Exact post-scheduling SINR analysis for be denoted by

orthogonal random beamforming systems

Chanhong Kim and Jungwoo Lee

Derived are the exact cumulative distribution functions (CDFs) of eachusers feedback signal-to-interference-plus-noise ratio (SINR), and thepost-scheduling SINR in orthogonal random beamforming systemswith M transmit antennas and K single-antenna users consideringboth user feedback and scheduling. A key contribution is to derivethe exact CDF of the post-scheduling SINR by direct integration, andit is veried by Monte-Carlo simulations. It is also shown that theexisting approximate CDF is different from the exact distribution forSINR smaller than 0 dB.

Introduction: Since orthogonal random beamforming (ORBF) was rstproposed in [1], it has been considered a practical scheme for a Gaussianbroadcast channel with M transmit antennas and K single-antenna users.To analyse the performance of ORBF systems such as sum-rate,throughput, and average BER, it is necessary to know the statisticaldistribution of the scheduled users signal-to-interference-plus-noiseratio (SINR). In the literature [15], however, performance analysishas been done with an approximate cumulative distribution function(CDF) of the post-scheduling SINR, which assumes an unrealisticfeedback scheme.This Letter studies the exact statistical distribution of the post-sche-

duling SINR in the orthogonal random beamforming system with Mtransmit antennas and K single-antenna users considering both userfeedback and scheduling. The probability distributions of each usersfeedback SINR and the post-scheduling SINR are derived rigorouslyby direct integration and multinomial distribution. It is also shownthat the derived CDF of the post-scheduling SINR happens to be thesame as the the existing approximate CDF for SINR higher than 0 dB.

System model: Let us consider a Gaussian broadcast channel with Mtransmit antennas (M orthonormal random beams) and K single-antenna users. If the transmit power is equally allocated to M beams,the received signal at the kth user is given by

yk =NameMeNameMeNameMe1

M

Mm=1

hTkwmsm + nk , k = 1, . . . ,K (1)

where hk is the kth users M 1 channel vector composed of IIDcomplex Gaussian random entries with distribution CN(0, 1), wms areorthonormal random beamforming vectors, sm is the transmittedsymbol at the mth beam, and nk is an IID complex additive whiteGaussian noise with distribution CN(0, N0).

M SINRs of the kth user can be calculated as [1]

SINRk,m = |hTkwm|2

M/r+ l=m

|hTkwl|2,m = 1, . . . ,M (2)

where r Es/N0 and Es E[|sm|2]. It is assumed that all the usershave the same average SNR (r). Each receiver feeds back itsmaximum SINR hk along with the corresponding index ik. Thus, hkand ik can be written as hk max1m M SINRk,m and ik argmax1m M SINRk,m, respectively. The transmitter assigns sm to theuser with the largest SINR among candidates whose feedback index ism. The post-scheduling SINR gm can then be written as gm maxk[Km hk (m 1, . . . , M ) where Km is the user index set of themth beam given by Km {k|ik m, k 1, . . . , K}.

CDF of each users feedback SINR: In (2), let |hkTwm|2 zm, M/r c,and SINRk,m xm for convenience. Since SINRk,m is IID over k,the subscript k is omitted. Now (2) can be rewritten as

xm = zm/c+

l=1,l=m(m = 1, 2, . . . ,M ). Since zms are IID with x2(2)

and variance of 1/2 per each degree of freedom, the joint probabilitydensity function (JPDF) of zms is denoted by f (z1, z2, . . . , zM) e 2

Mm1zm (zm 0, zm). The CDF of each users feedback SINR can

ELECTRONICS LETTERS 1st March 2012 Vol. 48Fhk (x) = Pr{xm x, xm}

= Pr zm x c+Ml=1l=m

zl

, zm

(3)

where we use the fact that the denominator (c+l1,l = mM zl ) is alwayspositive.Since xms are not independent from each other, Fhk (x) cannot

be easily obtained from simple order statistics [6]. Instead, we try to

directly integrate the region R = {(z1, z2, . . . , zM )|0 zm x(c+Ml=1,l=m

zl), m} using the JPDF given of zms. Fhk(x) can be written inan integral form as

Fhk (x) =

R

eMm=1

zmdz1dz2 dzM (4)

