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Capacity analysis of Interlaced Clustering in aDistributed Transmission System with/without CSITVishnu V. Ratnam, Student Member, IEEE, Andreas F. Molisch, Fellow, IEEE, and Giuseppe Caire, Fellow, IEEE

Abstract—With growing base-station density and decreasingfrequency reuse factor, inter-cell interference and low cell-edgeuser rates are becoming serious problems. Legacy solutions likeFractional Frequency Reuse are simple to implement but aresuboptimal. In this paper we investigate interlaced clustering asa solution to the edge user problem for a general distributedcellular transmission system. In interlaced clustering severaldifferent coverage patterns co-exist on disjoint parts of thespectrum. We demonstrate how various previously suggestednetwork architectures can be interpreted as special cases ofinterlaced clustering. We then characterize the downlink userthroughputs at the proportional fairness operating point of therate region for both of the cases that the transmitter does, or doesnot, have channel state information. Based on this derivation,we develop a novel algorithm to solve the resource allocationproblem for systems with interlaced clustering. Simulations basedon practical cell parameters show that interlaced clustering canprovide, on an average, a 100% gain on edge user rate withoutappreciable loss in rates elsewhere. We also verify that this resultis robust to irregular deployment of the remote antenna units.

Index Terms—Cloud-RAN, Distributed Antenna System, Dis-tributed MIMO, Network Resource Allocation, Interlaced Clus-tering, Cell-edge performance.

I. INTRODUCTION

With the proliferation of wireless devices and with theadvent of high data-rate applications (like video streaming, on-line gaming) a tremendous increase in demand for cellular datathroughput is observed, and predicted to continue in the nearfuture [2]. Over the last decade, the use of classical techniquesto cope with the rising data demand, like shrinking of cell sizeand reduction of frequency reuse factor have led to high inter-cell interference. In the currently used co-located transmissionsystems1, this severely degrades the user rates especially at thecell edge, as illustrated in Fig.1a.

One solution to boost spectral efficiency while also cateringto the edge users is distributed transmission [3]–[8]. In 3GPPjargon, this network architecture is more commonly referred toas cloud-RAN [9], [10]. In a distributed transmission system,the cellular service region is divided into cooperation clusterssuch that the remote antenna units (for example base stations)within a cooperation cluster jointly transmit signals to theusers in their common coverage region (see Fig.1b). The intra-cluster interference can then be mitigated thereby reducing the

Part of this work will be presented at IEEE ICC 2015 [1]. This workwas supported by the Ming Hsieh Institute (MHI) and the Dean’s office ofthe Viterbi School of Engineering at the University of Southern California.The authors are with the Department of Electrical Engineering, University ofSouthern California, Los Angeles, CA, 90089 USA (e-mail: ratnam@usc.edu,molisch@usc.edu, caire@usc.edu)

1Here, we shall refer to a cellular system with no base-station cooperationas a co-located transmission system

overall interference in the interior of the cluster. However, atthe cluster-edge, the inter-cluster interference power is stillsignificant (see Fig.1b). Therefore, distributed transmission isonly partially successful in solving the edge user problem [11]and additional techniques to cater to the cell edge problemare essential. Some Coordinated Multi-Point (CoMP) schemes

(a) Colocated (b) Distributed

Fig. 1. Edge user problem in wireless downlink, for the example case ofuniversal frequency reuse.

like Dynamic Cell Selection or Coordinated Scheduling [12],[13] can possibly be used in conjunction with distributedtransmission systems to introduce partial cooperation amongthe cooperating clusters and thereby reduce the inter-cellinterference. One similar scheme with inter-cluster coordi-nation via block diagonalization was introduced in [14]. Analternate strategy, where the pattern of cooperation clusters isdynamically changed based on scheduled user locations waspresented in [15]. A combination of the above two strate-gies was considered in [16]. However, with these solutions,the edge user problem might not be completely eliminatedand, additionally, inter-cluster timing synchronization, channelstate information at the transmitter (CSIT) and/or back-haulinformation exchange across multiple cooperation clusters arerequired. Also, the dynamic clustering procedures may imposesignificant computational load on the transmitters. A goodreview of such schemes is available in [17]. Alternate schemes,which are enabled in 3GPP-LTE and WiMAX standards, likeFractional Frequency Reuse (FFR) boost the cell-edge userrates at the cost of the interior users. For example, both in thecase of soft-FFR [18] and hard-FFR [19], the transmit powerto interior users is reduced and/or universal frequency reuseis not possible. Some other techniques in the literature, likeoverlapping coverage [20], though simple, do not permit fullfrequency reuse and hence are suboptimal.

In the current paper, we explore the possibility of interlacedclustering as a solution to mitigate the cell edge problem in

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distributed transmission systems. This type of network archi-tecture was first considered, by one of the authors, in [21]–[24], wherein two scenarios of interlaced clustering, namelyoverlapped clustering and expanded cellular system, wereanalysed in the specific context of a distributed transmissionsystem with CSIT. Here we shall show that the concept isalso applicable to the cases without CSIT. Also, as shallbe discussed later, many other network architectures, e.g.,centralized tiered networks or even fractional frequency reusecan also interpreted as (possibly suboptimal) instances ofinterlaced clustering. Thus, in this work we analyse interlacedclustering from a more general perspective and in more detail,and we shall show that the idea can be generalized to any typeof distributed transmission system.

As mentioned before, in distributed transmission the RemoteAntenna Units (RAUs) are grouped into disjoint cooperationclusters, thus forming a cluster pattern (CP). In interlacedclustering, multiple such CPs operate on orthogonal frequencybands. These CPs may differ from each other in a varietyof ways. In one important case, called overlapped clustering[21], the CPs are offset from each other (by less than a clustersize). As an illustration, Fig.2a and Fig.2b depict two suchpossible cluster patterns (CP1 and CP2) that create a differentgrouping of the RAUs into clusters. In interlaced clustering,these cluster patterns CP1 and CP2 operate, in parallel, onorthogonal frequency bands ω1 and ω2 such that the totalsystem bandwidth BW = ω1 + ω2 (see Fig.2c). In someother forms of interlaced clustering, the cluster patterns maydiffer in the sets of active RAUs (for example, expandedcellular systems [22] and heterogeneous networks) or havedifferent frequency reuse factors (e.g. FFR). These cases shallbe discussed in more detail in section II-A). Typically, allclusters in a CP use the whole frequency band allocated tothat CP.2 For example, each cluster in CP1 uses the wholeof ω1. The rationale in favour of interlaced clustering is thata user who is located at a cluster edge (and hence has lowSINR) in one CP, is in the interior of a cluster on a differentCP with a high probability. Intelligent scheduling of thenetwork resources (power, time-frequency blocks) across theCPs can then eliminate the notion of edge users and improvethroughput performance. Another important advantage of hav-ing interlaced clusters is that it improves resilience to irregulardeployment of RAUs. The network planner does not need toworry about shadowed areas with low SINR as any point isserved by multiple CPs, thereby reducing the probability that ithas a bad SINR on all CPs. Unlike CoMP, dynamic-clusteringor inter-cluster block-diagonalization, CSIT is not necessarilyrequired, no inter-cluster cooperation is required and the edgeusers can be effectively eliminated. Furthermore, unlike FFR,interlaced clustering allows full use of spectrum and power atany geographical point, thereby, leading to higher throughputs.One disadvantage of interlaced clustering is that it requires anincreased amount of back-haul connectivity. As opposed to nointerlacing, where each RAU is a member of a single cluster,in interlaced clustering each RAU is a member of a differentcluster in each CP. There thus need to be connections between

2One exception is the case of FFR (see section II-A)

(a) CP 1

(b) CP 2

(c) Interlaced

Fig. 2. An example layout for Interlaced clustering

all the corresponding RAUs (see Fig. 2a and 2b). Note howeverthat the total ”amount” of data exchange required is still thesame, because, each CP operates only on a fraction of thewhole system bandwidth.

