4492 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 65, NO. 10, OCTOBER 2017

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1 4492 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 65, NO. 10, OCTOBER 2017 OFDM-Based Interference Algnment n Sngle-Antenna Cellular Wreless Networks Huacheng Zeng, Member, IEEE, YSh,Senor Member, IEEE, Y. Thomas Hou, Fellow, IEEE, Wenng Lou, Fellow, IEEE, Xu Yuan, Member, IEEE, Rongbo Zhu, Member, IEEE, and Jannong Cao, Fellow, IEEE Abstract Interference algnment (IA) s wdely regarded as a promsng nterference management technque n wreless networks. Despte ts rapd advances n cellular networks, most results of IA are lmted to nformaton-theoretc exploraton or physcal-layer sgnal desgn. Lttle progress has been made so far to advance IA n cellular networks from a networkng perspectve. In ths paper, we am to fll ths gap by studyng IA n large-scale cellular networks. For the uplnk, we propose an OFDM-based IA scheme and prove ts feasblty at the physcal layer by showng that all data streams n the IA scheme can be transported free of nterference. Based on the IA scheme, we develop a cross-layer IA optmzaton framework that can fully translate the benefts of IA to throughput gan n cellular networks. Furthermore, we show that the IA optmzaton problem n the downlnk can be solved n the exactly same way as that n the uplnk. Smulaton results show that our OFDM-based IA scheme can sgnfcantly ncrease the user throughput and the throughput gan ncreases wth user densty n the network. Index Terms Interference algnment, cellular networks, crosslayer optmzaton, throughput maxmzaton. I. INTRODUCTION INTERFERENCE algnment (IA) s a promsng nterference management technque n wreless networks as t may yeld much hgher throughput than we thought before. The basc dea of IA s to ontly construct sgnals at the transmtters wth the am of squeezng nterferng sgnals nto a reduced-dmensonal subspace at each recever, thereby leavng larger subspace for the recepton Manuscrpt receved January 30, 2017; revsed Aprl 23, 2017 and June 6, 2017; accepted June 6, Date of publcaton June 12, 2017; date of current verson October 16, Ths work was supported n part by NSF under Grants , , , , , and ONR Grant N H. Zeng s work was partally supported by EVPRI nternal research grant from the Unversty of Lousvlle. R. Zhu s work was supported by Natonal Scence Foundaton of Chna (Grant No ), Fundamental Research Funds for the Central Unverstes (Grant No. CZP17043), and Youth Elte Proect of State Ethnc Affars Commsson of Chna. J. Cao s work was supported by a grant from the Innovaton and Technology Commsson of the HKSAR Government to the Hong Kong Branch of Natonal Ral Transt Electrfcaton and Automaton Engneerng Technology Research Center (Proect Code: K-BBY1). The assocate edtor coordnatng the revew of ths paper and approvng t for publcaton was D. Nyato. (Correspondng author: Huacheng Zeng.) H. Zeng s wth the Unversty of Lousvlle, Lousvlle, KY USA (e-mal: huacheng.zeng@lousvlle.edu). Y. Sh, Y. T. Hou, and W. Lou are wth Vrgna Polytechnc Insttute and State Unversty, Blacksburg, VA USA. X. Yuan s wth the Unversty of Lousana at Lafayette, LA, USA. R. Zhu s wth South-Central Unversty for Natonaltes, Wuhan , Chna. J. Cao s wth the Hong Kong Polytechnc Unversty, Hong Kong. Color versons of one or more of the fgures n ths paper are avalable onlne at Dgtal Obect Identfer /TCOMM of desred sgnals. It was shown by Cadambe and Jafar n [1] that IA makes t possble for the K -user nterference channel to acheve K /2 degrees of freedom (DoFs), ndcatng that the aggregate DoFs of the nterference channel ncrease lnearly wth the number of users. Gven ts huge potental, IA has ganed tremendous momentum n the research communty and been appled to a varety of networks (see, e.g., [2] [5]). Along wth ts success n theory, IA n cellular networks (and WLAN) has attracted sgnfcant attenton due to ts ndustral potentals. Research efforts have produced a floursh lne of results that deepen our understandng of IA n cellular networks. For example, Suh et al. [6], [7] showed that the use of IA can completely elmnate nter-cell nterference f the number of users n each cell s suffcently large. Morales- Cespedes et al. n [8] developed an IA scheme usng reconfgurable antennas to remove nterference for partally connected cellular networks. Results of IA n cellular networks also nclude spatal IA desgn (see, e.g., [9] [11]), blnd IA desgn (see, e.g., [12], [13]), and channel state nformaton (CSI) analyss (see, e.g., [14], [15]). Although there s a large body of work on IA n cellular networks, most of them are lmted to nformaton-theoretc exploraton or physcal-layer sgnal desgn. It remans open how to develop an IA scheme that can be ncorporated wth upper-layer user schedulng algorthm to maxmze networklevel throughput. Ths stagnaton underscores the techncal challenges n the exploraton of cross-layer IA desgn from a networkng perspectve, whch we descrbe as follows. Frst, developng an IA scheme for a generc cellular network wth an arbtrary number of base statons (BS) and users s not a trval problem, as t requres complex sgnal desgn at transmtters and onerous sgnal detecton at recevers. Second, n a large-scale network envronment, IA desgn s coupled wth upper-layer user schedulng. An solated desgn of IA at the physcal layer s prone to yeld an nferor performance, and thus cannot fully harvest the benefts of IA for throughput maxmzaton. Therefore, a cross-layer IA scheme s needed. However, developng a cross-layer IA scheme together wth user schedulng can easly become ntractable and s a challengng task. In ths paper, we study OFDM-based IA n large-scale cellular networks from a networkng perspectve. We consder a network that conssts of a set of grd-deployed BSs and a set of randomly dstrbuted users. Each BS has a fxed servce area and provdes servce to the users wthn ts servce area. A user may fall nto the servce areas of multple BSs and IEEE. Personal use s permtted, but republcaton/redstrbuton requres IEEE permsson. See for more nformaton.

