Performance Analysis of an OFDM System in the Presence of Carrier Frequency Offset, Phase Noise and

Timing Jitter over Rayleigh Fading Channels

Shankhanaad Mallick, Member IEEE and Satya Prasad Majumder, Member IEEE

Department of Electrical & Electronic Engineering, Bangladesh University of Engineering & Technology, Dhaka-1000, Bangladesh

Email: shankhanaadmallick@hotmail.com , spmajumder@eee.buet.ac.bd

Abstract – A theoretical analysis for evaluating the performance of an Orthogonal Frequency Division Multiplexing (OFDM) under the combined influence of Carrier Frequency Offset (CFO), phase noise and timing jitter over rayleigh fading channels is presented. An exact closed form expression for the Signal-to-Interference plus Noise Ratio (SINR) is derived and the combined effects of these synchronization impairments are exhibited by the Bit Error Rate (BER) performances of a BPSK-OFDM system over rayleigh fading channels. Results show that OFDM system suffers significant SINR penalty due to CFO and jitter, however, the effect of phase noise is the dominant one.

I. Introduction Being very efficient in combating multipath fading as well as Inter Symbol Interference (ISI) and in the use of available bandwidth, Orthogonal Frequency Division Multiplexing (OFDM) has been widely adopted and implemented in wire and wireless communications, such as Digital Subscriber Line (DSL), European Digital Audio Broadcasting (DAB), Digital Video Broadcasting-Terrestrial (DVB-T) and its handheld version DVB-H, and IEEE 802.11a/g standards for Wireless Local Area Networks (WLANs) [1]-[2] etc. Unfortunately OFDM is very much sensitive to the synchronization errors such as Carrier Frequency Offset (CFO), phase noise or timing jitter [3]. The CFO arises mainly due to the Doppler shifts introduced by the channel which causes frequency difference between the transmitter and receiver oscillators. The deleterious effects caused by the CFO are the reduction of the signal amplitude and introduction of Inter-Carrier-Interference (ICI) from the other carriers which are then no longer orthogonal to the filter [4]. Phase noise results from the imperfections of the Local Oscillators (LO) used for the conversion of a baseband signal to a passband (or vise-versa). Phase noise has two effects on an OFDM system: rotation of the symbols over all subcarriers by a Common Phase Error (CPE) and the occurrence of ICI which introduces a blurring of the constellation like thermal noise [5]. Timing errors would occur either when the clock signal is not correctly recovered, or when sampling is not performed at precise sampling instants. Because of the non-ideal nature of the sampling circuit the amplitude

of the signal is affected by timing jitter and it introduces additional source of additive noise [6]. The individual effects of CFO and phase noise have been analyzed by several authors and the degradation introduced in the system has been characterized for some particular cases in [3]-[5], [7]-[11]. The effect of timing jitter on the performance of discrete multitone system was also investigated in [12] and in [13]. However, a closed form analytical result that shows the exact quantitative effect of the combination of these three impairments even for Additive White Gaussian Noise (AWGN) channels has not been well addressed. The purpose of this paper is to analyze, via mainly an analytical approach, the impact of the combined effects of CFO, phase noise and timing jitter to the performance of OFDM systems in rayleigh fading environment. The exact Signal to Interference plus Noise Ratio (SINR) expression in a closed form is derived which provides a quantitative understanding of how system behavior changes with certain parameters. We evaluate the Bit Error Rate (BER) performances of a BPSK-OFDM system over rayleigh fading channels considering the combined influence of these synchronization impairments. The rest of the paper is organized as follows: In Section II, CFO, along with phase noise and timing jitter process is reviewed and the OFDM system model is given in the presence of CFO, phase noise and timing jitter over rayleigh fading channel. In Section III the exact SINR expression for the combined effects is derived. Section IV gives the results of system performance analysis and finally Section V finishes the paper by giving conclusion.

II. System Model and Description Consider the thm symbol of an N -subcarrier OFDM system in the presence of normalized CFO, ε , phase noise, )(nmϕ , and timing jitter, nξ as shown in Fig. 1. A. Carrier Frequency Offset (CFO) Model The absolute value of the actual CFO εf , is either an integer multiple or a fraction of fΔ , or the sum of them.

