Diversity Gains of Generalized Distributed AntennaSystems with Cooperative Users

Yifan ChenNanyang Technological University

SingaporeEmail: yfchen@ntu.edu.sg

Chau YuenInstitute for Inforcomm Research

SingaporeEmail: yuen@ieee.org

Yan ZhangSimula Research Laboratory

NorwayEmail: yanzhang@ieee.org

Zhenrong ZhangGuangxi University

ChinaEmail: zzr76@gxu.edu.cn

Abstract— A geometry-based channel model is proposed todescribe the topology of a generalized distributed antenna systemwith cooperative diversity (GDAS-CD). The system architecturecomprises a number of largely separated access points (APs)each with multiple antennas within an AP at one side of thelink, and several geographically closed user terminals (UTs) eachhaving multiple antennas within a UT at the other side. TheUTs are assumed to be cooperative devices. The average cross-correlation of signals received from non-collocated APs and UTsis derived based on the system topology and the empirical modelsproposed by Sørensen. Subsequently, we investigate the diversitygains obtainable from the GDAS-CD based on the proposedmodel, which would provide insight into the degree of possibleperformance improvement when combining multiple copies ofthe received signal.

I. INTRODUCTION

In recent years, considerable attention has been drawn tomulti-input multi-output (MIMO) communication techniquesdue to the prospect of significant improvements of systemdiversity gains (DIV) [1], [2]. This idea has also been extendedto a generalized distributed antenna system, which refers toan MIMO system comprising multiple antennas collocatedat one user terminal (UT) and M geographically scatteredaccess points (APs) each with multiple antennas collocatedwithin an AP at the other side [3], [4]. Such system hasthe advantage of macrodiversity that is inherent to the widelyspaced antenna, and therefore offers the capability to enhancesignal quality and improve coverage. This system employs asectorized architecture where each AP has a separate feeder toa central base station (BS) [5], [6], which ensures that variousbenefits of an MIMO system such as diversity combining andinterference reduction can be realized. In this paper, we willconsider a more general scenario where N(N ≥ 1) spatiallyscattered UTs may cooperate with one another to transmit orreceive signals from M APs, which will be referred to asthe generalized distributed antenna system with cooperativediversity (GDAS-CD).

As the system diversity level is dependent on fading cor-relation of the channel matrix in the GDAS-CD, it is ofspecial interest to investigate the cross-correlation, ρ, betweentwo communication links from two ports at one side of thelink to two ports at the other side. There is currently nowell-agreed model for ρ. Two key parameters have beenconsidered in the past studies: 1) the angle-of-arrival difference

(AAD) between two AP-UT paths, and 2) the relative distancedifference. In most cases, the AAD has the strongest influenceon the measured cross-correlation [7], [8]. However, modelsconsidering both AAD and relative distance difference havealso been proposed [9]. In the current work, only the AAD-dependency is taken into account. With a geometry-basedchannel model, network level simulations of GDAS-CD couldbe facilitated by generating position-dependent correlationcoefficients. The theoretical result of the average correlationvalue will be derived, which provides critical insight into thequality of fading wireless links. Subsequently, a parsimoniousapproach is employed to investigate the DIV of GDAS-CDbased on the proposed channel model.

The paper is organized as follows. Section II presents theGDAS-CD topology and cross-correlation models. Section IIIdiscusses a parsimonious method to estimate the DIV of thesystem. Finally, some concluding remarks are given in SectionIV. The following notations will be used throughout thepaper. E(·) is the expectation operator. We will use boldfaceletters for position vectors. For a vector V, |V| represents itsmagnitude. For a set S, |S| denotes its Lebesgue measure.

