On Interference Alignment for Multi-hop MIMO Networks

Transcription

1 013 Proceedngs IEEE INFOCOM On Interference Algnment for Mult-hop MIMO Networks Huacheng Zeng Y Sh Y. Thomas Hou Wenng Lou Sastry Kompella Scott F. Mdkff Vrgna Polytechnc Insttute and State Unversty, USA U.S. Naval Research Laboratory, Washngton, DC, USA Abstract Interference algnment (IA) s a maor advance n nformaton theory. Despte ts rapd advance n the nformaton theory communty, most results on IA reman pont-to-pont or sngle-hop and there s a lack of advance of IA n the context of mult-hop wreless networks. The goal of ths paper s to make a concrete step toward advancng IA technque n mult-hop MIMO networks. We present an IA model consstng of a set of constrants at a transmtter and a recever that can be used to determne a subset of nterferng streams for IA. Based on ths IA model, we develop an IA optmzaton framework for a multhop MIMO network. For performance evaluaton, we compare the performance of a network throughput optmzaton problem under our proposed IA framework and the same problem when IA s not employed. Smulaton results show that the use of IA can sgnfcantly decrease the DoF consumpton for IC, thereby mprovng network throughput. I. INTRODUCTION Interference algnment (IA) s wdely regarded as a maor advance n nformaton theory n recent years [10]. The concept of IA refers to the constructon of sgnals at transmtters so that these sgnals overlap at non-ntended recevers whle they reman resolvable at ntended recevers. It was shown n [1] that by usng IA, each user n the K-user nterference channel can obtan 1/ nterference-free channel capacty regardless of the number of users. That s, the aggregate user capacty scales lnearly wth K/. Gven ts potental n capacty mprovement n wreless networks, IA has become a central research theme n the nformaton theory communty (see, e.g. [], [19]). Despte ts rapd advance n the nformaton theory communty, most results on IA reman pont-to-pont or snglehop and there s a lack of advance of IA n the context of mult-hop wreless networks. Ths s manly due to the complex nterference pattern nherent n a mult-hop network envronment (see Secton IV). As a result, exstng (snglehop) IA schemes cannot be easly extended nto mult-hop wreless networks. In [13], L et al. attempted to explore IA n a mult-hop MIMO networks. The dea of IA was descrbed n several examples n the paper to llustrate ts benefts. However, the key concept of IA (.e., the constructon of sgnals at transmtters such that these sgnals overlap at nonntended recevers whle they reman resolvable at ntended recevers) was not ncorporated nto ther problem formulaton and thus was absent n the fnal soluton. In another recent effort n [7], the authors used IA n ther paper ttle although they only consdered transmtter-sde zero-forcng technque. The lack of results of IA n mult-hop networks underlnes both the techncal barrer n ths area and the crtcal need to close ths gap by the research communty. The goal of ths paper s to make a concrete step toward advancng IA technque n mult-hop MIMO networks. We consder IA as the constructon of transmt data streams so that () they overlap at recevers where they are consdered as nterferng streams and () they are resolvable at ther ntended recevers (not to be overlapped by ether nterferng streams or other data streams). The constructon of transmt data streams s equvalent to the desgn of transmt vector for each data stream at each transmtter. Snce the nterferng streams are overlapped at a recever, one can use fewer number of DoFs to cancel these nterferng streams. As a result, the DoF resources consumed for IC wll be reduced and thus more DoF resources become avalable for data transport. The man contrbutons of ths paper are summarzed as follows. We model IA for a transmtter and a recever n a multhop MIMO network. Our model conssts of a set of constrants at a transmtter to determne whch subset of nterferng streams can be used for IA and a set of constrants at a recever to determne whch subset of nterferng streams. Based on the proposed IA model, we develop a set of constrants across multple layers of a mult-hop MIMO network. Collectvely, these constrants form an IA optmzaton framework for a mult-hop MIMO network. Under ths framework, IA can be exploted to the fullest extent for a target network performance obectve. For performance evaluaton, we compare the performance of a network throughput optmzaton problem under our proposed IA framework and the same problem when IA s not employed. We show that the use of IA can reduce DoF consumpton for IC at recevng nodes n the network and acheve hgher throughput obectve than the case when IA s not employed. The remander of ths paper s organzed as follows. Secton II presents related work on IA. Secton III offers some essental background on IA n MIMO networks. Secton IV dscusses the challenges of applyng IA n mult-hop networks. In Secton V, we model IA at both transmtter and recever. In Secton VI, we develop an IA optmzaton framework for a mult-hop MIMO network. In Secton VII, we apply the IA optmzaton framework to evaluate the benefts of IA n a /13/$ IEEE 1354

