National Mobile Communications Research Laboratory - New ......WIRELESS COMMUNICATIONS AND MOBILE COMPUTING Wirel. Commun. Mob. Comput. 2011; 11:829–842 Published online 30 October

WIRELESS COMMUNICATIONS AND MOBILE COMPUTINGWirel. Commun. Mob. Comput. 2011; 11:829–842

Published online 30 October 2009 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/wcm.864

SPECIAL ISSUE PAPER

New deterministic and stochastic simulation modelsfor non-isotropic scattering mobile-to-mobileRayleigh fading channels†

Xiang Cheng1∗, Cheng-Xiang Wang1, David I. Laurenson2, Sana Salous3 andAthanasios V. Vasilakos4

1 Joint Research Institute for Signal and Image Processing, School of Engineering & Physical Sciences, Heriot-Watt University,Edinburgh, EH14 4AS, U.K.2 Joint Research Institute for Signal and Image Processing, Institute for Digital Communications, University of Edinburgh, Edinburgh,EH9 3JL, U.K.3 Center for Communication Systems, School of Engineering, University of Durham, Durham, DH1 3LE, U.K.4 Department of Computer and Telecommunications Engineering, University of Western Macedonia, GR 50100 Kozani, Greece

ABSTRACT

For the practical simulation and performance evaluation of mobile-to-mobile (M2M) communication systems, it is desirableto develop accurate M2M channel simulation models for more realistic scenarios of non-isotropic scattering. In this paper,by using a ‘double-ring’ concept to describe M2M non-isotropic scattering environments, we propose new deterministicand stochastic sum-of-sinusoids (SoS) based simulation models. The proposed simulation models extensively consider thedistributions of the angle of arrival (AoA) and the angle of departure (AoD), and thus provide a good approximation to thedesired statistical properties of the reference model. Copyright © 2009 John Wiley & Sons, Ltd.

KEYWORDS

mobile-to-mobile communications; sum-of-sinusoids channel simulator; Rayleigh fading; non-isotropic scattering

*CorrespondenceXiang Cheng, Joint Research Institute for Signal and Image Processing, School of Engineering & Physical Sciences, Heriot-WattUniversity, Edinburgh, EH14 4AS, U.K.E-mail: [email protected]

1. INTRODUCTION

In recent years, mobile-to-mobile (M2M) communica-tions have received increasing attention due to some newapplications, such as wireless mobile ad hoc networks[1], relay-based cellular networks [2], and dedicated shortrange communications for intelligent transportation sys-tems (IEEE 802.11p) [3]. Such M2M systems consider thatboth the transmitter (Tx) and receiver (Rx) are in motionand are equipped with low elevation antennas. To success-fully analyze and design such M2M systems, it is necessaryto have proper mathematical reference models for theunderlying propagation channels. Many M2M mathemati-cal reference models have been proposed for both isotropicscattering environments, e.g., the models in References [4]and [5], and more realistic environments of non-isotropicscattering, e.g., the models in References [6--10].

†This paper was presented in part at VehiCom’09, co-located with IWCMC’09, Leipzig, Germany, 21–24 June 2009.

Besides the modeling of a M2M channel and theinvestigation of its statistical properties, the developmentof accurate M2M channel simulation models also plays amajor role in the practical simulation and performance eval-uation of M2M systems. However, in the literature, mostchannel simulation models for wireless communicationswere developed for conventional fixed-to-mobile (F2M)cellular radio systems [11--17]. As mentioned in Reference[18], simulation models for a F2M channel cannot bereadily used to simulate a M2M channel due to the dif-ferences of their statistical properties. Wang and Cox [19]were the first to propose a simulation model for isotropicscattering M2M Rayleigh fading channels by using theline spectrum filtering method. Due to the application ofline spectrum filtering, the model has high complexityand severe performance deficiencies as pointed out inReference [18]. Therefore, the sum-of-sinusoids (SoS)

Copyright © 2009 John Wiley & Sons, Ltd. 829

New deterministic and stochastic simulation models X. Cheng et al.

method was used in References [18] and [20] to developbetter isotropic scattering M2M Rayleigh fading simulationmodels. Generally speaking, the SoS method allows us toapproximate the underlying random fading processes bythe superposition of a finite number of properly weightedharmonic functions. An SoS based simulation model can beeither deterministic or stochastic in terms of the underlyingparameters (gains, frequencies, and phases) [16]. For adeterministic model, all the parameters are fixed for allsimulation trials. In contrast, a stochastic model has atleast one parameter (gains and/or frequencies) as a randomvariable which varies for each simulation trial. Therefore,the relevant statistical properties of a stochastic modelvary for each simulation trial but converge to the desiredones when averaged over a sufficient number of trials. InReference [18], the authors used the ‘double-ring’ conceptto derive two SoS based simulation models (deterministicand stochastic models) for isotropic scattering M2Mchannels. The stochastic model in Reference [18] wasfurther improved in Reference [20] by choosing orthogonalfunctions for in-phase and quadrature components of thecomplex fading envelope. More recently, in Reference[21] the Rayleigh M2M stochastic simulation model inReference [18] was extended to include a line-of-sightcomponent, i.e., for Ricean fading channels. However, it isworth noting that all the aforementioned simulation modelslimit their applications to isotropic scattering environments.The simulation models for M2M Rayleigh fading channelsunder a more realistic scenario of non-isotropic scatteringare scarce in the current literature.

To the best of the authors’ knowledge, only onestochastic SoS based simulation model [22] has beenproposed for the simulation of non-isotropic scatteringM2M Rayleigh fading channels. However, this model onlyconsidered the symmetrical property of the distributionsof the angle of arrival (AoA) and angle of departure(AoD) and was designed based on the acceptance rejectionalgorithm (ARA). This leads to the ARA model havinga notable difficulty in reproducing the desired statisticalproperties of the reference model and a comparativelyhigh computational complexity. Furthermore, it is worthnoting that accurate deterministic simulation models fornon-isotropic scattering M2M Rayleigh fading channelsare not available in the current literature.

To fill the above gap, in this paper, based on the ‘double-ring’ concept in Reference [18], we first propose a newM2M deterministic SoS based simulation model. By allow-ing at least one parameter (frequencies and/or gains) to bea random variable, our deterministic model can be furthermodified to be a stochastic model. It is worth noting thatthe proposed simulation models incorporate the probabilitydensity functions (PDFs) of the AoA and AoD, and thus canapproximate the desired statistical properties of the refer-ence model for any non-isotropic scattering M2M Rayleighfading channel. Moreover, compared to the ARA stochasticmodel in Reference [22], our stochastic model presents bet-ter approximation to the desired properties of the referencemodel with an even smaller number of harmonic functions.

The paper is structured as follows. Section 2 gives abrief description of the reference model for non-isotropicscattering M2M Rayleigh fading channels. In Section3, we propose two new SoS based simulation models(deterministic and stochastic models). The performanceevaluation of our models is presented in Section 4. Finally,conclusions are drawn in Section 5.

