CDMA cellular systems performance with fading, shadowing, and imperfect power control

450 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 47, NO. 2, MAY 1998

CDMA Cellular Systems Performance with Fading,Shadowing, and Imperfect Power Control

Giovanni Emanuele Corazza, Giovanni De Maio, and Francesco Vatalaro

Abstract—This paper addresses capacity estimation for cel-lular code-division multiple-access (CDMA) systems, assumingthe IS-95 standard as a reference. Extending the analyticalmethod from [6], we obtain a sequence of bounds on capacity,and then we introduce an accurate approximation to reducecomputation complexity. The analysis accounts for interferenceinternal and external to the reference cell, fading, shadowing,and imperfect power control. Outage probability is expressed interms of the characteristic functions (cf’s) of the interferenceand imperfect power control random variables (rv’s). The in-terference contributions are computed on the basis of a Poissondistribution for the number of users in a lognormally shadowedchannel. Results are provided for different channel conditionsand are validated against Monte Carlo simulations. A comparisonagainst previously published CDMA capacity estimates is carriedout, aimed at clarifying some controversial issues. It is alsoconfirmed that large system capacity is achievable under tightpower control.

Index Terms—Cellular systems, code-division multiple access,lognormal channels, outage probability, power control, spread-spectrum systems.

I. INTRODUCTION

T HE code-division multiple-access (CDMA) technique ex-hibits some well-known theoretical advantages for wire-

less personal communications [1]. An outstanding featureof a CDMA cellular system is universal frequency reuse,which gives the potential for large spectrum efficiency, yieldsmaximum flexibility in resource assignment, and allows im-plementation of soft handover algorithms. Universal frequencyreuse is accomplished by proper use of signature sequences ina classical direct-sequence spread-spectrum transmission. Inthe forward link (base station to mobile users), it is possible,within a cell, to have synchronous CDMA. Conversely, inthe return link, asynchronous CDMA is usually adopted. Thereturn link is generally considered to limit system capacity [2],i.e., the average number of users per cell which can be servedwith a specified quality. We will only consider the return linkin the following.

Since all users in a cell share the same frequency band,a stronger signal (possibly sent from a terminal nearer tothe base station) virtually uses a larger part of the availableresource, thus lowering the total number of users that a basestation can serve. This is usually identified as thenear–farproblem. Therefore, power control must be implemented to

Manuscript received September 19, 1996; revised February 27, 1997.The authors are with the Dipartimento di Ingegneria Elettronica, Universita

di Roma Tor Vergata, 00133, Rome, Italy (e-mail: [email protected]).Publisher Item Identifier S 0018-9545(98)03292-7.

ensure that all received signals have very closely the samepower level. However, power control regulates interferencefrom users within the same cell, i.e., those connected withthe same base station (internal interference), but interferencefrom other cells (external interference) will not, in general, bepower controlled.

Another feature of CDMA is that any improvements on thereceiver performance have an immediate and direct influenceon system capacity. In other words, if the threshold value forthe energy per bit-to-noise power spectral density ratio can belowered, then more users can be accommodated without anychange in system architecture. Measures such as diversity andchannel coding are used for this purpose in the recently in-troduced IS-95 CDMA cellular standard [3]. In addition, thereis a widespread interest in interference-reduction techniques.In particular, IS-95 adopts voice activity detection, multiratespeech coding, and 120sectorized base-station antennas.In the future, further advantages may be obtained throughadaptive antennas and multiuser estimation techniques at thebase station.

Estimation of capacity for a CDMA system requires that alarge number of cells is considered, each loaded with a userpopulation of given spatial distribution. This is in contrastwith the analysis of frequency- or time-division multiple-access systems for which worst case conditions can easily beidentified involving only a small number of users. The twokey aspects that need to be properly accounted for in CDMAcapacity estimation arecell selectionand power control. Cellselection by the generic user entails a search for the returnlink with least attenuation, which, through power control, alsocorresponds to least interference to/from surrounding cells.It follows that external interference not only depends on theattenuation in the paths from the interferers to the base stationserving the desired user, but also on the attenuation in the pathsfrom the interferers to their own serving base stations. Powercontrol imperfections and shadowing must be considered forany analysis to be realistic.

