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JOURNAL OF COMMUNICATIONS AND NETWORKS, VOL. 21, NO. 2, APRIL 2019 85

Large-Scale Cloud Radio Access Networks:Fundamental Asymptotic Analysis

Kyung Jun Choi and Kwang Soon Kim

Abstract: Large-scale cloud radio access network (LS-CRAN) isa highly promising next-generation cellular network architecturewhereby many base stations (BSs) equipped with a massive an-tenna array are connected to a cloud-computing based centralprocessor unit via digital front/backhaul links. This paper stud-ies the asymptotic behavior of downlink (DL) performance ofa LS-CRAN. As an asymptotic performance measure, the scal-ing exponent of the signal-to-interference-plus-noise-ratio (SINR)is derived for interference-free (IF), maximum-ratio transmission(MRT), and zero-forcing (ZF) operations. Our asymptotic analysisreveals four fundamental operating regimes and the performancesof both MRT and ZF operations are fundamentally limited by theUL transmit power for estimating user’s channel state information,not the DL transmit power. We obtain the conditions that MRTor ZF operation becomes interference-free, i.e., order-optimal. Ashigher UL transmit power is provided, more users can be associ-ated and the data rate per user can be increased simultaneouslywhile keeping the order-optimality.

Index Terms: Asymptotic performance, cloud-RAN, massive-MIMO, ultra-dense network.

I. INTRODUCTION

RECENTLY, mobile data traffic is explosively and contin-uously rising due to smart phone and tablet users and it

is expected that next-generation (i.e., the 5th generation) cel-lular networks will offer a 1000x increase in network capacityas well as a 1000x increase in energy-efficiency in the follow-ing decade to meet such an excessively high user demand [1].The most prominent method to increase the network capacity isthe network densification [2] by either implementing more an-tennas at base stations (BSs) called large-scale antenna system(LSAS) [3] or adding more small cells at hot spot areas calledultra-dense network (UDN) [4]. By using a lot of antennas ateach BS, the LSAS can exploit massive spatial dimension togenerate a sharp beam and thus it can provide higher spectralefficiency and also lower power consumption [5]. However, theperformance of the LSAS is highly limited by the accuracy ofchannel state information (CSI) and varies over the geographical

Manuscript received December 3, 2018; approved for publication by EIC,Division I Editor, December 5, 2019.

This work was supported in part by ICT R&D program of MSIT/IITP 2014-0-00552, Next Generation WLAN System with High Efficient Performance, and2018-0-00923, Scalable Spectrum Sensing for Beyond 5G Communication,and in part by Basic Science Research Program through the National ResearchFoundation of Korea (NRF) funded by the Ministry of Education, Science andTechnology (NRF-2014R1A2A2A01007254, NRF-2015K2A3A1000189).

K.J.Choi and K.S.Kim are with the Department of Electrical and ElectronicEngineering, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul 120-749,Korea, email:kjchoi,[email protected].

K.S.Kim is the corresponding author.Digital Object Identifier: 10.1109/JCN.2019.000008

locations of users. On the other hand, an UDN exploits spatialreuse obtained from deploying more small BSs so that it can of-fer geographically uniform performance to each user due to thereduction of the access distance between users and BSs [6], [7].However, its performance is limited by uncoordinated out-of-cell interference so that an efficient interference management isa key challenge for an UDN, while managing the costs in termsof the front/backhaul overhead and computational complexityfor a network operator.

Large-scale cloud radio access network (LS-CRAN) is re-cently regarded as a novel wireless cellular network architectureto unify the above two architectures and is able to implement theinterference handling mechanisms introduced for the long-termevolution (LTE) or LTE-Advanced, such as the enhanced inter-cell interference coordination [8] and the coordinated multi-point transmission [9], [10]. In the LS-CRAN, lots of BSs eachwith a massive antenna array are connected to a virtualized cen-tral processing unit (CPU) via dedicated front/backhaul link andsome of the baseband processing functionality of each BS is mi-grated to the CPU. As a result, it is expected that the perfor-mance of the LS-CRAN is much higher than that of a conven-tional network.

Although the performance of an LS-CRAN has been widelyinvestigated in literature [9], [11] via simulations, an intuitiveanalytic result considering practical limitations is not availableyet. The exact distribution of the signal-to-interference-plus-noise ratio (SINR) of the MRT or ZF operation is derived in[12] for the case of two distributed BSs and its Laplace approx-imated distribution is derived in [13] for a general LS-CRAN.In [14], a simpler form of the SINR distribution is derived byapproximating the SINR as a Gamma random variable. As par-allel works, the average achievable rate of the MRT or ZF oper-ation is derived in [15] for a single BS case, and in [16] for thecase of fully-distributed or fully-colocated antennas as a func-tion of the BS and user locations. In order to understand thenetwork behavior more intuitively, stochastic geometry has beenrecently adopted to remove such dependency and describe im-portant network metrics in a probabilistic way. In [17]–[19], theSINR distributions of cooperative transmission schemes are pro-vided by adopting stochastic geometry, but the analysis resortson multiple numerical integrals. In [20], an asymptotic analy-sis on the tail of the SINR distribution is provided by applyinglarge-deviation theory and stochastic geometry. In [21], the av-erage achievable rate of the MRT and ZF successive interferencecancellation is investigated and its scaling laws are obtained ina dense random network with multiple receive antennas.

The main objective of this paper is to characterize the asymp-totic behavior of the LS-CRAN. The scaling exponents ofthe signal-to-interference-plus-noise ratio (SINR) defined and

1229-2370/19/$10.00 c© 2019 KICS

86 JOURNAL OF COMMUNICATIONS AND NETWORKS, VOL. 21, NO. 2, APRIL 2019

BS 1

BS 2

BS 3

user 1

user 2

user 5

user 6

user 4

CPU

1x

user 3

11

22

33

2x

3x

11

11

1x

2x

BS 3

3

(a)

CPU

BS l

Users

UL channel estimation

UL pilot transmission

Over air

Over front/

backhaul

Fronthaul Backhaul exchange

Non-cooperative

Fronthaul exchange Backhaul

Cooperative

DL reception

UT phase Cooperative DT phase

Precoder computation

Frame length = coherence interval

Non-cooperative DT phase

ll ll

exchange

exchange

(b)

Fig. 1. System model and DL frame structure for LS-CRAN: (a) System model and (b) DL frame structure.

drived as a function of the numbers of BS, BS antennas, andsingle-antenna users, and the UL/DL transmit power for theinterference-free (IF), MRT, and ZF operations, which are statedin Theorems 1–3. Based on the derived scaling exponents,four fundamental regimes are distinguished according to the ULtransmit power (or the CSI acquisition error): (i) The extremelyhigh (UL transmission) power regime (EH), (ii) the high powerregime (H), (iii) the medium power regime (M), and (iv) the lowpower regime (L). The scaling exponents of the three operationsare derived according to the four regimes with insightful discus-sions.

The reminder of this paper is organized as follows. In Sec-tion II, the LS-CRAN system model is described and the scalingexponent of the SINR is defined as a network performance mea-sure. Section III provides some candidate operations suitablefor practical implementation and the corresponding scaling ex-ponents are analyzed to provide an intuitive understanding onthe LS-CRAN in Section IV. Finally, conclusion is given in Sec-tion V.

Matrices and vectors are respectively denoted by boldface up-percase and lowercase characters. The superscript (·)∗, (·)T and(·)H denote the conjugate, transpose and conjugate transpose,respectively. |·|, E[·], Tr(·), and δ(n) stand for the cardinality ofa set, statistical expectation, trace of a square matrix, and Kro-necker delta function, respectively. Also, 1A×1 and 0A×1 aretheA×1 all one vector and all zero vector, respectively. Further,diag(a1, a2, · · ·, an) denotes the diagonal matrix whose (k, k)thelement is ak, and CN (µ, σ2) denotes the circularly symmetriccomplex Gaussian distribution with mean µ and variance σ2.For a C × 1 vector xab, [xab]

A,Ba=1,b=1 denotes the AC × B ma-

trix[[xH11, · · ·,xHA1]

H, · · ·, [xH1B , · · ·,xHAB ]H

].

