IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 10, OCTOBER 2018 6887

Interference Coordination for 3-DBeamforming-Based HetNet Exploiting

Statistical Channel-State InformationXiao Li , Member, IEEE, Chaosong Li , Shi Jin , Senior Member, IEEE, and Xiqi Gao , Fellow, IEEE

Abstract— In this paper, we investigate the inter-tier interfer-ence coordination for a heterogeneous network (HetNet), whereboth macro and small cells work in the frequency-divisionduplexing mode and share the same spectrum. A 2-D large-scaleantenna array is deployed at a macro-base station (BS), whileconventional multiple antenna array is deployed at each smallcell. First, we derive an approximation of macro-users’ signal-to-interference-plus-noise ratio (SINR), under the assumptionof only statistical channel-state information (CSI) at macro-BS,and a 3-D beamforming transmission scheme is applied formacro-users. A lower-bound of small-cell users’ expected SINRis also derived using the property of complex Wishart matrix.Based on these, simple metrics are derived to measure the inter-tier interference. Then, new interference coordination algorithmsare proposed to achieve a good tradeoff for the performanceand traffic between macro-cells and small cells. The proposedalgorithms require only a few additional statistical CSI. More-over, we derive tractable ergodic rate approximation of both themacro-cell and small-cell users which are shown to match wellwith the Monte Carlo results.

Index Terms— 3-D beamforming, massive MIMO, HetNet,statistical CSI.

I. INTRODUCTION

AS ADVANCED communication devices and data-hungryapplications are adopted, wireless networks continue to

face significant growth in user data demands. A straight-forward and traditional approach, which is also an energy-

Manuscript received February 10, 2018; revised May 30, 2018 andAugust 1, 2018; accepted August 4, 2018. Date of publication August 20,2018; date of current version October 9, 2018. The work of X. Li wassupported in part by the National Natural Science Foundation of Chinaunder Grant 61571112 and Grant 61831013 and in part by the Founda-tion for the Author of National Excellent Doctoral Dissertation of Chinaunder Grant 201446. The work of S. Jin was supported in part by theNational Science Foundation for Distinguished Young Scholars of Chinaunder Grant 61625106 and in part by the National Natural Science Foun-dation of China under Grant 61531011. The work of X. Gao was sup-ported by the National Natural Science Foundation of China under Grant61320106003 and Grant 61521061. This paper was presented in part at theIEEE International Workshop on Signal Processing Advances in WirelessCommunications, Sapporo, Japan, July 2017. The associate editor coor-dinating the review of this paper and approving it for publication wasD. W. K. Ng. (Corresponding author: Xiao Li.)

X. Li, S. Jin, and X. Gao are with the National Mobile CommunicationsResearch Laboratory, Southeast University, Nanjing 210096, China (e-mail:li_xiao@seu.edu.cn; jinshi@seu.edu.cn; xqgao@seu.edu.cn).

C. Li was with the National Mobile Communications ResearchLaboratory, Southeast University, Nanjing 210096, China. He is now withHuawei Technologies Co., Ltd, Chengdu 611731, China (e-mail:lichaosong1@huawei.com).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TWC.2018.2864976

efficient way [1], [2], to increase the network capacity isnetwork densification [3], i.e., to make the cells small and addnew cell sites. However, network expansion using a macro-cellular infrastructure is costly and is usually restricted byzoning. A rising alternative is the deployment of hetero-geneous networks (HetNets) [4], [5], in which various lowpower nodes, often referred to as small cells, are distributedthroughout the macro cell network. Therefore, macro basestations (BSs) could ensure outdoor coverage and serve highlymobile users, while small cells act as the main capacity-driverfor indoor and outdoor hotspots [6]. Massive multiple-inputmultiple-output (MIMO) [7] is another way to achieve highdata rates and coverage gains. It is shown that massive MIMOenables large gains in spectral and energy efficiency comparedto conventional MIMO systems [8], [9]. Furthermore, simplelinear signal processing approaches, such as matched-filtering(MF), can be used to achieve these advantages. Other benefitsof massive MIMO include extensive use of inexpensive low-power components, reduced latency, simplification of the mul-tiple access (MAC) layer, and robustness against intentionaljamming [7]. The promise of these benefits makes HetNetand massive MIMO be recognized as key technologies ofthe future 5G wireless systems [2], [10], and the coexistenceof HetNet and massive MIMO is attracting more and moreresearch interests [6], [11], [12].

Although it has potential large gains in spectral and energyefficiency, there remains several challenges to build massiveMIMO HetNets in practice. Since the downlink training rep-resents a significant bottleneck and the corresponding channelstate information (CSI) feedback yields an unacceptably highoverhead for the uplink in frequency division duplexing (FDD)systems, most of the research works considered time divisionduplexing (TDD) systems [8]. However, as FDD is generallyconsidered to be more effective for systems with symmetrictraffic and delay-sensitive applications [13] and most cellu-lar systems today employ FDD, there remains considerableinterest in making massive MIMO work for FDD systems.An alternative approach for FDD systems is to exploit thesecond-order channel statistics [14]–[16], which varies muchslowly and therefore can be obtained by the BS through long-term feedback. In [15] and [16], it was shown that when thequality of instantaneous CSI is not good, the transmissionalgorithm exploiting only statistical CSI could even outper-form the algorithms exploiting instantaneous CSI. Moreover,user scheduling using instantaneous CSI is also a difficult taskwhen the number of users is large. It is especially the truth

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6888 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 10, OCTOBER 2018

for multi-cell and HetNet systems, since it will cause a lot ofcomputation and data exchange between BSs and tiers. Usingstatistical CSI is also an effective way to reduce the frequencyand amount of computation and data exchange.

Another main challenge is that the number of antennas thatcan be equipped at the BS is limited. If deployed in a horizon-tal line, only a limited number of antennas can be accommo-dated due to form factors [17]. Placing the antennas in a two-dimensional (2D) grid is an effective way to reduce the antennapanel size, which is referred to as three-dimensional MIMO(3D-MIMO) or full-dimension MIMO (FD-MIMO). It can beimplemented using active antenna system (AAS) [18], [19],which combines the 2D antenna array with a transceiver unitarray. In this way, one or a few analog beams with adaptabledowntilt can be generated, providing the ability of dynamicbeam steering in the 3D space. Another way to implement3D-MIMO, which can fully exploit the potential of largenumbers of antennas, is to connect a dedicated transceiverunit to each antenna element [19]. Such a design allows digitalprecoding to be performed jointly across all antenna elements,so that the degree of freedom (DoF) of both horizontal andvertical directions can be exploited by generating multipledigital beams in both directions. Based on this concept,Adhikary et al. [14] proposed a 3D beamforming plus jointspatial division and multiplexing transmission algorithm expl-oiting both statistical and instantaneous CSI, [15], [16], [20]further exploited the statistical 3D beamforming transmissionalgorithm for downlink 3D-MIMO systems which has onlystatistical CSI at the BS. In [21], it was shown that, althoughthe conventional uniform linear array (ULA) outperforms the3D-MIMO under the same total amount of antenna elements,the 3D-MIMO outperforms the ULA given the same numberof horizontal antenna elements.

Furthermore, interference management among coexistingmassive MIMO systems and small cells is a critical issue.Macro BS with a large number of antennas should be ableto communicate with its own macro users without interferingwith users in small cells. A precoding method for reversedTDD system was proposed in [22]. In this approach, the macroBS estimates the null space to the small cells during theirdownlink transmission and then projects its downlink datatransmission into the null space of the small cells, to eliminatethe interference caused by macro BS to small cells. Nullingwas also used in [23] for TDD systems to avoid interferencefrom macro BS to small cell users. Assuming that users areconcentrates at certain areas in the cell, Adhikary et al. [24]developed three interference coordination strategies based onspatial blanking method.

Motivated by the above observations, we investigatethe downlink interference coordination of a HetNet where3D-MIMO with a dedicated transceiver unit connected to eachantenna element is employed at macro BS, while conventionalLTE-like MIMO (no more than 8 antenna elements [25])is deployed at each small cell. The HetNet works in FDDmode where only statistical CSI is available at macro BS.A 3D beamforming downlink transmission scheme similar tothe one proposed in [16] is applied in macro tier, while zero-forcing (ZF) beamforming is employed at each small cell.

We derive an approximation of the signal-to-interference-plus-noise ratio (SINR) of macro users and a lower-bound of theexpected SINR of small cell users. Based on these, simplemetrics using only statistical CSI to evaluate the interferencebetween macro BS and small cells are derived. Then, inter-tierinterference coordination algorithms are proposed to balancethe performance of macro and small cells. In the proposedalgorithms, the small cells which cause strong interferenceto macro users are either shut down or working at lowertransmit power, and the macro users which cause strong inter-ference to small cells are either removed from the schedulinglist or offloaded to the nearest small cells. Moreover, we deriveclosed-form ergodic rate approximations for both macro andsmall cell users in the considered HetNet, which are shown tomatch well with the Monte Carlo results.