In spite of the vast literature on cellular CDMA, someimportant issues are not fully assessed as yet. Gilhousenet al.[2] clearly posed the problem of CDMA capacity estimation,but their results are limited by the following assumptions:1) cell selection is governed by minimum distance ratherthan minimum attenuation; 2) power control is perfect; 3) thenumber of users per cell is fixed; and 4) the overall interferingpower is modeled as a Gaussian random variable (rv). Theminimum distance criterion leads to overestimation of interfer-ence due to errors in cell selection, for shadowing can easily

0018–9545/98$10.00 1998 IEEE

CORAZZA et al.: CDMA CELLULAR SYSTEMS PERFORMANCE 451

produce lower attenuation toward a farther base station. Tosolve this problem, a weight function was introduced in [2] tocut the occurrence of these errors, and the result was claimed tobe an upper bound on the interference power. This procedurewas shown to be not rigorous by Chebaro and Godlewski [4].However, the same authors later on claimed that the boundshould hold for the average external interference [5]. Viterbiet al. carried out an exact analysis involving a minimumattenuation criterion among a fixed number of closest cellsand provided results for [6]. However, their analysisconcerned average external interference only and not systemcapacity. No results appear to be presently available in theliterature for the interference variance. Therefore, the claim in[2] that a lower bound is therein provided for system capacityis still lacking justification. Additionally, as we will show later,the claim in [5] does not have a general value. Extending themethod in [6] and following an approach first introduced in[7], in this work we carry out a complete system capacityevaluation considering the minimum attenuation criterion andimperfect power control, letting the number of users in eachcell be a Poisson rv rather than fixed and modeling thetotal interfering power as a chi-square () rv instead ofGaussian. We derive a sequence of bounds which are tighteras is increased. However, for some channel conditions ofpractical significance, the value of needed for accuracyturns out to be very large, leading to high computationalcomplexity. This was not evident in [6], in which only averageexternal interference was considered. To solve this problem,we additionally introduce an approximation which is accurateeven for relatively small values of . The analytical resultsare validated against Monte Carlo simulations.

The paper is structured as follows. In Section II, we intro-duce the analytical formulation for outage probability, basedon the characteristic function (cf) of both normalized interfer-ence and imperfect power control variables. Then, we definesystem spectrum efficiency and system capacity. In Section III,we introduce the user statistics and the channel model adoptedin the analysis. Then, in Section IV we provide the interferencemodel used to obtain the bounds on system capacity and itsapproximation. In Section V, we extend the analysis to thecase of imperfect power control on all transmitted signals.In Section VI, we provide numerical results in both casesof perfect and imperfect power control. Finally, Section VIIcontains the paper conclusions.

II. OUTAGE PROBABILITY AND SPECTRUM EFFICIENCY

The transmission quality specification for a CDMA cellularsystem may be given in terms of the energy per bit over totalnoise power spectral density averaged over Rayleighfading

(1)

where is the bandwidth, is the single-user informationbit rate, is the nominal received power with ideal powercontrol, is a zero-mean Gaussian rv with variance, whichaccounts for imperfect power control on the desired user,is

the total internal interference power, is the total externalinterference power, is the thermal noise power (AWGN),and . We consider the link to be in outage if

[2].1 The threshold value depends on channelcharacteristics, modulation, and coding. Letting ,

, , and , the outageprobability may be expressed as

(2)

where is the lognormal probability density function(pdf) of the imperfect power control variable andis the pdf of the normalized interference power, which, ingeneral, accounts for both shadowing and imperfect powercontrol effects. Based on a suggestion in [9] to resort to cf’s,the following equivalent expression for the outage probabilitymay be derived:

(3)

where , , anddenotes expectation. Being the internal and external

interference independent, . In thespecial case of perfect power control ( ), we have

.We define the system spectrum efficiency of a cellular

system as the ratio of the overall average data rate in theservice area to the overall bandwidth allocated to the service[10]

(4)

where is the number of cells in the service area, isthe average number of users per cell (system capacity),is the cluster size, and is the bandwidth per cell. Being

for CDMA and fixing both and , the servicebandwidth, then spectrum efficiency increases independentlywith both and . An increase with means thatfor a constant service area, is inversely proportional to

, being the cell radius. System capacity is linked to thetransmission quality specification, i.e., , and tochannel and system characteristics.

