Technical Report on Efficient Integration of Dynamic TDD with Massive MIMO

T echnical Report on Efficient Integration of Dynamic TDD with Massive MIMO Y an Huang ∗ , Brian Jalaian † , Stephen Russell † , and Hooman Samani ‡ The Bradley Department of Electrical and Computer Engineer ing, V irginia T ech, Blacksburg, V A, USA ∗ U.S. Army Research Laborator y , Adelphi, M D , USA † The Departmen t of Electr ical En gineering , National T aipei University , T aipei, T aiwan ‡ Abstract Recent advances in massi ve m ultiple-inpu t multiple-outp ut (MIMO) co m munication show th at equippin g base stations (BSs) with large arrays of antenna can significantly improve the performan ce o f cellu lar ne twork s. Massi ve MIMO has th e potential to mitigate the interference in the network and enhance th e av erage throug hput per u ser . On the other hand, dyn amic time di v ision duplexing (T DD), which allo ws neighb oring cells to operate with different uplink (UL) and downlink (DL) sub-frame configu rations, is a prom ising enhancement for the conv entional static TDD. Compar ed with s tatic TDD, dynamic TDD can offer mor e flexibility to accommodate various UL and DL traffic patter ns acro ss different cells, but m a y resu lt in addition al interferen ce among cells transmitting in different directions. Based on the u nique character istics and pr operties of massive MIMO an d dynamic TDD, we p r opose a marriag e of these two techniqu es, i.e., to have massi ve MIMO addr e ss the limitation of dynamic TDD in macro cell (M C) ne twork s. Sp ecifically , we advocate that th e b enefits o f dyn amic TDD c an be fully extracted in MC n etworks eq uipped with m assi ve MI MO, i.e., the BS-to- BS interference can be effectively removed by in c r easing the number of BS an tennas. W e provide detailed analysis using rando m m atrix th eory to show that the effect of the BS-to-BS interferen ce on uplink transmissions vanishes as th e numbe r of BS an tennas per-user grows infinitely large. Last but not least, we validate our analysis by num e rical simulations. K eyword s Massi ve MIMO, dynamic TDD, interfere nce suppress ion, pilot c ontamina tion, random m atrix theory . 1 Introd uction Massi ve multiple-i nput multiple-outp ut (MIMO) has attracted much attention from research community in recent years[ 1 ]–[10]. Massi ve MIMO is typically co nsider ed as a c ommunicat ion system cons isting of a base stati on (BS) equipp ed with a very lar ge ante nna array and a numbe r of sing le-ant enna or multiple-ante nna user te rminals (UTs). The number of antenna s at the BS is usua lly much lar ger than the number of U T anten nas. In prac tical scenario s, 1 there might be hundreds of BS antennas serving tens of UTs within one cell simultaneou sly [1]. An appea ling feature of massiv e MIMO is its ability of mitigat ing the interfere nce in the system and enhan cing the av erage per -user throug hput enable d by the lar ge antenna arr ay a t th e BS. In [5], M arzetta shows that the impacts of the unc orrela ted noise a nd t he i ntra-ce llular interfe rence are vanish ing with t he increasing number of BS antennas per -UT . Ho we ve r , the pilot cont aminatio n effec t is still remaining . Moreov er , pre viou s studies sho w that the transmit po wer per -antenna at the BS s could be reduced significa ntly in massi ve MIMO systems [2, 3, 4]. This featu re is appealin g to future communica tion netw orks wher e gr een iss ues wo uld be a major co ncern [11]. On the other hand, time di visio n duple xing (TDD) is widely adopted in cellular communication systems due to its fl exibility and promising features. A ke y feature of TDD is the ability to accommodate v ariou s patterns of uplink (UL) and do wnlink (DL) traf fic in cellu lar netw orks (which is not as easy in frequenc y di vision duplexi ng (FDD) syste m) [12]. There are two main T DD schemes: (i) stati c TDD and (ii) dynamic TDD. In stat ic TDD, all the neighbor ing cells must transmit in the same directi on (UL or DL) on each sub-frame accor ding to a predefined schedu le [13], [14]. W hile with dynamic TDD , each cell is allo wed to configur e UL and DL sub-frames adapti v ely based on its traf fic condit ion, s uch that adjacent cells in a netw ork are not necess arily operating in the same direct ion at a gi ve n ti me instant. The latter approa ch is also referred to as “cell-spec ific traffic adaptatio n” and “d ynamic UL- DL configuration” [13]-[19]. Dynamic T D D can potentially achie ve a highe r spectrum ef ficienc y (by intellig ently schedu ling UL and D L sub-frames for each cell based on its traf fic conditi on) compared with static TDD , where a uniqu e UL-DL c onfigura tion is employed ac ross all cells i n the netw ork [14]. Prior work has shown that the DL performance of a cellular network can be improve d by employ ing dynamic TDD [1 5]. The reason is that the DL sig nal-to -inter ference-plus-noise (SINR) of a cell would in crease when neigh- boring cells c hange their tran smission s from DL to U L, which results in a reduc tion of interfere nce experien ced by the consid ered cell (since tra nsmit power of UT s is much lower than th at of BSs). The key challe nge for imple ment- ing dynamic TDD in reali stic networks is the interfere nce managemen t for U L transmission s. In a dynamic T D D netwo rk, when a BS is receiv ing UL transmissi ons from UTs, the DL trans missions from BSs in neighbo ring cells would strongl y impact its UL SINR . Such interferen ce is called BS-to-BS in terfer ence , as it is from a BS transmittin g in DL to another BS receiv ing in U L [15]. If the BS-to-BS interf erence cannot be properly mitigated, the resul ted loss in U