2. REFERENCE MODEL

Using Akki and Haber’s mathematical model [4] andconsidering the directions of movement of the Tx andRx, we can express the complex fading envelope ofour reference model, under a narrowband non-isotropicscattering M2M Rayleigh fading assumption, as

h(t) = hi(t) + jhq(t) = limN→∞

1√N

N∑n=1

ejψnej[2πfTmax t cos(φnT−γT )+2πfRmax t cos(φnR−γR)]

(1)

where hi(t) and hq(t) are in-phase and quadrature compo-nents of the complex fading envelope h(t), respectively,j = √−1, N is the number of harmonic function represent-ing propagation paths (effective scatterers), fTmax and fRmaxare the maximum Doppler frequency due to the motionof the Tx and Rx, respectively. The Tx and Rx move indirections determined by the angles of motion γT and γR,respectively. The random AoA and AoD of the nth path aredenoted by φnR and φ

nT , respectively, and ψn is the random

phase uniformly distributed on [−π, π). It is assumed thatφnR, φ

nT , and ψn are mutually independent random variables.

Since the number N of effective scatterers in thereference model h(t) tends to infinity, the discrete expres-sions of the AoA φ(n)R and AoD φ

(n)T can be replaced

by the continuous expressions φR and φT , respectively.Note that since h(t) describes a non-isotropic scatteringM2M Rayleigh fading channel, the AoA φR and AoD φTexhibit non-uniform distributions. In the literature, manydifferent non-uniform distributions have been proposed tocharacterize the AoA φR and AoD φR, such as Gaussian,wrapped Gaussian, and cardioid PDFs [23]. In this paper,the von Mises PDF [24] is used, which can approximateall the aforementioned PDFs and match well the measureddata in References [7] and [24]. The von Mises PDF isdefined as f (φ) � exp[k cos(φ − µ)]/[2πI0(k)], whereφ ∈ [−π, π), I0(·) is the zeroth-order modified Besselfunction of the first kind, µ ∈ [−π, π) accounts for themean value of the angle φ, and k (k ≥ 0) is a real-valuedparameter that controls the angle spread of the angle φ. Fork = 0 (isotropic scattering), the von Mises PDF reducesto the uniform distribution, while for k > 0 (non-isotropicscattering), the von Mises PDF approximates differentdistributions based on the values of k [24]. Applying

830 Wirel. Commun. Mob. Comput. 2011; 11:829–842 © 2009 John Wiley & Sons, Ltd.DOI: 10.1002/wcm

X. Cheng et al. New deterministic and stochastic simulation models

the von Mises PDF to the AoA φR and AoD φT , weobtain f (φR) � exp[kR cos(φR − µR)]/[2πI0(kR)] andf (φT ) � exp[kT cos(φT − µT )]/[2πI0(kT )], respectively.

The autocorrelation function (ACF) of the referencemodel h(t) can be expressed as [10]

ρhh(τ) = E[h (t) h∗ (t − τ)]

= 1I0(kT )I0(kR)

I0

[(A2T + B2T

)1/2]× I0

[(A2R + B2R

)1/2](2)

with

AT = kT cos(µT ) + j2πτfTmax cos(γT ) (3a)BT = kT sin(µT ) + j2πτfTmax sin(γT ) (3b)AR = kR cos(µR) + j2πτfRmax cos(γR) (3c)BR = kR sin(µR) + j2πτfRmax sin(γR) (3d)

where τ is the time separation, (·)∗ denotes the complexconjugate operation, and E[·] is the statistical expectationoperator.

3. NEW SIMULATION MODELS

In this section, based on the reference model introduced inSection 2, we propose the corresponding deterministic andstochastic SoS based simulation models.

3.1. New deterministic simulation model

Before proposing a new deterministic simulation model,which needs only one simulation trial to obtain the desiredstatistical properties, we would like to first introducethe ‘double-ring’ concept [18] to model non-isotropicscattering M2M scenarios. In such a case, the ‘double-ring’model defines two rings of effective scatterers, one aroundthe Tx and the other around the Rx, as shown in Figure 1.

RυTυ

Rγ

Tγ

XT XR

Figure 1. Mobile-to-mobile scattering environment.

To model non-isotropic scattering environments, theeffective scatterers are assumed to be non-uniformlydistributed. The Tx and Rx move with speeds υT and υRin directions determined by the angles of motion γT andγR, respectively. Based on the ‘double-ring’ model, wecan then design a new deterministic simulation modelas

h̃ (t) = h̃i (t) + jh̃q (t) (4)

h̃i (t) = 1√NiMi

Ni,Mi∑ni,mi=1

cos[ψ̃nimi

+ 2πfTmax t cos(φ̃miT − γT )+ 2πfRmax t cos(φ̃niR − γR)

](5)

h̃q (t) = 1√NqMq

Nq,Mq∑nq,mq=1

sin[ψ̃nqmq

+ 2πfTmax t cos(φ̃mqT − γT )+ 2πfRmax t cos(φ̃nqR − γR)

](6)

where h̃i(t) and h̃q(t) are in-phase and quadrature compo-nents of the complex fading envelope h̃(t), respectively,Ni/q is the number of effective scatterers located on the ringaround the Rx, Mi/q is the number of effective scattererslocated on the ring around the Tx, the AoA φ̃

ni/qR and the

AoD φ̃mi/qT are discrete realizations of the random variables

φR and φT , respectively, the phases ψ̃ni/qmi/q are the singleoutcomes of the random phases ψn in Equation (1). Itis assumed that φ̃

ni/qR , φ̃

mi/qT , and ψ̃ni/qmi/q are mutually

independent. Note that the AoA φ̃(ni/q)R and the AoD φ̃

(mi/q)T

remain constant for different simulation trials due to thedeterministic nature of the proposed simulation model.It is worth stressing that the single summation in thereference model (1) is replaced by a double summation dueto the double reflection incorporated by the ‘double-ring’model. However, the channel essentially remains the samefor the following two reasons: (1) the assumption of theindependence between the AoA and AoD is used in both thereference model (1) and the deterministic simulation model(4); and (2) for both the reference model (1) and the deter-ministic simulation model (4), each path will be subject to adifferent Doppler shift due to the motion of both the Tx andRx.

From the above proposed deterministic simulationmodel, it is obvious that the key issue in designing aM2M deterministic simulation model is to find the setsof AoAs {φ̃ni/qR }Ni/qni/q=1 and AoDs {φ̃

mi/qT }Mi/qmi/q=1 that make

the simulation model reproduce the desired statisticalproperties of the reference model as faithfully as possiblewith reasonable complexity, i.e., with a finite numberof Ni/q and Mi/q. Under the condition of non-isotropicscattering environments, the PDFs of the AoA φR and AoD

Wirel. Commun. Mob. Comput. 2011; 11:829–842 © 2009 John Wiley & Sons, Ltd. 831DOI: 10.1002/wcm


φT should be used to design the sets of AoAs {φ̃ni/qR }Ni/qni/q=1and AoDs {φ̃mi/qT }Mi/qmi/q=1 that guarantee the uniqueness of thesine and cosine functions related to the AoA φ(n)R and AoDφ

(n)T in the reference model (1). This means that the design

of the sets of AoAs and AoDs for the reference model h(t)in Equation (1) should meet the following two conditions:

(1) cos(φ̃ni/qR − γR) �= cos(φ̃

n′i/q

R − γR), ni/q �= n′i/q; and (2)cos(φ̃

mi/qT − γT ) �= cos(φ̃

m′i/q

T − γT ), mi/q �= m′i/q.Considering the non-isotropic scattering M2M environ-

ments, we now design the sets of AoAs and AoDs of ourdeterministic simulation model. Via extensive investigationof the PDFs of the AoA φR and AoD φT , we find that unlikeisotropic scattering M2M environments [18], it is difficultto obtain sets of AoAs and AoDs that meet the two condi-tions for any non-isotropic scattering M2M environments.Therefore, we divide the non-isotropic scattering M2Menvironment into the following three categories in terms ofthe mean AoA µR and mean AoD µT (i.e., the PDFs of theAoA and AoD), and the angles of motion γR and γT andthen design the sets of AoAs and AoDs separately for thesethree cases: (1) Case I: the main transmitted and receivedpowers come from the same direction as or oppositedirection to the movements of the Tx and Rx, respectively,i.e., |µT − γT | = |µR − γR| = 0◦ or π; (2) Case II: themain transmitted and received powers come from thedirections that are perpendicular to those of the move-ments of the Tx and Rx, respectively, i.e., |µT − γT | =|µR − γR| = 90◦; and (3) Case III: different from Cases Iand II.