III. U SER STATISTICS AND CHANNEL MODEL

We assume that the mobile users are represented by points( ) randomly located on a plane region according to auniform distribution. Therefore, as shown in Appendix I fora CDMA system the number of users,in any given limitedregion (cell, sector, etc.) can be assumed to be Poissondistributed [9]

(5)

where indicates the average number of users in the region.In Appendix I, we also note that, according to this model,

1Other definitions can be assumed for the outage event, e.g., see [8].


the number of users in a region is statistically independentfrom the number of users in any disjoint region. In particular,this model allows to account for instantaneous unequal cellloading, while loading is equal in average. Taking voiceactivity into account, each user being active with probability, it can be easily shown that the number of active users in

any given region is still Poisson distributed with average.We also assume a lognormal shadowing model, so that

the local average channel attenuationbetween the mobileterminal and theth base station may be expressed as

(6)

where is a constant, is the mobile-to-base stationdistance, is the path-loss exponent, is thedB spread, and

is a zero-mean unit-variance Gaussian shadowing variableassumed to be spatially white [2].

We adopt the criterion that a user is connected to the basestation toward which the local average attenuation is minimum,i.e., it is connected to the th base station if

(7)

where, in principle, all base stations must be taken intoaccount. The statistics of the interference may be determinedwith reasonable accuracy if we assume that the minimizationin (7) only involves a finite number of base stations nearest tothe user [6]. However, the probability that the attenuation islower toward a farther station is never null due to shadowing,so the analysis for a finite value of provides an upperbound to the interference power.

IV. I NTERFERENCEANALYSIS

In this section, we address the problem of interferencecharacterization under the assumption of perfect power control.Generalized results will be provided in the next section.

A. Interference Problem Formulation

The normalized interference variableis the sum of the con-tributions coming from all interfering users. The single-usernormalized interference contribution during activity periods is

, where is the local average attenuation towardthe base station to which we assume the desired user is linked(BS0). By means of (6) and (7), the interference rv may alsobe expressed as , where

(8)

having introduced the normalized distance ,.

We denote as the region of the plane which contains allthe points having BS0 included among the nearest stations.As an example, Fig. 1 shows the regions associated to

, , , , and . Computer simulations for differentvalues of show that may be a complex polygonal region.In general, the area of is equal to , where is thearea of the hexagonal cell. When sectorization is adopted, the

Fig. 1. Cellular layout: enlarged borders separate regionsS0 and S0 forNc = 1, 2, 3, 4, and 9.

region is reduced accordingly (e.g., see Fig. 1 for 120sectors).

In order to evaluate the cf of the normalized interference, we consider a large, but limited region of the plane,

say , , containing cells. The region shouldbe sufficiently large so that interference at BS0 coming fromusers outside is negligible. Furthermore, we denote asthe region complementary to within (see Fig. 1). Thesingle-user normalized interferencehas different statisticalbehavior depending on the user belonging to or to .Users belonging to are external interferers and contribute to

. Conversely, an interferer in may be either externalor internal. Therefore, we divide the set of users belongingto into a subset zero, including the desired user and theinternal interferers which contribute to , and a subset, including external interferers contributing to .

B. External Interference

Let indicate the cf associated to a single-userexternal interferer. Assuming that interferers are independentand identically distributed, is the conditional cf forthe interference from users. The unconditional cf of the totalnormalized external interference can be achieved by averagingover the Poisson pdf

(9)

where , with the average number ofusers per cell, and we temporarily assume continuous voiceactivity. In practice, the number of interferers will be limited.However, the contribution in (9) from values ofmuch largerthan is negligible [11], and the same applies to muchsmaller values (note that is in the order of thousands inour computations). For the significant values of, the centrallimit theorem could be applied to approximate witha Gaussian cf, with mean and variance , where and

are the single-user external interference mean and standarddeviation. However, the Gaussian approximation yields a tailfor negative values of the interference power, leading tooptimistic results particularly for low values of . A more


appropriate alternative is given by the one-sided form of thecentral limit theorem [12], which uses the distribution asthe limiting pdf. The corresponding cf is given by

(10)

Therefore, the cf of the total normalized external interferencecan be approximated as

(11)

so that we only need the mean and the variance of the single-user external interference. Considering that the probability of auser to belong to is and that the average probabilityof a user within to be connected to BS0 is 1/, we canshow that

(12)

(13)

where and denote the th moment of the exter-

nal interference within region and region , respectively.To evaluate these moments, we must average over the spatialdomain in which the interference variable is defined

(14)

(15)

where denotes statistical average conditioned on theuser location ( ) and is the probability distributionof a user located in to be an external interferer. As shownin Appendix II, the conditional moments and

, are, respectively, given by

(16)

(17)

where . In , it is assumed that for each point( ) the nearest stations are numbered for .In , the subscript zero is reserved to BS0, and the remaining

stations are numbered for . is thestandard Gaussian probability function

(18)

Note that (14) and (15) are consistent with [6, eqs. (6) and(7)], respectively, for and provide generalizations for

1, with a scaling of on the dB spread due to thedifferent shadowing model adopted.