5th International Conference on Electrical and Computer EngineeringICECE 2008, 20-22 December 2008, Dhaka, Bangladesh

978-1-4244-2015-5/08/$25.00 (c)2008 IEEE 205

Fig. 1 OFDM system model (receiver) in the presence of CFO, phase noise and timing jitter over rayleigh fading channel

If εf is normalized to the subcarrier spacing fΔ , then the resulting normalized CFO of the channel can be generally expressed as

∈+=Δ

= δε ε

ff (1)

where δ is an integer and 5.0≤∈ .The influence of an integer CFO on OFDM system is different from the influence of a fractional CFO. In the event that

0≠δ and 0=ε , symbols transmitted on a certain subcarrier, e.g., subcarrier k , will shift to another subcarrier δk , 1mod −+= Nkk δδ (2) As we focus on the ICI effect, we will consider

normalized CFO, =∈Δ

=f

fεε since no ICI is caused by

an integer CFO. We assume relative CFO ( ε ) to be a Gaussian process, statistically independent of the input signal, with zero mean and variance 2

εσ .

B. Phase Noise Model Phase noise )(nmϕ , generated at both transmitter and receiver oscillators, can be modeled as [5]

∑ ∑

∑+++

= =

−=−

++==

+++−=

nNNNm

i

n

imm

n

Nigmm

gg

g

iTuCiu

iNNmuNn

)(

0 0

1

)()(

])([)1()( ϕϕ (3)

where mC and mT are defined by ∑−++

=

1)(

0)(

gg NNNm

iiu and

gg NNNm ++ )( respectively. gN is the length of cyclic prefix and siu )'( denote mutually independent Gaussian random variables having zero mean and variance

RNTu /2/22 πβπβσ == , where β denotes the two-sided 3-dB linewidth of the Lorentzian power density spectrum of the free running carrier generator [8], T and R denote OFDM symbol period and the transmission data rate, respectively.

C. Timing Jitter Model In the sampling circuit at the receiver additional error may occur in the determination of the best sampling phase. This means that the sampling instants are non-ideal and is given by [12]

nn nTt ξ+= (4)

where nξ is the timing jitter of the thn sampling instant normalized by symbol duration T . Timing jitter can be modeled as a stationary Gaussian random process statistically independent of the input signal with zero mean and variance 2

ξσ [6].

D. OFDM System Model As shown in Fig. 1, the transmitted OFDM signal for the

thm symbol is given by the N point complex modulation sequence

∑−

=

=1

0

2

)()(N

k

nkN

j

mm ekXnxπ

(5)

where n ranges from 0 to 1−+ gNN .

After passing through a rayleigh fading channel and LO, the received signal impaired by AWGN and phase noise can be modeled as

)(])()([)( )(1

0

)(2

nweekHkXny mnj

N

k

knN

j

mmmm += ∑

−

=

+ ϕεπ (6)

or, )()()( )( nwensny mnj

mmm += ϕ (7)

where, ∑−

=

+=

1

0

)(2

)()()(N

k

knN

j

mmm ekHkXnsεπ

In (6) )(kHm is the transfer function of the rayleigh fading channel at the frequency of the thk carrier and )(nwm is the complex envelope of AWGN with zero mean and variance 2σ .

Assuming )(nmϕ is small [9] so that

)(1)( nje mnj m ϕϕ +≈ (8)

Rayleigh Fading Channel

Frequency offset

n

Nj

eεπ2

)(nsm Down

Conversion (LO)

Phase Noise

)(nj me φ

)(nymA/D

S/P Remove CP DFTP/S

Demodulation & Detection

Output data

)(~ nym

non ideal sampler, nn nTt ξ+=

)(nxm

AWGN )(nwm

206

Substituting (8) into (7) yields

)()()()()( nwnjnsnsny mmmmm ++= ϕ (9)

After DFT and by dropping the subscript ‘m ’ (9) yields

)()()()()( kWkjkSkSkY +Θ⊗+= (10)

where, )(kS , )(kΘ and )(kW are the DFT responses of )(nsm , )(nmϕ , and )(nwm respectively and⊗ denotes the

circular convolution operation.

III. Exact SINR Expression for the Combined Effect

Let, )()()( 11 kIkSkS +=

where

)1(

1

0

2

1

)sin(

)sin()()(

)()(1)(

NNj

N

n

Nnj

e

NN

kHkX

ekHkXN

kS

−

−

=

=

= ∑πε

επ

πεπε

(11)

The first component, )(1 kS , is the modulation value )(kX modified by the channel transfer function. It

experiences an amplitude reduction and phase shift due to CFO,( ε ). As N is always much greater than ( πε ),

)sin(N

N πε is replaced by (πε ).

