388 ieee journal on selected areas in communications, vol....

388 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 30, NO. 2, FEBRUARY 2012

Cooperative MIMO Channel Modeling andMulti-Link Spatial Correlation Properties

Xiang Cheng, Member, IEEE, Cheng-Xiang Wang, Senior Member, IEEE, Haiming Wang, Xiqi Gao, SeniorMember, IEEE, Xiao-Hu You, Senior Member, IEEE, Dongfeng Yuan, Senior Member, IEEE, Bo Ai, Senior

Member, IEEE, Qiang Huo, Ling-Yang Song, Member, IEEE, and Bing-Li Jiao Member, IEEE

Abstract—In this paper, a novel unified channel model frame-work is proposed for cooperative multiple-input multiple-output(MIMO) wireless channels. The proposed model framework isgeneric and adaptable to multiple cooperative MIMO scenar-ios by simply adjusting key model parameters. Based on theproposed model framework and using a typical cooperativeMIMO communication environment as an example, we derivea novel geometry-based stochastic model (GBSM) applicable tomultiple wireless propagation scenarios. The proposed GBSMis the first cooperative MIMO channel model that has theability to investigate the impact of the local scattering density(LSD) on channel characteristics. From the derived GBSM, thecorresponding multi-link spatial correlation functions are derivedand numerically analyzed in detail.

Index Terms—Cooperative MIMO channels, geometry-basedstochastic model, spatial correlation, non-isotropic scattering.

Manuscript received 15 February 2011; revised 20 July 2011. The authorswould like to acknowledge the support from the RCUK for the UK-ChinaScience Bridges Project: R&D on (B)4G Wireless Mobile Communications.The work of X. Cheng was jointly supported by the National NaturalScience Foundation of China (Grant no. 61101079), the Science Founda-tion for the Youth Scholar of Ministry of Education of China (Grant no.20110001120129), the State Key Laboratory of Rail Traffic Control and Safety(RCS 2010K009), Beijing Jiaotong University, the Key Laboratory of WirelessSensor Network & Communication (Grant no. 2011004), Shanghai Instituteof Microsystem and Information Technology, Chinese Academy of Sciences,Important National Science & Technology Specific Project of China (Grant no.2010ZX03001-002), Major Program of National Natural Science Foundationof China (60830001), Program for Changjiang Scholars and InnovativeResearch Team in University (IRT 0949), Program for New Century ExcellentTalents in University (NCET-09-0206), and State Key Laboratory of RailTraffic Control and Safety (RCS 2008ZZ006), Beijing Jiaotong University. C.-X. Wang would also like to acknowledge the support by the Scottish FundingCouncil for the Joint Research Institute in Signal and Image Processing withthe University of Edinburgh, as part of the Edinburgh Research Partnership inEngineering and Mathematics (ERPem), and by the Opening Project of KeyLaboratory of Cognitive Radio and Information Processing (Guilin Universityof Electronic Technology), Ministry of Education (Grant No.: 2011KF01).The work of H.-M. Wang was supported in part by the National Science andTechnology Major Project of China under Grant 2009ZX03007-003-02.X. Cheng, Q. Huo, S.-L. Yang, and B.-L. Jiao are with the School

of Electronics and Computer Science, Peking University, Beijing 100871,China (e-mail: {xiangcheng, qiang.huo, lingyang.song, jiaobl}@pku.edu.cn).X. Cheng is also with the State Key Laboratory of Rail Traffic Controland Safety, Beijing Jiaotong University, and Key Laboratory of WirelessSensor Network & Communication, Shanghai Institute of Microsystem andInformation Technology, Chinese Academy of Sciences.C.-X. Wang is with the Joint Research Institute for Signal and Image

Processing, School of Engineering & Physical Sciences, Heriot-Watt Uni-versity, Edinburgh EH14 4AS, UK (e-mail: [email protected]).(corresponding author)H. Wang, X. Gao, and X.-H. You are with the School of Information

Science and Engineering, Southeast University, Najing, 210096, China (e-mail: {hmwang, xqgao, xhyu}@seu.edu.cn).D. Yuan is with the School of Information Science and Engineering,

Shandong University, Jinan, 250100, China (e-mail: [email protected]).B. Ai is with the State Key Laboratory of Rail Traffic Control

and Safety, Beijing Jiaotong University, Beijing 100044, China (e-mail:[email protected]).Digital Object Identifier 10.1109/JSAC.2012.120218.

I. INTRODUCTION

CONVENTIONAL multiple-input multiple-output(MIMO) technology, known as point-to-point MIMO,

has been widely used in many standards [1]– [3] due to itsability to significantly enhance the performance of wirelesscommunication systems [4]– [5]. Research on cooperativeMIMO technologies has recently received much attention[6]– [8]. Cooperative MIMO, also known as virtual MIMOor distributed MIMO, groups multiple radio devices toform virtual antenna arrays so that they can cooperatewith each other by exploiting the spatial domain of mobilefading channels. As a new emerging technology, manyresearch challenges in cooperative communications haveto be addressed before the wide deployment. Detailedknowledge about the underlying propagation channels andthe corresponding channel models are the fundamental tomeet those challenges for the better design of cooperativeMIMO systems [9].Several papers have reported measurements of various sta-

tistical properties of cooperative MIMO channels for differentscenarios. A few indoor cooperative channel measurementswere reported in [10], where the cooperative nodes are allstatic. Mobile multi-link measurements were presented in [11]for indoor cooperative MIMO channels. Outdoor cooperativeMIMO channel measurements were addressed in [12] and[13] for static nodes and mobile nodes, respectively. All thesemeasurement campaigns concentrated on the investigation ofthe channel characteristics of individual links for differentscenarios, such as path loss, shadow fading, and small scalefading. Unlike conventional point-to-point MIMO systems,cooperative MIMO systems consist of multiple radio linksthat may exhibit strong correlations, e.g., base station (BS)-BS, BS-relay station (RS), RS-RS, RS-mobile station (MS),BS-MS, and MS-MS links. The correlation of multiple linksexists due to the environment similarity arising from commonshadowing objects and scatterers contributing to different linksand can significantly affect the performance of cooperativeMIMO systems. The investigation of the correlations betweendifferent links is rare in the current literature.The multi-link correlation consists of large scale fading

correlation and small scale fading correlation. Only a fewpapers have analyzed and modeled large scale fading cor-relations, including shadow fading correlation, delay spreadcorrelation, and azimuth correlation. The 3rd GenerationPartnership Project (3GPP) Spatial Channel Model (SCM)[14], the Wireless World Initiative New Radio Phase II(WINNER II) channel model [15], and the IEEE 802.16j