L perfo rmance may off set the potential benefits of dynamic TDD. Pre vious works in [15], [16], and [17] sho w that small cell (SC) depl oymen ts can actually benefit from emplo ying dynamic TDD. T his is due to the fact that the BS -to-BS interf erence is negligi ble as the SC BSs typica lly operate at lower transmit power compared to the m acro cell (MC) BS s and are usually bett er mutual ly isol ated. H owe ve r , employ ing dyna mic TDD in MC is not recommend ed without appropriat e BS -to-BS interference mitigati on tech nique s. The limita tion o f dy namic TDD in MC is a ttrib uted to h igher transmit po wer of MC BS s and the li ne-of-s ight (LoS) characteris tic of c hanne ls between MC BS s (causing stro ng BS-to-BS couplings )[15 , 18]. Based on the unique character istics and prope rties of massiv e MIMO and dynamic TDD, we propose a marriage 2 of these two techniq ues, i.e., to ha v e massi ve MIMO address the limitation of dynamic TDD in MC networks. Specifically , we advoca te that the benefits of dyna mic TDD can be fully extrac ted in MC netwo rks equippe d with massi v e MIMO, i.e., the BS -to-BS interfere nce can be effecti vely re mov ed w hen the number of BS antennas is very lar ge. The main contri b ution s of this paper are as foll o ws. • T o the best of our kno wledge, this is the first paper that considers emplo yin g massi ve MIMO to reap the benefits of dynamic TDD and address its limitation s in M C networks. Pre vious works on massi ve MIMO mainly considered static TDD, which cannot accommodate une ve n UL and DL traf fic demands across diffe rent MCs. On the other hand, studies on dynamic TDD con cluded that MC netwo rks is not suitable for operati ng with dynamic TDD. In contras t, our analysi s rev eals the potential benefits from the marriage of both techn iques to MC netwo rks. • By using the random matrix m ethod s, we deri ve deter ministic approximation s of the B S-to-BS interfe rence and per -user achi e v able rate in dynamic TDD networks . Based on the result, we sho w that the impact of the BS-to-BS inter ferenc e on UL transmis sions van ishes as the numbe r of BS antenn as per -UT gro ws infinitely lar ge. • W e sho w th at d ynamic TDD in massi ve MIMO can inc rease t he p er -use r av erag e achie vable rate s in both UL and D L. • W e conduct numerical simulatio ns to veri fy that a dynamic TDD netw ork with massi v e MIMO achie ves higher throug hput in both DL and UL compared with a st atic TDD netwo rk. The remaind er of this paper is org anized as follo ws. In Section 2, we expla in our system model in detai ls. In Section 3 , we deriv e a dete rministic appro ximation of the power of BS-to-BS interferenc e and sho w tha t th is po wer decrea ses to zero as the n umber of BS antennas pe r UT in crease s in finitely . In section 4, we valida te o ur analysis by numerica l simulat ions. S ection 5 conc ludes this paper . 2 System Mo del For notation , the opera tors tr( · ) , E [ · ] , E [ ·|· ] , ( · ) ⊤ and ( · ) † repres ent trace, e xpecta tion, condition al expectati on, transp ose and H ermitian transpo se, respecti vely . lim M denote s lim M →∞ . The notation C N ( 0 , R ) stands for the circula r symmetric complex G aussia n distri b ution with mean 0 and cov ariance matrix R . T he notatio n“ a . s . − − − − → M →∞ ” repres ents almost sure con ver gence as M → ∞ . The cellular networ k consists of L MCs w ith a freque ncy reuse factor of one. Each cell contain s K single- antenn a UTs and an MC BS equipped with M antenna s. All channels are assumed to be flat-fa ding. In the time 3 C h a n n e l t r a i n i n g U L D L D L D L D L U L D L D L D L D L C h a n n e l t r a i n i n g U L U L D L D L D L U L U L D L D L D L C h a n n e l t r a i n i n g U L U L U L D L D L U L U L U L D L D L C e l l 1 C e l l 2 C e l l 3 B o t h U L a n d D L t r a n s m i s s i o n s c o e x i s t i n t h e n e t w o r k C h a n n e l c o h e r e n c e t i m e Figure 1: D ynamic cell-specific UL-DL configurations . domain, we emplo y a block -fad ing chan nel model, i.e., channe l fading is consta nt w ithin th e coheren ce time. Each indi vidual coheren ce time interv al is partitione d into two phases that are allocated for channel training and data trans- mission, respecti vely . T he tr aining and transmission phases of all L cells are sup posed to b e perfectly sy nchro nized, such that d uring a ch annel t rainin g phase , BSs woul d on ly r ecei ve trai ning si gnals from UTs. During a transmission phase, each cell can sc hedule UL and DL su b-frames based on its o wn traffic condi tion (mor e U L sub-frames wo uld be schedu led if UL traf fic is heavie r th an DL t raf fic, and vice v ersa). Thus on a giv en sub- frame, s ome of the L cells may be opera ting in U L while other s are in DL. W e exempl ify suc h ra dio fra me stru cture in Fig.1. The foc us of our analys is in this pape r is on sub -frames wher e tra nsmissio ns of b oth UL and DL coex ist in the n etwork . 