The second term, )(1 kI , is the ICI caused only by the CFO and is given by

)()1(1

0

1

0

21

01

))(

sin(

)sin()()(

)()(1)(

Nkrj

NNjN

krr

N

n

NnjN

krr

ee

Nkr

NrHrX

erHrXN

kI

−−−−

≠=

−

=

−

≠=

+−=

=

∑

∑∑

ππε

επ

εππε (12)

Assuming 0][ =kXE and rkrk XXXE δ2*][ = and average

channel gain 22}{ HHE r = we can obtain from (11)

)(])([ 22221 πεcnisHXkSE = (13)

])([ 21 kIE is evaluated in [4] as

5.0;)(sin5947.0])([ 22221 ≤≤ επεHXkIE (14)

Let, )()()(2 kjkSkI Θ⊗= (15)

In )(2 kI , the term that causes ICI due to the joint effect of phase noise and CFO is given by

)())sin(

)sin()()(

)())()(()(

1

0

1

0

21

02

rk

NN

rHrXj

rkerHrXNjkI

N

krr

N

n

NnjN

krr

−Θ=

−Θ=′

∑

∑∑

−

≠=

−

=

−

≠=

πεπε

επ

(16)

In (Appendix) the energy of )(rΘ is given by

)(sin2])([

2

22

NrN

rE uπ

σ=Θ (17)

Thus ])([2

2 kIE ′ can be evaluated as

∑−

=

=′1

1

2

2222

2

2

)(sin2)(])([ N

r

u

NrN

cnisHXkIEπ

σπε (18)

As channel SNR chγ is defined by,

2

0

2

2

22

])([αγ

αγ in

inch N

E

kWE

HX=== , where

0NEin

in =γ is

the input SNR, inE is the averaged transmitted energy of

the individual carriers, 2

0N is the power spectral density

of the AWGN in the fading transmission channel andα is the rayleigh fading channel attenuation/gain parameter. Therefore the SINR expression in the presence of CFO and phase noise in rayleigh fading environment may be expressed as

5.0;

}])(sin

1)(2

{)(sin5947.0[1

)}({),,(

1

1 2

22

22

222

≤

++≥

∑−

=

ε

ππεσπεαγ

πεαγασεN

r

uin

inu

Nr

cnisN

cnisSINR

(19)

In the non-ideal sampling circuit, the amplitude of the samples is affected by a random timing jitter (ξ ) which ultimately causes inE to degrade by a factor of ( ξ−1 ) over a time slot and on the other hand it increases additive noise energy by ξinE . Considering the effect of jitter the SINR expression of (19) is modified as follows

1,5.0;

}])(sin

1)(2

{)(sin5947.0)[1(1

)}(){1(),,,(

21

1 2

22

22

222

≤≤

++−+

−≥

∑−

=

ξε

ξαγππεσπεξαγ

πεξαγαξσεin

N

r

uin

inu

Nr

cnisN

cnisSINR

(20)

207

(20) indicates that, in the presence of CFO, phase noise, timing jitter and rayleigh fading, several parameters affect OFDM system performance, resulting in severe performance degradation which is unacceptable in practice. In the absence of CFO ( 0=ε ) the SINR expression of (20) reduces to

1;

}])(sin

12

)[{1(1

)1(),,(2

1

1 2

22

22

≤

+−+

−≥

∑−

=

ξ

ξαγπσξαγ

ξαγαξσin

N

r

uin

inu

NrN

SINR

(21)

which reflects the combined influences of phase noise and timing jitter in rayleigh fading environment. In the case of perfect carrier-phase synchronization, )(rΘ becomes a Dirac delta function and 2

uσ approaches zero, while the SINR expression of (21) reduces to

1;1

)1(),(

2

2

≤+

−≥ ξ

ξαγξαγαξ

in

inSINR (22)

Finally for an ideal sampling circuit ( 0=ξ ) and for a non-fading environment ( 1=α ) SINR expression of (22) becomes input SNR, inγ .

From the SINR expression of (20), the conditional probability of bit error, ),,|( αξεePb conditioned on a given value of ε ,ξ andα , for a given input SNR, inγ and phase noise variance, 2

uσ can be obtained as [14]

)),,((21),,|( αξεαξε SINRerfcePb = (23)

Then the average BER for BPSK-OFDM system over rayleigh fading channels can be evaluated as

αξεαξεαξε αξε dddPPPePBER b )()()(),,|(∫ ∫ ∫∝

∝−

∝

∝−

∝

∝−

= (24)

here the probability density functions (PDFs) )(εεP and

)(ξξP are assumed Gaussian whereas the PDF of α ,

)(ααP is rayleigh.