The rest of the paper is organized as follows. Section IIdescribes the system model. In Section III, we introduce the3D beamforming downlink transmission scheme for macrousers, and derive an approximation of the SINR of macrousers and a lower-bound of the expected SINR of smallcell users. After that, in Section IV, we propose some inter-tier interference coordination algorithms based on the derivedSINR approximation and lower-bound. Moreover, the ergodicrate of macro and small cell users are analyzed. Simulationresults are presented in Section V, and we conclude the paperin Section VI.

We adopt the following notation: Vectors are represented ascolumns and are denoted in lower-case bold-faced characters,and matrices are represented in upper-case boldfaced. Thesuperscript (·)T , (·)∗, and (·)H indicate the matrix transpose,conjugate, and conjugate transpose operation respectively. Thecomplex number field is represented by C, and E{·} denotesthe expectation.

II. SYSTEM MODEL

Consider the downlink of a HetNet formed by a singlemacro BS coexisting in the same coverage area with Ns smallcells, as shown in Fig. 1. The macro BS ensures the overallcoverage, while small cells serve hotspots. Both tiers workin FDD mode and share the same time-frequency resourceblock. The macro BS is equipped with a massive MIMOantenna array placed in a 2D grid with Mh � 1 antennasin each row and a total number of Mv � 1 rows in thevertical dimension. There are L single-antenna macro usersto be scheduled, the total number of users that can be servedsimultaneously by macro BS is Km, and the set of scheduledmacro users is denoted by M. Without loss of generality, eachsmall cell is equipped with a LTE-like MIMO of Ms antennas,where Ms is no more than 8. Small cell i devotes its availableresources to Ksi single-antenna users, where Ksi < Ms, andthe set of users served by small cell i is denoted by Si.We assume that NsMs � 1 and transmissions across the tiersare perfectly synchronized. All transmissions are assumed totake place over flat fading channels.

A. Small Cell Setup

We assume that the transmit power of small cell i is Psi ,and the small cells are not fully loaded. The channel vector

LI et al.: INTERFERENCE COORDINATION FOR 3-D BEAMFORMING-BASED HetNet 6889

Fig. 1. A HetNet formed by a macro cell overlaid with small cells.

between small cell i and user k, denoted as gTsi,k

(gsi,k ∈CMs×1), can be modeled as

gTsi,k =

√βsi,kh

Tw,si,k, (1)

where βsi,k represents the large-scale fading coefficientbetween small cell i and user k, and hw,si,k is a randomvector with zero mean and unit variance independent identicaldistribution (i.i.d.) complex Gaussian elements. We assumethat βsi,k is given by

βsi,k = wnsi,k/dα

si,k, (2)

where dsi,k is the distance between small cell i and user k,α is the path-loss exponent, w is the wall penetration loss, andnsi,k is the number of walls between small cell i and user k.According to [24], we assume that 1

nsi,k =

⎧⎪⎨

⎪⎩

0, k ∈ Si,

1, k ∈ M,

2, k ∈ Sj , j �= i.

(3)

Note that users in the same small cell are usually locate closeto and around the small cell, and the small cells are distributedrandomly in macro cell whose radius is much larger than smallcells. Therefore, the distance between different small cells isusually much larger than the distance between each small celland its own users, and the distance dsi,k for k ∈ Sj andi �= j can be reasonably approximated as dsi,k ≈ dsi,sj ,where dsi,sj is the distance between small cell i and j.Then, βsi,k for k ∈ Sj and i �= j can be approximated asβsi,k ≈ βsi,sj = w2/dα

si,sj.

Moreover, note that the number of antenna elementsemployed at each small cell is relatively small and users insmall cells usually move at relatively low speed. Therefore,we assume that each small cell has the instantaneous CSI ofits own users and applies column-normalized zero-forcing (ZF)beamforming transmission.

B. Macro Tier Setup

We assume that the total transmit power of macro BS is Pm.At every transmission resource block, the macro BS schedules

1This models the case that small cell users are indoor users clustered insmall environments, such as airport lounges or coffee shops, with the smallcells located at the center of its users, and macro users are distributed inoutdoor macro cell area. Although we made this assumption, the analysis andthe proposed algorithm could also work for other different assumptions.

at most Km users out of L users. The distances between twoadjacent antenna elements in a row and a column are both halfof the carrier wavelength.

The channel vector between the Mv × Mh antennas of themacro BS and user k, denoted as fT

m,k (fm,k ∈ CMvMh×1),can be written as

fTm,k =

√βm,khT

m,k, (4)

where βm,k represents the large-scale fading coefficient, andhm,k represents the small-scale fading coefficient vector. Sim-ilar to (2), we assume that βm,k is given by

βm,k = wnm,k/dα

m,k, (5)

where dm,k is the distance between macro BS and user k, andnm,k is the number of walls between macro BS and user k.We assume that [24]

nm,k =

{0, k ∈ M,

1, k ∈ Si, i = 1, · · · , Ns.(6)

According to [16], hm,k can be written as

hm,k = vec (Hm,k), (7)

where

Hm,k = R1/2V,m,kHw,m,kR

1/2H,m,k, (8)

the (i, j)-th element of Hm,k is the small-scale channelfading coefficient between the antenna element on the i-throw j-th column of macro BS antenna array and user k,Hw,m,k ∈ CMv×Mh is a random matrix with zero meanand unit variance i.i.d. complex Gaussian elements, Hw,m,k1

and Hw,m,k2 are independent when k1 �= k2, RV,m,k ∈CMv×Mv and RH,m,k ∈ CMh×Mh are the vertical andhorizontal channel covariance matrix of user k respectively.Substitute (7) into (4), we have that

fTm,k =

√βm,khT

w,m,k

[R1/2

H,m,k ⊗ (R1/2

V,m,k

)T ], (9)

where hw,m,k = vec (Hw,m,k).For macro BS, to reduce the overhead of CSI feedback

and exchange, we assume that it has only statistical CSIof each user, i.e., βm,k, βsi,k, RV,m,k, RH,m,k, and Psi ,at most. Therefore, macro BS performs scheduling, precoding,and inter-tier interference coordination based only on thesestatistical CSI.

As in [14], we consider the one-ring scattering model todetermine the channel covariance matrix. Both RV,m,k andRH,m,k are modeled according to [14, eq. (3)] (also see [26]).We define that

ΛV,m,k � FHMv

RV,m,kFMv ,

ΛH,m,k � FHMh

RH,m,kFMh, (10)

where, FMv ∈ CMv×Mv and FMh∈ CMh×Mh are both

unitary DFT matrices. Assume that the i-th diagonal ele-ments of ΛV,m,k and ΛH,m,k are λ

(i)V,m,k and λ

(i)H,m,k, and

the maximum diagonal elements of ΛH,m,k and ΛV,m,k areλmax

H,m,k = λ(hk)H,m,k and λmax

V,m,k = λ(vk)V,m,k, respectively. Accord-

ing to [14], for ULA of large dimension, the eigenvectors

6890 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 10, OCTOBER 2018

of its correlation matrix can be approximated by a unitaryDFT matrix. Therefore, ΛV,m,k and ΛH,m,k become diagonalmatrices as Mh, Mv → ∞.

Note that users in the same small cell are usually concen-trated over an area much smaller than the macro cell, thescattering environment that determines the angular distributionof the propagation between such users and the macro BSare almost identical for all such users [24]. Therefore, it isreasonable to assume that RV,m,k ≈ RV,si and RH,m,k ≈RH,si for k ∈ Si. Moreover, since the radius of the macrocell is much larger than the small cells, the distances betweenthe macro BS to the users in the same cell are almost thesame and can be reasonably approximated as dm,k ≈ dm,si

for k ∈ Si, where dm,si is the distance between macro BSand small cell i. Then, βm,k for k ∈ Si can be approximatedas βm,k ≈ βm,si = w/dα

m,si. We also define that ΛV,si �

FHMv

RV,siFMv , ΛH,si � FHMh

RH,siFMh, and λ

(l)V,si

and

λ(l)H,si

are the l-th diagonal elememnts of λV,si and λH,si ,respectively.

