Degrees of Freedom of Full-Duplex Multiantenna Cellular Networks

Transcription

1 Degrees of Freedom of Full-Duplex Multantenna Cellular etworks Sang-Woon Jeon, Member, IEEE, Sung Ho Chae, Member, IEEE, and Sung Hoon Lm, Member, IEEE Abstract arxv: v [cs.it] 3 Jan 205 We study the degrees of freedom (DoF) of cellular networks n whch a full duplex (FD) base staton (BS) equpped wth multple transmt and receve antennas communcates wth multple moble users. We consder two dfferent scenaros. In the frst scenaro, we study the case when half duplex (HD) users, parttoned to ether the uplnk (UL) set or the downlnk (DL) set, smultaneously communcate wth the FD BS. In the second scenaro, we study the case when FD users smultaneously communcate UL and DL data wth the FD BS. Unlke conventonal HD only systems, nter-user nterference (wthn the cell) may severely lmt the DoF, and must be carefully taken nto account. Wth the goal of provdng theoretcal gudelnes for desgnng such FD systems, we completely characterze the sum DoF of each of the two dfferent FD cellular networks by developng an achevable scheme and obtanng a matchng upper bound. The key dea of the proposed scheme s to carefully allocate UL and DL nformaton streams usng nterference algnment and beamformng technques. By comparng the DoFs of the consdered FD systems wth those of the conventonal HD systems, we establsh the DoF gan by enablng FD operaton n varous confguratons. As a consequence of the result, we show that the DoF can approach the two-fold gan over the HD systems when the number of users becomes large enough as compared to the number of antennas at the BS. Index Terms Cellular network, degrees of freedom, full duplex, nterference algnment, multantenna technque. I. ITRODUCTIO Current cellular communcaton systems operate n half-duplex (HD) mode by transmttng and recevng ether at dfferent tme slots or over dfferent frequency bands. The system s desgned n such a way that the downlnk (DL) and uplnk (UL) traffcs are structurally separated by tme dvson duplexng (TDD) or frequency dvson duplexng (FDD). The advantage of such desgn prncple s that t avods the hgh-powered self-nterference that s generated durng smultaneous transmsson and recepton. Recent results [] [6], however, have demonstrated the feasblty of full-duplex (FD) wreless communcaton by suppressng or cancellng self-nterference n the RF and baseband level. Varous practcal desgns to realze self-nterference cancellaton have been proposed n the lterature, ncludng addng addtonal antennas [2], addng auxlary transmt RF chans [3] or auxlary receve RF chans [4], usng polarzaton [3], [4], employng balun crcuts [5], and many more. For more detals, see [6], [7] and the references theren. By enablng smultaneous transmsson and recepton, FD rado s expected to double the spectral effcency of current HD systems [7], and s consdered as one of the key technologes for next generaton communcaton systems. Evdently, n stuatons where the base staton (BS) and the user smultaneously transmt bdrectonally as shown n Fgure (a), enablng FD doubles the overall spectral effcency. Ths pont-to-pont bdrectonal communcaton example, however, s just one nstance of how a FD cellular system wll functon. Ths work has been supported by the Basc Scence Research Program through the atonal Research Foundaton of Korea (RF) funded by the Mnstry of Educaton, Scence and Technology (MEST) [RF-203RAA064955]. The materal n ths paper was presented n part at the IEEE Global Communcatons Conference (GLOBECOM), Austn, TX, December 204 and has been submtted n part at the IEEE Internatonal Symposum on Informaton Theory (ISIT), Hong Kong, Chna, June 205. S.-W. Jeon s wth the Department of Informaton and Communcaton Engneerng, Andong atonal Unversty, Andong, South Korea (e-mal: swjeon@anu.ac.kr). S. H. Chae, the correspondng author, s wth the Dgtal Meda & Communcatons (DMC) Research Center, Samsung Electroncs, Suwon, South Korea (e-mal: sho.chae00@gmal.com). S. H. Lm s wth the School of Computer and Communcaton Scences, Ecole Polytechnque Fédérale de Lausanne (EPFL), Lausanne, Swtzerland (e-mal: sung.lm@epfl.ch).

2 2 HD user 2 FD user 2 FD base staton FD user (a) Bdrectonal full-duplex. FD base staton HD user (b) Full-duplex at the BS only. FD base staton FD user (c) Full-duplex at both the BS and the users. Fg.. Full-duplex network confguratons. In some practcal cases, the system may have to support HD users whch do not have FD rado due to extra hardware burden on moble devces. In such case, the FD BS can smultaneously communcate wth two sets of users, one recevng DL data from the BS and the other transmttng UL data to the BS (Fgure (b)). In another confguraton shown n Fgure (c), for nstance, when the BS has many more antennas compared to each user, the FD BS may wsh to smultaneously communcate wth multple FD users usng mult-user multple-nput and multple-output (MIMO) technques. Snce the BS s smultaneously transmttng and recevng, there s potental to double the overall spectral effcency compared to the conventonal HD only systems. However, the confguratons shown n Fgures (b) and (c) nduce a new source of nterference that does not arse n HD only networks. In Fgure (b), snce user s transmttng to the BS whle user 2 s recevng from the BS, the transmsson from user causes nterference to user 2. Smlarly, n Fgure (c), the UL transmsson of the users causes nterference to the DL recepton to each other. In cases where ths type of nterference s strong and proper nterference mtgaton technques are not appled, the gan of havng FD rados can be severely lmted even when self-nterference s completely removed. To manage nter-user nterference and fully utlze wreless spectrum wth FD operaton, n ths paper we employ sgnal space nterference algnment (IA) schemes optmzed for FD networks ncludng the cases n Fgure. Intally proposed by the semnar works n [8] [0], IA s a codng technque that effcently deals wth nterference and s known to acheve the optmal DoF for varous nterference networks [] [22]. Especally, t s shown that IA can be successfully appled to mtgate nterference n varous cellular networks, such as two-cell cellular networks [], [2] and multantenna UL DL cellular networks [20]. Furthermore, the dea of IA can also be appled to the (mult-user) bdrectonal cellular network wth ergodc phase fadng [2], n whch the achevable scheme s based on the ergodc IA scheme proposed n [22]. Motvated by the aforementoned prevous works related to IA, we propose the optmal transmsson schemes that attan the optmal sum DoFs for two confguratons: ) a cellular network wth a multantenna FD BS and HD users (Fgure (b)); 2) a cellular network wth a multantenna FD BS and FD users (Fgure (c)). The key dea of the proposed schemes s to carefully allocate the UL and DL nformaton streams usng IA and beamformng technques. The UL data s sent to the BS usng IA such that the nter-user nterference s confned wthn a tolerated number of sgnal dmensons, whle the BS transmts n the remanng sgnal dmensons va zero-forcng beamformng for the DL transmsson. Wth the proposed schemes, our prmary goal s to answer whether f FD operaton can stll double the overall spectral effcency even n the presence of nter-user nterference. We answer ths queston by provdng matchng upper bounds wth the proposed achevable schemes, completely charactersng the sum DoFs of the consdered networks. As a consequence of the result, even n the presence of nter-user nterference, we show that the overall DoF can approach the two-fold gan over HD only networks when the number of users becomes large as compared to the number of antennas at the BS. We further provde the DoF gan of the FD systems by consderng varous confguratons (see Sectons III and VI.). A. Prevous Works In [0], Cadambe and Jafar proposed a novel nterference management technque called nterference algnment (IA), whch acheves the optmal sum DoF of K 2 for the K-user nterference channel (IC) wth tme-varyng

