arXiv:1602.08836v1 [cs.IT] 29 Feb 2016 Aswt admylctdRH.I 1] ioilpoint binomial C- a of [10], In performance RRHs. located the randomly analyze with RANs to geometry stochastic from LI distributed. for be phenomenon will effective RRHs simple since a suppression serves path naturally combined, are loss C-RAN ante and and full-duplex use When antennas the separation. directional as shields, such electromagnetic techniques passive of using domain antenna the o iiae usatal,cncuesgicn perform [ significant degradation This cause (LI can input. interference substantially, the loopback mitigated the not to called transceiver interference, the of of form output sign the the ch from is major implementation leakage a full-duplex end, in overcome this To to lenge [8]. ex- [7], a and solution operation theory practical full-duplex efficient both make on to years design few hardware been last perimental has in There made channel. same progress the rapid on C- nodes receive radio and with Full-duplex transmit [6]. approach can [5], systems complementary implementation 5G wireless a for 4G as RAN current promise of high efficiencies shows spectral the ing unit optica baseband speed high located a centrally via backbone. processing a sophisticated with of capable to operatio them radio transmission/collection is remote connect called signal C-RAN and units for of radio (RRHs) idea distributed heads main of pool The a [4]. researchers deploy to operators interest of mobile topic and a become reason has these capital/operati concept For C-RAN deployment. reduce network cellular and ar- for expenses utilization transmissio C-RAN energy-efficiency spectral [1]–[3]. high technology component improved provide access integral can radio key chitecture 5G a future become of to acclaimed paradigm emomn eina ela h R soito scheme gains. full-duplex association the the RRH determining of in the choice role as critical the well a while mode plays as mode, full-duplex half-duplex design the and the beamforming using UL to by the compared achieved of as performanc be density significant that can and show improvements function antennas results a RRHs Numerical as RRHs. of DL schemes number association the among com of rate to sum useful average expressions the analytical derive ra- We zero-forcing/MRT(ZF/MRT) combining/maximal processing. and ratio beam- (MRC/MRT) maximum distributed transmission of tio with form the schemes in association partic forming all RRH a consider nearest We reflect distribut process. and point and Poisson to antennas a multiple to a order with according RRHs equipped (DL) In be downlink to and user. assumed (UL) uplink full-duplex the scenario, a realistic with (RR heads radio communicate remote distributed spatially where (C-RAN) hr aebe eea tde hthv anse tools harnessed have that studies several been have There nprle,fl-ulxcmuiaincpbeo boost- of capable communication full-duplex parallel, On network new a is (C-RAN) network access radio Cloud Abstract ulDpe lu-A ihUplink/Downlink with Cloud-RAN Full-Duplex § ∗ eateto lcrcladEetoi niern,Univ Engineering, Electronic and Electrical of Department eateto lcrcladCmue niern,Univer Engineering, Computer and Electrical of Department Ti ae osdr lu ai cesnetwork access radio cloud a considers paper —This ? .Taiinly Isprsini efre in performed is suppression LI Traditionally, ]. oamdl Mohammadi Mohammadali † aut fEgneig hheodUiest,Ia (e-ma Iran University, Shahrekord Engineering, of Faculty .I I. eoeRdoHa Association