0733-8716/12/$25.00 c© 2012 IEEE

CHENG et al.: COOPERATIVE MIMO CHANNEL MODELING AND MULTI-LINK SPATIAL CORRELATION PROPERTIES 389

channel model [16] all investigated and modeled large scalefading correlations of different links for multiple scenarios.However, as mentioned in [17], these correlation models arenot consistent and a unified correlation model for large scalefading is necessary. Recently, in [18] a unified frameworkthat can investigate both static and dynamic shadow fadingcorrelations was proposed for indoor and outdoor-to-indoorscenarios. There are even fewer papers available investigatingsmall scale fading correlations. In [19], the authors proposed amultiuser MIMO channel model focusing on the investigationof the impact of surface roughness on spatial correlations.In [17], a preliminary investigation on spatial correlationsfor coordinated multi-point (CoMP) transmissions was re-ported. The investigation on spatial correlations of multi-link propagation channels in amplify-and-forward (AF) relaysystems was reported in [20]. However, all the aforementionedinvestigations on multi-link spatial correlations are scenario-specific. For example, [19] only modeled the scenario wherescatterers are located in streets, [17] only focused the CoMPscenario, and [20] only investigated the AF relay scenario.A unified channel model framework to investigate multi-link small scale fading correlations for different scenarios istherefore highly desirable.

To fill the aforementioned gap, this paper proposes a unifiedchannel model framework for cooperative MIMO systems andinvestigates spatial correlations of different links in multiplescenarios. The main contributions and novelties of this paperare listed as follows.

1) We propose a wideband unified channel model frame-work that is suitable to mimic different links in coop-erative MIMO systems, such as the BS-BS/RS/MS link,RS-RS/MS link, and MS-MS link. Due to different localscattering environments around BSs, RSs, and MSs, ahigh degree of link heterogeneity or variations is expectedin cooperative MIMO systems. In this paper, we areinterested in various cooperative MIMO environmentswhich can be classified based on the physical scenariosand application scenarios. The physical scenarios includeoutdoor macro-cell, micro-cell, pico-cell, and indoor sce-narios. Each physical scenario further includes 3 appli-cation scenarios, i.e., BS cooperation, MS cooperation,and relay cooperation. Therefore, 12 cooperative MIMOscenarios are considered in this paper and the proposedframework can be adapted to the 12 scenarios by simplyadjusting key model parameters.

2) Taking a cooperative relay system, which includes threelinks (BS-RS, RS-MS, and BS-MS), as an example, weshow how to apply the proposed channel model frame-work and derive a novel geometry-based stochastic model(GBSM) for multiple physical scenarios. The proposedGBSM is the first cooperative MIMO channel model thathas the ability to mimic the impact of the local scatteringdensity (LSD) on channel characteristics.

3) From the proposed GBSMs and taking the spatial cor-relation between the BS-RS link and the BS-MS linkas an example, we further derive the multi-link spatialcorrelation functions.

II. A UNIFIED COOPERATIVE MIMO CHANNEL MODEL

FRAMEWORK

Cooperative MIMO channel measurements [18], [21] haveclearly demonstrated that the degree of link heterogeneityin cooperative MIMO systems is highly related to localscattering environments around different devices. Therefore,the cooperative MIMO model framework needs to reflect theinfluence of different local scattering environments on thelink heterogeneity for different scenarios while keeping theacceptable model complexity.Let us now consider a general wideband cooperative MIMO

system where all nodes are surrounded by local scatterers anda link between Node A and Node B is presented as shownin Fig. 1. It is assumed that each node can be in motion andis equipped with L antenna elements. The proposed unifiedchannel model framework expresses the channel impulse re-sponse (CIR) between the pth antenna in Node A and the qthantenna in Node B as the superposition of line-of-sight (LoS)and scattered rays

hpq (t, τ) = hLoSpq (t, τ) +I∑i=1

fI (i)∑g=1

higpq (t, τ) (1)

where I ≥ 1 is the number of related local scattering areas,fI(i) = I!

(I−i)!·i! denotes the total number of i-bounced com-ponents, and higpq (t, τ) represents the gth scattered componentconsisting of i-bounced rays. For example, h21

pq (t, τ) denotesthe first double-bounced component. It is worth noting thatthe parameter fI(i) is obtained not purely based on thenumber of related local scattering areas, but also accordingto the following practical criterion: the i-bounced waves arealways bounced by i scatterers located in different localscattering areas from far to near relative to the receiver. Basedon this practical criterion, some i-bounced components arenot necessarily to be considered, which makes the proposedmodel more practical. For the cooperative communicationenvironment shown in Fig. 1 with I=4, the proposed modelframework consists of the LoS component, f4(1)=4 single-bounced components, f4(2)=6 double-bounced components,f4(3)=4 triple-bounced components, and f4(4)=1 quadruple-bounced component. While other multi-bounced componentsthat violate the aforementioned practical criterion are notincluded in the proposed model framework, such as theAp − SB − SC −Bq double-bounced component.In the proposed model framework (1), the LoS component

of the CIR is deterministic and can be expressed as [22]

hLoSpq (t, τ)=

√KpqΩpqKpq + 1

e−j2πλ−1χpqej2πf

Amaxt cos(αLoS

pq −γA)

×ej2πfBmaxt cos(φLoS

pq −γB)δ(τ − τLoS) (2)

where χpq is the travel path of the LoS waves through thelink between Ap and Bq (Ap−Bq link), τLoS denotes the LoStime delay, and λ is the wave length with λ = c/f where c isthe speed of light and f is the carrier frequency. The symbolsKpq and Ωpq designate the Ricean factor and the total powerof the Ap−Bq link, respectively. Parameters fAmax and fBmax


.

