THE pressing need to improve the efficiency of wireless

SUBMIED O IEEE RANSACIONS ON SIGNAL PROCESSING, MANUSCRIP, OCOBER 04 Bind Interference Aignment for Ceuar Networks Máximo Moraes-Céspedes, Student Member, IEEE, Jorge Pata-Chaves, Member, IEEE, Dimitris oumpakaris, Member, IEEE, Syed A Jafar, Feow Member, IEEE, and Ana García Armada, Senior Member, IEEE Abstract We propose a bind interference aignment scheme for partiay connected ceuar networks he scheme cances both intrace and interce interference by reying on receivers with one reconfigurabe antenna and by aowing users at the ce edge to be served by a the base stations in their proximity An outer bound for the Degrees of Freedom is derived for genera partiay connected networks with singe-antenna receivers when knowedge of the channe state information at the transmitter is not avaiabe It is demonstrated that for symmetric scenarios this outer bound is achieved by the proposed scheme On the other hand, for asymmetric scenarios the achievabe Degrees of Freedom are not aways equa to the outer bound However, the penaty is typicay sma, and the proposed scheme outperforms other bind interference aignment schemes Moreover, significant reduction of the supersymbo ength is achieved compared to a standard bind interference aignment strategy designed for fuy connected networks Index erms Bind Interference Aignment, Ceuar Networks, Degrees of Freedom I INRODUCION HE pressing need to improve the efficiency of wireess systems has ed to the intensive study of interference and its effect on communication Unti fairy recenty, the typica design approach was to avoid interference as much as possibe Latey, there has been a gradua shift to operating in the presence of interference Interference Aignment IA) is based on this approach he aim of IA is to ensure that, at each receiver, a interference is contained in a signa subspace with the smaest possibe dimension It is then possibe to Máximo Moraes Céspedes and Ana García Armada are with the Department of Signa heory and Communications, Universidad Caros III de Madrid, Avda Universidad 0, 89, Leganés Madrid), Spain emai: maximo@tscucmes, agarcia@tscucmes) his work has been partiay funded by research projects COMONSENS CSD008-0000) and GRENEC0-9006-C0-0) Jorge Pata-Chaves is with the Department of Eectrica Engineering ESA- SCD/ SISA), Kathoieke Universiteit Leuven, B-00 Leuven, Begium emai: jpata@esatkueuvenbe) his research work was party carried out at the ESA Laboratory of KU Leuven in the frame of the Begian Programme on Interuniversity Attractive Poes Programme initiated by the Begian Science Poicy Office: IUAP P7/ Begian network on stochastic modeing anaysis design and optimization of communication systems BESCOM) 0-07 Dimitris oumpakaris is with the Department of Eectrica & Computer Engineering, University of Patras, 6500, Rio Achaias, Greece emai: dtouba@upatrasgr) he work of D oumpakaris was supported by the European Union European Socia Fund - ESF) and Greek nationa funds through the Operationa Program Education and Lifeong Learning of the Nationa Strategic Reference Framework through the Research Funding Program haes - Investing in knowedge society through the European Socia Fund Syed A Jafar is with the Department of Eectrica Engineering and Computer Sciences, University of Caifornia, Irvine, CA, 9697 USA e-mai: syed@uciedu) he work of Syed Jafar was supported in part by NSF grants CCF-904 and CCF-75 Copyright c) 04 IEEE Persona use of this materia is permitted However, permission to use this materia for any other purposes must be obtained from the IEEE by sending a request to pubs-permissions@ieeeorg cance the effect of interference by projecting the received signa onto the orthogona subspace of the subspace containing the interference he concept of Degrees of Freedom DoF) is often empoyed to characterize the performance of variants of IA in the high SNR regime, 4 It has been shown that, for severa scenarios, IA attains the optima DoF Severa variants of IA exist, depending on the amount of channe knowedge that is avaiabe at the transmitter, the scenario over which IA is appied, and the channe statistics An overview of IA is given in 5 An important assumption of the first IA schemes that were proposed was perfect Channe State Information is avaiabe at the ransmitter CSI) his requirement is often chaenging or even impossibe to satisfy in a reaistic impementation 6 Recenty, a technique caed Bind Interference Aignment BIA) was proposed for the Mutiuser Mutipe-Input Singe-Output MU-MISO) Broadcast Channe that achieves a growth in DoF compared with orthogona techniques such as DMA or FDMA, 7, 8 As demonstrated in 7, if the transmitter is equipped with N t antennas that serve K tot singe-antenna users, the sum DoF that is achieved by BIA is N tk tot N t+k tot, which is aso the outer bound for this setting 9 As wi be discussed in more detai in Section II, the BIA scheme of 7 requires that the channe not change during one supersymbo herefore, coherence time or bandwidth is important when determining whether BIA can be used his motivates the search for BIA schemes that require short supersymbos Moreover, each user needs to be equipped with a reconfigurabe antenna whose function is to switch its radiation pattern among a set of preset modes 0 Athough this adds compexity to the receiver, there has been active interest and recent progress in the area, which makes it ikey that such receivers may be affordabe in the future he BIA scheme of 7, which wi be caed standard BIA sbia) from now on, was devised for one mutipe-antenna base station BS) Ceary, it is of interest to investigate how the scheme can be appied to ceuar systems and what the achievabe rates are he performance of sbia in ceuar and custer systems was anayzed in It was shown that the rates of the users ocated at the ce edge can be poor because of interce interference In, ways to appy sbia to ceuar scenarios such as Frequency Reuse FR) were proposed and were compared to Linear Zero Forcing Beamforming LZFB) taking into account the cost of CSI One interesting observation in and was that, if the BIA codes of the BSs of neighboring ces are synchronized, interce interference can be reduced consideraby In addition to coordination among the BSs, the authors in 4 derive a scheme that reies

