Page 1

Hindawi Publishing Corporation

EURASIP Journal on Wireless Communications and Networking

Volume 2007, Article ID 89780, 9 pages

doi:10.1155/2007/89780

ResearchArticle

On Sum Rate and Power Consumption of Multi-User

Distributed Antenna System with Circular Antenna Layout

Jiansong Gan, Yunzhou Li, Limin Xiao, Shidong Zhou, and Jing Wang

Department of Electronic Engineering, Tsinghua University, Room 4-405 FIT Building, Beijing 100084, China

Received 18 November 2006; Accepted 29 July 2007

Recommended by Petar Djuric

We investigate the uplink of a power-controlled multi-user distributed antenna system (DAS) with antennas deployed on a circle.

Applying results from random matrix theory, we prove that for such a DAS, the per-user sum rate and the total transmit power

both converge as user number and antenna number go to infinity with a constant ratio. The relationship between the asymptotic

per-user sum rate and the asymptotic total transmit power is revealed for all possible values of the radius of the circle on which

antennas are placed. We then use this rate-power relationship to find the optimal radius. With this optimal radius, the circular

layoutDAS(CL-DAS)isprovedtoofferasignificantgaincomparedwithatraditionalcolocatedantennasystem(CAS).Simulation

results are provided, which demonstrate the validity of our analysis.

Copyright © 2007 Jiansong Gan et al. This is an open access article distributed under the Creative Commons Attribution License,

which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. INTRODUCTION

Information theory suggests that for a system with a large

number of users, increasing the number of antennas at the

base station leads to a linear increase in sum-rate capacity

without additional power or bandwidth consumption [1].

However, previous studies have mostly focused on scenar-

ios with all antennas colocated at the base station. Suppose

antennas are connected but placed with geographical separa-

tions, each user will be more likely to be close to some an-

tennas, and the transmit power can therefore be saved. This

is the concept of distributed antenna system (DAS) which

was originally introduced for coverage improvement in in-

door wireless communications [2].

Recent interests in DAS have shifted to advantages in ca-

pacityorsumrate.Thechannelcapacityofasingle-userDAS

and the sum rate of a multi-user DAS were investigated and

compared with those of co-located antenna systems (CAS)

using Monte Carlo simulation in [3] and [4], respectively,

where significant improvements have been observed. How-

ever, these works did not provide theoretical analysis to char-

acterize the exact gain that a DAS offers over a CAS. Nei-

therdidtheypresentoptimalparametersforantennadeploy-

ment.

Another line of work introduces coordination between

base stations and suggests an architecture quite similar to

DAS[5].Sumrateofsuchasystemhasbeenstudiedin[6,7].

However,unlikeinvestigationsinDASwhichevaluateperfor-

manceimprovementbyscatteringco-locatedantennas,these

works mainly assess performance enhancement by introduc-

ing coordination between base stations. In addition, analysis

in [7], for example, assumes a large number of antennas co-

located at the base station, which differs from ideas of DAS.

In this study, we demonstrate the advantage of scattering

co-located antennas in an analytical way. Though there are

different ways to scatter antennas and different antenna lay-

outs may result in different performances, investigating all

possible layouts is rather difficult. For analytical tractability

we only consider a special DAS with antennas deployed on

a circle. A similar model has been used in [8] to study the

capacity of CDMA system with distributed antennas. Since

distributed antennas are relatively cheap, it is feasible to de-

ploy a large number of antennas, which makes application of

random matrix theory possible. Applying recent results from

this theory, we prove that for a circular-layout DAS (CL-

DAS), the per-user sum rate and the total transmit power

both converge as user number and antenna number go to

infinity with a constant ratio. Then, the relationship between

the asymptotic per-user sum rate and the asymptotic total

transmit power is disclosed for all possible values of the ra-

dius of the circle on which antennas are deployed. We fur-

ther show how the rate-power relationship can be used to

find the optimal radius. A CL-DAS with this optimal radius

Page 2

2 EURASIP Journal on Wireless Communications and Networking

R0

a

b

R

Antenna

Central processor

User

Figure 1: Illustration of CL-DAS.

is proved to offer a significant gain over a traditional CAS.

Though the maximum achievable gain that a general layout

DAS provides over a CAS has not been found yet, it can be

lower bounded by the presented gain for the optimized CL-

DAS (OCL-DAS). Hence, we demonstrate the possibility of

greatperformanceenhancementbyscatteringthecentralized

antennas.

The remainder of this paper is organized as follows.

