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3360 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 12, DECEMBER 2006

SNR Analysis of OFDM Systems in the

Presence of Carrier Frequency Offset for Fading Channels

Jungwon Lee,Member, IEEE, Hui-Ling Lou, Member, IEEE, Dimitris Toumpakaris, Member, IEEE,

and John M. Cioffi, Fellow, IEEE

Abstract This letter analyzes the effect of the carrier fre-quency offset on orthogonal frequency division multiplexing(OFDM) systems for multipath fading channels. A simple ap-proximate expression for the average signal-to-noise ratio (SNR)is derived. This approximate expression is shown to be an upperbound of the average SNR for flat fading channels and an exactexpression for the AWGN channel. The approximate averageSNR expression is validated using Monte Carlo simulation forboth flat fading channels and frequency-selective fading channels.

Index Terms carrier frequency offset, orthogonal frequency

division multiplexing (OFDM), synchronization, fading channel

I. INTRODUCTION

ORTHOGONAL Frequency Division Multiplexing

(OFDM) is an important modulation technique for

high-speed communications through frequency selective

channels [1]. It can easily remove inter-symbol interference

(ISI) and can be implemented using the computationally

efficient fast Fourier transform (FFT). Partly because of

these advantages, OFDM has been chosen as the modulation

scheme of various wireless and wireline communication

systems such as wireless LAN, digital audio broadcasting

(DAB), digital video broadcasting (DVB), and digital

subscriber lines (DSL).Although OFDM has many advantages [1], it is well known

that it is susceptible to carrier frequency offset, which results

from the Doppler shift or from the mismatch between the

oscillator frequencies of the transmitter and the receiver.

Various carrier frequency synchronization methods have been

developed [2], [3], and the effect of the carrier frequency

offset has been analyzed quite extensively [4][11]. In [4][6],

the signal-to-noise ratio (SNR) degradation due to the carrier

frequency offset was derived for the additive white Gaussian

noise (AWGN) channel. SNR has also been analyzed for atime-invariant multipath channel [7], a shadowed multipath

channel [8], and a general multipath fading channel [9].

Moreover, in [10], [11], symbol error rate (SER) expressions

have been derived.This letter analyzes the effect of the carrier frequency offset

on the average SNR for general multipath fading channels.

Manuscript received January 17, 2004; revised September 6, 2005; acceptedDecember 8, 2005. The associate editor coordinating the review of this letterand approving it for publication was A. Conti. Part of this work was presentedat IEEE GLOBECOM, Nov. 29 - Dec.3, 2004, Dallas, Texas.

J. Lee, H. Lou, and D. Toumpakaris are with Marvell Semiconduc-tor, Inc., 5488 Marvell Lane, Santa Clara, CA 95054 (e-mail: jung-

[email protected],{hlou,dimitris}@marvell.com).J. Cioffi is with the Department of Electrical Engineering at Stan-ford University, 350 Serra Mall, Room 363, Stanford, CA 94305 (e-mail:[email protected]).

Digital Object Identifier 10.1109/TWC.2006.04867.

DEMUX

MUX

IFFT

FFT

ym[N +Ng - 1]

ym[Ng - 1]

Ym[N- 1]

Xm[N- 1]

Xm[1]

Xm[0]

xm[0]

xm[N +Ng - 1]

ym[0]

ym[n]

xm[n]

hm[n]

Ym[1]

Ym[0]

zm[n]

ym[n]

~

~

ej[2f[n + m(N+Ng)]T+ 0]

Fig. 1. Baseband equivalent model of an OFDM system.

An approximate average SNR expression is derived, and it is

compared with the lower bound of the approximate average

SNR in [9]. It is shown that this approximate average SNR

expression is an upper bound of the average SNR for flat

fading channels and equals the exact expression for the AWGN

channel. The approximate average SNR expression is simple

enough to demonstrate clearly the relationship between the

SNR and various system parameters in the presence of carrier

frequency offset.

I I . SYSTEM M ODEL

An OFDM system transmits information as a series of

OFDM symbols [1]. As is shown in the baseband equivalent

model of Fig. 1, the inverse fast Fourier transform (IFFT) is

performed on the transmit symbolsXm[k],k = 0, 1, , N

1, to produce the time-domain samples xm[n] of the m-thOFDM symbol:

xm[n] =

1N

N1k=0 Xm[k]e

j2k(nNg)/N,for 0 n N+Ng 1

0, otherwise,(1)

where N andNg are the numbers of data samples and cyclicprefix samples, respectively.

