Optimizing RRH Placement Under a Noise-Limited Point-to-Point Wireless Backhaul
Hussein A. Ammar, Raviraj Adve, Shahram Shahbazpanahi, Gary Boudreau
OOptimizing RRH Placement Under aNoise-Limited Point-to-Point Wireless Backhaul
Hussein A. Ammar ∗ , Raviraj Adve ∗ , Shahram Shahbazpanahi † , and Gary Boudreau ‡∗ University of Toronto, Dep. of Elec. and Comp. Eng., Toronto, Canada † University of Ontario Institute of Technology, Dep. of Elec. and Comp. Eng., Oshawa, Canada ‡ Ericsson Canada, Ottawa, CanadaEmail: { ammarhus, rsadve } @ece.utoronto.ca Abstract —In this paper, we study the deployment decisionsand location optimization for the remote radio heads (RRHs) incoordinated distributed networks in the presence of a wirelessbackhaul. We implement a scheme where the RRHs use zero-forcing beamforming (ZF-BF) for the access channel to jointlyserve multiple users, while on the backhaul the RRHs areconnected to their central units (CUs) through point-to-pointwireless links. We investigate the effect of this scheme on thedeployment of the RRHs and on the resulting achievable spectralefficiency over the access channel (under a backhaul outageconstraint). Our results show that even for noise-limited backhaullinks, a large bandwidth must be allocated to the backhaul toallow freely distributing the RRHs in the network. Additionally,our results show that distributing the available antennas on moreRRHs is favored as compared to a more co-located antennasystem. This motivates further works to study the efficiency ofwireless backhaul schemes and their effect on the performanceof coordinated distributed networks with joint transmission.
Index Terms —Cooperative distributed network, distributedantenna system, RRH placement, wireless backhaul \ fronthaul. I. I
NTRODUCTION
Cooperation between the network transmitters is imperativeto control interference. In a distributed network, this canbe achieved by deploying two key modules: a central unit(CU) that executes protocols to schedule the transmission andthe reception of the signals, and remote radio heads (RRHs)deployed throughout the network’s coverage area to cooperateand jointly serve users [1]. However, a key component for sucha scheme is the backhaul links which connect RRHs to theirCUs (also called fronthaul of the CU). Notably, this scheme isa distributed implementation of the 5G next-generation NodeBspecified in the New Radio specification [1].The backhaul links carry the heaviest communication loadsince they carry the data for all users served by the RRHsand hence are the bottleneck to throughput. The works in [2],[3] modeled the reliability of dedicated backhaul links asa Bernoulli distribution to study their effect on the systemoutage. The aim from these studies is to derive optimalstrategies for the assignment of the backhaul links. Importantissues such as power allocation [4], cooperation strategies [5],beamforming design [6], resource allocation [7] and formationof serving clusters [8] have been investigated under the themeof a limited-capacity backhaul. Additionally, the work in [9] analyzed backhaul signal compression as a mean to minimizethe impact of the limited capacity of the backhaul.As an alternative, several investigations have considered theproblem of transmitter placement. The work in [10] studiedRRHs placement without accounting for the backhaul. Fur-thermore, the study in [11] optimized the placement of relaysin in-band or out-of-band non-cooperative cellular networksby modeling users traffic using queueing theory; their resultsshow that the latter provides better performance. The workin [12] also optimized the deployment of relays, while thework in [13] studied inter-site distance for RRHs deploymenton a highway road scenario, where the outage probabilitywas derived. Nonetheless, these works about transmitters de-ployment did not consider the effect of the backhaul on thedeployment decisions in a coordinated distributed network.The backhaul capacity strongly affects the network perfor-mance and the flexibility of the deployment of the RRHs.Motivated by this fact, we herein investigate the effect ofusing point-to-point