For Case I, the in-phase component hi(t) and quadraturecomponent hq(t) of the reference model h(t) in Equation(1) are correlated. Therefore, to meet this correlation, weset Ni = Nq = N and Mi = Mq = M and thus the AoAφ̃

ni/qR and AoD φ̃

mi/qT can be replaced by the φ̃

nR and φ̃

mT ,

respectively. Inspired by the modified method of equal area(MMEA) proposed in Reference [25] for non-isotropic scat-tering F2M channels, we design the AoA and AOD of ourmodel as

n − 1/4N

=∫ φ̃n

R

φ̃n−1R

f (φ̃nR)dφ̃nR, φ̃

nR ∈ [−π, π)

n = 1, 2, . . . , N (7a)

m − 1/4M

=∫ φ̃m

T

φ̃m−1T

f (φ̃mT )dφ̃mT , φ̃

mT ∈ [−π, π)

m = 1, 2, . . . , M (7b)

where φ̃0R = −π and φ̃0T = −π, f (φ̃nR) and f (φ̃mT ) denotethe PDFs of the AoA φ̃nR and the AoD φ̃

mT , respectively. The

cumulative distribution functions (CDFs) of the AoA φ̃Rand the AoD φ̃T are defined as FR(x) =

∫ x−∞ f (φ̃

nR) dφ̃

n

R and

FT (x) =∫ x

−∞ f (φ̃mT )dφ̃

m

T , respectively. If the inverse func-

tions F−1R (·) of FR(·) and F−1T (·) of FT (·) exist, (7a) and (7b)

become

φ̃nR = F−1R(

n − 1/4N

)(8a)

φ̃mT = F−1T(

m − 1/4M

)(8b)

Note that the value of 1/4 used in Equations (8a) and(8b) guarantees that the designed sets of AoAs andAoDs can meet the aforementioned two conditions(φ̃nR �= −φ̃(n

′)R + 2γR, n �= n′ and φ̃mT �= −φ̃m′T + 2γT ,

m �= m′) for Case I. The proof of the above statement isomitted here since one can easily prove it by following theprocedure provided in Reference [25]. The performance ofthis design will be validated in Section 4.

Under the condition of Case II, the cross-correlationbetween the in-phase component hi(t) and quadraturecomponent hq(t) of the complex fading envelope h(t)in Equation (1) is equal to zero. Therefore, by settingNi �= Nq and Mi �= Mq in this case, we can directly use theexpression of our simulation model itself to guarantee theaforementioned cross-correlation is equal to zero ratherthan through the design of the AoAs and AoDs. This makesfor a more efficient use of the number of harmonic func-tions and thus results in better performance of the model.Following the similar parameter computation methodof Case I, we design the AoA and AoD of our modelas

φ̃ni/qR = F−1R

(ni/q − 1/2

Ni/q

), φ̃

ni/qR ∈ [−π, π)

ni/q = 1, 2, . . . , Ni/q (9a)

φ̃mi/qT = F−1T

(mi/q − 1/2

Mi/q

), φ̃

mi/qT ∈ [−π, π)

mi/q = 1, 2, . . . , Mi/q (9b)

Note that the value of 1/2 is applied in Equations (9a)and (9b) instead of the value of 1/4 in Equations (8a) and(8b). The reason is that unlike Case I, for Case II it isdifficult to design sets of AoAs and AoDs that meet thetwo conditions. Although it is difficult to find one valuethat can guarantee the designed sets of AoAs and AoDshave the best approximation to the two conditions. Basedon simulations and motivated by the modified method ofexact Doppler spread (MEDS) in Equations (37) and (38)in Reference [18] for the simulation of isotropic scatteringM2M channels, we found that with the value of 1/2, theproposed simulation model presents better performancethan the one with other values, e.g., 1/4. The performanceof this design will be validated in Section 4.

For Case III, since the in-phase and quadrature com-ponents of the reference model h(t) in Equation (1) arecorrelated (similarly to Case I) we set Ni = Nq = N andMi = Mq = M as well and thus the AoA φ̃ni/qR and AoDφ̃

mi/qT can be replaced by the φ̃

nR and φ̃

mT , respectively.



Following the similar parameter computation method ofCase I, we design the AoA and AoD of our model as

φ̃nR = F−1R(

n − 1/2N

), φ̃nR ∈ [−π, π)

n = 1, 2, . . . , N (10a)

φ̃mT = F−1T(

m − 1/2M

), φ̃mT ∈ [−π, π)

m = 1, 2, . . . , M (10b)

Note that the value of 1/2 is used in Equations (10a) and(10b) due to the same reason as Case II.

The correlation properties of the proposed deterministicsimulation model must be analyzed by using time averagerather than statistical average. The time-average ACF of theproposed simulation model h̃(t) is defined as

ρ̃h̃h̃(τ) =〈h̃(t)h̃∗(t − τ)〉 (11)

where 〈·〉 denotes the time average operator. SubstitutingEquation (4) into Equation (11), we have

ρ̃h̃h̃ (τ) = 2ρ̃h̃ih̃i (τ) − 2jρ̃h̃ih̃q (τ) (12)

where

ρ̃h̃ih̃i (τ) =1

2NiMi

Ni,Mi∑ni,mi=1

cos[2πfTmaxτ cos(φ̃

miT − γT ) + 2πfRmaxτ cos(φ̃niR − γR)

](13a)

ρ̃h̃ih̃q (τ) =

− 12NM

N,M∑n,m=1

sin[2πfTmaxτ cos(φ̃

mT − γT ) , Ni = Nq = N and Mi = Mq

+ 2πfRmaxτ cos(φ̃nR − γR)] = M (Case I and III)

0, Ni �= Nq and Mi �= Mq (Case II)

(13b)

In Appendix A, we give a brief outline of the derivations ofEquations (13a) and (13b). From Equation (13b), and basedon the corresponding derivation in Appendix A, it is clearthat by setting Ni �= Nq and Mi �= Mq, the cross-correlationbetween the in-phase component h̃i(t) and quadrature com-ponent h̃q(t) of the proposed simulation model h̃(t) is equalto zero no matter how the sets of AoAs and AoDs aredesigned. When N (Ni) and M (Mi) tend to infinity, it isstraightforward that the time-average ACF in Equation (12)matches the ensemble average ACF in Equation (2). Thisallows us to conclude that for {N(Ni), M(Mi)} → ∞, oursimulation model can represent the correlation propertiesof the reference model.