Section 2 describes the system model. Sum rate, power con-

sumption,andtheirrelationshipareanalyzedinSection 3.In

Section 4,weshowhowtousetherate-powerrelationshipfor

antenna deployment optimization and how much gain can

be obtained. Simulation results can be found in Section 5.

Finally, concluding remarks are given in Section 6.

2. SYSTEM MODEL

Beforeproceedingfurther,wefirstexplainthenotationsused

in this paper. All vectors and matrices are in boldface, XT

and XHare the transpose and the conjugate transpose of X,

respectively, Xi,jis the (i, j)th element of X, Xi,:is the ith row

of X, X:,j is the jth column of X, and E is the expectation

operator.

As illustrated in Figure 1, an isolated coverage area of ra-

dius R is considered. To describe antenna and user distribu-

tions, we use polar coordinates (r,θ) relative to the center of

thecoveragearea.TheCL-DASunderstudyconsistsofN an-

tennas which are independent and uniformly distributed on

the circle with r = a. (We do not assume deterministic de-

ployment scheme here, considering that the complex terrain

may make deploying a large number of antennas with de-

terminate positions difficult.) These antennas are connected

to the central processor via optical fibers. K single-antenna

users are mutually independent and uniformly distributed in

the coverage area excluding the radius R0neighborhood of

each antenna [9]. To describe user distribution, b is used to

denote user polar radius in the following analysis.

2.1.Signalmodel

Let xkand pkbe the transmitted signal with unit energy and

the transmit power of the kth user, respectively. Let hk ∈

CN×1denote the vector channel between the kth user and the

distributed antennas. Then, the received signal y ∈ CN×1can

be expressed as

y =

K?

k=1

hk

?pkxk+n = HP1/2x +n, (1)

where x = [x1,x2,...,xK]Tis the transmitted signal vec-

tor, P = diag(p1, p2,..., pK) is the transmit power matrix,

n ∈ CN×1is the noise vector with distribution CN(0,σ2

and H = [h1,h2,...,hK] is the channel matrix. Since anten-

nas are geographically separated, to model DAS channel, we

should encompass not only small-scale fading but also large-

scale fading. Here, we model H as

nIN),

H = L ◦ Hw, (2)

where “◦” is the Hadamard product or element-wise prod-

uct, Hw, a matrix with independent and identically dis-

tributed(i.i.d.),zeromean,unitvariance,circularlysymmet-

ric complex Gaussian entries, reflects the small-scale fading,

and L represents large-scale fading between users and anten-

nas. Adding shadowing to path loss model used in [3], we

model entries of L as

?

where Dn,kand Sn,kare independent random variables rep-

resenting the distance and the shadowing between the nth

antenna and the kth user, respectively, γ is the path loss ex-

ponent. {Sn,k | n = 1,2,...,N, k = 1,2,...,K} are i.i.d.

random variables with probability density function (pdf),

Ln,k=

D−γ

n,kSn,k,

R0≤ Dn,k< 2R ∀n,k, (3)

fS(s) =

1

√2πλσssexp

?

−(lns)2

2λ2σ2

s

?

,

s > 0, (4)

where σsis the shadowing standard derivation in dB and λ =

ln10/10. Since these Sn,ks are i.i.d., we will not distinguish

them in the following analysis and simply use S instead.

We note that the system model used in this study differs

from that in [6, 7] where the large-scale fading part L is as-

sumed to be fixed. A fixed L is applicable for performance

comparisonbetweensystemswithandwithoutcoordination,

since coordination does not impact L. However, this fixed

L does not apply to performance comparison between CAS

and DAS, since it cannot fully reflect the large-scale fading

between antennas and users for different antenna layouts.

Therefore, a stochastic L must be included, which makes our

work quite different from analyses in [6, 7].

Power allocation policy impacts system performance to a

great extent. To investigate performance of DAS, we assume

a power control scheme widely used in CDMA systems, with

Page 3

Jiansong Gan et al.3

which all users are guaranteed to arrive at the same power

level. The power control scheme is given by

pk

N?

n=1

L2

n,k= PR

∀k, (5)

where PRis the required receiving power level.

2.2. Distancedistributions

As distances between users and antennas impact the chan-

nel directly, a key problem for the performance analysis is to

investigate their characteristics. Since they are random vari-

ables, we characterize them by presenting their distribution

functions.

