IEEE COMMUNICATIONS LETTERS, VOL. 12, NO. 5, MAY 2008 365

A Closed Form Approximation of theSum Rate Upperbound of Random Beamforming

Yohan Kim, Student Member, IEEE, Janghoon Yang, Member, IEEE, and Dong Ku Kim, Member, IEEE

Abstract— In this letter, a closed form expression of the sumrate upperbound is derived for random beamforming. Theproposed analytic solution provides a good approximation ofthe ‘actual’ sum rate performance, for which the conventionalasymptotic analysis is less meaningful. Moreover, our result leadsto an implication of the asymptotic growth rate of M log log K.

Index Terms— Random beamforming, sum rate upperbound,closed form approximation, asymptotic performance.

I. INTRODUCTION

RANDOM beamforming has been considered as an effi-cient scheme for the multiple antenna broadcast chan-

nel because of its asymptotic growth rate M log log K (orM log

(1 + ρ

M log K)

reflecting the received SNR ρ), whichis equal to that of the optimal dirty paper coding with Mtransmit antennas and K users [1][2]. This asymptotic analysisprovided a good growth rate result when K → ∞, but it wasnot able to show the actual sum rate performance. Therefore,the asymptotic result is not useful for practical system designconsidering a finite number of users. Nevertheless, there havebeen few attempts to derive the exact sum rate performancefor random beamforming to the best of our knowledge, sincethe distribution of the received SINR of random beamformingis somewhat hard to deal with. Thus, there has been noanalytic tool yet to predict the sum rate performance of randombeamforming.

In this letter, we derive a closed form approximation ofthe sum rate upperbound of random beamforming for anarbitrary number of users. Even though the approximation isreluctantly intended for relatively high received SNR(≥15dB),the proposed analytic result provides a good match to thesimulation result of the sum rate upperbound for both highand low SNR. Moreover, our result also shows the asymptoticgrowth rate of M log log K. As a result, this work offers auseful analytic tool to estimate the sum rate performance ofrandom beamforming for given K, as well as ρ and M .

II. SYSTEM MODEL

Let us consider a base station with M transmit antennas(or equivalently M orthonormal random beams) and K userswith single antenna. If the transmit power is equally allocated

Manuscript received January 30, 2008. The associate editor coordinatingthe review of this letter and approving it for publication was R. Blum. Thiswork was supported by LG Electronics, Korea.

The authors are with the Department of Electrical and Electronic En-gineering, Yonsei University, Seoul, Korea (e-mail: {john5958, jhyang00,dkkim}@yonsei.ac.kr).

Digital Object Identifier 10.1109/LCOMM.2008.080150.

to M beams, the received signal at the kth user is written by

yk =M∑

m=1

hkwmsm + nk, k = 1, · · · ,K (1)

where hk is an 1 × M complex channel vector for the userk, nk is an additive noise, and the elements of hk and nk

are i.i.d. complex Gaussian with zero mean and unit variance.The received SNR ρ is assumed to be the same for all usersand the transmit power of each transmit signal sm is givenby ρ

M . M × 1 orthonormal random beams wm are generatedaccording to an isotropic distribution [1]. The received SINRfor each random beam at the kth user was given in [1] as

SINRm,k =|hkwm|2

M∑j=1,j �=m

|hkwj |2 + Mρ

(2)

for m = 1, · · · ,M and k = 1, · · · ,K. In [1], the pdf ofreceived SINR in (2) is given by

fs(z) =e−

Mρ z

(1 + z)M

{M

ρ(1 + z) + M − 1

}, (3)

and its cdf is as follows:

Fs(z) = 1 − e−Mρ z

(1 + z)M−1, (4)

where z ≥ 0.

III. MAIN RESULT

The scheduler of random beamforming in [1] selects a userwith the maximum SINR among K users for each randombeam, leading to M selected users for M beams, respectively.Thus, the cdf and the pdf of the maximum SINR are given by

F (x) = (Fs(x))K

f(x) = K (Fs(x))K−1fs(x)

(5)

by using (3) and (4). The average sum rate of randombeamforming is

E {Rsum} = E

{M∑

m=1

log2

(1 + max

1≤k≤KSINRm,k

)}

(a)≈ ME

{log2

(1 + max

1≤k≤KSINR1,k

)}∆= ME {log2 (1 + x)} ,

(6)

where x∆= max

1≤k≤KSINR1,k and (a) holds since the probabil-

ity that multiple beams might have been assigned to one useris actually negligible by Lemma 2 in [1]. A straightforward

1089-7798/08$25.00 c© 2008 IEEE

366 IEEE COMMUNICATIONS LETTERS, VOL. 12, NO. 5, MAY 2008

way to calculate (6) is the direct integration E {Rsum} =M

∫ ∞0

log2 (1 + x) f(x)dx. However, the pdf f(x) in (5) istoo complicated to deal with this integration. Moreover, theintegration of log2 (1 + x) f(x) gives us lots of burden forcalculation. Thus, we focus on the upperbound of E {Rsum}.By applying the Jensen’s inequality to (6), we easily have

E {Rsum} ≤ M log2 (1 + E {x}) . (7)

For the calculation of E {x}, we use the following alternativeformulation in [3]:

E {x} = K

∫ 1

0

F−1s (u)uK−1du, (8)

where F−1s is the inverse function of the distribution function

in (4). We introduce the following lemma.