SUBMIED O IEEE RANSACIONS ON SIGNAL PROCESSING, MANUSCRIP, OCOBER 04 on data sharing when transmitting data to ce-edge users his way, interce interference is competey canceed Athough this scheme improves significanty the rates of ce-edge users at ow SNRs, there is a oss in DoF because of the identica data that are sent by a BSs that transmit to ce-edge users In order to maximize the achievabe DoF over the entire ceuar system when using BIA, a straightforward approach woud be to appy a fuy coordinated BIA scheme cbia) among a BSs in the system Assuming N BS BSs each with N t antennas, the tota number of antennas is equa to M N BS N t If K tot is the number of active users in the MK tot M+K tot entire network, DoF coud potentiay be achieved Ceary, such an approach woud be costy in practice because of the need for a BSs to share data of a users Moreover, because of the arge number of antennas and users, the ength of the supersymbo coud be arge, meaning that arge channe coherence time or bandwidth woud aso be necessary Last but not east, fu connectivity woud be required in the system, which is generay not true in severa practica scenarios Due to the partia connectivity 5, ony signas of a sma number of BSs can be decoded at each user Users at the ce edge can receive data with an acceptabe Signa-to-Noise ratio In contrast, for users ocated near a BS, the signas from other BSs are weaker and their decoding is strongy handicapped by the noise power At first sight, it may appear that partia connectivity eads to a oss in DoF Interestingy, this is not the case A major objective of this paper is to demonstrate that, owing to the partia connectivity, use of BIA can actuay ead to more DoF than if the system were fuy connected In retrospect, this is not surprising he same way that arge path oss can hep increase spectra efficiency by aowing frequency reuse, partia connectivity aows simutaneous transmission of more data streams compared to a fuy connected network As an exampe, in 6 it is shown that, in a K-user interference channe, there exist scenarios where treating interference as noise achieves a points in the capacity region up to a constant gap, namey it is DoF-optima In this paper, a network BIA nbia) scheme is proposed for partiay connected ceuar networks he scheme differentiates between private users near the BSs who treat interce interference as noise and shared users at the ce edge who are connected to a BSs in their proximity Unike 4, the BSs do not share data Instead, each BS handes the transmission of part of the overa data stream For the symmetric scenario, where the number of private users, K p, in each ce is the same, it is shown that the proposed scheme is DoF-optima Moreover, as wi be shown, owing to the partiay connected topoogy, fewer reconfigurabe modes are needed for the private users Finay, the nbia supersymbo is shorter than cbia his reaxes the requirements for the coherence time or bandwidth, and renders the scheme attractive for practica impementation he remainder of this paper is organized as foows In Section II the system mode is presented Section III introduces a toy exampe to provide an overview of cbia and, at the same time, motivates our work Section IV presents the network BIA nbia) scheme for a symmetric ceuar network with Fig Ceuar system with partia connectivity and N BS BSs Each BS is equipped with N t,n antennas and serves K p,n private users as we as K sh shared users together with the other BSs partia connectivity In Section V, we provide an outer bound for the sum-dof in a partiay connected network From this outer bound, we show that nbia is DoF-optima for symmetric scenarios An extension of the nbia scheme for asymmetric user distributions is presented in Section VI In Section VII cosed-form expressions are derived for the rates achieved by nbia Section VIII shows severa simuation resuts where the performance of nbia is compared to other BIA schemes Finay, Section IX provides concuding remarks II SYSEM MODEL We consider a set of N BS Base Stations BSs) N {,,, N BS } that want to send a set of messages to K tot users in a partiay connected ceuar network as shown in Fig Each BS n, n N, has N t,n transmit antennas and wishes to send data to a set of private users K p,n {p,n,, p Kp,n,n} as we as a set of shared users K sh {sh,, sh Ksh } ocated on the edge of a N BS ces Each private user is equipped with one reconfigurabe antenna that can switch among N t,n preset modes, whereas each shared user can switch among M N BS n N t,n modes herefore, if m pk,n i denotes the antenna mode of private user p k,n of BS n at time i, the signa received by the user at time i can be written as y p k,n i h p k,n m p k,n i) xi + z p k,n i, ) where z pk,n i CN 0, ) is additive white Gaussian noise AWGN), xi x i x i x NBS i C M, ) In practice, in a network with user mobiity, each user shoud be abe to switch among M preset modes, since it may transition from being private to being shared and vice versa

MORALES-CESPEDES et a: BLIND INERFERENCE ALIGNMEN FOR CELLULAR NEWORKS and h pk,n h p k,n, h p k,n,n h p k,n,n BS 0 a, h p k,n,n 0 b, C M, ) with a n n N t,n, b N BS n n+ N t,n and 0 c, is a vector of zeros of dimension c In ), x n i C Nt,n is the signa sent by BS n at time i, whereas in ), h pk,n,n m) h p k,n,n m) h p k,n,n N t,n m) C N t,n contains the channe coefficients between the N t,n antennas of BS n and the singe antenna of private user p k,n when its radiation pattern is set to mode m {,,, N t,n } As can be seen in ), we mode the situation where the K p,n K p,n private users of ce n are cose to BS n, and assume that signas received from any other BS n n are negigibe hus, no data sharing among the BSs is required to serve the private users, and x n i does not contain data intended to any private user p k,n K p,n, n n Simiar to the mode for the private users, the signa received by shared user sh k at time i can be written as y sh k i h sh k m sh k i) xi + z sh k i, 4) where, xi is as defined in ) and h sh k h sh k, h sh k,n BS C M, 5) with M N BS n N t,n and h sh k,n m) C Nt,n denoting the channe between the N t,n antennas of BS n and shared user sh k for mode m We use index k instead of k to distinguish from private users It is assumed that shared users can receive signas from a BSs because of their ocation in the network As a resut, the task of sending data to the shared users can be jointy undertaken by the N BS BSs We aso assume that the channe input is subject to an average power constraint E { x n i } P for a i and n N Furthermore, the channes between each user, whether private or shared, and the BSs are considered to be drawn from a continuous distribution and, therefore, are ineary independent amost surey We aso assume that the switching pattern functions m pk,n i and m sh k i are predetermined and are known to everyone in the system On the contrary, we assume that the transmitters do not have any channe state information CSI) Moreover, we assume that the physica channes stay constant across a sufficient number of time or frequency sots For simpicity, we focus