IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 5, MAY 2007 1581

Multi-User Opportunistic Scheduling usingPower Controlled Hierarchical Constellations

Md. Jahangir Hossain, Student Member, IEEE, Mohamed-Slim Alouini, Senior Member, IEEE,and Vijay K. Bhargava, Fellow, IEEE

Abstract— In this letter, we study an application of hierar-chical constellations (known also as embedded, multi-resolution,or asymmetrical constellations) for multi-user opportunisticscheduling. The key idea is to rely on hierarchical constellationsto transmit information to two or more best users simultaneouslyin each transmission. The transmit power as well as the constel-lation parameter is changed according to the link qualities of theselected users in a way that a given target bit error rate (BER)is satisfied. The expressions for the average transmit power,and the outage probability with truncated channel inversionpower control are presented. We also analyze and comparebuffer distribution, average buffer occupancy, and packet lossprobability of different schemes via queuing analysis. We finallycompare the hierarchical scheme for multi-user scheduling witha uniform constellation-based time-slotted scheme.

Index Terms— Multi-user diversity, opportunistic scheduling,hierarchical constellation, and power control.

I. INTRODUCTION

ACCORDING to the classical opportunistic scheduling(see, for example, [1], [2]), the scheduler at the base

station selects a single “best” user in each time slot based onavailable channel state information and transmits informationto this selected user. Although this single best user oppor-tunistic scheduling offers overall throughput gain (which isknown as multi-user diversity gain), the frequency of channelaccess of a given user can be quite low. As a consequence, thedelay experienced by the packets at the transmission buffercan be high and also the packet loss due to buffer overflow for a memory limited system can increase consider-ably. This motivates the design of opportunistic schedulingschemes which can offer more frequent channel access andpacket transmission to the users but still exploit the multiuserdiversity gain. Recently, the concept of multiple best userscheduling, which assigns different number of orthogonalcodes to two selected users in each transmission, has been

Manuscript received April 7, 2005; revised March 15, 2006; acceptedSeptember 9, 2006. The associate editor coordinating the review of this letterand approving it for publication was S. Aissa. This work was presentedin part at the IST Mobile and Wireless Communication Summit, Dresden,Germany, and is supported in part by the Natural Sciences and EngineeringCouncil of Canada (NSERC), in part by the Hui Memorial Fellowship:a University of British Columbia graduate fellowship, and in part by theCenter of Transportation Studies (CTS) through the Intelligent TransportationSystems (ITS) Institute, Minneapolis, Minnesota, USA.

M. J. Hossain and V. K. Bhargava are with the Department of Electricaland Computer Engineering, University of British Columbia, Vancouver, V6T1Z4, Canada (email: {jahangir, vijayb}@ece.ubc.ca).

M.-S. Alouini is with the Department of Electrical and Computer Engi-neering, Texas A&M University, Qatar, Education City, Doha, Qatar (email:alouini@qatar.tamu.edu).

Digital Object Identifier 10.1109/TWC.2007.05228.

proposed for code division multiple access (CDMA) systems[3], [4].

Hierarchical constellations consist of non-uniformly spacedsignal points (see, for example, [5], [6]). Due to the capabilityof providing different levels of protection, this type of con-stellations has been proposed in many applications includingmultimedia transmission [7], downlink multiplexing [8] andsuperposing bits from different users in the same sub-carrierof an orthogonal frequency division multiple (OFDM) system[9].

The above mentioned power efficient hierarchical multiplex-ing schemes motivates us to study an application of hierarchi-cal constellations in multi-user opportunistic scheduling whichtransmits information to multiple best users simultaneouslyrather than a single best user in a given time slot. Thisway the users are given more frequent information access inorder to reduce queuing delay and loss rate of the packetsat the transmission buffer. Contrary to the orthogonal codeallocation-based two user scheduling schemes [3], [4], wepropose in this letter a modulation assisted multiple best useropportunistic scheduling scheme. The detailed description ofour proposed scheme will be given in Section II-B.

