On the Effect of Wiener Phase Noise in OFDM Systems.pdf

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580 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 5, MAY 1998

On the Effect of Wiener Phase Noise in OFDM SystemsLuciano Tomba, Member, IEEE

Abstract Multicarrier modulation exhibits a significant sen-sitivity to the phase noise of the oscillator used for frequency

down-conversion at the portable receiver. For this reason, itis important to evaluate the impact of the phase noise on thesystem performance. In this letter we present an accurate methodto determine the error probability of an orthogonal frequency-division multiplexing (OFDM) system in the presence of phasenoise. In particular, four modulation schemes are analyzed andtheir performances are compared.

Index Terms OFDM, phase noise.

I. INTRODUCTION

ORTHOGONAL frequency-division multiplexing

(OFDM) has been suggested in a recent paper [1]

for the transmission of asynchronous transfer mode (ATM)

data packets over indoor radio channels. The main advantage

of OFDM is to split the available signal bandwidth into

a large number of independent narrow-band subchannels.

By selecting the subchannel bandwidth smaller than the

coherence bandwidth of the frequency-selective channel, each

subchannel experiences flat fading [2].

This work limits itself to consider the effect of Wiener phase

noise in OFDM systems. Some analysis of the effect of phase

noise in OFDM systems has been presented in [3]. However,

in [3] the authors do not derive exact error probability formulas

because they approximate the contribution of the phase noise

with an additional noise component. Their approach is accurate

enough in cases where the linewidthsymbol period productis sufficiently small (and the signal-to-noise ratio large) such

that the error rate degradation is mainly determined by the

intercarrier interference. Instead, the analysis presented in this

letter takes into account the exact statistic of the phase noise

to evaluate the error probability. Lastly, the analysis is used to

compare the performance of four modulation schemes [binary

phase-shift keying (BPSK), quarternary phase-shift keying

(QPSK), differential BPSK (DBPSK), and differential QPSK

(DQPSK)].

II. SYSTEM MODEL

Basically, an OFDM signal [2], [4] consists of the superpo-sition of sinusoidal subcarriers with frequency spacing ;

each subcarrier is modulated by symbols with period equal

to the inverse frequency spacing. The modulated subcarriers

Paper approved by O. Andrisano, the Editor for Modulation for FadingChannels of the IEEE Communications Society. Manuscript received February13, 1997; revised June 15, 1997 and January 15, 1998. This work wassupported in part by the Italian Research Ministry (MURST) and by theConsiglio Nazionale delle Ricerche (CNR), Rome, Italy.

The author is with the Dipartimento di Elettronica e Informatica, Universit adi Padova, 35131 Padova, Italy (e-mail: [email protected]).

Publisher Item Identifier S 0090-6778(98)03855-0.

overlap spectrally, but since they are orthogonal within a

symbol duration (the th carrier frequency is ,

where is some reference frequency and )

the signal associated with each sinusoid can be recovered

as long as the channel does not destroy their orthogonality.

In practice, the samples of the OFDM signal are generated

by taking the inverse discrete Fourier transform (IDFT) of a

discrete-time input sequence and passing the transform

samples through a pulse-shaping filter; at the receiver side,

dual transformations are implemented.

By assuming an additive white Gaussian noise (AWGN)

channel, after frequency down-conversion, the received signal

is given by [3], [4]

(1)

where ( is the imaginary unit),

is the additive noise, and is the multiplicative

error term due to the oscillator phase noise.1 As it is usual in

theoretical papers on the subject (e.g., [3] and [5]), to accountfor the phase noise is assumed to be a continuous-pathBrownian motion (or WienerLevy process) with zero mean

and variance (the parameter represents the two-sided

3-dB linewidth of Lorentzian power density spectrum of the

oscillator). The above mathematical model is accurate enough

for modeling the phase noise process and it is suitable to get

a quantitative measure of the system performance degradation

due to the phase noise by analytical methods. However, it mustbe noted that this approach is valid as long as the baseband

conversion of the OFDM signal is performed by a free-

running oscillator.2 Instead, when the baseband conversion is

performed by means of an oscillator which is properly locked

to the carrier of the received signal, the variance of the phase

error does not increase without limit and the numerical results

presented in this letter do not apply.

