Cooperative User Scheduling in Massive MIMO Systems2018/04/24 · INDEX TERMS Massive MIMO,...

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Cooperative User Scheduling in MassiveMIMO SystemsXIANG CHEN1,2, FENG-KUI GONG1, (MEMBER, IEEE), HANG ZHANG2, AND GUO LI1,(Student Member, IEEE)1State Key Lab. of ISN, Xidian University, Shaanxi Province (710071), China2Science and Technology on Communication Networks Laboratory, Shijiazhuang (050081), China

Corresponding author: Feng-Kui Gong (e-mail: [email protected]).

The work was supported in part by the National High-tech R&D Program of China (2014AA01A704), Joint Fund of Ministry of Edulationof China (6141A02022338) and the Opening Project of Science and Technology on Communication Networks Laboratory(KX162600027).

ABSTRACT Taking advantage of distributed computation and the characteristic of Device-to-Device(D2D) communication among users, a cooperative user scheduling (CUS) scheme is firstly proposed.Through reducing the number of users which feed back their channel state information (CSI) to the basestation (BS), the problem of huge feedback in massive multiple-input multiple-output (MIMO) systems canbe, to a large extent, solved. The property of D2D allows users to exchange information, so the users cancalculate their correlation coefficients with the selected best user locally. Then, these coefficients are used tofilter out these strongly correlated users without greatly affecting the system sum rate. Specially, to decreasethe computational complexity and reduce the number of feedback users, the cell users are divided intoseveral groups randomly. The users in the same group can determine by themselves which need to feed backtheir CSIs, so the global optimal selection problem is decomposed into several local optimization problems.Through theoretical analysis on the CUS scheme, we obtain the lower bound expression of system sum rate.Simulation results indicate that, the uplink feedback resources can be greatly saved with the CUS scheme,whilst the proposed scheme’s influence on the system sum rate is negligible.

INDEX TERMS Massive MIMO, cooperative user scheduling, D2D, user filtering, huge feedback.

I. INTRODUCTION

RECENTLY, the next-generation wireless communica-tion becomes the focus of attention, worldwide re-

searchers have achieved numerous achievements on the fifth-generation (5G) wireless communication. Since proposed,massive antenna array is regraded as one of the key tech-niques in the 5G communication. In massive multiple-inputmultiple-output (MIMO) systems, the base station (BS) isequipped with dozens or even hundreds of antennas to servemultiple users. Through deeply developing the wireless re-sources in the spatial dimension, the large-scale antennaarray has the potential to significantly improve the spectralefficiency and power efficiency. However, the technique alsofaces many challenges, such as pilot contamination, the chan-nel modelling and the huge feedback in user scheduling [1],[2].

In many user scheduling algorithms [3], [4], the BS needsthe channel state information (CSI) of all the users in cellto select the optimal user set to maximize the system sum

rate. For example, in the traditional MIMO systems, the au-thors in [5] propose a low-complexity semi-orthogonal userselection (SUS) algorithm based on the exact CSI. The SUSalgorithm iteratively selects the user with the greater channelnorm and the correlation coefficient. The SUS algorithm caneffectively decrease the computational complexity, while itneeds the exact CSIs of all the users. The limitation makes thealgorithm difficult to be applied to massive MIMO systems.Since, in the frequency division duplexing (FDD) mode, thefeedback load is positively correlated with the numbers of theBS antennas and users, the huge amount of feedback wouldoccupy a great part of spectral resources, which, in turn,decreases the system spectral efficiency when the numbersof the BS antennas and users increase sharply. In additionto the huge feedback overhead, it is also extremely difficultfor the BS to select the optimal user set from a large numberof users. Nowadays, many researchers have come up with agreat variety of solutions to ease the problems of high com-putational complexity and huge feedback [3], [6]–[10]. The

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X. Chen et al.: Cooperative User Scheduling in Massive MIMO Systems

authors in [3] propose a greedy user scheduling algorithmbased on the rate allocation in vector perturbation precod-ing systems, which reduces the computational complexitythrough removing the insignificant users from the candidateuser set. Considering that the principle of user schedulingis similar with that of antenna selection, the authors in[6] propose a joint strategy which simultaneously performsantenna selection and schedules the optimal users with theobjective of maximizing the system sum rate and reducingthe computational complexity of user scheduling and antennaselection. Owing to the limited space of the BS antennaarray, the adjacent antennas are unavoidably correlated witheach other when the number of antennas increases sharply.The authors in [7], [8] utilize the correlations among theadjacent antennas to group the correlated antennas, thus thedimensionality of the antennas at the BS and the feedbackoverhead are reduced. Similarly, taking advantage of thecorrelations among the users to group the users with thesimilar covariance eigenvectors [9], [10], the joint spatialdivision and multiplexing (JSDM) algorithm can reduce thedimensionality of the effective channels, simplify the systemoperations, and reduce the system feedback through ran-domly selecting part of users to feedback their CSIs.

For the articles above-mentioned, the user scheduling, thecompression and recovery of CSI and the user grouping areaccomplished at the BS, which would increase the feed-back overhead of the uplink channel and the computationalcomplexity at the BS. If part of the task of user schedulingis allocated to users, the computational complexity at theBS and the feedback load would be effectively decreased.Motived by the above viewpoints, we propose a cooperativeuser scheduling scheme for Device-to-Device (D2D) com-munication systems [11]. The key of the proposed schedulingscheme is to utilize the D2D communication among users tofilter out those non-significant users and decrease computa-tional complexity at the BS via the distributed computation.The proposed cooperative scheduling scheme consists oftwo phases: user filtering and user scheduling. During theuser filtering phase, the cell users are divided into severalgroups firstly. Through a timer, the user with the best channelcondition in each group is found, then the optimal userbroadcasts its CSI to the other users in the same group. Thus,the residual users can calculate their correlation coefficientswith the best user locally. Only the user with the best channelcondition and the users whose correlation coefficients aregreater than the filtering threshold can feed back their CSIs tothe BS. During the second phase, the BS selects the user setthrough the popular user scheduling algorithms. Consideringthe similarity that both SUS algorithm [5] and the proposeduser filtering algorithm simplify the candidate user set basedon the correlation coefficients, we combine SUS algorithmwith user filtering for maximizing the system sum rate.

In this paper, we analyze the system sum rate and feedbackoverhead of the proposed scheme in massive MIMO systems.According to the performance analysis, we conclude that thecooperative user scheduling scheme can reduce the compu-

tational complexity at the BS and the system feedback over-head without greatly decreasing the sum rate performance.The main contributions of this paper include:

• To ensure the low feedback and efficient transmission inmassive MIMO systems, we propose a cooperative userscheduling scheme based on the cooperation amongusers. Through filtering out the non-significant usersand simplifying the candidate user set before CSI feed-back, the cooperative user scheduling (CUS) schemecan greatly reduce the number of feedback users. As faras we know, it is the first time to apply the concept of“user filtering” in user scheduling. Besides the advan-tage of reducing the system feedback, the user filteringis accomplished at users, so the proposed scheme canreduce the search complexity at the BS. Furthermore,through combining SUS algorithm with the user filter-ing, the CUS scheme can achieve excellent sum rateperformance through selecting the appropriate user setfrom the permitted feedback users.

