λ under a hgh SNR regme. Based on Lemma 1, we show the gap between (3) and (4) s sma by showng E H [ d =1 og λ ] s neggby sma. We state the resut n the foowng theorem and gve a proof n Appendx A. Theorem 1. Under a hgh SNR regme, for a MIMO nk wth Gaussan random channe matrx H of rank d, the approxmaton gap ncurred by the smpe mode n (4) s cose to zero. To offer some quanttatve nsghts on ths gap, n Tabe I, we show the normazed gap between (3) and (4) for a MIMO channe under ρ = 20 db and ρ = 30 db, respectvey. We vary the number of antennas from 2 to 8 (range for practca MIMO systems). We can see that the gap between the approxmaton and the exact capacty s ndeed neggby sma. For exampe, wth 4 antennas under 30 db, the gap s ony 2.39%. III. LINK LAYER MODELING FOR MULTI-HOP MIMO NETWORKS At the nk ayer, MIMO opens up new opportuntes n space doman to mtgate nterference. In Secton III-A, we frst descrbe zero forcng beamformng (ZFBF), whch s a powerfu MIMO nterference mtgaton technque. We aso dscuss ts benefts and chaenges n mut-hop network settng. In Secton III-B, we propose a space-tme schedung scheme caed OBIC and n Secton III-C, we construct ts mathematca mode. A. Zero-Forcng Beamformng: Benefts and Chaenges In ceuar MIMO systems, one of the most powerfu nterference mtgaton technque s caed ZFBF [16], [17]. ZFBF uses mut-antenna arrays to steer beams toward the ntended

4 Ths fu text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for pubcaton n the IEEE INFOCOM 2010 proceedngs Ths paper was presented as part of the man Technca Program at IEEE INFOCOM recever to ncrease SNR, whe formng nus to unntended recevers to avod nterference. In MIMO ceuar systems, however, ZFBF s usuay performed at the transmtter sde. In ths paper, we generaze ZFBF to mut-hop networks by aowng beamformng to be performed on both transmtter and recever sdes. To see how the generazed ZFBF can be used n mut-hop MIMO networks, consder a network havng L nks among whch L 0 nks are actve. We denote Ī the set of nks that nterfere wth the recepton of nk s ntended recever, = 1, 2,...,L 0. We denote H Tx(m),Rx() the nterference channe gan matrx from transmttng node of nterference nk m (Tx(m)) to recevng node of nk (Rx()). To extract the transmtted sgna through a MIMO channe, a transmt beamformng matrx and a receve beamformng matrx are empoyed on the channe. Thus, the receved sgna at nk can be wrtten as y = ρ α R Rx() H T Tx() x }{{} Desred sgna part + ρm, α m R Rx() H Tx(m),Rx()T Tx(m) x m m Ī +n,, }{{} Interference part where ρ m, denotes the nterference-to-nose rato (INR) from node Tx(m) to node Rx(). By expotng the mut-antenna array at each node, t s possbe to cance out a nterferences by judcousy confgurng T s and R s. Specfcay, we can confgure T s and R s n such a way that R Rx() H Tx(m),Rx()T Tx(m) = 0,, m Ī. (5) Note that f there exst non-trva soutons to (5) (.e., R Rx() 0, T Tx(m) 0,, m Ī), then t means that a L 0 nks can be actve smutaneousy n an nterference-free envronment. Moreover, the ranks of T Tx() and R Rx() determne the maxmum number of data streams z that can be transmtted over nk,.e., z mn{rank(t Tx() ), rank(r Rx() )}. Athough ZFBF s benefts are appeang, there reman sgnfcant chaenges to empoy t n mut-hop networks. Ths s because fndng an optma set of T s and R s satsfyng (5) requres sovng a arge number of bnear equatons. Unke near equaton systems, a genera souton to bnear equaton systems remans unknown [18]. Thus, t becomes an ntractabe probem to desgn schedung schemes based on sovng (5). B. OBIC: Basc Idea We fnd that the specfc eement confguratons n T s and R s are more cosey ted to beamformng desgn than to nk ayer schedung. Therefore, nstead of focusng on sovng (5), we propose to reposton ourseves to expot matrx dmenson constrants that are suffcent for (5) to hod. By dong so, we can characterze the nk ayer schedung performance wthout entangng wth the detas of beamformng desgns. Lnk m z m Rx(m) Tx() Fg. 3. z m R Rx(m) H Tx(),Rx(m) T Tx() A two-nk exampe. 