C. Signal Model

For the considered HetNet, the signal received by macrouser k can be written as

rm,k =∑

j∈M√

Pm,jfTm,kbm,jxj

+∑Ns

i=1

∑

l∈Si

√Psi,lg

Tsi,kbsi,lsi,l + nk, (11)

where, xj is the data symbol intended for macro user j withE{|xj |2

}= 1, bm,j ∈ CMvMh×1 is the unit-norm beamform-

ing vector of macro user j, Pm,j is the transmit power formacro user j with total power constraint

∑j∈M Pm,j = Pm,

si,l is the data symbol transmitted by small cell i for its user lwith E

{|si,l|2}

= 1, bsi,l ∈ CMs×1 is the column-normalizedZF beamforming vector of small cell i for its user l satisfyingbH

si,lbsi,l = 1, Psi,l is the transmit power of small cell i for its

user l with total power constraint∑

l∈SiPsi,l = Psi , and nk ∼

CN(0, σ2

k

)is the complex additive white Gaussian noise.

Let us denote Bsi =[bsi,l1 ,bsi,l2 , · · · ,bsi,lKsi

], Gsi =

[gsi,l1 ,gsi,l2 , · · · ,gsi,lKsi

]T, and Qsi = GH

si

(GsiG

Hsi

)−1 =[qsi,l1 ,qsi,l2 , · · · ,qsi,lKsi

], where

{l1, l2, · · · , lKsi

} ∈ Si.Then, we have

bsi,l =1

‖qsi,l‖qsi,l, l ∈ Si. (12)

For small cells, since we assume that they have perfect CSIof its own users and apply ZF precoding, the received signalof user k in small cell i, i.e., k ∈ Si, is given by

rsi,k =√

Psi,kgTsi,kbsi,ksi,k +

∑

j∈M√

Pm,jfTm,kbm,jxj

+∑Ns

p=1,p�=i

∑

l∈Sp

√Psp,lgT

sp,kbsp,lsp,l + nk. (13)

In this paper, we assume equal power allocation among macroand small cell users, i.e., Pm,j = Pm/Km, j ∈ M andPsi,l = Psi/Ksi , l ∈ Si.

III. MACRO TIER DOWNLINK TRANSMISSION

AND SINR ANALYSIS

In this section, we first briefly describe the downlink trans-mission scheme applied in macro tier. Then, we analyze theSINR of macro and small cell users, since it relates closelyto the achievable rate and is frequently used to optimize thetransmissions, user scheduling, and interference coordination.However, optimization based on instantaneous SINR causeslarge amount of feedback for FDD systems and data exchangebetween BSs and tiers. Therefore, we derive an approximationof the SINR of macro users under large numbers of transmitantennas, and a lower-bound of the expected SINR of smallcell users. Both of the derived approximation and lower-bounddepend only on the statistical CSI of each user.

A. Downlink Transmission Scheme

With the assumption that only statistical CSI is available atmacro BS, Li et al. [16] proposed a 3D beamforming down-link transmission scheme for single-cell 3D-MIMO systemwhich is almost optimal when Mh, Mv → ∞. In this paper,we assume that a downlink transmission scheme similar to the3D beamforming scheme in [16] is employed for macro users.The details of the shcme are listed in Algorithm 1.

Algorithm 1 Macro-Tier Downlink Transmission Scheme

Step 1: Choose M̄h + 1 integers ni, i = 0, · · · , M̄h

satisfying 0 = n0 < n1 < · · · < nM̄h= Mh,

so that they divide the horizontal space into M̄h

blocks (referred to [16, Fig. 1]).Step 2: Choose M̄v + 1 integers mi, i = 0, · · · , M̄v

satisfying 0 = m0 < m1 < · · · < mM̄v= Mv,

so that they divide the vertical space into M̄v

blocks.Step 3:Divide the macro users into M̄hM̄v groups, so that

user k in group (i, j) satisfies mi−1 +1 � vk � mi

and nj−1 + 1 � hk � nj , that is the principlediagonal element of ΛV,m,k belongs to the i-thvertical block, and the principle diagonal elementof ΛH,m,k belongs to the j-th horizontal block.

Step 4: Select the user with the largest λmaxH,m,kλmax

V,m,k ineach group.

Step 5:Assume the number of users selected in step 4) isK̃m. If K̃m > Km, select Km users with the largestλmax

H,m,kλmaxV,m,k from the users selected in step 4); if

else, go to the next step.Step 6: Performing beamforming transmission for the

scheduled user. For user k in the selected users,its beamforming vector is

bm,k = (FMh)hk

⊗ (F∗

Mv

)vk

. (14)

The main purpose of Step 1 - Step 3 is to dividethe users into M̄hM̄v groups, so that users in differentgroups are orthogonal in the dominant diagonal element ofeither ΛH,m,k or ΛV,m,k and inter-user interference can bedecreased. It can be seen that only three scalars, i.e., hk, vk,

LI et al.: INTERFERENCE COORDINATION FOR 3-D BEAMFORMING-BASED HetNet 6891

and λmaxH,m,kλmax

V,m,k, are required at the macro BS for eachmacro user to perform this downlink transmission scheme.

B. Macro-Tier SINR Analysis

From (11), the SINR of macro user k can be written as (15),as shown at the top of the next page. The following theoremprovides an approximation of the SINR of macro user kin (15).

Theorem 1: Assume NsMs = μMhMv, where 0 <μ < ∞. When MhMv → ∞, we have

1MhMv

(SINRm,k − SINRDm,k)

p→ 0, where

SINRDm,k =

Pmβm,kαk,k/Km

∑

j∈M,j �=k

Pmβm,kαk,j/Km +Ns∑

i=1

Psiβsi,k + σ2k

,

(16)

αk,j = bHm,j

(RH,m,k ⊗ RT

V,m,k

)bm,j, (17)

andp→ represents convergence in probability. Under the 3D

beamforming downlink transmission scheme in Section III-A,(16) can be further written as SINRD

m,k = SINRD,3Dm,k , where

SINRD,3Dm,k is expressed as (18), as shown at the top of the next

page.Proof: See Appendix A.

Note that (16) holds for any beamforming vector bm,j thatdepends only on statistical CSI, and (18) only holds for the3D beamforming vector (14) in Section III-A.

From Theorem 1, it can be seen that we could use SINRDm,k

to approximate SINRm,k when MhMv � 1, and designtransmission and interference coordination strategies based onit. From (16), it can be seen that, to maximize SINRD

m,k

so that the achievable rate of macro user k is maximized,the numerator Pmβm,kαk,k should be as large as possible,and Pmβm,kαk,j , j ∈ M, j �= k in the denominator shouldbe as small as possible. Note that ΛV,m,k and ΛH,m,k

become diagonal matrices as Mh, Mv → ∞. Therefore,from (17), we have Pmβm,kαk,k ≤ Pmβm,kλmax

H,m,kλmaxV,m,k

when Mh, Mv → ∞, with the equality holds if and only ifbm,k = (FMh

)hk⊗ (F∗

Mv

)vk

. Moreover, Adhikary et al. [14]showed that only a few adjacent eigenvalues, i.e., from the i1-th to the i2-th eigenvalues, of the correlation matrix of a large-scale ULA are non-zero. Therefore, ΛH,m,k and ΛV,m,k alsohave the similar property. Based on this, when Mh, Mv → ∞,we have that Pmβm,kαk,j ≥ 0 with the equality holds if andonly if bm,j = (FMh

)h̄k⊗(F∗

Mv

)v̄k

, where h̄k and v̄k satisfy

that λ(h̄k)H,m,k = 0 or λ

(v̄k)V,m,k = 0.

From the above analysis we can get that, to maximizeSINRD

m,k, the beamforming vector of the scheduled user kshould be bm,k = (FMh

)hk⊗ (

F∗Mv

)vk

, which is the 3Dbeamforming vector in Algorithm 1. Meanwhile, the scheduleduser should satisfy λ

(hk)H,m,j = 0 or λ

(vk)V,m,j = 0 for j ∈

M, j �= k, and Step 1 - Step 3 in Algorithm 1 are trying tosatisfy this constraint. Moreover, from (18), we can find thatunder the 3D beamforming method in Algorithm 1, SINRD,3D

m,k

increases as λmaxH,m,kλmax

V,m,k increases. Therefore, to achievehigh achievable rate, we should schedule the user with the

largest λmaxH,m,kλmax

V,m,k, that is the user whose dominant eigen-direction in space captures most of its channel power. This isconsistent with the main idea of the Step 4 in Algorithm 1.Then, we can get that, to achieve high rate for macro users,it is reasonable to employ the 3D beamforming transmissionscheme in Algorithm 1 for macro users.

C. Small Cell SINR Analysis

For small cell users, from (13), the SINR of user k insmall cell i can be written as (19), as shown at the top ofthe next page. The following theorem gives an lower-boundof the expected SINR of user k in small cell i.

Theorem 2: The expectation of the SINR of user k insmall cell i in (19) can be lower-bounded as E [SINRsi,k] ≥SINRL

si,k, where

SINRLsi,k

=(Ms − Ksi)Psiβsi,k/Ksi

∑

j∈MPmβm,siαsi,j/Km +

Ns∑

p=1,p�=i

Pspβsp,si + σ2k

,

(20)

and

αsi,j = bHm,j

(RH,si ⊗ RT

V,si

)bm,j. (21)

Under the 3D beamforming downlink transmission schemein Section III-A, (20) can be further written as SINRL

si,k =SINRL,3D

si,k, where SINRL,3D

si,kis expressed as (22), as shown

at the top of the next page.Proof: See Appendix B.

