1

Bit loading using imperfect CSIT for predictionbased resource allocation in mobile OFDMA

Jorge F. Schmidt, Student Member, IEEE, Juan E. Cousseau, Senior Member, IEEE, Risto Wichman,and Stefan Werner, Senior Member, IEEE

Abstract— We present a prediction-based resource allocationalgorithm (RA) for orthogonal frequency division multiple access(OFDMA) downlink, where inaccuracies in the wireless channelpredictions are accounted for in the problem formulation. Asthe prediction quality degrades significantly with the predictionhorizon, we propose a solution based on the histogram of the pre-diction error. This characterization also enables different mobilestations (MSs) to use different channel predictors as it does notrely on a specific prediction scheme. Using this characterizationof the prediction error and based on classical resource allocationstrategies we derive an algorithm that incorporates imperfectchannel prediction information of future time slots. We evaluatethe proposed algorithm using a practical low complexity channelpredictor, suitable for implementation at the MSs. Simulationresults show that the proposed algorithm outperforms previousprediction-based RA strategies without the characterization ofthe prediction error, and the system throughput is comparable tothe case with perfect channel state information in the transmitter.

I. INTRODUCTION

OFDMA is a multiple access technique capable of ex-ploiting multiuser diversity in a frequency selective fadingscenario. Because of its orthogonal structure, OFDMA allowsmultiple users to transmit simultaneously on the differentsubcarriers of one OFDM symbol. Considering time-varyingenvironments, the problem of allocating specific subcarriersto users over successive time slots, taking into account overalldata throughput as well as fairness constraints, has received alot of attention over the last years [1]-[8].

Predicted channel state information in the transmitter(CSIT) plays an important role on efficient sharing of channelresources in time varying wireless channels. For example, inlong-term evolution (LTE) downlink [9], the physical layerscheduler allocates channel resources between mobile stations(MSs) in 1 ms resolution. The RA is based on achievable ratevalues reported by the MSs through a feedback channel. In thiscontext, channel prediction has already been considered forcompensating feedback latency in rapidly varying scenarios,

Copyright (c) 2011 IEEE. Personal use of this material is permitted.However, permission to use this material for any other purposes must beobtained from the IEEE by sending a request to pubs-permissions@ieee.org.

Jorge F. Schmidt and Juan E. Cousseau are with Universidad Nacional delSur, IIIE-CONICET, Av. Alem 1253, Bahıa Blanca, Argentina. {schmidt,jcousseau}@uns.edu.ar. Risto Wichman and Stefan Werner are withAalto University School of Electrical Engineering, P.O. Box 13000, 00076Aalto, Finland. {risto.wichman, stefan.werner}@aalto.fi.

This work was partially supported by the Academy of Finland, SmartRadios and Wireless Research (SMARAD) Center of Excellence, AgenciaNacional de Promocion Cientifica y Tecnologica PICT 2008-0182 and Univ.Nacional del Sur, Argentina, Project 24/K043

see, e.g., [1], [2]. However, the use of channel predictors toimprove RA algorithms has not been fully explored yet.

With the development of long range channel predictors[10]-[14], it becomes feasible to include information aboutthe channel state on future time slots in the RA algorithm.However, in order to exploit this new information, predictionaccuracy has to be taken into account. Specifically, in highmobility scenarios, channel prediction degrades appreciably asthe prediction horizon increases [10], and the assumptions ofperfect CSIT or constant prediction error are no longer valid.Several RA algorithms for OFDMA downlink consideringimperfect CSIT have been proposed recently [1]-[4]. A mod-ification of the proportional fair scheduler is proposed in [4]to predict the allocation for several time slots. This method isshown to improve fairness among users when comparing to thecase without prediction. However, in high mobility scenariosand with practical implementation of channel predictors, twomain issues related to prediction-based RA in rapidly varyingchannels can be identified: 1) The trade-off between predictionhorizon and prediction accuracy, and; 2) Characterization ofthe prediction error to improve system performance.

In this paper, we propose a prediction-based RA algorithmfor high mobility scenarios. The main concern is to character-ize the prediction error associated to practical channel predic-tors. Imperfect CSIT due to prediction error has an importantimpact on the overall system performance as achievable ratesof MSs are computed from CSIT by evaluating the channelpower on each subchannel.

The main contributions of this paper are: 1)The derivation ofa prediction-based RA algorithm, which is aware of predictionerror and capable of providing close to perfect CSIT systemthroughput at the system BER constraint, and 2) A statisticaldescription of the prediction error that does not assume aspecific error model (usually associated to a specific predictiontechnique), and is thus feasible to be applied to generallong range channel predictors. To the best of our knowledge,general theoretical models for long range predictors have notbeen developed yet.