BS

DS

CS

AS

A

B

C

D

A

B

C

D

p

q

LoS Component : ( , )LOS

p q pqA B h t

i=1, Single-bounced

Components

11 12

13 14

: ( , ) : ( , )

: ( , ) : ( , )

p B q pq p C q pq

p D q pq p A q pq

A S B h t A S B h t

A S B h t A S B h t

i=2, Double-bounced

Components

21 22

23 24

25 26

: ( , ) : ( , )

: ( , ) : ( , )

: ( , ) : ( , )

p A B q pq p A C q pq

p A D q pq p D B q pq

p D C q pq p C B q pq

A S S B h t A S S B h t



i=3, Triple-bounced

Components

31 32

33 34

: ( , ) : ( , )

: ( , ) : ( , )

p A D B q pq p A D C q pq

p A C B q pq p D C B q pq

A S S S B h t A S S S B h t

A S S S B h t A S S S B h t

i=4,

Quadruple-bounced

Components

41: ( , )p A D C B q pqA S S S S B h t

Fig. 1. Geometry of a unified cooperative MIMO channel model framework.

are the maximum Doppler frequency with respect to Node Aand Node B, respectively, γA and γB are the angles of motionwith respect to Node A and Node B, respectively, and αLoSpq

and φLoSpq denote the angles of arrival/departure of the LoSpath with respect to Node A and Node B, respectively. Thescattered component of the CIR in (1) can be shown as [22]

higpq (t, τ)=

√ηigpqΩpqKpq + 1

lim{Ng

k}ik=1→∞

{Ngk}i

k=1∑{ng

k}ik=1=1

1√∏ik=1N

gk

×ej„ψ{n

gk}i

k=1−2πλ−1χ

pq,{ngk}i

k=1

«

×ej2πfAmaxt cos

„α

pq,{ngk}i

k=1−γA

«

×ej2πfBmaxt cos

„φ

pq,{ngk}i

k=1−γB

«δ(τ − τ{ng

k}ik=1

) (3)

where Ngk is the number of effective scatterers in the kth local

scattering area with respect to the gth i-bounced component,{χpq,ng

k}ik=1 is the travel path of the gth i-bounced waves

through the Ap−Bq link, {τngk}ik=1 denotes the time delay of

the multipath components. The phases {ψnk}ik=1 are indepen-

dent and identically distributed (i.i.d.) random variables withuniform distributions over [−π, π) and determined by scatter-ers {SnK}ik=1, {Xnk

}ik=1 represents Xn1 , Xn2 , Xn3 , ..., Xni ,and {αpq,ng

k}ik=1 and {φpq,ng

k}ik=1 denote angles of ar-

rival/departure of a i-bounced path with respect to Node A andNode B, respectively. Here, ηigpq is a energy-related parameterspecifying how much the gth i-bounced rays contribute to thetotal scattered power Ωpq/(Kpq+1). Note that energy-relatedparameters satisfy

∑Ii=1

∑fI (i)g=1 ηigpq=1.

It is clear that the proposed unified channel model frame-work in (1) can naturally include the impact of local scatteringarea on channel characteristics with the help of properlychoosing fI(i) i-bounced components. Note that parameterfI(i) is related to the number of related local scatteringareas I , which is determined by physical environments, out-door macro-cell, micro-cell, pico-cell, and indoor. This meansby properly adjusting the parameter I , the proposed model


framework is suitable for different basic cooperative environ-ments. Furthermore, the proposed model framework can modelmultiple links with different degrees of link heterogeneitydue to different application scenes (e.g., BS cooperation, MScooperation, and relay cooperation) for a typical cooperativeenvironment by simply adjusting the Ricean factor Kpq andenergy-related parameters ηigpq . How to properly set these keymodel parameters, i.e., I , Kpq , ηigpq , will be explained in thenext section.

III. A NEW MIMO GBSM FOR COOPERATIVE RELAYSYSTEMS

Without loss of generality, this section considers a wide-band cooperative relay communication environment that in-cludes three different links: BS-RS, RS-MS, and BS-MS, toimplement the proposed cooperative MIMO channel modelframework. Note that the designed cooperative MIMO GBSMcan be easily extended to other cooperative MIMO scenarioswith multiple relays. The RS can be another BS for BScooperation or another MS for MS cooperation. In order topropose a generic cooperative MIMO GBSM that is suitablefor the aforementioned 12 cooperative scenarios, we assumethat the BS, RS, and MS are all surrounded by local scatterers.Fig. 2 shows the geometry of the proposed cooperative MIMOGBSM, combining the LoS components and scattered compo-nents. To keep the readability of Fig. 2, the LoS componentsare not shown. It is assumed that the BS, RS, and MS are allequipped with AB = AR = AM = 2 uniform linear antennaarrays. The local scattering environment is characterized bythe effective scatterers located on circular rings. Supposethere are N1 effective scatterers around the MS lying ona circular ring of radius R1n1 ≤ ξMn1

≤ R1n2 and then1th (n1 = 1, ..., N1) effective scatterer is denoted by Sn1 .Similarly, assume there are N2 effective scatterers around theRS lying on a circular ring of radius R2n1 ≤ ξRn2

≤ R2n2

and the n2th (n2 = 1, ..., N2) effective scatterer is denotedby Sn2 . For the local scattering area around BS, N3 effectivescatterers lie on a circular ring of radius R3n1 ≤ ξBn3

≤ R3n2

and the n3th (n3 = 1, ..., N3) effective scatterer is denoted bySn3 . The parameters in Fig. 2 are defined in Table I.As this paper only focuses on the investigation of multi-

link spatial correlations (not time or frequency correlations),we will neglect t and τ in (1) for the proposed channel modelframework to simplify notations. Due to the page limit, thespatial correlation properties between the BS-RS link and BS-MS link are chosen as an example for detailed investigation.Therefore, in the following, we will show the channel gains ofthe BS-RS link and BS-MS link for the proposed cooperativeMIMO GBSM.

A. BS-RS(MS) link

The channel gain of the BS-RS(MS) link between antennap3 at the BS and antenna p2 at the RS(p1 at the MS) can beexpressed as

hp3p2(p1) = hLoSp3p2(p1)+

3∑i=1

f3(i)∑g=1

higp3p2(p1)(4)

TABLE ITABLE I. DEFINITION OF PARAMETERS IN FIG. 2.