Head Radio Remote NTRODUCTION † ia .Suraweera A. Himal , ipate ance pare ,if ), nna Hs) al- ng ed ns re n, al s, n e l riyo eaeia r ak emi:[email protected]. (e-mail: Lanka Sri Peradeniya, of ersity Rscmuiaewt uldpe srt upr si- are support contributions follows: to Our as transmissions. summarized user DL full-duplex and UL a equipped multaneous antenna with multiple communicate which in RRHs case a consider We sion. operation, full-duplex assumed. of was However, cancellation aspect LI DL [13]. important perfect or in namely, an proposed UL neglected operation was only it Full-duplex antennas [11], considered. distributed analyzed. [10], with been been [4], have has works performance selection previous these antenna transmission In transmit ratio or maximal antenna (ZF). with multiple (MRT) a forcing network of C-RAN zero transmission equipped (DL) regularized downlink the has assuming maximization [4], constraints In out sum-rate power carried transmit weighted and been average selection [12], antenna under In antenn [3]. compare was distributed selection in In station of base performance characterized. and RRH beamforming outage the arrays, were with investigate the C-RANs achieved to [2], order for In capacity distributed strategies [11]. ergodic multi-cell in association the a studied and been of probability shadowing has capacity and system ergodic fading RRH with The best The selection analyze effects. C-RAN. channel to a framework and of analytical antenna distributions considered an was user developed (PPP) and authors process antenna point model Poisson to a and process eoeteEciennr n ojgt rnps operator transpose conjugate respectively; and norm Euclidean the denote cf fteRV functi the distribution cumulative of and (cdf) (pdf) function density aibe(RV) variable odlwrcs etr odnt vectors. denote to letters case lower bold Notation: iyo let,Cnd emi:[email protected] (e-mail: Canada Alberta, of sity hspprcniesaCRNwt uldpe transmis- full-duplex with C-RAN a considers paper This • • • xrsin o h vrg LadD aeo the of rate association RRH DL UL/DL scheme. and single (SRA) the UL for average user full-duplex the for tractable and expressions and exact decoding derive combining we ZF/MRT, linear and ratio (MRC)/MRT maximum DL namely, and schemes, precoding UL different Assuming u sn R ceerslsi oeblne rate balanced more a in DL, results or region. UL scheme toward SRA biased strongly using is but that region results rate scheme a (ARA) in association RRH all that show We cee oso h eet fteformer. the of benefits the show SRA and to ARA schemes under modes half-duplex of main- and performance full-duplex UL/DL the and the compare we between between rate Moreover, fairness transmission. balance sum of level a average acceptable power, taining system ensure LI of the can value maximizing scheme fixed a ZF/MRT for the that reveal findings Our l [email protected]) il: euebl pe aeltest eoematrices, denote to letters case upper bold use We § n hnh Tellambura Chintha and , E { x x ; } X f tnsfrteepcaino h random the of expectation the for stands X respectively; , ( · ) and F X ( · ) M eoeteprobability the denote X ∗ ( s ) stemoment the is k · k lk) ) and ( · on ) d a † , generating function (MGF) of the RV, X; Γ(a) is the Gamma We point out that in the case of full-duplex transmission, function; Γ(a, x) is upper incomplete Gamma function [14, selection of a nearest RRH is also a practical assumption, mn a1 ap Eq. (8.310.2)]; and G z | ··· denotes the Meijer G- since transmitting high power signals towards (from) distant pq b bq 1··· periphery UL (DL) RRHs in order to guarantee a quality- function [14, Eq. (9.301)].  