1 z Fg. 4. Lnk z 3 3 DoF regon of two-nk exampe. To understand how we can extract the matrx dmenson constrants for ZFBF-based schedung, we frst use a smpe two-nk network shown n Fg. 3 as an exampe. In ths network, nk has 3 antennas on each sde and nk m has 5 antennas on each sde. For ths smpe network, we can frst choose a T Tx() arbtrary wthout consderng nk m s exstence. Suppose that T Tx() s fu-rank (.e., 3 data streams beng transmtted). Next, we choose an R Rx(m) to cance the nterference from nk, whe recevng data streams from ts desred transmtter. Ths s equvaent to sovng (H Tx(),Rx(m) T Tx() ) R Rx(m) = 0 wth T Tx() aready determned. Ths mpes that a coumn vectors n R Rx(m) have to e n the nu space of (H Tx(),Rx(m) T Tx() ). The dmenson of the nu space n ths case s dm(nu((h Tx(),Rx(m) T Tx() ) )) = 5 3=2, meanng that Rx(m) can receve up to 2 streams. Note that nks and m are both actve n an nterference-free envronment. Further, by varyng the ranks of T Tx() and R Rx(m), t s easy to verfy that the achevabe DoF regon under ths ZFBF-based scheme s the trapezod shown n Fg Observe that the schedung scheme n ths two-nk exampe s performed n an ordered fashon: we arbtrary choose a T Tx() frst, and then choose an R Rx(m) such that the nterference can be emnated. We now extend ths order-based nterference canceaton dea to a three-nk exampe shown n Fg. 5, whch s more compcated than the prevous exampe. Here, each recevng node of a nk s beng nterfered by the transmttng nodes of other nks. For ths exampe, we can start wth a schedung order for the sx nodes. Such a schedung order w be subject to an optmzaton n Secton III-C. Suppose the schedung order for the sx nodes s Tx() Rx(m) Rx() Tx(m) Tx(n) Rx(n). Then, through the foowng schedung decson, we can show that the stream combnaton (1, 1, 2) s achevabe. 1 Note that the achevabe DoF regon n Fg. 4 concdes wth the maxmum DoF regon descrbed n [19, Theorem 2]. Thus, for ths two-nk exampe, the proposed schedung scheme s an optma schedung scheme.

5 Ths fu text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for pubcaton n the IEEE INFOCOM 2010 proceedngs Ths paper was presented as part of the man Technca Program at IEEE INFOCOM antenna 1 antenna z =1 Tx() Rx() Lnk Tx(m) z m =1 2 antennas Lnk m R Rx() R Rx(m) Rx(m) 2 antennas For a gven ordered node st, start from the 1 st node A nodes schedued? Y N Move to the next node n the ordered node st Current node a Tx/RX node? Rx Nether Tx Nu ts nterference to Rx nodes schedued before tsef Suppress nterference from Tx nodes schedued before tsef T Tx(n) z n =2 Tx(n) Rx(n) 4 antennas Lnk n 4 antennas Fg. 5. A three-nk exampe. 1) Tx(): Snce nodes Tx() s the frst to be schedued, t does not have any nterference to concern about. Aso, snce Tx() has ony 1 antenna, we can et Tx() transmt 1 data stream; 2) Rx(m): Snce Rx(m) s schedued after Tx(), t needs to suppress the nterference from Tx(),.e., sovng (H Tx(),Rx(m) T Tx() ) R Rx(m) = 0. We have dm(nu((h Tx(),Rx(m) T Tx() ) )) = 2 1=1,.e., we can et Rx(m) receve 1 stream n ths case; 3) Rx(): Snce no nterferng transmttng node s schedued before Rx(), Rx() does not need to concern about any nterference. Gven Rx() has ony 1 antenna, we can et t receve 1 stream; 4) Tx(m): Foowng the smar argument as for Rx(m), we can et Tx(m) transmt 1 stream; 5) Tx(n): Snce Tx(n) s transmsson shoud not nterfere wth Rx() [ and Rx(m), t foows that T Tx(n) R Rx() shoud satsfy H ] Tx(n),Rx() R Rx(m) H T Tx(n) = 0. Snce Tx(n),Rx(m) ( [ R Rx() dm nu H ]) Tx(n),Rx() R Rx(m) H =4 (1+1) = 2, we Tx(n),Rx(m) can schedue 2 data streams at Tx(n); 6) Rx(n): Foowng a smar anayss