2 z 1 data streams z data streams TABLE I NOTATION. T 1 R 1 T R Fg. 1. SM and IC n MIMO. mult-hop MIMO network. Secton VIII concludes ths paper. II. RELATED WORK The concept of IA was coned n a semnar paper by Jafar and Shama for the two-user X channel [1]. Snce then, results for IA have been developed for a varety of channels and networks n ncreasngly sophstcated forms, such as the K- user nterference channel [1], the X network wth arbtrary number of users [], the cellular network [19], ergodc capacty n fadng channel [8]. A dstrbuted IA scheme was proposed by Gomadam et al. n [5]. The feasblty of IA n sgnal vector space for K-user MIMO nterference channel was studed by Yets et al. n [1], and blnd IA (no CSI at transmtter) was studed n [9]. A tutoral on IA from nformaton theory perspectve s [10]. In wreless communcatons and networkng communtes, efforts on IA have been manly lmted to valdatons on small toy networks [3], [4], [14]. In [3], El Ayach et al. dd an expermental study of IA n MIMO-OFDM nterference channels and showed that IA acheves the theoretcal throughput gans. In [4], Gollakotta et al. demonstrated that the combnaton of IA and IC ncreases the average throughput by 1.5 on the downlnk and on the uplnk n a MIMO WLAN. In [14], Ln et al. proposed a dstrbuted random access protocol (called 80.11n + ) based on IA and demonstrated that the system can double the average network throughput n a small network wth three pars of nodes. III. PRELIMINARIES: IA IN MIMO In ths secton, we revew MIMO s DoF resources for spatal multplexng (SM) and nterference cancellaton (IC). We also revew how IA can help reduce the number of DoFs requred for IC. Table I lsts the notaton used n ths paper. MIMO s DoF Resources for SM and IC. The number of DoFs of a node s typcally assumed to be the same as the number of antennas at the node and represents the total avalable resources at the node for SM and IC [11], [18], [0]. SM refers to the use of one or multple DoFs (both at transmttng and recevng nodes) for data transport, wth each DoF correspondng to one ndependent data stream. IC refers to the use of one or more DoFs to cancel nterference from other nodes, wth each DoF beng responsble for cancellng one nterferng stream. IC can be done ether at a transmt node (to cancel nterference to another node) or a receve node (to cancel nterference from another node). For example, consder two lnks n Fg. 1. To transmt z 1 data streams on lnk (T 1, R 1 ), both nodes T 1 and R 1 need to consume z 1 DoFs for SM. Smlarly, to transmt z data streams on lnk (T, R ), both nodes T and R need to consume z DoFs for SM. The nterference from T to R 1 can be cancelled by ether R 1 or Symbol Defnton A The set of nterferng streams from transmtter T to unntended recever R B The subset of nterferng streams n A that are algned to other nterferng streams at R c k An arbtrary nonzero number e k Unt vector wth 1 n the k-th entry and 0 n all others e k The nterferng stream from transmtter T to recever R that corresponds to transmt vector u k F The number of sessons n the network H Channel matrx between transmtter and recever I The set of nodes wthn node s nterference range L The number of lnks n the network L The set of lnks n the network L n The set of ncomng lnks at node L out The set of outgong lnks at node M The number of antennas at each node N The number of nodes n the network N The set of nodes n the network N r The number of recevng nodes n the network N t The number of transmttng nodes n the network r mn The mnmum data rate among all sessons n the network r(f) The data rate of sesson f r l (f) The amount of rate on lnk l that s attrbuted to sesson f Rx(l) The recever of lnk l R The -th recevng node n the network T The -th transmttng node n the network u k The transmt vector for stream s k at transmtter T x (t) A bnary varable to ndcate whether node s a transmtter for some lnk n tme slot t y (t) A bnary varable to ndcate whether node s a recever for some lnk n tme slot t z l (t) The number of data streams on lnk l n tme slot t α (t) The cardnalty of A n tme slot t β (t) The cardnalty of B n tme slot t λ The number of outgong data streams at transmtter T µ The number of ncomng data streams at recever R T. If R 1 cancels ths nterference, t needs to consume z DoFs. If T cancels ths nterference, t needs to consume z 1 DoFs. IA n MIMO. In the context of MIMO, IA refers to the constructon of transmt data streams so that () they overlap at recevers where they are consdered as nterferng streams and () they are resolvable at ther ntended recevers (not to be overlapped by ether nterferng streams or other data streams) [1], [4]. The constructon of transmt data streams s equvalent to the desgn of transmt vector (weghts) for each data stream at each transmtter. Snce the nterferng streams are overlapped at a recever, one can use fewer number of DoFs to cancel these nterferng streams. As a result, the DoF resources consumed for IC wll be reduced and thus more DoF resources become avalable for data transport. We use the followng example to llustrate the benefts of IA n MIMO networks. Consder the 4-lnk network shown n Fg.. A sold lne wth arrow represents drected lnk whle a dashed lne wth arrow represents drected nterference. Assume that each node s equpped wth three antennas. Suppose that there are data streams on lnk (T 1, R 1 ), data streams on lnk (T, R ), and 1 data stream on lnk (T 3, R 3 ). Denote u k as the transmt vector for the k-th data stream s k 1355

3 R1 R R3 [u 1 1 u 1 ] H4u 1 [u 1 3] T1 [u 1 u ] T T3 H41u 1 1 H43u 1 3 R4 H4u Fg.. An llustraton of IA at node R 4. H4u 1 on lnk (T, R ) and H as the channel matrx between T and R. When IA s not employed, R 4 needs to consume 5 DoFs to cancel the nterference from transmtters T 1, T, and T 3 [11], [18]. Snce there are only 3 DoFs avalable at R 4, t s not possble to cancel all 5 nterferng streams, let alone to receve any data stream from T 4. But when IA s used (see Fg. ), we can algn the 5 nterferng data streams nto dmensons, whch can be cancelled by R 4 wth DoFs. Therefore, R 4 stll has 1 DoF remanng, allowng t to receve 1 data stream from T 4. We now show one possble approach to construct the 5 transmt vectors at T 1, T, and T 3 so that ther 5 nterferng streams are algned nto dmensons at recever R 4. Frst, we construct the transmt vectors at T 1 ndependently by lettng u 1 1 = e 1 and u 1 = e, where e k s a unt vector wth the k-th entry beng 1 and other entres beng 0. For the two transmt vectors [u 1 u ] at T, we can algn the nterferng stream correspondng to u 1 to the nterferng stream correspondng to u 1 1 at recever R 4. Ths can be done by lettng H 4 u 1 = H 41 u 1 1 and thus u 1 = H 1 4 H 41 u 1 1. Smlarly, we can algn the nterferng stream correspondng to u to the nterferng stream correspondng to u 1 at recever R 4. Ths s done by lettng H 4 u = H 41 u 1 and thus u = H 1 4 H 41 u 1. Fnally, for the transmt vector u 1 3 at T 3, we can algn ts nterferng stream to the nterferng stream correspondng to u 1 1 at recever R 4. Ths s done by havng H 43 u 1 3 = H 41 u 1 1 and thus u 1 3 = H 1 43 H 41 u 1 1. As a result of IA, the 5 nterferng streams are algned nto only dmensons and can be cancelled by R 4 wth DoFs (nstead of 5). IV. APPLYING IA IN MULTI-HOP NETWORKS: WHERE ARE THE CHALLENGES As dscussed n Secton II, although there s a floursh of nformaton theoretc research on IA at the physcal layer, results on applyng IA n mult-hop networks reman very lmted. Ths s because there are a number of new challenges for applyng IA n mult-hop MIMO networks, whch we summarze as follows. How to perform IA among a large number of nodes n the network s a very hard problem. In partcular, for each par of nodes, one needs to determne whch subset of nterferng streams for IA and how to algn them successfully at the recever. Whle performng IA, one T4 also has to ensure that the desrable data streams at each recever reman resolvable (wthout beng overlapped by ether nterferng streams or other data streams). The answers to these questons requre the development of new IA constrants at both transmtter and recever. In MIMO networks, IA, IC and SM are coupled together through each node s DoF resources. Ths makes t dffcult to perform IA at each node whle the node s DoF s also beng used for SM and IC. The answer to ths queston requres the development of new DoF constrants for SM, IC, and IA at both transmtter and recever. In a mult-hop envronment, an IA scheme s also coupled wth the upper layer schedulng and routng algorthms. The upper layer algorthms determne the set of transmtters, the set of recevers, the set of lnks, and the number of data streams on each lnk, whch are dfferent n each tme slot. Thus, an IA scheme must be ontly desgned wth upper layer schedulng and routng algorthms, whch s agan a new and challengng problem. V. MODELING IA FOR A TRANSMITTER AND A RECEIVER In ths secton, we develop a set of constrants for IA n a mult-hop MIMO network. Assume that each node has M antennas. In a gven tme slot, suppose that we have a set of lnks L. Denote {T : 1 N t } and {R : 1 N r } as the sets of transmtters and recevers of L, respectvely. For transmtter T, denote λ as the number of outgong data streams and thus we have λ = l L s the out set of outgong lnks from T and z l as the number of data streams on lnk l L. 1 Smlarly, for recever R, denote µ as the number of ts ncomng data streams and thus we have µ = l L z n l, where L n nto R. At transmtter T, denote s k z l, where L out s the set of ts ncomng lnks as ts k-th outgong data stream and denote u k as the transmt vector of data stream s k Ḋenote I as the set of nodes wthn node s nterference range. Consder