Although (20) is a lower-bound of the expected SINR, it canreveal the relative level of SINR between different users. FromTheorem 2 it can be seen that the interference each user insmall cell i received from the macro BS, which can be revealedby∑

j∈M Pmβm,siαsi,j/Km, is affected by the transmissionscheme employed by macro tier. From (22), it can be seen thatunder the 3D beamforming downlink transmission scheme,users in small cell i receive strong interference from macro BSwhen small cell i is close to the macro BS, i.e., βm,si is large,and its dominant direction relative to macro BS is close to thescheduled macro users’ dominant direction, i.e., λ

(hj)H,si

λ(vj)V,si

islarge. Moreover, when the radius of each small cell remainsunchanged, the increase in small cell density decreases theachievable rate of each small cell user. Since it causes thedecrease in the distance between small cells, and thereforeincreases the interference they cause to each other.

IV. INTERFERENCE COORDINATION

In the above section, an approximation of macro users’SINR and a lower-bound of small cell users’ expected SINRare derived for the considered HetNet. From Theorem 1 and 2,it can be seen that the approximation (18) and lower-bound(22) for the users in the 3D beamforming based HetNetare determined only by statistical CSI and can be calculatedeasily. Based on them, we will propose some inter-tier interfer-ence coordination algorithms for the 3D beamforming basedHetNet.

6892 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 10, OCTOBER 2018

SINRm,k =Pm,k

∣∣fTm,kbm,k

∣∣2

∑j∈M,j �=k Pm,j

∣∣fT

m,kbm,j

∣∣2 +

∑Ns

i=1

∑l∈Si

Psi,l

∣∣gT

si,kbsi,l

∣∣2 + σ2

k

. (15)

SINRD,3Dm,k =

Pmβm,kλ(hk)H,m,kλ

(vk)V,m,k

/Km

∑j∈M,j �=k Pmβm,kλ

(hj)H,m,kλ

(vj)V,m,k

/Km +

∑Ns

i=1 Psiβsi,k + σ2k

. (18)

SINRsi,k =Psi,k

∣∣gT

si,kbsi,k

∣∣2

∑j∈M Pm,j

∣∣fT

m,kbm,j

∣∣2 +

∑Ns

p=1,p�=i

∑l∈Sp

Psp,l

∣∣gT

sp,kbsp,l

∣∣2 + σ2

k

. (19)

SINRL,3Dsi,k

=(Ms − Ksi)Psiβsi,k/Ksi

∑j∈M Pmβm,siλ

(hj)H,si

λ(vj)V,si

/Km +

∑Ns

p=1,p�=i Pspβsp,si + σ2k

. (22)

A. Interference From Small Cells

First, let us consider the interference caused by small cellsto macro users. From the approximation (18), it can be seenthat the interference power macro user k received from smallcells can be approximated as

I(s)m,k =

∑Ns

i=1I(si)m,k, (23)

where I(si)m,k = Psiβsi,k is the interference power received

from small cell i. The useful signal power received by macrouser k can be approximated as

Sm,k = Pmβm,kλ(hk)H,m,kλ

(vk)V,m,k

/Km. (24)

To guarantee the quality of service (QoS) of macro users,we would like that J

(si)m,k � I

(si)m,k

/Sm,k is below a certain

threshold, i.e.,

J(si)m,k =

Psiβsi,kKm

Pmβm,kλ(hk)H,m,kλ

(vk)V,m,k

� δ1, i = 1, · · · , Ns,

(25)

where δ1 is the interference threshold.There are some ways to deal with the small cells that do not

satisfy the constraint (25). One is to turn off these small cells,another one is to turn down their transmit power. However,turning off the small cells increases the burden of macro BS,and decreasing the transmit power of small cells reduces theQoS of small cell users. An applicable tradeoff between themacro cell burden and the QoS of small cell users is to set aminimum transmit power, i.e., Pmin

s , for small cells. Assumethat the transmit power of each small cell is initiated to theirmaximum transmit power Pmax

s . Then, turn down its transmitpower to the maximum value that satisfying (25), which is

Psi =Pmβm,kλ

(hk)H,m,kλ

(vk)V,m,k

δ1βsi,kKm, (26)

when the interference caused by small cell i is strong, i.e., theconstraint (25) cannot be satisfied. If the transmit power in (26)is below the minimum transmit power Pmin

s , then turn off thissmall cell. Therefore, based on this concept, we propose thefollowing Algorithm 2 to control the interference from smallcell i.

Algorithm 2 Controlling Interference From Small Cell

Step 1: Calculate J(si)m,k, k ∈ M using equation (25), and

find ki = arg maxk∈M

J(si)m,k .

Step 2: If J(si)m,k � δ1, end the scheme. If else, set the

transmit power of small cell i to the power in (26),i.e., Psi = Pmβm,kiλ

(hki)

H,m,kiλ

(vki)

V,m,ki

/δ1βsi,kiKm.

Step 3: If Psi � Pmins , end the scheme for small cell i.

If else, turn off small cell i.

It can be seen that, to perform this interference coordinationalgorithm, some additional information for each scheduledmacro user k is needed, i.e., βm,k and βsi,k where i =1, · · · , Ns. Note that, in practical systems, only a few smallcells which are located near a macro user can cause stronginterference to that macro user, and the interference form othersmall cells can be ignored. Therefore, each macro user k couldonly feedback βsi,k of the small cells which is close to it,and the macro BS can treat the βsi,k of the other small cellsas zero. In this way, the feedback overhead can be furtherreduced.

B. Interference From Macro Cell

Now, let us consider the interference coordination algorithmto control the interference caused by macro BS to small cellusers. To maintain the QoS of small cell users, we wouldlike that the SINR of each small cell user is above somecertain threshold. Note that interference coordination basedon instantaneous CSI requires large amount of data exchange.Therefore, we exploit the lower-bound (22) which containsonly statistical CSI.

Let us define

J(mj)si,k

�KsiPmβm,siλ

(hj)H,si

λ(vj)V,si

Km (Ms − Ksi) Psiβsi,k, k ∈ Si, j ∈ M.

(27)

From (22), it can be seen that SINRL,3Dsi,k

increases as J(mj)si,k

decreases. Therefore, to control the interference from macroBS so as to maintain some certain SINRL,3D

si,k, we would like

LI et al.: INTERFERENCE COORDINATION FOR 3-D BEAMFORMING-BASED HetNet 6893

that J(mj)si,k

is below a certain threshold, i.e.,

J(mj)si,k

� δ2, k ∈ Si, j ∈ M, (28)

where δ2 is the interference threshold.Note that similar to the correlation matrices of

the macro users, only a few adjacent eigenvaluesof ΛV,si and ΛH,si are non-zero, i.e., ΛH,si =diag

{0, · · · , 0, λ

(ai)H,si

, · · · , λ(bi)H,si︸ ︷︷ ︸

non−zero

, 0, · · · , 0}

and ΛV,si =

diag{0, · · · , 0, λ

(ci)V,si

, · · · , λ(di)V,si︸ ︷︷ ︸

non−zero

, 0, · · · , 0}

, when

Mh, Mv → ∞. Therefore, we can get that J(mj)si,k

→ 0when hj /∈ [ai, bi] or vj /∈ [ci, di]. This means that the signaltransmitted by the macro BS for macro users j causes nointerference to small cell i, when the strongest diagonalelement of ΛV,m,j is orthogonal to the non-zero diagonalelements of ΛV,si , or the strongest diagonal element ofΛH,m,j is orthogonal to the non-zero diagonal elementsof ΛH,si . In this case, no interference coordination is needed.In practice, there might exist some diagonal elements ofΛV,si and ΛH,si which are close to but not exactly equal tozero. We could set a threshold δ3, and treat λ

(j)V,si

or λ(j)H,si

as

zero if λ(j)V,si

/λmax

V,si< δ3 or λ

(j)H,si

/λmax

H,si< δ3, where λmax

V,si

and λmaxH,si

are the maximum diagonal elements of ΛV,si andΛH,si respectively.