Althoughradiusofthecircleforantennadeploymentcan

beoptimized,itisaconstantoncechosen.Sointhefollowing

analysis, we first consider an arbitrary a and establish a rate-

powerrelationshipbetweentheasymptoticper-usersumrate

and the asymptotic total transmit power. Then, we optimize

atogetthebestperformance.Asantennasaremutuallyinde-

pendent and uniformly distributed on the circle, their polar

angles Θa1,Θa2,...,ΘaNare i.i.d. random variables with pdf,

fΘa

?θa

?=

1

2π,0 ≤ θa< 2π.

(6)

Since users cannot fall into the radius R0neighborhood

of each antenna, when there are a large number of anten-

nas, users cannot fall into the area such that the polar radius

b satisfies |b − a| < R0. As users are mutually independent

and uniformly distributed, the polar radiuses of the K users

B1,B2,...,BKare i.i.d. random variables with pdf,

fB(b) =

b

?

r∈Br dr,

b ∈ B,(7)

where B = {b | 0 ≤ b ≤ R, |b−a| ≥ R0} is the effective cov-

erage area. The polar angles of the K users Θu1,Θu2,...,ΘuK

are i.i.d. random variables with pdf

fΘu

?θu

?=

1

2π,0 ≤ θu< 2π.

(8)

We characterize the distance distributions from two as-

pects: the perspective of a user and the perspective of an an-

tenna. Consider a user indexed k, we assume its polar radius

is b and polar angle is θu. Then, the distance between this

user and the nth antenna can be expressed as

Dn,k=

?

a2+b2−2abcos?Θan−θu

?

∀n.

(9)

Since Θa1,Θa2,...,ΘaN are i.i.d. random variables with

pdf (6), distances from this user to all antennas D1,k,

D2,k,...,DN,karei.i.d.randomvariableswithcumulativedis-

tribution function (cdf):

FD|B(d | b) = Pr?D ≤ d | B = b?

⎧

⎪⎪⎪⎪⎪⎩

=

⎪⎪⎪⎪⎪⎨

0 if d < |a −b|,

if d ≥ a+b,

otherwise.

1

1

πarccos

?a2+b2− d2

2ab

?

,

(10)

Since antennas are symmetric, distance distributions for

all antennas are the same. Without loss of generality, we con-

sider an antenna indexed n. When the polar radius of the kth

user is b, distribution of Dn,kis the same as (10). Averaging

the distribution over all possible user polar radius, we get the

cdf of Dn,k,

?

FD(d) =

b∈BFD|B

?d | b?fB(b)db.

(11)

Since users are mutually independent and uniformly dis-

tributed, distances from the nth antenna to all users

Dn,1,Dn,2,...,Dn,Kare i.i.d. random variables with cdf (11).

3. ASYMPTOTIC ANALYSIS

When instantaneous channel state information is available at

the receiver side but not at the transmitter side, the sum rate,

normalized by the number of users, can be expressed as [10]

?

C =1

Klog2det

IN+1

σ2

nHPHH

?

.

(12)

Unfortunately, (12) depends on distances, shadowing, and

small-scalefadingbetweenantennasanduserswhichareran-

dom variables. Hence, it is random and difficult to evaluate.

To assess the cost of the system, we introduce a metric

called total transmit power which can be defined as

E ?

K?

k=1

pk=

K?

k=1

PR

?N

n=1L2

n,k

, (13)

where the second equality follows from (5). Equation (13) is

also a random variable depends on distances and shadowing

between antennas and users.

In this section, we prove as N,K → ∞ with K/N → β,

both the per-user sum rate and the total transmit power con-

vergetotheirrespectiveasymptoticvalues.Toprovethiscon-

vergence, we first cite some definitions and results from ran-

dom matrix theory.

3.1.Definitionsandpreliminaryresults

Definition 1. Given a vector v = [v1,v2,...,vN], its empirical

distribution function (edf) is defined as

FN(x) ?1

N

N?

n=1

I[vn,∞)(x), (14)

Page 4

4 EURASIP Journal on Wireless Communications and Networking

where IA(w) is the indicator function taking value 1 if w ∈ A

and 0 otherwise. If FN(·) converges as N → ∞, its limit (the

asymptotic edf) is denoted by F(·).

The following proposition is a reformulation of some

theorems presented in [11, 12], which provides a theoretical

support for our analysis.

Proposition 1. Consider an N × K matrix H = L ◦ Hwwith

L and Hwindependent N ×K random matrices. Entries of Hw

are independent, zero mean, unit variance, circularly symmet-

ric complex Gaussian variables. Let G = L ◦ L be the power

gain matrix of H. If for all k, edf of (G:,k)Tconverges as N → ∞

to the cdf of a random variable with expectation 1 and all mo-

ments bounded, and for all n, edf of Gn,:converges as K → ∞ to

the cdf of a random variable with expectation 1 and all mo-

ments bounded, the sum rate normalized by the number of

transmit antennas converges almost surely as N,K → ∞ with

K/N → β:

1

Klog2det

?