• To avoid occupying huge system resources during thefeedback phase, we design an efficient CSI feedbackstrategy to improve the system spectral efficiency. Byutilizing the cooperation among users, each user canobtain the correlation coefficient with the strongest user,which is found through a timer. Through comparingthese coefficients with a filtering threshold, each usercan locally determine which users can feed back theirCSIs.

• Through analyzing the probability that each user isselected and the probabilities under different numbers ofselected users, we obtain the lower bound of the systemsum rate of the proposed CUS scheme. According to thesimulation results, we verify the validity of the lowerbound of sum rate under different antenna numbers, andconclude that the lower bound would get tighter whenthe number of BS antennas increases.

The remaining parts of this paper are organized as follows.In Section II, a multi-user MIMO system is considered andthe section also quantifies the effect of zero-forcing (ZF)beamforming on effective signal to interference plus noiseratio (SINR). In Section III, the proposed cooperative userscheduling scheme is discussed in detail. Section IV givesan elaborate theoretical analysis on the sum rate of thescheduling algorithm. The simulation results are shown inSection V and the final concluding remarks are provided inSection VI.

As for notations, the uppercase boldface and lowercaseboldface are used to denote matrices and vectors. (·)H de-notes the conjugate transpose of a matrix or vector. ‖·‖ de-notes the Euclidean vector norm. card(·) is the number of theelements in set. E (·) denotes the mathematical expectationoperator. Ak,j denotes the element in the k-th row and j-thcolumn of matrix A.

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II. SYSTEM AND CHANNEL MODELSA single cell massive MIMO broadcast system is consideredin this paper, where a BS transmits space information to Nusers, as shown in Fig. 1. In the system, the BS is equippedwith Nt transmitting antennas, and each user is equippedwith one antenna. We consider frequency division duplex-ing (FDD) in this paper, where the downlink and uplinkchannels occupy different frequency bandwidths, and the BSobtains the downlink CSI via the uplink feedback channel.It is assumed that the downlink (DL) channel is quasi-staticRayleigh block fading channel, which means that the channelcondition changes from one frame until the next frame butremains constant within one frame. The uplink (UL) channelis assumed to be perfect and free from the channel noise.

FIGURE 1. System model of the massive MIMO system.

As shown in Fig. 1, the BS transmits the data x to theseusers belonging to the user set S. All the scheduled users’received data y can be expressed as

y = Hx + n =

h1

h2

...hk...

hK

x +

n1

n2

...nk...nK

, k ∈ S, (1)

where y ∈ CK×1 is the received signal, in which K isthe number of the selected users in the user set S, that is,K = card(S) ≤ N . x ∈ CNt×1 denotes the transmit-ted signal from the BS. H =

[hT1 ,h

T2 , . . . ,h

Tk , . . . ,h

TK

]Tdenotes the channel gain matrix, in which hk ∈ C1×Nt

is the channel condition vector of the k-th user. The j-thentry in hk is the channel fading coefficient between thej-th BS antenna and the k-th user, and it obeys Rayleighdistribution. As [12] considered, the antennas of BS areindependent. n ∈ CK×1 is the additive white noise vector,where nk is the noise interfering the k-th user’s receivingsignal, and n ∼ CN(0, I), I ∈ CK×K denotes the unitary

matrix. The power constraint for the transmitted signal isE{xHx

}= P̄T . Since the noise variance is unit, P̄T also

is the transmitting signal-to-noise ratio (SNR).In order to effectively eliminate the interference between

users, the original signal s needs to be preprocessed. Weadopt the zero-forcing beamforming (ZFBF) transmitter [13]for data preprocessing, so the post-processed signal x can beexpressed as

x =K∑k=1

√P̄kwksk, k ∈ S, (2)

where sk is the original transmitting signal of the k-th user,and s = [s1, s2, . . . , sk, . . . , sK ]

T . wk ∈ CNt×1 is the zero-forcing beamforming vector of the k-th user. P̄k is the

power allocated to the k-th user, andK∑k=1

P̄k = P̄T . Let

W̃ = HH(HHH

)−1= [w̃1, w̃2, . . . , w̃K ] [14] be the

beamforming matrix. For satisfying the power constraint,the beamforming vector of each user has to be normal-ized, i.e., wi = w̃i

‖w̃i‖ , so W = [w1,w2, . . . ,wK ] =[w̃1

‖w̃1‖ ,w̃2

‖w̃2‖ , . . . ,w̃K

‖w̃K‖

]. Thus, the k-th user’s received sig-

nal yk can be denoted as

yk =√P̄khkwkxk +

K∑j=1,j 6=k

hk

√P̄jwjxj + nk, k, j ∈ S,

(3)where nk is the noise received by the k-th user. According to(3), the effective receiving SINR of the k-th user is

γk =P̄k|hkwk|2

1 +K∑

j=1,j 6=k

P̄j |hkwj |2. (4)

Without loss of generality, equal power allocation is as-sumed, i.e., P̄k = P̄T

K .

III. COOPERATIVE USER SCHEDULING (CUS) SCHEMEIn this section, we would introduce the proposed cooperativeuser scheduling scheme in detail. The CUS scheme is dividedinto two parts: user filtering based on D2D communicationand user scheduling.

A. USER FILTERING BASED ON THE D2DCOMMUNICATIONThe first step of the proposed CUS scheme is to filter outsome users via the cooperation among users, which meansthat not all users in cell can feed back their CSIs. Thus thescheme can decrease the system feedback overhead.

For reducing the computational complexity and savingthe hardware cost, we need to assign the ultra-high dimen-sion matrix computation of BS to multiple user groups.Considering that most of the existing grouping algorithmshave strong dependency on CSI or channel correlation in-formation [15]–[17], and need extra function evaluations,we adopt random user grouping algorithm [15] to ease the

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computational complexity and reduce the time delay at BS.Furthermore, In the proposed scheme, we cannot adopt thegrouping algorithms which depend on the instant channelinformation, since the grouping is accomplished at the usersbefore the CSI feedback. The cell users are randomly dividedinto M groups. For simplicity, it is assumed that the numberof users in each group is same, namely, NG =

⌈NM

⌉. The

users in different groups work in the different time slots, sothese groups are independent with each other, and there is nointer-group interference.

Focusing on the optimal path selection in relay networks,the authors in [18] propose a cooperative protocol to find thebest relay, and its idea of using a timer can be extended toour scheme. Before feedback, each user is set a timer and thetimer’s start point is directly correlated with the user’s CSI[18]. The timer of the user having the best end-to-end channelcondition will expire firstly. The start point of the timer canbe set as

Ts =λ

‖hs‖, s = 1, 2, . . . , NG, (5)

where Ts is an initial value of the s-th user’s timer. λ is a timeconstant. The start point Ts is inversely proportional to thechannel state vector norm ‖hs‖, so the user with the strongestchannel would expire firstly. Here, We assume that all thetimers in the same group start at the same time.

