Resource Allocation and Scheduling in Non-coherent User-centric Cell-free MIMO
Hussein A. Ammar, Raviraj Adve, Shahram Shahbazpanahi, Gary Boudreau, Kothapalli Srinivas
RResource Allocation and Scheduling inNon-coherent User-centric Cell-free MIMO
Hussein A. Ammar ∗ , Raviraj Adve ∗ , Shahram Shahbazpanahi †∗ , Gary Boudreau ‡ , and Kothapalli Srinivas ‡∗ 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, Canada
Abstract —We study the problem of user-scheduling andresource allocation in distributed multi-user, multiple-inputmultiple-output (MIMO) networks implementing user-centricclustering and non-coherent transmission. We formulate aweighted sum-rate maximization problem which can provideuser proportional fairness. As in this setup, users can be servedby many transmitters, user scheduling is particularly difficult.To solve this issue, we use block coordinate descent, fractionalprogramming, and compressive sensing to construct an algorithmthat performs user-scheduling and beamforming. Our resultsshow that the proposed framework provides an - to -foldgain in the long-term user spectral efficiency compared to bench-mark schemes such as round-robin scheduling. Furthermore, wequantify the performance loss due to imperfect channel stateinformation and pilot training overhead using a defined area-based pilot-reuse factor. Index Terms —User-centric clustering, cell-free, user-scheduling, resource allocation, distributed MIMO, distributedantennas system, fairness, imperfect CSI.
I. I
NTRODUCTION
Deploying user-centric clustering in distributed multiple-input multiple-output (MIMO) networks enhances the perfor-mance of the conventional cell-edge users by placing eachuser at the effective center of its serving cluster [1], [2]. User-centric clustering can outperform general cell-free networksthat assume all the remote radio heads (RRHs) can servethe users [3]. Recently, resource allocation under cell-freeMIMO has attracted significant attention. The studies in [4],[5] optimize beamforming design by minimizing a weightedsum mean square error (MSE) utility, which is easier to tacklethan weighted sum rate (WSR) maximization problems butsuffers a penalty in terms of sum-rate [4].The work in [3] considers optimizing power allocationto maximize lower bounds for sum-rate and minimum rate.Similarly, [6] optimizes the beamforming to maximize theminimum rate. Note that, max-min rate solutions do not pro-vide flexibility to control the fairness. Moreover, the authorsin [7] consider a near-optimal power control algorithm usingzero-forcing (ZF) and conjugate beamforming that is sim-pler than the max–min power approach for cell-free massiveMIMO networks. Furthermore, the work in [8] optimizesthe beamforming by using a lower-bound for the logarithmfunction of the rate to obtain a local optimum.A crucial component in optimizing the WSR is user-scheduling. In conventional networks, techniques that havebeen investigated include dual decomposition and the gradient method [9], where the scheduling variables are relaxed frombeing binary. Notably, this relaxation is optimal for a largenumber of subcarriers [10]. Furthermore, the investigationsin [11], [12] use fractional programming to perform resourceallocation in conventional networks, where the user-schedulingpart is performed using a combinatorial search.In summary, the main limitation of these studies is eithernot specifically addressing the user-centric clustering scheme,or ignoring the user-scheduling step for the users, that is,the scheduled users are assumed to be preselected . In thispaper, we optimize user-scheduling and resource allocation ina user-centric cell-free MIMO network through formulatinga WSR problem. We study the non-coherent transmissionmode, which does not require the RRHs to strictly synchronizetheir transmissions, but it prevents from directly using theweighted minimum mean square error (WMMSE) [13]. Thescheduling part of the problem cannot be solved efficientlyusing a combinatorial search algorithm because each user canbe served by many RRHs with overlapping serving clusters.To tackle this, we employ tools from block coordinate descent,fractional programming, and compressive sensing, which allowthe construction of an algorithm that guarantees convergenceof