In this paper, we only deal with a scaling exponent, s, of a ran-dom measure, f(n), as n → ∞ in a probabilistic sense, whichis mathematically defined as follows. The scaling exponent off(n) is s in probability if

limn→∞

Pr

(∣∣∣∣ log f(n)

log n− s∣∣∣∣ < ε

)= 1, (1)

holds for any positive finite ε. Also s = ±∞ is used iflimn→∞ Pr

(∣∣∣ log f(n)logn

∣∣∣ > t)

= 1 for any finite t. Let s and t

be the scaling exponents of f(n) and g(n), respectively. Then,the following modified order notations are used through this pa-per.1) f(n) = o(g(n)) or f(n) g(n), if s < t,2) f(n) = O(g(n)), if s ≤ t,3) f(n) = ω(g(n)) or f(n) g(n), if s > t,4) f(n) = Ω(g(n)), if s ≥ t, and5) f(n) = Θ(g(n)) or f(n) g(n), if s = t.

II. SYSTEM MODEL

Consider an LS-CRAN system as illustrated in Fig. 1 (a).Suppose that L BSs with M antennas and K users with a sin-gle antenna are uniformly distributed on a finite region R. Thesets of BSs and users are denoted as X = X1, X2, · · ·, XLand U = U1, U2, · · ·, UK, respectively, and they are assumedto be independent homogeneous Poisson point processes (PPPs)over R.1 With a slight abuse of notations, Xl and Uk are usedas the locations of BS l and user k, respectively. It is assumedthat each user is associated with neighboring BSs and the set ofthe BSs serving user k is defined as

Xk = Xl ∈ X ||Uk −Xl| ≤ Rth, (2)

whereRth is the association range and the set of users associatedwith BS l is denoted as Ul = Uk ∈ U|Xl ∈ Xk. To guaranteeeach user is served by at least one BS, Xk needs to be not emptyfor all k. Note that Ul and Ul′ are not necessarily disjoint andtypical users are associated with multiple BSs for being servedcooperatively. In this paper, a network is called fully associated,if Xk = X for all k. Otherwise, the network is called partiallyassociated.

The BSs are connected to the CPU via high-speed dedicatedfront/backhaul link for enabling a cooperative transmission op-eration. For inter-signaling between the BSs and the CPU, thesets of front/backhaul information are defined as Fl and Bl,

1Typical BS deployment in practice would be better represented by a pointprocess with a mutual interaction. However, such a mutual interaction typicallybecomes weaker as a deployment becomes denser and more random so that anasymptotic analysis using a PPP asumption is not only mathematically tractablebut also enough for capturing good insights on the asymptotic behaviors of largenetworks.

CHOI et al.: LARGE-SCALE CLOUD RADIO ACCESS NETWORKS: FUNDAMENTAL ... 87

where Fl is the fronthaul information set (from CPU to BS l)and Bl is the backhaul information set (from BS l to CPU). Theelements of Fl and Bl are closely related according to a specifictransmission operation, which will be described in Section III.

The DL frame structure operating in time-division duplex(TDD) mode is shown in Fig. 1(b) and has three phases as fol-lows. UL training (UT) phase, non-cooperative data transmis-sion (DT) phase, and cooperative DT-phase. In the UT-phase,the instantaneous CSI of each user is estimated by receivinguser’s pilot (or reference) signal at each associated BS inde-pendently. After the UT-phase, the non-cooperative DT-phaseis performed first and followed by the cooperative DT-phase. Inthe non-cooperative DT-phase, each BS separately transmits thedata symbols to the associated users without any front/backhaulexchange. In the cooperative DT-phase, the BSs jointly transmitthe data symbols to the associated users with the aid of the ex-changed information via the front/backhaul link. For simplicity,the cooperative DT-phase is only focused in this paper.

Let gHlk denote the 1×M flat-fading DL channel vector fromBS l to user k, which can be written as2

gHlk =√βlkh

Hlk, (3)

where hlk ∈ CM×1 is the short-term CSI whose elements areindependent and identically distributed (i.i.d.) CN (0, 1) andβlk(≥ 0) is the long-term CSI depending on the path-loss andshadowing. The long-term CSI between BS l and user k is mod-eled as βlk = |Xl − Uk|−α, where α(> 2) is the wireless chan-nel path-loss exponent.3

It is assumed that the short-term CSI of each user remainsconstant within a given frame but independent across differentframes, while the long-term CSI does not vary during a muchlonger interval. Further, it is assumed that the long-term CSIsamong all BSs and users are perfectly known at the CPU throughan infrequent feedback with a negligible overhead. Additionally,we assume perfect TDD reciprocity calibration so that the ULchannel vector is just a transpose of the DL channel vector (i.e.,the UL channel vector from user k to BS l is denoted as g∗lk).

Let s = [s1, · · ·, sK ]T be the K × 1 information symbol vec-tor with E[ssH ] = IK , where sk denotes the information sym-bol for user k. Let x = [xT1 , · · ·,xTL]T be the LM × 1 globaltransmitted signal vector, where xl is the local transmitted sig-nal vector of BS l, given by

x = FΥ(QΥ

) 12 s, (4)

where QΥ , diag(QΥ1 , · · ·, QΥ

K) denotes the K ×K power al-location matrix and QΥ

k (≥ 0) denotes the power allocated touser k, FΥ = [fΥ

lk ]L,Kl=1,k=1 denotes the LM × K precodingmatrix and fΥ

lk denotes the M × 1 precoding vector of user kfor BS l, and Υ denotes the cooperative transmission operationused in the LS-CRAN. Since the BS serves the associated usersonly, fΥ

lk = 0M×1 for ∀Xl /∈ Xk (or equivalently ∀Uk /∈ Ul).Note that the transmitted signal vector of BS l can be written

2A narrow-band flat-fading channel is assumed because wideband frequency-selective channels may be decomposed into multiple narrow-band channels us-ing modulation schemes such as the orthogonal frequency division multiplexing.

3By using the random displacement theorem similarly as in [22], this modelcan include the shadowing effect.

by xl =∑Uj∈Ul f

Υlj

√QΥj sj . Then, the DL transmit power for

user j, P dlj , is given as

P dlj = QΥ

j

∑Xl∈Xj

∥∥fΥlj

∥∥2, (5)

and the total DL transmit power is given by P dlΣ ,

∑Uj∈U P

dlj .

Let y , [y1, · · ·, yK ]T be the K × 1 aggregated received sig-nal vector, given by

y = Gx + n

= GFΥ(QΥ

) 12 s + n,

(6)

where G =(

[glk]L,Kl=1,k=1

)Hdenotes the K × LM channel

matrix among all users and all BSs and n , [n1, · · ·, nK ]T ∼CN (0, IK) denotes the K × 1 noise vector. Then, the receivedsignal at user k can be expressed as

yk = ψΥkk

√QΥk sk +

∑Uj∈U\Uk

ψΥkj

√QΥj sj + nk, (7)

where ψΥkj =

∑Xl∈Xj g

Hlkf

Υlj is the effective channel seen at

user k.

III. LS-CRAN OPERATIONS AND PERFORMANCEMEASURE

In this section, the ideal IF operation is introduced as a refer-ence system and practical cooperative operations are reviewed.Note that a comprehensive review on the operations is beyondthe scope of this paper so that two well-known practical opera-tions, MRT operation [23] and ZF operation [24], are focused.

A. Cooperative Transmission Operations

A.1 Ideal IF Operation

As a reference, the ideal IF operation is considered, where theinterference term in (7) is removed by Genie perfectly withoutany cost while the desired signal power is maximized by usingthe IF precoding matrix,

Fif = GH . (8)

Obviously, the ideal IF operation provides an upper-bound onthe performance of any practical operation. Since this operationcannot be realizable, Bifl and F if

l are not defined.

A.2 MRT Operation

MRT operation tries to maximize the received signal powerof a desired user without considering the effect of interferenceto undesired users [23]. This operation is regarded as a goodcandidate as the number of BS antennas increases due to its low-computational complexity and low-overhead requirement [25].The precoder for MRT operation is given by

Fmrt = GH . (9)


Since the precoder of MRT operation does not need infor-mation exchange among BSs, the backhaul and fronthaul in-formation sets can be expressed as Bmrt

l = ∅, and Fmrtl =√

Qmrtj sj |Uj ∈ Ul

, respectively.

A.3 ZF Operation

ZF operation can cancel the interference term in (7) (perfectly,provided that the perfect channel estimation is available at theCPU) at the expense of the desired signal power loss [24]. Theprecoder for ZF operation is given by

Fzf = GH(GGH

)−1

. (10)

Note that ZF operation can be used only when the number of an-tennas in the system is larger than or equal to that of users, i.e.,LM ≥ K and the estimated channel matrix, G, has full-rank.To construct the ZF precoder, the CPU requires to know the esti-mated channel matrix G so that the short-term CSIs of all asso-ciated users at each BS need to be conveyed to the CPU via thededicated front/backhaul link. Thus, the backhaul and fronthaulinformation sets can be expressed as Bzfl =

hlj |Uj ∈ Ul

and

F zfl =

f zflj

√Qzfj sj |Uj ∈ Ul

, respectively.

