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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 8, AUGUST 2000 2389

Blind Channel and Carrier Frequency OffsetEstimation Using Periodic Modulation Precoders

Erchin Serpedin, Antoine Chevreuil, Georgios B. Giannakis, Fellow, IEEE, and Philippe Loubaton, Member, IEEE

Abstract—Recent results have shown that blind channel esti-mators, which are resilient to the location of channel zeros, colorof additive stationary noise, and channel order overestimationerrors, can be developed for communication systems equippedwith transmitter-induced cyclostationarity precoders. The presentpaper extends these blind estimation approaches to the more gen-eral problem of estimating the unknown intersymbol interference(ISI) and carrier frequency offset/Doppler effects using such pre-coders. An all-digital open-loop carrier frequency offset estimatoris developed, and its asymptotic (large sample) performance isanalyzed and compared to the Cramér–Rao bound (CRB). Asubspace-based channel identification approach is proposed forestimating, in closed-form, the unknown channel, regardless ofthe channel spectral nulls. It is shown that compensating for thecarrier frequency offset introduces no penalty in the asymptoticperformance of the subspace channel estimator. Simulationsare presented to corroborate the performance of the proposedalgorithms.

Index Terms—Communication channels, equalizers, estimation,frequency estimation, mobile communications.

I. INTRODUCTION

RECENT works [7], [14], [28], [34] have shown thatblind identification of single-input single-output (SISO)

finite impulse response (FIR) communication channels canbe achieved from second-order statistics of the received datawithout imposing any restriction on the channel zeros, colorof additive noise, and channel order overestimation errors andwithout introducing excess redundancy in the transmitted se-quence. The underlying channel identification approach relieson the output second-order cyclostationary (CS) statistics,which are induced by precoding the input symbol stream witha periodically time-varying precoder, which is referred to as atransmitter-induced cyclostationarity (TIC) precoder.

The purpose of the present paper is to extend these channelidentifiability results to the more general problem of blind esti-mation of the carrier frequency offset and the channel, assumingknowledge only of the output CS-statistics induced at the trans-mitter by a TIC precoder that is implemented by modulating

Manuscript received January 8, 1999; revised April 20, 2000. This work wassupported by the Office of Naval Research under Grant N00014-93-1-0485. Theassociate editor coordinating the review of this paper and approving it for pub-lication was Dr. Alle-Jan van der Veen.

E. Serpedin is with the Department of Electrical Engineering, Texas A&MUniversity, College Station, TX 77843 USA.

A. Chevreuil and P. Loubaton are with the Laboratoire Systèmes de Commu-nication, Université de Marne-la-Vallée, Marne-la-Vallée, France.

G. B. Giannakis is with the Department of Electrical and Computer Engi-neering, University of Minnesota, Minneapolis, MN 55455 USA (e-mail: [email protected]).

Publisher Item Identifier S 1053-587X(00)05969-9.

the input symbol stream with a strictly periodic and determin-istic sequence. In this paper, algorithms for batch estimation andcompensation for the unknown time-invariant channel and fre-quency offset effects from knowledge only of the received dataare developed, and their asymptotic performance is analyzedand compared with the Cramér–Rao bound (CRB).

Blind equalization of frequency-selective channels withcarrier frequency offset (FO) and/or Doppler effects is wellmotivated when it comes to compensating for local oscillatordrifts. Thus far, only a limited number of approaches havebeen proposed for blind estimation of frequency–selectivechannels in the presence of carrier frequency offset. In [3],[19], and [22], the approach consists of two steps: First, thechannel is equalized by minimizing a certain nonlinear costfunction [constant modulus algorithm (CMA)], and then, thecarrier phase is tracked from the equalized output. In general,these algorithms require a large number of samples to achieveconvergence, and in the presence of residual inter-symbolinterference (ISI) at the equalizer output, the performanceof corresponding frequency offset estimators may degrade.In the present approach, the influence of residual ISI on thecarrier offset estimate is alleviated, and consistency of thefrequency offset estimators is guaranteed even in the presenceof residual ISI. In addition, the present TIC framework enablesdevelopment of a subspace channel identification algorithmthat offers a closed-form expression for the unknown channel.Thus, the potential local minima or loss of convergence as-sociated with [3], [19], and [22] in the presence of noise arealso avoided. Furthermore, the TIC-based subspace approachis robust to channel overestimation errors, location of channelzeros, and color of additive stationary noise. It is shownthat in the presence of FO, the proposed channel estimationapproach achieves the same asymptotic performance as thecorresponding subspace algorithm that assumes perfect carriersynchronization. In other words, compensating for FO doesnot introduce asymptotically any loss in the performanceof the channel estimation approach. An additional featureof the proposed subspace channel estimation approach isguaranteed identifiability, irrespective of the channel zerolocations. Simulations illustrate that equalization of channelswith possibly spectral nulls within the TIC framework requiresan order of magnitude less samples than a CMA-based equal-izer. Recently, two additional approaches have been proposedfor blind channel estimation and equalization of SISO FIRcommunication channels in the presence of carrier FO [], [32].Based on iterative minimization of certain nonlinear functions,the algorithms proposed in [] and [32] may be computationallycomplex and prone to divergence in the presence of noise.