Specifically, it is shown that based on the typical parametersof practical OFDMA systems, a set of histograms can be cal-culated and updated periodically to approximate the statisticsof the prediction error. To corroborate the effectiveness of theproposed algorithm in practical OFDMA systems, we considerrealistic system parameters as well as a realistic channel modelfor each MS. Also, as OFDMA downlink channel predictionis performed at mobile stations (MS), we consider in this worka computationally inexpensive channel predictor, practical for

2

implementation at the MSs [10].The outline of the paper is as follows. In Section II we

define the system model, introduce the notation, review theprediction-based RA literature and describe a long rangechannel predictor suitable for application on prediction basedRA schemes. The error aware prediction-based RA is derivedin Section III, where the effect of imperfect predicted CSITis analyzed and an empirical characterization is proposed tocompensate the prediction error. An analysis of the computa-tional cost associated with the proposed scheme is presentedin Section IV and the performance of the proposed predictioncorrection scheme is evaluated in Section V. Finally, SectionVI provides our conclusions.

II. PROBLEM STATEMENT

A. System model

We consider an OFDMA downlink transmission whereresources are allocated on subcarrier basis. On the otherhand, in LTE and mobile WiMAX, resource blocks consist ofseveral subcarriers, either contiguous or distributed within theOFDMA band, and the signaling of rate values correspondingto the resource blocks is vendor specific. Nevertheless, wechose to minimize the system assumptions and study a genericcase where a resource block consists of one subcarrier. How-ever, the concept can be adapted to other types of resourceblocks as well.

The system under consideration has N available subchan-nels, K active MSs, and resources are allocated on a time slotbasis, where a time slot consists of M consecutive OFDMsymbols. For each time slot s, the base station (BS) schedulerdecides which MS is assigned to a particular subchanneln. More than one subchannel might be assigned to a MSdepending on its rate requirement or channel state.Pt symbols carrying pilot subcarriers are evenly distributed

over a time slot and Pf pilot subcarriers are evenly dis-tributed in frequency over subcarriers on the pilot symbolsto aid channel estimation and prediction at the MSs. Pilotsymbols are QPSK while data symbols can be taken from aset of available QAM constellations [β1 . . . βK]. Furthermore,convolutional coding of the data symbols is used to allowa finer grid of possible bit rates. The base station transmitson each subchannel using one of the modulation and codingschemes subject to the system BER constraint. MSs areassumed to experience independent channel fading with thesame statistics. Regarding the time selectivity of the channel,we assume that the channel varies significantly from one timeslot to the next. As the RA is performed on a time slot basis,we assume that the channel variation over a time slot can beneglected. Thus, the channel frequency response for time slots is given by

Hk(s, n) ≈L∑l=1

hk(s, l)e−j2πlnN . (1)

which is a complex Gaussian random variable with zero meanand σ2

Hkvariance. For l = 1, . . . , L, hk(s, l) denotes the

l-th channel tap for user k over time slot s, being L thelength of the channel impulse response. CSIT is estimated

for current time slot and predicted for the following W timeslots. There are two alternatives to acquire CSIT at the BS.The prediction can be performed either in BS or in mobilestations. The latter case requires a larger overhead in thefeedback channel as W estimates are fed back for each slot.In the former case, the overhead is less at the expense ofa high computational load at the BS. The first alternative isimplemented in this paper, while the derivations apply in bothcases. In the following, transmitted power is assumed to bethe same in all subchannels to emphasize the bit loading. Wedenote the rate achieved by user k on time slot s by Rk(s),which is given as the sum rate over all subchannels assignedto user k as Rk(s) = M

∑n∈Ik rk(m,n), being Ik the index

set for the subchannels assigned to user k and rk(m,n) therate for user k at symbol time m on subchannel n. This way,the overall system throughput for time slot s is given by

R(s) =

K∑k=1

Rk(s). (2)

The design goal is to maximize the system throughputdefined in (2) with constraints on total transmit power, systemBER and fairness among users.

B. Prediction-based RA

A fair and efficient sharing of radio resources among users isan important design factor in wireless networks. Proportionalfair scheduler (PFS) is a popular solution to provide a fairdistribution of resources among users when CSIT is available.Denoting the kth MS average data rate by Rk, based on afairness reasoning, it aims to maximize MSs Rk over time,based on allocations on previous time slots. That is, U(s+1) =arg maxU

∑Kk=1 log(Rk(s)), where U(s + 1) is the utility

function to be maximized for allocating time slot s+ 1 usinginformation up to time slot s and U is the set of all possibleallocations for the considered time slot. The algorithm keepstrack of the average throughput Rk(s) of each MS in anexponentially weighted window of length τ .

The performance of this scheduling algorithm can be im-proved, if channel predictions from MSs are available forfuture time slots. However, long-range channel predictors havenot been fully exploited within resource allocation algorithms.An extension of the PFS to include CSIT prediction has beenderived in [4]. This prediction-based PFS (P-PFS) calculatesthe achievable rates in a prediction window from slot s+ 1 tos + W , and the average rates at the end of this window aremaximized to allocate time slot s + 1. The prediction-basedallocation scheme can be expressed as

U(s+W ) = arg maxU

K∑k=1

log(RWk ), (3)

where RWk denotes the average rate of user k at the end ofthe next W time slots and is computed as

RWk =

(1− 1

τ

)WRk(s)+

1

τ

W∑w=1

(1− 1

τ

)W−wRk(s+ w), (4)

3

where Rk indicates a “virtual” allocated rate, as time slots s+2to s + W have not been allocated yet. A practical algorithmto implement (3) is also given in [4], which will be used inthis paper.