II. SYSTEM TOPOLOGY AND CROSS-CORRELATION

MODEL

Fig. 1(a) shows the topology of a GDAS-CD wirelessnetwork. The deployment cell has a number of distributedAPs spaced apart by a large distance. Each AP itself is anantenna array and is connected to the central BS througheither optical fibers, coaxial cable or radio link. To synthesizethe randomness in the AP placement owing to the complexlandform of real environments, it is assumed that these APsare uniformly distributed in a circular cell with radius R. Thistopology model is similar with the random antenna layoutin [10]. However, the UTs may not locate at the center ofthe circular region, which is different from the structure in[10]. It is worth emphasizing that in real-life deployment,there could be multiple cells to serve the region of interest. Inthis initial study, we will only focus on a single-cell scenario.At the UT site, it is supposed that the message knowledgecan be ideally shared among the UTs with the help of afictitious genie if the UTs are operating in the cooperativemode and are geographically close to each other [11], [12].Eventually, they can fully cooperative with each another during

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Fig. 1. (a) Pictorial illustration of a GDAS-CD network; (b) networkarchitecture in the numerical examples

the process of transmission and reception. In the followingdiscussion, the antenna layout of the UTs will be analyzedusing a deterministic approach as the number of cooperativeUTs is usually small.

Let us consider the system geometry pictured in Fig. 1(a).The first step is to find the mean value of cross-correlationcoefficient at APs, which roughly reflects the overall degreeof correlation. Let us define the spatial locations of APsas A1,A2, · · · ,AM ∈ CA and the locations of UTs asU1,U2, · · · ,UN ∈ CU , where CA and CU represent thedeployment cell and the working cluster in a wireless network,respectively (see also Fig. 1(a)). We will derive the probabilitydensity function (pdf) of the AAD between two paths from oneUT, Un, to any two APs, Am1 and Am2 , belonging to CA.

The piecewise continuous model proposed by Sørensen isbased on the 900 MHz measurements conducted in Aarhus,Denmark [7]. The measurement site is characterized by almostuniform building height of about 4 to 5 storeys, and a gentlyrolling terrain. The independent non-parametric data boot-strap’s procedure has been applied to estimate a confidenceinterval for the correlation coefficient. The shape of cross-correlation against AAD is summarized as

ρ(∆φ) =

0.78 − 0.0056 · |∆φ|, if 0◦ ≤ |∆φ| ≤ 15◦

0.48 − 0.0056 · |∆φ|, if 15◦ ≤ |∆φ| ≤ 60◦

0, if |∆φ| ≥ 60◦(1)

where ∆φ is the AAD. This model describes the main char-acteristics that are likely to be observed in urban areas. Notethat though the actual ρ against ∆φ may change for a differentenvironment, the general methodology presented below is still

applicable.As the APs are uniformly distributed within a circle, the pdf

of angle-of-arrival (AOA) viewed from Un (n = 1, 2, · · · , N)when the UT locates outside the deployment cell is given by[13]

Case I :R

Dn≤ 1

fφ,n(φ) =

2Dn cos φ√

D2n cos2 φ−D2

n+R2

πR2 ,

− sin−1(

RDn

)≤ φ ≤ sin−1

(R

Dn

)0, otherwise

(2)

where Dn is the distance of separation between the centralBS and Un as depicted in Fig. 1(a). When the UT lies withinthe cell, R

Dn> 1. The pdf of AOA can be derived as (the

derivation is given in Appendix)

Case II :R

Dn> 1

fφ,n(φ) =R2 +D2

n + 2RDn cos(sin−1 Dn sin φ

R + φ)

2πR2

(3)

Subsequently, the pdf of AAD is obtained as [14]

f∆φ,n(∆φ) ={ ∫∞0fφ,n(∆φ+ φ)fφ,n(φ)dφ, ∆φ ≥ 0∫∞

−∆φfφ,n(∆φ+ φ)fφ,n(φ)dφ, ∆φ < 0 (4)

and the mean value of cross-correlation at any two APs in thedeployment cell is

ρA = E(ρ) =1N

N∑n=1

∫∆φ

ρ(∆φ)f∆φ,n(∆φ)d(∆φ) (5)

where the subscript A denotes that the correlation is observedat the AP site.