a node par (T, R ). For the transmsson of data stream s k on T, f R s not the ntended recever of ths data stream, then we call ths data stream as an nterferng stream, denoted as e k, at node R. Denote A as the set of nterferng streams from transmtter T to unntended recever R and denote α as the cardnalty of A. Note that wthout IA, recever R needs to expend α DoFs to cancel the nterference from transmtter T. Also, note that one data stream may be consdered as an nterferng stream by multple recevers. To reduce DoF consumpton for IC at a recever R, we can algn a subset of ts nterferng streams to the other nterferng streams by properly constructng ther transmt vectors. Among the nterferng streams n A, denote B (wth β = B ) as the subset of nterferng streams that are algned to the other nterferng streams at recever R. Then the effectve cardnalty of nterferng streams at recever R 1 The actvty of lnk l s determned by z l. If z l > 0, then lnk l s actve. If z l = 0, then lnk l s nactve. 1356

4 A 1,B 1 R 1 T 1 A 1,B 1 [u 1 u u λ T ] A,B A,B A Nr,B Nr R R T T A,B A,B A Nt,B Nt R A set of nterferng streams Fg. 3. IA constrants at transmtter T. R Nr T Nt A set of nterferng streams Fg. 4. IA constrants at recever R. s decreased from α to α β, resultng n a savng of β DoFs for IC. The queston to ask s then how to perform IA among the nodes n the network so that (C-1): each nterferng stream n B s s algned successfully; (C-): each data stream at ts ntended recever remans resolvable (not to be overlapped by ether nterferng streams or other data streams). Sectons V-A and V-B answer ths queston by mposng constrants at a transmtter and a recever, respectvely. A. IA Constrants at A Transmtter Based on the defntons of β and α, we have the followng constrants at transmtter T : β α, I. (1) Constrant (1) gves an upper bound for each β. At transmtter T, there are λ transmt vectors correspondng to λ outgong data streams. Each of the λ transmt vectors may correspond to multple nterferng streams, each for a dfferent unntended recevers. However, one can construct each transmt vector so that only one of ts correspondng nterferng streams s successfully algned to a partcular drecton for IA at ts recever. Gven that λ = l L z out l, we have the followng constrants at transmtter T : z l. () I β Constrant () ensures that (C-1) holds at transmtter T. As an example, let s consder transmtter T shown n Fg. 3. Transmt vector u k corresponds to the set of nterferng streams {e k : I }. For the set of nterferng streams {e k : I }, only one of them can be successfully algned to some drecton for IA by constructng u k. Thus, among those nterferng streams n I A (.e., all nterferng streams from transmtter T ), at most λ nterferng streams can be successfully algned to some drecton for IA at ther recevers. Therefore, the number of nterferng streams n I B s bounded by λ (.e., l L z out l ). B. IA Constrants at A Recever To ensure (C-1) and (C-) at recever R (see Fg. 4), we have the followng three condtons on IA. The frst condton s that each nterferng stream n I B can only be algned to an nterferng stream n I (A \B ). The second condton s that any nterferng stream n B cannot be algned to an nterferng stream n A. To show ths s true, suppose that e k n B s algned to e k n A at R. Then, we have u k = ck H 1 H u k = c k uk s a nonzero number), mplyng that transmt vectors (c k u k and uk streams s k and s k recever. are lnearly dependent. Ths means that data are not resolvable at ther ntended The thrd condton s that any two nterferng streams n B cannot be algned to the same (a thrd) nterferng stream. To show ths s true, suppose that both e k and e k n B are algned to e l r at R. Then, we have u k = c k H 1 H ru l r and u k = c k H 1 H ru l r. Based on these two equatons, we have u k = ck u k, ndcatng c k that transmt vectors u k and u k are lnearly dependent. Ths means that data streams s k and sk are not resolvable at ther ntended recever. The followng lemma gves necessary and suffcent condton for the exstence of IA scheme that meets the above three condtons at a recever. Lemma 1: There exsts an IA scheme that meets the above three condtons at recever R f and only f β k (α k β k ), I. (3) PROOF. We frst show the f part by constructon and then show the only f part by contradcton. Suffcent condton: We frst propose an algorthm based on (3) to obtan an IA scheme at R, and then show that the IA scheme obtaned by the proposed algorthm satsfes the three condtons at R. The proposed IA algorthm s as 1357