of-service can cause overwhelming LI at U (interference II. SYSTEM MODEL between UL and DL RRHs). Similar C-RAN association We consider a C-RAN, consisting of baseband unit (BBU) schemes in context of a half-duplex user can also be found and a group of spatially distributed RRHs to jointly support in [2], [4]. a full-duplex user, denoted by U for both DL and UL transmissions. We assume that each RRH, is equipped with C. Uplink/Downlink Transmission M ≥ 1 antennas, and the full-duplex user is equipped DL Transmission: We assume that all DL RRHs transmit with two antennas: one receive antenna and one transmit with power Pb as in [4]. Hence, according to the ARA antenna. The locations of the RRHs are modeled as a scheme, the received signal at the user can be expressed as homogeneous PPP Φ = {x } with density λ in a disc k yd = Pbℓ(xi)hi†wt,isd + PuhLIsu + nd, (1) D, of radius R. We assume that p% of the RRHs, are i Φd b(o,R) deployed to assist the DL communication and (1 − p)% ∈ X∩ p p where b(o, R) denotes a ball of radius R centered at the for UL communication. Therefore, the set of DL RRHs is M 1 origin, wt,i ∈ C × is the transmit beamforming vector denoted as Φd = {x ∈ Φ: B (p)=1} where B (p) are k k k at DL RRH i, P is the user transmit power and s is the independent and identically distributed (i.i.d.) Bernoulli RVs u u user signal satisfying E s s = 1, and nd denotes the with parameter p associated with x . Similarly, the set of u u† k additive white Gaussian noise (AWGN) with zero mean. We UL RRHs is a PPP with density (1 − p)λ and is denoted  proceed with all noise variances set to one. hLI denotes the LI as Φu = {x ∈ Φ: B (p)=0}. Therefore, the number of k k channel at the user. In order to mitigate the adverse effects of DL RRHs, Nd and the number of UL RRHs, Nu in D are the LI on system’s performance, an interference cancellation Poisson distributed as Pr(N ) = µNi e µi /Γ(N + 1), with i i − i scheme (i.e. analog/digital cancellation) can be used at the i ∈{u, d}, µd = πpλR2 and µu = π(1 − p)λR2. full-duplex user and we model the residual LI channel with Rayleigh fading assumption since the strong line-of-sight A. Channel Model component can be estimated and removed [?], [7]. Since each Signal propagation is subject to both small-scale multipath implementation of a particular analog/digital LI cancellation fading and large-scale path loss. We denote the DL channel scheme can be characterized by a specific residual power, a M 1 vector from RRH i to U as h ∈ C and the UL channel 2 2 i × parameterization by hLI satisfying E |hLI| = σ allows C1 M aa vector from U to RRH i as gi† ∈ × , respectively. These these effects to be studied in a generic way [?]. channels capture the small-scale fading and are modeled By invoking (4), the DL signal-to-interference-plus-nois e as Rayleigh fading such that gi and hi ∼ CN (0M , IM ), ratio (SINR) for the user is given by where CN (·, ·), denotes a circularly symmetric complex 2 P ℓ(x )|h†w | Gaussian distribution. The path loss model is denoted by A i Φd b(o,R) b i i t,i SINRd = ∈ ∩ . (2) 2 + 2 ℓ(·): R → R . We consider a non-singular path loss model P Pu|hLI| +1 1 with ℓ(x1, x2) = α where α > 2 is the path loss Moreover, with the SRA scheme, the received SINR at the ǫ+ x1 x2 exponent and ǫ > 0 kis the− k reference distance. Further, as in user can be established as 2 [4] we assume that there exist an ideal low-latency backhaul Pbℓ(xq)|h†wt,i| SINRS q (3) d = 2 . network with sufficiently large capacity (e.g. optical fiber) Pu|hLI| +1 connecting the set of RRHs to the BBU, which performs all UL Transmission: Let us denote w ∈ CM 1 as the receive the baseband signal processing and transmission scheduling r,j × beamforming vector at the UL RRH, j. According to the for all RRHs. ARA scheme, received signal at the BBU is given by