C. Channel prediction

To be able to use information on future time slots, achiev-able rates must be estimated within the prediction window.In what follows we briefly describe a low-complexity channelpredictor [10] that will be used both as a motivating exampleand for testing the proposed prediction-based RA scheme.

To simplify the notation, we focus in this subsection ononly one MS and drop MS index k. It is shown in [10]that the time variation over M (several times larger than M )OFDM symbols of the channel coefficient corresponding toone subcarrier can be well described in terms of a size Mdiscrete cosine transform (DCT) truncated to its first G� Mbasis functions. The basis dimension G is determined from themaximum expected channel Doppler shift and the DCT energycompression characteristics. By interpreting the DCT basisfunctions as the impulse responses of ideal bandpass filterscentered at the cosines frequencies it is also shown that anapproximation H(FB)(m,n) of H(m,n) for m = 1, . . . , Mcan be obtained by recursive filtering of H(m,n) with asecond order IIR filter bank based on a normalized allpasslattice realization, which frequency response is given by

HFB(ejω) =0.5(1− s20)(1 + e−j2ω)

1− s20e−j2ω

+

G−1∑i=1

0.5(1− s2i)(1− e−j2ω)

1 + (s2i + 1)s1ie−jω + s2ie−j2ω,

(5)

where s1i = − cos(πiM

)is the lattice parameter defining the

filters central frequencies, and s2i (0 < s2i < 1) is related tothe 3 dB bandwidth of each narrow band filter. By definingxi(m,n), 0 ≤ i ≤ G − 1, the state vector for each passbandfilter on subcarrier n, the filter bank structure of (5) can beexpressed in state space form as

xi(m+ 1, n) = Aixi(m,n) + biH(m,n)

H(FB)i(m,n) = cixi(m,n) + diH(m,n)

H(FB)(m,n) =∑G−1i=0 H(FB)i(m,n),

(6)

where, Ai, bi, ci and di are the state space description of thefilters in (5). Using (6), the optimum algorithm for estimatingH(m,n) in AWGN is a set of G scalar Kalman filters forthe filter bank of (6). Further, as the DCT basis functions arefixed for the whole transmission, the Kalman gains have asteady state solution resulting in a low complexity limitingKalman filter. Having the Kalman estimate H(m,n) as input,a long range channel predictor can be obtained by decimatingon T (determined according to the maximum channel Dopplershift such that the decimated channel is sampled above theNyquist frequency) the input estimates, scaling up in frequencythe passband filters by the same factor T and using L stepextrapolation on the limiting Kalman filter. In this manner,

the following limiting Kalman predictor can be obtained as

ep(m+ `, n) =H(m+`−T, n)− Hp(m+`−T, n)xpi`(m+ `, n) =(Ap

i )Lxpi`(m+ `− TL, n)

+kpi ep(m+ `, n)

Hpi (m+ `, n) =cpix

pi`(m+ `, n)

Hp(m+`+T(L−1),n) =∑G−1i=0 Hp

i (m+ `, n),(7)

where ` = 0, . . . , (T × L) − 1 are the state vector samplesused at each iteration, xpi` is the predictor state vector, Ap

i

and cpi are the transition matrix and output vector of thepassband filters scaled up in frequency by T , and kpi are thecorresponding steady state Kalman gains. Finally, to obtain thepredicted channel over all subcarriers, one limiting Kalmanfilter predictor is run over each of the Pf available pilotsubcarriers and interpolated for all other subcarriers usingDCT interpolation in frequency.

III. ERROR AWARE PREDICTION-BASED RA

In this section we evaluate achievable rates rk(s+w, n) forfuture time slots (input to the P-PFS of Section II-B) whenimperfect CSIT due to prediction errors is available. Whenperfect CSIT is available, extending [17] for coded modula-tion, the instantaneous BER for subchannel n in symbol timem can be approximated by

Pe(s, n) ≈ c1 exp

{−c2γδ(s, n) |H(s, n)|2

2β(s,n) − 1

}, (8)

where c1 = 0.2, c2 = 1.6, β(s, n) is the number of bits persymbol of the QAM constellation used for a subchannel andtime slot and δ(s, n) is the coding gain, with respect to theuncoded case, for the corresponding QAM constellation. SNRvalue γ accounts for the path loss while small scale fadingeffects are represented by H(s, n). By evaluating (8) for thepossible pairs of β(s, n) and δ(s, n), the one that satisfies thesystem BER constraint is selected and determines the numberof bits per symbol r(s, n) for transmission.