Based on the above analysis, interference coordination isneeded for macro user j only when hj ∈ [ai, bi], vj ∈[ci, di], and J

(mj)si,k

> δ2. In this case, we use an algorithmwhich is referred to as OFFLOAD algorithm. If the smallcell nearest to macro user j can serve it with acceptableQoS, then offload macro user j to this small cell. Otherwise,remove macro user j from the scheduled user set M. Theremaining problem is how to determine whether the QoS isacceptable or not. For simplicity, we assume that the QoSof small cell is acceptable if S̄si,j/S̄m,j � δ4, where δ4 isa designed threshold to control the fraction of macro usersbeing offloaded, S̄si,j = Psi

∣∣gT

si,jbsi,j

∣∣2/Ksi is the signal

power received by user j when it is associated to small cell i,and S̄m,j = Pm

∣∣fT

m,jbm,j

∣∣2/Km is the signal power received

by user j when it is associated to macro BS. However, in thispaper, to reduce the feedback overhead, the macro BS has onlystatistical CSI of each user. Using Mullen’s inequality [27],we have

E[S̄si,j/S̄m,j

]� E

[S̄si,j

]/E

[S̄m,j

]. (29)

It can be obtained that

E[S̄m,j

]= Pmβm,jλ

(hj)H,m,jλ

(vj)V,m,j/Km. (30)

Substituting (30) and (78) into (29), we have that

E

[S̄si,j

S̄m,j

]� Δsi,j =

Km (Ms − Ksi)Psiβsi,j

KsiPmβm,jλ(hj)H,m,jλ

(vj)V,m,j

. (31)

Note that Δsi,j is determined only by statistical CSI. There-fore, we change the metric from S̄si,j/S̄m,j � δ4 to

Δsi,j =Km(Ms − Ksi) Psiβsi,j

KsiPmβm,jλ(hj)H,m,jλ

(vj)V,m,j

� δ4. (32)

If condition (32) is satisfied, then offload the macro user jto small cell i. Otherwise, remove macro user j from thescheduled user list. Then, we summarize the overall algorithmof controlling the interference that macro BS causes to theusers of small cell i in Algorithm 3.

Algorithm 3 Controlling Interference From Macro BS

Step 1: Treat λ(p)V,si

or λ(q)H,si

as zero if λ(p)V,si

/λmax

V,si<

δ3 or λ(q)H,si

/λmax

H,si< δ3.

Step 2: Find all the macro users j ∈ M satisfying theconstraint hj ∈ [ai, bi] and vj ∈ [ci, di]. The set ofthese macro users is denoted by Ui.

Step 3: For macro user j ∈ Ui, calculate J(mj)si,k

using(28), where small cell user k is the one with thesmallest large-scale fading coefficient in small celli, i.e., k = arg min

k′∈Si

βsi,k′ .

Step 4: If J(mj)si,k

� δ2, go to Step 7). Otherwise, go to thenext step.

Step 5: Find the nearest small cell to macro user j, anddenote it as small cell r, i.e., r = arg max

i=1,··· ,Ns

βsi,j .

Calculate Δsr ,j using (32).Step 6: If Δsr ,j � δ4, then offload macro user j to small

cell r. Otherwise, remove macro user j from M.Step 7: Turn to the next user in Ui and repeat Step 3).

It can be seen that, to perform this interference coordinationalgorithm, some additional statistical information for smallcell i is needed, i.e., Ksi , min

k′∈Si

βsi,k′ , the nonzero diagonal

elements of λV,si and λH,si , and their corresponding indexes.Note that, compared with macro users, the environment aroundsmall cells changes much more slowly. The statistical CSI ofsmall cells, i.e., the nonzero diagonal elements of λV,si andλH,si , and their corresponding indexes, may remain the sameover a long time. Therefore, the feedback frequency for smallcells can be even lower than the frequency for macro users,thus further reduce the feedback overhead.

C. Achievable Rate

In this section, we analyze the achievable data rate of macroand small cell users in the 3D beamforming based HetNet withthe proposed interference coordination algorithms. Assumethat the number of scheduled users and active small cells afterthe interference coordination are K̄m and N̄s respectively,the number of active users in small cell i is K̄si .

First, let’s consider the macro users. The ergodicrate of macro user k can be rewritten as Rm,k =E [log2 (1 + SINRm,k)]. The following theorem provides anapproximation of the ergodic rate of macro user k.

Theorem 3: The ergodic rate of macro user k in the consid-ered 3D beamforming based HetNet can be approximated as

Rm,k ≈ R̄m,k

=∑

j∈Mk

eτk,jE1 (τk,j)

ln 2∏

l∈Mk,l �=j

(1 − λ

(hl)H,m,kλ

(vl)V,m,k

λ(hj )H,m,kλ

(vj)V,m,k

)λ

(hk)H,m,kλ

(vk)V,m,k

λ(hj)H,m,kλ

(vj)V,m,k

,

(33)

6894 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 10, OCTOBER 2018

where, E1 (x) =∫∞1

e−xu

u du is the exponential integral,

Mk ={

j∣∣∣j ∈ M, λ

(hj)H,m,kλ

(vj)V,m,k > 0

}, and

τk,j =K̄m

(σ2

k +∑N̄s

i=1 Psiβsi,k

)

Pmβm,kλ(hj)H,m,kλ

(vj)V,m,k

. (34)

Proof: See Appendix C.Note that, in practice, the λ

(hj)H,m,kλ

(vj)V,m,k for some user j

may be very close to but not exactly equal to zero. Thatmight cause the calculation of R̄m,k inaccurate, or evenoverflow. To overcome this, we set a threshold δ5 forλ

(hj)H,m,kλ

(vj)V,m,k

/λ

(hk)H,m,kλ

(vk)V,m,k, and treat λ

(hj)H,m,kλ

(vj)V,m,k as zero

if λ(hj)H,m,kλ

(vj)V,m,k

/λ

(hk)H,m,kλ

(vk)V,m,k < δ5. Therefore, in practice,

we modify the Mk in Theorem 3 to

Mk ={j∣∣∣j ∈ M, λ

(hj)H,m,kλ

(vj)V,m,k � δ5λ

(hk)H,m,kλ

(vk)V,m,k

}.

(35)

Then, let us turn to the small cell users. The ergodic rateof user k served by small cell i can be written as Rsi,k =E [log2 (1 + SINRsi,k)]. The following theorem provides anapproximation of the ergodic rate Rsi,k.

Theorem 4: The ergodic rate of user k served by smallcell i in the considered 3D beamforming based HetNet canbe approximated as Rsi,k ≈ R̄si,k, where R̄si,k is expressedas (36), as shown at the bottom of this page.

Proof: Applying Jensen’s inequality [28], i.e.,E {log (a)} � log (E {a}) where a is positive,to Rsi,k = E [log2 (1 + SINRsi,k)], we get

Rsi,k � log2 (1 + E [SINRsi,k]) . (37)

From Theorem 2, we obtain that E [SINRsi,k] can belower-bounded as (38), as shown at the bottom of thispage. Then, substituting (38) into (37), we can get theapproximation (36).

From (36), it can be observed that the achievable rate ofuser k in small cell i increases as βsi,k/βm,si increases. Thissuggests that, for the considered HetNet, small cells shouldbe placed at the cell edge, and the users located at the celledge should be associated to the nearby small cell, to achievehigher throughput.

V. NUMERICAL RESULTS

In this section, we present numerical results to validate andinvestigate our proposed interference coordination algorithms.We define the SNR as SNR = Pm/σ2. For all results,we assume that the noise level of macro and small cell usersare the same, i.e., σ2

k = σ2, the distance between macro

TABLE I

SIMULATION PARAMETERS

BS and each macro user/small cell is from 100m to 1500m,the macro users and small cells are distributed uniformly inmacro cell, the distance between each small cell and its ownuser is from 3m to 20m, the small cell users are uniformlydistributed in their serving small cells, the horizontal andvertical angular of arrival of macro users and small cellsare distributed uniformly in (−90◦, 90◦), the horizontal andvertical angular spread of each macro user are distributeduniformly in (5◦, 10◦), the horizontal and vertical angularspread of each small cell are distributed uniformly in (5◦, 15◦).The rest of the parameters are listed in Table I. The perfor-mance of two interference coordination algorithms referredto as “ON/OFF” and “OFFLOAD” algorithms similar tothe algorithms proposed in [14] are also presented. In the“ON/OFF” algorithm, small cells are shut down whenever thecross-tier interference they receive from macro BS or causeto macro users is strong, i.e., the constraint (25) or (28) isnot satisfied. In the “OFFLOAD” algorithm, macro users thatreceive or cause strong interference, i.e., the macro users thatcan not satisfy the constraint (25) or (28), are offloaded totheir neighboring small cells if the constraint (32) is satisfied.Otherwise, shut down the correspond small cell.

Fig. 2 shows the tradeoff between the average sum rate ofmacro cell and small cells for different interference coordi-nation algorithms. The tradeoff curves are obtained by lettingthe parameter Km equals to 2k, and varying k from 0 to 6,i.e., the leftmost marker in each plot corresponds to Km = 1and the rightmost to Km = 64. In this figure, SNR = 20 dB.