1053–587X/00$10.00 © 2000 IEEE

2390 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 48, NO. 8, AUGUST 2000

Fig. 1. Communication channel with frequency offset.

A different class of algorithms is proposed in [35] forblind separation/estimation of a nonconvolutive mixture ofBPSK sources with residual carriers using the spatial diversityprovided by an array of receiving antennas. The present paperassumes a different framework: a single-user channel withboth convolutive and residual carrier effects and no spatialdiversity condition but one that adopts a periodic precoderat the transmitter. Extension of the present single-user TICapproach to a multiuser scenario with both convolutive andresidual carrier effects is possible when multiple users can beseparated based on their distinct cyclic frequencies [9]. Theproposed FO estimation setup can be viewed as an extensionto frequency–selective channels of the FO estimation approachproposed for flat-fading channels in [18].

The rest of the paper is organized as follows. Section II intro-duces the communication channel setup and the concept of peri-odically time-varying TIC precoder. Sections III and IV proposechannel and frequency offset estimation approaches based onthe information provided by the received CS-statistics. In Sec-tions V and VI, the asymptotic performance of the frequencyoffset and subspace channel estimators is investigated. Com-puter simulations are conducted in Section VII, and last, con-clusions are drawn in Section VIII.

II. PRELIMINARIES

We consider the general communication channel setupdepicted in Fig. 1, where the independently and identicallydistributed (i.i.d.) input symbol stream is modulatedby the periodic sequence , .The modulation of with the periodic sequencerepresents the nonredundant periodic TIC precoder, and itsrole is to induce cyclostationarity in the transmitted sequence

(symbol stands for “is defined as”).It can be easily checked that the second-order statistics of

are periodic. By defining the time-varying correlation ofat time and lag as ,

(superscript stands for complex conjugation) , it fol-lows that ,where . Thus, exhibits CS-statisticsof period . Sequence is converted by the dig-ital-to-analog (D/A) converter into the continuous-timewaveform , which is pulseshaped by the baseband transmit filter and modu-lated by the carrier . The resulting signal, whichis denoted by , propagates through the unknownband-pass channel , (the underline denotes a bandpassrepresentation), and at the receiver, it is demodulated witha carrier frequency offset , which is due to the local os-cillator drifts. Defining the equivalentbasebandcomponents

, ,, where stands for convolu-

tion, the input/output (I/O) channel response can be expressedin the form

(1)

where the complex-valued noise is assumed circularlyGaussian distributed with zero mean and independently dis-tributed of . Sampling the continuous-time waveformat the symbol rate , the following discrete-time model isobtained from (1):

(2)

Considering , of support ,, and , from (2), we arrive at the

equivalent discrete-time model shown in Fig. 2, which isdescribed by the I/O relation

(3)

In the next sections, a two-step approach is proposed for esti-mating the unknown channel and frequency offset from the in-formation provided by the output CS-statistics. In the first step,the channel taps are recovered, within a scaling factor dependenton the unknown carrier frequency offset, using the second-orderCS statistics of the output. In the second step, the carrier fre-quency offset is estimated from the second- or fourth-order CSstatistics of the output and then used to remove the FO-depen-dent scaling factor present in the channel estimate.

III. B LIND CHANNEL IDENTIFICATION

The channel model with FO (3) can be rewritten in the equiv-alent form of a channel without FO, as is depicted in Fig. 3, butwith an additional modulation superimposed on

(4)

where , and. Since , it

follows that exhibits CS statistics as well. Using (4), itcan be easily shown that the output time-varying correlation

is also

SERPEDINet al.: BLIND CHANNEL AND CARRIER FREQUENCY OFFSET ESTIMATION 2391

periodic. Hence, it admits a Fourier series (FS) decomposition,whose coefficients are referred to as the cyclic correlations.The output cyclic correlation at cycle and lag is given by[c.f. (4)]

(5)

where ,, and . For a fixed cycle ,

the cyclic spectrum corresponding to cycleis defined as theZ-transform of the cyclic correlation sequence ,i.e., , and substituting, from(5), we arrive at

(6)

where ,, and . Henceforth, only

nonzero cycles are considered, which implies that thecontribution of the additive stationary noise is cancelledout in (6). This key property is used next to derive a channelestimation approach that is resilient to the color of .

It is well-known that the transfer function can be ex-tracted as the greatest common divisor (gcd) of the family ofcyclic spectra , , provided that theperiod of satisfies [14]. By taking advantageof the structure of (6), a subspace channel estimation approachthat is also robust to channel order overestimation errors may bedeveloped, as in [8], [10], and [28]. In order to shorten the ex-position of the subspace approach, we summarize here the mainresults presented in [8] and [10].