3.2. New stochastic simulation model

Our deterministic model can be further modified to astochastic simulation model by allowing both the phases

and frequencies to be random variables. Unlike the deter-ministic model, the properties of the stochastic model varyfor each simulation trial, but will converge to the desiredones when averaged over a sufficient number of simula-tion trials. A hat is used to distinguish this model from thedeterministic one, thus

ĥ (t) = ĥi (t) + jĥq (t) (14)

ĥi (t) = 1√NiMi

Ni,Mi∑ni,mi=1

cos[ψ̂nimi

+ 2πfTmax t cos(φ̂miT − γT )+ 2πfRmax t cos(φ̂niR − γR)

](15)

ĥq (t) = 1√NqMq

Nq,Mq∑nq,mq=1

sin[ψ̂nqmq

+2 πfTmax t cos(φ̂mqT − γT )+ 2πfRmax t cos(φ̂nqR − γR)

](16)

where ĥi(t) and ĥq(t) are in-phase and quadrature compo-nents of complex fading envelope ĥ(t), respectively, φ̂

ni/qR

and φ̂mi/qT denote the AoA and AoD of our stochastic simula-

tion model, respectively, and the phases ψ̂ni/qmi/q are randomvariables uniformly distributed on the interval [−π, π).Parameters φ̂

ni/qR , φ̂

mi/qT , and ψ̂ni/qmi/q are independent of each

other. Note that unlike the AoA and AoD in a determin-istic simulation model, the AoA φ̂

ni/qR and AoD φ̂

mi/qT are

random variables and thus vary for different simulation tri-als. The fundamental issue for the design of the sets ofAoAs {φ̂ni/qR }Ni/qni/q=1 and AoDs {φ̂

mi/qT }Mi/qmi/q=1 is how to incor-

porate a random term into the AoA and AoD. In this paper,to deal with this fundamental issue, we apply the methodproposed in Reference [12] for the simulation of isotropicscattering F2M channels. Based on the comparative anal-ysis in Reference [26], we can conclude that the smaller(but sufficient) range on which the AoA φ̂

ni/qR and AoD

φ̂mi/qT are designed, the better performance the stochastic

model that will be obtained. Based on the extensive inves-tigation of the PDFs of the AoA φR and AoD φT , we findthat unlike isotropic scattering M2M environments [18], theappropriate range on which the AoA and AoD are designedvaries for different non-isotropic scattering M2M environ-ments. Therefore, similarly to our deterministic model, wedesign the sets of AoAs and AoDs of our stochastic model



separately for the following three cases: (1) Case I: the maintransmitted and received powers come from the same direc-tion as the movements of the Tx and Rx that is along x axis,i.e., µT = γT = µR = γR = 0◦ or π; (2) Case II: the maintransmitted and received powers come from the directionsthat are perpendicular to those of the movements of the Txand Rx, respectively, i.e., |µT − γT | = |µR − γR| = 90◦;and (3) Case III: different from Cases I and II.

For Case I, due to the same reason as the design of ourdeterministic model for Case I we have Ni = Nq = N andMi = Mq = M and thus the AoA φ̂ni/qR and AoD φ̂mi/qT canbe replaced by the φ̂nR and φ̂

mT , respectively. In this case, we

found that the PDFs of the AoA and AoD are symmetricwith respect to the origin. Therefore, the appropriate rangesfor the design of both AoA and AoD are from 0 to π becausethe rest of the range is redundant (i.e., the range [−π, 0)provides us with the same infomation as [0, π)). Inspiredby the method in Reference [12] for isotropic scatteringF2M channels, we design the AoA and AoD of our modelas

φ̂nR = F−1R(

n − 1/2 + θRN

), φ̂nR ∈ [0, π)

n = 1, 2, . . . , N (17a)

φ̂mT = F−1T(

m − 1/2 + θTM

), φ̂mT ∈ [0, π)

m = 1, 2, . . . , M (17b)

where θR and θT are random variables uniformly distributedon the interval [−1/2, 1/2) and are independent to eachother. It is worth stressing that the interval [−1/2, 1/2) andthe constant value of 1/2 are chosen to guarantee that thedesign of the AoA and AoD is based on the desired range(here is [0, π)). This indicates that any interval and thecorresponding constant value can be chosen only if theycan fulfill the aforementioned guarantee (e.g., the intervalis [0, 1) and constant value is 1). Note that the introductionof the random terms θR and θT in the AoA φ̂nR and AoDφ̂mT , respectively, leads to the AoA and AoD being randomvariables and thus varying for different simulation trials.φ̃

ni/qR and AoD φ̃

mi/qT of our deterministic simulation model,

which are constant for different simulation trials.For Case II, analogous to our deterministic model, we

impose Ni �= Nq and Mi �= Mq in our stochastic modelto guarantee the cross-correlation between the in-phasecomponent ĥi(t) and quadrature component ĥq(t) of the pro-posed stochastic model ĥ(t) for one simulation trial is equalto zero. In this case, since the PDFs of the AoA and AoD areasymmetric with respect to the origin, the full range (i.e.,from −π to π) is needed for the design of the AoA andAoD. Therefore, we can express the AoA and AoD as

φ̂ni/qR = F−1R

(ni/q − 1/2 + θR

Ni/q

), φ̂

ni/qR ∈ [0, π)

ni/q = 1, 2, . . . , Ni/q (18a)

φ̂mi/qT = F−1T

(mi/q − 1/2 + θT

Mi/q

), φ̂

mi/qT ∈ [−π, π)

mi/q = 1, 2, . . . , Mi/q (18b)

For Case III, due to the same reason as Case I we haveNi = Nq = N and Mi = Mq = M and thus the AoA φ̂ni/qRand AoD φ̂

mi/qT can be replaced by the φ̂

nR and φ̂

mT , respec-

tively. In this case, analogous to Case II, we know that thefull range (i.e., from −π to π) is necessary for the designof the AoA and AoD. Therefore, the AoA and AoD can bedesigned as

φ̂nR = F−1R(

n − 1/2 + θRN

), φ̂nR ∈ [−π, π)

n = 1, 2, . . . , N (19a)

φ̂mT = F−1T(

m − 1/2 + θTM

), φ̂mT ∈ [−π, π)

m = 1, 2, . . . , M (19b)

Unlike our deterministic simulation model, the time ACFof our stochastic simulation model should be computedaccording to ρ̂ĥĥ(τ) = E[ĥ(t)ĥ∗(t − τ)]. It can be shownthat our stochastic model exhibits correlation properties ofthe reference model irrespective of the values of Ni/q andMi/q, i.e., for any Ni/q and Mi/q. Appendix B outlines thederivation of the ACF ρ̂ĥĥ(τ) for the model ĥ(t).

It is worth noting that the proposed stochastic modelshows better performance and has lower complexity thanthe ARA model [22]. Since in Reference [22] the authorsdid not give the detailed explanation on how to generate theAoAs and AoDs for their model by using the ARA, it isimpossible to reproduce this model. Therefore, to validatethe above statement, in Figure 2 we compare the ACF ofthe real part of the ARA model obtained from Figure 5 inReference [22] with the one of our model. For a fair com-parison, the same parameters as those used in Figure 5 in

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Time delay, τ

Aut

ocor

rela

tion,

real

par

t

Reference modelThe proposed stochastic modelARA stochastic model

Figure 2. Comparison between the proposed stochastic modeland the ARA stochastic model proposed in Reference [22].