the network sum-rate through a smooth non-decreasingpattern. In summary, the contributions of this paper are: • Formulating the WSR problem for the non-coherent cell-free multiuser MIMO setting • Using fractional programming to optimize beamformingand employing compressive sensing to solve the schedul-ing problem • Developing and implementing robust beamforming toaccount for channel estimation errorsThe rest of the paper is organized as follows. Section IIpresents our system model, while Section III formulates theoptimization problem. Section IV presents our proposed re-source allocation algorithm. Finally, Sections V and VI reportour simulation results and conclusion, respectively.II. S
YSTEM M ODEL
A. Network Model
As shown in Fig. 1, we consider the downlink of a time-division duplex (TDD) system comprising several RRHs,represented by the set B , each equipped with M antennas andjointly serving the active users, represented by the set U . Both a r X i v : . [ c s . I T ] F e b U RRH
User 𝑢𝐶 𝑢 𝐶 𝑢 ′ User 𝑢 ′ Front-haul
Fig. 1: Serving cluster using user-centric clustering.RRHs and users are randomly located in 2D space. The RRHsare controlled by a single control unit (CU), and as in [3], weassume a relaxed front-haul constraint, which can be realizedthrough technologies like the radio stripes system [14].For each user u ∈ U , we define a cluster C u that includesthe RRHs that potentially can be selected to serve the useraccording to user-centric clustering. Specifically, C u comprisesthe RRHs with strong average channels, i.e., C u = { r | ( ψ ru L ( d ru )) ≥ ρ } , where ψ ru denotes the shadowing, L ( d ru ) accounts for the path loss; here, d ru is the distance betweenRRH r and user u . If no RRH meets this criterion, the C u forthe user comprises the RRH with largest ( ψ ru L ( d ru )) . Finally,we represent the users that may be served by RRH r as E r . B. Channel Estimation
Channel estimation is performed through an uplink pilot-training phase of length τ p . During this phase, we can writethe signal Y r ∈ C M × τ p received at RRH r as Y r = (cid:88) u ∈U √ p u h ru Φ u + Z r , (1)where Φ u ∈ C × τ p is the unit norm ( Φ u Φ Hu = 1 ) pilotsequence used by user u , p u is the transmit power of the user,and Z r is the additive white Gaussian noise (AWGN) withentries ∼ CN (cid:0) , σ Z (cid:1) ; h ru ∈ C M × , the channel betweenRRH r and user u is modeled as h ru (cid:44) (cid:112) ψ ru L ( d ru ) g ru ,where g ru ∼ CN ( , I M ) accounts for small-scale fading.As in [2], we assume knowledge of the users’ transmitpowers and large-scale fading. Hence, using ˘y r = vec { Y r } ∈ C Mτ p × and linear MMSE, the channel estimate ˆh ru , ∀ u ∈E r can be obtained as ˆh ru = R ru R − r ˘y r , with R ru = √ p u ψ ru L ( d ru ) ( Φ ∗ u ⊗ I M ) and R r = (cid:88) u ∈U p u ψ ru L ( d ru ) (cid:0) Φ Tu Φ ∗ u ⊗ I M (cid:1) + σ z I Mτ p . When the number of users |U| ≥ τ p , the available pi-lot sequences need to be reused by the users, adding pilotcontamination. This results in the estimated channel ˆh ru ∼CN ( , Ψ ru ) , with the error covariance matrix given by [15] Ψ ru (cid:44) D ru (cid:32) (cid:88) u (cid:48) ∈U u D ru (cid:48) + σ Z p u I M (cid:33) − D ru , (2)where D ru ∈ C M × M is a diagonal matrix with diagonalentries [ D ru ] mm (cid:44) ψ ru L ( d ru ) , and U u is the set of users employing the same pilot sequence as user u (including user u ). It is known from MMSE that the channel estimation error e ru = h ru − ˆh ru is uncorrelated with ˆh ru and can be modeledas e ru ∼ CN ( , Θ ru ) , where Θ ru (cid:44) D ru − Ψ ru .The downlink signal received at user u can be modeled as y u = (cid:88) r ∈C u √ s ru (cid:16) ˆh Hru + e Hru (cid:17) w ru x ru + (cid:88) r (cid:48) ∈B (cid:88) u (cid:48) ∈E r (cid:48) ,u (cid:48) (cid:54) = u √ s r (cid:48) u (cid:48) (cid:16) ˆh Hr (cid:48) u + e Hr (cid:48) u (cid:17) w r (cid:48) u (cid:48) x r (cid:48) u (cid:48) + z u (3)where { x ru : r ∈ C u } are the symbols transmitted by theserving RRHs for user u with E {| x ru | } = 1 , w ru ∈ C M × is the precoding vector used by RRH r to serve user u , and z u ∼ CN (0 , σ z ) is the AWGN. C. Pilot Assignment (PA) Policy
Properly assigning the pilots to the users is clearly pivotalto decrease pilot contamination.