The next step is to derive the average cross-correlationamong the UTs. The AAD between two propagation pathsfrom any AP, A ∈ CA, to two UTs, Un1 and Un2 , can beobtained from the elementary geometry shown in Fig. 1(a)

∆ψn1,n2(A) = cos−1

(|Un1A|2 + |Un2A|2 − |Un1Un2 |2

2 |Un1A| |Un2A|

)(6)

The average cross-correlation coefficient can be expressed as

ρU =∫CA

ρ(∆ψn1,n2(A))fA(A)dA (7)

where the subscript U denotes that ρ is observed at the UTsite. ρ(∆ψn1,n2(A)) is calculated by substituting (6) into (1).fA(A) is the distribution function of APs, which can be moreconveniently expressed in polar coordinates (r, ϑ) as illustratedin Fig. 1(a). Eq. (7) can thus be reformulated as

ρU =∫ R

0

∫ 2π

0

ρ (∆ψn1,n2(r, ϑ)) fr,ϑ(r, ϑ)dϑdr

=∫ R

0

∫ 2π

0

ρ (∆ψn1,n2(r, ϑ))r

πR2dϑdr

(8)

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Fig. 2. Mean value of cross-correlation against the ratio of |oo′| to R

If the channel correlation matrix is given by the Kroneckerproduct of the transmitter and receiver correlation matrices(i.e., the widely used “Kronecker” model in conventionalMIMO systems, [15], [16]), the mean correlation betweenpaths Am1 → Un1 and Am2 → Un2 can be calculated as

ρAU = ρA × ρU (9)

Finally, to gauge the access qualities of the GDAS-CDsystem, we characterize the mean path gain as

PG =1N

N∑n=1

E(|UnA|−2)

=1N

N∑n=1

∫CA

|UnA|−2fA(A)dA

(10)

As to be shown in Section III, PG is an important parameterthat determines the system diversity gains.

In the following example, two UTs, U1 and U2, are coop-erating together to transmit and receive signals as illustrated inFig. 1(b). The radius of the deployment cell is 100 m and thedistance of separation between U1 and U2 is 10 m. We shallrepresent the centers of the circular cell and U1U2 as o and o′,respectively. It is further assumed that oo′⊥U1U2. Fig. 2 plotsthe mean cross-correlation against |oo′|

R . As |oo′|R increases, the

UT moves away from the center of the cell. The angular rangeof the AOA at the UT decreases. Therefore, ρA, ρU , and ρAU

will increase. Furthermore, ρU is much larger than ρA due tothe smaller spacing between the UT ports. It is also observedthat all the three curves feature steep transition slope when theUTs are near the boundary of the deployment cell (i.e., when|oo′|R = 1). Finally, it can be seen that as |oo′|

R increases further,the mean value of cross-correlation will eventually approachan upper bound.

Fig. 2 also plots the values of |oo′|−2

PG, which compares the

mean path gain of GDAS-CD and the conventional collocatedantenna system. Obviously, the ratio is smaller than 1 formost of the values of |oo′|

R , which implies that GDAS-CDwould achieve higher received signal-to-noise ratio (SNR).

Fig. 3. Double-directional channel impulse responses for two wireless linksν1 and ν2. Some scattering elements are common to both links, which causethe cross-correlation between ν1 and ν2

Furthermore, the minimum |oo′|−2

PGis achieved in the proximity

of |oo′|R = 1 and eventually approaches to 1 as |oo′|

R increasesfurther.

Armed with the aforementioned model preliminaries, theDIV in a GDAS-CD can be estimated as to be discussed inthe following section.

III. ANALYSIS OF THE DIVERSITY GAIN

DIV is commonly defined as the negative exponent ofSNR in the probability of error expression in the high-SNRregion [2]. To shed a light on the nature of this measure,DIV can be thought of as the redundancy of the transmittedsignal in a particular communication system. We will nowseek a parsimonious approach to estimate the diversity levelin a GDAS-CD which should be simple and yet have soundphysical basis.