wireless backhaul on the placement ofthe RRHs, where the aim is to enhance the access channelspectral efficiency while meeting a backhaul outage constraint.Our work adds to the literature on both the limited-capacitybackhaul and transmitters placement for distributed networksand provides insights into RRH deployment decisions. The restof the paper is organized as follows: in Section II, we presentour system model. While in Section III, we formulate theRRHs location optimization problem. In Sections IV and V,we present our proposed solution and results, respectively.Finally, we present our conclusions in Section VI.II. S
YSTEM M ODEL
A. Network Model
We consider a cooperative distributed network comprising Q cells. Assuming joint transmission with disjoint clustering, theRRHs found in each cell q jointly serve the users inside theircell boundary B q [14]. Each cell employs N RRHs to serve K users on the same time-frequency resource block using zero-forcing beamforming (ZF-BF). Each RRH uses M antennasto serve the users, where N M > K . In each cell, a singleCU, located at the cell center, controls the transmissions ofthe RRHs through point-to-point backhaul links. The users inthe cell may be concentrated in (but not restricted) hotspots a r X i v : . [ c s . I T ] F e b ith a priori known traffic pattern. The backhaul and accesslinks are spectrally orthogonal. To focus on network design,we assume perfect channel state information (CSI) is available.Our aim is to deploy the RRHs in the network so that we obtainthe highest spectral efficiency over the access channel whilecontrolling backhaul outage. B. Access Channel Transmission Scheme
The signal received at a typical user k located in cell q is r qk = N (cid:88) n =1 h Hqn,qk w qnk s qk (cid:124) (cid:123)(cid:122) (cid:125) useful signal + (cid:88) n = { ,...,N } ,k (cid:48) = { ,...,K } ,k (cid:48) (cid:54) = k h Hqn,qk w qnk (cid:48) s qk (cid:48) (cid:124) (cid:123)(cid:122) (cid:125) intra-cluster interference + (cid:88) q (cid:48) = { ,...,Q } ,q (cid:48) (cid:54) = q,n (cid:48) = { ,...,N } ,k (cid:48) = { ,...,K } h Hq (cid:48) n (cid:48) ,qk w q (cid:48) n (cid:48) k (cid:48) s q (cid:48) k (cid:48) (cid:124) (cid:123)(cid:122) (cid:125) inter-cluster interference + z qk (1)where s qk ∈ C is the power-limited signal sent to user k by the serving RRHs in cell q , that is E { s q s Hq } = p I K for s q = [ s q , . . . , s qK ] , with p being the total power budget forthe RRHs in the cell, which is referred as vector normaliza-tion [15].The vector h qn,qk ∈ C M denotes the channel betweenRRH n in cell q and user k in cell q and accounts forthe small-scale and large-scale fading. We have h qn,qk = (cid:112) (cid:96) k ( x qn , y qn ) g qn,qk , where g qn,qk ∼ CN (0 , I M ) is a whitecomplex Gaussian random vector representing the Rayleighfading, while (cid:96) k ( x qn , y qn ) = (1 + d qn,qk /d ) − α is the pathloss of the signal, with d qn,qk = (cid:112) ( x qn − (cid:101) x k ) + ( y qn − (cid:101) y k ) being the distance between RRH n and user k , d is thereference distance, and α is the path loss exponent. The terms x qn and y qn are the x, y -coordinates of RRH n and will beour optimization variables. Furthermore, w qnk ∈ C M is thelinear precoding vector used by RRH n to serve user k in itscell q , and z qk ∼ CN (0 , σ z ) is the independent additive whiteGaussian noise (AWGN), with σ z being the noise power.The RRHs use ZF-BF to serve the K users in each cell.To construct the beamformer, we denote the concatenationof the channel matrix for all the users in cell q as H q =[ h q . . . h qK ] ∈ C NM × K with presumably linearly indepen-dent rows, with h qk = [ h Tqk,qk . . . h TqN,qk ] T ∈ C NM being theconcatenation of the access channels for user k . We can write h qk = D / qk g qk , where D qk ∈ C NM × NM is a diagonal matrixrepresenting the large-scale fading between all the RRHs inthe network and user k . Hence, [ D qk ] mm = (cid:96) k ( x qn , y qn ) where n = (cid:100) m/M (cid:101) for m = { , . . . , N M } . Furthermore, g qk ∈ C NM represents the vector of the small-scale fadingcoefficients between the