CDMA systems in fading channels: admissibility, network capacity

962 IEEE TRANSACTIONS ON INFORMATION THEORY, VOL. 46, NO. 3, MAY 2000

CDMA Systems in Fading Channels: Admissibility,Network Capacity, and Power Control

Junshan Zhang, Student Member, IEEE,and Edwin K. P. Chong, Senior Member, IEEE

Abstract—We study the admissibility and network capacityof imperfect power-controlled Code-Division Multiple Access(CDMA) systems with linear receivers in fading environments.In a CDMA system, a set of users isadmissibleif their simulta-neous transmission does not result in violation of any of theirQuality-of-Service (QoS) requirements; thenetwork capacityis themaximum number of admissible users. We consider a single-cellimperfect power-controlled CDMA system, assuming knownreceived power distributions. We identify the network capacitiesof single-class systems with matched-filter (MF) receivers forboth the deterministic and random signature cases. We alsocharacterize the network capacity of single-class systems withlinear Minimum-Mean-Square-Error (MMSE) receivers forthe deterministic signature case. The network capacities can beexpressed uniquely in terms of the users’ signal-to-interferenceratio (SIR) requirements and received power distributions. Formultiple-class systems equipped with MF receivers, we find anecessary and sufficient condition on the admissibility for therandom signature case, but only a sufficient condition for thedeterministic signature case. We also introduce the notions ofeffective target SIRand effective bandwidth, which are useful indetermining the admissibility and hence network capacity of animperfect power-controlled system.

Index Terms—Admissibility, CDMA, deterministic signature,fading channel, matched filter, MMSE, network capacity, powercontrol, random signature, scale family.

I. INTRODUCTION

T HE last fifteen years have witnessed a tremendous growthof wireless networks. Due to the fast-growing demand for

network capacity in wireless networks, it is essential to utilizeefficiently the limited resources. The characterization of net-work capacity is therefore a fundamental and pressing issue inwireless network research. In this paper, we consider a modelfor the uplink of a single-cell symbol-synchronous Code-Divi-sion Multiple Access (CDMA) system in fading channels. Thenetwork therein consists of numerous mobile subscribers com-municating with one base station, which is typically intercon-nected to a backbone network via a wired infrastructure.

Two approaches have been studied extensively to achieveefficient utilization of network resources in CDMA systems:

Manuscript received September 16, 1998; revised October 12, 1999.This work was supported in part by the National Science Foundation underGrant ECS-9501652 and by the U.S. Army Research Office under GrantDAAH04-95-1-0246. The material in this paper was presented in part at theAllerton Conference, Monticello, IL, 1998 and at IEEE Infocom, New York,NY, 1999.

The authors are with the School of Electrical and Computer Engineering,Purdue University, West Lafayette, IN 47907-1285 USA (e-mail: {junshan}{echong}@ecn.purdue.edu).

Communicated by M. L. Honig, Associate Editor for Communications.Publisher Item Identifier S 0018-9448(00)03088-1.

multiuser detectionand power control. Multiuser detectionrefers to the process of demodulating one or more user datastreams from a nonorthogonal multiplex and is concernedwith designing good receivers to achieve efficient interferencesuppression. In particular, among multiuser receivers, linearreceivers have attracted a large amount of attention becausethey are practically appealing (see, e.g., [13]–[15], [30]). Powercontrol, on the other hand, is implemented at the transmitter andis concerned with allocating powers to fulfill individual users’Quality-of-Service (QoS) requirements (see, e.g., [7], [9], [28],[36]). Since both linear receivers and power control are em-ployed to suppress interference effectively and utilize networkresources efficiently, it is natural to ask how linear receiversperform in power-controlled systems. In [27], Tse and Hanlycharacterized the network capacities for several importantlinear receivers via a notion ofeffective bandwidth, assumingusers have random signatures. (See also earlier work in [9].)Viswanath, Anantharam, and Tse [31] studied the joint opti-mization problem of signature allocation and power control, andobtained simple characterizations of the network capacities ofsingle-cell systems with linear Minimum-Mean-Square-Error(MMSE) receivers. However, both [27] and [31] assumedperfect power control in characterizing the network capacity. Ina practical wireless communication system, due to delays anderrors in power control and time-varying channel conditions, itis difficult to implement perfect power control, and the receivedpowers typically fluctuate around the desired levels. Therefore,it is more appropriate to model the received powers as random.However, little work has been done on characterizing networkcapacity of imperfect power-controlled CDMA systems withlinear receivers in fading channels.

In this paper, we focus primarily on the admissibility andnetwork capacity ofimperfectpower-controlled CDMA sys-tems with matched filter(MF) receivers in fading channels.Throughout this paper, we assume MF receivers unless speci-fied otherwise. Each user in the system is assigned a signatureonto which the user’s information symbols are spread. Everyuser also has a minimum signal-to-interference (SIR) require-ment. Roughly speaking, a set of users is said to beadmissibleif their simultaneous transmission does not result in violationof any of their SIR requirements; thenetwork capacityis themaximum number of admissible users. Following the approachof [27], [30], we formulate the problem in an asymptotic settingin which we allow the number of users and the degrees offreedom (length of the signatures) to grow, while keepingtheir ratio fixed. The results are stated in terms of this ratioof number of users per degree of freedom. A feature thatdistinguishes this work from [27] is that the received power for

0018–9448/00$10.00 © 2000 IEEE

ZHANG AND CHONG: CDMA SYSTEMS IN FADING CHANNELS 963

Fig. 1. A simplified block diagram of the uplink of a CDMA system with linear receivers.

each user in our model is random. The SIR requirements in oursetting are also therefore probabilistic, unlike those of [27].

We treat separately systems with single class of users andsystems with multiple classes of users. In each case, we con-sider both deterministic and random signatures. For a single-class systems with MF receivers, we identify the network ca-pacities for both deterministic and random signature cases. Wealso characterize the network capacity of a single-class systemwith linear MMSE receivers for the deterministic signature case,which turns out to be exactly the same as that of the corre-sponding system with MF receivers. For multiple-class systemswith MF receivers, we provide a necessary and sufficient con-dition for the random signature case, but only a sufficient con-dition for the deterministic signature case, for a set of users perdegree of freedom to be admissible. The analysis in the deter-ministic signature case involves Welch-bound-equality (WBE)signature sets and inequalities, using results of [16], [34], as in[31]. Based on the results on the admissibility, we introduce thenotions ofeffective target SIRandeffective bandwidth, whichare useful in determining the admissibility and hence networkcapacity of an imperfect power-controlled system.

