IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 3, MARCH 2008 497

Iterative ML-Based Estimation ofCarrier Frequency Offset, Channel Impulse

Response and Data in OFDM TransmissionsFilippo Zuccardi Merli and Giorgio Matteo Vitetta, Senior Member, IEEE

Abstract—This paper provides an approximate closed formsolution to the problem of data-aided joint maximum likelihoodestimation of the carrier frequency offset and of the channelimpulse response in an Orthogonal Frequency Division Mul-tiplexing transmission over a multipath fading channel. Thisresults in a novel pilot-aided feedforward estimator, that canbe employed for pilot-based receiver training. It is also shownhow the novel joint estimation strategy can be exploited in aniterative (turbo) receiver structure to track the fast channel andfrequency offset changes occurring during the transmission ofinfomation symbols. The performance of this structure is assessedby computer simulations and is compared with that provided byother estimation/detection strategies.

Index Terms—Orthogonal frequency division multiplexing,joint maximum likelihood estimation, fading channel, carrierfrequency offset.

I. INTRODUCTION

IT is well known that the use of the orthogonal frequencydivision multiplexing (OFDM) [1] technique in mobile

communication systems requires an accurate acquisition andtracking of both the carrier frequency offset (CFO) and thechannel impulse response (CIR) at the receiver [2]. Differentpilot-aided strategies have been proposed in the technicalliterature to solve these problems. A traditional approach toCFO recovery and CIR estimation in OFDM transmissionsis represented by various schemes, in which these two tasksare separate functions. In this case, channel independentestimation of the CFO [3], [4], [5] precedes channel esti-mation, which is accomplished after the frequency correctionof received signal samples [6], [7]. A more refined approachto this problem is offered, however, by data-aided joint es-timation of CIR and CFO. This problem has been recentlytackled by various researchers in a maximum likelihood (ML)perspective [8]-[11], but only approximate solutions have beenfound, because of the complicated nonlinear dependence ofthe joint ML (JML) function on the CFO. In fact, in all thesepapers, CFO (CIR) estimation algorithms are derived underthe assumption of a known CIR (CFO), so that some form of

Paper approved by S. N. Batalama, the Editor for Spread Spectrum andEstimation of the IEEE Communications Society. Manuscript received January11, 2006; revised July 11, 2006 and October 18, 2006. This paper waspresented in part at the IEEE Global Telecommunications Conference, SanFrancisco, CA, USA, 27 Nov.-1 Dec 2006.

F. Zuccardi Merli and G. M. Vitetta are with the Dept. of InformationEng., University of Modena and Reggio Emilia, 41100 Modena, Italy (e-mail:{zuccardimerli.filippo, vitetta.giorgio}@unimo.it).

Digital Object Identifier 10.1109/TCOMM.2008.060023.

iterative processing is needed to approach the ML solution.In particular, the use of a least-squares (LS) approach to CIRestimation combined with a steepest descent algorithm forCFO estimation is proposed in [8], where the use of thesetechniques for CFO & CIR decision-directed tracking is alsoconsidered. A conceptually related approach is proposed in[9], where the expectation-maximization (EM) algorithm isused to assess CIR in its E step (under the assumption ofa known CFO); then, the resulting CIR estimate is exploitedin its M step to update the CFO estimate. A substantiallylarger complexity is required by the estimator illustrated in[10], that consists of a coarse synchronization step, followedby ML timing/frequency synchronization and by ML chan-nel estimation. The training of the proposed algorithm isaccomplished, like in [8], processing pilot OFDM symbolsregularly interspersed in the data symbol sequence, but itsuse for parameter tracking is not suggested. It is also worthnoting that the initial estimate of the CFO is set to zero inthe algorithms of [8]-[10], and this may entail poor trackingperformance in the presence of sudden changes in the resid-ual CFO due to Doppler and/or local oscillator instabilities.Finally, in [11] pilot tones embedded in each OFDM symbolare exploited to evaluate an initial channel estimate; then, adecision-directed algorithm is employed to refine the data,CIR and CFO estimates iteratively. The tracking speed ofthis algorithm is limited by the fact that, when estimating theCFO, both data and CIR are assumed ideally known, whereas,in the estimation of each of the other two sets of unknownparameters (i.e., CIR and data), the roles of the remainingones are interchanged.

The considerations illustrated above suggest that the tech-niques of [8]-[11], while providing satisfactory performancein the presence of negligible CFO & CIR changes betweenconsecutive OFDM symbols, can suffer from quick time vari-ations in CIR/CFO, since OFDM is sensitive to both frequencysynchronization and CIR estimation error [12], [13], [14],[15]. We believe that, in such scenarios, improved trackingperformance should be provided by truly joint data-aided CIRestimation & CFO synchronization strategies employed, in adecision-directed fashion, within an iterative or turbo receiver[16], refining the quality of both parameter estimates and datathrough an exchange of information between the detectionand the channel estimation & synchronization sections of thereceiver. Our claim is supported by the fact that, generallyspeaking, joint estimators offer better tracking capabilities and

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498 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 3, MARCH 2008

faster acquisition speed than techniques handling the sametasks in a separate fashion [17], [18].

The purpose of this paper is twofold. First of all, theproblem of data-aided joint ML estimation of residual CFOand CIR in OFDM signals is revisited. It is proved that, if thereceived signal model is linearized with respect to the CFO, anexact closed form solution to this problem can be found. Thisresults in a novel open-loop estimation algorithm, consisting oftwo distinct steps. Its first step produces a channel independentestimate of the CFO that is exploited, in its second one, toestimate the CIR. Secondly, it is shown how the proposed algo-rithm can be used in an iterative receiver structure, periodicallytrained by pilot OFDM symbols, like the estimators describedin [8], [10], and progressively refining data and CIR/CFOestimates during the transmission of information symbols.Numerical results evidence that this structure can approachideal performance after only one full iteration, even in thepresence of a strong frequency instability and of appreciablevariations in the communication channel.

This paper is organized as follows. Signal and channelmodels are described in Section II. In Section III the problemof JML estimation of CIR & CFO, in the presence of knowndata, is analyzed and a closed form solution is derived; then,the computational complexity of the resulting algorithm isassessed. The use of the novel estimation technique in aniterative receiver structure is described in Section IV. Someperformance results are illustrated in Section V, where theproposed solution is compared with that of [8] and with thatof a coherent receiver endowed with an ideal knowledge ofCIR and CFO. Finally, Section VI offers some conclusions.