B. Association Schemes yu = Puℓ(xj )wr,j† gj xu (4) For the system under consideration, we investigate the j Φu b(o,R) ∈ X∩ q performance of the following two RRH association schemes: ji + Pbℓ(xj , xi)w† Hudwt,isd + w† nj , All RRH Association (ARA) Scheme: All corresponding r,j r,j • i Φd b(o,R) q DL RRHs cooperatively transmit the signal, sd to the ∈ X∩  ji CM M full-duplex User, U. Moreover, all the corresponding UL where Hud ∈ × is the channel matrix between the RRHs deliver signals from U to the BBU. DL RRH i and UL RRH j consists of complex Gaussian Single Nearest RRH Association (SRA) Scheme: The distributed entries with zero mean and unit variance, nj ∼ • full-duplex User, U associates with the nearest DL RRH CN (0M , IM ) denotes the AWGN vector at the UL RRH j. and the nearest UL RRH, respectively. Without loss of Therefore, the SINR can be expressed as 2 generality, we assume that the full-duplex user, U is † A j Φu b(o,R) Puℓ(xj )|wr,j gj | SINR ∈ ∩ (5) located at the origin of D. Therefore, the associated u = 2 , Iud + kwr,jk UL RRH p and DL RRH q for user U are given by P where p = argmaxi Φu ℓ(xi) and q = argmaxi Φd ℓ(xi), 1 ∈ ∈ ji 2 respectively. Iud = Pbℓ(xj , xi)|wr,j† Hudwt,i| . j Φu b(o,R) i Φd b(o,R) 1Our results can also be easily extended to an N nearest RRH association ∈ X∩ ∈ X∩ scheme, where User U associates with the N nearest DL and UL RRHs According to the SRA scheme only one UL (nearest) RRH among the total Nd (Nu) DL (UL) RRHs. and one DL (nearest) RRH are selected to assist the full- duplex user. Let the sub-indexes p and q correspond to the conduct our analysis in an amicable way to present useful active UL and DL RRH, respectively. Therefore, the SINR insights into the performance of the considered network. at the BBU is given by A similar assumption can be found in [2]. The following P ℓ(x )|w g |2 proposition establishes an upper bound to the average DL SINRS u p r,p† p u = pq . (6) rate, R¯d for non-singular and standard singular path loss P ℓ(x , x )|w† H w |2 + kw k2 b p q r,p ud t,q r,p models. In the next section, we consider different processing schemes for transmit and receive beamforming vectors and Proposition 2. The average DL rate achieved by the ARA characterize the system performance using the UL and DL scheme with MRT processing can be upper bounded as M average sum rate given by ∞ ∞ zPb − ¯d FD R = 1−exp −2πpλ 1− 1+ α dx R = Ru + Rd, (7) ǫ+kxk sum Z0 Z0   ! !! i i where Ru = E ln 1+ SINRu , Rd = E ln 1+ SINRd exp(−z) × dz. (9) with i ∈ {A, S} are the spatial average UL and DL rates, z(1 + P σ2 z) 

