Kalman filter-based channel estimation and ICI suppression for high-mobility OFDM systems

INTERNATIONAL JOURNAL OF COMMUNICATION SYSTEMSInt. J. Commun. Syst. 2008; 21:1075–1090Published online 23 June 2008 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/dac.938

Kalman filter-based channel estimation and ICI suppression forhigh-mobility OFDM systems

Prerana Gupta∗,† and D. K. Mehra

Department of Electronics & Computer Engineering, Indian Institute of Technology Roorkee, Roorkee 247667,Uttarakhand, India

SUMMARY

The use of orthogonal frequency division multiplexing (OFDM) in frequency-selective fading environmentshas been well explored. However, OFDM is more prone to time-selective fading compared with single-carrier systems. Rapid time variations destroy the subcarrier orthogonality and introduce inter-carrierinterference (ICI). Besides this, obtaining reliable channel estimates for receiver equalization is a non-trivial task in rapidly fading systems. Our work addresses the problem of channel estimation and ICIsuppression by viewing the system as a state-space model. The Kalman filter is employed to estimate thechannel; this is followed by a time-domain ICI mitigation filter that maximizes the signal-to-interferenceplus noise ratio (SINR) at the receiver. This method is seen to provide good estimation performance apartfrom significant SINR gain with low training overhead. Suitable bounds on the performance of the systemare described; bit error rate (BER) performance over a time-invariant Rayleigh fading channel serves asthe lower bound, whereas BER performance over a doubly selective system with ICI as the dominantimpairment provides the upper bound. Copyright q 2008 John Wiley & Sons, Ltd.

Received 26 April 2007; Revised 7 March 2008; Accepted 7 March 2008

KEY WORDS: OFDM; inter-carrier interference (ICI); channel estimation; Kalman filtering

1. INTRODUCTION

Orthogonal frequency division multiplexing (OFDM) is a popular modulation scheme employedfor high data rate communication over wireless channels. This is primarily because of its ability toconvert a frequency-selective fading channel into a flat fading one, thereby reducing the equalizerto a single tap filter. The operation of OFDM systems relies on the orthogonality of subcarriers.However, mobility-induced Doppler spread perturbs the orthogonality between subcarriers andleads to inter-carrier interference (ICI), wherein the symbol received on each subcarrier is affected

∗Correspondence to: Prerana Gupta, Department of Electronics & Computer Engineering, Indian Institute of Tech-nology Roorkee, Roorkee 247667, Uttarakhand, India.

†E-mail: [email protected]

Copyright q 2008 John Wiley & Sons, Ltd.

1076 P. GUPTA AND D. K. MEHRA

by symbols transmitted on all the subcarriers. Several methods have been proposed in the literatureso far for ICI mitigation and effective equalization.

Zhao and Haggman [1] have described a self-cancellation method for OFDM systems, in whichtransmit data symbols are coded across multiple subcarriers. The signals received on adjacentsubcarriers are linearly combined with appropriate coefficients to minimize the residual ICI. It isshown that by using just two or three adjacent symbols, the method provides substantial improve-ment in the carrier-to-interference power ratio compared with a system without ICI cancellation.However, the method incurs a bandwidth penalty and requires the use of higher-order modulationschemes to avoid it. Choi et al. [2] have proposed the time-domain channel estimation usingfrequent pilot symbols interspersed between data. Symbol detection is done based on variouscriteria and the best performance is achieved using minimum mean-squared error (MMSE) crite-rion with successive detection. In [3], the authors have analyzed ICI and inter-symbol interference(ISI) for large delay spread systems, with insufficient cyclic prefix (CP), and presented an iterativemethod for joint ICI and ISI mitigation. The method offers substantial gains within two to threeiterations, provided the channel state information (CSI) is available at the receiver. Chen and Yao[4] have proposed a technique for ICI mitigation and channel estimation based on pilot tonesassuming that the channel taps vary linearly over an OFDM symbol.

Seyedi and Saulnier [5] have proposed a general ICI self-cancellation technique based on time-domain windowing. The method performs well for systems affected by frequency offset as well asDoppler spread. However, it assumes the availability of CSI and perfect equalization. Stamouliset al. [6] have developed a model for the analysis of ICI in single input single output and multi-input multi-output (MIMO)-OFDM systems. The signal-to-interference plus noise ratio (SINR)metric is defined for these systems and a time-domain ICI mitigation filter is designed based onthe maximization of this SINR. The efficiency of this filter depends on the reliability of channelestimates available at the receiver. Pilot tones have been employed for channel estimation and theiroptimal placement within an OFDM symbol has been studied. In [7], a two-stage ICI removalscheme is proposed assuming linear variation of the CIR over a symbol period. In the first stage,a set of pre-filters are used to compensate for multiplicative distortion and form estimates that areused by the parallel interference cancellation equalizer in the second stage. Huang and Wu [8]proposed a time-varying MMSE-based ICI coefficient estimator, together with an MMSE-basedfrequency-domain Q-tap equalizer (which reduces the complexity compared with conventionalMMSE) to suppress ICI for a better detection performance.