5 follows: For the nterferng streams n each B, we algn them to those nterferng streams n k (A k \B k ) wthout repetton. Snce β k (α k β k ) accordng to (3), we know that every nterferng stream n B can be algned to an nterferng stream n ths algorthm. We now show that the IA scheme obtaned by ths algorthm satsfes the three condtons at R. In ths algorthm, every nterferng stream n B s algned to an nterferng stream n k (A k \B k ). Thus, we know that the frst condton s satsfed. After performng ths algorthm at recever R, t s easy to see that any nterferng stream n B wll not be algned to an nterferng stream n A and that any two nterferng streams n B wll not be algned to the same (a thrd) nterferng stream. Thus, the second and thrd condtons are satsfed. Therefore, the f part of Lemma 1 s proved. Necessary condton: Consder any IA scheme at R. Suppose that β > k (α k β k ) for some I. Then for node par (T, R ), n order to meet the frst condton, the nterferng streams n B must be algned to the nterferng streams n k I (A k \B k ). In order to meet the second condton, the nterferng streams n B must be algned to the nterferng streams n k (A k \B k ). However, snce the cardnalty of B s greater than the cardnalty of k (A k \B k ) (.e., β > k (α k β k )), there exst two nterferng streams n B that are algned to the same (a thrd) nterferng stream n k (A k \B k ). Ths leads to a contradcton to the thrd condton. Ths completes the proof of the only f part of Lemma 1. VI. AN OPTIMIZATION FRAMEWORK In ths secton, we develop an optmzaton framework for IA n mult-hop MIMO networks. Consder a mult-hop MIMO network consstng of a set of nodes N (wth N = N ), each of whch s equpped wth M antennas. Denote L as the set of lnks n the network, wth L = L. Denote F the set of sessons n the network, wth F = F. Denote r(f) as the data rate of sesson f F. Denote src(f) and dst(f) as the source node and the destnaton node of sesson f F, respectvely. To transport data flow f from src(f) to dst(f), we allow flow splttng nsde the network for better load balancng and network resource utlzaton. For schedulng, we assume tme s slotted and a tme frame conssts of T tme slots. Half Duplex Constrants. We assume that a node cannot transmt and receve n the same tme slot. Denote x (t) (1 t T ) as a bnary varable to ndcate whether node N s a transmtter n tme slot t,.e., x (t) = 1 f node s a transmtter n tme slot t and 0 otherwse. Smlarly, denote y (t) (1 t T ) as another bnary varable to ndcate whether node N s a recever n tme slot t. Then the half duplex constrants can be wrtten as x (t) + y (t) 1, (1 N, 1 t T ). (4) Node Actvty Constrants. Denote z l (t) as the number of data streams on lnk l L n tme slot t. If node s a transmtter, then we have 1 l L z out l (t) M. Otherwse (.e., node s ether a recever or nactve), then we have l L z out l (t) = 0. Combnng the two cases, we have the followng constrants: x (t) z l (t) M x (t), (1 N, 1 t T ). (5) Smlarly, by consderng whether or not node s a recever, we have the followng constrants: y (t) z l (t) M y (t), (1 N, 1 t T ). l L n (6) General IA Constrants at a Node. In Secton V, we developed IA constrants for a transmtter and a recever. Here we generalze those constrants at a node that can be ether a transmtter, recever, or dle. Suppose that node s wthn the nterference range of node,.e., I. If node s a recevng node n tme slot t (.e., y (t) = 1), then α (t) (the number of nterferng streams from node to node n tme slot t) s Rx(l) z l (t), where Rx(l) s the recever of lnk l. Otherwse (.e., y (t) = 0), we have α (t) = 0 based on the defnton of α (t). In general, we have the followng constrants: Rx(l) α (t) = y (t) z l (t), ( I, 1 N, 1 t T ). (7) For β (t), f node s a transmtter, then based on (1), we have β (t) α (t), I. Otherwse (node s ether a recever or dle), we have β (t) = 0 and α (t) = 0 for each I based on ther defntons. Combnng these two cases, we have the followng constrants: β (t) α (t), ( I, 1 N, 1 t T ). (8) If node s a transmtter, based on (), we have I β (t) l L z out l (t). Otherwse (node s ether a recever or dle), we have I β (t) = 0 and l L z out l (t) = 0. Combnng these two cases, we have the followng constrants: β (t) z l (t), (1 N, 1 t T ). (9) I If node s a recever, based on (3), we have β (t) k (α k (t) β k (t)) for each I. Otherwse (node s ether a transmtter or dle), we have β (t) = 0 and α (t) = 0 for each I based on ther defntons. Combnng these two cases, we have the followng constrants: β (t) k [α k (t) β k (t)], ( I, 1 N, 1 t T ). (10) DoF Consumpton Constrants. Although an nterference can be cancelled at ether ts transmttng node or ts recevng node, we only consder the case where IC s done at a recevng 1358