2.1 Uplink Reception On a sub-f rame when transmissi ons of both UL and D L coexist in the networ k, a BS oper ating in UL rece i ves not only transmissi ons fro m UTs in UL cells, bu t also signals from BS s transmittin g in DL . Denote the set o f the indices of cells in UL by S u and cells in DL by S d . Let y ul j ∈ C M repres ents the instantaneo us recei v ed base -band signa l vec tor at BS j ∈ S u , then y ul j = √ p ul X l ∈S u K X m =1 h j lm x lm + √ p dl X n ∈S d G j n W n z n + n ul j , (1) where h j lm ∈ C M denote s the channel v ector from UT m in cell l to BS j , G j n ∈ C M × M denote s the chan nel matrix from BS n to BS j , x lm ∼ C N (0 , 1) is the independ ent transmit signal of UT m in cell l , z n ∈ C K ∼ C N ( 0 , I K ) is the inde pende nt DL data symbol v ector at BS n for the K UTs in cell n [20], n ul j ∈ C M ∼ C N ( 0 , I M ) is the recei ver noise ve ctor that is uncorrelate d with the channels and data signals , W n ∈ C M × K is the p recodi ng matrix used for th e DL tr ansmissi ons at BS n , and p ul , p dl models th e UL and DL SNR s, respecti v ely . The chann el vector h j lm is giv en as [6] h j lm = ¯ R j lm v j lm ∈ C M , (2) 4 where the determini stic H ermitian- symmetric positi ve definite matrix R j lm = ¯ R j lm ¯ R † j lm ∈ C M × M models the antenn a correlation at the recei ve r and lar ge-sca le fadi ng effect s, and v j lm ∼ C N ( 0 , I M ) is an independen t R ayleig h fad ing cha nnel v ector [21]. The chann el matrix G j n is mod eled as [7] G j n = ˘ G j n + ¯ G j n = ¯ C j n V j n ¯ T j n + ¯ G j n ∈ C M × M , (3) where ˘ G j n ∈ C M × M and the determin istic matrix ¯ G j n ∈ C M × M respec ti ve ly corr espon d to the non-line -of-sig ht (NLoS) and the line-of-sigh t (LoS) compon ents of the channel, C j n = ¯ C j n ¯ C † j n ∈ C M × M and T j n = ¯ T † j n ¯ T j n ∈ C M × M are determini stic matrices similar to R j lm that character ize the larg e-scale fadin g and spatial correlat ion structu res at recei ving a nd transmitting antenna arrays , respecti vely , and V j n ∈ C M × M is a statistic ally independen t random matrix whose entrie s are Gaussia n and indepe ndent and identi cally distrib uted (i.i.d.) with zero mean and unit varian ce. 2.2 Downlink Recep tion W e assume perfect channel reciprocit y in this paper , i.e., DL chan nels are simply Hermit ian transpose s of UL chann els. The UTs in a DL cell recei ve signals transmitted from all BSs operatin g in DL and UTs in U L cells. Let y dl ik ∈ C M denote the instan taneo us base-band recei ve sign al at UT k in cell i , where i ∈ S d . Then y dl ik = √ p dl X n ∈S d h † nik W n z n + √ p ul X l ∈S u K X m =1 g ik lm x lm + n dl ik , (4) where g ik lm ∼ C N (0 , α ik lm ) denot es the channe l response from UT m in ce ll l to UT k in cell i , α ik lm repres ents the transmit SNR and lar ge-scale fadin g on g ik lm , and n dl ik ∼ C N ( 0 , 1) is the recei v er nois e. 2.3 Channel Estimation W e consider uplin k channel estimat ion in this paper . The channe ls h j lk ’ s are estimate d by B Ss through UL pilot signal ing from UTs. Under perfe ct chan nel reciprocity , DL channe l estimate s are simply Hermitian transp oses of UL channel estimates. In each cohe rence time interv al, the chann el training phases of all L cells are assumed to be perfect ly synchro nized (see Fig.1). Thus during a channel training phase, the BS s would only recei ve training signal s from U Ts. Assume t hat a set of K mutually ort hogon al pilot sequ ences is re used in all L c ells. By c orrela ting the recei ved pilot signal with the pilot sequence cor respon ding to UT k , BS j obtains y tr j k = X l ∈S u h j lk + 1 √ p tr n tr j k , (5) 5 where the noise vec tor n tr j k ∈ C M ∼ C N ( 0 , I M ) and p tr denote s the SN R in the up link training phase. F rom y tr j k , BS j can furth er make an minimu m mea n sq uare er ror (MMSE) estimate ˆ h j j k for the cha nnel ve ctor h j j k , which is gi ve n as [6] ˆ h j j k = R j j k Q j k L X l =1 h j lk + 1 √ p tr n tr j k ! , (6) where Q j k is defined as Q j k =  P L l =1 R j lk + 1 p tr I M  − 1 . Denote Φ j lmk = R j lk Q j k R j mk . It can be sho wn that ˆ h j j k is Gaussian distrib uted as ˆ h j j k ∼ C N ( 0 , Φ j j j k ) . W ith the abo ve MMSE estimate, the chann el h j j k can be decompo sed as h j j k = ˆ h j j k + ˜ h j j k , (7) where ˜ h j j k ∼ C N ( 0 , R j j k − Φ j j j k ) is the esti mation error vector and uncorrelate d with ˆ h j j k . Furthermore, ˆ h j j k and ˜ h j j k are s tatisti cally indepe ndent becaus e the two v ectors are jointly Gaussia n distr ib uted . The correl ation matrices R j lk , C j l and T j l can be estimated by using standard co v arianc e es timation techn iques and therefore are supposed to be perfectly kno wn by BSs. The chan nel training between BSs for estimating the chann el matrices G j l is not considere d in this work , which means that the instanta neous information of B S -to-BS chann els are u nkno wn to bo th ends. 2.4 Uplink Detection and Downlink Pr ecoding In this paper , we cons ider MMSE detection and precoding for U L and DL transmission s, respec ti v ely . Denote the MMSE detection v ector c orresp ondin g to the UT k in cel l j as [6] a j k = 1 M Λ ul j ˆ h j j k = 1 M 1 M K X i =1 ˆ h j j i ˆ h † j j i + 1 M F ul j + ϕ ul j I M ! − 1 ˆ h j j k , (8) where ϕ ul j > 0 is a reg ulariz ation parameter and F ul j ∈ C M × M is a Hermitian nonne gativ e definite matrix. The MMSE precoding matrix used for DL tran smission s at BS n i s defined as W n = p λ n Ω n = p λ n K X i =1 ˆ h nni ˆ h † nni + F dl n + M ϕ dl n I M ! − 1 ˆ H nn , (9) 6 where ˆ H nn = h ˆ h nn 1 . . . ˆ h nnK i , ϕ dl n > 0 and F dl n ∈ C M × M are regu lariza tion parameters similar to those in (8 ), and λ n normaliz es the expecta tion of the transmit power per UT of B S n and is defined as λ n = K E h tr Ω n Ω † n i . (10) The parameters ϕ ul j , F ul j , ϕ dl n and F dl n could be optimized according to certain criterion , whic h is not addresse d in this work. The setting for these quantities is arbitrary and has no impact on our analy sis in Section 3. For exa mple, follo wing stan dard appr oache s for derivi ng the MMSE detecto r and precode r , one could set ϕ ul j = 1 M p ul , ϕ dl n = 1 M p dl , an d F ul j and F dl n as cov arian ce matrices of the interferenc e and erro r ter ms [6]. 