Fig. 3 shows two communication links ν1 and ν2 connectingUT n1 to AP m1 and UT n2 to AP m2, respectively. A double-directional channel impulse response for path ν1 is taken tobe

hν1(θT , θR) =Lν1∑l=1

β(ν1)l δ

(θT − θ

(ν1)T,l

)δ(θR − θ

(ν1)R,l

)(11)

where {θ(ν1)T,l } and {θ(ν1)

R,l } are the angles-of-departure (AODs)and AOAs. The channel is characterized by the coefficientsβ

(ν1)l that couples the multiple transmit angles with multiple

receive angles. We will assume that the discrete model in(11) has considered the finite dimensionality of the spatialsignal space due to finite number of antenna elements andfinite array aperture (i.e., the propagation paths have beenpartitioned into resolvable AOD/AOA bins [17]). Therefore,Lν1 defines the diversity level of path ν1 and {β(ν1)

l } areindependently distributed random variables. Nonetheless, inreal-life scenarios, a diversity link will be broken if thereceived signal strength is below a power threshold Υ. We

assume that(β

(ν1)l

)2

are identical lognormal fading signals

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with pdf

f

((β

(ν1)l

)2

;µν1 , sν1

)

=1(

β(ν1)l

)2

sν1

√2π

×e−[ln(

β(ν1)l

)2−µν1

]2/2s2

ν1(12)

where µν1 and sν1 are the location and shape parameters ofthe lognormal pdf, respectively. Note that the mean path gain,PGν1 , is related to the two parameters through the followingrelationship PGν1 = eµν1+s2

ν1/2. The outage probability is

simply equal to the cumulative density function of the outputsignal

Pν1,out = Pr((

β(ν1)l

)2

≤ Υ)

=12

+12erf[ln Υ − µν1

sν1

√2

](13)

Effectively, the number of diversity paths for ν1 should bescaled by (1 − Pν1,out) to account for the outage loss.

Consider another radio link ν2 illustrated in Fig. 3. Thedouble-directional channel impulse response can be writtenfollowing the similar procedure with (11). When propagationpaths ν1 and ν2 are correlated, a number of scattering elementswill overlap between the two paths. The discrete channelmodels for both ν1 and ν2 can thus be rewritten as

hν1(θT , θR) =Lν1,ν2∑

l=1

β(ν1,ν2)l δ

(θT − θ

(ν1)T,l

)δ(θR − θ

(ν1)R,l

)

+Lν1∑

l=Lν1,ν2+1

β(ν1)l δ

(θT − θ

(ν1)T,l

)δ(θR − θ

(ν1)R,l

)(14)

hν2(θT , θR) =Lν1,ν2∑

l=1

β(ν1,ν2)l δ

(θT − θ

(ν2)T,l

)δ(θR − θ

(ν2)R,l

)

+Lν2∑

l=Lν1,ν2+1

β(ν2)l δ

(θT − θ

(ν2)T,l

)δ(θR − θ

(ν2)R,l

)(15)

The scattering coefficients associated with the first term ofsummations in (14) and (15) are identical to both paths,whereas those with the second terms are distinct and unique tothe relevant path. By introducing the following two parameters

χ(ν1)l =

(β

(ν1)l

)2

− E

((β

(ν1)l

)2)

and χ(ν2)l =

(β

(ν2)l

)2

−

E

((β

(ν2)l

)2)

, we can express the cross-correlation between

the signal fading on the two paths as

ρν1,ν2 =E

(∑Lν1l=1 χ

(ν1)l ×∑Lν2

l=1 χ(ν2)l

)√

E

[(∑Lν1l=1 χ

(ν1)l

)2]√

E

[(∑Lν2l=1 χ

(ν2)l

)2] (16)

Substituting (14) and (15) into (16) yields

ρν1,ν2 =

∑Lν1,ν2l=1

(σ

(ν1,ν2)l

)2

√∑Lν1l=1

(σ

(ν1)l

)2

×√∑Lν2

l=1

(σ

(ν2)l

)2(17)

where {σ(ν1)l } and {σ(ν2)

l } are the standard deviations oflognormally-distributed {χ(ν1)

l } and {χ(ν2)l }, respectively.