Based on the assumption that the users can estimate theirCSIs exactly through channel estimation techniques, eachuser calculates its timer start point according to (5) inde-pendently. As shown in Fig. 2, the timer of the user withthe strongest channel will expire first. Then the best usertransmits a short flag packet [18] to signal its presence. Theflag packet also contains the best user’s channel state vector.All the users in the same group, while waiting for their owntimers to reach zero, are in listening mode (each user cannotoverhear the flag packet of the users in other groups). As soonas they hear one user in their own group flagging its presence(the best user), they back off immediately. In this way, we caneasily find the user with the strongest channel in each group(the user with the greatest channel norm).

The other users in each group, obtain the best user’schannel vector hb. In the following parts, we assume that theindex of the best user is b. Then they calculate the correlationcoefficients between their own channel vectors and the b-thuser’s channel vector, that is

αsb =

∣∣hshHb ∣∣‖hs‖ ‖hb‖

, 1 ≤ s, b ∈ Z ≤ NG, s 6= b, (6)

where s and b are the indices of the users which are in thesame group. Z denotes the integer set. Only the user with thebest channel condition and the users satisfying the filteringthreshold can feed back theirs CSIs to the BS. That is, ifαsb ≤ α, user s can feed back its CSI to the BS, otherwise,user s is filtered out and will not communicate with the BS inthis frame. α is a given filtering threshold, which is same for

FIGURE 2. User timer and packet broadcast in one group, the user thatexpires firstly broadcasts a flag packet to other users in the same group

all the groups. Thus, after the user filtering, the user set canbe denoted as

Sf=

M⋃m=1

Sf,m

=M⋃m=1

{s,b∈Gm

∣∣∣∣maxb∈Gm

‖hb‖,αsb=∣∣hshHb ∣∣‖hs‖ ‖hb‖

≤α,b 6=s

}, (7)

where Sf,m denotes the permitted user set of the m-th group,which can feed back its CSI to the BS. Gm is the initial userset containing all the users of the m-th group before the userfiltering, and card(Gm) = NG.

B. USER SCHEDULING

After the user filtering phase, the BS can get the mostwanted users’ CSIs from all the groups. Through combiningwith the semi-orthogonal user scheduling algorithm [19], theappropriate user set can be selected from all the feedbackusers for data transmission.

As shown in Tab. 1, nf denotes the total of the feedbackusers, M ≤ nf ∈ Z ≤ N . Sc,i and Si respectively denotethe candidate user set and the selected user set at the i-thiteration. π(i), i = 1, ..., Nt, denotes the index of the selecteduser at the i-th iteration. Step 2 is carried out to select theuser with the strongest channel as the first scheduled user,and Step 3 is used for obtaining the projection vectors ofthe candidate users on the orthogonal complement space ofthe selected user set. In Step 4, the candidate user with thestrongest projection vector joins in the selected user set. Step5 is used to simplify the candidate user set, and in this stepthe correlation coefficients between the selected users and thecandidate users are obtained through the projected vector gnand g(i). After the SUS scheduling phase, the BS obtains thefinal user set S for data transmission.

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TABLE 1. Semi-orthogonal user scheduling algorithm in massive MIMOsystems

Step1: initialization;

selected user set S0 = Si = φ, i = 0,

candidate user set Sc,0 ={1, 2, . . . , nf

};

Step2: π (i) = arg maxn∈Sc,i

{‖hn‖},

i = i+ 1,

Si = Si−1 ∪ π (i),

Sc,i = Sc,i−1/π (i);

Step3: gn = hn −card(Si)∑i=1

hngH(i)

‖g(i)‖2g(i), n ∈ Sc,i;

Step4: π (i) = arg maxn∈Sc,i

{‖gn‖},

i = i+ 1,

Si = Si−1 ∪ π (i),

Sc,i = Sc,i−1/π (i),

g(i) = gπ(i);

Step5: if card(Si) = Nt, end,

otherwise, Sc,i =

{n ∈ Sc,i

∣∣∣∣∣∣∣∣gngH

(i)

∣∣∣‖gn‖‖g(i)‖

≤ β, i ∈ Si

},

if Sc,i = φ, end,

otherwise, return to Step 3.

IV. SUM RATE ANALYSIS OF THE COOPERATIVE USERSCHEDULINGIn this section, we analyze the achievable sum rate of theproposed scheme with ZFBF transmitters. The sum rate ofthe massive MIMO system can be denoted as

CCUS =

Nt∑K=1

PS (K)K∑k=1

E {log2 (1 + SINRk)} (8)

=

Nt∑K=1

PS (K)

K∑k=1

∫ ∞0

log2 (1 + γk)Pk (γk)dγk (9)

=

Nt∑K=1

PS (K)Czf,SUS(K), (10)

where PS(K) is the probability that the number of theselected users is K. Pk(γk) is the probability that the k-th user’s SINR is γk. As ZFBF can effectively suppress theinterference, SINRk = SNRk, so according to (4), we haveγk = P̄k|hkwk|2

1+K∑

j=1,j 6=k

P̄j |hkwj |2= P̄k|hkwk|2. Czf,SUS(K) denotes

the system sum rate with the ZFBF transmitters and the SUSalgorithm.

According to (10), the sum rate analysis is divided into twoparts, one is about PS(K), the other is about Czf,SUS(K).

A. PROBABILITY ANALYSIS ABOUT THE DIFFERENTSELECTED USER NUMBERS

1) User filteringAfter the user filtering, the probability Pnf

that the remaininguser number is nf can be expressed as

P{Nfilt=nf}∆= Pnf

=M∏m=1

P{Nm = nm} (11)

=M∏m=1

(NG − 1nm − 1

)Pnm−1filt (1−Pfilt)NG−nm , (12)

where Pfilt denotes the probability that the correlationcoefficient between user s and the best user is less thanα. P {Nm = nm},m = 1, 2, . . . ,M, denotes the probabilitythat the remaining user number of the m-th group is nm,which can be calculated as

P {Nm=nm}=(NG−1nm−1

)Pnm−1filt (1−Pfilt)NG−nm . (13)

After the user filtering, the g-th (1 ≤ g ≤M ) group’s userset can be expressed as

Wg = Sf,g/b =

{s

∣∣∣∣∣∣∣hshHb ∣∣‖hs‖ ‖hb‖

≤ α, s, b ∈ Gg

}. (14)

The probability that user s belongs to the the set Wg can bedescribed as [5]

P {sg ∈Wg}∆= Pfilt = F2j,2(Nt−j)

(Nt − jj

α2

1− α2

)= Iα2 (j,Nt − j) , (15)

where sg denotes the user s in the g-th group. Fn,m(x) isthe cumulative density function (CDF) of the F distribution.Iz (a, b) = Bz (a, b)/B (a, b) is the regularized incompletebeta function, in which Bz(a, b) is the incomplete beta func-tion, andB(a, b) is the complete beta function. Note that j =1 in our case, since we only need to calculate the correlationcoefficient with one vector. Due to the fact that each groupis i.i.d., we have, P {s1 ∈W1}= P {s2 ∈W2} = · · · =P {sM ∈WM} = Pfilt.