on the tempora dimension, without oss of generaity Hence, from now on each symbo extension i corresponds to a time sot he appication of the scheme aong frequency sots is straightforward III FULLY COOPERAIVE BLIND INERFERENCE A A fuy cooperative scheme ALIGNMEN he sbia scheme can be extended to a ceuar system in a straightforward way by creating a fuy cooperative BIA cbia) scheme where, as in a network MIMO system, fu connectivity and fu data sharing among a BSs is assumed If M N BS n N t,n antennas transmit to a K tot users, which can switch among M reconfigurabe modes, foowing the scheme in 7 a supersymbo that contains M ) Ktot aignment bocks per user, each providing M DoF to the user, can be buit over M ) Ktot + K tot M ) Ktot symbo extensions A generic cbia supersymbo is shown in Fig In the supersymbo, user k switches among a M preset modes during each aignment bock, whie the channes h k m) of a other users, k k, remain in a specific preset mode For exampe, in Fig the first aignment bock of user is composed by the first M ) symbo extensions of Bock pus symbo extension M ) Ktot + the first symbo extension of Bock ) herefore, if we ignore the noise, a typica signa vector Y k y k y k M received by user k in a given aignment bock is given by h k ) h k ) u k k k Y k u k + h k M ) h k M ) u k, } h k M) {{ } H k k k } 0 {{ } interference 6) where H k C M M is a fu-rank matrix, u k C M and, for simpicity, the tempora index refers to the position of the symbo extension in the aignment bock instead of its position in the supersymbo In the BIA scheme of 7, the interference term in 6) can be removed by measuring it in appropriate sots of Bock hen as ong as the {h k m)} M m are ineary independent, the M data streams u k can be decoded by inverting the resuting inear system Ỹk H k u k, where Ỹ k is the received signa after interference subtraction Since each of the K tot users achieves M DoF in each of its M ) Ktot aignment bocks, which are distributed over a supersymbo of M ) Ktot + K tot M ) Ktot symbo extensions, the sum DoF per symbo extension of cbia is DoF cbia MK tot M + K tot, 7) where K tot N BS n K p,n +K sh For the symmetric scenario for which N t,n N t and K p,n K p for a n, 7) reduces to DoF cbia,symm N BSN t N BS K p + K sh ) N BS N t + N BS K p + K sh 8) For iustrative purposes, we consider the toy exampe shown in Fig, where each BS is equipped with N t antennas Each ce contains K p private user, whereas K sh shared user is ocated in the inter-ce area Hence, the system has a tota of K tot K p + K sh users For this setting, cbia achieves DoF per symbo extension by empoying a supersymbo comprising 54 symbo extensions B Moving to partiay connected networks he cbia scheme reies on the assumption of fu connectivity, which does not hod in a typica ceuar system By reinstating the assumption that private users ony receive signas

4 SUBMIED O IEEE RANSACIONS ON SIGNAL PROCESSING, MANUSCRIP, OCOBER 04 M! ) M! ) K tot! h) h) hm-) h) hm-) h) hm-) h) hm-) hm) hm) h) hm-) h) hm-) h) h) h) h) hm-) h) hm-) h) hm-) h) hm-) hm) hm) h) hm-)! Ktot h) h) h) h) h) h) h) hm-) hm-) h) hm-) h) hm-) hm) hm) M! ) K tot K tot M! ) K tot! Fig Supersymbo for the cbia scheme appied to M N BS n Nt,n antennas serving Ktot users 4 5 6 7 p,/p, sh h) h) h) h) h) h) h) h) h) h) h) h) h) h4) Fig 4 Supersymbo of the nbia scheme for the toy exampe N t, K sh, and K p, K p, Fig oy exampe: downink scenario with fu connectivity he BSs are equipped with N t antennas each, and serve K tot users from their respective BS, the DoF in 7) are no onger achievabe if cbia is appied In a scenario with partia connectivity, as the one shown in Fig, the channe between the BSs and private user p k,n at ce n can be approximated as shown in ), ie, h pk,n m) 0 a, h pk,n,n m) 0 b, Consequenty, the channe matrix H p k,n h p k,n ) h p k,n M) corresponding to private user p k,n is no onger fu-rank Because of this, in 6) private user p k,n cannot decode the data streams sent by BSs n, n n herefore, even if fu data sharing is aowed between the BSs, cbia fais to achieve the DoF given by 7) because of the ack of fu connectivity wo questions that arise naturay are whether it is possibe to devise a scheme that works in a scenario with partia connectivity and M transmitters and what the achievabe DoF are In this paper we propose a network BIA scheme nbia) that not ony aows the appication of BIA to partiay connected networks, but aso attains more DoF than 7) o achieve this, we everage the partia connectivity as a resource that aows to obtain more DoF, and decrease the number of modes used by the private users as we as the ength of the supersymbo, which is one of the major imitations when appying cbia in practica systems IV A NEWORK BLIND INERFERENCE ALIGNMEN SCHEME FOR PARIALLY CONNECED CELLULAR NEWORKS In this section, we present the network BIA nbia) scheme for partiay connected networks We first describe the key idea of nbia using the toy exampe of Section III hen, for the sake of an easy exposition we describe nbia for the symmetric scenario with N BS BSs, each equipped with N t antennas serving K p private users and K sh shared users 9) A he key to Bind Interference Aignment in ceuar systems Consider again the toy exampe of Fig his time, as shown in the figure, partia connectivity is assumed he shared user receive data from both BS On the other hand, each private user, ie p, and p,, can ony be served by its corresponding BS, BS and BS, respectivey hus, user p k,n does not decode the data sent by any other BS n, n n As a positive counterpart of this ack of connectivity, private users of a given BS are not subject to interference by any other BS, at east in theory Since cbia does not take into account the ack of fu connectivity, it does not achieve 8) For the toy exampe, cbia achieves 4 DoF when there is partia connectivity, which is ess than the DoF attained in a fuy connected system As an aternative to cbia, consider the supersymbo of Fig 4 and the beamforming matrices I 0 0 0 I 0 0 0 I X I 0 0 0 I 0 0 0 I 0 0 0 p, u p, u p, u p, where xi C 4, u sh and u p, u p, I 0 0 0 I 0 0 0 I + I 0 0 0 I 0 0 0 I 0 0 0 p, u p,, 0, u p,, u sh, u p, u p, u p, I I I + 0 0 0 I sh u sh,, 0, u sh, he vectors u p,n,n 0) C and u sh,n C contain the symbos transmitted by BS n to p,n and sh, respectivey, and I and 0 are the 4 4 identity and zero matrix, respectivey Let us first focus on the transmission of data to shared user sh As is expained in 7, since sh is served by both BSs,