u aa respectively. Moreover, for ǫ → 0 (i.e., the standard singular path loss model) the average DL rate can be upper bounded as III. PERFORMANCE ANALYSIS ∞ δ k (G(M, δ, pλ)Pb ) 12 2 1−δk, 0 R¯d = G P σaa , (10) In this section, UL/DL average rates provisioned under the Γ(k + 1) 21 u 0 k considered RRH association schemes are evaluated. We also X=1   δπpλ present UL/DL average rates for a half-duplex user, which where G(M, δ, pλ)= Γ(M) Γ (M + δ) Γ (− δ). serves as a benchmark for performance comparison and to Proof: By using [15, Lemma 1], Rd can be expressed illustrate the gains due to full-duplex operation. as ∞ exp(−z) A. Average Downlink Rate Rd = MY (z) (1 −MX (z)) dz, (11) z Z0 We consider MRT processing at the DL RRHs and set 1 h where MY (s)= 2 and w = i . In the sequel, we will investigate the average 1+Puσaas t,i hi k k 2 DL rate for the ARA and SRA schemes. E Pbkhik MX (s)= Φd exp −s α ARA Scheme: In this case, the received SINR at U is ǫ + kxik 2 ( i Φd !) given by (2). For notational convenience, we denote δ = α , X∈ 2 2 X = P X with X = ℓ(x )kh k and Y = (a) sPbkhk b xi Φd b(o,R) i i i i E E 2 ∈ ∩ = Φd h exp − α Pu|hLI| . The following proposition provides the average DL ǫ + kxik (i Φd   ) P Y∈ rate achieved by the full-duplex user with the ARA scheme M (b) ∞ sP − and MRT processing. = exp −2πpλ 1− 1+ b dx . (12) ǫ + kxkα Proposition 1. The average DL rate achieved by the ARA Z0   ! ! 2 2 scheme with MRT processing can be expressed as In (12) (a) follows from the fact that khik are i.i.d and Nd Nd Nd also independent from the point process Φd and (b) holds ∞ (−1)k Rd = ··· (8) due to the probability generating functional (PGFL) for a k! 2 Nd=1 k=1 n1=1 nk=1 PPP [16] and by using the MGF of khk which is chi-square X X X X distributed with 2M degrees of freedom.3 n1=n2 =nk 6 ···6 By converting the integral from Cartesian to polar coordi- k Nd d ∞ exp(−z) | {z } µd exp(−µ ) nates, MX (s) in (12) for ǫ→ 0 can be further simplified as × 2 Xnℓ (Pbz)dz , z(1 + Puσaaz) Γ(Nd + 1) Z0 ℓ=1 ! Y M (s)= exp G(M, δ, pλ)(sP )δ . (13) where X b M 1 i i j Accordingly, by substituting (13) into (11) we obtain δ − i ǫ − s
δ Xℓ(s)= 2 i (j+δ)+1 ∞ 1−exp G(M, δ, pλ)(zPb) exp(−z) R j Γ(i + 1) (s + 1) − ¯ (14) i=0 j=0  Rd = 2 dz. X X 0 z(1+Puσaaz) Rα j + δ − i, 1 Z

× G12 . In order to simplify (14), we adopt a series expansion of 22 s +1 j + δ, 0   the exponential term. Substituting the series expansion of Proof: See Appendix A. exp G(M, δ, pλ)(zP )δ into (14) and then using 1 = b 1+cxk We remark that the average DL rate is an increasing 11 k 0 G11 cx | 0 , yields function of the cell radius. This follows from the fact that
∞ δ k
(Pb G(M, δ, pλ)) increasing the cell radius also increases the effective density R¯d = (15) Γ(k + 1) (and consequently the number) of the RRHs which serve the k X=1 user. However, the gains become marginal after a certain ∞ δk 1 11 2 0 value of R, since the received power from distant RRHs × z − exp(−z)G11 Puσaaz dz. 0 0 becomes negligible. In fact, it can be shown that as R Z   To this end, using [14, Eq. (7.813.1)] we obtain the closed- attains a large value, the average DL rate saturates and form expression for R¯d as given in (10). becomes independent of R (cf. Section IV). Therefore, we SRA Scheme: For this scheme, the average DL rate, Rd is let R → ∞, which corresponds to case where all DL RRHs given in the following proposition. of Φd participate in DL transmissions, since it allows us to 3 2 In what follows, we will use the notation x ∼ χ2K to denote that x is 2 The average DL rate is zero for the case of Nd = 0. a chi-square distributed RV with 2K degrees-of-freedom. Proposition 3. The average DL rate achieved by the SRA Proof: The proof is omitted due to space limitations. scheme with MRT processing can be expressed as ∞ Nd R α M−1 α Proposition 4. The average UL rate achieved by the SRA µ µd r +ǫ d exp(− ) P r +ǫ Rd = e b AnEM−n scheme with MRC/MRT processing