Let the number of all the remaining users, after the user

filtering, be nf , that is,M∑m=1

nm = nf . (12) can be further

denoted as

Pnf=

M∏m=1

(NG−1nm−1

)Pnm−1filt (1−Pfilt)NG−nm

= Pnf−Mfilt (1−Pfilt)N−nf

M∏m=1

(NG−1nm−1

). (16)

2) Semi-orthogonal user scheduling (SUS)As Step 5 shown in Tab. 1, the probability that the absolutevalue of the correlation coefficient between two vectors isless than β is defined as the “survival probability”. The“survival probability” at the i-th iteration of SUS can beexpressed as [19]

Pi (β) = 1− (1−√β)Nt−i. (17)

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The final probability Ps that each cell user is selected fordata transmission can be calculated as

Ps =

Nt∑i=1

Ps,i, (18)

where Ps,i, 1 ≤ i ≤ Nt, denotes the probability that user s isselected at the i-th iteration.

The probability Ps,1 that user s is selected at the firstiteration of the SUS algorithm is

Ps,1 =1

M

N∑nf=M

Pf1,M∑m=1

nm = nf , 1 ≤ nm ≤ nf , (19)

wherePf1 =Pf,g1M∏

m=1,m6=g

(NG − 1nm − 1

)Pnm−1filt (1− Pfilt)NG−nm .

Pf,g1 denotes the probability that user s is the best user of theg-th group and the remaining user number of the g-th groupis ng . That is, Pf,g1 = P (Ng = ng, s ∈ Sf,g, s /∈Wg), it’sdetails will be given in the Appendix A. The probability Ps,2that user s is selected at the second iteration of the SUSalgorithm is expressed as

Ps,2 = Ps,21 + Ps,22 , (20)

where Ps,21denotes the probability that user s is selected

at the second iteration and it is the best user of the g-thgroup. Ps,22 denotes the probability that user s is selectedat the second iteration and it is not the best user of the g-thgroup. The probability Ps,3 that user s is selected at the thirditeration of the SUS algorithm is

Ps,3 ≥P{card(Sc,1)=ns,1, s ∈ Sc,1}ns,1 − 1

ns,1ns,2

·(ns,1−2ns,2−1

)P2

ns,2(β)(1−P2(β))ns,1−ns,2−1

,

(21)

where P{card(Sc,1) = ns,1, s ∈ Sc,1} denotes the probabil-ity that, at the first iteration of SUS algorithm, the candidateuser number is ns,1 and user s is in the candidate user setSc,1. ns,2 is the candidate user number at the second iterationof SUS algorithm.

The probability Ps,k, 3 ≤ k ≤ Nt that user s is selected atthe k-th iteration of the SUS algorithm is

Ps,k≥1

ns,k−1P{card(Sc,k−1)=ns,k−1, s ∈ Sc,k−1}, (22)

where, similarly, P{card(Sc,k−1) = ns,k−1, s ∈ Sc,k−1}denotes the probability that, at the k− 1-th iteration, the can-didate users number is ns,k−1 and user s is in the candidateuser set Sc,k−1. The details of the final probability of eachuser being selected are given in Appendix A.

As (19)-(22) shown, we can obtain the exact probabilityof each user being selected at the first iteration of SUSalgorithm, and the lower bound of the probability that users is selected at the k-th (k ≥ 2) iteration. Thus, accordingto (18), we obtain the lower bound of the probability of eachuser being selected.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Correlation coefficient β

Pro

babili

ty

Simulation result

Theoretical analysis

FIGURE 3. The probability of each user being selected

Fig. 3 depicts the probability density function (PDF) thateach user is selected. In this simulation, the BS is equippedwith 4 antennas, and there are 20 users which are dividedinto 2 groups. The channels between the BS and the usersobey Rayleigh fading. The filtering threshold α is 0.5, and thecorrelation coefficient β is in the range of (0, 0.45). In reality,for achieving a better scheduling user set, the parametersα and β should be smaller than the values given in thesimulation (it will be proved in Section V), which meansthat the selected users would be less correlated. Here, we justconsider the conventional antenna array instead of massiveMIMO systems due to its huge computational complexity ofFig. 3 and Fig. 4.

Just as Fig. 3 shown, we obtain a lower bound of theprobability that each user is selected, which coincides withthe analysis of (18)-(22). The theoretical lower bound hasan excellent performance. Especially, when the coefficientefficient β is in the range of (0, 0.33), the result of theoreticalanalysis is almost equivalent to the simulation result. Inreality, when there are thousands of users in cell, the optimalvalue of β is just in this interval (0, 0.33) [5], thus thetheoretical lower bound (18) that each user is selected canbe regard as the actual probability.

Based on the probability of each user being selected, wegive a probability analysis about the different selected usernumbers after the SUS algorithm. According to (16) and (42),the probability PS(1) of the selected user number being onecan be calculated as

PS(1)=1

M

N∑nf=M

Pnf

M∑i=1

(1−P1,c)ni−1

M∏m=1,m6=i

{1−P1(β)}nm, (23)

Following this, the probability PS(k) of the selected usernumber being k, 2 ≤ k ≤ Nt can be expressed as

PS(k)≈P{card(Sc,k−1)=ns,k−1}{1−Pk(β)}ns,k−1−1, (24)

where P{card(Sc,k−1) = ns,k−1}, ns,k−1 ≥ 1 representsthe probability that the candidate user number at the k − 1-

6 VOLUME 4, 2016




th iteration is ns,k−1, in which P{card(Sc,1) = ns,1} can becalculated as

P {card (Sc,1) = ns,1} =1

M

N∑nf=M

Pnf

·M∑m=1

P{card(Sc,1m)=ns,11}P{

card(S̄c,1m)=ns,12

},

(25)

where P {card(Sc,1m) = ns,11

} denotes the probabilitythat the m-th group’s remaining user number is ns,11

,

and P {card(Sc,1m) = ns,11} =

(nm − 1ns,11

)Pns,111,c

(1−P1,c)nm−ns,11

−1. The probability P{

card(S̄c,1m)

= ns,12} denotes that the other groups’ (except the m-th group) remaining user number is ns,12 , and the prob-ability can be expressed as P

{card(S̄c,1m

) = ns,12

}=(

nf − nmns,12

)Pns,121 (β) (1− P1 (β))

nf−nm−ns,12 . We

have ns,11+ ns,12

= ns,1, 1 ≤ ns,1 ≤ nf , Sc,1m∪ S̄c,1m

=Sc,1. At the k-th iteration (2 ≤ k ≤ Nt), the probabilityP{card(Sc,k) = ns,k} of the candidate user number beingns,k can be expressed as