6 node n ths paper. Then the DoF consumpton for SM and IC at a node can be summarzed as follows. Transmttng Node. The number of DoFs consumed for SM at a transmttng node s equal to the number of ts outgong data streams. Furthermore, there s no DoF consumpton for IC at a transmttng node, as t s not responsble for IC. Recevng Node. The DoF consumpton at a recevng node conssts of two parts: for SM and for IC. The number of DoFs consumed for SM at a recevng node s equal to the number of ts ncomng data streams, whle the number of DoFs consumed for IC at a recevng node s equal to the dmenson of ts nterference subspace (.e., I (α β ) for R ). Suppose that node s a transmtter n tme slot t. Then the number of DoFs t consumes s l L z out l (t) M. Otherwse, we have l L z out l (t) = 0. Combnng these two cases, we have the followng constrants: z l (t) M x (t), (1 N, 1 t T ). (11) Suppose that node s a recever n tme slot t. Then t consumes l L z n l (t) DoFs for SM and I [α (t) β (t)] DoFs for IC. Snce the number of DoFs consumed for SM and IC cannot exceed the total number of avalable DoFs at a node, then we have the followng DoF constrant at node : z l (t)+ I [α (t) β (t)] M. Otherwse (node l L n s ether a transmtter or dle), we have z l (t) = 0 for l L n and α (t) = β (t) = 0 for I based on ther defntons. Combnng these two cases, we have the followng constrants: z l (t) + l L n I [α (t) β (t)] M y (t), (1 N, 1 t T ). (1) Lnk Capacty Constrants. Denote r l (f) as the amount of data rate on lnk l that s attrbuted to sesson f F. For smplcty, we assume that one data stream n one tme slot corresponds to one unt data rate. 3 Then the average rate of lnk l over T tme slots s 1 T T t=1 z l(t). Snce the aggregate data rates cannot exceed the average lnk rate, we have F r l (f) 1 T z l (t), (1 l L). (13) T f=1 t=1 Flow Routng Constrants. At each node, flow conservaton must be observed. At a source node, we have r l (f) = r(f), ( = src(f), 1 f F ). (14) At an ntermedate relay node, we have r l (f) = r l (f), (1 N, src(f), l L n dst(f), 1 f F ). (15) The case where IC can be done at both transmttng and recevng nodes wll be nvestgated n our future work. 3 We assume fxed modulaton and codng scheme (MCS) n ths paper. At a destnaton node, we have r l (f) = r(f), ( = dst(f), 1 f F ). (16) l L n It can be easly verfed that f (14) and (15) are satsfed, then (16) s also satsfed. Therefore, t s suffcent to nclude only (14) and (15). VII. PERFORMANCE EVALUATION In ths secton, we apply the IA optmzaton framework for mult-hop MIMO networks that we developed n the prevous secton. In partcular, we use t to study a network throughput maxmzaton problem, and compare ts performance to the case where IA s not employed. A. A Throughput Maxmzaton Problem In a mult-hop MIMO network, suppose that the obectve s to maxmze the mnmum rate among all sessons, denoted as r mn. 4 Then we have the followng constrants: r mn r(f), 1 f F. (17) Accordng to the constrants developed n Secton VI, we have the followng formulaton: Max r mn s.t. Half duplex constrants: (4); Node actvty constrants: (5), (6); IA constrants: (7), (8), (9), (10); DoF consumpton constrants: (11), (1); Lnk capacty constrants: (13); Flow routng constrants: (14), (15); Mn rate constrants: (17). Among all these constrants, only (7) s nonlnear. We lnearze (7) by employng reformulaton lnearzaton technque (RLT) [17]. By analyzng the relatonshp between α (t) and Rx(l) z l (t) n (7), we construct two new sets of constrants (18) and (19). It can be verfed that the combnaton of (18) and (19) s equvalent to (7). and 0 Rx(l) z l (t) α (t) (1 y (t)) B, ( I, 1 N, 1 t T ), (18) 0 α (t) y (t) B, ( I, 1 N, 1 t T ), (19) where B s a constant nteger (e.g., B = M). By replacng nonlnear constrant (7) wth (18) and (19), we have the followng problem formulaton: OPT-IA Max r mn s.t. (4), (5), (6), (8), (9), (10), (11), (1), (13), (14), (15), (17), (18), (19). 4 Note that problems wth other obectves such as maxmzng sum of weghted rates or a proportonal ncrease (scalng factor) of all sesson rates belongs to the same category and can be solved followng the same token. 1359

7 N13 N1 N45 N3 N4 N39 N9 N3 N36 N N18 N37 N46 N9 N19 N10 N N47 N14 N6 N33 N5 N4 N5 N34 N7 N1 N11 N30 N40 N0 N N36 N39 N33 N13 N9 N47 N N1 N19 N45 N9 N14 N18 N3 N10 N5 N6 N4 N37 N3 N46 N N4 N5 N34 N7 N1 N11 N30 N40 N0 N N31 N48 N38 N49 N43 N0 N15 N6 N4 N8 N8 N17 N44 N16 N1 N41 N35 N N0 N6 N43 N17 N44 N35 N38 N49 N8 N16 N7 N31 N48 N15 N4 N8 N1 N (a) A 50-node network topology. (b) Tme slot N13 N39 N36 N N9 N47 N33 N5 N4 N N13 N39 N36 N N9 N47 N33 N5 N4 N N1 N45 N9 N3 N4 N3 N18 N37 N46 N19 N14 N10 N5 N6 N N7 N1 N11 N30 N N45 N3 N4 N1 N19 N9 N18 N14 N10 N6 N37 N3 N5 N7 N0 N1 N11 N30 N N6 N44 N43 N17 N0 N35 N38 N16 N31 N49 N8 N8 N7 N0 N3 N15 N1 N48 N4 N N6 N44 N43 N17 N46 N0 N35 N38 N16 N N31 N49 N8 N8 N7 N3 N15 N1 N48 N4 N (c) Tme slot. (d) Tme slot 3. Fg. 5. Transmsson/recepton pattern, nterference pattern, and IA scheme n each tme slot. In (b)-(d), a sold arrow lne represents a drected transmsson lnk (wth the number of data streams on ths lnk shown n a box). A dashed arrow lnk represents an nterference, wth the total number of nterferng streams and the number of subset nterferng streams chosen for IA shown n a box,.e., (α, β ). where x (t) and y (t) are bnary varables; z l (t), α (t), and β (t) are non-negatve nteger varables; r(f) and r l (f) are non-negatve varables; M, N, L, F, T, and B are constants. OPT-IA s a mxed nteger lnear programmng (MILP). Although the theoretcal worst-case complexty of solvng a general MILP problem s exponental [15], there exst hghly effcent