3 3 channel coeffcents. In addton, for the case n whch all channel coeffcents are constant, Motahar et al. [23], [24] proposed a dfferent type of IA scheme based on number-theoretc propertes of ratonal and rratonal numbers and showed that the optmal DoF of K 2 s also achevable. Later, alternatve methods of algnng nterference n the fnte sgnal-to-nose regme has been also proposed n [22], [25] [27]. The concept of IA has been successfully adapted to varous network envronments, e.g., see [3] [9] and the references theren. The DoF of cellular networks has been frst studed by Suh and Tse for both UL and DL envronments, where nter-cell nterference exsts [], [2]. It was shown that, for two-cell networks havng K users n each cell, the 2K sum DoF of K+ s achevable for both UL and DL. Thus, multple users at each cell are benefcal for mprovng the DoF of cellular networks. These models were further extended to more general cases n terms of the number of users and the number of antennas at each BS [28] [33]. In addton, recently, the DoF of the multantenna UL DL cellular network consstng of DL and UL cells has been studed n [20], [34]. For a cellular network wth FD operaton n the absence of self-nterference, the DoF of the (mult-user) bdrectonal case has been studed n [2] for ergodc phase fadng settng. B. Paper Organzaton The rest of ths paper s organzed as follows. In Secton II, we descrbe the network model and the sum DoF metrc consdered n ths paper. In Secton III, we present the man results of the paper and ntutvely explan how FD operaton can ncrease the DoF. In Sectons IV and V, we provde the achevablty and converse proofs of the man theorems, respectvely. In Secton VI, we dscuss the mpacts of self-nterference and schedulng on the DoF. Fnally, we conclude n Secton VII. otatons: We wll use boldface lowercase letters to denote vectors and boldface uppercase letters to denote matrces. Throughout the paper, [ : n] denotes {,2,,n}, 0 n denotes the n all-zero vector, and I n denotes the n n dentty matrx. For a real value a, a + denotes max(0,a). For a set of vectors {a }, span({a }) denotes the vector space spanned by the vectors n {a }. For a vector b, b span({a }) means that b s orthogonal wth all vectors n span({a }). For a set of matrces {A }, dag(a,,a n ) denotes the block dagonal matrx consstng of {A }. II. PROBLEM FORMULATIO For a comprehensve understandng of the DoF mprovement by ncorporatng FD operaton, we consder two types of network models: the frst network model conssts of a sngle FD BS whch smultaneously transmts to a set of DL users (n HD mode) and receves from a set of UL users (n HD mode); the second model conssts of a sngle FD BS communcatng wth a set of FD users. Unless otherwse specfed, we smply denote BS for FD BS n the rest of ths paper. A. etwork Model In ths subsecton, we formally defne the network models for the two cases mentoned above. ) FD-BS HD-user cellular networks: Ths network model conssts of a mxture of a FD BS and HD users. The HD users are parttoned nto two sets, n whch one set of users are transmttng to the BS, and the other set of users are recevng from the BS smultaneously. Ths cellular network s depcted n Fgure 2. We assume that the FD BS s equpped wth M transmt antennas and M 2 receve antennas. On the user sde, we assume that there are DL users and UL users, each equpped wth a sngle antenna. Here, each user s assumed to operate n HD mode. The BS wshes to send a set of ndependent messages (W [d] same tme wshes to receve a set of ndependent messages (,,W[u],,W[d] ) to the DL users and at the ) from the UL users. For [ : ], the receved sgnal of DL user at tme t, denoted by y [d] (t) R, s gven by (t) = g (t)x [bs] (t)+ h j (t)x [u] j (t)+z[d] (t) () y [d] j=

4 4 (W [d],,w[d]) (Ŵ[u],,Ŵ[u]) 2 M tx antennas M 2 rx antennas g g f 2 f FD BS h h DL user h 2 UL user h 2 DL user UL user 2 Fg. 2. The (M,M 2,,) FD-BS HD-user cellular network. and the receved sgnal vector of the BS at tme t, denoted by y [bs] (t) R M2, s gven by y [bs] (t) = f j (t)x [u] j (t)+z[bs] (t), (2) j= where x [bs] (t) R M s the transmt sgnal vector of the BS at tme t, x [u] j [t] R s the transmt sgnal of UL user j at tme t, g (t) R M s the channel vector from the BS to DL user at tme t, h j (t) R s the scalar channel from UL user j to DL user at tme t, and f(t) R M2 s the channel vector from UL user j to the BS. The addtve noses z [d] (t) R and z [bs] (t) R M2 are assumed to be ndependent of each other and also ndependent over tme, and s dstrbuted as z [d] (t) (0,) and z [bs] (t) (0 M2,I M2 ). We assume that channel coeffcents are drawn..d. from a contnuous dstrbuton and vary ndependently over tme. It s further assumed that global channel state nformaton (CSI) s avalable at the BS and each UL and DL user. The BS and each UL user s assumed to satsfy an average transmt power constrant,.e., E [ x [bs] (t) 2] P and E [ x [u] j (t) 2] P for all j [ : ]. In the rest of the paper, we denote ths network as a (M,M 2,, ) FD-BS HD-user cellular network. Remark : We assume perfect self-nterference suppresson wthn the BS durng FD operaton. Hence there s no self-nterference for the nput output relatons n () and (2). We wll dscuss how mperfect self-nterference suppresson effects the DoF n Secton VI-A. 2) FD-BS FD-user cellular networks: In ths model, we consder the case where both the BS and users have FD capablty (depcted n Fgure 3). As before, we assume that the BS s equpped wth M transmt antennas and M 2 receve antennas. However, unlke the FD-BS HD-user cellular network, there s a sngle set of FD users, each equpped wth a sngle transmt and a sngle receve antenna, that smultaneously transmts to and receves from the BS. The BS wshes to send a set of ndependent messages (W [d],,w[d] ) to the users and at the same tme wshes to receve a set of ndependent messages (,,W[u] ) from the same users. For [,], the receved sgnal of user at tme t s gven by y (t) = g (t)x [bs] (t)+ j=,j h j (t)x j (t)+z (t) (3)

