Adaptive resource allocation in OFDMA systems with fairness and QoS constraints

EUROPEAN TRANSACTIONS ON TELECOMMUNICATIONSEur. Trans. Telecomms. 2007; 18:549–562Published online 15 June 2007 in Wiley InterScience(www.interscience.wiley.com) DOI: 10.1002/ett.1228

Adaptive resource allocation in OFDMA systems with fairnessand QoS constraints†

Liang Chen∗, Brian Krongold and Jamie Evans

ARC Special Research Centre for Ultra-Broadband Information Networks (CUBIN),Department of Electrical and Electronic Engineering, The University of Melbourne, VIC 3010, Australia

SUMMARY

This paper describes several practical and efficient adaptive subchannel, power and bit allocation algorithmsfor orthogonal frequency-division multiple-access (OFDMA) systems. Assuming perfect knowledge ofchannel state information (CSI) at the transmitter, we look at the problem of minimising the total powerconsumption while maintaining individual rate requirements and QoS constraints. An average signal-to-noise ratio (SNR) approximation is used to determine the allocation while substantially reducing thecomputational complexity. The proposed algorithms guarantee improvement through each iteration andconverge quickly to stable suboptimal solutions. Numerical results and complexity analysis show that theproposed algorithms offer beneficial cost versus performance trade-offs compared to existing approaches.Copyright © 2007 John Wiley & Sons, Ltd.

1. INTRODUCTION

Demand for high-rate wireless communication services hasgrown tremendously in recent years. In order toincrease throughput and allow more users to transmitsimultaneously, advanced modulation technologies havebeen developed to reduce both inter-symbol interference(ISI) and inter-channel interference (ICI) in wirelesschannels. Orthogonal frequency-division multiplexing(OFDM) has emerged as perhaps one of the most promisingsolutions for its ability to compensate the impairment ofmultipath fading.

The original idea of OFDM dates back to the mid 1960s[1]. Later technologies like the fast Fourier transform (FFT)[2] and cyclic prefix (CP) [3] have further refined OFDM’sability to combat frequency distortion and time-delayspread.

As different subchannels experience different fades,users can benefit from adaptive resource allocation tooptimise the power usage and/or system throughput,

* Correspondence to: Liang Chen, ARC Special Research Centre for Ultra-Broadband Information Networks (CUBIN), Department of Electrical andElectronic Engineering, The University of Melbourne, VIC 3010, Australia. E-mail: [email protected]†A previous edition of the paper has been presented in the 12th European Wireless Conference (EW 2006), Athens, Greece.

i.e. instead of allocating extra power on the deep fadedsubchannels, it is more efficient to boost up the power onthe other subchannels. Many papers show that adaptivemodulation [4] and dynamic resource allocation [5]significantly increase throughput and allow more users totransmit simultaneously.

For a single-user system, the combination of OFDMand adaptive resource allocation can utilise the advantagesof both. A practical and optimal bit loading algorithm tomaximise throughput and/or minimise power in a single-user OFDM system is studied in Reference [6].

For multi-user system, two classes of resource allocationschemes exist: predetermined and adaptive allocation. Onecan use existing classic static MAC schemes, such asTDMA, CDMA or FMDA [7], which assign an independentdimension (e.g. a predetermined time slot or subchannel, ora unique code to each user) to apply OFDM with adaptivemodulation. However, these approaches fail in exploitingfull diversity as different users experience different andindependent fades, i.e. a subchannel which appears in deep

Copyright © 2007 John Wiley & Sons, Ltd. Accepted 1 December 2006

550 L. CHEN, B. KRONGOLD AND J. EVANS

fade to one user may not be in deep fade for other users.Multi-user OFDM systems with an adaptive subchannel,power and bit allocation scheme or simply referred asorthogonal frequency-division multiple-access (OFDMA)can therefore, achieve higher multiplexing and diversitygain with lower power consumption.

Allocation strategies for OFDMA are still not fullyexplored. Although a simple multi-user water-fillingalgorithm [8] can be shown to approach the capacity regionfor Gaussian multiple-access channels in the presence ofISI [9], it does not guarantee fairness among users. Thoseusers at the cell boundary whose channel conditions arenormally disadvantaged would experience a very highoutage probability. Thus, to ensure a fair service deliveryand QoS among users, instead of following the usualapproach of information theoretic studies, which is tomaximise the sum capacity under a power constraint, inthis paper, we focus on minimising the total power usagewhile maintaining the service grades for different users withminimum rate and BER requirements. Existing methodsdevised for this problem either guarantee optimality, but areslow to converge [10, 11], or are computationally efficient,but far from optimal [12].

In this paper, we present a class of practical andefficient adaptive subchannel, power and bit allocationalgorithms for OFDMA systems with individual rate andQoS requirements. These algorithms use an average-SNR approximation to perform iterative allocations whichguarantee improvement in each iteration and converge tosuboptimal solutions. They also provide stable solutionsno matter what the subchannel correlation condition is.Furthermore, when adjacent subchannels have similarfading characteristics, assigning a contiguous frequencyband at a time can further reduce the overhead.

