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

Transcription

1 A Tractabe and Accurate Cross-Layer Mode for Mut-Hop MIMO Ad Hoc Networks Ja Lu Y Sh Cunhao Gao 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 wreess ad hoc networks. Athough there has been sgnfcant progress on MIMO at the physca ayer or snge-hop communcaton over the years, advances n the theory of MIMO for mut-hop ad hoc 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 cross-ayer research for mut-hop MIMO ad hoc networks. In partcuar, at the physca ayer, we deveop a smpe mode for MIMO nk 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 SUCCINCT that sgnfcanty advances the exstng zeroforcng beamformng (ZFBF) to hande nterference n a muthop network settng. The proposed SUCCINCT scheme empoys smpe numerc computaton on matrx dmensons to smpfy ZFBF n a mut-hop ad hoc network. As a resut, we can characterze nk ayer schedung behavor wthout entangng wth beamformng detas. Fnay, we appy both our physca and nk ayer modes n cross-ayer performance optmzaton for a mut-hop MIMO ad hoc network. I. INTRODUCTION Snce ts ncepton [5], [21], 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), 4G ceuar networks (LTE), etc. Athough there has been extensve studes on MIMO at the physca ayer for pont-to-pont and ceuar communcatons over the past decade (see, e.g., [2] for an overvew), fundamenta understandng and resuts on MIMO n mut-hop ad hoc networks reman mted, partcuary from a crossayer 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 ad hoc network optmzatons. Due to these chaenges, most efforts on mut-hop MIMO ad hoc networks to date [1], [4], [8], [10], [12], [15] [17], [20] 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 formuaton under ths approach s key to become ntractabe due to the heavy burden from the underyng modes. For exampe, Km et a. studed a maxmn optmzaton probem n [12] 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 [12] s based on mnmum mean square error (MMSE) beamformng. In [4], Chu and Wang aso studed cross-ayer agorthms for MIMO ad hoc networks where MMSE sequenta nterference canceaton technque (MMSE-SIC) s 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 [12] and [4] 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 many mportant characterstcs of MIMO for crossayer desgn opportuntes and thus ead to resuts far from MIMO s achevabe performance. In [1], [8], 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 appropratey, resutng n an unnecessary sma DoF regon and nferor throughput performance. Aso, n [10], [15] [17], [20], 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 cross-

2 2 ayer 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 requre compex matrces computaton and the 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 SUCCINCT (abbrevaton of successve nterference canceaton). The proposed SUC- CINCT empoys smpe numerc computaton on matrx dmensons to smpfy ZFBF n a mut-hop ad hoc network. Moreover, by carefuy arrangng the canceaton order among the nodes, SUCCINCT does not waste unnecessary DoF resources on nterference mtgaton, thus offerng superor throughput performance than those n [1], [8]. As an appcaton, we use the proposed new modes to study cross-ayer utty maxmzaton probems for MIMO ad hoc networks. We show that the resutng optmzaton probems no onger nvove compex matrx varabes and operatons. Further, the formuated probems share a ot of smartes wth those crossayer optmzaton probems under snge-antenna ad hoc networks, whch have been actvey studed n recent years. Ths suggests that new soutons to MIMO ad hoc 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 SUCCINCT. In Secton IV, as an appcaton of our new modes, we study a cross-ayer optmzaton probem n a mut-hop MIMO ad hoc 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. Before proceedng to the detas of our proposed MIMO ad hoc network modeng, we frst summarze a the key notaton that appear n ths and the subsequent sectons n Tabe I. 