The organization of the rest of this paper is as follows. In thenext section, we provide the model description of an imperfectpower-controlled CDMA system in fading channels. In SectionIII, we study the admissibility and identify the network capaci-ties of single-class systems with MF receivers. We also identifythe network capacity of single-class systems with linear MMSEreceivers for the deterministic signature case. Then we addresssome issues on power control and introduce the notion ofef-fective target SIR. In Section IV, we extend the study to mul-tiple-class systems, and explore the notion ofeffective band-width for imperfect power-controlled CDMA systems. Finally,we draw some conclusions in Section V.

II. SYSTEM MODEL

Consider a -user CDMA system where the users transmitdata over a fading channel. Fig. 1 depicts a simplified blockdiagram of the uplink of a CDMA system with linear receivers.Based on [27], [28], [30], we have the following discrete-time

model for the uplink of a synchronous CDMA system: for anysymbol interval, the baseband received signal at the front end ofthe receiver is

(1)

where and are user 's signature, informationsymbol, and received power, respectively, . Thevector is background noise that comes from thesampling of the ambient white Gaussian noise with positivevariance . Note that different processing gains correspond tosystems with different bandwidths, in which users may havedifferent received powers. To keep our model more general, weuse to denote user 's received power to emphasize thedependence of the received power on the processing gain.Since our purpose is not to evaluate or optimize modulationperformance, for simplicity, we assume that the modulation isantipodal, i.e., . This assumption is not crucial,but simplifies the analysis.

The signatures provide diversity gain and the correspondingmodel is of considerable interest. Following [27], [29], we as-sume that when the processing gain is, takes value in

. We study the admissibility and network ca-pacity for the following two cases:

• The deterministic signature case:In this case, the users’signatures are jointly designed and are deterministic (see,e.g., [16], [23], [31]). Our objective for the deterministicsignature case is to find the tightest upper bound on thenumber of admissible users over all possible choices ofsignatures regardless of practical constraints.

• The random signature case: In this case, the userschoose their signatures randomly and independently,which is applicable to several practical scenarios [15],[27], [30] (we will elaborate further on this point inSection III). Our objective for the random signature caseis to characterize the number of admissible users in thismore realistic setting.


We assume that each user is capable of decentralized powercontrol. One common approach to implementing power controlin practical CDMA systems is to drive the received powers forall the users having the same QoS requirements to be a fixedpredetermined value all the time, namely,power balancing(see,e.g., [7], [32]). In fact, as illustrated in various chapters in [29],equal received power is the optimal power allocation for usershaving same QoS requirements in most modulation/demodula-tion systems. Because of fading effects and power control errors,however, the received powers typically fluctuate around the de-sired levels in a practical system. Accordingly, we assume thatthe received powers are random (due to imperfect power con-trol), and that the received powers across different users are in-dependent. Throughout this paper, we also assume that powercontrol is good enough to ensure that the fluctuation of receivedpowers around their expectations is uniformly bounded (say by) with probability one, which is reasonable in any practical

system.Let denote the support of the received powers in the lim-

iting regime (as ). Define . For tech-nical reasons, we assume that is connected. (By [22, Propo-sition 4.12], can be either an interval or a single point.) Weassume throughout that the distributions of the received powersare known. This assumption is reasonable because the distribu-tions of the received powers can usually be obtained throughmeasurements (see, e.g., [21]).

Typically, since the output interference in a large CDMAsystem can be approximated as Gaussian (see, e.g., [30]), itis reasonable to take the QoS requirement as meeting the SIRconstraint (see, e.g., [36], [27], [30]). We define the SIR to bethe ratio of the desired signal power to the sum of the noise andmultiple-access interference (MAI) powers at the output of thereceiver in a symbol interval [14]. Because of the randomnessof the received powers and/or signatures, the SIR is random aswell. Therefore, we adopt a probabilistic model for the users’QoS requirements as follows (cf., [18], [19]):

SIR

where SIR is the achieved SIR of userwhen the processinggain is the target SIRof user , and

.Our results are asymptotic in nature, with both and

going to infinity. As is standard (see, e.g., [27], [30]),is taken to be fixed as goes to infinity. In fact, we only need

; however, it is more convenient to fix thisratio in the following discussions.

III. SINGLE-CLASS SYSTEMS

Typically, fading channel gains are assumed to be stationaryand ergodic (see, e.g., [8], [11]), and all the users are assumedto have independent and identically distributed (i.i.d.) channelgains (see, e.g., [11]). Accordingly, we model the receivedpowers for all users in one class (a set of users having thesame QoS requirements) to be independent and identicallydistributed in a large network.

In this section, we consider a single-class system. We useto denote the received power distribution when the pro-

cessing gain is , and the received power distribution in thelimiting regime (as ). We assume that the 's and

are continuous on , and that as we scale up the system,converges pointwise to . We let and denote the

expectations corresponding to and , respectively.The SIR achieved by the matched-filter receiver is

SIR (2)

Because all the users have the same QoS requirements, we adoptthe following probabilistic model for their SIR requirements:

SIR

where . We assume that , becauseotherwise it is impossible to meet the users’ SIR requirementseven when the output MAI vanishes.

A. Admissibility and Network Capacity

1) The Deterministic Signature Case:In this case, the users’signatures are deterministic, and the signatures of all theusersform a signature set when the processing gain is. As ,we have a sequence of signature sets. Letdenote the collec-tion of sequences of signature sets satisfying the following con-dition:

as (3)

where and are the signatures for userand when theprocessing gain is . Roughly speaking, thecondition given in (3) requires that the crosscorrelation of anytwo users’ signatures goes to zero at a rate faster thanas .

We observe that the condition in (3) is a veryweakregularitycondition on signature sets since the rate at whichgoes to is very slow. Because low crosscorrelation is essentialfor taking advantage of statistical multiplexing in CDMA sys-tems (see, e.g., [10], [34]), we expect thatcontains a large col-lection of interesting sequences of signature sets. For example,suppose the signatures are binary random spreading sequences.Then we claim that for any

(4)

That is, for almost every realization, the crosscorrelation of anychosen two users’ signatures goes to zero at the rate of,which is considerably faster than . The above claimcan easily be shown as follows: for any


The matrix is of rank . Clearly, the spectral radius ofis for all . Appealing to [26, Lemma 3.1], we have that

where is some constant. Since , we usethe first Borel–Cantelli Lemma [2] to conclude that (4) holds,thereby verifying our claim.

We also note that, in general, the condition in (3) cannot beweakened further because otherwise the output MAI power maydiverge (see [5]). In what follows, we confine ourselves to se-quences in .