The organisation of this paper is as follows. We first detailthe system and mathematical model of this optimisationproblem. An overview of some previous approaches isgiven in Section 2. In Section 3, we give the details of thealgorithms and an analysis of computational complexity.Simulation results are presented in Section 4 for theproposed algorithms and compared to some of existingapproaches. Finally, we draw conclusions in Section 5.

2. SYSTEM MODEL AND PROBLEMFORMULATION

A block of the multi-user adaptive OFDM system is shownin Figure 1. We assume zero-delay constraint and anindependent uniform distribution of user locations within

a single cell. Perfect channel state information (CSI) isassumed to be known† through channel estimation and thehelp of feedback channels. The system can support upto K users with individual rate requirements of Rk bitsper user per OFDM symbol, which implies the fairnessconstraint. There are N subchannels available within thesystem; each having a bandwidth that is much smallerthan the coherence bandwidth of the channel. The systemdoes not allow sharing in either time or frequency; eachsubchannel is assigned to one user exclusively at any time.

At the transmitter, instantaneous CSI information is usedby a subchannel, power and bit allocation scheme to assignthe data rate and corresponding power budget for each of theK users. We define ck,n to be the transmit rate on subchanneln for user k. As there is no subchannel sharing among users,clearly we have ck′,n = 0 for all k′ �= k, if ck,n �= 0. Weassume the allowable sets of subchannel rates (constellationsizes) are the same for all users and all subchannels, and thata maximum rate, Rmax, exists. Thus, there is no additionaltotal power constraint in our scenario.

The complex symbols on N subchannels are thentransformed into a time-domain signal via the inverse fastFourier transform (IFFT). A CP is attached and used as aguard interval to help preserve orthogonality after channeldispersion resulting from transmission through a frequency-selective fading channel. Let Hk,n be the magnitude ofnth subchannel gain for user k and σ2

k,n be the respectivenoise power. The resulting unit power signal-to-noiseratio (SNR) for user k on subchannel n is defined to beα2

k,n = H2k,n/σ

2k,n.

Given a universal QoS constraint, let Pk,n be theminimum transmit power to support rate ck,n on subchanneln for user k. The relationship between these quantities is

Pk,n = fk,pe (ck,n)

α2k,n

(1)

where pe denotes some error-probability/QoS constraintand fk,pe (·) denotes the minimum received power requiredfor a subchannel to support a particular rate for user k on aunit-SNR subchannel. To ensure the fairness, instantaneousrate must be guaranteed at all time and this can be easilyextended to proportional fairness by setting Rk’s to differentvalues. On the receiver side, we assume the subchanneland bit allocation information is known, and appropriatedemodulation can be performed.

† In practice, estimated or predicted CSI, although not perfect, can be usedto provide performance gains. In this paper, however, we look at the idealcase to evaluate the methodology.

Copyright © 2007 John Wiley & Sons, Ltd. Eur. Trans. Telecomms. 2007; 18:549–562DOI: 10.1002/ett

ADAPTIVE RESOURCE ALLOCATION IN OFDMA SYSTEMS 551

Figure 1. Block diagram of a multi-user adaptive OFDM system.

The objective is to develop a fast and efficient resourceallocation scheme for a multi-user OFDM system, in whichsubchannels, power and rate are all adaptively assignedwith regard to individual rate requirements. This schemeshould guarantee each user’s QoS, ensure ‘fairness’ amongusers (a guaranteed minimum instantaneous rate), and alsominimise power usage.

Mathematically, we formulate this problem as

PT = minck,n∈D

N∑n=1

K∑k=1

1

α2k,n

fk,pe (ck,n) (2)

subject to:∑n

ck,n = Rk, ∀ k ∈ {1, 2, . . . , K}, and

For each n ∈ {1, . . . , N}, if ck,n �= 0, then ck′,n = 0for all k′ �= k,

where D is the set of rates (constellations) available to eachsubchannel.

We further note that this problem is, in general, anon-convex, NP-hard‡ optimisation problem, regardless ofwhether or not fk,pe (ck,n) is a convex function of ck,n.

2.1. Some previous approaches

2.1.1. Pre-determined allocation

In the OFDM–FDMA scheme, a system simply allocatesfixed bandwidth to users proportional to their individual raterequirements. Although subchannels assigned to a givenuser can be frequency interleaved to improve diversity, theoverall performance is limited.

2.1.2. Lagrangian relaxation

By allowing users to time-share subchannels, the originalproblem in Equation (2) is relaxed to obtain a convex

‡ Following the approach used in Reference [11], this problem can beconverted to an integer programming (IP) problem, which is typically NP-hard.



optimisation problem. A standard method is used inReference [10] to search for the optimal set of Lagrangianmultipliers {λk} that solve an equivalent dual optimisationproblem. Convergence accuracy and speed can becontrolled by varying the minimum increment �λ in eachiteration. However, accuracy and speed negatively affecteach other making it very difficult to reach an optimalor good suboptimal solution. The main limitations of thisalgorithm are:

� It does not converge smoothly as it does not guaranteeimprovement in each iteration. Thus, even if it is forcedto stop after a large number of iterations, the result is stillquite unpredictable and may not be close to the optimalsolution.