as n 5), t can be shown that 2 streams can be schedued at Rx(n). The dea n the three-nk exampe can be syntheszed for a genera mutpe-nk settng. The essence of ths schedung scheme s to perform nterference canceaton successvey based on an ordered node st: If a node s transmttng, then t s ony necessary to ensure that ts transmssons do not nterfere wth prevousy schedued recevng nodes n the ordered node st. It does not need to expend precous DoF resources to nu ts nterference to those recevng nodes to be schedued after tsef n the node st. If a node s recevng, t ony needs to suppress nterference from transmttng nodes schedued before tsef n the node st. It does not need to concern nterferng transmttng nodes to be schedued after tsef. The nterference canceaton behavor descrbed above offers the basc dea for a node-based schedung scheme. For easy reference, we ca ths schedung scheme OBIC (order-based nterference canceaton). Addtona quanttatve constrants on DoF on each transmttng and recevng node (as shown n ast two exampes) w be dscussed n the next Fg. 6. Stop The fow chart of OBIC schedung scheme z m 1 z Fg. 7. Achevabe DoF regon comparson between OBIC and CM for the exampe n Fg. 3. secton. Fg. 6 shows the fow chart of the OBIC scheme. Remark 1. In [6], Hamdaou and Shn proposed severa nterference avodance schemes based on ZFBF. For the socaed CM scheme (the best among the proposed schemes n [6]), the authors aso recognzed that nterference can be canceed by ether the transmttng or the recevng node of an nterference nk. However, wthout empoyng a node-based sequenta schedung, t s mpossbe to know whch node shoud perform nterference mtgaton. As a resut, the CM scheme requres both the transmttng and recevng nodes of an nterference nk to expend precous DoFs for nterference canceaton (c.f. [6, Eq. (10)]). Ths approach adversey eads to a much smaer DoF regon. As an exampe, we compare the performance of OBIC and the CM mode on the smpe twonk exampe n Fg. 3. Under the CM mode, t s not dffcut to verfy that the achevabe DoF regon s the shaded trange n Fg. 7, representng the convex hu of the whte dots, whch are the DoF combnatons drecty achevabe under CM. It can be seen that ths regon s smaer than the achevabe DoF regon by OBIC. In genera, t can be shown that the DoF regon acheved under the CM mode s aways a subset of that under the OBIC [15]. Remark 2. We pont out that for the three-nk network exampe n Fg. 5, a arger DoF regon can be acheved by nterference agnment (IA) [20]. The basc dea of IA s that by agnng the nterference from two nterferng transmtters at each recevng node, the nterference from dfferent nodes becomes dependent, mpyng a smaer rank of the effectve nterference channe. Ths n turn eads to a hgher dmensona nu space that can be expoted for data transmssons. We note, however, that IA aso has ts mtatons. IA was proposed for the cassca nterference channe n the context of network nformaton theory [21], where the channe possesses

6 Ths fu text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for pubcaton n the IEEE INFOCOM 2010 proceedngs Ths paper was presented as part of the man Technca Program at IEEE INFOCOM a fuy-connected structure. That s, each nk n the network s nterfered by a remanng nks. For genera mut-hop network topoogy where the fu connectvty may not hod, the achevabe DoF regon of IA remans an open probem. C. OBIC: A Mathematca Mode Havng ntroduced the basc dea of OBIC, we now deveop ts mathematca mode. We represent the topoogy of a muthop MIMO network by a drected graph, denoted by G = {N, L}, where N and L are the sets of nodes and a possbe MIMO nks, respectvey. Suppose that the cardnates of the sets N and L are N = N and L = L, respectvey. In ths paper, we assume that schedung operates n a frameby-frame system wth T tme sots n each frame. We