SINR and Throughput Analysis for Random Beamforming Systems With Adaptive Modulation

1460 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 12, NO. 4, APRIL 2013

SINR and Throughput Analysis for RandomBeamforming Systems with Adaptive Modulation

Chanhong Kim, Member, IEEE, Soohong Lee, and Jungwoo Lee, Senior Member, IEEE

AbstractIn this paper, we derive the exact probabilitydistribution of post-scheduling signal-to-interference-plus-noiseratio (SINR) considering both user feedback and scheduling.We also develop an optimized adaptive modulation schemein orthogonal random beamforming systems with M transmitantennas and K single-antenna users. The exact probabilitydistributions of each users feedback SINR and the exact post-scheduling SINR are derived rigorously by direct integrationand multinomial distribution. It is also shown that the derivedcumulative distribution function (CDF) of the post-schedulingSINR happens to be identical to the the existing approximateCDF for SINR higher than 0 dB. The closed form expressions ofsystem performance, such as average spectral efficiency (ASE)and average bit error ratio (A-BER), are derived using the CDFof the post-scheduling SINR. The optimal SINR thresholds thatmaximize the ASE with a target A-BER constraint are solvedusing the derived closed form CDF and a Lagrange multiplier.Key contributions of this paper include the derivation of theexact CDF of post-scheduling SINR by direct integration, andits application to an optimized adaptive modulation based ona Lagrange multiplier. Simulations show the correspondencebetween theoretical and empirical CDFs, and the performanceimprovement of the proposed adaptive modulation method interms of ASE.

Index TermsAdaptive modulation, average spectral effi-ciency, cumulative distribution function, orthogonal randombeamforming, post-scheduling SINR.

I. INTRODUCTION

TODAYS wireless communication systems demand highdata rate, which poses stringent requirements on spectralefficiency. One way to cope with the demand for high spectralefficiency is a multiple-input multiple-output (MIMO) tech-nique, where data rate improvement can be achieved throughspatial multiplexing scheme [1], [2], and it has been furtherextended to multiuser (MU) schemes, in which simultaneoustransmission to multiple users is used. Especially, in mul-tiple antenna broadcast channels, MU-MIMO systems suchas space division multiple access offer higher throughputthan single user (SU) MIMO time division multiple access

Manuscript received September 7, 2011; revised December 27, 2011 andAugust 30, 2012; accepted January 20, 2013. The associate editor coordinatingthe review of this paper and approving it for publication was Y. Jing.

C. Kim and J. Lee are with the Department of Electrical Engineering andComputer Sciences, Seoul National University, Seoul, 151744, Korea (e-mail: [email protected], [email protected]).

S. Lee is with the Department of Mathematical Sciences, Seoul NationalUniversity, Seoul, 151744, Korea (e-mail: [email protected]).

This research was supported in part by the Basic Science Research Pro-gram (20100013397) and Mid-career Researcher Program (20100027155)through the National Research Foundation of Korea (NRF) funded by theMinistry of Education, Science and Technology, Seoul R&BD Program(JP091007, 042320090051), the Institute of New Media & Communications(INMAC), and the BK21 program.

Digital Object Identifier 10.1109/TWC.2013.022113.111663

systems [3]. Another way is a link adaptation technique, wheretransmission parameters such as modulation and coding aredynamically adapted to varying channel conditions [4]. Atypical link adaptation technique is adaptive modulation inwhich an adequate modulation level is selected according tothe current signal-to-noise ratio (SNR).

The capacity region of a Gaussian MIMO broadcast channelcan be achieved by an optimal transmission strategy calleddirty paper coding (DPC) [5], and it has also been shown thatthe sum capacity of that channel with DPC scales at a rate ofM log logK as K tends to infinity, where M is the number oftransmit antennas and K is the number of users [6]. However,the DPC scheme requires prohibitively high implementationcomplexity, so several low complexity MU-MIMO schemessuch as zero-forcing beamforming (ZFBF) and orthogonalrandom beamforming (ORBF) have been proposed [7], [8].These two schemes have the same scaling law as the sum-rate capacity of DPC even with limited feedback [8], [9].Although ZFBF can achieve a large fraction of DPC capacitywith reduced complexity, it also requires accurate channelstate information at the transmitter (CSIT) as in DPC. Instead,ORBF only requires a partial CSIT. In an ORBF system, Morthogonal random beams are constructed and information istransmitted to the users with the highest signal-to-interference-plus-noise ratios (SINRs). Since each receiver is only requiredto feed back its maximum SINR along with the correspondingbeam index, the amount of feedback is small. Due to thisadvantage, a modified ORBF system, which is called per userunitary and rate control (PU2RC) in the industry, was proposedto the next-generation wireless standards [10].