� It is computationally intensive, requiring a large numberof iterations.

2.1.3. Bandwidth Assignment Based on SNR andAmplitude-Craving Greedy (BABS+ACG)

Another fast suboptimal solution [12] uses a flat fadingassumption to reduce the computational complexity andbreak the problem into two parts: bandwidth allocation(BA) and subchannel assignment (SA). Simple greedyalgorithms are used on both parts to decide how manysubchannels are assigned to each user as well as whichsubchannels are assigned to which user. The majordrawbacks of this algorithm are:

� The result depends on how one labels subchannels andusers. Thus, it is not unique and stable.

� There is no further refinement to fix the problem broughton by the invalid flat fading assumption.

3. SUBOPTIMAL ITERATIVE RESOURCEALLOCATION

The problem raised in Equation (2) is non-convex,combinatorial, and computationally intractable because ajoint decision of subchannel, bit and power allocation has tobe made.§ Although exclusive subchannel assignment (SA)prevents any greedy algorithm to be optimal in the multi-user environment, the single-user case gives us the idea thatif we could break this joint allocation problem into a set ofsubproblems (BA, SA and power/bit loading), that once theSA is done, the rest of the problem becomes much easier.

§ Notice if there is only one user in the system, the original problembecomes the well-investigated single-user OFDM system which isgenerally convex and tractable.

In this paper, we focus on finding a low complexity,suboptimal resource allocation scheme which has regardfor both subchannel gains and rate/QoS requirements fordifferent users. Furthermore, this scheme should give aunique, stable, suboptimal solution, which means thatit does not depend on how one labels the subchannelsand users. In addition, it can cater for different channelcoherence times and guarantees improvement throughfurther processing.

The main body of this proposed iterative scheme can beintuitively separated into 3 stages:

1. Bandwidth allocation: Based on subchannel allocationinformation, number of subchannels each user has willbe decided.

2. Subchannel assignment: Based on output numbers ofBA, which subchannel goes to which user will bedecided.

3. Iterative refinement: Iterative rearranging to amend theproblem brought in by the initial flat fading assumption,and to further reduce the total power usage.

An average-SNR approximation and fast greedy searchesare used in all stages. SA information is used as both inputand output throughout each iteration.

3.1. Bandwidth allocation

Results from Reference [10] show that giving extrabandwidth to users with worse channel conditions usuallyhelps to reduce total power consumption. In the first stage ofthe algorithm, the number of subchannels each user shouldhave will be decided. We rearrange the rate constraint inEquation (2) into

Rk =∑n

f−1k,pe

(α2

k,nP∗k,n

), ∀ k ∈ {1, 2, . . . , K} (3)

where P∗k,n denotes the optimal distribution of transmit

powers among subchannels. Note that Equation (3) isfrequently used as a check function throughout theoptimisation. In order to reduce the complexity, wefollow the average-SNR approach in Reference [12] andmomentarily assume each user experiences flat fading overits assigned frequency band. This approximation turnsEquation (3) into an overall SNR–Power relationship

Rk ≈ Skf−1k,pe

(α2kPk,n), ∀ k ∈ {1, 2, . . . , K} (4)



where Sk is the total number of subchannels allocated touser k, and α2

k represents the average-SNR approximation.

α2k = 1

N

∑n

α2k,n, ∀ k = 1, 2, . . . , K (5)

When users experience flat fading, the optimal power-rate allocation for each user would be equally distributingpower and bits among assigned subchannels. The minimumvalue of Sk is Rk/Rmax and the transmission rate oneach subchannel is Rk/Sk. The new optimisation problembecomes [12]

P ′T = min

Sk

K∑k=1

Sk

α2k

fk,pe

(Rk

Sk

)(6)

subject to:∑k

Sk = N, and

Sk � Rk

Rmaxfor all k ∈ {1, . . . , K}

Consider the subchannels as N resource blocks, andnodes with α2

k as K orthogonal transmission carriers.This new optimisation problem is very similar to a singleuser discrete water-filling problem with minimum raterequirements. We assume a feasible solution exists, andthe optimal, greedy algorithm, which we call ‘BandwidthAllocation based on Iterative Queuing’ (BAIQ) is asfollows:

1. Set all Sk to �Rk/Rmax� to ensure the feasibility.2. While

∑k Sk � N, block the user with the smallest rate

requirement.3. Compute the �Pk’s, the potential power reductions if

one more subchannel is given to user k.4. Generate a permutation/queue of user indexes, which is

based on the descending order of their �Pk’s.5. Assign one more subchannel to the first entry k∗ in the

queue with maximum �Pk∗ .6. Update �Pk∗ and insert k∗ back into the right position

of the queue.7. Pick the first entry in the index queue again if more

subchannels are available.