2 ZENG et al.: OFDM-BASED IA IN SINGLE-ANTENNA CELLULAR WIRELESS NETWORKS 4493 wll choose one of them as ts servce provder. We assume the transmsson s based on OFDM modulaton and the set of avalable subcarrers n OFDM s gven. For such a network, smlar to the IA scheme n [6] and [7], we study IA n the frequency doman by proectng the weghted transmt sgnals onto OFDM subcarrers. Dong so allows us to develop an IA scheme that can be easly appled to a network wth heterogeneous antenna confguratons. Gven that uplnk and downlnk are ndependent n both TDD and FDD networks, we consder them separately. Our obectve s to develop an OFDM-based IA scheme that can be ontly optmzed wth user schedulng to maxmze the uplnk/downlnk user throughput n cellular networks. The contrbutons of ths paper are summarzed as follows: For the uplnk, we develop an OFDM-based IA scheme for the data transmssons from users to ther servng BSs. Specfcally, at each user, we propose an approach to determne whch subset of ts nterferng streams should be selected for algnment; at each BS, we propose a procedure to algn nterferng streams so that the desred data streams can be decoded free of nterference. For the proposed IA scheme, we develop a set of IA constrants for each user and BS, and show that f the IA constrants are satsfed, the IA scheme s always feasble at the physcal layer. Based on the OFDM-based IA scheme, we develop a cross-layer IA optmzaton framework to maxmze the user throughput for the uplnk of cellular networks. To reduce the complexty of the optmzaton framework, we elmnate ts nonlnear constrants through reformulaton wthout compromsng ts optmalty. The resultng optmzaton framework s n a form that can be easly handled by commercal off-the-shelf (COTS) optmzaton solvers. We show the uplnk-downlnk dualty of the IA scheme. Specfcally, we show that the uplnk IA scheme can be appled to the downlnk by smply swtchng the roles of user and BS. Further, the downlnk IA optmzaton problem has the same formulaton as the uplnk and therefore can be solved n the exactly same way. We evaluate the throughput performance of our IA scheme va smulaton. We compare t aganst two other schemes: no-ia scheme and crude-ia scheme. Smulaton results show that our IA scheme has a sgnfcant throughput gan over no-ia and crude-ia schemes. Further, the gan of our IA scheme ncreases wth user densty of the network. The remander of ths paper s organzed as follows. Secton II presents related work. Secton III offers a prmer of our OFDM-based IA scheme. In Secton IV, we develop an OFDM-based IA scheme and prove ts feasblty. In Secton V, we develop an IA optmzaton framework to maxmze network throughput. In Secton VI, we establsh the uplnk-downlnk dualty of our IA scheme. Secton VII presents numercal results to show the effcacy of the IA scheme and Secton VIII concludes ths paper. II. RELATED WORK The dea of IA frstly appeared n [16] and the termnology of IA was created by Jafar and Shama n a semnar paper for the two-user X channel [17]. Snce ts emergence, the dea has ganed tremendous momentum n both ndustry and academa (see, e.g., [1], [5], [18] [24]). Snce there s an overwhelmng large amount of work on IA, we cannot survey all the IA papers and therefore focus our lterature survey on IA n cellular networks. In [6] and [7], Suh et al. proposed a frequency-doman IA scheme, called subspace nterference algnment, for both uplnk and downlnk of cellular networks. They further showed that ther IA scheme can acheve K /( G 1 K + 1) G 1 DoFs for each cell, where G s the number of cells and K s the number of users n a cell. If the number of users per cell s large enough, each cell can acheve one DoF. Ths result s sgnfcant, as t ndcates that nterference may not be the domnant factor n cellular networks. Despte ts sgnfcance, the IA scheme n [6] and [7] cannot be used n practcal networks due to ts underlyng assumptons, ncludng () only one data stream per user, () dentcal user number for each BS, () restrcted relatonshp between subcarrer and user numbers, and (v) a sngle nterference collson doman. In contrast, the OFDM-based IA scheme n ths paper does not rely on these assumptons, thereby makng a concrete step towards ts practcal applcatons n cellular networks. In addton to ts advances n the frequency doman, IA was also studed n the spatal doman (usng multple antennas) for cellular networks [9] [11], [15], [25] [27]. In [9], Zhuang et al. nvestgated the feasblty of IA n MIMO cellular networks and proposed a max-sinr algorthm to desgn IA solutons. In [10], Shn et al. proposed an IA scheme to desgn transmt and receve beamformng vectors for a twocell MIMO network and showed that ther IA algorthm can acheve the optmal DoF. In [11], Ntranos et al. studed spataldoman IA n cellular MIMO networks. They showed that ther IA scheme can acheve 1/2 DoFs per antenna n the uplnk of a three-sector cellular network wth one actve user per sector when both the user and the sector have M antennas. Another research lne of IA n cellular networks s focused on addressng ts CSI problem. In [8], Morales-Cespedes et al. studed blnd IA for partally-connected cellular networks and developed a blnd IA scheme based on reconfgurable antenna to remove ntra-cell and nter-cell nterference. In [13], Wang et al. developed a blnd IA scheme for the downlnk of cellular clustered networks wth reconfgurable antennas. In [12], Jose et al. studes the combnaton of IA and opportunstc schedulng to facltate algnment n the cellular downlnk whle not requrng CSI at transmtters. Rao and Lau [14] frst quantfed CSI feedback for IA n MIMO cellular networks, and then derved closed-form tradeoff between the CSI feedback and IA performance. Tresch and Gullaud [15] studed the sum mutual nformaton acheved by IA n cellular networks and derved ts upper and lower performance bounds n the scenaros wth mperfect channel knowledge. Whle there s a large amount of IA results n cellular networks, most of them were focused on nformaton-theoretc

3 4494 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 65, NO. 10, OCTOBER 2017 TABLE I NOTATION Fg. 1. Schematc dagram of IA at user. exploraton or physcal-layer sgnal desgn. Lttle progress has been made to advance our understandng of IA from a networkng perspectve. Ths paper flls ths gap by developng an IA scheme whch can be ontly optmzed wth upper-layer user schedulng to maxmze the throughput of cellular networks. III. OFDM-BASED IA IN CELLULAR NETWORKS: APRIMER IA s a promsng nterference management technque n wreless networks. Its basc dea s to ontly desgn sgnals at transmtters usng lnear precodng technques, wth the am of proectng nterferng sgnals nto a reduced-dmensonal subspace and keepng the desred sgnals resolvable at each recever. Generally speakng, IA can be done n three domans: spatal (MIMO), spectral (OFDM), and temporal (tme slots). In ths paper, we consder OFDM-based IA n the frequency doman n cellular networks. We assume that the transmsson s usng OFDM modulaton and the set of subcarrers avalable for IA s gven. At each transmtter, as shown n Fg. 1, IA s acheved by proectng ts outgong data streams onto the subcarrers usng lnear precodng technque. As such, the core of IA s a constructon of precodng vectors for the outgong data streams at the transmtters. Suppose that there are K (e.g., 64) subcarrers avalable n the network. Then the precodng vector for each data stream s a K 1 complex vector. At each transmtter, we am to desgn a precodng vector for each of ts data streams so that ts transmtted sgnals overlap as much as possble at ts unntended recever(s) whle remanng resolvable at ts ntended recever(s). Table I lsts the notaton that we use n the paper. A. An Example Consder a small network wth 2 BSs and 4 users as shown n Fg. 2, where a sold arrow lne represents a drected lnk and a dashed arrow lne represents a drected nterference. For both BSs and users, each of them has a sngle antenna. We assume that CSI s avalable at both BSs and users. To show the benefts of IA, let s start wth a smple example by assumng 3 subcarrers avalable for data transmsson (.e., K = 3). Note that we take K = 3 only for ease of llustraton and we wll consder a larger value of K later. Wth K = 3, we show that by usng IA, a total of 4 data streams can be transmtted from the users to ther respectve BSs, wth 1 data stream from each user. To show ths, we frst ntroduce the notaton. For vectors a and b, denote a := b f there exsts a nonzero complex number c such that a = cb,.e.,a and b are n the same drecton. Denote {ub 1 uk b } as a set of lnearly ndependent bass vectors wth dmenson K 1 and nonzero entres. The ndependence requrement ensures that the data streams from the same user reman resolvable at the BS; and the nonzero requrement ensures that all subcarrers are fully used. Denote H as the frequency-doman channel matrx between user to BS. Due to the orthogonalty of the subcarrers, H s a dagonal square matrx. Denote u k as the precodng vector for the kth outgong stream at user. We construct the precodng vectors at user 1 and user 2 as follows: let u1 1 := u1 b and let u2 1 := H 1 22 H 21u1 1. As a result, at BS 2, the nterferng stream from user 1 s algned to the nterferng stream from user 2, as shown n Fg. 2(a). Lkewse, we construct the precodng vectors at user 3 and user 4 as follows: let u3 1 := u2 b and let u4 1 := H 1 24 H 23u3 1. Then at BS 1, the nterferng stream from user 3 s algned to the nterferng stream from user 4, as shown n Fg. 2(a). By usng the above precodng vectors at the 4 users, the receved data and nterferng streams at each