optmal and approxmaton algorthms (e.g., branchand-bound wth cuttng planes [16]) and heurstc algorthms (e.g., sequental fxng algorthm [6]). Another approach s to employ an off-the-shelf solver such as CPLEX []. Snce the goal of ths paper s to develop an IA optmzaton framework for mult-hop MIMO networks (rather than developng a soluton procedure for a specfc problem), we wll employ CPLEX solver n ths performance evaluaton. B. Smulaton Settng Wthout loss of generalty, we normalze all unts for dstance, data rate, bandwdth, tme and power wth approprate dmensons. We consder a randomly generated mult-hop MIMO network wth 50 nodes, whch are dstrbuted n a square regon. Each node n the network s equpped wth four antennas. We assume that all nodes have the same transmsson range 50 and nterference range 500. C. A Case Study As a case study, we nvestgate a network nstance n Fg. 5(a) wth the above settng. There are four actve sessons n the network (N 10 to N 43, N 3 to N 47, N 30 to N 16, and N to N 7 ). For ease of llustraton, we assume that there are only 3 tme slots n a tme frame. By solvng OPT-IA, we obtan 1360

8 TABLE II A COMPARISON BETWEEN P (N ) AND Q(N ). P (N ) IS THE TOTAL NUMBER OF INTERFERING STREAMS AT NODE N AND Q(N ) IS THE TOTAL NUMBER OF DOFS THAT ARE CONSUMED FOR IC AT NODE N. Tme slot 1 Tme slot Tme slot 3 Rx P (Rx) Q(Rx) Rx P (Rx) Q(Rx) Rx P (Rx) Q(Rx) N 5 4 N 7 4 N 6 4 N 18 6 N 16 N 3 4 N 8 4 N 19 6 N 31 4 N 43 N 0 4 N 37 4 N 47 4 N 3 4 the optmal obectve (.e., the maxmum throughput) of Fg. 5(b) (d) show the transmsson/recepton pattern, nterference pattern, and IA scheme n each tme slot. Specfcally, a sold arrow lne represents a drected transmsson lnk (wth the number of data streams on ths lnk shown n a box). A dashed arrow lnk represents an nterference, wth the total number of nterferng streams and the number of subset nterferng streams chosen for IA shown n a box,.e., (α, β ). For example, n Fg. 5(b), on the dashed lne between N 6 and N 18 ), (, ) represents that α 6,18 = and β 6,18 =,.e., there are two nterferng streams from node N 6 to node N 18 and both of these nterferng streams are selected for IA at node N 18 n our soluton. As an example to llustrate how IA s performed n a network, let s take a look at N 18 n tme slot 1 (Fg. 5(b)). At node N 18, there s a total of 6 nterferng streams (from transmttng nodes N 19, N 6, and N 3 ). In our soluton, we fnd that for the nterferng streams from node N 19, both of them are algned to the nterferng streams from node N 3. Smlarly, the nterferng streams from node N 6 have also been algned to the nterferng streams from node N 3. That s, among the 6 nterferng streams at node N 18, 4 of them have been successfully algned to the remanng nterferng streams. As a result, node N 18 only needs to consume DoFs for IC. Table II summarzes the savngs of DoFs n IC due to IA at each recevng node n each tme slot. To abbrevate notaton n the table, denote P (N ) as the total number of nterferng streams at node N,.e., P (N ) = I α. Denote Q(N ) as the total number of DoFs that are consumed by node N for IC,.e., Q(N ) = I (α β ). Then the dfference between P (N ) and Q(N ) s the savng n DoFs at node N due to IA. Note that savngs n DoFs drectly translate nto mprovement n network throughput. Comparson to OPT-base. To compare the case when our IA framework s not appled, we formulate the same network throughput optmzaton problem (wth only MIMO s SM and IC) as OPT-base, whch s gven n the appendx. By solvng OPT-base wth CPLEX, we have that the obectve s only 0.33 (comparng to 0.67 under OPT-IA). D. Complete Results The prevous secton gves results for one 50-node network nstance. In ths secton, we perform the same drll for 50 network nstances, each wth 50 nodes randomly deployed n TABLE III A COMPARISON OF OBJECTIVE VALUES BETWEEN OPT-IA AND OPT-BASE. Index OPT-base OPT-IA Index OPT-base OPT-IA the square. Agan, there are four sessons n each network nstance, wth each sesson s source and destnaton nodes beng randomly selected among the nodes. Here, a tme frame has sx tme slots. Table III lsts the obectve values under OPT-IA and OPT-base. The average percentage ncrease n obectve value (over 50 nstances) s 43.4%. VIII. CONCLUSIONS The goal of ths paper s to make a concrete step forward n advancng IA technque n mult-hop MIMO networks. We developed an IA model consstng of a set of constrants for each transmtter and recever n a mult-hop MIMO network. Based on ths IA model, we developed an optmzaton framework for IA n a mult-hop MIMO network. We antcpate that ths framework (or varants of t) wll be wdely adopted by the networkng communty to study IA n a mult-hop network envronment. As an applcaton of ths optmzaton framework, we studed a network throughput optmzaton problem and compared performance obectves wth our IA model and that wthout IA. Smulaton results showed that the use of IA n a multhop MIMO network can sgnfcantly reduce DoF consumpton for IC at the recevers, thereby mprovng network throughput. ACKNOWLEDGMENT The work of Y.T. Hou and W. Lou was supported n part by NSF grants (Hou), (Hou), (Lou) and (Lou). 1361