For convenience, let denote the signature set. We define admissibility for a class of users

in the deterministic signature case as follows: when theprocessing gain is users per degree of freedom of theprocessing gain areadmissiblein the system if there existsa signature set such that when the signatures in areassigned to the users, the users’ SIR requirements are satisfied,that is,

SIR

which is equivalent to

SIR (5)

Of particular interest is the maximum number of users admis-sible by the system, which is defined to be thenetwork capacity,denoted by , in terms of the number of users per degreeof freedom of the processing gain, that is,

SIR (6)

Theasymptotic network capacity is defined as the limitof the sequence of network capacities :

(We shall show in the proof of Theorem 3.1 that the limit ofexists.)

Given a distribution function , for , we define (cf.,[3, p. 53])

Our main result on the network capacity of a system with MFreceivers for the deterministic signature case is essentially asfollows (the formal statement is given in Theorem 3.1).

Deterministic signature case, MF:The asymptotic networkcapacity is

We note that in the special case of perfect power control, theasymptotic network capacity is simply , which is asgiven in [31].

As has been observed in [31], under perfect power control,the network capacity of a system with linear MMSE receivers

is exactly the same as that of the corresponding system withMF receivers. Then a natural question to ask is, “Is the networkcapacity of a system with linear MMSE receivers still identicalto that of the corresponding system with MF receivers whenthe power control is imperfect?” To answer this question, wefirst need to construct the linear MMSE receiver in an imperfectpower-controlled system.

We assume that the MMSE receiver for userhas knowledgeof user 's instantaneous received power , which can be ob-tained through channel estimation. Because the received powersmay vary from symbol to symbol due to imperfect power con-trol, it is difficult to obtain knowledge of the other users’ instan-taneous received powers. As noted before, however, the distri-butions of the received powers can usually be obtained throughmeasurements. Therefore, we assume that the MMSE receiversfor each user has knowledge of the other users’ mean receivedpowers. In particular, the mean received powers are in thesingle-class case.

The linear MMSE receiver for usergenerates an output ofthe form of , where is chosen to minimize the mean-square error

It is easy to see that

and the SIR achieved by the MMSE receiver for useris (cf.,[14], [27])

SIR (7)

where

Along the same lines as in the MF receiver case, we definethe network capacity of a system with linear MMSE receiversas the maximum number of users admissible by the system, thatis,

SIR (8)

Theasymptotic network capacity is defined as

Our main result on the network capacity of a system withMMSE receivers for the deterministic signature case is essen-tially as follows (the formal statement is given in Theorem 3.1).

Deterministic signature case, MMSE:The asymptotic net-work capacity is


As would be expected, the asymptotic network capacity in theMMSE case is identical to that of the MF case. This observationis a generalization of a result in [31].

The proofs of our results on the asymptotic network capacitymake heavy use of WBE signature sets (see [23] for a survey).For ease of reference, we restate the definition of WBE signaturesets as follows [16].

Definition 1: Suppose . A WBE signature setis a setof vectors (of length ) with unit norm satisfyingthe following equality:

An important property of WBE signature sets is

, where .

The achievability of the asymptotic network capacity, in boththe MF and linear MMSE receiver cases, relies on the existenceof WBE signature sets for each . For signatures whosecomponents are real, the existence and construction of WBEsignature sets for each is provided in [31]. In practicalcommunication systems, signature sets are usually constructedfrom linear codes. An interesting construction method of WBEsignature sets from linear codes for each can be foundin [23].

Under the choice of a WBE signature set, it is easy to showthat

That is, in an imperfect power-controlled system, when theusers’ signatures form a WBE signature set, the MMSEreceiver is just a scaled version of the MF receiver (cf., [31]),which reveals the underlying reason why the network capacityof a system with MMSE receivers is the same as that of thecorresponding system with MF receivers.

To achieve the asymptotic network capacity, we need to findsequences of WBE signature sets in. It has also been shownin [23] that there exist large collections of WBE signature setswhose maximum crosscorrelations are equal to or only slightlylarger than

(which is the Welch lower bound on the maximum crosscor-relation of a signature set), for example, WBE signature setscorresponding tosmall Kasami codesor Kerdock codes. It isstraightforward to see that the condition in (3) is satisfied by se-quences of such WBE signature sets, which approach this lowerbound asymptotically. In Appendix B we give two examples ofsequences of WBE signature sets by exploiting some known re-sults on Hamming codes and Bose–Chaudhuri–Hocquenghem(BCH) codes. These sequences of WBE signature sets easilysatisfy the condition in (3) (i.e., they are in). However, forgeneral , the existence of sequences of WBE signaturesets in is open.

We are now ready to state formally our results on the asymp-totic network capacity for the deterministic signature case.

Theorem 3.1 (Deterministic Signature Case):The asymp-totic network capacity of a system with MF receivers satisfies

Moreover, if there exists a sequence of WBE signature sets incorresponding to some less than the right-handside of the above inequality, then . The same resultholds for the asymptotic network capacity of a systemwith linear MMSE receivers.

Roughly speaking, by Theorem 3.1, in a large system the net-work capacity can be approximated by the quantity

. Assuming the existence of WBE signature sets with “low”crosscorrelations, this network capacity is achieved by such asignature set. To be precise, if for any ,there exists a sequence of WBE signature sets inwith

, then the asymptotic network capacity is exactly

The same holds for the MMSE receiver case. Because the proofof Theorem 3.1 is rather technical, we defer the details to Sec-tion III-C.

In a practical wireless system, it is often more reasonableto assume that the signatures are randomly and independentlychosen because of the time-varying channel distortion, arbitrarytime delays, etc. Therefore, it is of considerable interest to char-acterize the network capacity of a system with random signa-tures, which we consider in the next subsection.

2) The Random Signature Case:In this case, the userschoose their signatures randomly and independently. The modelfor random signatures is as follows: ,where the 's are i.i.d. with mean zero and variance,

and . For technical reasons, weassume that is finite. The random signature model isapplicable to several practical systems, for example, systemsemploying long pseudo-random spreading sequences, andsystems where the signatures are picked randomly and inde-pendently when the users are admitted into the system initially[15], [27], [30].

We assume that the received powers and the signatures areindependent, which is valid in many practical scenarios, for ex-ample, in a system employing long pseudo-random spreadingsequences.

Because the received powers are i.i.d. and the signaturesare i.i.d., the users’ SIR’s are identically distributed. Hence,

SIR does not depend on. Without loss ofgenerality, we study user 1.

We define admissibility for a class of users in the random sig-nature case as follows: when the processing gain is, usersper degree of freedom of the processing gain areadmissibleinthe system if

SIR


Similar to the deterministic signature case, thenetwork capacityis defined as the maximum number of users (per degree

of freedom of the processing gain) that are admissible in thesystem, that is,

SIR

Theasymptotic network capacity is defined as the limitof

We have the following result on the network capacity for therandom signature case.

