IEEE SIGNAL PROCESSING LETTERS, VOL. 16, NO. 6, JUNE 2009 493

Inter-Carrier Interference Estimation in OFDMSystems With Unknown Noise Distributions

Jaechan Lim and Daehyoung Hong, Member, IEEE

AbstractThere are a number of approaches to estimating car-rier frequency offset (CFO) that causes inter-carrier interference(ICI) in orthogonal frequency division multiplexing (OFDM) sys-tems: self-cancelation method, the extended Kalman filter (EKF),particle filter, filter (HF), etc. In particular, the HF is of in-terest because prior statistical noise information is not necessarilyrequired in its application. Cost reference particle filter (CRPF),newly developed in the particle filtering framework, has the samefeature as HF; it also does not require the prior noise informationof the state and the measurement equation. In this letter, we com-pare and analyze the performances of two similar methods. Thesimulation results show that CRPF outperforms HF, particularlywhen the bit energy to noise ratio of the measurement is low. There-fore, CRPF is very effective and robust, especially when the noisestatistics are unknown with a low bit energy to noise ratio.

Index Terms filter, carrier frequency offset (CFO), costreference particle filter (CRPF), inter-carrier interference (ICI),orthogonal frequency division multiplexing (OFDM).

I. INTRODUCTION

T HE ORTHOGONAL frequency-division multiplexing(OFDM) scheme is popularly employed for modernwide-band digital communications such as digital television,wireless networking, broadband internet access, etc. due to itsvarious advantages. However, it also has a few disadvantages,e.g., carrier frequency offset (CFO), sensitivity to frequencysynchronization, etc. In this letter, we focus on the first one,i.e., CFO which causes inter-carrier interference (ICI) in thesystem.

For many problems in engineering or statistical science, wecan model and describe them by the dynamic state system (DSS)model where the states of interest are correlated in time or space.Based on the DSS model, there are numerous approaches to esti-mating the states of interest dynamically with time or space. TheCFO estimation problem also can be well described by the DSSmodel, and a number of approaches are proposed to estimateCFO to combat ICI [1][3]. The Kalman filtering, specificallythe extended Kalman filtering (EKF) is an effective and popularmethod to combat ICI in the literature. Particle filtering andfilter (HF) are also employed besides the EFK [2], [3]. In partic-ular, HF is of interest because we can apply it without the noise

Manuscript received January 21, 2009; revised February 19, 2009. First pub-lished March 16, 2009. Current version published April 24, 2009. The associateeditor coordinating the review of this manuscript and approving it for publica-tion was Dr. Z. Jane Wang.

The authors are with the Department of Electronic Engineering, Sogang Uni-versity, Seoul, Korea (e-mail: jaechan@gmail.com; dhong@sogang.ac.kr).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/LSP.2009.2017571

information of the state and the measurement equations. Ref-erences, [4] and [5] show that HF outperforms the EKF whenthe noise distributions are unknown. The cost reference particlefilter (CRPF), recently developed in the particle filtering frame-work [6], has the same feature as HF; the prior noise distribu-tions are not necessarily required when we apply it. In this letter,we compare and analyze the performances of two methods thathave the common feature, for estimating CFO in OFDM sys-tems. Simulation results show that CRPF outperforms HF, es-pecially when the bit energy to noise ratio ( where isthe bit energy, and is the noise power spectral density.) islow. Therefore, CRPF is effective and robust, particularly whenthe noise information is not available with a low level of .

II. OFDM SYSTEM MODELThe binary bits are mapped to symbols on the complex signal

constellation space. Grouped symbols are modulated onto sub-carriers in the form of inverse fast Fourier transform (IFFT) asfollows:

(1)

where is a symbol, and is the number of sub-carriers.The cyclic prefix is added up to the signal to mitigate the in-tersymbol interference, and removed at the receiver. Then thereceived signal is expressed in the time domain as follows:

(2)

where is a normalized CFO (meaning the relative offset from acarrier frequency); is the -tap channelimpulse response; and is unknown additive noise. We wantto estimate based on the received signals, and then decode thesymbols.