4 ZENG et al.: OFDM-BASED IA IN SINGLE-ANTENNA CELLULAR WIRELESS NETWORKS 4495 Fg. 3. The uplnk transmsson n a cellular network. sgnfcant as the number of users ncreases. In the two-bs cellular network where each BS has n users, the nterferences at each BS can be algned to the same drecton. The spectrum effcency at each BS s n/(n + 1) and the total spectrum effcency at the two BSs s 2n/(n + 1). Snce the network wthout IA s 1, the gan of IA s (n 1)/(n + 1). Fg. 2. An example of IA n the frequency doman. IV. AN OFDM-BASED IA SCHEME AND ITS FEASIBILITY In ths secton, we develop an OFDM-based IA scheme for the uplnk communcaton n a sngle-antenna cellular wreless network. The IA scheme ncludes IA constrants at each user and BS, as well as how to construct the precodng/decodng vectors for each stream. In what follows, we frst present the OFDM-based IA scheme and then prove ts feasblty at the physcal layer. BS are on 3 dfferent drectons. Therefore, 3 subcarrers are suffcent to support 4 data streams. However, f IA s not used, 3 subcarrers can support only 3 data streams from the four users (wth any combnatons), snce puttng more than one data stream on a subcarrer wll nevtably cause nterference on that subcarrer. When the network has 6 subcarrers (.e., K = 6), we show that by usng IA, 8 data streams can be transmtted from the users to ther BSs, wth 2 data streams at each user. We construct the precodng vectors at user 1 and user 2 as follows: let [u1 1 u2 1 ]:=[u1 b u2 b ] and let [u1 2 u2 2 ]:=H 1 22 H 21[u1 1 u2 1 ].Asa result, at BS 2, the two nterferng streams from user 1 are algned to the two nterferng streams from user 2, as shown n Fg. 2(b). Lkewse, we construct the precodng vectors at user 3 and user 4 as follows: let [u3 1 u2 3 ] := [u3 b u4 b ] and let [u4 1 u2 4 ] := H 1 24 H 23[u3 1 u2 3 ]. As a result, at BS 1, the two nterferng streams from users 3 are algned to the two nterferng streams from user 4, as shown n Fg. 2(b). By usng those precodng vectors at the 4 users, the receved data and nterferng streams at each BS are on 6 drectons. Therefore, 6 subcarrers are suffcent to support 8 data streams. However, f IA s not used, 6 subcarrers can support only 6 data streams from the four users (wth any combnatons), snce puttng more than one data stream on a subcarrer wll lead to nterference. Followng the same token, when the network has 256 subcarrers, t can transport 340 data streams usng IA, n contrast of 256 data streams n the case wthout IA. It s easy to see that the gan of IA for ths network s 1/3. It s worth pontng out that the gan of IA becomes more A. An OFDM-Based IA Scheme Consder a cellular network shown n Fg. 3, where each node (BS or user) has a sngle antenna. Denote N as the set of users n the network wth N = N. Denote M as the set of BSs n the network M = M. Denote T usr as the set of users who choose BS as ther servce provder. Denote I usr as the set of users that nterfere wth BS,.e., BS s wthn the nterference range of these users and BS s not the servce provder of these users. Denote I bs as the set of BSs that are nterfered wth by user,.e., these BSs are wthn the nterference range of user but are not chosen by user as ts servce provder. In our study, we use Rayleght fadng as the channel fast fadng model. For the channel realzatons, we assume that the CSI s avalable at both BSs and users. Consder user that nterferes wth BS,.e., I bs. Denote S ={s k : 1 k σ } as the set of streams from user, wheres k s the kth stream and σ s the number of streams at user (.e., σ = S ). Then each stream n S s an nterferng stream for BS. AtBS, we wsh to algn as many nterferng streams as possble to some predefned nterference drectons. Among the nterferng streams n S, denote A as the subset of nterferng streams that can be algned to some predefned nterference drectons at BS. Denote α as the cardnalty of A,.e.,α = A. Then, at BS, the number of drectons occuped by the nterferng streams s reduced from σ to σ α, resultng n a savng of α drectons at BS. Among the streams n S, there may be a subset B of streams that are

5 4496 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 65, NO. 10, OCTOBER 2017 Fg. 4. An example of llustratng IA constrants at user and BS I bs. not algned to any predefned nterference drecton at the BSs n I bs. Denote β as the cardnalty of B,.e., β = B. Thus we have B = S \ ( I bsa ). 1) Constrants at User: Consder one of the σ outgong data streams at user as shown n Fg. 4(a). Ths outgong data stream nterferes wth all the BSs n I bs (BS 1, 2, and 3 n the fgure). We wsh to algn ts produced nterference at as many BSs as possble. Now the queston s that the nterference produced by ths data stream can be algned at how many BSs n I bs. As we showed n the example n Secton III, the constructon of ths data stream s precodng vector can guarantee ts produced nterference to be algned at one BS. Note that n some extreme crcumstances (e.g., when all the channels are exactly the same), the constructon of ths data stream s precodng vector can algn ts produced nterference at multple BSs. But n a general case, the constructon of ths data stream s precodng vector can guarantee the algnment of ts nterference at only one BS. We now consder all the σ outgong data streams at user. Snce each of them guarantees that one of ts produced nterferng streams can be successfully algned at the correspondng BS. The constructon of the σ outgong data streams precodng vectors guarantees that σ of ther produced nterferng streams can be successfully algned at the correspondng BSs. To ensure the feasblty of IA, we mpose constrant I bs α σ. Recall that β s a non-negatve nteger optmzaton varable. The constrant can be equvalently translated to β + α = σ, for N. (1) I bs 2) Constrants at BS: At BS (see Fg. 4(b) for example), we need to algn the nterferng streams n A (for each I usr ) to some predefned nterference drectons. To do so, we have the followng two questons: () what should be the set of predefned nterference drectons at BS ; () how to algn the nterferng streams n A to the set of predefned nterference drectons. There may be many possble solutons to the above two questons. Here, we show one soluton for whch we can offer a feasblty proof (see Secton IV-B). In our soluton, for the frst queston, we use I usr B as the set of predefned nterference drectons at BS. That s, each nterferng stream n A wll be algned to an nterferng stream n I usr B. For the second queston, we algn each nterferng stream n A, I usr, to a unque nterference stream n k = k I usr B k. That s, each nterferng stream n A s algned unquely n the nterference subspace formed by the unon of B k over k I usr except ts own B. Here, unquely means that any two nterferng streams n A wll not be algned to the same nterferng stream n k = k I usr B k. Based on our proposed soluton to questons () and (), we have the followng constrants at BS : α k = k I usr β k, for I usr, M. (2) 3) Dmenson Constrants: At BS, the total number of ts desred data streams s T usr σ, whle the number of ts unalgned nterferng streams s I usr (σ α ). Snce the number of drectons for desred data streams and unalgned nterferng streams cannot exceed the number of avalable subcarrers, we have the followng constrants at BS : σ + (σ α ) K, for M. (3) T usr I usr So far we have derved three constrants for IA to characterze ts capablty (.e., the number of data streams that can be sent by each user to ts servng BS) wthout rgorous argument. In the next subsecton, we show that as long as these three constrants are satsfed, there always exst precodng/decodng vectors so that σ data streams can be sent from user to ts servng BS free of nterference, N. B. Feasblty of the IA Scheme Consder an IA scheme π wth DoF vector (σ 1,σ 2,,σ N ). For each stream s k n π, denote u k as ts precodng vector at user and v l as ts decodng vector at ts ntended BS. WesayIAschemeπ s feasble f there exst precodng and decodng vectors so that user can send σ data streams to ts ntended BS free of nterference, N. Then we have the followng defnton. Defnton 1: An IA scheme π s feasble at the physcal layer f there exst precodng and decodng vectors that meet (v l )T H u k = 1; (4) (v l )T H u k = 0, T usr I usr, 1 k σ,(, k ) = (, k); (5)