In the receive unit the signal is multiplied by tones

with the same frequency as in the transmitter and sent to the

input of integrate-and-dump (I&D) filters with integration

time . In conclusion, the signal at the output of the th I&D

filter is [3]

(2)

1 In principle, the phase noise of both of the oscillators (respectively, at thebase station and at the portable terminal) should be accounted for. However, inpractice, the oscillator used at the base station is sufficiently stable to disregardits phase noise.

2 The numerical results presented in this letter (and the formulas ofAppendixes A and B) have been obtained by assuming the initial condition .

00906778/98$10.00 1998 IEEE


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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 5, MAY 1998 581

where

(3)

and are samples of an additive Gaussian noise. In (2) the

information quantity is multiplied by which depends

on the phase noise, while it is independent of the particular

subcarrier index. The summation in (2), instead, represents the

interbin interference [3] which is zero when . Notethat this interference depends on data from all of the different

subchannels.

III. PERFORMANCE EVALUATION

To compare different modulation schemes, the performance

measure considered in this work is the probability of bit error

(BER). In particular we consider BPSK and QPSK, and their

differential versions (DBPSK and DQPSK, respectively). All

symbols are assumed independent and identically distributed.

The following analysis makes use of some basic results

presented in [5] and in [6] which are reported, for convenience

sake, in the appendixes.BPSK: In this case the decision variable for the th sub-carrier is given by

(4)

The statistic of is given in Appendix A. Nevertheless, it

is not easy to find the statistic of . For this reason, we

approximate by a complex Gaussian random variable (RV)

whose mean and variance are evaluated in Appendix B. By

using the Smirnov test, by simulation it can be verified that

the significance levels are, approximately, between 0.70.9. In

conclusion, is a zero-mean Gaussian RV withvariance and the BER is3

(5)

Each term between curved brackets of (5) can be evaluated

by using the moment generating function (MGF) of the RV

involved

(6)

In (6) is the MGF of conditional to ,

namely

(7)

where

(8)3

1 and 1 , as well as the subscripts and , stand for real andimaginary part, respectively.

The saddlepoint approximation [7] is a fast and accurate

procedure to calculate integrals like (6) and it is used in this

work.

QPSK: The decision variable is

(9)

To get the BER it is helpful to use the union bound[8]; hence,

we get

(10)

where denotes the decision region for the symbol .

Once again, use of the MGF yields

(11)

where

(12)

The analytical expression of is given by (17),

while is the same as with replaced by

. Finally, the BER is

(13)

DBPSK: One possible implementation of this modulation

scheme makes use of the samples with the same subcarrier

index in the previous discrete Fourier transform (DFT) block

as a phase reference. In the presence of phase noise this method

does not represent a good solution because the phase rotation

on the information data is not the same in different DFT

blocks. To overcome this problem, another technique consists

in performing detection of one symbol transmitted on a given

subcarrier using an adjacent subcarrier in the same DFT block

as [2]. Let be the differentially encoded data from the source

bits and ; the decision variable is ( stands for complex

conjugate)

(14)

From (14) it is apparent that the phase noise does not produce

any phase rotation on the product of the information symbols

and , but it modifies only its magnitude. The BER is

(15)

The decision variable involves RVs which are assumed

independent of each other. The error probability conditional

to a pair and is known [8]. The average with

respect to is simply obtained by using the method of the

Gaussian quadrature rules because the moments of can

be calculated using the MGF reported in Appendix A.


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582 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 5, MAY 1998

Fig. 1. BER of BPSK for the AWGN channel in the presence of phase noisefor different values of .

Fig. 2. BER of QPSK for the AWGN channel in the presence of phasenoise for different values of .