Since adaptive modulation can be applied to most of wire-less communication systems, it has been proposed for single-input single-output systems [11][13], and further extended toSU-MIMO systems [14][21] to adjust the modulation orderbased on the channel condition. In [14], a unified expressionfor the approximate BER was proposed, which has been usedin order to derive closed form expressions of system perfor-mance, such as average spectral efficiency (ASE) and averagebit error ratio (A-BER) [15], [18][21]. Especially, in [18] and[21], the authors tried to obtain the optimal SNR thresholds forselecting modulation order using a Lagrange multiplier basedon the distribution of the post-processing SNR. In MU-MIMOsystems, the probability distribution of the post-schedulingSINR, which corresponds to the post-processing SNR of SU-MIMO systems, has not been known yet except for ORBF.Moreover, even in the ORBF literature [8], [22][25], theapproximate cumulative distribution function (CDF) of thepost-scheduling SINR has been used.

1536-1276/13$31.00 c 2013 IEEE

KIM et al.: SINR AND THROUGHPUT ANALYSIS FOR RANDOM BEAMFORMING SYSTEMS WITH ADAPTIVE MODULATION 1461

In this paper, we try to apply adaptive modulation tech-niques in [18] and [21] to an ORBF system with M transmitantennas and K single-antenna users. At first, we derive theexact CDF of the post-scheduling SINR by direct integration,and apply it to derive the ASE and the A-BER. We thenobtain the optimal SINR thresholds for adaptive modulation.Although the aggregated throughput derived in [24] is similarto the ASE in this paper, in order to obtain the optimal SINRthresholds based on the A-BER approach, new expressions ofthe ASE and the A-BER are derived. Unique contributions ofthis paper include the following.

1) The CDF of each users feedback SINR is derived bydirect integration.

2) The CDF of the post-scheduling SINR is derived froma multinomial approach. It turns out that the existingapproximate CDF [8] happens to be exact for SINRhigher than 0 dB.

3) Optimal adaptive modulation is developed to maximizethe ASE performance of an ORBF system with a finiteset of modulation order.

The remainder of this paper is organized as follows. Sec-tion II describes the signal model and the user schedulingmethod. In Section III, the exact CDF of each users feedbackSINR and the exact CDF of the post-scheduling SINR arederived. In Section IV, adaptive modulation schemes withinstantaneous BER (I-BER) and A-BER constraints are de-scribed. The closed form expressions for the ASE and the A-BER are derived as well. Simulations show the correspondencebetween theoretical and empirical CDFs, and performance im-provement in terms of ASE in Section V. Finally, Section VIconcludes the paper.

II. SYSTEM OVERVIEW

A. Signal Model

We consider a multiple-antenna Gaussian broadcast channelconsisting of a transmitter equipped with M antennas, and Kreceivers with one antenna. M orthonormal complex M 1vectors wm (m = 1, . . . ,M ) for random beamforming aregenerated according to an isotropic distribution. Althoughwms are chosen randomly per given channel use, they arefixed during that period. All of wms are known at thetransmitter and the receivers. Let sm be the mth transmittedsymbol, which is the scheduled users QAM symbol at themth beam. Let Es be the symbol energy, i.e. E [|sm|2] = Es.At every symbol duration, the mth vector is multiplied by themth transmitted symbol, so that the transmitted signal vectoris formed by the sum of all the symbols. Let the transmittedsignal vector be s which can be expressed by

s =

1M

Mm=1

wmsm, (1)

where

1M is a normalization factor to fix the average

transmitted power to Es, i.e. E [sHs] = Es. Equal powerallocation is also assumed. Note that the M elements of theM 1 vector s will be transmitted from the M transmitantennas simultaneously.

Let the received signal of the kth user be yk, which is givenby

yk = hTk s+ nk, k = 1, . . . ,K, (2)

where hk is an M1 flat fading channel vector which has i.i.d.complex Gaussian random entries with distribution CN (0, 1),and nk is an i.i.d. complex additive white Gaussian noise(AWGN) with distribution CN (0, N0). Note that ()T meansa matrix transpose.

Our system model is equivalent to that of [8] except forpower normalization. It is also assumed that all the users havethe same average SNR EsN0 , and that the kth receiver has perfectestimation of hk.

B. User Scheduling and Post-scheduling SINR

Assuming that sm is the kth users desired signal and theother sls (l = m) are interference, yk can be written as

yk =

1M h

Tkwmsm +

1M

l =m

hTkwlsl + nk. (3)

Now we can calculate M SINRs of the kth user as [8, Eq.(5)]

SINRk,m =

E[ 1MhTk wmsm2

]

N0 +

l =m E[ 1MhTk wlsl2

]

=

hTk wm2M +

l =m

hTkwl2 , m = 1, . . . ,M,(4)

where is the average SNR defined by = EsN0 .Each receiver feeds back its maximum SINR denoted by k

along with the corresponding index ik. Thus, k and ik arewritten as

k = max1mM

SINRk,m, (5)

ik = argmax1mM

SINRk,m. (6)

The transmitter assigns sm to the user with the highest SINRamong candidates whose feedback index is m. Then, the post-scheduling SINR, denoted by m, can be written as

m = maxkKm

k, m = 1, . . . ,M, (7)

where Km is the user index set defined by

Km ={k ik = m, k = 1, . . . ,K}. (8)

Finally, the transmitter sends the data to the M active userssimultaneously.