Associate with an arbitrary vector thematrix defined as

......

.... . .

. . ....

......

.... . .

......

......

......

......

(7)

Denote by the set of polynomials in over the com-plex field and by the set of -dimensional polyno-mial vectors , with ,

. Denoting by the gcd of the set of polynomials

, , the irreducible part of is defined as[1]. By

direct substitution, it follows that

(8)

The subspace approach that we borrow from [8] and [10] relieson (8). We collect all the cyclic spectra (6) corresponding to allnonzero cycles in the vector

(9)

Due to (6), in (9) can be factorized as

(10)

where is a vector given by

(11)

Because the condition is assumed, is theirreducible part of . Taking into account (8), we thus have

(12)

Suppose that , where stands for anupper bound (or estimate) of the unknown channel order. Inorder to write the polynomial vector equation (12) in the timedomain, several additional notations are introduced next. Thepolynomial vector given by (9) admits the power series

expansion , where we set

(13)

Let be the vector defined as

(14)

The unknown polynomial vector can be written as

, where

In addition, define the vectors ,, and .

Equation (14) can be rewritten in the equivalent matrix form

(15)

where , and denotes theblock Toeplitz

matrix with the first column and row given by the vectors


and , respec-tively. Parameterization of given by (11) provides, in thetime domain, the relation

(16)

where the matrix is block-diagonal with its th diagonal entrygiven by

... (17)

Using (16), (15) can be written in the form

(18)

Identifiability of the subspace approach has been established in[7], [8], and [10] and is summarized by the following result.

Proposition 1 (Structured Subspace Identifiability Re-sult): Assuming that the frequency offset is known,

, and , (18) admits a unique solution(within a scaling factor ambiguity) spanned by thevector .

Now, let us denote by andthe impulse response vectors corre-

sponding to the transfer functions and , respectively.Since , it follows that

(19)

where is a diagonal matrix defined asdiag . In practice, the cyclic

correlations involved in have to be estimated from a finitenumber of samples. A consistent estimate for the cycliccorrelation at cycle and lag is given by [12]

(20)

Denote by an estimate for the vector of cyclic correlations,and assume that the channel orderis known. Plugging (19)into (18), it follows that

(21)

where . Because all blind channel identifica-tion approaches estimate the unknown channel within a scalingfactor ambiguity, nonuniqueness of the channel estimate pre-vents us from computing the asymptotic performance. We re-move the phase ambiguity by fixing . Thus, (21) canbe rewritten as

(22)

where matrix consists of the last columns of ,vector corresponds to the first column of , and

. In addition, define the vectors ofcyclic correlations ,

and thenormal-ized asymptotic covariance matrix

. By replacing the cyclic correlations with

their sample estimates and the unknown FOwith the estimate[which is provided by (32) or (34)], (22) can be rewritten as

(23)

Since transfer function can be estimated without anyknowledge of from the set of equations (18), the I/O relation(4) can be deconvolved, assuming the existence of perfectinverse , to yield

(24)

We observe from (19) and (24) that recovery of andrequires estimation of . Before proceeding any further, wesuppose that estimation and deconvolution of have beenperformed, and we concentrate next on the frequency offset es-timation problem relying on .

IV. FREQUENCYOFFSETESTIMATION

Estimating from in (24) amounts to retrieving acomplex exponential in multiplicative noise [ is random]and additive noise : a problem originally addressedin [16] and [36]. Two different frequency offset estimatorsproposed herein rely on the results of [16] and [36]. The firstFO-estimator applies to input symbol constellations that satisfythe moment condition . For such constellations,recovery of is shown to be possible from the output secondorder CS statistics. Real-valued constellations (e.g., BPSK,PAM) are typical examples of input constellations for which themoment condition holds. The second estimator ap-plies to input constellations that satisfy the moment conditions

, . For such constellations,it is shown that the output fourth order CS statistics provideenough information for estimating . Complex–valued con-stellations [e.g., QAM(4) QAM(32)] are examples of inputconstellations for which these moment conditions are satisfied.

A. BPSK Constellations

It is known that for real-valued input constellations (e.g.,BPSK, PAM), which satisfy , the second-ordercon-jugatecyclic correlations of allow recovery of [16],[36]. Consider that is a BPSK sequence, and define theconjugate time-varying correlation

(25)

where . Being al-most periodically varying, the generalized Fourier series coeffi-cient of , which is termed the conjugate cyclic cor-relation, is given by [c.f. (25)]

(26)

where .Since is strictly periodic, consists of Kroneckerdeltas located at the harmonics , with being an integer;


i.e., . From (26), it followsthat also consists of Kronecker deltas (or spectrallines) located at cycles , where is an integer.An estimate of the FO can be obtained by measuring the lo-cation of these spectral lines. Consider the following nonlinearleast-squares (NLS) estimator

(27)

(28)

where , with the real-valuedvariables , , , and

, where the last inequality is imposed to ensureidentifiability of FO .