5 5 (W [d],,w[d] ) (Ŵ[u],,Ŵ[u] ) M tx antennas M 2 rx antennas g g f f FD BS h h User User 2 User 2 2 W[u] Fg. 3. The (M,M 2,) FD-BS FD-user cellular network. and the receved sgnal vector of the BS at tme t s gven by y [bs] (t) = f j (t)x j (t)+z [bs] (t). (4) j= As before, we assume that self-nterference at the BS and each user s completely suppressed, whch s reflected n the nput output relatons n (3) and (4). The rest of the assumptons are the same as those of the (M,M 2,, ) FD-BS HD-user cellular network. In the rest of the paper, we denote ths network as a (M,M 2,) FD-BS FD-user cellular network. B. Degrees of Freedom For each network model, we defne a set of length n block codes and ts achevable DoF. ) FD-BS HD-user cellular networks: Let W [d] and j be chosen unformly at random from [ : 2 nr[d] [ : 2 nr[u] j ] respectvely, where [ : ] and j [ : ]. Then a (2 nr[d],,2 nr [d],2 nr[u],,2 nr [u] conssts of the followng set of encodng and decodng functons: Encodng: For t [ : n], the encodng functon of the BS at tme t s gven by x [bs] (t) = φ t (W [d],,w[d],y [bs] (), y [bs] (t )). For t [ : n], the encodng functon of UL user j at tme t s gven by where j [ : ]. x j (t) = ϕ t ( j ), Decodng: Upon recevng y [bs] () to y [bs] (n), the decodng functon of the BS s gven by Ŵ [u] j = χ j (y [bs] (),,y [bs] (n),w [d],,w[d] ) for j [ : ]. Upon recevng y () to y (n), the decodng functon of DL user s gven by where [ : ]. = ψ (y (),,y (n)), ] and ;n) code

6 6 A rate tuple (R [d],,r[d],,r[u] ) s sad to be achevable for the FD-BS HD-user cellular network f there exsts a sequence of (2 nr[d],,2 nr [d],2 nr[u],,2 nr [u] ;n) codes such that Pr(Ŵ[d] W [d] ) 0 and Pr(Ŵ [u] j j ) 0 as n ncreases for all [ : ] and j [ : ]. Then the achevable DoF tuple s gven by (d [d],,d[d],r [u],d [u],,d[u] ) = lm P ( ) R [d] [d] [u] [u] logp,, R R 2 2 logp, logp,, R 2 2 logp. (5) We further denote the maxmum achevable sum DoF of the FD-BS HD-user cellular network by d Σ,,.e., d Σ, = max d [d] + d [u] j, (6) (d [d],,d [d],d [u],,d [u] ) D where D denotes the DoF regon of the FD-BS HD-user cellular network. 2) FD-BS FD-user cellular networks: Smlar to the FD-BS HD-user cellular network, we can defne an achevable DoF tuple of the FD-BS FD-user cellular network. The key dfference s that each user also operates n FD mode for ths second model. Specfcally, the encodng functon of user at tme t [ : n] s gven by x (t) = ϕ t (,y (),,y (t )) and the decodng functon of user s gven by Ŵ[d] = ψ (y (),,y (n), ), where [ : ]. Then the defnton of an achevable DoF tuple (d [d],,d[d],d[u],,d[u] ) s the same as that of the FD-BS HD-user cellular network. Smlarly, we denote the maxmum achevable sum DoF of the FD-BS FD-user cellular network by d Σ,2. = j= III. MAI RESULTS In ths secton, we state the man results of ths paper. We completely characterze the sum DoFs of both the (M,M 2,, ) FD-BS HD-user cellular network and the (M,M 2,) FD-BS FD-user cellular network. Theorem : For the (M,M 2,, ) FD-BS HD-user cellular network, d Σ, = mn { M +M 2,max(, ),max ( M + ( M ),M 2 + ( M 2 ) )}. (7) Proof: The achevablty proof s gven n Secton IV and the converse proof s gven n Secton V. We demonstrate the utlty of Theorem by the followng example. Example (Symmetrc FD-BS HD-user cellular networks): Consder the (M, M,, ) FD-BS HD-user cellular network,.e., M = M 2 = M and = =. For ths symmetrc case, d Σ, = mn(2m,) from Theorem. On the other hand, f the BS operates n HD mode, we can easly see that the sum DoF s lmted by mn(m,). By comparng the sum DoFs, we can see that there s a two-fold gan by operatng the BS n FD mode when we have enough number of users n the network,.e., 2M. Fgure 4 plots d Σ, wth respect to when M = 5. As shown n the fgure, FD operaton at the BS mproves the sum DoF as ncreases and eventually the sum DoF s doubled compared to HD BS for large enough. For the FD-BS FD-user cellular network, we have the followng theorem. Theorem 2: For the (M,M 2,) FD-BS FD-user cellular network, d Σ,2 = mn(m +M 2,). (8) Proof: From the network model and the DoF defnton n Secton II, any achevable sum DoF n the (M,M 2,,) FD-BS HD-user cellular network s also achevable for the (M,M 2,) FD-BS FD-user cellular network. In partcular, the encodng functons at the BS are the same for both network models, and the BS also receves the same sgnal as shown n () and (3). Comparng the user encoders, we can see that the user encodng functon for the FD-BS FD-user cellular network s more general than the encodng functon for the FD-BS HDuser cellular network. Furthermore, we can easly see that the receved sgnal (4) s better than the receved sgnal for the FD-BS HD-user cellular network (3), n that t has less nterference (self-nterference s suppressed for the FD user case). Hence, from Theorem, the sum DoF of mn(m + M 2,) s achevable for the (M,M 2,) FD-BS FD-user cellular network, whch concdes wth d Σ,2 n (8). The converse proof s gven n Secton V. We demonstrate the utlty of Theorem 2 by the followng example.