In a practical scenario, it is impossible to have perfectCSIT and the determination of r(s, n) is based on an es-timate H(s, n) of H(s, n). In this case, the use of (8) nolonger guarantees that the BER constraint is satisfied. Incase of imperfect CSIT [17], an average BER Pe(s, n) =E|H(s,n)||H(s,n)

{Pe(s, n)} is considered instead of the in-stantaneous BER of (8), where the expectation is evaluatedover |H(s, n)||H(s,n) . By defining the random variable α =

|H(s, n)||H(s,n) it results in

Pe(s, n) =

∫ ∞0

Pe(s, n)f(α)dα

=

∫ ∞0

c1 exp

{−c2γδ(s, n)α2

2β(s,n) − 1

}f(α)dα,

(9)

where f(α) is the probability density function (pdf) of|H(s, n)||H(s,n) . When H(s, n) is an estimate or slightlydelayed estimate of the true channel coefficient H(s, n),several assumptions hold regarding f(α) [17], [18], [2] andclosed form solutions can be derived for evaluating (9).

4

Unfortunately, for the prediction-based RA application con-sidered here, H(s, n) is a prediction of H(s, n) based onsignificantly older CSIT and a derivation of f(α), besidescomplex to obtain, depends on the characteristics of thechannel predictor used to obtain H(s, n). In the followingparagraphs we reformulate (9) in terms of the prediction errorinstead of α, as the former is the available observation in caseof the prediction-based RA.

For a prediction-based RA algorithm considering CSIT upto W time slots in the future, W different pdf should be evalu-ated. Let us define for each considered prediction horizon the

random variables ew =

(αw −

∣∣∣H(s+ w, n)∣∣∣|H(s,n)

)/σ2

H

such that the pdfs of αw and ew are related by

fαw(αw) =1

σ2H

few(ew). (10)

Replacing (10) in (9), the estimated BER for each predictionhorizon can be evaluated in terms of the defined normalizedprediction error ew using

Pew(s+ w, n) = Pew(s+ w, n)× ρw(n), (11)

where Pew(s + w, n) denotes the evaluation of (8) for theavailable predicted value of the channel coefficient

Pew(s+w, n)=c1exp

−c2γδ(s+w, n)

∣∣∣H(s+w, n)∣∣∣2|H(s,n)

2β(s+w,n) − 1

,

(12)and ρw(n) is the correction factor compensating the imperfectCSIT through the normalized prediction error

ρw(n)=1

σ2H

∫ ∞−∞

exp

{−c2γδ(s+w, n)θ(s+w, n)

2β(s+w,n) − 1

}few(ew)dew,

(13)where θ(s + w, n) = 2σ2

H

∣∣∣H(s+ w, n)∣∣∣|H(s,n)

ew + σ4He

2w.

It is clear that for perfect CSIT ρw(n) = 1. We derive next anestimator of few(ew) in (13) to evaluate this correction factor.

Assuming that the prediction error ew is identically dis-tributed over the system subchannels and is independent ofH(s, n), the number Q of prediction error samples for thedifferent subchannels can be used to construct a histogram toestimate few(ew). Following [19, Ch. 5.5], to find few(ew)we use the empirical pdf. We expect that as Q→∞, fε(ew)approaches the true density few(ew).

To derive the estimator, let the ith histogram interval becentered on εi, and let ω be its width. Hence the ith intervalwill be denoted by ∆i =

{ew : εi − ω

2 < ew ≤ εi + ω2

}, so

that the histogram fε(ew) is defined by

fεi(ew) =QiQω

for ew ∈ ∆i, (14)

being Qi the number of observations for which ew ∈ ∆i.The interval width ω, together with the setting of a maximumerror threshold εQ, determine the number of probability masspoints of the estimator. The bias of fε(ew) as an estimator offew(ew) can be shown to be for each interval b(fεi(ew)) ≈f

′′

ew(εi)ω2

24 , where f′′

ew(εi) is the second derivative of few(εi),

and the normalized standard error σε for estimating few(εi) isσε =

√1−ωfew (εi)Qωfew (εi)

. It can be noted that the effect of the cellwidth ω on σε is the opposite to that on the bias such that anappropriate choice of ω is to set ω ∝ 1/

√Q [19].