Based on the absence or availability of CSIT, this paperconsiders the following two scenarios of deployment for adistributed transmission system:

1) Distributed Antenna Systems (DAS) - Without CSIT:Initially proposed for covering shadowed regions or dead-spots, the potential benefits of DAS for improving spectralefficiency were soon recognized and widely investigated [3],[4], [25]–[30]. The MISO downlink rates for the case of

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a single-user per cell, both with single RAU transmissionand with blanket transmission were studied in [3] and latergeneralized to other schemes in [4]. The case with multiplescheduled users per cell was considered in [25]. The optimalplacement of the RAU’s was considered in [26]. The cellaveraged ergodic capacities, mean and outage capacities for theMIMO scenario with multiple mobile station antennas werederived in [27]–[29]. Optimal transmit power allocation in amulti-cell, multi-user setting is considered in [30]. However ananalysis on performance of a DAS in an interlaced cluster typesetting has not been considered in the literature, to the bestof the authors’ knowledge. A preliminary study on interlacedclustering in a Distributed Antenna System was presented byus in the conference paper [1].3

2) Distributed MIMO (DMIMO) - With CSIT: The capacityof the MIMO Broadcast channel with no common messageand the optimality of Dirty Paper Coding was shown in[31]. Significant amount of work on spectral efficiency ofMu-MIMO has followed suit. The multi-cell setting withWyner model was proposed in [32] and extended to partialcooperation scenarios in several other works (see [33] andreferences therein). Recent work in [7], [34] consider morerealistic channel models, clustered cooperation among BaseStations and derive the ergodic rates at the fairness operatingpoint of the capacity region in the large system limit. Apreliminary throughput analysis of DMIMO with two specific,albeit important, scenarios of interlaced clustering using Gaus-sian single-user codes, zero-forcing pre-coding, equal powerallocation and treating all interference as noise was discussedin [22] and some simulation results were presented in [21].However, an information theoretic analysis of the achievablerates of a DMIMO system with interlaced clustering, underpossible non-linear pre-coding and optimal power allocationunder some fairness constraint has not been considered yet inliterature, to the best of the authors’ knowledge.

We would like to clarify that the terms DAS and DMIMOhave often been used interchangeably in literature. There areseveral papers on DAS with full/partial CSIT as well ason DMIMO without CSIT (see [35] and references therein).However, we follow the aforementioned nomenclature here fordisambiguation.

The contributions of this paper are as follows: A verygeneralized form of interlaced cluster network is proposed.We also show how many previously suggested network ar-chitectures fall within this framework. The paper character-izes the proportional-fairness operating point of the downlinkachievable-rate region, for a DAS (without CSIT) and aDMIMO system (with CSIT) with interlaced clustering inthe large antenna limit.4 In the process, novel algorithmsfor finding the optimal allocation of resources are proposed.We also present decentralized practical scheduling schemesthat can achieve these theoretically obtained results. Thebenefits of interlaced clustering are studied, both in termsof the edge-user and over all cell throughput performance,

3Mentioned references only correspond to the no-CSIT case. DAS with fullCSIT shall here be referred to as DMIMO

4We consider Gaussian single-user coding followed by possible multi-userpre-coding at transmitter and treat interference as noise at receiver

in practically relevant simulation settings. The resilience toirregular deployment of RAUs and impact of the number ofcluster patterns are also studied via simulations.

The rest of the paper is organized as follows: Section IIdiscusses the general assumptions, the channel model, thenetwork utility maximization problem and some examples ofinterlaced clustering. In section III and IV achievable rate-region and the PF operating point for DAS and DMIMOare characterized, respectively. Practical resource allocationstrategies to achieve these rates are discussed in section V.An evaluation of interlaced clustering in terms of edge userperformance, resilience to irregular RAU deployment andresilience to rising user demand, performed in a variety ofpractical simulation settings, are presented in section VI. Theconclusions are discussed in section VII.

Notation used in this paper is as follows: scalars are repre-sented by light case letters; vectors and matrices by bold caseletters; sets are represented by calligraphic letters; ⊗ denotesthe Kronecker product, [A]i,j represents the (i, j)

th element ofa matrix A, [A]j represents the jth element of a vector or setA, {A}j represents the jth element of a set A taken in somearbitrary but prefixed order, |A| denotes the cardinality of aset A or the determinant of a matrix A depending on context,E{} denotes the expectation operator, ‖.‖ denotes the l2 norm,{}† denotes the conjugate transpose operation, PX{} denotesprobability density/mass function of random variable X , andPerm{.} defines the set of all permutations of a vector.

II. GENERAL ASSUMPTIONS AND CHANNEL MODEL

The cellular system contains F = {1, ..., F} differentCPs operating on disjoint frequency bands (ω1, ..., ωF ). Eachcluster is represented as 〈z, f〉 where z ∈ Z is the cluster indexand f ∈ F represents the frequency band/cluster pattern. Let Brepresent the set of all RAUs, each having a unique RAU indexb ∈ B, and let Bz,f = {b ∈ B : RAU-b ∈ 〈z, f〉} represent thesubset of RAUs assigned to a cluster 〈z, f〉. Similarly, let Ube the set of all users, each with a unique user index u, andlet Uz,f ⊂ U be the subset of users assigned to a cluster〈z, f〉. For example in Fig.2c, {u1, u2, u

′3, u′4} ⊂ U1,1,

{u2, u3} ⊂ U1,2 and {u3, u4} ⊂ U3,1. Also we define thefunction:

g : U × F → Z ×F s.t. g(u, f) = (z, f) iff u ∈ Uz,f (1)

The paper focuses on the downlink scenario (DL) as it isgenerally the bottleneck [36]. Each RAU has Nt transmitantennas and each mobile-station/user is assumed to have asingle receive antenna. The DL channel for any Tx-Rx pair isassumed to be composed of a distance based path-loss, log-normal shadow fading component and a small-scale Rayleighfading component. For ease of analysis, the DL channel isassumed to be independent across the users and RAU pairs,and across antennas (at an RAU), i.e., we assume independentsmall scale fading in the spatial domain. We consider the caseof a large array of antennas at the RAUs, which not onlyfollows the current trend in cellular and WiFi standardization,but also eases theoretical analysis, since in the large-systemlimit the instantaneous (per-fading block) user rates converge

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to deterministic limits that can be characterized by compactalmost closed-form expressions. With such a large numberof Tx antennas, the channel fading, and correspondinglyergodic user rates, are identically distributed across frequency.Therefore, without loss of generality, we consider a frequencyflat block-fading channel and we restrict analysis to having asingle frequency resource block per CP.