Proposition 1.
We propose to use a heuristic low-overheadlocation-based PA policy. Our policy assigns non-orthogonalpilots for users that are far from each other by using thehierarchical agglomerative clustering (HAC) algorithm [16].The HAC creates a tree to cluster the users into many groupseach containing a number of users less than or equal tothe number of available orthogonal pilot sequences. We thenassign each group the available orthogonal sequences. Thealgorithm can be constructed as follows:1) Treat each active user as a cluster head.2) Combine the two nearest clusters into one using an av-erage linkage, e.g., Ward’s minimum variance criterion.3) Repeat Step 2 until you reach the root of the tree whereall the users are in the same cluster.4) While backtracking the tree starting from the root, defineeach cluster when its number of users is less than orequal τ p .5) Assign the orthogonal pilots to each cluster randomly.The HAC algorithm is more consistent than the K-meansand Gaussian mixture models, and it is not sensitive to thechoice of the used distance-metric [16]. Also, as this algorithmdoes not require selecting the number of clusters needed, itallows us to easily define the cluster based on an upper limitof the number of users belonging to it, i.e., relate it to τ p .III. P ROBLEM F ORMULATION
A. Problem Definition
To decode the data streams from the RRHs, the usersemploy successive interference cancellation (SIC). Under theassumption of perfect SIC, the effective achievable rate for This expression is based on using the famous Jensen’s Inequality to writedown a lower-bound for the data rate with an expectation over the unknowninstantaneous channel state information (CSI) error { e ru : r ∈ B , u ∈ E r } ,i.e., E e { log (1 + (cid:101) γ u ) } ≥ log (cid:16) / E e (cid:110)(cid:101) γ − u (cid:111)(cid:17) , then using SIC todecode the received data streams at the user. Note that this expression isonly used to perform the resource allocation, however, when we plot theperformance, we use the actual achievable rate using the actual channels. ser u can be modeled by the CU as [13] R u = ( τ d − τ p ) τ d log (cid:32) (cid:80) r ∈C u s ru | ˆh Hru w ru | A u ( S , W ) (cid:33) , (4)with A u ( S , W ) = (cid:88) r (cid:48) ∈B (cid:88) u (cid:48) ∈E r (cid:48) ,u (cid:48) (cid:54) = u s r (cid:48) u (cid:48) (cid:12)(cid:12)(cid:12) ˆh Hr (cid:48) u w r (cid:48) u (cid:48) (cid:12)(cid:12)(cid:12) + (cid:88) r (cid:48) ∈B (cid:88) u (cid:48) ∈E r (cid:48) s r (cid:48) u (cid:48) w Hr (cid:48) u (cid:48) Θ r (cid:48) u w r (cid:48) u (cid:48) + σ z (5)where τ d is the channel coherence time, S = { s , . . . , s |B| } is the set of binary scheduling variables at the RRHs with s r = [ s ru . . . s ru |E r | ] T ∈ B |E r |× , i.e, if s ru = 1 ,user u is scheduled by RRH r , else it is not. Similarly,the set of the beamformers is W = { W , . . . , W |B| } with W r = [ w ru , . . . , w ru |E r | ] ∈ C M ×|E r | . The term Θ ru (cid:48) = E { e ru (cid:48) e Hru (cid:48) } is the covariance of the estimation errorof the channel between RRH r and user u (cid:48) , and including itin the model allows to construct a robust beamforming.We formulate the following WSR problem on the CU (P1) max S , W (cid:88) u ∈U δ u log (1 + γ u ) (6a)s.t. (cid:88) u ∈E r s ru ≤ M, r ∈ B (6b) (cid:88) u ∈E r (cid:107) w ru (cid:107) ≤ p, r ∈ B (6c) γ u = (cid:80) r ∈C u s ru (cid:12)(cid:12)(cid:12) ˆh Hru w ru (cid:12)(cid:12)(cid:12) A u ( S , W ) , u ∈ U (6d) s ru ∈ { , } r ∈ B , u ∈ E r (6e)where δ u denotes the proportional fair weights for user u . Theterm A u is defined in (5). Problem (6) optimizes the decisionvariables S and W which determine the user-scheduling andbeamforming weight vectors, respectively, such that the totalutility in (6a) is maximized. We ignore the pre-log pilot train-ing overhead factor because it is a constant. Constraints (6b)prevent the RRHs from simultaneously serving more than M users on the same channel. Constraints (6c) satisfy the powerbudget of the RRHs, and (6e) show that a user u can bescheduled or not. Constraints (6d) define the effective signalto interference and noise ratio (SINR) as an auxiliary variable.Problem (6) is a mixed-integer non-convex problem andobtaining a global optimum is mathematically prohibitive. B. Problem Analysis
The beamforming vectors are constructed for users that areactually scheduled on the channel and hence s ru = {(cid:107) w ru (cid:107) } = (cid:13)(cid:13)(cid:13) (cid:107) w ru (cid:107) (cid:13)(cid:13)(cid:13) (7)where (cid:107)·(cid:107) is the (cid:96) -norm. Using the literature of compressivesensing, the (cid:96) -norm of a vector x can be approximatedas a weighted convex (cid:96) -norm (cid:107) x (cid:107) (cid:39) (cid:80) m α m | x m | = (cid:107) α x (cid:107) [17], where α m are positive weights that penalizethe nonzero coefficients x m , and α = diag { α , α , . . . } isa diagonal matrix. For our case, x = (cid:107) w ru (cid:107) which is scalar. We can construct an iterative process to find these weights ateach iteration i as suggested in [17] α ( i +1) ru = 1 (cid:13)(cid:13)(cid:13) w ( i ) ru (cid:13)(cid:13)(cid:13) + (cid:15) , (8)where (cid:15) > provides stability and ensures that a zero-valuedcomponent in (cid:107) w ru (cid:107) does not strictly prohibit a nonzeroestimate at the update in the next iteration.As a result, our problem can be formulated as follows (P2) max W (cid:88) u ∈U δ u log (1 + γ u ) (9a)s.t. (cid:88) u ∈E r α ru (cid:107) w ru (cid:107) ≤ M, r ∈ B (9b) (cid:88) u ∈E r (cid:107) w ru (cid:107) ≤ p, r ∈ B (9c) γ u = (cid:80) r ∈C u (cid:12)(cid:12)(cid:12) ˆh Hru w ru (cid:12)(cid:12)(cid:12) B u ( W ) , u ∈ U (9d)where B u ( W ) = A u ( , W ) . We use the Lagrangian for theequality constraints in (9d) L ( W , γ , ν ) = (cid:88) u ∈U δ u log (1 + γ u ) − (cid:88) u ∈U ν u (cid:32) γ u − (cid:80) r ∈C u (cid:12)(cid:12)(cid:12) ˆh Hru w ru (cid:12)(cid:12)(cid:12) B u ( W ) (cid:33) (10)When W is fixed, we evaluate the first optimality conditionof the SINR auxiliary variable γ u by setting the derivative of(10) with respect to γ u to zero, which results in a value for ν u that satisfies this optimality. Substituting ν u back into (10): f ( W , γ ) = (cid:88) u ∈U δ u (log (1 + γ u ) − γ u )+ (cid:88) u ∈U δ u (cid:32) (1 + γ u ) (cid:80) r ∈C u (cid:12)(cid:12)(cid:12) ˆh Hru w ru (cid:12)(cid:12)(cid:12) (cid:88) r (cid:48) ∈B (cid:88) u (cid:48) ∈E r (cid:48) w Hr (cid:48) u (cid:48) (cid:16) ˆh r (cid:48) u ˆh Hr (cid:48) u + Θ r (cid:48) u (cid:17) w r (cid:48) u (cid:48) + σ z (cid:33) (11)Setting the derivative of (11) to zero, we obtain the expectedoptimal formula for γ u in (9d), which means they are equiv-alent.Hence, our new reformulated problem can be written as (P3) max W , γ f ( W , γ ) (12)s.t. (9b) and (9c)Note that we are not writing the dual problem here, but ratherwe are introducing SINR auxiliary variables γ that act as aproxy to account for the changes of the other variables. Proposition 2.