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 TABLE I NOTATION. t() The transmttng node of nk. r() The recevng node of nk. H A MIMO matrx channe. H A MIMO matrx channe for nk. H,j A MIMO matrx channe from t() to r(j). T n Transmttng beamformng matrx for node n. R n Recevng beamformng matrx for node n. g n (t) Node n s transmsson status n tme sot t. h n (t) Node n s recepton status n tme sot t. A n The number of antennas at node n. N The set of nodes n the network. L The set of nks n the network. N The number of nodes n the network. L The number of nks n the network. L Out n The set of nks orgnatng from node n. L In n The set of nks comng nto node n. I n The set of nodes wthn the nterference range of node n. π mn(t) The orderng reatonshp between nodes m and n n tme t. z (t) The number of data streams on nk durng tme sot t. C The DoF regon of a ZFBF-based schedung scheme. r (f) The amount of fow of sesson f on nk. r f The end-to-end sesson rate of sesson f. src(f) The source node of sesson f. dst(f) The destnaton node of sesson f. C (t) The capacty of nk durng tme sot t. α (t) The power aocated on nk durng tme sot t. x n t antennas Tx() Transmttng Node... Fg. 1. H n r antennas... A MIMO channe ustraton. Rx() Recevng Node 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 n Secton IV, α s usefu to mode the power aocaton at each node f 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 [23] wth zero mean and unt varance. To compute MIMO channe capacty, one needs to dagonaze H so that the channe s transformed nto a set of parae spata channes. More specfcay, by snguar vaue decomposton (SVD), the channe mode n (1) can be transformed as y = ρ α U Λ V x +n, where U and V y

3 3 x Fg. 2. n t antennas Tx() Transmttng Node... Λ 1 2 z n r antennas... Rx() Recevng Node The equvaent parae scaar channes after transformaton. 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 rewrtten as ỹ = ρ α Λ x + ñ, (2) whch s equvaent to the channe n Fg. 2. The number of non-zero snguar vaues (.e., non-zero dagona entres n Λ ) s z mnn 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. By smpe matrx manpuatons, (2) can be re-wrtten n the foowng form: ỹ = ρ α R Rx() H T Tx() x + ñ, (3) where Tx() and Rx() denote the transmttng and recevng nodes of nk, respectvey, and matrces T Tx() = V and R Rx() = U. In MIMO communcatons terature (see, e.g., [22]), the matrces T Tx() and R Rx() are usuay referred to as transmt beamformng matrx and receve beamformng matrx. As ater shown n Secton III, transmt and receve beamformng matrces w pay an mportant roe n MIMO networks nterference mtgaton at the nk ayer. The capacty for the set of parae channes n (2) can be found by the water-fng power aocaton agorthm [21]: C (wf) = max W og 2 det(i + ρ α H Q H Q ) z = W (og 2 (ρ α µλ )) +, =1 where W represents the bandwdth of the channe; Q = Ex x } s the nput covarance matrx representng the power aocaton of sgna x ; det( ) represents matrx determnant; I represents an 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 z =1 (µ (ρ α λ ) 1 ) + = 1. Further, snce H s a random matrx, the ergodc capacty of such a fadng MIMO channe can be computed as [6]: C (wf),ergodc = E H [C (wf) ] = W z =1 ỹ E λ [ (og2 (ρ α µλ )) + ] = W z =1 (og2 (ρ α µλ )) + f λ (λ)dλ, (4) where E λ [ ] represents the expectaton taken over the dstrbuton of λ and f λ ( ) denotes the dstrbuton of λ. Athough (4) s the exact formua for computng MIMO channe capacty, there are some ssues that prevent (4) 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, even for a cubc or quartc poynoma equaton (correspondng to 3 3 and 4 4 MIMO channes), the formua for roots computaton s cumbersome to use and the poynoma equaton s often soved approxmatey by numerca methods nstead. Further, due to the compexty of computng λ from H H, t s even harder to determne the dstrbuton f λ ( ) from 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 property of the water-fng agorthm, 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 (4) 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. (4) 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 by usng (4), we propose a smpe mode to approxmate the MIMO nk capacty computaton as: C (sm) = W z og 2 ( 1 + ρ α z ), (5) The constructon of (5) to approxmate the MIMO channe capacty s based on the foowng ntuton. Frst, note that n (4), 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 z ), thus emnatng the expectaton computaton. Wth such a smpfcaton, t s easy to verfy that the optma waterfng 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 (5). The man beneft of (5) s 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. Aso, note that when z = 1, (5) s reduced to Shannon formua. We now formay examne the accuracy of (5). Frst, we quantfy the gap between (4) and (5) for a snge channe reazaton. We have the foowng emma and ts proof s gven n Appendx A.