7 Sum DoF FD-BS HD-user HD-BS HD-user umber of UL or DL users () Fg. 4. Sum DoFs when M = M 2 = 5 and = =. Example 2 (Symmetrc FD-BS FD-user cellular networks): Consder the (M, M, ) FD-BS FD-user cellular network,.e., M = M 2 = M. For ths symmetrc case, d Σ,2 = mn(2m,) from Theorem 2, whch concdes wth the sum DoF of the symmetrc FD-BS HD-user cellular network n Example. Agan, f both the BS and the users are lmted to operate n HD mode, then the sum DoF s lmted by mn(m,). To be far, the (M,M,) FD-BS FD-user cellular network n Example 2 has been consdered n [2] under the ergodc fadng settng assumng that the phase of each channel coeffcent n {h j (t)},j [:], j s drawn ndependently from a unform phase dstrbuton. For ths case, t has been shown n [2, Theorem ] that the achevable DoF tuple satsfes: = d [d] + = j= j= d [d] mn(m,) d [u] j mn(m,) d [u] j mn(2m,), (9) where (9) characterses the sum DoF. Ths result n [2] s general n that t provdes a general achevable DoF regon, whle our result n Theorem 2 generalzes the sum DoF result n [2] by consderng arbtrary number of transmt and receve antennas at the BS, and also extends to any..d. generc channel settng ncludng the ergodc fadng settng. In Secton VI, we dscuss n detal regardng the DoF mprovement by enablng FD operaton, and also the effect of mperfect self-nterference suppresson. IV. ACHIEVABILITY In ths secton, we prove that the sum DoF d Σ, n Theorem s achevable. To better llustrate the man nsght of the codng scheme, we frst consder the achevablty of Theorem for the case = n Secton IV-A. The

8 8 v [d] v [d] 2 M2 F v [u] M2 F v [u] F 2 v [u] 2 F 2 v [u] 2M2 M tx antennas M 2 rx antennas Ḡ F 2 F FD BS H DL user H 2 UL user Ḡ v [d] 2 M2 H v [u] = = H 2 v [u] 2 UL user v [u] v [u] M2 Ḡ v [d] v [u] 2M2 H v [u] M2 = = H 2 v [u] 2M2 v [u] 2 Fg. 5. Transmt beamformng for the (M,M 2,,) FD-BS HD-user cellular network when M 2. man component of the scheme utlzes IA va transmt beamformng wth a fnte symbol extenson. For general, nterference from multple UL users should be smultaneously algned at multple DL users, whch requres asymptotc IA,.e., an arbtrarly large symbol extenson. In Secton IV-B, we ntroduce transmt beamformng adoptng such asymptotc IA for the general network confguraton. A. The Case = For the (M,M 2,, ) FD-BS HD-user cellular network, { f M 2, d Σ, = (0) M M2 f M 2 from Theorem. For the proof on how (0) can be evaluated from (7) for the case =, we refer to the proof n Lemma. In the followng, we show that d Σ, n (0) s achevable by consderng two cases, M 2 and M 2. For the frst case M 2, we can easly acheve d Σ, = by smply utlzng only the UL transmsson,.e., the BS receves from the UL users wth M 2 receve antennas. ow consder the second case where M 2, whch we explan wth the help of Fgure 7. For ths case, communcaton takes place va transmt beamformng over a block of tme slots,.e., symbol extenson. Denote Ḡ = dag(g (), g ( )) R 2 M2, H j = dag(h j (),,h j ( )) R 2 2, F j = dag(f j (),,f j ( )) R M22 2, () where j [ : ]. The BS sends M 2 nformaton symbols to the DL user va the M beamformng vectors { v [d] k } k [: M 2]. On the other hand, UL user j [ : ] sends M 2 nformaton symbols to the BS va the beamformng vectors { v [u] jk } k [:M 2].

9 9 ( ) W [d],,w[d] DL user DL BS wth M antennas DL user UL user (Ŵ[u] ),,Ŵ[u] 2 UL BS wth M 2 antennas 2 UL user Fg. 6. Two-cell multantenna cellular networks n whch the frst and second cells operate as DL and UL respectvely. We frst construct { v [d] k } k [: M 2] as a set of M 2 lnearly ndependent random vectors. ext, we construct lnearly ndependent { v [u] jk } k [:M 2],j [:] such that for each k [ : M 2 ], all the nformaton symbols that are ndexed wth k [ : M 2 ] are algned at the DL user,.e., satsfyng the IA condton H v [u] k = = H 2 v [u] k for all k [ : M 2 ]. Specfcally, we frst construct { v [u] k } k [:M 2] as a set of M 2 lnearly ndependent random vectors. Then, for a gven { v [u] k } k [:M 2], we construct v [u] jk = ( H j ) H v [u] k for all k [ : M 2],j [2 : ]. By such constructon, the resultng { v [u] jk } k [:M 2],j [:] are lnearly ndependent almost surely. We now move on to the decodng step at the DL user. Due to the prevous IA procedure of the UL users, the number of dmensons occuped by the nter-user nterference sgnals s gven by M 2. Furthermore, the DL sgnals sent by the BS occupy M 2 dmensons and are lnearly ndependent of the nter-user nterference sgnals almost surely. Hence, the DL user s able to decode ts ntended nformaton symbols achevng one DoF each. ext, consder decodng at the BS. Snce { v [u] jk } k [:M 2],j [:] are lnearly ndependent, { F j v [u] jk } k [:M 2],j [:] are also lnearly ndependent almost surely. Hence, the BS s able to decode the M 2 nformaton symbols. Fnally, from the fact that a total of M 2 +M 2 nformaton symbols are communcated over tme slots, d Σ, = M M2 s achevable for the case M 2. B. General Case Followng the ntuton n the prevous subsecton, wth IA, we would lke to confne the nterference sgnals transmtted from multple UL users nto a preserved sgnal subspace at each DL user, leavng the rest of subspace for the ntended sgnals sent from the BS. For general, ths requres arbtrarly large number of symbol extensons [0]. For ths purpose, a recently developed IA technque n [20] for the multantenna UL DL cellular network can be appled for the (M,M 2,, ) FD-BS HD-user cellular network. To show how the scheme n [20] fts nto our problem, we begn wth a bref overvew of ther network model. In [20], the authors consder a UL DL cellular network (Fgure 6), where two cells co-exst (each cell conssts of one BS and a set of users). In one cell, a BS wth M antennas transmts to a set of DL users, whle n the other cell a set of UL users transmt to a BS wth M 2 antennas. Thus, the network models the case when t can schedule each cell n DL or UL phase separately. The structural smlarty wth our FD-BS HD-user cellular network s apparent, and the key dfference between them s that there s no nter-cell nterference between the DL BS and UL BS (snce n the FD-BS HD-user cellular

