Full Duplex in Massive MIMO Systems: Analysis and Feasibility

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1 Fll Dplex in Massive MIMO Systems: Analysis and Feasibility Radwa Sltan Lingyang Song Karim G. Seddik andzhan Electrical and Compter Engineering Department University of oston TX USA School of Electrical Engineering and Compter Science Peking University Beijing China Electronics and Commnications Engineering Department American University in Cairo New Cairo Egypt Abstract In this paper we stdy fll dplex FD massive mltiple-inpt mltiple-otpt MIMO systems which are expected to reslt in a significant increase in the network s efficiency. We consider a base station eqipped with massive MIMO which can operate either in FD or half dplex D. We start by stdying the performance of the network when the nmber of antennas grows asymptotically. We derive bonds for both the plink and downlink capacities. Afterwards we find the necessary conditions to be satisfied for the base station to work in FD in terms of the nmber of transmitting antennas co-channel interference and selfinterference cancellation. Simlations reslts validate the derived capacity bonds and thresholds obtained for FD commnication. I. INTRODUCTION Next generation commnication networks are reqired to provide a significant increase in the achieved data rates. Accordingly deploying different techniqes that help give the necessary performance boost is essential. Theoretically speaking fll dplex FD can doble the data rate achieved by half dplex D transmission becase it allows each node to simltaneosly transmit and receive at the same time and freqency resorces. owever the main hrdle that limits the capacity gain is the high self-interference SI cased by the node s transmission on the node s reception. Recently the evoltion in the SI cancellation has revived the attention to FD commnication [] [3]. Additionally large scale mltipleinpt mltiple-otpt MIMO or massive MIMO [4] [7] has recently emerged as a promising techniqe that increases the spectral efficiency and commnication reliability than the traditional MIMO. The core idea of massive MIMO is to achieve network densification by largely increasing the nmber of active antennas. It is particlarly fascinated that with a large nmber of antennas the simplest form of ser detection beamforming becomes optimal. owever it mst be remarked that massive MIMO is feasible only in time division dplexing TDD systems becase of the large nmber of channels that need to be estimated for ser detection or beamforming. The advantage of TDD systems is that we only need to estimate the plink channels and obtain the downlink channels assming channel reciprocity. Conseqently it is expected that tilizing both FD and massive MIMO can greatly enhance the commnication network s capacity. owever as a reslt of the different factors that affect the FD performance we mst be very decisive when otlining the needed conditions to be satisfied for FD massive MIMO to be more efficient than D massive MIMO. Actally stdying the performance of FD and massive MIMO networks has attracted mch recent research work. In [8] an investigation of the effect of deploying massive MIMO in two-tier celllar networks is presented. Frthermore a resorce allocation problem is formlated to find the optimal sers biasing that maximizes the total system capacity. In [9] the tilization schemes of the spatial resorces which can be sed to enhance the plink UL performance are considered for large SI on the UL reception. In [0] the ratio of the transmit antennas and receive antennas in FD massive MIMO is analyzed. It is shown that the optimal antenna ratio converges to the ratio between the nmbers of DL sers and UL sers. In this paper we consider a single cell network with one massive MIMO base station BS which can operate in FD or D and mltiple downlink DL and UL sers. To the best of or knowledge all the previos work that addressed FD massive MIMO was concerned abot deriving a ratio between the transmitting and receiving antennas that maximizes either the UL or DL FD capacity [0] []. Therefore or motivation is to compare both the D and FD massive MIMO performance and determine the necessary conditions to be satisfied nder which it will be beneficial for the network to operate in the FD mode. Or contribtions in this paper are smmarized as follows: Investigating the difference between the different transmission modes in terms of the received power co-channel interference CCI and mltiser interference MUI. Analyzing the asymptotic network performance when the nmber of antennas grows. Deriving pper and lower bonds for the DL and the UL capacities respectively for both FD and D commnications.