II. SIGNAL AND CHANNEL MODELS

The block diagram of an OFDM system is illustrated in Fig.1. The input data stream is mapped into a sequence of inde-pendent and identically distributed channel symbols belongingto an M -PSK constellation. This sequence is partitioned intoblocks, each consisting of Nu

.= 2Nα +1 symbols, where Nu

represents the number of useful subcarriers1. In the followingthe channel symbols forming the l-th block are grouped in thevector a(l) .= [a(l)

−Nα, ..., a

(l)0 , ..., a

(l)Nα

]T , where (·)T denotesthe transpose operator. Then Nvc

.= N−Nu virtual carriers areadded to a(l) and the resulting N dimensional vector is sentto an N -th order inverse discrete Fourier transform (IDFT).The IDFT output vector is periodically extended appending acyclic prefix of Ncp channel symbols to it and, after seriesto parallel conversion, feeds a transmission filter, having animpulse response p(t), at a rate 1/Ts, where Ts is the channelsymbol interval. In the following we assume that p(t) is timelimited to the interval (−NpTs, NpTs) and that its Fouriertransform P (f) is the root of a raised cosine with roll-off α,so that Nα = int[N(1 − α)/2] [19], [20].

The OFDM signal is transmitted over a wide sense station-ary uncorrelated scattering (WSS-US) Rayleigh fading chan-nel with time varying frequency response (impulse response)H(f, t) (h(t, τ)) [21]. In the following we assume that: (a)

1The parameter Na is defined below.

S/PVirtualCarriersInsertion

IDFTAdd

CyclicPrefix

tps t

12N N

( )la ( )lb

DataIn

P/SMapper

SamplerS/PRemoveCyclicPrefix

DFTVirtualCarriers

Remotion

Equaliz.&

DetectionP/S

12N

( )lR

Up-Converter

Channel

Filter &Down-

Converter

r t

( )la

cpN N

NcpN N

DataOut

Fig. 1. Block diagram of an OFDM system.

the impulse response of the multipath fading channel is

h(t, τ) =L−1∑n=0

hn (t) δ(τ − nTs) (1)

where L is the number of distinct echoes and {hl(t), l = 0, 1,..., L− 1} are mutually independent, WSS Gaussian complexrandom processes, having zero mean and statistical powers{σ(n) .= E{|hn(t)|2}, n = 0, 1, ..., L − 1}; (b) the channelcan be deemed static over each OFDM symbol interval (quasistatic channel), so that in the l-th OFDM symbol interval theCIR (1) can be approximated as

h(t, τ) ∼= h(l) (τ) .=L−1∑n=0

hn [l] δ(τ − nTs) (2)

where h(l)(τ) and hn[l] denote the CIR and the value takenby the n-th tap gain hn(t) (with n = 0, 1, ..., L − 1) of (1)in the l-th OFDM symbol interval, respectively; (c) Ncp ≥2Np +L−1, so that no inter-block interference (IBI) is foundin the detection of OFDM symbols. Then, if the CFO doesnot exceed the subcarrier spacing and changes negligibly overeach OFDM interval, it can be shown that the received signal,after down-conversion and low pass filtering, can be expressedas [19], [20]

r(t) = s(t − lNT Ts, a(l),H(l), ν(l)

)+ w(t) (3)

for t ∈ [lNT Ts, (lNT + N)Ts], where

s(t, a(l),H(l), ν(l)

).= exp

(j2πν(l)t/NTs

) 1√NTs

Nα∑k=−Nα

a(l)k H

(l)k exp(j2πfkt)

(4)

is the useful component of r(t). Here fk � k/NTs is thek-th subcarrier frequency, and H(l) .= [H(l)

−Nα, H

(l)−Nα+1, ...,

H(l)Nα

]T , with H(l)n

.= H(l)(fn) (n = −Nα, −Nα +1, ..., Nα),is a Nu dimensional vector collecting the values taken on bythe channel frequency response H(l)(f) (i.e., the continuousFourier transform of h(l)(τ)) at the subcarrier frequencies in

ZUCCARDI MERLI and VITETTA: ITERATIVE ML-BASED ESTIMATION OF CARRIER FREQUENCY OFFSET, CHANNEL IMPULSE RESPONSE 499

the l-th OFDM symbol interval. In addition w(t) is complexadditive white Gaussian noise (AWGN) with two-sided powerspectral density 2N0 over the OFDM signal bandwidth andν(l) is the carrier frequency offset, normalized to the OFDMsymbol rate 1/NTs, affecting the demodulated signal in thel-th OFDM symbol interval.

At the receive side the signal r(t) (3) is sampled2 at theinstants t

(l)n

.= lNT Ts + nTs, with n = −Ncp, −Ncp + 1, ...,N − 1, when detecting the l-th OFDM symbol, but the firstNcp samples are discarded since they are associated with thecyclic prefix. This generates the set of N samples collectedin the vector r(l) .= [r(l)

0 , r(l)1 , ..., r

(l)N−1]

T (with r(l)n

.= r(t(l)n ),n = 0, 1, ..., N − 1), feeding an N -th order DFT. Removingthe virtual carriers from the DFT output produces the Nu-dimensional vector R(l) .= [R(l)

−Nα, R

(l)−Nα+1, ..., R

(l)Nα

]T . Itcan be shown that [3], [8], [22]

R(l)i = m(ν(l)) a

(l)i H

(l)i +

Nα∑k=−Nα

k �=i

g(k− i, ν(l)) a(l)k H

(l)k +n

(l)i

(5)with i = −Nα,−Nα + 1, ..., Nα, where

m(ν) .=1N

exp(

jπνN − 1

N

)sin(πν)

sin(πν/N)(6)

represents, for any i, the attenuation on the i-th subcarrier dueto the residual CFO, and

g(x, ν) .=1N

exp(

jπ (x + ν)N − 1

N

)sin [π(x + ν)]

sin [π(x + ν)/N ](7)

with x = k− i ( i, k = −Nα, −Nα +1, ..., Nα), accounts forthe intercarrier interference generated by the k-th subcarrier onthe i-th one. Moreover, the noise contribution n(l) .= [n(l)