IN+ρ

KHHH

?

a.s.

− → C(β,ρ)

?ρ

?ρ

= log2

?

1+ρ

β−1

?

4F

β,β

??

−log2e

??

4ρ

F

?ρ

β,β

?

+1

βlog2

1+ρ −1

4F

β,β

,

(15)

where ρ is the signal-to-noise ratio (SNR) and F (·,·) is de-

fined as

F (x,z) ?

??

x?1+√z?2+1 −

?

x?1 −√z?2+1

?2.

(16)

3.2. Proofoftheconvergenceandderivationof

therate-powerrelationship

We have shown that though antenna placement and user dis-

tribution are performed in a random manner, distributions

of distances between antennas and users are known. With

such information, we investigate the distributions of entries

of channel matrix H. Since its small-scale fading part Hwis

well modeled as with independent, zero mean, unit variance,

circularly symmetric complex Gaussian entries, the rest is to

investigate its large-scale fading part. Denote the power gain

matrix of H by G with Gn,k= L2

the distributions of elements of G from two perspectives: a

user’sperspectiveandanantenna’sperspective.From(4)and

(10), we learn that for a user indexed k, G1,k,G2,k,...,GN,k

are i.i.d. random variables with cdf:

n,k= D−γ

n,kS, we characterize

FG|B

?g | b?=

?∞

0

?

1 −FD|B

??g

u

?−1/γ????b

??

fS(u)du,

(17)

given Bk = b. Then, the arithmetical average 1/N?N

can be writen as

G(a,b) ? E?G | B = b?= exp

?1

π

n=1Gn,k

?1

2λ2σ2

s

?

· E?D−γ| B = b?

(18)

= exp

2λ2σ2

s

?1

?π

0

?a2+b2−2abcosθ?−γ/2dθ,

(19)

according to (4) and (10).

From the perspective of an antenna indexed n, Gn,1,

Gn,2,...,Gn,Kare i.i.d. random variables with cdf

?∞

0

FG(g) =

?

1 −FD

??g

u

?−1/γ??

fS(u)du,(20)

according to (4) and (11).

From (5), we have

pk=

PR

?N

n=1L2

n,k

=

PR

?N

n=1Gn,k

∀k.

(21)

To evaluate (12), we rewrite it to (15)’s form and get the

power-controlled equivalent channel HE = H√KP1/2. Ac-

cording to property of HEand Proposition 1, we present the

following proposition:

Proposition 2. The uplink per-user sum rate of a CL-DAS

converges almost surely, as N → ∞,K → ∞ with K/N → β,

to

?

Proof. See appendix.

C

β,βPR

σ2

n

?

.

(22)

To characterize power consumption of a CL-DAS, we in-

vestigate the asymptotic behavior of E in (13) when there are

a large number of antenna and users. By the strong law of

large numbers and the distributions of G’s elements, we have

1

K

K?

k=1

K/N

n=1Gn,k

1/N?N

a.s.

− → β

?

b∈B

1

G(a,b)fB(b)db, (23)

as N,K → ∞,K/N → β, where G(a,b) is defined in (18).

Let

?

b∈B

we have

P(a) ?

1

G(a,b)fB(b)db, (24)

E

a.s.

− → βPRP(a).

(25)

The total transmit power represents the cost of a system,

and the per-user sum rate stands for the output. To com-

pare cost-output relationships for different antenna configu-

rations, we establish the relationship between the asymptotic

per-user sum rate and the asymptotic total transmit power

according to (22) and (25):

?

C = C

β,

E

P(a)σ2

n

?

.

(26)

Page 5

Jiansong Gan et al.5

4. DEPLOYMENT OPTIMIZATION AND

PERFORMANCE ENHANCEMENT

We learn from (26) that different a results in different rate-

power relationship. In CAS, a is fixed to be 0, while it may

vary from 0 to R for CL-DAS. Therefore, choosing the opti-

mal radius and achieving the best rate-power relationship is

possible. In this section, we will show how to decide the opti-

mal radius and how much gain an optimized CL-DAS offers

compared with a CAS.

As C(β,ρ) is a monotonic increasing function of ρ, the

optimal radius that minimizes the power consumption for a

given sum-rate requirement is the a that minimizes (24). In

the same way, this a also leads to the maximum sum rate for

a given total transmit power. So the a that minimizes (24) is

optimal for antenna deployment in the sense of both power

consumption and sum rate.