Due to the nonlinear minimization involved by the NLS-esti-mator (27) and (28), it is of interest to obtain a computationallyefficient algorithm. Straightforward manipulations show that1

Re

(29)

(30)

Substituting (29) into (28) and taking into account (30), it fol-lows that

(31)

Thus, asymptotically, the NLS-estimator coincides with the es-timator (see also [36])

(32)

The FO-estimator (32) can be implemented efficiently by con-sidering the fast Fourier transform (FFT) of the squared data.Zero-padding of the data to a sufficiently large number of points( ) prior to the FFT is necessary so that the frequency bins

1By a(T ) = O(T ), we mean thatlim a(T )=T = c, where theconstantc satisfies0 < jcj < 1. The notationa(T ) = o(T ) stands forlim a(T )=T = 0.

are small enough to allow accurate estimation of the FO. Forthe FFT bin to be compatible with this limit, it is necessary toensure a zero-padding on the order of CRB (seee.g., [31, pp. 147–150]).

B. QAM Constellations

A similar carrier frequency-offset estimator can bedeveloped when the input belongs to a QAM con-stellation. Suppose now that is drawn from aQAM constellation that satisfies the moment conditions

, , and . Definethe fourth-order time-varying correlation

. It can be easily checked that, with. The

generalized Fourier series coefficient of can beexpressed as

(33)

with .Since consists of a sequence of Kronecker deltas locatedat harmonic cycles , also consists ofa sequence of Kronecker deltas located at ,

. Similar to (32), an FFT-based estimatorcan be obtained as

(34)As before, the assumption is necessary to ensureuniqueness of the FO in (34). In the next section, the asymp-totic performance of FO-estimator (32) is established. Becauseanalysis of the FO-estimator (34) entails similar steps, we omitits derivation. We conclude this section by summarizing themain steps of the proposed channel and FO estimation algo-rithm.

Step 1) Compute the cyclic correlation estimates[using e.g., (20)].

Step 2) Find from the sample estimate of (18).Step 3) Equalize the channel (4).Step 4) Find using (32) or (34).Step 5) Find using (19).

V. ASYMPTOTICPERFORMANCE OFFO-ESTIMATORS

We will assume that the input is BPSK ( ) and that thezero-mean additive noise satisfies the mixing condition

(35)where cum

stands for the st-order cumulant of .Condition (35) refers to the absolute summability of cumulants


of any order and is a reasonable assumption in practice sinceit is satisfied by all time series of weak memory, i.e., anasymptotically vanishing span of dependence of the availabletime series samples [4, pp. 8, 25–27]. Assumption (35) willprove useful in establishing the consistency of our estimationalgorithms and in facilitating calculation of the asymptoticperformance of the proposed estimators. From (24), we obtainfor [since is assumed BPSK ,

]

(36)

(37)

Letting and substitutinginto (36), we obtain

Since estimators (32) and (27) coincide asymptotically, it suf-fices to establish the asymptotic performance of the NLS-esti-mator (27). We follow the lines of proof presented in [5], [20],and [21]. After some lengthy calculations, whose details are pre-sented in Appendix A, the following expression is obtained forthe FO-estimator’s asymptotic variance, as shown in (38) at thebottom of the page, with , , and

standing for the cyclic spectrum of CS-noiseevaluated at cycle and frequency .

Relation (38) does not allow any simple interpretation for theasymptotic variance, except the fact that it depends inverselyproportional on the sum of squares of the magnitude of all spec-tral lines and is directly proportional to the cyclic spectra ofadditive CS noise . However, it is important to recognizethe rapid rate of convergence of the FO estimator ( ).This observation will prove useful in the next section, where itis shown that the operation of compensating for the FO does notintroduce any penalty in the asymptotic performance of the sub-space channel estimator. This fast rate of convergence ( )of the proposed FO estimator will be maintained even in thepresence of residual ISI effects; the only modification in (38)consists of including in the cyclic spectra ofthe contributions due to residual ISI left uncompensated.

To benchmark performance of our FO estimator, the perfor-mance of the stochastic maximum likelihood method is assessedhere by deriving its CRB. The vector of unknown parametersis denoted by , and the unknown but randomtransmitted symbols are grouped together in the vector

. Computation of the CRB requires knowl-edge of the second-order statistics of additive noise (whichmay not be available). To simplify the calculations, additivenoise is assumed white and normally distributed. The CRBfor an unbiased estimatoris bounded below by the inverse ofthe Fisher information matrix (FIM) :

where , ,, and denotes the th entry of ,

and stands for the likelihood function of. Since inputis assumed unknown but random, the conditional likeli-

hood has to be averaged over all input sequences:. The exact CRB is given by

(39)

Since the evaluation of is computationally in-tractable, it is common to adopt (see, e.g., [], [13], and [17])a looser bound, which is referred to as the stochastic CRB,which is not as tight as the exact CRB, but it is computationallyeasier to evaluate. Due to the concavity of thefunction andJensen’s inequality, we obtain from (39) the following validCRB bound:

(40)

Direct calculations show that ,and

. Thus, we obtain

(41)

VI. A SYMPTOTICPERFORMANCE OF THESUBSPACECHANNEL

ESTIMATOR

The asymptotic performance of the least-squares (LS)channel estimate in (23) can be established using standardapproaches (see, e.g., [24, pp. 96 and 97]) and is given by thefollowing result.