Maximizing the second term in the objectivefunction in (12) is equivalent to maximizing the resulting |C u | terms if we expand the numerator, where |C u | is the size of theserving cluster for user u , i.e., the number of possible servingRRHs. If we decouple these terms and reorganize them withrespect to each RRH r , we can restructure (11) as f ( W , γ ) = (cid:88) u ∈U δ u (log (1 + γ u ) − γ u ) + (cid:88) r ∈B f ( r ; W , γ ) , (13)here for each RRH r we have f ( r ; W , γ ) = (cid:88) u ∈E r δ u (cid:32) (1 + γ u ) (cid:12)(cid:12)(cid:12) ˆh Hru w ru (cid:12)(cid:12)(cid:12) (cid:88) r (cid:48) ∈B (cid:88) u (cid:48) ∈E r (cid:48) w Hr (cid:48) u (cid:48) (cid:16) ˆh r (cid:48) u ˆh Hr (cid:48) u + Θ r (cid:48) u (cid:17) w r (cid:48) u (cid:48) + σ z (cid:33) (14)This restructuring follows from the fact that (cid:80) u ∈U (cid:16) a u (cid:80) r ∈C u A ru B u (cid:17) = (cid:80) r ∈B (cid:80) u ∈E r (cid:16) a u A ru B u (cid:17) , whereeach term in the summation in (14) is the fraction of theuseful signal received at user u from RRH r over the totalsignals received at this user (including the useful signals).Using fractional programming [11, Corollary 1] over (14),we can define the following function. f ( r ; W , γ , β r ) = (cid:88) u ∈E r (cid:32) Re (cid:110) β ∗ ru (cid:112) δ u (1 + γ u ) w Hru ˆh ru (cid:111) − | β ru | (cid:88) r (cid:48) ∈B (cid:88) u (cid:48) ∈E r (cid:48) w Hr (cid:48) u (cid:48) (cid:16) ˆh r (cid:48) u ˆh Hr (cid:48) u + Θ r (cid:48) u (cid:17) w r (cid:48) u (cid:48) + σ z (15)where vector β r ∈ C |E r | is introduced as a new auxiliaryvariable, and Re {·} is the real part. The function (15) isconcave in β r . Also, it can be shown to be equivalent to (14)in the same way as was done with (11), i.e., by setting thepartial derivative with respect to β ∗ ru to zero, then substitutingthe value of β ru in (15) which yields (14).Then, our objective function in (12) can be written as f ( W , γ , β ) = (cid:88) u ∈U δ u (log (1 + γ u ) − γ u ) + (cid:88) r ∈B f ( r ; W , γ , β r ) , (16)where β = (cid:104) ( β ) T . . . (cid:0) β |B| (cid:1) T (cid:105) is the concatenation of theauxiliary variables β r ∈ C |E r | introduced in (15) for eachRRH r . IV. R ESOURCE A LLOCATION
A. Optimal Expressions
When the variables other than β r are fixed, the optimalvalue of the auxiliary variable β ru can be obtained from itscorresponding first-order optimality condition from (16) as β ru = (cid:112) δ u (1 + γ u ) w Hru ˆh ru (cid:80) r (cid:48) ∈B (cid:80) u (cid:48) ∈E r (cid:48) w Hr (cid:48) u (cid:48) (cid:16) ˆh r (cid:48) u ˆh Hr (cid:48) u + Θ r (cid:48) u (cid:17) w r (cid:48) u (cid:48) + σ z (17)Similarly for the beamformers w ru , we can write theLagrangian formulation using the new objective function (16)and the constraints in (12), then evaluating the correspondingfirst-order optimality condition to write w ru as w ru = (cid:112) δ u (1 + γ u ) β ∗ ru (cid:18) (cid:88) r (cid:48) ∈B (cid:88) u (cid:48) ∈E r (cid:48) | β r (cid:48) u (cid:48) | (cid:16) ˆh ru (cid:48) ˆh Hru (cid:48) + Θ ru (cid:48) (cid:17) + ( µ r + λ r α ru ) I M (cid:19) − h ru (18) where the Lagrangian multipliers λ r ≥ and µ r ≥ correspond to the capacity (9b) and power (9c) constraints.Importantly, both these constraints relate to the power usedat RRH r , i.e., both cannot be tight simultaneously. Fromcomplementary slackness, therefore, one of these Lagrangemultipliers, both corresponding to RRH r , must be zero.Unfortunately, we do not know a priori which constraintwill remain tight. As we will see in our algorithm section,we propose a heuristic that, at each iteration of the algorithm,checks for whether the capacity constraint is satisfied (allowing λ r = 0 ); if it is not satisfied, we update set λ r to a small valueand update µ r using a bisection search to meet the powerconstraint. Our results show that after a few iterations, λ r always converges to zero; we will comment on this in theresults section. B. Optimization Algorithm
Algorithm 1:
