Cross-Correlation Analysis of GeneralizedDistributed Antenna Systems with Cooperative

DiversityYifan Chen

Nanyang Technological UniversitySingapore

Email: 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— In this paper, geometry-based channel models areproposed to describe the topology of generalized distributedantenna systems with cooperative diversity (GDAS-CD). Thesystem architecture comprises M largely separated access points(APs) at one side of the link, and N geographically closeduser terminals (UTs) at the other. The UTs are assumed to beoperating in cooperative mode to enhance the system diversity.The average cross-correlation of signals received from non-collocated APs and UTs is derived based on the network topologyand the correlation models derived from the empirical data. Theanalysis is also extended to more general scenarios when theAPs are placed in a clustered form. The presented results wouldprovide useful insight into the degree of possible performanceimprovement from the GDAS-CD.

I. INTRODUCTION

Multi-input multi-output (MIMO) communication has beena subject of growing interest in recent years due to theprospect of significant improvements of system diversity gains(DIV) [1]. The concept has been generalized to a distributedantenna system, where multiple antennas are collocated at oneuser terminal (UT) and M geographically scattered accesspoints (APs) each with multiple antennas are collocated withinan AP at the other side [2], [3]. Such system employs asectorized architecture where each AP has a separate feeder toa central base station (BS) [4], [5], 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 thusimportant to investigate the cross-correlation, ρ, between twocommunication links from two APs at one side of the linkto two UTs at the other side. There is currently no well-established model for ρ. Two key parameters have beenconsidered in the past studies: 1) the angle-of-arrival differ-ence (AAD) and 2) the relative distance difference (RDD)between two AP-UT paths. In most cases, the AAD has

the strongest influence on the measured cross-correlation [6],[7]. However, models considering both AAD and RDD havealso been proposed [8]. With geometry-based channel models,network level simulations of GDAS-CD could be facilitated bygenerating position-dependent correlation coefficients derivedfrom the empirical data in [6] and [8]. The theoretical result ofthe average correlation value can be derived, which providescritical insight into the quality of fading wireless links.

In Section II of the paper, the GDAS-CD topology andcross-correlation models will be discussed. Firstly, we considera degenerate scenario when the APs are placed in a single de-ployment cell and ρ is only dependent on AAD. Subsequently,more comprehensive discussions (i.e., the cross-correlation isboth AAD- and RDD-dependent and the APs are scatteredin a clustered form) are provided. Numerical examples willbe presented in Section III to exemplify the usefulness of theproposed model. Finally, some concluding remarks are givenin Section IV. Throughout the paper, we will use boldfaceletters for position vectors. For a vector V, |V| represents itsmagnitude.

II. SYSTEM TOPOLOGY AND CROSS-CORRELATION

ANALYSIS

A. AAD-dependent Correlation

In this section, we consider the simplest scenario whenthe APs locate in a single cell and the correlation is AAD-dependent only. Fig. 1(a) shows the topology of a GDAS-CDwireless network. A number of distributed APs spaced apartby a large distance are placed in the deployment cell. EachAP itself is an antenna array and is connected to the centralBS through either optical fibers, coaxial cable or radio link.To synthesize the randomness in the AP placement owing tothe complex landform of real environments, it is assumed thatthese APs are uniformly distributed in a circular cell withradius R. This topology model is similar with the randomantenna layout in [9]. However, the UTs may not locate atthe center of the circular region, which is different from thestructure in [9]. At the UT site, it is supposed that the messageknowledge can be ideally shared among the UTs with the helpof a fictitious genie if the UTs are cooperative devices and are

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geographically close to each other [10], [11]. Eventually, theycan fully cooperative with each another during the processof transmission and reception. In the following, the antennalayout of the UTs will be analyzed using a deterministicapproach as the number of cooperative UTs is usually small.

Consider the system geometry pictured in Fig. 1(a). First ofall, we want to determine 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 respectively, as shownFig. 1(a). We will derive the probability density function (pdf)of the AAD between two paths from one UT, Un, belongingto CU , 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 [6]. A vehicle with a roof mounted omnidirec-tional antenna provided simultaneous measurements of re-ceived signals from three APs located in the downtown area.The measurement site was characterized by almost uniformbuilding height of about 4 to 5 storeys, and a gently rollingterrain. A number of measurement routes with an averagelength 400 m were defined by driving around a single or agroup of apartment blocks. The independent non-parametricdata bootstrap’s procedure has been applied to estimate aconfidence interval for the correlation coefficient. The shapeof cross-correlation against AAD is summarized as [6]

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 stillapplicable.