remark that ths tme-sotted assumpton s not a necessary restrcton. For ths work, there s no fundamenta dstncton between tme and frequency dmensons. Thus, the tme sot ndex t n the mathematca modeng descrbed beow coud equvaenty be used to descrbe frequency sots or even a tme-frequency tupe. Modeng an Ordered Node Lst. Refer to Fg. 6 and the dscusson n Secton III-B. Before we start schedung on a node, we must have an ordered node st, whch w be subject to an optmzaton. To mode an ordered node st that can be optmzed, we defne the foowng bnary varabe. For, j N, j, we et π j (t) =1f node j s schedued after node n tme sot t and 0 otherwse. It s easy to see that π j (t) must satsfy the foowng two propertes. ) Mutua excusveness: If node j s schedued after node (.e., π j (t) =1), then t aso mpes that node s before node j (.e., π j (t) =0). Ths reatonshp can be modeed as π j (t)+π j (t) =1,, j N : j, t. (6) ) Transtvty: If node j s schedued after (.e., π j =1) and node k s schedued after node j (.e., π jk =1), then t mpes that node k s schedued after node (.e., π k =1). To mode ths transtvty property, we have the foowng emma and refer readers to [15] for the detas of the proof due to space mtaton. Lemma 2. Let Ω( ) be a one-on-one mappng from each eement n the set N to a natura number n {1, 2,...,N}. For nodes, j, k N wth Ω() < Ω(j) < Ω(k), the foowng two constrants are suffcent to descrbe the transtvty reatonshp among nodes node trpet, j, and k: 1 π j (t)+π jk (t)+π k (t) 2. (7) The constrant n (7) can be nterpreted n a ogca sense. It s easy to see that 1 π j (t)+π jk (t)+π k (t) 2 mpes that at east 1 and at most 2 π-varabes can be equa to one. If not, then we have ether π j (t) =π jk (t) =π k (t) =0or π j (t) =π jk (t) =π k (t) =1, both of whch cannot be true snce the orderng for, j, and k woud then form a oop. Modeng the Transmttng Node Behavor. Next, we mode the bock n Fg. 6 where a node s schedued to be a transmttng node. Note that n each tme sot t, 1 t T, due to haf-dupex, each node ether transmt, receve, or be de. To mode haf-dupex, we ntroduce two groups of bnary varabes g (t) s and h (t) s as foows. g (t) =1f node s transmttng n tme sot t and 0 otherwse; h (t) =1f node s recevng n tme sot t and 0 otherwse. Then, the haf-dupex constrant can be characterzed by g (t)+h (t) 1,, t. (8) We assume that scatterng s rch enough n the envronment such that a channe matrces are of fu-rank. As a resut, the number of data streams that a node can transmt or receve s mted by ts number of antennas and we have the foowng two constrants: g (t) z (t) g (t)a, (9) L Out h (t) L In z (t) h (t)a, (10) where L Out and L In represent the sets of outgong and ncomng nks at node, respectvey; z (t) denotes the number of data streams over nk n tme sot t, and A represents the number of antennas at node. From Fg. 6, we see that the data streams transmtted by node shoud not nterfere wth those recevng nodes schedued prevousy. Ths s equvaent to sayng that the transmsson beamformng vectors n T shoud e n the nu space of the stacked matrx formed by stackng a R j H,j matrces, where j denotes a prevousy schedued recevng node that coud be nterfered by node. That s, T nu. R j H,j. j I and j s, schedued before (.e., π j =1), (11) where I represents the set of nodes wthn the nterference range of node. For convenence, we et S denote the stacked matrx n (11). Note that L z Out (t) s the number of data streams that node transmts n tme sot t. Thus, from (11), we have that L z Out (t) shoud be ess than or equa to the nuty of S,.e., L z Out (t) nu(s). Aso, note that the rank of S s j I π j (t) Tx() :Rx()=j z (t). Therefore, accordng to rank-nuty theorem [22] (.e., rank(s)+nu(s) s equa to the number of coumns n S), we can mode the dmensona constrant as foows: for a, j N and for a t [1,...,T], z (t)+ π j (t) z (t) A +(1 g (t))m.(12) j I L Out :Rx()=j Tx() In (12), M s a suffcenty arge number (e.g., we can set M = j I A ). When node s a transmsson node (.e., g (t) =1), then (12) s reduced to the rank-nuty condton wth respect to S. Otherwse, f node s schedued to be a