In our formulation, the mean and variance of the totalexternal normalized interference are related to the firstand second moment of the single-user interference as follows:

(19)

Var (20)

Equations (19) and (20) can be proved through applicationof the moment theorem to the exponential form of .This is a difference with respect to the assumption of a fixednumber of users per cell [2], which leads to a total externalnormalized interference variance simply proportional to.

It is now appropriate to analyze the behavior ofand Var as a function of

the dB spread to gain insight on the accuracy that can beexpected from the above approach and to compare withpreviously published data. The results are reported for ,no sectorization, and continuous voice activity in Figs. 2 and3, respectively, where the contributions from regionsand

, and the overall results are separately shown forand . Fig. 2 also contains the average external interferencecomputed according to the method proposed in [2], and theresults from [6, Table I], the latter for the value andwith the above-mentioned scaling on the dB spread. First,note the agreement between our results for and thosefrom [6], as expected. Further, it is interesting to note thatthe average interference estimated in [2] is higher than thatprovided by our upper bound for for values of overabout 3.8 dB, while it is appreciably lower than the boundbelow this value. This is in agreement with [6], which statesthat the analysis provided in [2] is an approximation ratherthan a bound. This also clarifies that the conclusion presentedin [5] based on numerical results for dB does notpossess general value.2 Turning to an analysis of the momentsestimated by our model, note that for small values of, thecontribution from is predominant over that from , whilethe situation reverses for large values of. For a given valueof , as increases, the contribution from lowers. Thisbehavior can also be seen on the variance in Fig. 3, wherethe knee on the contribution is even more dramatic.

We have found that when the contribution from prevails,the bound on capacity is generally too pessimistic. This meansthat in order to have accurate results, for a fixed value of thedB spread, one should select a value for large enoughto make the contribution predominant. Unfortunately, for

2Although the dB spread should not be converted from decibels, it appearsthat the value�S = 10

0:8= 6:3 was actually used for the numerical

computations in [2], as confirmed later [5].


Fig. 2. Normalized meanEfIext=Wg=N as a function of�s, the dB spreadin the channel (Nc = 4 and9, = 4, no sectorization, and continuous voiceactivity). Results from [2] and [6] are shown for comparison.

Fig. 3. Normalized variance VarfIext=Wg=N as a function of�s, the dBspread in the channel (Nc = 4 and9, = 4, no sectorization, and continuousvoice activity).

typical values of the dB spread (i.e., around 8 dB), the valueneeded for becomes very large. To solve this problem,we introduce here an alternative method for the computationof the external interference moments in , which yields avery accurate approximation even for large dB spread withrelatively small values of . In essence, the modificationconsists in adding BS0 to the set of nearest stations to agiven point in . When the attenuation is lowest toward BS0,

Fig. 4. Normalized meanEfIext=Wg=N for Nc = 1, 2, 3, 4, 9, and70as a function of the dB spread in the channel�s ( = 4, no sectorization,and continuous voice activity). The result of the approximation forNc = 9

is also reported.

the contribution to the external interference is nulled. This issomewhat similar to the weight function approach from [2], buthere we have a much larger number of base stations, whichare checked before the contribution is neglected. Followingthis approach, (16) must be substituted with

(21)

where, as before, it is assumed that for each point () thenearest stations are numbered for . Notethe strong similarity of (17) and (21). In Figs. 4 and 5, wereport and Var as a function ofthe dB spread for , with as a parameter, for boththe bounding and the approximation approaches. Contrastingthe approximation for against the bound for ,it is apparent that (21) can be used to match the results ofthe bounding approach obtained for extremely large, evenfor large values of the dB spread. In particular, this applies tothe evaluation of the external interference variance, which, ingeneral, needs large values to be accurately estimated.