In this letter, we also present expressions for the averagetransmit power, and the outage probability with truncatedchannel inversion power control for a fading environmentwhere the users’ channels are identically and independentlydistributed (i.i.d). We also analyze higher layer performances,for example, buffer distribution, average buffer occupancy, andloss probability of the packets at the transmission buffer viaqueuing analysis. Using these expressions, we explore thetrade-offs between these higher layer performances and thepower efficiency of multi-user opportunistic scheduling usinghierarchical constellations. Some selected numerical examplesshow that the multi-user scheduling using hierarchical con-stellation reduces the average buffer occupancy (consequentlythe average queuing delay) and the packet loss probability(PLP) at the expense of a certain transmit power compared tothe classical opportunistic single-user scheduling. Interestinglyenough, these numerical examples show also that this addi-tional power requirements decreases to a small value as thenumber of users increases in the system. We finally comparein this letter the hierarchical multi-user scheduling schemewith a time-slotted multi-user opportunistic scheduling. In thistime-slotted scheme two or more best users are scheduled fortransmission in a mini time-slotted fashion. This comparisonshows that although the uniform constellation based timeslotted scheme offers the same average queuing delay and

1536-1276/07$25.00 c© 2007 IEEE

1582 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 5, MAY 2007

11

00

0010

00

d1

d2

10 00

11 01

10

01 11 11 01

10 00

01

10

01 11

d’

Fig. 1. Hierarchical 4/16-QAM constellation.

PLR, it requires higher average transmit power than ourproposed hierarchical scheme.

II. MULTI-USER OPPORTUNISTIC SCHEDULING

A. Hierarchical Constellations

Throughout this letter, we limit ourselves two best useropportunistic scheduling using hierarchical 4/16-QAM con-stellation which is constructed with two levels of hierarchy.However, our approach can easily be generalized for thescheduling of more than two best users using higher orderhierarchical constellations [6].

A hierarchical 4/16-QAM constellation (see, for example,[5]), is shown in Fig. 1 and can be modelled as follows. Weassume that there are two streams of data where one streamrequires higher level of protection than the other. For everychannel access two bits are chosen from each level. The twobits requiring the highest level of protection are assigned tothe most significant bit (MSB) position in the inphase (I) andquadrature (Q) phase channels. Consequently, the two bits thatrequire the least protection are assigned to the least significantbit (LSB) position in the I and Q channels.

In Fig. 1, the fictitious black symbols represent a 4-QAMconstellation (referred to as the first hierarchy and denoted byH1). The distance between the symbols in the first hierarchyis represented by d1. The actual transmitted symbols are thewhite symbols and they represent a 16-QAM constellation.This is the second level of hierarchy (denoted by hierarchyH2). One of these white symbols surrounding a selected blacksymbol in the first hierarchy is selected by the two bits thatrequires the least protection. The distance between the symbolsin the second hierarchy is denoted by d2.

The exact bit error rate (BER) expressions over additivewhite Gaussian noise (AWGN) channel for this hierarchical4/16-QAM constellation were derived in [6]. The BER equa-tions in [6] are not invertible in terms of transmit power andthe constellation priority parameter for given other parameters.However, the well known SNR-gap approximation [11] can beeasily inverted in terms of the transmit power and this leadsto a simple design of hierarchical constellations (see [9]). As

such, in this letter, we use this SNR-gap approximation toevaluate the average transmit power analytically. Accordingto the SNR-gap approximation, the average power of kthhierarchy is given by [9], [11]

Pk =d2

k(2m − 1)6

(1)

where dk is the distance between the constellation symbolsin the kth hierarchy and m is the number of bits assigned tokth hierarchy which is equal to two in the case of hierarchical4/16-QAM. The distances d1 and d2 as shown in Fig. 1 arecalculated recursively as follows. The distance between theactual constellation (16-QAM) symbols, d2 is calculated as

d2 =

√6 × 2N0Γ

α2, (2)

where α2 is the received power channel gain of the userassigned to second hierarchy, N0 is the noise power spectraldensity, and Γ is the SNR-gap of the uncoded QAM constel-lation and given by

Γ =13

[Q−1

(BER0

4

)]2

, (3)

where Q−1(·) is the inverse standard Gaussian Q-function andBER0 is the target BER. The distance between the actualsymbols in different quadrant d′ as shown in Fig. 1, can becalculated as

d′ =

√6 × 2N0Γ

β2, (4)

where β2 is the received channel fading power gain of theuser assigned to the first hierarchy. Now the distance betweenfictitious symbols (4-QAM) in the first hierarchy can bewritten as (see Fig. 1)

d1 = d′ + d2. (5)

Using Eqs. (1)-(5), the total transmit power with hierarchical4/16-QAM constellation (assuming that the second hierarchyis assigned to a user with channel gain α2 and that the firsthierarchy is assigned to a user with channel gain β2) can beexpressed as

P (α, β) =C2

α2+

C2

α2β2(α2 + 2αβ + β2) (6)

where the constant C2 = 2N0Γ(22 − 1).