Note that it is well-known that the performance of ZF oper-ation is better than that of MRT operation if the network hassufficient transmit power. But, if not, ZF operation can be in-ferior to MRT operation. In this paper, we will show that ZFoperation is always superior or identical to MRT operation inthe viewpoint of the scaling exponent of the SINR regardless ofthe operating downlink or uplink transmit power.

B. Pilot Allocation and Channel Estimation

The UL channel is estimated during the dedicated UT-phaseand then the DL channel is obtained by the TDD channelreciprocity. It is assumed that the length of the UT-phaseis T and there are T orthonormal pilot signals, denoted asψi ∈ CT×1, i = 1, 2, · · ·, T , where ψHi ψj = δ(i − j). Then,

user j transmits√P ulj ψ

Tπj during the UT-phase of length T ,

where P ulj is the UL transmit power of user j and πj is the in-

dex of the pilot signal allocated to user j. The total UL transmitpower is denoted as P ul

Σ =∑Uj∈U P

ulj . Then, the M × T re-

ceived signal matrix at BS l during the UT-phase can be writtenas

Yl =∑Uj∈U

√P ulj βljh

∗ljψ

Tπj + Vl, (11)

where Vl denotes the M × T noise matrix whose elementsare i.i.d. CN (0, 1). Using the minimum mean-square error(MMSE) channel estimator [26], the estimated short-term CSIof user k ∈ Ul at BS l can be written as

hlk = ϑlkY∗l ψπk

= φlkkhlk +∑

Uj∈U\Uk

φljkhljδ(πk − πj) + ϑlkvlk,

(12)

where

ϑlk =

√P ulk βlk∑

Ui∈UP uli βliδ(πk − πi) + 1

,

φljk =

√P ulj P

ulk βljβlk∑

Ui∈UP uli βliδ(πk − πi) + 1

,

vlk = V∗lψπk with [vlk]m ∼ CN (0, 1). In (12), the first term isthe desired user’s channel, the second term is the leakage fromthe other users’ channels, called the pilot contamination (PC) ef-fect due to the pilot signal reuse, and the third term is the noisepart. Invoking the orthogonality principle of the MMSE estima-tor [26], hlk can be decomposed as hlk = hlk + hlk, wherehlk ∼ CN (0, φlkkIM ) and hlk ∼ CN (0, (1 − φlkk)IM ) aremutually independent. Note that the estimated version of glk atBS l is given as glk =

√βlkhlk for ∀k ∈ Ul or glk = 0M×1

for ∀k /∈ Ul and thus the estimated version of G is given as

G =(

[glk]L,Kl=1,k=1

)H.

C. Performance Measure

When operation Υ is used, the signal-to-noise ratio (SNR),SNRΥ

k , the signal-to-interference ratio (SIR), SIRΥk , and the

signal-to-interference-plus-noise ratio (SINR), SINRΥk , are re-

spectively defined as

SNRΥk = QΥ

k

∣∣ψΥkk

∣∣2,SIRΥ

k =QΥk

∣∣ψΥkk

∣∣2∑Uj∈U\Uk

QΥj

∣∣∣ψΥkj

∣∣∣2 ,

SINRΥk =

QΥk

∣∣ψΥkk

∣∣2∑Uj∈U\Uk

QΥj

∣∣∣ψΥkj

∣∣∣2 + 1.

Obviously, SNRΥk , SIRΥ

k , and SINRΥk are random variables de-

pending on the realization of the short-term fading and the long-term fading (i.e., realization of locations of users and BSs).

One of the main objectives of this paper is to characterize theasymptotic behavior of SINRΥ when the key network parame-ters such as the number of BSs L, the number of users K, andthe number of BS antennas M are scaled up. To do this, wedefine an auxiliary parameter N = LM as the network size (orequivalently, the total number of antennas in the network) andmake the following relations:

L = Θ(Nηbs), M = Θ(Nηant), K = Θ(Nηuser), (13)

where ηbs, ηuser, and ηant denote the scaling exponents of thenumbers of BSs, users, and BS antennas, respectively.4 Notethat we only consider the case where 0 ≤ ηbs, ηant, ηuser ≤ 1and ηbs + ηant = 1 by definition. Then, the asymptotic perfor-mance of the network can be characterized as follows.

4Note that the network behavior when the network size (N ) is sufficientlylarge is of interest in this paper.


ulr

IF

MRT

ZF

0bs

2

ah-

bs ant

2

ah h- -

,EHHML

¡

¡

snr

sir

(a)

ulrbs

2

ah- 0

bs ant

2

ah h- -

user

2

ah-

IF

MRT

ZF

EHHML,¡

¡

snr

sir

(b)

Fig. 2. SNR (dashed line) or SIR (solid line) scaling exponents for IF (o-marker), MRT (+-marker), or ZF (x-marker) operations according to ρul: (a) ηuser > ηbsand (b) ηuser ≤ ηbs.

Definition 1 (Performance Measure) The scaling exponentof the SINR of operation Υ, sinrΥ, is the order of growth ofthe SINR of a randomly selected user as N increases such that,for any ε > 0,

limN→∞

Pr

(∣∣∣∣∣ log SINRΥk

logN− sinrΥ

∣∣∣∣∣ < ε

)= 1. (14)

Manipulating (14), we can also obtain

limN→∞

Pr(N sinrΥ−ε < SINRΥ

k < N sinrΥ+ε)

= 1,

which implies that the SINR of a randomly selected user isbounded by [N sinrΥ−ε, N sinrΥ+ε] as N increases. Invoking ourorder notations, we can simply write SINRΥ

k = Θ(N sinrΥ)and the sum-rate served by a network can also be written asCΣ = Θ(Nηuser log2(1 + N sinrΥ)). Similarly, we define snrΥ

and sirΥ as the scaling exponents of the SNR and SIR, respec-tively. Note that it is sufficient to find snrΥ and sirΥ and thensinrΥ = minsnrΥ, sirΥ is obtained since SINRΥ

k is equalto the harmonic mean of SNRΥ

k and SIRΥk , i.e, (SINRΥ

k )−1 =(SNRΥ

k )−1 + (SIRΥk )−1.

IV. SCALING EXPONENTS OF THE SINR

The major merit of the LS-CRAN is that it can decrease thetransmit power consumption so that it is suitable for an energy-efficient communication system. So, the total transmit power isa key constraint in a future cellular system. In order to limit thetotal transmit power of uplink or downlink, similarly as in (13),we make an additional asymptotic relationship as

P ulj = Θ(Nρul

) for UL and P dlj = Θ(Nρdl

) for DL, (15)

where ρul and ρdl denote the scaling exponents of the UL andDL transmit powers, respectively. Also, the total transmit poweris PΣ =

∑Kj=1(P ul

j + P dlj ) = Θ(Nηuser+maxρul,ρdl)5.

5Note that the power constraint used in this paper can include both the totalpower constraint and the per-node power constraint so that the derived asymp-totic result is valid even if tighter per-node power constraint is assumed.

In order to quantify the effect of the CSI accuracy accord-ing to the UL transmit power, we consider the following fourregimes.• Case EH (ρul ≥ 0): the UL transmit power is sufficiently

high so that every BS can acquire the accurate CSI of a ran-domly selected users.

• Case H (−α2 ηbs ≤ ρul < 0): each BS can acquire the ac-curate CSI of a randomly selected users within a distance ofΘ(Nρul/α).

• Case M (−α2 ηbs − ηant ≤ ρul < −α2 ηbs): randomly se-lected user’s CSI is erroneous even at the nearest BS but isstill meaningful for providing an array gain.

• Case L (ρul < −α2 ηbs − ηant): randomly selected user’s CSIbecomes quite poor even at the nearest BS so that no arraygain can be provided.Theorem 1: Suppose that IF operation is used with a full as-

sociation and no pilot reuse. Then, the scaling exponents arerespectively given by

snrif = ρdl +α

2ηbs + Ξ, (16)

sirif =∞, (17)

sinrif = snrif , (18)

where Ξ =(ρul + α

2 ηbs + ηant

)+−(ρul + α2 ηbs

)+denotes the

array gain and (x)+ = maxx, 0.