Re

(38)


Theorem 1: The asymptotic estimation errorhas a limiting zero-mean complex Gaussian distribution withasymptotic covariance

(42)

where is the matrix with its th column given by

(43)

and denotes theth entry of cyclic vector .Proof: See Appendix D.

Next, we show that the same asymptotic covariance matrixis obtained for the subspace channel estimator in the pres-ence and/or absence of carrier frequency offset whenever theinput source is circularly distributed. We first show that theasymptotic mean-square error of the channel estimate,corresponding to the FO-channel model of Fig. 2, is equal tothe asymptotic mean-square error of the subspace estimatecorresponding to the channel model depicted in Fig. 4, whichassumes perfect carrier frequency offset synchronization butthe channel instead of . Keeping in mind (42) andthat the FO-channel model of Fig. 2 can be represented asin Fig. 3, it suffices to show that the asymptotic covariancematrix is the same for the channels represented in Figs. 3and 4. From the proof of Proposition 3, it will follow that theasymptotic covariance of the cyclic correlation coefficientsis not dependent on the modulating sequence thatis present in [ ; see Fig. 3].This claim relies on the circularity assumption on .

Proposition 3: Assuming that is circularly distributed,the asymptotic covariance matrix does not depend on themodulating sequence present in . For given cy-cles , , , the following asymptotic resultholds:

(44)

where

(45)

and

(46)

where , stands forthe kurtosis of , and

.Proof: See Appendix E.

Fig. 2. Equivalent discrete-time model.

Fig. 3. Equivalent discrete-time model: Without FO.

Fig. 4. Equivalent asymptotic model.

Fig. 5. Equivalent asymptotic model.

We show next that the same asymptotic distributionholds for the subspace channel estimates corresponding tothe channels depicted in Figs. 4 and 5. In Fig. 4, the un-known FO is supposed to be given by the FO estimator(34). Let and denote the output cyclicspectra corresponding to the channels depicted in Figs. 4and 5, respectively. Since , itfollows that . Thus, anestimate of can be obtained from an estimate of

using the relation

(47)

where stands for a consistent estimate of ,

and denotes the estimate (34) of. Let and denotethe subspace channel estimates corresponding to the channelsrepresented in Figs. 4 and 5, respectively, which are obtainedusing (21). It then holds that

(48)

(49)

Interestingly, (47) implies the following link between and:

(50)

wherediagonal matrix: ;

identitymatrix, diag

;Kronecker product.


Plugging (50) into (48), we obtain

(51)

Comparing (49) and (51) and repeating the arguments used inthe proof of Theorem 1, it follows that the channel estimates

and have the same asymptotic distribution. We have thusestablished the following theorem.

Theorem 2: Assuming the input is circularly distributed,compensating for the carrier FO does not introduce any penaltyin the asymptotic performance of the subspace channel esti-mator.

An intuitive explanation of this result may be given on thegrounds that the frequency estimator has a much faster rateof convergence than the subspace channel estimator

. Thus, for sufficiently large , the FO estimatemay be considered to be equal to its true value and can be con-sidered compensated exactly without affecting the channel esti-mate.

It is interesting to note that in general, two frequency offsetestimation setups are possible: post-equalization FO estimator,which estimates the FO from knowledge of equalized sam-ples (the setup assumed in this paper) and pre-equaliza-tion FO estimator, which estimates the FOdirectly from thereceived samples . Pre-equalization frequency offset esti-mators can be similarly derived by replacing within the previous developments, e.g., the estimator (32) takes theform

(52)

A natural question that arises is: Which one of the two fre-quency offset/channel estimation setups is preferable in prac-tice? We discuss separately the two estimation problems: fre-quency offset and channel. From a frequency offset estimationviewpoint, it is clear that asymptotically ( ) and for largesignal-to-noise ratios (SNR’s) , the post-equalizationsetup is preferable since the asymptotic variance (38) attains thevalue zero. This is not the case for the pre-equalization setup,where even in the absence of additive noise , the asymp-totic variance (38) is nonzero due to residual channel ISI effectsthat are left uncompensated. At very low SNR’s and in the pres-ence of channel spectral nulls, it is expected that pre-equaliza-tion will cause noise amplification. This may affect the varianceof and the bit error rate. In such a scenario, the pre-equaliza-tion estimation of FO may appear as an alternative. Due tothe very intricate expressions that are involved, this analysis isreported in the companion paper [11], where it is shown that es-timation and compensation of the carrier frequency offset canbe performed first, assuming the presence of an unknown fre-quency selective channel and with no loss in performance rel-ative to methods where the channel is first pre-equalized, andthen, the frequency offset is compensated afterwards.