D1, D2, D3 distances of BS-MS, RS-MS, and BS-RS, respectivelyR1n1 , R2n1 ; R1n2 , min and max radii of the circular rings aroundR2n2 ; R1n3 , R2n3 the MS, RS and BS, respectively

θ, θ′ angles between the RS-MS link and BS-MS link, andbetween the BS-RS link and BS-MS link, respectively

δ1, δ2, δ3 antenna element spacings of MS, RS and BS, respectivelyβ1, β2, β3 orientations of the MS, RS and RS antenna arrays in

the x-y plane (relative to the x-axis), respectivelyα1ni

, α2ni, azimuth angles of Sni

-MS, Sni-RS, and Sni

-BS linksand α3ni

in the x-y plane (relative to the x-axis), respectivelyξB

n1, ξB

n2, ξB

n3distances d(BS,Sn1 ), d(BS,Sn2 ), andd(BS,Sn3), respectively

ξRn1, ξR

n2, ξR

n3distances d(RS,Sn1 ), d(RS,Sn2), and

d(RS,Sn3), respectivelyξM

n1, ξM

n2, ξM

n3distances d(MS,Sn1 ), d(MS,Sn2), andd(MS,Sn3), respectively

εping (εngpi), distances d(pi, Sng ), d(pi, pj), and

εpipj, and εngnk

d(Sng , Snk), respectively

where hLoSp3p2 and hLoSp3p1 denote the LoS component of the BS-

RS link and the BS-MS link, respectively, and higp3p2 and higp3p1

represent the gth i-bounced component of the BS-RS link andthe BS-MS link, respectively, with the following expressions

hLoSp3p2(p1)=

√Kp3p2(p1)Ωp3p2(p1)

Kp3p2(p1) + 1e−j2πλ

−1χp3p2(p1) (5)

h1gp3p2(p1)

=

√√√√η1gp3p2(p1)

Ωp3p2(p1)

Kp3p2(p1) + 1lim

Ng→∞

Ng∑ng=1

1√Ng

×ej(ψng−2πλ−1χp3p2(p1),ng ) (6)

h2gp3p2(p1)

=

√√√√η2gp3p2(p1)

Ωp3p2(p1)

Kp3p2(p1) + 1lim

Ng1 ,Ng2→∞

Ng1 ,Ng2∑ng1 ,ng2=1

× 1√Ng1Ng2

ej

“ψng1 ,ng2

−2πλ−1χp3p2(p1),ng1 ,ng2

”(7)

h31p3p2(p1)

=

√η31p3p2(p1)

Ωp3p2(p1)

Kp3p2(p1) + 1lim

N1,N2,N3→∞

N1,N2,N3∑n1,n2,n3=1

× 1√N1N2N3

ej(ψn1,n2,n3−2πλ−1χp3p2(p1),n1,n2,n3) (8)

where g=1, 2, 3, {g1, g2}={3, 2} for g=1, {g1, g2}={3, 1}for g=2, and {g1, g2}={1, 2} for g=3. In (5)–(8), χp3p2(p1)=εp3p2(p1), χp3p2(p1),ng

=εp3ng+εngp2(p1), χp3p2(p1),ng1 ,ng2=

εp3ng1+εng1ng2

+εng2p2(p1), and χp3p2(p1),n1,n2,n3 = εp3n3+εn3n1(n2)+εn1n2 +εn2(n1)p2(p1) are the travel times of thewaves through the link Bp3−Rp2(Mp1), Bp3−Sng−Rp2(Mp1),Bp3 −Sng1

−Sng2−Rp2(Mp1), and Bp3 −Sn3−Sn1(Sn2)−

Sn2(Sn1)−Rp2(Mp1), respectively. The symbols Kp3p2(p1)

and Ωp3p2(p1) designate the Ricean factor and the total powerof the BS-RS(MS) link, respectively. Parameters η1g

p3p2(p1),

η2gp3p2(p1)

, and η31p3p2(p1)

specify how much the single-, double-,and triple-bounced rays contribute to the total scattered powerΩp3p2(p1)/(Kp3p2(p1)+1) with

∑3g=1(η

1gp3p2(p1)

+η2gp3p2(p1))+

η31p3p2(p1)

=1. The phases ψng , ψng1 ,ng2, and ψn1,n2,n3 are i.i.d.

random variables with uniform distributions over [−π, π).From Fig. 2 and based on the normally used assumption

min{D1, D2, D3} � max{δ1, δ2, δ3} [22] and the application


.

BS

RS

MS

Fig. 2. The proposed cooperative MIMO GBSM.

of the law of cosines in appropriate triangles, the distances in(5)–(8) can be expressed as εp3p2 ≈ D3 − δ3

2 cos(β3 − θ′) +δ22 cos(β2 − θ′), εp3p1 ≈ D1−δ3

2 cosβ3 + δ12 cos(β1), εp3ng ≈

ξBng− δ3

2 cos(β3 − α1ng ), εngp2 ≈ ξRng− δ2

2 cos(β2 − α2ng ),εngp1 ≈ ξMng

− δ12 cos(β1 − φng ), εn1n2 = [(ξRn1

)2+(ξRn2)2−

2ξRn1ξRn2

cos(α2n1−α2n2)]1/2, and εn3n� = [(ξBn3)2+(ξBn�

)2−2ξBn3

ξBn�cos(α3n3−α1n�)]1/2, where ξBn1

= [D21 + (ξMn1

)2 +2D1ξ

Mn1

cosα1n1 ]1/2, ξRn1

= [D22+(ξMn1

)2+2D2ξMn1

cos(α1n1+θ)]1/2, ξBn2

=[D23+(ξRn2

)2+2D3ξRn2

cos(α2n2 − θ′)]1/2, ξRn3=

[D23+(ξBn3

)2−2D3ξBn3

cos(α3n3 − θ′)]1/2, ξRn2∈ [R1n2 , R2n2 ],

ξBn3∈ [R1n3 , R2n3 ], � = 1, 2, and g = 1, 2, 3. Note that

the AoD α3n1 , α3n2 , α3n3 and AoA α2n1 , α2n2 , α2n3

are independent for double- and triple-bounced rays, whilethey are interdependent for single-bounced rays. It is worthhighlighting that scatterers Sng around MS, RS, and BS arerelevant to the angles α1n1 , α2n2 , and α3n3 , respectively.Therefore, all other AoDs and AoAs have to be related tothe aforementioned three key angles. By following the generalmethod given in [22], the relationship of the key angles withother AoAs and AoDs of BS-RS link can be obtained as:sinα3n1 =