The review of recent work emphasizes that the use of OFDM in rapidly time-varying environ-ments is limited by ICI and reliable CSI should be available at the receiver in order to effectivelyemploy any ICI mitigation scheme. Several techniques have been studied in the literature for theestimation of time-varying channels. Most of them are training-based techniques making use ofpilot tones and/or symbols [9–11]; there are also semi-blind techniques, such as expectation-maximization [12, 13], that make an initial training-based estimate and improve upon it; and thereare blind techniques such as subspace-based methods that rely totally on the statistics of the receivedsignal. However, for rapidly time-varying systems, the pilot-based estimation schemes requirefrequent retraining and are wasteful of bandwidth, whereas the semi-blind and blind approachesinvolve large computational complexity and latency.

Cai and Giannakis [14] derived a matched-filter bound for OFDM over doubly selective Rayleighfading channels, which benchmarks the best possible performance with complete ICI cancellation.ICI power leakage onto different subcarriers is studied and low-complexity MMSE and decisionfeedback approaches for ICI cancellation are proposed. However, availability of reliable channel

Copyright q 2008 John Wiley & Sons, Ltd. Int. J. Commun. Syst. 2008; 21:1075–1090DOI: 10.1002/dac

KALMAN FILTER-BASED CHANNEL ESTIMATION AND ICI SUPPRESSION 1077

estimates is assumed. Al-Gharabally and Das [15] analyzed the performance of OFDM systemsin time-varying channels with channel estimation error. An analytical expression for the averageprobability of error is obtained by finding the joint probability density function (PDF) of thechannel and its estimate, followed by averaging the conditional error probability over the jointPDF; this usually requires solving a threefold integral. The authors prove the asymptotic Gaussiannature of the ICI component and propose a simple approach to derive the bit error rate (BER) forOFDM systems in time-varying channels with channel estimation error. Kaioukov et al. [16] haveproposed a pilot-based channel estimator for MIMO-OFDM systems in rapid TV environments.The work includes Doppler frequency estimation, delay path profile estimation and interpolationin time and frequency domains. The method when tested over the pilot structure of IEEE 802.16eprovides good performance at high Doppler spreads but involves a large latency and computationalcomplexity.

Estimation of a rapidly varying channel and equalization at the receiver are two main hurdlesencountered when applying OFDM to high-mobility systems. It is evident that channel estimationis a non-trivial task for high-mobility systems, unless some simplifying assumption, such as linearvariation of channel taps, is made. This paper addresses the joint problem of channel estimationand ICI suppression in high-mobility OFDM systems. In our work, the Kalman filter, which isknown to provide the optimum solution to linear filtering problem, has been employed for channelestimation. The channel estimates so obtained are used to design an ICI mitigation filter. Themethod is seen to provide better performance in terms of SINR gain compared with [6]. Themotivation for this comes from [17], where the use of the Kalman filter for tracking Ricean fadingMIMO channels has been studied and an adaptive decision feedback equalizer (MIMO-DFE)architecture is proposed.

We have used the following notation in this paper. Boldface letters are used to denote matricesand vectors. Superscripts * and H denote complex conjugate and Hermitian operations, respectively.IL denotes an identity matrix of dimensions L×L , whereas OL×M denotes an L×M matrix ofall zeros. E[ ] denotes the expectation operation. The paper is organized as follows. Section 2describes the system model followed by a discussion of the proposed technique in Section 3.Appropriate bounds on the performance of the system are described and these comprise Section 4.Validation of the proposed technique is carried out through Monte Carlo simulations, which formsSection 5. Finally, the paper is concluded in Section 6.

2. SYSTEM MODEL

We consider an OFDM system as shown in Figure 1 with N tones or equivalently N/2 parallelsubchannels. Input data are buffered, converted to a parallel form and QPSKmodulated into symbolsXk , where Xk denotes the kth symbol and 0�k�N−1. OFDM modulation is accomplished bytaking N -point IDFT of the block X=[X0X1 . . . XN−1]T. A CP of length gi is appended to formthe transmitted block as x′ =[x−gi x−gi+1 . . . x−1x0x1 . . . xN−1]T, where x−i = xN−i for 1�i�gi .The signaling interval is taken as Ts and the useful OFDM symbol duration is T =NTs.