Under these assumptions, the DL received signal at a useru on CP f can be described as:

yfu =∑

b∈Bg(u,f)

βfu,bHfu,bX

fb + nfu (2)

where Xfb is the Nt × 1 transmit signal, Hf

u,b ∼ CN (0, INt)

is the 1 × Nt small scale fading coefficient vector, nfu ∼CN (0, 1) is the normalized noise plus interference signal,βfu,b =

√PL(dfu,b)/σ

2u,f is the average link quality, PL(dfu,b)

is the large scale fading component5, dfu,b is the distance fromthe RAU-b to user u and σ2

u,f is the power of noise plusinter-cluster interference. In the absence of cooperation, wecan’t exploit the inter-cluster interference diversity and so wereplace the noise plus interference power by its long termmean:

σ2u,f = σ2

n +∑

b/∈Bg(u,f)

P fb,avPL(dfu,b) (3)

P fb,av ≈ P fsum/|Bz,f |, ∀b ∈ Bz,f (4)

where, P fb,av is the average transmission power of RAU b onCP f and P fsum is the per-cluster sum power constraint on CPf . A sum power constraint is more appropriate than a per RAUpower constraint for a Distributed Transmission system. Thisis because the RAU power amplifiers typically have a widedynamic range and therefore power imbalance among RAUscan be implemented. 6 The power constraints mostly stemeither from interference regulations or from the operationalexpenditure both of which depend on the per-cluster sumtransmit power. Though (4) is valid when we have a singleCP, it does not strictly hold for interlaced clustering. This isbecause, intuitively, an RAU spends more transmit power onthe frequency band/ CP in which it is in the cluster interior thanon other CPs.7 Hence on any CP, most of the the interferencepower comes from the cluster interiors. Therefore, assumption(4) is conservative and it underestimates the performanceof interlaced clustering when compared to other schemes.However, it is required for analytical tractability.

When we have multiple interlaced cluster patterns, eachuser can be served by either of them and hence, a jointresource allocation has to be considered. The proportionalfairness metric (which is most widely used in practice) is usedto allocate the network resources (power, time or frequencyslots) to the users. Let mf

u be the amount of resource (power

5Both pathloss and a shadow fading are included in the PL(dfu,b), and areassumed to be identical for all RAU antenna elements

6A large number of parallel transmissions going on within each cluster (ondifferent frequency sub-carriers) ensure that the power imbalance among theRAUs is not significantly large.

7Refer to [1] for a more extensive explanation in the case of a DAS

and/ or fraction of T-F slots) allocated to a user u by cluster〈g(u, f)〉. Correspondingly, let Cfu

(mf)

represent the ergodiccapacity of user u on frequency band ωf for given resourceallocations mf = [mf

1 , ...,mf|U|]. The proportional fairness

boundary point of the rate region is then given by the ratevector R = [R1, .., R|U|] that maximizes:

max

{∑u∈U

log2 [Ru]

}s.t. (5)

Ru ≤∑f

Cfu(mf)∀u ∈ U∑

u∈Uz,f

mfu ≤Mf ∀〈z, f〉

where, Mf is the per cluster, sum resource constraint (powerand/ or T-F slots) on CP f . Such a joint/centralized resourceallocation problem may seem against our motive of havingno cooperation among the clusters and the cluster patterns.However we shall later see (in section V) practical schedulingstrategies that can achieve similar performance in a decentral-ized manner.

A. Other examples of interlaced clustering

A heterogeneous cellular network with a single base-stationper cell and an underlying femto-cell tier operating on or-thogonal frequency bands ω1, ω2, respectively, is an exampleof interlaced clustering. As depicted in Fig.3, CP1 is formedby the base-stations with each base-station solely forming acooperation cluster. CP2 on the other hand is formed justby the femto-cells, with the femto-cells in each cell togetherforming a cooperation cluster. In this case, typically, CP2 haslimited (or no) intra-cluster cooperation. Note that here, asthe CPs are not shifted versions of each other, the edge-userproblem persists. Nevertheless the analysis presented in thecurrent paper can also be applied to this scenario. Similarly,fractional frequency reuse can also be treated as an exampleof interlaced clustering with 2 CPs. Here, CP1 creates somegrouping of RAUs into clusters, for example, as depictedin Fig.2a. Each of these clusters use the whole availablebandwidth ω1 (reuse 1). CP2 has the same layout as CP1but has a higher frequency reuse factor Λ. Therefore eachcluster in CP2 uses only a fraction of the bandwidth ω2/Λ. Itis clear that here we do not have universal frequency reuse andhence FFR is a simple albeit sub-optimal form of interlacedclustering. Similarly, the sector offset approach [37] can alsobe treated as a special case of interlaced clustering whenthere is no inter-RAU cooperation. Here the sectors at eachRAU constitute a cluster and the different sector patterns withangular offset form the different cluster patterns.

III. ACHIEVABLE RATES IN DAS DOWNLINK

The DL signal and interference for a DAS are as givenin (2) and (3), respectively. In the absence of CSIT, themultiple antennas at an RAU are used only to improve signaldiversity and the transmit power is allocated equally acrossthem. Though schemes exist that allow in any cluster 〈z, f〉,the scheduling of up to |Bz,f | users on the same T-F slot,

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(a) Femto-Layout

(b) CP 1

(c) CP 2

Fig. 3. Femto-cell architecture as an example of interlaced clustering

one per RAU (refer to [25]), this complicates the analysis ofthe interlaced clustering and hence we limit the analysis toone scheduled user per T-F slot in a cluster i.e. orthogonalintra-cluster access. Therefore, in this case, the total clusterpower P fsum is allocated only to the scheduled user. Also,in the absence of CSIT, we consider equal power allocationacross the antennas at each RAU. Though instantaneous CSITis unavailable, we assume that all the RAUs b ∈ Bz,f knowthe average link quality Dg(u,f) = {βfu,b : b ∈ Bg(u,f)} to

the users in their cluster.8 The achievable rate of a user u inpattern f can then be expressed as:

Cfu (tfu)

= tfuωf maxP f

u,:

E

log2

1 +∑

b∈Bg(u,f)

(βfu,b)2P fu,bH

fu,bH

fu,b

†

Nt

s.t:

∑b∈Bg(u,f)

P fu,b = P fsum

≤ tfuωf max∑b P

fu,b=P f

sum

{log2

(1 +∑

b∈Bg(u,f)

(βfu,b)2P fu,b

)}(6a)

= tfuωf log2

(1 + P fsum max

b{(βfu,b)

2})

= tfuωf log2

1+P fsum maxb{PL(dfu,b)}

σ2n+∑b′ /∈Bg(u,f)

P fsumPL(dfu,b′)/Bb,f

︸ ︷︷ ︸

Cfu

(6)

where, tfu is the fraction of T-F resource allocated to anyuser u by cluster 〈g(u, f)〉, ωf is the bandwidth of CP-f ,P fu,b = E{‖Xf

b ‖2} is the transmit power to u from RAU-b

and Bb,f = |Bz,f | for 〈z, f〉 such that RAU-b ∈ Bz,f . Though(6a) is an upper bound, obtained via Jensen’s inequality, forNt � 1 it is pretty tight (from law of large numbers). From(6), as expected for Nt � 1, the channel hardens and thepower is allocated to the RAU b with the best channel linkquality to the user (this is similar to transmission strategy in[3]). Therefore for a user scheduled on a T-F slot, the powerdistribution among the RAUs is very skewed with a per clusterpower constraint. However considering that a large number ofusers are scheduled in parallel on different sub-carriers andacross different CPs, the power allocation is more regular overthe whole bandwidth.