MORALES-CESPEDES et a: BLIND INERFERENCE ALIGNMEN FOR CELLULAR NEWORKS 5 to send M N t, + N t, 4 distinguishabe data streams, the BSs need to transmit u sh repetitivey during 4 symbo extensions over which the antenna of sh switches through M 4 different modes At the same time, the beams need to be aigned into one dimension at the users that are subject to interference by the signa sent to sh herefore, during these symbo extensions, p, and p, maintain the same mode For instance, by ooking at the supersymbo of Fig 4 and the beamforming matrix of 0), we can check that symbo extensions,, and 7 constitute an aignment bock for sh If we ignore the noise, the signa received at user sh is y sh h sh ) x y sh y sh h sh ) x h sh ) x ) y sh 7 h sh 4) x7 ) h sh ) h sh ) u p, + u p, ) h sh ) h sh ) u sh h + sh ) u p, + u p, ) h sh 4) h sh ) u p, + u p, 0 H sh interference Since the channes h sh m), m {,,, 4} are generic, once the second term associated with the interference has been removed, user sh can decode the 4 data streams u sh Now, if we consider the signa received at the private users y p,n h p,n,n ) x n + 0,x n y p,n y p,n h p,n,n ) x n + 0,x n h p,n,n ) x n + 0,x n y p,n 7 h p,n,n ) x n 7 + 0,x n 7 h p,n,n ) u p,n,n h p,n,n ) ) h p,n,n ) u p,n,n h p,n,n ) u p,n,n + h p,n,n ) h p,n,n ) 0 h p,n,n ) desired signas interference u sh,n, n {, }, n n, we can observe that the four transmissions of u sh u sh, u sh, are aigned into one dimension at the private users his way, since during symbo extension 7 BS n ony transmits u sh,n, by appying zero forcing based on y p,n 7, p,n can subtract the interference during symbo extensions, and Next, we concentrate on the transmission to private user p, Unike shared user sh, p, can ony be served by BS o send N t, distinguishabe symbos, u p,,, to user p, in the absence of CSI, BS repeatedy transmitts u p,, during symbo extensions over which the antenna of p, switches through N t, modes Moreover, to aign the two transmissions of u p,, into one dimension at the users subject to interference because of the transmission to user p,, the affected users shoud keep the same radiation pattern However, due to the partia connectivity of the network, sh is now the ony user subject to interference herefore, the radiation pattern of its antenna is the ony one that has to be kept constant to project the interference caused by the transmissions of u p,, into one dimension From the supersymbo of Fig 4, we can easiy check that the pairs of symbo extensions {, 4}, {, 5} and {, 6} satisfy a the previous conditions Each of these pairs constitutes an aignment bock for private user p, For instance, consider the aignment bock formed by symbo extensions {, 4} Ignoring the noise, the signa received by the private user p, is y p, h p,, ) y p, x + 0,x 4 h p,, ) x 4 + 0,x 4 h p,, ) h p,, ) u p,, h p,, ) + u sh, ) 0 H p,, interference Private user p, appies zero forcing based on y p, 7 to remove the interference at time instants, and see )) Consequenty, due to the fact that the channes h p,, m), m {, }, are generic, the symbos in u p,, can be decoded he same procedure can be foowed to decode the data steams u p,, and u p,, transmitted repetitivey over the pairs of symbo extensions {, 5} and {, 6}, respectivey Reca that, as can be seen from ), the transmission of data to private users of a specific ce does not cause interference to private users of other ces Consequenty, p, can reuse the same radiation pattern and the same beamforming matrix as p,, as can aso be verified from 0) and Fig 4 his way, each pair of symbo extensions {, 4}, {, 5} and {, 6} aso constitutes an aignment bock of p, Moreover, note that in ) the interference associated with the repeated transmissions of u p,, by BS aong the - th aignment bock of p, is aigned into the same singe dimension as the transmissions of u p,, by BS aong the -th aignment bock of p, Hence, in ) the interference term associated with the transmission of u p,, and u p,, can be removed if user sh appies zero forcing based on the signas received during the time sot over which it is not receiving data For instance, if sh appies zero ) forcing based on y sh 4 h sh ) u p, + u p,, it can remove a interference during symbo extension Simiary, sh can remove the interference during symbo extensions and by appying zero forcing based on y sh 5 and y sh 6, respectivey In summary, using a reconfigurabe antenna with N t modes, each private user achieves 6 DoF, DoF per aignment bock At the same time, using a singe antenna with 4 modes, shared user sh achieves 4 DoF over ony one aignment bock herefore, a tota of 6 DoF are achieved aong 7 symbo extensions, which yieds 6/7 DoF per symbo extension Note that the new scheme improves upon the DoF per symbo extension achieved by cbia in a network where there is fu connectivity and where a BSs share data intended to every user of the system Furthermore, the improvement is achieved using a supersymbo of 7 instead of 54 symbo extensions o concude, we note that the key of nbia ies on the generaization of the definition of an aignment bock to a communication system with partia connectivity If a user k

6 SUBMIED O IEEE RANSACIONS ON SIGNAL PROCESSING, MANUSCRIP, OCOBER 04 N t! ) M! ) K sh group group group ength M! ) K sh User: sh k! h) h) h) h) h) h) hm-) hm-) hm-) M! ) k" p, n!h)"!h)"!hnt -) "!h)"!hnt -) "!h)"!hnt -) "! # # # # pkp, n!h)"!h)"!h)"!h)"!h)"!hnt -) "!hnt -) " Shared users Bock Ksh Bock Ksh Bock Ksh Bock Ksh Bock Ksh Bock Ksh Bock Ksh sub-bock M! ) k "! M! ) k "! M! ) k "! M! a) Buiding bock of shared user sh k M! ) K sh N t! ) K p a) S-Bock of the nbia scheme for a symmetric ceuar scenario with partia connectivity Bock K sh is shown in Fig 6 M! ) sh h) h) hm-) h) h) hm-) h) h) hm-) User: p k,n sub-bock group h) h) h) h) h) h) hnt -) hnt -) hnt -) b sh N t! ) k b sh N t! ) k! group group b sh N t! ) k! b sh N t! ) k! N t! b) Buiding bock of private user p k,n ength!!!!! shksh h) h) h) h) h) h) hm-) hm-) hm-) Fig 6 Buiding bocks of the private and the shared users M! ) K sh b) Bock of the cbia scheme for transmission to the K sh shared users Fig 5 S-Bock of the nbia scheme can be served by N k transmit antennas, then an aignment bock for this user consists of N k symbo extensions over which it can receive N k distinguishabe data streams At the same time, these beams are ony aigned into one dimension at a users subject to interference On one hand, to decode N k distinguishabe data streams, the channe state of user k has to switch through N k different modes, one per symbo extension of the aignment bock As was seen in the toy exampe and as is described in 7, to aign the aforementioned beams into one dimension at a users subject to interference, their channe state has to be maintained constant over the N k symbo extensions that form the aignment bock of the desired user k he data streams intended to a specific user need ony be aigned into