2.5 Ergodic Achiev able Rates In the UL transmission phase, BS j proc esses its r ecei ved signal y ul j with the lin ear detection vector a j k to obtain an estimate ˆ x j k for t he t ransmitte d signal x j k , i.e., ˆ x j k = a † j k y ul j = √ p ul a † j k ˆ h j j k x j k + √ p ul a † j k ˜ h j j k x j k + √ p ul X l ∈S u K X m =1 m 6 = k a † j k h j lm x lm + √ p ul X l ∈S u l 6 = j a † j k h j lk x lk + √ p dl X n ∈S d a † j k ˘ G j n W n z n + √ p dl X n ∈S d a † j k ¯ G j n W n z n + a † j k n j , (11) where the decompo sition s of G j n and h j j k follo w fro m (3) and (7), respecti vely . The associat ed UL SINR γ ul j k tak es the form γ ul j k = S ul j k I ul , (1) j k + I ul , (2) j k + I ul , (3) j k + I ul , (4) j k + I ul , (5) j k , (12) 7 where S ul j k = p ul | a † j k ˆ h j j k | 2 , (13) I ul , (1) j k = p ul E  a † j k ˜ h j j k ˜ h † j j k a j k     ˆ H j j  , (14) I ul , (2) j k = p ul X l ∈S u K X m =1 m 6 = k E  a † j k h j lm h † j lm a j k     ˆ H j j  , (15) I ul , (3) j k = p ul X l ∈S u l 6 = j E  a † j k h j lk h † j lk a j k     ˆ H j j  , (16) I ul , (4) j k = p dl X n ∈S d ( E  a † j k ˘ G j n W n W † n ˘ G † j n a j k     ˆ H j j  + E  a † j k ¯ G j n W n W † n ¯ G † j n a j k     ˆ H j j  ) , (17) I ul , (5) j k = a † j k a j k . (18) The terms I ul , (1) j k , I ul , (2) j k , I ul , (3) j k , I ul , (4) j k and I ul , (5) j k stand for dif feren t categ ories of inter ferenc e and noise. Specifi- cally , I ul , (1) j k corres ponds to the chan nel est imate erro r . I ul , (2) j k and I ul , (3) j k are t he inter -UT inte rferenc e from all other UTs transmittin g in UL in the netwo rk. I ul , (5) j k is the recei ver noise. I ul , (4) j k charac terizes the BS-to-BS interfe rence caused by DL transmiss ions fro m neighb oring BSs. I ul , (4) j k is unique to dynamic TDD and will disapp ear w hen static TDD is emplo yed. The U L er god ic achie vable rat e R ul j k of UT k in cell j is defined as [6], [8] R ul j k = E h log 2  1 + γ ul j k i . (19) 8 As f or th e DL , we deco mpose th e DL recei ved signal y dl ik as y dl ik = √ p dl E h h † iik w ik i z ik + √ p dl  h † iik w ik − E h h † iik w ik i z ik + √ p dl K X m =1 m 6 = k h † iik w im z im + √ p dl X n ∈S d n 6 = i h † nik W n z n + √ p ul X l ∈S u K X m =1 g ik lm x lm + n dl ik , (20) where w ik is the k th column of W i and z ik is the k th entry of z i . By treating E h h † iik w ik i as the a v erage ef fecti ve chann el and assuming that it is perfectly kn o wn at the U T , we can writ e the DL ergod ic achie vable rat e as R dl j k = log 2  1 + γ dl j k  , (21) where the DL S INR γ dl j k = S dl j k I dl , (1) j k + I dl , (2) j k + I dl , (3) j k + I dl , (4) j k + 1 , (22) where S dl j k = p dl    E h h † iik w ik i    2 , (23) I dl , (1) j k = p dl v ar h h † iik w ik i , (24) I dl , (2) j k = p dl K X m =1 m 6 = k E h h † iik w im w † im h iik i , (25) I dl , (3) j k = p dl X n ∈S d n 6 = i E h h † nik W n W † n h nik i , (26 ) I dl , (4) j k = p ul X l ∈S u K X m =1 α ik lm . (27) I dl , (4) j k repres ents the UT -to-UT interfer ence from all UTs tr ansmittin g in UL in the net work. 9 3 The F easibility of Dynamic TDD in Massive MI MO Network s In th is sect ion, we study the f easibi lity of dyn amic TDD in massi ve MIMO MC networ ks by ta king a closer loo k at the interference exp erienc ed by e ach U L or DL link. In fa ct, there would be a trade-of f among dif feren t categori es of inter ferenc e ef fects w hen employi ng dynamic TDD in a cellular network . That is, the interf erence in one category increases while it decreases in the other . T o illustr ate this trade- off, consider a cell operatin g in U L. W ith dynamic TDD, the BS-to-BS in terfer ence I ul , (4) j k from adjace nt DL cells increases w hile the interferen ce I ul , (2) j k , I ul , (3) j k decrea ses. Similarly , for a cell operating in DL with dynamic TDD, the UT -to-UT interferen ce I dl , (4) j k from adjacen t UL cells increases while th e interfere nce I dl , (3) j k decrea ses. Practical ly , the trans mit po wer of UTs is m uch lower than that of BSs. In addition, the channe ls between UTs from diffe rent cells are more likely to be NLoS compared with inter -cel l BS-to-UT channels. Thus when using dynamic TDD, the increase of I dl , (4) j k would typ ically be less than the redu ction of I dl , (3) j k , whic h means that the DL SINR (also the DL achie vable rate) of dyna mic TDD systems could be impro v ed ov er that of static T D D systems. This will be verified by the simulation resu lts in Section 4. In UL, howe ver , becau se of the lar ge BS trans mit power and the high probabi lity that channels between BSs ha v e LoS componen ts, the interference I ul , (4) j k might be much more significant than the redu ction of I ul , (2) j k and I ul , (3) j k . That is, the UL SINR might dete riorate with the a pplica tion of dynamic TDD. In order to in vestigat e the UL performan ce of massi ve MIMO network s with dynamic TDD, we present an asympto tic analys is of the BS-to-BS interferen ce. The fo llo wing a ssumptio ns are m ade: Assumption 1: The antenn a correlat ion matrices R j lk , C j l and T j l are non neg ati ve definite an d ha ve uni formly bound ed spectral norms on M , i.e., lim sup M k R j lk k ≤ R < ∞ ∀ j, l, k , lim sup M k C j l k ≤ C < ∞ ∀ j, l , and lim sup M k T j l k ≤ T < ∞ ∀ j, l . Assumption 2 : The traces of the antenna correlat ion matric es R j lk , C j l and T j l are scaled up with r espect to M , such that lim inf M 1 M tr R j lk ≥ ǫ R > 0 ∀ j, l , k , lim inf M 1 M tr C j l ≥ ǫ C > 0 ∀ j, l and li m inf M 1 M tr T j l ≥ ǫ T > 0 ∀ j, l . Assumption 3: The prod uct L oS channel matrices 1 M ¯ G † j n ¯ G j n ha v e uniformly bounded spectra l norms on M , i.e., li m sup M    1 M ¯ G † j n ¯ G j n    ≤ G < ∞ ∀ j 6 = n . Assumption 4: The parameter matrices F ul j and F dl n of the MMSE detector and the R ZF precoder are Hermi- tian nonne gativ e definite and ha ve unifo rmly bounded spectra l norms on M , i.e., lim sup M k F ul j k < ∞ ∀ j and lim sup M k F dl n k < ∞ ∀ n . 