The PF operating point of the rate region can then becharacterized by the rate vector R maximizing:

maxR,t

∑u∈U

log(Ru)

s.t: (7)

Ru ≤∑f∈F

Cfu (tfu) ∀u ∈ U ,∑

u∈Uz,f

tfu = 1 ∀〈z, f〉

tfu ≥ 0 ∀u ∈ U ,∀f ∈ F

Note that this is the same as (5) with the resource being the T-Fslots. As the cluster power is allocated to a single user at anytime, the network resource allocation only involves T-F slotallocation. It is easy to verify that (7) is a convex optimizationproblem. Instead of using Interior point methods [38] whichare relatively slower, we propose a novel algorithm to solve itas explained below. Using Lagrangian dual variables W, we

8As large scale fading changes very slowly with time, these values caneasily be tracked

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can reformulate (7) as:

minW≥0

{−∑u∈U

(logWu + 1) + maxt{∑u∈U

∑f∈F

tfuWuCfu}}

s.t.∑

u∈Uz,f

tfu = 1 ∀〈z, f〉, tfu ≥ 0 ∀u ∈ U ,∀f ∈ F

≡ minW≥0

{−∑u

(logWu + 1) +∑〈z,f〉

maxv∈Uz,f

{WvCfv }}

(8)

where Cfu is as defined in (6). This dual objective function isnon-differentiable, but it reduces the number of optimizationvariables by a factor of the number of CPs |F|. We propose anovel gradient search algorithm for solving the problem, whichis presented in Appendix A. Though it is difficult to find acomplexity bound on the algorithm (as in the case of simplexalgorithm), we observe that it performs very well in practice.For small values of |F| (which are expected in practice), weobserve that it converges to the solution much more quicklythan interior point methods. With the optimal dual variablesW∗ computed, the optimal user rates can be computed as:R∗u = 1/W ∗u .

IV. ACHIEVABLE RATES FOR DMIMO SYSTEM

In the case of DMIMO, for analytical tractability (similarto the approach in [7]), on any CP f we discretize the userlocations by clubbing the users into user-groups consisting ofN co-located users each. Users in a user-group are assumed tohave identical large scale fading but independent small-scalefading. This is reasonable since large scale fading changesslowly with location. The channel model and assumptionsin Section II are also applicable here, but with the slightnotational change that u refers to the index of a user groupinstead of an individual user. Similarly, U is the set of all usergroups in the geographical area and |Uz,f | is number of usergroups in cluster 〈z, f〉. Under this notational change, the DLreceived signal and the noise-plus-interference power at a usergroup u in cluster pattern f can again be expressed as in (2)and (3), respectively. However, yfu, nfu ∼ CN (0, IN ) are nowN×1 vectors corresponding to the N users in the user group uand Hf

u,b is the N×Nt small scale fading matrix from RAU bin cluster 〈g(u, f)〉 to user group u with i.i.d. CN (0, 1) entries.The DL signal can further be written compactly in matrix formas:

yfu = Hu,f ×√Nt

Xf

1...

Xf|Bg(u,f)|

︸ ︷︷ ︸

Xg(u,f)

+nfu, where: (9)

Hu,f ,1√Nt

[βfu,1H

fu,1 . . . βfu,|Bg(u,f)|H

fu,|Bg(u,f)|

]and for ease of representation, we denote βfu,m = βfu,b,Hfu,m = Hf

u,b and Xfm = Xf

b if RAU-b is the m-th RAUin cluster 〈g(u, f)〉 i.e. b = {Bg(u,f)}m.

As this is an example of a broadcast channel, we considerthe use of the capacity achieving Dirty Paper Coding (DPC) inthe DL transmission from RAUs to user groups. This allows

for use of Gaussian single user coding and non-linear multiuser pre-coding at the transmitter. By the Broadcast-MACDuality [39], any set of user rates achievable in a DL Broadcastchannel for a cluster using DPC (with sum power constraint)can also be attained in the dual uplink (UL) multiple accesschannel (MAC) using Successive Interference Cancellation(SIC) with the same sum power constraint on the dual transmitpowers. In the dual MAC channel for a cluster 〈z, f〉, let Qfu,πfu denote the allocated dual MAC power and the dual MACSIC order for user group u respectively (i.e., ∀u, v ∈ Uz,f ifπfu < πfv , then u is decoded before v in the UL MAC channel).Let us define:

Kz,f ,[πf{Uz,f}1

. . . πf{Uz,f}|Uz,f |

](10)

Qz,f ,[Qf{Uz,f}1

. . . Qf{Uz,f}|Uz,f |

](11)

as the SIC order in the cluster 〈z, f〉 and the corresponding al-located dual transmit powers, respectively. Note that Kz,f canbe expressed as a permutation of the vector [1, 2, ..., |Uz,f |].We consider joint decoding of all users within the sameuser group u in the dual UL MAC channel and hence πfucorresponds to a user group and not a particular user. In thecorresponding BC channel, this assumption is equivalent totaking the average (over a user group) of the user rates. As weare concerned with the long term ergodic user rates (which areindependent of small scale fading) all users in the same groupare equivalent and hence joint decoding in the dual MACchannel is acceptable here. The per user ergodic achievablerate in a user group u in DL on frequency f for a given powerallocation Qg(u,f) and SIC order Kg(u,f) can be expressed as(see [7] for details):

Cfu (Kg(u,f),Qg(u,f))

=ωfN

E

log

∣∣∣I +∑i∈{u}∪J f

uH†i,fHi,fQ

fi

∣∣∣∣∣∣I +∑k∈J f

uH†k,fHk,fQ

fk

∣∣∣ (12)

where, J fu = {v ∈ Ug(u,f) : πfv > πfu}, ωf is the bandwidthof CP f and Hu,f is as defined in (9).