6 ZENG et al.: OFDM-BASED IA IN SINGLE-ANTENNA CELLULAR WIRELESS NETWORKS 4497 for N and 1 k σ. Note that H, the frequency-doman channel matrx between user and BS, s a dagonal complex matrx wth the kth dagonal entry beng channel coeffcent of the kth subcarrer. The followng theorem s the man result of the rest of ths subsecton. Theorem 1: For uplnk IA scheme π, f ts DoF vector (σ 1,σ 2,,σ N ) satsfes (1), (2), and (3), then t s feasble at the physcal layer. Theorem 1 provdes a suffcent condton to verfy the feasblty of an IA scheme. Instead of constructng precodng and decodng vectors satsfyng (4) and (5) n Defnton 1, Theorem 1 allows to verfy the feasblty of an IA scheme through smple calculaton n (1), (2), and (3). The rest of secton wll be devoted to provng Theorem 1. Here s our road map. Frst, we construct precodng vectors for the data streams at each user. Second, we gve two lemmas to characterze the dmensons of such precodng vectors. Fnally, based on those two lemmas, we show that there always exsts a decodng vector for each stream so that (4) and (5) n Defnton 1 are satsfed. 1) Constructon of Precodng Vectors: Denote E S ={u k : 1 k σ } as the set of precodng vectors for the streams n S at user. Among the precodng vectors n E S, denote E A as the subset of precodng vectors that correspond to the nterferng streams n A ; denote E B as the subset of precodng vectors that correspond to the nterferng streams n B. Snce we defne a unque precodng vector for each stream, we have E A =α, for M, I usr ; E B =β, for N ; E B = E S \ ( I bse A ), for N ; E A 1 E A 2 =, for N, 1, 2 I bs, 1 = 2. We defne E A = N, I bse A and E B = N E B.Then we have N E S = E A E B. We frst construct the precodng vectors n E B and then construct the precodng vectors n E A. Denote {ub k : 1 k K } as a set of lnear ndependent complex vectors wth dmenson K 1andnonzero entres. Then, for the precodng vectors n E B, we construct each of them as follows: u k := u k b. (6) Now we construct the precodng vectors n E A. Recall that n IA scheme π, each nterferng stream n A s algned to an nterferng stream n k = k I usr B k. Therefore, for each u k E A, we defne u k := H 1 H uk. (7) where u k s an precodng vector n E B (.e., u k := ub k ) and =. 2) Propertes of Precodng Vectors: Denote dm(e S ) as the dmenson of the subspace spanned by the vectors n set E S. Then we have the followng lemma: Lemma 1: At each user N, the constructed precodng vectors E S are lnearly ndependent,.e., dm(e S ) = E S. A proof of Lemma 1 s gven n Appendx A. At BS, denote Q T as the set of ts drectons for ts desred data streams and Q I as the set of drectons for ts nterferng streams. Mathematcally, we have Q T Q I = T usr {H u k : u k E S }, = I usr {H u k : u k E S }. Then, we have the followng lemma: Lemma 2: At each BS M, each of ts desred data streams occupes an ndependent drecton,.e., dm(q T Q I ) = T usr σ + dm(q I ), for M. (8) A proof of Lemma 2 s gven n Appendx B. 3) Exstence of Decodng Vectors: So far we have constructed precodng vectors for the streams at each transmtter and showed two mportant propertes of the constructed. For the decodng vectors at recevers, we have the followng proposton: Proposton 1: If the constructed precodng vectors satsfy (8), then there exsts a decodng vector for each stream so that constrants (4) and (5) are satsfed. A proof of Proposton 1 s gven n Appendx B. Ths completes the proof of Theorem 1. V. A THROUGHPUT OPTIMIZATION FRAMEWORK: COMBINING IA WITH USER SCHEDULING A. Problem Statement Our goal s to explot the benefts of OFDM-based IA to ncrease user throughput n cellular networks from a networkng perspectve. We consder a network that conssts of a set of grd-deployed BSs and a set of randomly dstrbuted users (see e.g., Fg. 7). Each BS has a fxed servce area (a dsk wth radus of ts transmsson range) and t only provdes servce to the users wthn ts servce area. A user may fall nto the servce areas of multple BSs and wll choose one of them as ts servce provder. In ths secton, we focus on the IA optmzaton for the uplnk. The downlnk wll be consdered n the next secton. In the uplnk, a user s transmtter and t wll nterfere wth the BSs wthn ts nterference range other than ts servce provder. We assume the transmsson uses OFDM modulaton and the set of subcarrers for IA s gven. Wthn the set of subcarrers, we am to ontly optmze IA and user schedulng so that the uplnk user throughput can be maxmzed at network level. B. Our Approach To solve ths problem, we develop a cross-layer IA optmzaton framework wth the obectve of maxmzng user throughput. In the prevous secton we developed an IA scheme and showed that as long as IA constrants (1) (3) are satsfed, user N can send σ data streams to ts servng BS free of nterference. Constrants (1) (3) defne a feasble IA desgn space for a cellular network that s readly used for throughput optmzaton. However, constrants (1) (3) were derved under the assumpton that each user s servng BS

7 4498 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 65, NO. 10, OCTOBER 2017 Fg. 5. Sets llustraton at BS and user. s gven a pror and therefore can only be appled to a network wth statc user access. In a practcal network, as stated earler, a user may be wthn the servce area of multple BSs and can choose any of them as ts servng BS. Ths freedom (.e., user schedulng at each BS) provdes another dmenson for IA optmzaton space to mprove user throughput and should be ncorporated nto the IA desgn space. In what follows, we frst model the user schedulng and then ncorporate the user schedulng nto our developed IA constrants. Fnally, we formulate a user throughput optmzaton framework and elmnate ts nonlnear constrants wthout loss of optmalty. C. Combnng IA Constrants Wth User Schedulng As shown n Fg. 5(a), denote C usr as the set of users wthn the servce area of BS ; denote O usr as the set of users that are outsde the servce area of BS but can stll nterfere wth BS. As shown n Fg. 5(b), denote C bs as the set of BSs that user can choose as ts servce provder; denote O bs as the set of BSs whose servce areas do not cover user but are stll nsde the nterference range of user. To effcently use IA and IC capabltes, channel qualty (path loss and slow fadng) should be taken nto account when constructng these sets (.e., C usr, O usr, C bs,ando bs ). Denote x as a bnary varable to ndcate whether or not user chooses BS C bs as ts servce provder. Specfcally, x = 1fuser chooses BS as ts servce provder and 0 otherwse. Snce user can choose only one BS as ts servce provder, we have x = 1, N. (9) C bs Denote y as the complementary bnary varable of x. That s, y = 1fuser does not choose BS C bs as ts servce provder and 0 otherwse. Then we have the followng constrants. x + y = 1, C bs, N. (10) We now show that the above BS selecton (user schedulng) varables can be ncorporated nto (1), (2), and (3) n our IA scheme. To ncorporate BS selecton varables n (1), we need to frst clarfy I bs,.e., the set of BSs that are nterfered wth by user. Based on the defntons of O bs, C bs,andy,wehave I bs = O bs { : y = 1, C bs }. Then, (1) can be rewrtten as: β + α + α y = σ, N. (11) O bs C bs Lkewse, for (2), we need to frst clarfy I usr,.e., the set of users that are nterferng wth BS. Based on the defntons of O usr, C usr,andy,wehave I usr = O usr Dependng on whether user n O usr rewrtten as: α k = k O usr α y k O usr β k + β k + { : y = 1, C usr }. (12) k C usr k = k C usr β k y k, β k y k, or C usr, (2) can be O usr, M, (13) C usr, M, (14) Fnally, for (3), we need to frst clarfy T usr,.e., the set of users that choose BS as ther servce provder. Based on the defntons of C usr and x,wehave T usr ={ : x = 1, C usr }. Then, (3) can be rewrtten as: σ x + (σ α ) K M. C usr I usr whch s equvalent to σ x + (σ α ) y + (σ α ) K, M, C usr C usr O usr (15) based on I usr n (12). Constrants (9) (15) defne a feasble IA desgn space when user schedulng s ontly consdered. In the sequel, we employ ths IA space to study an uplnk user throughput maxmzaton problem n a cellular network. D. User Throughput Optmzaton Framework For smplcty, we assume that fxed modulaton and codng scheme (MCS) s used for each data stream and that each data stream corresponds to one unt data rate. The goal s to maxmze the mnmum rate among all the users. Denote r mn as the mnmum rate among all users. Then we have σ r mn, N. (16) Based on the constrants n Secton V-A, the user throughput maxmzaton problem can be formulated as follows: OPT-IA raw : Max r mn S.t. User schedulng: (9), (10); IA constrants: (11), (13), (14), (15); Mnmum rate constrants: (16).