From a channel estimation viewpoint, it is important to notethat both pre- and post-equalization FO estimators are asymptot-ically ( ) similar. This is because the pre-equalization FO

estimators also converge also at a rate and by repeatingthe arguments involved in proving Theorem 2, one reaches theconclusion that asymptotically pre-equalization channel estima-tion schemes behave similarly, as if there is no carrier FO. How-ever, the comparison is difficult to perform when only a finitenumber of samples is available. This problem is beyond thescope and length of the present paper and is to be reported ina future paper.

In the next section, several simulation experiments are con-ducted in order to test the performance of the proposed algo-rithms.

VII. SIMULATIONS

Throughout our simulations, we considered a channelwith the baseband channel impulse response

, frequency offset , and periodic precoder, ,

. The input was drawn from an equiprobable QPSKor BPSK constellation, and the additive noise was whiteand normally distributed, with zero mean. As channel estima-tion performance measures, the normalized root-mean-squareerror (RMSE) and the average bias (Avg. Bias) of channelestimates are computed via (see also, e.g., [23] and [33])

RMSE

Avg. Bias

wherechannel estimate in theth realization;number of Monte Carlo trials;Euclidean norm of .

The SNR is defined at the cyclostationary input of the equalizer

as SNR , wherestands for the noiseless channel output.

Experiment 1—Estimation of Frequency Offset:In order toverify the efficacy of the proposed FO estimation algorithms,we have run Monte Carlo simulations with BPSK symbols andcomputed the root-mean square error ofthe estimator (32) and compared it with the asymptotic limit(38) and the stochastic CRB (41). In Fig. 6(a) and (b), the ex-perimental RMSE values of FO estimator (32), which is repre-sented by the dash-dotted line, are plotted assuming the SNRrange: [0, 40] dB and two different sample sizes and

samples, respectively. The amount of zero padding was. The experimental RMSE values were obtained

by averaging over Monte Carlo trials. The channeloutput wasa priori equalized using a minimum mean squareequalizer (MMSE) with 50 taps [28]. The asymptotic limit (38)is denoted by the dashed line, whereas the CRB (41) is repre-sented by the solid line. Fig. 6(b) shows that the asymptotic limit(38) is achieved for a limited number of samples and ata low SNR (10 dB). Although the proposed algorithm does not


Fig. 6. Standard deviation of FO estimator (BPSK).

Fig. 7. TIC versus CMA: Comparisons (QPSK).

attain the stochastic CRB, Fig. 6(a) reveals the excellent perfor-mance of the FO estimator (32). For as low as samplesand SNR dB, it achieves a variance of .

Experiment 2—Comparison of TIC with CMA:In this ex-periment, we assess the performance of the proposed TIC setup and compare it with the CMA setup of [19] in terms ofequalizing the unknown channel and estimating the car-rier frequency offset . Since the channel has two spectralnulls, for both simulation setups, the equalizer length (50 taps)is chosen sufficiently long to compensate well the ISI effects.Input is drawn from a QPSK constellation, additive noise

is white and normally distributed, and as before, the fre-quency offset . The fourth-order moment CMA cri-terion of [19] is tested, assuming two different sample recordsavailable during each Monte Carlo trial: samples withadaptation step size and samples withadaptation step size . In both cases, the step sizewas optimally chosen using trial searches. For TIC setup, duringeach Monte Carlo trial, the unknown channel is first es-timated using the subspace approach, and its estimate is thenplugged into the expression of the MMSE equalizer [28]. Forthe CMA setup, during each Monte Carlo run, a linear equal-izer is estimated using the algorithm [19]. Both setups assumethat frequency offset estimation is performed using (34) on theequalized channel output. The parameters of FO estimator (34)

are and . In Fig. 7(a), we plot the stan-dard deviation of the frequency estimator (34) versus SNR cor-responding to CMA and TIC setups. In Fig. 7(b), the probabilityof symbol error versus SNR is plotted, assuming an averagingof Monte Carlo runs per SNR point. Both figures sug-gest that the TIC setup presents almost one order of magnitudefaster convergence than the CMA. We performed quite exten-sive simulation experiments using the fast implementation of theblock-based CMA proposed in [2] and the fractionally spacedCMA (FS-CMA), using a reduced number of samples[see Fig. 10(a)]. The poor performance of the CMA algorithmscalls for thorough performance and convergence analyses of theblock/FS-CMA algorithms and for developing fast convergingimplementations before drawing any definite conclusion.

Experiment 3—Performance of Subspace Channel Esti-mator: In Fig. 8, the RMSE and average bias of the subspacechannel estimates are plotted versus SNR in the presenceand absence of FO. A number of Monte Carloruns are averaged, assuming two different sample records

and samples, respectively. Input wasdrawn equiprobable from a QPSK constellation. The solid linerepresents the experimental values achieved by the subspaceapproach (23) in the case of a channel with FO, whereas thedashed line depicts the experimental values achieved by thesubspace approach, assuming perfect carrier synchronization.