2.1. Doubly selective fading channel

Doubly selective fading channel (selective in both time and frequency domains) following wide-sense stationary uncorrelated scattering (US) is assumed. The continuous time-varying channel



Figure 1. Block schematic of the proposed OFDM system.

impulse response (CIR) is given by hc(t,�), which denotes the channel response at time t to animpulse applied at instant t−�. The concept of wide-sense stationarity (WSS) in the time variableimplies that the statistical description of the channel is independent of the absolute time, so that theautocorrelation function is only a function of the time lag. Similarly, US implies that the scattererscontributing to the different multipath components are mutually uncorrelated. This amounts toWSS in the frequency variable, i.e. the autocorrelation function is independent of the absolutefrequency; only the frequency separation counts. It is useful to discretize the continuous-time CIRby sampling it every Ts seconds. The discrete-time CIR is thus given by

h(n, l)=hc(t,�)|t=nTs,�=lTs

Assuming that the channel delay spread to be bounded by �max, the discrete number of channeltaps L is determined by ��max/Ts�. Thus, the channel tap vector at each instant of time is denotedby hn =[h(n,0)h(n,1) . . .h(n, L−1)], where h(n, l) is the lth tap at time instant n.

The classical time autocorrelation function, according to Jakes’ model, at a time lag of �t is�t (�t)= J0(2� fd�t), where J0(x) is the zeroth-order Bessel’s function of the first kind and fdis the maximum Doppler frequency. The classical Doppler spectrum obtained through Fouriertransform thus becomes ⎧⎨⎩(�

√f 2d − f 2)−1 for f � fd

0 otherwise(1)

In actual practice, the aforesaid autocorrelation function is difficult to achieve. An autoregressive(AR) process of order p [17–19] may be used to approximate the Bessel autocorrelation. Thisamounts to solving the system of p linear Yule–Walker equations. It is found that for an order aslow as 2, a good autocorrelation matching is achieved for time lags up to 20. This is more thansufficient for time-selective fading channels, especially for the high Doppler spread considered inour work. Thus, the classical Doppler spectrum for each of the L channel taps is approximated byan independent AR-2 process.

We therefore have

h(n, l)=−a1h(n−1, l)−a2h(n−2, l)+v(n, l) (2)

where a1 and a2 are the AR-2 coefficients and v(n, l) is the modeling noise for the lth tap at timeinstant n.



Equating the autocorrelation functions of Jakes’ model and the AR-2 model (at a discrete lagof m sampling intervals), we have

�(m)=E[h(n, l)h∗(n−m, l)]= J0(2� fdmTs) (3)

The above equation on expanding with Taylor series and equating the first few terms gives thepoles at p=(1−�d/�)e±j0.8�d .

It is found that a second-order AR process with a parameter f ′d is approximately equal to a

fading process with a maximum Doppler shift of approximately√2 f ′

d. The AR-2 coefficients a1and a2 are found as

a1=−2rd cos(0.8�d) and a2=r2d (4)

where rd=(1−�d/�) and �d=2� fdTs.For a close approximation between the AR-2 and Jakes’ model, the pole radius rd should be as

close to unity as possible.

2.2. ICI analysis

As already defined, the lth channel tap at any time instant n is represented as h(n, l). As thediscrete time index n varies from 0 to N−1 over an OFDM symbol, the channel taps h(n, l) ateach instant of time can be represented by an N×N matrix H for convenience, whose structureis as follows:

H=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

h(0,0) 0 . . . h(0, L−1) . . . h(0,1)

h(1,1) h(1,0) . . . . . . . . . h(1,2)

......

......

......

......

... h(N−2,0) 0

0 0 . . . h(N−1,1) h(N−1,0)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(5)

The above matrix representation takes into account the effects of a cyclic prefix (CP) as well.The symbol received at any time instant is corrupted by the fading channel and AWGN as

y(n)=L−1∑l=0

h(n, l)xn−l +z(n) (6)

Gathering the received symbols over an entire useful OFDM symbol duration, as the discrete timeinstant varies from 0 to N−1, we obtain the vector form as

y=HQHX+z (7)

where Q is the standard N -point DFT matrix, y and z are the received symbol and noise vectorsof dimensions N×1 over a symbol, respectively.