B. Hierarchical Constellation-based Multi-user OpportunisticScheduling

To describe our proposed multi-user opportunistic schedul-ing let us consider a base station (BS) transmitting to K usersopportunistically. The users have to meet a predeterminedtarget BER, BER0 and the channel variation among the users isassumed to be i.i.d. The channel fading amplitude of a givenuser is Rayleigh faded which corresponds to the followingprobability density function (pdf) for the channel fading powergain g

pg(g) =1g0

e−g/g0 ; g ≥ 0, (7)

where g0 is the average channel gain.

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 5, MAY 2007 1583

We now describe the scheduling as well as the correspond-ing adaptation process.

• Step 1: Based on users’ channel qualities, the schedulerat the BS ranks the users and picks up the first best user(denoted by U1) and the second best user (denoted byU2).

• Step 2: If the channel fading power gain of the secondbest user, g2 is above a threshold g2

th , go to the next step.Otherwise declare a transmission outage for that time stepand restart from step 1 in the next time slot.

• Step 3: Since the user, U1 requires less protection thanuser U2, hierarchy H2 is assigned to the user U1 whereashierarchy H1 is assigned to the user U2. With thisassignment, the transmitter adapts the transmission poweras well as the relative distances between the symbols(d1, d2 and d′) such as the target BER, BER0 is metwith the corresponding channel fading power gains g1

and g2.The position of the bits in a transmitted symbol for theselected users are sent via a feed-forward channel. Thus theselected users receive the symbols and look only for the bitsin particular positions within the symbol. Since the averagetransmit power with channel inversion power control policy onRayleigh faded channel is infinite [10], a truncation thresholdg2

th is used in the channel inversion power control. This thresh-old g2

th is set to a value such that a pre-specified transmissionoutage probability is met. The expression for the outageprobability will be given in closed-form in Section III. In Step3, the required power as well as the constellation parametercan be obtained numerically solving BER expressions of thehierarchical 4/16-QAM [6]. However, as mentioned above foranalytical tractability, we use the approximations in Eq. (6).

C. Comparison with Uniform QAM Constellation-basedScheme

Multi-user scheduling is also possible using uniform QAMmodulation in a time division fashion. Basically, the wholetransmission slot is divided into a number of mini-slots andin each mini-slot the data of a particular user is transmitted. Intwo user scheduling, the transmission slot is divided into twomini-slots. The first best user is scheduled in the first mini-slot whereas the second best user is scheduled in the secondmini-slot. The transmit power in each mini-slot is adjustedaccording to the corresponding selected user’s channel gainso that the bits are transmitted below the target BER usinguniform 16-QAM over an AWGN channel.

III. PERFORMANCE ANALYSIS AND RESULTS

In this section, we derive analytical expressions for theaverage transmission power and the outage probability. Wealso analyze the buffer distribution function, the average bufferoccupancy, and the PLP for all the schemes under considera-tion. Some selected numerical results obtained via computersimulations are also presented to compare the performance ofthe schemes under consideration. We use an uncoded targetBER, BER0 of 10−4 and a pre-specified outage probability of10−3. We assume channel variation among the users is i.i.dwith an average channel fading power gain of 5 dB.

A. Outage Probability

For two-user opportunistic scheduling a transmission outageoccurs if the channel fading power gain of the second best user,g2 is less than a threshold g2

th . Therefore, the outage probabilityof two-user scheduling, O2

out can be written as

O2out =

∫ g2th

0

∫ +∞

g2

pg1g2(g1, g2)dg1dg2 (8)

where pg1g2(g1, g2) is the joint pdf of the largest two channelgains and can be expressed as [12]

pg1g2(g1, g2) =K(K − 1)

(g0)2e−g1/g0e−g2/g0

×(1 − e−g2/g0

)K−2

; g1 ≥ g2 ≥ 0. (9)

Using Eq. (9) we can evaluate the integral in Eq. (8) asfollows:

O2out =

∫ g2th

0

∫ +∞

g2

K(K − 1)(g0)2

e−g1/g0e−g2/g0

×(1 − e−g2/g0

)K−2

dg1dg2

= 1 −K−2∑i=0

(K − 2

i

)(−1)i K(K − 1)

i + 2e−(i+2)g2

th/g0.