ξMn1ξB

n1sinα1n1 , sin(α2n1 + θ) =

ξMn1ξR

n1sin(θ + α1n1),

sin(α3n2 − θ′) =ξR

n2ξB

n2sin(α2n2 − θ′), and sin(α2n3 − θ′) =

ξBn3ξR

n3sin(α3n3 − θ′). The channel gain of the RS-MS link can

be similarly obtained and omitted here for brevity.In the literature, different scatterer distributions have been

proposed to characterize the key angles α1n1 , α2n2 , and α3n3 ,such as the uniform and Gaussian PDFs [23]. In this paper,the von Mises PDF [24] is used, which is more generic andcan approximate all the aforementioned PDFs [22]. The vonMises PDF is defined as f(φ) Δ= exp[k cos(φ−μ)]/[2πI0 (k)],where φ∈ [−π, π), I0(·) is the zeroth-order modified Besselfunction of the first kind, μ∈[−π, π) accounts for the mean

value of the angle φ, and k (k≥0) is a real-valued parameterthat controls the angle spread of the angle φ. In this paper,for the key angles, i.e., the α1n1 , α2n2 , and α3n3 , we useappropriate parameters (μ and k) of the von Mises PDF as μ1

and k1, μ2 and k2, and μ3 and k3, respectively.

B. Adjustment of Key Model Parameters

The proposed cooperative MIMO GBSM is adaptable tothe above mentioned 12 cooperative scenarios for this in-terested typical cooperative MIMO environment by adjustingkey model parameters. From previous section, we know thatthese important model parameters are the number of localscattering environment I , Ricean factors Kp3p2 , Kp3p1 , Kp2p1

that denotes the Ricean factor of the RS-MS link, and energy-related parameters ηigp3p2 , η

igp3p1 , and η

igp2p1 that is the energy-

related parameter of the RS-MS link. The parameter setting ofI is basically based on basic scenario. For outdoor micro-cell,pico-cell, and indoor scenarios, we assume that the BS, RS,and MS are all surrounded by local scattering area as shownin Fig. 2 and thus I = 3 in this case. For outdoor Marco-cell scenario, the BS is free of scatterers and thus I = 2.In this case, the channel model can also be obtained fromthe proposed model by setting the energy-related parametersrelated to the local scatterers around BS equal to zero, e.g.,for BS-RS link, the channel model can be obtained from (4)by setting η13

p3p2 = η21p3p2 = η22

p3p2 = η31p3p2 = 0. For outdoor

macro-cell BS cooperation scenario, RS actually representsthe other BS, symbolled as BS2, and thus is free of scatterersas well. In this case, we have the currently most maturecooperative MIMO scheme: CoMP and the number of localscattering area I = 1. Similarly, the channel model with I = 1can also be obtained from the proposed model by settingthe energy-related parameters related to the local scatterersaround BS and RS(BS2) equal to zero. It is clear that the


proposed GBSM can be adaptable to different basic scenariosby setting relevant energy-related parameters equal to zero.Therefore, the key model parameters of the proposed GBSMactually are reduced as the Ricean factors and energy-relatedparameters. The basic criterion of setting these key modelparameters is summarized as following: the longer distanceof the link and/or the higher the LSD, the smaller the Riceanfactors and the larger the energy-related parameters of multi-bounced components, i.e., the multi-bounced components bearmore energy than single-bounced components. Since the localscattering area is highly related to the degree of link hetero-geneity in cooperative MIMO systems as presented in [18],[21], the LSD significantly affects the channel characteristicsand should be investigated. In general, the higher the LSD,the lower the possibility that the devices (BS/MS/RS) sharethe same scatterers. In this case, the cooperative MIMOenvironments present lower environment similarity. Therefore,the higher the LSD, the lower the environment similarity.For macro-cell scenarios, the Ricean factor Kp3p1 is very

small or even close to zero due to the large distance D1.Under the condition of BS cooperation scenes, Ricean factorKp2p1 is similar to Kp3p1 due to the similar distances of D1

and D2. While the BS-RS link is disappeared and replacedby wired link. In this case, we only have single bouncedcomponents, i.e., η11

p3p1 and η11p2p1 . For MS/RS cooperation

scenes, RS/MS actually represents the other MS/RS and issymbolled as MS2/RS2. In this case, Ricean factor Kp3p2

is similar to Kp3p1 due to the similar distances of D1 andD3. Since the large value of distances D1 and D3, for theBS-MS/BR2 link and BS-MS2/RS link, the impact of LSDon channel characteristics is small and in general the double-bounced rays bear more energy than single-bounced rays, i.e.,{η21p3p1 , η

21p3p2} > {η11

p3p1 , η12p3p1 , η

11p3p2 , η

12p3p2}. While for the

MS/RS2-MS2/RS link, due to the small distance of D2 theimpact of LSD is significant. For a low LSD, the scatterersare sparse and thus more likely single-bounced rays ratherthan double-bounced rays, i.e., {η11

p2p1 , η12p2p1} > η21

p2p1 , andRicean factor Kp2p1 is large. For a high LSD, the double-bounced components bear more energy than single-bouncedcomponents, i.e., η21

p2p1 > {η11p2p1 , η

12p2p1} and Ricean factor

Kp2p1 is smaller than that in the low LSD. Similar to macro-cell scenarios, the key model parameters setting for other 9cooperative scenarios with the consideration of different LSDscan be easily obtained by following the aforementioned basiccriterion and thus omits here for brevity.