can be expressed as Γ(Nd + 1) Pb 2n N =1 0 n=0 2R Xd Z X   ∞ ut 2nς ∆(2n, 0), ∆(1, ∆(m, 0)), 1 1 1 Ru =µ Gvu Puσaa2 0 0 z ∆(1, ∆(2n,M)), 0 + e B0E1 2 fkxq k(r)dr, (16) ! Puσaa Z Z     2 α M 32 σaar 0, 1,M,M exp(−z) where En(·) is exponential integral [14, Eq. (8.211)], An = n × G44 fdud (r)drdz, (22) rα ǫ dn α 1 + −1 Pbz 1, 1,M, 0 z lim (r+ǫ) n 2 , and B0 = 1 − z→− n! Pb dz z(1+Puσaaz)    Pb √nπ −M where , , , and Pb 1    µ =2ζ (2π) n t = m +2n v = t +1 u =2n +1 α −1 . Moreover, Pu σaa2 (r +ǫ) fdud (r) is given in (19). Nd−1  2Nd  r 2 r 2 f x (r)= 1− , 0≤ r ≤ R, (17) Proof: See Appendix B. k q k r R R       The MRC/MRT scheme does not take into account the im- Proof: The proof is omitted due to space limitations. pact of the interference between the UL/DL RRHs. Therefore, the system performance suffers under the impact of strong in- B. Average Uplink Rate terference. Motivated by this, we now study the performance In this subsection, we investigate the average UL rate with of a more sophisticated linear combining scheme, namely the MRC/MRT and ZF/MRT processing respectively. In case of ZF/MRT scheme. the ARA scheme, deriving the statistics of the UL SINR in (5) SRA Scheme with ZF/MRT Processing: We can adopt with MRC/MRT and ZF/MRT appears intractable. Hence, ZF beamforming at the UL RRH to completely cancel the in order to evaluate the average UL rate, we have resorted interference between the UL/DL RRHs. To ensure this is to simulations in Section IV. In the sequel, we consider possible, the number of the antennas equipped at the UL RRH the standard singular path loss model and obtain analytical should be greater than one, i.e., M > 1. After substituting expressions for the average UL rate. wMRT = hq into (5), the optimal receive beamforming t,q hq k k SRA Scheme with MRC/MRT Processing: MRC processing vector at the UL RRH wr,p can be obtained by solving the for the UL with MRT processing for the DL is the optimal following problem: transmit-receive diversity technique since it can maximize 2 max |wr,pgp| the SNR. Although MRC/MRT processing is not optimal in wr,p =1 k k presence of interference between the UL/DL RRHs, it could pq s.t. wr,p† Hudhq =0. (23) be favored in practice, because it can balance the performance Hence, the optimal combining vector wr,p can be obtained as and system complexity. pq † pq† ZF Ag Hud hq h Hud MRC gp MRT p , q Substituting w = and w into (6), the re- wr,p = , where A I − pq 2 . Accordingly, r,p gp t,q Agp Hud hq k k k k k k ceived SINR at the BBU can be expressed as substituting wr,p into (6) the received SNR at the BBU can 2 Puℓ(xp)kgpk be expressed as SINRu = , (18) 2 M SNRu = Puℓ(xp)kg˜pk , (24) Pbℓ(xp, xq) i=1 Zi +1 MRC† 2 2 pq 2 where kg˜pk ∼ χ2(M 1). where Zi = UiVi with Ui = P|wr,p hudi| and Vi = − MRT 2 pq Hpq With the SNR, (24) in hand, we now study the average (wt,q,i ) where hudi is the ith column of ud (i.e., Hpq hpq hpq hpq MRT UL rate of the SRA scheme with ZF processing and for any ud = [ ud1, ud2, ··· , udM ]) and wt,q,i is the ith element m MRT arbitrary value of α = n > 2 with gcd(m,n)=1. of wt,q . For the notational convenience, let us denote 2 M α Proposition 5. The average UL rate achieved by the SRA W = Puℓ(xp)kgpk , Z = Pbℓ(xp, xq) i=1 Zi, and dud− = ℓ(xp, xq). As we assume the UL and DL RRHs are randomly scheme with ZF/MRT processing can be expressed as P positioned in the disk with radius R, the pdf fdud (r) is given st ∆(m, 0), ∆(2n, 0), 1 Ru = κG ς , (25) by [17] vs ∆(2n,M − 1), ∆(2n, 0)   M− 2 (2n) 1 2m 2r 2 1 r r r where κ = m+2n and s =4n +1. f (r)= cos− − 1− , (19) 2Γ(M 1) (2π) dud R2 π 2R πR 4R2 − r !! Proof: The proofq is omitted due to space limitations. for 0