one dimension at those users where the power of the interference created by the aforementioned data streams is high enough, and therefore cannot be treated as noise B he network BIA scheme We now describe the nbia scheme for the genera symmetric scenario of a partiay connected network First, an sbia scheme is impemented by each BS to send data to its set of K p private users As shown in 7, this strategy aows each private user to remove interference caused by transmission to a other private users in its ce he sbia scheme is reused by a N BS BSs owing to the partia connectivity Furthermore, a BSs of the system jointy impement a cbia scheme to send data to the K sh shared users of the system and to et them cance the interference among them Finay, to obtain the supersymbo shown in Fig 5a), the two schemes are combined appropriatey in order to remove the interference that the transmission of data to private users causes to the shared users and vice-versa ) Design of S-Bock of nbia: We first consider the design of Bock of the supersymbo of the nbia scheme, which wi be denoted as Super-Bock S-Bock ) It comprises M ) K sh N t ) Kp symbo extensions As shown in Fig 5a), the symbo extensions of the shared users are formed concatenating N t ) Kp Bocks of a cbia scheme for K sh users see Fig 5b)) As potted in Fig 6a), the buiding bock of sh k is formed by M sub-bocks comprising M ) k symbo extensions During the m- th sub-bock, m {,,, M }, the receiver of sh k maintains the m-th reconfigurabe mode Hence, the tempora correation function of sh k in the entire S-Bock is f shk i) h sh k m) if mod i, M ) k ) I sh m), with i {,,, M ) K sh N t ) Kp }, and { I sh m) m )M ) k +,, m M ) k, mod m M ) k, M ) k )} 4) As can be seen in Fig 5a), Bock- of the private users is cosey based on Bock of a cbia scheme aimed at transmitting to K p users using N t antennas he mode of p k,n is periodic with the buiding bock shown in Fig 6b), which is repeated N t ) Kp k times to form S-Bock he buiding bock is now composed of N t sub-bocks, each with ength b sh N t ) k, where b sh M ) K sh As in the sub-bocks associated with sh k, the m-th mode is used in the m-th subbock, m {,, N t } his way, during each Bock of Fig 5b), each private user maintains a fixed mode Hence, the tempora correation function for private user p k,n for any ce n {,,, N BS } is f pk,n i) h pk,n m) if mod i, b sh N t ) k) I p m) with i {,,, M ) K sh N t ) Kp } and I p m) { m ) b sh N t ) k +,, m b sh N t ) k, mod m b sh N t ) k, b sh N t ) k)} 5) For instance, in a two-ce scenario where N t, K p and K sh S-Bock has the form shown in Fig 7 ) ransmission strategy and beamforming matrices for S- Bock : he key for the design of the beamforming matrices

MORALES-CESPEDES et a: BLIND INERFERENCE ALIGNMEN FOR CELLULAR NEWORKS 7 4 5 6 7 8 9 0 p, n h) h) h) h) h) h) h) h) h) h) sh h) h) h) h4) h5) h) h) h) h4) h5) Fig 7 Structure of S-Bock when K p and K sh in a two-ce scenario where each BS is equipped with N t antennas 4 5 6 7 is to create aignment bocks that take into account the partia connectivity p, ofn the h) network h) h6) h6) Each h6) h6) aignment h6) bock of a shared or private user corresponds to one bock coumn in the sh h6) h6) h) h) h) h4) h5) corresponding beamforming matrix Since each shared user sh k is served by a BSs, ie M antennas, each bock coumn of its beamforming matrix is obtained by pacing an M M identity matrix, I M, at the rows corresponding to the symbo extensions of the aignment bock he remaining bocks are fied with M M zero matrices, 0 M o obtain the signas transmitted from the BSs to shared user sh k, the beamforming matrix is mutipied by u sh k u sh k, sh u k,n sh u k,n BS where u sh k C N BSN t,,,, M ) Ksh N t ) Kp and u sh k,n contains the N t symbos transmitted from BS n to sh k during aignment bock he same procedure is appied to obtain the beamfroming matrix for each private user at any ce n However, p k,n is ony served by the N t antennas of BS n Reca that the signas x n i transmitted by BSs n do not contain data intended to any private user p k,n, n n herefore, each bock coumn of the beamforming matrix is formed as for the shared users However, to obtain the signas transmitted from the BSs to p k,n the corresponding beamforming matrix is mutipied by u p k,n 0 Ntn ), u p k,n,n 0 NtN BS ), and,,, N t ) Kp M ) K sh o maintain the data beams of one aignment bock distinguishabe at the user for which they are intended, the channe between the transmit antennas and the user shoud change at each symbo extension of each aignment bock Moreover, during these symbo extensions, each of the affected users shoud maintain a constant channe so that interference be aigned As is shown in Sections IV-B and IV-B4, in S-Bock both decodabiity and interference aignment requirements can be satisfied by using groups Each group consists of the first M or N t symbo extensions of the aignment bock of a shared or private user, respectivey In particuar, we can group the -th symbo extension in each one of the M sub-bocks within one buiding bock as shown in Fig 6a) for shared user sh k Since each sub-bock consists of M ) k symbo extensions, a tota of M ) k groups can be buit within one buiding bock As was mentioned above, each of these groups wi be associated with a specific aignment bock of sh k Simiary, as shown in Fig 6b), for private user p k,n, the -th symbo extension in each of the N t sub-bocks of one buiding bock can be grouped Since each sub-bock of p k,n is now composed of b sh N t ) k symbo extensions, a tota of b sh N t ) k groups can be formed within one buiding bock Recaing that S-Bock of sh k consists of N t ) Kp M ) K sh k buiding bocks of M ) k symbo extensions, the -th group in the p -th buiding bock of sh k comprises symbo extensions {p )M ) k + κm ) k + } M κ0 6) where {,,, M ) k } and p {,,, M ) K sh k N t ) Kp } Anaogousy, taking into account that S-Bock of p k,n, n {,,, N BS } is formed by N t ) Kp k buiding bocks of b sh N t ) k symbo extensions, the -th group in its p-th buiding bock consists of symbo extensions {p ) b sh N t ) k + κ b sh N t ) k + } Nt κ0, 7) where p {,,, N t ) Kp k }, and {,, b sh N t ) k } For instance, particuarizing to our iustrative scenario with ces, N t, K p and K sh, during S-Bock I 6 0 6 I 6 0 6 I 6 0 6 X I0 I 0 u p, u p, u p, u p, 4 u p, 5 } {{ } to p, I0 + I 0 u p, u p, u p, u p, 4 u p, 5 } {{ } to p, I 6 0 6 + I 6 0 6 0 6 I 6 0 6 I 6 0 6 I 6 0 6 I 6 0 6 I 6 to sh u sh u sh, 8) where u p, u p,, p 0,, u, 0, u p,, sh and u u sh, u sh, Each private user has 5 groups formed by symbo extensions Specificay, for both private users these groups are formed by the pairs of symbo extensions {, 6}, {, 7}, {, 8}, {4, 9} and {5, 0} On the contrary, shared user sh has two groups, each composed of 5 symbo extensions, ie {,,, 4, 5} and {6, 7, 8, 9, 0} ) Achieving decodabiity and interference aignment at the shared users: First, reca that the channe switching pattern for each shared user is created by concatenating N t ) Kp identica Bocks associated with a cbia scheme aimed at transmitting data to K sh users his way, based on the resuts in 7, it is straightforward to show that each group of each user sh k is formed by M symbo extensions over which the mode of its antenna changes whie the mode of a other shared users remains constant Consequenty, the data sent by a BSs to each user sh k over each of its aignment bocks can be decoded and the interference induced to the other shared users is aigned into one dimension of their signa space Note that the private users are aso subject to interference