For finding the ergodic capacity region, all the users areconsidered as simultaneously transmitting (in the dual MACchannel) and hence also feature in the SIC order. This isequivalent to setting tfu = 1 ∀u, f and therefore, the jointresource allocation only involves power allocation. Now, asexplained before in Section II, the PF operating point is givenby the rate vector R maximizing:

maxR,Q,K

∑u∈U

log(Ru) s.t: (13)

Ru ≤∑f

Cfu (Kg(u,f),Qg(u,f)) ∀u ∈ U∑u∈Uz,f

Qfu ≤ P fsum ∀〈z, f〉, Qfu ≥ 0 ∀u ∈ U , ∀f ∈ F

Kz,f ∈ Perm {[1, 2, ..., |Uz,f |]} ∀〈z, f〉

where Kz,f , Qz,f are as defined in (10) and (11), respectivelyand K = {Kz,f |z ∈ Z, f ∈ F}, Q = {Qz,f |z ∈ Z, f ∈ F}are the power allocation and SIC-order sets respectively. Note

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that this is same as (5) with the resource being the clusterpower. Defining Wu as the dual variables, the Lagrangian dualfor (13) can be rewritten as:

minW≥0

maxR,Q,K

∑u∈U

[log(Ru)−WuRu

+∑f∈F

WuωfN

E

log

∣∣∣I+∑i∈{u}∪J f

uH†i,fHi,fQ

fi

∣∣∣∣∣∣I +∑k∈J f

uH†k,fHk,fQ

fk

∣∣∣]

(14)

s.t.∑

u∈Uz,f

Qfu ≤ P fsum ∀〈z, f〉, Qfu ≥ 0 ∀u ∈ U ,∀f ∈ F

Kz,f ∈ Perm {[1, 2, ..., |Uz,f |]} ∀〈z, f〉J fu =

{v ∈ Ug(u,f) : πfv > πfu

}∀u ∈ U , f ∈ F

The objective function in (14) can further be reduced to:

minW≥0

[maxR

S(W,R)

+∑〈z,f〉

maxQz,f ,Kz,f

{ ∑u∈Uz,f

WuCfu (Kz,f ,Qz,f )

}︸ ︷︷ ︸

Inner WSRM

](15)

where, S(W,R) =∑u∈U log(Ru)−WuRu.

The objective function above can be interpreted as follows:for each choice of weights W, the inner WSRM problem findsthe optimal SIC order and the optimum allocation of powercorresponding to a boundary point of the user achievable rateregion. The outer minimization step helps set these weightscorresponding to the proportional fairness point. To removedependence on instantaneous channel realizations, we allowNt = γN and N → ∞ (large antenna limit). This is acommon assumption that is often used in literature and hasbeen shown to accurately match the ergodic user rates, evenfor small dimensions such as N = 2 [7], [40], [41].

A. Inner Weighted Sum Rate Maximization (WSRM) Problem:

The inner WSRM on cluster 〈z, f〉 is given by:

maxQz,f ,Kz,f

{ ∑u∈Uz,f

WuωfN

E log|I+

∑i∈{u}∪J f

uH†i,fHi,fQ

fi |

|I +∑k∈J f

uH†k,fHk,fQ

fk |

}s.t.

∑u∈Uz,f

Qfu ≤ P fsum, Qfu ≥ 0 ∀u ∈ Uz,f

Kz,f ∈ Perm {[1, 2, ..., |Uz,f |]}

Note that this inner optimization only involves a single cluster(on a particular cluster pattern). Just for this subsection, forease of notation, let Uz,f = {1, ..., U}, Bz,f = {1, .., B}and κf be the inverse mapping of πf{:} over {1, ..., U}, i.e.if πfu = j, then κfj = u for any u ∈ {1, ..., U}. Without lossof generality, dropping the reference to the cluster 〈z, f〉, a

general abstraction of the WSRM problem is then given as:

I(W) :

maxQ,κ1:U

{U∑j=1

Wκjω

NE log

|I +∑Ui=j H†κi

HκiQκi ||I +

∑Uk=j+1 H†κkHκk

Qκk|

}

s.t.U∑i=1

Qκi≤ Psum, Qu ≥ 0 ∀u ∈ {1, ..., U}[

κ1 . . . κU]∈ Perm {[1, 2, ..., U ]}

It is well known that the SIC rate region above is a poly-matroid and the optimal decoding order is in the increasingorder of the weights W i.e. for any u, v ∈ {1, ..., U}, π∗u > π∗vif Wu > Wv and correspondingly κ∗i = u if Wu is the ith

maximum weight within the cluster. The algorithm for findingthe optimal power allocation Q∗ under the large antenna limitwas derived in [7] which is reproduced here (with requiredchanges to the notations) in Algorithm 1, for convenience.

Algorithm 1: Input power optimization for Weighted SumRate Maximization

Inputs: γ, Psum, βi,m and Wu

∀u ∈ {1, .., U},∀m ∈ 1, .., BInitialize Qk(0) = Psum/U for all k = 1, .., UInitialize κ∗i = u if Wu is the ith max weightfor j = 1 to U do

∆j = Wκ∗j−Wκ∗j−1

end forwhere, we let Wκ∗−1

= 0.for i = 0, 1, 2, ... do

Iterate until following solution settles for j = 1, ..U :

Qj(i+ 1) = Psum

∑jk=1 ∆k(1−Υ

(j)k:U (i))∑U

l=1

∑lk=1 ∆k(1−Υ

(l)k:U (i))

Here, Υ(j)k:U (i) = 1/(1 + Γ

(j)k:U (i)), and Γ

(j)k:U (i) is

obtained as a solution of the following system offixed-point equations:

Γ(j)k:U (i)=

B∑m=1

γ[βκ∗i ,m

]2Qκ∗j (i)

1+∑Ul=k(β

2κ∗l,m

Qκ∗j )(i)/(1+Γ(l)k:U (i))

where, j ≥ kend forDenote by Γ

(j)k:U (∞), Υ

(j)k:U (∞) and by Qj(∞) the

fixed-points reached by the iterations above. If thecondition:

Psum

j∑k=1

∆kΓ(j)k:U (∞) ≤

U∑l=1

l∑k=1

∆k(1−Υ(l)k:U (∞))

is satisfied for all j such that Qj(∞) = 0, then stop.Otherwise, go back to the initialization step and setQj(0) = 0 and repeat above procedure from the newinitialization.