Using the estimated pdf for the prediction error, (13) canbe rewritten in terms of (14) as

ρw(n)=1

σ2H

εQ∑−εQ

exp

{−c2γδ(s+ w, n)θ(s+ w, n)

2β(s+w,n) − 1

}fεi(εi),

(15)being θ(s+w, n) = 2σ2

H

∣∣∣H(s+ w, n)∣∣∣|H(s,n)

εi+σ4Hε

2i . This

last expression is then inserted into (11) replacing ρw(n) andused to evaluate Pew(s+w, n) on each subchannel such that

Pew(s+ w, n) = Pew(s+ w, n)× ρw(n). (16)

The pair β∗(s + w, n), δ∗(s + w, n) satisfying (16) deter-mines the achievable rate r(s + w, n) on subchannel n. Itis worth noting that as the number of subcarriers in typicalOFDMA systems is usually large (above 512 subcarriers), thebias and standard error will be small, and the W histogramsbuilt for each MS will describe the statistics of the predic-tion error accurately. Further, the histograms for estimatingfew(εi) are constructed assuming perfect CSIT at the MS forcurrent time slot. In a practical implementation Hk(s, n) isestimated at the MS for symbol detection. Assuming that theestimation error is much lower than the prediction error inthe prediction range, which is a reasonable assumption forpractical estimators/predictors (as will be shown in Section V),the former can be used as a reference for the characterizationof the latter without affecting much the resulting fεi(εi). Also,if we further assume that the statistics of ew are slowly timevariant, then the histograms can be periodically improved withthe incorporation of new data points. That is

f (s)εi (εi) = ξf (s)

εi (εi) + (1− ξ)f (s−W )εi (εi), (17)

where 0.5 ≤ ξ ≤ 1 is a tuning parameter selected tocompensate for the possibly time-varying statistics of ew.Finally, the resulting values of rk(s + w, n) correspondingto each MS are fed to the prediction-based RA algorithm.

IV. COMPLEXITY ANALYSIS

The proposed prediction error characteriza-tion/compensation scheme involves an additionalcomputational cost when compared with the baseline scenarioof uncompensated imperfect CSIT. For the uncompensatedcase, the computational cost for the bit loading is given bythe evaluation of (8) for the possible pairs of β and δ. Usingthe binary search algorithm to this end, the average number ofrequired evaluations of (8) for this search is log2(K)− 1 andat most log2(K) probes in the worst case, being K the numberof available β-δ pairs, which is a small number for practicalsystems. On the other hand, for the proposed scheme, thecomputational cost involved in the histogram constructionof (14) and in the evaluation of the correction factor ρw(n)of (15) should be considered besides the evaluation of Pewsimilar to (8).

5

The constructed histograms have 2εQ + 1 bins (probabilitymass points). The value of Qi in (14) for each probability masspoint can also be evaluated using the binary search algorithmrequiring on average log2(2εQ + 1) trials for each sample.Thus, the computational cost associated to the W histogramsconstruction is evaluated as O(WQ log2(2εQ + 1)), and as-suming that the histograms are updated every W time slots itreduces to O(Q log2(2εQ+1)). Regarding the computation ofthe correction factor ρw(n) of (15), it must be evaluated forall possible pairs of β and δ to be then inserted into (16). Theevaluation of (15) involves O(2εQ + 1) complex operations,which leads to a computational cost ofO(K(2εQ+1)) complexoperations.

Summarizing the discussion from the previous paragraphs,the additional computational cost of the proposed scheme,when compared to the baseline of the uncompensated case,is O(Q log2(2εQ + 1) + K(2εQ + 1)). Thus, in practice thecomplexity is dominated by the number of error samples Q,which is much larger than the number of available modula-tion/coding pairs K.

V. NUMERICAL EVALUATION

In this section we evaluate the impact of the prediction erroraware bit loading presented in Section III to the performanceof the prediction-based RA algorithm of [4]. We seek toquantify the performance improvement achieved with theproposed empirical characterization of the prediction error.

We consider the downlink of an OFDMA system based on3GPP specification [9] operating at a carrier frequency fC =2GHz with 10 MHz bandwidth, 15KHz subcarrier spacing and5µs cyclic prefix. FFT size is 1024 and N = 600 subcarriersare in use occupying 90% of the bandwidth. The length ofthe time slot is M = 15 OFDMA symbols, equivalent to1ms and the system uses 5% pilot ratio. 4, 16 and 64-QAMconstellations are available for data transmission. Also, thesystem employs convolutional coding for data transmissionusing the (133,171) rate 1/2 code with puncturing to obtain2/3 and 3/4 code rates. A bit loading function similar to [8]is used corresponding to a system target BER of 1 · 10−3. Allthe simulation results are obtained by averaging over time inorder to average the channel fading statistics.

Based on the derivations on Section III, the parameters forthe construction of the histograms are chosen as Q = 600(all system subchannels used), ω = 0.078 and the limits forthe histogram set to ±0.8. Finally, parameter ξ for updatingof the histograms is set to ξ = 0.7. To evaluate a realisticpropagation environment, we consider independent wirelessfading channel for each MS following the ITU-Vehicular Amodel, which results in a 27-tap frequency selective channelfor the given system parameters. Each channel tap varies intime according to Jakes Doppler spectrum, and we employthe practical channel estimator/predictor structure of [10].Following [10], the parameters for this channel predictor areM = 12M such that s1i = − cos

(πiM

). It is shown in [10] that

the performance is robust to the typical values of the selectivityparameter s2i such that its value is set to 0.998. Also, for theMSs speeds of interest a value of G = 8 results for the basis

expansion dimension. Finally, the extrapolation factor for theKalman predictor is fixed to L = 3 and the decimation factorT adjusted to meet the different prediction horizons.