10 These assump tions are general in studies based on rand om matrix method s [6, 7, 8, 10 ]. Assumption 1 implies the conser v ation of cha nnel ener gy [10], a nd ena bles the ap plicat ion of random matr ix method s. T he p hysica l meaning of Assumption 2 is that the ranks of correlatio n matrices, which represe nt degr ees of freed om (DoF) of chann els, increa se at least propo rtiona lly w ith respect to M . Assu mption 3 is necessary for the asymptotic analysis of the BS-to-BS interferenc e, and holds true when the channel model for LoS chan nel compon ents and design criteri a of antenn a ar rays propose d in [22], [23] and [24] are applied. Under Assumption 3, we model the LoS mat rices ¯ G j n as  ¯ G j n  r,c = α 1 / 2 e i φ r c r , c ∈ { 1 , 2 , . . . , M } , (28) where α represent s lar ge-sca le fadin g, and φ r c is the phase of the ( r, c ) th entry . It has been sho wn that in certain condit ions the columns of ¯ G j n can be orthogon al, such that ¯ G † j n ¯ G j n = αM I M and    1 M ¯ G † j n ¯ G j n    = α < ∞ . In this case, ¯ G j n ’ s are full-r ank M × M matrice s. A ssumption 4 holds since the matrices F ul j and F dl n are typica lly linear combinations of R j lk , T j l , C j l and Φ j j j k (see, e.g., Eq. (12) in [6]) whose spectral norms are unif ormly bound ed on M under Assumpti on 1. 3.1 Deterministic A ppr o ximation of the BS-to-BS interfer ence In the follo wing proposit ion, we prov ide a deter ministic approximat ion ¯ I ul , (4) j k of the BS -to-BS interfere nce I ul , (4) j k based o n random matrix methods Lemma 1 an d Lemma 2 (s ee A ppendi x B). Pr oposition 1: Let Assumpti ons 1-4 hold and M → ∞ while K/ M < ∞ . T hen, I ul , (4) j k − ¯ I ul , (4) j k a . s . − − − − → M →∞ 0 , where ¯ I ul , (4) j k is gi ven as ¯ I ul , (4) j k = p dl  1 + 1 M tr Φ j j j k Ψ j  2 × X n ∈S d ¯ λ n M ( 1 M K X m =1 1 M tr ¯ G † j n ¯ Ψ ′ j k ¯ G j n ¯ Γ ′ nm  1 + 1 M tr Φ nnnm Γ n  2 + " K X m =1 1 M tr Φ nnnm Γ ′ j n  1 + 1 M tr Φ nnnm Γ n  2 # 1 M tr Φ j j j k Ψ ′ j n ) , (29) where ¯ λ n = K P K m =1 1 M tr Φ nnnm ¯ Γ ′ n ( 1+ 1 M tr Φ nnnm Γ n ) 2 , (30) 11 and 1. Γ n = Γ  ϕ dl n  is giv en by Lemma 1 for R i = Φ nnni ∀ i an d S = F dl n / M , 2. Γ ′ j n = Γ ′  ϕ dl n  is giv en by Lemma 2 for R i = Φ nnni ∀ i , S = F dl n / M and Θ = T j n , 3. ¯ Γ ′ n = Γ ′  ϕ dl n  is gi ven by Lemma 2 for R i = Φ nnni ∀ i , S = F dl n / M and Θ = I M , 4. ¯ Γ ′ nm = Γ ′  ϕ dl n  is giv en by Lemma 2 for R i = Φ nnni ∀ i , S = F dl n / M and Θ = Φ nnnm , 5. Ψ j = Γ  ϕ ul j  is giv en by Lemma 1 for R i = Φ j j j i ∀ i an d S = F ul j / M , 6. Ψ ′ j n = Γ ′  ϕ ul j  is gi ven by Lemma 2 for R i = Φ j j j i ∀ i , S = F ul j / M and Θ = C j n , 7. ¯ Ψ ′ j k = Γ ′  ϕ ul j  is giv en by Lemma 2 for R i = Φ j j j i ∀ i , S = F ul j / M and Θ = Φ j j j k . Follo wing similar approaches, we can deri v e de terminis tic approximati ons for all other terms in (12), which are omitted in this paper to sav e space since our focus is on the BS-to-BS interferenc e. R eaders w ho are interes ted in those results m ay refer to [6]. By repla cing all terms in (12) with their determini stic appro ximatio ns, we obtain an approx imation of the UL S INR denoted by ¯ γ ul j k . Then the UL er godic achiev able rate R ul j k can be app roximate d by ¯ R ul j k = log 2  1 + ¯ γ ul j k  . In S ection 4, we will use nu merical results to v erify the accurac y of ¯ I ul , (4) j k and ¯ R ul j k . 3.2 Pote ntial Impro vemen t of UL Perf ormance The channel model consider ed in the abov e analysis is general in the sense that vario us prop agatio n charact eristic s and per -cha nnel spatial correlations can be encompasse d. Howe ve r , it is difficu lt to physically understand and to gain insigh ts fr om (29) under such chann el mod el. Next, we consider a simplified case where all channels are assumed to be uncorrelate d. Specifically , let ¯ R j j k = I M , ¯ R j lk = √ α I M , ¯ C j l = √ α I M , ¯ T j l = I M for ∀ l 6 = j , where α ∈ (0 , 1 ] repres ents the relati ve strength of the inter- cell interfe rence. Additional ly , the LoS m atrices ¯ G j l ’ s are modeled as (28), which sat isfies 1 M tr ¯ G † j l ¯ G j l = αM . W ith these ass umption s, results of L emma 1 an d Lemma 2 can be gi ven in closed form [8], and we ha ve th e fo llo w ing pro positi on: Pr oposition 2: Under the simplified channel m odel F dl n = 0 ∀ n , F ul j = 0 ∀ j and ϕ ul j = ϕ > 0 ∀ j , ¯ I ul , (4) j k is gi ve n as ¯ I ul , (4) j k = 2 p dl L dl αη τ ′ K (1 + τ ) 2 M , (31) 12 where L dl is the number of cells operating in DL, and η = 1+ α ( L − 1)+ 1 p tr , τ = 1 − ϕη − κ + √ ϕ 2 η 2 +( κ − 1) 2 +2 ϕη ( κ +1) 2 ϕη , τ ′ = (1+ τ ) 2 κ 2 − κ + ϕ 2 η 2 (1+ τ ) 2 +2 ϕηκ (1+ τ ) , with κ = K M . W e can see from (31) that the BS-to-BS interfere nce ¯ I ul , (4) j k is a function of t he ratio K/ M . If K and M increase with a fixed ra tio c ∈ (0 , 1) , i.e., the number of BS antenn as per -UT is a constant c , t he BS-to-BS interference does not v anish as M grows, i.e., ¯ I ul , (4) j k → 2 p dl L dl c αη ¯ τ ′ (1+ ¯ τ ) 2 , where ¯ τ and ¯ τ ′ are obt ained by subs titutin g c for κ in τ and τ ′ . Howe ver , if K is fi xed or incr eases much more slo wly than M such that K / M → 0 , we ha ve ¯ I ul , (4) j k → 0 , i.e., the B S-to-BS interferen ce can be remov ed by increa sing M to infinity . In f act, the v anishing ef fect of the BS-to-BS interferenc e also hol ds und er the general cha nnel mod el. Pr oposition 3: Under the gene ral channe l model, if there e xists some δ > 0 such that lim inf M ϕ ul j > δ > 0 f or all j ∈ S u , then both NL oS and LoS componen ts of the B S-to-BS interfer ence I ul , (4) j k con ver ge to zero almost surely as M → ∞ while K kee ps con stant. Pr oof. The pr oof of Propositi on 3 is gi ve n in Appendi x A. Pre vious work on m assi ve MIMO systems has shown that as the number of antennas grows to infinity , all ef fects of channel estimati on error I ul , (1) j k , inter ferenc e I ul , (2) j k and noise I ul , (5) j k v anish except for the inter- cell interfere nce I ul , (3) j k due to pilot contaminati on [5], [6]. T his means that the interferenc e from a UL cell cannot be entirely remov ed by increasing the number of BS antennas. Howe ve r , our analysis sho ws that the BS-to-BS inte rferen ce from a DL cell wo uld v anish completely when K/ M → 0 . Thus for a massiv e MIMO network, dy namic TDD can potentially impro ve achie vable rat es o f UL cel ls a s their a djacen t inter fering cell s may change transmissio n directio ns from UL to DL and their recei ved int erferen ce would decrease. 4 Numerical Results In this sectio n, we v alidat e our analysis in S ection 3 by simulation results. The netwo rk setting is as follo w s. An inter -cell interfer ence coef ficient α = 0 . 1 is used to model the lar ge-sc ale fadi ng between neig hborin g cells. The lar ge-s cale fad ing between a BS an d UT s in i ts cell is normalized to one . W e employ the e xponential mod el [25, 26] for antenna correlation matrices. The correlation between adjace nt antennas is denoted by β . W e consider L = 7 and K = 10 . Other pa rameters are set a s p tr = p ul = p dl = 6 dB , ϕ ul j = 1 /p ul , ϕ dl n = 1 /p dl , and F ul j = F dl j = 0 . In Fig. 2, we compare DL per -user achiev able rates w ith dyn amic TDD and s tatic TD D. W e ca n se e that the DL perfor mance is the wors t w hen static TDD is emplo yed (all cells are in DL). It is impro ve d as the number of UL cells increase s. In additio n, the gap among rates with dynamic T DD and static TDD becomes large r as M increases. 13 50 100 150 200 250 300 350 400 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 Number of antennas M Average rate per−user (b/s/Hz) K=10, α =0.1, β =0.4, L=7, p tr =p ul =p dl =6dB 4 DL, 3 UL 5 DL, 2 UL 6 DL, 1 UL 7 DL, 0 UL Figure 2: D L per-u ser ach ie v able rat es und er di f ferent UL/DL cell assignments with antenna corre lation β = 0 . 4 . 50 100 150 200 250 300 350 400 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9 3.1 3.3 Number of antennas M Average rate per−user (b/s/Hz) Approx.−7 UL, 0 DL Approx.−6 UL, 1 DL Approx.−5 UL, 2 DL Approx.−4 UL, 3 DL Simul.−7 UL, 0 DL Simul.−6 UL, 1 DL Simul.−5 UL, 2 DL Simul.−4 UL, 3 DL K=10, α =0.1, β =0.4, L=7, p tr =p ul =p dl =6dB (a) β = 0 . 4 50 100 150 200 250 300 350 400 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9 3.1 3.3 3.5 Number of antennas M Average rate per−user (b/s/Hz) Approx.−7 UL, 0 DL Approx.−6 UL, 1 DL Approx.−5 UL, 2 DL Approx.−4 UL, 3 DL Simul.−7 UL, 0 DL Simul.−6 UL, 1 DL Simul.−5 UL, 2 DL Simul.−4 UL, 3 DL K=10, α =0.1, β =0.2, L=7, p tr =p ul =p dl =6dB (b) β = 0 . 2 50 100 150 200 250 300 350 400 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 Number of antennas M Average rate per−user (b/s/Hz) Approx.−7 UL, 0 DL Approx.−6 UL, 1 DL Approx.−5 UL, 2 DL Approx.−4 UL, 3 DL Simul.−7 UL, 0 DL Simul.−6 UL, 1 DL Simul.−5 UL, 2 DL Simul.−4 UL, 3 DL K=10, α =0.1, β =0, L=7, p tr =p ul =p dl =6dB (c) β = 0 (uncorrelated) Figure 3: Comparis on of UL per -use r achie vab le rates under dif ferent UL/DL cell assi gnments and antenna corre- lation s. 50 100 150 200 250 300 350 400 450 500 550 600 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 Number of antennas M Power of the interference Approximations Simulations K=10, α =0.1, β =0.4, L=7, p tr =p ul =p dl =6dB BS−to−BS interference from a DL cell Inter−cell interference from a UL cell (a) β = 0 . 4 50 100 150 200 250 300 350 400 450 500 550 600 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Number of antennas M Power of the interference Approximations Simulations K=10, α =0.1, β =0.2, L=7, p tr =p ul =p dl =6dB BS−to−BS interference from a DL cell Inter−cell interference from a UL cell (b) β = 0 . 2 50 100 150 200 250 300 350 400 450 500 550 600 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 Number of antennas M Power of the interference Approximations Simulations K=10, α =0.1, β =0, L=7, p tr =p ul =p dl =6dB BS−to−BS interference from a DL cell Inter−cell interference from a UL cell (c) β = 0 (uncorrelated) Figure 4: Comparison be tween the B S-to-BS i nterfe rence from a DL cell and the inter -cell interference from a U L cell under dif fere nt ante nna corre lations . These results support our conclusi on that dynamic TDD can achie v e better DL performance compar ed with static TDD. Fig. 3 sho ws the UL per -user achiev able rates and their determinist ic approximatio ns as functions of M under dif fere nt ante nna correlations . W e can see that when M is relati ve ly small, the UL performance of static TDD 14 (all adjacent cells operatin g in UL) out performs that of dynamic TDD (some adjacent cells oper ating in UL while others operating in DL ). The reason is that in this reg ion, the B S-to-BS inter ferenc e from a DL cell is still more significa nt than the inter -cell int erferen ce from a UL cell. Thus the more adjac ent DL ce lls t here are , the poorer UL perfor mance the conside red cell would hav e. As M increase s, the gaps among rates of dynamic TDD and static TDD are shrinking. The intersection of the rates in each figure corresponds to the point where dynamic TDD and static TDD achie v e the same UL performance . At those inters ection s, the BS-to-BS interfer ence from a DL cell is identica l to the inter -cell interfere nce from a UL cell. Thus it makes no dif feren ce whether neigh borin g cells are operating in UL or DL. Denote the the number of anten nas at the intersecti on as M ∗ . From the figures, we observ e that M ∗ is smaller with lo w er anten na correlation β . Specifically , M ∗ = 380 , 240 , 200 for β = 0 . 