C. Internal Interference

To derive the cf of the overall normalized internal interfer-ence, we first consider that the probability of havinginternalinterferers is connected to the probability of having userswithin the region , given by (5), conditioned on having one


Fig. 5. Normalized variance VarfIext=Wg=N for Nc = 1, 2, 3, 4, 9, and70 as a function of the dB spread in the channel�s ( = 4, no sectorization,and continuous voice activity). The result of the approximation forNc = 9

is also reported.

desired user. Taking also into account that the mean of theinternal interference Poisson distribution is equal to, wehave

internal interferersone desired userusers

at least one user connected to BS0

(22)

Then, the cf of the internal interference is evaluatedas the cf of the interference from users averagedover

(23)

With perfect power control, the cf for an internal interferer issimply , while under imperfect power control,

is evaluated in the following section.

V. IMPERFECT POWER CONTROL

In our analysis, we assume that imperfect power controlis such as not to modify the selection of the serving basestation by any mobile terminal. This is equivalent to assumingperfect estimation of received pilot tones power and an error

in transmitted power at the mobile. The impact of imperfectpower control is twofold. First, it varies the power level of thedesired user, possibly causing the received signal at BS0 tofall below the quality threshold. Second, it alters the statisticaldistribution of both internal and external interference. The firsteffect is prevailing and is explicitly taken into account in (3)through , the cf of a lognormal variate, which is related tothe zero-mean Gaussian variable having variance [see(1)]

(24)It is now necessary to see how and are

modified by imperfect power control. All interfering signalsare assumed to be affected by power control impairments withthe same statistics as the desired signal. Therefore, the single-user internal interference variable again follows the lognormaldistribution, so that given by (24). Accordingto our modeling, to characterize the external interference, weonly need the first and second moments, conditioned on userlocation ( ), of the single-user interference variables inregions and . Taking into account the independencebetween the lognormal imperfect power control variable andthe lognormal interference variable, it can be shown that themoments of any order may be simply evaluated as the productof the moment of the same order of the interference variableunder perfect power control and that of a lognormal rv withparameter [13]

(25)

So modified, the first- and the second-order moments ofthe normalized external interference rv are then used in theexpressions for and , respectively, provided by (12) and(13).

VI. ESTIMATION OF SYSTEM CAPACITY

In this section, we provide outage probability results, com-puted numerically on the basis of our formulation, and validatethem through Monte Carlo simulations. All the results havebeen obtained over a region comprising cells.

For system capacity calculations, the values of the normal-ized moments in Figs. 4 and 5 must be multiplied by theaverage number of users per cell, by the voice activityfactor, and divided by the number of sectors to obtain thecorresponding moments to be used in (19) and (20) forany practical case. All the following results refer to 120sectorization and voice activity factor of 3/8. Also, we set

kb/s, MHz, and dB [2]. First,we assess the influence of 1) channel parametersand ; 2)

threshold ; and 3) imperfect power control standarddeviation . In these cases, the bounding technique is usedwith , which is large enough to ensure accuracy forthe considered values of .

1) Assuming dB, dB, and dB,Fig. 6 shows outage probability as a function of theaverage number of users per cell for different values


Fig. 6. Outage probability for different values of the path-loss exponent

and �s = 6 dB as a function of the average number of users per cellN(Nc = 4, � = 7 dB, 120� sectorization, voice activity factor= 3=8, andideal power control).

of the path-loss exponent. Outage probability increasesvery significantly with decreasing . The agreementwith simulation results is very satisfactory. Analogously,Fig. 7 provides system capacity results for dB.Comparing Figs. 6 and 7, it can be noted that for smallervalues of , the capacity loss due to higher dB spreadis larger.

2) It is interesting to note the influence of on outageprobability, as reported in Fig. 8: at ,every 1-dB decrease translates into a capacity gain ofat least 20 users per cell. This fact motivates the use ofcoding, diversity, and interference cancellation as muchas receiver complexity allows.

3) Releasing the ideal power control assumption, Fig. 9shows outage probability with the power control errorstandard deviation as a parameter for ,dB, and dB. At , every 1-dB increasetranslates into a capacity loss of at least 20 users percell: a specification of dB for the power controlprocedure seems necessary. This specification forisin substantial agreement with that reported for the IS-95system based on experimental evidence [14].3

Referring to a dB spread of dB, Fig. 10 comparesunder different assumptions the average number of users per120 sectorized cell, , which was the parameter assumedin [2] for CDMA capacity estimates. The dotted curve cor-responds to the so-called “surrounding cells full” case in[2]. For the sake of comparison, we modified our modelingaccording to the assumptions of Gaussian interference powerand fixed number of users. By doing so, we find excellent

3Note that the measured standard deviation in [14] is not directly compa-rable to�p for it contains other contributions too.