User-scheduling and resource allocation Initialize W and weights α ru for all users. while NOT converged do Update γ using (9d). Update β using (17). Update W using (18). Update { µ r , λ r : r ∈ B} as described usingcomplementary slackness. Update weights α using (8). end We construct Algorithm 1 to allocate the resources for theusers. The algorithm initializes some variables (Step 1) (e.g.,conjugate beamforming to initialize w ru ). Then, it updatesthe variables γ , β , W , and α iteratively one at a time untilconvergence.The complexity of updating γ , β , and α is O ( |U| ) , O ( |U| C avg ) , and O ( |U| C avg ) , respectively, where C avg is theaverage cluster size per user, and it is affected by both thedensity of the active users and the large scale fading threshold ρ . The complexity of the beamforming using a weightedMMSE [18] is O (cid:0) |U s | M + |U s | M (cid:1) , where U s is the setof scheduled users. Hence, leading to a total algorithm com-plexity of at most O (cid:0) M |B| + M |B| + |U| C avg (cid:1) , where thenumber of the scheduled users is at most |U s | ≤ M |B| .V. N UMERICAL R ESULTS AND A NALYSIS
To eliminate network borders, we consider a wrap-aroundstructure consisting of Q = 7 hexagonal virtual cells eachhaving an inner radius
500 m and containing N RRHs that areuniformly distributed in each virtual cell. Users are randomlydistributed with a density λ users and a circular exclusion regionof
20 m around each RRH. We average our results usingMonte Carlo simulations over both network realizations andtime slots (TSs), and we include the effect of the users fairness We create these virtual cells to allow for wrap-around; the cells have nophysical meaning.
ABLE I: Simulation parameters.
Parameter Value
Cell config. Q , N , M , λ users , , , users/km Power, Imperfect CSI p , τ d , ( τ p ) , p u
30 dBm , , (16 , , ,
20 dBm
Noise spectral density,Noise figure S z , F z , Band-width − / Hz , ,
180 KHz
Others σ shadowing , ρ , η , (cid:15) , L (0 . , . , . pM by simulating TSs and averaging the results over the lastallocated TSs, representing steady state performance .We use the COST231 Walfish-Ikegami [19] to model thepath loss at MHz, resulting in L | dB ( d ru ) = − . −
38 log ( d ru ) where d ru is in km . In Table I, we summarizethe parameters used.The proportional fairness weight, δ u , for user u is theinverse of the achieved long-term average rate over an ex-ponentially decaying window; in time slot t we set δ u as [20] δ ( t ) u = 1¯ R ( t ) u , (19)where δ ( t ) u is the value of δ u at time slot t , and ¯ R ( t ) u is theuser exponentially weighted rate averaged over previous timeslots, and it is updated as ¯ R ( t ) u = ηR ( t ) u + (1 − η ) ¯ R ( t − u witha forgetting factor ≤ η ≤ , where R ( t ) u is the rate achievedby user u at time t , and it can be defined as [13] R ( t ) u = ( τ d − τ p ) τ d log (cid:80) r ∈C u s ru (cid:12)(cid:12) h Hru w ru (cid:12)(cid:12) (cid:88) r (cid:48) ∈B (cid:88) u (cid:48) ∈E r (cid:48) ,u (cid:48) (cid:54) = u s r (cid:48) u (cid:48) (cid:12)(cid:12) h Hr (cid:48) u w r (cid:48) u (cid:48) (cid:12)(cid:12) + σ z (20)In Fig. 2(a), we plot the evolution of the allocated powerof the beamformer’s weights for a typical RRH in a typicalnetwork as a function of the algorithm iterations. It is clearthat after a few iterations, the power constraint (9c) is tighterthan the capacity constraint (9b) which becomes deactivated,as previously discussed. In Fig. 2(b), we illustrate the conver-gence of the algorithm for several channel realizations.In Fig. 3, we plot the long-term performance results underideal CSI, i.e., the channels are known and there is no pilottraining overhead. Fig. 3(a) shows the long-term network sumof spectral efficiency (SE) as a function of the algorithmiterations. The evolution of the curve shows that the algorithmconverges smoothly with a non-decreasing fashion. Also, theresults show a huge performance gain from using our approachcompared to the ZF and the conjugate beamforming schemeswith a round-robin scheduling. Compared to these schemesrespectively, we obtain about a . -fold and . -fold im-provement. Additionally, we plot the resulting network sum SEwhen using the ZF beamforming scheme with the optimizeduser-scheduling obtained from our proposed approach. The We emphasize that plotting the results from allocating a single TS woulddefinitely give much higher performance, because the users with the bestchannel’s conditions would be served, i.e., fairness is equal for all the users.Nonetheless, we are interested in studying the effect of the scheme on thelong-term. (a) Allocated power for the users’beamformers on a typical RRH.
