1322 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 2, MARCH 2008

where x ≥ 0. That is, the conditional pdf Prml(rml|y) can be ex-

pressed as

Prml(rml|y)=

∞∫0

Prml(rml|x)Px (x|y) dx

=1

2[ε(1−λ2)σ2

s1+σ2

w

] exp

{− rml+y2

2[ε(1−λ2)σ2

s1+σ2

w

]}

× I0

[ √rml · y

ε(1 − λ2)σ2s1

+ σ2w

]. (A6)

Recall that Py(y) is also a Rician distribution and can be ex-pressed as

Py(y) =y

12σ2

s1[1 + λ2ε + 2λ

√ε cos(φs3)]

× exp

{− y2 + a2

J

σ2s1

[1 + λ2ε + 2λ√

ε cos(φs3)]

}

× I0

{y · aJ

12σ2

s1[1 + λ2ε + 2λ

√ε cos(φs3)]

}. (A7)

Thereby, Prml(rml|S) can be expressed as

Prml(rml|S) =

∞∫0

Prml(rml|y)Py(y)dy

=1

2σ2g

exp

(−rml + a2

J

2σ2g

)I0

(√rml · aJ

σ2g

)U(rml).

(A8)

Applying the same technique, we can also gain Prml(rml|NS) as

Prml(rml|NS) =

1

2σ2w

exp

(−rml + a2

J

2σ2w

)

× I0

(√rml · aJ

σ2w

)U(rml). (A9)

REFERENCES

[1] K. C. Teh, A. C. Kot, and K. H. Li, “Performance analysis of anFFH/BFSK product-combining receiver with multitone jamming overRician-fading channels,” in Proc. IEEE Veh. Technol. Conf., May 2000,vol. 2, pp. 1508–1512.

[2] K. C. Teh, A. C. Kot, and K. H. Li, “Error probabilities of an FFH/BFSKself-normalizing receiver in a Rician fading channel with multitone jam-ming,” IEEE Trans. Commun., vol. 48, no. 2, pp. 308–315, Feb. 2000.

[3] Y. S. Shen and S. L. Su, “Performance analysis of an FFH/BFSK receiverwith ratio-statistic combining in a fading channel with multitone interfer-ence,” IEEE Trans. Commun., vol. 51, no. 10, pp. 1643–1648, Oct. 2003.

[4] K. C. Teh, A. C. Kot, and K. H. Li, “Performance study of a maximum-likelihood receiver for FFH/BFSK systems with multitone jamming,”IEEE Trans. Commun., vol. 47, no. 5, pp. 766–772, May 1999.

[5] Y. Han and K. C. Teh, “Performance study of suboptimum maximum-likelihood receivers for FFH/MFSK systems with multitone jamming overfading channels,” IEEE Trans. Veh. Technol., vol. 54, no. 1, pp. 82–90,Jan. 2005.

[6] G. Li, Q. Wang, V. K. Bhargava, and L. J. Mason, “Maximum-likelihooddiversity combining in partial-band noise,” IEEE Trans. Commun.,vol. 46, no. 12, pp. 1569–1574, Dec. 1998.

[7] S. G. Glisic, Adaptive WCDMA Theory and Practice. Hoboken, NJ:Wiley, 2003.

[8] N. C. Beaulieu, “Generation of correlated Rayleigh fading envelopes,”IEEE Commun. Lett., vol. 3, no. 6, pp. 172–174, Jun. 1999.

[9] R. N. McDonough and A. D. Whalen, Detection of Signals in Noise,2nd ed. San Diego, CA: Academic, 1995.

QAM Symbol Error Rate in OFDM SystemsOver Frequency-Selective Fast Ricean-Fading Channels

Rainfield Y. Yen, Hong-Yu Liu, and Wei K. Tsai

Abstract—For digital data transmission using an orthogonal frequency-division multiplexing (OFDM) over fading channels, the interchannel-interference (ICI) term caused by the Doppler spread is correlated withthe desired signal term. Nonetheless, many researchers have assumed theICI term to be uncorrelated with the desired signal term. In this paper,we present the symbol-error-rate (SER) performance for the M-quadraticamplitude-modulation OFDM systems over Ricean fading using the exactcorrelated ICI model. The results are then compared with the uncorrelatedICI results. We find that the two results present some differences. Thedifferences tend to increase as the Ricean factor is increased.