15 8 ZF/MRT P =23 dBm 10 u 7 MRC/MRT α=3, P =23 dBm Simulation u α=3, P =10 dBm 5 ARP−Analytical 6 u SRA−Analytical Simulation 5 0 −50 −40 −30 −20 −10 0 10 20 σ2 (dBm) aa 4

15 3 P =10 dBm u 2 10 Average Downlink Rate (nats/sec/Hz) Average Uplink Rate (nats/sec/Hz) ARP−Analytical 1 α=4, P =10 dBm SRA−Analytical u Simulation 5 0 −50 −40 −30 −20 −10 0 10 −10 0 10 20 30 40 σ2 (dBm) P (dBm) aa b 2 Fig. 1. Average DL rate of SRA and ARA schemes versus σaa (Pb = 46 Fig. 2. Average UL rate of the SRA scheme with MRC/MRT and ZF/MRT dBm, M = 2, and λ = 0.001). processing (M = 2, p = 0.5, and λ = 0.001).

12 where τ is a fraction of the time slot duration of T , used † 2 SNRd P ℓ x h w for DL transmission, = i∈Φd∩b(o,R) b ( i)| i t,i| 10 † 2 and SNRu = Puℓ(xj )|w gj | . In this case, the j∈Φu∩b(o,R) P r,j average sum rate achieved by the ARA scheme can be P 8 obtained from (10). DL RRH power = P b Corollary 2. The average sum rate of the half-duplex user 6 Total DL RRHs power = P achieved by the ARA scheme can be approximated as b ∞ (P δG(M, δ, pλ))k 4 RHD ≈ τ b Γ (δk) sum FD−SRA(ZF/MRT)

Γ(k + 1) Average Uplink Rate (nats/sec/Hz) k=0 2 FD−SRA(MRC/MRT) X FD−ARA(ZF/MRT) δ k ∞ (Pu G(M, δ, (1 − p)λ)) FD−ARA(MRC/MRT) + (1 − τ) Γ (δk). (27) HD−SRA(MRC/MRT) Γ(k + 1) 0 k=0 0 2 4 6 8 10 X Average Downlink Rate (nats/sec/Hz) Proof: The proof is omitted due to space limitations Fig. 3. Rate region of the ARA and SRA schemes for full-duplex and Note that since half-duplex transmissions does not suffer half-duplex modes of operation (M = 3, α = 3, and λ = 0.001). from LI and interference, the DL and UL SNR with SRA scheme can be found from (24). Therefore, the average DL increases the average UL rate of the system, (cf. Fig. 2) it and UL rate achieved by the SRA scheme can be obtained degrades the average DL rate. by replacing M and 1 − p by M +1 and p in (25). Fig. 2 compares the average UL rate of the SRA scheme with MRC/MRT and ZF/MRT processing and under different IV. NUMERICAL RESULTS AND DISCUSSION cases of user power and path loss exponent values. It can be In this section, we investigate the system performance and observed that the analytical curves are in perfect agreement confirm the derived analytical results through comparison with the simulations. In addition, the average UL rate due with Monte Carlo simulations. The simulations adopt pa- to the MRC/MRT processing degrades when the interference rameters of a LTE-A network [18]. The maximum transmit power from the DL RRH becomes stronger (i.e., when Pb power of the DL RRHs and the full-duplex user are set to increases), while the average UL rate due to ZF/MRT pro- 46 dBm and 23 dBm, respectively. The receiver noise has a cessing remains the same regardless of the interference power power spectral density of −120 dBm/Hz or −50 dBm over level. Moreover, we see that the MRC/MRT outperforms the entire bandwidth of 10 MHz. ZF/MRT in the low interference power regime. 