(10)

For single-user scheduling a transmission outage occurs ifthe channel gain of the first best user is below a threshold g1

th

and thus the outage probability of single-user scheduling, O1out

can be written as

O1out =

∫ g1th

0

pg1(g1)dg1 (11)

where pg1(g1) is the pdf of the largest channel gain and isknown to be given by [12]

pg1(g1) =K

g0e−g1/g0

(1 − e−g1/g0

)K−1

. (12)

Using Eq. (12) in Eq. (11), the outage probability for singleuser scheduling can be written in the desired closed-form

O1out = 1 −

K−1∑i=0

(K − 1

i

)(−1)i K

i + 1e−(i+1)g1

th/g0. (13)

Solving Eqs. (10) and (13) for a given transmission outage, wecan find the values of the thresholds g2

th and g1th , respectively.

B. Average Transmit Power

Averaging Eq. (6) over the channel gain joint pdf of the twobest users, the average transmit power in two-user hierarchicalconstellation-based scheduling, P2

H can be expressed as

P2H =

∫ +∞

g2th

∫ +∞

g2

[C2

g1+

C2

g1g2(g1 + 2

√g1g2 + g2)

]× pg1g2(g1, g2)dg1dg2. (14)

1584 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 5, MAY 2007

Substituting Eq. (9) in Eq. (14), the expression of P2avg can be

simplified as

P2H =

C2K(K − 1)(g0)2

∫ +∞

g2th

∫ +∞

g2

[2g1

+ 21√g1g2

+1g 2

]

×e−g1/g0e−g2/g0

(1 − e−g2/g0

)K−2

dg1dg2

=C2K(K − 1)

(g0)2

K−2∑i=0

(K − 2

i

)(−1)i

[∫ +∞

g2th

2

×[

E1

(g2

g0

)+ E1/2

(g2

g0

)]e−(i+1)g2/g0dg2

+g0E1

((i + 2)g2

th

g0

)], (15)

where the function En(·) is defined as

En(x) =∫ +∞

1

e−xt

tndt. (16)

Similarly the average power in uniform 16-QAMconstellation-based two user scheduling, P2

U can be written insimplified form as

P2U =

C4K(K − 1)2(g0)2

K−2∑i=0

(K − 2

i

)(−1)i

[∫ +∞

g2th

E1

(g2

g0

)

× e−(i+1)g2/g0dg2 + g0E1

((i + 2)g2

th

g0

)], (17)

where the constant C4 = 2N0Γ(24 − 1). Unfortunately,the integrals in Eqs. (15) and (16) can not be evaluated insimple closed-form. However, it can be evaluated numericallyand consequently the average transmit power for two-userscheduling can be obtained.

The average transmit power for single-user scheduling canbe expressed as

P1 =∫ +∞

g1th

C4

g1pg1(g1)dg1. (18)

Substituting Eq. (12) in Eq. (18), P1 can be simplified to

P 1 =∫ +∞

g1th

C4

g1

K

g0e−g1/g0

(1 − e−g1/g0

)K−1

dg1

=K−1∑i=0

C4K

i + 1

(K − 1

i

)(−1)i E1

((i + 1)g1

th

g0

). (19)

The average power for different schemes is plotted in Fig.2. There are two observations from this figure. First, thetwo user scheduling always requires more power than thesingle user scheduling. This is due to the inclusion of thesecond best user in the transmission. Second, the requiredadditional power decreases as the number of users in thesystem increases. This can be explained by the fact that asthe number of users increases in the system, the frequency ofhaving higher difference between the first largest channel gaing1 and the second largest channel gain g2 decreases. Therefore,the average power required to transmit to the second bestuser decreases. On the other hand, the hierarchical schemesalways require less power than the uniform constellation-basedmini-slotted scheme. This is expected due to the superposition

5 10 15 20 25 306

8

10

12

14

16

18

20

22

Number of users in the system

Nor

mal

ized

ave

rage

tran

smit

pow

er

Two−best user scheduling (mini−slotted scheme)Two−best user (hierachical scheme)Single best user scheduling (classical scheme)

Fig. 2. Normalized (by N0) average transmit power versus the number ofusers.

of the information bits for two different users in the samemessage symbol. It is also obvious from Fig. 2 that therequired additional power with mini-slotted scheme decreasesas the number of users increases in the system. This is againexpected, as the frequency of having higher difference betweeng1 and g2 decreases with the number of users in the system.