8 ZENG et al.: OFDM-BASED IA IN SINGLE-ANTENNA CELLULAR WIRELESS NETWORKS 4499 OPT-IA raw s a mxed nteger nonlnear programmng (MINLP). To elmnate the nonlnear terms n the constrants, we employ the Reformulaton-Lnearzaton Technque (RLT) n [28]. Specfcally, to elmnate the nonlnear term α y n the constrants, we defne λ = α y.ths replacement requres to add the followng two constrants: 0 λ α, C bs, N, (17) α (1 y ) K λ y K, C bs, N. (18) Smlarly, to elmnate the nonlnear term β y n the constrants, we defne μ = β y. Ths replacement requres to add the followng two constrants: 0 μ β, C bs, N, (19) β (1 y ) K μ y K, C bs, N. (20) By replacng λ = α y and μ = β y n the IA constrants (11), (13), (14), (15), we have the followng lnear IA constrants: β + α + λ = σ, N, (21) α O bs k = k O usr λ k O usr C bs β k + β k + k C usr k = k C usr μ k, μ k, O usr, M, (22) C usr, M, (23) (σ λ ) + (σ α ) K, M. (24) C usr O usr Then, OPT-IA raw s reformulated as follows: OPT-IA: Max r mn S.t. User schedulng: (9), (10); IA constrants: (17), (18), (19), (20), (21), (22), (23), (24); Mnmum rate constrants: (16); where N, M, C bs, O bs, C usr, O usr,andk are known; x and y are bnary varables; r mn, σ, α, β, λ,andμ are non-negatve nteger varables. OPT-IA s a mxed nteger lnear programmng (MILP). Although the theoretcal worst-case complexty to a general MILP problem s exponental [29], [30], there exst hghly effcent optmalty/approxmaton algorthms (e.g., branch-andbound wth cuttng planes [31]) and heurstcs (e.g., sequental fxng algorthm [32]). Another approach s to employ an offthe-shelf solver such as IBM CPLEX optmzaton solver [33], whch can successfully handle a moderate-szed network. As the man goal of ths paper s to study IA from a networkng perspectve rather than developng a specfc soluton to an optmzaton problem, we wll employ the IBM CPLEX optmzaton solver to obtan numercal results n the next secton. VI. DUALITY BETWEEN UPLINK AND DOWNLINK In the prevous sectons, we studed an IA scheme for the uplnk of a cellular network. We now consder the downlnk Fg. 6. Downlnk communcaton n a cellular network. case. We wll show that the IA scheme developed for uplnk can also be appled to downlnk and therefore the downlnk user throughput maxmzaton problem can be solved n the same way as the uplnk problem. Consder the downlnk communcaton as shown n Fg. 6, whch has the same settng as the uplnk n Fg. 3. For the downlnk, denote ˆπ as ts IA scheme wth DoF vector ( ˆσ 1, ˆσ 2,, ˆσ N ), where ˆσ s the number of desred data streams at user. Atuser, denote Ŝ ={ŝ k : 1 k ˆσ } as the set of ts desred data streams and ˆv k as the decodng vector of ts stream ŝ k.atuser s ntended BS, denote ûl as the precodng vector of user s stream ŝ k. Then we have the followng theorem: Theorem 2: For downlnk IA scheme ˆπ, f ts DoF vector ( ˆσ 1, ˆσ 2,, ˆσ N ) satsfes (1), (2), and (3), then t s feasble at the physcal layer. A proof of Theorem 2 s gven n Appendx D. Based on Theorem 2, we have the followng observatons on π and ˆπ: For user N, f t can send σ data streams to ts BS n the uplnk transmsson, then t can receve σ data streams from BS n the downlnk transmsson, and vce versa. For the downlnk problem, the precodng and decodng vectors n ˆπ are the same as the correspondng decodng and precodng vectors n π, respectvely. That s, stream ŝ k s precodng vector s stream sk s decodng vector and stream ŝ k s decodng vector s stream sk s precodng vector. Snce the uplnk and downlnk have the same IA desgn space, the downlnk user throughput maxmzaton problem has the same formulaton as the uplnk problem. Therefore, the downlnk has the same optmal user throughput as the uplnk. VII. PERFORMANCE EVALUATION In ths secton, we frst use a case study to llustrate how IA scheme works n a cellular network. Then, we compare the user throughput performance of our IA scheme aganst two other schemes: no-ia scheme and crude-ia scheme. In no-ia scheme, a subset of subcarrers s allocated to each user for ts data transmsson, wth each data or nterferng stream occupyng a unque subcarrer at each BS. That s, there s a complete absence of overlappng of nterferng streams on any subcarrer. We denote the user throughput maxmzaton problem under no-ia scheme as OPT-noIA and ts formulaton s gven n Appendx E. In crude-ia scheme,

9 4500 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 65, NO. 10, OCTOBER 2017 Fg. 7. Cellular network nstances wth 4, 9, and 16 BSs. a subset of subcarrers s allocated to each user for ts data transmsson so that at a BS, each of ts desred data streams s on a unque subcarrer whle the nterferng streams are allowed to overlap. Ths problem s smlar to ours except that each data stream n our IA scheme s proected onto all subcarrers and there s an optmzaton on the desgn of drectons for ntended data streams and nterferng data streams. In lght of ths key dfference, we denote the user throughput maxmzaton problem under crude-ia scheme as OPT-crudeIA and ts formulaton s gven n Appendx F. A. Smulaton Settng Wthout loss of generalty, we normalze all unts for dstance, tme, bandwdth, and data rate wth approprate dmensons. We consder cellular networks wthn a area for three cases: () 4 BSs wth 100 users; () 9 BSs wth 100 users; and () 16 BSs wth 100 users. Fg. 7 llustrates our BS deployment for the three cases. In each case, to mmc the BS deployment n real-world cellular networks, we place the BSs n grd. The 100 users are randomly dstrbuted n the area wth a unform probablty. A user can be n actve or nactve state, wth equal probablty. When actve, a user has a persstent traffc for transmsson; when nactve, a user s not served by any BS. For any comparson study, the state of a user s the same under all three schemes. To assure that each user s wthn the servce area of at least one BS, we set the transmsson range to 360 for the 4-BS case, 240 for the 9-BS case, and 180 for the 16-BS case. The nterference range s twce of the transmsson range and the number of subcarrers avalable for data transmsson s 256, unless otherwse specfed. B. A Case Study We use the network nstance n Fg. 7(b) to llustrate how IA works to mprove throughput. Among the 100 users, 55 of them are actve and 45 of them are nactve (nactve users are not shown n the fgure). The number of subcarrers avalable for IA s 256. By solvng the OPT-IA problem for ths network nstance, we obtan the optmal obectve value of 13. We then solve the OPT-noIA problem for ths network Fg. 8. User schedulng at each BS and nterference pattern. nstance, we obtan the optmal obectve value of 6. Ths ndcates that our IA scheme can ncrease the user throughput by 117% as compared to the no-ia scheme. We also solve the OPT-crudeIA problem for ths network nstance, we obtan the optmal obectve value of 9. Ths ndcates that our IA scheme can ncrease the user throughput by 44% as compared to the crude-ia scheme. We now show how our IA scheme works n ths network nstance. Fg. 8 shows the user schedulng result and nterference pattern n the IA soluton, where a sold arrow lne represents an establshed lnk from a user to a BS and a dashed lne represents an nterference. Table II summarzes the IA behavor at each BS. In ths table, the frst column lsts the BSs n the network; the second column lsts the number of users that choose ths BS as ther servce provder; the thrd column lsts the number of desred data streams at ths BS, where each user has 13 data streams to ts BS; the fourth column lsts the dmenson of the subspace for the nterferng streams at ths BS, whch s 256 mnus the number n the thrd column; the ffth column lsts the number of undesred nterferng streams (from neghborng nterferng users) at ths

10 ZENG et al.: OFDM-BASED IA IN SINGLE-ANTENNA CELLULAR WIRELESS NETWORKS 4501 TABLE II IA BEHAVIOR AT EACH BS IN THE CASE STUDY Fg. 9. Throughput gan of our IA scheme over crude-ia scheme. Fg. 10. Throughput gan of our IA scheme over no-ia scheme. BS; the sxth column lsts the nterference overlappng rato, whch s the rato of the ffth column to the fourth column. In the sxth column, a value greater than 1 ndcates the exstence of nterference overlappng. The large the rato s, the more IA has been acheved at the correspondng BS. Now let s take a look at the row for BS 5 n Table II as an example. As shown n Fg. 8, BS 5 s used as servce provder by 12 users. Snce each user has 13 outgong data streams, the number of desred data streams at BS 5 s 156. Thus, the dmenson of the subspace for the nterferng streams s upper bounded by 100 (.e., ). As shown n Fg. 8, BS 5 s beng nterfered by 17 users and thus has 221 (.e., 17 13) nterferng streams. Therefore, the nterference overlappng rato at BS 5 s 221/100 = 2.21 (as shown n the table). As ndcated n Table II, nterferng streams are squeezed n a reduced-dmensonal subspace at most BSs n ths network nstance, makng t possble to transport more data streams than the no-ia and crude-ia schemes. C. Throughput Gan of Our IA Scheme To study throughput gan of our IA scheme over crude- IA and no-ia schemes, we generate 200 randomly network nstances wth 256 subcarrers. For each network nstance, we solve ts OPT-IA, OPT-crudeIA, and OPT-noIA formulatons usng CPLEX optmzaton solver and obtan ther optmal obectve values. Fg. 9 presents the throughput gan of our IA scheme over crude-ia scheme n three cases (4-BS, 9-BS, 16-BS), where x-axs s the throughput gan n percentage (.e. the rato of the optmal obectve value from OPT-IA to that from OPT-crudeIA mnus one and tmes 100) and the y-axs s the cumulatve probablty. From the fgure we can see that the throughput gan ranges from 10% to 70% n the three cases, ndcatng that our IA scheme always outperforms crude-ia scheme. On average, the throughput gan of our IA scheme over crude-ia scheme s 40.1% n the 4-BS case, 34.7% n the 9-BS case, and 30.8% n the 16-BS case. In the same format, Fg. 10 presents the throughput gan of our IA scheme over no-ia scheme. On average, the throughput gan of our IA scheme over no-ia scheme s 96.1% n the 4-BS case, 82.0% n the 9-BS case, and 69.9% n the 16-BS case. From Fg. 9 and Fg. 10, we can see that our IA scheme has hgher throughput gan over no-ia scheme than over crude-ia scheme. Ths s not surprsng, as the no-ia scheme does not allow any algnment whereas the crude-ia scheme allows algnment of nterference on each ndvdual subcarrer. 1) Impact From the Number of Subcarrers: We now study the mpact of subcarrer number on the throughput gan of our IA scheme. We generate 200 randomly network nstances wth the number of subcarrers varyng from 32, 64, 128, 256, 512, to Fg. 11 presents the throughput gan of our IA scheme (averaged over the 200 network nstances) versus the number of subcarrers n 4-BS, 9-BS, and 16-BS cases. For

