Accuracy of geometric channel-modeling methods

82 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 53, NO. 1, JANUARY 2004

Accuracy of Geometric Channel-Modeling MethodsYifan Chen and Vimal K. Dubey, Senior Member, IEEE

Abstract—When constructing a propagation channel model, asubstitute is often created by giving an arbitrary shape or form toscatterer distributions based on its intuitive appeal for a certainradio environment. However, such models do not necessarilyrepresent the actual propagation process and may yield inaccurateresults. The main objective of this paper is to provide an insightinto the underlying relationship between geometric models andthe particular physical propagation process they represent. Theworkhorse is the semi-geometrically based statistical (SGBS)model and the two heuristic rules. The SGBS model definesthe distribution of dominant scatterers contributing to the lastreradiation of multipath components to the base station. Theearlier multiple-reflection process is modeled using the compositeNakagami/log-normal probability density function. Two param-eters are then introduced; namely, the effective path length andthe normalized space-dependent intensity measure. Using these twometrics, two heuristic rules are subsequently proposed to providethe missing link between the canonical models and the physicalchannel. The rules are then applied to revisit several widely usedgeometric models in macro- and microcellular environments. As aworking example, the Gaussian scatterer density model is furtherextended using such an approach. Important channel parameterssuch as power azimuthal spectrum, power delay spectrum, andazimuthal and delay spreads are then calculated and comparedwith simulation results.

Index Terms—Compound spatial point process, geometricchannel models, multipath channels, propagation aspects, wirelesscommunications.

I. INTRODUCTION

I N recent years, the growing demand to increase the ca-pacity of wireless channels by using smart antennas for

antenna/space diversity has motivated researchers to investigatethe propagation phenomena in the space domain [1]–[3]. Oneof the most commonly used directional channel models isbased on a geometric description of the scattering process[3], [4], which focuses on the detailed internal constructionor realization of the channel. The model assumes a statisticaldistribution of scatterers around the two ends of the wirelesslink and channel properties are derived from the positions ofthe scatterers by applying the fundamental laws of propagationmechanism of electromagnetic waves. Most of the existinggeometric channel models (GCMs) take into account onlythe local scattering cluster [5]–[7], which is always locatedaround the mobile unit with few available models defining theshape and distribution of far clusters [8]–[10]. The GCM iswell suited for simulations requiring a complete model of the

Manuscript received June 25, 2002; revised December 18, 2002, April 14,2003, July 1, 2003, and September 4, 2003.

The authors are with the School of Electrical and Electronic Engineering,Nanyang Technological University, Singapore 639798, Singapore (e-mail: [email protected]; [email protected]).

Digital Object Identifier 10.1109/TVT.2003.821999

wireless channel due to its ray-tracing nature. However, theshape and size of the spatial scatterer density function requiredto achieve a reliable simulation of the propagation phenomenonis subject to debate. Other authors have proposed stochasticchannel models, which make certain assumptions about theform of the received signal vector [11]–[14]. In [11], the VonMises angular distribution has been introduced to model thenonisotropic angle of arrival of multipath components. Astatistical model based on different environment types is alsodescribed in [12] and [13], which uses empirical input suchas power delay spectrum (PDS), power azimuthal spectrum(PAS), and shadow fading. A similar model [14] derives thejoint distribution of path gain and delay spread and providesfor the performance study of various cellular systems. Finally,the most comprehensive attempt ought to be the COST 259directional channel model [8], which includes several modelsin the generic framework. An overview of the recent modelingattempts involving the multiple-input–multiple-output radiochannel can be found in [2], [3], and [15].

Generally, the aforementioned GCMs have two main draw-backs. First, only a single specular reflection is accounted forand neither scattering, diffraction, nor multiple bounces are con-sidered. Second, the geometry of scattering areas in the existingGCMs is obtained on a rather undefined plane and is, thus, ques-tionable. The inadequacy of the former argument is apparent,although the latter requires further elaboration.

In the analysis of a GCM, one subject is often overlooked byresearchers—the actual constitution of a realistic GCM. Theaxioms corresponding to a priori properties of the channel thatis modeled using geometric approach must represent realisticwave propagation. A GCM is one type of geometric model,which simulates a process or phenomenon that has a geometricanalog [16]. In a GCM, a shape (spatial pattern of scatterers)is defined by a set of logical relationships satisfying a set ofaxioms (wave propagation obeying diffraction, reflection, andscattering laws), which are in turn interpreted as true statementsabout the model. Subsequently, one can safely infer the processbehavior from the model behavior if and only if the modelclosely corresponds to the process. However, the shape andscatterer density of the scattering areas in the existing GCMshave been defined with a certain level of ambiguity, with neitherrigorous mathematical statement nor sound physical reasoningavailable to support the validity of the various models.