Index Terms—Frequency-selective fading channels, interchannelinterference (ICI), orthogonal frequency-division multiplexing (OFDM),Ricean-fading channels.

I. INTRODUCTION

Orthogonal frequency-division multiplexing (OFDM) employs par-allel transmission of data carried by orthogonal subcarriers over over-lapping subbands to avoid high-speed equalization, to combat impulsenoise, to mitigate multipath distortion, and to fully use the availablebandwidth [1]. Many error-rate performance studies of the OFDMsystems can be found in the literature [2]–[7]. The complexity ofthe error-rate analysis depends on the employed digital-modulationmethod and the used channel-fading model. By far, the most commondata-modulation method employed for the OFDM is either M -phase-shift keying (PSK) or M -quadratic amplitude modulation [8], [9].Popular fading models include Rayleigh [2]–[4], Ricean [5], [6], andNakagami-m [7]. Fading can be frequency-selective and nonselective.A closed-form error-rate expression may or may not be obtainable. Forcomplicated cases where closed-form error rates are difficult to obtain,one must resort to numerical evaluations. Coding and interleavingare usually used in the OFDM to improve performance and mitigateburst errors caused by deep fades [1], [10]. However, different codingschemes call for different formulations. Thus, many error-rate analysesin the literature do not consider coding.

It is well known that the OFDM system will lose subcarrier orthogo-nality when the Doppler spread is a significant portion of the subcarrierspacing [11]. A destruction of the subcarrier orthogonality will resultin intercarrier or interchannel interference (ICI) (other than the ICIcaused by timing and frequency offsets [12], [13]), thus degradingerror-rate performance. Strictly speaking, the ICI is correlated withthe desired signal term. However, in analyzing quadratic amplitude-modulation (QAM) performance in uncoded OFDM systems overfrequency-selective Rayleigh-fading channels under the influence ofthe Doppler spread, Wang et al. [2] used a linear approximation forthe time-varying subchannel frequency responses to show that the ICI

Manuscript received December 18, 2003; revised October 23, 2006,May 28, 2007, and June 12, 2007. The review of this paper was coordinated byDr. W. Zhuang.

R. Y. Yen is with the Department of Electrical Engineering, TamkangUniversity, Taipei 25137, Taiwan, R.O.C. (e-mail: rainfieldy@yahoo.com).

H.-Y. Liu is with the Department of Computer and CommunicationEngineering, Dahan Institute of Technology, Hualien 97145, Taiwan, R.O.C.(e-mail: hongyu.liu@msa.hinet.net).

W. K. Tsai is with the Department of Electrical and Computer Engineering,University of California—Irvine, Irvine, CA 92697 USA (e-mail: wtsai@uci.edu).

Digital Object Identifier 10.1109/TVT.2007.906371

0018-9545/$25.00 © 2008 IEEE

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 2, MARCH 2008 1323

can be modeled as uncorrelated with the desired signal term. Theuncorrelated ICI model has also been used in [3] for binary PSK andin [4] for 16-QAM over the Rayleigh-fading channels. The OFDMperformance in Ricean and Nakagami-m can be found in the papersof Glavieux et al. [5], Lu et al. [6], and Du et al. [7]. However, theyonly considered slow fading (the Doppler spread is not significant, andfading remains constant over at least one OFDM block), and hence,the ICI was not included.

In this paper, we shall analyze symbol-error-rate (SER) performancein uncoded OFDM systems employing M -QAM over frequency-selective Ricean fading. Unlike the previously mentioned studies, ourcontribution is to use the more exact correlated ICI model for fast-fading channels (fading rate is governed by the Doppler spread ormaximum Doppler frequency fM = (υ/c)fc, where υ is the mobilespeed, c is the speed of light, and fc is the carrier frequency). Then,we will show the discrepancies in the results between the correlatedand uncorrelated ICI models.

This paper is organized as follows. Section II describes the OFDMsignal and the frequency-selective Ricean-fading-channel model.Section III analyzes the QAM SER performance of OFDM in fastfrequency-selective Ricean fading using the exact correlated ICImodel. Then, Section IV presents the simulation results. Comparisonsare made between the correlated and uncorrelated ICI models. Finally,Section V draws conclusions.

