An Analytical Model for Interference Alignment in Multi-hop MIMO Networks

Transcription

1 1 An Analytcal Model for Interference Algnment n Mult-hop MIMO Networks Huacheng Zeng, Student Member, IEEE, Y Sh, Senor Member, IEEE, Y. Thomas Hou, Fellow, IEEE, Wenng Lou, Fellow, IEEE, Sastry Kompella, Senor Member, IEEE, Scott F. Mdkff, Senor Member, IEEE Abstract Interference algnment (IA) s a powerful technque to handle nterference n wreless networks. Snce ts ncepton, IA has become a central research theme n the wreless communcatons communty. Due to ts ntrnsc nature of beng a physcal layer technque, IA has been manly studed for pont-to-pont or sngle-hop scenaro. There s a lack of research of IA from networkng perspectve n the context of mult-hop wreless networks. The goal of ths paper s to make such an advance by brngng IA technque to mult-hop MIMO networks. We develop an IA model consstng of a set of constrants at a transmtter and a recever that can be used to determne IA for a subset of nterferng streams. We further prove the feasblty of ths IA model by showng that a DoF vector can be supported free of nterference at the physcal layer as long as t satsfes the constrants n our IA model. Based on the proposed IA model, we develop an IA desgn space for a mult-hop MIMO network. To study how IA performs n a mult-hop MIMO network, we compare the performance of a network throughput optmzaton problem based on our developed IA desgn space aganst 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. Index Terms Interference algnment, modelng and optmzaton, mult-hop MIMO network. 1 INTRODUCTION Interference management s a fundamental problem n wreless networks. Interference algnment s a maor advance n nterference management n recent years that offers a new drecton to handle mutual nterference among dfferent users. The basc dea of IA s to construct sgnals at transmtters so that these sgnals overlap (algn n the same drecton) at ther unntended recevers whle they are resolvable at ther ntended recevers. It was shown n [3] that IA can acheve K/ degrees of freedom (DoF) n the K-user nterference channel based on the assumpton of arbtrary large tme or frequency dversty. It was also shown n [6], [16] that IA can sgnfcantly ncrease the user throughput n practcal MIMO WLAN. Gven ts huge potental n ncreasng network DoFs, IA has brought tremendous attenton n the wreless communcatons communty. Due to ts ntrnsc nature of beng a physcal layer technque, most of the IA results are lmted to pont-topont or sngle-hop scenaro. There s a lack of advance of IA technque from networkng perspectve, especally n the context of mult-hop wreless networks. Extendng IA from a sngle-hop to mult-hop network does not appear to be straghtforward, as the transmsson and nterference patterns n a mult-hop network are much An abrdged verson of ths paper appeared n the Proc. of IEEE INFO- COM 013, Turn, Italy. H. Zeng, Y. Sh, Y.T. Hou, W. Lou, and S.F. Mdkff are wth Vrgna Tech, Blacksburg, VA, 4061 USA. E-mal: {zeng, ysh, thou, wlou, mdkff}@vt.edu. (Correspondng author: Y.T. Hou) S. Kompella s wth the US Naval Research Laboratory, Washngton, DC 0375, USA. E-mal: sastry.kompella@nrl.navy.ml. Manuscrpt receved December 9, 013; revsed February 9, 015; accepted February 3, 015. more complex and can easly become ntractable. In [15], L et al. made the frst attempt to explore IA n a mult-hop MIMO network. There, the dea of IA was dscussed n several example scenaros to llustrate ts benefts. However, the key concept of IA (.e., constructng sgnals at transmtters so that these sgnals overlap at ther unntended recevers whle remanng resolvable at ther ntended recevers) was not ncorporated nto ther problem formulaton and soluton procedure. In [8] and [9], Zeng et al. studed IA n cellular and multhop networks from networkng perspectve. But ther IA results were lmted to sngle-antenna networks and cannot be appled to mult-hop MIMO networks. The lack of results of IA n mult-hop MIMO networks underscores both the techncal barrer n ths area and the crtcal need to close ths gap. The goal of ths paper s to make a concrete step toward advancng IA technque n mult-hop MIMO networks. We study IA n ts most basc form [3],.e., the constructon of transmt data streams so that () they overlap at ther unntended recevers and () they reman resolvable at ther ntended recevers. The constructon of transmt data streams requres the desgn of precodng vector for each data stream at ts transmtter. Snce the nterferng streams are overlappng at the recevers, one can use fewer DoFs to cancel these nterferng streams. As a result, the DoF resources consumed for IC can be reduced and thus more DoF resources can be avalable to transport data streams. Although our IA desgn may not be mmedately used n practcal mult-hop networks due to ts requrement of central controller and global channel state nformaton (CSI), t can serve as a proof of concept and offer theoretcal nsghts and gudance n the desgn of practcal IA schemes. The man contrbutons

2 z 1 data streams z data streams T 1 R 1 T R Fg. 1. SM and IC n MIMO. A sold lne wth arrow represents ntended transmsson whle a dashed lne wth arrow represents nterference. of ths paper are summarzed as follows. We develop an analytcal IA model for a multhop MIMO network. Our model conssts of a set of constrants at each transmtter to determne the subset of nterferng streams used for IA and a set of constrants at each recever to determne the algnment pattern of the nterferng streams. We prove the feasblty of the proposed analytcal IA model. Specfcally, we show that a DoF vector can be supported at the physcal layer as long as t satsfes the constrants n our IA model. Ths s done by constructng the precodng and decodng vectors for each data stream such that all data streams n ths DoF vector can be transported free of nterference. Based on the analytcal IA model, we develop a set of constrants across multple layers of a multhop MIMO network. Collectvely, these constrants characterze an IA desgn space for a mult-hop MIMO network. Based on ths space, IA can be ontly exploted wth upper-layer schedulng for a target network performance obectve. To evaluate the performance of our IA desgn space, we study a network throughput optmzaton problem and compare the results aganst those for the same problem when IA s not employed. We show that the use of IA can conserve DoF resources n the network and ncrease throughput sgnfcantly. The remander of ths paper s organzed as follows. Secton offers some essental background on IA n MIMO networks. Secton 3 dscusses the challenges of applyng IA n mult-hop networks. Secton 4 presents a new analytcal IA model for MIMO networks and Secton 5 proves the feasblty of ths model. In Secton 6, we apply the IA model to a mult-hop MIMO network and develop a cross-layer desgn space for IA. In Secton 7, we apply our IA desgn space to study a throughput maxmzaton problem and demonstrate the benefts of IA n a mult-hop MIMO network. Secton 8 presents related work and Secton 9 concludes ths paper. PRELIMINARIES: IA IN MIMO In ths secton, we revew MIMO n terms of ts DoF resources for spatal multplexng (SM) and nterference cancellaton (IC). We also revew how IA can help conserve DoF consumpton requred for IC. The notaton n ths paper s lsted n Table (n supplemental materal). MIMO s DoF Resources for SM and IC. The concept of DoF was orgnally defned to represent the maxmum multplexng gan of an MIMO channel by the nformaton theory communty (see e.g., [7]). It was then extended by the networkng research communty to characterze a node s spatal freedom provded by ts multple antennas (see e.g., [8], [1], [4]). Typcally, the total number of DoFs at a node s equal to the number of antennas at ths node, and represents the total avalable resources at ths node that can be used for SM and IC. SM refers to the use of one or multple DoFs (both at transmt and receve nodes) for data stream transmsson/recepton, wth each DoF correspondng to one ndependent data stream. IC refers to the use of one or more DoFs to cancel nterference, wth each DoF beng responsble for cancelng one nterferng stream. IC can be done ether at a transmt node (to cancel nterference to a receve node) or a receve node (to cancel nterference from a transmt node). For example, consder the 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 canceled by ether R 1 or 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 a constructon of data streams at transmtters so that () they overlap (algn) at ther unntended recevers and () they reman resolvable at ther ntended recevers [3]. The constructon of transmt data streams s equvalent to the desgn of precodng vector for each data stream at each transmtter. Snce the nterferng streams are overlapped at a recever, one can use fewer 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. To perform IA n an MIMO network, CSI s requred at both transmtter and recever sdes. We use the followng example to llustrate the benefts of IA n MIMO networks. Consder the 4-lnk network as shown n Fg.. 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 ). At transmtter T, denote u k as the precodng vector for ts outgong data stream k. Denote H as the channel matrx between recever R and transmtter T. We assume that H s of full rank. When IA s not employed, R 4 needs to consume 5 DoFs to cancel the nterference from transmtters T 1, T, and T 3 [8], [4]. Snce there are only 3 DoFs avalable at recever 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 streams nto dmensons, whch can be canceled by R 4 wth only DoFs. Hence, R 4 stll has 1 remanng DoF, allowng t to receve 1 data stream from transmtter T 4. We gve one possble approach to construct the 5 precodng vectors at T 1, T, and T 3, respectvely. To show that the 5 nterferng streams can ndeed be algned nto dmensons at recever R 4, we denote a := b f there