Demodulation involves taking DFT of the received block after removing the CP to get[X̂0 X̂1 . . . X̂ N−1]. The demodulated signal at the kth tone thus becomes [20]

X̂k = 1√N

N−1∑n=0

y(n)e−j2�nk/N

= 1√N

N−1∑n=0

L−1∑l=0

h(n, l)xn−le−j2�nk/N +

N−1∑n=0

z(n)e−j2�nk/N (8)

Using synthesis equation for symbols xn−l in (8) and interchanging the order of summation,we obtain

X̂k = 1

N

N−1∑m=0

N−1∑n=0

L−1∑l=0

Xmh(n, l)e−j2�ml/N ej2�n(m−k)/N +Zk (9)

The time-varying frequency response of the channel at time instant n is defined as

H(n,k)= 1√N

L−1∑l=0

h(n, l)e−j2�kl/N , 0�k�N−1 (10)

Substituting (10) in (9), we obtain

X̂k = 1

N

N−1∑m=0

N−1∑n=0

XmH(n,m)ej2�n(m−k)/N +Zk (11)

where Zk is the DFT of noise vector z at the kth tone.The above may be rewritten as

X̂k =N−1∑m=0

ICI(k,m)Xm+Zk (12)

where

ICI(k,m)= 1

N

N−1∑n=0

H(n,m)ej2�(m−k)n/N (13)

Separating the terms in (12), we find that the demodulated symbol at any tone has a contributionnot only from the desired symbol, but also from the symbols transmitted on all the other tones,apart from additive noise. Thus, (12) may be rewritten as

X̂k =g(k)Xk+N−1∑

m=0,m �=kICI(k,m)Xm+Zk (14)

where g(k)= ICI(k,k) is the gain of the desired symbol.

3. PROPOSED TECHNIQUE

The channel tap vector hn at each instant of time needs to be estimated. In our work, the systemis viewed as a state-space model, with channel vector hn comprising the unknown state of the



system. It can be seen that (2) provides the transition from one time instant to the next. Hence, itprovides the basis for forming the process equation (15) of the state-space model

Sn =ASn−1+Vn (15)

Here, the state to be estimated at time instant n, Sn comprises the channel tap vectors at twoconsecutive instants of time Sn =[hn−1 hn]T and is a 2L×1 vector.

Transition matrix

A=[OL×L IL

a2IL a1IL

]2L×2L

and the process noise vector

Vn =[O1×L...v(n,0)v(n,1) . . .v(n, L−1)]T

is a 2L×1 vector where v(n, l) is the modeling noise as in (2).Similarly, (6) describes the received symbol in terms of the transmitted symbol and the time-

varying channel taps. Thus, it provides the measurement equation (16) for the state-space model

y(n)=XnSn+z(n) (16)

Here, the measurement matrix

Xn =[O1×L... xnxn−1 . . . xn−L+1]1×2L

uses L transmitted symbols that affect the received symbol at any time instant. The measurementnoise vector z(n) comprises the AWGN sample at time instant n.

A Kalman filter is then employed to estimate the unknown state of the system, i.e. the channel tapsin the time domain. The received signal y(n) at each time instant is given as an input observationto the Kalman filter algorithm [19, 21]. The time-update equations (17) and (18) project the currentstate estimate ahead to the next time instant. The measurement-update equations (19)–(21) adjustthe projected estimate by the actual observation at that instant

Sn|n−1 =ASn−1 (17)

Kn|n−1 =AKn−1AH +Q1 (18)

Gn =Kn|n−1XHn [XnKn|n−1X

Hn +Q2]−1 (19)

Sn = Sn|n−1+Gn[y(n)−XnSn|n−1] (20)

Kn = [I2L −GnXn]Kn|n−1 (21)

where Gn is the 2L×1 Kalman gain vector at time instant n, Sn|n−1 is the state estimate at timeinstant n, given the observations up to time n−1, Q1 and Q2 are the 2L×2L and 1×1 sizecovariance matrices of Vn and z(n), respectively, and Kn is the covariance matrix of estimationerror. The state vector and covariance matrix for the Kalman filter are initialized as

S0|−1 = E[S0]K0|−1 = E[[S0−E[S0]][S0−E[S0]]H ]



When the channel taps are of zero mean, the above conditions simplify to

S0|−1 = 02L×1

K0|−1 = E[S0SH0 ]=�2hI2L

where �2h is the variance of channel taps.The time-domain channel taps for each instant of time are extracted from the state estimate at

that instant. Applying the above algorithm for time instants 0�n�N−1, channel tap estimatesover an entire OFDM symbol are obtained. These are used to form the channel tap matrix H, asin (5), followed by the design of a time-domain ICI mitigation filter [6].