{σ(ν1,ν2)l } are the standard deviations for those scatterers

common to both wireless links. If σ(ν1)l = σ

(ν2)l = σl (i.e.,

identical and independent faders), (17) can be simplified as

ρν1,ν2 =Lν1,ν2√Lν1Lν2

(18)

which leads to Lν1,ν2 = ρν1,ν2

√Lν1Lν2 . Along the lines

of derivation for (16)-(18), we can introduce a new measurecalled as generalized cross-correlation coefficient among threeradio links ν1, ν2, and ν3

�ν1,ν2,ν3

= E

Lν1∑

l=1

χ(ν1)l ×

Lν2∑l=1

χ(ν2)l ×

Lν3∑l=1

χ(ν3)l

× 1

3

√E

[(∑Lν1l=1 χ

(ν1)l

)3] × 1

3

√E

[(∑Lν2l=1 χ

(ν2)l

)3]

× 1

3

√E

[(∑Lν3l=1 χ

(ν3)l

)3]

=∑Lν1,ν2,ν3

l=1 γ(ν1,ν2,ν3)l

3√∑Lν1

l=1 γ(ν1)l × 3

√∑Lν2l=1 γ

(ν2)l × 3

√∑Lν3l=1 γ

(ν3)l

(19)

where {γ(ν1)l }, {γ(ν2)

l }, and {γ(ν3)l } are the 3rd moment

of the lognormal faders associated with the relevant paths.{γ(ν1,ν2,ν3)

l } are the 3rd moment for those scatterers commonto all three links. If γ(ν1)

l = γ(ν2)l = γ

(ν3)l = γl, (19) can be

simplified as

�ν1,ν2,ν3 =Lν1,ν2,ν3

3√Lν1Lν2Lν3

(20)

Thus we have

Lν1,ν2,ν3 = �ν1,ν2,ν33√Lν1Lν2Lν3 (21)

In general, further extension of (19)-(21) to the scenario ofK (K > 3) wireless links is very difficult. However, it isreasonable to conjecture that for K > 3, the number ofscattering elements universal to all the links would be smalland thus can be ignored. We note that no empirical datahave been reported in the literature to characterize the valuesof �ν1,ν2,ν3 . Nevertheless, this parameter is critical to theattainable DIV in a GDAS-CD network as to be made clearin the following analysis.

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Fig. 4. Venn diagrams for DIV estimation

We will now represent the redundant copies of receivedsignals via paths ν1, ν2, and ν3 as sets Dν1 , Dν2 , and Dν3

as in Fig. 4 (Venn diagrams). Apparently, the overall DIV isgiven by the number of elements in the union of the three sets

DIV = |Dν1 ∪ Dν2 ∪ Dν3 |= |Dν1 | + |Dν2 | + |Dν3 |− |Dν1 ∩ Dν2 | − |Dν1 ∩ Dν3 | − |Dν2 ∩ Dν3 |+ |Dν1 ∩ Dν2 ∩ Dν3 |

(22)

Subsequently, the Lebesgue measure of each “diversity-paths”set can be computed as

|Dν | = (1 − Pν,out)Lν , ν ∈ {ν1, ν2, ν3}|Dνa

∩ Dνb| = (1 − max{Pνa,out, Pνb,out})×Lνa,νb

,{νa, νb} ⊂ {ν1, ν2, ν3}

|Dν1 ∩ Dν2 ∩ Dν3 |= (1 − max{Pν1,out, Pν2,out, Pν3,out})×Lν1,ν2,ν3

(23)

The final result of DIV can be obtained by substituting (23)into (22).

Extending (22) to the general situation with MN linksyields

DIV =

∣∣∣∣∣∣⋃

1≤ν≤MN

Dν

∣∣∣∣∣∣=

MN∑ν=1

|Dν | −∑

1≤(νa,νb)≤MN

|Dνa∩ Dνb

|

+∑

1≤(νa,νb,νc)≤MN

|Dνa∩ Dνb

∩ Dνc|

=MN∑ν=1

(1 − Pν,out)Lν −∑

1≤(νa,νb)≤MN

(1−

max{Pνa,out, Pνb,out})× ρνa,νb

√Lνa

Lνb

+∑

1≤(νa,νb,νc)≤MN

(1 − max{Pνa,out, Pνb,out

, Pνc,out})× �νa,νb,νc

3√Lνa

LνbLνc

(24)

It is worth emphasizing that though (24) correctly countsthe available DIV and yields a substantially correct intuitive

Fig. 5. (a) Upper and lower bounds of the average generalized cross-correlation coefficient when M takes different values and two syntheticmodels with M = 20; (b) diversity gains for the two synthetic models

picture, a large number of measurement data are required tofacilitate a realistic implementation of the formulation (e.g.,parameterizations of cross-correlations, estimation of diversitylevel at individual links etc).