−Nα,

n(l)−Nα+1, ..., n

(l)Nα

]T to R(l) consists of independent and iden-tically distributed complex Gaussian random variables, eachhaving zero mean and variance σ2

n = 2N0/Ts. Expressing theresult (5) in matrix form yields

R(l) = P(ν(l)

)A(l) H(l) + n(l) (8)

whereP(ν) .= toeplitz{pn(ν);pp(ν)} (9)

is the Nu × Nu toeplitz matrix having the vector

pn(ν) .= [m(ν), g(−1, ν), g(−2, ν), ..., g(−Nα, ν)]T (10)

as its first column and

pp(ν) .= [m(ν), g(1, ν), g(2, ν), ..., g(Nα, ν)] (11)

as its first row, and A .= diag{a(l)i , i = −Nα, −Nα + 1, ...,

Nα} is a Nu ×Nu diagonal matrix containing all the channelsymbols of a(l) along its main diagonal. It is useful to notethat the channel vector H(l) can be expressed as [6], [23]

H(l) = FL h(l) (12)

where h(l) .= [h0[l], h1[l], ..., hL−1[l]]T and FL is a Nu × LDFT matrix with [FL]p,q = exp[−j2πpq/N ] ( p = −Nα, ...,0, ..., Nα and q = 0, 1, ..., L − 1).

2Ideal timing recovery is assumed at the receive side.

Since the signal processing algorithms described in thefollowing sections operate on a block-by-block basis, thedependence of the signal model (8)-(12) on the block index lis dropped in the following to simplify the notation.

III. DATA-AIDED JOINT MAXIMUM LIKELIHOOD

ESTIMATION OF CIR AND CFO

In this Section ML methods are exploited to devise a noveldata-aided joint estimator of the CFO and the CIR. Then, itscomputational complexity is assessed and compared with thatof the algorithm proposed in [8].

A. Derivation of the Algorithm

Given the received signal model expressed by (8) and (12),the data-aided JML estimates νML and hML of the normalizedCFO ν and of the CIR vector h, respectively, maximize thelikelihood function [8]

l(R|h, ν,A

).= f

(R|h, ν,A

)

=1

(πσ2n)2Nα+1

· exp

⎡⎢⎣−

∥∥∥R − P (ν) AFL h∥∥∥2

σ2n

⎤⎥⎦(13)

where f(R|h, ν, A) denotes the joint probability density func-tion of R, conditioned on the trial values h and ν of h and ν,respectively, and the data matrix A is assumed known. Takingthe logarithm of (13) and dropping irrelevant terms leads to theequivalent problem of minimizing the log-likelihood function

L(R|h, ν,A

).= −2Re

{RH P (ν) AFL h

}

+hH FHL AH P (ν)H P (ν) AFL h

(14)

Therefore, νML and hML are the solutions of the nonlinearsystem of equations{

ddν L(R|h, ν,A) = 0

∇hL(R|h, ν,A) = 0L(15)

whered

dνL(R|h, ν,A) = −2Re

{RH P(ν)AFL h

}

+hH FHL AH

{d

dν

[P (ν)H P (ν)

]}

·AFL h (16)

∇hL(R|h, ν,A) = −2�{FH

L AH P(ν)H R}

+ 2

·Re{FH

L AH P(ν)H P(ν)AFL h}

(17)

∇ is the complex gradient operator and P(ν) is the first deriva-tive of P(ν) (9). If m(ν) and g(x, ν) denote the first deriva-tives, both with respect to ν, of m(ν) (6) and of g(x, ν) (7), re-spectively, then we have that P(ν) = toeplitz{pn(ν); pp(ν)},with pn(ν) .= [m(ν), g(−1, ν), g(−2, ν), ..., g(−Nα, ν)] and

500 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 3, MARCH 2008

pp(ν) .= [m(ν), g(1, ν), g(2, ν), ..., g(Nα, ν)]. Note that fromthe second equation of (15) and from (17) it is easily inferredthat

h =[FH

L AH P (ν)H P (ν) AFL

]−1

FHL AH P (ν)H R

(18)Then, substituting (18) into (16) and (16) into the first equationof (15) leads to a scalar equation in the unknown ν. Anexact closed form solution of this equation cannot be easilyevaluated, because of the nonlinear dependence of P(ν) (9)on the CFO, but an approximate solution can be found if νis small with respect to unity3, as shown in the following. Tobegin, we note that, if ν is small, a simple, but accurate, rep-resentation of P(ν) is provided by its Taylor series truncatedto the second order4. This representation can be derived asfollows. The Taylor series of m(ν) (6) and that of g(x, ν) (7),both truncated to the second order, are given by

m(ν) ∼= 1 + m1ν + m2ν2 (19)

andg(x, ν) ∼= n1(x) ν + n2(x) ν2 (20)

respectively, where

m1.= jπ

N − 1N

, m2.= −2π2 N − 1

N(21)

n1(x) .=1N

exp (jy0)[jπ

N − 1N

sin α

sin β

+π cosα sinβ − (π/N) sinα cosβ

sin2 β

](22)

and

n2(x) .=1

2Nexp (jy0)

{−π2 (N − 1)2

N2

sinα

sinβ

+j2πN − 1

N

π cosα sin β − (π/N) sin α cosβ

sin2 β

+sinα sin

[β π2

(1 − N2

)/N2

]sin2 β

sin4 β−[π cosα sin β − (π/N) sin α cosβ]

· (π/N) sin(2β)sin4 β

}(23)

with y0.= πx(N − 1)/N , α

.= πx and β.= πx/N . Then,

substituting (19) and (20) into (10)-(11) and (10)-(11) into (9)yields

P (ν) ∼= INu + P1 ν + P2 ν2 (24)

where INu is the Nu × Nu identity matrix, P1.=

toeplitz{p1,n;p1,p}, P2.= toeplitz{p2,n;p2,p}, p1,n

.=[m1, n1(−1), n1(−2), ..., n1(−Nα)], p1,p

.= [m1, n1(1),n1(2), ..., n1(Nα)], p2,n

.= [m2, n2(−1), n2(−2), ...,n2(−Nα)] and p2,p

.= [m2, n2(1), n2(2), ..., n2(Nα)].From the approximate model (24) it can be easily inferred

3A coarse synchronizer, like that proposed in [5], can be used to correctthe integer part of the CFO, so that, in any case, |ν| ≤ 0.5.

