Fairness Maximization in Multi-Antenna BroadcastChannels Using Random Beamforming

Alireza Borhani, Soroush AkhlaghiFaculty of Engineering, Shahed University, Tehran, Iran

Emails: {aborhani, akhlaghi}@shahed.ac.ir

Abstract— A Multi-Input Multi-Output (MIMO) BroadcastChannel (BC) with large number (n) of users is considered.It is assumed each user either receives the minimum rateconstraint of Rmin or remains silent. Accordingly, for the caseof random beamforming, a user selection strategy together witha proper power allocation method is proposed, showing themaximum number of active users scales as M ln(ln(n))

Rmin− θ(1)

in the asymptotic case of n→∞, where M represents thenumber of transmit antennas. Noting the asymptotic sum-rateof such channel is M ln(ln(n)), the proposed method is able toapproach the asymptotic sum-rate capacity within a constant gap.

Index terms — MIMO broadcast channel, Throughput,Random beamforming, Minimum rate constraint.

I. INTRODUCTION

The sum-rate capacity of MIMO broadcast channel (BC) isaddressed in [1], [2], [3]. This is achieved through using anelegant coding strategy, called Dirty Paper Coding (DPC) [4].However, DPC is less likely to find a practical implicationas it is too complex to deal with. On the other hand, it isdemonstrated that random beamforming is able to approachthe sum-rate capacity of such channel, for the case of manyusers [5].

In random beamforming strategy, considering the trans-mitter is equipped with M antennas, M data streams aresent along M random directions, such that each data streamcorresponds to the user which yields the maximum Signalto Interference-plus-Noise Ratio (SINR) along this direction.Then, a user selection strategy is proposed to effectively selectthe best user for each direction. In this method, a partialknowledge of channel gains at the transmitter is adequate toachieve the sum-rate capacity of such channel. Accordingly, itis shown the asymptotic sum-rate capacity of such channel forlarge number (n) of users scales as M ln(ln(n)).

Although, the random beamforming strategy is able toapproach the sum-rate capacity of MIMO BC, a few numberof users are selected simultaneously (at most M active users),and a great deal of information should be sent for each selecteduser. Thus, this method dramatically increases the waiting listand renders an excess delay to happen. Noting above, thismethod is practically infeasible, specifically for the case ofdelay limited applications.

However, in most of practical applications, it is desiredthat each user is supported by a constant rate. Motivated bythis, in [6] for the case of single antenna BC and assumingthe transmitter makes use of a multi-layer superposition code

for having concurrent transmissions to many users, an elegantpower allocation among layers is proposed. Accordingly, it isshown this method is able to support ln(ln(n))

Rminusers at the same

time, where n and Rmin are, respectively, the number of usersand the minimum rate constraint. Noting, the sum-rate capacityof such channel is ln(ln(n)), this method is able to achieve thesum-rate capacity of such channel within a negligible constantgap.

This paper concerns fairness maximization in MIMO BCby the use of M random beamforming vectors, assuming thetransmitter employs M multi-layer superposition codes, eachalong one direction. This is achieved through selecting a setof appropriate users along each direction for which a properpower allocation strategy similar to what is proposed in [6] isapplied to the corresponding superposition code.

It is worth mentioning that the method of [6] can notbe directly adopted across the layers of the aforementionedsuperposition codes. This is due to the inter-beam interferenceterm arising from other M−1 directions. However, it is arguedthat by using a proper user selection strategy which aims atreducing the inter-beam interference term, one can readilydecompose the MIMO BC channel into M parallel inter-beaminterference-free channels. Although, the corresponding chan-nel gains are no longer Rayleigh distributed, it is demonstratedthat the method proposed in [6] is again applicable in suchchannel.

The remainder of this paper is organized as follows. Sec-tion II, describes the problem formulation. The proposed userselection strategy and the maximum number of active usersunder random beamforming strategy are given in section III.Then, section IV provides some simulations to confirm theanalytical results, and finally section V summarizes findings.