The OFDM symbolxm[n] is transmitted through a channelhm[n] and is affected by additive zero-mean Gaussian noisezm[n]. The channelhm[n] is assumed to be block-stationary,i.e., time-invariant over each OFDM symbol. With this as-

sumption, the outputym[n] of the channel can be representedas follows:

ym[n] =hm[n] xm[n] + zm[n], (2)

1536-1276/06$20.00 c 2006 IEEE


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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 12, DECEMBER 2006 3361

where is the convolution operator, i.e., hm[n] xm[n]

r= hm[r]xm[n r], and zm[n] is white Gaussian noisewith variance 2

Z and independent of the transmit signal and

the channel. When the channel length is less thanNg+1, therewill be no inter-symbol interference, and hm[n] xm[n] =

Ngr=0 hm[r]xm[n r]. Although the channel taps can be cor-

related, in general, depending on the design of the transmit and

receive filter, it is assumed here that the transmit and receive

filters are ideal such that the channel taps are uncorrelated:

E[hm[n]h

m[p]] =[n p]E[|hm[n]|2]. (3)

When the oscillator of the receiver is not perfectly matched

to the carrier of the received signal, a carrier frequency offset

fand a phase offset0will appear. The frequency offsetfcan be represented with respect to the subcarrier bandwidth

1/NTby defining the normalized frequency offset as

fNT , (4)

where T is the sampling period. Using the normalized fre-quency offset , the received sampleym[n] is expressed as

ym[n] =N cm(, n)(hm[n] xm[n]) +zm[n]. (5)

where= NgN , zm[n] =N cm(, n)zm[n], and

cm(, n) 1

Nej2n/Nej2m(1+)ej0 . (6)

The noise zm[n] is a zero-mean complex Gaussian randomvariable with variance 2Z=

2Z

.

III. EFFECT OF C ARRIERF REQUENCYO FFSET

In this section, the effect of the carrier frequency offset is

analyzed based on the system model in Section II. The relative

frequency offset of (4) can be divided into an integer part land a fractional part such that 1/2 < 1/2:

= l+ . (7)

The effects of the integer frequency offset l and the fractionalfrequency offset are explained below.

The discrete Fourier transform (DFT) ofym[n] is equal to

Ym[k] = Cm(, k) (Hm[k]Xm[k]) +Zm[k] (8)

whereCm(, k),Hm[k], andZm[k]are the DFTs ofcm(, n),hm[n], and zm[n], respectively. The DFT ofcm(, n) can beexpressed as

Cm(, k) =

sin(( k))

Nsin(( k)/N)ej(k)(11/N)

ej[2m(1+)+0], (9)

and the magnitude ofCm(, k) is always less than or equal to1 since

|Cm(, k)| N1n=0

cm(, n)ej2nk/N = 1 (10)

Assuming that Hm[k] and Xm[k] are periodic with periodN for notational convenience, the received symbol at the(k +l)-th subcarrier can be shown to be

Ym[k+l] = Cm(, 0)Hm[k]Xm[k]

+N1

r=1

Cm(, r)Hm[k r]Xm[k r]

+Zm[k+l], (11)

From (11), it can be seen that the desired signal Hm[k]Xm[k]is affected by the frequency offset in the following ways:

The desired signalHm[k]Xm[k]is received by the(k+l)-th subcarrier instead of the k-th subcarrier.

The magnitude of Hm[k]Xm[k] is attenuated by

|Cm(, 0)|= sin()Nsin(/N) .

The phase ofHm[k]Xm[k]is increased by(11/N)+2m(1 +) +0.

The signal Hm[k]Xm[k] is subject to ICI in addition to

the background noise.

As a result, the carrier frequency offset reduces the SNR

and increases the SER. Since the integer carrier frequency

offset results in the cyclic shift of the subcarriers, it should be

corrected perfectly for the proper operation of the receiver.

Thus, in the following, the effect of the fractional carrier

frequency offset on the SNR is analyzed assuming that there

is no integer carrier frequency offset.

IV. SNR ANALYSIS

In this section, the effect of the carrier frequency offset

on the SNR is analyzed for multipath fading channels. Theaverage SNR provides useful information about the receiver

performance when appropriate coding and interleaving is used.