Algorithm 1. BAIQ Algorithm.

Sk = �Rk/Rmax�, ∀k = 1, 2, . . . , K

while∑

k Sk � N dok̂ = arg min

kSk

Sk̂ = 0

end while�Pk = 1

α2k

{Skfk,pe

(Rk

Sk

)− (Sk + 1)fk,pe

(Rk

Sk+1

)}, ∀k

while∑

k SK < N doA = {

bk ∈ {1, 2, . . . , K}|�Pbk� �Pbk+1 , ∀k

}k∗ = b1Sk∗ = Sk∗ + 1update �Pk∗ , update A

end while

The BAIQ algorithm can easily be shown to optimallysolve the BA problem raised in Equation (6) by mathematicinduction when a feasible solution exists and the power-ratefunction fk,pe (·) is convex and monotonically increasing,which is true for most modulation techniques.

To ensure the feasibility of the problem, we require thealgorithm to start with∑

k

Sk � N (7)

Otherwise, the user with minimum rate requirement will bedropped in turn to ensure a higher delivery ratio.

When fk,pe (·) is not a continuous function but rathera set of operating points {(ci; pi)} that exist in the non-negative quadrant, the continuous convexity does not applyany more. If we define the gradient of the discrete power-rate function as

f ′k,pe

(ci) = fk,pe (cj) − fk,pe (ci)

cj − ci

(8)

where ci, cj are any two adjacent achievable rates. Thenotion of ‘discrete convexity’ implies that the gradients,{f ′

k,pe}, at all rate points are non-decreasing with the

increase of the transmit rate.Comparing to the BABS algorithm introduced in Refer-

ence [12], this algorithm requires a much lower executiontime in general as shown later in the complexity section.

3.2. Subchannel assignment

The BA algorithm gives the number of subchannels eachuser should have. Since a flat fading assumption is used toreduce complexity, all the subchannels are essentially thesame to each user at this stage. Therefore, in the BA stage,BA, subchannel assignment information is not available.We now propose another new greedy algorithm to allocatesubchannels to users.

Results from Reference [13] show that using a flattransmit power spectrum density (PSD) may only resultin a small throughput reduction in OFDMA systems(vs. adaptive modulation), especially when the fading is



less variant. But with this marginal loss, the allocationcomplexity can be significantly reduced. Under the flat PSDassumption, the assignment problem can be converted intoa classical matching problem which has polynomial timesolutions, e.g. the cost-scaling method [14] with O(N3) orHungarian method [15] with O(N4).

In order to fully exploit the flexibility and diversityof multi-carrier modulation, subchannel numbers in anOFDM system are usually quite large. Although the optimalalgorithms have polynomial time complexities, it is stillprohibitively expensive for implementation.

We perform a ‘Subchannel-Oriented Search’ (SOS) andwish to choose the best possible subchannels for eachuser by giving every subchannel to its best potential user.Although this can be achieved by greedy searching amongall possible α2

k,n for the maximum [16], the stable solutionis found with a much higher complexity compared to ACG[12]. Therefore, a ‘best user’ list is introduced in SOS tosimplify the two-dimensional search into one dimensionand consequently reduce the complexity.

Assume each user is allowed to have Sk subchannels afterthe BA, and let ρk,n = 1 indicate user k getting subchanneln. Once

∑n ρk,n = Sk, no more subchannels will be given

to user k. The greedy SOS algorithm works as follows:

1. Find the best user with the highest SNR on eachsubchannel.

2. Search within the best user list for the overall best user–subchannel pair.

3. Allocate one subchannel each time, starting from thatoverall best.

4. Once user k has Sk subchannels, move this user toinactive list.

5. Once a user becomes inactive, it can no longer be thebest user on any remaining subchannel. Update the bestuser list for remaining subchannels.

6. Go back and search for best user subchannel pair if moresubchannels are available.

Algorithm 2. SOS Algorithm.

ρk,n = 0, ∀k, n

B = {(k, n)|(k, n) = arg maxk

α2k,n, ∀n}

while B �= φ do(k∗, n∗) = arg max

(k,n)∈Bα2

k,n

while∑n

ρk∗,n = S∗k do

α2k∗,n = 0, ∀n = 1, 2, . . . , N

update B(k∗, n∗) = arg max

(k,n)∈Bα2

k,n

end while

ρk∗,n∗ = 1Sk∗ = Sk∗ + 1B = B\(k∗, n∗)

end while

After this simple algorithm has finished, subchannelsassigned to a given user should be less variant than theentire bandwidth of subchannels, which better fits the initialflat PSD assumption. Through a simple inductive argument,the SOS algorithm can be shown to provide the same stableoutcome as greedy search in Reference [16], but with muchlower complexity. This approach differs from the ACGapproach in Reference [12] as the ordering of subchannelsand/or users does not affect the resulting allocation.Furthermore, the SOS gives much better performance interms of the total power consumption compared to the ACGapproach.