Throughout the paper, vectors are bold face lower case,the Hermitian operation is denoted by (.)H , and accordingto the Kunth’s notation [7], for any functions f(n) andg(n); f(n) = O(g(n)) and f(n) = θ(g(n)) are equivalent, re-spectively, to limn→∞| f(n)

g(n) | < ∞, and limn→∞| f(n)g(n) | =

c, where c is a constant value.

II. PROBLEM FORMULATION

A MIMO broadcast channel with M transmit antennas andn single-antenna users is considered. Assuming the transmittedinformation vector at time instant t is s(t), the received signalat the i’th receiver can be represented as follows ,

yi(t) = hTi s(t) + wi(t) i = 1, 2, ..., n , (1)

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25th Biennial Symposium on Communications

where hi is the M × 1 channel gain vector of the i’th usercomposed of M independent, identically distributed (i.i.d.) el-ements which are drawn from a circularly symmetric gaussiandistribution, i.e. CN (0, 1

M ), and are assumed to be perfectlyavailable at the affiliated receiver. Moreover, it is assumedthe channel gains are constant across the coding block, andvaries for each block. wi(t) represents a scaler additive whitegaussian noise with variance σ2, i.e., wi(t) ∼ CN (0,σ2).Further more, it is assumed the average transmit power perantenna is unit, i.e., E{sH(t)s(t)} = M .

It is assumed the transmitter makes use of random beam-forming by employing a random unitary matrix, whosecolumns, Φ(l) for l = 1, . . . , M , correspond to M beam-forming vectors [8]. As a result, the transmitted vector canbe represented as follows,

s(t) =M∑l=1

Φ(l)s(l)(t) , (2)

where s(l)(t) denotes the transmitted signal to be sent alongthe l’th direction, Φ(l), and entails the transmitted signals ofusers which are assigned to the l’th direction, and can beexpressed as follows,

s(l)(t) =ml∑k=1

√pπl(k)xπl(k)(t) for l = 1, . . . ,M

Π(l) = {πl(1), πl(2), . . . , πl(ml)}, (3)

where πl(k) for k = 1, . . . , ml denotes the index of k’thuser in the set Π(l), the set of assigned users to the l’thdirection, which is assumed is of size ml. Also, pπl(k) andxπl(k)(t) denote, respectively, the allocated power and thetransmitted signal of the k’th user in the set Π(l). As a result,by substituting (2) into (1), the received signal of the i’th userin the j’th set Π(j), can be extracted as,

yπj(i)(t) = hTπj(i)

( M∑l=1

Φ(l)s(l)(t))

+ wπj(i)(t) , (4)

or equivalently, after some manipulations and consider-ing (3), it follows,

yπj(i)(t) = hTπj(i)

( M∑l=1

ml∑k=1

Φ(l)√pπl(k)xπl(k)(t))+wπj(i)(t)

= √pπj(i)h

Tπj(i)

Φ(j)xπj(i)(t) + Iπj(i),1(t)+ Iπj(i),2(t) + wπj(i)(t) , (5)

where,

Iπj(i),1(t) = hTπj(i)

mj∑k=1, k �=i

Φ(j)√pπj(k)xπj(k)(t)

Iπj(i),2(t) = hTπj(i)

M∑l=1,l �=j

Φ(l)s(l)(t) .

Note that, the terms Iπj(i),1(t) and Iπj(i),2(t) can be treated asthe intra-beam interference and inter-beam interference terms,

respectively. Assuming the corresponding information for thei’th user is sent through the j’th direction, thus the receivedSINR of the i’th user becomes,

SINRπj(i)=pπj(i)|hT

πj(i)Φ(j)|2

σ2 + E{|Iπj(i),1(t)|2} + E{|Iπj(i),2(t)|2}(6)

Moreover, assuming the transmit power of each beamformingdirection is one ( the total transmit power of M directions isM ) and noting E{|xπj(i)(t)|2}=1, it follows E{|s(l)(t)|2} =∑mj

i=1 pπj(i)=1, where mj denotes the total number of usersthat receive their information from the j’th direction.