Theorem 3.2 (Random Signature Case):The asymptotic net-work capacity of a system with MF receivers is

We note that in the special case of perfect power control, theasymptotic network capacity is simply , which agrees with[27].

We have the following heuristic interpretation of Theorems3.1 and 3.2. Suppose the received power distribution isin alarge system. Roughly speaking, by optimally allocating signa-tures, at most users can be admitted intothe system without sacrificing their QoS requirements; if theusers choose their signatures randomly and independently, thenat most users can be admitted into the systemwithout sacrificing their QoS requirements. We conclude thatin the imperfect power control case, the network capacity of asystem with optimally allocated signatures is precisely one userper degree of freedom greater than that of a system with randomsignatures, which indicates that the MF receiver is sensitive tothe choice of signature sets. This observation is consistent withthat of [31], which is for the perfect power control case.

Alternatively, we have a second approach to study the ad-missibility and network capacity, which is different in principlefrom the approach above (our first approach). More specifically,we first define asymptotic admissibility as follows. For the de-terministic signature case, we say thatusers per degree offreedom of the processing gain areasymptotically admissibleif

SIR

For the random signature case, we say thatusers per degree offreedom of the processing gain areasymptotically admissibleif

SIR

For each case, the asymptotic network capacity is defined as themaximum number of asymptotically admissible users.A priori,it is not clear if the two different approaches lead to the same ex-pression for the asymptotic network capacity. It turns out that forboth deterministic and random signature cases, the asymptoticnetwork capacities obtained by the two approaches are indeedthe same.

Fig. 2. The distribution that achieves the supremum ofF (1� a)=�.

B. Power Control and Effective Target SIR

Based on Theorems 3.1 and 3.2, we have that for a givenreceived power distribution , in the deterministic signaturecase, users per degree of freedom is asymptotically admis-sible if ; in the random signature case,

users per degree of freedom is asymptotically admissible if

Assuming that a received power distribution of any desiredform can be achieved by power control, we now ask a morefundamental question: how many users (per degree of freedom)can be made asymptotically admissible through power controlfor given SIR requirements? Specifically, we are interested infinding the largest value of over all possible receivedpower distributions. Loosely speaking, the above problem canbe regarded as the dual of the near–far resistance problem forMF receivers in the following sense: on one hand, near–far resis-tance is concerned with the worst case performance [29]; on theother hand, optimum power-controlled capacity is concernedwith the best case performance.

First we study the deterministic signature case. Define

Based on Theorem 3.1, we have that

(9)

where is a distribution with mean . Then the calcula-tion of boils down to finding the supremum of the ratio of

to over all possible distributions with mean. Itis straightforward to see (e.g., using Markov's Inequality [2, p.283]) that

(10)

where the supremum is achieved by the distribution (denoted as) as shown in Fig. 2. Note that the corresponding to

is simply the set , which is connected.Combining (10) with (9), we conclude that . Be-

cause , the asymptotic network capacity for any


distribution is strictly less than . Therefore, in the determin-istic signature case, there exists a distribution function for which

users (per degree of freedom) are admissible if and only if.

Observe that the optimal power control is in the form of“bang-bang” control. We have the following heuristic interpre-tation. Suppose the number of users (per degree of freedom) isless than (and close to) . Because each user’s QoS require-ment is that SIR , we can implement powercontrol such that the SIR of each user is greater than (andclose to) with probability and equals with probability

. Thus very little power is wasted, i.e., the power controlis efficient. Accordingly, when , perfect power controlis no longer the best in this context. The reason lies in the factthat we have loosened the users’ QoS requirements. Whenapproaches , the optimum power control strategy becomesperfect power control, and , which agrees with theexpression given in [31].

For the random signature case, it can be shown that

Thus in the random signature case, there exists a distributionfunction for which users (per degree of freedom) are admis-sible if and only if . When approaches , the op-timum power control strategy becomes perfect power control,and , which is as given in [27].

In a fixed wireless environment, the received power distri-butions are more or less of the same type (with different pa-rameters) when we scale up/down the transmitted powers; forexample, lognormal distributions due to obstacles in the signalpaths [33]. Suppose users per degree of freedom are not ad-missible for a given received power distribution. Then an inter-esting question to ask is whether or not it is possible to makeadmissible by scaling up the received power and when is it pos-sible. The answer is yes and the underlying intuition is that itis possible to make the users admissible at the expense of morepower. To be more specific, we first define ascale family(cf.,[3, p. 118]).

Definition 2: Let be any distribution function. Thenfor any , the family of distribution functions ,indexed by the parameter, is called thescale family with stan-dard distribution function .

Note that for any is a “scaled version” of, andvice versa. A very special (degenerate) example of the

scale family is

which essentially represents the totality of distribution functionsunder perfect power control (here, we use the notationtodenote the indicator function of the set).

Given a scale family , we define , where. An important observation is that is fixed

for the scale family . That is, is invariant over all the dis-tributions in . It follows that is also invariant over all thedistributions in . Therefore, is a property of the whole scalefamily. We call the effective target SIRfor . For example,

we have for the scale family of distributions that havethe form as shown in Fig. 2.

Observe that in a given scale family, there exists a one-to-onecorrespondence between distribution and mean. Letanddenote the corresponding distribution and mean, respectively.We have the following proposition for a given scale familywith standard distribution function .

Theorem 3.3:

a) (Deterministic signature case)There exists a finite positivevalue that can be designated as the meansuch that users perdegree of freedom are asymptotically admissible forif andonly if . Moreover, the minimum value for is

(11)

b) (Random signature case)There exists a finite positivevalue that can be designated as the meansuch that usersper degree of freedom are asymptotically admissible forifand only if . Moreover, the minimum value for is

(12)

The proof of Theorem 3.3 follows directly from Theorems 3.1and 3.2.

Theorem 3.3 is particularly useful in the scenarios where theusers’ received powers are random due to imperfect power con-trol. (The SIR requirements are therefore probabilistic.) Specif-ically, in a given scale family users per degree of freedomare asymptotically admissible in the deterministic (or random)signature case for any distribution with mean no less than(or

); conversely, users per degree of freedom are not asymptot-ically admissible for any distribution with mean less than(or

). Let denote the distribution function corresponding to(or ). An easy observation is thatusers are asymptoticallyadmissible when we “scale up” by any constant greater than

(i.e., for any distribution function ).

Theorem 3.3 also allows us to conclude that even ifusersper degree of freedom are not asymptotically admissible for agiven distribution , it is still possible to make users per de-gree of freedom asymptotically admissible by scaling uptothe extent that the meanexceeds (or ). This is becausethe effective target SIR is invariant for all distributions in ,and the asymptotic network capacity corresponding to a partic-ular distribution in the deterministic (or random) signaturecase is simply (or ). By scaling updecreases, and the asymptotic network capacity approaches thepower-unconstrained network capacity (or ).