antennas of the RRHs and user k .The precoding matrix W q is designed for each cell q assuming CSI at the RRHs is available. We have W q =[ v q . . . v qK ] ∈ C NM × K with v qk = [ w Tq k . . . w TqNk ] T ∈ C NM , where w qnk ∈ C N is the precoding vector used atRRH n , and is given by W q = (cid:102) W q µ q = ( H q ) † µ q = H q (cid:0) H Hq H q (cid:1) − µ q (2) Here, µ q ∈ C K × K is a diagonal matrix which satisfies thepower budget E (cid:8) tr (cid:8) W q W Hq (cid:9)(cid:9) = p , i.e., using averagepower normalization [16], and its k th entry is µ qk = [ µ q ] k,k = (cid:113) pK − E {(cid:107) (cid:101) v qk (cid:107) } − (3)with (cid:101) v qk = [ H q (cid:0) H Hq H q (cid:1) − ] .k is k th column of the matrix. C. Backhaul Transmission Scheme
We consider point-to-point single-input single-output(SISO) noise-limited wireless transmission between the CUsand their RRHs (representing an upper-bound for the perfor-mance of a SISO solution). We assume that the CU imple-ments directional antennas, providing parallel
Rician channels between the CU and the N RRHs.The downlink average achievable rate between CU q andits served RRHs is defined as R ( b ) qn ( x qn , y qn ) = E (cid:110) log (cid:16) ρ c | ¯ g qn | ¯ (cid:96) q ( x qn , y qn ) (cid:17)(cid:111) (4)where ρ c = p c σ , ¯ (cid:96) q ( x qn , y qn ) = (cid:0) d qn /d (cid:1) − α is thepath loss on the backhaul link, and it depends on thedistance between CU q and RRH n denoted as ¯ d qn = (cid:112) ( x qn − ¯ x q ) + ( y qn − ¯ y q ) , where (¯ x q , ¯ y q ) are the coordi-nates of the CU. The term ¯ g qn is the small-scale fading of thechannel, which follows a Rician fading model. This Ricianchannel is characterized by the Rician parameter (the RicianK-factor) K = η /η , where η and η represent the powerof the line-of-sight (LoS) and NLoS components respectively.III. P ROBLEM F ORMULATION
We aim to find the optimal deployment for the RRHlocations in the presence of specific traffic distributions, inter-cell interference, and most importantly, under limited capacitywireless backhaul. These important factors will be character-ized in the next subsections.
A. Traffic Distribution
We use a traffic probability density function (PDF) thatis a combination of both a uniform distribution, chosenwith probability P , and a number N h of bivariate normaldistributions, representing hotspots (a choice of small P means more dense hotspots). The number of these hotspotsis uniformly distributed, i.e., N h ∼ U (cid:0) N min h , N max h (cid:1) . Thisnumber can be generated for the whole network or foreach cell separately. These hotspots are centered at locations [ (cid:101) x h , (cid:101) y h ] = (cid:104) [ (cid:101) x h , . . . , (cid:101) x hN h ] T , [ (cid:101) y h , . . . , (cid:101) y hN h ] T (cid:105) ∈ R N h × ,and they have equal variances σ h on both the x and y axis,without any correlation between the two axis. Therefore, thecell traffic distribution PDF bounded by the cell boundary B q is defined as f q ( (cid:101) x k , (cid:101) y k ) = f (cid:18) P (cid:18) B q (cid:19) + (1 − P ) N h πσ h N h (cid:88) i ∈ (cid:32) exp (cid:32) − ( (cid:101) x k − (cid:101) x h i ) + ( (cid:101) y k − (cid:101) y h i ) σ h (cid:33)(cid:33)(cid:33) (5)here B q (different from cell boundary B q ) is the area of cell q , f is a normalizing factor that can be calculated numericallyto normalize the PDF. This traffic model is flexible in the sensethat it can be constructed from a traffic survey that identifiesthe locations of hotspots in the network. B. Access Channel Spectral Efficiency