4Note that a similar simplification has been also adopted in [9], which,however, has appeared after the submission of our manuscript.

that (a) P(ν) ∼= P1 + 2P2ν and that (b) PH(ν)P(ν) ∼=INu + (P2 + PH

1 P1 + PH2 )ν2 since P1 + PH

1 = 0Nu×Nu ,where 0Nu×Nu is the Nu×Nu null matrix. To further simplifythe derivation of our estimation algorithm, we retain only theconstant and linear terms in approximating the matrix productP(ν)H P(ν) and its first derivative in (16) and (17), so that

P (ν)H P (ν) ∼= INu (25)

andd

dν

[P (ν)H P (ν)

] ∼= 2(P2 + PH1 P1 + PH

2 )ν (26)

Then, substituting (25) and the expression (24) truncated tothe first order into (18) leads to the simplified result

h ∼= EAH(I + PH

1 ν)

R (27)

where E .= BFHL and B .= (FH

L FL)−1. It is interesting tonote that in (27), unlike (18), the only matrix to be inverted(i.e. B) does not depend on both data and the CFO, so that itneeds to be computed only once.

Following a similar line of reasoning and exploiting (26),(16) can be simplified as

d

dνL(R|h, ν,A) ∼= −2Re

{RH P1AFL h

}

−4Re{RH P2 AFL h

}ν

+2νhH FHL CFL h (28)

where C .= AH(P2 + PH1 P1 + PH

2 )A. Finally, substituting(27) into (28) and (28) into the first equality of (15) yields,after some manipulation, the channel independent CFO esti-mate

ν =Re {S(R,A)}

T (R,A) − 2Re {U(R,A)} − 2Re {V (R,A)} (29)

where S(R,A) .= RH P1 ADAH R, T (R,A) .=RH ADCDAH R , U(R,A) .= RH P1 AD AH PH

1 R,V (R,A) .= RH P2 ADAH R and D .= FL BFH

L . Eqs.(29) and (27) form a two step data-aided algorithm for theevaluation of an approximate solution to the ML estimationproblem expressed by (15)-(17). In fact, given the receivedvector R and the data vector A, a CFO estimate ν can becomputed from (29); then setting ν = ν in (27) produces theestimate

h = EAH(I + PH

1 ν)

R (30)

of the CIR vector h.

Establishing the properties of the joint estimation strategybased on (29) and (30) is not easy, mainly because of thecomplicated dependence of the CFO estimate ν (29) on R.However, since ν (29) and h (30) approach the JML estimatesνML and hML as ν → 0, it can be easily inferred that, ifν = 0, as the data record R increases, i.e. as Nu → +∞, theCFO estimate tend to become unbiased, to achieve the Cramer-Rao lower bound and to have a Gaussian distribution [27]. Forfinite values of Nu, our simulation results have evidenced that,for any signal-to-noise ratio (SNR) of interest, the estimationbias is so small to have a negligible impact on the systemperformance and that the linearity range of the estimator (29)

ZUCCARDI MERLI and VITETTA: ITERATIVE ML-BASED ESTIMATION OF CARRIER FREQUENCY OFFSET, CHANNEL IMPULSE RESPONSE 501

TABLE ICOMPUTATIONAL COMPLEXITY PER OFDM SYMBOL OF THE (DD) JML-CFE, OF THE (DD) MKS-MLE AND OF THE (DD) LHK-EME TECHNIQUES.

Algorithm Overall number of complex multiplies and complex additions

JML-CFE14N2

u + (L + 4)Nu mult.6N2

u + (L + 5)Nu add.

MKS-MLE(K + 1) · [(L + 4)N2

u + (L2 + 4L + 3)Nu + L2 + L3] mult.(K + 1) · [(L + 4)N2

u + (L2 + 4L + 3)Nu + L2 + L3] add.

LHK-EME(K + 1) · [(L + 1)N2

u + (L2 + 4L + 5)Nu + L3] mult.(K + 1) · [LN2

u + (2L2 + 2L + 1)Nu + L3] add.

DD JML-CFE(Nit + 1) · [14N2

u + (L + 4)Nu + 3N2 log2 N ] mult.

(Nit + 1) · [6N2u + (L + 5)Nu + 3N log2 N ] add.

DD MKS-MLE(K + 1) · [(L + 4)N2

u + (L2 + 4L + 3)Nu + L2 + L3] + 3N2 log2 N mult.

(K + 1) · [(L + 4)N2u + (L2 + 4L + 3)Nu + L2 + L3] + 3N log2 N add.

DD LHK-EME(K + 1) · [(L + 1)N2

u + (L2 + 4L + 5)Nu + L3] + 3N2 log2 N mult.

(K + 1) · [LN2u + (2L2 + 2L + 1)Nu + L3] + 3N log2 N add.

is roughly the interval [−0.2, 0.2].In the following the data-aided algorithm based on (29)

and (30) is dubbed Joint Maximum Likelihood Channel andFrequency Estimator (JML-CFE, briefly). As illustrated inSection IV, this algorithm, if employed in a decision directedfashion, can be exploited to devise a new iterative strategy,estimating data, CFO and channel jointly. However, beforeillustrating this application, it is important to assess its com-putational complexity and to compare it with that of otherexisting strategies.

B. Computational Complexity

In this paragraph, the computational complexity of the JML-CFE is compared with that of the CIR & CFO estimationstrategies of [8] and [9]5, since the last two techniques canbe employed in place of the first one with the transmissionformat described in Section IV. Other algorithms for jointestimation of CIR & CFO, like those described in [10]and in [11], are not taken into consideration here for thefollowing reasons6. The technique of [10], originally proposedfor pilot-based estimation, is overly complicated for decision-directed tracking. The strategy of [11] requires the use ofpilot subcarriers embedded in each OFDM symbol, i.e. theadoption of a transmission format substantially different fromthat exploited by both the JML-CFE and the MKS-MLE inthis work.