3 3 [u 1 1 u 1 ] H 4u 1 R 1 T 8 R 1 T 1 [u 1 u ] H 41 u 1 1 H 43 u 1 3 H 4 u H 4 u 1 T 1 R R 9 R 10 T 9 R T R 4 T 4 T 7 R 8 T 10 R 11 [u 1 3] R 3 T R 3 T 3 Fg.. An llustraton of IA at node R 4. A sold lne wth arrow represents ntended transmsson whle a dashed lne wth arrow represents nterference. T 3 R 5 R 4 R 7 T 6 T 4 R 6 T 5 R 1 T 11 exsts a nonzero complex number c for vectors a and b such that a = c b. We begn by constructng the precodng vectors at transmtter T 1 by lettng u 1 1 := e 1 and u 1 := e, where e k s a vector wth the k-th element beng 1 and all the other elements beng 0. For the two precodng vectors [u 1 u ] at transmtter T, we 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 4 H 41u 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 4 H 41u 1. Fnally, for the precodng vector u 1 3 at transmtter 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 lettng H 43 u 1 3 := H 41 u 1 1 and thus u H 41u 1 1. As a result of IA, the 5 nterferng streams are algned nto only dmensons and can be canceled wth DoFs (nstead of 5 DoFs) by recever R 4. Note that n ths example, we only llustrate how to acheve IA at one recever. In a mult-hop network, the goal s to accomplsh IA at as many recevers as possble so as to maxmally harvest the benefts of IA. Ths requres careful coordnaton at network level and s a much harder problem, as we elaborate n the next secton. 3 IA IN MULTI-HOP NETWORKS: WHERE ARE THE CHALLENGES As dscussed n Secton 1, although there s a floursh of research on IA n the pont-to-pont or sngle-hop scenaros, results on extendng IA to a mult-hop network reman very lmted. Ths s because there are a number of new challenges, whch we summarze as follows. () How to coordnate IA among a large number of nodes n an MIMO network s a very hard problem. In partcular, for each par of nodes, one needs to decde whch subset of nterferng streams for IA and how to algn them successfully at the recever. Whle performng IA, one must also ensure that the desred data streams at each ntended recever reman resolvable. The answers to these questons Fg. 3. An MIMO network. A sold lne wth arrow represents ntended transmsson whle a dashed lne wth arrow represents nterference. requre the development of a new IA model, as we shall present n ths paper. () Provng the feasblty of an IA model s not trval task. One must show that any DoF vector can be supported n the network as long as t satsfes the constrants n the underlyng IA model. Specfcally, one needs to show that for each data stream characterzed by the DoF vector, there exst a precodng vector at ts transmtter and a decodng vector at ts recever so that ths data stream can be transported free of nterference. As we wll see n Secton 5, constructng such a precodng vector and decodng vector for each data stream s very challengng. () 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 vary from each tme slot. Thus, an IA scheme must be ontly desgned wth upper layer schedulng and routng algorthms, whch s agan a challengng problem. In ths paper, we address challenges () and () n Sectons 4 and 5, respectvely. Challenge () s addressed n Secton 6. 4 MODELING IA IN MIMO NETWORKS Consder a mult-hop MIMO network n Fg. 3. Each node s equpped wth N A antennas. We assume that the network s statc and the CSI s avalable at both transmtter and recever sdes. We also assume that schedulng s done n the tme doman, wth each tme frame havng K tme slots. To develop an IA model, we focus on one tme slot t (1 t K). Denote N T as the number of transmtters and N R as the number of recevers n the tme slot. 1 Denote L as the set of lnks n 1. When there s no ambguty, we omt the tme slot ndex t n ths secton and the next secton.

4 4 the network wth L = L. Denote φ = (z 1, z,, z L ) as the DoF vector n the network, where z l s the number of data streams on lnk l L. At a transmtter, ts dfferent data streams may go to dfferent recevers (see, e.g., T 1 n Fg. 3). For transmtter T, denote λ as the number of outgong data streams and thus we have λ = l L z out l, where L out s the set of outgong lnks from transmtter T. Smlarly, at a recever, t may receve desred data streams from multple transmtters (see, e.g., R 1 n Fg. 3). 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 s the set of ts ncomng lnks to recever R. Consder a node par (T, R ). Denote s k as the transmsson of stream k (1 k λ ) from transmtter T to recever R. If ths stream k s ntended to recever R, then s k s a data stream for recever R. Otherwse, stream s k s an nterferng stream for recever R. Denote S as the set of data streams from transmtter T to recever R, wth σ = S. Denote A as the set of nterferng streams from transmtter T to recever R, wth α = A. Thus, we have σ + α = λ. Note that wthout IA, recever R needs to consume α DoFs to cancel the nterferng streams from transmtter T [8], [4]. For recever R, to reduce ts DoF consumpton for IC, we can algn a subset of ts nterferng streams to the other nterferng streams by properly constructng ther precodng vectors (as llustrated by Fg. ). Among the nterferng streams n A, denote B as the subset of nterferng streams that are algned to the other nterferng streams at recever R, wth β = B. Then the number of effectve nterferng streams from transmtter T to recever R s decreased from α to α β. The queston to ask s how to perform IA among the nodes n the network so that (C-1): each nterferng stream n B s can be successfully algned at the unntended recevers; (C-): each data stream remans resolvable at ts ntended recever. Sectons 4.1 and 4. address ths queston by explorng constrants at a transmtter and at a recever, respectvely. 4.1 IA Constrants at a Transmtter Consder a transmtter T as shown n Fg. 4. Based on the defntons of A and B, we know B A. Thus, we have the followng constrants at transmtter T : β α, I, 1 N T, (1) where I s the set of nodes wthn the nterference range of node. At transmtter T n Fg. 4, there are λ precodng vectors correspondng to λ outgong streams. Snce each outgong stream nterferes wth all the unntended recevers wthn transmtter T s nterference range, the correspondng precodng vector determnes the drecton. The actvty of lnk l s determned by the value of z l. When z l = 0, lnk l s nactve. [u 1 u u λ T ] A 1,B 1 A,B A,B A Nr,B Nr A set of nterferng streams R 1 R R R Nr Fg. 4. IA constrants at transmtter T. In ths fgure, data streams from T to the recevers are not shown. of one nterferng stream for each of those recevers. For nstance, precodng vector u 1 determnes the drectons of the outgong stream at recevers R 1, R,, R Nr, one of whch s the ntended recever and the rest are unntended recevers. However, among the N r 1 drectons for nterferng streams, only one of them can be successfully algned to a partcular drecton for IA by constructng u 1. Therefore, for the nterferng streams from transmtter T, at most λ nterferng streams can be successfully used for IA at ther recevers, snce there are λ precodng vectors at transmtter T. Mathematcally, we have the followng constrants at transmtter T : β λ, 1 N T. () I At transmtter T, the DoF consumpton s only for SM. Specfcally, the number of DoFs consumed at transmtter T s equal to the number of ts outgong data streams (.e., λ ). Snce the DoFs consumed at a node cannot exceed ts total DoFs, we have the followng constrants at transmtter T : λ N A, 1 N T. (3) 4. IA Constrants at a Recever Consder a recever R n Fg. 5. To ensure (C-1) and (C-) at recever R, we have the followng condtons: Based on our defnton of B, the nterferng streams n each B should not occupy effectve drectons at recever R. Therefore, at recever R, each nterferng stream n I B can only be algned to an nterferng stream n I (A \B ). To ensure the resolvablty of the data streams at each recever, we must have that any nterferng stream n B cannot be algned to an nterferng stream n A. To show the reason, suppose that s k n B s algned to s k n A at recever R. Then, we have u k H u k := u k. Ths means that u k and u k are lnearly dependent and consequently these two streams are not resolvable at ther ntended recevers.