Thus, an approximate average SNR expression is derived for

multipath fading channels and is examined for the special case

of flat fading channels and the additive white Gaussian noise

(AWGN) channel. For the sake of notational simplicity, the

OFDM symbol numberm is omitted in this section.

A. Multipath Fading Channel

From (11) and under the assumption ofl = 0, the received

signal in the presence of carrier frequency offset is

Y[k] =C(, 0)H[k]X[k] +I[k] +Z[k], (12)

where

I[k] =N1r=1

C(, r)H[k r]X[k r]. (13)

A coherent receiver should be able to estimate the phase of

C(, 0)H[k] in order to decode the received symbol Y[k]correctly, as can be seen from (12). Let C(, 0)H[k] =|C(, 0)H[k]|ej[k], and assume that the estimate of the phase[k] is perfect. Then, the decision metric M[k] for coherentdemodulation is obtained by multiplying the received symbol

Y[k] by ej[k]:

M[k] =ej[k]Y[k] =|C(, 0)H[k]|X[k] +I[k] +Z[k],(14)


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where the ICI I[k] is

I[k] =ej[k]N1r=1

C(, r)H[k r]X[k r] (15)

and the noise Z[k] is

Z

[k] =ej[k]

Z[k]. (16)

Since the ICII[k]is a linear combination of the X[r]for r =k, the ICI is uncorrelated with the signal |C(, 0)H[k]|X[k]when the data symbols in each subcarrier are uncorrelated with

those in other subcarriers.

Then, for a given channel gain H[k] for k = 0, , N1, the SNR of the k-th subcarrier in the presence of carrierfrequency offset is

SNR(k) =E[|C(, 0)H[k]X[k]|2]

E[|I[k] +Z[k]|2] . (17)

The power of the desired signal is equal to

E[|C(, 0)H[k]X[k]|2] =|C(, 0)H[k]|22X , (18)

where 2X is the power of the transmit signal. Since thebackground noise has zero mean and is independent of the

ICI,

E[|I[k] +Z[k]|2] =E[|I[k]|2] +E[|Z[k]|2]. (19)

The power of the ICI I[k] can be expressed as

E[|I[k]|2]

=N1r=1

N1s=1

C(, r)C(, s)H[k r]H[k s]

E[X[k r]X[k s]]

=

N1r=1

|C(, r)H[k r]|22X . (20)

The (k r)-th subcarrier contributes|C(, r)H[r]|22X to thepower of the ICI that affects the k-th subcarrier. It can beshown from (9) that

|C(, 1)| |C(, 1)| |C(, 2)| |C(, 2)|

|C(,N/2)| |C(, N/2)| (21)

for 0 12 . Thus, the subcarriers close to the k-thsubcarrier contribute more to the ICI I[k]than those far fromthek-th subcarrier when the received signal power is the samefor all subcarriers in the absence of a carrier frequency offset.

Since Z[k] is a rotated version ofZ[k], E[|Z[k]|2] =2Z.Thus, the instantaneous SNR of the k-th subcarrier for a

given channel is equal to

SNR(k) = |C(, 0)H[k]|22X

N1r=1 |C(, r)|

2|H[k r]|22X+2Z

(22)

The average SNR in the presence of carrier frequency offsetcan be calculated by averaging the above instantaneous SNR

expression over the distribution of the channel gains. However,

the calculation of the average SNR using multiple integration

is too complex. Thus, in this letter, an approximate average

SNR expression is derived by taking the average of the

numerator and the denominator of (22) separately:

SNR(k) E

|C(, 0)H[k]|2

2X

EN1

r=1 |C(, r)|2|H[k r]|22X +

2Z

, (23)where the expectation is over the channel gains h[n]. Since

it was assumed that the channel response h[n] for delay nis uncorrelated with the response h[p] for delay p = n,E[|H[k]|2] is

E[|H[k]|2] = E

Ngn=0

Ngp=0

h[n]h[p]ej2(np)k/N

=

Ngn=0

E[|h[n]|2] (24)

for any k. By defining

Ngn=0E[|h[n]|

2] = , the numeratorof (23) can be written as

E[|C(, 0)H[k]X[k]|2] =|C(, 0)|22X , (25)

whereas the denominator is equal to

E

N1r=1

|C(, r)|2|H[k r]|22X+ 2Z

=N1r=1

|C(, r)|22X+ 2Z

= (1 |C(, 0)|2)2X+ 2Z, (26)

where the following fact is used

N1r=0

|C(, r)|2 =NN1n=0

|c(, n)|2 = 1, (27)

Then the approximate average SNR of the k-th subcarrierfor carrier frequency offset is

SNRapprox(k) = f2N()SNR0

(1 f2N())SNR0+ 1, (28)

where SNR0 is the average SNR in the absence of a carrier

frequency offset, i.e., SNR0 = 2X2Z

, and

fN() |C(, 0)|= sin()Nsin(/N) (29)

from (9). The approximate average SNR decreases as the

normalized carrier frequency offset increases from0 to1/2because fN() is a monotonically decreasing function of for 0 < 1/2 1. Moreover, the SNR also depends on thenumber Nof subcarriers because fN() depends on N.

