A Cluster-Based Transmit Diversity Scheme for Asynchronous Joint Transmissions in Private Networks

MITSUBISHI ELECTRIC RESEARCH LABORATORIES https://www.merl.com

Kim, Kyeong Jin; Guo, Jianlin; Orlik, Philip V.; Nagai, Yukimasa; Poor, H. Vincent TR2021-075 June 12, 2021

Abstract In this paper, a multiple cluster-based transmission diversity scheme is proposed for asynchronous joint transmissions (JT) in private networks, in which the use of multiple clusters or small cells is preferable to increase transmission speeds, reduce latency, and bring transmissions closer to the users. To increase the spectral efficiency and coverage, and to achieve flexible spatial degrees of freedom, a distributed remote radio unit system (dRRUS) is installed in each of the clusters. When the dRRUSis disposed in the private environments, it will be associatedwith multipath-rich and asynchronous delay propagation. Taking into account of this unique environment of private networks, asynchronous multiple signal reception is considered in the development of operation at the remote radio units to make an intersymbol interference free distributed cyclic delay diversity (dCDD) scheme for JT to achieve a full transmit diversity gain without full channel state information. A spectral efficiency of the proposed dCDD-based JT is analyzed by deriving the closedform expression, and then compared with link-level simulations for non-identically distributed frequency selective fading over the entire private network.

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A Cluster-Based Transmit Diversity Scheme for Asynchronous Joint Transmissions in Private Networks Kyeong Jin Kim, Jianlin Guo, Philip V. Orlik, Yukimasa Nagai, and H. Vincent Poor

Abstract—In this paper, a multiple cluster-based transmission transmission speeds and capacity, reduce latency, and make diversity scheme is proposed for asynchronous joint transmissions the signal closer to the users, a dense deployment of small (JT) in private networks, in which the use of multiple clusters or cells or clusters is expected [2]. small cells is preferable to increase transmission speeds, reduce latency, and bring transmissions closer to the users. To increase To increase the spectral efficiency and coverage, a dis- the spectral efficiency and coverage, and to achieve flexible tributed antenna system (DAS) [3], [4], in which antennas spatial degrees of freedom, a distributed remote radio unit system are installed in a distributed manner over a coverage area of (dRRUS) is installed in each of the clusters. When the dRRUS the base station (BS), is a promising approach for private is disposed in the private environments, it will be associated networks. When each antenna operates as a BS, the DAS with multipath-rich and asynchronous delay propagation. Taking into account of this unique environment of private networks, can be recognized as coordinated multiple point (CoMP) [5], asynchronous multiple signal reception is considered in the [6]. Its core concept is to provide simultaneous communi- development of operation at the remote radio units to make an cations by a plurality of BSs to a single or multiple users intersymbol interference free distributed cyclic delay diversity to improve the rate over a whole communication region. As (dCDD) scheme for JT to achieve a full transmit diversity gain major approaches of CoMP, coordinated beamforming and without full channel state information. A spectral efficiency of the proposed dCDD-based JT is analyzed by deriving the closed- joint processing that includes joint transmissions (JT) are form expression, and then compared with link-level simulations available. However, since we do not assume full channel state for non-identically distributed frequency selective fading over the information at the transmitter (CSIT), we will focus on JT entire private network. in this paper. Geographically placed BSs make the system Index Terms—Private networks, distributed cyclic delay diver- properly counters path loss and shadowing [7]. However, it sity, joint transmission, spectral efficiency. is challenging to collect full CSIT in the distributed system. Although a very reliable channel estimate can be obtained by the user, the feedback overhead will be overwhelming for large I.INTRODUCTION number of BSs. A conventional codebook based feeding back A private network is a promising new connectivity model may not be working for CoMP due to significant differences offering previously unavailable wireless network performance in received signal strength [8]. to businesses and individuals. The owners can optimize ser- In addition, a tight clock synchronization among BSs is vices at their specific areas by planning and installing their own required. Its mismatch will cause interference at the user due to networks, and ensure reliable communications by an exclusive the difference in signal arrival times from all the BSs. When use of available resources. Since they have complete control Global Navigation Satellite System (GNSS) signals are not over every aspect of the network, they can determine how available in the private area, it is possible to achieve a desired resources are utilized, how traffic is prioritized, how a specific clock synchronization by the precision time protocol (PTP) [9]. security standard is deployed, and so on. Potential applications When a plurality of BSs transmit simultaneously, the existence to industries, businesses, utilities, and public sectors have of interference is an intrinsic problem [3] as well. We are gravitated towards 5G wireless networks with increasingly investigating the following four problems in this paper. stringent performance requirements, in terms of availability, 1) When a DAS is installed in the private environments, it reliability, latency, device density, and throughput [1]. The will be affected by multipath-rich propagation. deployment of private networks can be feasible in the shared 2) Since the distance between each of the remote radio units spectrum or unlicensed spectrum. Furthermore, to increase (RRUs) varies with respect to the receiver (RX), a received symbol timing cannot be aligned at the RX due to a K. J. Kim, J. Guo, and P. V. Orlik are with Mitsubishi Electric Research Laboratories (MERL), Cambridge, MA 02139 USA path dependent propagation delay [10]. Note that RRU Y. Nagai is with Mitsubishi Electric Corp., Japan has only a single antenna and fixed power for simple H. V. Poor is with the Department of Electrical Engineering, Princeton processing, so that a system comprising a plurality of University, Princeton, NJ 08544 USA This work was supported in part by the U.S. National Science Foundation RRUs, called the distributed remote radio unit system under Grant CCF-1908308. (dRRUS), is recognized as the DAS [11]. 3) CSIT-dependent precoders are usually employed to mini- PNS mize interference caused by simultaneous multiple trans- Coordinator missions. Thus, it is necessary to apply an interference-free