This work provides a unified analysis to circumvent the twoproblems of existing GCMs. The semi-geometrically based sta-tistical (SGBS) model is first presented. It is parameterized insuch a way that only the distribution of scatterers contributingto the last reradiation is emulated, while the preceding mul-tiple-bounce is modeled as a stochastic process, which has alognormal shadowing with Nakagami fading [1], [17]. Such a

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CHEN AND DUBEY: ACCURACY OF GEOMETRIC CHANNEL-MODELING METHODS 83

modeling approach has two distinct merits. First, it reproducesthe basic stochastic processes1 of the received signal vector overspace and time (e.g., PAS, PDS, azimuthal, and temporal disper-sions) and is useful in studying the higher order moments aboutthe origin of the stationary processes (e.g., variation of powerreceived in the azimuth and delay domains). Second, it pavesthe way for the postulation of the two suppositions to be intro-duced in this paper. An attempt is then made to unveil the under-stated assumptions made in arbitrary GCMs involving the phys-ical propagation process, which is achieved by the introductionof two parameters: the effective path length (EPL) and the nor-malized space-dependent intensity measure (NSDIM). The EPLis related to the large-scale path loss suffered by multipath com-ponents (MPCs) via the scatterers, whereas the NSDIM bearsrelation to the spatial patterns of scattering points. With thesetwo parameters, two heuristic rules are subsequently proposedfrom the wave-propagation viewpoint to establish a relationshipbetween the geometry and the physical channel. To demonstratethe accuracy and applicability of these two rules, the a prioriassumptions made in several widely used GCMs are further ex-ploited, thereby revisiting the validity of these models.

The organization of this paper is as follows. Section IIpresents the SGBS model and describes the employed parame-ters, while Section III gives the expressions of EPL and NSDIMwith an explanation of their physical significance. The twoheuristic rules are presented in this section as well. Section IVthen proceeds to deal with the accuracy of a few widely usedGCMs. As a working example, a popular urban macrocellmodel is extended to include multiple scattering, as shown inSection V, where the spatial pattern of scatterers is circularlysymmetric and follows a bivariate Gaussian distribution [4],[7], [9]. The important channel parameters based on theextended Gaussian scatterer density model (EGSDM) are thencalculated. Finally, the conclusion is presented in Section VI.

II. SEMIGEOMETRICALLY BASED STATISTICAL MODEL

A. General Remarks

The wireless channel is typically a multipath channel due tothe natural and man-made objects, which are situated betweenthe base station (BS) and mobile station (MS), as depicted inFig. 1. At the receiver, plane waves arriving from different di-rections are combined vectorially to produce a composite re-ceived signal. If only one discrete scatterer is present, an expres-sion can be easily found for the signal at the receiver by usingfundamental wave propagation and scattering laws. However, ifmany objects are present, things become much more compli-cated, since the interaction between different objects must beaccounted for. Consequently, a complex model would result ifmultiple bounces are to be accurately modeled. By placing ascattering object at the last reradiation and approximating thepreceding scattering as a stochastic process, some of the prop-erties of multiple bounces can be retained while providing for amuch simpler model.

1Only received power is studied in the subsequent discussion for the sakeof simplicity. The other two basic stochastic processes involving the complexreceived voltage and the received envelope are not considered in this work.

Fig. 1. Typical multipath propagation environment.

When this approach is applied to a GCM, a compoundrandom point field would first be defined2 [18], [19]. This fieldis determined by scatterer positions witha real valued scatterer response process correspondingto each position. The points are samples from a spatial pointprocess, which will be described as a nonhomogeneous Poissonprocess in Section III. The represent the instantaneouspower contributed by individual scatterers at the receiver,which are statistically independent and wide sense stationary.It would be useful to model this function as a shadowedNakagami-fading process, since the Nakagami distribution ismathematically convenient and is able to closely approximate aRicean distribution when a specular path is present in the radiopropagation environment.

Although scatterers are buildings, cars, and other pertinentstructures in real-life situations, rather than delta functions, theyare indistinguishable from spatial delta functions as their dimen-sions are small relative to the spatial resolution of the system, asnoted by Braun and Dersch [20]. Moreover, the reradiation froma scatterer may be considered as narrowband3 for the majority ofexisting communication systems [21], i.e., the difference in thescattering properties within the frequency band is negligible dueto the small bandwidth of the system as compared to the carrierfrequency (usually much less than 1% such as the HiperLAN/2system, which operates on the 5-GHz band with a channel band-width of 20 MHz [22]).

Nonetheless, recent advancement in wireless communi-cations has generated a renewed interest in ultrawide-band(UWB) systems. As defined by the Federal CommunicationCommission, such systems operate either with a fractionalbandwidth greater than 20% or a 10 dB bandwidth that isbeyond 500 MHz [23]. In such cases, when a wave reflectsoff an object or penetrates through a material during multipathpropagation, the effects are frequency sensitive, thus resultingin the waveform being filtered to some extent. The pulse shapeassociated with a propagation path is dependent upon that path,with each path having a unique impulse response or frequencytransfer characteristic [24]. Hence, the distribution of scattererswould have to be defined in a three-dimensional (3-D) space,

2It has been assumed that the number of scattering objects in the environmentis large enough to form an ensemble of the point process. Furthermore, the finite-dimensional distribution of the points satisfies the symmetry and consistencycondition as stated in Kolmogorov theorem [18].

3Note that the above assumption is different from the other narrow-band as-sumption, which regards the system bandwidth as relative to the delay spreadof the channel. A system may be narrowband in the former case, but not in thelatter.


Fig. 2. Pictorial representation of scattering from a scatterer S.

with two dimensions representing the Euclidean space and onedimension representing the frequency domain [25]. This is asimple extension of the methodology introduced in this paper toa higher dimension. In the discussions that ensue, focus wouldbe placed solely on the narrow-band scenario, with spatialpattern of the point process considered to be indifferent acrossthe whole spectrum. Thus, the effect of frequency variationswill not be taken into account.