7 Ths fu text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for pubcaton n the IEEE INFOCOM 2010 proceedngs Ths paper was presented as part of the man Technca Program at IEEE INFOCOM recevng node or n de status (.e., g (t) =0), then (12) trvay hods due to the arge vaue of M. We note that the nonnear terms π j (t) Tx() :Rx()=j z (t) n (12) coud compcate optmzatons. To remove these nonnear terms, we can ntroduce a new nteger varabe φ j (t) and reformuate (12) as foows: z (t)+ φ j (t) A +(1 g (t))m, (13) j I L Out where φ j (t), N, j I, t [1,...,T], satsfes the foowng constrants: φ j (t) Tx() :Rx()=j z (t), (14) φ j (t) A π j (t), (15) φ j (t) A π j (t) A + Tx() :Rx()=j z (t). (16) It s easy to verfy that ths set of new constrants (13) (16) s equvaent to (12). Modeng the Recevng Node Behavor. Smary, for the bock n Fg. 6 where a node s schedued to be a recevng node, we can derve the foowng constrants: for a, j N and for a t [1,...,T], L z In (t)+ j I ϕ j (t) A +(1 h (t))m,(17) ϕ j (t) Rx() :Tx()=j z (t), (18) ϕ j (t) A π j (t), (19) ϕ j (t) A π j (t) A + Rx() :Tx()=j z (t). (20) Lnk Capacty Computaton under OBIC. In Secton II, we proposed a smpe and accurate physca ayer mode to approxmate the capacty of a snge MIMO nk. We are now ready to further extend (4) to approxmate the capacty of each MIMO nk under OBIC. Frst, we note that there s no nterference among nks due to ZFBF. Thus, as the snge nk case n Secton II, the capacty of each nk under OBIC s not affected by nterference. However, compared wth the snge MIMO nk case, a key dfference n OBIC s that each actve nk now transmts z data streams nstead of d and z d. In ths case, one may conjecture that a smpe way to modfy (4) for OBIC s to repace d wth z. Indeed, the foowng theorem says that such an extenson s correct and ts rgorous proof s provded n [15]. Theorem 2. Under OBIC, each MIMO nk s capacty n tme sot t can be approxmated as ( C (t) =W z (t) og 2 1+ ρ ) α (t). (21) z (t) Moreover, under a hgh SNR regme, the approxmaton gap ncurred by (21) s neggbe. IV. APPLICATION IN MULTI-HOP NETWORKS In Sectons II and III, we have deveoped two modes for the physca ayer and the nk ayer n mut-hop networks, respectvey. In ths secton, we w show how to appy them for cross-ayer optmzaton n mut-hop MIMO networks. We consder a generc utty maxmzaton probem nvovng a set of sessons, F, n an ad hoc network. Denote src(f) and dst(f) the source and destnaton nodes of sesson f F, respectvey. Denote r(f) the fow rate of sesson f and r (f) the fow rate on nk that s attrbuted to sesson f F, respectvey. Denote C (t) the capacty of nk n tme-sot t. For stabty, we have the foowng constrants on the fow rates: f F r (f) 1 T T t=1 C (t),. (22) At the network ayer, dfferent routng schemes can be adopted. Under any routng scheme, the fow baance constrants must hod at each node N. L Out L Out L In r (f) L In r (f) = L In r (f) L Out r (f) =r(f), f =src(f), (23) r (f), f src(f), dst(f), (24) r (f) =r(f), f =dst(f). (25) It can be easy verfed that f (23) and (24) are satsfed, then (25) s automatcay satsfed. As a resut, t s not necessary to st (25) n probem formuaton once we have both (23) and (24). When a node s transmttng smutaneousy on more than one outgong nk, t s necessary to consder power aocaton at node. Reca that α (t) [0, 1] represent a fracton of transmt power aocated to nk n tme-sot t. Then, for each node n tme-sot t, wehave among L Out L α Out (t) g n (t), n, t. (26) The constrant n (26) ensures that the sum of transmt power of a outgong nks at node does not exceed the power mt. In the case when node s not n transmsson mode, then g (t) =0and α (t) =0for a L Out. Consder a utty functon for each sesson, u ( r(f) ), whch we assume s concave. Then a genera MIMO network utty maxmzaton (MIMO-NUM) probem can be formuated as foows. MIMO-NUM F max f=1 u( r(f) ) s.t. Network ayer fow-baance routng constrants n (23) and (24); Lnk capacty constrants n (22); OBIC based nk ayer constrants n (6), (7), (8), (9), (10) and (13) (20); Smpfed MIMO physca ayer mode n (21) and (26). Two remarks are n order. 