IEEE COMMUNICATIONS LETTERS, VOL. 11, NO. 6, JUNE 2007 507

Trade-Off Between Frequency Diversity Gain and Frequency-SelectiveScheduling Gain in OFDMA Systems with Spatial Diversity

Seung Joon Lee, Senior Member, IEEE

Abstract— OFDMA systems inherently take advantages of bothfrequency diversity gain and frequency-selective scheduling gain.The former is achieved by allocating a user the subcarrierswidely scattered over an entire frequency band, while the latteris achieved by allocating a user the subcarriers consecutivelylocated within a subband of a limited bandwidth which is themost favorable to the user among many subbands in the entirefrequency band. In this letter, the effect of user mobility isquantitatively investigated in relation to a trade-off between thefrequency diversity gain and the frequency-selective schedulinggain in OFDMA systems. As a performance measure, outagecapacity is considered. Also studied is how the spatial diversityresulting from multiple antenna techniques and the outageprobability affect the dependence of the performance trade-offon the user mobility.

Index Terms— Frequency diversity gain, frequency-selectivescheduling gain, OFDMA.

I. INTRODUCTION

ORTHOGONAL frequency division multiple access(OFDMA) is a very popular transmission scheme for 3G

and 4G wireless communication systems [1]–[5]. In OFDMA,a large total bandwidth is split into many orthogonal sub-carriers, among which several subcarriers are allocated to auser. Since the total bandwidth is usually much larger thana coherent bandwidth, the channel gains of the orthogonalsubcarriers are frequency-selective. Wireless communicationsystems based on OFDMA take advantage of this frequency-selective property in two different ways. The first approach isto allocate a user the orthogonal subcarriers widely scatteredover an entire frequency band, achieving frequency diversitygain. The frequency diversity gain denotes the performanceimprovement obtained by using multiple subcarriers whosepath gains are independently faded rather than using sub-carriers whose path gains are equally or similarly faded.The second approach is to allocate a user several subcarriersconsecutively located within a subband of a limited band-width which is the most favorable to the user among manysubbands. This approach achieves frequency-selective oppor-tunistic scheduling gain. The frequency-selective schedulinggain represents the performance improvement obtained byusing subcarriers selected to be favorable to a specific userbased on channel state information rather than using sub-carriers predefined without channel knowledge. For example,these two approaches are realized by the forms of diversity

Manuscript received February 12, 2007. The associate editor coordinatingthe review of this letter and approving it for publication was Biao Chen. Thiswork was supported in part by the Post-doctoral Fellowship Program of KoreaScience & Engineering Foundation (KOSEF).

S. Lee is with the Electronics & Telecommunications Research Institute,Daejeon 305-700, Korea (email: s.j.lee@ieee.org).

Digital Object Identifier 10.1109/LCOMM.2007.070233.

subchannelization and band adaptive modulation and coding(AMC) subchannelization, respectively, in [2], [4] and bythe forms of frequency distributed mapping and frequencylocalized mapping, respectively, in [5].

It is commonly considered that use of frequency-selectivescheduling gain is preferred for low mobility users, whileuse of frequency diversity gain is preferred for high mobilityusers. However, no quantitative analysis is available for thisclaim. So in this letter, the effect of user mobility is quanti-tatively analyzed in relation to a trade-off between frequencydiversity gain and frequency-selective scheduling gain. Alsoinvestigated is how spatial diversity order and required outageprobability affect the dependence of the performance trade-offon the user mobility.

The remainder of this letter is organized as follows. InSection II, a system model is described. In Section III,outage capacities are derived for the scheme achieving the fre-quency diversity gain and the scheme achieving the frequency-selective scheduling gain, depending on user mobility. In Sec-tion IV, numerical examples are presented showing the trade-off between the frequency diversity gain and the frequency-selective scheduling gain. We draw conclusions in Section V.

II. SYSTEM MODEL

We assume the following system model: There are Ksubbands available to a user. The bandwidth of each subbandis less than a coherent channel bandwidth, so that the channelgain for the k-th subband (k = 1, 2, · · · ,K) at time t isrepresented by NRX × NTX matrix Hk(t), where NRX andNTX denote the number of receiving antennas and the numberof transmitting antennas, respectively. The value of NTX is1 or 2. If NTX = 2, a space-time processing scheme withmaximal transmit diversity, such as [6], is considered. Allcomponents in the matrix Hk(t) are independent complexGaussian random variables with mean 0 and variance 1.Hence, the squared Frobenius norm ‖Hk(t)‖2 has a centralchi-square distribution with the number of degrees of freedomL = 2 · NTX · NRX. The channel gain matrices {Hk(t), k =1, 2, · · · ,K} for all K subbands are mutually independent.