Lemma 1: Let us define a function as

y = Fs(z) = 1 − e−Mρ z

(1 + z)M−1.

Then, the inverse function of y is given by

F−1s (z) =

ρ (M − 1)M

W

(Me

Mρ(M−1)− 1

M−1 log(1−z)

ρ (M − 1)

)− 1

(9)

for M �= 1. W (·) is the Lambert W function [6].Proof: We introduce only the idea for the proof due

to lack of space. For an equation in the form of v = wew,the solution of this equation is w = W (v). Inspired by

this, one can reformulate y = 1 − e− M

ρz

(1+z)M−1 such that w =M

ρ(M−1) (1 + z) and v = Mρ(M−1)e

Mρ(M−1)− 1

M−1 log(1−y).By using (9), we rewrite (8) as follows:

E {x} =Kρ (M − 1)

M

×∫ 1

0

W

(Me

Mρ(M−1)− 1

M−1 log(1−u)

ρ (M − 1)

)uK−1du − 1.

(10)

Unfortunately, the closed form of the integration∫ 1

0W (f(x)) xµ−1dx for an arbitrary function f(x) is

not known yet to the best of our knowledge. Thus, weintroduce the following approximation for W (t):

W (t) ≈ a log (t + b) + c, (11)

where a, b, and c are constants. The discussion on thisapproximation will be given in Section IV. For convenience,we let ξ = M

ρ(M−1)eM

ρ(M−1) . Then, (10) can be rewritten by

E {x} ≈ aKρ (M − 1)M

×∫ 1

0

log(

1 +ξ

b(1 − u)−

1M−1

)uK−1du

+ρ (M − 1)

M(a log b + c) − 1.

(12)

We now take a look at∫ 1

0log

(1 + ξ

b (1 − u)−1

M−1

)uK−1du.

Consider the following manipulation:∫ 1

0

log(

1 +ξ

b(1 − u)−

1M−1

)uK−1du

= − 1M − 1

∫ 1

0

log (1 − u) uK−1du + logξ

b

∫ 1

0

uK−1du

+∫ 1

0

log(1 + bξ−1 (1 − u)

1M−1

)uK−1du.

(13)

By [4], it is known that∫ 1

0

log (1 − u) uK−1du = − 1K

K∑k=1

1k

. (14)

Thus, (13) is rewritten by∫ 1

0

log(

1 +ξ

b(1 − u)−

1M−1

)uK−1du

=1

K (M − 1)

K∑k=1

1k

+1K

logξ

b

+∫ 1

0

log(1 + bξ−1 (1 − u)

1M−1

)uK−1du.

(15)

Let us denote bξ−1 by δ. After some manipulation, we have∫ 1

0

log(1 + δ (1 − u)

1M−1

)uK−1du

=δ

K

K∑k=0

(Kk

)(−1)k

× 2F1 (1, k (M − 1) + 1; k (M − 1) + 2;−δ)k (M − 1) + 1

,

(16)

where 2F1 (α, β; γ; z) denotes the Gauss hypergeometic func-tion. For the purpose of brevity, we define H(K, b, ξ,M) =K∑

k=0

(Kk

)(−1)k 2F1(1,k(M−1)+1;k(M−1)+2;− b

ξ )k(M−1)+1 . Finally,

replacing E {x} in (7) with (12) inserted by (15) and (16),we have the following closed form approximation:

E {Rsum} ≤ M log2

(aρ

M

K∑k=1

1k

+abρ2 (M − 1)2

M2e−

Mρ(M−1) H (K, b, ξ,M)

+aρ (M − 1)

Mlog

M

ρ (M − 1)

+a +ρ (M − 1)

Mc

).

(17)

The first term in (17) can be rewritten by

K∑k=1

1k

= C + log K + o

(1K

), (18)

where C is the Euler constant [4]. Moreover, we easily see thatH (K, b, ξ,M) ≤ ξ

b log K + ξb log

(1K + b

ξ Γ(1 + 1

M−1

)), so

that no term inside the log2 in (17) exceeds the growth rate oflog K. Thus, our approximation implies the asymptotic resultof M log log K derived in [1].