We consider that the ICI suppression filter is expressed in the form of an N×N matrix W andis applied to the received symbol in the time domain prior to demodulation. We may then expressthe demodulated symbol using (7) as

X̃=QWHQHX+QWz (22)

We denote QWHQH by G and separate terms as in (14) corresponding to the desired symbol andremaining symbols (which contribute to ICI). Thus, for the kth tone, we obtain

X̃k =G(k,k)Xk+N−1∑

m=0,m �=kG(k,m)Xm+ Z̃k (23)

where Z̃=QWz.The autocorrelation function of Z̃ is found as

RZ̃Z̃ = E[Z̃Z̃H ]=QWE[zzH ]WHQH

= �2QWWHQH (24)

The variance of Z̃ is thus given by

�2Z̃ = �2

Ntr(QWWHQH )

= �2

Ntr(WWH ) (25)

Using (23) and (25), we define the SINR at the mth tone [6] as

SINRm = Ex |G(m,m)|2

Ex∑

n �=m |G(m,n)|2+ �2

Ntr(WWH )

(26)

where Ex is the input symbol energy per tone (assuming equal distribution over all subcarriers), thenumerator and the first term in the denominator denote the energy in the signal and ICI component,respectively, whereas the second term in the denominator gives the noise variance.



Expressing the above in terms of unit vectors, we obtain

= Ex |eHmGem |2�2

Ntr(WWH )+Ex

∑n �=m |eHmGen|2

where em is the mth unit vector of dimensions N×1, consisting of all zeros and 1 as the mth entry.The above simplifies to

SINRm = wHm hmhH

mwm

wHm

(1

SNRIN +HH

H −hmhHm

)wm

(27)

where

hm =HQHem and wm =WHQHem

Optimizing the SINR expression in (27) and using matrix inversion lemma, the optimum filtercoefficients [6] for each frequency bin 0�m�N−1 are obtained as

w̃m =R−1yy HQHem (28)

where Ryy is the autocorrelation matrix of the received symbols, obtained from (8) as

Ryy = E[yyH ]

= 1

SNRIN +HH

H(29)

The coefficients so obtained are scaled to unit norm to obtain wm,o. The expression in (28) showsthat the filter coefficients for each frequency bin are a function of the channel tap matrix H. Thisillustrates the significance of obtaining reliable channel estimates at each instant; the better theestimate of H, the more reliable the resulting ICI removal process. The filter coefficient vectorwm,o obtained for each tone is stacked to form an N×N matrix W as follows:

W=QH

⎡⎢⎢⎢⎢⎢⎢⎣

wH0,o

wH1,o

...

wHN−1,o

⎤⎥⎥⎥⎥⎥⎥⎦ (30)

This filter matrix W is used in (22) for ICI mitigation. Thus in the proposed system, the receivercarries out channel estimation and ICI mitigation tasks (Figure 1) prior to demodulation. Whenthe optimum filter coefficients wm,o are employed for equalization, the corresponding SINR forthe mth bin is found to be

SINRm,o= hHmR−1

yy hm

1−hHmR−1

yy hm(31)



Denoting scalar hHmR−1

yy hm by p, we may express

SINRm,o= p

1− p

However, this SINR is a function of channel tap matrix H. With an estimated channel tap matrixavailable at the receiver, the resulting SINR may be expressed as

SINRm,e= p̂

1− p̂

in which

p̂ = [eHmQ(H+�H)H ][

1

SNR+(H+�H)(H+�H)H

][(H+�H)QHem]

= p+�p

where

�p = hHmR−1

yy �HQHem+hHm (H�H

H +�HHH +�H�H

H)(H+�H)QHem

+eHmQ�HH[

1

SNR+(H+�H)(H+�H)H

](H+�H)QHem

and �H is the estimation error in H.The SINR resulting from the estimated channel matrix is therefore

SINRm,e= p+�p

1−(p+�p)

and it differs from the optimum SINR by an amount

SINRm,o−SINRm,e = p

1− p− p+�p

1−(p+�p)

= − �p

(1− p)(1−(p+�p))(32)

The accuracy of channel estimates thus determines the effective design of the ICI removal filter.During the training phase, the transmitted symbols are known to the receiver and form the input

to the channel estimator. Subsequently, the operation is switched to the decision-directed mode,where a decision on the transmitted symbols is made using the one-step-ahead prediction givenby the Kalman filter. To avoid error propagation in decision-directed mode, especially when theDoppler spread is very high, periodic retraining is used and the estimates are refreshed.