Scheduling decision (i.e., selecting a subband to use fortransmission) is calculated based on the measurement {Hk(t−τ), k = 1, · · · ,K} at time t−τ but is applied to transmissionat time t. The channel variation between times t and t− τ isdescribed by the following Markov model

Hk(t) = ρHk(t − τ) +√

1 − ρ2Wk(t) (1)

where ρ is the time-autocorrelation for each component ofHk(t) defined as

ρ = E[hk,ij(t)h∗

k,ij(t − τ)]

(2)

1089-7798/07$25.00 c© 2007 IEEE

508 IEEE COMMUNICATIONS LETTERS, VOL. 11, NO. 6, JUNE 2007

and where hk,ij(t) denotes the i-th row and j-th columncomponent of Hk(t). The value of ρ can be related to theuser velocity v by ρ = J0

(2πfc

vc τ

)where J0(·) denotes

the zeroth-order modified Bessel function of the first kind,fc is the carrier frequency, and c is the speed of light [7].In (1), the components of Wk(t) are independent, identicallydistributed (i.i.d.) complex Gaussian with zero mean and unitvariance, and it holds that Wk1(t1) is independent of Wk2(t2)for k1 �= k2 or t1 �= t2 and is also independent of Hk2(t2)for any k1, k2, t1 and t2.

III. OUTAGE CAPACITIES

All K subbands are used for transmission without priorknowledge of channel state when achieving the frequencydiversity gain. On the other hand, for obtaining the frequency-selective scheduling gain, only one subband is selected fortransmission, which is estimated to be the best among Ksubbands. The estimation is based on prior knowledge of theprevious channel. Here those two schemes are analyticallycompared in terms of outage capacity [10] per used bandwidth.

A. Outage Capacity with Frequency Diversity Gain

The capacity of the k-th subband in [bits/second/Hz] isgiven by

Ck(t) = log2

(1 + ‖Hk(t)‖2 SNRTX

NTX

)(3)

and its probability density function (PDF) is obtained as

fCk(t)(x) =2x ln 2

SNRTX/NTX· f‖Hk(t)‖2

(2x − 1

SNRTX/NTX

)(4a)

f‖Hk(t)‖2(x) =x

L2 −1e−x(L2 − 1

)!

(4b)

where (4b) represents a central chi-square distribution withL degrees of freedom [8]. Since all K subbands are usedwithout knowledge of the channel state on a transmitter side,the average capacity of the scheme achieving the frequencydiversity gain is obtained as

C (D)(t) =1K

K∑k=1

Ck(t) (5)

and its PDF is derived as

fC(D)(t)(x) = K · fC1(t)(Kx) ∗ fC2(t)(Kx) ∗ · · · ∗ fCK(t)(Kx)(6)

where ∗ denotes a convolution operation.However, it is difficult to evaluate the convolution of (6) for

K > 2 even with numerical methods. Alternatively, we canuse the following approximation by applying the central limittheorem to i.i.d. random variables of {Ck(t), k = 1, · · · ,K}

fC(D)(t)(x) � 1√2πσC

exp{− (x − mC)2

2σ2C

}(7a)

mC = E [Ck(t)] (7b)

σ2C =

E[C2

k(t)] − E2 [Ck(t)]K

. (7c)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0 1 2 3 4 5 6 7

Prob

abili

ty d

ensi

ty fu

nctio

n

x

(a)(b)(c)

Fig. 1. Comparison of probability density functions: (a) fCk(t)(x), (b)fC(D)(t)(x), and (c) Gaussian approximation of fC(D)(t)(x) when NTX =NRX = 1, K = 2, and SNRTX = 10 dB.

Fig. 1 presents an example comparing (6) and (7) (togetherwith (4) as a reference) and shows that the PDF of C (D)(t)already resembles a Gaussian distribution, even when K = 2.

If we let the cumulative distribution function (CDF) ofC (D)(t) be denoted by FC(D)(t)(x):

FC(D)(t)(x) =∫ x

0

fC(D)(t)(y)dy, (8)

then the outage capacity with outage probability po for thescheme achieving the frequency diversity gain is given by

C (D)po

= F−1C(D)(t)

(po). (9)

B. Outage Capacity with Frequency-Selective SchedulingGain

Let kt−τ denote the best subband for a specific user at timet − τ . Then kt−τ is determined by

kt−τ � arg max1≤k≤K

‖Hk(t − τ)‖. (10)

In order to derive the PDF of ‖Hkt−τ(t − τ)‖, we define

Hi,j � maxi≤k≤j

‖Hk(t − τ)‖ (11)

where it holds that

‖Hkt−τ(t − τ)‖ = max

1≤k≤K‖Hk(t − τ)‖ = H1,K . (12)

Since H1,j = max (H1,j−1,Hj,j) for j > 1, the PDF of‖Hkt−τ

(t−τ)‖ can be obtained by using the following relation[9]

fH1,j(x) = fH1,j−1(x)FHj,j

(x) + FH1,j−1(x)fHj,j(x) (13a)

fHm,m(x) � f‖Hm(t−τ)‖(x) =

2xL−1e−x2

(L2 − 1

)!