b1 b2 transmission scheme to achieve a full transmit diversity. C1

4) Feeding back overhead from the RX increases in propor- RRU 1,4 C2 tion to the number of RRUs. CM1 b1,1 RRU1,3 CM2 b2,4 b1,2 b2,1 RRU2,3 b1,3 RRU RRU2,1 Taking into four mentioned problems, we can summarize the 1,1 b2,2 RRU1,2 h1,4 b2,4 RRU2,4 h1,1 h2,1 following two key contributions comparing with existing work. h1,3 h2,3 RRU2,2 RRU1,5 h2,2 h2,4 1) A new multi-cluster-based distributed remote radio unit h1,5 system (MC-dRRUS): To provide a greater throughput for RX the private network, we propose a new MC-dRRUS, in which the private network server (PNS) provides transmis- Fig. 1. Illustration of the proposed MC-dRRUS with ﬁve RRUs and four RRUs in each of the clusters. Two nodes highlighted in blue are assigned as sion signals and synchronization to the respective cluster CMs to control the RRUs within its cluster. D1 D2 masters (CMs). Within non-overlapping clusters, each CM θ t1 Fig. 2. Two-way packet ex- Sync forms an individual dRRUS. df change for synchronization. A 2) Distributed asynchronous cyclic delay diversity-based JT Follow-up t2 (t1, t2) ﬁlled circle dot denotes a p θ (dACDD-JT) scheme: It is desired to increase the number timestamp in the event mes- t3 (t1, t2, t3) dr sage recorded at its transmis- Delay-Req of RRUs to cover larger territory and to increase channel (t1, t2, t3, t4) t4 sion and reception. The pro- p diversities. However, such an increase would result in cessing time taken at all the