The dynamic state system (DSS) that describes the hiddenstate and observed measurement with additive noise pro-cesses of and at time is expressed as follows:

(3)(4)

where and are the state transition and the observation func-tion, respectively, which are known. Then, the correspondingDSS and the measurement equations for the problem can be ex-pressed as follows:

(5)(6)

1070-9908/$25.00 2009 IEEE

494 IEEE SIGNAL PROCESSING LETTERS, VOL. 16, NO. 6, JUNE 2009

where we can assume CFO is a constant within the data frame( is from 0 to ) since each OFDM frame is short enough.Then, from (2), we obtain

(7)

Now, we are ready to apply filter, CRPF, and other methodsto estimating CFO with known preamble symbols ; wealso assume the channel impulse response is known.

III. SOLUTIONS

A. Filtering

According to the game theory approach to filtering, thecost function is defined as follows with the performance bound

which assists to make it tractable to minimize the cost function[4]:

(8)

where , , , and are the weight parameters thatare positive definite matrices; is the number of total timesteps; denotes the vector norm; and implies

. Because the state equation does not have the ad-ditive process noise in (5), (8) is modified for the problem asfollows:

(9)

Then, minimax problem is to solve

(10)

where . Then, the following steps are constructedafter the measurement equation is linearly approximatedby the Talyor expansion [5].

1) Initialize the performance bound ; the initial esti-mate of CFO ; the weight parameters , , and

where ; and computewhere .

2) For , recursively take the following steps: Compute . Compute

where denotes complex conjugate, and.

Update the estimate.

B. Cost Reference Particle FilterCRPF also has the same feature as HF; it does not require

the noise information about the state and the measurementequations [7]. From DSS equation (3) and measurement (4), ifwe rewrite them for the sake of convenience with the time index

instead of here, we obtain

(11)(12)

The cost function in CRPF, which is recursive additivestructure and corresponds to the weight in standardparticle filtering (SPF) is defined as:

where is the forget-ting factor which makes it possible to adaptively change theamount of contributions of past particles in evaluating costfunction, and is the incremental cost function whichindicates the accuracy of the estimate of given . The costfunction is a measure of estimate quality like the weightas in SPF. Similarly to SPF, the cost-based random measureis represented by a set of particles and associated costs as:

where stands for ,is the particle index, and is the number of particles. Be-

sides the cost function, the risk function is defined in CRPFas:

where ; a good choice of the risk func-tion is . The risk function measuresthe adequacy of the estimate given the observation ,and is also a prediction of the cost increment ;the cost increment can be computed by .Based on these definitions, the sequential algorithm proceedswith time, recursively repeating the steps of risk evaluation,resampling, particle propagation, and updating the cost.The steps of CRPF algorithm are summarized in Table I.CRPF can be easily adopted to the estimation of CFO basedon (5) and (6); is an identity function in the problem; and

. The rest of thesteps are straightforward following Table I.

C. The Extended Kalman FilterWe compare the performance of the extended Kalman filter

against those of HF and CRPF as well; noise statics are knownfor EKF. The steps of the EKF for estimating CFO are straight-forward; however, it should be noted that the correction step attime is specified as follows [2]:

(13)

where is the estimate of at time given observations upto the time , and is the Kalman gain at that time.

D. Recursive Least-Squares (RLS) MethodWe also compare the performance of the RLS method with

those of other methods because RLS method is a traditional ap-proach which also has the common feature that we do not have toconsider the noise information. It is very advantageous to applythe RLS method to estimating CFO in the problem because

LIM AND HONG: ICI ESTIMATION IN OFDM SYSTEMS 495

TABLE ICOST REFERENCE PARTICLE FILTER ALGORITHM

does not vary with time, which makes it easily tractable to com-pute that minimizes the squared errors even though it does notguarantee a satisfactory performance. We define the squared er-rors up to the time as