On the access channel, the ZF beamformer is formed percell q . Hence, the intra-cluster interference found in (1) will becompletely canceled. As a result, the mean achievable spectralefficiency over the access channel for user k in cell q is R ( a ) qk ( x , y ) = E log µ qk (cid:80) n (cid:48) = { ,...,N } ,q (cid:48) (cid:54) = q,k (cid:48) = { ,...,K } | h Hq (cid:48) n (cid:48) ,qk w q (cid:48) n (cid:48) k (cid:48) | + σ z (6)The concatenated form of the RRHs’ locations in thenetwork is written as [ x , y ] , where the the x-coordinates are x = [ x T , . . . , x Tq , . . . , . . . , x TQ ] T ∈ C NQ , for q ∈ { , . . . , Q } (similarly for y ). These will form our optimization variables.We use [ x q , y q ] to refer to the locations of the RRHs in cell q and [ x − q , y − q ] to denote the locations of all the RRHs inthe network except those in cell q .The spectral efficiency over the access channels depends onthe distances between all the RRHs found in the network andthe user k , which in its turn depends on the locations of allthe RRHs. Given the distances to the serving and interferingRRHs, the lower-bound of spectral efficiency can be writtenas [10] R ( a ) qk ( x , y ) = log (cid:32) γ k ( x − q , y − q ) − N (cid:88) n =1 (cid:96) k ( x qn , y qn ) (cid:33) (7)with γ k ( x − q , y − q ) = NK ( NM − K ) ρ MρK Q (cid:88) q (cid:48) =1 ,q (cid:48) (cid:54) = q ICI q (cid:48) k ( x q (cid:48) , y q (cid:48) ) + 1 (8)ICI q (cid:48) k ( x q (cid:48) , y q (cid:48) ) = N (cid:88) l =1 (cid:96) k ( x q (cid:48) l , y q (cid:48) l ) K (cid:88) j =1 (cid:96) j ( x q (cid:48) l , y q (cid:48) l ) ξ ( q (cid:48) , j ) (9)where ρ = pσ z , and ICI q (cid:48) k ( x q (cid:48) , y q (cid:48) ) denotes the inter-cellinterference (ICI) contributed by cell q (cid:48) to the user under testand ξ ( q (cid:48) , j ) = tr { D q (cid:48) j } = M (cid:80) Nm =1 (cid:96) j ( x q (cid:48) m , y q (cid:48) m ) . As canbe seen, this interference depends on the locations of the usersin the interfering cells because the beamformers in each cellis designed based on the channels of the users in these cells.Using the traffic distribution PDF, f q ( (cid:101) x k , (cid:101) y k ) , in (5), wefurther average this achievable rate over f q (cid:48) ( (cid:101) x k , (cid:101) y k ) in theinterfering cells by writing the ICI term asICI q (cid:48) k ( x q (cid:48) , y q (cid:48) ) = N (cid:88) l =1 (cid:96) k ( x q (cid:48) l , y q (cid:48) l ) K E (cid:101) x j , (cid:101) y j (cid:26) (cid:96) j ( x q (cid:48) l , y q (cid:48) l ) ξ ( q (cid:48) , j ) (cid:27) = N (cid:88) l =1 (cid:96) k ( x q (cid:48) l , y q (cid:48) l ) K (cid:90) (cid:90) (cid:101) x j , (cid:101) y j ∈B q (cid:48) (cid:96) j ( x q (cid:48) l , y q (cid:48) l ) ξ ( q (cid:48) , j ) f q (cid:48) ( (cid:101) x j , (cid:101) y j ) d (cid:101) x j d (cid:101) y j (10)where, as indicated earlier, B q (cid:48) is the boundary of cell q (cid:48) . C. Problem Definition
We define our problem of optimizing the locations of theRRHs in the network asmax x , y E (cid:101) x k , (cid:101) y k (cid:110) R ( a ) qk ( x , y ) (cid:111) , ∀ q (11a)s.t. P (cid:110) ω c R ( b ) qn ( x qn , y qn ) ≤ Kω E (cid:101) x k , (cid:101) y k (cid:110) R ( a ) qk ( x , y ) (cid:111)(cid:111) ≤ (cid:15), n = 1 , . . . , N (11b)where E (cid:101) x k , (cid:101) y k (cid:110) R ( a ) qk ( x , y ) (cid:111) = (cid:90) (cid:90) (cid:101) x k , (cid:101) y k ∈B q R ( a ) qk ( x , y ) f q ( (cid:101) x k , (cid:101) y k ) d (cid:101) x k d (cid:101) y k (12)The terms ω , ω c are the bandwidth allocated for the accesschannel and the backhaul, respectively, and (cid:15) is the allowedbackhaul outage probability for each CU-RRH link.The formulation in (11) maximizes the average spectralefficiency of typical user k in the network, and this spectralefficiency is averaged over the traffic distribution in the user’scell as shown in (11a), which means it maximizes the spectralefficiency for the system. Additionally, the N constraintsin (11b) place an upper bound on the sum of the rates over theaccess channel with respect to the backhaul achieved capacity.If this constraint is not respected the backhaul will experiencean outage. Proposition:
The probability of outage, P qn ( x , y ) is given by P qn ( x , y ) = P (cid:110) ω c R ( b ) qn ( x qn , y qn ) ≤ Kω E (cid:101) x k , (cid:101) y k (cid:110) R ( a ) qk ( x , y ) (cid:111)(cid:111) = 1 − Q (cid:32) √ η η , (cid:112) ζ qn ( x , y ) η (cid:33) (13)where Q ( . ) is the Marcum Q -function, and ζ qn ( x , y ) = 1 ρ c ¯ (cid:96) q ( x qn , y qn ) (cid:18) exp (cid:18) K ωω c E (cid:101) x k , (cid:101) y k (cid:110) R ( a ) qk ( x , y ) (cid:111)(cid:19) − (cid:19) (14) Proof.
