196 IEEE COMMUNICATIONS LETTERS, VOL. 14, NO. 3, MARCH 2010

Oversampling to Reduce the Effect of Timing Jitter onHigh Speed OFDM Systems

Lei Yang, Student Member, IEEE, and Jean Armstrong, Senior Member, IEEE

AbstractThe impairments caused by timing jitter are asignificant limiting factor in the performance of very high datarate OFDM systems. In this letter we show that oversamplingcan reduce the noise caused by timing jitter. Both fractional over-sampling achieved by leaving some band-edge OFDM subcarriersunused and integral oversampling are considered. The theoreticalresults are compared with simulation results for the case of whitetiming jitter showing very close agreement. Oversampling resultsin a 3 dB reduction in jitter noise power for every doubling ofthe sampling rate.

Index TermsTiming jitter, OFDM, oversampling.

I. INTRODUCTION

ORTHOGONAL frequency division multiplexing(OFDM) is used in many wireless broadbandcommunication systems because it is a simple and scalablesolution to intersymbol interference caused by a multipathchannel. Very recently the use of OFDM in optical systemshas attracted increasing interest (see [1] and the referencestherein). Data rates in optical fiber systems are typicallymuch higher than in RF wireless systems. At these veryhigh data rates, timing jitter is emerging as an importantlimitation to the performance of OFDM systems. A majorsource of jitter is the sampling clock in the very high speedanalog-to-digital converters (ADCs) which are required inthese systems. Timing jitter is also emerging as a problemin high frequency bandpass sampling OFDM radios [2]. Theeffect of timing jitter has been analyzed in [3], [4]. Thesepapers focus on the coloured low pass timing jitter whichis typical of systems using phase lock loops (PLL). Theyconsider only integral oversampling.

In OFDM, fractional oversampling can be achieved byleaving some band-edge subcarriers unused. In this letterwe investigate both fractional and integral oversampling.Weextend the timing jitter matrix proposed in [5] to analyze thedetail of the intercarrier interference (ICI) in an oversampledsystem. Very high speed ADCs typically use a parallel pipelinearchitecture not a PLL [6] and for these the white jitter whichis the focus of this paper is a more appropriate model.

II. SYSTEM MODEL AND TIMING JITTER MATRIX

Consider the high-speed OFDM system shown in Fig. 1.The OFDM symbol period, not including the cyclic pre-fix, is . At the transmitter, in each symbol period, up to

Manuscript received October 6, 2009. The associate editor coordinating thereview of this letter and approving it for publication was S. Galli.

This work was supported under an Australian Research Councils Discoveryfunding scheme (DP 0772937).

The authors are with the Department of Electrical and ComputerSystems Engineering, Monash University, Victoria, Australia (e-mail:lei.yang@eng.monash.edu.au).

Digital Object Identifier 10.1109/LCOMM.2010.03.091963

De-QAM

+P/S

QAM+

P/SIFFT

FFT

P/S

S/P A/D

h(t)

High speed data

D/A

Transmitter

ReceiverKt

Xk

Yl

xn

yn

x(t)

Sampler

Fig. 1. OFDM block diagram.

Timing jittered signal

Timing jitter

Ts 2Ts 3Ts 4TsWn

0(a)

(b)

Fig. 2. Definition of timing jitter.

complex values representing the constellation points areused to modulate up to subcarriers. Timing jitter canbe introduced at a number of points in a practical OFDMsystem but in this letter we consider only jitter introduced atthe sampler block of the receiver ADC. Fig. 2 shows howtiming jitter is defined. Ideally the received OFDM signal issampled at uniform intervals of / . The dashed lines in Fig.2(a) represent uniform sampling intervals. The solid arrowsrepresent the actual sampling times. The effect of timingjitter is to cause deviation between the actual samplingtimes and the uniform sampling intervals. Fig. 2(b) shows thediscrete timing jitter for this example. In OFDM systemswhile timing jitter degrades system performance, a constanttime offset from the ideal sampling instants is automaticallycorrected without penalty by the equalizer in the receiver.