1) Tractabty. Reca that exstng MIMO cross-ayer optmzaton nvoves many matrx varabes n the capacty cacuaton and ZFBF schedung, makng network eve research qute chaengng. Wth our smpe modes, matrx varabes no onger appear n the MIMO- NUM probem, whch sgnfcanty smpfes formuaton and reduces computatona compexty. 2) Sovabty. By usng our smpe modes, the MIMO-NUM probem s reduced to a smar mathematca form as a NUM probem for sngeantenna ad hoc networks. Note that athough the OBIC part

8 Ths fu text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for pubcaton n the IEEE INFOCOM 2010 proceedngs Ths paper was presented as part of the man Technca Program at IEEE INFOCOM (m) 1500 N32 N11 N28 N38 N17 N40 N10 N45 N13 N23 N14 N47 N3 N N21 N12 N31 N1 N4 N41 N29 N50 N33 N43 N27 N39 N16 N26 N N2 N30 N37 N34 N9 N36 N15 N20 N5 N24 N44 N25 N49 N22 N8 N6 N35 N7 N46 N48 0 N (m) (m) 1500 N32 N11 N28 N38 N17 N40 1:1 2:1 N10 N45 N13 N23 N14 N47 N3 N19 0:1 3: N12 N31 N21 N1 N4 N41 1:2 2:1 N29 N50 N33 N43 N27 L317 0:1 N39 3:1 1:1 2:2 N16 N26 N2 N N30 N37 1:2 N34 2:2 1:1 3:1 N9 N36 N44 L65 N15 N20 N5 N24 N6 N22 N25 0:3 3:2 0:2 N49 1:1 2:1 3:2 N8 0:1 2:1 N7 N46 N48 0 N18 N (m) Fg. 8. A 50-node 5-sesson mut-hop MIMO network. s unque, t s of near form and can be handed easy. Ths suggests that we can take advantage of the exstng experences ganed from snge-antenna ad hoc networks n the terature to deveop soutons. As an exampe to ustrate the sovabty of our MIMO- NUM formuaton, we consder a mut-hop MIMO network consstng of 50 nodes that are unformy dstrbuted n a square regon of 1500m 1500m (see Fg. 8). Each node n the network s equpped wth 4 antennas and the maxmum power for each node s 100 mw. The channe bandwdth s 20 MHz. The path-oss ndex s 3.5. There are 5 sessons n the network: N26 to N19, N44 to N18, N24 to N15, N48 to N2, and N9 to N32, respectvey. Suppose that mnmumhop routng s empoyed at the network ayer. The objectve s to maxmze the sum of the end-to-end sesson rates,.e., u ( r(f) ) = r(f). Suppose that there are 4 tme sots n each tme frame,.e., T = 4. Gven these parameters and network settngs, the MIMO-NUM probem s now competey specfed. We can use CPLEX sover to obtan an optma souton. The optma schedung orderng for each node n each tme sot s sted n Tabe II. In ths tabe, each coumn gves the node orderng for schedung n a gven tme sot of a frame. For exampe, n the frst tme sot, the optma orderng of the nodes s N19 N18... N2. Fg. 9 shows the routng paths for each sesson and optma schedung souton (shown n shaded boxes). As an exampe, the shaded box next to the nk from N6 to N18 contans 1:1 2:1 3:2, whch means that n tme sots 1, 2, 3, there are 1, 1, and 2 streams on ths nk, respectvey. In tme sot 4, the nk s not transmttng. Based on the number of streams, the smpe physca ayer mode (21) and the nk capacty constrant n (22), the optma sesson rates (n Mb/s) are: 60.4 for N26 N19, 151 for N44 N18, 102 for N24 N15, 36.6 for N48 N2, and 57.2 for N9 N32. Fg. 9. Schedung resut on each nk. TABLE II OPTIMAL NODE ORDERING IN EACH TIME SLOT OF A FRAME. Tme Sot 1 Tme Sot 2 Tme Sot 3 Tme Sot 4 1st N19 N48 N24 N24 2nd N18 N27 N22 N32 3rd N44 N32 N44 N26 4th N15 N15 N9 N48 5th N26 N9 N15 N18 6th N22 N19 N27 N24 7th N5 N44 N32 N9 8th N48 N26 N26 N22 9th N6 N18 N19 