Fig. 7. Outage probability for different values of the path-loss exponent and �s = 4 dB as a function of the average number of users per cellN(Nc = 4, � = 7 dB, 120� sectorization, voice activity factor= 3=8, andideal power control).

Fig. 8. Outage probability for different values of theEb=N0 threshold� as afunction of the average number of users per cellN (Nc = 4, = 4, �s = 6

dB, 120� sectorization, voice activity factor= 3=8, and ideal power control).

agreement between the result from [2] with this modifiedbound calculated for (dashed–dotted curve). If weincrease the number of tested base stations, all other conditionsbeing maintained, the capacity bound is increased, gainingabout two users per sector for (dotted curve), thusshowing that previous estimates could be improved. However,when we return to the assumption of interference power


Fig. 9. Outage probability for different values of imperfect power controlstandard deviation�p as a function of the average number of users per cellN (Nc = 4, � = 7 dB, = 4, �s = 6 dB, 120� sectorization, and voiceactivity factor = 3=8).

Fig. 10. Outage probability as a function of the number of users per sectorNs. Comparison of results from [2] with our model forNc = 4 and9 anddifferent assumptions on the number of users and external interference powerdistributions (�s = 6:3 dB, � = 7 dB, = 4, 120� sectorization, and voiceactivity factor = 3=8).

and Poisson distribution of users, as the solid curve in Fig. 10shows, the bound is decreased, providing a loss of about threeusers per sector at .

As a final result, in Fig. 11 we provide capacity estimatesfor dB. The dotted curve shows the bound on systemcapacity when tested base stations are assumed—the

Fig. 11. Outage probability as a function of the number of users per cellN .Comparison of bounding technique forNc = 9 and 70 and approximationfor Nc = 9 (�s = 8:0 dB, � = 7 dB, = 4, 120� sectorization, and voiceactivity factor = 3=8).

capacity estimate is still far from its actual value as we canshow by evaluating the bound for an extremely large number oftested stations (e.g., , see solid curve in Fig. 11). Thesolid curve is practically coincident with the curve we achievewith the approximation (21) and a much more reasonable valueof tested stations (e.g., , see dotted curve in Fig. 11).

VII. CONCLUSIONS

This paper presented a new approach to the evaluationof system capacity for a CDMA cellular system. Havingextended to higher moments the analysis provided in [6], ourapproach allows a complete system capacity evaluation. Thiswas carried out here on the basis of a sequence of boundsand introducing an approximation which is often advisable toreduce computation complexity of the former technique.

Our approach takes into account both internal and externalinterference, fading, shadowing, and imperfect power control.The approach is characterized by several features which distin-guish it from previously proposed methods. The main featuresare:

1) adoption of a minimum attenuation criterion overbase stations to derive the interference statistics ratherthan the minimum distance criterion ( );

2) use of a model for the number of users through a Poissondistribution, which is more realistic than that based ona fixed number of users per cell;

3) use of the one-sided form of the central limit theorem,so that the interference power pdf is always null fornegative values, as opposed to the case of the Gaussianapproximation;


4) inclusion of a lognormal variable to account for powercontrol inaccuracy on both the desired user and all theinterfering users.

Validation via Monte Carlo simulation was carried outyielding very good agreement. Comparison with previouslypublished results was also provided, so clarifying some con-troversial issues on CDMA system capacity. In conclusion, theanalysis provided in the paper confirmed the high performanceexpected in the presence of relatively tight power control.

APPENDIX ISTATISTICAL DISTRIBUTION OF THE

NUMBER OF USERS IN A REGION

Let and indicate the areas of two planar regionsand such that . Assume the users’ locations

( ) to be independent identically distributed uniform rv’sinside region . Users are located in regionwith probability

. Considering users in , the probability of havingusers in is given by

(26)

Extending both the region and the number of users toinfinity under the condition

constant (27)

it is possible to show [11] that tends to a Poissondistribution with parameter. In practice, for a CDMA system,

is finite, but very large such that the use of the Poissondistribution is justified.

Furthermore, if we consider two disjoint regions andinside , under the same limiting condition for and asabove, it can be shown [11] that the Poisson distributions forthe numbers of users in and are statistically independent.