KIM et al.: A CLOSED FORM APPROXIMATION OF THE SUM RATE UPPERBOUND OF RANDOM BEAMFORMING 367

5 10 15 20 25 30 35 401

2

3

4

5

6

7

8

9

Users

Ave

rage

Sum

Rat

e (b

ps/H

z)

Numerical Sum RateNumerical UpperboundAnalytic Result

30dB

10dB

5dB

0dB

Fig. 1. Comparison of analytic and simulation results of sum rate upperboundas a function of the number of users when M = 4, ρ = 0, 5, 10 and 30dB.

IV. DISCUSSION ON APPROXIMATION CONSTANTS

The approximation of the Lambert W function W (t) in(11) is based on the approximation claimed in [5]. Ourmodel has been further modified to simplify the integrationin (10). Approximation constants for (11) are a = 0.665,b = 0.6728, and c = 0.2636, respectively, which are foundby the Levenberg-Marquardt curve-fitting method with 95%confidence interval and the normalized mean squared errorof 4.029 × 10−5 when t ≤ 0.05. Basically, it is not easy tobuild an approximation of W (t) for a large range of t. Inour approximation, the range of t of W (t) depends on thebehavior of t = ξe−

1M−1 log(1−u) ≤ ξ. However, ξ abruptly

goes to infinity for low ρ and rapidly goes to zero withhigh ρ for any M , leading to a quite wide range. Thus, weconsidered relatively high SNR region corresponding to quitesmall argument t in W (t). Actually, t ≤ 0.05 correspondsto SNR higher than 15dB for any M . However, our modelheuristically happens to approximate W (t) with a small erroreven for t > 0.05, so that it is applicable to SNR lower than15dB.

V. NUMERICAL RESULTS AND CONCLUSION

Fig. 1 and 2 show the sum rate performance based onsimulation and the analytic result in (17), which is denotedby ‘Analytic Result’. ‘Numerical Sum Rate’ and ‘NumericalUpperbound’ stand for the simulation results of (6) and (7),respectively. They are compared in Fig.1 for various valuesof ρ when the number of users increases and M = 4.Despite the fact that the approximation discussed in SectionIV holds only for high SNR, the analytic result at low SNRapproximates the numerical upperbound within 0.5 bps/Hzdifference, which is a small fraction compared to the sumrate value. This gap becomes negligible as ρ grows, sinceconstants in (11) are estimated for relatively high SNR. Fig.2 plots comparison as a function of ρ value for 2, 4, and 6transmit antennas (beams) when K = 20. The analytic resultsapproach the numerical upperbound for all M values withincreasing ρ. M log2

(1 + ρ

M log K)

in [2] is also depictedfor the asymptotic result when M = 4, but it shows too largevalue compared to the actual sum rate.

0 5 10 15 20 25 302

4

6

8

10

12

14

SNR (dB)

Ave

rage

Sum

Rat

e (b

ps/H

z)

Numerical Sum RateNumerical UpperboundAnalytic Result

M=2

M=4

M=6

M log2 (1+ ρ/M logK)

(M=4)

Fig. 2. Comparison of analytic and simulation results of sum rate upperboundas a function of the received SNR when M = 2, 4, and 6 for K = 20

TABLE I

COMPARISON OF ASYMPTOTIC RESULT AND CLOSED FORM

RESULT WITH NUMERICAL UPPERBOUND(N.U.)

(M = 4, ρ = 10dB)

User (K) 10 20 30 40Asymptotic result(%)a 227.6 211.7 201.9 196.4

Closed form(%)b 106.7 105.4 104.2 103.9a M log2

(1 + ρ

Mlog K

)/ (N.U.) × 100

b (17)/ (N.U.) × 100

Table 1 shows the accuracy of our result and the asymptoticresult for M = 4 and 10dB SNR. The asymptotic result showsaround twice as much as the numerical upperbound, that is, itclearly overestimates the actual sum rate performance. On thecontrary, the closed form solution (17) estimates the numericalupperbound within 7 % error. This error becomes negligiblewhen SNR increases as shown in Fig. 1 and 2.

Therefore, we observe that our result provides a goodapproximation for the sum rate upperbound of random beam-forming for an arbitrary number of users as well as ρ andM , leading to dealing with the ‘actual’ sum rate performancewhich has been out of sight of the conventional asymp-totic analysis, while implying the asymptotic performance ofM log log K. One of the promising applications of this workis ‘mode optimization’, in which one can optimally vary thenumber of antennas or beams M to maximize the sum rateperformance for given ρ and K based on our result. This topicis worth investigating as a future work.

REFERENCES

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[3] H. A. David and H. N. Nagaraja, Order Statistics, 3rd Edition. John Wiley& Sons, Inc., 2003.

[4] I. S. Gradsbteyn and I. M. Ryzhik, Table of Intergrals, Series, andProducts, 6th Edition. Academic Press, 2000.

[5] A. Ringwald and F. Schrempp, “QCDINS 2.0-A Monte Carlo generatorfor instanton-induced processes in deep-inelastic scattering,” Comp. Phy.Comm., vol. 132, pp. 267-305, Nov. 2000.

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