In the following example, we assume the same systemtopology and architecture with the example in Section II (seealso Fig. 1(b)). There are M APs and 2 UTs in the wirelessnetwork. The number of diversity paths associated with eachindividual AP-UT link is assumed to be 5. For the lognormalfading at each link, s = 4 dB (cf. Eq. (12)) and the outagethreshold is taken to be 20 dB below the mean receivedsignal at |oo′|

R = 5. Subsequently, we derive the upper andlower bounds of the average generalized cross-correlation, �,by noting that the minimum and maximum values of overallDIV are 5(1 − Pout) and 10M(1 − Pout), respectively. Asshown in Fig. 5(a), � will increase with |oo′|

R for differentvalues of M . Similar with Fig. 2, a steep transition in theproximity of |oo′|

R = 1 is observed for all curves correspondingto different M . As could be expected, the value of � is muchless than ρ predicted in Fig. 2. Furthermore, � will decreaseand the bounds become much tighter when M increases, which

This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2007 proceedings.

802

signifies an increased sensitivity of system performance withrespect to �.

To illustrate the impact of generalized cross-correlation onDIV, two piecewise-linear synthetic models for � when M =20 have been plotted in Fig. 5(a) as well. These two modelssynthesize the various levels of curve steepness in differentranges of |oo′|

R . The corresponding DIV estimates are shownin Fig. 5(b). It can be seen that a slight difference in the shapeof � against |oo′|

R will introduce remarkable changes in thediversity gains.

IV. CONCLUSIONS

The cross-correlation model and diversity analysis for aGDAS-CD network have been presented in this paper. Startingwith a simple network topology and system architecture,the average correlation coefficients have been derived. Wehave also provided a parsimonious method for calculatingthe system diversity levels. To facilitate the estimation ofDIV, a new parameter called as generalized cross-correlationhas been introduced, which is shown to have a significantinfluence on the attainable system diversity. The proposedmodel can also serve as a convenient network simulation toolfor the performance analysis of a GDAS-CD. In future, moreempirical data are required to describe the cross-correlationagainst AAD in various service environments. Specifically,characterization of the generalized cross-correlation coefficientwould be of significant practical interest.

APPENDIX

DERIVATION OF EQUATION (3)

Fig. 6 illustrates the geometry when the UT lies within thedeployment cell. Let us consider the differential strip betweenα = φ and α = φ + ∆φ. Since the APs are assumed to beuniformly distributed within the circular cell, the area withinthe strip is proportional to the probability of AOA at the UT.With some manipulations, the area can be expressed as

A(φ ≤ α ≤ φ+ ∆φ)

=12

√R2 +D2 + 2RD cos

(sin−1 D sinφ

R+ φ

)×

√R2 +D2 + 2RD cos

(sin−1 D sin(φ+ ∆φ)

R+ φ+ ∆φ

)× sin ∆φ

(25)

Subsequently, the probability of AOA at UT is derived as

Pr(φ ≤ α ≤ φ+ ∆φ) =A(φ ≤ α ≤ φ+ ∆φ)

πR2(26)

and the pdf of AOA is given by

fφ(φ) = lim∆φ→0

Pr(φ ≤ α ≤ φ+ ∆φ)∆φ

=R2 +D2 + 2RD cos

(sin−1 D sin φ

R + φ)

2πR2

(27)

This completes the derivation.

Fig. 6. Illustration of the geometry for deriving the pdf of AOA at the UTwhen UT lies within the deployment cell

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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2007 proceedings.

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