IV. MULTI-LINK SPATIAL CORRELATION FUNCTIONS

In this section, based on the proposed cooperative MIMOGBSM and considering the spatial correlation between theBS-RS link and the BS-MS link as an example, we willderive the multi-link spatial correlation functions for non-isotropic scattering cooperative MIMO environments. Thenormalized spatial correlation function between any two linkscharacterized by channel gains hpq and hp′q′ , respectively, isdefined as [25], [26]

ρpq,p′q′ =E

[hpqh

∗p′q′

]√

ΩpqΩp′q′(9)

where (·)∗ denotes the complex conjugate operation, E [·] isthe statistical expectation operator, p, p′ ∈ {1, 2, ...,MT}, andq, q′ ∈ {1, 2, ...,MR}. Substituting (4) into (9), we have thecorrelation function between BS-RS link and BS-MS link as

ρp3p2,p′3p1=ρLoSp3p2,p′3p1

+3∑g=1

(ρ1gp3p2,p′3p1

+ρ2gp3p2,p′3p1

)+ρ31p3p2,p′3p1

(10)

with

ρLoSp3p2,p′3p1=

√Kp3p2Kp′3p1

(Kp3p2 + 1)(Kp′3p1 + 1

)ej2πλ−1“χp′

3p1−χp3p2

”

(11)

ρ1gp3p2,p′3p1

=

√√√√ η1gp3p2η

1gp′3p1

(Kp3p2 + 1)(Kp′3p1 + 1

) ∫ π

−π

∫ R2ng

R1ng

×ej2πλ−1“χp′

3p1,g−χp3p2,g

”Qgf(∅g)d∅gdg

(12)

ρ2gp3p2,p′3p1

=

√√√√ η2gp3p2η

2gp′3p1

(Kp3p2 + 1)(Kp′3p1 + 1

) ∫ π

−π

∫ π

−π

∫ R2ng1

R1ng1∫ R2ng2

R1ng2

Qg1g2f(∅g1)f(∅g2)ej2πλ−1χp′

3p1,g1,g2

×e−j2πλ−1χp3p2,g1,g2d∅g1d∅g2dg1dg2 (13)

ρ31p3p2,p′3p1

=

√√√√ η31p3p2η

31p′3p1

(Kp3p2 + 1)(Kp′3p1 + 1

) ∫ π

−π

∫ π

−π

∫ π

−π∫ R2n1

R1n1

∫ R2n2

R1n2

∫ R2n3

R1n3

Q123ej2πλ−1

“χp′

3p1,1,2,3−χp3p2,1,2,3

”

×f(∅1)f(∅2)f(∅3)d∅1d∅2d∅3d1d2d3 (14)

where {∅g}3g=1 = {φ1, α2,2, α1,3} are the continu-

ous expressions of the discrete expressions of anglesα1n1 , α2n2 , α3n3 , respectively, and εp′3ng

≈ ξBng+

δ32 cos(β3 − α1ng ), χp′3p1,g = χp′3p1,ng

, χp3p2,g =χp3p2,ng , χp′3p1,g1,g2 = χp′3p1,ng1 ,ng2

, χp3p2,g1,g2 =χp3p2,ng1 ,ng2

, χp′3p1,1,2,3 = χp′3p1,n1,n2,n3 , and χp3p2,1,2,3 =χp3p2,n1,n2,n3 with α1n1 , α2n2 , and α3n3 being replacedby φ1, α2,2, and α1,3, respectively. Parameters f(∅g) =exp[kg cos(∅g−μg)]/[2πI0 (kg)], {g}3

g=1 = {ξMn1, ξR2n2

, ξB1n3},

Qg = 2�ng

R22ng

−R21ng

, Qg1g2 =4�ng1

�ng2(R2

2ng1−R2

1ng1)(R2

2ng2−R2

1ng2),

Q123 = 8�n1�n2�n3(R2

2n1−R2

1n1)(R2

2n2−R2

1n2)(R2

2n3−R2

1n3), and parameters

g, g1, and g2 are the same as the ones in (5)–(8). Note that in(27) other correlation terms are equal to zero and thus omitted.These omitted correlation terms contain the integral of randomphases ψng , ψng1 ,ng2

, or ψn1,n2,n3 . Since the random phasesfulfill the uniform distribution over the range of [π,−π), theintegral of the random phases in the range of [π,−π) is equalto zero. Therefore, other correlation terms with the value ofzero are omitted in (27). Other two multi-link correlationfunctions can be similarly derived and thus omitted here forbrevity.


0 20 40 60 80 100 120 140 1600

0.5

1

(a)

0 20 40 60 80 100 120 140 1600

0.5

1

(b)Angle θ’ between the BS−RS link and BS−MS link

Spa

tial c

orre

latio

n pr

oper

ties

ρ21p

1p

2,p’

1p

3

ρ22p

1p

2,p’

1p

3

ρ23p

1p

2,p’

1p

3

ρ31p

1p

2,p’

1p

3

ρ11p

1p

2,p’

1p

3

ρ12p

1p

2,p’

1p

3

ρ13p

1p

2,p’

1p

3

Fig. 3. Absolute values of spatial correlation functions between the BS-RS link and BS-MS link for (a) the single-bounced components and (b) thedouble- and triple-bounce components.

V. NUMERICAL RESULTS AND ANALYSIS

In this section, the derived spatial correlation propertiesbetween the BS-RS link and BS-MS link will be numericallyanalyzed in detail. The parameters for the following numericalresults are listed here or specified otherwise: f = 2.4GHz,D1 = D3 = 100m, R1n1 = R1n2 = R1n3 = 5m,R2n1 = R2n2 = R2n3 = 50m, δ3 = δ2 = δ1 = 0, β3 = 30◦,β2 = β1 = 60◦, Kp3p2 = Kp′3p1 = 0, k1 = k2 = k3 = 10,μ1 = 120◦, μ2 = 300◦, and μ3 = 60◦.In Fig. 3, we present the spatial correlation properties of

all scattered components in (10) with parameters δ3 = δ2 =δ1 = 3λ and k1 = k2 = k3 = 3. Fig. 3 clearly depictsthat the spatial correlation properties vary significantly fordifferent scattered components. More importantly, we noticethat the scattered component that includes more bounced raysexpresses lower spatial correlation properties. This is becausewith more bounced rays, the component is related to morelocal scattering areas and thus easier experiences higher degreeof link heterogeneity, resulting in lower link similarity for thiscomponent.To validate the utility of the proposed cooperative MIMO