In [9], a lower bound of the SNR was claimed to be

derived for multipath fading channels. However, the SNR in

[9] was defined as the average signal power divided by the

average noise and interference power, which corresponds to

the approximate average SNR for multipath fading channels in

1fN

() = (Ncos() sin(

N)sin() cos(

N))

N2 sin2(N

) 0, which is equivalent

to Ntan(N

) tan(). The function h(x) =x tan(x

) is an increasingfunction for x 1 since h(x) = tan(

x) + x

cos2(x

) 0 for x 1.

Hence, h(N) h(1) and fN

() 0.


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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 5, NO. 12, DECEMBER 2006 3363

this letter. Thus, strictly speaking, the lower bound expression

in [9] provides a lower bound for the approximate average

SNR (28):

SNRapprox,LB(k) = sinc2()SNR0

0.5947 sin2()SNR0+ 1. (30)

The accuracy of this lower bound is evaluated numerically in

Section V.

B. Special case of flat fading and AWGN channels

For the special case of flat fading channels, the channel gain

is the same for all subcarriers, i.e. H[k] = h[0]. From (22),the instantaneous SNR for a given channel gain h[0] is

SNR(, ) = f2N()

2X

(1 f2N())2X+

2Z

, (31)

for all subcarriers, where = |h[0]|2 is the instantaneouschannel power. Then the average SNR can be calculated by

SNR() =

0

f2N()w2X

(1 f2N())w2X+

2Z

p(w)dw, (32)

wherep(w)is the probability density function of the squaredmagnitude of the channel gain. It is interesting to see that

the approximate average SNR expression (28) is an upper

bound of the average SNR for flat fading channels. By taking

the derivative of (31) with respect to , it can be seen thatSNR(, ) is a concave function offor a given . Then, byJensens inequality, E[SNR(, )] SNR(, E[]). Thus,

SNR() f2N()SNR0

(1 f2N())SNR0+ 1, (33)

where SNR0 = 2X2Z

. The tightness of this upper bound

depends on the actual distribution of the channel gain and

is evaluated by simulation in Section V.

When the channel is the AWGN channel, the approximate

average SNR expression (28) becomes equal to the exact SNR.

The SNR in the presence of carrier frequency offset for the

AWGN channel is

SNR() = f2N()SNR0

(1 f2N())SNR0+ 1, (34)

where SNR0 = 2

X

2Z

.

V. SIMULATIONR ESULTS

In this section, Monte Carlo simulation is performed to

evaluate the average SNR in the presence of carrier frequency

offset. The numberN of subcarriers is chosen as 64.Fig. 2 is a plot of the average SNR for flat fading channels in

the presence of carrier frequency offset. Rayleigh and Rician

flat fading channels are considered, and the approximate

average SNR using (28) is also plotted. The K-factor of the

Rician fading channel is chosen as 3. As can be seen from the

figure, the SNR degradation increases as the carrier frequencyoffset increases and as the nominal SNR increases for both

fading channels. It can also be seen that the Rayleigh fading

channel exhibits larger SNR degradation than the Rician

fading channel. However, the difference is not significant, and

0 5 10 15 20 25 305

0

5

10

15

20

25

Average SNR in the absence of carrier frequency offset (dB)

AverageSNRi

nthepresence

ofcarrierfrequencyoffset(dB)

= 0.05, Approximate SNR

= 0.05, Simulation, Rayleigh, Frequency Selective

= 0.05, Simulation, Rician, Frequency Selective

= 0.05, Lower Bound of Approximate SNR





Fig. 2. Average SNR in the presence of carrier frequency offset for flatfading channels: Rayleigh and Rician fading channels are considered.