Please see Appendix A.IV. O
PTIMIZING
RRH P
LACEMENT U NDER B ACKHAUL C ONSTRAINTS
In the next subsections, we solve the optimization problemin (11) using two different approaches.
A. Direct Approach
For notational simplicity we define the following term. J qn ( x , y ) = (cid:112) ζ qn ( x , y ) η = J qn ( x qn , y qn ) × J q ( x , y )= η (cid:115) ρ c ¯ (cid:96) q ( x qn , y qn ) × (cid:115)(cid:18) exp (cid:18) K ωω c E (cid:101) x k , (cid:101) y k (cid:110) R ( a ) qk ( x , y ) (cid:111)(cid:19) − (cid:19) (15)We can write the Lagrangian formulation of our problem as L ( x , y , λ ) = − E (cid:101) x k , (cid:101) y k (cid:110) R ( a ) qk ( x , y ) (cid:111) + N (cid:88) n =1 λ n ( P qn ( x , y ) − (cid:15) ) (16)here the λ (cid:60) denotes the vector of Lagrange multipliers.For a specific RRH m ∈ { , . . . , N } in cell q , we differentiatethe Lagrangian formulation with respect to the x-coordinate x qm of RRH m , set it to zero, and obtain an iterative formulafor x qm that can be written as x ( i +1) qm = (1 − A − A ) (cid:90) (cid:90) (cid:101) x k , (cid:101) y k ∈B q (cid:101) x k A ( (cid:101) x k , (cid:101) y k ) f q ( (cid:101) x k , (cid:101) y k ) d (cid:101) x k d (cid:101) y k + A ¯ x q (1 − A − A ) (cid:90) (cid:90) (cid:101) x k , (cid:101) y k ∈B q A ( (cid:101) x k , (cid:101) y k ) f q ( (cid:101) x k , (cid:101) y k ) d (cid:101) x k d (cid:101) y k + A (17)The same formulation applies by differentiating the La-grangian with respect to the y-coordinate. y ( i +1) qm = (1 − A − A ) (cid:90) (cid:90) (cid:101) x k , (cid:101) y k ∈B q (cid:101) y k A ( (cid:101) x k , (cid:101) y k ) f q ( (cid:101) x k , (cid:101) y k ) d (cid:101) x k d (cid:101) y k + A ¯ y q (1 − A − A ) (cid:90) (cid:90) (cid:101) x k , (cid:101) y k ∈B q A ( (cid:101) x k , (cid:101) y k ) f q ( (cid:101) x k , (cid:101) y k ) d (cid:101) x k d (cid:101) y k + A (18)where A ( (cid:101) x k , (cid:101) y k ) = γ k ( x − q , y − q ) − (1 + d qm,qk /d ) − − α d qm,qk (cid:32) γ k ( x − q , y − q ) − N (cid:88) n =1 (1 + d qn,qk /d ) − α (cid:33) (19) A = K ωω c λ m J qm ( x , y ) exp (cid:16) K ωω c E (cid:101) x k , (cid:101) y k (cid:110) R ( a ) qk ( x , y ) (cid:111)(cid:17) η (cid:114) ρ c (cid:16) exp (cid:16) K ωω c E (cid:101) x k , (cid:101) y k (cid:110) R ( a ) qk ( x , y ) (cid:111)(cid:17) − (cid:17) × exp (cid:32) − (cid:32) η η + (cid:0) J qm ( x , y ) (cid:1) (cid:33)(cid:33) F (cid:18) ; 1; η η (cid:0) J qm ( x , y ) (cid:1) (cid:19) (cid:18) d qm d (cid:19) α (20) A = λ m J qm ( x , y ) (cid:114)(cid:16) exp (cid:16) K ωω c E (cid:101) x k , (cid:101) y k (cid:110) R ( a ) qk ( x , y ) (cid:111)(cid:17) − (cid:17) η √ ρ c × exp (cid:32) − (cid:32) η η + (cid:0) J qm ( x , y ) (cid:1) (cid:33)(cid:33) F (cid:18) ; 1; η η (cid:0) J qm ( x , y ) (cid:1) (cid:19) (cid:32)(cid:0) d qm /d (cid:1) α − ¯ d qm (cid:33) (21) A = K ωω c exp (cid:18) K ωω c exp (cid:18) − (cid:18) η η + ( J qm ( x , y ) ) (cid:19)(cid:19)(cid:19) η (cid:115) ρ c (cid:18) exp (cid:18) K ωω c exp (cid:18) − (cid:18) η η + ( J qm ( x , y ) ) (cid:19)(cid:19)(cid:19) − (cid:19) × (cid:88) n ∈D q ,n (cid:54) = m λ n J qn ( x , y ) exp (cid:32) − (cid:32) η η + (cid:0) J qn ( x , y ) (cid:1) (cid:33)(cid:33) × F (cid:18) ; 1; η η (cid:0) J qn ( x , y ) (cid:1) (cid:19) (cid:18) d qn d (cid:19) α (22) Proof.