channel model, Fig. 4 investigates the spatial correlationproperties of the proposed cooperative MIMO GBSM in (10).Without loss of any generality, the outdoor macro-cell MScooperation scenario is chosen for further investigation. Asshown in Fig. 4, three different LSD conditions are consideredwith parameters δ3 = δ2 = δ1 = 3λ, i.e., high LSD, low LSD,and mixed LSD.For outdoor macro-cell MS cooperation scenario, the BS

is free of scatterers and the RS actually represents the otherMS, symbolled as MS2. Therefore, we have the energy-relatedparameters related to the local scatterers around BS to be equalto zero, i.e., η13

p3p2 = η13p′3p1

= η21p3p2 = η21

p′3p1= η22

p3p2 =η22p′3p1

= η31p3p2 = η31

p′3p1= 0, and assume D1 = D3 = 1500m

and Kp3p2 = Kp′3p1 = 0 due to the large distance amongBS, MS, and MS2. Considering the basic criterion of settingkey model parameters expressed in Section III, we choose theother energy-related parameters as: η11

p3p2 = η11p′3p1

= η12p3p2 =

η12p′3p1

= 0.05 and η23p3p2 = η23

p′3p1= 0.9 for high LSD, and

−40 −30 −20 −10 0 10 20 30 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Angle θ’ between the BS−RS link and BS−MS link

Spa

tial c

orre

latio

n pr

oper

ties

High LSD Low LSD Mixed LSD

Fig. 4. Absolute values of spatial correlation functions between the BS-RSlink and BS-MS link for the outdoor macro-cell MS cooperation scenariowith different LSDs.

η11p3p2 = η11

p′3p1= η12

p3p2 = η12p′3p1

= 0.2 and η23p3p2 = η23

p′3p1=

0.6 for low LSD. While for mixed LSD case, we considerthe scenario that the local scattering area around MS presentslow LSD and the one around MS2 shows high LSD, and thusassume the energy-related parameters as: η11

p′3p1= η12

p′3p1=

0.2, η23p′3p1

= 0.6, η11p3p2 = η12

p3p2 = 0.1, and η23p3p2 = 0.8. In

general, the higher the LSD, the more distributed and largersize the local scattering area, and thereby the smaller the valueof kg and the larger the value of R2ng − R1ng (g = 1, 2, 3).Therefore, we have the following environment parameters as:k1 = k2 = 1, R1n1 = R1n2 = 5m, and R2n1 = R2n2 = 200mfor high LSD; k1 = k2 = 10, R1n1 = R1n2 = 5m, andR2n1 = R2n2 = 20m for low LSD; and k1 = 10, k2 = 2,μ1 = 60◦, μ2 = 120◦, R1n1 = R1n2 = 5m, R2n1 = 20m, andR2n2 = 100m for mixed LSD.Fig. 4 clearly shows that the LSD significantly affects the

spatial correlation properties. It is observed that the higherthe LSD, the lower the spatial correlation properties. Thisis because that with a higher LSD, the local scattering areais more distributed and presents larger size, resulting in thereceived power comes from many different directions. Moreimportantly, from the observation in Fig. 3 and based on theconstraints of the energy-related parameters for cooperativescenarios with different LSDs, we know that with a higherLSD, the multi-bounced components bear more energy thansingle-bounced components and thus the corresponding coop-erative environment has a higher possibility to reveal a highdegree of link heterogeneity, i.e., a low degree of environmentsimilarity. Therefore, the above conclusion based on Fig. 4 isconsistent with our intuition that a low degree of environmentsimilarity results in low multi-link spatial correlations.

VI. CONCLUSIONS

This paper has proposed a novel unified cooperative MIMOchannel model framework, from which a novel GBSM hasbeen further derived. The proposed multiple-ring GBSM issufficiently generic and adaptable to a wide variety of co-operative MIMO propagation scenarios. More importantly,the proposed GBSM is the first model that is capable ofinvestigating the impact of LSD on channel statistics. From theproposed GBSM, the multi-link spatial correlations have beenderived and numerically evaluated. Numerical results have


shown that the LSD has great impact on multi-link spatialcorrelation properties. It has also been demonstrated that ahigh multi-link spatial correlation may exist if the underlyingpropagation environments have low LSDs.

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[26] C.-X. Wang, D. Yuan, H.-H. Chen, and W. Xu, “An improved de-terministic SoS channel simulator for efficient simulation of multipleuncorrelated Rayleigh fading channels,” IEEE Trans. Wireless Commun.,vol. 7, no. 9, pp. 3307–3311, Sep. 2008.

Xiang Cheng (S’05-M’10) received the BSc andMEng degrees in communication and informationsystems from Shandong University, China, in 2003and 2006, respectively, and the PhD degree fromHeriot-Watt University and the University of Edin-burgh, Edinburgh, UK, in 2009, where he receivedthe Postgraduate Research Thesis Prize.Dr. Cheng has been a lecturer at Peking Univer-

sity, Beijing, China since 2010. His current researchinterests include mobile propagation channel mod-eling and simulation, multiple antenna technologies,

mobile-to-mobile communications, and cooperative communications. He haspublished more than 30 research papers in journals and conference proceed-ings.Dr. Cheng was awarded the “2009 Chinese National Award for Outstanding

Oversea PhD Students” for his academic excellence and outstanding perfor-mance. He served as a Symposium Vice-Chair for CMC 2010 and CMC2011.