APPENDIX IIMOMENTS OF THEEXTERNAL INTERFERENCECONDITIONED

ON USER LOCATION WITHIN REGIONS AND

In this Appendix, we extend to the moments of any order aresult provided in [6] for the average only. We start evaluatingthe moment of order of the single-user interference in thedomain , conditioned on a certain location ( ). From (8)we have

(28)

where we let , and we made use of the assumptionthat and ( ) be independent rv’s. The first factor iseasily found

(29)

To evaluate the second factor in (28), we need the expressionfor the pdf of the minimum among independent rv’s. Let

, be the pdf of , andsimilarly let be the pdf of . Itholds

(30)

where , is the cumulative distributionfunction (cdf) of . Therefore, we have

(31)

It holds

(32)

(33)

where and are the pdf and the cdfof the Gaussian variable conditioned on the point ( ),respectively. We have

(34)

and by means of straightforward manipulations on (34), takinginto account (29), we have (16).

Now, we consider the moments of the single-user externalinterference rv of the user located within the region .This interference may again be expressed as ,see (8), with

(35)

where the minimum is evaluated among the nearest basestations, excluding BS0. We note that for 1,otherwise the user would be controlled by BS0. Therefore, wehave

(36)

where , and we let. By noting that is zero-mean Gaussian with


variance , if we interchange in (36) integration with respectto and , we have

(37)

Substituting (30) in (37) and after some manipulations similarto those performed before, we achieve (17).

ACKNOWLEDGMENT

The authors gratefully acknowledge cooperation from Dr.F. Ceccarelli in performing computer simulations and thankthe reviewers for their constructive criticism which led tosignificant improvements in the paper.

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Giovanni Emanuele Corazzawas born in Trieste,Italy, in 1964. He received the Dr.Ing. degree inelectronic engineering in 1988 from the Universityof Bologna, Italy, and the Ph.D. degree in 1995 fromthe University of Rome Tor Vergata, Italy.

From 1989 to 1990, he was with the Canadianaerospace company COM DEV, Ontario, wherehe worked on the development of microwave andmillimeter-wave components and subsystems. In1991, he joined the Department of Electronic En-gineering, University of Rome Tor Vergata, where

he is presently a Research Associate. During 1995, he visited ESA/ESTEC,The Netherlands, under a research fellowship granted from the EuropeanCommunity. During 1996, he visited the Communications Sciences Institute,University of Southern California, Los Angeles. His research interests are inthe areas of communication theory, personal communications systems, spread-spectrum techniques, and synchronization.

Dr. Corazza received the Marconi International Fellowship Young ScientistAward in 1995 (ex-aequo). He serves as an Editor for spread spectrum for theIEEE TRANSACTIONS ON COMMUNICATIONS.

Giovanni De Maio was born in Rome, Italy, in1968. He received the electronic engineering degreefrom the University of Rome Tor Vergata, Rome,Italy, in 1995.

In 1997, he joined CoRiTel, Rome, a researchconsortium specializing in telecommunication dis-ciplines, where he currently deals with architecturedefinition and performance evaluation of proposalsfor the third-generation universal mobile telecom-munications systems (UMTS’s) radio interface. Hispresent research interests include multiple-access

schemes for wireless personal communications and radio channel modeling.

Francesco Vatalaro received the Dr.Ing. degreein electronics engineering from the University ofBologna, Italy, in 1977.

From 1977 to 1980, he was with FondazioneUgo Bordoni at Pontecchio Marconi, Italy. Then, hewas with the FACE Standard Central Laboratory,Pomezia, Italy, from 1980 to 1985. While withSelenia Spazio, Rome, Italy, he was Group Leaderof Satellite Ground Segment System Engineering.In 1987, he became an Associate Professor of RadioSystems at the University of Roma Tor Vergata. He

is presently a Visiting Professor at the University of Southern California,Los Angeles, holding a course on spread spectrum for the spring semesterof 1998. He has been responsible for several research contracts both atItalian national and European levels. He is the author of more than 60scientific papers in international and national journals and in internationalconference proceedings. His research interests include mobile and personalcommunications, spread-spectrum communications, and satellite systems.

Dr. Vatalaro was a cowinner of the 1990 “Piero Fanti” INTEL-SAT/Telespazio international prize. He is a member of the AEI.

CDMA cellular systems performance with fading, shadowing, and imperfect power control

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