In [5] we showed that the effect of timing jitter can bedescribed by a timing jitter matrix. The compact matrix formfor OFDM systems with timing jitter is

Y =WHXT +N (1)

where X, Y and N are the transmitted, received and additivewhite Gaussian noise (AWGN) vectors respectively, H is thechannel response matrix and W is the timing jitter matrix

whereY =

[/2+1 0 /2

]TH =diag

(/2+1 0 /2

)X=

[/2+1 0 /2

]1089-7798/10$25.00 c 2010 IEEE

YANG and ARMSTRONG: OVERSAMPLING TO REDUCE THE EFFECT OF TIMING JITTER ON HIGH SPEED OFDM SYSTEMS 197

and

W =

2 +1,2 +1 2 +1,0 2 +1,2...

. . .... . .

. ...0,2 +1 0,0 0,/2

... . .. ...

. . ....

2 ,2 +1 2 ,0 2 ,2

(2)Timing jitter causes an added noise like component in thereceived signal. From (1)

Y = HXT + (W I)HXT +N (3)where I is the identity matrix. The first term in (3) isthe wanted component while the second term gives the jitternoise. In [5] it was shown that the elements of the timing jittermatrix W are given by

, =1

/2=/2+1

2

2 () (4)

where is the time index, is the index of the transmittedsubcarrier and is the index of the received subcarrier.

III. THEORETICAL ANALYSIS OF TIMING JITTER

We now analyze the effect of both fractional and integraloversampling in OFDM and show that either or both can beused to reduce the degradation caused by timing jitter. Toachieve integral oversampling, the received signal is sampledat a rate of / , where is an integer. For fractionaloversampling some band-edge subcarriers are unused in thetransmitted signal. When all subcarriers are modulated,the bandwidth of the baseband OFDM signal is /2 , sosampling at intervals of / as shown in Fig. 2 is Nyquistrate sampling. If instead, only the subcarriers with indicesbetween and + are non zero, the bandwidth of thesignal is (+ )/2 . in this case sampling at intervals of/ is above the Nyquist rate. the degree of oversamplingis given by ( + )/ . In the general case, where bothintegral and fractional oversampling are applied, the signalsamples after the ADC in the receiver are given by

=

(

)

=1

=

(2

)+

(

)(5)

where is the oversampled discrete time index and is theAWGN. With integral oversampling, the -point FFT in thereceiver is replaced by an oversized -point FFT. Theoutput of this FFT is a vector of length with elements.

=1

1

/2=/2+1

(2

)(6)

where is the index at the output of the point FFT.Then combining (4), (5) and (6), we obtain the modifiedweighting coefficients for the oversampling case,

, =1

/2=/2+1

2

2 ( ) (7)

By using the approximation = 1 + for small , as in[5], (7) becomes

, 1

/2=/2+1

(1 +

2

)

2 ( )

(8)which can be simplified to give

,

1

/2=/2+1

2

2 ( ) =

1 + 1

/2=/2+1

2 =

(9)So for = the variance of the weighting coefficients isgiven by

{ ,2

}(

2

)2

{ } 2 ()

(10)where = . When the timing jitter is white, then{ } = 0 for = 0 so

{ ,2

}(

1

)(2

)2{2

} = (11)

From (11) it can be seen that white timing jitter { ,2

}is inversely proportional to so increasing the integeroversampling factor reduces the intercarrier interference (ICI)due to timing jitter. Note also that

{ ,2

}depends on

2 but not on , so higher frequency subcarriers cause moreICI, but the ICI affects all subcarriers equally. Similarly

{ , 2

} 1 +

(1

)(2

)2{2

} =

(12)We now calculate the average jitter noise power for eachsubcarrier for the case of white jitter. From (3), (6) and (9),

= +

=

( , ,) +N()(13)

where the second term represents the jitter noise. In thefollowing, we consider a flat channel with, = 1 , andassume that the transmitted signal power is distributed equallyacross the used subcarriers so that for each used subcarrier{2}= 2 . Then the average jitter noise power, () to

received signal power of th subcarrier is given by

()

2=

=

(, ,)2

2

=

=

{, ,2

}(14)

For a system in which there are equal numbers of unused sub-carriers at each band-edge so = /2, = /2+1

198 IEEE COMMUNICATIONS LETTERS, VOL. 14, NO. 3, MARCH 2010

, where is number of used subcarriers. Then (14) becomes

()

2=

/2=/2+1

{, ,2

}(15)