B. Model Description

Based on the abovementioned methodology, the followingSGBS model is proposed by taking into account multiplebounces. Fig. 2 shows the multipath propagation from the MSto the BS. In the sequel, it is assumed that the signals received atthe BS are plane waves arriving from the horizon. In addition,the following terms are introduced:

Scatterer contributing to the finalreradiation of incoming MPCs to theBS.Distance between the BS (MS) andscatterer .Local objects around , scatteringfrom which results in local subpathsto .Objects around the MS that causethe MPCs to undergo multiple reflec-tions and/or diffractions.Scattering coefficient of .Directive gain of the antenna at theBS (MS) that satisfies the normaliza-tion function fortwo-dimensional (2-D) propagation.

It is further assumed that the transmitter-to-scatterer path hasa shadowed Nakagami fading [1], [17]. Subsequently, the nor-malized area-mean power received at has the general form

(1)

with being the path-loss exponent. Assuming shadowing tobe superimposed on (1), the local-mean power is log-normallydistributed about the area mean, viz.

(2)

where is the standard deviation of the shadowing, ex-pressed in natural units. Multipath reception at the scatterercauses Nakagami fading; with the squared envelope having aGamma-log-normal density function

(3)

For the special case when the multipath is Rayleigh-distributed, (3) reduces to a composite exponential/lognormal

probability density function (pdf).As shown in [1], the composite pdf can be approximated by

a lognormal distribution with mean and variation

new (4)

new (5)

where is the Euler psi function and is the Riemann’szeta function.

Let new new , by applying(1) and (4) to the equation, we obtain

new (6)

where is defined as the Nak-agami-fading loss factor (NFLF). It is observed that the effectof Nakagami fading in (3) would decrease the meanand increase the variance. However, such an effect reduces asthe shape factor increases [1].

Apparently, the received power reradiated by the scattererat the BS is also log-normally distributed with mean and stan-dard deviation

(7)

new (8)

with . The operator representseither the product or sum of the modulus of its arguments, de-pending on whether scattering or specular reflection dominates.As lognormals tend to remain lognormal under addition andsubtraction [26], the locally averaged total path gain at a given


Fig. 3. Artificial coordinate (r, l).

location thus tends to be lognormal, as commonly observed inmany field trials [1].

In order to fully develop the model, the following additionalassumptions are made.

1) Attenuation loss is a constant within each identifiablescattering cluster, which is dependent upon the reflectionloss pertaining to the material of the scatterer surface [27].

2) Signals reradiated from a scatterer region possess thesame fading statistics, which implies that the NFLF isa constant within each cluster.

III. UNDERLYING CONNECTION BETWEEN THE SPATIAL

PATTERN OF SCATTERERS AND ACTUAL

WAVE-PROPAGATION PROCESS

A. Effective Path Length and Normalized Space-DependentIntensity Measure

Assuming that the spatial position of a scattering point is de-noted by the vector with components ( , ) related to an ar-tificial coordinate in a specified Borel set of 2-D Euclideanspace , where

is a continuous and differentiable function with the modulus ofthe arguments defined in Section II (or refer to Fig. 3). is apoint at the oriented curve : with being an arbi-trarily defined starting point and the positive sense on beingclockwise, as illustrated by the arrow in Fig. 3. In our formu-lation of the GCM, the function is defined such that the non-homogeneous Poisson process in could be constructed usingthe intensity measure at any vector .

First, let be the intensity of independent identi-cally distributed (i.i.d.) homogeneous Poisson process4 on dis-crete curves : , : . Sinceis a continuous function, the number of points on a fixed line

will be zero with probability one. Thus, the lines should befattened up to strips

4It is assumed that the intensity measure of the Poisson process defined hereonly depends on r. Nevertheless, this conjecture holds an intuitive appeal (allthe factors affecting the distribution of scatterers have been embedded in thearguments of r) and has been implicitly assumed in many existing GCMs.

as shown in Fig. 4. The points in different strips (nonoverlap-ping differential regions) are independent, which is essential forthe uncorrelated scattering properties assumed in many channelmodels. A more formal realization of this process can be de-scribed below, following from [19]. Let be a positive func-tion of the vector with dimension area . Let be a smallneighborhood of of area (Lebesgue measure) with all thepoints of the process occurring outside denote by . If

denote the number of points in a region , it is re-quired that

pr

pr

pr

where pr denotes the probability. Since the intensity measuredepends only on , it is rewritten as in the subsequent

discussion. It can be easily shown that if is an arbitraryregion of area , has a Poisson distribution of mean

[18], [19].When the total number of points in the bounded region is

normalized to 1, actually represents the variation of scat-terer density at any position on the curve , which isuniform along the curve due to the fundamental characteristicsof a homogeneous Poisson process on . Such a parameter isthus defined as the normalized space-dependent intensity mea-sure (NSDIM). Apparently, the distribution of can be obtainedusing the line integral of over . viz.

(9)

where denotes the length of the curve .Next, the wave propagation in the wireless link is considered.

The EPL is defined as

(10)

with being the corresponding Nakagami-fading loss factor inthe targeted scattering area. As shown in (7), the expectationof received power at the BS from any scattering point is in-versely proportional to . Therefore, the EPLis a measure of the mean path loss suffered by MPCs, whereasthe NSDIM is relevant to the geometry of canonical models.

B. Heuristic Rules

The following heuristic rules governing the relationshipbetween the geometric distribution of scatterers ( and

) and the power distribution are proposed. Apictorial representation of the two rules is shown in Fig. 5.

1) Bijective mapping rule: Let denote the set of all curvesin the region with each curve representing a collectionof scatterers with the same value of and let denotethe set of all curves with each representing a collectionof scatterers in the same region, which contribute to aconstant average received power at the BS. The mappingf: defines a one-to-one correspondence between

and through the function .