II. OFDM SIGNAL AND CHANNEL MODEL

For an OFDM system, the N complex data symbols Xk over atime interval T constitute a data block (or an OFDM symbol block),where k = 0, 1, . . . , N − 1. Thus, each data symbol occupies a sym-bol interval ∆t = T/N . The signal bandwidth is 1/∆t. The data aretransmitted one block at a time (block rate = 1/T , and symbol rate =1/∆t). Before a block is transmitted, the N symbols {Xk} in thatblock are first passed through an N -point inverse discrete Fouriertransformer to produce N parallel complex outputs given as

xn =1

N

N−1∑k=0

Xkej2πnk/N , n = 0, 1, . . . , N − 1. (1)

The parallel {xn}’s are then converted to a serial sequence overthe block by a parallel-to-serial converter. Next, the serial sequenceis transformed into an analog form by analog-to-digital (A/D) con-version. The A/D conversion is equivalent to letting fk = k/T , andt = n∆t = nT/N . The analog form is

x(t) =1

N

N−1∑k=0

Xkej2πfkt, 0 < t < T. (2)

Equation (2) only represents the first block. If the ith block is spokenof, we should replace t by t − iT , and the range of t should beiT < t < (i + 1)T . In view of (2), x(t) can be viewed as a sum ofcomplex subcarriers at frequencies {fk = k/T} with amplitudes Xk,k = 0, 1, . . . , N − 1; each subcarrier occupies a subband of width∆f = 1/T . With this subband spacing, it is readily shown that allsubcarriers are orthogonal to one another. Now, the complex signalx(t) is frequency-upshifted to fc for transmission in the channelpassband. In other words, the transmitted passband signal is of theform s(t) = Re[x(t)ej2πfct], where x(t) is simply the equivalentlow-pass signal [14].

For the channel, we shall assume a frequency-selective fadingchannel whose equivalent low-pass discrete-time channel response can

be modeled by a time-varying tapped delay line [14], i.e.,

Hk =

ν−1∑m=0

hme−j2πmk/N , k = 0, 1, . . . , N − 1 (3)

where hm is the discrete-time baseband-channel impulse response ortap gain, and ν is the channel dispersion length. For most fading media,{hm} can be assumed spatially uncorrelated [14], i.e., E[(hm −hm)(hm′ − hm′)] = 0, m �= m′, where E[·] denotes expectation,and hm = E[hm]. For the fading process, we use the Ricean-fadingmodel adopted by the International Telecommunication Union [15].The {hm}’s are the uncorrelated complex Gaussian random variables(RVs). The first tap gain h0 corresponding to the shortest path delaycontains scattered paths plus a specular or line-of-sight component.Thus, h0 is a nonzero-mean Gaussian RV, and the amplitude |h0| isa Ricean RV. The remaining {hm,m �= 0} are zero-mean GaussianRVs, and {|hm|,m �= 0} are Rayleigh RVs. By virtue of (3), weeasily find that Hk is a complex Gaussian RV with mean E[Hk] = h0

and variance V [Hk] =∑ν−1

m=0σ2

m − |h0|2, where σ2m = V [hm] is the

variance of hm.

III. SER ANALYSIS

Suppose we append a cyclic prefix of length G ≥ ν to each block;then, the transmitted signal sequence of (1) is given as

xn =1

N

N−1∑k=0

Xkej2πnk/N , −G ≤ n ≤ N − 1 (4)

where the prefix symbols are xn = xN+n (where n = −G, . . . ,−1).Then, after frequency downshift, A/D conversion, and removal of theprefixed symbols at the receiver, the noiseless received discrete-timesignal is given by

yn =

ν−1∑m=0

hm(n)xn−m, n = 0, 1, . . . , N − 1. (5)

Note that we have now used hm(n) to include a time index n toaccount for the time-varying response due to fast fading. Also, after theremoval of the prefix, only N samples of yn are taken. The convolutionsum in (5) is circular, i.e., the index n − m for xn−m is of modulo N .

Then, the serial sequence of yn is converted to a parallel form andis discrete-Fourier-transformed to yield

Yk =

N−1∑n=0

yne−j2πnk/N , k = 0, 1, . . . , N − 1. (6)

Using (4) and (5), we can rearrange (6) as

Yk =

N−1∑n=0

ν−1∑m=0

hm(n)xn−me−j2πnk/N

=1

N

N−1∑n=0

Hk(n)Xk +1

N

N−1∑n=0

N−1∑p=0p�=k

Hp(n)Xpej2πn(p−k)/N .