Number of iterations O b j ec ti v e (b) Convergence plot for manychannel realizations. Fig. 2: Evolution of the algorithm.results still show a . -fold improvement. Clearly, this gap isdue to the fact that the ZF beamformers are constructed at eachRRH using only the channels of the served users, and eachuser is being allocated equal power irrespective of the channelconditions. Moreover, to quantify the effect of optimized user-scheduling, we compare the round-robin scheduling with thatof the optimized one for the ZF beamforming. The resulthighlights the importance of optimized scheduling, where a . -fold improvement is achieved. Scheme0100200300400500600 L ong - t e r m N e t S u m S E ( n a t s / s / H z ) Proposed approachZF, optimized schedulingZF, round-robinConjugate, round-robin (a) Network sum SE.
Long-term SE (nats/s/Hz/user) C D F Proposed approachZF, optimized schedulingZF, round-robinConjugate, round-robin (b) CDF of SE per user.
Fig. 3: Long-term results, N = 10 .In Fig. 3(b), we plot the cumulative density function (CDF)of the long-term SE of the users, where an to -fold gainis observed in the median long-term user spectral efficiencyfor our approach compared to round-robin scheduling. Weemphasize that this plot presents the long-term average ratethat accounts for the user scheduling. Users may not bescheduled in every time slot; this is determined by theirchannels and their weights as defined in (19). The gainsfor the 10 th -percentile rate, is clear. The proposed approachresults in about -fold and -fold improvement in the cell-edge long-term rate compared to round-robin scheduling andZF beamforming with optimized scheduling respectively.To quantify the performance of imperfect CSI, we comparethe following cases: • PI : Our proposed approach using ideal channels, whereno channel estimation phase is accounted for. PEAR : Our proposed approach when the algorithm isusing the estimated channel and using robust beamform-ing, i.e., accounting for the estimation error. However,when plotting the results, we plot the actual networkperformance, i.e., using (20).Since we use user-centric clustering, we define an area-based pilot-reuse factor (not cell-based) as ξ p (cid:44) τ p /λ users . Forexample, ξ p = 0 . , means that on-average one-quarter of theusers found in an area of × are using orthogonalpilots. Under the user density specified in Table I, the pilotsequence lengths τ p = 64 , , produce on-average ξ p =0 . , . , . respectively.In Fig. 4, we plot the long-term network sum SE of thedifferent studied cases using, but this time using N = 5 RRHsper virtual cell. The results show a drop of performance for
PEAR by . , , and . percent compared to the idealchannel case ( PI ). If we are to quantify the performance dropdue to only imperfect CSI, we obtain . , , and . percent drop in the performance compared to the ideal case.From the results, using τ p = 32 provides the highest sum SE,i.e., it is a good compromise between the pilot contaminationand the pilot-training overhead. Number of iterations L ong - t e r m N e t S u m S E ( n a t s / s / H z ) PIPEAR, p = 64PEAR, p = 32PEAR, p = 16 Fig. 4: Evolution of the Long-term sum of SE, N = 5 .VI. C ONCLUSION
This paper optimized user-scheduling and resource alloca-tion in a distributed cell-free MIMO system under the user-centric clustering scheme and the non-coherent transmissionmode using a weighted sum rate problem formulation. We usedtools from block coordinate descent, fractional programming,and compressive sensing to provide closed-form expressionsfor the optimized variables, while keeping the other variablesfixed. This allowed us to construct an iterative optimizationalgorithm that converges smoothly in non-decreasing fashion.Our key contribution is optimized user-scheduling, which isneglected in most of the literature. The numerical results showthat our optimized resource allocation boosts network perfor-mance, both in terms of sum-rate and long-term proportionalfair rates, compared to conventional round-robin schemes,where an to -fold gain in the long-term user spectralefficiency is observed. 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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