P{card(Sc,k) = ns,k}≈P{card(Sc,k−1) = ns,k−1}

·(ns,k−1−1ns,k

)Pns,k

k (β)(1−Pk(β))ns,k−1−ns,k−1.(26)

Here, we obtain an approximated result based on the assump-tion that each user is i.i.d.. Since the relationships wouldbecome weak and complicated after the second iteration dueto the projection in Step 3 of Tab. 1. According to (24), (25)and (26), we can obtain the probability PS(k) of the selecteduser number being k, 2 ≤ k ≤ Nt.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Correlation coefficient β

Pro

babili

ty

theory analysis

simulation resultn=1n=2

n=4

n=3

FIGURE 4. The probability analysis of different selected user numbers

Fig. 4 depicts the PDF that the numbers of the finalselected users are one, two, three, and four. The simulationconditions are same as those of the Fig. 3. As seen from Fig.4, an exact result is obtained when the selected user number

is one. When the user numbers are two, three or four, we onlyobtain the approximated results, while the theoretical resultsare very close to the simulation results. Concluding from Fig.3 and Fig. 4, we can verify the correctness of the theoreticalanalysis about the probability of different selected numbers,which accomplishes the first part of (10).

B. SUM RATE ANALYSISTheorem 1 : The lower bound of the sum rate of the proposedCUS algorithm can be denoted as

CCUS =

Nt∑K=1

PS(K)Czf,SUS(K) (27)

≥ log(e)

Nt∑K=1

PS(K)eK/P̄TKNt−K+2

P̄Nt−K+1T

(28)

·Nt−K+1∑n=1

(P̄TK

)nΓ

(n−Nt +K − 1,

K

P̄T

),

where Γ(a, x) =∞∫x

ta−1e−tdt is the upper incomplete

gamma function [13].Proof : The effective SINR of user k, 1 ≤ k ≤ K with

ZFBF can be expressed as

γk =P̄k[

(HHH)−1]kk

=P̄T

K[(HHH)

−1]kk

, (29)

where K is the total number of the effective users. Set

Z = HHH , (30)

where Z is termed as a complex Wishart distribution [13],and Z ∼WK (Nt,Λ), Λ = I.

According to [20], γk in (29) is a Chi-squared distributionrandom variable with degrees of freedom 2(Nt −K + 1), sothe PDF of γk can be calculated as

fγk(γk) =Ke−Kγk/P̄T

P̄T (Nt −K)!

(K

P̄Tγk

)Nt−K

, γk ≥ 0. (31)

Note that the constraint Nt ≥ K (the constraint of Wishartdistribution) is satisfied because of the SUS algorithm.

Given the user number K, the sum rate [21]–[23] with theZFBF transmitters when the cell users are independent is

Czf (K)=K∑k=1

E{log(1 + γk)}=K∑k=1

∞∫0

fγk(γk) log(1 + γk)dγk

= K

∞∫0

fγk(γk) log(1 + γk)dγk

= log(e)eK/P̄TKNt−K+2

P̄Nt−K+1T

(32)

·Nt−K+1∑n=1

(P̄TK

)nΓ

(n−Nt +K − 1,

K

P̄T

),

VOLUME 4, 2016 7




TABLE 2. The normalized relative differences between simulation results and theoretical results

SNR 0 dB 2 dB 4 dB 6 dB 8 dB 10 dB 12 dB 14 dB 16 dB 18 dB 20 dBNt=4 0.450 0.416 0.385 0.356 0.319 0.289 0.264 0.242 0.226 0.207 0.199Nt=8 0.318 0.290 0.264 0.237 0.214 0.198 0.178 0.164 0.149 0.137 0.129Nt=16 0.220 0.198 0.174 0.157 0.147 0.133 0.113 0.110 0.097 0.096 0.077

where (32) is obtained from the integral identity [13]∞∫

0

log(1 + t)e−µttn−1dt =

(n− 1)!eµn∑i=1

Γ(i− n, µ)

µi, n = 1, 2, . . . .

(33)

Then, we will consider the effect of correlation on the sumrate. The sum rate Czf (K) in (32) is obtained based on theassumption of independent users. However, in the proposedCUS scheme, users are selected based on the correlation co-efficients. The selected users are semi-orthogonal with eachother instead of being randomly selected. So, the elements offthe diagonal of Z in our scheme are smaller compared withthose of independent users.

Given Z′k,k = Zk,k and Z′k,j > Zk,j , k 6= j, we

have(Z′−1)k,k

>(Z−1

)k,k

. Thus, according to (29), the

effective γk would be greater after the SUS procedure. So,

Czf,SUS(K) ≥ Czf (K). (34)

Thus, we obtain the lower bound of the sum rate with the SUSalgorithm and the ZFBF transmitters, which accomplishesthe second part of (10). According to the known channel-hardening [24] property in massive MIMO systems, as thenumber of the BS antennas increases sharply, the differentusers’ channels are nearly orthogonal. Thus the non-diagonalelements in Z would get smaller further, and the lower boundwill get tighter. Combining the sum rate Czf,SUS and theprobability of different selected user numbers, we obtain thelower bound of the sum rate of the CUS scheme according to(10), (24) and (34), which can be calculated as

CCUS =

Nt∑K=1

PS(K)Czf,SUS(K)

≥ log(e)

Nt∑K=1

PS(K)eK/P̄TKNt−K+2

P̄Nt−K+1T

(35)

·Nt−K+1∑n=1

(P̄TK

)nΓ

(n−Nt +K − 1,

K

P̄T

).

So we obtain the sum rate expression in Theorem 1. Here,we give a detailed performance analysis of the proposedalgorithm step by step. In [5], the authors just give an asymp-totic sum rate expression when the users number approachesinfinity, and do not give an exact probability analysis aboutthe selected user number. Except obtaining the probabilitiesof different selected user numbers, our formulation obtainsan elaborate lower bound of the sum rate. And the theoretical

lower bound would get closer to the simulation results as theBS antenna number increases.

0 5 10 15 200

5

10

15

20

25

30

35

40

45

SNR (dB)

Su

m R

ate

(b

it/s

/HZ

)

Simulation result

Theoretical analysisNt=16

Nt=8

Nt=4

FIGURE 5. The sum rate comparisons between the simulation results and thetheoretical results

Fig. 5 depicts the sum rate comparisons of the simulationresults and theoretical results. In the simulation, there are 32users, they are randomly divided into two groups. For ana-lytical tractability, the numbers of the BS transmit antennasare set equal to 4, 8, and 16. The filtering threshold α andcorrelation coefficient β are 0.3 and 0.25. The simulationresults of the sum rate are obtained over 105 independentchannel realizations. Concluding from Fig. 5, we obtain thelower bound of the system sum rate, which coincides withthe analysis in (35).