5 5 T 1 T T T Nt A 1,B 1 A,B A,B A Nt,B Nt A set of nterferng streams Fg. 5. IA constrants at recever R. Data streams from the transmtters to recever R are not shown. To ensure the resolvablty of the data streams at each recever, we must ensure that any two nterferng streams n B cannot be algned to the same (a thrd) nterferng stream. To show the reason, suppose that both s k and sk n B are algned to s l at recever R. Then, we have u k := H 1 H ul and u k H ul. We therefore have uk := uk, ndcatng that u k and u k are lnearly dependent. Ths means that these two streams are not resolvable at ther ntended recevers. We shall show that the above three condtons are all satsfed f the followng constrants are satsfed at each recever R : β k k I (α k β k ), I, 1 N R. (4) At each recever R, ts DoFs are consumed for SM and IC. Specfcally, the number of DoFs consumed for SM s equal to the number of ts ncomng data streams (.e., µ ); the number of DoFs consumed for IC s equal to the number of effectve nterferng streams at ths recever (.e., I (α β )). Snce the DoFs consumed for SM and IC cannot exceed ts total DoFs, we have the followng constrants at recever R : µ + k I (α k β k ) N A, 1 N R. (5) Collectvely, constrants (1) (5) characterze an analytcal IA model for an MIMO network. A queston about ths model s ts feasblty: For a DoF vector φ = (z 1, z,, z L ) that meets these IA constrants, s t also feasble? We answer ths queston n the followng secton. R transmtter T and v l s ts decodng vector at recever R. Then, DoF vector φ = (z 1, z,, z L ) s feasble f there exst precodng vector u k and decodng vector v l for each stream k from transmtter T, 1 N T, 1 k λ, such that (v l ) T H u k = 1, (6a) (v) l T H u k = 0, (6b) for 1 k λ, I, (, k ) (, k). Note that (6a) and (6b) are blnear constrants and how to develop a general soluton to a set of blnear equatons remans open [13]. Smply put, we say a DoF vector s feasble f there exst precodng and decodng vectors for each stream so that the stream can be decoded at ts ntended recever free of nterference. The followng theorem s the man result of ths secton. Theorem 1: A DoF vector φ = (z 1, z,, z L ) s feasble f t satsfes constrants (1) (5) n the IA model. It s worth pontng out that for a gven DoF vector φ = (z 1, z,, z L ), the values of α, λ, and µ n the IA constrants are fxed (.e., λ = and α = Rx(l) z l, µ = l L z n l, z l ), whle the values of β depend on the specfc IA scheme that one desgns. In the rest of ths secton, we prove Theorem 1 by constructon. 5.1 Proof of Theorem 1: A Roadmap As for notaton, we use callgraphc uppercase letter to denote a set of data/nterferng streams and use boldface uppercase letter to denote the set of ts correspondng precodng vectors. For a set of data streams n S, denote S as the correspondng set of precodng vectors. For a set of nterferng streams n A, denote A as the correspondng set of precodng vectors. For a set of nterferng streams n B, denote B as the correspondng set of precodng vectors. Mathematcally, we have S = {u k : s k S }, A = {u k : s k A }, B = {u k : s k B }. Accordngly, we have S = σ, A = α, and B = β. Consder recever R shown n Fg. 5. Denote D S as the set of data stream drectons at recever R. Denote D I as the set of nterferng stream drectons at recever R. Then we have D S = I {H u k : u k S }, D I = I {H u k : u k A }. 5 FEASIBILITY OF THE IA MODEL To prove the feasblty of the proposed analytcal IA model, we must frst clarfy what we mean by feasblty. The followng defnton clarfes ths ssue. Defnton 1: Suppose that a stream k from transmtter T s ntended to recever R. u k s ts precodng vector at The followng lemma shows a suffcent condton for DoF vector φ to be feasble. Lemma 1: A DoF vector φ = (z 1, z,, z L ) s feasble f there exsts a precodng vector u k for each stream k from transmtter T (1 N T, 1 k λ ), such that dm(d S D I ) = µ + dm(d I ), 1 N R, (7)