4 4 Normazed gap SNR=20dB 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()). Reca from Eq. (3) that, when extractng 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 (for smpcty, we drop the tdes for x, y, and n): SNR=30dB Number of antennas Fg. 3. Normazed gap vs. number of antennas. Lemma 1. For a MIMO nk wth nstantaneous channe gan H of rank z, C C (wf) C (sm) W z =1 og 2 λ under hgh SNR regme. Based on Lemma 1, we show the gap between (4) and (5) s sma by showng E H [ z =1 og λ ] s neggby sma. We state the resut n the foowng theorem and gve a detaed proof n Appendx B. Theorem 1. Under hgh SNR regme, for a MIMO nk wth Gaussan random channe matrx H of rank z, the approxmaton gap ncurred by the smpe mode n (5) s cose to zero. To offer some quanttatve nsghts on ths gap, we pot n Fg. 3 the normazed gap between (4) and (5) for a MIMO channe under 20dB and 30dB of SNRs, 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 20dB, the gap s ony 2.3%. III. LINK LAYER MIMO MODELING FOR AD HOC 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 ad hoc networks appcatons. In Secton III-B, we propose a space-tme schedung scheme caed SUCCINCT and n Secton III-C, we construct ts mathematca mode. A. Zero-Forcng Beamformng: Benefts and Chaenges In MIMO pont-to-pont or ceuar MIMO systems, one of the most powerfu nterference mtgaton transcever desgns s caed zero-forcng beamformng (ZFBF) [3], [19]. ZFBF uses mut-antenna arrays to steer beams toward the ntended recever to ncrease SNR, whe formng nus to unntended recevers to avod nterference. To see how ZFBF can be used n MIMO ad hoc networks, consder an ad hoc 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 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 +n,,(6) m Ī } } 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. (7) Note that f there exst non-trva soutons to (7) (.e., R Rx() 0, T Tx(m) 0,, m Ī), then t means that a L 0 nks can be actve smutaneousy n an nterferencefree envronment. Moreover, the ranks of T Tx() and R Rx() determne the number of data streams z that can be transmtted over nk,.e., z = mnrank(t Tx() ), rank(r Rx() )}. Athough ZFBF s benefts are appeang, there reman sgnfcant chaenge to empoy t n arge ad hoc networks. Ths s because fndng an optma set of T s and R s satsfyng (7) 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 (7). B. SUCCINCT: 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 (7), we propose to reposton ourseves to expot matrx dmenson constrants that are suffcent for (7) to hod. By dong so, we can characterze the nk ayer schedung performance wthout entangng wth the detas of beamformng desgns. To understand how we can extract the matrx dmenson constrants for ZFBF-based schedung, we frst use a smpe two-nk one-nterferng-node network shown n Fg. 4 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 schedue the transmssons of nks and m as foows. Frst, we arbtrary choose a transmt beamformng matrx T Tx() wthout consderng nk m s exstence. Suppose that T Tx() s fu-rank, meanng that 3 data streams are

5 5 Lnk m z m Rx(m) Tx() H Tx(),Rx(m) Lnk z Move to the next node n the ordered node st 5 5 Fg. 4. R Rx(m) T Tx() 3 3 A two-nk one-nterferng-node exampe. For a gven ordered node st, start from the 1 st node A nodes schedued? Y N Current node a Tx node? N Y Nu ts nterference to Rx nodes schedued before tsef Suppress nterference from Tx nodes schedued before tsef 5 zm Stop Fg. 5. DoF regon of two-nk onenterferng-node exampe. z 1 antenna 1 antenna z = 1 Tx() Rx() Lnk Tx(m) z m = 1 2 antennas Lnk m T Tx(n) R Rx() R Rx(m) Rx(m) 2 antennas z n = 2 Tx(n) Rx(n) 4 antennas Lnk n 4 antennas Fg. 6. A three-nk mutpenterferng-node exampe. beng transmtted. Next, we choose a R Rx(m) to cance the nterference from nk, whe recevng data streams from ts desred transmtter. To do ths, we need to sove for R Rx(m) such that (H Tx(),Rx(m) T Tx() ) R Rx(m) = 0, where T Tx() s aready determned. From basc near agebra, we know that a the receve beamformng vectors n R Rx(m) have to e n the nu space of (H Tx(),Rx(m) T Tx() ), denoted by nu((h Tx(),Rx(m) T Tx() ) ). Note that the dmenson of the nu space n ths case s dm(nu((h Tx(),Rx(m) T Tx() ) )) = 5 3 = 2. Ths mpes that Rx(m) can receve up to 2 streams from Tx(m) (or nk m can have 2 DoFs for recevng). Note that n ths case 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 our ZFBF-based schedung scheme s the trapezod shown n Fg Observe that the schedung scheme n the above two-nk one-nterferng-node exampe s performed n a sequenta fashon: we arbtrary choose a T Tx() frst, and then choose a R Rx(m) such that the nterference can be emnated. We now extend ths successve nterference canceaton dea to a three-nk mutpe-nterferng-node exampe shown n Fg. 6, whch s much more compcated than the earer exampe. Here, each recevng node of a nk s beng nterfered by the transmttng nodes of other nks. To schedue transmsson/recepton, we can start wth a schedung order for the sx nodes. Such schedung order w be subject to optmzaton n Secton III-C. For exampe, suppose the schedung order for the sx nodes s Tx() Rx(m) Rx() Tx(m) Tx(n) Rx(n). Then the foowng schedung decsons w take pace. 