Fig. 4. Realization of the spatial nonhomogeneous Poisson process.

Fig. 5. Pictorial representation of the two heuristic rules.

2) Tapering-off NSDIM rule: If defines a monotonicallyincreasing function, the NSDIM gradually tapers off asincreases where the maximum NSDIM corresponds to theline-of-sight (LOS) path between the BS and the MS inthe radio link. Thus, as the average received power at theBS due to reradiation from a scatterer at a certain locationdecreases, the intensity measure at that location decreasesas well.

A useful analogy can be used to establish the relationship be-tween the two rules and the rules governing the equipotentiallines in electromagnetic filed theory. In the geometric channel-modeling problems, the set in the first rule resembles the vi-sualization of an electrical field and the set is similar to thecollection of equipotential lines. The electrical potential is con-stant on each equipotential line, as depicted in Fig. 6. (In a sim-ilar manner, scatterers are uniformly distributed on each curvein the GCMs.) These curves will not intersect each other (bijec-tive mapping rule). Assuming the potential at infinite distanceto be zero, the further the lines are to the charges, the smallerthe absolute value of the potential (tapering-off NSDIM rule).

The two heuristic rules have provided a venue to exploit theunderstated assumptions made in an arbitrary GCM concerningthe power distribution in the wireless link. Nevertheless, it hasto be noted that they are only a first approximation in a spatiallyregular scattering environment, where the waves arriving at thereceiver stem from the same propagation mechanism and/orscattering process [28], thereby specifying a unique nonho-mogeneous Poisson process. For the case of a more-complexwireless channel where the propagation of signals could begoverned by various mechanisms, the distribution of scattererscan be classified into a few geometric clusters, as distinguishedby their respective spatial point processes [29]. This taxonomyneed not imply separable topography and the channel impulseresponses due to different clusters may be overlapping in theazimuth-delay domain. A detailed discussion of multicluster

Fig. 6. Equipotential lines due to a pair of configuration charges.

environments would be beyond the scope of this paper and,thus, will not be included herein.

IV. REVISITING SOME MOSTLY USED

GEOMETRIC CHANNEL MODELS

A. Macrocellular Environments

In what follows, it is assumed that omnidirectional antennasare used at both ends of the connection, i.e.,

. The EPL can, thus, be simplified as

(11)

The circular scattering model (CSM) is shown in Fig. 7,where the scatterer density is assumed to be uniform in azimutharound the MS. Such models have been suggested for macrocelltype of environments with moderate values of angular spreadand large values of delay spread. It has been assumed that in


Fig. 7. Circular scattering model.

macrocellular environments, there will be no signal scatteringfrom locations near the BS, since the antenna heights are rela-tively large. Furthermore, it is also assumed that the scattererssurrounding the MS are about the same height or higher thanthe mobile.

The curve representing an aggregate of scatterers with thesame value of is a circle defined by : . Applyingthe bijective mapping rule, the EPL is a bijection of ,which is valid when the following relationship is satisfied:

(12)

Equation (12) implies

if is defined as sum (13.1)if is defined as product (13.2)

with being a constant. In this case, the EPL oris a monotonically increasing function with respect to

, thereby showing that the condition for the tapering-offNSDIM rule is fulfilled. First, the CSM implies that when spec-ular reflection dominates, the transmitted signal has severe Nak-agami fading around the MS, as suggested by (13.1). Further-more, when scattering is the principal mechanism of interactionbetween the electromagnetic field and the material surfaces, themajority of scattering elements are densely clustered around theMS and the width of the scattering area is much less than the dis-tance between the BS and the MS (13.2). These two statementsare generally valid in a dense urban macrocell [3].

Three distributions of that are currently available in theliterature will be considered in this section. It should be notedthat only the scatterers contributing to the last reradiation ofMPCs are emulated in the models in order for them to be an-alyzed in the proposed framework.

A linear distribution

elsewhere(14)

This pdf leads to the GBSBM model [5], where the scatterers areuniformly distributed within a circle with radius . Employing(9), the NSDIM can be obtained as

(15)

which is a constant and does not abide by the tapering-offNSDIM rule. In fact, (15) implies that as the received power

due to rebouncing from a scatterer at a certain location de-creases, the scatterer density at that location remains constant.Apparently, this assumption of the GBSBM model does notaccurately describe the observed phenomena (average power ofsignal decreases systematically with increasing distance) andrequires further investigation.

A normal distribution of the - and the - coordinate mea-sured from the position of the MS leads to a Rayleigh distribu-tion for .

(16)

where is the standard deviation of the local scatterer density.This pdf leads to the Gaussian scatterer density model (GSDM)[4], [7], [9], where the spatial distribution of the scatterers is ap-proximated by the bivariate Gaussian pdf. Similarly, the NSDIMis given by

(17)It is apparent from (17) that the NSDIM decreases monoton-

ically as the EPL increases, which abides by the second rule.Therefore, implementing this model to resemble the physical re-alities actually implies that as the averaged power at the BS dueto reradiation from a scatterer at a certain location decreases,the intensity measure at that location decreases as well, whichholds an intuitive appeal.

A one-sided Gaussian distribution is represented using

(18)

with being the standard deviation of the scatterer density.This distribution leads to the generic MIMO channel model[30]. Subsequently, the NSDIM follows from (9) as

(19)

This model also abides by the tapering-off NSDIM rule. Acomparison of (17) and (19) reveals that as increases [byobserving in the denominator of (19)], the mean numberof relevant scatterers in the one-sided Gaussian distributionmodel decreases more significantly as compared to the pre-ceding model. Although an exact mapping of the average signalpower to the NSDIM is difficult to justify, the two heuristicrules nonetheless reveal a close relationship between the spatialpattern and the physical propagation process.