C. User Scheduling Example

A user scheduling example may be helpful to understandthe overall system. An ORBF system with M = 4 and K =10 is considered. It is assumed that all the users SINRs arecalculated as Table I from (4). In the table, U and B denotethe user index and the beam index, respectively. Let us denoteeach users feedback information as Uk(k, ik). From (5) and(6), Uk(k, ik)s are obtained as follows:


- U1(10.08, 4), U2(2.78, 3), U3(5.43, 2), U4(26.73, 2),U5(14.94, 2)

- U6(3.15, 2), U7(2.18, 4), U8(18.29, 3), U9(7.33, 1),U10(15.38, 2)

Now at the transmitter, the user index set at each beam isobtained from (8) as K1 = {9}, K2 = {3, 4, 5, 6, 10}, K3 ={2, 8}, and K4 = {1, 7}. By the max-SINR rule of (7), a userto be assigned at each beam is determined by B1U9, B2U4,B3U8, and B4U1. The post-scheduling SINR correspondingto each beam is obtained as 1 = 7.33, 2 = 26.73, 3 =18.29, and 4 = 10.08.

III. PROBABILITY DISTRIBUTION OF THEPOST-SCHEDULING SINR

In order to analyze the performance of the system in termsof sum-rate, throughput, and average BER, it is necessary toknow the probability distribution about the scheduled usersSINRs, i.e., ms. Unlike existing literature, the CDFs of kand m are derived exactly by direct integration, and the finalresult is compared to the existing results.

A. Cumulative Distribution Function of Each Users FeedbackSINR

In (4), lethTk wm2 = zm, M = c, and SINRk,m = xm

for convenience. Since SINRk,m is i.i.d. over k, the subscriptk is omitted. Now (4) can be rewritten as

xm =zm

c+M

l=1,l =m zl, (m = 1, 2, . . . ,M). (9)

Since wm is an orthonormal complex vector, W =[w1 wM ] is a unitary matrix. As mentioned in Section II-A, hk is assumed to have i.i.d. complex Gaussian randomentries with distribution CN (0, 1). Thus, hTkW is a vectorwith i.i.d. CN (0, 1) entries. This implies that |hTkwm|2 arei.i.d. over m and k with 2(2) and variance of 12 pereach degree of freedom. Hence, the joint probability densityfunction (JPDF) of zms is denoted by

f(z1, z2, . . . , zM )

=

{e

Mm=1 zm , zm 0, zm (m = 1, 2, . . . ,M)

0, otherwise.

(10)

Since the kth users feedback SINR is chosen to be themaximum value among the M SINRs as in (5), the CDFof k can be denoted by

Fk(x) = Pr{max(x1, x2, . . . , xM ) x

}= Pr

{xm x, for m = 1, 2, . . . ,M

}= Pr

{zm

c+M

l=1,l =m zl x, for m = 1, 2, . . . ,M

}

= Pr

{zm x

(c+

Ml=1l =m

zl

), for m = 1, 2, . . . ,M

},

(11)

where the last equality comes from the fact that the denomi-nator (c+

Ml=1,l =m zl) is always positive. Since xms are not

independent from each other, Fk(x) cannot be easily obtained

from order statistics. Instead, we try to integrate the region{(z1, z2, . . . , zM )

0 zm x(c + Ml=1,l =m zl), m =1, 2, . . . ,M

}directly using the JPDF given in (10). Let the

integration region be R(c, x). Now Fk(x) can be denoted by

Fk(x) =

R(c,x)

eM

m=1 zmdz1dz2 dzM . (12)

For M = 2, Fk(x) is obtained easily from geometry as

Fk(x) =

{1 2ecx1+x , x 11 2ecx1+x +

1x1+xe

2cx1x , 0 x < 1, (13)

whose proof is shown in Appendix A. For general M , wederived a closed form expression of Fk(x) with directionintegration by substitution as

Fk(x) =11

(1 + x)M1

min(M1, 1x 1)r=0

(1)r(

M

r + 1

)(1 rx)M1e

(r+1)cx1rx ,

(14)

where 1x is the smallest integer not less than1x . The detailed

derivation is rather complicated, and it is given in AppendixB. Substituting M = 2 into (14), we can also obtain (13) asexpected.

B. Cumulative Distribution Function of the Post-SchedulingSINR

Let pm be the probability that each users feedback beam-forming vector index is equal to m and dm be the cardinalityof Km. In other words, pm = Pr{ik = m}, dm = |Km|,and

Mm=1 dm = K . Since k is independent over k, the

probability that |Km| = dm for m = 1, . . . ,M has amultinomial distribution as

Pr{|K1| = d1, |K2| = d2, . . . , |KM | = dM

}=

K!Mm=1 dm!

Mm=1

pmdm .

(15)

Under the assumption that a certain user index set Km isgiven, the conditional CDF (denoted by Fm|Km) of the post-scheduling SINR can be easily obtained from order statistics[26, Eq. (2.1.1), p. 9] as

Fm|Km(x) ={Fk(x)

}dm. (16)

Since ms are chosen from mutually exclusive users, m isindependent over m. Thus, the conditional joint CDF (JCDF)of ms given |Km| = dm for m = 1, . . . ,M can be given by

F1,2,...,M |K1,K2,...,KM(x1, x2, . . . , xM

d1, d2, . . . , dM)=

Mm=1

{Fk(xm)

}dm.