Fig. 8. Channel Avg. RMSE/bias versus SNR.

Fig. 9. Channel Avg. RMSE/frequency variance versus SNR. Exact/overestimated channel order.

In Fig. 8(a), the asymptotic variance (42) of the subspaceapproach is represented by the solid line with circles. Fig. 8confirms our theoretical finding that the performance of thesubspace approach in the presence of FO achieves, asymptot-ically, the performance of the subspace approach, assumingperfect carrier synchronization. In Fig. 9(a), the RMSE valuesof the subspace channel estimates are plotted versus SNRassuming samples, Monte Carlo runs, andtwo different modulating precoders ( and )in the presence of exact and overestimatedchannel orders. In Fig. 9(b), the variance of the frequencyestimator (32) is plotted versus SNR in the presence of exactand overestimated channel orders ( ). Fig. 9(a) and(b) shows that the subspace channel and FO estimators arequite robust to channel order overestimation errors.

Experiment 4 —Equalization Performance:In this experi-ment, we test if compensating for the carrier frequency offsetaffects the symbol error rate. In Fig. 10(b), we have plotted theprobability of symbol error versus SNR for two scenarios: in thepresence (solid line) and absence (dashed line) of FO, respec-tively. An MMSE linear equalizer with 50 taps is used for bothscenarios. For each one of the Monte Carlo runs,samples are used in order to compute the subspace channel esti-mate. During each Monte Carlo trial, the MMSE linear equalizerof [28] is constructed using the subspace channel estimate. We

have repeated this experiment for three different data records500, 1000, and 2000 samples. The set of plots represented

in Fig. 10(b) suggests that the same symbol error curves areachievable in the presence or absence of FO. From an equal-ization viewpoint, carrier frequency offset compensation affectsperformance minimally.

VIII. C ONCLUSIONS

A blind channel and carrier frequency offset estimatorhas been proposed. Unlike existing approaches, the subspacechannel estimator is resilient to order overestimation errors,color of additive stationary noise, and location of channel zerosand does not require minimization of a nonlinear criterion. Acomputationally simple FO estimator, whose performance isrobust to residual ISI, has also been proposed. It has been shownthat compensating for FO does not introduce any penalty in theasymptotic performance of the subspace channel estimationapproach when compared with a perfect carrier synchronizationscenario. Extension of the proposed work to the more generalclass of time- and frequency-selective channels describedin [15] represents an important future research direction.Preliminary investigations show that the present results can beextended to compensate possibly different Doppler effects thatare induced by the relative motion of the mobile.


Fig. 10. Probability of symbol error.

APPENDIX AASYMPTOTIC ANALYSIS OF FO-ESTIMATOR

We follow the lines of proof presented in [5], [20], and [21].Define the new variables:

(53)

(54)

(55)

(56)

(57)

(58)

It is easy to check that

. Theasymptotic distribution of is performed by first establishingthe asymptotic consistency2 of , namely, ,

, and and then by derivingthe asymptotic distribution via a Taylor series expansion of theNLS-criterion in the neighborhood of the true values. Becausethe consistency of the NLS estimates can be established similarto [21, pp. 113–115] or [5, Lemma and Th. 1a], we defer itspresentation. Considering the Taylor series expansion of thegradient vector around the true value, we obtain

(59)

Since , from (59), we obtain

(60)

2If a andb denote two arbitrary sequences of random variables, the no-tationsa = o (1) andb = O (1) denote thata converges to zero withprobability one and thatb is bounded with probability one, respectively.

where

Re (61)

Re

Re

(62)

and stands for conjugate transposition. Define the matrix

Re

(63)Plugging (61) and (62) back into (60) and taking into account(63), we obtain

Re (64)

Define the block diagonal matrixdiag . From (64), we deduce that

Re (65)

In Appendix B, it is shown that the second term in the expansionof [see (63)] is asymptotically negligible as

, i.e., .


From (65), we obtain

Re

Re Re

Re (66)

Since , (66) can be expressed equivalently as

Re

Re Re

Re (67)

To compute the individual factors in the right-hand side of (67),we appeal to the following result [21].

Proposition 2: With denoting a positive integer anddenoting Kronecker delta, it holds that

(68)

Using (68), some direct computations show that the symmetricmatrix Re is given by

Re

...

......

.... . .

. . ....

...(69)

To compute the second factor in the right-hand side of (67), thefollowing identity is used:

Re Re

Re Re (70)

In Appendix C, a generic entry of the matrixin (70) is computed

explicitly. Using similar arguments, it can be shown that.

The asymptotic covariance of the FO estimatorcan now be calculated and is

given by the entry of the right-hand sidematrix in (67). The computations are eased by noting that aclosed-form expression for the inverse of matrix (69) can besimply obtained by using the block-matrix inversion formula[24, p. 413, (a.14)]. After some straightforward calculations(whose details do not present any interest), (38) is obtained forthe FO estimator’s asymptotic variance.