Proof: Please see Appendix B. 2

Theorem 1 is also illustrated in Figs. 2 and 3 according toρul and ηbs, respectively. Intuitively, the SNR of IF operation iscomposed of the three parts as

SNRifk Nρdl︸︷︷︸

DL transmit power

× Nα2 ηbs︸︷︷︸

Densification gain

× NΞ︸︷︷︸Array gain

, (19)

where the first part is the DL transmit power, the second partis the densification gain which comes from the decrease of theaccess distance of Θ(N−

12ηbs) and the last part is the array gain


of a coherent transmission which depends on the CSI accuracyand thus the UL transmit power.

Remark 1 (SNR behavior of IF operation) Fig. 2 reveals howthe UL transmit power affects on the SNR behavior. In EH andH, the full array gain (Ξ = ηant) is achieved so that an addi-tional UL transmit power does not improve the quality of DLservice in the network, i.e., is wasteful. In M, a partial arraygain (0 ≤ Ξ < ηant) depending on the UL transmit power is ob-tained so that the network total power needs to be consumed byconsidering both the DL transmit power and the CSI accuracy.In L, no array gain (Ξ = 0) is obtained due to poor CSI accuracyso that the quality of DL service becomes irrelevant to the ULtransmit power and its performance is identical to the randombeamforming without small-scale CSIs in [27].

Fig. 3 shows the SNR behavior according to the BS scalingexponent. It turns out that additional BSs (even with smallernumber of BS antennas at each BS) are always beneficial butthe slope of snrif (vs. ηbs) varies according to ρul. The slope be-comes α/2 in L, α−1 in M, and α/2−1 in H or EH, which im-plies that additional BSs (while keeping the network size fixed)become the most effective in M because the additional BSs im-prove not only the densification gain but also the array gain andthe least effective in H or EH because only the densification gainis improved.

Theorem 2: Suppose that MRT operation is used with a fullassociation and no pilot reuse. Then, the scaling exponents arerespectively given by

snrmrt = snrif , (20)sirmrt = snrmrt −∆mrt, (21)

sinrmrt = snrmrt −(∆mrt

)+, (22)

where ∆mrt = ρdl + α2 min ηbs, ηuser+ (ηuser − ηbs)

+.

Proof: Please see Appendix C. 2

Theorem 2 is also illustrated in Figs. 2 and 3 according to ρul

and ηbs, respectively. Interestingly, although snrmrt is identicalto snrif , the gap between sirmrt and snrmrt, denoted by ∆mrt,changes according to the sign of ηbs−ηuser. From the definitionof ∆mrt, the interference caused in MRT operation at a randomlyselected user k can be represented as

Ik N∆mrt

= Nρdl︸︷︷︸DL transmit

power

×N α2 minηbs,ηuser︸︷︷︸

Received power ofa dominant interferer

×N (ηuser−ηbs)+︸︷︷︸

# of dominantinterferers

, (23)

where the second and third parts depend on the sign of ηuser −ηbs. When ηuser ≥ ηbs, one BS should simultaneously serveΘ(Nηuser−ηbs) users apart by Θ(N−

12ηbs) so that there are

Θ(Nηuser−ηbs) interferer whose received power is Θ(Nα2 ηbs).

When ηuser < ηbs, the dominant interference comes from theBS apart by Θ(N−

12ηuser) so that the received power of the dom-

inant interference is Θ(Nα2 ηuser).

Remark 2 (Asymptotical optimality of MRT operation) In or-der for MRT operation to behave as IF operation asymptotically

(i.e., ∆mrt ≤ 0), the DL transmit power should be limited as

ρdl ≤

−α2 ηuser, if ηbs ≥ ηuser,

−α2 ηbs − (ηuser − ηbs), if ηbs < ηuser,(24)

which gives us the following insights.

• For MRT operation, the IF optimality condition depends onlyon the numbers of users (ηuser) and BSs (ηbs) and the DLtransmit power (ρdl), but is independent to the number of BSantennas (ηant) and the UL transmit power (ρul).6

• When the number of antennas in a BS is much larger thanthe total number of users in a multi-cell network, MRT op-eration becomes interference-free asymptotically as expectedin literature [3]. However, this is of little interest becausethe network size is too large (or the number of users is toosmall). Instead, Theorem 2 and (24) gives more insightful IFoptimality condition for MRT operation being asymptoticallyinterference-free in a multi-cell network.

• The above aymptotical optimality may become meaningful(i.e., positive SNR scaling exponent) in case of an ultra-densedeployment of BS (ηbs ≥ ηuser or ρul + (α/2)ηbs ≥ 0).

Remark 3 (SIR behavior of MRT operation) As can be seenfrom Fig. 3, additional BSs are beneficial (while fixing N ) inmost cases and the slope of sirmrt (vs. ηbs) can be 0, 1, α/2− 1and α/2. Interestingly, when −(2/α)ρul ≤ ηbs ≤ ηuser, theslope of sirmrt becomes 0, which means that additional BSs arewasteful as long as ηbs is within that interval. On the other hand,the slope becomes α/2 in M and in L if ηbs > ηuser, in whichadditional BSs are the most effective.

Theorem 3: Suppose that ZF operation is used with a fullassociation and no pilot reuse. Then, the scaling exponents arerespectively given by

snrzf = snrif , (25)

sirzf = snrzf −∆zf (26)

sinrzf = snrzf −(∆zf)+, (27)

where ∆zf = ∆mrt−(1− (2/α))((α/2) min ηbs, ηuser+ ρul

)+−2α

(ρul)+

.

Proof: Please see Appendix D. 2

Theorem 3 is also illustrated in Figs. 2 and 3 according to ρul

and ηbs, respectively. Although the SNR exponent is identicalto that in MRT operation, the interference is reduced as the CSIaccuracy improves so that the gap between snrzf and sirzf , ∆zf ,varies as ρul increases. Again, from the definition of ∆zf , theinterference in MRT operation at a randomly selected user k for

6Such an independence on the number of BS antennas or the UL transmitpower comes from the fact that the array gain is already reflected both on snrmrt

and sirmrt.


bsh0 1

L

userh

slope=1

slope=2

a

slope=2

a

,¡

¡

snr

sir

(a)

( )ul21

2r

a- +

-user

hul2r

a-

HML

slope=2

a

slope=1

slope=2

a

slope= 12

a-

slope=0

slope= 12

a-

slope= 12

a-

slope= 1a -

bsh10

,¡

¡

snr

sir

(b)

ul2r

a-

slope= 1a -

slope=2

a

slope= 12

a-

slope= 12

a-

slope= 12

a-

slope=0

HM

userh bs

h10

,¡

¡

snr

sir

(c)

EH

userh

slope=0

slope= 12

a-

slope= 12

a-

slope= 12

a-

bsh10

,¡

¡

snr

sir

(d)

Fig. 3. SNR (dashed line) or SIR (solid line) scaling exponents for IF (o-marker), MRT (+-marker), or ZF (x-marker) operations according to ηbs: (a) ρul < −α2

,(b) −α

2≤ ρul < −1, (c) −1 ≤ ρul < 0, and (d) 0 ≤ ρul.

ηuser ≥ ηbs can be represented as

Ik N∆zf

=

× N

α2 ηbs × Nηuser−ηbs , L, M,

Nρdl × N−ρul × N

2αρ

ul+ηuser , H,

︸︷︷︸DL transmit

power

× N−ρul︸︷︷︸

Received power ofa dominant interferer

× Nηuser︸︷︷︸# of dominant

interferers

, EH.

(28)

The CSI of the users whose access distance is o(Nρul/α) isaccurately estimated at the nearest BS, while the CSI of theusers whose access distance is Ω(Nρul/α) is poor. So, whenρul < −(α/2)ηbs, i.e., L or M, the CSIs of all users are poorlyestimated at all BSs because the access distance is Ω(N−

12ηbs).