4 , 0 . 2 , 0 , respecti vely . In the rang e of M > M ∗ , the redi rection of UL transmissions in adjacent cells can reduce the ov erall interferen ce po wer to the conside red cell, such that dynamic TDD can improve the UL perfor mance. When M = M ∗ , dyna mic TDD is still more benefici al to the network than static TDD since it can offe r the flexibility to accommodate diff erent UL/DL traffic patterns acr oss cel ls. W e no w tak e a closer look at the interfe rence from a singl e cell. Fig . 4 compares the post-d etectio n po wers of the BS-to-BS interf erence from a D L cell and the inter -cell interference from a U L cell under dif ferent antenna correla tions. W e can observ e the follo wing: (i) the po w ers of the BS-to-BS interferen ce decline dramatical ly with the increa se of M , which valid ates Propositi on 2 and 3, (ii) When M is rel ati v ely small, the BS-to-BS interfer ence is much more significant than the UL inter -cell interferen ce, bu t the gaps between the rates are decreasing rapidly with the growth of M , and (iii) When M is large enough, the BS-to-BS interfer ence becomes weaker than the UL inter -cell interferen ce, and the gaps grow large r as M increas es. The reason is that in contrast to the BS-to- BS interfe rence which van ishes with infinite M , some componen ts of the UL inter -cel l interfe rence that represen t the pilot contaminatio n effec t would not vani sh as M gro ws [5 ]. In addition , in each case of antenna corre lation , the number of antennas at the point where inter ferenc e po wers intersect perfec tly matches M ∗ . This explain s the interse ctions of rates in F ig. 3. 5 Conclusion In this w ork, we proposed a marr iage between mass i ve MIMO and dynamic TDD. Massiv e MIMO has t he potential to address ke y limitati ons of dynamic TDD , i.e., the poten tial increase in interferen ce and loss of UL perfo rmance. W e pro vided detailed analysi s based on random matrix theory to show that the eff ect of the BS-to-BS interfere nce on uplink transmissions v anish es ef fectiv ely as the number of BS an tennas increases to infinity . Numerical simul ations ver ified that a dynamic TDD networ k with massiv e MIMO achie v es higher throughpu t in both DL and UL compared with a stat ic TDD network. 15 A Proof of Pr oposition 3 Pr oof. W e will sho w that both NLoS and LoS compone nts of I ul , (4) j k v anish asymptotic ally as M → ∞ while K kee ps constant. In addition to Assumption 1-4, we assume that lim inf M ϕ ul j > δ > 0 , which is vali d for most MMSE detection desi gns. Please refer to Appendix B for us eful lemmas used in the followin g proo f. A.1 The NLoS components of the BS-to-B S interferen ce Each NL oS component of I ul , (4) j k can be written as E  a † j k ˘ G j n W n W † n ˘ G † j n a j k     ˆ H j j  = a † j k Σ j n a j k , (32) where Σ j n = E h ˘ G j n W n W † n ˘ G † j n i . It can be bounded as a † j k Σ j n a j k = 1 M 2 ˆ h † j j k Λ ul j Σ j n Λ ul j ˆ h j j k ( a ) 6    Λ ul j Σ j n Λ ul j    1 M 2 ˆ h † j j k ˆ h j j k 6 k Σ j n k    Λ ul j    2 1 M 2 ˆ h † j j k ˆ h j j k ( b ) = k Σ j n k n min eig h ( Λ ul j ) − 1 io 2 1 M 2 ˆ h † j j k ˆ h j j k , (33) where Λ ul j is defined in ( 8), (a) fol lo ws from Lemma 5, and (b ) from Lemma 8. In ( Λ ul j ) − 1 = 1 M P K i =1 ˆ h j j i ˆ h † j j i + 1 M F ul j + ϕ ul j I M , because all terms 1 M ˆ h j j i ˆ h † j j i ’ s and 1 M F ul j are posit i ve semi-definite (Assumpt ion 4), their summation is also positi v e se mi-definite and thus al l its eigen valu es are nonne gati ve. From Lemma 9, we hav e eig h ( Λ ul j ) − 1 i > ϕ ul j > δ , and therefore min eig h ( Λ ul j ) − 1 i > δ. (34) Then we can obtai n, a † j k Σ j n a j k 6 1 δ 2 k Σ j n k 1 M 2 ˆ h † j j k ˆ h j j k . (35) The ne xt tw o p ropos itions would sh o w that 1 M 2 ˆ h † j j k ˆ h j j k con ver ges to zero almost su rely , and the s pectra l norm k Σ j n k is bound ed on M , respect i ve ly . 16 Pr oposition 4: Let Assumpti on 1 hold. Then, 1 M 2 ˆ h † j j k ˆ h j j k a . s . − − − − → M →∞ 0 . Pr oof. 1 M 2 ˆ h † j j k ˆ h j j k = 1 M  1 M ˆ h † j j k ˆ h j j k − 1 M tr Φ j j j k  + 1 M 2 tr Φ j j j k . (36) In (36 ), from Lemma 4 we ha ve 1 M ˆ h † j j k ˆ h j j k − 1 M tr Φ j j j k a . s . − − − − → M →∞ 0 . As for the second ter m, 1 M 2 tr Φ j j j k ( a ) 6 1 M k Φ j j j k k = 1 M k R j j k Q j k R j j k k 6 1 M k R j j k k 2 k Q j k k ( b ) = 1 M k R j j k k 2 min eig( Q − 1 j k ) , (37) where ( a ) follo ws from Lemma 6, and ( b ) fro m Lemma 8. In the e xpres sion Q − 1 j k = P L l =1 R j lk + 1 p tr I M , beca use each matrix R j lk is positi ve semi-definite under Assumption 1, the sum P L l =1 R j lk is posit i ve semi-definite as well, and therefore all eigen va lues of P L l =1 R j lk are nonne gati ve. Then from Lemma 9, we ha ve eig  Q − 1 j k  = eig( P L l =1 R j lk ) + 1 p tr > 1 p tr , an d a lso m in eig( Q − 1 j k ) > 1 p tr . Then, 1 M 2 tr Φ j j j k 6 1 M k R j j k k 2 min eig( Q − 1 j k ) 6 p tr R 2 M , (38) and therefore 1 M 2 tr Φ j j j k → 0 , (39) as M → ∞ , which brings 1 M 2 ˆ h † j j k ˆ h j j k a . s . − − − − → M →∞ 0 . (40) Pr oposition 5: Let Assumpti on 1 hold. Then th e sp ectral norm k Σ j n k is bou nded by K C T . 