Follows from the derivative chain rule and has beenskipped due to lack of space.The terms A ( (cid:101) x k , (cid:101) y k ) , A , A and A depend on the RRHslocations, but we do not write the RRHs x and y coordinatesas parameters to minimize the notation. Here F (; . ; . ) is theregularized confluent Hypergeometric function, and we can write it in an alternate form as a function of the modifiedBessel function of first kind as F (cid:16) ; 1; η η (cid:0) J qm ( x , y ) (cid:1) (cid:17) = I (cid:16) √ η η J qm ( x , y ) (cid:17) , i.e., F (; 1; z ) = I (2 √ z ) .The derivative of L ( x , y , λ ) in (16) with respect to λ m is L λ m ( x , y , λ ) = ∂ L ( x , y , λ ) ∂λ m = (cid:32) − Q (cid:32) √ η η , (cid:112) ζ qm ( x , y ) η (cid:33) − (cid:15) (cid:33) (23)Using the batch gradient descent, we can obtain an iterativeformula for λ ( i +1) m as λ ( i +1) m = (cid:104) λ ( i ) m + ν L λ m ( x ( i ) , y ( i ) , λ ( i ) ) (cid:105) + (24)where [ · ] + = max ( · , , and ν ∈ R + is a step size cho-sen small enough to guarantee convergence. Based on thisanalysis, we can construct Algorithm 1 to obtain the optimallocations of the RRHs in the network as described below. Algorithm 1:
RRHs Locations Optimization Generate random locations for the RRHs in all cells Define d max big enough while d max > d cvg do for q ∈ { , . . . , Q } do Wrap-around cells to make cell q at center for m ∈ D q do Update x qm , y qm , λ m using eq. (17), (18), (24) end d q = max n (cid:26) max x,y (cid:110) | x ( i +1) qn − x ( i ) qn | , | y ( i +1) qn − y ( i ) qn | (cid:111)(cid:27) end d max = max q d q end The algorithm starts by choosing random locations for theRRHs (Step 1). Then, we define a d max as a distance largeenough to start the locations update. For the RRHs locationupdate in each algorithm iteration, we perform a cell wrap-around (Step 5) to place the cell of these RRHs at the center,hence eliminating network border effect. After that, we updatethe locations of the RRHs in this cell using the indicatedequations in Step 7, and we calculate the maximum RRHslocation change. We do one iteration for each cell at a timeuntil we iterate through all the network cells. Hence, the totalnumber of iterations will be the same for all the cells, and theupdated RRHs locations in the interfering cells will be used,which is very reliable. At last, the algorithm convergence isdetermined when the maximum change in the RRHs locations( d max in Step 11) is lower than a small distance d cvg . We notethat our scheme is sub-optimal mainly because of the complextraffic distribution shown in (5). B. Distance-based Approach
Choosing an appropriate step size ν for λ update in (24)can be tricky especially when the constraint (11b) is tight. Toddress this issue, we define an equivalent approach that isdistance-based to solve problem (11) by formulating it asmax x , y E (cid:101) x k , (cid:101) y k (cid:110) R ( a ) qk ( x , y ) (cid:111) , ∀ q (25a)s.t. ¯ d qn ¯ d out ≤ , n = 1 , . . . , N (25b)where ¯ d out is the maximum allowed backhaul distance toguarantee that the outage is lower than (cid:15) . When the achievablespectral efficiency over the access channel is fixed, ¯ d out can beeasily obtained using a bisection search to obtain P qn ( x , y ) = (cid:15) defined in (13). Once ¯ d out is found, we can write an iterativeformula for x qm and y qm as x ( i +1) qm = αd (cid:90) (cid:90) (cid:101) x k , (cid:101) y k ∈B q (cid:101) x k A ( (cid:101) x k , (cid:101) y k ) f q ( (cid:101) x k , (cid:101) y k ) d (cid:101) x k d (cid:101) y k + λ m ¯ d out ¯ d qm ¯ x qαd (cid:90) (cid:90) (cid:101) x k , (cid:101) y k ∈B q A ( (cid:101) x k , (cid:101) y k ) f q ( (cid:101) x k , (cid:101) y k ) d (cid:101) x k d (cid:101) y k + λ m ¯ d out ¯ d qm (26)Similarly for the y-coordinates: y ( i +1) m = αd (cid:90) (cid:90) (cid:101) x k , (cid:101) y k ∈B q (cid:101) y k A ( (cid:101) x k , (cid:101) y k ) f q ( (cid:101) x k , (cid:101) y k ) d (cid:101) x k d (cid:101) y k + λ m ¯ d out ¯ d qm ¯ y qαd (cid:90) (cid:90) (cid:101) x k , (cid:101) y k ∈B q A ( (cid:101) x k , (cid:101) y k ) f q ( (cid:101) x k , (cid:101) y k ) d (cid:101) x k d (cid:101) y k + λ m ¯ d out ¯ d qm (27)To update λ im , we perform another bisection search, such that ¯ d qm = ¯ d out . This method eliminates the need for a step size ν to update λ , and at the same time (26) and (27) guarantee thatthe RRHs will be placed in the locations that maximize (25a)as we will see in the results section. Consequently, we canobtain the optimal locations of the RRHs in our network usingAlgorithm 1, but with replacing Step 7 with two steps; one thatobtains ¯ d out using bisection search and the other updates x qm and y qm using (26) and (27) respectively.V. N UMERICAL R ESULTS
We consider a network of Q = 9 cells with wrap-aroundand cell dimension of × meters. The cells havesquare shapes, but any other preferred shape can be used ifneeded, e.g., circular. Moreover, we consider a system of resource blocks (RBs), where each RB has a bandwidth (BW)of KHz. We use the traffic distribution in (5) to model thelocations of users. We summarize the rest of the simulationparameters in Table I, where the available system bandwidthis divided between the backhaul (of BW ω c ) and the accesschannel (of BW ω ).In Figure 1, we show a typical generated traffic distribution(equation (5)), and we present the resulted optimized locationsof the RRHs when we have ω = 6 RBs (i.e., ω c = 19 RBs). We note that the users exist also in locations outside thehotspots (hotspots are represented with yellow areas) with aprobability P shown in Table I. We include the no constraintcase for comparison purpose. The results show that even atthis high bandwidth allocation for the backhaul compared tothe access channel, the backhaul constraint is still the limitingfactor in the deployment of RRHs in each cell. Parameter Value
Cell config. Q , N , M , K , , , Power p , p c dBm, dBmBandwidth RB, ω , ω c KHz, RBs, RBsNoise spectral density S z ,noise figure F z − dBm/Hz, dBmHotspots P , σ h ; N min h , N max h . , meters;Per network: Q , Q Path loss, Fading d , α , K , η , η . meters, . , dB, , √ Algorithm (cid:15) , d cvg , ν . , meter, TABLE I: Simulation parameters. -1500 -1000 -500 0 500 1000 1500
Meters -1500-1000-500050010001500 M e t e r s -7 CUs RRHs, no constraintCell boundary RRHs, constraint
Fig. 1: Typical generated network hotspots showing the opti-mal RRHs location when ω = 6 RBs, and at no constraint.In Figure 2, we plot the spectral efficiency on the accesschannel as a function of different RBs allocations betweenthe access channel and the backhaul. A ratio of