B. Micro- or Picocellular Environments

The elliptical scattering models (ESMs) [31] (Fig. 8) are pro-posed for the pico- or microcell, where antenna heights are rel-atively low and the BS and the MS are the foci of the ellipticalarea. With low antennas, the BS will receive MPCs from scat-teres distributed around the mobile as well as itself. All anglesin the azimuth plane are involved in this case and the maximum


Fig. 8. Elliptical scattering model.

delay is determined by the size of the ellipse. In the physicalsense, it is possible to ignore components with larger delays,since signals with longer delays have relatively low power ascompared to those with shorter delays, due to the longer pathlength traveled.

The curve representing a collection of scatterers with thesame value of is an ellipse defined by : .Applying the bijective mapping rule, the EPL represents aone-to-one correspondence of . This statement is justifiedwhen the following relationship is satisfied:

(20)

which implies

and the operator designates sum (21)

In such a case, the EPL consistently increases with . Theexpression in (21) implies that, for the ESM’s, specular reflec-tion is the dominant mechanism of interaction between the elec-tromagnetic waves and the surfaces of scattering objects. Othermechanisms, such as diffraction and diffuse scattering, can beignored. Moreover, the transmitted signal has mild short-termfading around the MS. Such observations are generally valid inindoor propagation [32].

In the GBSBEM model [31], the scatterers are uniformlydistributed within an ellipse with semi-major axis and semi-minor axis , as illustrated in Fig. 8. The pdf ofcan be found as

(22)

The NSDIM following from (9) is

(23)

where

(24)It can be shown from (24) that the pdf has a maximum and an

integral singularity at , where the peak corresponds to theLOS path. The GBSBEM model thus abides by the tapering-off

Fig. 9. Parabolic scattering model.

NSDIM rule and is a good approximation to the physical realityfrom a wave-propagation viewpoint.

C. Additional Comments

Apart from the abovementioned scenarios, the Nakagami-fading loss factor could also be a direction-sensitive variable,whereby the closer the scatterers are to the LOS path betweenthe BS and the MS, the less severe the Nakagami fading is. Sucha phenomenon could occur when the BS antenna is mounted onthe rooftop of a roadside building along a straight road on whicha MS is located. This represents a street-dominated environ-ment, as reported in [33]. A parabolic scattering model (PSM),as shown in Fig. 9, would fit into the situation mentioned aboveand is very convenient for the analysis of channel characteristics[34].

A parabolic line can be parameterized as (assuming that theEPL

(25)

which can be rewritten as

for small (26)

where is half of the latus rectum of the parabolic line. From

(26), and increases as increases.A brief summary of this section is listed in Table I. It has

been demonstrated that both the circular and elliptical scatteringmodels can be investigated using the proposed analytical tool.The validity of the models has also been reexamined using thetwo rules proposed. The analysis clearly points out that it maybe necessary to reconsider the understated assumptions in someof the widely used GCMs.

V. EXTENDED GAUSSIAN SCATTERER DENSITY MODEL:A WORKING EXAMPLE

A. General Remarks

As shown in Table I, the commonly used Gaussian scattererdensity model [4], [7], [9] is an appropriate first approximationof the urban macrocellular environments based on the two rules.As suggested by [9], the scatterers could be organized in clus-ters (Fig. 10), which are special cases of the geometric clus-ters as mentioned in Section III-B (see [34] for more details).The local cluster corresponds to waves, which travel directly tothe BS after being scattered around the MS. On the other hand,


TABLE ISUMMARY OF IMPLICATION OF THE TWO RULES

Fig. 10. Cluster structure of multipath scattering in the EGSDM model.

the apparent clusters represent waves, which, after being scat-tered around the MS, are reflected by an extended object faraway from the MS. In this section, this model would be ex-tended based on the modeling approach presented in Section II,for which the scatterers are assumed to contribute solely to thelast reradiation of the incoming waves. The power azimuthaland power-delay spectrums are then calculated using this modeland compared with empirical and simulation results. We furtherdefine the azimuthal spread (AS) and the delay spread (DS),which are measures of azimuthal and temporal dispersions ofthe channel, respectively. This section would also present theequations for describing the variance relationship of small-scalereceived power fluctuations. Only the properties of the localcluster are being considered here, since the properties of ap-parent clusters can be analyzed in a similar manner.

B. PAS at the BS

Consider the triangle BS- -MS in Fig. 7, it can be shown that

(27)

As described in Section II, the instantaneous power received atthe BS due to reradiation from the scatterer with the coordinates( , ) is log-normally distributed. Thus, the joint pdf of

instantaneous power and angle of arrival (AOA) , whichis dependent upon the area mean power , can be expressed as

(28)

where is the standard deviation of the lognormal shadowing.is related to the EPL by with being the

average attenuation loss of the scatterers or as shownin (13) and representing the path loss exponent.

The next step lies in determining the joint pdf of andconditioned on , which can be evaluated using the fol-

lowing transformation:

(29)

where is the Jacobian transformation given by.