Before addressing the performance of P-PFS, Fig. 1 illus-trates the virtual rates Rk(s+w) in (4) (input to P-PFS) whenimperfect CSIT is available. The figure shows the averagecomputed Rk(s+w) for a single MS, over 3 successive timeslots based on the predicted CSIT with a SNR of 25dB. Theinput to the P-PFS of (3) is the average rate RWk in (4) of user kat the end of the W = 3 time slots considered computed basedon these virtual rates Rk(s + w). In general, it noticed thatimperfect CSIT degrades the virtual rates. However, even withimperfect CSIT the prediction increases system throughput aswill be seen later on. The proposed bit loading based on (16)is compared against the uncompensated case (8), and the caseof perfect CSIT is also shown for reference. As expected, theaverage bit loading is independent on the prediction horizonwhen perfect CSIT is available. Two bars for imperfect CSITare shown for the proposed bit loading when the true channel(Perf. Ref.) and channel estimates (Est. Ref.) are employed forthe histograms construction. It can be noted that the proposedscheme effectively reduces the loading gap between the perfectand imperfect cases obtaining a reduction from 0.35, 0.75 and1.1 to 0.1, 0.35 and 0.6 bits/subchannel for w = 1, 2 and 3time slots prediction respectively. Although not used by P-PFS, bit loading without prediction (w = 0) is also plotted forreference.

0 1 2 30

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Prediction horizon (TSs)

Av

erag

e b

it l

oad

ing

(bit

s/su

bch

ann

el)

Perfect CSIT

Corrected CSIT (Perf. Ref)

Corrected CSIT (Est. Ref)

Imperfect CSIT

Fig. 1. Average bit loading per subcarrier for no prediction andfor w = 1, 2 and 3 time slot prediction for a single MS. Resultsare averaged over 2000 time slots and also over MSs and systemsubchannels. Mobile speed is set to 50km/h, SNR is 25 dB and thesystem target BER is 1 · 10e−3.

Figure 2 shows, for a 10 MSs, 50 km/h and 25dB SNRscenario, the improvement in system throughput obtained bythe P-PFS as the prediction window increases (0 stands forthe conventional PFS). It can be noted that for all predictionwindow lengths the performance is significantly improvedusing the proposed scheme (16), while the improvement bythe corrected CSIT is larger for shorter prediction windows asthe associated prediction error is smaller. This figure clearlyshows the capabilities of the prediction based approach. At

6

the same time the impact of the degraded virtual rates dueto imperfect CSIT Rk(s+ w) (illustrated in Figure 1) on theperformance of P-PFS is evident.

0 2 4 6 8 10 12 145

5.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9

6

Length of prediction window (TSs)

Av

erag

e sy

stem

th

rou

ghp

ut

(bit

s/su

bch

ann

el)

Perfect CSIT

Corrected CSIT

Imperfect CSIT

Fig. 2. System throughput vs. prediction window length for P-PFS.Results are averaged over 1000 time slots and also over MSs andsystem subchannels to obtain system level results. Mobile speed isset to 50km/h, SNR is 25 dB and the system target BER is 1 ·10e−3.

In Figure 3, system throughput is evaluated for a predictionwindow of length W = 3 and up to 20 MSs moving at 25km/hin a high and low SNR scenarios. Channel power for MSs isuniformly distributed between -3 and 0dB. The 25dB and 10dBSNR settings result in channel estimate references with MSEof -60.15dB and -33.64dB respectively for the constructionof the histograms. In both scenarios, the results for PerfectCSIT are plotted as a best performance bound. As the RA forthe next time slot is based on the computation of the averagerates achieved by MSs at the end of the prediction window, ifthe computed achievable rates for future time slots differ toomuch from the true achievable rates, the prediction-based RAperformance will degrade in terms of system throughput orin terms of the system BER constraint. P-PFS with imperfectCSIT is not able to exploit the available resources to full extentand the system throughput is reduced, in particular for the caseof few MSs. On the other hand, it can be seen that the proposedcorrection scheme gives close to perfect CSIT performance,which demonstrates its effectiveness. This happens because,as the histograms constructed in (14) are not restricted to havezero mean, the correction factor of (15) has the capability tocompensate a bias on the predicted CSIT. As expected, in thislow SNR regime, system throughput is reduced as channelquality for the MSs is degraded. However, the proposed errorcorrection scheme still reduces effectively the throughput gapfor both references used in the histograms construction. Itis worth noting that the performance improvement obtainedwith the proposed scheme is larger in this last case, asthe uncompensated bit loading cannot approach the perfectCSIT case even for a large number of users (where theprobability of having a user with good channel state at eachtime slot increases). This latter conclusion justifies the addedcomplexity of the proposed scheme.