If the symbol period is unchanged, reducing the number ofsubcarriers will reduce the overall data rate, the transmit powerand the bandwidth. To make a fair comparison the symbolperiod is also reduced so that = / where isthe symbol period for used subcarriers and for used subcarriers. Substituting (11) (12) into (15), gives thefollowing

()

2=

/2=/2+1

1

(2

)2{2}

(16)

Using/2

=/2+1 2= 112(2 +

2 ), we obtain

()

2=

2

3{2}(

)(

)2

+1

3{2}(3

)(

)2(17)

Since the first term in (17) is very small, it can be ignored.Then (17) becomes with some rearranging

()

2=

2

3

(

2

){2}

(18)

If there is no integral oversampling or fractional oversampling, = 1 and = , therefore

()

2=

2

3

(2

2

){2}

(19)

Comparing (18) and (19) it can be seen that the combinationof integral oversampling and fractional oversampling reducesthe jitter noise power by a factor of / .

IV. SIMULATIONS

We now present simulation results for 2000 OFDM sym-bols, = 512 and jitter variance

{2}= (0.3/)

2 .Note that the jitter variance is not changed when oversamplingis applied, so the jitter represents a larger fraction of thesampling period for the oversampled systems. Fig. 3 showsthe variance of the noise due to jitter as a function of receivedsubcarrier index when band-edge subcarriers are unused. Itshows that the power of the jitter noise is not a function ofsubcarrier index and that removing the band-edge subcarriersreduces the noise equally across all subcarriers.

Fig. 4 shows both the theoretical and simulation results foraverage jitter noise power as a function of the oversamplingfactor. There is close agreement between theory and simu-lation. Increasing the sampling factor gives a reduction of10log10 (/) in jitter noise power, so every doublingof the sampling rate reduces the jitter noise power by 3 dB.

300 200 100 0 100 200 3006.5

6

5.5

5

l (Subcarrier index)

P j (dB

) (Av

erage

jitter

noise

powe

r)

Nv=512

Nv=492

Nv=472

Nv=452

Nv=400

Fig. 3. Average jitter noise power versus subcarrier index when leavingband-edge subcarrier unused.

1 2 3 4 5 6 7 8 9 1016

14

12

10

8

6

4

Oversampling factor

P J (dB

) (Av

erage

jitter

noise

powe

r)

TheorySimulation

Fig. 4. Average jitter noise power versus sampling factor.

V. CONCLUSIONS

It has been shown both theoretically and by simulation thatoversampling can reduce the degradation caused by timingjitter in OFDM systems. Two methods of oversampling wereused: fractional oversampling achieved by leaving some ofthe band-edge subcarriers unused, and integral oversamplingimplemented by increasing the sampling rate at the receiver.For the case of white timing jitterboth techniques result ina linear reduction in jitter noise power as a function ofoversampling rate. Thus oversampling gives a 3 dB reductionin jitter noise power for every doubling of sampling rate. Itwas also shown that in the presence of timing jitter, highfrequency subcarriers cause more ICI than lower frequencysubcarriers, but that the resulting ICI is spread equally acrossall subcarriers.

REFERENCES

[1] J. Armstrong, OFDM for optical communications, J. Lightwave Tech-nol., vol. 27, no. 1, pp. 189-204, Feb. 2009.

[2] V. Syrjala and M. Valkama, Jitter mitigation in high-frequency bandpass-sampling OFDM radios, in Proc. WCNC 2009, pp. 1-6.

[3] K. N. Manoj and G. Thiagarajan, The effect of sampling jitter in OFDMsystems, in Proc. IEEE Int. Conf. Commun., vol. 3, pp. 2061-2065, May2003.

[4] U. Onunkwo, Y. Li, and A. Swami, Effect of timing jitter on OFDMbased UWB systems, IEEE J. Sel. Areas Commun., vol. 24, pp. 787-793,2006.

[5] L. Yang, P. Fitzpatrick, and J. Armstrong, The Effect of timing jitter onhigh-speed OFDM systems, in Proc. AusCTW 2009, pp. 12-16.

[6] L. Sumanen, M. Waltari, and K. A. I. Halonen, A 10-bit 200-MS/sCMOS parallel pipeline A/D converter, IEEE J. Solid-State Circuits,vol. 36, pp. 1048-55, 2001.