Estimation of Wiener Phase Noise by the Autocorrelation of the ICI weighting Function in OFDM Systems

Estimation

of

Wiener

Phase

Noise

by

the

Autocorrelation

of

the

ICI

Weighting

Function

in

OFDM

Systems

Yi-Ching Liao and

Kwang-Cheng Chen

Institute of Communication

Engineering, College of Electrical Engineering

National Taiwan

University, Taipei, Taiwan,

10617, R.O.C.

E-mail: ycliao@

santos.ee.ntu.edu.tw, chenkc@cc.ee.ntu.edu.tw

Abstract- PerformancedegradationduetoWienerphasenoise which causesbothcommonphaseerror(CPE)andinter-carrier interference (ICI) is a crucial challenge to the implementation of OFDM systems. In this paper, we theoretically employ the Lorentzian model to investigate the autocorrelation function of the ICI weighting function which is the discrete Fourier trans-form oftheexponentialphase noise process. This autocorrelation function canbe shown tobe the kernel of the covariance of the ICI. Based on this kernel, a pilot-aided decision-directed CPE estimatorisproposedaccordingtomaximum-likelihoodcriterion. Different from conventionalmaximum likelihoodapproachwhich ideally assumes the ICI observed on different subcarriers to be independentidentically distributed, we systematically derive the covariances among carriers and practically utilize them to enhance the estimation. Finally, three conventional CPE estimatorsarecomparedwith theproposedschemebycomputer simulation, the numerical results illustrate the effectiveness of the proposed algorithm.

Index Terms-OFDM, Phase Noise and Maximum Likelihood Estimation.

I. INTRODUCTION

Orthogonal frequency division multiplexing (OFDM) has been attracting considerable research interests as apromising candidate tohigh dataratecommunications since its resistance

toimpairments such as frequency selective fadingand

impul-sive noise. However, OFDM is tremendously more sensitive tocarrier frequency offset and phase noise than single carrier systems [1]. Mainly resulting from the instability of local

oscillator, phase noise can be classified into two categories. When the system is onlyfrequency-locked, the resulting phase noise is modeled as a zero-mean Wiener process. When the system is phase-locked, the resulting phase noise is modeled as a zero-mean stationary random process [2]. Here, we concentrate onthesuppression of Wiener phase noise.

Methodsfor thecompensation of the effects of phase noise

have been proposed by several authors. The conventional

ap-proachescanbecategorizedinto decision directed approaches and pilot-aided approaches. The decision directed approaches

[3], [4], estimate the CPE using the averaged phase rotation of the observed symbols from the ideal constellation points. Asymbol by symbol pre-compensation is necessarytoensure this rotationnotexceedingthedecisionboundary. Asfor pilot-aidedapproach[5]-[7], theaverageorweighted average of the

phase differences between the transmitted and received pilot

symbols are used toestimate the CPE.

In this paper, by using the Lorentzian model, the

autocor-relation function of the ICI weighting function which can

be shown to be the kernel of the second order statistics of

the ICI is systematicallyderived. Different fromconventional

maximum likelihood (ML) schemes [4], [7] which ideally

assume the ICI observed on different subcarriers to be

inde-pendentidentically

distributed,

we

practically

investigate their

covariances bytheautocorrelation function of theICI

weight-ing function and combine pilot-aided and decision-directed approaches toyieldageneralized maximum likelihood estima-tion scheme. Simulaestima-tion results demonstrate that the proposed algorithm outperforms the conventional approaches

The rest of this paper is organized as follows: Section II presents the phase noise corrupted OFDM signal model and thephase noise model respectively. Based on that, Section III

first investigate the statistical characteristics of the sufficient

statistics, then, it proceed to

derive

the maximum likelihood estimator for the CPE. Performance evaluation via computer

simulation is addressed in Section IV. Finally, Section V discusses and concludes thispaper.

II. BACKGROUND

A. Phase Noise CorruptedOFDMSignalModel

ConsideringageneralOFDM systemusing N-point inverse

fast Fourier transform

(IFFT)

for modulation. Assume the

frequency domain subcarrier index set is composed of three

mutually exclusive subsets defined by

D-{d1,d2,

,dNd}

P

{pl1,P2

..*

PNP

}

(1)

X-={N,

n2, ---

',nN,}

I

where D denotes the set of indices for Nd data-conveying

subcarriers,Pis thesetof indices for

Np

pilot subcarriers and X stands forNn virtual subcarriers. Then, the set of indices forNu useful subcarriers U can be defined as

A-{Ul,U2,* * * UN,,,,

I

=DUP,

(2)

whereNu = (Nd+

Np).