As the APs are uniformly distributed within a circle, thepdf of angle-of-arrival (AOA) viewed from the nth UT (n =1, 2, · · · , N) when the UT locates outside the deployment cellis given by [12]

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 central BSand the nth UT as depicted in Fig. 1(a). When the UT lieswithin the cell, the pdf of AOA can be derived as [13]

Case II :R

Dn> 1

fφ,n(φ) =R2 +D2

n + 2RDn cos(sin−1 Dn sin φ

R + φ)

2πR2

(3)

Fig. 1. (a) Pictorial illustration of a GDAS-CD; (b) network architecture inthe numerical examples

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 is

ρA =1N

N∑n=1

∫∆φ

ρ(∆φ)f∆φ,n(∆φ)d(∆φ) (5)

where the subscript A denotes that the average correlationis observed at the AP site. The next step is to derive theaverage cross-correlation among the UTs. The AAD betweentwo propagation paths from any AP, A ∈ CA, to two UTsUn1 and Un2 can be obtained from the elementary geometryshown 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 the correlation is observedat the UT site. ρ(∆ψn1,n2(A)) is calculated by substituting(6) into (1). fA(A) is the distribution function of APs, whichcan be more conveniently expressed in polar coordinates (r, ϑ)(see also 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)

310

Fig. 2. Contour plots of the correlation models derived from (a) the predictiontable in [8] and (b) Eq. (12)

If the channel correlation matrix is given by the Kroneckerproduct of the transmit and receive correlation matrices (i.e.,the widely used “Kronecker” model in conventional MIMOsystems, [15], [16]), the mean correlation between pathsAm1 → 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 (assuming thatthe pathloss exponent is 2)

PG =1N

N∑n=1

∫CA

|UnA|−2fA(A)dA (10)

As discussed in [13], PG is an important parameter thatdetermines the system diversity gains.

B. AAD/RDD-dependent Correlation

Similar with [8], [17], we introduce the following variable χto denote the distance ratio between two links with propagationdistances d1 and d2, respectively

χ =∣∣∣∣10 log10

(d1

d2

)∣∣∣∣ [dB] (11)

A cross-correlation prediction table is presented in [8], whichis based on the 900 MHz measurements performed in a small-cell urban environment in Mulhouse, French. Two APs withomnidirectional antennas were set up, where the antennas wereabout 30 m above ground level, and 10 m over the skyline ofsurrounding buildings. A vehicle equipped with roof mountedquarter-wavelength antenna measured the signals from both

APs simultaneously. The measurements were performed in thesurrounding area, up to 3 km far from the APs. A contour plotof the predicted correlation values in [8] with respect to AADand RDD are shown in Fig. 2(a) for illustration purpose.

To ensure a backward compatibility of the AAD/RDD-dependent model with the AAD-only model in (1), a generalexpression that includes (1) as a special case needs to bedefined. Two collocated APs (χ = 0 dB) have fully corre-lated shadow fading, and the correlation level decreases withincreasing χ. At some threshold point χ = χth, the correlationcoefficient reaches the minimum, which may not be zero dueto the common local scattering around the UT. The followingmodel is thus introducedρ(∆φ, χ) =

g(χ, χth, α) (0.78 − 0.0056|∆φ| + σ1) + σ2,if 0◦ ≤ |∆φ| ≤ 15◦

g(χ, χth, α) (0.48 − 0.0056|∆φ| + σ1) + σ2,if 15◦ ≤ |∆φ| ≤ 60◦

g(χ, χth, α)σ1 + σ2, if |∆φ| ≥ 60◦

(12)

where

g(χ, χth, α) =

{ (1 − χ

χth

)α

, if χ ≤ χth

0, if χ > χth

(13)

and χth denotes the point where the RDD-dependent corre-lation reaches its minimum value. α controls the shape ofthe distance-dependent function. σ1 is related to the standarddeviation of the correlation at each AAD, which was foundto be approximately 0.1 on average in [6]. σ2 quantifies theminimum correlation coefficient due to the common localscattering around the UT. A contour plot of the functionin (12) is depicted in Fig. 2(b) for α = 1, χth = 4 dB,σ1 = 0.1, and σ2 = 0.1. Parallel trend with the pattern shownin Fig. 2(a) can be observed from Fig. 2(b). To further verifythe applicability of the proposed model, we have performedsimulation studies by considering a similar cell layout withthe measurement area in [6] (see also Fig. 3(a) and 3(c)). Thefollowing model parameters have been assumed: α = 0.5,χth = 7 dB, σ1 = 0.1, and σ2 = 0. Comparing Fig. 3(b) toFig. 3(d) shows that, in general the model in (12) can capturethe main characteristics of scattered correlations at each AAD.Finally, it is worth noting that (12) reduces to (1) for α = 0,σ1 = 0, and σ2 = 0.

Subsequently, the average correlation between two pathsfrom Un to any two APs, Am1 and Am2 , can be derivedasρA

=1N

N∑n=1

∫∫ρ (∆φ (Am1 ,Am2 ,Un) , χ (Am1 ,Am2 ,Un))

×fAm1(Am1) fAm2

(Am2)dAm1dAm2

=1N

N∑n=1

1(πR2)2

∫∫∫∫ρ(∆φ(x1, y1, x2, y2, xn, yn),

χ(x1, y1, x2, y2, xn, yn))dx1dy1dx2dy2(14)

311

Fig. 3. (a) Relative positions of BSs and measurement area and (b) measured correlation coefficients in [6]; (c) hypothetical measurement site and (d) thecorresponding correlation coefficients generated from Eq. (12) in the simulation study

where (x1, y1), (x2, y2), and (xn, yn) are the Cartesian coor-dinates of Am1 , Am2 , and Un respectively.