N5 10th N27 N2 N18 N6 11th N24 N24 N6 N15 12th N32 N22 N5 N19 13th N3 N5 N48 N2 14th N1 N3 N1 N27 15th N9 N1 N3 N1 16th N2 N6 N2 N3 V. CONCLUSION Exstng modes for MIMO suffer from ether ntractabty or naccuracy when they are empoyed to study mut-hop MIMO networks. We proposed a tractabe and accurate mode for MIMO that s amenabe for cross-ayer anayss n muthop settng. Our contrbutons ncuded a mode at the physca ayer and a mode at the nk ayer. At the physca ayer, we proposed a smpe mode to compute MIMO channe capacty that captures the essence of spata mutpexng and transmt power mt wthout nvovng compex matrx operatons and the water-fng agorthm. We proved that the approxmaton gap n ths physca ayer mode s neggbe. At the nk ayer, we proposed a schedung scheme caed OBIC that s based on ZFBF nterference mtgaton. The proposed OBIC scheme cuts through the compexty assocated wth beamformng desgns n a mut-hop network by usng smpe agebrac computaton. Ths aows us to expore the nk ayer schedung performance wthout entangng wth beamformng detas. By appyng the proposed cross-ayer mode to a genera NUM probem, we vadate ts effcacy n practce. The resuts n ths paper offer an mportant anaytca too to fuy expot

9 Ths fu text paper was peer revewed at the drecton of IEEE Communcatons Socety subject matter experts for pubcaton n the IEEE INFOCOM 2010 proceedngs Ths paper was presented as part of the man Technca Program at IEEE INFOCOM the potenta of MIMO n mut-hop networks. ACKNOWLEDGEMENT The work of Y.T. Hou, J. Lu, and Y. Sh was supported n part by the Natona Scence Foundaton (NSF) under Grant CNS and Offce of Nava Research (ONR) under Grant N REFERENCES [1] I. E. Teatar, Capacty of mut-antenna Gaussan channes, European Trans. Teecomm., vo. 10, no. 6, pp , Nov [2] G. J. Foschn, Layered space-tme archtecture for wreess communcaton n a fadng envornment when usng mut-eement antennas, Be Labs Tech. J., vo. 1, no. 2, pp , [3] E. Bger, R. Caderbank, A. Constantndes, A. Godsmth, A. Pauraj, andh.v.poor,mimo Wreess Communcatons. Cambrdge Unversty Press, Jan [4] S.-J. Km, X. Wang, and M. Madhan, Cross-ayer desgn of wreess muthop backhau networks wth mutantenna beamformng, IEEE Trans. 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San Dego: Academc Press, APPENDIX A PROOF OF THEOREM 1 From Lemma 1, we have d d E H [ΔC ] W E λ [og 2 λ ] W og 2 E λ [λ ]. =1 where the ast nequaty foows from the concavty of the og functon and Jensen s nequaty. The Mar cenko-pastur theorem [23] says that for a matrx H wth nr n t = β, the mtng p.d.f. of the egenvaues of the correspondng Wshart matrx H H as n t,n r s: ( f β λ (x) = 1 1 ) (x )+ (u x) + δ(x)+, (27) β + 2πβx where =(1 β) 2, u =(1+ β) 2, and ( ) + = max(0, ), and δ(x) s the Drac deta functon. Moreover, even for sma vaues of n t and n r, the p.d.f functon n (27) can be used to serve as an exceent approxmaton [13]. Now, et us frst consder the case when β 1. In ths case, the p.d.f. can be smpfed to f β λ (x) = (x )+ (u x) + /2πβx. Snce a egenvaues are..d. dstrbuted, we have d =1 og 2 E λ [λ ]=d og 2 E[λ] = d og 2 ( 1 2πβ u =1 ) x2 + 2(1 + β)x (1 β) 2 dx. For convenence, et R(x) = x 2 + 2(1 + β)x (1 β) 2.By usng [24, Eq ], we can derve that u 2x 2(1 + β) u R(x)dx = R(x) + 16β u dx. 4 8 R(x) (28) Note that the frst term n the summaton n (28) s zero. Then by usng [24, Eq ], we can further derve that u ( ) 2x + 2(1 + β) u R(x)dx =2βarcsn 16β = 2β(arcsn( 1) arcsn(1)) = 2πβ. It then foows that d ( ) 2πβ E H [ΔC ] W og 2 E λ [λ ]=Wd og 2 =0. 2πβ =1 For the case when β>1, the frst term n the p.d.f. functon n (27) becomes non-zero. Thus, we need to further evauate the expectaton of the frst term. In ths case, t s easy to see that u ( x 1 1 ) ( δ(x)dx = 1 1 ) u xδ(x)dx =0. β β Combnng two cases, we have E H [ΔC ] 0 for a β, and the proof s compete.