10 0 λ T( ǫ T ) streams (for each DL user)! # " # $ # {v [d] k } {v [d] 2k } {v [d] k } λ 2 T( ǫ T ) { F v [u] k }{ F 2 v [u] 2k } M tx antennas M 2 rx antennas Ḡ Ḡ F 2 F FD BS H DL user H 2 H UL user {Ḡv [d] k } {Ḡv [d] 2k } {Ḡv [d] k } λ T( ǫ T ) λ 2 T { H v [u] k } { H 2 v [u] 2k } {Ḡ v[d] k } DL user λ T( ǫ T ) λ 2 T H 2 {Ḡv [d] {Ḡv [d] k } 2k } { H v [u] k } { H 2 v[u] 2k } UL user {v [u] 2k } {v [u] k } λ 2 T( ǫ T ) streams Resdual nterference after DL nterference nullng UL nterference algnment Fg. 7. Conceptual llustraton of transmt beamformng for the (M,M 2,,) FD-BS HD-user cellular network, where for convenence we assume λ λ 2 n the fgure. network, UL and DL s performed wth a sngle FD BS). Accordngly, the transmt sgnal vector of the DL BS n the UL DL model (Fgure 6) can also be used as the transmt sgnal vector of the FD BS n the FD-BS HD-user cellular network (Fgure 2), and the transmt sgnal of each UL user n the UL DL model (Fgure 6) can also be used by each UL user n the FD-BS HD-user cellular network (Fgure 2). Therefore, the IA scheme stated n [20, Secton IV-E] s applcable to the (M,M 2,, ) FD-BS HD-user cellular network. However, due to the self-nterference suppresson capablty n the FD BS case, the performance resultng from ths scheme wll be dfferent for the two networks, and our contrbuton for achevablty les n the analyss of the sum DoF of the scheme for the FD-BS HD-user cellular network. For completeness and better understandng, we brefly summarze how the IA scheme n [20, Secton IV-E] can be adapted to the (M,M 2,, ) FD-BS HD-user cellular network. We then gve the analyss of ts achevable sum DoF. ) DL nterference nullng and UL nterference algnment: Communcaton takes place over a block of T tme slots,.e., T symbol extenson. Denote Ḡ = dag(g (), g (T)) R T MT, H j = dag(h j (),,h j (T)) R T T, F j = dag(f j (),,f j (T)) R M2T T, (2) for [ : ] and j [ : ]. Each nformaton symbol s transmtted through a length-t tme-extended beamformng vector. Fgure 7 s a conceptual llustraton for ths transmt beamformng. We refer to [20, Secton IV-E] for the detaled constructon of beamformng vectors. Suppose that λ,λ 2 (0,] and ǫ T 0 as T ncreases.

11 λ 2 λ 2 λ 2 M2 M2 2 2 ( M2 2, M2 2 ) M2 ( M M, ) 2 (a) M, M 2 λ 2 M λ M M λ λ (b) M, M 2 (c) M, M 2 λ 2 ( M2 2, M2 2 ) M2 2 ( M M, ) M2 2 M λ (d) M, M 2, M +M 2 M M λ (e) M, M 2, M +M 2 M Fg. 8. Feasble (λ,λ 2) regon and the extreme ponts attanng the maxmum sum DoF. For [ : ], the BS sends λ T( ǫ T ) nformaton symbols to DL user usng the set of T tme-extended beamformng vectors {v [d] k } k [:λ T( ǫ T)]. Smlarly, UL user j sends λ 2 T( ǫ T ) nformaton symbols to the BS usng the set of T tme-extended beamformng vectors {v [u] jk } k [:λ 2T( ǫ T)], where j [ : ]. As seen n Fgure 7, the set of beamformng vectors transmtted from each UL user s set to algn ts nterference [u]} at each DL user. More specfcally, by applyng asymptotc IA for { v jk j [:],k [:λ 2T( ǫ T)], we can guarantee ( that span { Hj v [u] } ) jk j [:],k [:λ 2T( ǫ T)] occupes at most λ 2 T dmensonal subspace n T dmensonal sgnal space for all [ : ] almost surely n the lmt of large T, where ǫ T 0 as T ncreases, see also [20, Lemma 2]. Then the set of beamformng vectors transmtted from the BS s set to null out ( ts nterference at each) DL user. More specfcally, { v [d] k } [d] [: ],k [:λ T( ǫ T)] s set to satsfy Ḡ v jk {Ḡ v span [d] k }k [:λ T( ǫ T)] for all,j [ : ] satsfyng j and k [ : λ T( ǫ T )],.e., zero-forcng s performed usng M transmt antennas. In order to apply such DL nterference nullng, M T λ T( ǫ T )( ) λ T( ǫ T ) (3) should be satsfed. Agan, as seen n Fgure 7, for relable decodng at each DL user achevng one DoF for each nformaton symbol, λ T( ǫ T )+λ 2 T T (4) should be satsfed. Smlarly, for relable decodng at the BS achevng one DoF for each nformaton symbol, λ 2 T( ǫ T ) M 2 T (5) should be satsfed. Therefore, the proposed scheme s able to delver ( λ + λ 2 )T( ǫ T ) nformaton symbols over T tme slots under the constrants (3) to (5). Fnally, from the fact that ǫ T 0 as T ncreases, ts achevable sum DoF s represented by the followng optmzaton problem: max λ +λ 2 λ M λ 2 M 2 { λ + λ 2 }. (6)