2 hence it will consider the BS receiving antennas as additional transmission directions on which the transmitter shold limit the interference. Based on the previos assmptions we are going to illstrate the difference between both the FD and D operations in the massive MIMO setting. A. FD Transmission Fig.. System Model. Determining the necessary conditions to be satisfied for the FD mode to otperform the D mode. The remainder of this paper is organized as follows. The system model is presented in Section II. The asymptotic performance analysis is presented in Section III. Nmerical reslts are presented in Section IV. Finally the paper is conclded in Section V. II. SYSTEM MODEL We consider a single cell network 2 with N antennas BS which operates either in FD or D and serves mltiple single antenna sers. In the case of the FD transmission the BS will assign N t antennas for the DL transmission and N r antennas for the UL reception sch that N = N t + N r. On the other hand in the case of D transmission the BS will assign all of its N antennas either for transmission or reception. Additionally when operating in FD each time slot will be assigned to simltaneosly serve K d DL sers and K UL sers. owever when operating in D the system will operate in TDD in which the first time slot will be assigned for the DL transmission of 2K d DL sers and the second time slot will be assigned for the UL transmission of 2K UL sers. In other words regardless of the transmission mode the network will serve 2K d DL sers and 2K UL sers in two sccessive time slots. The system model is shown in Fig.. For massive MIMO deployment it is assmed that N t K d and N r K. Frthermore it is assmed that in DL transmission the BS deploys the ZF precoder. owever it mst be mentioned that in case of deploying FD the ZF precoder will accont for the large SI from the BS transmitting antennas and 2 In this paper we consider single cell networks. In the ftre work we are extending the analysis to consider mlticell networks after taking into accont additional sorces of interferences that will be introdced from neighboring cells [2] [3]. As mentioned before in the case of FD transmission each time slot will simltaneosly serve the transmission of K d DL sers and K UL ser. Accordingly the received signal-tointerference noise ratio SINR of the kd th DL ser is given by Γ DL FD = σ 2 + P d h kd w kd 2 d P d h kd w j 2 + j= k = j =k d P h kd k 2 where P d is the DL transmission power h kd C Nt is the channel vector between the kd th DL ser and the BS 3.All channel coefficients are assmed to be independent and identically distribted i.i.d. zero mean complex Gassian random variables with nit variance i.e. Rayleigh flat fading channel model. w kd C Nt is the precoding vector for the kd th DL ser s data. In this paper we consider the ZF beamforming in which the precoding matrix W ZF C Nt Nr+Kd is given by W ZF = t t t 2 [ ] T where t = d a C N r+k d N t is the channel matrix which is composed of the DL channel matrix d = [h h 2 h Kd ] T C K d N t and the channel between the transmitting and receiving antennas a C Nr Nt. Frthermore σ 2 is the additive white Gassian noise variance. The second term in the denominator of the last expression is the MUI on the kd th DL ser s transmission from other DL transmissions. Finally the last term in the denominator of the last expression is the CCI on the kd th DL ser s transmission from the UL transmissions where P is the UL transmission power and h kd k is the channel coefficient between the k th DL ser. Similarly the received SINR from the kth the kd th ser is given by UL ser and UL Γ UL FD = σ 2 + K P g k 2 P g i 2 + CP d a W a Throghot the paper h x denotes the channel vector between the x DL ser and the BS