(17)


TABLE IAN EXAMPLE OF SINRk,m IN THE ORTHOGONAL RANDOM BEAMFORMING SYSTEM WITH M = 4 AND K = 10.

U1 U2 U3 U4 U5 U6 U7 U8 U9 U10

B1 8.05 2.58 0.18 2.08 0.02 2.82 0.36 11.72 7.33 8.25B2 0.34 1.94 5.43 26.73 14.94 3.15 1.44 0.15 4.59 15.38B3 0.15 2.78 5.27 1.77 0.77 0.04 0.96 18.29 3.69 0.15B4 10.08 0.23 0.04 0.14 2.88 1.18 2.18 0.81 0.06 5.83

Since the JCDF of ms is given by

F1,2,...,M(x1, x2, . . . , xM

)=

m, dmZ+{0}M

m=1 dm=K

F1,2,...,M |K1,K2,...,KM(x1, . . . , xM

d1, . . . , dM)

Pr{|K1| = d1, |K2| = d2, . . . , |KM | = dM

}.

(18)

Substituting (15) and (17) into (18), the JCDF can be obtainedas

F1,2,...,M(x1, x2, . . . , xM

)=

{ Mm=1

pmFk (xm)

}K.

(19)Since the channel is assumed to be i.i.d., all the pms are

identical, i.e., m, pm = 1M . By setting xm to x, and theothers to infinity, the CDF of m, which is the marginal CDF,can be obtained as

Fm(x) = F1,2,...,M(,, . . . , xm = x, . . . ,

)=

{Fk(x)

M+ 1 1

M

}K.

(20)

Substituting (14) into (20), Fm(x) is finally given by

Fm(x) =

{1 1

M(1 + x)M1

min(M1, 1x 1)r=0

(1)r(

M

r + 1

)(1 rx)M1e

(r+1)Mx(1rx)

}K.

(21)

When M = 2, from (13), we have

Fm(x) =

(1 e

2x

1+x

)K, x 1(

1 e 2

x

1+x +1x

2(1+x)e 4x(1x)

)K, 0 x < 1

.

(22)

C. Comparison with the existing results

In [8], in order to evaluate the lower and upper bounds ofthe throughput, the CDF of SINRk,m denoted as Fs(x) isshown to be [8, Eq. (15)]

Fs(x) = 1e

M x

(1 + x)M1(x 0), (23)

and the CDF of max1kK SINRk,m is obtained as

Fs(x) =

{1 e

M x

(1 + x)M1

}K(x 0). (24)

This result also can be thought as the SINR of the scheduleduser is selected from E [|Km|] M 1 K candidates, where

E [|Km|] is the average cardinality of the user index set, and itcan be approximated as Kpm = KM [22]. In the literature [22],[23], [27], Fs(x) is exploited to obtain the asymptotic resultand the closed form bounds of sum-rate capacity. However,Fs(x) is not exact. Fs(x) is derived based on the assumptionthat all the SINRs of users are fed back to the transmitter. Inthat feedback structure, an event that a certain user is selectedfor more than one signal may occur. But in the original ORBFsystems, that event never occurs because each user feeds backonly one SINR, which is the largest among the M SINRs.Instead, there exists an event with small probability that somebeams are not used in the feedback, i.e., m, Km = [28].It is observed that Fm(x) = Fs(x) when x 1. For 0 x < 1, however, Fs(x) is not valid any more. In (21), it canbe seen that Fm(x) has different number of terms dependingon 1x.

IV. ADAPTIVE MODULATION

Adaptive modulation is commonly used in wireless com-munication systems in order to increase the throughput of thesystem while satisfying a given target BER. In a rate-adaptiveORBF system, an appropriate modulation can also be selectedper transmitted symbol sm. Since adaptive modulation forsingle user MIMO systems has been studied in the literature[14], [18], [19], [21], we take a similar approach for multi-user MIMO systems. For analysis, a discrete rate adaptivesystem is considered where modulation levels are restrictedto a finite set, M = {M0,M1, ,ML}, with Gray codedquadrature amplitude modulation (QAM). Each Ml denotesthe constellation size and l, Ml1 < Ml. For example, if M1corresponds to the BPSK modulation, M1 = 2. Especially, M0denotes no transmission, and is set to be 1 for mathematicalmanipulation. The SINR range is subdivided into (L+1) binsbounded by the switching threshold l (l = 0, 1, . . . , L + 1)where 0 = 0 and L+1 = . The modulation correspondingto Ml is selected whenever l m < l+1. If m < 1,data transmission is suspended for the corresponding channelbecause the target BER constraint cannot be satisfied.

A. SNR Thresholds for Instantaneous BER Constraint

An easy way to set the switching thresholds ls is to usethe instantaneous BER (I-BER). In this approach, the BERof every reception has to be less than or equal to the targetBER 0. In order to meet the constraint, the BER for a QAMin AWGN channels can be used. Although the exact BERexpressions for M-QAM are shown in [29], they are not easilyinverted with respect to the SINR, so that a numerical methodis necessary. Instead, in the adaptive modulation literature[18], [19], [21], an exponential function form is used, which


TABLE IICONSTELLATION SPECIFIC CONSTANTS FOR BER APPROXIMATION IN

AWGN CHANNELS [21].