IV. System Performance Analysis In the presence of CFO, phase noise and timing jitter over fading channels, (24) and (20) indicate that BER and SINR are functions of these three impairments as well as some critical system parameters. These relations are depicted respectively in Figs. 2-7. If not mentioned in the

figures the following parameters are used for computation in this section:

Table 1: System and Channel Parameters

No. of Subcarriers ( N ) 2048

Cyclic Prefix Length ( gN ) 128

Modulation BPSK

Data Rate( R ) 64/7 MHz

Channel type Rayleigh fading

Fading variance of channel 0.33

Input SNR 20 dB

Fig. 2 illustrates the catastrophic effect of CFO, phase noise and timing jitter on the BER performance of a BPSK-OFDM system over rayleigh fading channel. As shown, the performance degrades with the increase of the combinational variances of 22 , uσσε and 2

ξσ . At high SNRs there exists BER floors resulting from the combined ICI effects lowering the effective SNRs, and that BER floor runs up when variance level increases. It also implies that OFDM systems with high SNR are more sensitive to ICI, though higher SNR leads to better performance.

0 5 10 15 20 25 30 35 40 45 50

10-4

10-3

10-2

10-1

100

BE

R

SNR (dB)

var: jtr=0.01,CFO= 0.03,phnoise=0.0001var: jtr=0.06,CFO= 0.03,phnoise=0.0001var: jtr=0.01,CFO= 0.09,phnoise=0.0001var: jtr=0.06,CFO= 0.09,phnoise=0.0001var: jtr=0.01,CFO= 0.03,phnoise=0.0005var: jtr=0.06,CFO= 0.03,phnoise=0.0005var: jtr=0.01,CFO= 0.09,phnoise=0.0005var: jtr=0.06,CFO= 0.09,phnoise=0.0005

Fig. 2 BER performance of an OFDM system over rayleigh fading channel for different combination of variances (var) of CFO, phase noise (phnoise) and jitter (jtr).

Fig. 3, 4 & 5 demonstrates the SINR penalty suffered by the system as a function of 2

εσ , 2ξσ and 2

uσ respectively for 20dB input SNR.

In Fig. 3, first of all, we consider the variance of CFO ( 2

εσ ) as variable and 2ξσ , 2

uσ as constants. When the variances of jitter and phase noise are zero, ICI occurs only due to CFO. With the increase of 2

εσ , SINR penalty

208

increases and we can see that more than 7dB penalty is suffered by the system for 1.02 =εσ . If either or both jitter and phase noise are present along with fading then the system suffers a penalty even at 02 ≈εσ . The SINR penalty curve shifts upwards with the increase of the constant variances of jitter and phase noise.

10-4

10-3

10-2

10-1

2

4

6

8

10

12

14

16

18

SIN

R p

enal

ty,S

NR

(dB

)-S

INR

(dB

)

variance of CFO

Const Variance: jitter=0 & phase noise=0Const Variance: jitter=0.01 & phase noise=1e-5Const Variance: jitter=0.02 & ph noise=2e-5

Fig. 3 SINR penalty as a function of the variance of CFO for different Constant Variances of jitter and phase noise.

Similar curves are obtained in Fig. 4, for considering 2ξσ

as variable and 2εσ , 2

uσ as constants. From Fig. 3 & 4, we can see that nearly 1-2dB additional penalty is suffered by the system for jitter than CFO when their respective constant variances have relatively low values.

In Fig. 5 we see that SINR penalty drastically changes for a very small change in the variances of phase noise. It is also noticeable that SINR is strongly dependent on the number of sub-carriers ( N ) as ICI due to phase noise is a function of N . Larger number of N leads to shorter subcarrier spacing distance, hence more sensitive to phase noise and as a result of that SINR penalty increases. From Fig 3, 4 & 5 we can conclude that OFDM is more sensitive to phase noise than CFO and jitter.

10-4

10-3

10-2

10-1

2

4

6

8

10

12

14

16

18

SIN

R p

enal

ty,S

NR

(dB

)-S

INR

(dB

)

variance of relative timing jitter

Const Variance: CFO=0 & phase noise=0Const Variance: CFO=0.01 & phase noise=1e-5Const Variance: CFO=0.02 & phase noise=2e-5

Fig. 4 SINR penalty as a function of the variance of relative timing jitter for different Constant Variances of CFO and phase noise.