additional interference, difficulty in time synchronization, nodes is assumed to be p. Delay-Resp and undesirable feedback overhead. With an available (t1, t2, t3, t4) distribution for propagation delays over the whole paths, the CM is able to consider its variant in assigning the CDD clocks, which have multiple PTP ports to interact with other delay at each of its RRUs, which is necessary to remove in- clocks. The RX is represented as the ordinary clock. Very tersymbol interference (ISI) cased by asynchronous signal reliable main backhaul links, {b1,b2}, are configured to pro- reception at the RX [12]. vide backhaul access to the clusters via the coordinator that resides at the PNS2. Other very reliable secondary backhaul Notation: IN denotes an N × N identity matrix; 0 denotes an all-zero matrix of with an appropriate size; links, {bi,j,i = 1, 2, j = 1,...,K}, provide backhaul access and CN µ, σ2 denotes a complex Gaussian distribution to RRUs via the CM. The CM controls all RRUs and is with mean µ and variance σ2. The binomial coefficient is responsible for transmitting the signals. Every nodes in the △ cluster is assumed to be equipped with a single antenna. denoted by n = n! . For a vector a, L(a) denotes the k (n−k)!k! A frequency selective fading channel from the kth RRU, cardinality; and its lth element is denoted by a(l). For another deployed in the ith cluster, to the RX is denoted by h with vector a with the second subscript defining its cardinality, i,k i,N L(h )= N . A distance-dependent large scaling fading is sum(a ) = c denotes the sum for all set of positive i,k i,k i,N denoted by α . For a distance d from RRU to the RX, indices of a a satisfying N a ; i,k i,k i,k { i(1),..., i(N)} j=1 i(j) = c α is defined by α = (d )−ǫ, where ǫ denotes the path and the binomial coefficient becomesP the multinomial i,k i,k i,k c △ c! loss exponent. The RX is placed at a specific location with coefficient as follows: a = a a a . i,N ( i,N (1))!( i,N (2))!...( i,N (N))! respect to the RRUs, and, thus, independent but non-identically a △ a a a n1,...,nJ = n2=1 ... nJ =1 . distributed (i.n.i.d.) frequency selective fading channels from n1=1 P n16=n26=...6=nJ P P n26=n1 P nJ 6=n1,...,nJ 6=nJ−1 the RRUs to RX are assumed. The RX is assumed to have For a set of continuous random variables, {x , x ,...,x }, 1 2 N knowledge of the number of multipath components of the x denotes the ith smallest random variable, so that it hii channels connected to itself by either sending a training becomes the ith order statistic. For these order statistics, the sequence or adding a pilot as the suffix to each symbol spacing statistics, {y ,...,y }, are obtained by changing the 1 N block. To reduce the feedback overhead, the RX first computes variables as yi = xhii − xhi−1i with y1 = xh1i. △ Nmax=max{Ni,k, ∀i, k}. After then, it feeds back Nmax to the PNS, so that it is not necessary for the CMs to use X2 II. SYSTEM AND CHANNEL MODELS interface to exchange their channel relevant parameters. Fig. 1 illustrates the considered MC-dRRUS with two non- To estimate the clock offset, θ, and propagation delay, overlapping and co-located clusters. Each cluster is recognized d, PTP [9] specifies four event messages, such as Sync, as an individual dRRUS. The PNS works as the grand master Delay-Req, Pdelay-Req, and Pdelay-Resp, within which an clock by PTP, so that it can compute propagation delay to the accurate hardware timestamp is generated and recorded at RX passing over a particular CM1 and RRU. In contrast, the transmission and reception of its respective messages. Thus, CM and RRUs work as the boundary clock and transparent 2Interested reader refer to [13] that investigates unreliable backhaul in the 1Access point can be work as the CM in IEEE 802.11be. similar system setup. Thus, this paper does not consider unreliable backhaul. after exchanging two-way packets between D1 and D2, four III. DACDD-JT FOR CP-SC TRANSMISSIONS hardware timestamps, (t1,t2,t3,t4), are available at D1 and Without loss of generality, we assume that RRU1,1’s signal D2. Based on four timestamps, d and θ are respectively arrives first at the RX in the considered private networks. Since (t4+t2)−(t3+t1) (t3−t1)−(t4−t2) computed as: d ≈ 2 and θ ≈ 2 , the PNS has propagation delay