Thus, the interference cancellation is not effective so that theinterference in ZF operation is asymptotically the same to thatin MRT operation. However, when ρul ≥ −(α/2)ηbs, i.e., Hor EH, accurate CSI is available so that the cancellation opera-tion becomes effective. In H, the dominant interference comesfrom the BS apart by Θ(Nρul/α) so that the received power of

the dominant interference is Θ(N−ρul

). The dominant inter-ferers are located in the doughnut of radii of Θ(Nρul/α) andΘ(Nρul/α+ε) with an arbitrarily small ε so that the number of in-terferers is Θ(N

2αρ

ul+ηuser). In EH, the number of dominant in-terferers becomes Θ(Nηuser), while the received power of themis still Θ(N−ρ

ul

) which can be explained as follows. Once aBS acquires sufficiently accurate CSI of a user, the interferencecaused from the user becomes proportional to the channel es-timation error, i.e., inversely proportional to the UL transmitpower. Thus, in H and EH, the slopes of the reduction in ZFoperation are (1 − 2/α) and 1 with respect to ρul, respectively,as shown in Fig. 3. In EH, every BS can acquire all user’sCSI with a sufficiently good accuracy so that the slope becomes1. In H, however, BSs far from each user (outside the circlewith radius of Θ(Nρul/α)) cannot obtain its CSI accurately sothat ZF operation does not reduce the interference from thoseBSs. Note that sirzf does not improve as ρul increases within[−(α/2)ηbs,−(α/2)ηuser] in H or EH if ηuser ≥ ηbs because noactual interference reduction happens during that interval due tothe low user density.

Remark 4 (Asymptotical optimality of ZF operation) In or-der for ZF operation to behave as IF operation asymptotically


(i.e., ∆zf ≤ 0), the DL transmit power should be limited as fol-lows.

If ηbs ≥ ηuser,

ρdl ≤

ρul − ηuser, if EH,

−α2 ηuser +(1− 2

α

) (α2 ηuser + ρul

)+, if H,

−α2 ηuser, if M or L,

or if ηbs < ηuser

ρdl ≤

ρul − ηuser, if EH,

−ηuser +(1− 2

α

)ρul, if H,

−α2 ηbs − (ηuser − ηbs), if M or L,

which gives us the following insights.• Unlike MRT operation, the IF condition of ZF operation de-

pends on the UL transmit power (ρul) as well as ηbs, ηuser,and ρdl, but is still independent to the number of BS antennas(ηant).

• ∆zf = ∆mrt in M and L while ∆zf ≤ ∆mrt in EH or H, whichimplies that multi-cell cooperation using ZF operation is use-less without a sufficient CSI accuracy. As the UL transmitpower increases, ZF operation begins to further reduce theinterference caused from other BSs’ users as shown in Fig. 2.Remark 5 (SIR behavior of ZF operation) As can be seen

again from Fig. 3, additional BSs (while fixing N ) are alwaysbeneficial and although the slope of sirzf vs. ηbs is identical tothat of sirmrt vs. ηbs in L or M, it remains α/2 − 1 in H or EH,unlike MRT operation.

V. CONCLUSION

This paper presented a comprehensive and rigorous asymp-totic analysis on the performance of the large-scale cloud radioaccess network (LS-CRAN). As a main performance measure,the scaling exponent of the signal-to-interference-plus-noise ra-tio (SINR) was defined and derived as a function of key net-work parameters, 1) the number of BS, L = Θ(Nηbs), 2)the number of BS antennas M = Θ(Nηant), 3) the numberof single-antenna users K = Θ(Nηuser), 4) the uplink (UL)transmit power, P ul

k = Θ(Nρul

), 5) the DL transmit power,P dl = Θ(Nρdl

), and the distance-dependent pathloss expo-nent α, when interference-free (IF), maximum ratio transmis-sion (MRT), or zero-forcing (ZF) operation is applied.

Then, we show that when MRT or ZF operation becomesinterference-free, i.e., order-optimal with the three practical con-straints. By considering limited transmit power only, MRT op-eration is shown to become interference-free only when the DLtransmit power is less then a threshold, but the threshold is toolow to be meaningful. Also, ZF operation is shown to becomeinterference-free at meaningful DL transmit power so that ashigher UL transmit power is provided, more users can be asso-ciated and the data rate per user can be increased simultaneouslywhile keeping the order-optimality.

APPENDIX APRELIMINARY RESULTS

Here, we provide key preliminary results to prove theoremsin this paper. First, we state the method to derive the scalingexponent of a random measure given by a sum of infinitely manyi.i.d. random variables.

Lemma 1. Let x1, x2, · · · be a sequence of i.i.d. real-valuedrandom variables with common distribution function of F (x)and h(x) be a non-negative integrable function. Suppose thath(n) has a finite scaling exponent. Then, as n→∞, we have

n∑k=1

h(xk) = Θ

(log n

∫ 1

0

nth(F−1

(n−1+t

))dt

)

= Θ

(n

∫ F−1(1)

F−1( 1n )

h(u)f(u)du

).

(29)

where F−1(x) is an inverse (or quantile) function of F (x) andf(x) = d

dxF (x) is the probability density function.

Proof: Suppose that y1, y2, · · ·yn are i.i.d. uniform randomvariables on [0, 1]. Then, for any positive small ε, the followingholds:

Pr

(min

1≤k≤nyk >

1

n1+ε

)=

(1− 1

n1+ε

)n→ 1, as n→∞,

(30)

which implies that all of y1, · · ·, yn are included in the inter-val (n−1−ε, 1], but not included in the interval [0, n−1−ε] inprobability. By dividing the interval (n−1−ε, 1] into dg(n)e in-tervals, S1 = (n−1−ε, nu1 ],S2 = (nu1 , nu2 ], · · ·,Sdg(n)e =

(nudg(n)e−1 , 1], where ui = −1+ idg(n)e for i = 1, 2, · · ·, dg(n)e,

d·e denotes the ceiling operation, and g(n) is a positive functionof n. Let Yi = yj |yj ∈ Si for i = 1, 2, · · ·, dg(n)e. Then, forg(n) = Θ(log(n)),

|Yi| = Θ(n1+ui

)= Θ

(n

idg(n)e

)(31)

and

1

dg(n)e

dg(n)e∑i=1

|Yi| =n

dg(n)e

= Θ

(n

log n

).

(32)

On the other hand, the left side hand of (32) can be rewritten as

1

dg(n)e

dg(n)e∑i=1

|Yi| 1

dg(n)e

dg(n)e∑i=1

ni

dg(n)e , (33)

where the right hand side may be considered as a Riemann sum(although not guaranteed because the integrand is also a func-tion of n) and if it is, it becomes

∫ 1

0ntdt = Θ

(n

logn

), which

coincides with (32). Thus, in this particular case, the scalingexponent of the left hand side of (32) can be obtained by first


changing the summation into an integral form as if it is a Riem-man sum and then taking the scaling exponent from the integralform. Also, it is easily seen that this property still holds whenq(n) i.i.d. random variables are considered instead of n i.i.d.random variables as long as q(n) has a finite scaling exponent.For example, when nr i.i.d. random variables are considered,the right hand side of (32) becomes Θ

(nr−1+ i

dg(n)e

)so that

both the direct calculation as in (32) and the above method yieldthe same result, Θ

(nr

logn

).

Now, assume that h(x) is a monotonically decreasing non-negative integrable function and h(n) has a finite scaling expo-nent. First, we consider the upper-bound:

1

dg(n)e

n∑k=1

h(yk) =1

dg(n)e

dg(n)e∑i=1

∑yk∈Yi

h(yk)

≤ 1

dg(n)e

dg(n)e∑i=1

|Yi|h (nui−1)

= Θ

n 1dg(n)e

dg(n)e

dg(n)e∑i=1

ni−1dg(n)eh

(n−1+ i−1

dg(n)e

)= Θ

(n

1dg(n)e

∫ 1

0

nth(n−1+t

)dt

),

(34)

where the last equality comes from the discussion in the previ-ous paragraph. Similarly, we obtain the lower-bound:

1

dg(n)e

n∑k=1

h(yk) ≥ Θ

(∫ 1

0

nth(n−1+t

)dt

)(35)

so that the two bounds are asymptotically tight (tight in terms ofthe scaling exponent) because 1/dg(n)]e is ignorable for g(n) =Θ(log n). Finally, by using the continuous mapping theorem[28] and changing the variable u = F−1(n−1+t), we obtain(29). 2

Lemma 2. Let Φ and Ψ be two independent homogeneousPPPs over a finite region R ⊂ R2 with densities λ and µ =Θ(λδ) for δ > 0, respectively. Assume that an arbitrarily chosenpoint Y0 ∈ Ψ is given. Then, the following holds for Φp(Z) ,X ∈ Φ ||X − Z| = O(λp),

1) For α > 2 and −1/2 < p ≤ 0, as λ→∞,∑X∈Φp(Y0)

|X − Y0|−α = Θ(λα2

). (36)