(14)

where is a weighting factor. If we express the complex-number measurement as , then the that mini-mizes can be computed from the following equation:

(15)

E. Particle FilteringWe may also apply standard particle filter (SPF) [8] to the

problem, and compare its performance with those of previouslymentioned methods even though SPF requires noise informa-tion. However, there is a technical problem to apply SPF if weuse the prior density ( , where denotes the particleindex) as the proposal distribution. In the DSS equation of theproblem, the state is assumed to be a deterministic constant,and does not vary with time; consequently, the prior density isthe dirac delta function of . Therefore, identical parti-cles are always generated, and do not converge to the true stateonce particles initially start from a wrong state (e.g., 0.1 as inthe simulations of the other methods in later section.) There-fore, in order to apply SPF, a proposal distribution that dependson not only the previous state but also the current measurementhas to be used, other than the prior density. However, it is oftennot easily tractable to use a proposal distribution other than theprior density. Therefore, we do not consider SPF in this letter.

Fig. 1. BER performances of the methods.

Fig. 2. BER performances in linear scale.

IV. SIMULATIONSBy computer simulations, we illustrate the performance com-

parison of the methods, especially CRPF and HF. We also com-pare the performances of the EKF and RLS methods againstthose of the previously mentioned techniques. We consider asingle antenna, 64-subcarrier OFDM system; the binary phaseshift keying (BPSK) scheme is employed for the simulation.Various additive Gaussian noise power levels are applied to themeasurement equation for different ratios of . The nor-malized CFO is set to 0.35 that we estimate. The parametersfor HF are: the initial state estimate ; ; ;

; and . The parameters for CRPF are: the numberof particles is 500; ; ; ; ; and theinitial variance for the Gaussian propagation density is0.0053. The 500 identical, initial estimates are generated forCRPF. A similar result is obtained when we apply 200 parti-cles for CRPF, and it may not considerably affect the perfor-

496 IEEE SIGNAL PROCESSING LETTERS, VOL. 16, NO. 6, JUNE 2009

Fig. 3. RMSE when estimating CFO .

mance anymore when we increase the number of particles morethan 500. The initial error covariance for the EKF is 0.1. Theweighting factor , and the nonlinear equation (15) isiteratively solved by using MATLAB function fsolve for theRLS method. 3000 independent, 64 preamble symbols, whichare known at the receiver, are generated for the performanceevaluation of the methods. The bit error rate (BER) is computedand depicted along with various for each method inFigs. 1 and 2: except for the RLS method, all the other methodsshow almost identical BER performances in Fig. 1; however,we find that CRPF outperforms HF, particularly at the levels of

below 12 dB if we convert the figure in linear scale asshown in Fig. 2. The simulation result of root mean squared error(RMSE) is shown in Fig. 3: the result of CRPF is significantlylower than that of HF under up to 12 dB of ; and the resultof CRPF approaches to that of EKF very closely throughout theentire range of the levels. Fig. 4 shows a particular real-ization of a simulation for each method when is 30 dB,where the RLS method shows relatively poor converging per-formance. Obviously, the RLS method shows the worst perfor-mance in the simulation result.

V. SUMMARY AND CONCLUSION

In this letter, we compared the performances of two methods(HF and CRPF) that have the common feature that they do notrequire the noise statistics when we estimate CFO which causesICI in OFDM systems. The simulation results show that both HFand CRPF estimate CFO very effectively with similar BER per-formances. However, CRPF outperforms HF particularly when

Fig. 4. Realization of each method when estimating CFO where in the realization.

the level of is low. Therefore, we can conclude thatCRPF is a very robust estimator when the noise statistics arenot available with a low level. Nonetheless, there are im-portant issues to take into account in the application of these ap-proaches; both methods are very sensitive to the parameter ini-tialization depending on the nature of the problem under studyor the scale of the parameters we want to estimate. Especially,the weighting factors , and have to be properly selectedfor HF. The initial variance of the propagating density hasto be very carefully selected when applying CRPF. These issueshave to be studied and investigated carefully in the future.

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