3.3. Iterative refinement

The theoretical idea behind the decomposition procedure isthe branch-and-bound search method [17]. The term branchrefers to the partition process where groups of solutions withthe same {Sk} numbers, but not the same set of subchannels,{ρk,n}, are combined into a branch. The term bound refersto the availability of an efficient algorithm for calculating alower bound inside a branch.

The ‘BA’ step solves the branch problem and helps us tofocus on one branch which is most likely to be optimal. Thesecond step, ‘SA’, solves the ‘bound’ problem by providinga fast algorithm that finds the local sub-optimal solution onthat branch.

A fast suboptimal allocation is already readily availableafter these first two stages of the proposed scheme, But wecan also further improve the performance if computationalcomplexity limits allow. The Sk’s are not the best choicein general due to the (invalid) flat fading assumption madeearlier. Even if the numbers are right, the Pk,s’s may not bethe best choice. Furthermore, the average SNR is calculatedover the entire bandwidth rather than those subchannels as-signed to that user, which also leaves room for improvement.

The global optimal value can be achieved by searchingamong all the local optima. Unfortunately, it is stillcomputationally prohibitive to go through all branchesone by one. Therefore, two more intelligent and efficientsearching methods are proposed in this section tosearch for better Sk and SAs, i.e. to move fromone locally optimal/suboptimal point to another locallyoptimal/suboptimal point in order to help alleviate theproblem brought in by the flat fading assumption and furtherreduce the power consumption.



Figure 2. Iteration number distribution for SDS.

3.3.1. Adding/Dropping Searching

The main objective of this proposed scheme is to ensurefairness among users with lowest cost/power. This can berelaxed to keep the power per bit as low as possible butstill meet all individual rate constraints. In the proposedalgorithm, average SNR is first updated to 1

Sk

∑n α2

k,nρk,n,i.e. averaging over those subchannels assigned to thatuser rather than the whole frequency band. The sameapproximation is used to evaluate power consumption perbit

Pk

Rk

=Skfk,pe

(Rk

Sk

) /α2

k

Rk

(9)

We start with looking at the user i with highest power perbit and search among all subchannels not assigned to i. Byusing the average-SNR approximation, for any subchanneln belong to user j rather than i, the power change ofthis subchannel involved in the possible add/drop �Pi,n isdefined as:

Pk,n = P ′k − Pk ≈ S′

k

α2k

′ fk,pe

(Rk

S′k

)− Sk

α2k

fk,pe

(Rk

Sk

)

(10)

where the new averaged SNR after add/drop subchannel n

is

α2k

′ =

α2k + α2

k−α2

k,n

S′k

if S′k = Sk − 1

α2k + α2

k,n−α2

k

S′k

if S′k = Sk + 1

(11)

If the total power increment �Pi,j,n = P ′i + P ′

j − Pi − Pj

is less then 0 when subchannel n is given from j to i, thissubchannel n is dropped from j and given to i. Inspiredby the steepest decent method [18] for unconstrainedconvex optimisation problem, the greedy adding/droppingalgorithm, which we call ‘Steepest Decent Searching’(SDS) works as follows:

1. Find the worst user k∗ with the highest power per bit.2. Search among all those subchannels not assigned to this

user for the largest potential power saving �Pk∗,k′,n′ .3. If user k′ has less than �Rk/Rmax� + 1 subchannels,

no more subchannels will be dropped from k′. Set allpotential power saving related to k′ to infinity.

4. Otherwise, subchannel n′ is dropped from k′ and givento k∗.

5. Repeat this procedure until any change leads to anincrease in total power.

6. Move user k∗ into the inactive list and move to the nextuser with highest power per bit.

Algorithm 3. SDS Algorithm.

K = {1, 2, . . . , K}while K �= φ do

k∗ = arg maxk∈K

Pk

Rk

N = {n|ρk∗,n = 0, ∀n}C = {(k, n)|ρk,n = 1 ∀n ∈ N}while min

(k,n)∈C�Pk∗,k,n � 0 do

(k′, n′) = arg min(k,n)∈C

�Pk∗,k,n

while Rk

Sk′−1 > Rmax do

�Pk∗,k′,n = ∞, ∀(k, n) ∈ C(k′, n′) = arg min

(k,n)∈C�Pk′,k,n

end while



Figure 3. Flow chart of IDS algorithm.

if min(k,n)∈C

�Pk∗,k,n � 0 do

Sk∗ = Sk∗ + 1; Sk′ = Sk′ − 1ρk∗,n′ = 1; ρk′,n′ = 0C = C\(k′, n′)update α2

k∗ ; α2k′

update �Pk∗,k′,n,end if

end whileK = K\k∗

end while

This algorithm is guaranteed to converge because forevery adding/dropping of a subchannel, the average powerper bit

∑Pk/

∑Sk and worst user with highest power per

bit Pk/Sk both decrease. These two values are also boundedby the best user with lowest Pk/Sk, which is non-decreasing.Thus, the algorithm must converge.

3.3.2. Iterative rearranging

Although the SDS algorithm can achieve impressiveperformance improvement over SOS, it is relativelycomputationally intensive as finding the steepest descentadjustment is time consuming, and thus, cannot be appliedin a fast-varying channel environment.