estimates for its whole network, where we assume that the forward propagation delay, df , it can compute the distribution for relative propagation delays is almost equal to the backward propagation delay, dr, i.e., with respect to RRU1,1’s signal. As an initial interactive df ≈ dr, and clocks are perfectly synchronized in frequency process between the PNS and RX, the PNS transmits dref to and phase. Applying the same procedure, D1 can estimate the the RX via clusters. propagation delay to another node, D3, which supports PTP, so After the removal of the CP signal and applying post- that it can be synchronized to D2. This clock synchronization processing by d1,1, the RX receives a composite signal from is accomplished via main and secondary backhaul links. two clusters as follows: Referring to Figs. 1 and 2, the PNS has a set of propagation M delay estimates {d1,k}k=1,...,K over the first cluster, C1. For r = Π1,hK−M+miH1,hK−M+miP1,hK−M+mis + other clusters, the PNS can estimate propagation delay as well. X h J1 m=1 Thus, we can assume that a complete set of propagation delays, Π {d } , is available at the PNS by employing 2,hK−M+miH2,hK−M+miP2,hK−M+mis + z (5) i,k i=1,...,2,k=1,...,K J2 i PTP. According to this set, the PNS computes the propagation where [·] and [·] respectively represent signals transmitted delay for the signal that arrives first at the RX as follows: J1 J2 from C1 and C2. In addition, Hi,hK−M+mi is right circulant △ (1) matrix determined by PT αi,hK−M+mihi,hK−M+mi with dref = min({di,k}i=1,...,2,k=1,...,K ) p PT denoting the transmission power for single carrier trans- after then a relative propagation delay with respect to d , ref missions. Note that Πi,hK−M+mi is right circulant and orthog- △ onal permutation matrix determined by a relative propagation δdi,k=di,k − dref , for i =1,..., 2, k =1,...,K. (2) delay, δdi,hK−M+mi. Since full CSIT is not available in the The distributed CDD (dCDD) scheme was proposed by considered system, the same PT is assigned to all the RRUs. [11], [14] for distributed cyclic-prefixed single carrier (CP-SC) By shifting down IQ by δdi,hK−M+mi rows, Πi,hK−M+mi transmissions to achieve transmit diversity without full CSIT. can be obtained. An additional set of permutation matrices, Depending on the block size, Q, of the transmission symbol, {Pi,hK−M+mi, ∀i,m}, will be defined later. The additive Q×1 0 2 s ∈ C , and the cyclic-prefix (CP) length, NCP, which is vector noise is denoted by z ∼ CN( , σz IQ). For proper set to Nmax, the maximum number of RRUs that achieves ISI- operation, we assume that 0 ≤ di,hK−M+mi ≤ NCP, so that free reception at the RX is determined by M = ⌊Q/NCP⌋, we have 0 ≤ δdi,hK−M+mi ≤ NCP [10]. where ⌊·⌋ denotes the floor function. When the ith dRRUS is overpopulated with the RRUs, i.e., Ki > M, the CMi needs A. dACDD for JT to select only M RRUs for dCDD operation. Thus, the RX needs to feed back necessary information to the PNS. Based on Using the properties of the right circulant matrix, (5) can available channel estimates, the RX rearranges them according be rewritten as: to their strength, from smallest to largest, as follows: M r = H1,mˆ Π1,mˆ [P1,mˆ s]J + H2,mˆ Π2,mˆ [P2,mˆ s]J Xm=1 h 3 4 i 2 2 αi,h1ikhi,h1ik ≤ ... ≤ αi,hKiikhi,hKiik . (3) +z (6) According to (3), the RX forms a list specifying the strength △ where mˆ =hK − M + mi. Furthermore, [·]J and [·]J corre- D △ D 3 4 order, that is, i=(h1i,..., hMi), and then feeds back i spond to local operations respectively performed at RRU1,mˆ D to the PNS. Furthermore, CMi can have i via the main and RRU2,mˆ . To make ISI-free reception at the RX, it is backhaul communications over bi, from which CMi selects required that Π1,mˆ P1,mˆ and Π2,mˆ P2,mˆ are orthogonal and M RRUs indexed by the last M elements of Di, that is, right circulant matrices, and meet the CDD delay assignment RRUi,hK−M+1i,..., RRUi,hKi. The remaining K −M RRUs for RRUi,mˆ . Accordingly, we can readily obtain δTi,mˆ that are controlled by CMi to be idle from communications. meets the condition: ∆mˆ = δdi,mˆ + δTi,mˆ . For operation For the chosen M RRUs, CM assigns the CDD delay to i [·]J3 , RRU1,mˆ assigns δT1,mˆ as its CDD delay rather than RRUi,hK−M+mi as follows: ∆mˆ . By circularly shifting down the transmission symbol s by