2 Fig. 1 shows that average DL rate versus σaa for M = 2 Fig. 3 shows the rate region of the ARA and SRA schemes and for the SRA and ARA schemes. We plot the average respectively for both full-duplex and half-duplex modes of DL rate for two different power constraints (Pb, Pu) = network operation. In this figure, we have set Pu = 23 2 (46 dBm, 23 dBm) and (Pb, Pu) = (46 dBm, 10 dBm) and dBm, σaa = −30 dBm and change p from 0 (i.e. only UL 5 let the LI power vary between −50 dBm and Pu dBm. The transmission) to 1 (i.e. only DL transmission). For a fair analytical upper bounds for the average DL rate of SRA comparison between the ARA and SRA schemes, we have and ARA scheme are also included which are sufficiently also included the case where the same total transmit power tight. As we observe when the Pu is low, the ARA scheme constraint is imposed on the DL such that the transmit power consistently outperforms the SRA scheme in all regimes of LI of the single DL RRH in SRA scheme (Pb = 23 dBm) is strength. However, it is clear that the gap between the ARA equally divided among all the DL RRHs in the ARA scheme. and SRA scheme decrease when the LI strength increase (i.e., For the ARA scheme with ZF/MRT processing we assume 2 both σaa and Pu are high) and becomes negligible when that each UL RRH adjusts its receive beamforming vector in no LI cancellation is applied. On the other hand, although such a way that the interference from its nearest DL RRH is increasing the transmit power of the full-duplex user Pu canceled. These results reveal that the ARA scheme results in a rate region that is strongly biased towards UL or DL, 5 With Pu dBm we mean that no LI cancellation is applied at the full- but using the SRA scheme results in a more balanced rate duplex user. Employing different passive and digital cancellation methods, some practical full-duplex radios can essentially cancel the LI almost to the region. For this setup, SRA scheme with ZF/MRT processing noise floor [6], [8]. can achieve up to 30% and 39% average sum rate gains 2R z as compared to the half-duplex SRA and full-duplex ARA ∞MZ (z) (1−MW (z)) e− Ru = fdud (r)drdz. (31) scheme counterparts, respectively. z Z0 Z0 Therefore, we need to compute the Laplace transforms V. CONCLUSION MZ (s) and MW (s) to derive the average UL rate. Note that In this paper, we studied the average sum rate of a C-RAN M α MZ (s) = i=1 MZi (Pbdud− s). Using the differentiation with randomly distributed multiple antenna UL and DL RRHs property of Laplace transform i.e., MZi (s) = sL(FZi (x)) communicating with a full-duplex user. Specifically, the per- Q and FZi (x) from Lemma 1 and then applying the integral formance of two RRH association schemes, namely, ARA equality [19, Eq. (3.40.1)] we obtain 2 and SRA with MRC/MRT and ZF/MRC processing were σaa 0, 1,M,M M (s)= G32 . (32) studied and analytical expressions for the average UL and Zi 44 s 1, 1,M, 0   DL rates were derived. The SRA scheme achieves a superior Using the differentiation property of Laplace transform and performance as compared to the ARA scheme. We found that Lemma 1, MW (s) can be obtained as for a fixed value of LI power, the SRA scheme with ZF/MRT 2n 4t 2nς ∆(2n, 0), ∆(1, ∆(m, 0)), 1 processing can ensure a balance between maximizing the MW (s)=1−µGu4 . s ∆(1, ∆(2n,M)), 0 ! average sum rate and maintaining an acceptable fairness   (33) level between UL/DL transmissions. Our results show that full-duplex transmissions can achieve higher data rates as To this end, substituting (32) and (33) into (31) yields the compared to half-duplex mode of operation, if proper RRH desired result in (22), thus completing the proof. association and beamforming are utilized and the residual LI is sufficiently small. REFERENCES APPENDIX A [1] G. Wang, Q. Liu, R. He, F. Gao, and C. 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