2) For α0, α1 > 2, as λ→∞,

∑Y ∈Ψ\Y0

∑X∈Φ\(Φp(Y0)

⋃Φp(Y ))

|X − Y0|−α0 |X − Y |−α1

= Θ(λs),

(37)

0( )py00y

0y

0( )qy00qy

x

( )p

lQ

y

( )q

lQ

max , ( )

p qlQ

ÎF

ÎY

2

0( ) ( )

p

py

dl

+= Q

00y

1 2

0( ) ( )

q

qy l

+= Q

00qy

Fig. 4. Conceptual illustration for the proof of Lemma 2.

where s is given by

s =

αmax

2+αmin

2δ, if z < − 1

2 ,

1 + (2− αmax) z +αmin

2δ, if − 1

2 ≤ z < −12δ,

1 + δ + (4− αmax − αmin) z, if z ≥ − 12δ,

for δ < 1, or

s =

αmax + αmin

2+ δ − 1, if z < − 1

2 ,

1 + δ + (4− αmax − αmin) z, if z ≥ − 12 ,

for δ ≥ 1, and∑Y ∈Ψ\Y0

∑X∈Φp(Y0)

⋂Φp(Y )

|X − Y0|−α0 |X − Y |−α1 = Θ(λt),

(38)

where t is given by

t =

αmax

2+αmin

2δ, if z > − 1

2δ > −12 ,

αmax + αmin

2+ δ − 1, if z > − 1

2 > −12δ,

αmax + αmin

2+ δ + 2z, if − 1

2δ < z < − 12 ,

−∞, otherwise ,

for αmax = maxα0, α1 and αmin = minα0, α1.

Proof: 1) To prove the first part of Lemma 2, we constructthe lower-bound and upper-bound as follows.

maxX∈Ap(Y0)

|X − Y0|−α ≤∑

X∈Ap(Y0)

|X − Y0|−α

≤∑X∈Φ

|X − Y0|−α.


Here, the lower-bound is obviously Θ(λα/2) because the min-imum distance is minX∈Ap(Y0) |X − Y0| = Θ(λ−1/2). Tocomplete the proof of the first part, it is sufficient to prove∑X∈Φ

|X − Y0|−α = Θ(λα/2). Let h(x) = x−α. For a ran-

domly selected X ∈ Φ, its contact distribution function is givenby

F (x;Y0) = Pr(|X − Y0| < x) =v (b(Y0, X)

⋂R)

v(R),

where b(a, r) = b ∈ R2||a − b| ≤ r and v(·) denotes theLebesgue measure (i.e. area measure). Since v(R) is finite andindependent to x, F (x) = c′x2 + o(x2) and thus h(F−1)(x) =cx−

α2 + o(x−

α2 ), where c is independent to x. From Lemma 1,

we haven∑k=1

h(xk) = Θ

(log n

∫ 1

0

n(1−α2 )t+α2 dt

)= Θ (n) + Θ

(nα2

)= Θ

(nα2

),

(39)

where the last equality comes from α > 2 and the fact n = Θ(λ)completes the proof of the first part.

2) Define two sets Ap(Z) , X ∈ Φ||X − Z| p= Θ(λp)and Bq(Z) , y ∈ Ψ||Y − Z| p= Θ(λq) for a given pointZ. Since Φ and Ψ are PPPs with density λ and µ, respec-tively, E|Ap(Z)| = Θ(λ1+2p) and E|Bq(Z)| = Θ(λδ+2q).Note that Pr |Ap(Z)| > 0 → 1 if and only if − 1

2 < p ≤ 0

and Pr |Bq(Z)| > 0 → 1 if and only if − δ2 < q ≤ 0. As-sume that Y1 ∈ Ψ is given with |Y0 − Y1| = Θ(λd) andα0 ≥ α1. For any X ∈ Ap(Y0), |X − Y1| = Θ(λmaxp,d)

so that |X − Y0|−α0 |X − Y1|−α1 = Θ(λ−α0p−α1 maxp,d).

Then, we have∑Y ∈Bd(Y1)

∑X∈Ap(Y0)\(Φp(Y0)

⋃Φp(Y ))

|X − Y0|−α0 |X − Y1|−α1

= Θ(λδ+2d+1+2p−α0p−α1 maxd,p

),

(40)

with the constraints of maxz,− 1

2

< p ≤ 0 and− δ2 ≤ d ≤ 0.

Similarly, we have∑Y ∈Bd(Y1)

∑X∈Ap(Y0)

⋂(Φp(Y0)

⋂Φp(Y ))

|X − Y0|−α0 |X − Y1|−α1

= Θ(λδ+2d+1+2p−α0p−α1 maxd,p

),

(41)

with the constraints of − 12 < p ≤ min z, 0 and − δ2 ≤ d ≤ 0.

Then, the second part is easily given by obtaining the supre-mums of (40) and (41) under the above constraints.

APPENDIX BPROOF OF THEOREM 1

Since the interference is removed by the Genie without anycost, sirif = ∞. Inserting Fif = GH into (7), ψmrt

kk =

∑Xl∈X‖ βlkh

Hlkhlk and the square of which is given by

∣∣ψifkk

∣∣2 ∑Xl∈Xk

β2lk

∣∣∣hHlkhlk∣∣∣2=∑Xl∈Xk

(P ulk

P ulk βlk + 1

)2

β4lk

∣∣hHlkhlk∣∣2

+∑Xl∈Xk

√P ulk

P ulk βlk + 1

2

β3lk

∣∣hHlkvlk∣∣2,(42)

where the first asymptotic equality comes from the fact thatthe cross term is asymptotically ignorable and the last equalitycomes from inserting (12).

Define Lk as the set of BSs sufficiently near user k, given by

Lk = Xl ∈ Xk|P ulk βlk = Ω(1). (43)

Since P ulk βlk = Ω(1) ⇔ |Xl − Uk| = O

(Nρul/α

), |Xk| =

O(Nηbs+

2αρ

ul)

, which means that Lk is a non-empty set if and

only if ρul ≥ −α2 ηbs and Lk = Xk if ρul ≥ 0. Then, we have

∣∣ψifkk

∣∣2 =∑Xl∈Lk

β2lk

∣∣hHlkhlk∣∣2 +1

P ulk

∑Xl∈Lk

βlk∣∣hHlkvlk∣∣2

+(P ulk

)2 ∑Xl∈Xk\Lk

β4lk

∣∣hHlkhlk∣∣2+ P ul

k

∑Xl∈Xk\Lk

β3lk

∣∣hHlkvlk∣∣2.(44)

Case EH (ρul ≥ 0): in this case, Lk = Xk and Xk\Lk = ∅.Then, we have∣∣ψif

kk

∣∣2 =∑Xl∈Xk

β2lk

∣∣hHlkhlk∣∣2 +1

P ulk

∑Xl∈Xk

βlk∣∣hHlkvlk∣∣2

Nαηbs+2ηant +N−ρul+α

2 ηbs+ηant

Nαηbs+2ηant ,

(45)

where the second asymptotic equality comes from Lemma 2 andthe last asymptotic equality comes from the fact that the firstterm is always dominant for ρul ≥ 0.

Cases M and L (ρul < −α2 ηbs): in this case, Lk = ∅ andXk\Lk = Xk, we have

|ψifkk|2 =

(P ulk

)2 ∑Xl∈Xk

β4lk

∣∣hHlkhlk∣∣2 + P ulk

∑Xl∈Xk

β3lk

∣∣hHlkvlk∣∣2 N2ρul+2αηbs+2ηant +Nρul+ 3α

2 ηbs+ηant

N2ρul+2αηbs+2ηant , if M,

Nρul+ 3α2 ηbs+ηant , if L,

(46)

where the second asymptotic equality comes from Lemma 2 andthe third asymptotic equality comes from the fact that the firstterm is only dominant if ρul > −α2 ηbs − ηant.


Case H (−α2 ηbs ≤ ρul < 0): The scaling is simply derivedby combining the results in (45) and (46) so that it is given by∣∣ψif

kk

∣∣2 Nαηbs+2ηant . (47)

Define an auxiliary variable ν such that Qifk = Θ (Nν) for

all k. The final step is to find the scaling exponent of the DLtransmit power for user k, which is given by

P dlk = Qif

k

∑Xl∈Xk

βlk‖hlk‖2

MQifk

∑Xl∈X‖

P ulk β

2lk

P ulk βlk + 1

Nν+α2 ηbs+ηant , if EH or H,

Nν+ρul+αηbs+ηant , if M or L.