Squaring (27) and solving for yields

(30)

Differentiating (30) gives

(31)

Using (16) and (27)–(31), we can derive the PAS after somemanipulations as


(32)

where and are the normalization factors to ensure thatis a pdf, denotes expectation, is the integra-

tion region, and is given in (30). is a small constantto avoid integral singularity at . Interestingly, the PASobtained using the SGBS model is independent of the lognormalshadowing around the MS, which implies that the spatial patternof the random field has retained all the information required tocharacterize the first-moment statistics of the received power inthe azimuth domain. The AS is computed as the root secondcentral moment of the PAS [13].

The variance of the received power along the azimuth angleis derived as

(33)

The physical significance of is that it describes the averagereceived power fluctuations in a local area along the azimuthdirection .

C. PDS at the BS

Based on the elementary geometrical considerations (Fig. 7),the path delay can be expressed as

(34)

where m/s is the propagation speed of EM wave infree space. Differentiating (34) yields

(35)

The joint pdf of and conditioned on is found using

(36)

By averaging over the Rayleigh-distributed random variable, the PDS can be written as

(37)

where and are normalization factors. Similarly, the PDSis independent of the composite fading statistic around the MS,which means that the first-moment statistics of received powerin the delay domain is completely retained by the geometryof the channel model. The delay spread is derived as the rootsecond central moment of the PDS [13], while the variance ofthe received power associated with the delay is calculated as

(38)


Fig. 11. (a) PAS for the EGSDM. Curves for � = 4:5 are shifted by 2 inthe coordinate; otherwise, they would be masked. (b) PDS for the EGSDM.Curves for � = 4:5 are shifted by 10 in the coordinate; otherwise, they wouldbe masked.

As a final remark, it should be noted that the above deriva-tions are presented in a general way, which can be convenientlyapplied to other extended GCMs defined with different spatialpatterns of scattering points.

D. Comparison of Theoretical and Simulation Results

The main purpose of this section is threefold. First, the the-oretical derivations of PAS and PDS will be verified by com-paring them with the simulation results. As the closed-form ex-pressions for (32) and (37) are not tractable, the integral is henceevaluated using numerical integration. Second, the dependencyof the azimuthal and delay spreads on the ratio of the standarddeviation of scatterer distribution to base-to-mobile distance isstudied. Finally, the variances of the received power in the az-imuth and delay domains are investigated.

In the first case, the base-to-mobile distance is 2 km andthe standard deviation of the local scatterer density is 0.2 km,

Fig. 12. Plots of (a) azimuthal and (b) delay spreads as a function of ratio ofstandard deviation of the scattering area to base-to-mobile distance.

with the path loss exponent and the attenuation lossof the local scattering area . It is further assumed thatthe NFLF corresponding to a Nakagami-fadingshape parameter . Subsequently, PAS and PDS for twodifferent values of standard deviation of shadowing wouldbe compared. Simulated normalized histograms are generatedby creating bivariate normally distributed scatterers around theMS and the PAS and PDS are calculated by performing 10 000Monte Carlo trials. The theoretical and simulation resultsare plotted in Fig. 11. It can be observed that the normalizedhistograms closely match the theoretical graphs. Fig. 11(a)shows that the PAS using the EGSDM has a good agreementwith a Gaussian pdf of standard deviation of 5.5 , which hasbeen verified in certain field measurements for GSM systemsin rural and suburban environments [35], [36] and is used in thestudy of diversity antennas [37], [38] and smart antennas [39].Fig. 11(b) shows that the PDS tapers off monotonically forlarger delay. It is interesting to note that the slope for the PDS inthe semi-log plot is approximately constant, which implies thatthe PDS is almost exponential in shape. This observation is inagreement with the results reported in many papers (see [1] andreferences therein). Last, it is noted that the variations in thestandard deviation of shadowing do not introduce a significant


Fig. 13. Plots of variance of received power in the (a) azimuth and (b) delaydomains.

difference in the angular and delay spectrum, as predicted by(32) and (37).

In the second case, the standard deviations of scatterer dis-tribution are varied from 150 to 300 m with the remaining pa-rameters unchanged. It is observed from Fig. 12 that as the ratioof increases, both the AS and DS increase linearly, whichare comparable to the results obtained in other geometric models[5], [7]. This also illustrates that there is a positive linear crosscorrelation between the AS and DS, if each pairs of AS and DScould be considered as the medians for different environmentclasses (typical urban, bad urban, and sub urban, etc. [13]). Fur-thermore, these results indicate that the mechanisms leading toazimuthal and temporal dispersion are related as observed in thefield measurement [13] and the potential space and frequencydiversity gains are highly correlated.

In the third study case, the same channel condition as in case1 is assumed. It is apparent from Fig. 13(a) that the varianceof received power increases to a maximum at (directpath) and decreases as the arrived signal is further away fromthe direct path. One possible explanation for the observation is

that as approaches zero, the received signal, which is reflectedoff the local scattering elements close to the direct path, wouldfluctuate more severely due to the more pronounced aggregateshadowing effect. The same conjecture can be used to explainFig. 13(b), where the variance of the received power is foundto be larger at a lower path delay . Furthermore, Fig. 13 alsoshows that as the standard deviation of shadowing increases by0.5 unit, the variance increases by 40 dB. Hence, the lognormalshadowing has a substantial impact on the average power fluc-tuations in a local area for both the azimuth and delay domains.