Modulation BPSK QPSK 16-QAM 64-QAMal 0.1978 0.1853 0.1613 0.1351cl 1.0923 0.5397 0.1110 0.0270

is given byPe(,Ml) al exp(cl), (25)

where is SNR, al = 0.2 and cl is a constellation specificconstant defined as [14]

cl =

{6

52l4 for rectangular QAM (odd l, l L)3

2(2l1) for square QAM (even l, l L). (26)

If we want a more accurate form than the above approxima-tion, we can find the modulation specific constants al and clnumerically using a curve-fitting method [21]. Table II showsthose values of M-QAMs which are often used. Inverting (25)with respect to , the switching threshold is determined by

l =1

clln

(al0

). (27)

Although it is simple, I-BER approach keeps the instan-taneous BER at all time instants below the target BER 0.This is so conservative that the average BER (A-BER) tendsto be far below 0. In order to make the A-BER be equal to0, SNR thresholds should be lowered. Therefore, the ASEcan be improved by adjusting the switching threshold of eachmodulation.

B. SNR Thresholds for Average BER Constraint

1) Closed form expressions for average spectral efficiencyand average BER: As shown in (21), the probability distribu-tion of the post-scheduling SINR for each beam is identicalwith respect to m. Thus, the A-BER of the system is also thesame as the A-BER for one particular beam. Since the ASEof the system is M times that of one beam, we focus on theanalysis for one beam without loss of generality. The ASE forone beam, denoted as , is given by

=

Ll=0

bl pl, (28)

where bl = log2 Ml is the number of bits corresponding to thelth modulation. pl is the probability that the post-schedulingSINR m falls into the lth bin, given by

pl =

l+1l

fm(x)dx, (29)

where fm(x) is the probability density function (PDF) ofm, which is obtained by differentiating (21) with respect tox. Although Fm(x) is derived as a closed form expression as(21), (21) is still too complicated (even for M = 2) to derive aclosed form expression of the A-BER because Fm(x) has Mterms in the summation for the range of 0 x < 1. Instead, ifit is assumed that 1 > 1 (0 dB), we can use only the first termin the summation of (21), which corresponds to the CDF ofmax1kK SINRk,m. Even though BPSK is used for M1,

it is unlikely to satisfy a given target BER when the post-scheduling SINR is below 0 dB. Without loss of generality,we assume that the transmitter is turned off in the range 0 x < 1. Thus, it is enough to deal with fm(x) only for therange of x 1 in the analysis. By differentiating Fs(x) of(24), i.e. {Fs(x)}K with respect to x, we have

fm(x) = K

{1 e

M x

(1 + x)M1

}K1

eM x

(1 + x)M

{(M 1) + M

(1 + x)

}(x 1).

(30)

Now by using (30), the ASE can be obtained as

=

Ll=1

bl{Fm(l+1) Fm(l)

}

=

Ll=1

bl

[{1 e

M l+1

(1 + l+1)M1

}K

{1 e

M l

(1 + l)M1

}K].

(31)

The A-BER Pe is given by

Pe =NeNb

, (32)

where Ne is the average number of error bits and Nb is theaverage number of transmitted bits. It is observed that Nb = by the definition in (28) and Ne is given by

Ne =

Ll=1

bl Pe(l), (33)

where Pe(l) is the A-BER when the SINR falls into the lthbin, given by

Pe(l) =

l+1l

Pe(x,Ml)fm(x)dx. (34)

Using (25) and (30), Pe(l) can be obtained as (see Ap-pendix C)

Pe(l) =

alK

K1s=0

(K 1

s

)(1)sel,s(l,s)s

[(M 1)

{( s, (1 + l)l,s

)

( s, (1 + l+1)l,s

)}+ Ml,s{

(1 s, (1 + l)l,s

)

(1 s, (1 + l+1)l,s

)}],

(35)

where s = (M 1)(1+ s) and l,s = cl + M (1+ s). Using


(32), (33), and (35), Pe is finally obtained as

Pe =

K

Ll=1

albl

K1s=0

(K 1

s

)(1)sel,s(l,s)s

[(M 1)

{( s, (1 + l)l,s

)

( s, (1 + l+1)l,s

)}+ Ml,s{

(1 s, (1 + l)l,s

)

(1 s, (1 + l+1)l,s

)}].

(36)

2) Optimal SINR thresholds: In the A-BER approach, thegoal is to maximize the ASE under the constraint that theA-BER is lower than or equal to 0. Defining the set ofadjustable switching thresholds as =

{l | l = 1, 2, . . . , L

},

the optimization problem can be formulated as

o = argmax

, subject to Pe 0. (37)

This problem can be solved with a Lagrange multiplier. SincePe is defined by Ne , the constraint Pe 0 can be changedinto Ne 0 for convenience, and the Lagrangian of (37)is defined by

L(, ) = + (Ne 0). (38)

Differentiating (38) with respect to l and equating to zero,the following relationship for l = 1, 2, . . . , L is obtained as

=bl bl1

bl1Pe(l,Ml1) blPe(l,Ml) 0(bl1 bl).