12 2 2) Achevable sum DoF: In the followng, we prove that the sum DoF attaned by solvng (6) s gven as d Σ, stated n Theorem. The lnear program n (6) s dvded nto fve cases dependng on the feasble regon of (λ,λ 2 ) as depcted n Fgure 8. Obvously, one of the corner ponts, whch are marked as ponts n Fgure 8, provdes the maxmum sum DoF. Hence, the maxmum sum DoF attaned from (6) s gven by max( (, ) ) f M,M 2, max 2,M + 2( M) f M,M 2, ( ) max,m 2 + (2 M2) f M,M 2, (7) ( ) max M + 2( M),M 2 + (2 M2) f M,M 2,M +M 2, M +M 2 f M,M 2,M +M 2. The followng lemma then shows that (7) s represented as d Σ, n Theorem, whch completes the achevablty proof of Theorem. Lemma : The sum DoF n (7) s represented as mn { M +M 2,max(, ),max Proof: For notatonal smplcty, denote ( M + ( M ),M 2 + ( M 2 ) a = M + ( M ) = + M ( ), )}. (8) a 2 = M 2 + ( M 2 ) = + M 2( ). (9) Then denote a 3 = mn{m +M 2,max(, ),max(a,a 2 )}. In the followng, we show that for each of the fve cases n (7), a 3 s represented as n the correspondng DoF expresson n (7). Case I (M,M 2 ): Obvously, M + M 2 max(, ). For, max(a,a 2 ) a + ( 2) =. For, max(a,a 2 ) a 2 + 2(2 ) =. Hence max(a,a 2 ) max(, ). In concluson, a 3 = max(, ) for Case I. Case II (M,M 2 ): Frst consder the case where. Then M + M 2 M + M + 2( M) = a. Also max(, ) = = + ( 2) + M( 2) = a. Snce a 2 + 2(2 ) =, max(a,a 2 ) = a. Hence a 3 = a. ext consder the case where. Then M + M 2 and max(, ) =. Also max(a,a 2 ) a 2 + 2(2 ) =. Hence a 3 =. Fnally, from the relaton that a for and a for, a 3 = max(,a ) for Case II. Case III (M,M 2 ): From the symmetrc relaton wth Case II, a 3 = max(,a 2 ) for Case III. Case IV (M,M 2,M + M 2 ): The condton M + M 2 means that M (2 M2) and M 2 2( M). Hence M + M 2 (2 M2) + M 2 = a 2 and M +M 2 M + 2( M) = a, whch show M + M 2 max(a,a 2 ). For, = + ( 2) + M( 2) = a and a 2. Smlarly, a 2 and a for. Hence max(, ) max(a,a 2 ). In concluson, a 3 = max(a,a 2 ) for Case IV. Case V (M,M 2,M +M 2 ): For, (M +M 2 ) M +M 2 and then M +M 2. Smlarly, M +M 2 for. Hencemax(, ) M +M 2. The condton M +M 2 means that 2M M 2 and M2 M. Then a = M + 2( M) M + M2 M M = M + M 2 and a 2 = M 2 + (2 M2) M 2 + 2M M 2 M 2 = M + M 2. Hence max(a,a 2 ) M +M 2. In concluson, a 3 = M +M 2 for Case V. In concluson, a 3 s represented as the correspondng sum DoF n (7) for all fve cases, whch completes the proof.

13 3 M tx antennas rx antennas W Tx Rx Ŵ Output from Rx 2 tx antennas M 2 rx antennas W 2 Tx 2 Rx 2 Ŵ 2 W Fg. 9. Two-user MIMO Z-IC wth output feedback for encodng and message sde nformaton for decodng. V. COVERSE In ths secton, we prove the converse of Theorems and 2. Recall the encodng and decodng functons of the FD BS and each FD user n Secton II-B. The key observaton s that the receved sgnals avalable for encodng the DL messages at the FD BS and the DL messages avalable for decodng the UL messages at the FD BS cannot ncrease the sum DoF. Smlarly, the receved sgnals avalable for encodng ts UL message at each FD user and ts UL message avalable for decodng ts DL message at each FD user cannot ncrease the sum DoF. A. Converse of Theorem To prove the converse of Theorem, we ntroduce the two-user MIMO Z-IC wth output feedback for encodng and message sde nformaton for decodng depcted n Fgure 9. The receved sgnal vectors of recevers and 2 at tme t are respectvely gven by y (t) =H x (t)+h 2 x 2 (t)+z (t), y 2 (t) =H 22 x 2 (t)+z 2 (t), (20) where H R M, H 2 R 2, and H 22 R M2 2 denote the channel matrces from transmtter to recever, from transmtter 2 to recever, and from transmtter 2 to recever 2, respectvely. The rest of the assumptons are the same as those of the FD-BS HD-user cellular network n Secton II-A. Obvously, the capacty of the two-user MIMO Z-IC s an outer bound on the capacty of the (M,M 2,, ) FD-BS HD-user cellular network, snce t corresponds to the FD-BS HD-user cellular network wth full cooperaton among the DL users and among the UL users. Lemma 2: Consder the two-user MIMO Z-IC wth output feedback for encodng and message sde nformaton for decodng n Fgure 9. Then the DoF regon s gven by the set of all DoF pars (d,d 2 ) satsfyng d mn(m, ), =,2 (2) d +d 2 max(, ). (22) Proof: The achevablty mmedately follows from that n [7, Theorem ], whch corresponds to the two-user MIMO Z-IC wthout output feedback for encodng and message sde nformaton for decodng. ext, we show the converse. Obvously d mn(m, ) and also d 2 mn(m 2, ) snce sde nformaton of W at recever 2 cannot ncrease the DoF more than mn(m 2, ), whch gves (2). ow substtute antennas wth max(, ) antennas at recever. Assume that both recevers are able to recover W and W 2 respectvely wth arbtrarly small probabltes of error. Then, after subtractng x from y (x s obtaned from re-encodngw ), recever constructs y = H 2 x 2 + z, where H 2 Rmax(,2) 2. Snce recever 2 recovers W 2 from y 2 = H 22 x 2 + z 2, where