4. PERFORMANCE BOUNDS

4.1. Lower bound: frequency flat Rayleigh fading channel

The performance of OFDM systems over a time-invariant (TIV) Rayleigh fading channel serves asa lower bound for analyzing the performance of the proposed technique. The general expression



for the probability of a bit error for an M-ary QAM transmission over the AWGN channel is[22, 23]

PBE=(1− 1√

M

)·erfc

⎛⎝√ 3�2Ex

2(M−1)N0

⎞⎠ (33)

where erfc() is the complementary error function defined as

erfc(u)= 2√�

∫ ∞

ue−z2 dz

Here, Ex , the average input energy per tone, is also the average symbol energy, N0 is the powerspectral density of AWGN and � is the attenuation factor assuming the channel to be frequencynon-selective and TIV over a symbol transmission. The probability of a bit error for QPSK as wellas 4-QAM transmission is obtained as

PBE= 1

2erfc

⎛⎝√�2Ex

2N0

⎞⎠ (34)

For QPSK, the bit energy and symbol energy are related as Eb=Ex/2, and therefore we have,

PBE= 1

2erfc

⎛⎝√�2Eb

N0

⎞⎠Denoting the received signal-to-noise ratio (SNR) �2Eb/N0 by �, we obtain

PBE= 12 erfc(

√�)=Q(

√2�) (35)

where Q() is the Q-function defined as

Q(u)= 1

2erfc

(u√2

)Since � is Rayleigh distributed, �=�2Eb/N0 is chi-square distributed with two degrees of freedom.

Thus, the PDF of � is

p(�)= 1

�e−�/� for ��0 (36)

� being the average value of the received SNR.The average probability of bit error over Rayleigh fading channel is obtained by averaging the

error probability in (35) over the PDF of �, i.e.

PBE =∫ ∞

0PBE(�) · p(�)d�

= 1

�

∫ ∞

0Q(√2�)e−�/� d� (37)



Integrating the above by parts, we obtain

PBE= 1

2

(1−

√�

1+�

)(38)

which is the theoretical lower bound for the probability of error performance.

4.2. Upper bound: doubly selective channel

The aforesaid analysis provides the BER performance over a TIV Rayleigh fading channel thatserves as a lower bound for the performance of this system. A more realistic bound on theperformance of such systems may be obtained if the rapidly time-varying nature of the channel,and the ICI so introduced, are taken into account. For this purpose, the received SNR in (35) isreplaced by the instantaneous SINR defined in (39), since ICI is the dominant impairment in suchsystems. The received SINR on mth tone is defined from (14) as

�m,ICI=Ex |g(m)|2

Ex∑N−1

k=0,k �=m |ICI(m,k)|2+�2(39)

where g(m) and ICI(m,k) may be obtained from (13) and (14) and the fading channel taps h(n, l)are Rayleigh distributed. Thus, to evaluate the average probability of bit error, we replace � in (35)and (37) by �m,ICI, which in turn is a function of h(n, l). Also, the PDF p(�) is replaced byRayleigh density of the channel (instead of chi-square distribution mentioned in (36)). The limitsof the integral remain the same, i.e. from 0 to ∞. However, the closed-form solution for thisintegral is difficult to obtain. To achieve an approximation to this result, we have relied on anensemble-averaging approach through simulation; where, the probability of bit error

PBE(�ICI)= 12 erfc(

√�ICI) (40)

is a function of the randomly varying channel taps and

�ICI=(N−1∏m=0

�m,ICI

)1/N

The probability of error in (40) is evaluated and averaged over several independent realizations ofthe channel.

5. SIMULATION AND RESULTS

During the simulation we have assumed an OFDM system with 128 subcarriers. The modulationscheme used is 4-QPSK with equi-probable modulated symbols given by Xk = 1√

2(±1± j). A 4-tap

channel is generated where each tap is a complex Gaussian with zero mean and unit variance. Eachtap is independently governed by an AR-2 process and updated at each time instant in accordancewith (2). The system is affected by a complex additive white Gaussian noise with variance �2.A carrier frequency fc=2.4GHz is assumed and the useful OFDM symbol duration T is 0.5ms.Mobile velocity of 90 km/h thus corresponds to a Doppler spread fd=200Hz. High-mobility



systems are considered with a large Doppler spread ( fdT>0.01). Initially, one symbol is sent fortraining and subsequently the operation is switched to the decision-directed mode; the process isrefreshed by sending a training symbol periodically. The tracking behavior of the Kalman estimatorfor a channel tap is shown in Figure 2 for fdT =0.1 at a received SNR of 20 dB. The estimatoris seen to track the channel effectively at a high Doppler spread.