6 6 where µ s the number of ncomng data streams at recever R and µ = l L z n l for 1 N R. Proof: We show DoF vector φ s feasble by argung that f (7) s satsfed, then we can fnd a decodng vector v l for each stream k from transmtter T such that (6a) and (6b) are satsfed. Specfcally, we show that the followng lnear system s consstent f (7) s satsfed. (v l ) T H u k = 1, (v l ) T H u k = 0, I, 1 k λ, (, k ) (, k). where v l s varable vector and H s and u s are gven. Based on the defnton of D S and DI, we know D S D I = {H u k : I, 1 k λ }. It s easy to see that D S DI s the set of coeffcentvectors of ths lnear system. Moreover, ths system has N A free varables and at most N A lnearly ndependent equatons. If we can show that vector H u k s not a lnear combnaton of other vectors n D S DI, then ths system s consstent. We prove ths pont by contradcton. Suppose that H u k s a lnear combnaton of other vectors n D S DI. Snce H u k DS, we have dm(d S D I ) < D S + dm(d I ) = µ + dm(d I ). But ths contradcts the gven condton n (7). Therefore, we conclude that the lnear system s consstent. Intutvely, Lemma 1 tells us that at each recever R, f a data stream les n an ndependent drecton (.e., not wthn the subspace spanned by other data/nterferng streams), then ths data stream s resolvable. Lemma 1 offers another route for checkng the feasblty of a gven DoF vector: nstead of checkng the exstence of both precodng and decodng vectors that satsfy (6a) and (6b) n Defnton 1, one only needs to check the exstence of the precodng vectors that satsfy (7) n Lemma 1. We gve a roadmap for our proof of Theorem 1. Step 1 (Desgnng An IA Scheme): Based on the constrants n the IA model, we propose an IA scheme for the network. The obectve of ths scheme s to ensure that at each recever R, the nterferng streams n I B can be successfully algned to the nterferng streams n I A \B. We acheve ths obectve by addressng two questons: () How to select β nterferng streams from A for B at each transmtter T? () How to algn the nterferng streams n B to other nterferng streams at each recever R? Detals are gven n Secton 5.. Step (Constructng Precodng Vectors): Based on the IA scheme proposed n Step 1, we present an approach to construct the precodng vectors at the transmtters. Specfcally, we dvde the precodng vectors nto two groups: B and U\B. For a precodng vector u k n U\B, we set u k := e k. For the precodng vectors n B, we construct them based on the IA scheme n Step 1. Detals are gven n Secton 5.3. Step 3 (Resolvng Intended Sgnals): We show that the constructed precodng vectors n Step satsfy (7) n Lemma 1, thereby concludng that DoF vector φ = (z 1, z,, z L ) s feasble. Detals are gven n Secton Step 1: Desgnng An IA Scheme Based on the constrants n the IA model, we propose an IA scheme at a transmtter and a recever. The goal of ths IA scheme s that at each recever R, the nterferng streams n I B can be successfully algned to the nterferng streams n I A \B. We present the IA scheme by addressng the followng two questons: () At each transmtter T, how to select a subset of β nterferng streams for B from the α nterferng streams n A? () At each recever R, how to algn the nterferng streams n B to others nterferng streams? Selectng nterferng streams for B. Consder transmtter T n Fg. 4. To ensure that the nterferng streams n I B can be successfully algned to partcular drectons at ther recevers, each of them must be correspondng to a unque precodng vector at transmtter T. Mathematcally, ths requrement can be nterpreted as B 1 B =, 1, I, 1, 1 N T. (8) Then, we have the followng lemma. Lemma : For any β that meets constrants (1) and (), we can select β nterferng streams for B (from the α nterferng streams n A ) so that (8) s satsfed. Proof: Provng Lemma s equvalent to solvng the precodng vector selecton problem (PVS-Problem) as follows: PVS-Problem: For transmtter T and ts neghborng recevers as shown n Fg. 4, select β precodng vectors from U = {u k : 1 k λ } for B, I, such that B A, I T, (9a) B 1 B =, 1, I, 1. (9b) where U = k I S k, A = k k I S k, S = σ, I σ = λ, and I β λ. We solve the PVS-Problem by two steps. Frst, we propose a greedy algorthm to select precodng vectors for B (for each I ). Second, we show that the resultng B satsfes constrants (9a) and (9b). A greedy algorthm. Wthout loss of generalty, we ndex the recevers wthn I from 1 to J, where J = I. We select precodng vectors for B (1 J) sequentally. Specfcally, we frst select β 1 precodng vectors for B 1, and then select β precodng vectors for B, and so forth. In each teraton, we select β precodng vectors for B as follows: For each k wthn < k < J, we move ( S k J k =k+1 β k )+ precodng vectors from S k to B, where ( ) + = max{, 0}. After that, f B does not have enough precodng vectors, we move the precodng vectors from k k I S k to B untl B has enough precodng vectors. A pseudo-code for ths algorthm s gven n Fg. 6.

7 7 Algorthm: Solvng PVS-Problem at transmtter T. 1. J = I ;. for := 1 to J { 3. S = S ; B = ; β = β ; σ = σ ; } 4. for := 1 to J { 5. for k := + 1 to J 1 { f β == 0 {break;} d = ( σ k J k =k+1 k ) + ; 8. move mn{ β, d} precodng vectors from S k to B ; 9. β := β mn{ β, d}; 10. σ k := σ k mn{ β, d}; } 11. for k := 1 to J { 1. f β == 0 {break;} 13. f k == {contnue;} 14. move mn{ β, σ k } precodng vectors from S k to B ; 15. β := β mn{ β, σ k }; 16. σ k := σ k mn{ β, σ k }; }} Fg. 6. A pseudo-code for solvng PVS-Problem at transmtter T. Algorthm analyss. Two observatons on the algorthm are n order. Frst, the resultng soluton meets (9a), because all precodng vectors n B are selected from k k I S k and A = k k I S k. Second, the resultng soluton meets (9b), because each precodng vector n U s selected for only one B. Therefore, f we can show that the algorthm can successfully select β precodng vectors for B n each teraton, then the PVS-Problem s solved. Consder the precodng vector selecton for B n teraton. In our algorthm (see Fg. 6), any precodng vectors n k k I Sk can be moved to B. Therefore, f we can show that β k k I σ k at the begnnng of each teraton, then the PVS-Problem s solved. We now argue that ths s true n dfferent cases. Case I. σ J β k=+1 k 0 at the begnnng of teraton. In ths case, we have k σ k β = σ k σ β k I k I (a) 1 = λ β k σ β (b) = λ σ (c) J k=1 k=+1 β k σ (d) = J k=+1 k=1 β k σ 0, where (a) follows from the fact that k I σ k = k I σ k 1 k=1 β k = λ 1 k=1 β k at the begnnng of teraton ; (b) and (d) follow from the fact that β k = β k for k J at the begnnng of teraton ; (c) follows from constrant (). Case II. σ J β k=+1 k > 0 at the begnnng of teraton. In ths case, f there exsts a such that < and σ = J β k= +1 k, then t s easy to see that k k I σ k β σ β 0. β k Otherwse (.e., there does not exst such a ), all precodng vectors n 1 k=1 B k are from S. Then, we have k k I σ k β (a) = k k I σ k β (b) = α β (c) = α β 0, where (a) follows from the fact that σ k = σ k for I and k ; (b) follows from the fact that k k I σ k = λ σ = α ; (c) follows from constrant (1). Combnng the two cases, we conclude that the PVS- Problem s solved and Lemma s proved. Lemma ensures that each nterferng stream n B corresponds to a unque precodng vector and, therefore, each nterferng stream n B can be algned to any partcular drecton by constructng ts correspondng precodng vector. Algnng the nterferng streams n B. Consder recever R n Fg. 5. For each I, we use the followng algorthm to algn the nterferng streams n B at recever R. Algorthm 1: At recever R, each nterferng stream n B s algned to a unque nterferng stream n k k I (A k \B k ), I. Based on (4), we know that there are more nterferng streams n k k I (A k \B k ) than those n B. Therefore, each nterferng stream n B can be successfully algned to a unque nterferng stream n k k I (A k \B k ). For an nterferng stream s k B, n order to algn t to an nterferng stream s k k k I (A k \B k ) at recever R, we should construct ts correspondng precodng vector by u k := H 1 H uk, whch we denote as uk u k. The followng lemma shows that there s a unque mappng for each precodng vector n B. Lemma 3: For each u k n B, there exsts one and only one u k, such that uk u k wth uk A \B and. Lemma 3 s proved by the followng two facts. Frst, each nterferng stream n B s assocated wth a unque precodng vector (Lemma ). Second, each nterferng stream n B s algned to an nterferng stream n k k I (A k \B k ) (accordng to Alg. 1). 5.3 Step : Constructng Precodng Vectors We now explan how to construct the precodng vector for each stream based on the IA scheme n Secton 5.. Denote U as the set of all precodng vectors n the network. Denote B as the set of the precodng vectors that correspond to the nterferng streams for algnment. Mathematcally, we have U = {u k : 1 k λ, 1 N T }, B = I,1 N T B. To construct the precodng vectors n U, we dvde U nto two groups: B and U\B. We frst construct the precodng vectors n U\B and then construct the precodng vectors n B. For each precodng vector n U\B, we construct t as