10-8

10-7

10-6

10-5

10-4

5

10

15

20

25

30

35

40

45

50

55

60

SIN

R p

enal

ty,

SN

R(d

B)-

SIN

R(d

B)

variance of phase noise

Const Var: CFO=0,jitter=0 (N=64)Const Var: CFO=0.01,jitter=.01 (N=64)Const Var:CFO=0.02,jitter=.02 (N=64)Const Var:CFO=0,jitter=0 (N=2048)Const Var:CFO=0.01,jitter=.01 (N=2048)Const Var:CFO=0.02,jitter=.02 (N=2048)

Fig. 5 SINR penalty as a function of the variance of phase noise for different constant variances of CFO and jitter with

2048&64=N .

For low variances of CFO and jitter, SINR is strongly dependent on the values of β and N at higher values of SNR. Here, the sample rate, R is kept constant. As shown in Fig. 6, when β is very small compared to the

subcarrier spacing, i.e., RNβ is of the order of 410− or

less, the ICI due to phase noise is negligible. As a result, there is a constant SINR penalty due to the presence of CFO and timing jitter along with fading. Meanwhile, for high phase noise levels with 1≥

RNβ , SINR penalty

exceeds the value of SNR itself, which implying that the ICI overwhelms the desired signals. Higher transmission data rate, R results in better system performance.

10 15 20 25 30 35 4010

15

20

25

30

35

40

45

50

55

60

SIN

R p

enal

ty,

SN

R(d

B)-

SIN

R(d

B)

SNR (dB)

Const Variance: CFO=0.01 & jitter=0.01BN/R=10BN/R=1BN/R=0.1BN/R=10e-2BN/R=10e-3BN/R=10e-4BN/R=10e-5

Fig. 6 SINR penalty as a function of SNR for different

(RNβ )settings.

Fig. 7 reflects the BER performances of an OFDM system over rayleigh channels with different fading variances where the variances of all three synchronization impairments are assumed constant. It is noticeable that the BER performance improves with the increase of the fading variance. This is because attenuation of the input SNR decreases at high variances of rayleigh fading channels.

209

-10 -5 0 5 10 15 20 2510

-4

10-3

10-2

10-1

100

BE

R

SNR (dB)

variance: fade=0.1variance: fade=0.2variance: fade=0.33variance: fade=0.45

Fig. 7 BER performances of an OFDM system for different variances of rayleigh fading channel with

03.02 =εσ , 01.02 =ξσ , 52 101 −×=uσ

V. Conclusion In this paper, an analytical technique is provided for evaluating the performance of an OFDM system impaired by CFO, phase noise and timing jitter over rayleigh fading channels. An exact close-form expression for the SINR is derived and the BER performances of a BPSK-OFDM system are evaluated considering these effects. It is noticed that the OFDM system suffers significant penalty due to CFO and jitter, however, the effect of phase noise is the dominant one. It is shown by analysis that the system performance also depends on several critical parameters such as number of subcarriers, phase noise linewidth, transmission data rate, input SNR and the fading characteristics of the channel.

Appendix Energy of )(rΘ

The DFT response of )(nmϕ is given by

nrN

jN

nm

rN

j

N

n rN

j

nrN

j

m

N

n

N

ni

irN

j

m

N

n

nrN

jn

im

N

n

nrN

jn

imm

N

n

nrN

j

m

eNrnnTu

eNrN

e

enTuN

enTuN

eiTuN

eiTuCN

enN

r

π

π

π

ππ

π

ππ

ππ

ϕ

−−

=−

−

= −

−−

=

−

=

−

−

=

−

=

−

=

−

=

−

=

−

∑

∑∑ ∑

∑∑

∑ ∑∑

⋅+⋅

−=

−

−⋅+=+=

+=

++==Θ

1

0

1

02

21

0

1 2

1

0

2

0

1

0

2

0

1

0

2

)sin()()sin(

1

1

1)(1)(1

)(1

])([1)(1)(

Note that 01 1

0

2

=∑−

=

−N

n

nrN

je

N

π for 1,,2,1 −= Nr … . Hence

according to the mutual independence of Gaussian random variables )(iu ’s, we can calculate the energy of

)(rΘ as

)(sin2

)(sin)(sin

])([

2

2

1

0

2

22

22

NrN

Nrn

NrN

rE

u

N

n

u

πσ

ππ

σ

=

=Θ ∑−

=

Since for 0≠r and for even N , ∑−

=

=1

0

2

2)(sin

N

n

NNrnπ

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