III. PROPOSED METHOD

Generally, there are two main issues that should be ad-dressed when trying to increase the maximum number of activeusers in a MIMO broadcast channel: (i) the beamformingstrategy and (ii) the power allocation method. The formerbasically concerns to assign one of the existing beamformingvectors out of M available vectors to each user which isthought to provide a better channel strength, thereby reducingthe interference. The later, however, concerns to devise aproper power allocation strategy among users with a view toincreasing the number of active users. Intuitively, it can beargued that using an effective beamforming strategy makes itpossible to achieve the degrees of freedom in such channels.The power allocation strategy, on the other hand, affects thenumber of active users on each direction. In the following, theaforementioned issues are discussed and addressed in details.

A. Beamforming Strategy

In [5], it is proved that for the case of large users, randombeamforming is able to achieve the asymptotic capacity ofmulti-antenna broadcast channels. In this method, M orthog-onal random beams are sent through the channel and eachuser feeds back the index of beamforming vector for whichthe maximum SINR value is achieved. In other words, thetransmitter sends the corresponding information for the i’thuser through the j’th direction as long as the term |hT

πl(i)Φ(l)|2

is maximized for l = j. This is achieved through settingup an initialization phase, where in the transmitter broadcastsΦ(l) for l = 1, . . . ,M beamforming vectors. Then, each userindividually computes the received SINR for each direction,and feeds back the corresponding index of the direction whichyields the maximum gain. Finally, the transmitter selects thebest user that its SINR value over this direction outperformsthat of the others. Although the beamforming strategy in thecurrent study is similar to [5], each direction is assigned tofairly large number of users at a given time.

B. Power Allocation Method

As noted earlier, referring to (5), the interference term canbe divided into two main parts: (i) the inter-beam interferenceand (ii) the intra-beam interference terms. The first term is hardto deal with. However, we will later prove that for the caseof large users, one can simply discard those users that their

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25th Biennial Symposium on Communications

corresponding inter-beam interference term exceeds a certainthreshold at the expense of a marginal degradation in thenetwork throughput. The intra-beam interference term, how-ever, can be decreased through using the superposition code.The power allocation method basically attempts to allocatethe available power among users such that the correspondingSINR value exceeds a certain threshold, thereby achieving theminimum rate constraint of Rmin. Thus, the interference termplays an important rule for power allocation. In what follows,we first formalizes and then states the condition for which theinter-beam interference term can be canceled out.

Theorem 1: Let n denotes the number of users selected bythe aforementioned beamforming strategy for a given direc-

tion, then there are at least n = n( ζM−1 )

M(1 + O(

√ln(n)

n ))

users for which the corresponding inter-beam interferencepower is less than ζ.

Proof : The proof is given in the Appendix.Also, the following lemma discuses on the value of n, the

number of selected users for transmitting along a specificdirection, according to the beamforming strategy discussedin III-A.

Lemma 1: For sufficiently large number (n) of users, thesize of each subset, the total number of users for whichthe value of zl= |hT

πl(i)Φ(l)|2 is maximized along a specific

direction, say l = j, scales as nj = nM (1 + O(

√ln(n)

n )) withprobability approaching one.

Proof : It can be readily verified that the probability that aspecific direction out of M directions is assigned to a user is1M . Thus, the total number of users for which the j’th directionis assigned to them makes a Binomial distribution, i.e., nj ∼B(n, 1

M ). Thus, after some manipulations and following thesame approach as is used in the proof of Theorem 1, we getthe result.

Noting Theorem 1 and Lemma 1, one can verify that nj ,the minimum number of users for which the j’th direction isassigned to them and their corresponding inter-beam interfer-ence power fall below a certain threshold, say ζ, is equal tonjP , which scales as n

M ( ζM−1 )M (1+O

( ln(n)n

)). Considering

ζ�σ2, one can easily discard the inter-beam interference term,∑Ml=1,l �=j |hT

πj(i)Φ(l)|2, and hence the resulting SINRπj(i) be-

comes,

SINRπj(i) ≈pπj(i)

1/γπj(i) +∑nj

k=1, k �=i pπj(k)

, (7)

where γπj(i) =|hT

πj(i)Φ(j)|2

σ2 . This makes the original prob-lem can be thought as M parallel single antenna broad-cast channel with non overlapping subset of users, each ofsize n