0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.01

0.02

0.03

Normalized time delay, τ×fT

max

(b)

Squ

ared

err

or

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

2

4 x 10−3


max

(c)

Squ

ared

err

or

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

2

4

6 x 10−3


max

(a)

Squ

ared

err

or Case ICase IICase III

Case ICase III

Case ICase III

Figure 3. Squared error in the CF of the proposed deterministic simulation model with kT = kR = 1 for different non-isotropic scatteringM2M Rayleigh fading channels: (a) �T = �R = 110◦ and �T = �R = 20◦; (b) �T = �R = �T = �R = 0◦; (c) �T = 30◦, �R = 160◦, �T =

10◦, and �R = 20◦.

Reference [22] are used, namely fTmax = 100 Hz, fRmax =50 Hz, µT = π/4, µR = −π/4, kT = kR = 3, γT = γR =0◦, and the number of simulation trials Nstat = 10. Note thatthe number of harmonic functions used in the ARA model isP = 144, while in our model is Ni = Mi = 10. From Fig-ure 2, it is obvious that our model outperforms the ARAmodel with even smaller number of harmonic functions,i.e., Ni × Mi = 100 < P .

4. NUMERICAL RESULTS ANDANALYSIS

In this section, we first validate the newly proposed deter-ministic model by using the squared error between thecorrelation properties of the simulation model and thoseof the reference model. Then the validation of the pro-posed stochastic simulation model is performed by utilizingthe difference in the time average properties of a singlesimulation trial for the stochastic model from the desiredensemble average properties. Furthermore, the performanceevaluation of the proposed models is carried out by com-paring the correlation properties of the proposed simulationmodels with those of the reference model. Unless other-wise specified, all the results presented here are obtainedusing fTmax = fRmax = 100 Hz, Ni = Mi = Nq = Mq = 20for Cases I, III and Ni = Mi = 20, Nq = Mq = 21 for CaseII of the deterministic model, Ni = Mi = Nq = Mq = 10

for Cases I, III and Ni = Mi = 10, Nq = Mq = 11 for CaseII of the stochastic model, and the normalized samplingperiod fTmaxTs = 0.005 (Ts is the sampling period).

To validate our deterministic model, in Figure 3 we com-pare the difference in the ACF ρ̃h̃h̃(τ) from the desired ρhh(τ)using the squared error |ρ̃h̃h̃(τ) − ρhh(τ)|2 for different non-isotropic M2M scenarios. Similarly, to validate our stochas-tic model, Figure 4 compares the difference in the timeaveraged ACF of a single simulation trial ρ̆hh(τ) from thedesired ACF ρhh(τ) as E[|ρ̆hh(τ) − ρhh(τ)|2] for differentnon-isotropic M2M scenarios. As pointed out in Reference[26], this provides a measure of the utility of the stochasticmodel in simulating the desired channel waveform using afinite harmonic functions Ni/q and Mi/q. The results in Fig-ure 4 are obtained by averaging over 104 simulation trials foreach value of time delay τ. Note that for the sake of the read-ability of figures, the difference of our models for Case II isonly shown in Figures 3(a) and 4(a) since this is extremelylarge for other cases. In addition, for longer time delays,the deviation of our simulation model for all cases becomeextremely large due to the insufficient number of harmonicfunctions. To maintain readability of the figures, we haveremoved the longer time delays and only presented shortertime delays, which is typically of interest for most commu-nication systems [18]. From Figures 3 and 4, it is clear thatdue to the impact of non-isotropic scattering, no one param-eter computation method in our models consistently out-performs others for all non-isotropic M2M scenarios. This,



0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.5

1

1.5 x 10−4


max

(a)

Mea

n sq

uare

d er

ror

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

1

2 x 10−3


max

(b)

Mea

n sq

uare

d er

ror

0 0.5 1 1.5 2 2.5 3 3.5 4 4.50

0.01

0.02


max

(c)

Mea

n sq

uare

d er

ror

Case ICase IICase III

Case ICase III

Case ICase III

Figure 4. Mean squared error in the CF of the proposed stochastic simulation model with kT = kR = 5 for different non-isotropicscattering M2M Rayleigh fading channels: (a) �T = �R = 110◦ and �T = �R = 20◦; (b) �T = �R = �T = �R = 0◦; (c) �T = 20◦, �R = 10◦,

�T = 10◦, and �R = 20◦.

hence, validates the utility of our models that include threedifferent sets of model parameters rather than only one.

To evaluate the performance of our simulation models, inFigures 5–7 we give a comparison between the ACF of thereference model and the one of our simulation models forvarious values of kT , kR, µT , and µR. The results obtained

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


max

Abs

olut

e va

lue

of th

e C

F

Reference modelDeterministic simulation modelStochastic simulation model

µT=µ

R=10°

µT=µ

R=30°

µT=µ

R=100°

µT=µ

R=50°

Figure 5. Comparison between the CF of the reference modeland the one of the proposed simulation models with kT = kR = 6and �T = �R = 10◦ for various values of the mean AoD �T and

mean AoA �R .

for the stochastic model are averaged over Nstat = 10simulation trials. It is obvious that the deterministic modelprovides a fairly good approximation to the ACF of thereference model in a shorter normalized time-delay range,while the stochastic model presents much better approxi-mation even with a smaller number of harmonic functionsNi/q and Mi/q. Compared to our deterministic model, ourstochastic model has a higher complexity since it requiresseveral simulation trials to achieve the desired properties.Notice that the quality of the approximation between theACFs of our deterministic model and the reference modelcan be improved by increasing the values of Ni/q and Mi/q,while the quality of the approximation between the ACFof our stochastic model and the one of the reference modelcan be improved by increasing the values of Ni/q and Mi/q,and/or the value of Nstat . More interestingly, from Figure 5,we observe that the increase of the values of |µT − γT | and|µR − γR| decreases the difficulty of our simulation modelin approximating the correlation properties of the referencemodel. Similarly, Figure 6 shows that the difficulty of oursimulation model in approximating the correlation prop-erties of the reference model increases with the increaseof the values of kT and kR. More importantly, Figures 5–7show that when the values of |µT − γT | and |µR − γR| aresmall and/or the values of kT and kR are large, the proposeddeterministic simulation model cannot even approximatethe correlation properties of the reference model for short



0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1


max

Abs

olut

e va

lue

of th

e C

F

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1


max

Abs

olut

e va

lue

of th

e C

F



kT=k

R=4

kT=k

R=1

kT=k

R=20

kT=k

R=10

Figure 6. Comparison between the CF of the reference model and the one of the proposed simulation models with �T = �R = 60◦and �T = �R = 30◦ for various values of kT and kR .

time-delays. In the aforementioned situations, to obtain anacceptable approximation between the correlation proper-ties of our deterministic model and the ones of the referencemodel, a large number of harmonic functions (here Ni =Mi = Nq = Mq = 60 as depicted in Figure 7) is necessary.This allows us to conclude that the stochastic simulation

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


max

Abs

olut

e va

lue

of th

e C

F

Reference modelDeterministic simulation modelStochastic simulation modelDeterministic simulation model (N

i/q=M

i/q=60)

kT=k

R=10

µT=µ

R=0°

kT=k

R=5

µT=µ

R=20°

kT=k

R=3

µT=µ

R=40°

kT=k

R=1

µT=µ

R=90°

Figure 7. Comparison between the CF of the reference modeland the one of the proposed simulation models with �T = �R =0◦ for various values of kT , kR , the mean AoD �T , and mean AoA

�R .

model is more suitable than the deterministic model for sucha non-isotropic scattering M2M scenario where the differ-ences between the angles of motion γT and γR, and the meanAoA µT and AoD µR are small (i.e., the value of |µT − γT |and |µR − γR| are small) and/or the received powers aremore concentrated in one direction (i.e., the value of kT andkR are large).