Furthermore, we notice after adding/dropping subchan-nels that we are not only moving to a better branch, butalso moving to the new local suboptimal solution on that

new branch. This is inefficient as we already have a fastalgorithm for the ‘bound’ process. Thus, a more efficientway is to just to move to a better branch and run anotheriteration of the fast SA algorithm to find out the new locallysuboptimal solution. This idea can be verified throughsimulation as well since the BAIQ and SOS algorithmsperform better when using a smaller, but well selected, setof subchannels for generating the average SNR.

We propose another iterative searching algorithm whichonly moves from the current branch to a better branch andperforms another SA algorithm on the new branch to findthe new local suboptimal solution. The flow chart of thisalgorithm is shown in Figure 3.

The input of the BA is the average SNR of each user,α2

k; the output is number of subchannels each user has, Sk.The input of SA is Sk; the output is assignment informationρk,n. In contrast to the SDS algorithm, which always triesto make the steepest adjustment, this ‘Iterative DescentSearching’ (IDS) algorithm adjusts {Sk}, as long power isdescending:

1. Calculate the average SNR over the entire frequencyband. Use this as an initial input for BA.

2. Use the output of the BA to perform SA.3. Then, according to the SA information, the average

SNRs are updated.4. Feedback these new average SNRs and perform another

allocation of both bandwidth and subchannels.



5. Iterations will stop when the new average SNRs are equalto the previous ones, or the new SA uses more power thanprevious one.

Algorithm 4. Iterative Decent Search (IDS).

{ρk,n}(0) = {1}, ∀k, n

P(0)total = ∞

{α2k}(1) = {∑ α2

k,n/N}, ∀k

{Sk}(1) = BA[{α2k}(1)]

{ρk,n}(1) = SA[{Sk}(1)]calculate P

(1)total

η = 1while {ρk,n}(η) �= {ρk,n}(η−1) and P

(η)total � P

(η−1)total do

η = η + 1

{α2k}(η) = {∑ ρ

(η−1)k,n α2

k,n/S(η−1)k }

{Sk}(η) = BA[{α2k}(η)]

{ρk,n}(η) = SA[{Sk}(η)]calculate P

(η)total

end while

3.4. Algorithm complexity

For a real-time application, the most important constraintis computational complexity. In this section, computationalcomplexities of some algorithms are briefly studied as afunction of the number of subchannels N and number ofusers K.

3.4.1. Lagrangian relaxation

Each iteration requires N − Sk inversions and Sk + 1evaluations of fk,pe (·) [12], where Sk is the number ofsubchannels assigned to the user with smallest rate whois being evaluated in that iteration. As the convergencespeed is related to �λ, the total number of iterationsis quite unpredictable. But usually, it is computationallyhazardous.

3.4.2. BABS+ACG

To calculate arithmetic means for K users, NK additionsand K multiplications are needed. To decide the number ofsubchannels each user should have, the algorithm requiresN iterations, with K evaluations and K comparisons ofthe function fk,pe (·) at each iteration. To decide whichsubchannel goes to which user, the algorithm requires N

iterations as well, with K comparisons in each iteration.The total complexity is then O(NK).

3.4.3. BAIQ

In order to calculate the averaged SNRs, we need NKadditions and K multiplications. Initial sorting of thepotential power savings requires K log K comparisons. Forallocating the N subchannels, the algorithm requires atmost N iteration, with at most log K comparisons forreordering at each iteration. Therefore, the complexity forBAIQ algorithm is O(N log K), which is smaller than theBABS algorithm.

3.4.4. SOS

Determining which subchannel goes to which user firstrequires a search to be performed over all α2

k,n to findthe best user on each subchannel, and this requires NK

comparisons. Among these N best user–subchannel pairs,the user with the best overall subchannel gain will getthat subchannel. Removing that user–subchannel pair, thealgorithm then searches for the best of the remaining, andso on. The complexity of this operation is O(N log N),but since each user only requires Sk subchannels, onceuser k′ has been satisfied, all of the (k′, n)’s will beremoved from further allocation consideration. Thus, ifuser k′ is the best user on some remaining subchanneln′, the new best user then needs to be updated on thatsubchannel. Updating best users requires at most (NK2)comparisons, and the resulting total complexity for oneiteration would be O(max(NK log N, NK2)). It should benoted that this worst-case complexity order is extremelyunlikely. In practice, it will be closer to the N complexityorder per iteration.

3.4.5. SDS

For each of the ‘Add/Drop’ procedure, it requires 2(N −Sk) evaluations of fk,pe (·) and 2(N − Sk) evaluations of

α2k

′. Figure 2 shows simulation result of a distribution of

iteration numbers by using the proposed SDS algorithmover 1000 different channels of a 128-subchannel and16-user OFDM system. The total number of iterationsin this case is relatively small, which implies that thecomputational complexity of SDS is fairly affordable in thiscase. But SDS is not quite efficient when used in a systemwith large number of subchannels, as the total number ofiterations increases much faster than linear with a long andsmooth PDF plot.