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; 1 Note that the achevabe DoF regon n Fg. 5 concdes wth the maxmum DoF regon descrbed n [11, Theorem 2]. Thus, for ths two-nk exampe, the proposed schedung scheme s an optma schedung scheme. Fg. 7. The fow chart of SUCCINCT schedung agorthm. 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): Smar anayss as n 5) can be done for Rx(n) to show that 2 streams can be schedued at Rx(n). The dea n the three-nk mutpe-nterferng-node exampe can be syntheszed for a genera mutpe-nk settng. The essence of ths schedung scheme s to perform nterference canceaton successvey on each node n 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 resource 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 node-based schedung agorthm. For easy reference, we ca ths schedung scheme SUCCINCT (successve nterference canceaton). Addtona quanttatve constrants on DoF on each transmttng and recevng node (as shown n prevous two exampes) w be dscussed n the foowng secton. Fg. 7 shows the fow chart of SUCCINCT agorthm. Remark 1. In [8], Hamdaou and Shn proposed severa nterference avodance schemes based on ZFBF. For the socaed CM scheme (the best among the proposed schemes

6 6 n [8]), the authors aso recognzed that nterference can be canceed by ether the transmttng or the recevng node of an nterference nk. However, wthout empoyng 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. [8, Eq. (10)]). Ths approach adversey eads to a much smaer DoF regon. As an exampe, we compare the performance of SUCCINCT and the CM mode on the smpe two-nk one-nterferng-node exampe n Fg. 4. Under the CM mode, t s not dffcut to show that the achevabe DoF regon s the shaded trange n Fg. 8, whch s much smaer than the achevabe DoF regon by SUCCINCT. The detaed anayss on why ths s the case s provded n Appendx C. In genera, t can be shown that the DoF regon acheved under the CM mode s aways a subset of that under the SUCCINCT scheme [13]. The detas of the proof can aso be found n Appendx C zm Fg. 8. Achevabe DoFs comparson between SUCCINCT and CM for the exampe n Fg. 4. C. SUCCINCT: A Mathematca Mode Havng ntroduced the basc dea of SUCCINCT, we now deveop ts mathematca mode. We represent the topoogy of a MIMO ad hoc 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 on perod frameby-frame system wth T tme sots n each frame. Ordered Node Lst: An Optmzaton Mode. Referrng to Fg. 7 and the dscusson n Secton III-B, before we start schedung on a node, we must have an ordered node st, whch tsef s subject to optmzaton. To mode an ordered node st that can be optmzed, we defne the foowng bnary varabe. For, j N, j, we et 1 f node j s after node n tme sot t, π j (t) = 0 f node j s before node n tme sot t. To mode an ordered node st, π j (t) must satsfy the foowng two propertes. 1) Mutua excusveness: If node j s schedued after node (.e., π j (t) = 1), then t aso mpes that node s before z TABLE II ENUMERATING THE TRANSITIVITY RELATIONSHIP. π j (t) π jk (t) π k (t) ndefnte 0 1 ndefnte node j (.e., π j (t) = 0). Ths reatonshp can be modeed as π j (t) + π j (t) = 1,, j N : j, t. (8) 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). We enumerate a possbe cases for π j and π jk n Tabe II and show what π k w be. In Tabe II, ndefnte means that π k (t) cannot be determned by the current settngs of π j (t) and π jk (t). Mathematcay, the reatonshp n Tabe II can be modeed as π k (t) π j (t) + π jk (t),, j, k N, t, (9) π k (t) π j (t) + π jk (t) 1,, j, k N, t. (10) For exampe, for the second row n Tabe II, when π j (t) = 1 and π jk (t) = 0, (9) gves π k (t) 1 and (10) gves π k (t) 0. Gven that π k (t) s a bnary varabe, ths s equvaent to sayng that π k (t) remans to be determned (or ndefnte). Note that accordng to (9) and (10), we can wrte 12 dfferent constrants n 6 groups for three nodes, j, k N as foows. πj (t) π k (t) + π kj (t) (11) π j (t) π k (t) + π kj (t) 1, πj (t) π jk (t) + π k (t) (12) π j (t) π jk (t) + π k (t) 1, πk (t) π j (t) + π jk (t) (13) π k (t) π j (t) + π jk (t) 1, πk (t) π kj (t) + π j (t) (14) π k (t) π kj (t) + π j (t) 1, πjk (t) π j (t) + π k (t) (15) π jk (t) π j (t) + π k (t) 1, πkj (t) π k (t) + π j (t) (16) π kj (t) π k (t) + π j (t) 1. A coser ook at these 6 groups of constrants show that any one group can be used to derve the other 5 groups of constrants (see Appendx D). In other words, any one group from the sx groups n (11) (16) s suffcent to descrbe the transtvty property for a node trpet, j, k}. To seect ony one group out of (11) (16) to descrbe the transtvty property, we need to mantan certan consstency n such seecton. Our approach to acheve such consstency s to create a mappng Ω( ) : N N, where each node n N s mapped to an nteger number n N = 1, 2,..., N}. Now,