8 8 follows: u k := e k, for u k U\B, (10) where e k s a vector wth the k-th element beng 1 and all the others beng 0. For the precodng vectors n B, ther constructon s more complcated. We descrbe ther constructon as follows. Based on Lemma 3, we know that f u k 1 1 B, then there exsts a precodng vector u k such that u k (.e., u k1 1 need to construct u k 1 1 H 1 u k ). To construct u k1 u k 1, we frst. If u k U\B, we know that u k has already been constructed by (10). Otherwse (.e., u k B), we construct u k n the same way as u k 1 1,.e., there exsts a precodng vector u k 3 3 such that u k u k 3 3. Followng the same token, we can establsh a chan as follows: C : u k1 1 1 u k M u k M 1 M 1 M 1 u k M M, (11) where m m+1 for m = 1,,, M 1. Chan C termnates f any of the followng two cases occurs. Case I: u k M M has already been constructed. Case II: u k M M appears twce n chan C. It s easy to see that chan C wll termnate, ether by case I or case II. We now show how to construct the precodng vectors n chan C for the two cases, respectvely. Case I. In ths case, chan C termnates because u k M M has already been constructed. We can conclude: () All other precodng vectors n chan C have not been constructed. () All precodng vectors n ths chan are unque. Thus, we can construct the precodng vectors n chan C sequentally n the backward drecton as follows: u k M 1 M 1 u k M M M 1 M 1 H M 1 M u k M M. M M H M M 1 u k M 1 M 1. Followng the same token, we construct all the precodng vectors n chan C. Case II. In ths case, chan C termnates because u k M M appears twce. We can conclude: () All precodng vectors n chan C have not been constructed. () All precodng vectors n chan C are unque except u k M M. () There exsts ˆm such that ( ˆm, k ˆm ) = ( M, k M ) and 1 ˆm < M. To construct the precodng vectors n chan C, we dvde chan C nto two sub-chans C 1 and C : C 1 : u k1 1 1 u k ˆm k u ˆm 1 ˆm 1 ˆm 1 u k ˆm ˆm, C : u k ˆm ˆm ˆm u k ˆm+1 ˆm+1 ˆm+1 M u k M 1 M 1 M 1 u k M M, where ( ˆm, k ˆm ) = ( M, k M ). For these two sub-chans, we frst construct the precodng vectors n C and then construct the precodng vectors n C 1. Based on the relatonshps among the vectors n chan C, we have: u k ˆm ˆm ˆm ˆm H ˆm ˆm+1 u k ˆm+1 ˆm+1, u k ˆm+1 ˆm+1 u k M M u k M 1 M 1 ˆm+1 ˆm+1 H ˆm+1 ˆm+ u k ˆm+ ˆm+,. (1) M M H M M 1 u k M 1 M 1, M 1 M 1 H M 1 M u k M M. Gven that ( ˆm, k ˆm ) = ( M, k M ), we have u k M M := egvec u k ˆm ˆm = u k M M. (13) (1) and (13) form a lnear equaton system, where H s are gven matrces and u s are varables. It can be verfed that a soluton to u k M M n the system s ( M 1 ) m= ˆm (H 1 m m H m m+1 ), (14) where egvec( ) s an egenvector of the square matrx. Once we obtan u k M M, we can sequentally construct all of the other precodng vectors n sub-chan C by (1). After constructng the precodng vectors n sub-chan C, we construct the precodng vectors n sub-chan C 1. Snce u k ˆm ˆm has already been constructed, we can construct the other precodng vectors n sub-chan C 1 followng the same token n Case I. It s easy to see that, n the end, all precodng vectors n U wll be constructed. 5.4 Step 3: Resolvng Intended Sgnals We now show that the constructed precodng vectors n Step satsfy (7) n Lemma 1. Frst, we present the followng lemma. Lemma 4: The constructed precodng vectors at each transmtter are lnearly ndependent,.e., dm{u k : 1 k λ } = λ for 1 N T. Proof: Consder transmtter T and ts neghborng recevers n Fg. 4. Let U = {u k : 1 k λ } and B = I B. Then we dvde the precodng vectors n U nto two groups: U \B and B. Recall that n our precodng vector constructon, we construct u k by u k := e k f u k U \B and construct u k by u k H uk ( ) f u k B. Ths ndcates that the precodng vectors n U \B are ndependent of the channel matrces and the precodng vectors n B are dependent on the channel matrces. Gven that the channel matrces are ndependent Gaussan random matrces, we have dm(u ) = dm(u \B ) + dm(b ) = U \B + dm( I B ), (15) almost surely. Now we analyze the dmenson of I B. Consder two precodng vectors u k B 1 and u k B wth 1. In our precodng vector constructon, u k u k := H 1 H u k 1 1 and u k s set to u k for some 1, k 1,, and k. Hence, u k s set to H u k s dependent on H 1 and u k s dependent on H. Gven that H 1 and H are two ndependent Gaussan random matrces, we

9 9 have dm( I B ) = I dm(b ), (16) almost surely. We now analyze the dmenson of B. Based on (11), each precodng vector u k B s constructed n the form of ) u k = ( M 1 m=1 (H 1 m m H m m+1 ) u k M M, where ( 1, k 1 ) = (, k), M, and u k M M s constructed ether by (10) or (14). Let G k = M 1 m=1 (H 1 m m H m m+1 ). We call G k the effectve channel for u k. We dvde the precodng vectors n B nto subsets such that the precodng vectors n the same subset have the same effectve channel. Denote the subsets as B n, 1 n N. Snce H s are ndependent Gaussan random matrces, any the effectve channels are ndependent random matrces. Thus, we have N dm(b ) = dm(b n ). (17) n=1 For each u k B n, t s determned by ts correspondng precodng vector u k M M and u k M M s constructed ether by (10) or (14). Denote B n as the set of precodng vectors u k M M correspondng to the precodng vectors n B n. Then we have dm( B n ) = B n based on three facts: () the precodng vectors n B n are at the same transmtter; () the precodng vectors constructed by (10) are lnearly ndependent; () there are N A lnearly ndependent solutons (egenvectors) to (14). Thus, we have dm(b n ) = dm( B n ) = B n = B n, (18) where the frst equaton follows from the fact that the effectve channel has full rank. Based on (17) and (18), we have N dm(b ) = dm(b n ) = B n = B. (19) n=1 N n=1 Based on (15), (16), and (19), we have dm(u ) = U \B + dm( I B ) = U \B + I dm(b ) = U \B + I B = λ. Therefore, Lemma 4 s proved. Denote D I,eff as the set of effectve nterferng stream drectons at recever R. Denote D I,algn as the set of nterferng stream drectons for algnment at recever R. Mathematcally, we have D I,eff = I {H u k : u k A \B }, D I,algn = I {H u k : u k B }. Based on the precodng vector constructon procedure, we know that for each H u k D I,algn, there exsts a H u k DI,eff such that H u k := H u k. Thus we have span(d I,algn ) span(d I,eff ). (0) For the number of vectors n D S DI,eff, we have D S D I,eff = µ + I (α β ) N A, (1) where the nequalty follows from (5). The dmenson of sgnal and nterference space at recever R s: dm(d S D I ) (a) = dm(d S D I,eff ) = dm I {H u k : u k S A \B } (b) = I dm{h u k : u k S A \B } (c) = I dm(s A \B ) (d) = I S A \B (e) = I S + I A \B = µ + I (α β ), () where (a) follows from (0); (b) follows from two facts: () The number of elements n D S DI,eff s bounded by N A as shown n (1); () The matrces n {H : I } are Gaussan random matrces and are ndependent of each other; (c) follows from our assumpton that H s of full rank, whch s usually the case n practcal networks; (d) follows from Lemma 4; (e) follows from S A = and B A. Smlarly, the dmenson of nterference subspace at recever R s: dm(d I ) = dm(d I,eff ) = I (α β ). (3) Based on () and (3), we have dm(d S D I ) = µ + dm(d I ), (4) whch ndcates that the constructed precodng vectors satsfy (7) n Lemma 1. 6 AN IA DESIGN SPACE FOR MULTI-HOP NET- WORKS In ths secton, we apply ths new analytcal IA model to develop a set of cross-layer constrants that can characterze an IA desgn space for a mult-hop MIMO network. Denote N as the set of nodes n the network wth N = N, each of whch s equpped wth N A antennas. Denote F the set of sessons n the network wth F = F. Denote r(f) as the data rate of sesson f F. Denote