R̄si,k = log2

⎛

⎝1 +Psi

(Ms − K̄si

)βsi,k

/K̄si

∑j∈M Pmβm,siλ

(hj)H,si

λ(vj)V,si

/K̄m +

∑N̄s

p=1,p�=i Pspβsp,si + σ2k

⎞

⎠ . (36)

E [SINRsi,k] �Psi

(Ms − K̄si

)βsi,k

/K̄si

∑j∈M Pmβm,siλ

(hj)H,si

λ(vj)V,si

/K̄m +

∑Ns

p=1,p�=i Pspβsp,si + σ2k

. (38)

LI et al.: INTERFERENCE COORDINATION FOR 3-D BEAMFORMING-BASED HetNet 6895

Fig. 2. Throughput tradeoff cures for different interference coordinationstrategies, SNR = 20 dB.

In both the proposed algorithm and OFFLOAD algorithm,we consider the offloaded macro users as small cell users,and count their throughput towards the small cell throughput.From Fig. 2, it can be seen that the increase of Km resultsin the increase of macro cell throughput, since more macrousers are scheduled, and part of the small cells are turnedoff due to the increasing possibility of receiving interferencefrom small cells for macro users. However, this results in thereduction of the total throughput of small cells, since moresmall cells are turned down of turned off. All these threeinterference coordination algorithms increase the macro cellthroughput at the expense of small cell throughput. We cansee that the ON/OFF algorithm is the best in increasing macrocell throughput but the worst in achieving high small cellthroughput due to shutting down large numbers of small cells.On the contrary, the OFFLOAD algorithm is the worst inincreasing macro cell throughput but the best in achievinghigh small cell throughput. In contrast, the proposed algorithmprovides a good tradeoff between the throughput of macro andsmall cells.

Fig. 3 and Fig. 4 compare the average rate of macro andsmall cell users, respectively. In both figures, the curves arealso obtained by letting the parameter Km = 2k, and varyingk from 0 to 6, and SNR = 20 dB. From these figures, it canbe seen that all three coordination algorithms increase theaverage rate of macro users. The OFFLOAD algorithm hasthe best performance in term of increasing the average rate ofmacro users, since a number of macro users are offloaded tosmall cells, which decreases the interference between macrousers. However, the OFFLOAD algorithm performs worst inthe average rate of small cell users. The performance of theON/OFF and the proposed algorithms in the average rate ofmacro and small cell users are close to each other, whilethe ON/OFF algorithm is slightly superior to the proposedalgorithm. However, the ON/OFF algorithm turns off largenumbers of small cells, which increases the burden of themacro cell. The proposed algorithm can provide relativelygood performance for both macro and small cell users, whileachieve good balance between the traffic of macro and smallcells.

Fig. 3. Average rate of macro users under different coordination algorithms,SNR = 20 dB.

Fig. 4. Average rate of small cell users under different coordinationalgorithms, SNR = 20 dB.

Fig. 5 and Fig. 6 show the CDFs of macro and small celluser rate for different coordination algorithms, when SNR =20 dB and Km = 32. From Fig. 5, it can be seen thatthe uncoordinated algorithm performs the worst, and all threecoordination algorithms improve the cell edge performance alot, as shown in the zoomed area of Fig. 5. The OFFLOADalgorithm is the best in terms of achieving good macro userrate. This is consistent with the result in Fig. 3, since it reducesnot only the inter-tier interference but also the inter-user inter-ference received by macro users. It can also be seen that theproposed algorithm is slightly inferior to the OFFLOAD andON/OFF algorithms. From Fig. 6, it can be seen that ON/OFFalgorithm performs the best in terms of achieving good smallcell user rate, which is consistent with the result in Fig. 4, sincethe small cells which receive strong interference from macroBS are turned off and the interference between different smallcells are also reduced. The OFFLOAD algorithm performsthe worst in terms of the cell edge performance of smallcells, as shown in the zoomed area on the left-hand side ofFig. 6. The cell edge performance of the proposed algorithmis almost the same as the ON/OFF algorithm. For the highsmall cell user rate region, the performance of the proposedalgorithm is between the ON/OFF and OFFLOAD algorithms,and is slightly inferior to the ON/OFF algorithm. It can be

6896 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 10, OCTOBER 2018

Fig. 5. CDF of macro user rate for different interference coordinationalgorithms, SNR = 20 dB, Km = 32.

Fig. 6. CDF of small cell user rate for different interference coordinationalgorithms, SNR = 20 dB, Km = 32.

seen that the proposed algorithm can achieve a good trade-off between the performance of macro cell and small cells,and a good trade-off between the sum rate and cell edgeperformance.

Fig. 7 shows the ergodic sum rate of macro users in the3D beamforming based HetNet obtained by Monte Carlosimulation and the approximation (33) under Km = 8, 16,and 32 for one realization of the random user pool. In thisfigure, the threshold δ5 in (35) is set to 0.5 × 10−2. Fig. 8shows the corresponding ergodic sum rate of small cell usersobtained by Monte Carlo simulations and the approximation(36). From these figures, it can be seen that the ergodic sumrate approximation of macro users achieved by (33) is almostthe same as the Monte Carlo result, and the ergodic sumrate approximation of small cell users achieved by (36) is inclose agreement with the Monte Carlo simulations. It can alsobe seen that the ergodic sum rate of macro users increasesas Km increases, while the ergodic sum rate of small cellusers decreases as Km increases. This is because more andmore small cells need to decrease their transmit power or tobe turned off as Km increases. Moreover, both the ergodicsum rate of macro users and small cell users converge tosaturation values as the SNR increases, since the systembecomes interference-limited in the high SNR region, and both

Fig. 7. Comparison of Monte Carlo simulation result and approximation ofthe ergodic sum rate of macro users.

Fig. 8. Comparison of Monte Carlo simulation result and approximation ofthe ergodic sum rate of small cell users.

the desired signal and inter-user interference increases as theSNR increases.

VI. CONCLUSION

In this paper, we considered the downlink transmission,especially the inter-tier interference coordination, for a HetNetwith 2D large-scale antenna array at macro BS and con-ventional MIMO at each small cell, under the assumptionof only statistical CSI at macro BS. First, we derived anapproximation of the SINR of macro users and a lower-bound of the expected SINR of small cell users. Then, with3D beamforming transmission applied in macro tier, simplemetrics to evaluate the inter-tier interference were derivedbased on the derived SINR approximation and lower-bound,and interference coordination algorithms exploiting only partof the statistical CSI were proposed. Moreover, tractableergodic rate approximations of both macro and small cell usersin the 3D beamforming based HetNet were derived. Simula-tion results showed that the proposed interference coordina-tion algorithms work well and can achievable good tradeoffbetween the performance of macro and small cells, the derivedergodic rate approximations match well with the Monte Carloresults.

LI et al.: INTERFERENCE COORDINATION FOR 3-D BEAMFORMING-BASED HetNet 6897

APPENDIX APROOF OF Theorem 1

From (15) and (16), it can be obtained that

1MhMv

(SINRm,k − SINRD

m,k

)=

a1 + a2 + a3

MhMvb1b2, (39)

where

a1 = Pm,kσ2k

(∣∣fTm,kbm,k

∣∣2 − βm,kαk,k

), (40)

a2 = Pm,k

∣∣fT

m,kbm,k

∣∣2∑

j∈M,j �=kPm,jβm,kαk,j

−Pm,kβm,kαk,k

∑

j∈M,j �=kPm,j

∣∣fT

m,kbm,j

∣∣2, (41)

a3 = Pm,k

∣∣fT

m,kbm,k

∣∣2∑Ns

i=1Psiβsi,k

−Pm,kβm,kαk,k

∑Ns

i=1

∑

l∈Si

Psi,l

∣∣gT

si,kbsi,l

∣∣2,

(42)

b1 =∑

j∈M,j �=kPm,j

∣∣fT

m,kbm,j

∣∣2

+∑Ns

i=1

∑

l∈Si

Psi,l

∣∣gT

si,kbsi,l

∣∣2 + σ2

k, (43)

b2 =∑

j∈M,j �=k

Pm,jβm,kαk,j +Ns∑

i=1

Psiβsi,k + σ2k, (44)

First, let us consider a1. Substituting (9) into (40), and defining

Ak,j =[R1/2

H,m,k ⊗ (R1/2

V,m,k

)T ]bm,jbH

m,j

×[(

R1/2H,m,k

)H ⊗ (R1/2

V,m,k

)∗], (45)

it can be obtained that

a1

MhMv= Pm,kσ2

kβm,k

(hT

w,m,kAk,kh∗w,m,k

MhMv− αk,k

MhMv

).