A more interesting observation is that the power-uncon-strained network capacity of an imperfect power-controlledsystem is of the same form as that of a perfect power-controlledsystem, except with the target SIRreplaced by the effectivetarget SIR . Roughly speaking, the effective target SIRplays the same role in determining the network capacity of animperfect power-controlled system as that of the target SIRindetermining the network capacity of a perfect power-controlledsystem. Using the effective target SIR, it is also easy to take


into account power constraints that arise naturally in a physicalsystem. We will elaborate further on this point at the end ofSection IV.

C. Proofs of Theorems 3.1 and 3.2

The proofs of Theorems 3.1 and 3.2 make use of Egoroff'sTheorem [22, p. 73] repeatedly. For ease of reference, we restateEgoroff's Theorem here.

Theorem 3.4:If is a sequence of measurable functionsthat converge to a real-valued functionalmost surely on a mea-surable set of finite measure, then given any , there isa subset with such that converges to uni-formly on . (The symbol denotes the Lebesgue mea-sure of .)

1) Proof of Theorem 3.1, MF Case:Throughout this proof,the signatures are deterministic. As in [31], we want to find themaximum number of users supportable by the system by op-timally allocating signature sets. Therefore, in this subsection,we focus on the case , because when ,trivially, orthogonal signature sets suppress multiple-access in-terference. We begin with some technical lemmas.

Define

For any sequence of signature sets , if

is not finite, it is straightforward to see that at least one usercannot have its SIR requirements met. Therefore, without lossof generality, we assume that

is finite.

Lemma 3.1:a) For any sequence in almost

surely.b) If , then , and

equality is achieved when is a WBE signature set.Proof: See Appendix A.

Let , and. Define

We have the following lemma.

Lemma 3.2: If , then

Proof: See Appendix A.

We now complete the proof of Theorem 3.1.First we show that is upper-bounded by

. By Lemma 3.1, we have that converges toalmost surely. Fix . Appealing to Egoroff's Theorem, thereexists a measurable setsuch that andconverges to uniformly on . Then it follows thatfor fixed , there exists an integer such that for all

for every point in .Based on (2), we have that

SIR

Then for all

SIR

SIR SIR

(13)

Combining (13) with the definition of network capacity yieldsthat

By Lemma 3.1, we have that

(14)

Appealing to Lemma 3.2, it follows that

Because both and are arbitrary positive numbers, and, we conclude that

Next we show that if there exists a sequence ofWBE signature sets in corresponding to this , where

Let , where . Observe that

Moreover, based on Lemma 3.2, we have that

Because is continuous on , it follows that there exist, ,and such that for all


which implies that

Under the choice of a WBE signature set corresponding tothis , we have that for

Thus it follows that for all

SIR

SIR SIR

(15)

that is,

SIR

Hence, , which dictates that .

We conclude that if for any , thereexists a sequence of WBE signature sets inwith ,then .

Note that the quantity is rational by definition (thoughis not necessarily rational). To get across the main ideas,

we neglected this constraint in the proof. However, because ra-tionals are dense on the real line, our approach leads to the sameresult.

2) Proof of Theorem 3.1, MMSE Case:Throughout thisproof, the signatures are deterministic as well. In what follows,by matrix inequality , we mean that ispositive definite (semidefinite).

Lemma 3.3:Suppose and are symmetric ma-trices, and . Then for any


Recall that . We have the following lemmaon the eigenvalues of .

Lemma 3.4:Let be the eigenvalues of .Then it follows that

where is any positive constant.The proof of Lemma 3.4 follows essentially the same line as

that of [31, Theorem 4.1].

We are now ready for the proof of Theorem 3.1 for the linearMMSE case.

Let First we show that

(16)

To this end, we note that

(17)

Because , appealing to Lemma 3.3 yields that

Combining the above with (3), we conclude that

Using the same argument as in the proof of part a) of Lemma3.1, it can be shown that

(18)

Then the desired result (16) follows by combining (17) with(18).

Next we verify that is upper-bounded by

. Based on (16), we appeal to Egoroff'sTheorem to obtain that for any fixed , there exists ameasurable set such that andconverges to uniformly on . Then it follows thatfor fixed , there exists an integer such that for all

, and for every point in

In what follows we first establish the following two inequal-ities:

(19)

(20)

Observe that for all and for every point in

SIR

Then it follows that

SIR

SIR SIR


Combining the above with the definition of network capacity,we have that for

which yields that

Then it is straightforward to see that the inequality (19) holds.To show the second inequality (20), using the matrix inverse

lemma [4, Lemma 12.2], we have that

(21)

Observe that

Appealing to Lemma 3.4, we have that

(22)

Since the above inequality holds for any , it is easyto obtain the inequality (20) by combining (22) with (21).

Based on the inequalities (19) and (20), we have that

which yields that

Again, because bothand are arbitrary positive numbers, weconclude that

It remains to show that if there exist a se-quence of WBE signature sets in corresponding to this ,where . It is easy to show that under thechoice of a WBE signature set, for

Then for for all , we have that

SIR

SIR SIR

Using the same argument as in the proof for the MF receivercase, it can be shown that if

then

SIR

which implies that . Thus we have that, completing the proof.3) Proof of Theorem 3.2:Throughout the proof of The-

orem 3.2, the signatures are random. First we define

and

where

Based on (2), we have that

SIR (23)

We have the following lemma.

Lemma 3.5:When , converges to andconverges to almost surely.


In what follows, we complete the proof of Theorem 3.2. Be-cause the proof makes use of similar techniques to those in theproof of Theorem 3.1, we omit some details here.

Appealing to Lemma 3.5, we have thatand with probability one. Fix . ByEgoroff's Theorem, there exists a measurable setsuch that

and converges to uniformly on ,and there exists a measurable setsuch thatand converges to uniformly on .

Let and . Then for fixed ,there exists an integer such that for all , thequantity is less than or equal tofor every point in ;


there exists an integer such that for all , andevery point in

Let

Then we have all

SIR

SIR SIR

(24)

and

SIR

(25)

Combining (24) with the definition of network capacity

It then follows that

Similarly, based on (25), we have that for

Then we have that

completing the proof.

IV. M ULTIPLE-CLASS SYSTEMS

We have studied the admissibility and network capacity forsingle-class systems in the previous section. However, futurewireless systems will have to support multimedia services suchas voice, data, video, and fax. Therefore, it is essential to havea level of generality dealing with users having different QoSrequirements.