14 4 H 22 R M2 2, recever can also recoverw 2 from y from the fact that mn(,max(, )) mn(,m 2 ). As a result, recever s able to decode both W and W 2 wth max(, ) antennas. Because output feedback cannot ncrease the sum DoF of the MIMO multple-access channel (MAC), d + d 2 max(, ), whch provdes (22). In concluson, Lemma 2 holds. Snce the sum DoF of the (M,M 2,, ) FD-BS HD-user cellular network s upper bounded by the sum DoF of the two-user MIMO Z-IC, d Σ, mn(m +M 2,max(, )) from Lemma 2, whch s yet not enough to show the converse. In a more refned way of applyng Lemma 2, we prove the converse of Theorem n the followng. Denote d [d] Σ = = d[d] and d [u] Σ = j= d[u] j. Frst consder the case where. For ths case, choose a subset of DL users n A [d] [ : ] satsfyng card(a [d] ) =. Then, by applyng Lemma 2 only for the DL users n A [d] (and for the entre UL users), we have By summng (23) over all possble A [d] satsfyng card(a [d] ) =, we have Therefore, d Σ, A [d] d [d] +d [u] Σ. (23) d [d] Σ + d [u] Σ. (24) d [d] d [u] max Σ mn(m, ) mn(m2,2) Σ d [d] Σ +d[u] Σ 2 {d [d] Σ +d[u] Σ }. (25) ow consder the case where. For ths case, choose a subset of UL users n A [u] [ : ] satsfyng card(a [u] ) =. Then applyng Lemma 2 for all possble A [u] satsfyng card(a [u] ) = and summng them provdes the same upper bound n (24). As a result, (25) also holds for. By solvng the lnear program (25) n a smlar manner as n Secton IV, we have max( (, ) ) f M,M 2, max,m + 2( M) f M,M 2, ( ) d Σ, max,m 2 + (2 M2) f M,M 2, (26) ( ) max M + 2( M),M 2 + (2 M2) f M,M 2,M +M 2, M +M 2 f M,M 2,M +M 2. ote that the upper bound n (26) s exactly the same as n (7). Therefore, from Lemma, { d Σ, mn M +M 2,max(, ),max whch completes the converse proof of Theorem. B. Converse of Theorem 2 ( M + ( M ),M 2 + ( M 2 ) )}, (27) In ths subsecton, we prove the converse of Theorem 2. We frst show that d Σ,2 M +M 2 n Secton V-B and then show that d Σ,2 n Secton V-B2. Combnng the above two bounds, we have the desred bound d Σ,2 mn(m +M 2,), whch completes the converse proof. ) MIMO two-way network upper bound: By allowng full cooperaton among the users n the (M,M 2,) FD-BS FD-user cellular network, we obtan a MIMO two-way network depcted n Fgure 0. Clearly, the consdered MIMO two-way network provdes an upper bound on d Σ,2. Therefore, from the result n [35], we have d Σ,2 mn(m,)+mn(m 2,) M +M 2. (28)

15 5 M tx antennas rx antennas (W [d],,w[d] ) Downlnk (Ŵ[d],,Ŵ[d] ) FD BS Full cooperaton among users (Ŵ[u],,Ŵ[u] ) Uplnk (,,W[u] ) M 2 rx antennas tx antennas Fg. 0. MIMO two-way channel by allowng full cooperaton among the users. M tx antennas (W [d],,w[d] ) BS t User r User r 2 2 Output from BS r User t 2 2 Output from user r User t 2 Output from user r 2 User r M 2 rx antennas User t BS r (Ŵ[u],,Ŵ[u] ) Output from user r (W [d],,w[d] ) Fg.. Step : The equvalent two-cell network wth output feedback at the encoders, and message sde nformaton at the decoders. 2) Four-node X network upper bound: We now prove d Σ,2 by usng the result of four-node X networks n [3]. In order to apply the result n [3], we convert the orgnal (M,M 2,) FD-BS FD-user cellular network nto the correspondng four-node X network as follows: Step : We frst transform the (M,M 2,) FD-BS FD-user cellular network nto the equvalent two-cell cellular network consstng of one DL cell and one UL cell depcted n Fgure. Specfcally, the FD BS s decomposed nto the BS t and the BS r and FD user s decomposed nto user t and user r, where [ : ]. There exsts output feedback from the BS r to the BS t and from user r to user t for all [ : ], whch can be used as sde nformaton for encodng. In addton, (W [d],w [d] ) s avalable at the BS r and s avalable at user r for all [ : ], whch can be used as sde nformaton for decodng. We refer to the encodng and decodng functons n Secton II-B2. The channel coeffcents from the BS t to the BS r and from user t to user r are set to zeros due to perfect self-nterference suppresson n the orgnal network.

16 6 ode ode 3 User t User r BS r ode 2 User r 2 (Ŵ[u],Ŵ[u] ) BS t User r (W [d],w[d] ) User t 2 (W [d],w[d] ) ode 4 User t User t User r Output from ode 3 Fg. 2. Step 2: Cooperaton between BSs and users. The valdty of ths transformaton s also proved by [3, Lemma ]. Step 2: As shown n Fgure 2, we allow full cooperaton among user t and user r, among BS t, user t 2 to user t, among BS r, user r 2 to user r, and among user t and user r, each of whch s called odes,2,3, and 4 respectvely. Because of such cooperaton, the set of (W [d] 2,,W[d],W[u] 2,,W[u] ) s prorly known at ode 3 as sde nformaton, so that ode 3 s able to attan those messages wthout communcaton. Hence, we delete those messages n the fgure wthout loss of generalty. In the end, ode wshes to send and estmate W [d], ode 2 wshes to send (W[d],W[d] ) wth the help of output feedback from ode 3,.e., the set of all output sgnals receved by the components consstng of ode 3, ode 3 wshes to estmate (,W[u] ) wth the help of message sde nformaton (W[d],W[d] ), and ode 4 wshes to send and estmate W[d]. Snce the network n Fgure 2 assumes cooperaton between some nodes and allow more nformaton for encodng and decodng, t provdes an outer bound on the DoF regon of the network n Fgure. Step 3: We now focus on an upper bound on d [u] + d [d]. We frst elmnate all the messages except W[u] and W [d], whch does not decrease d[u] + d[d] [8]. Then we provde M + receve antennas at ode 2 and M 2 + transmt antennas at ode 3 and allow FD operaton at all nodes, whch creates more lnks llustrated as dashed lnks n Fgure 3. We further assume that output feedback from odes 3 and 4 s avalable at odes and 2. Obvously, addng more antennas at some nodes, allowng FD operaton, and provdng more output feedback for encodng do not decrease d [u] +d[d]. As a result, the converted network n Fgure 3 provdes an upper bound on d [u] +d[d] achevable by the orgnal (M, M 2, ) FD-BS FD-user cellular network. ote that the converted network n Fgure 3 corresponds to the four-node X network studed n [3] except the fact thatw [d] s provded to ode 3 through a gene. As stated n [3, Secton IV], provdng ths sde nformaton does not ncrease the sum DoF and, therefore, we have d [u] +d[d] The full cooperaton assumpton mples that both output feedback and message sde nformaton are avalable at odes and 4.