δT1,mˆ , operation [·]J3 can be accomplished. Similar operation ∆m = (m − 1)NCP, m =1,...,M. (4) is conducted for [·]J4 . Thus, P1,mˆ and P2,mˆ can be obtained Thus, for an overpopulated dRRUS, the PNS can achieve the from IQ by circularly shifting down respectively by δT1,mˆ and same objective as distributed MRT (dMRT) with using only δT2,mˆ . partial CSIT. In summary, for dCDD operation, CMi needs ISI caused by a variant propagation delay and multiple to know M, NCP, and Di, which are available at each of the transmissions can be removed by a series of circular shift- CMs by backhaul communications made by the PNS. ing operations that are respectively performed by the RRUs, N M −e˜k 1 −l and caused by propagation. Thus, dACDD is an extensive Γ(ek) Qk=1(M+1−k) Γ(˜ek ) Pl=0 δl(bI ) version of dCDD allowing the distribution of propagation Gd+l ((bI ) /Γ(Gd + l))ΓU (Gd + l,x/bI ) (9) delays over the private network. For proper dACDD operation, the PNS is required to know {di,mˆ }i=1,2;m=1,...,M and where Γ(·) and ΓU (·, ·) respectively denote complete gamma {δdi,mˆ }i=1,2;m=1,...,M . However, due to the use of PTP, an and incomplete upper-gamma functions. additional feedback is not necessary from the RX, which is Based on (9), the spectral efficiency (SE) of the proposed the key difference from [12]. MC-dACDD is given by the following theorem. Theorem 3: The achievable SE of the proposed MC-dACDD IV. SPECTRAL EFFICIENCY OF JT BY ASYNCHRONOUS is given by MC-DACDD 1 ^ \ M SE = (M +1 − k)−ek Using the properties of the right circulant matrix, the achiev- k=1 log(2) n ,...,nX X Y 1 M n˜1,...,n˜M n 6=...6=n able signal-to-noise ratio (SNR) realized by MC-dACDD 1 M n˜16=...6=ñM M based JT can be derived by the following Theorem 1. −e˜k Gd Γ(ek) (M +1 − k) Γ(˜ek)(bI ) Theorem 1: Even for asynchronous signal reception at the Yk=1 RX, ISI-free reception can be achieved via MC-dACDD. Thus, N1 δl 3,1 0, 1 G2,3 1/bI (10) the achievable SNR realized by JT is given by h Xl=0 Γ(G + l) Gd + l, 0, 0 i d 2 m,n a1, ..., an,an+1, ..., ap γJT = γJT,1 + γJT,2 = ρs/σz (7) where G t denotes the Meijer p,q b1, ..., bm,bm+1, ..., bq M 2 M 2 G-function [15, Eq. (9.301)]. where ρs = PT α1,mˆ kh1,mˆ k + α2,mˆ kh2,mˆ k Pm=1 Pm=1 Proof: We first express the functions of in terms △ 2 △ M 2 x with ρ=PT /σz and γJT,i=ρ m=1 αi,mˆ khi,mˆ k . −1 1,1 0 P of Meijer G-functions, i.e., (1 + x) = G1,1 x and Proof: When {h1,mˆ , ∀m} and {h2,mˆ , ∀m} are indepen- 0 2,0 1 dent of each other, ρ , realized at the RX, is determined by Γu(j, αx) = G αx . After this, applying [16, eq. s 1,2 j, 0 the summation of their squared Euclidean norms. (2.24.1,2)], we can derive (10). For overpopulated dRRUS, the CM selects only M RRUs by referring to the channel strength. Thus, the order statistics are V. SIMULATION RESULTS employed in the expression for the achievable SNR. Theorem 1 proves that by compensating different signal arrival times at We assume the following simulation setup. the RX, the MC-dACDD makes JT provide the same benefit 1) C1: Six RRUs are placed at (−1.2, 4.7), (0.7, 4.0), as dMRT without full CSIT at the PNS and CMs. (3.0, 3.0), (−2.5, 2.7), (−3.3, 0.4), and (−3.0, 3.5). The Theorem 2: Due to the use of JT, the proposed MC- first cluster master, CM1, is placed at (0, 2) in a 2-D dACDD results in the SNR, γJT realized at the RX, whose plane. approximation of the moment generating function (MGF) is 2) C2: Four RRUs are placed at (12.8, 3.3), (7.4, 2.5), given by (10.0, 4.6), and (9.0, 1.7). The second cluster master, ^ \ M CM2, is placed at (10.0, 3.0) in a 2-D plane. −ek Mγ (s)= (M +1 − k) Γ(ek) JT k=1 3) For CP-SC transmissions, we assume that Q = 32 and n ,...,nX X Y 1 M n˜1,...,n˜M n 6=...6=n 1 M n˜16=...6=ñM NCP = 8. Thus, the CMs can support up to four RRUs M N1 for dACDD operation, i.e., M =4. −e˜k −l (M +1 − k) Γ(˜ek) δl(bI ) Yk=1 Xl=0 4) RX is placed at (3, −3). −Gd−l (1/bI + s) (8) 5) In all scenarios, we fix PT = 1. A fixed path-loss exponent is assumed to be ǫ =2.09. △ 2M △ where Gd= Ek, bI = min(1/Q1,..., 1/Q2M ) 6) The relative time difference, δTi,m, between the arrival Pk=1 with Qk and Ek, respectively the kth elements time of the signal transmitted from RRUi,m with respect T of Q = [q1,...,qM , q˜1,..., q˜M ] and E = to RRUi,1 is represented as an integer value uniformly T [e1,...,eM , e˜1,..., e˜M ] . Furthermore, N1 denotes an generated between 0 and NCP. △ 1 l We consider several frequency selective fading channel pa- upper limit summation, and δl= l i=1 iriδl−i with δ0 = 1 2M j P and ri = Ej (1 − bI Qi) . Additional terms specified in rameters for two clusters depending on the respective num- Pj=1 (8) are defined in Appendix A. ber of RRUs, K1 and K2. For notation purpose, we use Proof: See Appendix A. H1 = {N1,j, j = 1,...,K1} for C1 and H2 = {N2,j, j = Theorem 2 provides the MGF expressed by the weighted sum 1,...,K2} for C2. −Gd−l of N1 +1 terms, proportional to (1/bI + s) . 1) X1: H1 = {2, 3, 4, 2, 3} and H2 = {3, 2, 3, 3}. Corollary 1: The CDF of γJT can be expressed by a finite 2) X2: H1 = {3, 4, 2, 3, 2} and H2 = {3, 2, 3, 3}. number of gamma distributions. 3) X3: H1 = {2, 3, 4, 2, 3, 4} and H2 = {3, 2, 3, 3, 4}. 4) X4: H1 = {2, 3, 4} and H2 = {3, 2, 3}. M −e Fγ (x)=1 − Q (M+1−k) k JT X n1,...,nM X n˜1,...,n˜M k=1 5) X5: H1 = {3, 4, 5, 3, 4, 5} and H2 = {5, 4, 5, 5, 5}. g n16=...6=nM d n˜16=...6=ñM We denote the analytically derived SE by An, whereas we of RRUs for dCDD exists due to different frequency denote the exact performance metric obtained by the link-level selective fading severity across deployed clusters. simulations by Ex in the sequel. Since there is no existing similar setup, i.e., CP-SC based MC-dACDD with JT, we 6 mainly focus on the proposed scheme in this paper.