In a wireless system, the received power distributions dependon time-varying fading and power control algorithms. In partic-ular, short-term variation of the signal power may be modeledby Rician fading when there is a direct line of sight betweentransmitter and receiver, and by Rayleigh fading when such apath does not exist (see, e.g., [12], [29]). Thus the users closeto the base station may have Rician fading while the users awayfrom the base station may have Rayleigh fading. Furthermore,

because different users may have different QoS requirements,different power control algorithms may be applied to differentusers. Therefore, to make our model more general, we assumethat different users may have different received power distribu-tions.

Suppose users can be classified intoclasses according totheir received power distributions (users in each class have thesame received power distributions). We use to denote thereceived power distribution of classwhen the processing gainis . We assume that as we scale up the system,

converges pointwise to , which denotes the receivedpower distribution of classin the limiting regime. We letand denote the expectations corresponding to and ,respectively. Moreover, we assume that the 's and 's arecontinuous on .

Let denote the set of users in class, and the cardi-nality of . Define , where istaken to be fixed when , as in the single-class case.Let . For convenience, we call the collec-tion agroup of received power distributions, and

themean power vector.The SIR achieved by the MF receiver for theth user in class

can be expressed as

SIR

(26)

where the 's and the 's are the signatures and receivedpowers, respectively. The probabilistic model for the users’ SIRrequirements are

SIR

where . We assume that.

A. The Deterministic Signature Case

Similar to the single-class case, we assume that every chosensequence of signature sets satisfies (3), and for

Then along the same lines as in the proof of Lemma 3.1, it canbe shown that

(27)

It is desirable to design the signature sets such that all users’QoS requirements are satisfied. When , orthogonalsignature sets null out completely multiple-access interference,and each single user transmits data as if it were in a single-userchannel. When , however, a simple closed-formsolution to the global optimization of the signature sets in this


case seems unattainable. We study a sufficient condition for theadmissibility of multiple-class systems in the following.

1) Asymptotic Admissibility:For convenience, first we letdenote the signature set of classwhen the processing gain

is . Given a group of received power distributions, we say atuple is asymptotically admissibleif

SIR

Following [31], we “channelize” the system and have the fol-lowing suboptimal scheme: a given processing gainis di-vided into parts and , where is the degree of

freedom assigned to class. Moreover, is taken tobe fixed as . Note that users in differentclasses do not interfere with one another under this scheme.

Let , where

Observing that for

SIR

we are motivated to minimize the maximum amongover possible partitions of the processing gain

and choices of signature sets. Intuitively, we want to suppressthe interference as much as possible for all the users simultane-ously.

Given a partition of the processing gain, we have that when

and equality holds if and only if the signature set for classis aWBE signature set. Moreover, if the signature set for classisa WBE signature set, the 's are the same for all the users inclass .

In what follows, we first solve the following optimizationproblem:

(28a)

subject to (28b)

Let and . By appealing to [6,Theorem 2.1, p. 114], we have the following lemma, which isused in Theorem 4.1.

Lemma 4.1:Let . Then the vectoris a minimax point for (28a) under the constraint

(28b). Moreover, we have that


In a practical spread-spectrum system, it is necessary to have. In the asymptotic setting, however, the impact

of the difference between and on the solution to theminimax problem disappears.

Now we are ready to give the following result on the asymp-totic admissibility of multiple classes in the deterministic signa-ture case.

Theorem 4.1 (Deterministic Signature Case):A tupleis asymptotically admissible if it satisfies

(29a)

(29b)

Theorem 4.1 provides a sufficient condition for the asymp-totic admissibility of multiple-class systems with MF receivers.It is clear that the above condition is also a sufficient conditionfor the asymptotic admissibility of multiple-class systems withlinear MMSE receivers because the SIR achieved by the MFreceiver is always no greater than that achieved by the linearMMSE receiver.

Proof: Given a processing gain , we partition it intoparts such that , where is as definedin Lemma 4.1. Observe that for

(30)

Therefore, under the above partitioning of the processing gain,we can choose a WBE signature set for each class, and users indifferent classes do not interfere with one another.

Let denote a random variable that has distribution. Because converges pointwise to , we have that

converges in distribution to (see [2, p.192]). Then under the choice of a WBE signature set for eachclass, we have that by Lemma 4.1, for and

(31)

Combining (26), (27), and (31), we conclude that

SIR

Based on the definition of asymptotic admissibility forit suffices to show that under our proposed

scheme of designing signature sets

In the following, we assume that is an interval. (When isa single point, the problem boils down to the admissibility under


perfect power control, which has been solved in [31].) First sup-pose is an open interval. Then appealing to [1, Theorem 2.1]yields that

SIR SIR

It follows that for

SIR


(32)

Combining the above approach with Egoroff's Theorem, it canbe shown that (32) holds when is a closed (or half-closed)interval. This completes the proof.

2) Power Control, Effective Target SIR, and Effective Band-width: As in the single-class case, it is possible to have theusers’ SIR requirements satisfied by scaling up the receivedpowers even if the users are not admissible for a given groupof distributions. For each class, we introduce the scale family(with standard distribution function )

Define , where

We note that is invariant over all the distributions in andis a property of We call the effective target SIRfor .For convenience, we call the collection a groupof scale families.

Using the notation of effective target SIR, (29b) can bewritten as

Define

Using the above notation, (29b) further boils down to

(33)

where the symbol denotes the operator for Hadamard product,which simply performs the element-wise multiplication of twomatrices [24]. Note that inequality of vectors is equivalent tocomponent-wise inequalities.

We want to study the feasibility of (33), that is, the condi-tion under which there exist positive vectorssatisfying (33).Based on [17], [19], [36], we draw the conclusion that a neces-sary and sufficient condition for the existence of a finite positivesolution to (33) is the existence of a finite positive solution to thesystem of equations obtained by setting all inequalities in (33)to equalities. This remarkable result is due to [25, Theorem 2.1]and a strong result called “The Subinvariance Theorem” [25,Theorem 1.6]. More specifically, we have the following lemma.

Lemma 4.2:There exists a finite positive vector satis-fying (33) if and only if

(34)

Moreover, theth component of the component-wise minimummean power vector satisfying (33) is

(35)

Proof: Observe that is a nonnegative irre-ducible matrix and has rank one. Thus based on [25, Theorem1.1], the Perron–Frobenius eigenvalue of is

. Appealing to [17, Proposition 2.1], we have thata necessary and sufficient condition for the existence of anonnegative nonzero solution to (33) is

Define

Then the solution is Pareto optimal in the following sense:any other feasible solution to (33) will have every componentnot less and at least one component greater than the solution

.It is straightforward to see that

(36)

Recalling the relationship between and the mean powervector, we have that theth component of the component-wiseminimum mean power vector is

completing the proof.Observe that in a given group of scale families, there exists

a one-to-one correspondence between group of receivedpower distributions and mean power vector. Given a groupof scale families , we let and

denote the corresponding group of distributionsand mean power vector, respectively. Based on Theorem 4.1and Lemma 4.2, we have the following result for a given groupof scale families .