VI. CONCLUSION

In this paper, we have introduced a semi-geometrically basedstatistical channel model for a multipath propagation environ-ment, which defines the distribution of principal scatterers con-tributing to the last reradiation of MPCs to the BS. The com-posite Nakagami log-normal pdf was used to model the pre-ceding multiple-reflection process. Two parameters, namely, theeffective path length and normalized space-dependent intensitymeasure, were introduced to better understand the relationshipbetween the geometry of the canonical models and the actualpropagation process. Subsequently, the methodology was usedto revisit several widely used GCMs for narrow-band stationaryenvironments with omnidirectional antennas employed at bothends of the communication link. It was shown that both the cir-cular and elliptical scattering models could be inspected usingthe proposed analytical tool. The investigation also reveals thatthe understated assumptions in some of the widely used GCMswere questionable. As a working example, the popular GSDMmodel that abides by the heuristic rules was further extendedwith important channel parameters such as the PAS, PDS, AS,DS, and variance of received power derived. The theoretical andsimulation results showed a normally distributed PAS centeredat the base-to-mobile path, as well as an approximately expo-nential decaying PDS. Both are consistent with many availablemeasurements. Furthermore, theoretical analysis revealed thatthe average received power fluctuations were more significantas the arriving signals become closer to the base-to-mobile path.

ACKNOWLEDGMENT

The authors are grateful to the anonymous reviewers for theirvaluable comments.

REFERENCES

[1] G. L. Stuber, Principles of Mobile Communication, 2nd ed. Norwell,MA: Kluwer, 2001, pp. 98–117.

[2] U. Martin, J. Fuhl, I. Gaspard, M. Haardt, A. Kuchar, C. Math, A. F.Molisch, and E. Bonek, “Model scenarios for direction-selective adap-tive antennas in cellular mobile communication systems—scanning theliterature,” Wireless Pers. Commun., vol. 11, no. 1, pp. 109–129, 1999.

[3] R. B. Ertel, P. Cardieri, K. W. Sowerby, T. S. Rappaport, and J. H. Reed,“Overview of spatial channel models for antenna array communicationsystems,” IEEE Pers. Commun., vol. 5, no. 1, pp. 10–22, Feb. 1998.

[4] J. Fuhl, A. Molisch, and E. Bonek, “Unified channel model for mo-bile radio systems with smart antennas,” Proc. Inst. Elect. Eng.—Radar,Sonar Navig., vol. 145, pp. 32–41, Feb. 1998.

[5] P. Petrus, J. H. Reed, and T. S. Rappaport, “Geometrical-based statis-tical macrocell channel model for mobile environments,” IEEE Trans.Commun., vol. 50, pp. 495–502, Mar. 2002.


[6] O. Norklit and J. B. Anderson, “Mobile radio environments and adap-tive arrays,” in Proc. IEEE Int. Symp. Personal Indoor Mobile RadioCommun. (PIMRC) ’94, vol. 2, 1994, pp. 725–728.

[7] R. Janaswamy, “Angle and time of arrival statistics for the Gaussianscatter density model,” IEEE Trans. Wireless Commun., vol. 1, pp.488–497, July 2002.

[8] L. Correia, Ed., Wireless Flexible Personalized Communications. NewYork: Wiley, 2001.

[9] “Proposal for a Unified Spatial Channel Model,” Lucent Technol.,Espoo, Finland, Tech. Rep. 3GPP TSG R1-01-0722, June 2001.

[10] “Micro-Cell Wideband Directional Channel Model,” IMST GmbH, NewYork, Tech. Rep. 3GPP TSG R1-01-0992 , Nov. 2001.

[11] A. Abdi, J. A. Barger, and M. Kaveh, “A parametric model for the dis-tribution of the angle of arrival and the associated correlation functionand power spectrum at the mobile station,” IEEE Trans. Veh. Technol.,vol. 51, pp. 425–434, May 2002.

[12] K. I. Pedersen, P. E. Mogensen, and B. H. Fleury, “A stochastic modelof the temporal and azimuthal dispersion seen at the base station in out-door propagation environment,” IEEE Trans. Veh. Technol., vol. 49, pp.437–447, Mar. 2000.

[13] A. Algans, K. I. Pedersen, and P. E. Mogensen, “Experimental analysisof the joint statistical properties of azimuth spread, delay spread, andshadow fading,” IEEE J. Select. Areas Commun., vol. 20, pp. 523–531,Apr. 2002.

[14] L. J. Greenstein, V. Erceg, Y. S. Yeh, and M. V. Clark, “A newpath-gain/delay-spread propagation model for digital cellular chan-nels,” IEEE Trans. Veh. Technol., vol. 46, pp. 477–485, May 1997.

[15] K. Yu and B. Ottersten, “Models for MIMO propagation channels, a re-view,” J. Wireless Commun. Mobile Comput., vol. 2, no. 7, pp. 653–666,2002.

[16] M. E. Motenson, Geometric Modeling, 2nd ed. New York: Wiley,1997, pp. 1–18.

[17] M. K. Simon and M. S. Alouini, Digital Communication Over FadingChannels—A Unified Approach to Performance Analysis. New York:Wiley, 2000, pp. 15–30.

[18] H. Cramer and M. R. Leadbetter, Stationary and Related Stochastic Pro-cesses. New York: Wiley, 1967.

[19] D. R. Cox and V. Isham, Point Processes. London, U.K.: Chapman andHall, 1980.

[20] W. R. Braun and U. Dersch, “A physical mobile radio channel model,”IEEE Trans. Veh. Technol., vol. 40, pp. 472–482, May 1991.

[21] T. Svantesson, “Physical Channel Modeling of Multi-Element AntennaSystems ,” Chalmers Univ. Technol., Göteborg, Sweden, Tech. Rep.R002/2001, Apr. 2001. [Online]: www.ee.byu.edu:8080/~tomaso/pub-lications.html.