11 4502 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 65, NO. 10, OCTOBER 2017 each BS. Therefore, our IA scheme s more sutable for a dense network. Fg. 11. Fg. 12. Impact of subcarrer number on the gan of our IA scheme. Impact of user densty on the gan of our IA scheme. example, when the network has 4 BSs and 64 subcarrers, the average throughput gan of our IA scheme s 65% over crude-ia scheme and 121% over no-ia scheme, as shown n Fg. 11. From the fgure we can see that, as the number of subcarrers ncreases, the throughput gans of our IA scheme over crude-ia scheme converge to 37.6%, 35.3%, 31.5% n 4-BS, 9-BS, and 16-BS cases; and the throughput gans of IA scheme over no-ia scheme converge to 86.7%, 76.6%, 64.7% n the three cases. It should be noted that when the number of subcarrers s less than 64, crude-ia and no-ia schemes yeld zero throughput for most network nstances. 2) Impact of User Densty: Fnally, we study the mpact of user densty on the throughput gan of our IA scheme over no-ia and crude-ia schemes. For each network nstance (see, e.g., Fg. 7), nstead of fxng user number to 100, we consder dfferent user denstes: 25 users, 50 users, 100 users, and 150 users (wthn the square area). For each user densty, we generate 200 network nstances and compute ther averaged gan of our IA scheme. Fg. 12 presents the throughput gan of our IA scheme n 4-BS, 9-BS, and 16-BS cases, where x-axs s the number of users n the square area and y-axs s the averaged gan of our IA scheme over the 200 network nstances. From the fgure we can see that the gan of our IA scheme over both no-ia and crude- IA schemes becomes more sgnfcant as the number of users ncreases. As we explaned n Secton III, ths s because more users can acheve more algnment at each BS, thereby leavng larger subspace avalable for desred data stream recepton at VIII. CONCLUSIONS AND FUTURE WORK Ths paper studed IA n cellular networks from a networkng perspectve. We developed an OFDM-based IA scheme for cellular networks and proved ts feasblty at the physcal layer. Specfcally, we showed that as long as the correspondng IA constrants are satsfed, there always exst precodng and decodng vectors so that each data stream can be transported free of nterference. Such an IA scheme allows us to study network-level throughput problems wthout gettng nvolved nto the onerous desgn of precodng and decodng vectors. By ncorporatng user schedulng nto our IA scheme, we developed an uplnk user throughput optmzaton framework and demonstrated the throughput gan of our IA scheme at network level. For the downlnk problem, we showed that the IA scheme developed for the uplnk can also be appled to the downlnk. Furthermore, the downlnk user throughput maxmzaton problem has the same formulaton as the uplnk problem and therefore can be solved n the same way. Although the IA scheme was desgned n the frequency doman, t s a general framework whch can also be used n the temporal and spatal domans. Whle the beneft of IA has been recognzed n theory, there are a number of ssues needed to be addressed n order to use t n practcal cellular networks, ncludng CSI acquston on the transmtter sde, transmsson/recepton coordnaton among the nodes n the network, and tmng and frequency synchronzaton among the transmtters. Obvously, these ssues wll compromse the throughput gan of the proposed IA scheme. In our future work, we wll develop practcal frequencydoman IA solutons that can address those ssues whle maxmally preservng the throughput gan of IA n real-world cellular networks. APPENDIX A PROOF OF LEMMA 1 Based on the defntons of E S, E B,andE A,wehave E S = E B ( I bse A ). Accordng to the precodng vector constructon procedure, we know that the constructed precodng vectors n E B are ndependent of any channel matrces (see (6)), whereas the constructed precodng vectors n E A are determned by the channel matrces (see (7)). Gven that the dagonal entres n the channel matrces are drawn from complex Gaussan dstrbuton, we have dm(e S ) = dm(e B ( I bse A )) = dm(e B ) + dm( I bse A ). (25) Based on (7), we know that the precodng vectors n E A s determned by the channel matrx H. Snce the channel matrces n {H : I bs } are randomly ndependent of each other, we have dm( I bs E A ) = I bs dm(e A ). (26)

12 ZENG et al.: OFDM-BASED IA IN SINGLE-ANTENNA CELLULAR WIRELESS NETWORKS 4503 To analyze dm(e A ), we dvde the precodng vectors n E A nto dfferent groups based on the ther correspondng value of n (7): {E A E A = = I usr E A,whereE A : I usr, = }. Thus we have := H 1 H E B wth E B E B. Based on the precodng vector constructon procedure, we have dm(e A ) (a) = dm(e B ) (b) = E B (c) = E A, (27) where (a) follows from our mld assumpton that channel matrx has full rank [34, Ch. 1]; (b) follows from the fact that the precodng vectors n E B are constructed by (6); (c) follows from the fact that n our IA scheme, each nterferng streams n A s algned to a unque nterferng stream n B wth =. Based on the defntons and (27), we have dm(e A ) = dm( = = = I usr I usr E A ) (a) = = I usr dm(e A ) E A = E A, (28) where (a) follows from the fact that the channel matrces {H : I usr, = } are drawn complex Gaussan dstrbuton and ndependent of each other. Based on (25), (26), and (28), we conclude dm(e S ) = dm(e B ) + dm( I bse A ) = E B + dm(e A ) I bs Ths complete the proof. = E B + I bs = E S. E A APPENDIX B PROOF OF LEMMA 2 Consder a BS M as shown n Fg. 4(b). Denote Q I,Al as the set of algned nterferng stream drectons. Denote Q I,Def as the set of predefned nterferng stream drectons. Denote Q I,Eff as the set of effectve nterferng stream drectons. Then we have Q I,Al Q I,Def Q I,Eff = I usr {H u k : u k E A }, = I usr {H u k : u k E B }, = I usr {H u k : u k E S \E A }. Snce E B E S \E A,wehaveQ I,Def Q I,Eff. Based on the precodng vector constructon procedure, we know that for each H u k Q I,Al, there exsts a H u k Q I,Eff such that H u k := H u k. Consequentally, we have span(q I,Al ) ). Thus we have span(q I,Def span(q I I,Eff ) = span(q Q I,Al ) = span(q I,Eff ). (29) We now argue that the sgnal subspace Q T s lnearly ndependent of the effectve nterference subspace Q I,Eff at BS. Ths s true for the followng two reasons. Frst, based on the gven constrant (3), we have Q T Q I,Eff = T usr σ + I usr (σ α ) K. Thus, the number of drectons n Q T Q I,Eff s bounded by the total avalable dmenson (.e., the number of subcarrers K ). Second, the channel matrces {H : T usr I usr } are frequency-selectve and are randomly ndependent of each other. These propertes of the channel matrces are attrbutve to the network envronment. For these two reasons, we have dm(q T Q I,Eff ) = dm(q T ) + dm(q I,Eff ). (30) To characterze the dmenson of the sgnal (desred data stream) subspace at BS, wehave dm(q T ) = dm( T usr {H u k : u k E S }) (a) = dm({h u k : u k E S }) T usr (b) = T usr (c) = T usr dm(e S ) σ, (31) where (a) holds due to the random ndependence of the channel matrces {H : T usr } and Q T K ; (b) follows from our mld assumpton that H has full rank; (c) follows from Lemma 1 and E S =σ. Based on (29) and (31), we have dm(q T Q I ) (a) = dm(q T Q I,Eff ) (b) = dm(q T ) + dm(q I,Eff ) (c) = T usr σ + dm(q I ), (32) where (a) and (c) hold due to (29); (b) holds due to (31). Combnng (32) and Lemma 1, we conclude that Theorem 1 holds. Ths completes the proof. APPENDIX C PROOF OF PROPOSITION 1 We show that f the precodng vectors satsfy constrant (8), then there exst a set of decodng vectors that satsfy (4) and (5) n Defnton 1. Specfcally, we argue that f constrant (8) s satsfed, then the followng lnear system s consstent (.e., the system has at least one feasble soluton): (v l )T H u k = 1, (v l )T H u k = 0, T usr I usr, 1 k σ,(, k ) = (, k) where v l s varable vector whle H s and u s are known. Based on the defnton of Q T and Q I, we know Q T Q I ={H u k : T usr I usr, 1 k σ }.