ToevaluateperformanceenhancementthatanOCL-DAS

provides over a CAS, we define two metrics. The first is the

power gain under the same sum rate constraint, defined as

?

with COCL-DAS= CCAS. The second is the per-user sum-rate

gain in the high SNR regime under the same total transmit

power constraint, defined as

Gp? 10log

ECAS

EOCL-DAS

?

,(27)

Gc∞? COCL-DAS− CCAS

(28)

with EOCL-DAS→ ∞, ECAS→ ∞, and EOCL-DAS= ECAS. As per

3dB SNR increase in the high SNR regime leads to a per-user

sum-rate increase of min(1,1/β)bits/s/Hz [12, 13], we have

Gc∞= Gp/3min(1,1/β).

4.1. Optimizationforγ = 4

As a uniform closed-form expression for G(a,b) in (19) is

hard to obtain for all γ, we deal with a typical path loss expo-

nent of 4 in this section. Then, (19) becomes:

?1

π

?1

G(4)(a,b) = exp

2λ2σ2

s

?1

?

?π

0

1

?a2+b2−2abcosθ?2dθ

a2+b2

??a2−b2??3.

= exp

2λ2σ2

s

(29)

TogetP(4)(a),wefurtherexpressthepdfofuserpolarradius

(7) as

fB(b)=2b

A,

b ∈?0,max?0,a−R0

??∪?min?R,a+R0

?,R?,

(30)

where A = (max(0,a − R0))2+ max(0,R2− (a + R0)2). Sub-

stituting (29) and (30) into (24), we have

P(4)(a) =1

A

?f(0) − f?max?0,a −R0

+ f(R) − f?min?R,a+R0

??

???,

(31)

where

f (t) = exp

?

−1

2λ2σ2

s

??t6

3−2a2t4+7a4t2

−8a6ln?a2+t2??

.

(32)

Since in a practical system R0? R, we can rewrite P(4)(a) in

(31) to the following piecewise form:

P(4)(a)

⎧

=

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

f (R) − f?a +R0

f (0) − f?a −R0

?

?+ f(R) − f?a+R0

R2−?a+R0

?a − R0

f (0) − f?a −R0

?2,

?2+R2−?a+R0

?

0 ≤ a < R0,

?

R0≤ a ≤ R −R0,

?2

,

?a −R0

?2

,

R −R0< a ≤ R.

(33)

A closed-form rate-power relationship is then obtained by

substituting (33) into (26).

For a path loss exponent of 4, the optimal ra-

dius for antenna deployment is the one that minimizes

(33). Therefore, a(4)

(∂2P(4)(a)/∂2a)|a(4)

intervals in (33). Given R and R0, we can find a(4)

cally. Consider R = 2000m and R0= 20m according to the

COST231-Walfish-Ikegami model [14], we find that a(4)

1352mandP(4)(a(4)

stituting P(4)(a(4)

OCL-DAS becomes

?

opt satisfies (∂P(4)(a)/∂a)|a(4)

opt> 0, or it is one of the endpoints of the

opt

= 0 and

optnumeri-

opt=

opt) = 5.988·1011·exp(−(1/2)λ2σ2

opt) into (26), the rate-power relationship of

s).Sub-

C(4)

OCL-DAS= C

β,E ·exp?(1/2)λ2σ2

5.988 ·1011σ2

s

?

n

?

.

(34)

CAS is a special case of CL-DAS with a = 0. According to

(33), we have P(4)(0) = 5.334 ·1012·exp(−(1/2)λ2σ2

?

5.334 ·1012σ2

Comparing (34) and (35), we find that OCL-DAS offers

a power gain of G(4)

p

= 10log(53.34/5.988) = 9.498dB or a

per-user sum-rate gain in high SNR regime of G(4)

min(1,1/β)bits/s/HzoverCAS.Wenotethatshadowingonly

impacts P(a) by a scalar multiplication of exp(−(1/2)λ2σ2

and thus does not impact the value of the optimal radius.

s) and

C(4)

CAS= C

β,E ·exp?(1/2)λ2σ2

s

?

n

?

.

(35)

c∞ = 3.166 ·

s)

4.2.Optimizationforanarbitraryγ

For most practical systems, the path loss exponent is not

an integer, which makes the closed-form expression for (24)

hard to obtain. To find the optimal radius for an arbitrary γ,

we present a numerical optimization method. This method