(39)According to the relationship, once 1 is chosen, all the otherls are uniquely determined. Thus, the optimal SNR thresh-olds can be found numerically by adjusting 1 only. Althoughthe A-BER approach has higher computational complexitythan the I-BER approach, the thresholds can be calculatedoff-line so that the A-BER approach is still practical. If theSINR range is quantized with a few bits, we can calculatethe thresholds for the representative SINR of each bin. Thequantized feedback issue is analyzed with more details in [24].

V. SIMULATION RESULTS

As for wms, the column vectors of the discrete Fouriertransform (DFT) matrix are used, which are given by

WM = [w1 w2 wM ] = (wij), (40)

where wij = ej 2M

ijM

(i, j = 0, 1, . . . ,M 1). In order toobtain the empirical CDF of the post-scheduling SINR andthe average BER, we use 105 different realizations of hk (k =1, 2, . . . ,K).

A. Cumulative Distribution Function

In order to verify (21) numerically, two theoretical CDFsand two empirical CDFs are plotted in Figure 1. M and K areset to be 2 and 30, respectively. Since the two theoretical CDFsare different each other only at 0 x < 1, the average SNR is set to be 10 dB. In Figure 1, Theoretical (21) denotes thecurve obtained from (21) and Theoretical (24) denotes thecurve obtained from (24). Simul. (s1) denotes the empirical

CDF of the post-scheduling SINR at the first beam, i.e., 1and Simul. (s2) denotes that of 2. Even though SINR rangeis below 0 dB, as shown in Figure 1(a), the two theoreticalCDFs are not distinguishable each other so that three enlargedfigures are added. As shown in Figure 1(b) and Figure 1(c),both of the two empirical CDFs match with the derived result(22) and not with the approximation {Fs(x)}. However, itis observed in Figure 1(d) that the two theoretical CDFs arealmost indistinguishable in the range higher than 6 dB. Thisis because as x gets close to 1, the third term of the secondline in (22) tends to zero as

limx10

1 x2(1 + x)

e4x

(1x) = 0, (41)

so that Fm(x) for 0 x < 1 converges to Fs(x). It is alsoobserved in the figures that the empirical CDFs of 1 and 2are almost identical, which is consistent with the assumptionof m, pm = 1M for m = 1, . . . ,M .

B. Average Spectral Efficiency

In the ASE simulations, modulations are restricted to M-QAM with M = {1, 2, 4, 16, 64}, where 1 means no trans-mission. The target BER 0 is set to be 102. The ASE isaveraged over 105 channel realizations under a block fadingchannel model, where the entries of channel matrix do notchange during the transmission of one signal vector. In theI-BER approach, the SINR thresholds are from (27) with theconstellation specific constants defined in Table II. In the A-BER approach, the constellation specific constants in Table IIare also used, and the SINR thresholds are obtained from (39)with numerical search.

Fig. 2 shows the performance comparison between the I-BER and the A-BER constraints in terms of ASE. M is fixedto 2, and K is set to 20 or100. It is observed in the figure thatthe A-BER approach has SNR gain of 15 dB compared to theI-BER method at the same ASE. It is also observed that as thenumber of users increases, the overall performance improvesdue to increased multiuser diversity. Fig. 3 shows the averageBER of I-BER method and A-BER method. M and K areset to 2 and 20, respectively. As shown in the figure, both ofthe two methods satisfy the BER constraint 0 0.01 but I-BER method is more conservative than A-BER method. Fig. 4shows the ASE performance with respect to the number oftransmit antennas (beams). K is fixed to 100, and M is set to2, 3, and 4. It is observed that the ASE performance degradesas M increases, which mismatches with the asymptotic result,M log logK which is valid when K . This is because,when K is fixed to a finite number (e.g., 100 or 200), theasymptotic scaling law does not hold any longer, and randombeamforming cannot benefit from multiplexing gain. It shouldalso be noted that we do not use beam selection in this paper,which means that all the M beams are active. Thus, thesystem becomes more interference-limited as M increases.The asymptotic sum-rate with converges to a functionwith the factor of MM1 (a decreasing function of M ), whichis already known in the existing literature [24], [27].


12 11 10 9 8 7 6 5 4 30

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SINR (dB)

Pr(

SIN

R 0, andn 0,

0

etat bndt ={n!ane

ba , a > 0

0, a 0. (53)

Proof of Lemma 3:


(a) x 1

(b) 0 x < 1

Fig. 5. The integration region for calculating Fk (x).

a > 0: From (50) in Definition 1, at b = 0 whent < ba . Thus, the integral is equal to

ba

et(at b)ndt.From (47) in Lemma 1, it is given by n!ane

ba .

a 0: Since at b < 0, at b = 0. Therefore, theintegral is equal to zero.

Definition 2: For non-negative real numbers a and b, the n-dimensional integral In(a, b) on the region Sn(a, b) is definedas follows:

In(a, b) :=

Sn(a,b)

1dx1dx2 dxn, (54)

where

Sn(a, b) := {(x1, x2, , xn) | 0 xk a k,b a x1 + x2 + + xn b} Rn.