5. CONCLUSIONS

In this paper, based on the ‘double-ring’ concept andthe comprehensive analysis of the PDFs of the AoA andAoD, new deterministic and stochastic SoS based simu-lation models have been proposed for non-isotropic M2MRayleigh fading channels. The performance of the proposedsimulation models has been verified in terms of the CFthrough the theoretical and simulation results. Results haveshown that compared to the proposed deterministic model,the proposed stochastic model provides better approxima-tion of the reference model even with a smaller number ofharmonic functions. Moreover, our analysis has revealedthat the proposed stochastic model performs better than theARA stochastic model. Finally, either the proposed deter-ministic (lower complexity) model or stochastic (higheraccuracy) model can be used for simulations of M2Msystems.



ACKNOWLEDGMENTS

X. Cheng, C.-X. Wang, and D. I. Laurenson acknowledgethe support from the Scottish Funding Council for the JointResearch Institute in Signal and Image Processing betweenthe University of Edinburgh and Heriot-Watt University-which is a part of the Edinburgh Research Partnership inEngineering and Mathematics (ERPem).

Appendix A: Derivation of Equation(13a) and (13b)

In this appendix, we first derive the time-average ACFρ̃h̃ih̃i (τ) of the in-phase component h̃i(t) (5) of the proposeddeterministic simulation model

ρ̃h̃ih̃i (τ) =〈h̃i (t) h̃i (t − τ)

〉 = limT→∞

1

2T

T∫−T

h̃i (t) h̃i (t − τ) dt

= limT→∞

1

2T

T∫−T

1

NiMi

Ni,Mi∑ni,mi=1

NiMi∑n′i,m′

i=1

cos[ψ̃nimi + 2πfTmax t cos(φ̃miT − γT )

+ 2πfRmax t cos(φ̃niR − γR)]

cos[ψ̃n′

im′

i+ 2πfTmax (t − τ) cos(φ̃

m′i

T − γT )

+ 2πfRmax (t − τ) cos(φ̃n′i

R − γR)]

dt

= limT→∞

1

2T

T∫−T

1

NiMi

Ni,Mi∑ni,mi=1

cos[ψ̃nimi + 2πfTmax t cos(φ̃miT − γT ) + 2πfRmax t cos(φ̃niR − γR)

]

× cos [ψ̃nimi + 2πfTmax (t − τ) cos(φ̃miT − γT ) + 2πfRmax (t − τ) cos(φ̃niR − γR)] dt(since ni �= n′i and/or mi �= m′i, ρ̃h̃ih̃i (τ) = 0)

= limT→∞

1

2T

T∫−T

1

2NiMi

Ni,Mi∑ni,mi=1

cos[2πfTmaxτ cos(φ̃

miT − γT ) + 2πfRmaxτ cos(φ̃niR − γR)

]dt (20)

The integral of Equation (20) is trivial as the integranddoes not contain the variable of integration. Therefore,the expression of ρ̃h̃ih̃i (τ) in Equation (13a) can be easilyobtained from Equation (20).

The derivation of the time-average cross-correlationfunction ρ̃h̃ih̃q (τ) between the in-phase component h̃i(t) (5)

and the quadrature component h̃q(t) (6) of the proposeddeterministic simulation model is outlined as follows:

ρ̃h̃ih̃q (τ) =〈h̃i (t) h̃q (t − τ)

〉 = limT→∞

1

2T

T∫−T

h̃i (t) h̃q (t − τ) dt

= limT→∞

1

2T

T∫−T

1√NiMi

√NqMq

Ni,Mi∑ni,mi=1

Nq,Mq∑nq,mq=1

cos[ψ̃nimi + 2πfTmax t cos(φ̃miT − γT )

+ 2πfRmax t cos(φ̃niR − γR)]

sin[ψ̃nqmq + 2πfTmax (t − τ) cos(φ̃mqT − γT )

+ 2πfRmax (t − τ) cos(φ̃nqR − γR)]

dt

=

− limT→∞

1

2T

T∫−T

1

2NM

N,M∑n,m=1

sin[2πfTmaxτ cos(φ̃

mT − γT ) Ni = Nq = N and Mi = Mq

+2πfRmaxτ cos(φ̃nR − γR)]

dt, = M (Cases I and III)0 Ni �= Nq and Mi �= Mq (Case II)

(21)

where at the third equality of Equation (21), setting Ni �= Nqand Mi �= Mq results in the integrand for Case II containingthe variable of integration t unlike the integrand for othercases.



Appendix B: Derivation of the CFρ̂ĥĥ(τ)

In this appendix, we derive the CF ρ̂ĥĥ(τ) for the stochasticsimulation model in Equation (14)

ρ̂ĥĥ (τ) = E[ĥ (t) ĥ∗ (t − τ)]

=

1

NM

N,M∑n,m=1

12∫

−12

12∫

−12

ej2πτ

{fTmax cos

[F−1

(m−1/2+θT

vM

)−γT

]+fRmax cos

[F−1

(n−1/2+θR

uN

)−γR

]}dθT dθR,

Ni = Nq = N and Mi = Mq = M (Cases I and III)

1

2NiMi

Ni,Mi∑ni,mi=1

12∫

−12

12∫

−12

cos

{2πτ

(fTmax cos

[F−1

(mi − 1/2 + θT

Mi

)− γT

]

+ fRmax cos[F−1

(ni − 1/2 + θR

Ni

)− γR

])}dθT dθR

+ 12NqMq

Nq,Mq∑nq,mq=1

12∫

−12

12∫

−12

cos

{2πτ

(fTmax cos

[F−1

(mq − 1/2 + θT

Mq

)− γT

]

+ fRmax cos[F−1

(nq − 1/2 + θR

Nq

)− γR

])}dθT dθR, (Case II)

=

NM

NM

N,M∑n,m=1

F−1( nuN )∫F−1

(n−1uN

)F−1( mvM )∫

F−1(

m−1vM

) ej2πτ

[fTmax cos(βmT −γT )+fRmax cos(βnR−γR)

]ekT cos(β

mT

−µT )+kR cos(βnR−µR)

4π2I0(kT )I0(kR)dβmT dβ

nR

Ni = Nq = N and Mi = Mq = M (Cases I and III)

NiMi

2NiMi

Ni,Mi∑ni,mi=1

F−1(

niNi

)∫F−1

(ni−1Ni

)F−1

(miMi

)∫F−1

(mi−1Mi

) cos{

2πτ[fTmax cos

(β

miT − γT

) + fRmax cos (βniR − γR)]}

× ekT cos(β

miT

−µT )+kR cos(βniR −µR)