(13b)

for j > 1 and m ≥ 1. The kt−τ -th subband is used fortransmission at time t. The squared Frobenius norm of thechannel matrix Hkt−τ

(t) for the kt−τ -th subband at time t

LEE: TRADE-OFF BETWEEN FREQUENCY DIVERSITY GAIN AND FREQUENCY-SELECTIVE SCHEDULING GAIN IN OFDMA SYSTEMS 509

0.83

0.84

0.85

0.86

0.87

0.88

0.89

0.9

0.91

1 2 4

SNR =10dB,TX p =0.01o

SNR =20dB,TX p =0.01o

SNR =10dB,TX p =0.02o

SNR =20dB,TX p =0.02o

Spatial diversity order, L/2

(N =N =1)TX RX (N =N =2)TX RX(N =2, N =1)TX RX

ρ c

Fig. 2. Crossover values ρc of time-autocorrelation of a channel ρ(such

that C(D)po = C(S)

po

)for the performance trade-off between frequency diversity

gain and frequency-selective scheduling gain when K = 6.

has a conditional PDF given by the following noncentral chi-square distribution [8]

f‖Hkt−τ(t)‖2

∣∣‖Hkt−τ(t−τ)‖(x) =

12σ2

G

( x

s2

)L−24

· exp(−s2 + x

2σ2G

)· IL

2 −1

(√x

s

σ2G

)(14a)

s = ρ‖Hkt−τ(t − τ)‖ (14b)

σ2G =

1 − ρ2

2(14c)

and an unconditional PDF given by

f‖Hkt−τ(t)‖2(x) =

∫ ∞

0

f‖Hkt−τ(t)‖2

∣∣‖Hkt−τ(t−τ)‖=y

(x)

·f‖Hkt−τ(t−τ)‖(y)dy (15)

where IL2 −1(·) denotes the

(L2 − 1

)-th order modified Bessel

function of the first kind.Defining the CDF of ‖Hkt−τ

(t)‖2 as

F‖Hkt−τ(t)‖2(x) =

∫ x

0

f‖Hkt−τ(t)‖2(y)dy, (16)

the outage capacity with outage probability po for the schemeachieving the frequency-selective scheduling gain is given by

C (S)po

= log2

(1 +

SNRTX

NTX· F−1

‖Hkt−τ(t)‖2(po)

)� g(ρ) (17)

where g(ρ) denotes the dependence of C (S)po

on ρ and will beused in the next section.

IV. NUMERICAL EXAMPLES

To present numerical examples, we consider there are sixindependent subbands (K = 6).1 Fig. 2 shows the crossovervalue of ρ at which the outage capacities for two schemes

1This value of K = 6 is motivated by the fact that popular simulationchannel models, such as the Pedestrian-B model [11] and the Typical Case ofUrban Channel [12], have 6 multipaths which cause 6 independent channelgains in the frequency domain.

are the same for various conditions, such as different SNRs,different outage probabilities, and different spatial diversityorders. The crossover value of ρ for the performance trade-offcan be represented as

ρc = g−1(C (D)

po

)(18)

and it means that, if the time-correlation of a channel is greaterthan the given crossover value, then the scheme achieving thefrequency-selective scheduling gain obtains a larger outagecapacity than the scheme achieving the frequency diversitygain; however, if the time-correlation of a channel is less thanthe given crossover value, then the former has a smaller outagecapacity than the latter. It is seen that the crossover value of ρfor the performance trade-off decreases as the SNR, the spatialdiversity order, or the outage probability increases.

V. CONCLUSION

Two advantages inherent in OFDMA systems, frequencydiversity gain and frequency-selective scheduling gain, havebeen analytically compared in terms of outage capacity. Thecrossover values of the time autocorrelation of a channel havebeen evaluated at which the preference switches between thefrequency diversity gain and the frequency-selective schedul-ing gain. It has been shown that the crossover values areaffected by the SNR, the spatial diversity order, and the outageprobability.

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