LetXm(k)bethe modulatedsymbol on the kth subcarrierof themth OFDM symbol. For k EU,

Xm(k)

is taken from someconstellation with zero mean and average power

4

2

E{IXm(k)12}.

The output of the IFFT

has aduration of T seconds which is equivalent to N samples. A

Ng-sample

cyclic prefix longer than the channel impulse response is preceded to eliminate theinter-symbol interference (ISI).

At the receiver, timing and frequency recovery is assumed tobe accomplished.Considering the multiplicative phase noise

and the additive white noise, the received nth sample of the

mth OFDM symbol can bewritten as

rm(n)

=

[xm(nl)

0

hm(n)]

ej(km(n)+O)

+

Tm(ln)

(3)

in which 0 is the circular convolution and

q.(n)

= q

(m(N

+

Ng)

+

Ng

+

n)

(4)

where

xrm(n), hm(n)

and

q,

(n)

represent the transmitted signal, the channel impulse response and the phase noise respectively, while

(m(n)

denotes the AWGN noise and 0 is

the initialphase of the phase noiseprocess. After removing the cyclic

prefix

and performing the FFT, the

frequency

domain symbolcan beexpressed by

Rm(k)

=4)m(0)Hm(k)Xm(k)

+ E

4'm(k

-I)Hm(l)Xm(l)

+Zm(k)

IEU_i$k

I,, (k)

where

Hm(k)

is thechannel

frequency

response and

Zm(k)

denotes the frequencydomain

expression

of

(m(n). 4.m(h)

is

the discrete Fourier transform of the

phase

noiseprocess

given

by

N-1

'Jm(h)

- I E

ej(em(n)+G)e-j2

(6)

n=O

And itcan be viewed as a

weighting

function on the trans-mittedfrequency domain

symbols.

In

particular,

whenh =

0,

4.m(0)

isthe time average of the

phase

noise process within

one OFDM

symbol

duration. This term is

usually

known as

the commonphase error

(CPE)

whichcauses the same

phase

rotation andamplitude distortiontoeachtransmitted

frequency

domainsymbol. On the other

hand,

when h=

0,

the second

termin

(5)

is theinter-carrier interference

(ICI) resulting

from

the contributions of other subcarriers

by

the

weighting

of

4.m(h)

due to the loss of

orthogonality.

In the rest of this

paper, weshall call

4m(h)

the ICI

weighting

function. Based on

(5),

thereceived

frequency

domainvector can be given by r. =,

4m(0)HmXm

+

im

+

Cm

=

4bm(O)HmXm

+ cmI where cm = tm +

(ml

Hm=diag(Hm(0),Hm(l),

m.

,H(N -1)) Xm =

[Xm(0)

Xm(1)

...

Xm(N- 1)]T

,bm-

[Im(O)

In(l)

..Im

(N

-1)]

(m -

[(mn(°)

(m(l)

(mn(N-

1)]

X

and

diag(.)

is adiagonal matrix. From now on, we shall add

a second subscript to one of the vector or matrix variables

definedin(7) and (8) to indicate its sub-vector or sub-matrix which is taken according to one of the subcarrier index sets

in (1) and (2). The second subscript may be chosen from

{p, d,n,u} which relates to {P, D,

K,

U} respectively. For example,

rm,p

=

[Rm(pi)

Rm(P2)

...

Rm(pNP)]T

(9)

stands for thereceived pilot vector.

Conventionally,

rm,p is utilized to obtain the channel

re-sponse and common phase error to carry out equalization on

ri,d,

then the equalized results are sent to the detection block to get the decisions. Sinceaccurate channel estimation in OFDM systems can be obtained by either preambles or pilot symbols

[81,

in the following sections we assume that the channel frequency response is acquired perfectly at the receiver.

B.

Phase

Noise

Model

Accuratemodeling of oscillator phase noise isakey factor to the analysis and simulation of the distortion caused by phase noise. For a classic model of phase noise,

On(n)

can be modeled as adiscrete-time Wiener process

[1]

with

E[qm(n)]

= 0

E[(qm(n

+

An)

-

Om(n))2]

=

4ir/3TIAnl/N,

(10)

where

,3

(Hz) denotes the one-sided 3 dB linewidth of the

Lorentzian power

density

spectrum ofthe

free-running

oscil-lator. The Lorentzian spectrum is the squared

magnitude

of a

first order lowpass filtertransfer function

[8].