8 SUBMIED O IEEE RANSACIONS ON SIGNAL PROCESSING, MANUSCRIP, OCOBER 04 User: sh Ksh User: p,n User: p k,n User: p k +,n h) h) hm-) h) h) hm-) h) h) hm-) M! ) K sh! h) h) h) h) h) h) hnt -) hnt -) hnt -) b sh M! ) K sh M! ) K sh! a) he buiding bocks of shared users sh Ksh and p,n b sh N t! ) k! b sh h) h) hnt -) h) h) hnt -) h) h) hnt -) b sh N t! ) k! h) h) h) h) h) h) hnt -) hnt -) hnt -) b sh N t! ) k b sh N t! ) k b sh N t! ) k b) he buiding bocks of private users p k,n and p k+,n Fig 8 Buiding bocks of the private and shared users because of the data sent by the BSs to the shared users o aso aign this interference, the M data streams sent to a shared user over one of its groups aso need to be contained into one dimension at a private users As is aso shown in Figs 5a) and 8a), the channe mode of a private users does not change during an entire Bock of shared users Moreover, reca that each group of shared user sh k is composed of symbos within a specific buiding bock, which beongs to one of the Bocks of sh k Hence, within each group of any shared user sh k the reconfigurabe modes of the antennas of a private users remain the same In concusion, the interference caused by transmission to sh k during each one of its groups is aigned into one dimension at a private users 4) Achieving decodabiity and interference aignment at the private users: We now concentrate on the private users First, we check that the channe state of each private user changes at each symbo extension within any of its groups Note that 7) specifies the symbo extensions of the -th group in the p-th buiding bock of private user p k,n Now, it can be easiy seen that, for a p {,,, N t ) Kp k }, the moduus of these symbo extensions with b sh N t ) k yieds {κ b sh N t ) k + } Nt κ0 9) with {,,, b sh N t ) k } Hence, from 5), the channe states of p k,n are h pk,n ), h pk,n ),, h pk,n N t ) during the symbo extensions that form each one of its groups Next, we focus on proving that the interference caused by the transmission to private user p k,n is aigned into one dimension at the signa space of the other private users in ce n First, consider private users {p j,n } k j Note that the remainder of the division of the symbo extensions in 7) by b sh N t ) j is the same, ie mod, b sh N t ) j ), for a specific group in the p-th buiding bock of p k,n and any j {,,, k } Hence, from 5), within each group of p k,n, the channe state of a other private users {p j,n } k j remains constant Now, consider private users {p j,n } Kp jk+ Notice that the engths of the sub-bocks of the private users in S-Bock are arger than b sh N t ) k, ie the ength of a buiding bock associated with private user p k,n Hence, since b sh the boundaries of the buiding bocks of p k,n are aigned with those of the sub-bocks of p j,n, j {k+, k+,, K p } see Fig 8b)), the channes of this ast sub-group of private users are the same within each group of p k,n herefore, from the structure of S-Bock we can concude that the data streams transmitted over the N t symbo extensions of the -th group of user p k,n are aigned into one dimension at a other private users of ce n Utimatey, we show that interference caused by transmission to user p k,n is aso aigned at the private users of the other ces n, n n as we as at the K sh shared users Due to partia connectivity, we ony need to verify that for each group of users p k,n the channe state of a shared users remains constant Consider any shared user sh k and the symbo extensions in 7), which form the -th group in the p-th buiding bock of p k,n Since b sh is an integer mutipe of M ) k, the remainders of the indices of the symbo extensions in 7) divided by M ) k are the same, ie mod, M ) k ) Consequenty, from 4), within each group of p k,n, the channe state of any user sh k is constant Hence, the requirements of decodabiity and aignment are satisfied in each group of each private user As expained previousy, the transmission of data from BS n, n n, to its private users does not impose any constraints on the design of the channe pattern and the beamforming of private user p k,n hus, private users {p k,n } N BS n can reuse the same beamforming matrix and the same channe pattern in S- Bock when receiving data from their corresponding BSs his can be seen in our iustrative scenario in 8) and Fig 7 More generay, the same fact can be verified in Fig 5a) and in 7) where the symbo extensions of the groups associated with private users {p k,n } N BS n are the same As a resut, not ony are the N t data beams transmitted within each group of one private user p k,n aigned into one dimension at each shared user, but aso a data beams transmitted to a private users {p k,n } N BS n within each group specified in 7) are projected into the same singe dimension at each shared user 5) Design of S-Bock : From the design of S-Bock- and the corresponding beamforming matrices, we can undertake the design of the switching pattern of a users during Bock of the nbia scheme, which wi be caed Super-Bock S-Bock ) he purpose of S-Bock is to compete the aignment bocks of a users so that each user can decode the data received aong its groups and cance the interference caused by the transmission of data to other users during S-Bock From 6) notice that the number of aignment bocks associated with each shared user is equa to M ) Ksh N t ) Kp Consequenty, to compete the aignment bocks of the K sh shared users, a tota of t sh K sh M ) K sh N t ) Kp 0) symbo extensions are needed in S-Bock As shown in Fig 9, these symbo extensions are L S B +, L S B +,, L S B + t sh, where L S B M ) K sh N t ) Kp is the ength of S-Bock Within the aforementioned symbo extensions, sub-bock {L S B +k ) t sh /K sh + } t sh/k sh )