3 where g k C Nr is the channel vector between the k th UL ser and the BS the second term in the denominator of the last expression is the MUI on the k th UL ser from other UL transmissions the last term in the denominator of the last expression is the self-interference power 0 C is the SI cancellation coefficient [9] [0] 4 and W a incldes all the precoding vectors corresponding to the BS receiving antennas. From-3 the total FD ergodic network DL and UL capacities per nit bandwidth are given respectively by C DL FD =2K d E{log 2 +ΓDL FD } 4 C UL FD =2K E{log 2 +ΓUL FD } It mst be mentioned that from 4 it is assmed that de to the large nmber of antennas all the DL sers SINRs will have the same ergodic capacity. Similarly we assme that all the UL sers have the same ergodic capacity [4]. B. D Transmission As mentioned before in the case of D transmission the first time slot will be assigned for 2K d DL transmissions and the second time slot will be assigned for 2K UL transmissions. Accordingly the received SINR at the kd th DL ser is given by Γ DL D = σ 2 + 2K d P d h D k d wk D d 2 P d h D k d j= j =k d w D j 2 5 where h D k d C N is the channel vector between the kd th DL ser and the BS in case of D transmission wk D d C N is the precoding vector for the kd th DL ser s data in the case of D transmission. The precoding matrix for the D transmission is given by W D ZF = D D D 6 where D C 2Kd N is the total DL channel matrix for the D transmission the smmation in the denominator of 5 is the MUI on the kd th DL ser s transmission from other DL transmissions. Similarly the received SINR at the k th UL ser is given by Γ UL D = σ 2 + 2K P g D k 2 P g D i 2 7 where gk D C M is the channel vector between the k th UL ser and the BS in the D transmission the smmation in the denominator of 7 is the mltiser interference on the k th UL 4 As the vale of C decreases better SI cancellation is achieved ser from other UL transmissions in the D transmission. From 5-7 the total D ergodic network DL and UL capacities per nit bandwidth are given by C DL D =2K d E{log 2 +ΓDL D } 8 C UL D =2K E{log 2 +ΓUL D } III. ASYMPTOTIC ANALYSIS OF FULL DUPLEX AND ALF DUPLEX CAPACITY In this section we will derive pper bonds for the DL capacities for the FD mode and the D mode. Additionally we will derive lower bonds for the UL capacities of both the FD and the D modes. Before we proceed we will recall some reslts from large random matrix theory. First let p [p p n ] T and q [q q n ] T be mtally independent n random vectors. The entries of p and q are i.i.d. zero-mean random variables sch that E{ p i 2 } = σp 2 and E{ q i 2 } = σq 2 i = n. Then n p p a.s n σ2 p n p q a.s 0. 9 n Second if B = AA wherea is an m n random matrix whose colmns are zero-mean independent complex Gassian vectors with covariance matrix I. Then for n m B is a central complex Wishart matrix with probability distribtion B Wn I wheren indicates the degrees of freedom. For a central Wishart matrix B Wn I with n m E [ tr{b } ] = m n m. 0 Finally assme B is an m n random matrix whose colmns are zeromean independent complex Gassian vectors with identity covariance matrix. It is shown in [5] that as m grows withot bond we have m B B a.s I n where I n is an n n identity matrix. Based on the identity described in the precoding matrices W ZF andwzf D can be approximated respectively as Proposition by Ŵ ZF = t Ŵ D ZF = D 2K d. 2 An pper bond for the FD DL rate ˆC DL FD is given 2 Nt ˆC DL FD =2K d log 2 λ d Ψλ 3 where λ d = P d σ 2 λ = σ2 P and Ψλ is given by [ λ Ψλ = K 2 e λ E i λ K! 4 + l! K l λ ] l l=