Given the optimal power allocation, when N →∞ we have:

8

(refer [7], [42] for proof)

1

NE

log

∣∣∣∣I +

U∑i=j

H†κ∗iHκ∗i

Q∗κ∗i

∣∣∣∣ =

U∑i=j

log

(1 + γQ∗κ∗i

B∑m=1

β2κ∗i ,m

φjm

)

+ γ

B∑m=1

log

(1 +

U∑i=k

β2κ∗i ,m

Q∗κ∗i θji

)

− γU∑i=j

B∑m=1

β2κ∗i ,m

Q∗κ∗i φjmθ

ji (16)

where βu,m is as defined in (13) and for each j = 1, .., U ,{φjm : m = 1, .., B} and {θji : i = j, .., U} are the uniquesolutions to the system of fixed point equations:

φjm =

1 +

U∑i=j

Q∗κ∗i β2κ∗i ,m

θji

−1

, m = 1, .., B (17a)

θji =

(1 + γ

B∑m=1

Q∗κ∗i β2κ∗i ,m

φjm

)−1

, i = k, .., U (17b)

Using algorithm 1 and (16), the asymptotic rates at theboundary point of the rate region, defined by W, can thenbe computed as:

Cκ∗j (K∗(W),Q∗(W)) =ω

NE

{log

∣∣∣∣I +

U∑i=j

H†κ∗iHκ∗i

Q∗κ∗i

∣∣∣∣− log

∣∣∣∣I +

U∑i=j+1

H†κ∗iHκ∗i

Q∗κ∗i

∣∣∣∣}

(18)

B. Outer minimization problemBringing back the dependence on frequency band, let

ru(W) =∑f C

fκ∗j

(Kf∗(W),Qf∗(W)

). Then (15) can be

reduced to:

minW≥0

{maxR{S(W,R)}+

∑u∈U

Wuru(W)︸ ︷︷ ︸L(W)

}(19)

For any W we can write:

L(W) = maxR{S(W,R)}+

∑u∈U

Wuru(W)

≥ S(W,R∗(W)

)+∑u∈U

Wuru(W)

≥ L(W) +∑u∈U

(Wu −Wu)(ru(W)−R∗u(W))

where, R∗(W) is the optimal rate vector for weights W.Therefore the outer minimization can be performed using asub-gradient descent approach with the update being given as:

Wu(n+ 1) =

Wu(n)− µ[ru(W(n)

)−R∗u

(W(n)

)]∀u ∈ U (20)

where µ controls the step size of the update and can either bekept constant or be updated based on any line search algorithm(for faster convergence). For a given W, the optimal R∗(W)for PF scheduling is given as R∗u(W) = 1/Wu. Therefore(20) can be rewritten as:

Wu(n+ 1) =

Wu(n)− µ[ru(W(n)

)− 1/Wu(n)

]∀u ∈ U (21)

V. DECENTRALIZED RESOURCE ALLOCATION

The aforementioned theoretical analysis considers a jointallocation of resources across the clusters and cluster patterns.However, for implementation, we require per-cluster resourceallocation schemes, to minimize the inter-cluster cooperation.In the current subsection we shall discuss such practicalscheduling strategies which can achieve the above theoreticalperformance but in a decentralized way. For any user u letT fu,avg represent the average rate that the user gets from CPf . We let the neighbouring clusters share these average ratesfor their common users. Since these average rates change veryslowly with time, they introduce a negligible sharing overhead.Referring back to (5), the decentralized resource allocationalgorithm in any cluster 〈z, f〉 is formulated as:

max

∑u∈Uz,f

log2 [Ru]

s.t. (22)

Ru ≤ C fu(mf)

+∑f 6=f

T fu,avg ∀u ∈ Uz,f ,∑

u∈Uz,f

mfu ≤M f

For an arbitrary initial allocation of resources, after a few iter-ations, the rates converge very quickly to the joint schedulingsolution as depicted in Fig.5. This decentralized optimizationproblem can be implemented in practice using the channelquality index feedback for DAS (see [1]) or using the idea ofvirtual queues for DMIMO (see [7] and references therein).

VI. SIMULATION RESULTS

The current section quantifies and compares the benefitsof interlaced clustering (IC) to Fractional Frequency Reuse(FFR) and no-interlacing (NIC) in some practical multi-cellsimulation layouts.9 The examples involve both the simple 1-D (linear) and a 2-D (planar) cellular layouts. The simulationparameters are based on the WiMAX system specification [19]and are also very similar to the specifications for LTE [43].Some of them are tabulated in table I for convenience. Weconsider a large simulation layout containing several clustersand consider a torus wrap-around of the layout to get ridof the edge effects. On any cluster pattern, for given clustercentres, the RAUs are assigned to the cluster whose clustercenter is closest to them. The users/user-groups, on the otherhand, are always associated to their closest RAU and thereforeto the cluster to which their closest RAU belongs. In thiswork we consider prefixed, and uniformly spaced locations

9Note that NIC and FFR are just a special cases of IC with one and twoCPs, respectively. Therefore the derived rate expressions easily extend forthese settings

9

for the cluster centres for each cluster pattern. The problem ofchoosing optimal cluster center locations is deferred for futureanalysis. For the depicted results, we also consider a uniformlyspaced pattern of users/user-groups.10

While the analysis presented above is applicable to any formof interlaced clustering, for sake of brevity, the simulations arerestricted to the important case of overlapped clustering [22]with equal partitioning of bandwidth among the cluster pat-terns. For FFR, the optimal bandwidth partitioning among thetwo CPs (refer section II-A) that maximizes the proportionalfairness metric is found numerically. Also for FFR, we assumethat CP1 and CP2 have frequency reuse factors of 1 and 2,respectively. The cluster patterns of NIC and FFR are assumedto have the same shape as the CP1 for IC. The remainingsimulation settings for NIC and FFR are similar to those ofIC.

TABLE ISIMULATION PARAMETERS FOR DAS AND DMIMO

DAS DMIMOCellular Layout 1-D 2-D 1-D 2-DLayout dimensions (km) 3 3×3 3 3×3Inter-Cluster spacing 500 mNo. of CPs (F ) 3 2 3 2Mean Inter-RAU spacing (m) 333 250 333 250Total BW 10 MHzNormalized Cluster Power (P f

tot/σ2n) 153 dB

Pathloss COST231∗

Users or User-groups 900 900 100 576γ 11 - - 11.1 64

A. Location specific comparison

For ease of comparison we first consider a linear layout asdepicted in Fig.4a. The corresponding user rates in a referencecluster 〈2, 1〉 as a function of user location are presented inFig.5a, 5b. Apart from NIC, IC and FFR, the rates from thedecentralized optimization problem in (22) are also depictedunder the name IC-DC. Results show that the decentralizedscheme achieves the same performance as the joint resourceallocation problem. We see that as expected, the edge userperformance improves significantly by using an IC pattern incomparison to NIC, FFR with negligible loss in throughput inthe cell interior. Notice that in case of DAS, performance ofFFR is superior to NIC (Fig.5a) however the reverse is truein case of DMIMO (Fig.5b). This is because FFR restrictsthe performance of multi-user precoding by not allowing edgeusers and interior users to be scheduled on same T-F slot. Forpractically relevant results, a planar deployment of RAUs isalso considered where we restrict comparison to IC and NIC.In any practical scenario, the RAUs are likely to be irregularlydeployed. To simulate such settings we first consider a uniformsquare grid of RAUs. A Gaussian process is then used to jitterthe RAU locations as:

(x, y)z = (x, y)z + (nz1, nz2) (23)

10Similar performance trends were observed with other user distributionse.g. generated from a Cox process [44]

11These values correspond to the case of having same number of users asin DAS and Nt = 100.