12 5.5

10 5

8 4.5

3.5 2 3 4 5 6 2 Fig. 5. SE for various values of M and different numbers of multipath 0 components. 0 5 10 15 20 25 30 At a fixed 18 dB SNR, Fig. 5 shows the SE for various Fig. 3. SE for various system and channel parameters. system and channel parameters. For underpopulated and over- We first verify the analytically derived SEs for two overpop- populated dRRUSs, the impact of M on the SE is investigated. ulated systems. The first system with X1 assumes that dACDD • For a given K1 and K2, as M increases, the dRRUS is supports only two RRUs, while five and four RRUs exist populated with less RRUs. Although the SE increases in in C1 and C2, respectively. For the second system with X2, proportion to M, the growth rate of the SE decreases. dACDD supports only four RRUs, while five and four RRUs • As K1 or K2 greater, the growth rate of the SE increases. exist in C1 and C2, respectively. Fig. 3 shows an accuracy For example, (K1 =6,K2 = 5) vs. (K1 =5,K2 = 4). of the analytically derived SEs comparing with the exact SEs. • As the number of multipath components increases, a This figure also shows that if N1 is not sufficiently large, an greater SE is obtained. For example, X3 vs. X5. approximation, which is used in Theorem 2, does not provide an accurate SE. Thus, in the sequel, we use a sufficiently large VI. CONCLUSIONS value for N1 without a specific description for it. In general, as M increases, a larger value for N1 is required to obtain In this paper, we have proposed a multiple cluster-based very reliable and accurate analytic SEs. transmit diversity scheme for asynchronous joint transmissions. To relax the requirement of full channel state informa-