4 where E i λ = λ e x /x dx. Proof: From 4 and by applying the Jensen s ineqality to the concave fnction we can get C DL FD ˆC DL FD { } =2K d log 2 +E ΓDL FD 2 Nt =2K d log 2 λ d { } 5 E + λ h kd k 2 k = 2 Nt =2K d log 2 λ d Ψλ where the second eqality is obtained by sbstitting 2 and 9 in. As a reslt the expectation of the main signal is given by } 2 E {P d h kd w kd 2 Nt = P d 6 and the mltiser interference { will vanish. } Afterwards it was proved in [0] that E =Ψλ defined Proof: By applying the Jensen s ineqality for the convex fnction log 2 + /x we can get that C UL FD ˆC UL FD =2K log 2 { } σ2 + MUI UL FD + SI UL FD E P g k 2 =2K log 2 +ΦΦ 2 =2K log 2 P D α k σ 2 + N rk P + Nr Nr N t Nr +K d 2 CP d where MUI UL FD is the MUI on the FD UL transmission SI UL FD is the SI on the FD UL transmission Φ = E { /P D α k g k 2} and Φ 2 = E { } σ 2 + MUI UL FD + SI UL FD. The second eqality is derived from the fact that g k is independent of g i i = k and a that inflences MUI UL FD and SI UL FD respectively. The expected vale of / g k 2 can be calclated sing the Wishart matrix property illstrated in 0. The expected vale of the mltiser interference power on the UL transmission can be calclated as follows 8 E{MUI UL FD } = = K P D K P E = N rk P α i E { g i 2} { ]} tr [gi g i 9 where the last eqality is obtained by applying the identity in 9. Finally the vale of E { SI UL FD } can be derived as follows E { } SI UL FD = CPd E { aw a 2} { ]} = CP d E tr [Wa a aw a { ]} = CP d M re tr [Wa W a = N rn t 2 CP d 20 where the last eqality is obtained by applying both and 2 Proposition 3 An pper bond of the D DL rate ˆC DL D and a + K lower bond for the D UL rate ˆC UL D are given respectively by h λ kd k 2 k = 2 in 4. N ˆC DL D =2K d log 2 λ d 2 Proposition 2 A lower bond for the FD UL rate ˆC 2K d UL FD is given by P N ˆC UL D =2K log P ˆC UL FD =2K log 2 N r σ 2 + N2K P. 7 σ 2 + N rk P + Nr N t Nr +K d 2 CP d Proof: The expressions in 2 and 22 can be proved following the same steps sed in Propositions and 2. From 3-20 it is noticed that FD DL transmission is still affected by the CCI. On the other hand the FD UL transmission is still affected by the SI represented by the cancellation parameter C. Therefore it is reqired to find the conditions nder which the FD operation can otperform the D operation. Since in the practical commnication network it is always reqired to achieve more capacity in the DL transmission than the UL transmission we start by defining the necessary conditions nder which the DL FD capacity calclated in 3 is higher than the DL D capacity calclated in 2. Proposition 4 In order to have ˆC DL FD ˆC DL D the following conditions mst be satisfied NN + K d N t N +2K d Ψλ NN + K d N r <N N +2K d Ψλ ζ 3σ 2 23 where Nt and N r are the nmber of transmission and receive antennas respectively that make ˆC DL FD ˆC DL FD and ζ = E{ K P h kd k 2 } is the expectation of the CCI on the FD DL k = transmission.

5 Proof: We begin or proof by calclating the nmber of transmission antennas N t that will make ˆC DL FD ˆC DL FD. Therefore we have N t 2 2 N λ d Ψλ λ d C DL FD Simlation C DL FD Upper Bond C UL FD Simlation C UL FD Lower Bond 2K d N t Ψλ N 2K d. 24 Capacity bits/s/z After some mathematical maniplations we can get the vales of Nt and N r defined in 24. owever it mst be noted that for 24 to be valid the vale of Nt mst be smaller than N. ence we get NN + K d N +2K N Ψλ d Ψλ Therefore from the definition of Ψλ =E{/+ CCI FD }where σ 2 CCI FD is the CCI in the FD transmission and Jensen s ineqality for convex fnction we can obtain the last constraint that ζ 3σ 2. Afterwards after setting N t and N r to vales satisfying N t and N r respectively we will derive the SI cancellation threshold C that cases the FD UL capacity to be higher than the D UL capacity Proposition 5 The SI cancellation threshold C that makes ˆC UL FD ˆC UL D is given by where C =min ˆC 26 ˆC = 2 Nr σ 2 + M D N σ 2 + M FD 27 P d N rn t N where M D = E{MUI