10 10 src(f) and dst(f) as the source and the destnaton nodes 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. We assume that a tme frame conssts of K tme slots. Half Duplex Constrants. We assume that a node cannot transmt and receve n the same tme slot. Denote x (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) as a 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 K). (5) 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, we have 1 l L z out l (t) N A. Otherwse (.e., node s ether a recever or nactve), we have l L z out l (t) = 0. Combnng the two cases, we have the followng constrants: x (t) z l (t) N A x (t), (1 N, 1 t K). (6) Smlarly, by consderng whether or not node s a recever, we have the followng constrants: y (t) z l (t) N A y (t), (1 N, 1 t K). l L n (7) General IA Constrants at a Node. In Secton 4 and 5, we developed IA constrants at a transmtter and a recever. Here, we can rewrte these constrants at a node based on the node status varables. Based on (1) n our IA model, f node s a transmtter and node s a recever n tme slot t, we have β (t) α (t) for each I. Otherwse (.e., x (t) = 0 or y (t) = 0), we have β (t) = 0 and α (t) = 0. Combnng these two cases, constrant (1) can be rewrtten as β (t) α (t), ( I, 1 N, 1 t K). (8) Based on () n our IA model, f node s a transmtter n tme slot t, we have I β (t) l L z out l (t) as λ = l L z out l (t). Otherwse (.e., x = 0), we have I β (t) = 0 and l L z out l (t) = 0. Combnng these two cases, constrant () can be rewrtten as β (t) z l (t), (1 N, 1 t K). (9) I Based on (3) n our IA model, f node s a transmtter n tme slot t, we have l L z out l (t) N A as λ = l L z out l (t). Otherwse (.e., x = 0), we have l L z out l (t) = 0. Combnng these two cases, constrant (3) can be rewrtten as z l (t) N A x (t), (1 N, 1 t K). (30) Based on (4) n our IA model, f node s a recever n tme slot t, we have β (t) k k I [α k (t) β k (t)] for each I. Otherwse (.e., y = 0), we have β (t) = 0 and α (t) = 0 for each I. Combnng these two cases, constrant (4) can be rewrtten as β (t) k k I [α k (t) β k (t)], ( I, 1 N, 1 t K). (31) Based on (5) n our IA model, f node s a recever n tme slot t, we have l L z n l (t) + I [α (t) β (t)] N A. Otherwse (.e., y = 0), we have z l (t) = 0 for l L n and α (t) = β (t) = 0 for each I. Combnng these two cases, constrant (5) can be rewrtten as z l (t) + [α (t) β (t)] N A y (t), l L n I (3) (1 N, 1 t K). Fnally, we characterze the relatonshp between α (t) and z l (t). If node s a transmtter and node s a recever n tme slot t, we have α (t) = Rx(l) z l (t), where Rx(l) s the recever of lnk l. Otherwse (.e., x (t) = 0 or y (t) = 0), we have α (t) = 0. In general, we have the followng constrants: Rx(l) α (t) = y (t) z l (t), ( I, 1 N, 1 t K). (33) Lnk Capacty Constrants. Denote r l (f) as the amount of data rate on lnk l that s attrbuted to sesson f F. For ease of calculaton, we assume that fxed modulaton and codng scheme (MCS) s used for data transmsson and one data stream n one tme slot corresponds to one unt data rate. Then the average rate of lnk l over K tme slots s 1 K K t=1 z l(t). Snce the aggregate data rates cannot exceed the average lnk rate, we have F r l (f) 1 K f=1 K z l (t), (1 l L). (34) 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 ). (35) 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 ). (36) At a destnaton node, we have r l (f) = r(f), ( = dst(f), 1 f F ). (37) l L n It can be easly verfed that f (35) and (36) are satsfed, then (37) s also satsfed. Therefore, t suffces to nclude only (35) and (36).

11 N45 N13 N4 N6 N44 N3 N1 N43 N39 N9 N3 N17 N N37 N0 N35 N36 N18 N46 N9 N10 N38 N16 N19 N N0 N N49 N14 N8 Fg. 7. A 50-node network topology. Collectvely, constrants (5) (36) defne an IA desgn space for cross-layer throughput maxmzaton problems n a mult-hop MIMO network. In partcular, (9), (3), and (33) are couplng constrants that nvolve both the physcal layer and the lnk layer; (34) s couplng constrants that nvolve the physcal layer, the lnk layer, and the network layer. N47 N6 N7 7 PERFORMANCE EVALUATION In ths secton, we apply our IA desgn space developed n Secton 6 to study a throughput maxmzaton problem n a mult-hop MIMO network. Our goals are twofold. Frst, we want to see how IA s performed n a network settng through a case study. Second, we want to have a quanttatve comparson between our IA desgn space and the case when IA s not used, thereby affrmng the sgnfcant advantage of explotng IA n a network envronment. N33 N31 N8 N5 N7 N15 N1 N5 N30 N34 N1 N11 N4 N4 N48 N40 N41 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; N A, N, L, F, K, and B are constants. Among the constrants, (33) s nonlnear. We can apply the Reformulaton Lnearzaton Technque (RLT) [0] to lnearze (33). By analyzng the relatonshp between α (t) and Rx(l) z l (t) n (33), we construct two new sets of constrants (39) and (40) to replace (33): 0 Rx(l) z l (t) α (t) (1 y (t)) B, ( I, 1 N, 1 t K), (39) 0 α (t) y (t) B, ( I, 1 N, 1 t K), (40) where B s a fxed nteger (e.g., B = N A ). It can be verfed that the combnaton of (39) and (40) s equvalent to (33) n terms of maxmzng r mn. By replacng nonlnear constrant (33) wth (39) and (40), we have the followng problem formulaton: OPT-IA: Max r mn S.t. (5) (3), (34) (36), (38) (40); 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; N A, N, L, F, K, 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 [5], [18], there exst hghly effcent approxmaton algorthms (e.g., branch-and-bound wth cuttng planes [19]) and heurstc algorthms (e.g., sequental fxng algorthm [9], [10]). For small to moderate network sze, an off-the-shelf solver such as CPLEX [30] may also be effectve. Snce the man goal of ths paper s to present a new analytcal IA model mult-hop MIMO networks (rather than developng a soluton procedure), we wll employ CPLEX solver for numercal results. 7.1 A Throughput Maxmzaton Problem We defne the obectve functon to be maxmzaton of the mnmum rate among all sessons. 3 Denote the obectve varable as r mn. Then we have r mn r(f), 1 f F. (38) We have the followng formulaton: OPT-IA raw : Max r mn S.t. Half duplex constrants: (5); Node actvty constrants: (6), (7); General IA constrants: (8) (33); Lnk capacty constrants: (34); Flow routng constrants: (35), (36); Mn rate constrants: (38); 3. Other obectve functons such as maxmzng sum of weghted rates or a proportonal scalng of all sesson rates belongs to the same category of lnear functon and can be solved followng the same token. 7. A Case Study 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 (see Fg. 7), whch are dstrbuted n a square regon. Each node n the network s equpped wth 4 antennas. We assume that all nodes have the same transmsson range 50 and nterference range 500. In ths network, there are 4 actve sessons: N 10 to N 43, N 3 to N 47, N 30 to N 16, and N to N 7 n Fg. 7. For ease of llustraton, we assume that there are 4 tme slots n a tme frame. By solvng OPT-IA, we obtan the optmal obectve (.e., the maxmum throughput) of Fg. 8 shows the transmsson/recepton pattern, nterference pattern, and IA scheme n each tme slot. Specfcally, a sold lne wth arrow represents a drected lnk (wth the number of data streams on ths lnk shown