9 REFERENCES [1] V.R. Cadambe and S.A. Jafar, Interference algnment and degrees of freedom of the K-user nterference channel, IEEE Trans. on Informaton Theory, vol. 54, no. 8, pp , Aug [] V.R. Cadambe and S.A. Jafar, Interference algnment and the degrees of freedom of wreless X networks, IEEE Trans. on Informaton Theory, vol. 55, no. 9, pp , Sep [3] O. El Ayach, S.W. Peters, and R.W. Heath, The feasblty of nterference algnment over measured MIMO-OFDM channels, IEEE Trans. on Vehcular Technology, vol. 59, no. 9, pp , Nov [4] S. Gollakotta, S. Perl, and D. Katab, Interference algnment and cancellaton, n Proc. of ACM SIGCOMM, vol. 39 no. 4, pp , Barcelona, Span, Oct [5] K. Gomadam, V.R. Cadambe, and S.A. Jafar, A dstrbuted numercal approach to nterference algnment and applcatons to wreless nterference networks, IEEE Trans. on Informaton Theory, vol. 57, no. 6, pp , June 011. [6] Y.T. Hou, Y. Sh, and H.D. Sheral, Spectrum sharng for mult-hop networkng wth cogntve rados, IEEE Journal on Selected Areas n Communcatons, vol. 6, no. 1, pp , Jan [7] D. Hu and S. Mao, Cooperatve relay wth nterference algnment for vdeo over cogntve rado networks, n Proc. IEEE INFOCOM, pp , Orlando, FL, March 01. [8] S.A. Jafar, The ergodc capacty of phase-fadng nterference networks, IEEE Trans. on Informaton Theory, vol. 57, no. 1, pp , Dec [9] S.A. Jafar, Blnd nterference algnment, IEEE Journal of Selected Topcs n Sgnal Processng, vol. 6, no. 3, pp. 16 7, June 01. [10] S.A. Jafar, Interference algnment: A new look at sgnal dmensons n a communcaton network, Foundatons and Trends n Communcatons and Informaton Theory, vol. 7, no. 1, pp , 011. [11] S.A. Jafar, M. Fakhereddn, Degrees of freedom for the MIMO nterference channel, IEEE Trans. on Informaton Theory, vol. 53, no. 7, pp , July 007. [1] S.A. Jafar and S. Shama, Degrees of freedom regon for the MIMO X channel, IEEE Trans. on Informaton Theory, vol. 54, no. 1, pp , Jan [13] L.E. L, R. Alm, D. Shen, H. Vswanathan, and Y.R. Yang, A general algorthm for nterference algnment and cancellaton n wreless networks, n Proc. IEEE INFOCOM, pp , San Dego, CA, March 010. [14] K. Ln, S. Gollakota, and D. Katab, Random access heterogeneous MIMO networks, n Proc. of ACM SIGCOMM, vol. 41, no. 4, pp , Toronto, Canada, Aug [15] A. Schrver, Theory of Lnear and Integer Programmng, Chapter 18, WleyInterscence, New York, NY, [16] S. Sharma, Y. Sh, Y.T. Hou, H.D. Sheral, and S. Kompella, Cooperatve communcatons n mult-hop wreless networks: Jont flow routng and relay node assgnment, n Proc. IEEE INFOCOM, pp , San Dego, CA, Mar., 010. [17] H.D. Sheral and W.P. Adams, A Reformulaton Lnearzaton Technque for Solvng Dscrete and Contnuous Nonconvex Problems, Chapter 8, Kluwer Academc Publshers, [18] Y. Sh, J. Lu, C. Jang, C. Gao, and Y.T. Hou, An optmal lnk layer model for mult-hop MIMO networks, n Proc. IEEE INFOCOM, pp , Shangha, Chna, Aprl, 011. [19] C. Suh, M. Ho, and D. Tse, Downlnk nterference algnment, IEEE Trans. on Commun., vol. 59, no. 9, pp , Sep [0] D. Tse and P. Vswanath. Fundamentals of Wreless Communcaton, Chapter 7, Cambrdge Unversty Press, May 005. [1] C.M. Yets, T. Gou, S.A. Jafar, and A.H. Kayran, On feasblty of nterference algnment n MIMO nterference networks, IEEE Trans. on Sgnal Processng, vol. 58, no. 9, pp , Sep [] IBM ILOG CPLEX Optmzer, software avalable at 01.bm.com/software/ntegraton/optmzaton/cplex-optmzer. APPENDIX A PROBLEM FORMULATION WITHOUT IA We formulate the same network throughput optmzaton (wth only MIMO s SM and IC). We have the same DoF consumpton constrant on the transmttng node as (11) n OPT- IA. However, wthout IA, the DoF consumpton constrant on the recevng node s dfferent from (1) n OPT-IA. If node s recever, then ts DoF consumpton conssts of two parts: for SM and for IC. The number of ts DoFs consumed for SM s µ = l I z n l. The number of ts DoFs consumed for IC s equal to the number of ts nterferng streams (.e., I α ). Thus, we have the followng DoF constrant at node. z l + α M. I l L n Otherwse (node s ether a transmtter or nactve), we know z l (t) = 0 for l L n and α (t) = 0 for I based on ther defntons. Combnng these two cases, we have the followng DoF consumpton constrant on the recevng node: z l (t)+ α (t) M y (t), (1 N, 1 t T ), I l L n (0) where α (t) s constraned by (7), whch s equvalent to the combnaton of (18) and (19). Now we formulate the problem as follows: OPT-base Max r mn s.t. Half duplex constrants: (4); Node actvty constrants: (5), (6); DoF consumpton constrants: (11), (18 0); Lnk capacty constrants: (13); Flow routng constrants: (14 15); Mn rate constrants: (17). where x (t) and y (t) are bnary varables; z l (t) and α (t) are non-negatve nteger varables; r(f) and r l (f) are nonnegatve varables; M, N, L, F, T, and B are constants. 136