Cheng-Xiang Wang (S’01-M’05-SM’08) receivedthe BSc and MEng degrees in Communicationand Information Systems from Shandong University,China, in 1997 and 2000, respectively, and the PhDdegree in Wireless Communications from AalborgUniversity, Denmark, in 2004.He has been with Heriot-Watt University, Edin-

burgh, UK, since 2005, first as a Lecturer, then asa Reader in 2009, and then as a Professor since2011. He was a Research Fellow at the Universityof Agder, Grimstad, Norway, from 2001-2005, a

Visiting Researcher at Siemens AG-Mobile Phones, Munich, Germany, in2004, and a Research Assistant at Technical University of Hamburg-Harburg,Hamburg, Germany, from 2000-2001. His current research interests includewireless channel modeling and simulation, green communications, cognitiveradio networks, vehicular communication networks, mobile femtocell net-works, cooperative (relay) MIMO communications, and (beyond) 4G wirelesscommunications. He has published 1 book chapter and more than 150 papersin refereed journals and conference proceedings. Dr Wang is serving as anAssociate Editor for IEEE Transactions on Vehicular Technology and servedas an Editor for IEEE Transactions on Wireless Communications (2007-2009).He was the leading Guest Editor for IEEE Journal on Selected Areas inCommunications, Special Issue on Vehicular Communications and Networks.He served or is serving as a TPC member, TPC Chair, and General Chair formore than 60 international conferences. He received the IEEE Globecom’10Best Paper Award in 2010 and the IEEE ICCT’11 Best Paper Awards in 2011.


Haiming Wang received the M.S. and Ph.D. degreesin electrical engineering from Southeast University,Nanjing, China, in 2002 and 2009, respectively. Hejoined the School of Information Science and Engi-neering, Southeast University, in April 2002. Nowhe is an associate professor. His current researchinterests include signal processing for MIMO wire-less communications and millimeter-wave wirelesscommunications. Dr. Wang received the first-classScience and Technology Progress Award of JiangsuProvince of China in 2009.

Xiqi Gao (SM’07) received the Ph.D. degree inelectrical engineering from Southeast University,Nanjing, China, in 1997. He joined the Departmentof Radio Engineering, Southeast University, in April1992. Since May 2001, he has been a professorof information systems and communications. FromSeptember 1999 to August 2000, he was a visitingscholar at Massachusetts Institute of Technology,Cambridge, and Boston University, Boston, MA.From August 2007 to July 2008, he visited theDarmstadt University of Technology, Darmstadt,

Germany, as a Humboldt scholar. His current research interests includebroadband multicarrier communications, MIMO wireless communications,channel estimation and turbo equalization, and multirate signal processingfor wireless communications. He serves as an Associate Editor for the IEEETransactions on Signal Processing and the IEEE Transactions on WirelessCommunications.Dr. Gao received the Science and Technology Progress Awards of the State

Education Ministry of China in 1998, 2006 and 2009.

Xiaohu You (SM’11) was born on August 25, 1962.He received his Master and Ph.D. degrees fromSoutheast University, Nanjing, China, in ElectricalEngineering in 1985 and 1988, respectively. Since1990, he has been working with National MobileCommunications Research Laboratory at SoutheastUniversity, where he held the rank of professor. Hisresearch interests include mobile communicationsystems, signal processing and its applications. Hehas contributed over 40 IEEE journal papers and2 books in the areas of adaptive signal processing,

neural networks and their applications to communication systems. From 1999to 2002, he was the Principal Expert of the C3G Project, responsible fororganizing China’s 3G Mobile Communications R&D Activities. From 2002to 2006, he was the Principal Expert of the national 863 Beyond 3G FuTUREProject. Now he is the member of China 863 expert committee responsiblefor telecommunications.

Dongfeng Yuan (SM’01) received the M.S. degreein Department of Electrical Engineering, ShandongUniversity, China, in 1988, and got the Ph.D. degreein Department of Electrical Engineering, TsinghuaUniversity, China in January 2000. Currently he isa full professor and dean in School of Informa-tion Science and Engineering, Shandong University,China. From 1993-1994, he was with Electrical andComputer Department at the University of Calgary,Alberta, Canada. He was with Department of Elec-trical Engineering in the University of Erlangen,

Germany from 1998 to 1999; with Department of Electrical Engineering andComputer Science in the University of Michigan, Ann Arbor, USA, from 2001to 2002; with Department of Electrical Engineering in Munich University ofTechnology, Germany in 2005, and with Department of Electrical EngineeringHeriot-Watt University, UK, in 2006. His current research interests includecognitive radio systems, cooperative (relay) communications, and 4G wirelesscommunications.

Bo Ai (M’00-SM’09) is now working in BeijingJiaotong University as a professor and Ph.D. Su-pervisor. He is vice director of State Key Lab. ofRail Traffic Control and Safety. He has authored5 books and published over 124 scientific researchpapers in his research area. He has hold 5 nationalinvention patents and 1 US patent. He has been theresearch team leader for 13 national projects. He haswon some scientific research prizes, such as the FirstGrade of Technology Advancement Award of ShanxiProvince. He is an IEEE senior member and serving

as the editorial member of IEEE Transactions on Consumer Electronics andWireless Personal Communications, a publication editor for Ubiquitous Com-puting and Communication Journal. His research interests include radio wavepropagation and wireless channel modeling, power amplifier pre-distortion,LTE-R system and Cyber-physical System.

Qiang Huo received the B.S. degree in electronicand information engineering from South China Uni-versity of Technology, China in 2009. Since Septem-ber 2009, he has been a Ph.D. candidate in PekingUniversity, China. His main research interests in-clude adaptive signal processing, estimation andoptimization theory, cooperative communications,physical network coding and wireless propagationchannel modeling and simulation.

Ling-Yang Song (S’03-M’06) received a PhD fromthe University of York, UK, in 2007, where hereceived the K. M. Stott Prize for excellent research.He worked as a postdoctoral research fellow at theUniversity of Oslo, Norway, until rejoining PhilipsResearch UK in March 2008. Now, he is a fullprofessor with School of Electronics Engineeringand Computer Science, Peking University, China.His main research interests include adaptive signalprocessing, cognitive and collaborative communi-cations. He is co-inventor of a number of patents

and author or co-author of over 100 journal and conference papers. Heis currently on the Editorial Board four International Journals, and servesas a member of Technical Program Committee and Co-chair for severalinternational conferences and workshops. He is a member of the IEEE andIEEE ComSoc.

Bing-Li Jiao received B.S. and M.S. degree fromPeking University, China in 1983 and 1988, re-spectively, and received Ph.D. from University ofSarrbruecken, F. R., Germany in 1995. Then, hebecame an associate professor and professor withPeking University in 1995 and 2000, respectively.His current interests include communication theoryand techniques and sensor design.

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