M ( ζM−1 )M (1+ O

( ln(n)n

)). For a given direction, say the

j’th direction, without loss of generality, we assume users arelabeled based on their corresponding channel strength in adescending order, i.e., γπj

(1) ≥ γπj(2) ≥ . . . ≥ γπj

(nj).It is demonstrated that the single antenna gaussian broadcastchannel is stochastically degraded [9]. As a result, by usingsuperposition code at the transmitter and applying successive

interference cancelation at the receivers, one can increase thereceived SINRπj(i) of the i’th user through canceling outthose interfering signals arising from users for which theircorresponding channel strength outperforms that of the i’thuser in the ordered list, i.e., the interfering signals arisingfrom users πj(k) for k = 1, . . . , i − 1. As a result, theresulting throughput under gaussian code book assumptionfor i=1, . . . ,nj becomes,

Rπj(i)= log(1 +

pπj(i)

1/γπj(i) +∑i−1

k=1 pπj(k)

), (8)

As a result, the aforementioned MIMO BC can be thoughtas M independent single antenna BC. This enables to incorpo-rate single antenna power allocation strategies for MIMO BCwhen random beamforming is applied at the transmitter. Morespecifically, any attempt to maximize the number of activeusers in single antenna case maybe applied to this channel.This motivated us to seek for proper power allocation strategiesfor the case of single antenna channel that can be adopted inthis work.

Recently, in [6] an elegant power allocation strategy forthe case of single antenna BC in a Rayleigh distributedenvironment is devised and is shown can approach the theoret-ical maximum number of active users. Although the channelmodel in our work is no longer Rayleigh distributed, in whatfollows it is proved that the power allocation strategy proposedin [6] can also be applied in our work. In the following, theaforementioned power allocation strategy is briefly discussedand then it is argued that this strategy can be readily adoptedin our model.

C. Fairness maximization for single antenna broadcast chan-nels

A single antenna Rayleigh fading BC with large (n) numberof single antenna users is considered. The problem is to findthe maximum number of active users, such that each canreceive at least the minimum rate constraint of Rmin ≥0, otherwise, if this is not applicable, this user remainssilent for the entire transmission (its power is set to zero).Accordingly, for the case of large number of users, in [6] anovel power allocation method at the transmitter is proposedand is shown can approach the maximum number of activeusers, that is1 ln(P ln(n))

Rmin, where P is the total transmit power.

It is worth mentioning that the corresponding channelstrength of each user is not longer exponential distributed.This is due to the fact that the beamforming strategy proposedin III-A aims to assign the beamforming direction out of Mpossible choices which yields the maximum channel strength

at the receiver. Thus, γπj(i) =|hT

πj(i)Φ(j)|2

σ2 in (8) is no longerexponential distributed. Noting γπj(i) is the maximum of Mi.i.d. random variables, each of exponential distribution withparameter σ2, i.e., Pr(x)=σ2e−σ2xu(x), thus the probabilitydistribution function of γπj(i) becomes Fγπj(i)(x) = (1 −

1It is worth mentioning that the asymptotic capacity of such channel isln(P ln(n))[6].

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25th Biennial Symposium on Communications

e−σ2x)M . From now on, for the sake of convenience, we de-note the corresponding p.d.f. of γπj(i) as Ψ, i.e. Ψ: fγπj(i)(x)=

Mσ2e−σ2x(1 − e−σ2x)M−1. Accordingly, throughout the pa-per, the aforementioned BC is called Ψ-BC. In what follows, itis argued that the proposed power allocation strategy in [6] canbe readily applied to Ψ-BC, resulting in fairness maximization.Although there is a discrepancy between the channel model ofBC and that of Ψ-BC, one can verify that the power allocationstrategy in both cases can be considered the same. In whatfollows, we first discuss this power allocation strategy andthen state the idea behind using this strategy in our work.