MORALES-CESPEDES et a: BLIND INERFERENCE ALIGNMEN FOR CELLULAR NEWORKS 9 M! ) K sh! N t! ) K p M! ) K sh! N t! ) K p N t! ) K p! M! ) K sh N t! ) K p! M! ) K sh p, n h) h) hnt -) h) h) hnt -) hnt ) hnt ) hnt ) h) h) hnt -)! pkp, n h) h) hnt -) h) h) hnt -) h) h) hnt -) hnt ) hnt ) hnt ) sh hm) hm) hm) h) h) hm-) h) h) hm-) h) h) hm-)! shksh h) h) hm-) hm) hm) hm) h) h) hm-) h) h) hm-) K sh M! ) K sh! N t! ) K p ) K p! M! ) K sh K p N t! Fig 9 S-Bock of the nbia supersymbo with k {,,, K sh }, provides the ast symbo extensions of the aignment bocks of sh k In particuar, each symbo extension specified in ) constitutes the ast eement of the -th aignment bock of sh k Hence, in order to be abe to decode the signas of interest over the aignment bock, user sh k empoys the M-th preset mode during each symbo extension in ) his way, if the N BS BSs repetitivey transmit u sh k C M within each symbo extension of the -th aignment bock of sh k, the user can decode u sh k after removing the interference Since the interference caused by the first M transmissions of u sh k during the -th group of sh k in S-Bock is aigned into one dimension at a other shared and private users, zero forcing can be appied to remove it Due to the fact that ony u sh k, {,,, t sh K sh }, is transmitted during each symbo extension of ), any shared user sh j sh k and a private users p k,n can measure the interference caused by the transmission of u sh k herefore, they can subtract the interference received in S-Bock if, during the symbo extensions given in ), they maintain the same channe state as the one used during the -th aignment group of sh k From 6) notice that the symbo extensions that form the -th group of shared user sh k are {p sh, k ) M ) k + κm ) k + sh, k )} M κ0 ) where sh, k ) mod, M ) k ) + and p sh, k ) M ) k Consequenty, during the -th symbo extension specified in ) the channe state of shared users sh j sh k equas ) p sh, k ) M ) k + sh, k ), f shj whereas the channe state for a private users {p k,n } N BS ) f pk,n p sh, k ) M ) k + sh, k ), k is with f shj and f pk,n given in 4) and 5), respectivey Next, we consider the design of S-Bock for the private users As we have seen in 7), the number of aignment bocks per private user equas b sh N t ) Kp Due to the partia connectivity, BSs n and n can transmit simutaneousy the data associated with a specific aignment bock of p k,n and p k,n, respectivey, without interfering with each other hus, one symbo extension of S-Bock can be reused by private users {p k,n } N BS n to compete one of their aignment bocks hus, since there are K p private users per ce, a tota of t p K p M ) K sh N t ) Kp ) symbo extensions are needed in S-Bock for a private users In order not to create any interference, simiar to the symbo extensions of S-Bock for the shared users, each BS n ony transmits data to one specific user in its ce However, this time the BSs do not transmit data to a specific shared user Instead, each BS n ony transmits data to a specific private user p k,n during each of the t p symbo extensions As shown in Fig 9, the t p symbo extensions of S-Bock are L S B + t sh +, L S B + t sh +,, L S B + t sh + t p Within these symbo extensions, the sub-bock {L S B + t sh + k ) t p /K p + } tp/kp 4) k {,,, K p }, provides the ast symbo extensions of the aignment bocks of private users {p k,n } N BS n Hence, during each symbo extension in 4) the private users have to keep the N t -th preset mode his way, if each BS n appies a repetition code to send u p k,n C Nt during each symbo extension within the -th aignment bock of {p k,n } N BS n, each user p k,n at any ce n can use the signas received during its -th aignment bock to decode u p k,n Continuing the design of the symbo extensions of S-Bock, notice that the simutaneous transmission of {u p k,n } N BS n during the -th group of private users {p k,n } N BS n are aigned into the same singe dimension of the signa space of each shared user sh k Hence, to remove the interference caused by these transmissions, sh k can appy zero forcing based on the interference signa measured in S-Bock o do so, the preset mode of sh k during the -th symbo extension in 4) has to be equa to the mode of sh k during the -th aignment group of private users {p k,n } N BS n, which consists of symbo extensions {p b, k) b sh N t ) k + κ b sh N t ) k + p, k)} Nt κ0 5) with p, k) mod, b sh N t ) k ) + and p p, k) b sh N t ) Mathematicay, during the -th k

0 SUBMIED O IEEE RANSACIONS ON SIGNAL PROCESSING, MANUSCRIP, OCOBER 04 symbo extension in 4) the channe state of sh k equas f shk pp, k) b sh N t ) k + p, k) ), where f shk is given in 4) Due to the fact that the transmitted u p k,n during the symbo extensions of S-Bock are aigned into one dimension at the signa space of any private user p j,n p k,n, each private user of ce n can appy the same technique as shared user sh k to remove the interference caused by the transmission of u p k,n Specificay, at private user p j,n p k,n, the interference is removed by appying zero forcing based on the signa received during the -th symbo extension in 4) with the mode of its antenna equa to f pj,n pp, k) b sh N t ) k + p, k) ), where f pj,n is given in 5) Finay, due to the partia connectivity, the transmission of u p k,n to any private user p k,n at ce n does not interfere with the communication between BS n and any user p j,n at ce n n As a resut, users p j,n p k,n do not need to cance the interference caused by the transmission of data to p k,n during S-Bock C Achievabe Degrees of Freedom With the nbia scheme, each shared user achieves M DoF per aignment bock, whereas each private user attains N t DoF per aignment bock Since the tota number of aignment bocks of each shared user is equa to t sh /K sh M ) Ksh N t ) Kp in the supersymbo of the nbia scheme, a tota of M t sh /K sh DoF per supersymbo are achieved for each shared user Foowing a simiar reasoning and recaing that each private user empoys M ) K sh N t ) Kp aignment bocks per supersymbo, a tota N t t p /K p DoF are attained by each private user in a supersymbo hus, since the ength of the supersymbo equas L S B +L S B symbo extensions where L S B t sh + t p is the number of symbo extensions in S-Bock see Figs 5a), 5b) and 9), when nbia is used for the symmetric scenario the achievabe sum DoF per symbo extension are DoF nbia K sh M t sh t K sh + N BS K p N p t K p L S B + L S B M K sh N t ) + K p M ) M )N t ) + K sh N t ) + K p M ) 6) As wi be shown in Section V, the sum DoF per symbo extension achieved by nbia for the symmetric scenario is equa to the information-theoretic sum-dof outer bound Remark An aternative design of the supersymbo of the nbia scheme can aso be obtained As shown in Fig 0, a Bock associated with an sbia scheme aimed at transmitting data to K p users is repeated M ) K sh times to construct