Figure 3 shows the BER performance of the proposed ICI mitigation scheme for different valuesof fdT , namely, 0.01,0.05,0.07 and 0.1. It is evident that the ICI removal scheme works wellin time-varying environments. At fdT =0.01, it suffices to use nearly 8% training, i.e. 1 pilotsymbol in 12 data symbols, whereas the requirement increases to 16% at fdT =0.1. The analytical

Figure 2. Channel tracking performance of the Kalman-filter-based estimator, fdT =0.1, SNR=20dB.

Figure 3. Probability of error performance of the proposed scheme at different Doppler frequencies.



Figure 4. Comparison of the SINR gain metric.

result for a TIV Rayleigh fading channel is obtained using (38). The curve is shown in Figure 3,which serves as a lower bound for the performance of this system. Besides this, the bit errorrate performance over a doubly selective channel in the presence of ICI (as described in Section4.2) is obtained by ensemble averaging over 100 independent channel realizations, and plotted inFigure 3 for a Doppler spread of fdT =0.1. The curve is seen to provide an upper bound for theperformance achieved by our scheme, except at high SNR, when it drops close to the actual curve.

The performance of the proposed scheme is also compared with the scheme in [6]. In thiscase, the simulation parameters are the same as those used in [6], namely, carrier frequencyfc=2.4GHz, symbol duration T = 1

3125 s and mobile velocity v=60mi/h. These correspond to aDoppler spread of fd=214Hz and equivalently fdT =0.07. In [6], the ICI mitigation filter makesuse of pilot-based estimates followed by different types of interpolations, whereas in our systemthe Kalman-based estimation is employed. The average SINR gain, as defined in (41), is used asa figure of merit for comparison:

SINR gain=(∏N−1

m=0 SINRWm∏N−1

m=0 SINRm

)1/N

(41)

where SINRWm and SINRm are the SINRs at the mth subcarrier with and without the ICI removal

filter, respectively. As Figure 4 shows, our method provides over 2 dB SINR gain at an SNR of25 dB over the method in [6], with identical simulation parameters; and this gain increases withSNR. The SINR gain curve obtained with ideal CSI is also given for comparison.

6. CONCLUSION

The paper considers the problem of estimating TV channels, and equalization under the detrimentaleffects of ICI for OFDM systems. The scheme combines the ideas of state-space modeling, Kalman



filter-based channel estimation, use of one-step-ahead prediction ability of the Kalman filter in thedecision-directed mode and time-domain ICI filtering. The scheme is shown to perform well interms of BER as well as SINR gain, and incurs a low training overhead. The results emphasize thatthe system performance is mainly impaired by ICI (due to Doppler spread). Performance boundsverify the results obtained through simulation. A comparison with the pilot-based estimation schemeof [6] (using identical parameters) brings out the strength of the proposed estimation method.However, performance limitation is observed in the high Doppler spread regime ( fdT>0.07), wherethe BER curves are seen to saturate at SNR above 20 dB. Design of suitable channel estimationtechniques for such scenarios is under investigation.

REFERENCES

1. Zhao Y, Haggman SG. Intercarrier interference self-cancellation scheme for OFDM mobile communicationsystems. IEEE Transactions on Communications 2001; 49(7):1185–1190.

2. Choi YS, Voltz PJ, Cassara FA. On channel estimation and detection for multicarrier signals in fast and selectiveRayleigh fading channels. IEEE Transactions on Communications 2001; 49(8):1375–1387.

3. Chen S, Zhu C. ICI and ISI analysis and mitigation for OFDM systems with insufficient cyclic prefix intime-varying channels. IEEE Transactions on Consumer Electronics 2004; 50(1):78–83.

4. Chen S, Yao T. Intercarrier interference suppression and channel estimation for OFDM systems in time-varyingfrequency-selective fading channels. IEEE Transactions on Consumer Electronics 2004; 50(2):429–434.

5. Seyedi A, Saulnier GJ. General ICI self-cancellation for OFDM systems. IEEE Transactions on VehicularTechnology 2005; 54(1):198–210.

6. Stamoulis A, Diggavi SN, Al-Dhahir N. Intercarrier interference in MIMO OFDM. IEEE Transactions on SignalProcessing 2002; 50(10):2451–2464.