Let p0 denotes the probability that the corresponding chan-nel of a given user is in good condition, meaning the equivalentnoise, i.e., Ni= 1

γπj(i), resides bellow a certain threshold, i.e.,

N0. Referring to (8) and noting Fγπj(i)(x) = (1 − e−σ2x)M ,this probability can be readily computed as p0 = 1 − (1 −e

−σ2N0 )M ≈ Me−

σ2N0 for N0 � 1. Assuming xi is a binary

random variable representing wether the i’th user is in goodcondition or not,

xi ={

1 with probability p0

0 with probability 1 − p0

Clearly, the number of users which are in good condition canbe considered as a random variable, i.e., X =

∑ni=1 xi, with

Binomial distribution as B(n, p0).It should be noted that if one could guarantee that the

corresponding SINR of each active user exceeds α − 1,then Rmin = ln(1 + SINR) = ln(α) would be achievable.Noting above, in [6] for the case of Rayleigh channels itis proved that Rmin = ln(α) for m best users is achiev-able if max1≤i≤mNi < P

αm , where P represents the totaltransmit power. Accordingly, assuming users are ordered ina descending order, the assigned power to the i’th user ispi = C

αm−i for i = 1, . . . ,m, where C is a constant value andis defined such that the total power constraint is satisfied, thusC = (1 − 1/α)P . In this case, it is shown that the maximumnumber of active users scales as m = ln(P ln(n))

Rmin[6]. It should

be noted that as is argued in [6], the proposed power allocationmerely depends on the tail of distribution function, hence, anyother distribution which follows the same properties, such asour proposed Ψ-distribution which is defined earlier, obeysdouble logarithmic scaling law for the maximum number ofactive users with minimum rate constraint of Rmin.

Noting above, one can easily verify that the maximumnumber of active users in the j’th decomposed Ψ-BC becomesmj = ln(ln(nj))

Rmin. Also using the results of theorem 1 and

lemma 1, nj scales as nM ( ζ

M−1 )M (1 + O( ln(n)

n

)). Conse-

quently, mj for j = 1, ...,M are asymptotically equal and

scale as mj =ln(ln( n

M ( ζM−1 )M (1+O

(ln(n)

n

))))

Rmin.

Noting above, one can easily observe that the total numberof active users over all beamforming directions becomes∑M

j=1 mj =M ln(ln( n

M ( ζM−1 )M (1+O

(ln(n)

n

))))

Rminwhich asymptot-

ically scales as M ln(ln(n))Rmin

− θ(1), for the case of large n.

1 2 3 4 5 6 7 8 9 10

x 105

13

14

15

16

17

18

19

20

21

The total number of users

Sum

−R

ate

Analytical result, Rmin=0.1,0.2Numerical result, Rmin=0.1Numerical result, Rmin=0.2

Fig. 1. The comparison result between the sum-rate and the achievablethroughput of the proposed method for two values of Rmin = 0.1, 0.2,M = 4 and SNR=15dB.

1 2 3 4 5 6 7 8 9 10

x 105

60

80

100

120

140

160

180

200

220

The total number of users

The

num

ber

of a

ctiv

e us

ers

Numerical result for the proposed method, Rmin=0.1The upper bound of the maximum number of active users, Rmin=0.1Numerical result for the proposed method, Rmin=0.2The upper bound of the maximum number of active users, Rmin=0.2

Fig. 2. The comparison result between the maximum number of active usersand that of the proposed method for two values of Rmin = 0.1, 0.2, M = 4and SNR=15dB.

As a result, as the minimum rate constraint for each user isset to Rmin, the achieved sum-rate for the proposed strategybecomes at least M ln(ln(n)) − θ(1). Noting the maximumsum-rate of the original MIMO BC scales as M ln(ln(n)) [5],it turns out that our method achieves this throughput within aconstant gap.

IV. SIMULATION RESULTS

In this section, we aim at providing some numerical resultsto compare the sum-rate and maximum number of activeusers to what is achieved through using the proposed strategy.Accordingly, for the case of SNR = P

σ2 = 15dB and variousnumber of users ranging from 105 to 106, the achievablesum-rate and the total number of active users are depicted,respectively, in Figs. 1 and 2, for two values of Rmin =0.1, 0.2. Moreover, the threshold is set to ζ = 0.1σ2. Figs. 1and 2 show, respectively, there is a constant gap betweenthe sum-rate and the maximum number of active users ascompared to what is achieved through using the proposedmethod.