S-Bock for the K p private users of each ce n For the shared users, S-Bock is formed by augmenting the ength of the sub-bocks that form Bock of a cbia scheme for a system with K sh shared users and M transmit antennas his time, the ength of the sub-bock of a shared user equas N t ) Kp M ) k symbo extensions Simiary to S-Bock of Fig 9, the aternative design for S-Bock is obtained by competing the aignment bocks whose groups form S-Bock sh!h)"!h)"!hm-)"!h)"!hm-)"!h)"!hm-)"! # # # # shksh M! ) N t! ) K p!h)"!h)"!h)"!h)"!h)"!hm-)"!hm-)" Private users Bock Kp Bock Kp Bock Kp Bock Kp Bock Kp Bock Kp Bock Kp ) K p M! ) K sh N t! Fig 0 Aternative design of S-Bock of the nbia supersymbo It can be easiy verified that the same DoF as in 6) can aso be achieved by the aternative structure of the supersymbo V INFORMAION-HEOREIC SUM-DOF OUER BOUND OF HE CELLULAR SCENARIO WIH PARIAL CONNECIVIY In this section we derive an outer bound for the sum DoF he bound appies to the genera partiay connected network of Fig, where the number of private users in each ce may be different he proof is deveoped aong the ines of 7 In the symmetric case where the number of private users is the same in a ces, this bound is the same as the DoF that are achieved by the proposed nbia scheme of Section IV; therefore, the scheme is DoF-optima For simpicity, the twoce scenario is considered However, the proof can be easiy extended to the case of N BS BSs Consider two BSs equipped with N t, and N t, antennas, which transmit to K p, and K p, private users, respectivey, whie K sh shared users are served simutaneousy by both BSs he messages and the rates of the users in ce n are denoted as W p,,n, W p,,n,, W p Kp,,n and R p,n, R p,n,, R p Kp,n, respectivey; the messages and the rates of the shared users are denoted as W sh, W sh,, W sh K sh and R sh, R sh,, R sh K sh, respectivey Accordingy, we express the sum rate as R Σ R Σp + R Σp + R Σsh Kp,n n k Rsh k We aso define the k Rpk,n + K sh message sets W pn {W } p,,n,, W p Kp,,n with n {, }, and W sh { W sh,, W sh K sh } Consider private user p, in ce, who desires message W p,, In particuar, consider N t, random reaizations of this user, each corresponding to a different reaization of the channe Because there is no CSI, and we require reiabe decoding probabiity of error approaching zero), each reaization of the user shoud aso have probabiity of error approaching zero According to ) the signa received by the m-th reaization of user p, at time i can be written as y p, m i h m p,, x m i + z m p,, 7) where m {,,, N t, } and the iid Gaussian noise terms have been normaized to have unit variance Appying Fano s inequaity to codebooks spanning n chan-

MORALES-CESPEDES et a: BLIND INERFERENCE ALIGNMEN FOR CELLULAR NEWORKS ne uses, we have nr p, I W p,, ; h y p, m y m p, ) n ) h ) n ) + on) ) n ) W p,, + on) y p, m n ogp ) + oogp ))) ) n h W p,,) + on), 8) y p, m where P is the tota transmit power constraint at each BS Since this is true for every m {,,, N t, }, in 9) we add the inequaities corresponding to a N t, reaizations For steps 0)-), we use ha, B) ha)+hb), ha, B C) ha BC) + hb C) and the independence between any pair of messages o justify step )-), first note that from y p,,, y p, N t, we have N t, inear equations in the N t, transmitted symbos x x,, x,n t,, each subject to additive noise whose variance does not depend on P Since the channe reaizations are random, these inear equations are amost surey ineary independent, ie, one can recover x from these equations, subject to noise distortion However, from x and noise the messages intended for the users in ce that originate at BS can be recovered hus, the remaining uncertainty is just due to noise, which is no more that oogp )) per channe use Moreover, in )-) we use the fact that conditioning cannot increase the entropy Proceeding simiary for private user p, in ce, nn t, R p, nn t, ogp ) nr Σp R p, ) ) n h y p,,, y p, N t, W p, W p) + on) + n oogp )) 4) Adding ) and 4), we obtain 5)-6) Step ) 5)- n 6) is justified as foows From y p,,, y p, N t, and ) n y p,,, y p, N t, we have Nt + N t, generic inear equations subject to noise distortion), which are amost surey ineary independent and can therefore be soved to recover N t, + N t, input symbos from both BSs, subject to noise distortion hus, we can recover a messages within an n oogp )) term due to noise distortion Moreover, we use ha, B) ha) + hb) Repacing p, and p, with any private users p k, and p j, in 6), respectivey, after dividing by n ogp ), taking first the imit n and then the imit P, a rearranging of the terms yieds the foowing DoF outer bound N t, )d p k, + N t, )d pj, + d Σ N t, + N t,, 7) where d Σ d Σsh + d Σp + d Σp Adding a these bounds, after other rearrangement we obtain K p, K p, + N t, )d Σp + K p, K p, + N t, )d Σp + K p, K p, d Σsh K p, K p, N t, + N t, ) 8) Next, consider the first shared user, who wants the message W sh Aso consider M N t, + N t, reaizations of this user For any reaization m, starting from Fano s inequaity, we go through a simiar series of steps, as foows ) n ) nr sh I W sh ; y m sh + on) ) n n ogp ) h W sh) y sh m + on) + n oogp )) 9) Adding the bounds for a M reaizations, ) n ) nmr sh nm ogp ) h y sh,, y sh M W sh + on) + n oogp )) nm ogp ) nr Σp + R Σp + R Σsh R sh ) + on) + n oogp )) 40) Hence, we obtain the DoF outer bound M )d sh + d Σp + d Σp + d Σsh M 4) If we now sum 4) over a shared users, we obtain K sh + M )d Σsh + K sh d Σp + d Σp ) K sh M 4) he fina DoF outer bounds that we need are 8) and 4) Speciaized to the symmetric setting where N t, N t, N t and K p, K p, K p, we have the sum-dof outer bound maximize subject to d Σsh + d Σp + d Σp K p d Σsh + K p + N t )d Σp + d Σp ) K p N t N t + K sh )d Σsh + K sh d Σp + d Σp ) K sh N t4) his inear program is easiy soved to obtain the sum-dof bound M K sh N t ) + K p M ) d Σ M )N t ) + K sh N t ) + K p M ), 44) which is achieved when N t K p M ) d Σpn M )N t ) + K sh N t ) + K p M ) MK sh N t ) d Σsh M )N t ) + K sh N t ) + K p M ) 45) Note that this is exacty the same DoF achieved by the nbia scheme proposed in Section IV for symmetric ceuar networks whereas the number of private users is the same at each ce VI ASYMMERIC PARIALLY CONNECED CELLULAR NEWORKS So far, a symmetric scenario has been considered In this section, the nbia scheme is extended to asymmetric ceuar networks where the number of private users can be different at each ce It wi be shown that there exist some settings for which the proposed extension achieves the sum-dof outer bound of Section V However, this is not generay the case, and therefore, the DoF optimaity of the proposed approach for asymmetric ceuar networks is sti an open probem