7. Hou WS, Chen BS. ICI cancellation for OFDM communication systems in time varying multipath fading channels.IEEE Transactions on Wireless Communications 2005; 4(5):2100–2110.

8. Huang X, Wu H-C. Robust and efficient intercarrier interference mitigation for OFDM systems in time-varyingfading channels. IEEE Transactions on Vehicular Technology 2007; 56(5, part 1):2517–2528.

9. van de Beek JJ, Edfors O, Sandell M et al. On channel estimation in OFDM systems. Proceedings of the IEEE45th Vehicular Technology Conference, Chicago, IL, July 1995; 815–819.

10. Negi R, Cioffi J. Pilot tone selection for channel estimation in a mobile OFDM system. IEEE Transactions onConsumer Electronics 1998; 44(3):1122–1128.

11. Hsieh M, Wei C. Channel estimation for OFDM systems based on comb-type pilot arrangement in frequencyselective fading channels. IEEE Transactions on Consumer Electronics 1998; 44(1):217–225.

12. Moon TK. The expectation-maximization algorithm. IEEE Signal Processing Magazine 1996; 13(6):47–60.13. Ma X, Kobayashi H, Schwartz SC. EM-based channel estimation for OFDM systems. IEEE Conference on

Signals, Systems and Computers 2002; 2:1642–1646.14. Cai X, Giannakis GB. Bounding performance and suppressing intercarrier interference in wireless mobile OFDM.

IEEE Transactions on Communications 2003; 51(12):2047–2056.15. Al-Gharabally M, Das P. On the performance of OFDM systems in time varying channels with channel estimation

error. IEEE International Conference on Communications, ICC 2006, Istanbul, Turkey, vol. 11, 11–15 June 2006;5180–5185.

16. Kaioukov IV, Manelis VB, Cleveland JR. Channel estimation for MIMO-OFDM systems in rapid time variantenvironments based on channel statistics estimation. IEEE Global Telecommunications Conference GLOBECOM2006, San Francisco, CA, U.S.A., 27 November–1 December 2006; 1–5.

17. Komninakis C, Fragouli C, Sayed AH, Wesel RD. Multi-input multi-output fading channel tracking and equalizationusing Kalman estimation. IEEE Transactions on Signal Processing 2002; 50(5):1065–1076.

18. H-Yu Wu, Duel-Hallen A. On the performance of coherent and noncoherent multiuser detectors for mobileradio CDMA channels. IEEE International Conference on Universal Personal Communications, Cambridge, MA,vol. 1, 29 September–2 October 1996; 76–80.

19. Haykin S. Adaptive Filter Theory (4th edn). Pearson Education: Delhi, India, 2002.20. Li J, Kavehrad M. Effects of time selective multipath fading on OFDM systems for broadband mobile applications.

IEEE Communications Letters 1999; 3(12):332–334.



21. Welch G, Bishop G. An introduction to the Kalman filter. Technical Report TR 95-041, Department of ComputerScience, University of North Carolina at Chapel Hill, 1995.

22. El-Tanany MS, Wu Y, Hazy L. OFDM uplink for interactive broadband wireless: analysis and simulation in thepresence of carrier, clock and timing errors. IEEE Transactions on Broadcasting 2001; 47(1):3–19.

23. Proakis JG. Digital Communications (4th edn). McGraw-Hill: New York, 2001.

AUTHORS’ BIOGRAPHIES

Prerana Gupta was born in Delhi, India on October 14, 1981. She received her BTechdegree in Electronics and Communications Engineering from Indira Gandhi Instituteof Technology (GGSIPU), Delhi, India. She is currently pursuing her PhD at theDepartment of Electronics and Computer Engineering, Indian Institute of TechnologyRoorkee. Her research interests include channel estimation and equalization techniquesin doubly selective systems, channel coding and MIMO-OFDM.

D. K. Mehra was born in Amritsar, India on May 25, 1946. He received the BE andME degrees in electrical and communications engineering from the Indian Institute ofScience, Bangalore, India in 1968 and 1970, respectively, and the PhD degree fromthe Indian Institute of Technology, Kanpur, India in 1978. In 1975, he joined theElectronics and Computer Engineering Department of Indian Institute of TechnologyRoorkee (formerly University of Roorkee), India, where he became professor in 1987.His main teaching and research interests are in the area of adaptive signal processingtechniques and their application for interference suppression in CDMA/OFDM systemsand digital communication over fading dispersive channels.


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Kalman filter-based channel estimation and ICI suppression for high-mobility OFDM systems