3.4.6. IDS

IDS can be viewed as an iterative version of BAIQ+SOS,where the number of required iterations is minimal. Figure 4



Figure 4. Iteration number distribution for IDS.

shows simulation results of the average and distribution ofiteration numbers per user over 1000 different channels with512 subchannels and various user numbers. Results showthat the average number of iterations seems to be a smallconstant which does not vary with the number of users. Thusthe complexity of IDS will have a similar computational costas the previous BAIQ+SOS method.

4. SIMULATION AND PERFORMANCECOMPARISON

We now analyse the performance of a multi-user OFDMsystem using different resource allocation schemes includ-ing: predetermined FDMA allocation without interleaving(OFDM-FDMA), Bandwidth Allocation Based on SNRwith Amplitude-Craving Greedy subchannel assignment(BABS+ACG), Bandwidth Allocation based on IterativeQueuing with Subchannel-Oriented Search subchannelassignment (BAIQ+SOS), BAIQ with SOS and SteepestDecent Search subchannel assignment (BABS+SOS+SDS)and BAIQ and SOS with Iterative Decent SearchingAssignment (BABS+SOS+IDS).

We simulated 1000 sets of 5-ray frequency-selectiveRayleigh fading channels and exponentially decaying

power profiles with a decay factor 0.1. Each ray hasindependent Rayleigh-distributed amplitude and uniformlydistributed phase. The total average received power at agiven position follows the inverse power law and log-normal shadowing with a path loss exponent and shadowingvariance equal to 3 and 7 dB, respectively. The channel gainis normalised so that the first multipath ray for the user at cellboundary will have unit Rayleigh-distributed amplitudes.

The inner limit of the far field for the transmit antennais set to be 10% of the cell radius. Users are uniformlydistributed within a single-unit circular cell. The proposedalgorithms do not depend on how one labels subchannelsor users. Figure 5 shows an example of a simulated channelprofile and its corresponding pseudo-uncorrelated channel,which is just a random shuffle of the subchannels. Thus, forthe proposed algorithms, they give the same exact result,which further validates the stability of proposed algorithms.

Each subchannel can support integer bit rates of up to6 bits/symbol. The background noise single-sided powerspectral density level is set to unity. Once a tested algorithmfinishes assigning subchannels to the users, an optimalsingle-user bit allocation algorithm [6] is used so that eachuser meets the target bit rate with minimal power.

Figures 6 and 7 show the normalised total transmissionpower for various schemes in various user numbers



Figure 5. Correlated and pseudo uncorrelated channels.

with BERs equal to 10−2 and 10−4, respectively. Themulti-user OFDM system has 512 subchannels andguarantees 20 total bits per symbol time for each user.Simulation results suggest that the performances ofproposed algorithms are quite stable throughout differentQoS requirements. Total transmit power for BAIQ+SOSis 5.5–11.9 dB less than OFDM-FDMA and 1.0–4.5 dBless than BABS+ACG. There is another 0.3–1.0 dB powersaving if IDS is used for iterative refinement. If thecomputational complexity allows, compared with theBAIQ+SOS, the SDS gives an extra 0.8–1.4 dB powersavings.

Generally, the power gain increases with the number ofusers. This is mainly because the larger the user numberis, in general, given N K, the more combinatorialsubchannel allocation solutions we have, and hence theproposed algorithm will perform more optimally than arandomly chosen predetermined allocation.

For a system with 16 users, if 20 total bits persymbol time for each user is guaranteed, the total transmit

power decreases when more subchannels are available asindicated in Figure 8. The power saving between staticallocation and adaptive allocation can be as much as 7.0 dBfor 100 subchannels and 12.6 dB for 250 subchannels.The SOS algorithm outperforms ACG algorithm bygiving 1.3–4.2 dB saving, while the SDS and IDS cangive additional 0.4–1.2 dB and 1.0–1.4 dB power saving,respectively.

If the number of subchannels and users in the simulatedsystem are fixed while the rate requirements of individualusers are increasing, simulation results show that the powergain achieved through the proposed algorithms are slowlyincreasing as well. An OFDM system with 16 users and512 subchannels is considered, which guarantees TrafficLoad (TL) · Rmax · �N/K� total bits per symbol time foreach user, where the TL is defined as:

TL =

K∑k=1

Rk

Rmax × N(12)



Figure 6. Transmit power versus number of users for a 512-subchannel OFDM system with BER = 10−2.

Figure 7. Transmit power versus number of users for a 512-subchannel OFDM system with BER = 10−4.



Figure 8. Transmission power versus subchannel number for a 16-user OFDM with BER = 10−4.

Simulation results in Figure 9 show that by using bothSOS and SDS as the SA algorithm, the total transmitpower is 8.0–11.0 dB less than that for predeterminedOFDM and 2.5–3.3 dB less than BABS+ACG. Upon

BAIQ+SOS, the IDS results 0.5–1.0 dB lower transmitpower without increasing the order of complexity a lot. TheSDS among ‘branches’ gives another additional 1.0–1.5 dBsaving.