12 N13 N39 N N36 N9 N47 N33 N5 N34 N N13 N39 N N36 N9 N47 N33 N5 N34 N N1 N19 N9 1 N45 (1,0) N18 (1,1) N3 (,1) (1,0) (1,1) 1 N10 (,1) (,) N4 N37 N3 (,0) N N6 N44 N46 N43 N0 N38 N17 N16 N35 (1,0) N7 N14 N1 (,0) (1,0) N11 N5 N6 N30 N40 N0 N3 N31 N48 N49 N15 N4 N8 N8 N1 N41 N N1 N19 N45 N9 N18 1 N14 (1,0) N3 (1,0) (1,0) N10 (1,1) N5 N37 1 (1,0) N6 (1,1) N4 N3 (1,0) (,1) (,1) 1 (1,0) N46 N N31 (,0) N38 N49 N6 N43 N0 N8 N8 N17 N44 N16 N35 N7 N7 N1 N11 N30 N40 N0 N3 N48 N15 N4 N1 N (a) Tme slot 1. (b) Tme slot N36 N4 N13 N39 N9 N33 N5 N47 N N34 N1 N19 N45 N9 N18 N7 N14 N1 N3 (,0) N10 N5 N11 (,) N6 N37 N30 N4 N40 N3 N (,0) N31 (,0) N3 N46 N0 N48 N49 (,0) N43 N0 N38 N6 (,) N15 N4 N17 N8 N8 N44 N16 N1 N41 N35 N N39 N13 N1 N45 N9 N3 N4 N3 1 N6 N43 N17 N44 N36 N9 N N19 N18 N10 N37 N46 N (1,1) N0 N38 N16 N35 N4 N33 N5 N47 N34 (,0) N7 N1 N14 (,0) N5 N11 N6 N30 (,) (,0) N40 (,) N0 N3 N48 N31 N15 N49 N4 N8 N8 N1 N41 N (c) Tme slot (d) Tme slot 4. Fg. 8. Transmsson/recepton pattern, nterference pattern, and IA scheme n each tme slot. A sold arrow lne represents a drected lnk (wth the number of data streams on ths lnk (.e., z l ) shown n a box). A dashed arrow lnk represents 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 ). n a box,.e., z l ). A dashed lne wth arrow represents 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. 8(a), on the dashed lne between N 5 and N 37, (, ) n the box represents that α 5,37 = and β 5,37 =,.e., there are two nterferng streams from node N 5 to node N 37 and both of them are selected for IA at node N 37 n our soluton. As an example to llustrate how IA s performed n the network, let s take a look at N 37 n tme slot 1 (see Fg. 8(a)). At node N 37, there s a total of 7 nterferng streams (from transmttng nodes N 5, N 18, N 3, and N 8 ). In our soluton, we fnd that for the nterferng streams from node N 5, both of them are algned to the nterferng streams from node N 8. Smlarly, for the nterferng stream from node N 18, t has been algned to an nterferng stream from node N 8. For the nterferng streams from node N 3, one of them has been algned to the nterferng streams from node N 8. That s, for the 7 nterferng streams at node N 37, 4 of them have been successfully algned to the remanng 3 nterferng streams. As a result, node N 37 only needs to consume 3 DoFs to cancel the 7 nterferng streams. Table 1 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 are drectly translated nto mprovement of network throughput. To compare to the case when IA s not appled n the network, we formulate a throughput maxmzaton

13 13 TABLE 1 A comparson between P (N ) and Q(N ). P (N ) s the number of nterferng streams at node N and Q(N ) s the total number of DoFs consumed for IC at node N. probablty Tme slot 1 Tme slot Rx P (Rx) Q(Rx) Rx P (Rx) Q(Rx) N 7 N N N N N N 9 N N Tme slot 3 Tme slot 4 Rx P (Rx) Q(Rx) Rx P (Rx) Q(Rx) N 18 N 5 4 N 0 N 31 5 N 8 4 N N 3 4 N Throughput gan of IA (percentage) Fg. 9. CDF of the comparson between OPT-IA and OPTbase. problem and denote t as OPT-base (see supplemental materal for detals). By solvng OPT-base wth CPLEX, we fnd that the obectve s only 0.5 (comparng to 0.50 under OPT-IA). 7.3 Complete Results The case study dscussed n the last secton gves results from one 50-node network nstance. In ths secton, we perform the same drll for 100 network nstances. Here, a tme frame has 6 tme slots. For each network nstance, we compute the throughput gan of IA by (ˆr mn r mn )/ r mn, where ˆr mn and r mn are the optmal obectve values of OPT-IA and OPT-base, respectvely. Fg. 9 presents the CDF of the throughput gan of IA. We see that the CDF curve s not smooth but follows starcase shape. Ths s because the optmal obectve values of OPT-IA and OPT-base are dscrete. On average over the 100 random network nstances, the throughput gan of IA s about 48%. It s worth pontng out that t s unfar to compare our results wth the K/ DoF result n [3]. Ths s because the K/ result n [3] was acheved based on the assumpton of nfntely large tme or frequency dversty whle the IA desgn n our paper s based on practcal (fnte) spatal dversty from multple antennas (wth no symbol extensons). It was shown by Bresler et al. n [] that the total DoFs avalable n the K-user nterference channel, usng only spatal dversty from multple antennas, s at most, whch s n sharp contrast to the K/ result n [3]. 8 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 [], [3], the cellular network [], [3], the MIMO Y channel [14], ergodc capacty n fadng channel [17], the X network wth arbtrary number of users, and the complex nterference channel. A dstrbuted IA scheme was proposed by Gomadam et al. n [7]. The feasblty of IA n sgnal vector space for K-user MIMO nterference channel was studed by Yets et al. n [6], and blnd IA (no CSI at transmtter) was studed n [5]. A tutoral on IA from nformaton theory perspectve s [11]. In wreless communcatons and networkng communtes, efforts on IA have been manly nvested n valdatons on small toy networks [1], [4], [6], [16]. In [1], Al-Al et al. studed IA n vehcular cogntve rado networks wth the obectves of reducng the overhead of drect database queres and mprovng the accuracy of spectrum sensng for moble vehcles. In [4], 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 [6], Gollakotta et al. demonstrated that the combnaton of IA and IC ncreases the average throughput by 1.5 tmes on the downlnk and tmes on the uplnk n a MIMO WLAN. In [16], 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. None of these pror efforts have made advances to extend IA technque n a network settng as we have done n ths paper. 9 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 analytcal IA model consstng of a set of constrants at a transmtter and a recever. We also proved the feasblty of the IA model by showng that each DoF vector satsfyng the constrants n the IA model s feasble at the physcal layer. We antcpate that ths IA model or ts varants wll be adopted by the networkng communty to study IA n a mult-hop network envronment. Based on ths IA model, we characterzed an IA desgn space for cross-layer throughput maxmzaton problems