It is worth noting, however, that, despite the possibility ofemploying the JML-CFE, the MKS-MLE and the LHK-EMEin the same scenario, these strategies are substantially differ-ent. In fact, the JML-CFE is an open-loop algorithm, whereasthe MKS-MLE and the LHK-EME are adaptive/iterative al-gorithms requiring multiple (K) iterations for convergence.This need arises from the fact that the MKS-MLE (LHK-EME) requires not only an estimate of the CFO to evaluatean estimate of the channel parameters, like the JML-CFE (see(30)), but also a channel estimate for computing the CFO viaa steepest-descent procedure (via a maximization step).

5These strategies are dubbed Ma-Kobayashi-Schwartz ML Estimator(MKS-MLE) and Lee-Han-Kim Expectation Maximization Estimator (LHK-EME), respectively, in the following.

6Algorithms for the separate estimation of CIR and CFO are also ignoredhere, since they perform poorly in the challenging communication scenariosdescribed in Section V.

The expressions of the overall number of complex multiplies(cms) and complex additions (cas) required by the above men-tioned algorithms, when processing a single OFDM symbol,are listed in the second and in the third rows of Table I. Wenote that (a) the complexity of all the algorithms is dominatedby matrix and vector products, leading to a dependence on N2

u;(b) the multiplicative coefficient of N2

u for the MKS-MLEand the LHK-EME increases linearly with both the numberof channel taps L and that of their iterations K; (c) the JML-CFE is substantially less complicated than the MKS-MLEand the LHK-EME, even for small values of L and K; (d)this complexity gap gets larger when L and K increase. Forinstance, if the parameters values adopted in our simulations(i.e., Nu = 441, L = 16 and K = 2) are selected, it isfound that ∼ 2.7 · 106 cms and ∼ 1.2 · 106 cas are requiredfor the JML-CFE, whereas ∼ 1.2 · 107 cms and ∼ 1.2 · 107

cas (∼ 1.04 · 107 cms and ∼ 1.01 · 107 cas) are needed forthe MKS-MLE (LHK-EME). Moreover, our simulation resultshave evidenced that, in the scenarios considered in SectionV, the MKS-MLE and the LHK-EME produce similar errorperformance. This is not surprising since, as already stated inthe Introduction, these two techniques are closely related froma conceptual viewpoint. For this reason, the LHK-EME is nottaken into consideration again in the following.

IV. ITERATIVE ESTIMATION OF DATA, CIR AND CFO

In OFDM mobile transmissions robust tracking of both CIRand CFO is usually needed. This target is usually achievedtransmitting equally spaced pilot symbols [24], so that a data-aided retraining of the frequency synchronization and channelestimation algorithms can be accomplished at the receive side[8]. In the following it is assumed that each pilot OFDMsymbol is followed by (D − 1) information symbols, so thatthe pilot symbol rate is equal to 1/D.

In pilot symbol intervals the use of a channel-independentopen-loop frequency synchronizer with a large linearity range,like that proposed in [5], is advisable. Then, the CIR canbe accurately estimated after the frequency correction of thereceived signal samples [6], [7]. In particular, if pilot data aretransmitted in the n-th OFDM symbol interval, this approachproduces the estimates ν(n) and h(n) of ν(n) and of h(n),respectively, that can be used for initializing decision-directed

502 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 3, MARCH 2008

DFT

( ) ( )0l lr r

CFOCorrection

( )1ˆ lk

( )lkR

( )1

ˆ lkh

( )1

ˆ lkH

( )1lkr

0k

0k

( )lkr

( )ˆ lka

Equalization&

Decoding

DiscardVirtualCarriers

BlockDeint.

BlockDeint.LF

BlockInterleaver

JML-CFEData Out

Fig. 2. Block diagram of an OFDM iterative receiver based on the DDJML-CFE technique.

CFO & CIR tracking. In fact, in the next interval they can beexploited to compensate for the residual CFO and for one-tapequalization in the frequency domain, so that a decision a(n+1)

can be taken on the data vector a(n+1). Then, data estimatescan be used to generate the new estimates ν(n+1) and h(n+1)

to be employed in the successive OFDM symbol interval. Thisprocedure is repeated in all the following intervals until a newpilot symbol is received and processed for retraining.

A more refined approach to decision-directed tracking,called decision-directed (DD) MKS-MLE (DD MKS-MLE) inthe following, is proposed in [8], where an iterative procedurefor generating accurate estimates of data, CIR and CFO ineach OFDM symbol interval is derived. It is important topoint out, however, that, in this case, frequency correctionof the received signal samples is accomplished only at thebeginning of the first iteration, so that a relevant increase inthe computational complexity, due to the need of multipleDFTs, is avoided. For this reason, abrupt changes of the CFO,due to Doppler and/or oscillator instabilities, cannot be easilycompensated for. To circumvent this problem, a novel iterativetechnique for DD tracking, based on the JML-CFE and dubbedDD JML-CFE in the following, is proposed here. Its useis exemplified here, for simplicity, in the case of a codedtransmission in which each codeword generated by a givenchannel encoder (e.g., a convolutional encoder) is transmitted,after block interleaving, over a single OFDM symbol. Theproposed receiver structure for this transmission is representedin Fig. 2, depicting the system status at the end of the k-thiteration occurring in the l-th OFDM interval. At the beginningof the first iteration (corresponding to k = 0), the receiver DFTis fed by r(l)

0 = r(l) and the final channel estimate H(l−1)Nit−1 =

FLh(l−1)Nit−1 generated in the previous OFDM symbol interval7,

where Nit denotes the overall number of iterations, is used forequalization and decoding, so that H(l)

0 = H(l−1)Nit−1. Then, the

resulting decision vector a(l)0 is processed by the JML-CFE to

produce the estimates ν(l)1 and h(l)

1 of ν(l) and of h(l). Theestimate ν

(l)1 is used to compensate for the residual CFO in

r(l)0 : this generates the vector r(l)

1 that feeds the loop at the

7If the pilot data are transmitted in this interval, data-aided estimates areused, as already explained above.

beginning of the second iteration and that is also preserved ina buffer. At the end of the second iteration the new estimatesν

(l)2 and h(l)

2 are available together with the vector resultingfrom the CFO correction (based on ν

(l)2 ) of r(l)

1 . Then, a newiteration can be started, if needed.