DQPSK: This modulation format differs from the above

DBPSK only in the size of the constellation. The formulas to

compute the error probability are a straightforward extension

of the one for QPSK and DBPSK; hence, they are omitted.

IV. COMMENTS ON THE RESULTS AND CONCLUSION

For each modulation scheme, the BER is shown in Figs.

14. Different curves correspond to different values of

which depends on the oscillator linewidth, on the number of

subcarriers, and on the symbol period. We have assumed ideal

removal of any phase and frequency offset at the receiver

side; hence, the transmitted symbol is impaired only by

the common phase noise, i.e., the equal multiplicative term

on each subcarrier.

Fig. 3. BER of DBPSK for the AWGN channel in the presence of phasenoise for different values of .

Fig. 4. BER of DQPSK for the AWGN channel in the presence of phasenoise for different values of .

It is apparent that the performance strongly degrades for

in the case of BPSK. As expected, even worse results

are obtained for QPSK. Fig. 3 shows the BER for DBPSK.

Although this modulation scheme is less efficient than BPSK,

it is robust enough against the common phase noise whencompared to BPSK (see Fig. 1). The performance degradation

is mainly due to the additional amount of additive noise

corresponding to the interbin interference . The performance

of DQPSK is better than QPSK but worse than DBPSK; hence,

DBPSK seems to be the most robust scheme against the phase

noise.

In conclusion, the methodology presented in this work is

general and accurate. Moreover, with proper changes, it can

be applied to different modulation schemes also in the pres-

ence of multipath fading channels (by using a semianalytical

approach).


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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 46, NO. 5, MAY 1998 583

APPENDIX A

MGF OF

Let

(16)

When [5], the exponential function in (16) can be

expanded in a power series as . The joint

MGF of the real and imaginary parts of (16) is given by [5]

(17)

where , , and .

APPENDIX B

MEAN AND VARIANCE OF

Because of the assumptions on the data , .

Hence

(18)

(19)

Moreover, by extending the results presented in [6], we get

(20)

where is the inverse Laplace transform of the

argument evaluated at .

ACKNOWLEDGMENT

The author would like to thank the anonymous reviewers

for their careful revision and important suggestions which

significantly helped to improve the presentation of this letter.

REFERENCES

[1] A. Chini, M. S. El-Tanany, and S. A. Mahmoud, Transmission of highrate ATM packets over indoor radio channels, IEEE J. Select. AreasCommun., vol. 14, pp. 460468, Apr. 1996.

[2] H. Sari, G. Karam, and I. Jeanclaude, Transmission techniques fordigital terrestrial TV broadcasting, IEEE Commun. Mag., vol. 33, pp.100109, Feb. 1995.

[3] T. Pollet, M. Van Bladel, and M. Moeneclaey, BER sensitivity of

OFDM systems to carrier frequency offset and Wiener phase noise,IEEE Trans. Commun., vol. 43, pp. 191193, Feb./Mar./Apr. 1995.[4] J. A. C. Bingham, Multicarrier modulation for data transmission: An

idea whose time has come, IEEE Commun. Mag., vol. 28, pp. 514,May 1990.

[5] G. J. Foschini and G. Vannucci, Characterizing filtered light wavescorrupted by phase noise, IEEE Trans. Inform. Theory, vol. 34, pp.14371448, Nov. 1988.

[6] G. L. Pierobon and L. Tomba, Moment characterization of phasenoise in coherent optical systems, J. Lightwave Technol., vol. 9, pp.9961005, Aug. 1991.

[7] C. W. Helstrom, Approximate evaluation of detection probabilities inradar and optical communications, IEEE Trans. Aerosp. Electron. Syst.,vol. AES-14, pp. 630640, July 1978.

[8] J. G. Proakis, Digital Communications, 2nd ed. New York: McGraw-Hill, 1989.

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On the Effect of Wiener Phase Noise in OFDM Systems.pdf