(46)

For the simplicity of representation, let us define

X =1

MhMvhT

w,m,kAk,kh∗w,m,k − αk,k

MhMv. (47)

Note that hw,m,k is a random vector with zero mean andunit variance i.i.d. complex Gaussian elements, and αk,k =tr (Ak,k). From [29, Lemma 3], we have that

E[|X |2] � 2C2v4

(MhMv)2 tr

(Ak,kAH

k,k

), (48)

where, C2 is a constant, E[∣∣h(i)

w,m,k/√

MhMv

∣∣4] � v4, and

h(i)w,m,k is the i-th element of hw,m,k. Note that 2

∣∣h(i)w,m,k

∣∣2 ∼χ2

2. Therefore,

E

[∣∣∣∣

h(i)w,m,k√MhMv

∣∣∣∣

4]

=E[(

χ22

)2]

4(MhMv)4 =

D(χ2

2

)+ E

[χ2

2

]2

4(MhMv)4

=2

(MhMv)4 , (49)

where D (x) represents the variance of random variable x.Then, we can set v4 as

v4 = 2(MhMv)−4

. (50)

From (45), it can be obtained that

tr(Ak,kAH

k,k

)= α2

k,k � (MhMv)2. (51)

Substituting (50) and (51) into (48), we can get that

E[|X |2] � 4C2(MhMv)

−4. (52)

Furthermore, from [30, Th. 3.5], we have that, for any ε > 0,

P (|X | � ε) � ε−2E[|X |2], (53)

where P (·) represents the probability. Substituting (52) into(53), we can get that

P (|X | � ε) � 4C2(MhMv)−4

ε−2. (54)

Note that 4C2(MhMv)−4

ε−2 → 0 as MhMv → ∞. Then,from (54), we have that

limMhMv→∞

P (|X | < ε) = limMhMv→∞

1 − P (|X | � ε) = 1,

(55)

for any ε > 0. Therefore, we can get that [31], Xp→ 0 and

a1/MhMvp→ 0, (56)

when MhMv → ∞.Then, let us consider a2. From (41), it can be rewritten

as (57), as shown at the top of next page. Using the similarmethod as that used in the above paragraph, it can be obtainedthat

(βm,kαk,j −

∣∣fT

m,kbm,j

∣∣2)/

MhMvp→ 0, (58)

when MhMv → ∞. Combining (58) and (57), and notingX

p→ 0 when MhMv → ∞, we can get that

a2/MhMvp→ 0, (59)

when MhMv → ∞.Next, let us consider a3. From (42), a3 can be rewritten as

(60), as shown at the top of next page. It can be obtained that∑Ns

i=1

∑

l∈Si

Psi,l

∣∣gT

si,kbsi,l

∣∣2

=∑Ns

i=1

Psi

Ksi

βsi,khTw,si,kBsiB

Hsih∗

w,si,k. (61)

For simplicity, let us define

B̄k = diag{

Ps1

Ks1

βs1,kBs1BHs1

,Ps2

Ks2

βs2,kBs2BHs2

,

· · · ,PsNs

KsNs

βsNs ,kBsNsBH

sNs

}, (62)

and

hw,s,k =[hT

w,s1,k,hTw,s2,k, · · · ,hT

w,sNs ,k

]T. (63)

Therefore, (61) can be further written as∑Ns

i=1

∑

l∈Si

Psi,l

∣∣gTsi,kbsi,l

∣∣2 = hTw,s,kB̄kh∗

w,s,k. (64)

Note that the non-zero eigenvalues of BsiBHsi

are all 1. Then,the spectral radius of B̄k is∥∥B̄k

∥∥ = max

1�i�MsNs

λi

(B̄k

)= max

1�i�Ns

Psi

Ksi

βsi,k < ∞,

(65)

6898 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 17, NO. 10, OCTOBER 2018

a2 = Pm,k

(∣∣fTm,kbm,k

∣∣2 − βm,kαk,k

)∑

j∈M,j �=kPm,jβm,kαk,j

+ Pm,kβm,kαk,k

∑

j∈M,j �=kPm,j

(βm,kαk,j −

∣∣fT

m,kbm,j

∣∣2). (57)

a3 = Pm,k

(∣∣fT

m,kbm,k

∣∣2 − βm,kαk,k

) Ns∑

i=1

Psiβsi,k + Pm,kβm,kαk,k

(Ns∑

i=1

Psiβsi,k −Ns∑

i=1

∑

l∈Si

Psi,l

∣∣gT

si,kbsi,l

∣∣2)

. (60)

where λi

(B̄k

)is the i-th eigenvalue of B̄k. Note that

hw,s,k/√

MsNs is a random vector of i.i.d. entries with zeromean, variance 1/MsNs. Denote the i-th element of hw,s,k

as h(i)w,s,k. It can be obtained that

E

[∣∣∣∣

h(i)w,s,k√MsNs

∣∣∣∣

8]

=E[(

χ22

)4]

24(MsNs)4 = O

(1

(MsNs)4

). (66)

Furthermore, hw,s,k is independent of B̄k. Therefore, from[29, Lemma 4], we can get that

1MsNs

hTw,s,kB̄kh∗

w,s,k − 1MsNs

tr(B̄k

)MsNs→∞→ 0 (67)

almost surely. From (62) and (12), it can be obtained that

tr(B̄k

)=∑Ns

i=1

Psi

Ksi

βsi,ktr(BsiB

Hsi

)

=∑Ns

i=1

Psi

Ksi

βsi,k

∑

l∈Si

bHsi,lbsi,l

=∑Ns

i=1Psiβsi,k. (68)

Combining (67), (68), and MsNs = μMhMv, we have that

1MhMv

∑Ns

i=1

∑

l∈Si

Psi

∣∣gT

si,kbsi,l

∣∣2

− 1MhMv

∑Ns

i=1Psiβsi,k

MhMv→∞→ 0 (69)

almost surely. Combining (60) and (69), and noting Xp→ 0

when MhMv → ∞, we can get that

a3/MhMvp→ 0, (70)

when MhMv → ∞.Finally, since b1 > 0 and b2 > 0, combining (39), (56), (59),

and (70), we have that 1MhMv

(SINRm,k − SINRD

m,k

) p→ 0,when MhMv → ∞. Substituting (14) into (16), we have (18).

APPENDIX BPROOF OF Theorem 2

Note that the numerator and denominator of (19)are independent. Then, applying Mullen’s inequality [27],i.e., E [X/Y ] ≥ E [X ]/E [Y ] if X and Y are independentrandom variables, we have (71), as shown at the top of nextpage.

First, let us consider the numerator of (71). Note that eachsmall cell has perfect CSI of its own users and applies column-normalized ZF beamforming. Then, from (12), we can get that

∣∣gT

si,kbsi,k

∣∣2 = ‖qsi,k‖−2 =

(qH

si,kqsi,k

)−1. (72)

It can be seen that qHsi,k

qsi,k is the (k, k)-th element

of QHsiQsi . Note that Qsi = GH

si

(GsiG

Hsi

)−1, therefore

qHsi,k

qsi,k equals to the (k, k)-th element of(GsiG

Hsi

)−1.

Rewrite Gsi as

Gsi = B̃1/2si

Hw,si, (73)

where,

B̃si = diag{βsi,l1 , βsi,l2 , · · · , βsi,lKsi

}, (74)

Hw,si =[hw,si,l1 ,hw,si,l2 , · · · ,hw,si,lKsi

]T, (75)

and{l1, l2, · · · , lKsi

} ∈ Si. It can be seen that Hw,si is arandom matrix with zero mean and unit variance i.i.d. complexGaussian elements. Therefore, GsiG

Hsi

is a complex Wishartmatrix [32] with Ms degrees of freedom and covariance matrixB̃si , denoted as GsiG

Hsi

∼ CWKs

(Ms, B̃si

). It has been

proved in [33] that E[A−1

]= (n − m)−1Σ−1 where A ∼

CWm (n,Σ). Therefore, we can get that

E[(

GsiGHsi

)−1]

=(Ms − Ksi

)−1B̃−1

si, (76)

and then

E[qH

si,kqsi,k

]=(Ms − Ksi

)−1β−1

si,k. (77)

Applying the Mullens inequality [27] and (77) to (72), we getthe following inequality

E[∣∣gT

si,kbsi,k

∣∣2]

� 1E[qH

si,kqsi,k

]=(Ms − Ksi)βsi,k. (78)

Then, let us consider the denominator of (71). The first termof the denominator can be obtained as

∑

j∈MPm,jE

[∣∣fT

m,kbm,j

∣∣2]

=Pm

Kmβm,si

∑

j∈Mtr (Ak,j)

=Pm

Kmβm,si

∑

j∈M αsi,j .