7 7 wthout oss of generaty, suppose that nodes, j, and k satsfy Ω() < Ω(j) < Ω(k). We w seect group constrant (11), whch s the same as seectng two constrants from (9) and (10) such that Ω() < Ω(j) < Ω(k). Usng such a mappng and the constrant Ω() < Ω(j) < Ω(k), we can consstenty and unquey dentfy a group out of 6 groups for any node trpet. Now, combnng the ordered mappng and (9) and (10), we have: π k (t) π j (t) + π jk (t),, j, k N : Ω() < Ω(j) < Ω(k), t, (17) π k (t) π j (t) + π jk (t) 1,, j, k N : Ω() < Ω(j) < Ω(k), t. (18) We further show n Appendx D that (17) and (18) can be wrtten n a more compact form: 1 π j (t)+π jk (t)+π k (t) 2. We summarze ths resut n the foowng emma. Lemma 2. For nodes, j, k N such that 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. (19) It s not hard to see that Lemma 2 w sgnfcanty reduce computatona compexty n fndng the optma spata schedung order. Aso note that oop s not aowed n the transtvty reatonshp n (19). Ths s because that (19) mpes that at east 1 and at most 2 π-varabes can be equa to one. The cases when oop occurs,.e., ether π j (t) = π jk (t) = π k (t) = 0 or π j (t) = π jk (t) = π k (t) = 1, are not aowed n (19). A Mode for Transmttng Node Behavor. Next, we mode the bock n Fg. 7 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) = 1 f node s transmttng n tme sot t and 0 otherwse; h (t) = 1 f 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. (20) 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, (21) L Out h (t) L In z (t) h (t)a, (22) 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. 7, 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), (23) where I represents the set of nodes wthn the nterference range of node. For convenence, we et S denote the stacked matrx n (23). Note that L z Out (t) s the number of data streams that node transmts n tme sot t. Thus, from (23), we have that L z Out (t) shoud be ess than or equa to the nuty of 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 [9] (.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, z (t) + π j (t) z (t) A + (1 g (t))m. (24) j I L Out :Rx()=j Tx() In (24), 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 (24) s reduced to the rank-nuty condton wth respect to S. Otherwse, f node s schedued to be a recevng node or n de status (.e., g (t) = 0), then (23) becomes rreevant due to the arge vaue of M. We note that the nonnear terms π j (t) Tx() :Rx()=j z (t) n (24) coud compcate optmzaton agorthms desgn. To remove these nonnear terms, we can ntroduce a new nteger varabe φ j (t) and reformuate (24) as foows: z (t) + φ j (t) A + (1 g (t))m, (25) j I L Out where φ j (t), N, j I, t, satsfes the foowng constrants: φ j (t) Tx() :Rx()=j z (t), (26) φ j (t) A π j (t), (27) φ j (t) A π j (t) A + Tx() :Rx()=j z (t). (28) It s easy to verfy that ths set of new constrants (25) (28) s equvaent to (24). A Mode for Recevng Node Behavor. Smary, for the bock n Fg. 7 where a node s schedued to be a recevng node, we can derve the foowng constrants: for a, j N

8 8 and for a t, L z In (t) + j I ϕ j (t) A + (1 h (t))m, (29) ϕ j (t) Rx() :Tx()=j z (t), (30) ϕ j (t) A π j (t), (31) ϕ j (t) A π j (t) A + Rx() :Tx()=j z (t). (32) IV. APPLICATION IN MULTI-HOP AD HOC NETWORKS In Sectons II and III, we have deveoped two modes for the physca ayer and the nk ayer n mut-hop MIMO ad hoc networks, respectvey. In ths secton, we w show how to appy them for cross-ayer optmzaton n mut-hop MIMO ad hoc 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 rate of sesson f and r (f) the amount of 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),. (33) At the network ayer, dfferent routng schemes can be adopted. But regardess of specfc routng schemes, the fow baance constrants must hod at each node N n the network. L r Out (f) = r(f), f = src(f), (34) L r Out (f) = L r In (f), f src(f), dst(f), (35) L r In (f) = r(f), f = dst(f). (36) It can be easy verfed that f (34) and (35) are satsfed, then (36) s automatcay satsfed. As a resut, there s not necessary to st (36) n a formuaton once we have both (34) and (35). When a node s transmttng smutaneousy on more than one outgong nks, t s necessary to consder power aocaton among L Out at node. Reca that α (t) [0, 1] represent a fracton of transmt power aocated onto nk n tme-sot t. Then, for each node n tme-sot t, we have L α Out (t) g n (t), n, t. (37) The constrant n (37) ensures that the sum of transmt power of a outgong nks at node does not exceed the transmt power mt. In the case when node s not n transmsson mode, then g (t) = 0 and α (t) = 0 for a L Out. Consder an objectve 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 max F f=1 u( r(f) ) s.t. Network ayer fow-baance routng constrants n (35) and (34); Coupng constrants between network ayer and nk ayer n (33); SUCCINCT based nk ayer modeng constrants n (20), (21), (22), (8), (19) and (25) (32); Smpfed MIMO physca ayer mode n (5) and (37). Two remarks for the MIMO-NUM probem are n order. 1) Tractabty. Reca that a 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 MIMO-NUM probem, whch sgnfcanty smpfes formuaton and reduce computatona compexty. 2) Sovabty. By usng our smpe modes, MIMO-NUM probem s reduced to a smar mathematca form as n NUM probems for snge-antenna ad hoc networks. Note that athough SUCCINCT part s unque, t s of near form and can be easy handed mathematcay. Ths suggests that new soutons to MIMO-NUM probems may be deveoped by drawng upon the rch experences ganed for snge-antenna ad hoc networks. As an exampe to ustrate the sovabty of our MIMO- NUM formuaton, we consder a MIMO ad hoc network consstng of 50 nodes that are unformy dstrbuted n a square regon of 1500m 1500m (see Fg. 9). 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 mnmum-hop 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 optma souton. The optma schedung orderng for each node n each tme sot s sted n Tabe III. In ths tabe, each coumn gves the node orderng for schedung n a gven tme sot of the frame. For exampe, n the frst tme sot, the optma orderng of the nodes s N19 N18... N2. Fg. 10 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 (5), and the constrant n (33), the optma sesson rates are found to be (n Mb/s): 60.4 for

9 9 (m) (m) N11 N10 N12 N45 N1 N19 N50 N43 N13 N28 N41 N27 N3 N32 N31 N39 N26 N38 N23 N N2 N30 N37 N34 N9 N36 N15 N20 N5 N24 N44 N25 N49 N22 N8 N6 N35 N7 N46 N48 N (m) Fg N10 N4 N21 N33 N17 N40 N29 N47 N16 N42 A 50-node 5-sesson MIMO-based ad hoc network. N11 N12 N45 N1 N19 N13 N28 N3 0:1 3:3 N41 1:2 2:1 N32 1:1 2:1 N31 N38 N23 N14 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 N18 N (m) Fg. 10. N4 N21 N17 N40 Schedung resut on each nk. N26 N19, 151 for N44 N18, 102 for N24 N15, 36.6 for N48 N2, and 57.2 for N9 N32. N29 N47 TABLE III 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 for mut-hop ad hoc network. In ths paper, we proposed a tractabe and accurate mode for MIMO that s amenabe for cross-ayer anayss n mut-hop ad hoc networks. 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 SUCCINCT that s based on ZFBF nterference mtgaton. The proposed SUCCINCT scheme cuts through the compexty assocated wth beamformng desgns n a muthop ad hoc network by usng smpe numerc computaton on matrx dmensons. As a resut, we can characterze the nk ayer schedung performance wthout entangng wth beamformng detas. By appyng the proposed cross-ayer mode to a genera network utty maxmzaton probem, we vadate ts effcacy n practce. The resuts n ths paper offer an mportant anaytca too to fuy expot the potenta of MIMO n mut-hop ad hoc networks. REFERENCES [1] R. Bhata and L. L, Throughput optmzaton of wreess mesh networks wth MIMO nks, n Proc. IEEE INFOCOM, Anchorage, AK, May 6-12, 2007, pp [2] E. Bger, R. Caderbank, A. Constantndes, A. Godsmth, A. Pauraj, and H. V. Poor, MIMO Wreess Communcatons. Cambrdge Unversty Press, Jan [3] L.-U. Cho and R. D. Murch, A transmt preprocessng technque for mutuser MIMO systems usng a decomposton approach, IEEE Trans. Wreess Commun., vo. 3, no. 1, pp , Jan [4] S. Chu and X. Wang, Opportunstc and cooperatve spata mutpexng n MIMO ad hoc networks, n Proc. ACM Mobhoc, Hong Kong SAR, Chna, May 26-30, 2008, pp [5] 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 , [6] A. Godsmth, S. A. Jafar, N. Jnda, and S. Vshwanath, Capacty mts of MIMO channes, IEEE J. Se. Areas Commun., vo. 21, no. 1, pp , Jun [7] I. S. Gradshteyn and I. M. Ryzhk, Tabe of Integras, Seres, and Products. San Dego: Academc Press, [8] B. Hamdaou and K. G. Shn, Characterzaton and anayss of muthop wreess MIMO network throughput, n Proc. ACM Mobhoc, Montréa, Québec, Canada, Sep. 2007, pp [9] R. A. Horn and C. R. Johnson, Matrx Anayss. New York: Cambrdge Unversty Press, [10] M. Hu and J. Zhang, MIMO ad hoc networks: Medum access contro, saturaton throughput, and optma hop dstance, Speca Issue on Mobe Ad Hoc Networks, Journa of Communcatons and Networks, pp , Dec [11] S. A. Jafar and M. J. Fakhereddn, Degrees of freedom for the MIMO nterference channe, IEEE Trans. Inf. Theory, vo. 53, no. 7, pp , Ju [12] S.