UL D } M FD = E{MUI UL FD } are the expectations of the mltiser interference on the UL transmission in the case of FD and D respectively. It mst be noted that C can not exceed the vale of one. Therefore we mst always pick the minimm vale between and ˆC. 5. Proof: The expression in 26 can be easily proved by from the expressions of ˆCUL FD and ˆC UL D. IV. NUMERICAL ANALYSIS In this section we validate the pper and lower bonds derived for the DL and UL capacities. Afterwards we verify the obtained thresholds in Propositions 4 and 5. Fig. 2 validates the vales of ˆC DL FD and ˆC UL FD derived in 3 and 7 respectively. Additionally it shows the variation of both the DL and UL FD capacities with the UL transmission power P. It can be anticipated that increasing P increases the CCI interference on the DL transmission and then C DL FD will decrease. On the other hand increasing P will increase C UL FD. This behavior can be verified from Fig. 2. owever it can be seen that increasing P cases a slight increase in C UL FD.This response can be explained from 7 as increasing P increases both 5 The reslts obtained in Preposition 5 indicate that the SI threshold is related to a weighted difference between the MUI experienced in D and FD. This relation can be easily conclded becase in order for FD to achieve better UL capacity the SI mst be canceled in a way that compensates for the difference between the MUI experienced in the two transmission modes Uplink Power P in mw Fig. 2. Variation of Fll Dplex Downlink and Uplink Capacities with Uplink Transmission Power C DL D bits/sec/z C DL D Simlation C DL D Upper Bond P d Downlink Power P d in mw C UL D bits/sec/z C UL D Simlation C UL D Lower Bond Fig. 3. Variation of alf Dplex Downlink and Uplink Capacities with Downlink Transmission Power the UL signal power and the MUI power as well. Similarly Fig. 3 validates the vales of ˆCDL D and ˆC UL D derived in 2 and 22 respectively. Frthermore it can be verified that increasing P d increases the vale of the D DL capacity. On the other hand it will have no impact on the vale of the D UL capacity. Fig. 4 validates the vale of Nt and ζ derived in 23. It shows the variation ˆC DL FD with increasing the nmber of transmission antennas N t for different vales of Ψλ. It can be seen that at Ψλ = which corresponds to the case in which there is no CCI ˆCDL FD eqals ˆC DL D at Nt = 253 which matches the vale calclated in 23. Similarly in the case of Ψλ =/2 the vale of N t = 279 matches the vale calclated in owever in the case of Ψλ =/6 which corresponds to the case in which ζ>3σ 2 it can be noticed that ˆCDL FD will be always smaller than ˆC DL D. This reslt verifies the derived threshold for CCI expectation in 23. Fig. 5 shows the variation of ˆCUL FD with SI cancellation for 6 Increasing the vale of CCI will degrade the DL SINR which will reqire increasing the transmission antennas in order to be able to compensate for that degradation.

6 Downlink Capacity bits/sec/z Uplink Capacity bits/sec/z Fll Dplex Transmission Antennas N t Fig. 4. Validating the vales of N t and ζ FD Ψ = /6 FD Ψ = /2 FD Ψ = D Cancellation Coefficient C FD P d = W FD P d = 4W FD P d = 0 W D Fig. 5. Validating the vale of C different vales of P d. The reslts shown validate the vale of C derived in 26. In case of P d = 0W the vale of C = 0.34 which matches the vale calclated in 26. Additionally decreasing P d to 4W is expected to increase ˆC UL FD and therefore the vale of C will be larger as the qality of SI cancellation can be decreased for small vales of P d. This behavior can be verified from Fig. 5 as decreasing P d from 0W to 4W increases the vale of C from 0.34 to owever frther decrease in the vale of P d will make ˆC UL FD always better than ˆC UL D and hence no