We change the variable zms as z1 y1,z2 y2, . . ., zM21 yM21,z1+ ...+ zM y. Since the determinant of the correspondingJacobian is equal to 1, dzl...dzM dyl...dy1...dyM21dy. Thus, (4) canbe rewritten as

Fhk (x) =10

ey

S

1dy1dy2 dyM1dy (5)

where, from (3),

S = (y1, y2, . . . , yM1|0 ym x(c+ y)1+ x , m,

{

andy cx1+ x y1 + + yM1 y

} (6)

To calculate the last integral, for a, b 0, let

In(a, b) :=

Sn(a,b)

1dx1dx2 dxn (7)

where

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

(8)

In(a, b) can be computed in a closed form of Lemma 1, which can beeasily proven with mathematical induction.

Lemma 1 For n 1,

In(a, b) = 1n!

n+1r=0

(1)r n+ 1r

( )kb raln (9)

where kxl x if x 0, otherwise kxl 0.Since S = SM1(x(c+ y)/1+ x, y), Lemma 1 can be applied to (5) as

Fhk (x) = 11

(1+ x)M1min(M1,1/x1)r=0

(1)r Mr + 1

( )

(1 rx)M1e(r+1)cx/1rx(10)

where x is the smallest integer not less than x.

CDF of post-scheduling SINR: By using the independence of over gmwith a multinomial distribution for |Km| dm for m 1, . . . ,M, wehave

Fg1,g2,...,gM (x1, x2, . . . , xM ) =Mm=1

pmFhk (xm){ }K

(11)

No. 5

Since the channel is assumed to be IID, all the pms are identical, i.e. m,pm 1/M. The marginal CDF Fgm(x) is nally obtained as

Fgm (x) = 11

M (1+ x)M1min(M1,1/x1)r=0

(1)r Mr + 1

( ){

(1 rx)M1e(r+1)M/r(1rx)}K

(12)

Numerical results: In the literature [15], the asymptotic behaviour andthe closed form bounds of sum-rate capacity are analysed by theapproximate CDF, which is given by

Fs(x) = 1 eM/rx

(1+ x)M1{ }K

(x 0) (13)

which is the CDF of max1k K SINRk,m [1]. However, Fs(x) cannotalways be the post-scheduling SINR because (13) assumes that all theSINRs of users are fed back to the transmitter. But in ORBF feedbacksystems, each user feeds back only one SINR. It is observed thatFgm(x) Fs(x) when x 1. For 0 x , 1, however, Fs(x) is notexact any more. To verify (12) numerically, two theoretical CDFs andtwo empirical CDFs are plotted in Fig. 1. Simul. (sm) denotes theempirical CDF of gm (m 1, 2). In Fig. 1, the two empirical CDFsmatch with the derived result (12), but not with (13) when 0 x 1.

Conclusion: A key contribution of this Letter is to derive the exactpost-scheduling SINR distribution for orthogonal random beamformingsystems by using direct integration. It is shown that the approximatedistribution in the existing literature happens to be exact whenSINR . 0 dB, and it is not when SINR , 0 dB. The exact distributioncan be used to obtain performance measures such as throughput, BER,and sum-rate more accurately, especially in the low SNR regime.

Acknowledgments: This research was supported in part by NRF pro-grammes (20100013397, 2010-0027155), the Seoul R&BD Program(JP091007, 0423-20090051), the KETEP grant (2011T100100151),the INMAC, and BK21.

# The Institution of Engineering and Technology 20123 September 2011doi: 10.1049/el.2011.2650One or more of the Figures in this Letter are available in colour online.

Chanhong Kim and Jungwoo Lee (School of Electrical Engineering,Seoul National University, 599 Kwanak-Ro, Kwanak-Gu, Seoul151-744, Korea)

E-mail: junglee@snu.ac.kr

References

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4 Vicario, J., Bosisio, R., Anton-Haro, C., and Spagnolini, U.: Beamselection strategies for orthogonal random beamforming in sparsenetworks, IEEE Trans. Wirel. Commun., 2008, 7, (9), pp. 33853396

5 Park, K.-H., Ko, Y.-C., and Alouini, M.-S.: Accurate approximations12.0 11.5 11.0 10.5 10.00

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SINR, dB

Pr(S

INR