(7)

The first term in (7) is the desired signal term, and the second termis the ICI. N is usually large such that the ICI term of (7) is a sum ofmany independent RVs. We may thus invoke the central limit theoryto assume that the ICI term can be approximated by a Gaussian RV[16]. For a fixed channel realization (all subband responses are held

1324 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 2, MARCH 2008

constant), the desired and random ICI terms of (7), respectively, havethe power given as

PS,k =σ2

X

N2

N−1∑n=0

N−1∑l=0

Hk(n)H∗k (l) (8)

PICI,k =σ2

X

N2

N−1∑p=0p�=k

N−1∑n=0

N−1∑l=0

Hp(n)H∗p (l)ej2π(n−l)(p−k)/N . (9)

Then, taking into account an additive white Gaussian-noise (AWGN)term Zk with zero mean and variance σ2

Z , the sum of ICI and noiseis still Gaussian, having the variance PICI,k + σ2

Z . Thus, for a fixedchannel realization, the received signal-to-interference-plus-noise ratio(SINR) is

γk =PS,k

PICI,k + σ2Z

. (10)

Next, we remove the condition of fixed channel realization and con-sider the subband responses Hk(n) and Hp(n) in (8) and (9) as RVs,thus treating the γk of (10) as an RV. The M -QAM SER for thekth subband is obtained by averaging the M -QAM SER expressionconditioned on a given γk under AWGN over all γk’s [14], [17] as

Pk =

∞∫0

{1−

[1−

2(√

M−1)

√M

Q(√

2gγk

)]2}p(γk)dγk (11)

where Q(x) is the Gaussian tail function, p(γk) is the probabilitydensity function (pdf) of γk, and g = 3/(2(M − 1)) [17]. For fastfading, this γk has a very complicated pdf not only because of the quo-tient form given by (10) but also because of the correlations betweensubband responses. Equation (11) can numerically be evaluated. Bynormalizing the signal power σ2

X = 1 and the channel power |h0|2 +∑ν−1

m=0σ2

m = 1, using the modified Jakes model [18] for Rayleighgains h′

0 = h0 − h0 and {hm, m = 1, 2, . . . , v − 1}, and using (3),we can obtain a sufficient number of the RVs {γk} given by (10)for any given Doppler spread fM . We then numerically perform theaveraging of (11) over {γk} to get Pk. The overall SER is given by

PM =1

N

N−1∑k=0

Pk. (12)

Such a simulation approach has also been adopted by Monsen [19]for single-carrier equalizers over slow-fading channels and byFalconer et al. [20], [21] for frequency-domain decision-feedbackequalizers for single-carrier OFDM systems over fading channels.

IV. NUMERICAL EXAMPLES

By using 16-QAM in OFDM with N = 16 at a fixed SNR =σ2

X/σ2n = 28 dB, Fig. 1 shows the simulation results of the overall

SER PM versus the averaged received SINR γ = (1/N)∑N−1

k=0γk

in fast frequency-selective Ricean-fading channels for various Riceanfactors KR = |h0|2/σ2

0 (KR = −∞ corresponds to Rayleigh fading),where the kth subband average SINR γk (exact) is defined as

γk = E

[PS,k

PICI,k + σ2Z

]. (13)

Note that γk is governed by the Doppler spread. As fM is increased,γk will decrease. We have assumed an exponentially decayed channel-power profile with ν = 4. For a given Doppler spread fM (given the

Fig. 1. SER versus averaged received SINR. SNR = σ2X/σ2

Z = 28 dB,16-QAM OFDM with N = 16, exponential power profile with v = 4.Curve (1): KR = −∞ dB, exact SINR. Curve (2): KR = −∞ dB,approximate SINR. Curve (3): KR = 11 dB, exact SINR. Curve (4):KR = 11 dB, approximate SINR. Curve (5): KR = 19 dB, exact SINR.Curve (6): KR = 19 dB, approximate SINR.