17 7 Output from odes 3 and 4 ode ode 3 Ŵ [u] W [d] ode 2 W [d] Output from odes 3 and 4 ode 4 Fg. 3. Step 3: Elmnate all the messages except and W [d] and create more lnks and output feedback. from the result n [3]. In the same manner, we can establsh d [u] +d [d] j (29) for,j [ : ] wth j. By summng (29) for all,j [ : ] wth j, we fnally have d Σ,2 = = d [u] + = VI. DISCUSSIOS d [d]. (30) In ths secton, we brefly dscuss about the mpacts of self-nterference and UL and DL schedulng on DoF. A. Impacts of Self-Interference on DoF Throughout the paper, we assumed that there s no self-nterference wthn the BS durng FD operaton. However, n a practcal FD BS, the amount of resdual self-nterference may not be neglgble due to nsuffcent selfnterference suppresson or mperfect self-nterference cancellaton from the prorly known message nformaton at the recever sde [7]. In ths subsecton, we wll dscuss the mpacts of such self-nterference on the sum DoF. ote that when there exsts self-nterference wthn the BS of the (M,M 2,, ) FD-BS HD-user cellular network, the sum DoF s gven by { +mn(m, )( ) + +mn(m 2, )( ) + mn, max(, ) } M +,M 2 +,max(m,m 2 ),max(, ) from the result of [20], by nterpretng nter-bs nterference n [20] as self-nterference wthn the BS. Obvously, f we restrct for the BS to operate ether UL or DL only, then the sum DoF s gven by (3) max(mn(m, ),mn(m 2, )). (32) To see the effect of self-nterference on the sum DoF, let us consder the case where M = 6, M 2 = 8, and = 2 as an example. We plot the sum DoFs as a functon of the number of total users = + n

18 Sum DoF 5 0 FD-BS FD-user, Theorem 5 FD-BS HD-user, Theorem 2 FD-BS HD-user wth self-nterference, Eqn. (3) HD-BS HD-user, Eqn. (32) umber of total users ( = + ) Fg. 4. Sum DoFs for M = 6, M 2 = 8, and = 2. Fgure 4. For comparson, we also plot the sum DoF of the FD-BS FD-user cellular network when the number of FD users s gven by. As shown n the fgure, the FD-BS HD-user cellular network s able to acheve the same sum DoF attaned by the FD-BS FD-user cellular network when s large enough. However, FD capablty at the user sde s benefcal to mprove the sum DoF for small. Interestngly, even when there exsts self-nterference, FD operaton at the BS alone can ncrease the sum DoF n a certan regme. However, the sum DoF collapses to that of the HD-BS HD-user cellular network when s large enough. ote that smlar tendences can be observed for general (M,M 2,, ). Therefore, from these observatons, self-nterference suppresson or cancellaton s of crucal mportance for fully utlsng the potental of FD networks. B. Effects of Schedulng on DoF In ths subsecton, we dscuss the effects of HD user schedulng on the sum DoF. Suppose that there exst total HD users and we are able to coordnate the operatonal mode of each of these users,.e., dvdng them nto DL users and UL users, where + =. Obvously, the sum DoF vares wth the values of and from Theorem. As an example, consder agan the case where M = 6 and M 2 = 8. Frst, we fx the total number of users (= + ) as 50 and plot the sum DoF of the FD-BS HD-user cellular network wth and wthout self-nterference suppresson as a functon of n Fgure 5. For comparson, we also plot the sum DoFs of the FD-BS FD-user cellular network and the HD-BS HD-user cellular network. As depcted n Fgure 5, except the FD-BS FD-user cellular network, the achevable sum DoFs vary wth, and we can maxmze the sum DoF of each network by optmally choosng and. ow, we plot the sum DoFs as a functon of the number of total users n Fgure 6. Here, for each, we choose and to acheve the optmal sum DoFs. As seen n Fgure 6, when there s no self-nterference, the optmal sum DoF of the FD-BS HD-user cellular network approaches to that of the FD-BS FD-user cellular network and reaches the same sum DoF when s large enough. However, when there exsts self-nterference, the optmal sum DoFs of the FD-BS HD-user cellular network and the HD-BS HD-user cellular network are the same for any. Ths statement s also true for general M and M 2 snce the optmal schedulng for the FD-BS HD-user cellular network wth self-nterference s to operate all HD users as ether UL or DL, whch can be easly verfed

19 Sum DoF 5 0 FD-BS FD-user, Theorem 5 FD-BS HD-user, Theorem 2 FD-BS HD-user wth self-nterference, Eqn. (3) HD-BS HD-user, Eqn. (32) umber of DL users ( ) Fg. 5. Sum DoFs for M = 6, M 2 = 8, and = Sum DoF 5 0 FD-BS FD-user, Theorem 5 FD-BS HD-user, Theorem 2 FD-BS HD-user wth self-nterference, Eqn. (3) HD-BS HD-user, Eqn. (32) umber of total users ( = + ) Fg. 6. Optmal sum DoFs for M = 6 and M 2 = 8. from (3). Therefore, for the case n whch the optmal schedulng s allowed, FD operaton at the BS s not requred n terms of DoF f there exsts self-nterference.

20 20 VII. COCLUSIO In ths paper, we have studed the sum DoFs of cellular networks wth a multantenna FD BS and HD moble users and wth a multantenna FD BS and FD moble users. For our man contrbuton, we have completely characterzed the sum DoFs of these networks. To be specfc, for achevablty, the key dea was to fully utlze the ntended sgnal dmensons by mnmzng the nter-user nterference dmensons va IA for the UL transmsson and by mnmzng the ntra-cell nterference dmensons va multantenna nullng for the DL transmsson. For converse, we have provded a matchng upper bound that shows the optmalty of the proposed scheme. As a consequence of the result, we have shown that even when nter-user nterference exsts, FD operaton at the BS can double the sum DoF over the HD only networks when the number of users becomes large enough as compared to the number of antennas at the BS, for both the FD-BS HD-user cellular network and the FD-BS FD-user cellular network. 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