Kωω c = 1 . corresponding to ω ≤ RBs for the access channel allowsdeploying the RRHs freely in the network for the typicalparameters specified in Table I. In such an allocation, thewireless backhaul is not a bottleneck and the RRHs can befreely deployed as if the backhaul has unlimited bandwidth.This constraint becomes even more relaxed if we use only M = 2 antennas at each RRH, which leads to a lower spectralefficiency over the access channel and hence an ω ≤ RBswould be enough for freely deploying the RRHs.In Figure 3, we plot the spectral efficiency as a functionof the number of the RRHs per cell, N . We show the resultswhen the number of antennas M per RRH is fixed ( M = 8 ),and when the total number of antennas per cell is fixed( N M = 80 ), i.e., as N is increased we get more distributednetwork. Interestingly, the results show that distributing avail-able antennas on more RRHs per cell is a good strategy toincrease the spectral efficiency even when we have a backhaulconstraint. On the other hand, we plot the obtained efficiencieswhen the backhaul bandwidth is further divided among theRRHs to provide frequency division among the RRHs links,i.e., the w c /N plots. For these plots, the backhaul constraintcannot be satisfied even when the RRHs are co-located atthe CU, which means distributing the RRHs in the cell willnot be possible, and other solutions should be taken to makeuch approach successful, e.g., decreasing K or provide morebandwidth for the backhaul. Access RBs number S pe c t r a l e ff i c i en cy ( na t s / s / H z ) Opt. with constraintOpt. without constraintRandom without constraint
Fig. 2: Achievable spectral efficiency as a function of differentRBs allocation, M = 8 (solid line) and M = 2 (dashed).
10 20 30 40 50 60 70 80 N S pe c t r a l e ff i c i en cy ( na t s / s / H z ) NM = 80M = 8M = 8, c /NM = 2, c /N Fig. 3: Access channel spectral efficiency at ω = 5 RBs.VI. C
ONCLUSION
We analyzed the effect of a limited capacity backhaulon the achievable rate in a coordinated distributed network.We used point-to-point noise-limited wireless backhaul linksbetween the CUs and the RRHs and analyzed its effect on thedeployment decisions of the RRHs. We used ZF-BF on theaccess channel to allow the RRHs to jointly serve the usersthat are distributed according to some traffic distribution. Ourresults show that we need to allocate a very large bandwidthfor the backhaul compared to the access channel to allowserving a large number of users and to allow free deploymentof RRHs in the network. Our work underlines the fundamentalrole the backhaul plays in the design of distributed networks.A
PPENDIX
A. Proof of outage on the Noise-limited SISO Backhaul
The backhaul outage probability can be written as P qn ( x , y ) = P (cid:26) R ( b ) qn ( x qn , y qn ) ≤ K ωω c E (cid:101) x k , (cid:101) y k (cid:110) R ( a ) qk ( x , y ) (cid:111)(cid:27) = P (cid:26) | ¯ g qn | ≤ ρ c ¯ (cid:96) q ( x qn , y qn ) (cid:18) exp (cid:18) K ωω c E (cid:101) x k , (cid:101) y k (cid:110) R ( a ) qk ( x , y ) (cid:111)(cid:19) − (cid:19)(cid:27) (28) where ¯ g qn is the small-fading parameter for both the LoSand NLoS components between CU q and RRH n , whichis assumed to be Rician fading, hence the probability den-sity function (PDF) of the fading power δ qn = | ¯ g qn | is f ( δ qn ) = η exp (cid:16) − η + δ qn η (cid:17) I (cid:16) η η (cid:112) δ qn (cid:17) , where I ( z ) = π (cid:82) π exp ( z cos( θ )) d θ is the modified Bessel function of firstkind. Let us denote the right side of the inequality in (28) as ζ qn ( x , y ) , hence P qn ( x , y ) = (cid:90) ζ qn ( x , y )0 f ( δ qn ) d δ qn = 1 − Q (cid:32) √ η η , (cid:112) ζ qn ( x , y ) η (cid:33) (29)where Q ( · ) is the Marcum Q-function.A CKNOWLEDGMENT
This work was supported in part by Ericsson Canada andin part by the Natural Sciences and Engineering ResearchCouncil (NSERC) of Canada.R
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