17 Pr oof. Denote th e i t h co lumn o f Ω n by ω n,i ∈ C M , where Ω n is de fined in (9). Then, k Σ j n k =    E h ˘ G j n W n W † n ˘ G † j n i    ( a ) =    ¯ C j n E h V j n ¯ T j n W n W † n ¯ T † j n V † j n i ¯ C † j n    ( b ) = E h tr ¯ T j n W n W † n ¯ T † j n i k C j n k = λ n E h tr ¯ T j n Ω n Ω † n ¯ T † j n i k C j n k = λ n E " K X i =1 ω † n,i T j n ω n,i # k C j n k , (41) where ( a ) follows from the decompositi on (3), and ( b ) is obtained by taking the exp ectati on w ith respect to V j n . By Lemma 5 we can obt ain K X i =1 ω † n,i T j n ω n,i 6 k T j n k K X i =1 k ω n,i k 2 2 = k T j n k tr Ω n Ω † n , (42 ) which holds with the mean as well, i.e., E " K X i =1 ω † n,i T j n ω n,i # 6 k T j n k E h tr Ω n Ω † n i . (43) Then, k Σ j n k 6 λ n E h tr Ω n Ω † n i k T j n kk C j n k ( a ) = K k T j n kk C j n k 6 K C T , (44) where ( a ) follo ws from th e de finition λ n = K/ E h tr Ω n Ω † n i . Combing P roposition 4 and 5, we can obtain a † j k Σ j n a j k 6 K C T δ 2 1 M 2 ˆ h † j j k ˆ h j j k , (45) and therefore a † j k Σ j n a j k a . s . − − − − → M →∞ 0 , (46) i.e., e ach N LoS component of the BS-to-BS interfer ence con ver ges to zero al most s urely a s M → ∞ . 18 A.2 The LoS components of the BS-to-BS interfer ence The L oS components of I ul , (4) j k will be analyz ed as follo ws. D efine Λ ul j k =    1 M K X i =1 i 6 = k ˆ h j j i ˆ h † j j i + 1 M F ul j + ϕ ul j I M    − 1 . (47) By using similar approaches as in th e pr oof of (34), w e can obtain min eig h ( Λ ul j k ) − 1 i > δ > 0 . (48) Then all eigen v alues of Λ ul j k are positi ve, and th erefore Λ ul j k is positi ve defini te. Each LoS component of I ul , (4) j k tak es the form E  a † j k ¯ G j n W n W † n ¯ G † j n a j k     ˆ H j j  = 1 M 2 ˆ h † j j k Λ ul j ¯ G j n E h W n W † n i ¯ G † j n Λ ul j ˆ h j j k ( a ) = 1 M 2 ˆ h † j j k Λ ul j k ¯ G j n E h W n W † n i ¯ G † j n Λ ul j k ˆ h j j k  1 + 1 M ˆ h † j j k Λ ul j k ˆ h j j k  2 ( b ) 6 1 M 2 ˆ h † j j k Λ ul j k ¯ G j n E h W n W † n i ¯ G † j n Λ ul j k ˆ h j j k = 1 M  1 M ˆ h † j j k Λ ul j k ¯ G j n E h W n W † n i ¯ G † j n Λ ul j k ˆ h j j k − 1 M tr Φ j j j k Λ ul j k ¯ G j n E h W n W † n i ¯ G † j n Λ ul j k  + 1 M 2 tr Φ j j j k Λ ul j k ¯ G j n E h W n W † n i ¯ G † j n Λ ul j k , (49) where ( a ) follo w s fro m Lemma 3, an d ( b ) holds true because Λ ul j k is positi ve definite such th at 1 M ˆ h † j j k Λ ul j k ˆ h j j k > 0 and  1 + 1 M ˆ h † j j k Λ ul j k ˆ h j j k  2 > 1 . In the last equation , by Lemma 4, the first term con ver ges to zero almost surely . 19 For the second term, 1 M 2 tr Φ j j j k Λ ul j k ¯ G j n E h W n W † n i ¯ G † j n Λ ul j k = 1 M 2 tr ¯ G † j n Λ ul j k Φ j j j k Λ ul j k ¯ G j n E h W n W † n i ( a ) 6 1 M 2    ¯ G † j n Λ ul j k Φ j j j k Λ ul j k ¯ G j n    tr E h W n W † n i = 1 M 2    ¯ G † j n Λ ul j k Φ j j j k Λ ul j k ¯ G j n    λ n E h tr Ω n Ω † n i ( b ) = K M 2    ¯ G † j n Λ ul j k Φ j j j k Λ ul j k ¯ G j n    6 K M 2    Λ ul j k    2    ¯ G † j n ¯ G j n    k Φ j j j k k ( c ) = K M 2 1 n min eig h ( Λ ul j k ) − 1 io 2    ¯ G † j n ¯ G j n    k Φ j j j k k ( d ) 6 p tr GR 2 K δ 2 M , (50) where ( a ) follo ws from Lemma 7, ( b ) fol lo ws from the definition o f λ n , ( c ) follows from Lemma 8, and ( d ) fol lo ws from Assumption 3, (48), (37) and (38). T hen, 1 M 2 tr Φ j j j k Λ ul j k ¯ G j n E h W n W † n i ¯ G † j n Λ ul j k → 0 (51) as M → ∞ , and thus we ha ve E  a † j k ¯ G j n W n W † n ¯ G † j n a j k     ˆ H j j  a . s . − − − − → M →∞ 0 , (5 2) i.e., e ach L oS component of the BS-to-BS interfe rence con ver ges to zero almost surel y as M → ∞ . B Useful Lemmas Lemma 1 (T heor em 1 [6]): Assume that D ∈ C N × N and S ∈ C N × N are Hermitian nonneg ati v e definite matrices and h i ∈ C N , i = 1 , . . . , n are random vect ors subjec t to C N  0 , 1 N R i  . Suppos e that spectra l norms of D and R i ’ s are u niformly bounded with respect to N . T hen for an y ρ > 0 , we ha ve 1 N tr D n X i =1 h i h † i + S + ρ I N ! − 1 − 1 N tr DΓ ( ρ ) a . s . − − − − → N →∞ 0 , 20 where Γ ( ρ ) = 1 N n X i =1 R i 1 + δ i ( ρ ) + S + ρ I N ! − 1 , and δ i ( ρ ) , i = 1 , . . . , n are defined as δ i ( ρ ) = lim t →∞ δ ( t ) i ( ρ ) , w here for t = 1 , 2 , . . . , δ ( t ) i ( ρ ) = 1 N tr R i 1 N n X m =1 R m 1 + δ ( t − 1) m ( ρ ) + S + ρ I N ! − 1 , with initi al va lues δ (0) i ( ρ ) = 1 /ρ, ∀ i . δ i ( ρ ) ’ s can be calcu lated efficien tly by the fi xed -point algori thm with guaran - teed con ver gence . Lemma 2 (Theor em 2 [6]): Let Θ ∈ C N × N be Hermitian no nne gati ve definite with unifo rmly bound ed spec tral norm with respect to N . Under the condi tions of Lemm a 1, we ha ve 1 N tr D ( n X i =1 h i h † i + S + ρ I N ) − 1 Θ ( n X i =1 h i h † i + S + ρ I N ) − 1 − 1 N tr DΓ ′ ( ρ ) a . s . − − − − → N →∞ 0 , where Γ ′ ( ρ ) = Γ ( ρ ) ΘΓ ( ρ ) + Γ ( ρ ) " 1 N n X i =1 R i δ ′ i ( ρ ) (1 + δ i ( ρ )) 2 # Γ ( ρ ) , Γ ( ρ ) and δ i ( ρ ) are gi v en by Lemma 1, and δ ′ ( ρ ) = [ δ ′ 1 ( ρ ) . . . δ ′ n ( ρ )] ⊤ which is gi ven by δ ′ ( ρ ) = ( I n − J ( ρ )) − 1 u ( ρ ) , where [ J ( ρ )] r,c = 1 N tr R r Γ ( ρ ) R c Γ ( ρ ) N (1 + δ c ( ρ )) 2 1 ≤ r , c ≤ n, [ u ( ρ )] r = 1 N tr R r Γ ( ρ ) ΘΓ ( ρ ) 1 ≤ r ≤ n. Lemma 3 (Matrix In versio n Lemma [6, 8]): Let A ∈ C N × N be Hermitian in vertible . Then, for any vecto r x ∈ C N and any sc alar τ ∈ C such that A + τ xx † is in vertible, x †  A + τ xx †  − 1 = x † A − 1 1 + τ x † A − 1 x . 21 Lemma 4 ([6, 8]): Let A ∈ C N × N and x , y ∼ C N  0 , 1 N I N  . 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