Proposition 4.1: There exists a finite positive vector thatcan be assigned as the mean power vector suchthat a tuple is asymptotically admissible for

if

and

Proof: Substituting (35) into (29a), we have that for

If the two conditions specified in Proposition 4.1 are satisfied,then by applying Theorem 4.1 and Lemma 4.2, the tuple

is asymptotically admissible if the power vectorwith the th component given in (35) is designated as the meanpower vector.

Observing the conditions given in Proposition 4.1, we follow[27] and define theeffective bandwidthof class in the deter-ministic signature case as degrees of freedom peruser. However, we are able to give only a sufficient conditionfor asymptotic admissibility of a tuple in terms ofeffective bandwidth. Further work is needed for this case.

It is easy to see that the effective bandwidth is a monotoni-cally increasing function of the effective target SIR. Under theoptimum power control stated in the single-class case,

. It follows that the conditions given in Propo-sition 4.1 become the following:

and

B. The Random Signature Case

As in our study of single-class systems with random signa-tures, we apply Lemma A.1 to conclude that

SIR as (37)

Without loss of generality, we study the first user in class.

1) Asymptotic Admissibility:Given a group of receivedpower distributions, we say a tuple is asymptot-ically admissibleif

SIR

(That the above limit exists is shown in the proof of Theorem4.2 below.)

We have the following result on the asymptotic admissibilityof multiple classes in the random signature case.

Theorem 4.2 (Random Signature Case):A tupleis asymptotically admissible if and only if

(38)

Proof: First suppose is an open interval. Appealing to[1, Theorem 2.1], we have that

SIR SIR

By definition, a tuple is asymptotically admis-sible if and only if for

SIR

Thus we have that for

SIR


(39)

Combining the above approach with Egoroff's Theorem, it canbe shown that (39) holds when is a closed (or half-closed)interval. The proof is completed.

Alternatively, Theorem 4.2 may also be proved by applying[27, Proposition 3.3], but the proof would be more involved.We note that different from the deterministic signature case (inwhich we provide only a sufficient condition on the admissi-bility of multiple-class systems), we can also follow our firstapproach developed in the single-class case to define thenet-work capacity regionin the random signature case as

SIR

Theasymptotic network capacity regionis defined as

(It turns out that the above limit exists.)


Following the same argument as in the single-class case, itcan be shown that the asymptotic network capacity region is thefollowing simplex:

which is consistent with Theorem 4.2. We omit the details here.2) Power Control, Effective Target SIR, and Effective Band-

width: Using matrix notation, the inequality (38) can be sim-plified to

(40)

We now state the following lemma regarding the feasibilityof (40) without proof because it follows the same line of proofas that of Lemma 4.2.

Lemma 4.3:There exists a finite positive vectorsatisfying(40) if and only if

(41)

Moreover, theth component of the component-wise minimummean power vector satisfying (40) is

(42)

Based on Lemma 4.3, we can easily obtain the following re-sult for a given group of scale families .

Proposition 4.2: There exists a finite positive vector thatcan be assigned as the mean power vector suchthat a tuple is asymptotically admissible for

if and only if

Moreover, theth component of the component-wise minimummean power vector is

The proof of Proposition 4.2 follows by combining the resultsof Lemma 4.3 and Theorem 4.2.

Proposition 4.2 is a generalization of a result given in [27].Similar to the deterministic signature case, we define theef-fective bandwidthof class for the random signature case as

degrees of freedom per user. Under the optimumpower control stated in the single-class case, the condition givenin (41) boils down to .

We note that in both the deterministic and random signa-ture cases, the effective bandwidth is a monotonically increasingfunction of the effective target SIR. As observed in the single-class case, the effective target SIR plays the same role in deter-mining the network capacity of an imperfect power-controlledsystem as that of the target SIR in determining the network ca-

pacity of a perfect power-controlled system. Hence, we con-clude that the effective bandwidth in terms of the effective targetSIR plays the same role in determining the admissibility of animperfect power-controlled system as that of the effective band-width in terms of the target SIR in determining the admissibilityof a perfect power-controlled system. More specifically, Propo-sition 4.2 tells us that for a given group of scale families, a tuple

can be made asymptotically admissible throughpower control if and only if the sum of effective bandwidth ofall classes is less than one.

In a practical wireless system, the transmitted powers andhence received powers naturally are constrained. Suppose theusers in class have an average power constraint that

. Given a group of scale families, it is straightfor-ward to see that there exists a finite positive vector (satisfyingthe power constraints) that can be assigned as the mean powervector such that is asymptoticallyadmissible for if and only if

V. CONCLUSION

We have studied single-cell symbol-synchronous CDMA sys-tems in fading channels, focusing primarily on the scenarioswhere MF receivers are employed. We adopt a probabilisticmodel for the users’ QoS requirements. For single-class systemswith MF receivers, we identify the network capacities for bothdeterministic and random signature cases. The network capacitycan be expressed uniquely in terms of the users’ SIR require-ments and the received power distribution. For multiple-classsystems with MF receivers, we provide a necessary and suffi-cient condition for the random signature case, but only a suffi-cient condition for the deterministic signature case, for a set ofusers per degree of freedom to be admissible.

We have explored the notions ofeffective target SIR andeffective bandwidth . In particular, given a scale family,

is invariant over all the distributions in, and plays the samerole in determining the admissibility of an imperfect power-con-trolled system as that of the target SIRin determining the ad-missibility of a perfect power-controlled system. The effectivebandwidth , a monotonically increasing function of, isparticularly useful when there are multiple classes of users in animperfect power-controlled system.

We have also characterized the network capacity of single-class systems with linear MMSE receivers for the deterministicsignature case. A similar characterization for the random signa-ture case turns out to be highly nontrivial; we are looking intothis problem currently.

Our results are useful for network-level resource-allocationproblems such as admission control and power control in a largenetwork. It is easy to generalize these results to obtain the net-work capacity of CDMA systems that employ the techniques ofsectorization and voice-activity monitoring (see [7]).

In this paper, we have focused on characterizing the networkcapacity so that we can determine how many users can be ac-commodated without sacrificing their QoS requirements. On


the other hand, another fundamental issue is thechannel ca-pacity. That is, how much information can be transmitted reli-ably through fading channels in CDMA systems equipped withlinear receivers? Some work along this line is already underway(e.g., [30]). Our own preliminary study shows that there exists atradeoff between network capacity and channel capacity. Intu-itively, the more users in the systems, the stronger the MAI thatis imposed by other users, and hence less information that canbe transmitted reliably over the channel.