-J. Km, X. Wang, and M. Madhan, Cross-ayer desgn of wreess muthop backhau networks wth mutantenna beamformng, IEEE Trans. Mobe Comput., vo. 6, no. 11, pp , Nov [13] J. Lu, Y. Sh, and Y. T. Hou, A tractabe and accurate cross-ayer mode for mut-hop MIMO ad hoc networks, Technca Report, Department of ECE, Vrgna Tech, Ju [Onne]. Avaabe:

10 10 [14] M. L. Metha, Random Matrces, 3rd ed. London, UK: Academc Press, [15] S. Y. Oh, M. Gera, and J.-S. Park, MIMO and TCP: A case for crossayer desgn, n Proc. IEEE MILCOM, Orando, FL, Oct , [16] S. Y. Oh, M. Gera, P. Zhao, B. Daneshrad, G. Pe, and J. H. Km, MIMO-CAST: A cross-ayer ad hoc mutcast protoco usng MIMO rados, n Proc. IEEE MILCOM, Orando, FL, Oct , [17] J.-S. Park, A. Nandan, M. Gera, and H. Lee, SPACE-MAC: Enabng spata reuse usng MIMO channe-aware MAC, n Proc. IEEE ICC, Seou, Korea, May 16-20, 2005, pp [18] S. Roman, Advanced Lnear Agebra. New York, NY: Sprnger, [19] Q. H. Spencer, A. L. Swndehurst, and M. Haardt, Zero-forcng methods for downnk spata mutpexng n mutuser mmo channes, IEEE Trans. Sgna Process., vo. 52, no. 2, pp , Feb [20] K. Sundaresan and R. Svakumar, Routng n ad hoc networks wth MIMO nks, n Proc. IEEE Internatona Conf. on Network Protocos, Boston, MA, U.S.A., Nov. 2005, pp [21] I. E. Teatar, Capacty of mut-antenna Gaussan channes, European Trans. Teecomm., vo. 10, no. 6, pp , Nov [22] D. Tse and P. Vswanath, Fundamentas of Wreess Communcaton. Cambrdge, UK: Cambrdge Unversty Press, [23] A. M. Tuno and S. Verdú, Random Matrx Theory and Wreess Communcatons. Hanover, MA: now Pubshers Inc., APPENDIX A PROOF OF LEMMA 1 Under hgh SNR regme, ρ α s arge enough such that a spata channes are actve wth hgh probabty,.e., z =1 (µ (ρ α λ ) 1 ) + = z =1 (µ (ρ α λ ) 1 ) = 1. Thus, we have z µ = 1 + z =1 (ρ α λ ) 1. (38) Aso, we have that C (wf) = W z =1 og 2(ρ α µλ ). On the other hand, we can re-wrte the capacty formua for the smpe mode as C (sm) = W ( z =1 og ρ α z ). As a resut, the capacty gap can be computed as C = C (wf) W z = W z =1 og 2 C (sm) = W z =1 og 2 ρ α µλ 1+ρ α /z =1 og 2 z µλ (39) ( 1 + (ρ α ) 1 z =1 λ 1 ) λ (40) W z =1 og 2 λ, (41) where (39) foows from the fact that n hgh SNR regme, ρ α /z 1 so that 1 can be gnored; (40) foows from substtutng (38) nto (39), and (41) foows from the fact that n hgh SNR regme, the term (ρ α ) 1 z =1 λ 1 1 (none of λ can be too sma snce H s a we-condtoned matrx) and can be gnored. Ths competes the proof. APPENDIX B PROOF OF THEOREM 1 Frst of a, from Lemma 1, we have z z E H [ C ] W E λ [og 2 λ ] W og 2 E λ [λ ]. =1 =1 where the ast nequaty foows from the concavty of the og functon and Jensen s nequaty. The Mar cenko-pastur theorem [14] 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) +, (42) 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 (42) can be used to serve as an exceent approxmaton [23]. Now, et us frst consder the case wth 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 z og 2 E λ [λ ] = z og 2 E[λ] =1 = z og 2 ( 1 2πβ u ) 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 [7, Eq ], we can derve that u R(x)dx = 2x 2(1 + β) 4 R(x) u + 16β 8 u dx R(x). (43) Note that the frst term n the summaton n (43) s zero. Then by usng [7, Eqn ], we can further derve that u ( 2x + 2(1 + β) R(x)dx = 2β arcsn 16β = 2β(arcsn( 1) arcsn(1)) = 2πβ. =1 ) u It thus foows that z ( ) 2πβ E H [ C ] W og 2 E λ [λ ] = W z og 2 = 0. 2πβ For the case where β > 1, the frst term n the p.d.f. functon n (42) becomes reevant. 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. (44) β β Combnng two cases, we fnay have E H [ C ] 0 for a β, and the proof s compete. APPENDIX C PERFORMANCE COMPARISON BETWEEN SUCCINCT AND THE H-S MODEL For the CM scheme n [8], two sets of nteger varabes are defned: For every par of mutuay-nterfered nks and j n tme sot t, θ j (t) represents the number of DoFs assgned by t() to nu ts nterference at r(j), and ϑ j (t) represents the number of DoFs assgned by r(j) to suppress the nterference comng from t(). The H-S nk ayer mode s gven as foows (n ths paper s notaton): L z Out (t) + I + θ (t) A t(), (45) L z In (t) + I ϑ (t) A r(), (46) z (t) ϑ j (t) + α t() (1 y (t)), (47) z j (t) θ j (t) + β r(j) (1 y j (t)). (48)