γk), over 50 000 channel realizations have been used for averaging (in[21], 20 000 channel realizations are used). The performance curves ofSER versus the approximate average SINR using the uncorrelated ICIare also shown in Fig. 1. The approximate average SINR γ is definedas (adopted in [3] and [4])

γ =1

N

N−1∑k=0

E[PS,k]

E[PICI,k] + σ2Z

. (14)

Using (8) and (9), it can readily be shown that

E[PS] =E[PS,k]

=σ2X |h0|2+

σ2X

N2

[ν−1∑m=0

σ2M

][N+2

N−1∑i=1

(N−i)ρi

](15)

E[PICI] =E[PICI,k]

=σ2X

ν−1∑m=0

σ2M

{1− 1

N2

[N+2

N−1∑i=1

(N − i)ρi

]}(16)

where the temporal correlation coefficient ρi = E[h′0(n)h′∗

0 (n −i)]/σ2

0 or ρi = E[hm(n)h∗m(n − i)]/σ2

m, with m = 1, 2, . . . , ν, isgiven by the classical expression for Rayleigh fading as [22]

ρi = J0(2πfMTi/N). (17)

Comparing the SER performances of the uncorrelated ICI modelwith those of the more exact correlated model, we can see that thereexists a gap (discrepancy) between the correlated and uncorrelatedresults. As KR is increased, the gap becomes wider. The gap issmallest for KR = −∞ (Rayleigh fading). The following two extremepoints need be noted: 1) As the Doppler spread approaches zero,ρi → 1; thus, the ICI approaches zero, and SINR = SNR = 28 dB.This accounts for the right gap closing at SINR = 28 dB. 2) Asthe Doppler spread approaches infinity, fading becomes so fast thatρi → 0; thus, the ICI approaches a constant. This accounts forthe left gap closing. For the case of KR = −∞, we also show

IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 2, MARCH 2008 1325

Fig. 2. Comparisons between different averaged received SIRs for Rayleigh

fading. Curve (1): Averaged received SIR = (1/N)∑N−1

k=0E[PS,k/PICI,k].

Curve (2): Averaged received SIR = E[PS]/E[PICI] = E[PS,k]/E[PICI,k].Curve (3): Average of [2, eq. (25)].

Fig. 3. SER versus SNR for various Doppler spreads. KR = 8 dB. 16-QAMOFDM with N = 16.

in Fig. 2 the three curves of average signal-to-ICI ratio (SIR) (orcarrier-to-ICI ratio) versus fM/∆f = fMT (the normalized Dopplerspread) corresponding to SIR = (1/N)

∑N−1

k=0E[PS,k/PICI,k],

SIR = E[PS]/E[PICI] = E[PS,k]/E[PICI,k], and the SIR givenby [2, eq. (25)]. We see that, for the Rayleigh-fading case, thesethree curves are quite close to each other. We conclude that theapproximation of the uncorrelated ICI model can only be used forthe Rayleigh fading but not for the Ricean fading with large specularcomponents (large KR). Next, we show in Fig. 3 the performancecurves of SER versus SNR = σ2

X/σ2Z with a fixed KR = 8 dB for

various Doppler spreads. Here, we see that the SER curves level offat large SNR. This is because, when the AWGN approaches zero, thetotal noise becomes dominantly ICI, and hence, the SER is no longeraffected by the SNR. Instead, the performance curves are seen toget lower as the Doppler spread is increased (γk is decreased). Suchsimilar situations are also observed in [2].

V. CONCLUSION

We present the QAM SER in an uncoded OFDM system overfrequency-selective Ricean-fading channels under the influence ofDoppler spread. We have used the exact model where the ICI causedby the Doppler spread is correlated with the desired signal term. TheSER results using the approximated uncorrelated ICI model are alsopresented for comparison with the exact model. We conclude that theapproximation of the uncorrelated ICI model can only be used for thecase of KR = −∞ (Rayleigh fading) but not for large KR.

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[5] A. Glavieux, P. Y. Cochet, and A. Picart, “Orthogonal frequency divi-sion multiplexing with BFSK modulation in frequency selective Rayleighand Rician fading channels,” IEEE Trans. Commun., vol. 42, no. 2/3/4,pp. 1919–1928, Feb.–Apr. 1994.

[6] J. Lu, T. T. Tihung, F. Adachi, and C. L. Huang, “BER performance ofOFDM–MDPSK system in frequency-selective Rician fading with diver-sity reception,” IEEE Trans. Veh. Technol., vol. 49, no. 4, pp. 1216–1225,Jul. 2000.

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[8] R. Van Nee and R. Prasad, OFDMWireless Multimedia Communications.Boston, MA: Artech House, 2000.

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