The computation complexity of the DD JML-CFE is listedand compared with that of the DD MKS-MLE in the last rowsof Table I. In this table, the parameters Nit and K denotethe overall number of iterations for DD JML-CFE and DDMKS-MLE, respectively, whereas the terms (3N log2 N)/2and 3N log2 N represent the overall number of complexmultiplications and that of complex additions, respectively,required by a single DFT. It is worth pointing out that thecomplexity of the DD JML-CFE is appreciably smaller thanthat of the DD MKS-MLE when a small number of iterationsis accomplished. For instance, for N = 512, Nu = 441and L = 16, if Nit = K = 2 is selected, it is found that∼ 8.2 · 106 (1.2 · 107) cms and ∼ 3.6 · 106 (1.2 · 107) cas arerequired by the DD JML-CFE (DD MKS-MLE). However, oursimulation results have evidenced that, unlike the DD MKS-MLE, one iteration can be sufficient for the JML-CFE, so thatthe complexity gap between these two techniques in a realisticscenario turns out to be substantially larger.

V. NUMERICAL RESULTS

In our work, both the performance of the data-aided JML-CFE and MKS-MLE algorithms (in the presence of an idealknowledge of the transmitted data) and that of the iterativereceiver structures, based on the DD JML-CFE and DD MKS-MLE techniques described in the previous Section, have beenassessed by computer simulation. Moreover, both coded anduncoded OFDM transmissions have been considered.

In all the following results it is always assumed that:(a) the channel symbols {a(l)

k } belong to a BPSK (QPSK)constellation in an uncoded (code) transmission; (b) the DFTorder is N = 512 and the roll-off factor is α = 0.15, so thatNα = 220 and the number of useful subcarriers is Nu = 441;(c) the length of the cyclic prefix and the duration of p(t) areNcp = 55 and 40Ts (i.e., Np = 20), respectively; (d) the SNRis defined as Eb/N0, where Eb is the average received energyper information bit; the channel model (1) is characterized byL = 16 taps and by the exponential power delay profile [19]

σ2(l) =1 − exp(−1/5)1 − exp(−L/5)

exp(−l/5) (31)

with l = 0, 1, ..., 15.The normalized mean square error (NMSE) for the CFO

and the CIR estimates are shown in Fig. 3 and in Fig.4, respectively, for the MKS-MLE (carrying out K = 2iterations) and the JML-CFE, both operating with a normalizedloop noise bandwidth BLNTs = 0.5. A static CIR over eachblock (quasi static channel) and independent CIR realizationsover distinct OFDM symbols are assumed here. Moreover,the estimation accuracies have been assessed in the followingdifferent conditions: (a) in the absence of a residual CFO in thereceived signal (ν(l) = 0 for any l); (b) when the residual CFOchanges in an independent fashion from symbol to symbol,because of the instability of the local oscillator or of an abruptvariation in the Doppler shift of the received signal. In the last

ZUCCARDI MERLI and VITETTA: ITERATIVE ML-BASED ESTIMATION OF CARRIER FREQUENCY OFFSET, CHANNEL IMPULSE RESPONSE 503

Fig. 3. CFO NMSE for the JML-CFE and the MKS-MLE. The MCRB isalso shown for comparison.

case two distinct scenarios have been considered; in the firstscenario (case #1) the CFO ν(l) ∈ U(−0.01, 0.01) for any l,i.e. it is uniformly distributed over the interval [−0.01, 0.01],whereas in the second one (case #2) ν(l) ∈ U(−0.1, 0.1).The results shown in Fig. 3 evidence that the MKS-MLEoutperforms the JML-CFE by about 3 dB when ν(l) = 0 forany l, and at low-medium SNRs when the CFO is small (e.g.,in case #1). This is due to the fact that the steepest descentalgorithm used for CFO synchronization in the MKS-MLE isinitialized assuming a residual CFO equal to zero. For thisreason, when the true CFO is close to this initial value, theMKS-MLE is already synchronized and does not need anyiteration. However, at large SNRs, even the presence of littlechanges in the CFO (e.g., with ν(l) ∈ [−0.01, 0.01]) appre-ciably affects the performance of the MKS-MLE (because ofits slow convergence), but does not influence that of the JML-CFE. Moreover, in practical applications, where such changesare even larger, the frequency NMSE of the former algorithmdevelops a floor, so that its gap with the latter one enlargessubstantially as Eb/N0 increases. The results in Fig. 4 showthat the JML-CFE outperforms the MKS-MLE in the channelestimation task for any SNR in all the considered scenarios.In Figs. 3-4 the modified Cramer-Rao bounds8 (MCRBs) forthe CFO (MCRB(ν)) and the CIR (MCRB(h)) [25], [26]are also shown as they represent useful benchmarks for the

8As already shown in [8], CIR estimation is less sensitive to an unknownCFO than CFO estimation to an unknown CIR. This explains why the CIRNMSE of the JML-CFE is close to the corresponding MCRB, but the samedoes not hold for CFO estimation.

Fig. 4. CIR NMSE for the JML-CFE and the MKS-MLE. The MCRB isalso shown for comparison.

investigated algorithms. It can be shown that [8]

MCRB(ν) =3Nuσ2

n

2π2(Nu − 1)(2Nu − 1)∑L−1

l=0 |hl |2(32)

when only the channel statistics are known in CFO estimationand that [6]

MCRB(h) = σ2n tr

{FH

L FL

}(33)

in the absence of a residual CFO, where tr {X} representsthe trace of a matrix X. It is interesting to note that, asalready shown in [8], CIR estimation is less sensitive to anunknown CFO than CFO estimation to an unknown CIR. Thisexplains why the CIR NMSE of the JML-CFE is close to thecorresponding MCRB (see Fig. 4), but the same does not holdfor CFO estimation (see Fig. 3).