C. Buffer Density and Packet Loss Probability

In this section, we analyze and compare the buffer distri-bution, average buffer occupancy, and packet loss probabilityof all the schemes under consideration. We assume that eachuser has a transmission buffer where packets are queued untilthey are served to the desired users. The system is assumedto be a homogenous system where all users have the same(i) buffer length, (ii) packet arrival statistics, and (iii) averagelink gain. Therefore, we are interested in the queuing analysisof a given user’s buffer. Other users’ buffer will have the sameperformance. The finite buffer size is assumed to be B packetsand the packet arrival process is assumed to be Bernoulli witharrival probability a in each time slot. We assume that a packetarriving during time slot n−1 cannot be transmitted until timeinterval n.

Since with the two-best user scheduling schemes (both hier-archical and mini-slotted schemes) two packets are transmittedfrom one of the selected buffers per time slot, the number ofpackets transmitted during time interval n is min(bn, 2), wherebn is the buffer occupancy at time slot n. For i.i.d. channelvariations among K users, the probability that a user will getan access to the channel, in given time slot, is 2/K . Therefore,the probability of serving a given queue, s2, for a pre-specifiedoutage probability Oout, is expressed as1

s2 =2K

(1 − Oout). (20)

The dynamics of the buffer with the proposed two-best userscheduling is shown in Fig. 3. The discrete time Markov chain

1For the sake of simplicity of the analysis we assume the fading gain of agiven user varies independently from one time slot to the next.

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 5, MAY 2007 1585

T =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

b a 0 0 0 · · · 0 0 0 0bs2 as2 + bv2 av2 0 0 · · · 0 0 0 0bs2 as2 bv2 av2 0 · · · 0 0 0 00 bs2 as2 bv2 av2 · · · 0 0 0 0...

......

......

. . ....

......

...0 0 0 · · · 0 0 bs2 as2 bv2 av2

0 0 0 · · · 0 0 0 bs2 as2 bv2 + av2

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(21)

Fig. 3. Markov chain representation of the buffer dynamic with two-bestuser scheduling.

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

z [packets]

Pro

babi

lity

[Buf

fer

occu

panc

y=z

pack

ets]

Two−best user scheduling (proposed scheme)Single best user scheduling (classical scheme)

zavg

=0.5547 (two best user scheduling)

zavg

=1.0597 (single best user scheduling)

Fig. 4. Buffer distribution and average buffer occupancy level.

describing the buffer dynamics has the transition matrix shownin (21), where b = 1 − a and v2 = 1 − s2. The steady-stateprobability b∞ = [π0 π1 · · · πB ] can obtained solving thefollowing equations

b∞T = b∞ (22)B∑

k=0

πk = 1. (23)

The steady-state probability can be obtained using the standardprocedures, such as the ones outlined in [13].

Similarly we can construct the Markov chain for the singlebest user scheduling. Since four packets are taken out of thebuffer per channel access, we assume that the number ofpackets transmitted during time interval n is min(bn, 4). Inthis case the probability s1 that a given queue will be servedin given time slot can be written as

s1 =1K

(1 − Oout). (24)

The corresponding steady-state buffer distribution can be

0.06 0.08 0.1 0.12 0.14 0.16 0.1810

−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

Packet arrival probability, a

Pac

ket l

oss

prob

abili

ty

Single best user scheduling (classical scheme)Two−best user scheduling (proposed scheme)

Fig. 5. Packet loss probability versus packet arrival probability.

obtained using standard procedures. The steady-state bufferdistribution function is plotted in Fig. 4. In the numericalexample we consider K = 10, B = 10, and a = 0.1. FromFig. 4 we can observe that the buffer occupancy with twobest user scheduling is concentrated in the low occupancyregion with higher probability compared to the single bestuser scheduling. Therefore, two best scheduling results into alower average buffer occupancy (consequently, lower averagedelay for the packets) which is also obvious from the markedpoints on the x-axis of Fig. 4. The buffer distribution which isconcentrated in the low occupancy region with two best userscheduling also results into a lower packet loss probabilitydue to the buffer overflow as we will see from the followinganalysis and example.