0 5 10 15 20 25 305

0

5

10

15

20

25

Average SNR in the absence of carrier frequency offset (dB)

AverageSNRi

nthepresenceofcarrierfrequencyoffset(dB)









Fig. 3. Average SNR in the presence of carrier frequency offset forfrequency-selective fading channels: Rayleigh and Rician fading channels areconsidered.

the average SNR for both fading channels matches quite well

with the approximate average SNR, which is an upper bound

of the average SNR in this flat-fading channel case.Fig. 3 shows the average SNR in the presence of car-

rier frequency offset for multipath frequency-selective fading

channels. An exponential power delay profile is used with

a root-mean-square (RMS) delay spread of 3 taps. For the

case of the Rician fading channel, the first tap of the channel

follows the Rician distribution with K-factor 3, while the

other taps follow the Rayleigh distribution. Compared to the

Rayleigh flat fading channel, the Rayleigh frequency-selective

fading channel shows slightly larger SNR degradation at

high nominal SNR. The Rician frequency-selective fading

channel has almost the same SNR degradation as the Rayleigh

frequency-selective fading channel. Although some differencein the SNR degradation can be found depending on the

channel, Figs. 2 and 3 indicate that the approximate average

SNR is quite close to the actual average SNR for all channel

scenarios considered in this letter. The lower bound of the


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approximate average SNR is also plotted, and it also predicts

the average SNR quite well, especially for small SNR.

V I . CONCLUSION

This letter presented an analysis of the effect of the carrier

frequency offset in OFDM systems for multipath fading chan-

nels. The frequency offset attenuates the desired signal and

causes inter-carrier interference, thus reducing the SNR. Theaverage SNR is approximated using a simple expression for

multipath fading channels. The approximate average expres-

sion was shown to be an upper bound for the average SNR for

the case of flat fading channels and to be an exact expression

for the AWGN channel. From the approximate average SNR

expression, it was found that the SNR degradation increases

monotonically as the frequency offset increases. Moreover, the

SNR degradation of a system operating at high SNR values is

larger than that of a system operating at low SNR values. It

was also shown that the approximate average SNR expression

and its lower bound match quite well with the simulation

results.

REFERENCES

[1] J. A. C. Bingham, Multicarrier modulation for data transmission: Anidea whose time has come, IEEE Commun. Mag., vol. 28, pp. 5-14,May 1990.

[2] J. -J. Beek, M. Sandell, and P. O. Borjesson, ML estimation of timeand frequency offset in OFDM systems, IEEE Trans. Signal Processing,vol. 48, pp. 1800-1805, July 1997.

[3] T. Schmidl and D. C. Cox, Robust frequency and timing synchroniza-tion for OFDM, IEEE Trans. Commun., vol. 45, pp. 1613-1621, Dec.1997.

[4] T. Pollet, M. Van Bladel, and M. Moeneclaey, BER sensitivity ofOFDM systems to carrier frequency offset and Wiener phase noise,IEEE Trans. Commun., vol. 43, pp. 191-193, Feb./Mar./Apr. 1995.

[5] C. Muschallik, Influence of RF oscillators on an OFDM signal, IEEETrans. Consumer Electronics, vol. 41, pp. 592-603, Aug. 1995.

[6] H. Steendam and M. Moeneclaey, Sensitivity of orthogonal frequency-division multiplexed systems to carrier and clock synchronisation er-rors, Signal Processing, vol. 80, pp. 1217-1229, July 2000.

[7] H. Nikookar and R. Prasad, On the sensitivity of multicarrier transmis-sion over multipath channels to phase noise and frequency offsets, inProc. IEEE PIMRC 96, 1996, pp. 68-72.

[8] W. Hwang, H. Kang, and K. Kim, Approximation of SNR degradationdue to carrier frequency offset for OFDM in shadowed multipathchannels, IEEE Commun. Letters, vol. 7, pp. 581-583, Dec. 2003.

[9] P. H. Moose, A technique for orthogonal frequency division multiplex-ing frequency offset correction, IEEE Trans. Commun., vol. 42, pp.2908-2914, Oct. 1994.

[10] K. Sathananthan and C. Tellambura, Probability of error calculation ofOFDM systems with frequency offset, IEEE Trans. Commun., vol. 49,pp. 1884-1888, Nov. 2001.

[11] X. Ma, H. Kobayashi, and S. C. Schwartz, Effect of frequency offseton BER of OFDM and single carrier systems, in Proc. IEEE PIMRC2003, 2003, pp. 2239-2243.

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