It must also be pointed out that our results are for single-cellsynchronous systems, as is the case in [27], [31]. Further workis needed to extend these results to multiple-cell asynchronoussystems.

APPENDIX APROOFS OFTECHNICAL LEMMAS

We use the following strong law for triangular arrays repeat-edly, which follows directly from [5, Lemma 2.4]:

Lemma A.1:Let be a tri-angular array of random variables defined on a common proba-bility space such that for each are inde-pendent, and let . If in probabilityand

a.s.

then almost surely.

A. Proof of Lemma 3.1

Proof for part a): Let . Then

By Lemma A.1, it suffices to show that convergesto in probability and

(Note that and is fixed.)By assumption, is bounded by with prob-

ability one, which implies that is bounded by .Because

we have that

where the last step follows from the fact that

Then it follows that converges to in probability be-cause mean-square convergence implies convergence in proba-bility.

Observe that

holds with probability one. We conclude thatalmost surely.

Proof for part b): By Welch's bound [16], [34], it follows that

(43)

which implies that

Then we have that

which yields that

(44)

where the equality in (44) is achieved when is a WBE sig-nature set.

B. Proof of Lemma 3.2

To show that

it suffices to verify that

and

First we consider the interior part of . Suppose

Choose such that

Then there exists such that


for all . Observing that is nonincreasing, wehave that

(45)

By the definition of , we have thatbecause is continuous on . Moreover, becauseconverges pointwise to on , we have that

and

Since is strictly increasing on , it then follows that isstrictly decreasing on . We take limit of (45) to conclude that

Then we have a contradiction, which dictates that

Using similar arguments, we have that

Therefore, it follows that

Along the same lines, it can be shown that the result holds atthe endpoints of when is closed (or half-closed).

C. Proof of Lemma 3.3

Because and are symmetric and , it follows that

Thus for any , it is straightforward to see that

That is,

Observe that and are positive semidefinitewith eigenvalues and ,respectively. Then it follows that

which implies that for any

thereby proving Lemma 3.3.

D. Proof of Lemma 3.5

Note that the signature is known to the MF receiver foruser 1. For a given , the 's are i.i.d., and

By the Strong Law of Large Numbers

Moreover, we have that conditioned on

Thus it follows that

which yields that .It what follows, we show almost surely by using

Lemma A.1. Observe that

(46)

where we have used the assumption that . Note thatand are independent. Then it follows that for

almost every realization of

(All statements involving conditional expectation hold withprobability one by default.)

Since , it follows that for almost everyrealization of , which impliesthat Thus converges to inprobability. Then it remains to show that

Observe that

By the Strong Law of Large Numbers, we have that for almostevery realization of


which implies that

Then it follows that for almost every realization of

Therefore, we have that almost surely, completingthe proof of Lemma 3.5.

E. Proof of Lemma 4.1

We use the following lemma, which follows directly from [6,Theorem 2.1, p. 114] and [6, Property III, p. 51]:

Lemma A.2:Let be continuously dif-ferentiable convex functions on a convex closed setin .Consider the problem

A point is a minimax point for the above problem (i.e.,a solution to the above optimization problem) if and only if

(47)

where denotes the inner product in , and

We are now ready to prove Lemma 4.1. Fixsuch that. Define

It is easy to verify that is a closed convex set and thatlies in .

Let . DefineWe note that is convex on . In the following we verifythat satisfies the condition specified in Lemma A.2. Let

. We have that

which yields that

It then follows that

Note that the expression on the left-hand side of (47) cannot bepositive since we can always take and get a zero innerproduct. We prove by contradiction that the condition in (47) issatisfied by .

Suppose (47) fails to hold for . Then there exists a pointsuch that

Because for , we have that

that is, for

Then it follows that

Since , we have that . Thus we havethat

which implies that is not in , a contradiction. Therefore,satisfies (47).

Because can be chosen arbitrarily between and, we conclude that is a minimax point for

(28a) over the constraint in (28b), and.

APPENDIX BEXAMPLES OF SEQUENCES OFWBE SIGNATURE SETS

SATISFYING (3)

In practical spread-spectrum systems, the processing gainis of the form . When is large, the contribution of onecomponent in the signatures to the crosscorrelation is negligible.Thus we consider of form without essential loss ofgenerality.

As in [16], we associate a binary sequencein GF with a signature

where the component-wise mapping is as follows:

ifif

Let and be two arbitrary signatures of length ,and and the corresponding binary sequences in GF .Then , where denotes theHamming distance between the indicated vectors.

Example 1: Let denote a maximum-length shift-registercode of length where , and its dualcode. Then is an linear code in which all codewordsexcept the all-zero codeword have identical weight[20, p. 435]. Moreover, is a Hamming code with minimum


distance . By [16, Corollary to Proposition 2 ], a binary signa-ture set corresponding to a binary linear code is a WBE signa-ture set if the minimum distance of its dual code is at least three.Thus the signature set (say) corresponding to is a WBE set.

Let and be two arbitrary and different signatures chosenfrom . Then , which implies that

. It follows that as

In this example, .

Example 2: Let denote a triple-error-correcting primitivenarrow-sense binary BCH code of length , andits dual code. First we construct a linear code(which is a -di-mensional subspace of ) by using the weight distribution of

[35, p. 185].Consider the case odd (the weight distribution of

for odd, is given in [35, Table8-2(a), p. 185]. Suppose we choose and codewordshaving weights , , ,

, and , respectively. More-over, it is required that so thatwe can construct an block code by using the above

nonzero codewords plus all-zero codeword. Using theMacWilliams Identity in [35, p. 90], we can get the weight enu-merator of the dual code . We can choose proper values for

and so that the coefficients and in the weightenumerator of are zero. In general, this can be achievedsince we have five variables but only three constraints. (Becausethe bounds on and given in [35, Table 8-2(a), p.185] are much larger than even when is moderately large,we can choose and as needed.) Therefore, thereis no codeword having weight or in the dual code . By[16, Corollary to Proposition 2], the signature set (say) cor-responding to is a WBE set.

Along the same line, for the case even, by ex-ploiting [35, Table 8-2(b), p. 185], we can construct a linear code

to which the corresponding signature set is a WBE set.Let and be two arbitrary signatures chosen from the WBE

set corresponding to , and and the corresponding binarysequences in GF . Based on [35, Tables 8-2(a) and 8-2(b) ,p. 185], we have that

which implies that as

In this example, can be chosen to be, ,etc.

ACKNOWLEDGMENT

The authors wish to thank the anonymous reviewers for theirhelpful comments that improved the presentation of the paper.

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CDMA systems in fading channels: admissibility, network capacity

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