14 14 n a mult-hop MIMO network. As an applcaton of ths IA desgn space, we studed a network throughput optmzaton problem and compared performance obectve wth our IA model aganst that wthout IA. Smulaton results showed that the use of IA n a mult-hop MIMO network can sgnfcantly reduce DoF consumpton for IC at the recevers, thereby mprovng network throughput. REFERENCES [1] A. Al-Al, Y. Sun, M. DFelce, J. Paavola and K.R. Chowdhury, Accessng spectrum databases usng nterference algnment n vehcular cogntve rado networks, IEEE Trans. Veh. Technol., vol. 64, no. 1, pp. 63 7, Jan [] G. Bresler, D. Cartwrght, and D. Tse, Feasblty of Interference Algnment for the MIMO Interference Channel, IEEE Trans. Inf. Theory, vol. 60, no. 9, pp , Sep [3] V.R. Cadambe and S.A. Jafar, Interference algnment and degrees of freedom of the K-user nterference channel, IEEE Trans. Inf. Theory, vol. 54, no. 8, pp , Aug [4] O.El Ayach, S.W. Peters, and R.W. Heath, The feasblty of nterference algnment over measured MIMO-OFDM channels, IEEE Trans. Veh. Technol., vol. 59, no. 9, pp , Nov [5] M.R. Garey and D.S. Johnson, Computers and Intractablty: A Gude to the Theory of NP-Completeness, W. H. Freeman and Company, New York, [6] S. Gollakotta, S. Perl, and D. Katab, Interference algnment and cancellaton, n Proc. ACM SIGCOMM, pp , Barcelona, Span, Oct [7] K. Gomadam, V.R. Cadambe, and S.A. Jafar, A dstrbuted numercal approach to nterference algnment and applcatons to wreless nterference networks, IEEE Trans. Inf. Theory, vol. 57, no. 6, pp , June 011. [8] B. Hamdaou and K.G. Shn, Characterzaton and analyss of mult-hop wreless MIMO network throughput, n Proc. ACM MobHoc, pp , Montreal, Quebec, Canada, Sep [9] Y.T. Hou, Y. Sh, and H.D. Sheral, Spectrum sharng for multhop networkng wth cogntve rados, IEEE J. Sel. Areas Commun., vol. 6, no. 1, pp , Jan [10] Y.T. Hou, Y. Sh, and H.D. Sheral, Optmal base staton selecton for anycast routng n wreless sensor networks, IEEE Trans. Veh. Technol., vol. 55, no 3, pp , May 006. [11] 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. [1] S.A. Jafar and S. Shama, Degrees of freedom regon for the MIMO X channel, IEEE Trans. Inf. Theory, vol. 54, no. 1, pp , Jan [13] C.R. Johnson and J.A. Lnk, Soluton theory for complete blnear systems of equatons, Numercal Lnear Algebra wth Applcatons, vol. 16, ssue 11 1, pp , Nov. Dec [14] N. Lee, J.B. Lm, and J. Chun, Degrees of freedom of the MIMO Y channel: Sgnal space algnment for network codng, IEEE Trans. Inf. Theory, vol. 56, no. 7, pp , July 010. [15] 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. [16] K. Ln, S. Gollakota, and D. Katab, Random access heterogeneous MIMO networks, n Proc. ACM SIGCOMM, pp , Toronto, Canada, Aug [17] B. Nazer, M. Gastpar, S.A. Jafar, and S. Vshwanath, Ergodc nterference algnment, n Proc. IEEE ISIT, pp , Seoul, South Korea, July, 009. [18] A. Schrver, Theory of Lnear and Integer Programmng, WleyInterscence, New York, NY, [19] 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. [0] H.D. Sheral and W.P. Adams, A Reformulaton Lnearzaton Technque for Solvng Dscrete and Contnuous Nonconvex Problems, Kluwer Academc Publshers, [1] 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. [] C. Suh and D. Tse, Interference algnment for cellular networks, n Proc. Allerton Conference on Communcaton, Control, and Computng, pp , Illnos, USA, Sep [3] C. Suh, M. Ho, and D. Tse, Downlnk nterference algnment, IEEE Trans. Commun., vol. 59, no. 9, pp , Sep [4] K. Sundaresan, R. Svakumar, M. Ingram, and T.-Y. Chang, Medum access control n ad hoc networks wth MIMO lnks: Optmzaton consderatons and algorthms, IEEE Trans. Moble Comput., vol. 3, no. 4, pp , Oct [5] C. Wang, T. Gou, and S.A. Jafar, Amng perfectly n the dark Blnd nterference algnment through staggered antenna swtchng, IEEE Trans. Inf. Theory, vol. 59, no. 6, pp , Jun [6] C.M. Yets, T. Gou, S.A. Jafar, and A.H. Kayran, On feasblty of nterference algnment n MIMO nterference networks, IEEE Trans. Sgnal Process., vol. 58, no. 9, pp , Sep [7] L. Zheng and D. Tse, Dversty and multplexng: A fundamental tradeoff n multple-antenna channels, IEEE Trans. Inf. Theory, vol. 49, no. 5, pp , May 003. [8] H. Zeng, Y. Sh, Y.T. Hou, W. Lou, X. Yuan, R. Zhu, and J. Cao, Increasng user throughput n cellular networks wth nterference algnment, n Proc. IEEE SECON, Sngapore, June, 014. [9] H. Zeng, Y.T. Hou, Y. Sh, W. Lou, S. Kompella, S.F. Mdkff, SHARK-IA: An nterference algnment algorthm for mult-hop underwater acoustc networks wth large propagaton delays, n Proc. ACM WUWNet, Rome, Italy, Nov [30] IBM ILOG CPLEX Optmzer, software avalable at 01.bm.com/software/ntegraton/optmzaton/cplex-optmzer. Huacheng Zeng (S 09) receved hs B.E. and M.S. degrees n Electrcal Engneerng from Beng Unversty of Posts and Telecommuncatons (BUPT), Beng, Chna, n 007 and 010, respectvely. He receved hs Ph.D. degree n Computer Engneerng from Vrgna Tech, Blacksburg, VA, n 015. He s currently a Senor System Engneer at Marvell Semconductor, Santa Clara, CA. He was a recpent of ACM WUWNET 014 Best Student Paper Award. Y Sh (S 0 M 08 SM 13) receved hs Ph.D. degree n Computer Engneerng from Vrgna Tech, Blacksburg, VA n 007. He s currently a Senor Research Scentst at Intellgent Automaton Inc., Rockvlle, MD, and an Adunct Assstant Professor at Vrgna Tech. Hs research focuses on optmzaton and algorthm desgn for wreless networks. He was a recpent of IEEE INFOCOM 008 Best Paper Award and the only Best Paper Award Runner-Up of IEEE INFOCOM 011. Y. Thomas Hou (S 91 M 98 SM 04 F 14) s Bradley Dstngushed Professor of Electrcal and Computer Engneerng at Vrgna Tech, Blacksburg, VA. He receved hs Ph.D. degree n Electrcal Engneerng from New York Unversty (NYU) Polytechnc School of Engneerng n Prof. Hou s research focuses on developng nnovatve solutons to complex problems that arse n wreless networks. He s an IEEE Fellow and an ACM Dstngushed Scentst. He s the Char of IEEE INFOCOM Steerng Commttee and a Dstngushed Lecturer of the IEEE Communcatons Socety.

15 15 Wenng Lou (S 01 M 03 SM 08 F 15) s a Professor n the Department of Computer Scence at Vrgna Tech, USA. She receved her Ph.D. degree n Electrcal and Computer Engneerng from the Unversty of Florda n 003. Prof. Lou s research nterests nclude cyber securty and wreless networks. She s on the edtoral boards of a number of IEEE transactons. She s an IEEE Fellow and the Steerng Commttee Char of IEEE Conference on Communcatons and Network Securty (CNS). Prof. Lou s currently on IPA assgnment as a program drector at the US Natonal Scence Foundaton. Sastry Kompella (S 04 M 07 SM 1) receved hs Ph.D. degree n Computer Engneerng from Vrgna Tech, Blacksburg, Vrgna, n 006. Currently, he s the head of Wreless Network Theory secton, Informaton Technology Dvson at the U.S. Naval Research Laboratory (NRL), Washngton, DC. Hs research focuses on complex problems n cross-layer optmzaton and schedulng n wreless and cogntve rado networks. Scott F. Mdkff (S 8 M 85 SM 9) s a professor n the Bradley Department of Electrcal and Computer Engneerng and currently serves as Vce Presdent for Informaton Technology and Chef Informaton Offcer at Vrgna Tech, Blacksburg, VA, USA. He receved hs Ph.D. degree n Electrcal Engneerng from Duke U- nversty, Durham, NC. Durng , he served as a program drector at the US Natonal Scence Foundaton. Hs research nterests nclude wreless networks, network servces for pervasve computng, and cyber-physcal systems.