13 4504 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 65, NO. 10, OCTOBER 2017 It s easy to see that Q T Q I s the set of coeffcentvectors of ths lnear system. Moreover, ths system has K free varables and at most K lnearly ndependent equatons. If we can show that vector H u k s not a lnear combnaton of other vectors n Q T Q I, then ths system s consstent. Next, we argue ths pont by contradcton. Suppose that H u k s a lnear combnaton of other vectors n Q T Q I.GvenH u k Q T,wehave dm(q T Q I )< Q T +dm(q I ) = I usr σ + dm(q I ). Ths contradcts the gven condton n (8). Therefore, we conclude that the lnear system s consstent. Ths completes the proof. APPENDIX D PROOF OF THEOREM 2 We prove t by constructon. Consder an uplnk IA scheme π wth ts DoF vector (σ 1,σ 2,,σ N ) = ( ˆσ 1, ˆσ 2,, ˆσ N ). Snce ( ˆσ 1, ˆσ 2,, ˆσ N ) satsfes (1), (2), and (3), based on Theorem 1, we know that the uplnk IA scheme π s feasble. Further, for each stream s k n uplnk IA scheme π, there exst a precodng vector u k and a decodng vector v l that satsfy (4) and (5). To show that downlnk IA scheme ˆπ s feasble, we construct each stream ŝ k s precodng and decodng vectors as follows: û l = vl and ˆv k = u k,wherevl and u k are decodng and precodng vectors n the uplnk IA scheme π and can been constructed n (6) and (7). Now we argue that by usng these precodng and decodng vectors, each steam ŝ k n downlnk IA scheme ˆπ can be transported free of nterference. We frst check the transfer functon of stream ŝ k as follows: (ˆv k )T H û l (a) = (u k )T H v l (b) =[(v l )T H u k (c) ]T = 1, (33) where (a) follows from û l = vl and ˆvk = u k ; (b) follows from the fact that H s a dagonal matrx,.e., (H ) T = H ;and (c) follows from (4). We then check whether the nterference can be completely canceled. For stream ŝ k, t suffers from nterference from the streams that correspond to precodng vectors v l wth I bs {}, 1 l T σ,and(, l ) = (, l). Based on (5), we have (ˆv k )T H û l = (uk )T H v l =[(vl )T H u k ]T = 0, (34) for I bs {}, 1 l T σ,and(, l ) = (, l). (33) and (34) assure that each stream ŝ k ( N, 1 k ˆσ ) can be transported free of nterference n the downlnk. Therefore, we conclude that IA scheme ˆπ s feasble for the downlnk. Ths completes the proof. APPENDIX E NETWORK THROUGHPUT OPTIMIZATION UNDER NO-IA SCHEME In the no-ia scheme, a subset of subcarrers s allocated to each user for ts data transmsson such that at each BS, each data or nterferng stream occupes a unque subcarrer. That s, there s a complete absence of overlappng of nterferng streams on any subcarrer. Denote K as the set of subcarrers n the network. Denote w k as a bnary varable to ndcate whether the kth subcarrer s used by user. Specfcally, w k = 1fthekth subcarrer s used for data transmsson at user and w k = 0 otherwse. Thus, the number of outgong streams from user can be expressed as σ = k K w k, N. (35) At BS M, a subcarrer k K can be used by only one user wthn ts transmsson range and nterference range. Otherwse, t wll cause nterference collson or overlappng. Thus, we have the followng constrants: T I w k 1, k K, M. (36) Therefore, the throughput maxmzaton problem under no- IA scheme can be formulated as follows: OPT-noIA: Max r mn S.t. (16), (35), (36); where w k and σ are varables; whle C usr, O usr, N, M, K are known a pror based on the network topology and settng. APPENDIX F NETWORK THROUGHPUT OPTIMIZATION UNDER CRUDE-IA SCHEME We formulate the same network throughput problem under the crude-ia scheme. In the crude-ia scheme, a subset of subcarrers s allocated to each user for ts data transmsson such that at each BS, each of ts desred data streams s on a unque subcarrer whle the nterferng streams are allowed to overlap. Recall that K s the set of subcarrers n the network and w k s a bnary varable ndcatng whether the kth subcarrer s used at user. Then, the number of outgong streams from user can be expressed as T usr σ = k K w k, N. (37) Consder an BS M and ts servng users (.e. users n T usr ). To avod transmsson conflct, at most one of the users n T usr can use the kth subcarrer for data stream transmsson. Thus we have T usr w k 1. If none of the users n T usr uses the kth subcarrer for data stream transmsson, then ths subcarrer can accommodate any amount of nterference (.e., 1 N I usr w k 1). Combnng these two cases, the nterference avodance scheme can be modeled by the followng constrant: w k + 1 w k 1, M, k K. (38) N I usr Recall that C usr s the set of users wthn the transmsson range of BS and O usr s the set of users wthn the nterference range of BS. A user may be wthn the transmsson