(79)

The second term of the denominator can be rewritten as∑Ns

p=1,p�=i

∑

l∈Sp

Psp,lE[∣∣gT

sp,kbsp,l

∣∣2]

=∑Ns

p=1,p�=i

Psp

Ksp

βsp,si tr(BspB

Hsp

). (80)

Note that tr(BspB

Hsp

)=∑

l∈SpbH

sp,lbsp,l = Ksp . Therefore,(80) can be further written as∑Ns

p=1,p�=i

∑

l∈Sp

Psp,lE[∣∣gT

sp,kbsp,l

∣∣2]

=∑Ns

p=1,p�=iPspβsp,si . (81)

LI et al.: INTERFERENCE COORDINATION FOR 3-D BEAMFORMING-BASED HetNet 6899

E [SINRsi,k] ≥ Psi,kE[∣∣gT

si,kbsi,k

∣∣2]

∑j∈M Pm,jE

[∣∣fTm,kbm,j

∣∣2]+

∑Ns

p=1,p�=i

∑l∈Sp

Psp,lE[∣∣gT

sp,kbsp,l

∣∣2]+ σ2

k

, (71)

Finally, substituting (78), (79), and (81) into (71), we canget E [SINRsi,k] ≥ SINRL

si,k. Moreover, substituting (14) into(20), we can have (22).

APPENDIX CPROOF OF Theorem 3

Using (15), Rm,k can be written as

Rm,k = Rm,k,1 − Rm,k,2, (82)

where

Rm,k,1 = E

[

log2

(1

MhMv

∑

j∈MPm,j

∣∣fTm,kbm,j

∣∣2

+1

MhMv

N̄s∑

i=1

∑

l∈Si

Psi,l

∣∣gT

si,kbsi,l

∣∣2 +

σ2k

MhMv

)]

, (83)

and

Rm,k,2 = E

[

log2

(1

MhMv

∑

j∈M,j �=k

Pm,j

∣∣fT

m,kbm,j

∣∣2

+1

MhMv

N̄s∑

i=1

∑

l∈Si

Psi,l

∣∣gTsi,kbsi,l

∣∣2 +σ2

k

MhMv

)]

. (84)

From (69), it can be seen that 1MhMv

N̄s∑

i=1

∑

l∈Si

Psi,l

∣∣gT

si,kbsi,l

∣∣2

converges to 1MhMv

∑N̄s

i=1 Psiβsi,k when MhMv → ∞.

Therefore, we use 1MhMv

∑N̄s

i=1 Psiβsi,k to approximate

1MhMv

∑N̄s

i=1

∑l∈Si

Psi,l

∣∣∣gT

si,kbsi,l

∣∣∣2

when MhMv � 1.Then, Rm,k,1 can be approximated as

Rm,k,1 ≈ E

[log2

(1

MhMv

∑

j∈M Pm,j

∣∣fT

m,kbm,j

∣∣2

+1

MhMv

∑N̄s

i=1Psiβsi,k +

σ2k

MhMv

)]. (85)

Moreover, noticing that the eigenvectors of RV,m,k andRH,m,k can be approximated by unitary DFT matricesFMv and FMh

when Mh � 1 and Mv � 1,we use FMvΛ̃V,m,kF

†Mv

and FMhΛ̃H,m,kF

†Mh

to approximateRV,m,k and RH,m,k respectively, where Λ̃V,m,k and Λ̃H,m,k

are diagonal matrices generated using the diagonal elementsof ΛV,m,k and ΛH,m,k respectively. Therefore,

∣∣fTm,kbm,j

∣∣2

can be approximated as

∣∣fT

m,kbm,j

∣∣2 ≈ βm,kλ

(hj)H,m,kλ

(vj)V,m,k

∣∣h((hj−1)Mv+vj)

w,m,k

∣∣2. (86)

For simplicity of representation, let us define hj =h

((hj−1)Mv+vj)w,m,k , j ∈ M. Then, substituting (86) into (85),

Rm,k,1 can be further approximated as

Rm,k,1 ≈ log2

(σ2

k

MhMv+

1MhMv

∑N̄s

i=1Psiβsi,k

)

+ E

[log2

(1 +

∑

j∈Mk

1τk,j

|hj |2)]

. (87)

Note that hj, j ∈ M are independent complex Gaussian vari-ables with zero mean and unit variance. From [34, Lemma 3],it can be obtained that

E

[log2

(1 +

∑

j∈Mk

1τk,j

|hj|2)]

=∑

j∈Mk

eτk,jE1 (τk,j)

ln 2∏

l∈Mk,l �=j

(1 − λ

(hl)H,m,kλ

(vl)V,m,k

λ(hj )H,m,kλ

(vj )V,m,k

) . (88)

Substituting (88) into (87), we can obtain that

Rm,k,1 ≈ log2

(σ2

k +∑N̄s

i=1 Psiβsi,k

MhMv

)

+∑

j∈Mk

eτk,jE1 (τk,j)

ln 2∏

l∈Mk,l �=j

(1 − λ

(hl)H,m,kλ

(vl)V,m,k

λ(hj )H,m,kλ

(vj )V,m,k

) . (89)

Using the similar method, we can obtain that

Rm,k,2 ≈ log2

(σ2

k +∑N̄s

i=1 Psiβsi,k

MhMv

)

+∑

j∈Mk,j �=k

eτk,j E1 (τk,j)

ln 2∏

l∈Mk,l �=k,j

(1 − λ

(hl)H,m,kλ

(vl)V,m,k

λ(hj )H,m,kλ

(vj )V,m,k

) . (90)

Substituting (89) and (90) into (82), and after some manipu-lation, we have that Rm,k ≈ R̄m,k.

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Xiao Li (S’06–M’10) received the Ph.D. degreein communication and information systems fromSoutheast University, Nanjing, China, in 2010.

She joined the School of Information Science andEngineering, Southeast University, after graduation,where she has been an Associate Professor of infor-mation systems and communications since 2014.From 2013 to 2014, she was a Post-Doctoral Fellowat The University of Texas at Austin, Austin, TX,USA. Her current research interests include massiveMIMO, 3-D beamforming, and multiuser MIMO.

Dr. Li was a recipient of the 2013 National Excellent Doctoral Dissertationof China for her Ph.D. dissertation.

Chaosong Li received the B.S. degree in commu-nications engineering from the Harbin Institute ofTechnology, Harbin, China, in 2015, and the M.S.degree from Southeast University, Nanjing, China,in 2018. He joined Huawei Technologies Co., Ltd,Chengdu, where he is currently an Engineer with theBaseband Development Department.

Shi Jin (S’06–M’07–SM’17) received the B.S.degree in communications engineering from theGuilin University of Electronic Technology, Guilin,China, in 1996, the M.S. degree from the Nan-jing University of Posts and Telecommunications,Nanjing, China, in 2003, and the Ph.D. degree ininformation and communications engineering fromSoutheast University, Nanjing, in 2007.

From 2007 to 2009, he was a Research Fellowwith the Adastral Park Research Campus, UniversityCollege London, London, U.K. He is currently a

Faculty Member with the National Mobile Communications Research Labo-ratory, Southeast University. His research interests include space time wirelesscommunications, random matrix theory, and information theory.

Dr. Jin and his co-authors received the 2011 IEEE Communications SocietyStephen O. Rice Prize Paper Award in the field of communication theory andthe 2010 Young Author Best Paper Award from the IEEE Signal ProcessingSociety. He serves as an Associate Editor for the IEEE TRANSACTIONS

ON WIRELESS COMMUNICATIONS, the IEEE COMMUNICATIONS LETTERS,and IET Communications.

Xiqi Gao (S’92–A’96–M’02–SM’07–F’15) receivedthe Ph.D. degree in electrical engineering fromSoutheast University, Nanjing, China, in 1997.

In 1992, he joined the Department of Radio Engi-neering, Southeast University. From 1999 to 2000,he was a Visiting Scholar with the MassachusettsInstitute of Technology, Cambridge, MA, USA, andalso with Boston University, Boston, MA, USA.From 2007 to 2008, he visited the Darmstadt Uni-versity of Technology, Darmstadt, Germany, as aHumboldt Scholar. Since 2001, he has been a Pro-

fessor of information systems and communications with Southeast University.His current research interests include broadband multicarrier communications,multiple-input multiple-output wireless communications, channel estimationand turbo equalization, and multirate signal processing for wireless commu-nications.

Dr. Gao was a recipient of the National Technological Invention Award ofChina in 2011 and the 2011 IEEE Communications Society Stephen O. RicePrize Paper Award in the field of communications theory. He receivedthe Science and Technology Award from the State Education Ministry ofChina in 1998, 2006, and 2009. He served as an Editor for the IEEETRANSACTIONS ON WIRELESS COMMUNICATIONS from 2007 to 2012 andthe IEEE TRANSACTIONS ON COMMUNICATIONS from 2015 to 2017. From2009 to 2013, he served as an Associate Editor for the IEEE TRANSACTIONS

ON SIGNAL PROCESSING.