[22] J. Khun-Jush, P. Schramm, U. Wachsmann, and F. Wenger, “Structureand performance of the HiperLAN2 physical layer,” in Proc. IEEE Ve-hicular Technology Conf., Sept. 1999, pp. 2667–2671.

[23] First Report and Order in the Matter of Revision of Part 15 of the Com-mission’s Rules Regarding Ultra-Wideband Transmission Systems, Fed-eral Communication Commission , Washington, DC, Apr. 22, 2002.

[24] R. C. Qiu, “A study of the ultra-wideband wireless propagation channeland optimum UWB receiver design,” IEEE J. Select. Areas Commun.,vol. 20, pp. 1628–1637, Dec. 2002.

[25] Y. Chen and V. K. Dubey, “A frequency domain geometric model fora wideband mobile radio channel,” in Proc. IEEE Int. Symp. PersonalIndoor Mobile Radio Commun. (PIMRC)’03, vol. 2, Sept. 2003, pp.1227–1231.

[26] L. F. Fenton, “The sum of log-normal probability distributions in scattertransmission systems,” IRE Trans. Commun. Syst., vol. CS-8, pp. 57–67,Mar. 1960.

[27] S. Ichitsubo, K. Tsunekawa, and Y. Ebine, “Multipath propagationmodel of spatio-temporal dispersion observed at base station in urbanareas,” IEEE J. Select. Areas Commun., vol. 20, pp. 1204–1210, Aug.2002.

[28] J. Laurila, K. Kalliola, M. Toeltsch, K. Hugl, P. Vainikainen, and E.Bonek, “Wide-band 3-D characterization of mobile radio channels inurban environment,” IEEE Trans. Antennas Propagat., vol. 50, pp.233–243, Feb. 2002.

[29] Y. Chen and V. K. Dubey, “A generic channel model in multi-clusterenvironments,” in Proc. IEEE Vehicular Technology Conf., vol. 1, Apr.2003, pp. 217–221.

[30] A. F. Molisch. A generic model for MIMO wireless propagationchannel. presented at Proc. IEEE Int. Conf. Commun., Available onlineat http://www.merl.com/papers/TR2003-42/

[31] R. B. Ertel and J. H. Reed, “Angle and time of arrival statistics for cir-cular and elliptical scattering models,” IEEE J. Select. Areas Commun.,vol. 17, pp. 1829–1840, Nov. 1999.

[32] H. Hashemi, “The indoor radio propagation channel,” Proc. IEEE, vol.81, pp. 943–968, July 1993.

[33] A. Kuchar, J.-.P. Rossi, and E. Bonek, “Directional macro-cell channelcharacterization from urban measurements,” IEEE Trans. AntennasPropagat., vol. 48, pp. 137–146, Feb. 2000.

[34] Y. Chen and V. K. Dubey, “Parabolic distribution of scatterers for street-dominated mobile environments ,” IEEE Trans. Veh. Technol., submittedfor publication.

[35] D. Aszetly, “On antenna arrays in mobile communication systems: Fastfading and GSM base station receiver algorithms,” Ph. D. Dissertation,Royal Inst. Technol., Stockholm, Sweden, Mar. 1996.

[36] U. Martin, “Spatio-temporal radio channel characteristics in urbanmacrocells,” Proc. Inst. Elect. Eng.—Radar, Sonar, Navig., vol. 145,no. 1, pp. 42–49, 1998.

[37] F. Adachi, M. T. Feeney, A. F. Williamson, and J. D. Parsons, “Crosscor-relation between the envelopes of 900 MHz signals received at a mobileradio base station site,” Proc. Inst. Elect. Eng., F, vol. 133, pp. 506–512,Oct. 1986.

[38] J. Luo, J. R. Zeidler, and S. Mclaughlin, “Performance analysis of com-pact antenna arrays with MRC in correlated Nakagami fading channel,”IEEE Trans. Veh. Technol., vol. 50, pp. 267–277, Jan. 2001.

[39] B. Ottersten, “Spatial division multiple access in wireless communica-tions,” in Proc. Nordic Radio Symp., Apr. 1995.

Yifan Chen received the B.Eng. degree in electricaland electronic engineering (first class honors)from Nanyang Technological University (NTU),Singapore, in 2002. He is currently working towardthe Ph.D. degree with the School of Electrical andElectronic Engineering, NTU.

His research interests include the general areasof wireless communications, with emphasis onmobile radio channel stochastic modeling, ultra-wide bandwidth signal propagation aspects, andmultiple-input–multiple-output channel characteri-

zation.

Vimal K. Dubey (M’88–SM’93) received the B.Sc. (Hons.) degree in mathe-matics from the University of Rajasthan, Jaipur, India, the B.E. and M.E. degreesin electrical communication engineering from the Indian Institute of Science,Bangalore, and the Ph.D. degree in electrical engineering from McMaster Uni-versity, Hamilton, ON, Canada.

He has been with various research and development laboratories in India formore than ten years. From 1972 to 1976, he was a Research Scientist withDLRL, Hyderabad, India. From 1976 to 1982, he was with DEAL, Dehradun,India, where he conducted research on spread-spectrum systems for satellitecommunications and transportable troposcatter communication system. From1982 to 1986, he was a Commonwealth Research Scholar at McMaster Univer-sity. He joined the School of Electrical and Electronic Engineering, NanyangTechnological University, Singapore, in 1988, where he is now an AssociateProfessor. His main research interests include the areas of digital communi-cations, specializing in coding, modulation, and spread-spectrum systems forsatellite and wireless communications.

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