Figs. 5-6 illustrate the bit error rate (BER) performanceof the DD receivers described in the previous Section in thecase of an uncoded transmission. In these results and in thefollowing ones it is always assumed that: (1) the CIR is staticover each OFDM symbol interval, but changes from symbolto symbol; (2) the discrete time random process hn[l] (withl = 0, 1, ..., L−1) in (2) is characterized by the autocorrelationfunction Rh[l] = J0(2πlBDNTs), where J0(x) is the zero-thorder Bessel function of the first kind and BD is the fadingDoppler bandwidth; (3) the normalized Doppler bandwidthBDNTs is equal to 10−2; (4) each transmitted frame consistsof 10 information symbols, preceded by one training symbolfor pilot-aided initial frequency and channel estimation, sothat the pilot symbol rate is Rp = 1/10; (5) for any SNR the

504 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 3, MARCH 2008

Fig. 5. BER performance of coherent receivers employing the DD JML-CFE and the DD MKS-MLE techniques with BDNTs = 10−2. The errorperformance of a coherent detector endowed with an ideal knowledge of bothCIR and CFO and that of a coherent detector exploiting pilot-based estimatesof these parameters is also shown for comparison.

overall number of transmitted frames is not smaller than 8000.Fig. 5 shows the BER performance of the DD MKS-MLE

and of the DD JML-CFE in a scenario characterized by aconstant CFO ν(l) = 0.2 over all the transmission. Theperformance of a coherent receiver endowed with an idealknowledge of both CIR and CFO and that of a coherentreceiver exploiting the last pilot-aided CIR & CFO estimateswithout any form tracking are also shown for comparison.These results evidence (a) the importance of parameter track-ing in this scenario and (b) a substantial difference betweenthe DD JML-CFE and the DD MKS-MLE in their dependenceon the overall number of iterations. In fact, on the one hand,the MKS-MLE performance benefits from an increase of K .For instance, the performance improvement achieved throughits first iteration is about 8 dB at a BER = 5 · 10−3 andan additional gain of 2 dB gain is provided by its seconditeration. On the other hand, with the JML-CFE, a 8 dB gainis obtained from the first iteration at a BER = 10−2, butno further gain is provided by the other ones at any SNR.This suggests that the choice Nit = 1 represents an excellenttradeoff between complexity and achievable performance inan iterative receiver based on the DD JML-CFE. It is alsoworth noting that the small energy difference between theideal detector performance and that of the DD JML-CFEperformance (before this develops a floor) is due not only toestimation inaccuracies, but also to the energy loss determinedby the use of pilot symbols.

Fig. 6. BER performance of coherent receivers employing the DD JML-CFEand the DD MKS-MLE techniques in different scenarios with BDNTs =10−2. The error performance of a coherent detector endowed with an idealknowledge of both CIR and CFO is also shown for comparison.

Similar comments apply to Fig. 6 illustrating the BERperformance of the DD MKS-MLE and of the DD JML-CFE in two distinct scenarios. The first one (case #1) ischaracterized by the residual CFO

ν(l) = 0.1 + (ν(l−1) − ν(l−1)) (34)

in the l-th OFDM symbol interval. In other words, the CFO isincreased by 0.1 at the beginning of every OFDM symbol in-terval, independently of the frequency correction accomplishedon the basis of the CFO estimate evaluated in the previousinterval. In the second scenario (case #2) in the l-th OFDMsymbol interval, the residual CFO ν(l) can take on one of thetwo equiprobable values {±0.12} , independently of its valuein all the other intervals.

Finally, an interleaved coded OFDM system employing arate 1/2, 4 state binary convolutional encoder with generatorsg1 = (7)8 and g2 = (2)8 and a 21 × 21 = 441 blockinterleaver9 and operating over the same two scenarios as thoseconsidered in Fig. 6 has been considered10. The BER resultsillustrated in Fig. 7 confirm the robustness of the DD JML-CFE in a coded system and its superiority over the DD MKS-MLE.

9The interlever contains all the coded symbols transmitted in a singleOFDM symbol excluding the virtual carriers and the cyclic prefix.

10Since the code rate is 1/2 and QPSK is employed over each subcarrier,the data rate is exactly the same as that of the uncoded system consideredabove.

ZUCCARDI MERLI and VITETTA: ITERATIVE ML-BASED ESTIMATION OF CARRIER FREQUENCY OFFSET, CHANNEL IMPULSE RESPONSE 505

Fig. 7. BER performance of coherent receivers employing the DDJML-CFE and the DD MKS-MLE techniques in different scenarios withBDNTs = 10−2. Convolutional channel coding and block interleaving areemployed. The error performance of a coherent receiver endowed with anideal knowledge of both CIR and CFO is also shown for comparison.

VI. CONCLUSIONS

In this paper a novel data-aided CIR and CFO estimatorfor OFDM signals has been proposed. Its derivation is basedon a ML approach and relies on a simplified model of thereceived signal. It has been also shown that the proposedalgorithm, if exploited in a decision-directed fashion, can beemployed in an iterative receiver, jointly estimating data, CIRand CFO. Simulation results have shown that this receiverstructure offers excellent performance, approaching that ofa receiver endowed with an ideal knowledge of the channelparameters and the CFO. This result has been achieved at theprice of a reasonable complexity, certainly smaller than thatrequired by the competing techniques illustrated in [8] and in[9].

REFERENCES

[1] J. A. C. Bingham, “Multicarrier modulation for data transmission: anidea whose time has come,” IEEE Commun. Mag., vol. 28, no. 5, pp.5-14, May 1990.

[2] E. Chiavaccini and G. M. Vitetta, “Performance analysis of OFDMsignals transmitted over doubly-selective Rayleigh fading channels,” inProc. IEEE Globecom 2000, vol. 2, pp. 975-979, San Francisco, CA,Nov. 2000.

[3] P. H. Moose, “A technique for orthogonal frequency division multiplex-ing frequency offset correction,” IEEE Trans. Commun., vol. 42, no. 10,pp. 2908-2914, Oct. 1994.

[4] T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchro-nization for OFDM,” IEEE Trans. Commun., vol. 45, no. 12, pp. 1613-1621, Dec. 1997.

[5] M. Morelli and U. Mengali, “An improved frequency offset estimatorfor OFDM applications,” IEEE Commun. Lett., vol. 3, no. 3, pp. 75-77,Mar. 1999.

[6] M. Morelli and U. Mengali, “A comparison of pilot-aided channelestimation methods for OFDM systems,” IEEE Trans. Signal Processing,vol. 49, no. 12, pp. 3065-3073, Dec. 2001.

[7] H. Senol, H. A. Cirpan, and E. Panayirci, “Pilot-aided Bayesian MMSEchannel estimation for OFDM systems: algorithm and performanceanalysis,” in Proc. IEEE Globecom 2004, vol. 4, pp. 2361-2365, Nov.2004.