As shown in Tab. 2, given the receiving SNR, the normal-ized relative differences between the simulation results andthe theoretical results decrease obviously with the increase ofantenna number, in which the normalized relative differenceis calculated as |Csim−Ctheory|

Csim, where Csim and Ctheory respec-

tively denote the simulation sum rate and the theoretical sumrate. Concluding from Tab. 2, the lower bound would gettighter when the antenna number increases, which coincideswith our previous analysis. So it is reasonable to believe thatthe theoretical analysis would obtain a closer approximation,when the theoretical results are applied in massive MIMOsystems. According to Fig. 3, Fig. 4, and Fig. 5, we can verifythe correctness of the theoretical lower bound of sum rate(35).

In the performance analysis, we give the figures about thetheoretical results and simulation results in the conventionalMIMO systems. Since the computation complexity of thetheoretical analysis would have an exponential growth whenthe number of antennas increase sharply due to the step-by-step theoretical analysis. While, as stated in our paper,

8 VOLUME 4, 2016




we analyze the results in the massive MIMO systems, andconclude that, in the massive MIMO scenario, the theoreticalresults would get closer to the simulation results. That is,in the massive MIMO, the theoretical results would obtaina better performance. Except that, in the Section V, we givethe figures of performance comparison in the massive MIMOsystems.

V. SIMULATION AND DISCUSSIONIn this part, we mainly focus on the sum rate performanceand the system feedback overhead of the proposed CUSscheme. The noise variance of each user is one. The sumrate is obtained over 104 channel realizations. The channelin the paper obeys the Rayleigh distribution with mean 0 andvariance 1.

0 5 10 15 2010

20

30

40

50

60

70

80

90

100

SNR (dB)

Sum

rate

(bit/s

/Hz)

Proposed CUS with α=β=0.10

Full feedback SUS with β=0.10





FIGURE 6. The system sum rate with different filtering thresholds α andcorrelation coefficients β

Fig. 6 shows the system sum rate performance of theCUS scheme. In this simulation, the BS is equipped with 32antennas, and there are 100 users which are randomly dividedintoM = 10 groups. α is equal to β. The problem about howto set filtering threshold α and correlation coefficient β willbe discussed in detail in the following parts. As shown in Fig.6, given the number of groups, the filtering threshold and thecorrelation coefficient, the proposed CUS scheme only hasslight sum rate degradation compared with the full feedbackSUS algorithm. Especially, with α = β = 0.22, the sum ratedegradation is nearly negligible. Benefited from filtering outthe non-significant users, the proposed scheme can greatlysave the system feedback resources. The details of reducingthe feedback user number are depicted in Tab. 3.

TABLE 3. The average numbers of transmitting users and filtering users withdifferent filtering thresholds α

average numberof transmitting

users

average numberof filtering

users

percentage ofreduction of

feedback usersα = 0.1 34 66 66%α = 0.15 55 45 45%α = 0.22 80 20 20%

According to Tab. 3, through setting α = β, the systemfeedback load can be greatly saved. Especially, with α =β = 0.15, only about half of the cell users need to feed backtheir CSIs, and the sum rate only has a slight degradation asdepicted in Fig. 6.

0 5 10 15 2010

20

30

40

50

60

70

80

90

100

SNR (dB)

Su

m R

ate

(b

it/s

/Hz)


Proposed CUS with α=0.22







β=0.22

β=0.15

FIGURE 7. The relationship between the system sum rate and filteringthreshold α

Fig. 7 depicts the effect of filtering threshold α on thesystem sum rate. The simulation conditions are same as thoseof Fig. 6. As seen in Fig. 7, the sum rate of the CUS schemeincreases with the filtering threshold α given the correlationcoefficients β. But if the threshold α is too large, it cannoteffectively reduce the feedback users, and if α is too small,it would greatly degrade the system sum rate. Consideringthis factor, we can set β − α = 0.04 according to curvesin Fig. 7. Since, in this interval, the scheme would notgreatly decrease system sum rate (less than 15%), and it caneffectively save the system resources. Thus the system canget a good trade-off between the sum rate and the systemfeedback load. As for the realization in practice, the systemcan determine the values of α and β beforehand according tothe system configuration [5], then BS informs all the users inthe way similar to the realization of SINR threshold in limitedfeedback [25].

Fig. 8 plots the curves of the average numbers of trans-mitting users with different user groups. The 100 users arerandomly divided into M = 10 groups and M = 20groups. The average receiving SNR is 10 dB. According tothe results of Fig. 7, α is ranging from 0.11-0.18 when β isin the range of [0.15, 0.22]. As seen from Fig. 8, the averagenumber of transmitting users is in the range of [35, 68] when0.11 ≤ α ≤ 0.18, Nt = 32, and M = 10. So the scheme cansave about 32% − 65% system feedback resources. Mean-while, the reduction would not greatly affect the system sumrate. Due to the channel-hardening characteristic, the filteringthreshold α and correlation coefficient β would decreasewith the increasing BS antennas in order to obtain highersum rate, so more users would be filtered out when the BSantenna increases further. According to Fig. 8, the numberof transmitting users increases with M slightly. So, we can

VOLUME 4, 2016 9




0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.410

20

30

40

50

60

70

80

90

100

Filtering threshold α

Avera

ge n

um

ber

of tr

ansm

itting u

sers

Nt=16

Nt=32

Nt=64

Nt=16

Nt=32

Nt=64

M=20

M=10

FIGURE 8. The average numbers of transmitting user with different usergroups

slightly decrease the number of the groups M to reduce thesystem feedback overhead. As shown in Tab. 4, M has littleinfluence on the system sum rate (the simulation conditionsare same as those of Fig. 6).

TABLE 4. The relationship between the sum rate and user group number M

SNR P=0 dB P=5 dB P=10 dB P=15 dB P=20 dBM = 5 16.6 25.3 34.6 44.0 53.6M = 10 16.5 25.1 34.2 43.5 53.0M = 20 16.5 25.1 34.2 43.6 53.0

40 50 60 70 80 90 10030

35

40

45

50

55

60

65

70

75

Number of users

Sum

Rate

(bit/s

/Hz)



ZFS

gZFDP

FIGURE 9. The sum rate comparison with two popular algorithms

In Fig. 9, we compare the proposed CUS scheme with twopopular user scheduling algorithms to give more insights onthe proposed scheme’s performance. In the simulation, weassume the BS has 32 antennas, and the users are divided into10 groups and the receiving SNR is 10 dB. Seen from Fig. 9,the proposed user scheduling scheme achieves higher sumrate compared with greedy zero-forcing dirty-paper (gZF-DP) algorithm [26]. According to the feedback schemes

of the two algorithms, our scheme occupies less systemresources. The zero-forcing with user scheduling (ZFS) [26]algorithm obtains the highest sum rate, since it exhaustivelysearches the optimal user set among all the users. Thus ZFSalgorithm has the highest computational complexity, whichis infeasible as the numbers of users and BS antenna arelarge. Comparably, our proposed user scheduling scheme candecrease the complexity of scheduling through reducing thenumber of feedback users. So, according to the figure, ourscheme can achieve an excellent sum rate, save the systemresources and decrease the computational complexity at thesame time.