4π2I0(kT )I0(kR)dβmiT dβ

niR +

NqMq

2NqMq

Nq,Mq∑nq,mq=1

F−1(

nqNq

)∫F−1

(nq−1Nq

)F−1

(mqMq

)∫F−1

(mq−1Mq

) cos{

2πτ

[fTmax

× cos (βmqT − γT ) + fRmax cos (βnqR − γR)]}

ekT cos(βmq

T−µT )+kR cos(βnqR −µR)

4π2I0(kT )I0(kR)dβ

mqT dβ

nqR (Case II)



=

1

2π2I0(kT )I0(kR)

π∫0

π∫0

ej2πτ[fTmax cos(βT −γT )+fRmax cos(βR−γR)]

× ekT cos(βmT −µT )+kR cos(βnR−µR) dβT dβR, (Case I)1

4π2I0(kT )I0(kR)

π∫−π

π∫−π

cos{

2πτ[fTmax cos (βT − γT ) + fRmax cos (βR − γR)

]}× ekT cos(βT −µT )+kR cos(βR−µR) dβT dβR, (Case II)

1

4π2I0(kT )I0(kR)

π∫−π

π∫−π

ej2πτ[fTmax cos(βT −γT )+fRmax cos(βR−γR)]

× ekT cos(βmT −µT )+kR cos(βnR−µR) dβT dβR (Case III)(22)

where at the second equality of Equation (22), theintegration variables θT and θR were replaced by

βmi/qT = F−1( mi/q−1/2+θTMi/q ), dθT =

Mi/qekT cos(β

mi/q

T−µT )

2πI0(kT )dβ

mi/qT ,

βni/qR = F−1( ni/q−1/2+θRNi/q ), dθR =

Ni/qekR cos(β

ni/q

R−µR )

2πI0(kR)

dβni/qR , β

mT = F−1( m−1/2+θTM ), dθT = Me

kT cos(βmT

−µT )2πI0(kT )

dβmT , βnR = F−1( n−1/2+θRN ), and dθR = Ne

kR cos(βnR

−µR )2πI0(kR)

dβnR. The tow single definite integrals in the lastequality of (22) can be solved by using the equality∫ π

−π ea sin c+b cos cdc = 2πI0(

√a2 + b2) [27]. After some

manipulation, the closed-form expression of the CF ρ̂ĥĥ(τ)can be obtained and is the same as ρhh(τ) in Equation (2).

REFERENCES

1. Kojima F, Harada H, Fujise M. Inter-vehicle communi-cation network with an autonomous relay access scheme.Transactions on Communications 2001; E83-B(3): 566–575.

2. Pabst R, Walke BH, Schultz DC, et al. Relay-baseddeployment concepts for wireless and mobile broadbandradio. IEEE Communications Magazine 2004; 42(9): 80–89.

3. IEEE P802.11p/D2.01. Standard for wireless local areanetworks providing wireless communications while invehicular environment. Technical Report, March 2007.

4. Akki AS, Haber F. A statistical model for mobile-to-mobile land communication channel. IEEE Transactionson Vehicular Technology 1986; 35(1): 2–10.

5. Akki AS. Statistical properties of mobile-to-mobile landcommunication channels. IEEE Transactions on Vehicu-lar Technology 1994; 43(4): 826–831.

6. Zajić AG, Stüber GL. Space-time correlated mobile-to-mobile channels: modelling and simulation. IEEETransactions on Vehicular Technology 2008; 57(2): 715–726.

7. Zajić AG, Stüber GL, Pratt TG, Nguyen S. WidebandMIMO mobile-to-mobile channels: geometry-based sta-tistical modeling with experimental verification. IEEETransactions on Vehicular Technology 2009; 58(2): 517–634.

8. Sen I, Matolak DW. Vehicle-vehicle channel models forthe 5-GHz band. IEEE Transactions on Intelligent Trans-portation Systems 2008; 9(2): 235–245.

9. Cheng X, Wang C-X, Laurenson DI, Vasilakos AV. Sec-ond order statistics of non-isotropic mobile-to-mobileRicean fading channels. Proceedings of IEEE ICC’09,Dresden, Germany, June 2009.

10. Cheng X, Wang C-X, Laurenson DI, Salous S, VasilakosAV. An adaptive geometry-based stochastic model fornon-isotropic MIMO mobile-to-mobile channels. IEEETransactions on Wireless Communications 2009; 8(9):4824–4835.

11. Jakes WC (ed.). Microwave Mobile Communications.IEEE Press: New Jersey, 1994.

12. Zheng YR, Xiao CS. Improved models for the genera-tion of multiple uncorrelated Rayleigh fading waveforms.IEEE Communication Letter 2002; 6(6): 256–258.

13. Youssef N, Wang C-X, Pätzold M. A study on the secondorder statistics of Nakagami-Hoyt mobile fading chan-nels. IEEE Transactions on Vehicular Technology 2005;54(4): 1259–1265.

14. Wang C-X. Pätzold M, Yuan D. Accurate and effi-cient simulation of multiple uncorrelated Rayleigh fadingwaveforms. IEEE Transactions on Wireless Communica-tions 2007; 6(3): 833–839.

15. Wang C-X, Pätzold M, Yao Q. Stochastic modellingand simulation of frequency correlated wideband fadingchannels. IEEE Transactions on Vehicular Technology2007; 56(3): 1050–1063.



16. Wang C-X, Yuan D, Chen HH, Xu W. An improved deter-ministic SoS channel simulator for efficient simulationof multiple uncorrelated Rayleigh fading channels. IEEETransactions on Wireless Communications 2008; 7(9):3307–3311.

17. Pätzold M, Wang C-X, Hogstad BO. Two new sum-of-sinusoids-based methods for the efficient generation ofmultiple uncorrelated Rayleigh fading channels, IEEETransactions on Wireless Communications 2009; 8(6):3122–3131.

18. Patel CS, Stüber GL, Pratt TG. Simulation ofRayleigh-faded mobile-to-mobile communication chan-nels. Transactions on Communications 2005; 53(11):1876–1884.

19. Wang R, Cox D. Channel modeling for ad hoc mobilewireless networks. Proceedings of IEEE VTC’02-Spring,Birmingham, USA, May 2002; pp. 21–25.

20. Zajić AG, Stüber GL. A new simulation model formobile-to-mobile Rayleigh fading channels. ProceedingsIEEE WCNC’06, Las Vegas, USA, April 2006; pp. 1266–1270.

21. Wang LC, Liu WC, YH Cheng. Statistical analysisof a mobile-to-mobile Rician fading channel model.IEEE Transactions on Vehicular Technology 2009; 58(1):32–38.

22. Zheng YR. A non-isotropic model for mobile-to-mobilefading channel simulations. Proceedings IEEE MCC’06,Washington, USA, October 2006; pp. 1–6.

23. Cheng X, Wang C-X, Laurenson DI. A generic space-time-frequency correlation model and its correspondingsimulation model for MIMO wireless channels. Proceed-ings of EuCAP’07, Edinburgh, UK, November 2007;pp. 1–6.

24. Abdi A, Barger JA, Kaveh M. A parametric model forthe distribution of the angle of arrival and the asso-ciated correlation function and power spectrum at themobile station. IEEE Transactions on Vehicular Technol-ogy 2002 51(3): 425–434.

25. Gutiérrez CA, Pätzold M. Sum-of-sinusoid-based simu-lation of flat fading wireless propagation channels undernon-isotropic scattering conditions. Proceedings of IEEEGLOBECOM’07, Washington, D.C., USA, November2007; pp. 3842–3846.