14 ZENG et al.: OFDM-BASED IA IN SINGLE-ANTENNA CELLULAR WIRELESS NETWORKS 4505 range of multple BSs and we use x to ndcate whch BS s servng for t. Thus we have T usr ={ : C usr, x = 1} and I usr = O usr { : C usr, x = 0}. Then the nterference avodance constrants n the network can be expressed as x w k + 1 (1 x )w k + 1 w k 1, N N C usr C usr O usr M, k K. (39) To elmnate the nonlnear term x w k n (39), we defne a new varable as follows: q k = x w k, M, C usr, k K. (40) Gven that both x and w k are bnary varables, t t easy to verfy that constrant (40) s equvalent to the combnaton of the followng three lnear constrants: q k x, C usr, M, k K. (41) q k w k, C usr, M, k K. (42) q k x + w k 1, C usr, M, k K. (43) Replacng x w k by q k n nterference avodance constrant (39), we have N 1 q k + 1 N N w k 1, M, k K. C usr C usr O usr (44) Therefore, the throughput maxmzaton problem under crude-ia scheme can be formulated as follows: OPT-crudeIA: Max r mn S.t. (9), (16), (44), (41), (42), (43), (37), where x, w k, σ, q k,andr mn are varables; whle C usr, O usr, N, N, M, K are known a pror based on the network topology and settng. ACKNOWLEDGMENT Part of W. Lou s work was completed whle she was servng as a Program Drector at the NSF. Any opnon, fndngs, and conclusons or recommendatons expressed n ths paper are those of the authors and do not reflect the vews of the NSF. The authors thank Vrgna Tech Advanced Research Computng for gvng them access to the BlueRdge computer cluster. REFERENCES [1] 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 [2] V. R. Cadambe and S. A. Jafar, Interference algnment and the degrees of freedom of wreless X networks, IEEE Trans. Inf. Theory, vol. 55, no. 9, pp , Sep [3] T. Gou and S. A. Jafar, Degrees of freedom of the K user M N MIMO nterference channel, IEEE Trans. Inf. Theory, vol. 56, no. 12, pp , Dec [4] 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 , Jul [5] S. A. Jafar, Interference algnment A new look at sgnal dmensons n a communcaton network, Found. Trends Commun. Inf. Theory, vol. 7, no. 1, pp , [6] C. Suh and D. Tse, Interference algnment for cellular networks, n Proc. IEEE Annu. Allerton Conf. Commun., Control, Comput., Sep. 2008, pp [7] C. Suh, M. Ho, and D. N. C. Tse, Downlnk nterference algnment, IEEE Trans. Commun., vol. 59, no. 9, pp , Sep [8] M. Morales-Céspedes, J. Plata-Chaves, D. Toumpakars, S. A. Jafar, and A. G. Armada, Blnd nterference algnment for cellular networks, IEEE Trans. Sgnal Process., vol. 63, no. 1, pp , Jan [9] B. Zhuang, R. A. Berry, and M. L. Hong, Interference algnment n MIMO cellular networks, n Proc. IEEE Int. Conf. Acoust., Speech Sgnal Process. (ICASSP), May 2011, pp [10] W. Shn, N. Lee, J.-B. Lm, C. Shn, and K. Jang, On the desgn of nterference algnment scheme for two-cell MIMO nterferng broadcast channels, IEEE Trans. Wreless Commun., vol. 10, no. 2, pp , Feb [11] V. Ntranos, M. A. Maddah-Al, and G. Care, Cellular nterference algnment, IEEE Trans. Inf. Theory, vol. 61, no. 3, pp , Mar [12] J. Jose, S. Subramanan, X. Wu, and J. L, Opportunstc nterference algnment n cellular downlnk, n Proc. IEEE Annu. Allerton Conf. Commun., Control, Comput. (Allerton), Oct. 2012, pp [13] C. Wang, H. C. Papadopoulos, S. A. Ramprashad, and G. Care, Desgn and operaton of blnd nterference algnment n cellular and clusterbased systems, n Proc. IEEE Inf. Theory Appl. Workshop (ITA), Feb. 2011, pp [14] X. Rao and V. K. N. Lau, Interference algnment wth partal CSI feedback n MIMO cellular networks, IEEE Trans. Sgnal Process., vol. 62, no. 8, pp , Apr [15] R. Tresch and M. Gullaud, Cellular nterference algnment wth mperfect channel knowledge, n Proc. IEEE Int. Conf. Commun. Workshops, Jun. 2009, pp [16] M. A. Maddah-Al, A. S. Motahar, and A. K. Khandan, Sgnalng over MIMO mult-base systems: Combnaton of mult-access and broadcast schemes, n Proc. IEEE Int. Symp. Inf. Theory (ISIT), Jul. 2006, pp [17] S. A. Jafar and S. Shama (Shtz), Degrees of freedom regon of the MIMO X channel, IEEE Trans. Inf. Theory, vol. 54, no. 1, pp , Jan [18] S. A. Jafar, Blnd nterference algnment, IEEE J. Sel. Topcs Sgnal Process., vol. 6, no. 3, pp , Jun [19] S. A. Jafar, The ergodc capacty of phase-fadng nterference networks, IEEE Trans. Inf. Theory, vol. 57, no. 12, pp , Dec [20] S. Gollakota, S. D. Perl, and D. Katab, Interference algnment and cancellaton, SIGCOMM Comput. Commun. Rev., vol. 39, pp , Aug [21] F.Adb,S.Kumar,O.Aryan,S.Gollakota,andD.Katab, Interference algnment by moton, n Proc. ACM MobCom, Sep. 2013, pp [22] O. El Ayach, S. W. Peters, and R. W. Heath, Jr., The feasblty of nterference algnment over measured MIMO-OFDM channels, IEEE Trans. Veh. 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15 4506 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 65, NO. 10, OCTOBER 2017 [31] 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, Mar. 2010, pp [32] Y. T. Hou, Y. Sh, and H. D. Sheral, Spectrum sharng for mult-hop networkng wth cogntve rados, IEEE J. Sel. Areas Commun., vol. 26, no. 1, pp , Jan [33] (2015). IBM ILOG CPLEX Optmzaton Solver, accessed on Dec. 15, [Onlne]. Avalable: ntegraton/optmzaton/cplex-optmzer [34] R. A. Horn and C. R. Johnson, Matrx Analyss. Cambrdge, U.K.: Cambrdge Unv. Press, Wenng Lou (F 15) receved the Ph.D. degree n electrcal and computer engneerng from the Unversty of Florda. She s currently a Professor wth the Computer Scence Department, Vrgna Tech. Her research nterests are n the broad area of wreless networks, wth specal emphases on wreless securty and cross-layer network optmzaton. Snce 2014, she has been servng as a Program Drector at the Natonal Scence Foundaton. She s the Steerng Commttee Char of the IEEE Conference on Communcatons and Network Securty. Huacheng Zeng (M 15) receved the Ph.D. degree n computer engneerng from Vrgna Tech, Blacksburg, VA, USA, n He s currently an Assstant Professor of Electrcal and Computer Engneerng wth the Unversty of Lousvlle, Lousvlle, KY, USA. Hs research focuses on developng practce solutons to advancng wreless communcaton systems and enablng nnovatve wreless applcatons. He was a recpent of the ACM WUWNET 2014 Best Student Paper Award. Xu Yuan (S 13 M 16) receved the B.S. degree from the Department of Informaton Securty, Nanka Unversty, n 2009, and the Ph.D. degree from the Bradley Department of Electrcal and Computer Engneerng, Vrgna Tech, Blacksburg, VA, USA, n From 2016 to 2017, he was a Post-Doctoral Fellow of Electrcal and Computer Engneerng wth the Unversty of Toronto, Toronto, ON, Canada. He s currently an Assstant Professor wth the School of Computng and Informatcs at the Unversty of Lousana at Lafayette, LA, USA. Hs research nterest focuses on cloud computng securty, algorthm desgn and optmzaton for spectrum sharng, coexstence, and cogntve rado networks. Y Sh (S 02 M 08 SM 13) s currently a Senor Research Scentst wth Intellgent Automaton Inc., Rockvlle, MD, USA, and an Adunct Assstant Professor wth Vrgna Tech. He has authored one book, fve book chapters, and more than 120 papers on wreless network algorthm desgn and optmzaton. Hs research focuses on optmzaton and algorthm desgn for wreless networks and socal networks. He has co-organzed several IEEE and ACM workshops, and he has been a TPC Member of many maor IEEE and ACM conferences. He was a recpent of the IEEE INFOCOM 2008 Best Paper Award, the IEEE INFOCOM 2011 Best Paper Award Runner-Up, and the ACM WUWNet 2014 Best Student Paper Award. He s an Edtor of the IEEE COMMUNICATIONS SURVEYS AND TUTORIALS.HehasnamedanIEEECOMMUNICATIONSSURVEYS AND TUTORIALS Exemplary Edtor n Rongbo Zhu (M 10) receved the Ph.D. degree n communcaton and nformaton systems from Shangha Jao Tong Unversty, Chna, n He was a Vstng Scholar wth Vrgna Tech from 2011 to He s currently a Professor wth the College of Computer Scence, South-Central Unversty for Natonaltes, Chna. Hs research nterests nclude performance optmzaton and protocol desgn of wreless networks. Y. Thomas Hou (F 14) receved the Ph.D. degree from the NYU Tandon School of Engneerng (formerly Polytechnc Unversty) n From 1997 to 2002, he was a member of the Research Staff wth the Futsu Laboratores of Amerca, Sunnyvale, CA, USA. He s currently a Bradley Dstngushed Professor of Electrcal and Computer Engneerng wth Vrgna Tech, Blacksburg, VA, USA, whch he oned n He has authored over 100 ournal papers and 130 conference papers n networkng related areas. He authored/co-authored two graduate textbooks: Appled Optmzaton Methods for Wreless Networks (Cambrdge Unversty Press, 2014) and Cogntve Rado Communcatons and Networks: Prncples and Practces (Academc Press/Elsever, 2009). Hs current research focuses on developng nnovatve solutons to complex scence and engneerng problems arsng from wreless and moble networks. Hs papers were recognzed by fve best paper awards from the IEEE and two paper awards from the ACM. He holds fve U.S. patents. He s also a Dstngushed Lecturer of the IEEE Communcatons Socety. He was/s on the edtoral boards of a number of IEEE and ACM transactons and ournals. He s the Steerng Commttee Char of the IEEE INFOCOM Conference and a member of the IEEE Communcatons Socety Board of Governors. Jannong Cao (F 15) receved the B.Sc. degree n computer scence from Nanng Unversty, Nanng, Chna, and the M.Sc. and Ph.D. degrees n computer scence from Washngton State Unversty, Pullman, WA, USA. He s currently a Char Professor and the Head of the Department of Computng, The Hong Kong Polytechnc Unversty, Hong Kong. He has co-authored three books, co-edted nne books, and authored over 300 papers n maor nternatonal ournals and conference proceedngs. Hs research nterests nclude parallel and dstrbuted computng, wreless networks and moble computng, bg data and cloud computng, pervasve computng, and fault tolerant computng. He s a Senor Member of the Chna Computer Federaton and a member of the ACM. He was the Char of the Techncal Commttee on Dstrbuted Computng of the IEEE Computer Socety from 2012 to He has served as an Assocate Edtor and a member of the edtoral boards of many nternatonal ournals, ncludng the ACM Transactons on Sensor Networks, the IEEE TRANSACTIONS ON COMPUTERS, the IEEE TRANSACTIONS ON PARALLEL AND DISTRIBUTED SYSTEMS, the IEEE NETWORKS, PERVASIVE AND MOBILE COMPUTING JOURNAL, and Peer-to-Peer Networkng and Applcatons. He has also served as a Char and a member of organzng/program commttees for many nternatonal conferences, ncludng PERCOM, INFOCOM, ICDCS, IPDPS, ICPP, RTSS, DSN, ICNP, SRDS, MASS, PRDC, ICC, GLOBECOM, and WCNC.

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