Figure 9. Transmit power versus traffic load for a 512-subchannel, 16-user OFDM system with BER = 10−4.



5. CONCLUSION

We proposed four computationally efficient, adaptivesuboptimal resource allocation algorithms for OFMDAsystems. Based on the instantaneous CSI, the algo-rithms use an average-SNR approximation to reducetotal computational complexity and to help makegreedy decisions about subchannel allocations. Thosealgorithms can be used in different channel environ-ments and guarantee improvement throughout everyiteration.

The BAIQ algorithm optimally solved the BA problemunder the flat fading assumption with a lower complexity.The SOS, SDS and IDS algorithms deal with the SAproblem. BAIQ+SOS offers a fast and beneficial solutionto the resource allocation problem we have considered.When the channel condition allows, following iterativerefinement can be used to further reduce the transmitpower. Compared to some existing algorithms, the proposedalgorithms are superior in performance with affordabletrade-offs in complexity.

ACKNOWLEDGEMENT

This work was supported by the Australian Research Council(ARC). The ARC Special Research Centre for Ultra-BroadbandInformation Networks (CUBIN) is an affiliated program of theNational ICT Australia.

REFERENCES

1. Chang RW. Synthesis of band-limited orthogonal signals formultichannel data transmission. Bell Systems Technical Journal 1966;45:1775–1796.

2. Weinstein SB, Ebert PM. Data transmission by frequency-divisionmultiplexing using the discrete Fourier transform. IEEE Transactionson Communications 1971; 19(5):628–634.

3. Peled A, Ruiz A. Frequency domain data transmission usingreduced computational complexity algorithm. Proceedings ofIEEE International Conference on Acoustics, Speech, and SignalProcessing, Vol. 5, April 1980; pp. 964–967.

4. Keller T, Hanzo L. Adaptive modulation techniques for duplex OFDMtransmission. IEEE Transactions on Vehicular Technology 2000;49:1893–1906.

5. Wahlqvist M, Olofsson H, Ericson M, Ostberg C, Larsson R.Capacity comparison of an OFDM based multiple access system usingdifferent dynamic resource allocation. Proceedings of IEEE VehicularTechnology Conference, Vol. 3, May 1997; pp. 1664–1668.

6. Krongold BS, Ramchandran K, Jones DL. Computationally efficientoptimal power allocation algorithm for multicarrier communicationsystems. IEEE Transactions on Communications 2000; 48(1):23–27.

7. Lawrey E. Multiuser OFDM. Proceedings of IEEE InternationalSymposium on Signal Processing and its Application, Vol. 2, August1999; pp. 761–764.

8. Tse DNC, Hanly SV. Multiaccess fading channels. I. polymatroidstructure, optimal resource allocation and throughput capacities. IEEETransactions on Information Theory 1998; 44(7):2796–2815.

9. Yu W, Cioffi JM. FDMA capacity of Gaussian multiple-accesschannels with ISI. IEEE Transactions on Communications 2002;50(1):102–111.

10. Wong CY, Cheng RS, Letaief KB, Murch RD. Multiuser OFDMwith adaptive subcarrier, bit, and power allocation. IEEE Journal onSelected Areas in Communications 1999; 17(10):1747–1758.

11. Kim I, Lee HL, Kim B, Lee YH. On the use of linear programmingfor dynamic subchannel and bit allocation in multiuser OFDM.Proceedings of IEEE Global Telecommunications Conference, Vol.6, November 2001; pp. 3648–3652.

12. Kivanc D, Li G, Liu H. Computationally efficient bandwidth allocationand power control for OFDMA. IEEE Transactions on WirelessCommunications 2003; 2(6):1150–1158.

13. Rhee W, Cioffi JM. Increase in capacity of multiuser OFDM systemusing dynamic subchannel allocation. Proceedings of IEEE VehicularTechnology Conference, Vol. 2, May 2000; pp. 1085–1089.

14. Ahuja RK, Magnanti TL, Orlin JB. Network Flows: Theory,Algorithms, and Applications. Prentice Hall: Upper Saddle River, NewJersey, USA, 1993.

15. Khun HW. The Hungarian method for the assignment problem. NavalResearch Logistics Quarterly 1955; 2:83–97.

16. Li Z, Zhu G, Wang W, Song J. Improved algorithm of multiuserdynamic subchannel allocation in OFDM system. Proceedings ofInternational Conference on Communication Technology, Vol. 2, April2003; pp. 1144–1147.

17. Papadimitriou CH, Steiglitz K. Combinatorial Optimization:Algorithms and Complexity (2nd edn). Dover Publications, Inc.:Mineola, New York, USA, 1998.

18. Boyd S, Vandeberghe L. Convex Optimization (1st edn). CambridgeUniversity Press: Cambridge, UK, 2004.


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Adaptive resource allocation in OFDMA systems with fairness and QoS constraints