The

single-sided

spectrum So

(f)

is

given by

2/7r/3

S,I(f)

=

1+f2

/z2'

and the Lorentzian spectra with different linewidth are shown in

Figure 1.

-(7)

a

(8)

10° 10 104 f(Hz)

(1

1) one-sided 3 dB 106

As for the initial phase 0, it can be modeled as a random variableuniformly distributed in [0, 27r)and isindependentof

bm(ln).

Later

analysis

and simulation will be based on this model.

111. PILOT-AIDED DECISION-DIRECTED CPE ESTIMATION Most OFDM systems employ pilots to facilitate receiver

synchronization since data-aided estimation gives better and steadier estimate.However,sincepilotscostsystemutilization,

the number of pilots should bekeptaslow aspossible which

confines the performance ofpilot-aided CPE estimation algo-rithms. Comparatively, decision-directed approaches

enjoy

a largerobservation space.Nevertheless, ensuringthe acceptable

correctness of the decisions is also critical.

Therefore,

use

pilotsto acquiretheCPEandprovideaninitial compensation,

thenenlarge the observation space by

including

the tentative

decisions as the sufficient statistics to perform the final esti-mate canbenefitfromtheadvantages ofbothapproaches.We

refer thismethod to the

pilot-aided-decision-directed

(PADD)

approach. In the following, we shall

investigate

the statistical

characteristics of the sufficient statistics

rmn,

and

systemati-cally derive the MLestimator in accordance with the PADD approach.

A. Statistical Characteristics of the

Sufficient

Statistics Considering the PADD approach, since the data and the

channel frequency response are acquired, the statistical

char-acteristics of rm,u depend on that of Lm,u and 'm,u The

AWGN noise on each subcarrier can be modeled as a zero mean complex Gaussian random variable with variance az.

Since the ICI on each subcarrier is composed of the data

symbols on other subcarriers, we can apply the central limit theorem to model it as a complexGaussian random variable.

In the following, we first show that the ICI has a zero mean,

then,thesecondorderstatisticsof the ICI will beinvestigated. 1) The Mean of the ICI: Since

Xm(k)

and

Hm(k)

are assumed tobe known in thePADDcircumstance, the mean of theICI depends on that of the ICI weighting function4m(h) which can beexpressed as

N-i

E[4Fm(h)]

= + E

[ej(O-(n)+O)J

eJ2IT. (12)

n=O

Since 0 is independentof

qm(n),

the expectationin (12) can be decomposed into the product of

E[ejO-(n)]

and

E[ej0].

Let

Io(w)

denotes thecharacteristic function of the uniform

random variable, we have

E[ej0]

=

To(w)

l

2

27wwjr

=2

sin(

7)ei

(13)

=0.

Therefore, the mean ofthe ICIbecomes zero.

2)

The Second Order Statistics

of

the ICI: In the PADD

scenario,

the sufficient statistics are thereceived useful sym-bols

given

by

rm,u =

4Pm(0)Hm,uXm,u

+tm,u+(m,u- (14) We maydenote thecovariance matrix of tm,u

by

C,nl

u,then

thecovariance matrix of

rm,u

can be

expressed

as

Crr u =CtLnu + hzI (15)

Letthe elementof

C,L

bedenoted

by cir(k1, k2),

itcanbe

expressed

as

o1(kl,k2) = 5 S

Hm(1I)H(192)Xm(11)Xm(l2)

I1EU 12Eu

ll=A-kl 12:A-k2

E

[4Dm(Ukj

-

l)

>mI (Uk2 -

12)].

(16)

Wecanobservethat

oj(k1,

k2) depends

ontheautocorrelation function ofthe ICI

weighting

function

4cm(h). Therefore,

we have the

following

proposition.

Proposition

1

(The

autocorrelation

fuinction of

J>m(h)):

For a Wiener

phase

noise with the Lorentzian spectrum,

given

the one-sided 3 dB linewidth

,B,

the autocorrelation function ofthe ICI

weighting

function

41m(h)

can bedefined

by

RD(h1,h2)

E[4Pm(hi)4P*

(h2)]

and

R+(h1,

h2) =

6(h,-h2)N

2N

+-

Nez

- Nu N2 2-ezi -e-z1

1-N

+

Nez2

_eNu1 + -Iez2 (-

6(h,-h2)N)

Nu ( N2

1-e(z1-Z2)

1ezi 1- + 1e22 1eZ2J

(17)

where

Z1-u+

jVl

--

[/3T

+

jhl]

Z2--U + 2=--

[/3T

+

jh2]

(18)

and

(-)N

denotes the moduloby N, z-represents thecomplex

conjugate

ofz.