(a) Linear

0 500 1000 1500 2000 2500 30000

500

1000

1500

2000

2500

3000

RAU

Cluster−center

(b) Planar, CP10 500 1000 1500 2000 2500 3000

0

500

1000

1500

2000

2500

3000

RAU

Cluster−center

(c) Planar, CP2

Fig. 4. Cluster patterns and simulation layouts for the linear and planarscenarios

1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 20000

0.5

1

1.5

2

2.5

3

3.5

4x 10

5

User location

User r

ate

s in

nats

/s

NIC

IC

FFR

IC−DC

(a) DAS

1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 20001

2

3

4

5

6

7x 10

7

User location

Us

er r

ate

s i

n n

ats

/s

NIC

IC

FFR

IC−DC

(b) DMIMO

Fig. 5. Comparison of user rates [Linear]: For the layout in Fig. 4a, (a) and(b) depict the rates for users in cluster 〈2, 1〉

10

(a) DAS: Rates with NIC

(b) DAS: Rates with IC

(c) DAS: Rate Improvement Factor RICu−RNIC

uRNIC

u

(d) DMIMO: Rates with NIC

(e) DMIMO: Rates with IC

(f) DMIMO: Rate Improvement Factor: RICu−RNIC

uRNIC

u

Fig. 6. Comparison of user rates for the planar setting in Fig. 4: (a)-(c) correspond to DAS and (d)-(f) correspond to DMIMO, respectively

11

where, nz1, nz2 ∼ N (0, σRAU). One such sample layout isdepicted in Fig.4b, 4c and the corresponding user rates aregiven in Fig.6. We observe from the results that, on anaverage, IC boosts the edge-user rates by a factor of 3 for DASand 2 for DMIMO, respectively, when compared to NIC. Wesee that the gains are partially lower in DMIMO as opposed toDAS. This is because, with multi-user precoding, in DMIMOeach user is always scheduled on the whole frequency band,albeit, with different allocated powers in each CP. With IC,the interference is low on some CPs but high on others,which partially reduces the gains in DMIMO. It is worthmentioning that, due to the cost acquiring CSIT, if each useris only allowed to be scheduled on a fraction of the frequencyband, the gains with IC can be much higher in DMIMO. Insuch cases, interlaced clustering can also alleviate the pilotcontamination problem with uplink training. Such an analysisis beyond the scope of this paper and is deferred to futurework.

B. Number of CPs and resilience to RAU locations

An important metric which quantifies cell-edge user perfor-mance is the non-outage fraction of cell users [45]. The non-outage fraction for a given desired rate Rdes is the fractionof cellular users whose ergodic throughputs are greater thanRdes.12 The user non-outage fractions for IC and NIC arecompared for increasing number of cluster patterns in Fig.7a.These results are averaged over many jittered RAU deploy-ments with σRAU = 125 (refer (23)). Considering the 10%outage rate as an estimate of the per-user throughput that thesystem can support, these results suggest that, IC improves themaximum supportable per-user throughput by a factor of 1.4for DAS and 1.2 for the case of DMIMO, respectively. We alsosee that there are diminishing gains with increasing number ofcluster patterns. Though this would be the general trend, therelative gains may be different for larger cluster sizes and/orwith intelligent placement of cluster centres. We speculate that4 CPs are sufficient for a generic deployment scenario. Thisintuition is explained as follows: A deployment layout can bedivided into arbitrary disjoint regions with the condition thateach region must belong to the interior of atleast one cluster.Interpreting these disjoint regions as countries on a map andeach CP as a color, by the 4 color map theorem, the numberof CPs needed to create such cluster regions is at maximum4.

As user rates are sensitive to RAU locations, resilienceto irregular RAU locations is a very useful trait for anypractical scheme. The non-outage user fractions, averaged overmany RAU deployments, for a given desired rate Rdes aredepicted with increasing skewness of deployment (σRAU) forDAS and DMIMO in Fig.7b. We observe that even though weuse the naive approach of uniformly spaced cluster centres,

12Note that contrary to the popular notion of outage where the taildistribution of user rates are characterized over both the small scale fading andthe user population, here outage refers to the tail distribution of the ergodicrates (averaged over small scale fading) taken over the user population only.This notion is more meaningful here because in modern transmission systemsfrequency diversity, interleaving, link rate-adaption etc. ensure averaging oversmall scale fading.

105

106

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

No

n−

ou

tag

e f

rac

tio

n

Rdes

DAS

DAS: 1 CP

DAS: 2 CP

DAS: 4 CP

DAS: 5 CP

107

108

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

No

n−

ou

tag

e f

rac

tio

n

Rdes

DMIMO

DMIMO: 1CP

DMIMO: 2CP

DMIMO: 4CP

DMIMO: 5CP

(a) Number of CPs

0 50 100 150 200 250 300 3500.7

0.75

0.8

0.85

0.9

0.95

1

σRAU

No

n−

ou

tag

e u

se

r fr

ac

tio

n

DAS, NIC

DAS, IC

DMIMO, NIC

DMIMO, IC

(b) RAU location resilience

Fig. 7. User non-outage fractions for (a) increasing number of CPs (σRAU =125, X-axis for RDMIMO

des is at top and is reversed) and (b) increasing RAUdeployment irregularity (RDAS

des = 100 Knats/s, RDMIMOdes = 20 Mnats/s and

IC ⇒ 2 CP)

IC significantly outperforms NIC even for highly skeweddeployments. This is because in interlaced clustering eachgeographical region is covered by multiple cluster patterns,thereby in a sense, providing ”diversity” against irregulardeployment.

VII. CONCLUSIONS

In this paper, interlaced clustering is explored as a networkarchitecture to solve the cell edge user problem for distributedtransmission systems. It is shown that IC is a generic networkarchitecture and many systems e.g. overlapped clusteringor hierarchical centralized networks like femto-cells can beinterpreted as special cases of it. The proportional fairnessoperating point of the achievable rate region for a DAS anda DMIMO system with IC have been characterized. Based onthis mathematical analysis, the location specific user rates inpractical cellular settings have been studied. The results sug-gest that IC mitigates co-channel interference at the cell edge

12

significantly without degrading the rates in the cell interior,thereby, improving the over-all cell throughput performance.Quantitatively, IC can outperform FFR and boost edge userrates by a factor of 2 when compared to NIC and 1.5 whencompared to FFR. We also conclude that the gains with IC arepartially higher in a DAS system as compared to a DMIMOsystem. The non-outage user fraction curves suggest that ICcan increase the maximum supportable per-user data rate by afactor of 1.4 (DAS) and 1.2 (DMIMO), respectively. There isa diminishing improvement in performance with the numberof CPs and 4 CPs should be sufficient. The less steep roll-offof the non-outage fraction curves suggests that with IC, thereis more parity in the distribution of user rates. As any locationis covered by multiple cluster patterns, we also conclude thatinterlaced clustering helps improve resilience to imperfectionsin the cellular deployment layout. Simulations results validatethis claim. The above results however are obtained with a naiveuniform layout of cluster centers. Higher gains can possiblybe achieved with the optimal design of cluster patterns fora given RAU deployment. Such an analysis is left for futurework.

APPENDIX AConsider a general abstraction of the problem in (8) as:

L(w) =∑u

(− logwu) +∑z,f

maxv∈Uz,f

{wvCfv } (24)

We define A , {〈z, f〉|z ∈ Z, f ∈ F}. We can formulate afew assertions required for optimality:

• Let us define Au ,{〈z, f〉|u ∈ argmaxv∈Uz,f {wvC

fv }}

.For optimality, ∀u, Au 6= φ.