10 tion at the private network server and deal with different prop-

9 agation over the paths of the distributed systems composed of

8 remote radio units, a dACDD scheme has been developed to

7 the private network. The multi-level hierarchical PTP allows to break the dependency of different dRRUS clusters from a 6 common PNS allowing each dRRUS cluster to use dACDD in- 5 dependently from other clusters. For i.n.i.d. frequency selective 4 fading channels, a new closed-form expression for the spectral 3 efficiency has been derived. Its accuracy has been also verified 2 comparing with the link-level simulations. By integrating a 1 transmission scheme, propagation delay estimation, and oper- 0 0 5 10 15 20 25 30 ation at the RRUs effectively, we have seen that the proposed multiple cluster-based asynchronous joint transmissions can X Fig. 4. SE for various over-popularity of dRRUSs with 3. achieve the desired spectral efficiency for various simulations scenarios. In generating Fig. 4, we mainly use X3 with various over- popularity of dRRUSs and assume that M = 4. From Fig. 4 we can extract the following facts: APPENDIX A: PROOF OF THEOREM 2 • As the dRRUS is populated with more RRUs, a greater SE can be achieved. To simplify the notation, let us define xm as △ 2 • As the number of clusters increases, a greater SE can be xm=ρα1,hK1−M+mikh1,hK1−M+mik . Note that the order achieved. However, a more tight restriction on the number statistics are implicitly involved in this definition. We mainly △ n ,...,n focus on the derivation of the distribution for γJT,1 in the defined as 1 M and with pî=N2,hK2−M+ˆnii, P n16=n26=...6=nM sequel. The MGF of γ can be defined as △ f △ ˆ pî △ JT,1 ˆ 1 ˆ M (bi) ˆ pî+1 bi= , AM = i=1 , λi= j=2 aî(j), ρα2,hK2−M+ˆnii Γ(ˆpi) K1 ∞ ∞ ∞ ∞ Q P Mγ (s)=P n ,...,n R R ... R R △ pˆ a △ pˆ +1 JT,1 1 M 0 x1 xM−2 xM−1 ˆ i î (j+1) ˜ i n16=n26=...6=nM θi= j=2(Γ(j)) , pî= j=3 (j − 2)aî(j), and K Q λˆ pˆ˜ P M pi pi−1 −(s+bi)xi 1 △ (−1) i (ˆbi) i Qi=1 (bi) /Γ(pi) (xi) e Qi=M+1 γl(pi,bix1)/Γ(pi) ˆ J6 Xi= for aîs. e˜k and q˜k are defined as those of ek θî dxM . . . dx2dx1 and qk. Since the SNR realized at the RX is the sum of two RVs, γJT,1 and γJT,2, the MGF is given by where γl(·, ·) denotes the incomplete lower-gamma function, ^ \ M −e △ △ n ,...,n k 1 MγJT (s)=P 1 M P n˜1,...,n˜M Qk=1(M+1−k) Γ(ek ) pi=N1,hK1−M+nii, and bi= . Next, we rewrite n16=...6=nM n˜ 6=...6=ñ ρα1,hK1 −M+nii 1 M M −e˜k M −ek M −e˜k γ(pi,bix1)/Γ(pi) as follows: Qk=1(M+1−k) Γ(˜ek ) Qk=1(s+qk ) Qk=1(s+˜qk ) . (A.4)

γ(pi,bix1) p Now applying the approach proposed by [17], we can derive 1 ˜i −λibix1 = P a Xix e (A.1) sum( i,p +1)=1 ai,p 1 Γ(pi) i i+1 (8).

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