(48)

By using P dlk = Θ(Nρdl

) for all k, and the results (45), (46),and (47), we obtain (16).

APPENDIX CPROOF OF THEOREM 2

Inserting Fmrt = GH into (7), ψmrtkj =

∑Xl∈Xj

√βlkβljh

Hlkhlj ,

and the square of which is given by∣∣ψmrtkj

∣∣2 ∑Xl∈Xj

βlkβlj

∣∣∣hHlkhlj∣∣∣2

=∑Xl∈Xj

(P ulj

P ulj βlj + 1

)2

βlkβ3lj

∣∣hHlkhlj∣∣2

+∑Xl∈Xj

√P ulj

P ulj βlj + 1

2

βlkβ2lj

∣∣hHlkvlj∣∣2,(49)

where the first asymptotic equality comes from the fact that thecross term is asymptotically ignorable and the third equality isobtained by inserting (12). Note that ψmrt

kk = ψifkk so that the

SNR scaling exponent of MRT operation is the same as that ofIF operation.

The interference power, denoted as Ik, can be written as

Ik =∑

Uj∈U\Uk

Qmrtj |ψmrt

kj |2

=∑

Uj∈U\Uk

Qmrtj

∑Xl∈Lj

βlkβlj∣∣hHlkhlj∣∣2

+1

P ulk

∑Uj∈U\Uk

Qmrtj

∑Xl∈Lj

βlk∣∣hHlkvlj∣∣2

+(P ulk

)2 ∑Uj∈U\Uk

Qmrtj

∑Xl∈Xj\Lj

βlkβ3lj

∣∣hHlkhlj∣∣2+ P ul

k

∑Uj∈U\Uk

Qmrtj

∑Xl∈Xj\Lj

βlkβ2lj

∣∣hHlkvlj∣∣2,

(50)

where Lj is defined in (43).

Define an auxiliary variable ν such that Qmrtk = Θ (Nν) for

all k and consider the four regimes EH, H, M, and L.Case EH (ρul ≥ 0): in this case, Lj = Xj and Xj\Lj = ∅

for all j. Then, we have

Ik =∑

Uj∈U\Uk

Qmrtj

∑Xl∈Xj

βlkβlj∣∣hHlkhlj∣∣2

+1

P ulk

∑Uj∈U\Uk

Qmrtj

∑Xl∈Xj

βlk∣∣hHlkvlj∣∣2

Nν+α2 (ηbs+ηuser)+(1−α2 )(ηuser−ηbs)

++ηant

+Nν−ρul+α2 ηbs+ηuser+ηant


++ηant ,

(51)

where the second asymptotic equality comes from Lemma 2 andthe third asymptotic equality comes from the fact that the firstterm is always dominant due to

(α2 − 1

) ((ηuser − ηbs)

+ − ηuser

)≤ 0.

Cases M and L (ρul < −α2 ηbs): In this case, Lj = ∅ andXj\Lj = Xj . Then, we have

Ik =(P ulk

)2 ∑Uj∈U\Uk

Qmrtj

∑Xl∈Xj

βlkβ3lj

∣∣hHlkhlj∣∣2+ P ul

k

∑Uj∈U\Uk

Qmrtj

∑Xl∈Xj

βlkβ2lj

∣∣hHlkvlj∣∣2Nν+2ρul+ 3α

2 ηbs+α2 ηuser+(1−α2 )(ηuser−ηbs)

++ηant

+Nν+ρul+αηbs+α2 ηuser+(1−α2 )(ηuser−ηbs)

++ηant

Nν+ρul+αηbs+α2 ηuser+(1−α2 )(ηuser−ηbs)

++ηant ,

(52)

where the second equality comes from Lemma 2 and the thirdasymptotic equality comes from the fact that the first term isalways dominant due to ρul < −α2 ηbs.

Case H (−α2 ηbs ≤ ρul < 0): The scaling exponent can besimply derived as

Ik Nν+α2 (ηbs+ηuser)+(1−α2 )(ηuser−ηbs)

++ηant , (53)

because the result in (52) at ρul = −α2 ηbs is the same to the re-sult in (51). Thus, by combining (51)-(53), the scaling exponentof the interference power is given by

Ik


++ηant , if EH or H,

Nν+ρul+αηbs+α2 ηuser+(1−α2 )(ηuser−ηbs)

++ηant , if M or L,(54)

and combining it with (16) and (48) results in (21), which com-pletes the proof.

APPENDIX DPROOF OF THEOREM 3

Inserting Fzf = GH(GGH

)−1

into (7), ψzfkj = δ(k − j) +

[G]k,:Fzfej , where ej denotes the jth column of the identity


matrix IK , and the square of which is given by∣∣ψzfkj

∣∣2 = δ(k − j) + 2Re

[G]k,:Fzfej

+ Tr

((Fzf)H

RkFzfeje

Hj

) δ(k − j) + Tr

((Fzf)H

RkFzfeje

Hj

),

(55)

where Rk = [G]Hk,:[G]k,: and the last asymptotic equalitycomes from the fact that (a + b)2 a2 + b2. Also, the DLtransmit power consumed for user k is given by

P dlk = Qzf

k Tr((

Fzf)H

FzfekeHk

). (56)

Note that we have[GGH

]i,j

=∑Xl∈X

√βliβljh

Hli hlj

=

M

∑Xl∈X

Pulj β

2lj

Pulj βlj+1

, if i = j,

√M

∑Xl∈X

√Puli β

2li

Puli βli+1

Pulj β

2lj

Pulj βlj+1

, if i 6= j,

(57)

and consider the four regimes.Case EH (ρul ≥ 0): In this case, Lk = Xk so that Rk

1Pulk

IN and

[GGH

]i,j

M

∑Xl∈X

βlj , if i = j,

√M

∑Xl∈X

√βliβlj , if i 6= j,

(58)

Inserting (58) into (55), we have

∣∣ψzfkk

∣∣2 1 +1

MP ulk

( ∑Xl∈Xk

βlk

)−1

1 +N−ρul−α2 ηbs−ηant

1

(59)

and ∑Uj∈U\Uk

∣∣ψzfkj

∣∣2 1

MP ulk

·∑

Uj∈U\Uk

( ∑Xl∈Xk

βlk

) ∑Xl∈Xj

βlj

−2

N−ρul−α2 ηbs−ηant+ηuser ,

(60)

where the last asymptotic equality comes from Lemma 2 andρul ≥ 0. Defining an auxiliary variable ν such that Qk =Θ(Nν), the DL transmit power can be written as

P dlk =

Qzfk

M

( ∑Xl∈Xk

βlk

)−1

Nν−α2 ηbs−ηant .

(61)

For a reasonable DL transmit power P dlk = Θ(Nρul

), ν = ρdl +α2 ηbs + ηant and thus

snrzf =ρdl +α

2ηbs + ηant,

sirzf =ρul +α

2ηbs + ηant − ηuser.

(62)

Case H (−α2 ηbs ≤ ρul < 0): Define U (d)k =

Uj ∈ U| |Uk − Uj | = Θ(Nd)

. Since Rk = diag θ1kIM ,

· · ·, θLkIM, where θlk 1/P ulk , if Xl ∈ Xk, or θlk βlk, if

Xl ∈ X\Xk, and[GGH

]i,j

is given as in (63). Then,∣∣∣ψzfkj

∣∣∣2 is

given as in (64), where the last asymptotic equality comes fromthe fact that∑Xl∈Lk

βlk + P ulk

∑Xl∈Xk\Lk

β2lk N

α2 ηbs +Nηbs+( 2

α−1)ρul

N α2 ηbs .

(65)

Here, t1, t2, t3, and t4 can be further derived from Lemma 2as follows. Suppose that ηbs ≥ ηuser and −α2 ηbs ≤ ρul <−α2 ηuser. In this case, Lk

⋂Lj = ∅ for all Uj ∈ U in probabil-

ity so that, by using Lemma 2, we obtain

t1 = 0,

t2 Nηbs+αηuser+2αρ

ul

,

t3 Nα2 ηbs+

α2 ηuser ,

t4 Nηbs+α2 ηuser+( 2

α−1)ρul

.