2 4 6 8 10 12 14 16 18 201

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

No. of mobile stations

Av

erag

e th

rou

ghp

ut

(bit

s/su

bch

ann

el)

10dB SNR

25dB SNR

Fig. 3. System throughput vs. the number of MSs for W = 3 P-PFS. Results are averaged over 500 time slots and also over MSsand system subchannels to obtain system level results and are shownfor the case where the predicted CSIT is error free (circle marker),the predicted CSIT is corrected (square marker) as proposed in (16),and for uncompensated predicted CSIT using (8) (diamond marker).Mobile speed is set to 25km/h and the system target BER is 1·10e−3.

Figure 4 shows again the system throughput for the highSNR setting of Figure 3, this time evaluated for a linearpredictor [12] and a sum of sinusoids predictor [11], whichwe refer to as MMSE and ESPRIT predictors respectively. Itis worth noting that the computational cost of these predictorsis significantly larger than that of the predictor of [10] sothat their implementation at the MSs may not be feasible.It is observed for both predictors, the proposed scheme alsogives close to perfect CSIT performance showing that it can beapplied to different prediction schemes as the characterizationof the prediction error is independent of the channel predictionscheme.

Regarding BER behavior using the proposed error correc-tion scheme, Figure 5 shows the BER results for the highSNR setting of Figure 3 and all channel predictors tested. Asthe bit loading function used is discrete, all curves are belowthe BER constraint as expected and thus, the Perfect CSITcase is used as a reference for performance comparison. Whenthe channel power is underestimated, the uncompensated bitloading algorithm assigns modulation/coding schemes thatmeet the BER constraint for this underestimated power. If thechannel state is in reality better, the BER attained with theassigned more robust modulation/coding pair will be lower,indicating that the system resources are not fully exploited.On the other hand, if the channel power is overestimated,the assigned modulation/coding scheme will not be supportedby the real channel state, thus leading to a higher BER,indicating a loss in transmission reliability. It is easy to seein Figure 5 that while the RBEM predictor of Section II-Cunderestimates the channel power, the MMSE and ESPRITpredictors overestimate the channel power. However, in allcases the proposed compensation scheme effectively reducesthe BER gap relative to the Perfect CSIT case. As observed

7

in Fig. 3, the constructed histograms take into account apossible under/overestimation of the channel power and thuscan compensate this effect in the bit loading. In the high SNRscenarios of Figures 3 and 4, for a large number of usersall schemes converge to the same total system throughput.This happens because as the number of users increases, theprobability of having a user with good channel conditions atall time slots increases, and so does the system throughput.However, achievable rates computed by the different schemesare not necessarily the same. Thus, MSs scheduled by P-PFS might not be the same in all cases leading to differentBER performance for each scheme. This figure shows that theproposed technique approaches the Perfect CSIT case for allchannel predictors used, demonstrating that it does not rely ona specific prediction structure.

2 4 6 8 10 12 14 16 18 204.2

4.4

4.6

4.8

5

5.2

5.4

5.6

5.8

6

No. of mobile stations

Av

erag

e th

rou

ghp

ut

(bit

s/su

bch

ann

el)

Perfect CSIT

Corrected CSIT ESPRIT

Imperfect CSIT ESPRIT

Corrected CSIT MMSE

Imperfect CSIT MMSE

Fig. 4. System throughput vs. the number of MSs for W = 3 P-PFSand different prediction schemes. Results are averaged over 500 timeslots and also over MSs and system subchannels to obtain systemlevel results. Mobile speed is set to 25km/h, SNR is 25 dB and thesystem target BER is 1 · 10e−3.

2 4 6 8 10 12 14 16 18 20

-410

-310

-210

No. of mobile stations

BE

R

Perfect CSITCorrected CSIT ESPRITImperfect CSIT ESPRITCorrected CSIT MMSEImperfect CSIT MMSEImperfect CSIT RBEMCorrected CSIT RBEM

Fig. 5. System BER vs. the number of MSs for W = 3 P-PFS usingdifferent prediction schemes. Mobile speed is set to 25km/h, SNR is25 dB and the system target BER is 1 · 10e−3.

A last simulation setting is shown in Figure 6 where wefurther investigate the performance dependence on the MSsspeed. We consider again a prediction window of W = 3time slots and 10 MSs moving at speeds from 50 to 200km/hwith 25dB SNR. This figure shows how the bit loading gapincrease for higher speeds (prediction error increases for fixedprediction range) results in a throughput loss for high MSsspeeds. In all cases the proposed scheme reduces this losssignificantly compared to the uncompensated case.

50 100 150 2005.5

5.6

5.7

5.8

5.9

6

6.1

6.2

MSs speed (km/h)

Av

erag

e th

rou

ghp

ut

(bit

s/su

bch

ann

el)

Perfect CSITCorrected CSIT Perfect Ref.Corrected CSIT Imperfect Ref.Imperfect CSIT

Fig. 6. System throughput for W = 3 prediction-based RA. Resultsare averaged over 2000 time slots and also over MSs and systemsubchannels to obtain system level results. There are 10 MSs movingat speeds ranging from 50 to 200km/h and a SNR of 25 dB.