VI. CONCLUSIONIn this paper, through making use of the cooperation amongusers to filter out the users whose correlation coefficients aregreater than the filtering threshold, a new cooperative userscheduling scheme was proposed based on the D2D com-munication among users. The cooperative user schedulingscheme could effectively ease the problem of huge feedbackin massive MIMO systems and decrease the computationalcomplexity of user scheduling at the BS, which can beapplied to the maritime communication systems. Meanwhile,we analyzed the influence of filtering threshold, correlationcoefficient, and the number of groups on the system sumrate performance and feedback load. According to the sim-ulations, the sum rate and feedback overhead could achieve agood balance, while the corresponding values of these param-eters were changeable according to specific needs of differentsystems. Through exhaustive analysis on the CUS scheme,we obtained the probability of each user being selected, theprobabilities of different selected user numbers and the lowerbound of the sum rate. For analytical tractability, we justplotted the curves of the theoretical results in conventionalMIMO systems. Whilst, as analyzed, when the number ofBS antennas increased, the theoretical analysis could obtainbetter performance, and the lower bound would get tighter.

.

APPENDIX A PROOF OF EQUATIONS (18)-(22)1) user filteringAfter the user filtering phase, the probability that the remain-ing user number is nf and user s belongs to the set Sf can becalculated as

P (Nfilt = nf , s ∈ Sf ) =M∏m=1

Pf,m

= P (Ng = ng, s ∈ Sf,g)M∏

m=1,m6=g

P (Nm = nm)

= (Pf,g1 + Pf,g2)M∏

m=1,m6=g

(NG−1nm−1

)Pnm−1filt (1−Pfilt)NG−nm

= Pf1 + Pf2,(36)

where Pf,m = P (Nm = nm) ,m = 1, 2, . . . ,M,m 6=g is the probability that the m-th group’s remaining usernumber is nm, that is, Pf,m = P{card (Sf,m) = nm} =

10 VOLUME 4, 2016




Ps,22≥ 1

ns,1P {card(Sc,1) = ns,1, s ∈ Sc,1} (43)

=M − 1

M

N∑n′f=M

Pf2

n′f−1∑ns,1=1

1

ns,1P{Ni,s1 = n′i,s1

}P{Ng,s1 = n′g,s1

} M∏m=1,m6=i,m6=g

P{Nm,s1 = n′m,s1

}︸︷︷︸

case 1

+1

M

N∑n′′f =M

Pf2

n′′f−1∑ns,1=1

1

ns,1P{Ng,s1 = n′′g,s1

} M∏m=1,m6=g

P{Nm,s1 = n′′m,s1

}︸︷︷︸

case 2

. (44)

(NG − 1nm − 1

)Pnm−1filt (1− Pfilt)NG−nm , and

M∑m=1

nm =

nf . Sf,m is the m-th group’s user set after the user fil-tering. Pf,g = P (Ng = ng, s ∈ Sf,g) denotes the proba-bility that the g-th group’s remaining user number is ngand user s is in the set Sf,g. Pf,g = Pf,g1 + Pf,g2, inwhich Pf,g1 denotes the probability that the g-th group’sremaining user number is ng and user s is the bestuser. That is, Pf,g1 = P (Ng = ng, s ∈ Sf,g, s /∈Wg),

and Pf,g1 = 1NG

(NG − 1ng − 1

)Png−1filt (1− Pfilt)NG−ng .

Pf,g2 is the probability that the g-th group’s remain-ing user number is ng and user s is not the bestone. That is, Pf,g2 = P (Ng = ng, s ∈Wg), and

Pf,g2 = NG−1NG

(NG − 2ng − 2

)Png−1filt (1− Pfilt)NG−ng .

Pf1 = Pf,g1

M∏m=1,m6=g

Pf,m, Pf2 = Pf,g2M∏

m=1,m6=g

Pf,m.

2) Semi-orthogonal user scheduling (SUS)

The probability that user s is selected for data transmissioncan be calculated as

Ps =

Nt∑i=1

Ps,i, (37)

where Ps,i denotes the probability that user s is selected atthe i-th iteration of the SUS algorithm. Ps,1 can be expressedas

Ps,1 =1

M

N∑nf=M

Pf1,M∑m=1

nm = nf , 1 ≤ nm ≤ nf . (38)

Since the selected user at the first iteration must be one ofthe best users of M groups, so we do not need to considerthe case that user s is not the best user of the g-th group. Theprobability Ps,21

that user s is selected at the second iteration

and is the best user of the g-th group, can be expressed as

Ps,21≥ 1

ns,1P{card(Sc,1) = ns,1, s ∈ Sc,1} (39)

=M − 1

M

N∑nf=M

Pf1

nf−1∑ns,1=1

1

ns,1P{Ni,s1 = ni,s1} (40)

·P{Ng,s1 = ng,s1}M∏

m=1,m6=i,m6=g

P{Nm,s1 = nm,s1},

where P{Nm,s = nm,s1},m = 1, 2, · · · ,M, denotes theprobability that, after the first iteration of the SUS algorithm,the remaining user number of the m-th group is nm,s1 .M∑m=1

nm,s1 = ns,1, and card(Sc,1) = ns,1. Here, it is

assumed that the selected user at the first iteration is comingfrom the i-th group, i 6= g.

P{Nm,s1 = nm,s1}=

(ni−1ni,s1

)Pni,s11,c (1−P1,c)

ni−ni,s1−1, m = i(

ng−1ng,s1− 1

)Png,s11 (β){1−P1(β)}ng−ng,s1 ,m = g(

nmnm,s1

)Pnm,s11 (β){1−P1(β)}nm−nm,s1 , others

(41)

when m = i, the “survival probability” P1,c of the candidateusers in the i-th group, at the first iteration, is a conditionalprobability instead of P1(β). Since the candidate users fromthe i-th group are already satisfied with the coefficient con-straint (14). Thus the “survival probability” can be expressedas

P1,c = P

{ ∣∣hshHb ∣∣‖hs‖ ‖hb‖

≤ β

∣∣∣∣∣∣∣hshHb ∣∣‖hs‖ ‖hb‖

≤ α

}

=

P

{|hsh

Hb |

‖hs‖‖hb‖ ≤ β}

P

{|hshH

b |‖hs‖‖hb‖ ≤ α

} =P1 (β)

Pfilt. (42)

Here, α ≥ β (when α ≤ β, P1,c = 1). The inequality (39)holds since the candidate users which are in the same group