Proof:

cf.

Appendix.

U

As for

3,

since itcanbeobtained by preamble signal and has been proposed in the literature

[91,

we may assume that az is known

by

the receiverhenceforth.

B. MaximumLikelihood CPE Estimator

From the above

discussion,

the sufficient statistics 7mu

can be modeled as a complex Gaussian random vector with meanvector l.m(O)Hm,uXm,u andcovariance matrixCr,. Hence, the log-likelihood function shall be

(4Pm

()) 2R

{XmHHHC

rmn,u(D(0)}

-xmumuC H

Tofindthe ML estimation of m(0),wemay first let4Dm(0) =

Ame4O-

andlet thederivatives of (19) w.r.t. Am and

4,m

equal to zero to get

Xr

m_

_{X$,,uHH{,uC-uHmn.uxm,u}

Hum

urrnHmuz,

Then, the ML estimation of the CPE will be

xH HH C-1 rm,

41m(O)

= m,u m,u

rmc

m,u

H% H_{m um}H_",

C21

_rm,v

Hm,uXm,u

T° a a A a (20) I

o,,

E10

(21)

10

IV. PERFORMANCE SIMULATION

Theproposed maximum-likelihoodCPE estimator are

eval-uated in frequency selective slowly fading channels with 50 nsand75nsrmsdelay spread [10].Channelimpulse response

remains static within a frame containing 16 symbols, but varies independently from frame to frame. Transmitted data is constructed according to the IEEE 802.1 la standard [11]. 16QAM and64 QAM which are more sensitive to phase noise than M-PSK, are used in the simulation. The phase noise is

simulated using theLorentzianmodelwith 3equals to 1 kHz and 2 kHz. Pilot-aided approaches based on averaged-phase [5], weighted-averaged-phase

[6]

andleast-square criterion[7]

are also simulated as a comparison. Each simulation point is conducted by 3-105 OFDM symbols. The probability of

symbol error (SER) with 16 QAM and 64 QAMin different

channels are shown in Figure 2 andFigure 3 respectively.

It is easy to observe that Wiener phase noise causes an

irreducible errorfloorto OFDMreceiverperformance, which

isunacceptable in practice. Abouttheeffect ofdifferent chan-nels, comparing Fig. 2(a)with Fig. 2(b),we canfind that the shorterrmsdelayspreadgivesthebetterperformancewhich is

also evident in

Fig

3. Observing Fig. 2 which corresponds to 16QAM modulation, in general, the

proposed

ML estimator and the LS estimator

greatly outperform

the

averaged-phase

and

weighted-averaged-phase

approach.

The

performance

gaps

between the

proposed

ML estimator and the LS estimator at

moderate SNRare1 dBunder1 kHzlinewidth and2 dBunder

2 kHz linewidth

respectively.

In Fig. 3 which corresponds to 64

QAM

modulation, we can observe that the

performance

of the three conventional approaches get closer and the

performance

gap between the

twoaveragebased

approaches

and the

proposed

MLestimator

become smaller

compared

to 16

QAM.

And the

performance

gapsbetween the

proposed

MLestimatorand theLSestimator

at moderate SNR are 2.5 dB under 1 kHz linewidth and 2 dB under 2 kHz linewidth respectively. This phenomenon is mainly caused by the shrink of the decision

boundary

when

employing highordermodulation. V. CONCLUSIONS

In this paper, a CPE estimator to effectively remove the complexgain caused by Wienerphasenoiseonthe frequency

domain transmitted symbols is proposed. We systematically

derive the autocorrelation function of the ICIweighting func-tionbasedontheLorentzian model. The secondorder statistics

\sus A,... ~ ~5, ~ X

I.A,

a . Ew s \s,-o 1015 20 25. 10 is 20 25 SNR(Eb/No,dB)