(55)

Lemma 4: For n = 1,

I1(a, b) = b 2b a+ b 2a. (56)

Proof of Lemma 4: Relationship between a and b canbe divided into four cases as follows:

b < 0: Since x1 0 and x1 b < 0 in (55) are con-tradictory each other, S1(a, b) = . Thus, I1(a, b) = 0.From (50), b = b a = b 2a = 0. Therefore, theright-hand side of (56) is also equal to zero.

0 b < a: From (55), S1(a, b) = [0, b]. Thus, I1(a, b) =b. From (50), b = b, ba = b2a = 0. Therefore,the right-hand side of (56) is also equal to b.

a b < 2a: Since S1(a, b) = [ba, a], I1(a, b) = 2ab.From (50), b = b, b a = b a, and b 2a = 0.Therefore, the right-hand side of (56) is also equal to2a b.

b 2a: From (55), S1(a, b) = [0, a] [b a, b]. Sincethis set is an empty set or a set having only one point(when b = 2a, S1(a, b) = {a}), I1(a, b) = 0. From (50),b = b, ba = ba, and b2a = b2a. Therefore,the right-hand side of (56) is also equal to zero.

Lemma 5: For n > 1,

In(a, b) =

a0

In1(a, b x)dx. (57)

Proof of Lemma 5: Let us consider T = Sn(a, b) {(x1, x2, , xn) | xn = x} for x [0, a]. Then, we can seethat

(x1, , xn1, x) T x1, , xn1 [0, a] & x1 + + xn1

[b a x, b x] (x1, , xn1) Sn1(a, b x).

Therefore,

In(a, b) =

a0

T

1dx1 dxn1dxn

=

a0

In1(a, b x)dx.(58)

Lemma 6: For n 1,

In(a, b) =1

n!

n+1r=0

(1)r(n+ 1

r

)b ran. (59)

Proof of Lemma 6: Mathematical induction can be used:

For n = 1: (59) is equal to (56). Hence, this has alreadybeen proved in Lemma 4.


Inductive step: If In1(a, b) holds,

In(a, b)

=

a0

In1(a, b x)dx

=

a0

1

(n 1)!

nr=0

(1)r(n

r

)b ra xn1dx

=1

(n 1)!

nr=0

(1)r(n

r

)[ 1

nb ra xn

]x=ax=0

= 1n!

nr=0

(1)r(n

r

)b (r + 1)an+

1

n!

nr=0

(1)r(n

r

)b ran

=1

n!

n+1r=0

(1)r{(

n

r 1

)+

(n

r

)}b ran

=1

n!

n+1r=0

(1)r(n+ 1

r

)b ran,

(60)

where the third equality comes fromb ra

xn1dx = 1n b ra xn, which can be easilyobtained from (51).

Now we derive Fk(x) by using previous results. In orderto calculate (12), we change the variable zms as follows:

z1 y1z2 y2

zM1 yM1

z1 + + zM y

Since the corresponding Jacobian matrix and its determinantare

det

1 0 0 00 1 0 0...

.... . .

......

0 0 1 01 1 1 1

= 1, (61)

dz1 dzM = dy1 dyM1dy. Thus, we can rewrite (12) asfollows:

Fk(x) =

R(c,x)

eydy1dy2 dyM1dy

=

0

ey

S

1dy1dy2 dyM1dy,(62)

where S = {(y1, y2, , yM1) | (y1, y2, , yM1, y) R(c, x)}. From (11), we can evaluate the following relation-

ship

0 zm x(c+

Ml=1l =m

zl

), m = 1, . . . ,M

0 (1 + x)zm x(c+

Ml=1

zl

), m = 1, . . . ,M

0 (1 + x)ym x(c+ y), m = 1, . . . ,M 1 &

0 (1 + x)(y

M1m=1

ym

) x(c+ y)

0 ym x(c+ y)

1 + x, m = 1, . . . ,M 1 &

y cx1 + x

y1 + + yM1 y.

Hence, S has the same form as (55) in Definition 2, i.e.

S = SM1(x(c+ y)

1 + x, y). (63)

Now we can apply (59) of Lemma 6 to (62) as follows:

Fk(x)

=

0

eyIM1(x(c+ y)

1 + x, y)dy

=

0

ey

(M 1)!

Mr=0

(1)r(M

r

)y rx(c + y)

1 + x

M1dy

=Mr=0

(1)r(M 1)!

(M

r

)

0

ey(

1 rx1 + x

)y rcx

1 + x

M1dy.

(64)

From (53) of Lemma 3, the last integral can be calculated as(M 1)!

(1 rx1+x

)M1e

rcx1+x

1 rx1+x , 1 rx1+x > 0

0, 1 rx1+x 0. (65)

Since the integral has a non-zero value only when r < 1+ 1x ,Fk(x) is finally obtained as

Fk(x)

=

min(M,1+ 1x 1)r=0

(1)r(M 1)!

(M

r

)(M 1)!

(1 rx

1 + x

)M1e

rcx1+x

1 rx1+x

=1 +1

(1 + x)M1

min(M,1+ 1x 1)r=1

(1)r(M

r

){1 (r 1)x}M1e

rcx1(r1)x

=1 1(1 + x)M1

min(M1, 1x 1)r=0

(1)r(

M

r + 1

)(1 rx)M1e

(r+1)cx1rx ,

(66)


where x is the smallest integer not less than x.