• If the value of the Lagrangian dual variable wu isincreased, so should the value of wv all users v s.t.Av ⊆ Au.

• Similarly if the value of wu is decreased, so should thevalue wv for all users v ∈ Uz,f for atleast one cluster〈z, f〉 ∈ Au.

Using these arguments, we formulate the minimization algo-rithm as in Algorithm 2.

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[2] CISCO, “Visual Networking Index Global Mobile Data Forecast,” 2013-2018.

[3] W. Choi and J. Andrews, “Downlink performance and capacity ofdistributed antenna systems in a multicell environment,” Wireless Com-munications, IEEE Transactions on, vol. 6, no. 1, pp. 69–73, Jan 2007.

[4] J. Park, J. Kim, and W. Sung, “Performance of distributed MISO systemsusing cooperative transmission with antenna selection,” Communicationsand Networks, Journal of, vol. 10, no. 2, pp. 163–174, June 2008.

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Algorithm 2: Find minw{L(w)}{Cfu} ∀u, f - inputInitialize w(u) = |U|

|A|maxf{Cfu)}

Initialize ∆ = 0.1;while solution not saturated do

for every subset A∗ ⊆ A doAu =

{〈z, f〉|u ∈ argmaxv∈Uz,f {wvC

fv }}

;Cz,f = maxv∈Uz,f {wvCfv };U incA∗ = {u ∈ U|Au ⊆ A∗};UdecA∗ = {u ∈ U|A∗ ∩ Au 6= φ};

if |U incA∗ | >

∑〈z,f〉∈A∗ Cz,f then

while no change in Au,∀u ∈ U incA∗ do

w(u) = w(u)[1 + ∆] ∀u ∈ U incA∗ ;

Au ={〈z, f〉|u ∈ argmaxv∈Uz,f {wvC

fv }}

;end while

else if |UdecA∗ | <

∑〈z,f〉∈A∗ Cz,f then

while no change in Au,∀u ∈ UdecA∗ do

w(u) = w(u)[1−∆] ∀u ∈ UdecA∗ ;

Au ={〈z, f〉|u ∈ argmaxv∈Uz,f {wvC

fv }}

;end while

end ifend for

end while

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[18] Huawei, “Soft Frequency Reuse Scheme for UTRAN LTE,” Huawei,R1-050507, Athens, Greece, Tech. Rep. M Series, May 2005.

[19] 3GPP, “3GPP TR 25.814, Physical Layer Aspects for Evolved UTRA(Release 7),” 3rd Generation Partnership Project (3GPP), Palm Springs,USA, Tech. Rep. v7.1.0, Sept 2006.

[20] T.-P. Chu and S. Rappaport, “Overlapping coverage and channel rear-rangement in microcellular communication systems,” Communications,IEE Proceedings-, vol. 142, no. 5, pp. 323–332, Oct 1995.

[21] S. A. Ramprashad, G. Caire, and H. Papadopoulos, “A Joint Scheduling

13

and Cell Clustering Scheme for MU-MIMO Downlink with LimitedCoordination,” in Communications (ICC), 2010 IEEE InternationalConference on, May 2010, pp. 1–6.

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Vishnu V. Ratnam (S’10) received his Bachelorof Technology (Hons.) in electronics and electricalcommunication engineering from Indian Instituteof Technology, Kharagpur in 2012. He graduatedas the Salutatorian for the class of 2012. He iscurrently pursuing a Ph.D in Electrical Engineeringat University of Southern California. His researchinterests are in: the design & analysis of wirelesscellular networks; network resource allocation incellular systems; and design & analysis of multiantenna systems - specifically, hybrid preprocessing

for massive MIMO.

Andreas F. Molisch (S’89–M’95–SM’00–F’05) re-ceived the Dipl. Ing., Ph.D., and habilitation degreesfrom the Technical University of Vienna, Vienna,Austria, in 1990, 1994, and 1999, respectively. Hesubsequently was with AT&T (Bell) LaboratoriesResearch (USA); Lund University, Lund, Sweden,and Mitsubishi Electric Research Labs (USA). Heis now a Professor of Electrical Engineering andDirector of the Communication Sciences Institute atthe University of Southern California, Los Angeles.His current research interests are the measurement

and modeling of mobile radio channels, ultra-wideband communicationsand localization, cooperative communications, multiple-inputmultiple-outputsystems, wireless systems for healthcare, and novel cellular architectures. Hehas authored, coauthored, or edited four books (among them the textbookWireless Communications, Wiley-IEEE Press), 16 book chapters, some 180journal papers, 260 conference papers, as well as more than 80 patents and70 standards contributions.

Dr. Molisch has been an Editor of a number of journals and specialissues, General Chair, Technical Program Committee Chair, or SymposiumChair of multiple international conferences, as well as Chairman of variousinternational standardization groups. He is a Fellow of the IEEE, Fellow of theAAAS, Fellow of the IET, an IEEE Distinguished Lecturer, and a member ofthe Austrian Academy of Sciences. He has received numerous awards, amongthem the Donald Fink Prize of the IEEE, and the Eric Sumner Award of theIEEE.

14

Giuseppe Caire (S’92–M’94–SM’03–F’05) wasborn in Torino, Italy, in 1965. He received theB.Sc. in Electrical Engineering from Politecnicodi Torino (Italy), in 1990, the M.Sc. in ElectricalEngineering from Princeton University in 1992 andthe Ph.D. from Politecnico di Torino in 1994. Hehas been a post-doctoral research fellow with theEuropean Space Agency (ESTEC, Noordwijk, TheNetherlands) in 1994-1995, Assistant Professor inTelecommunications at the Politecnico di Torino,Associate Professor at the University of Parma, Italy,

Professor with the Department of Mobile Communications at the EurecomInstitute, Sophia-Antipolis, France, and he is currently a professor of ElectricalEngineering with the Viterbi School of Engineering, University of SouthernCalifornia, Los Angeles and an Alexander von Humboldt Professor with theElectrical Engineering and Computer Science Department of the TechnicalUniversity of Berlin, Germany.

He served as Associate Editor for the IEEE Transactions on Communi-cations in 1998-2001 and as Associate Editor for the IEEE Transactions onInformation Theory in 2001-2003. He received the Jack Neubauer Best SystemPaper Award from the IEEE Vehicular Technology Society in 2003, the IEEECommunications Society & Information Theory Society Joint Paper Awardin 2004 and in 2011, the Okawa Research Award in 2006, the Alexander vonHumboldt Professorship in 2014, and the Vodafone Innovation Prize in 2015.Giuseppe Caire is a Fellow of IEEE since 2005. He has served in the Board ofGovernors of the IEEE Information Theory Society from 2004 to 2007, and asofficer from 2008 to 2013. He was President of the IEEE Information TheorySociety in 2011. His main research interests are in the field of communicationstheory, information theory, channel and source coding with particular focuson wireless communications.