11 11 Now, et us compare the performance usng the 2-nk exampe shown n Fg. 4. Ceary, when both nks and j are actve,.e., y (t) = y j (t) = 1, we have from (47) and (48) that z (t) ϑ j (t) and z j (t) θ j (t). In Secton III-A, we can see that z (t) = 3 and z j (t) = 2 can be schedued wthout causng any mutua nterference. However, z (t) = 3 and z j (t) = 2 ceary voates (45)-(48). Ths s because z (t) = 3 θ j = 0 (by (45)) and θ j = 0 z j (t) = 0 (by (48)), a contradcton to z j (t) = 2. Thus, we can see that SIM can schedue more data streams than CM. In fact, for the smpe 2-nk exampe n Fg. 4, we can derve the DoF regon under the H-S mode as foows. Frst, note that there s ony one nterference gong from nk to nk j. Thus, when both nks are actve, we can smpfy (45)- (48) as: z + θ j 3, z j + ϑ j 5, and z j θ j. Snce z j θ j s tghter than z j + ϑ j 5 (because θ j 3), the atter can be gnored. As a resut, we have DoF regon as z + z j z + θ j 3. We pot ths DoF regon (as squares) n Fg. 8. It can be seen that the H-S mode can ony acheve a porton of the entre DoF regon. To formay state the superor performance of SUCCINCT, we have the foowng fact. Fact 1. The achevabe DoF regon C SUCCINCT contans the achevabe DoF regon C HS acheved by the H-S mode,.e., C HS C SUCCINCT. Proof: Reca that n the SUCCINCT scheme, the number of data streams and the number of DoFs used for nterference mtgaton satsfy the foowng constrants: z (t) + π j (t) z (t) A + (1 g (t))m., (49) j I L Out :Rx()=j Tx() z (t) + π j (t) z (t) A + (1 h (t))m. (50) j I L In :Tx()=j Rx() On the other hand, the nk ayer constrants for transmssons and nterference mtgaton n the H-S mode can be re-wrtten as: z (t) + z (t) A + (1 g (t))m., (51) j I :Rx()=j L Out z (t) + j I L Out Tx() :Rx()=j Tx() z (t) A + (1 g (t))m. (52) It can see that the RHS are dentca. However, n the SUC- CINCT scheme, the number of terms n the summaton for nterference mtgaton for each node s strcty ess than that n the H-S mode except for the ast node to be schedued n the SUCCINCT scheme. Ths s because for each node n the network, t ony has to perform nterference mtgaton for the nodes that are schedued earer. Mathematcay, ths can be refected by the fact that ony some of γ-varabes n the summaton are equa to one. By contrast, n the H-S mode, each node has to perform nterference mtgaton task for a nodes wthn ts nterference range. As a resut, the number of terms n the summaton for nterference mtgaton s strcty arger than that n the SUCCINCT scheme. Mathematcay, ths s equvaent to settng a π-varabes equa to 1 n (49) and (50). Snce more DoF resources are used for nterference mtgaton n the H-S mode, we have that the number of DoF used for data transmssons s strcty ess than that n the SUCCINCT scheme, thus resutng n a strcty arger DoFs regon. APPENDIX D PROOF OF LEMMA 2 For smpcty, we drop the tme sot ndex t of π varabes n ths proof. We frst show that gven any one group n (11) (16), the remanng 5 are redundant. Wthout oss generaty, et us consder the two constrants n group (11),.e., πk π j + π jk π k π j + π jk 1. Mutpy 1 and addng 1 on both sdes of the above two nequates, we have 1 πk 1 π j π jk 1 π k 1 π j π jk + 1 πk π j + π kj 1 π k π j + π kj. Ths shows that the set of constrants n group (12) can be derved from group (11) and the mutua excusveness constrants. Thus, group (12) s not ndependent and can be removed. Foowng the same token, we can aso show that πj π k + π kj π j π k + π kj 1 πj π jk + π k 1 π j π jk + π k, and πjk π j + π k π jk π j + π k 1 πkj π k + π j 1 π kj π k + π j. That s, the groups (14) and (16) can derved from groups (13) and (15). So, they can aso be removed. Now, et us consder the remanng 6 constrants assocated wth groups (11), (13), and (15). Note that π k π j + π jk π k π j + 1 π kj π j π k + π kj 1 π k π j + π jk 1 π k π j π kj π j π k + π kj. Ths shows that group (13) can be derved from group (11) and can aso removed. Foowng exacty the same token, we can show that group (15) can be derved from group (11) and aso can be removed. So fnay, we have ony 2 remanng constrants assocated wth group (11). Next, we show that (11) can be wrtten n a more compact form. Note that π k π j + π jk 1 π k π j + π jk π j + π jk + π k 1 π k π j + π jk 1 1 π k π k + π jk 1 π j + π jk + π k 2, whch gve the constrants stated n Lemma 2. Ths competes the proof.