cancellation will be reqired i.e. C =. V. CONCLUSION In this paper we consider a single network with a single massive MIMO base station that can operate either in fll dplex or half dplex. We analyze the asymptotic behavior of the network when the nmber of BS antennas grows and investigate the necessary conditions nder which the network operates in fll dplex mode to achieve a higher capacity than the half dplex mode. It was shown that the fll dplex downlink rate exceeds that of half dplex only if the expected vale of the co-channel interference is below a certain vale and the nmber of transmitting antennas are set above a certain threshold. Frthermore after setting the nmber of transmitting antennas it is shown that the plink fll dplex rate will exceed that of half dplex when the selfinterference cancellation is below a certain limit. Finally we provide the nmerical analysis to validate the derived capacity bonds and the reqired conditions for fll dplex operation gains. VI. ACKNOWLEDGMENT Thanks to US NSF CPS ECCS CCF CNS ECCS-4052 and NSFC64280 REFERENCES [] M. Darte and A. Sabharwal "Fll-dplex wireless commnications sing off-the-shelf radios: Feasibility and first reslts" in IEEE Conference Record of the Forty Forth Asilomar Conference on Signals Systems and Compters ASILOMAR Pacific Grove CA Nov. 200 pp [2] D. Bharadia E. McMilin and S. Katti "Fll dplex radios" in SIGCOMM ong Kong China Ag. 203 pp [3] M. Jainy J. I. Choiy T. M. Kim D. Bharadia S. Seth K. Srinivasan P. Levis S. Katti and P. Sinha "Practical real-time fll dplex wireless" in MobiCom Las Vegas NV Sep. 20 pp [4] F. Rsek D. Persson B. K. La E. Larsson T.L.Marzetta O. Edfors and F. Tfvesson "Scaling p MIMO: Opportnities and challenges with very large arrays" IEEE Signal Processing Magazine vol. 30 no. pp Jan [5] T. L. Marzetta G. Caire M. Debbah I. Chih-Lin and S. K. Mohammed "Special isse on massive MIMO" Jornal of Commnications and Networks vol. 5 no. 4 pp Ag [6] J. oydis K. osseini S. ten Brink and M. Debbah "Making smart se of excess antennas: Massive MIMO small cells and TDD" Bell Labs Technical Jornal vol. 8 no. 2 pp. 5-2 Ag [7] L. Song R. Wichman Y. Li and Z. an Convex Optimization Fll- Dplex Commnication. Cambridge University Press UK in print. [8] R. Sltan L. Song K. G. Seddik and Z. an "Users association in small cell networks with massive MIMO" in 206 International Conference on Compting Networking and Commnications ICNC. Kaai I: IEEE Feb [9] Y. Jang K. Min S. Park and S. Choi "Spatial resorce tilization to maximize plink spectral efficiency in fll-dplex massive MIMO" in IEEE International Conference on Commnications ICC London UK Jn. 205 pp [0] K. Min Y. Jang S. Park and S. Choi "Antenna ratio for sm-rate maximization in MU-MIMO with fll-dplex large array BS" in IEEE International Conference on Commnications ICC London UK Jn. 205 pp [] B. Yin M. W C. Stder J. R. Cavallaro and J. Lilleberg "Fll-dplex in large-scale wireless systems" in IEEE Asilomar Conference on Signals Systems and Compters. Pacific Grove CA: IEEE Nov. 203 pp [2] S. Goyal P. Li S. S. Panwar R. A. DiFazio R. Yang and E. Bala "Fll dplex celllar systems: will dobling interference prevent dobling capacity?" IEEE Commnications Magazine vol. 53 no. 5 pp [3] R. K. Mngara and A. Lozano "Interference srge in fll-dplex wireless systems" in 49th Asilomar Conference on Signals Systems and Compters Pacific Grove CA 205 pp [4] S. W. R. C.. M. D.. D. T. M. Slock "Large system analysis of linear precoding in correlated miso broadcast channels nder limited feedback" IEEE Transactions on Information Theory vol. 58 no. 7 pp Mar [5] T. L. Marzetta "Noncooperative celllar wireless with nlimited nmbers of base station antennas" IEEE Transactions on Wireless Commnications vol. 9 no. pp Oct. 200.