The packet loss probability, PLP defined as the probabilitythat a packet is dropped due to buffer overflow correspondsto the probability that the buffer is full and a packet arrivesin the buffer. Therefore, the PLP can be written as

PLP = πBa. (25)

The packet loss probability with different schemes are shownin Fig. 5 for different values of packet arrival probability. Thisfigure shows that the two best user scheduling always offer lesspacket loss probability. The is because of the comparativelylower probability of buffer occupancy in the high bufferoccupancy region with two best user scheduling as mentionedabove.

1586 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 6, NO. 5, MAY 2007

IV. CONCLUSION

In this letter, we proposed and studied a new hierarchi-cal constellation-based multi-user opportunistic schedulingscheme. The multi-user scheduling provides lower averagebuffer occupancy (i.e., queuing delay) and loss probabilityfor the packets at the transmission buffer at the expenseof transmit power compared to the classical single userscheduling. The required additional power decreases as thenumber of users increases in the system. The hierarchicalscheme has been compared with a uniform constellation-based mini-slotted scheme. This mini-slotted scheme providessimilar higher layer performances as the hierarchical schemebut requires higher transmit power. This additional powerrequirement decreases as the number of user increases inthe system. Therefore, we can conclude that the uniformconstellation-based mini-slotted scheme can be used when thenumber of users in the system is large.

ACKNOWLEDGEMENT

The authors are thankful to the anonymous reviewers fortheir valuable comments and suggestions which have greatlyimproved the paper.

REFERENCES

[1] R. Knopp and P. A. Humblet, “Information capacity and power controlin single-cell multiuser communications,” in Proc. IEEE Int. Conf.Commun.(ICC’95), pp. 306-311.

[2] D. N. Tse, P. Viswanath, and R. Laroia, “Opportunistic beamformingusing dumb antennas,” IEEE Trans. Inf. Theory, vol. 48, pp. 1277-1294,June 2002.

[3] D. I. Kim, “Two-best user scheduling for high-speed downlink multicodeCDMA with code constraint,” in Proc. IEEE Conf. Global Commun.(Globecomm’ 04), pp. 2659-2663.

[4] S. Dost, M. O. Sunay, and V. K. Bhargava, “A feasibility study of twouser downlink transmission for IS-856 system,” in Proc. Personal andIndoor Commun. Conf. (PIMRC’04), pp. 2029-2033.

[5] M. Morimoto, M. Okada, and S. Komaki, “A hierarchical image transmis-sion system in fading channel,” in Proc. IEEE Int. Conf. Univ. PersonalCommun. (ICUPC’ 95), pp. 769-772.

[6] P. K. Vitthaladevuni and M.-S. Alouini, “A recursive algorithm for theexact BER computation of generalized hierarchical QAM constellations,”IEEE Trans. Inf. Theory, vol. 49, no. 1, pp. 297-307, Jan. 2003.

[7] M. B. Pursley and J. M. Shea, “Adaptive nonuniform phase-shift-keymodulation for multimedia traffic in wireless networks,” IEEE J. Sel.Areas Commun., vol. 18, no. 8, pp. 1394-1407, Aug. 2000.

[8] M. J. Hossain, P. K. Vitthaladevuni, M.-S. Alouini, and V. K. Bhargava,“Hierarchical modulations for multimedia and multicast transmission overfading channels,” in Proc. IEEE Veh. Technol. Conf. (VTC’03 Spring),pp. 2633-2637.

[9] S. Pietrzyk and G. J. M. Janssen, “Subcarrier and power allocation forQoS-aware OFDMA systems using embedded modulation,” in Proc. Int.Conf. Commun. (ICC’ 04), pp. 3202-3206.

[10] A. J. Goldsmith, “The capacity of downlink fading channels usingvariable power and variable rate,” IEEE Trans. Veh. Technol., vol. 46,no. 19, pp. 569-580, Aug. 1997.

[11] J. M. Cioffi, “A multicarrier premier,” November 1991, also availablein http://www.stanford.edu/group/cioffi/pdf/multicarrier.pdf

[12] T. Eng, N. Kong, and L. B. Milstein, “Selection combining schemesfor RAKE receivers,” in Proc. 4th Int. Conf. Univ. Personal Commun.(ICUPC’95), pp. 426-430.

[13] F. Gebali, Computer Communication Networks: Analysis and Design.NorthStar Digital Design, Inc., Victoria, BC, Canada, 2002.