26. Patel CS, Stüber GL, Pratt TG. Comparative analy-sis of statistical models for the simulation of Rayleighfaded cellular channels. Transactions on Communica-tions 2005; 53(6): 1017–1026.

27. Gradshteyn IS, Ryzhik IM. Table of Integrals, Series, andProducts (5th edn), Jeffrey A (ed.). Academic: San Diego,CA, 1994.

AUTHORS’ BIOGRAPHIES

Xiang Cheng (S’05) received B.Sc.and M.Eng. degrees in communica-tion and information systems fromShandong University, China, in 2003and 2006, respectively. Since Octo-ber 2006, he has been a Ph.D. studentat Heriot-Watt University, Edinburgh,U.K.

His current research interests includemobile propagation channel modeling,

and simulation, multiple antenna technologies, mobile-to-mobile communications, and cooperative communications.He has published approximately 30 research papers in ref-ereed journals and conference proceedings.

Mr Cheng was awarded the Postgraduate Research Prizefrom Heriot-Watt University in 2007 and 2008, respectively,for academic excellence and outstanding performance. Heserved as a TPC member for IEEE HPCC2008 and IEEECMC2009.

Cheng-Xiang Wang (S’01-M’05-SM’08) received the B.Sc. andM.Eng. degrees in communication andinformation systems from ShandongUniversity, China, in 1997 and 2000,respectively, and Ph.D. degree in wire-less communications from AalborgUniversity, Denmark, in 2004.

Dr Wang has been a lecturer atHeriot-Watt University, Edinburgh, U.K. since 2005. Heis also an honorary fellow of the University of Edinburgh,U.K., a guest researcher of Xidian University, China, andan adjunct professor of Guilin University of ElectronicTechnology, China. He was a research fellow at the Uni-versity of Agder, Grimstad, Norway, from 2001–2005, avisiting researcher at Siemens AG-Mobile Phones, Munich,Germany, in 2004, and a research assistant at Techni-cal University of Hamburg-Harburg, Hamburg, Germany,from 2000–2001. His current research interests includewireless channel modeling and simulation, error models,cognitive radio networks, vehicular communication net-works, green radio communications, cooperative (relay)communications, cross-layer design of wireless networks,MIMO, OFDM, UWB, and 4G wireless communicationsand beyond. He has published one book chapter and over120 papers in refereed journals and conference proceedings.

Dr Wang serves as an Editor for four internationaljournals: IEEE Transactions on Wireless Communications,Wireless Communications and Mobile Computing Journal(John Wiley & Sons), Security and Communication Net-works Journal (John Wiley & Sons), and Journal of Com-puter Systems, Networks, and Communications (Hindawi).



He is the leading Guest Editor for IEEE Journal on SelectedAreas in Communications, Special Issue on Vehicular Com-munications and Networks. He served or is serving as a TPCmember, TPC Chair, and General Chair for more than 40international conferences. Dr Wang is listed in ‘Dictionaryof International Biography 2008 and 2009’, ‘Who’s Who inthe World 2008 and 2009’, ‘Great Minds of the 21st Century2009’, and ‘2009 Man of the Year’. He is a Senior Memberof the IEEE, a member of the IET, and a Fellow of the HEA.

David I. Laurenson (M’90) is cur-rently a Senior Lecturer at TheUniversity of Edinburgh, Scotland. Hisinterests lie in mobile communications:at the link layer this includes mea-surements, analysis and modeling ofchannels, whilst at the network layerthis includes provision of mobilitymanagement and Quality of Service

support. His research extends to practical implementa-tion of wireless networks to other research fields, such asprediction of fire spread using wireless sensor networks(http://www.firegrid.org), to deployment of communicationnetworks for distributed control of power distribution net-works. He is an associate editor for Hindawi journals, andacts as a TPC member for international communicationsconferences. He is a member of the IEEE and the IET.

Sana Salous (M’95) received B.E.E.degree from the American Univer-sity of Beirut, Beirut, Lebanon, in1978 and M.Sc. and Ph.D. degreesfrom the University of Birmingham,Birmingham, U.K., in 1979 and 1984,respectively. She was an Assistant Pro-fessor with Yarmouk University, Irbid,Jordan, until 1988 and a Research

Associate with the University of Liverpool, Liverpool, U.K.,until 1989, at which point, she took up a lecturer post withthe Department of Electronic and Electrical Engineering,University of Manchester Institute of Science and Technol-ogy (UMIST), Manchester, U.K. In 2003 she took up theChair in Communications Engineering with the School ofEngineering, Durham University, Durham, U.K, where sheis currently the Director of the Centre for CommunicationsSystems.

Athanasios V. Vasilakos (M’89) iscurrently Professor at the Dept. ofComputer and TelecommunicationsEngineering, University of WesternMacedonia, Greece, and Visiting Pro-fessor at the Graduate Programme ofthe Department of Electrical and Com-puter Engineering, National TechnicalUniversity of Athens (NTUA). He is

coauthor (with W. Pedrycz) of the books Computa-tional Intelligence in Telecommunications Networks (CRCpress, USA, 2001), Ambient Intelligence, Wireless Net-working, Ubiquitous Computing (Artech House, USA,2006), coauthor (with M. Parashar, S. Karnouskos, W.Pedrycz) Autonomic Communications (Springer, to appear), Arts and Technologies (MIT Press, to appear), coau-thor (with Yan Zhang, Thrasyvoulos Spyropoulos) DelayTolerant Networking (CRC press, to appear), coauthor(with M. Anastasopoulos) Game Theory in Commu-nication Systems (IGI Inc, USA, to appear). He haspublished more than 200 articles in top internationaljournals (i.e IEEE/ACM Transactions on Networking,IEEE Transactions on Information Theory, IEEE JSAC,IEEE Transactions on Wireless Communications, IEEETransactions on Neural Networks, IEEE Transactions onSystems, Man, and Cybernetics, IEEE T-ITB, IEEE T-CIAIG, etc.) and conferences. He is the Editor-in-chiefof the Inderscience Publishers journals: InternationalJournal of Adaptive and Autonomous CommunicationsSystems (IJAACS, http://www.inderscience.com/ijaacs),International Journal of Arts and Technology (IJART,http://www.inderscience.com/ijart). He was or is at theeditorial board of more than 20 international journalsincluding: IEEE Communications Magazine (1999–2002and 2008-), IEEE Transactions on Systems, Man andCybernetics (TSMC, Part B, 2007-), IEEE Transactionson Wireless Communications(invited), IEEE Transactionson Information Theory in Biomedicine (TITB, 2009-) etc.He chairs several conferences, e.g., ACM IWCMC’09,ICST/ACM Autonomics 2009. He is a chairman of theTelecommunications Task Force of the Intelligent Sys-tems Applications Technical Committee (ISATC) of theIEEE Computational Intelligence Society (CIS). SeniorDeputy Secretary-General and fellow member of ISIBMwww.isibm.org (International Society of Intelligent Bio-logical Medicine (ISIBM)). He is member of the IEEE andACM.


Documents

National Mobile Communications Research Laboratory - New ......WIRELESS COMMUNICATIONS AND MOBILE COMPUTING Wirel. Commun. Mob. Comput. 2011; 11:829–842 Published online 30 October