APPENDIX CDERIVATION OF Pe(l)

Substituting (25) and (30) into (34), and using the binomialexpansion, Pe(l) is calculated as

Pe(l) =

l+1l

aleclxK

K1s=0

(K 1

s

)(1)s

eM (1+s)x

(1 + x)s(M1)+M

{(M 1) + M

(1 + x)

}dx

=alKK1s=0

(K 1

s

)(1)s

l+1l

e{cl+M (1+s)}x

(1 + x)s(M1)+M

{(M 1) + M

(1 + x)

}dx.

(67)

Since the upper incomplete Gamma function is defined as [30,Eq. 8.350.2, p. 899]

(, x) x

ett1dt, (68)

the integral form

ex(1+x) dx is given by

ex

(1 + x)dx = e1

(1 , (1 + )

). (69)

Let us denote s = (M 1)(1+ s) and l,s = cl+ M (1+ s)in (67). Now using (69), Pe(l) can be finally obtained as

Pe(l) =

alK

K1s=0

(K 1

s

)(1)sel,s(l,s)s

[(M 1)

{( s, (1 + l)l,s

)

( s, (1 + l+1)l,s

)}+ Ml,s{

(1 s, (1 + l)l,s

)

(1 s, (1 + l+1)l,s

)}].

(70)

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[1] I. E. Telatar, Capacity of multi-antenna Gaussian channels, AT & TBell Lab., Tech. Rep., 1995, #BL0112170-950615-07TM.

[2] G. J. Foschini, Layered space-time architecture for wireless commu-nication in a fading environment when using multiple aantennas, BellLabs. Tech. J., vol. 1, no. 2, pp. 4159, 1996.

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[15] A. Maaref and S. Assa, Adaptive modulation using orthogonal STBCin MIMO Nakagami fading channels, in Proc. 2004 IEEE Intern. Symp.Spread Spectrum Tech. App., pp. 145149.

[16] , Rate-adaptive M-QAM in MIMO diversity systems using space-time block codes, in Proc. 2004 IEEE Intern. Symp. Personal, IndoorMobile Radio Commun., pp. 22942298.

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[21] , On performance of adaptive modulation in MIMO systems usingorthogonal space-time block codes, IEEE Trans. Veh. Technol., vol. 58,no. 8, pp. 42384247, Oct. 2009.

[22] W. Zhang and K. Letaief, MIMO broadcast scheduling with limitedfeedback, IEEE J. Sel. Areas Commun., vol. 25, no. 7, pp. 14571467,Sept. 2007.

[23] Y. Kim, J. Yang, and D. K. Kim, A closed form approximation of thesum rate upperbound of random beamforming, IEEE Commun. Lett.,vol. 12, no. 5, pp. 365367, May 2008.

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[25] C. Kim and J. Lee, Adaptive modulation for multiuser orthogonalrandom beamforming systems, in Proc. 2011 IEEE SPAWC.

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[28] C. Kim, S. Jung, and J. Lee, Post-scheduling SINR mismatch analysisfor multiuser orthogonal random beamforming systems, IEEE Trans.Commun., vol. 59, no. 6, pp. 15091513, June 2011.

[29] M. P. Fitz and J. P. Seymour, On the bit error probability of QAMmodulation, Intern. J. Wireless Inf. Networks, vol. 1, no. 2, pp. 131139, Apr. 1994.

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Chanhong Kim received the B.S. degree and thePh. D. degree in electrical engineering from SeoulNational University, Seoul, Korea in 2004 and 2011.He currently works as a technical staff at SamsungElectronics, Suwon, Korea. His research interestsinclude link adaptation, modulation, and coding forwireless communications, with current emphasis onanalysis of multiuser MIMO techniques.


Soohong Lee will receive the B.S. degree in theDepartment of Mathematics from Seoul NationalUniversity, Seoul, Korea in 2013. Currently hisresearch interests are algebraic geometry, noncom-mutative geometry, and application of mathematicsin engineering. He received silver and gold medals atInternational Mathematical Olympiad in 2007, 2008respectively.

Jungwoo Lee was born in Seoul, Korea. He re-ceived a B.S. degree in Electronics Engineering fromSeoul National University, Seoul, Korea in 1988and M.S.E. degree and Ph.D. degree in ElectricalEngineering from Princeton University in 1990 and1994. He was a member of technical staff workingon multimedia signal processing at Sarnoff Corpora-tion from 1994 to 1999. He has been with WirelessAdvanced Technology Lab of Lucent Technologiessince 1999, and worked on W-CDMA base sta-tion algorithm development. His research interests

include wireless communications, signal processing, communications ASICarchitecture/design, multiple antenna systems, and wireless video. He holds12 U.S. patents. He was an associate editor for IEEE TRANSACTIONS ONVEHICULAR TECHNOLOGY (2008 to 2011), and he is an associate editor forthe Journal of Communications and Networks. He is also a director of KICSand a steering committee member of JCCI.

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SINR and Throughput Analysis for Random Beamforming Systems With Adaptive Modulation