VOLUME 4, 2016 11




P{card(Sc,1) = ns,1, s ∈ Sc,1}

=M − 1

M

N∑nf=M

Pf1

nf−1∑ns,1=1

P{Ni,s1 = ni,s1}P{Ng,s1 = ng,s1}M∏

m=1,m6=i,m6=g

P{Nm,s1 = nm,s1}

+M − 1

M

N∑n′f=M

Pf2

n′f−1∑ns,1=1

P{Ni,s1 = n′i,s1}P{Ng,s1 = n′g,s1}M∏

m=1,m6=i,m6=g

P{Nm,s1 = n′m,s1}

+1

M

N∑n′′f =M

Pf2

n′′f−1∑ns,1=1

P{Ng,s1 = n′′g,s1}M∏

m=1,m6=g

P{Nm,s1 = n′′m,s1}

a=M − 1

M

N∑nf=M

(Pf1 + Pf2)

nf−1∑ns,1=1

P{Ni,s1 = ni,s1}P{Ng,s1 = ng,s1}M∏

m=1,m6=i,m6=g

P{Nm,s1 = nm,s1} (50)

+1

M

N∑n′f=M

Pf2

n′f−1∑ns,1=1

P{Ng,s1 = n′g,s1}M∏

m=1,m6=g

P{Nm,s1 = n′m,s1}.

P {card(Sc,k) = ns,k, s ∈ Sc,k}

≈ P{card(Sc,k−1) = ns,k−1, s ∈ Sc,k−1}ns,k−1 − 1

ns,k−1

(ns,k−1 − 2ns,k − 1

)Pns,k

k (β)(1− Pk(β))ns,k−1−ns,k .

(51)

with the first selected user satisfy (14), which would lead to ahigh selection probability. While, in this analysis, we assumethat all the remaining users’ selection probability are P1(β),which causes that the theoretical probability are lower thanthe actual one.

The probability Ps,22that user s is selected at the second

iteration and is not the best user of the g-th group is calculatedin (43), in which the case 1 represents the selected user anduser s are in the different groups. The case 2 represents thefirst selected user is in the same group with the user s, inwhich

P{Nm,s1 = n′m,s1}=

(ni−1n′i,s1

)Pn′i,s11,c (1−P1,c)

ni−n′i,s1−1, m = i(

ng−1n′g,s1− 1

)Pn′g,s11 (β){1−P1(β)}ng−n′g,s1 ,m = g(

nmn′m,s1

)Pn′m,s11 (β){1−P1(β)}nm−n′m,s1 , others

(45)

P{Nm,s1 = n′′m,s1} =(nmn′′m,s1

)Pn′′m,s11 (β)(1−P1(β))

nm−n′′m,s1 , m 6= g(ng − 2n′′g,s1 − 1

)Pn′′g,s11,c (1−P1,c)

ng−n′′g,s1−1, m = g

(46)

whereM∑m=1

n′m,s1 = ns,1, 0 ≤ n′m,s1 ≤ ns,1.M∑m=1

n′′m,s1 =

ns,1, 0 ≤ n′′m,s1 ≤ ns,1. n′m,s1 and n′′m,s1 respectively

represent the remaining users number of the m-th group incase 1 and case 2, 1 ≤ m ≤M .

The probability Ps,2 that user s is selected at the seconditeration of the SUS algorithm is

Ps,2 = Ps,21+ Ps,22

. (47)

According to (39), (43), and (47), we obtain the lower boundof the probability that user s is selected at the second itera-tion.

Similarly, the probability that user s is selected at the k-th iteration of the SUS algorithm (3 ≤ k ≤ Nt) can dedescribed as

Ps,k≥1

ns,k−1P{card(Sc,k−1)=ns,k−1, s ∈ Sc,k−1} (48)

≥ P{card(Sc,k−2)=ns,k−2, s ∈Sc,k−2}ns,k−2 − 1

ns,k−2ns,k−1

(49)(ns,k−2−2ns,k−1 − 1

)Pns,k−1

k−1 (β){1−Pk−1(β)}ns,k−2−ns,k−1−1

where ns,k−1 and ns,k−2 respectively represent the remain-ing users number at the k − 1-th and k − 2-th iteration, thatis, card(Sc,k−1) = ns,k−1 and card(Sc,k−2) = ns,k−2.

According to (40) and (44), the probability P {card(Sc,1) = ns,1, s ∈ Sc,1} that the total survival user number,at the first iteration, is ns,1 and user s is in the candidate userset Sc,1 can be obtained as (50), in which the equality

a= is

used for simplifying the equation.

12 VOLUME 4, 2016




The probability P {card(Sc,k) = ns,k}, 2 ≤ k ≤ Nt that,at the k-th iteration of the SUS algorithm, the remainingusers number is ns,k and user s is in the set Sc,k can becalculated as (51). So, according to (38), (47), and (48), wecan obtain the lower bound of the probability of each userbeing selected, which also proves (18).

REFERENCES[1] L. Lu, G. Y. Li, A. L. Swindlehurst, A. Ashikhmin, and R. Zhang, “An

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XIANG CHEN was born in Hubei, China, in1992. He received the B.Sc. degree in communi-cations engineering from Dalian Maritime Univer-sity, Dalian, China, in 2014, and M.Sc. degree incommunications engineering from Xidian Univer-sity, Xian, China, in 2014. He is currently pursuingthe Ph.D. degree in communication and informa-tion system, Xidian University. His research inter-ests include Massive MIMO wireless communica-tions, cooperative communications, and NOMA.

FENG-KUI GONG (M’12) was born in Shan-dong, China, in 1979. He received the M.S.and Ph.D. degrees from Xidian University, Xian,China, in 2004 and 2007, respectively. From 2011to 2012, he was a Visiting Scholar with the De-partment of Electrical and Computer Engineer-ing, McMaster University, Hamilton, ON, Canada.He is currently a Professor with the Departmentof Communication Engineering, State Key Lab-oratory of Integrated Services Networks, Xidian

University. His research interests include cooperative communication, dis-tributed space-time coding, digital video broadcasting system, satellite com-munication, and 4G/5G techniques.

HANG ZHANG was born in Hebei, China, in1983. He received the B.Sc. degree in communi-cations engineering from Xidian University, Xian,China, in 2005 and M.Sc degree in electromag-netic field and microwave technology from the54th Research institute of CETC, Shijiazhuang,China, in 2008. Until then he has been workingin the Science and Technology on InformationTransmission and Dissemination in Communica-tion Networks Laboratory, China. He is currently

pursuing the Ph. D. degree in information and communication engineering,Northwestern Polytechnical University. His research interests include mi-crowave and millimeter wave communication wireless Ad Hoc network.

VOLUME 4, 2016 13




GUO LI (S’14) was born in Shaanxi, China, in1989. He received the B.Sc. and M.Sc. degreesin communications engineering from Xidian Uni-versity, Xian, China, in 2011 and 2014, respec-tively. He is currently pursuing the Ph.D. degreein communication and information system, XidianUniversity. His research interests include MIMOwireless communications, cooperative communi-cations, and large-scale antenna system.

14 VOLUME 4, 2016

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