Since α2 ηbs + α

2 ηuser−(ηbs + αηuser + 2

αρul)

=(α2 − 1

)(ηbs

−ηuser) − 2α

(ρul + α

2 ηuser

)≥ 0 and α

2 ηbs + α2 ηuser −(

ηbs + α2 ηuser +

(2α − 1

)ρul)

=(1− 2

α

) (α2 ηbs + ρul

)≥ 0,

t3 becomes always dominant and thus∑Uj∈U\Uk

∣∣ψzfkj

∣∣2 N−α2 ηbs+α2 ηuser−ηant . (66)

Now, suppose that ηbs ≥ ηuser and ρul ≥ −α2 ηuser. In thiscase, Lj = Xj if d ≤ ρul/α and Xj

⋂Lj = ∅ if d > ρul/α

for all Uj ∈ U (d)k in probability so that, by using Lemma 2, we

obtain

t1 Nα2 ηbs+ηuser+( 2

α−1)ρul

,

t2 Nηbs+ηuser+2( 2α−1)ρul

,

t3 Nα2 ηbs+ηuser+( 2

α−1)ρul

,

t4 Nηbs+ηuser+2( 2α−1)ρul

.

Since α2 ηbs+ηuser+

(2α − 1

)ρul−

(ηbs + ηuser + 2

(2α − 1

)ρul)

=(1− 2

α

) (α2 ηbs + ρul

)≥ 0, t1 or t3 becomes dominant and

thus ∑Uj∈U\Uk

∣∣ψzfkj

∣∣2 N( 2α−1)ρul−α2 ηbs+ηuser−ηant . (67)


[GGH

]i,j

M

( ∑Xl∈Lj

βlj + P ulj

∑Xl∈Xj\Lj

β2lj

), if i = j,

√M

∑

Xl∈Li⋂Lj

√βliβlj +

√P ulj

∑Xl∈Li\Lj

√βliβ2

lj

+√P uli

∑Xl∈Lj\Li

√β2liβlj +

√P uli P

ulj

∑Xl∈Xi∩Xj\Li∪Lj

√β2liβ

2lj

, if i 6= j.

(63)

∑Uj∈U\Uk

∣∣∣ψzfkj

∣∣∣2 ∑Uj∈U\Uk

1Pul

∑Xl∈Xk∩Xj

βlj +∑

Xl∈Xk\Xjβ2lj +

∑Xl∈Xj\Xk

βlkβlj + Pul∑

Xl∈X\(Xk∪Xj)βlkβ

2lj

M

( ∑l∈Xj

βlj + Pul∑

Xl∈X\Xjβ2lj

)2

1

Nαηbs+ηant

1

Pul

∑Uj∈U\Uk

∑Xl∈Xk∩Xj

βlj

︸︷︷︸t1

+∑

Uj∈U\Uk

∑Xl∈Xk\Xj

β2lj

︸︷︷︸t2

+∑

Uj∈U\Uk

∑Xl∈Xj\Xk

βlkβlj

︸︷︷︸t3

+Pul∑

Uj∈U\Uk

∑Xl∈X\(Xk∪Xj)

βlkβ2lj

︸︷︷︸t4

,

(64)

When ηbs ≤ ηuser, the similar derivation can be done and weobtain ∑

Uj∈U\Uk

∣∣ψzfkj

∣∣2 N( 2α−1)ρul−α2 ηbs+ηuser−ηant . (68)

Also,

∣∣ψzfkk

∣∣2 1 +

1Pul

∑Xl∈Lk

βlk + P ul∑

Xl∈Xk\Lkβ3lk

M

( ∑Xl∈Lk

βlk + P ulk

∑Xl∈Xk\Lk

β2lk

)2

1 +N−ρul−α2 ηbs−ηant

1,

(69)

where the second and last asymptotic equalities come fromLemma 2 and the fact −α2 ηbs ≤ ρul < 0, respectively. Sincethe DL transmit power can be written as

P dlk =

Qzfk

M

∑Xl∈Lk

βlk + P ul∑

Xl∈Xk\Lk

β2lk

−1

Nν−α2 ηbs−ηant

(70)

so that ν = ρdl + α2 ηbs + ηant for a reasonable DL transmit

power,snrzf = ρdl +

α

2ηbs + ηant, (71)

and sirzf is given as in (72).

Cases M and L (ρul < −α2 ηbs): In these cases, Rk diagβ1kIM , β2kIM , · · ·, βLkIM, and

[GGH

]i,j

MP ul

j

∑Xl∈Xj

β2lj , if i = j,√

MP uli P

ulj

∑Xl∈Xi

⋂Xj

√β2liβ

2lj , if i 6= j,

(73)

Inserting (73) into (55), we have

∣∣ψzfkj

∣∣2 1kj +1

MP ulj

∑Xl∈Xj

β2lj

−2 ∑Xl∈Xj

β2ljβlk, (74)

which is obtained in a similar way as before as∣∣ψzfkk

∣∣2 1 +N−ρul−α2 ηbs−ηant

1, if M,

N−ρul−α2 ηbs−ηant , if L,

(75)

and ∑Uj∈U\Uk

∣∣ψzfkj

∣∣2

1

MP ul

∑Uj∈U\Uk

∑Xl∈Xj

β2lj

−2 ∑Xl∈Xj

β2ljβlk

N−ρul−(α2 +1)ηbs−ηant+ηuser−(α2−1)(ηbs−ηuser)

+

.

(76)


sirzf =

(1− 2

α

) (ρul + α

2 ηuser

)++ α

2 (ηbs − ηuser) + ηant, if ηbs ≥ ηuser,(1− 2

α

)ρul + α

2 ηbs − ηuser + ηant, if ηbs < ηuser.(72)

Since the DL transmit power can be written as

P dlk =

Qzfk

MP ulk

( ∑Xl∈Xk

β2lk

)−1

Nν−ρul−αηbs−ηant (77)

so that ν = ρdl +ρul +αηbs +ηant for a reasonable DL transmitpower, we can obtain

snrzf =

ρdl + ρul + αηbs+ηant, if M,

ρdl + α2 ηbs, if L,

(78)

and

sirzf =

ρul + α

2 ηbs + ηant

+(ηbs − ηuser) +(α2 − 1

)(ηbs − ηuser)

+, if M,

(ηbs − ηuser) +(α2 − 1

)(ηbs − ηuser)

+, if L,

(79)

which concludes the proof. 2

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Kyung Jun Choi was born in Busan, Korea, on 7February. 1987. He received the B.S. and Ph.D. de-grees in Electrical and Electronic Engineering fromYonsei University, Seoul, Korea in February 2010 andFebruary 2017, respectively. He is currently workingwith WILUS Inc., Seoul, Korea. His current researchinterests include massive MIMO transceiver design,network performance analysis via stochastic geome-try and random matrix theory.

Kwang Soon Kim (S’95, M’99, SM’04) was born inSeoul, Korea, on September 20, 1972. He receivedthe B.S. (summa cum laude), M.S.E., and Ph.D. de-grees in Electrical Engineering from Korea AdvancedInstitute of Science and Technology (KAIST), Dae-jeon, Korea, in February 1994, February 1996, andFebruary 1999, respectively.

From March 1999 to March 2000, he was withthe Department of Electrical and Computer Engineer-ing, University of California at San Diego, La Jolla,CA, U.S.A., as a Postdoctoral Researcher. From April

2000 to February 2004, he was with the Mobile Telecommunication ResearchLaboratory, Electronics and Telecommunication Research Institute, Daejeon,Korea as a Senior Member of Research Staff. Since March 2004, he has been


with the Department of Electrical and Electronic Engineering, Yonsei Univer-sity, Seoul, Korea, now is an Associate Professor.

He is a Senior Member of the IEEE, served as an Editor of the Journal ofthe Korean Institute of Communications and Information Sciences (KICS) from2006-2012, as the Editor-in-Chief of the journal of KICS since 2013, as an Ed-itor of the Journal of Communications and Networks (JCN) since 2008, as anEditor of the IEEE Transactions on Wireless Communications since 2009.

He was a recipient of the Postdoctoral Fellowship from Korea Science andEngineering Foundation (KOSEF) in 1999. He received the Outstanding Re-searcher Award from Electronics and Telecommunication Research Institute(ETRI) in 2002, the Jack Neubauer Memorial Award (Best system paper award,IEEE Transactions on Vehicular Technology) from IEEE Vehicular TechnologySociety in 2008, and LG R&D Award: Industry-Academic Cooperation Prize,LG Electronics, 2013. His research interests are in signal processing, commu-nication theory, information theory, and stochastic geometry applied to wirelessheterogeneous cellular networks, wireless local area networks, wireless D2Dnetworks and wireless ad doc networks.

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