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25th Biennial Symposium on Communications

V. CONCLUSION

This paper aimed at investigating the fairness maximizationin MIMO broadcast channel, when the transmitter makesuse of random beamforming. It is assumed each user eitherreceives the minimum rate constraint of Rmin or remainssilent throughout the transmission (its power is set to zero).Accordingly, for the case of many users (n), it is shown themaximum number of active users scales as M ln(ln(n))

Rmin− θ(1),

where M represents the number of transmit antennas. Notingthe asymptotic sum-rate of such channel is M ln(ln(n)), theproposed method is able to approach the sum-rate capacity ofsuch channel within a constant gap.

VI. APPENDIX

Theorem: Let n denotes the number of users selected by thebeamforming strategy proposed in III-A for a given direction,

then there are at least n = n( ζM−1 )

M(1 + O(

√ln(n)

n ))

usersfor which the corresponding inter-beam interference power isless than ζ.

Proof: Assume P denotes the probability that the cor-responding inter-beam interference power of the user in-dexed by πj(i) in the j’th direction is less than ζ. Not-ing E[|Iπj(i),2(t)|2] =

∑Ml=1,l �=j |hT

πj(i)Φ(l)|2E[|s(l)(t)|2] =∑M

l=1,l �=j |hTπj(i)

Φ(l)|2, and setting zl = |hTπj(i)

Φ(l)|2, it fol-lows,

P = Pr{M∑

l=1,l �=j

zl < ζ}

Since Φ(l) for l = 1, . . . ,M are orthonormal vectors, thushT

πj(i)Φ(l) is an i.i.d random variable with the same distribu-

tion as the entries of hπj(i), i.e., Nc(0, 1). This implies thatzl = |hT

πj(i)Φ(l)|2 are i.i.d over l with χ2(2) distribution, i.e.,

Przl(x) = e−xu(x). According to the beamforming strategy

described in III-A, the j’th direction is assigned to the userindexed by πj(i) such that the corresponding channel gain onthis direction has the highest value,

zj = max1≤l≤Mzl . (9)

As a result, the distribution of zj is the distribution of themaximum value of M i.i.d random variables with χ2(2)distribution,

Fzj(x) = Fzmax

(x)=Pr(zmax ≤ x)= Pr(z1 ≤ x, z2 ≤ x, . . . , zM ≤ x)= Pr(z1 ≤ x)M=(1 − e−x)M

On the other hand, the probability that the inter-beam inter-ference power defined in (9) is lower than a certain threshold,i.e., ζ, can be lower bounded as follows,

P = Pr{M∑

l=1,l �=j

zl < ζ} ≥ Pr{(M − 1)zmax < ζ}

= Pr{zj <ζ

M − 1} = (1 − e−

ζM−1 )M (10)

Noting 1 − x + x2

2 ≥ e−x, thus one can simplifies the abovelower bound as follows,

P ≥ (1 − e−ζ

M−1 )M

≥ (ζ

M − 1)M

(1 −

ζ2

M − 1)M

≈ (ζ

M − 1)M for ζ � 1 . (11)

Consequently, the minimum number of users (n) for whichthe inter-beam interference term is lower than ζ makes aBinomial distribution, that is B(n, ( ζ

M−1 )M ) 2. Using gaussianapproximation for a binomial distribution when n is largeenough, we have

Pr{n(ζ

M − 1)M (1 − δ) < n < n(

ζ

M − 1)M (1 + δ)}

≈ 1 − 2Q( ( ζ

M−1 )Mδn√( ζ

M−1 )M (1 − (ζ

M−1 )M)n

)(12)

It can be verified that by setting δ =√

2(( M−1ζ )M−1) ln(n)

n and

noting the approximation Q(x) = 1√2πx

e−x22 for x 1, it

follows,

Pr{

λ(1 − O(

√ln(n)

n))

< n < λ(1 + O(

√ln(n)

n))}

∼ 1 − o(1n

) (13)

where λ = n( ζM−1 )M . It means for sufficient large number of

users, n scales as n( ζM−1 )M

(1+O(

√ln(n)

n ))

with probabilityapproaching one.

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25th Biennial Symposium on Communications