VI. CONCLUSIONS

We proposed a characterization of the prediction error forprediction-based resource allocation for OFDMA downlinkover mobile wireless channel when imperfect channel stateinformation is available. Based on the large amount of fre-quency data samples available in a typical OFDMA system,we derived an empirical approach based on histograms forthe characterization of the prediction error for the differentprediction horizons considered in the prediction window.

We evaluated the proposed scheme under realistic channelconditions, system parameters and a practical channel predic-tor feasible for implementation at mobile stations. Simulationresults indicate that the proposed scheme outperforms similarprediction-based resource allocation algorithms that disregardthe prediction error.

REFERENCES

[1] S. Stefanatos and N. Dimitriou, “Downlink OFDMA resource allocationunder partial channel state information,” in IEEE ICC, Jun. 2009, pp.1–5.

[2] I. Wong and B. Evans, “Optimal resource allocation in the OFDMAdownlink with imperfect channel knowledge,” IEEE Trans. Commun.,vol. 57, no. 1, pp. 232–241, Jan. 2009.

[3] A. Falahati and M. Ardestani, “An improved low-complexity resourceallocation algorithm for OFDMA systems with proportional data rateconstraint,” in IEEE ICACT, vol. 1, Feb. 2007, pp. 606 –611.

[4] J. Hajipour and V. C. M. Leung, “Proportional fair scheduling in multi-carrier networks using channel predictions,” in IEEE ICC, May 2010,pp. 1–5.

8

[5] S. Najeh, H. Besbes, and A. Bouallegue, “Predictive approach forOFDMA resource allocation over fixed wireless channels,” in IEEEICSCS, Nov. 2009, pp. 1–5.

[6] D. Ngo, C. Tellambura, and H. Nguyen, “Efficient resource allocationfor OFDMA multicast systems with spectrum-sharing control,” IEEETrans. Veh. Technol., vol. 58, no. 9, pp. 4878 –4889, Nov. 2009.

[7] Y. Zhang and C. Leung, “Subchannel power-loading schemes in mul-tiuser OFDM systems,” IEEE Trans. Veh. Technol., vol. 58, no. 9, pp.5341 –5347, Nov. 2009.

[8] D. Kivanc, G. Li, and H. Liu, “Computationally efficient bandwidthallocation and power control for OFDMA,” IEEE Trans. WirelessCommun., vol. 2, no. 6, pp. 1150–1158, Nov. 2003.

[9] 3GPP, “Physical layer aspects for evolved UTRA,” 3GPP technicalreport, TR 25.814, Ver. 1.0.3, Feb. 2006.

[10] J. Schmidt, J. Cousseau, R. Wichman, and S. Werner, “Low-complexitychannel prediction using approximated recursive DCT,” Accepted forpublication at IEEE Transaction on Circuits and Systems 1.

[11] I. Wong and B. Evans, “Low-complexity adaptive high-resolution chan-nel prediction for OFDM systems,” in IEEE GLOBECOM, Dec. 2006,pp. 1–5.

[12] D. Schafhuber and G. Matz, “MMSE and adaptive prediction of time-varying channels for OFDM systems,” IEEE Trans. Wireless Commun.,vol. 4, no. 2, pp. 593–602, Mar. 2005.

[13] S. Semmelrodt and R. Kattenbach, “Investigation of different fadingforecast schemes for flat fading radio channels,” in IEEE VTV, vol. 1,Oct. 2003, pp. 149–153.

[14] A. Duel-Hallen, “Fading channel prediction for mobile radio adaptivetransmission systems,” Proceedings of the IEEE, vol. 95, no. 12, pp.2299–2313, Dec. 2007.

[15] N. Mokari, M. Javan, and K. Navaie, “Cross-layer resource allocation inOFDMA systems for heterogeneous traffic with imperfect CSI,” IEEETrans. Veh. Technol., vol. 59, no. 2, pp. 1011 –1017, Feb. 2010.

[16] M. Awad, V. Mahinthan, M. Mehrjoo, X. Shen, and J. Mark, “A dual-decomposition-based resource allocation for OFDMA networks withimperfect CSI,” IEEE Trans. Veh. Technol., vol. 59, no. 5, pp. 2394–2403, Jun. 2010.

[17] S. Ye, R. Blum, and L. Cimini, “Adaptive OFDM systems with imperfectchannel state information,” IEEE Trans. Wireless Commun., vol. 5,no. 11, pp. 3255–3265, Nov. 2006.

[18] Y. Yao and G. Giannakis, “Rate-maximizing power allocation in OFDMbased on partial channel knowledge,” IEEE Trans. Wireless Commun.,vol. 4, no. 3, pp. 1073–1083, May 2005.

[19] M. Jeruchim, P. Balaban, and K. Shanmugan, Simulation of Communi-cation systems. New York: Plenum Press, 1992.