[8] X. Ma, H. Kobayashi, and S. C. Schwartz, “Joint frequency offset andchannel estimation for OFDM,” in Proc. Globecom 2003, vol. 1, pp.15-19, Dec. 2003.

[9] J. H. Lee, J. C. Han, and S. C. Kim, “Joint carrier frequency syn-chronization and channel estimation for OFDM systems via the EMalgorithm,” IEEE Trans. Veh. Technol., vol. 55, no. 1, pp. 167-172, Jan.2006.

[10] H. Minn, V. K. Bhargava, and K. B. Letaief, “A combined timing andfrequency synchronization and channel estimation for OFDM,” in Proc.IEEE ICC 2004, vol. 2, pp. 872-874, June 2004.

[11] T. Cui and C. Tellambura, “Robust joint frequency offset and channelestimation for OFDM systems,” in Proc. IEEE Veh. Tech. Conf. 2004,vol. 1, pp. 603-607, Sept. 2004.

[12] T. Pollet, M. Van Bladed, and M. Moeneclaey, “BER sensitivity ofOFDM systems to carrier frequency offset and Wiener phase noise,”IEEE Trans. Commun., vol. 43, no. 2/3/4, pp. 191-193, Feb./Mar./Apr.1995.

[13] L. Wei and C. Schlegel, “Synchronization requirements for multi-userOFDM on satellite mobile and two-path Rayleigh fading channels,”IEEE Trans. Commun., vol. 43, pp. 887-895, Feb./Mar./Apr. 1995.

[14] A. Dowler, A. Nix, and J. McGeehan, “Data-derived iterative channelestimation with channel tracking for a mobile fourth generation widearea OFDM system,” in Proc. IEEE Globecom 2003, vol. 2, pp. 804-808,Dec. 2003.

[15] I. Harjula and A. Mammela, “Channel estimation algorithm for space-time block coded OFDM systems,” in Proc. IEEE Globecom 2003, vol.1, pp. 143-147, Dec. 2003.

[16] R. Otnes and M. Tuchler, “Iterative channel estimation for turboequalization of time-varying frequency-selective channels,” IEEE Trans.Wireless Commun., vol. 3, no. 6, pp. 1918-1923, Nov. 2004.

[17] Raheli, A. Polydoros, and C. K. Tzou, “Per-survivor processing: ageneral approach to MLSE in uncertain environments,” IEEE Trans.Commun., vol. 43, pp. 354-364, Feb.-Apr. 1995.

[18] A. N. D’Andrea, U. Mengali, and G. M. Vitetta, “Approximate MLdecoding of coded PSK with no explicit carrier phase reference,” IEEETrans. Commun., vol. 42, no. 2/3/4, pp. 1033-1039, Feb./Mar./Apr. 1994.

[19] E. Chiavaccini and G. M. Vitetta, “Maximum likelihood frequencyrecovery for OFDM signals transmitted over multipath fading channels,”IEEE Trans. Commun., vol. 52, no. 2, pp. 244-251, Feb. 2004.

[20] M. Luise, R. Reggianini, and G. M. Vitetta, “Blind equaliza-tion/detection for OFDM signals over frequency-selective channels,”IEEE J. Select. Areas Commun., vol. 16, no. 8, pp. 1568-1578, Oct.1998.

[21] P. A. Bello, “Characterization of randomly time-variant linear channels,”IEEE Trans. Commun. Syst., vol. 9, pp. 360-393, Dec. 1963.

[22] M. Luise, M. Marselli, and R. Reggianini, “Low-complexity blind carrierfrequency recovery for OFDM signals over frequency-selective radiochannels,” IEEE Trans. Commun., vol. 50, no. 7, pp. 1182-1188, July2002.

[23] Y. Li, N. Seshadri, and S. Ariyavisitakul, “Channel estimation forOFDM systems with transmitter diversity in mobile wireless channels,”IEEE J. Select. Areas Commun., vol. 17, no. 3, pp. 461-471, Mar. 1999.

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[25] M. Moeneclaey, “On the true and the modified Cramer-Rao boundsfor the estimation of a scalar parameter in the presence of nuisanceparameters,” IEEE Trans. Commun., vol. 46, no. 11, pp. 1536-1544,Nov. 1998.

[26] A. N. D’Andrea, U. Mengali, and R. Reggianini, “The modified Cramer-Rao bound and its application to synchronization problems,” IEEETrans. Commun., vol. 42, no. 2/3/4, pp. 1391-1399, Feb./Mar./Apr. 1994.

[27] S. M. Kay, Fundamentals of Statistical Signal Processing - EstimationTheory. Englewood Cliffs, NJ: Prentice Hall Inc., 1993.

506 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 56, NO. 3, MARCH 2008

Filippo Zuccardi Merli was born in Correggio,Italy, in July 1980. He received the Dr. Ing. degree inelectronic engineering (cum laude) in 2004 from theUniversity of Modena and Reggio Emilia, Modena,Italy, where he is currently working toward the Ph.D.degree.

In 2006-2007 he spent one year as a VisitingScholar at the Department of Electrical Engineeringof the Columbia University, New York, NY. Hisresearch interests lie in the areas of wireless com-munications techniques, with particular emphasis on

multicarrier modulation, synchronization, multiuser detection and statisticalsignal processing.

Giorgio M. Vitetta (S’89-M’91-SM99) was born inReggio Calabria, Italy, in April 1966. He receivedthe Dr. Ing. degree in electronic engineering (cumlaude) in 1990, and the Ph.D. degree in 1994, bothfrom the University of Pisa, Pisa, Italy.

In 1992/1993, he spent a period at the Universityof Canterbury, Christchurch, New Zeland, doingresearch for digital communications on fading chan-nels. From 1995 to 1998, he was a Research Fellowwith the Department of Information Engineering,University of Pisa. From 1998 to 2001, he was an

Associate Professor of Telecommunications with the University of Modenaand Reggio Emilia, Modena, Italy, where he is currently a full Professorof Telecommunications. His main research interests lie in the broad areaof communication theory, with particular emphasis on coded modulation,synchronization and channel equalization.

Dr. Vitetta is serving as an Editor of both the IEEE TRANSACTIONS ON

COMMUNICATIONS and the IEEE TRANSACTIONS ON WIRELESS COMMU-NICATIONS.