(a)rmsdelayspread=50 ns 10° 2 u a a. 10 0-II .I

,1

I1 5., \,, 1012 10 is 20 SNR(Eb/No,dB) Nocorrection --2k Hz, avg. 3=2kHz,w-avg. 3=2kHz,LS - -2kHz,ML 3=1kHz, aVg. - I=1kHz,w-avg. + f3=lkHz,LS * 3=1kHz,ML * Ideal I -I,.2 30 3 -a-Nocorrection 032kHz, avg. 1-32kHz,w-avg. ..=2kHz.LS -12kHz, ML [3=lkHz.avg. a 1=1kHz,w-avg. 33=1kHz, LS I =lkHz, ML Ideal (b) rmsdelay spread=75 ns

Fig. 2. SER Performance of the CPE correction schemes with 16 QAM of the ICI can be obtained by this autocorrelation function. Different from theconventional ML

approaches

whichdo not

considerthestatistic of Wiener

phase noise,

we

investigate

the

covariances between theICIobservedondifferent

subcarriers

to yield a generalized maximum likelihood estimator. The

effectiveness oftheproposed

algorithm

ismanifested

by

simu-lations and is shown to

outperform

the conventional schemes.

APPENDIX PROOF OF PROPOSITION 1

By definition, considering (6), the autocorrelation function of

4)(h)

canbeexpressed as

N-1 N E

[eJ(0.(ni)-O.(n2))]

h2n2-N2

ns=Ofl30

(22) By(10),

q.m(ni)

-

4)m(n2)

canbe treatedasaGaussian

ran-dom variable withzeromeanand variance

4r3TInj

-n2l

/N.

Therefore,

E[e

j(e

(nh)-O.¢(n2))1

=

e-27r3T

N (23)

6 aG

,5

D -9 DQ 10 10l E '.: 'o 10 5 10 15 20 25 30 35 SNR(Eb/No,dB)

(a) rmsdelayspread= 50ns

V.,5 ^--b-0- ^Noc c {correction 332kHz, avg. ..-2kHz, w-avg. .=2kHz, LS - =2kHz,ML .=1kHz, avg. "':4--, * 33l~~~~~~~~~-kHz, w-avg. P-=1kHz,LS \, 4--33=lkHz, ML Ideal 5 1 20 25 30 35 _* 5 10 15 20 25 30 35 SNR(Eb/N,.dB) (b)rmsdelay spread=75ns

Fig. 3. SER Performance of the CPE correctionschemes with 64QAM

Substitute (23) into (22), we have

N-1 N-1 2 3T 'N- 272 h2n2-hl

R4)(hl,h2)= -27r

N2i=

e (24)

nl=On2=o

Let

N2RD(hl,

h2) bedenoted by SI, then divide the double summation intotwo parts as follows

N-1 N-1nl-i

SI = ei N + 5e-2r3T

n=O nl=In2=0 (25)

(e N +2e& Ne),

It is easy to show that the first summation in (25) is

N6(hl-h2)N-

Let the double summation in (25) be denoted by S2 and use changeofvariableby letting t =ni - n2, we

have

N-iN-1-t

S2 N e-j2 +ej2 e N

t=i n2=O

(26)

Carry

out

the

summation

of

n2, itbecomes

-N-1

S2

=6(hl-h2)N

E

(N

-

t)

(e-j2ir4

+

ei2-

4)

t=1 (1

-6(hl-h2)

-ej2i

e) i +

ej27rha.

h; L e322rT ±ej2~~

e27r4

) 7 -27r)3Tt e-

N-(27)

In (27),we maydenote the first andthe second summation by

Q,

and Q2 respectively. Since They are composed of

geometric series, by the definition in (18), after some

ma-t

nipulation,

L5

1-N+Nez -eNu I- N+Nez2- Nu

Qi- 2-ez1 -e-z 2-ez2 - e-z2 (28)

and

Q2= (1-Nu) ( 1 -=fezi + 1-eZ2 1 (29)

Finally,

Rp

(hi, h2) can beiteratively evaluatedandgiven by

S1

N`(hl-h2)N

+S2 Ro(

=N2

= N2

6(hs-h2)N(N

+Qi) (1-

6(ho-h2)N)Q2

N2 N2(1ez -Z2) (30) Q.E.D. REFERENCES

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10 e---s--- 4--o---4---- 2 -iaEa-- No correction 1 =2kHz, avg. . -- ,B=2k Hz,w-avg -=2k Hz, LS -33=2kHz, ML X.=1kHz, avg. 2\ ^>\>,<=* 3=1kHz,w-avg. a-4-.,=1kHz, LS ; 3:=lkHz, ML Ideal X4 *X*,* 2 I -1 i

f