The mean cross-correlation among UTs is calculated in thesame manner as (8) with inclusion of RDD-dependent cross-correlation:

ρU =∫ R

0

∫ 2π

0

ρ (∆ψn1,n2(r, ϑ), χn1,n2(r, ϑ)) fr,ϑ(r, ϑ)dϑdr

=∫ R

0

∫ 2π

0

ρ (∆ψn1,n2(r, ϑ), χn1,n2(r, ϑ))r

πR2dϑdr

(15)

with χn1,n2 being the distance ratio between the paths A →Un1 and A → Un2 .

C. Multi-Cell Scenario

In real-life deployment, there could be multiple cells toserve the region of interest. For example, the candidate lo-cations of APs may be constrained by the office layout orbuilding structure, which causes clustered placement of APs.In order to assess the effect of multiple deployment cells on theaverage cross-correlation, a hypothetical system architecture isconsidered in this paper, as depicted in Fig. 4(a). The APs areuniformly distributed in a number of circular microcells, andeach AP is connected to a central BS through a separate feeder.Fig. 4(b) illustrates some simplified network topologies, wherethe vertices of each regular polygon (equilateral triangle,square, regular pentagon, etc) correspond to the centers ofcircular microcells. To derive the values of ρA, ρU , and ρAU ,

we simply change the circular integration region in (5), (8),(14), and (15) to the multiple circular regions in Fig. 4(b).

III. NUMERICAL EXAMPLES AND DISCUSSIONS

A. Network Architecture with Single Deployment Cell

Fig. 4. (a) Pictorial illustration of a multi-cell network; (b) network topologyin the numerical example

312

Fig. 5. (a) Mean value of cross-correlation against the ratio of |oo′| to R.Curves marked with triangles correspond to the mean cross-correlation derivedfrom the AAD/RDD-dependent model in (12); (b) influence of various multi-cell structures where the x-axis indicates the index of different topologicalmodels shown in Fig. 4(b): 1. Triangle, 2. Square 3. Pentagon, 4. Hexagon,5. Heptagon, 6. Octagon

In the following example, two UTs, U1 and U2, are coop-erating together to transmit and receive signals as illustratedin Fig. 1(b). The radius of the deployment cell is 100 m andthe distance of separation between U1 and U2 is 10 m. Weshall represent the centers of the circular cell and U1U2 aso and o′, respectively. It is further assumed that oo′⊥U1U2.Fig. 5(a) plots the mean cross-correlation against |oo′|

R for both

the AAD-only and AAD/RDD-dependent models. As |oo′|R

increases, the UTs move away from the center of the cell. Theangular range of the AOA at the UT decreases. Therefore, ρA,ρU , and ρAU will increase. Furthermore, ρU is much largerthan ρA due to the smaller spacing between the UT ports. It isalso observed that all the three curves feature steep transitionslope when the UTs are near the boundary of the deploymentcell (i.e., when |oo′|

R = 1). Another important observation isthat, with inclusion of the RDD dependency, the mean cross-correlation will increase for both ρA and ρU . Finally, it can beseen that as |oo′|

R increases further, the mean correlation willeventually approach an upper bound.

Fig. 5(a) also shows the values of |oo′|−2

PG, which compares

the mean path gain of GDAS-CD and the conventional col-located antenna system. Obviously, the ratio is smaller than1 for most of the values of |oo′|

R , which implies that GDAS-CD would achieve higher received signal strength as comparedto the collocated antenna system. Furthermore, the minimum|oo′|−2

PGis achieved in the proximity of |oo′|

R = 1 and eventually

approaches to 1 as |oo′|R increases further.

B. Network Architecture with Multiple Deployment Cells

In this section, we investigate the influence of multiplecells on the system correlation level. The network topologiesconsidered in the numerical example are illustrated in Fig.4(b) with L = 100 m, l = 20 m, and oo′ = 100 m. Note thatL quantifies the overall spread of the AP distribution and ldefines the size of each microcell. As can be seen from Fig.5(b), the impact of various topological structures on the meancorrelation is insignificant. This phenomenon is also observedfor other values of oo′. Following from the results in Figs. 5(a)

and 5(b), it is expected that |oo′|L would be the most dominant

design parameter among others such as the number and sizeof microcells.

IV. CONCLUSIONS

We have analyzed the cross-correlation of a GDAS-CDnetwork. Starting with a simple network topology and systemarchitecture, the average correlation coefficients have beenderived. We have also addressed more general scenarios whenthe APs are grouped into multiple clusters and the cross-correlation is both AAD- and RDD-dependent. The proposedmodel can also serve as a convenient network simulation toolfor the performance analysis of a GDAS-CD network.

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