APPENDIX BPROOF OF

A generic entry of the matrixhas the form

(71)

where , and . Since satisfiesthe mixing condition (35), has finite moments, andhas finite memory, it follows that [which is defined in (37)]satisfies the mixing condition

(72)

where . We show next that. Straightforward computations show

that

(73)

Using the inequality , ,we deduce from (73) the following relations:

which proves the assertion.


APPENDIX CCALCULATION OF

Since a generic entry of the vector is of the form,

with , and , it followsthat the calculation of an arbitrary entry of the matrix

reduces to the evaluation ofthe limit

(74)

where , and . Consideringthe change of variables and , (74) can beexpressed equivalently as

(75)

where . Next, the right-handside double sum in (75) is decomposed into three sums, and eachsum will be evaluated individually:

(76)

Denote the first term in the right-hand side of (76) by. By in-terchanging the order of summation,can be further expressedas

(77)

Let us now suppose thatis fixed, and evaluate the inner termof (77)

(78)

Using Proposition 2 and the mixing condition (35), it is easy tocheck that

(79)

Considering the binomial expansion of in (78) andkeeping in mind (79), it follows that the limit (78) reduces tothe calculation of

(80)

Considering the change of variables , ,and , (80) can be expressed as

(81)

Plugging the binomial expansion of into (81),taking into account (79), and interchanging the order of sum-mation, (81) reduces to

(82)

where stands for the cyclic correlation coeffi-cient [see (5)]. Plugging (82) back into (77), it follows that

(83)

Next, we show that the second term on the right-hand side of(76) is equal to zero. Denote this term by

(84)

Due to the mixing condition (35), for any arbitrary , thereexists an integer such that

(85)


We split the sum (84) in two terms

(86)

We next show that both terms on the right-hand side of (86)converge to zero as . Since for

and , we can upper bound the firstterm on the right-hand side of (86) as follows:

(87)

where in deriving the last two inequalities in (87), we used thechange of variables , , and (85).Similarly, we can upper bound the second term on the right-handside of (86) as

(88)

Thus, . In a similar way, the third term onthe right-hand side of (76) is shown to converge to zero. Thisconcludes the proof.

APPENDIX DPROOF(SKETCH) OF THEOREM 1

The LS-solution of (23) is given by

(89)

Since , from (89), it follows that

(90)

Since , (90) can be expressed as

(91)

Because matrix depends linearly on theentries of the vector , there exists a selection matrixwhose entries are not dependent on such that

(92)

Simple calculations show that is indeed given by (43). In-serting (92) in (91), we obtain

(93)

It has been shown that has a limiting zero meanGaussian distribution with covariance matrix[12]. It followsthat . However, if ,then for an arbitrary continuous function , we have

[24, th. C1,2, pp. 422 and 423].It follows that asymptotically, the estimates and canbe substituted by their true values in (93) without affectingthe asymptotic distribution of . From (93), theasymptotic covariance (42) is obtained.

APPENDIX EPROOF OFPROPOSITION3

Since

(94)

it follows that the computation of the asymptotic covariance (44)resumes to computing

The following decomposition holds

(95)

The contribution of the first term on the right-hand side termof (95) corresponds to the product of means, and it cancelsout, considering all the terms in the entire decomposition on


the right-hand side of (94). The third term on the right-handside of (95) is zero due to the circularity assumption on( , , ). Next, we compute the contribu-tion of the second term on the right-hand side of (95). We havethe following relations:

(96)

Plugging

into (96), we can express as follows:

(97)

Considering the change of variables and, we can decompose the right-hand side of (97) into the sum

(98)

where

(99)

(100)

Next, we compute the term . We distinguish two cases. If, we then have

and using well-known results on Césaro sums, we obtain that

(101)

If sinceis uniformly bounded in , and

, we have that .Similar computations lead to

(102)

From (101) and (102), we obtain that

(103)

and using the Parserval formula, the right-hand side of (103)reduces to [see (45)] [6].

Next, we compute the fourth term on the right-hand side of(95), i.e., the cumulant term, and show it to be equal asymptot-ically with [see (46)]. Define

(104)

Since is white and independently distributed of , wehave

cum

cum

Since is a periodic sequence, we can consider the Fourierseries decomposition

Due to the linearity of cumulants, we further expresscum as

cum

cum

(105)

The cumulant term cumin (105) is nonzero only if its indices are

equal, i.e., , , and . Itfollows easily that the general term on the right-hand side of


(104) is equal to

(106)

where cum denotes the kur-tosis of input . It follows easily that

Henceforth, we obtain that

(107)

Applying Parserval’s formula, it can be easily shown that[6]. It is also transparent that the asymptotic

covariance matrix is not dependent on the additional modu-lation factor superimposed on the input . This isdue to the fact that all the entries of the asymptotic covariancematrix contain only terms of the form .

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Erchin Serpedin received the Diploma of electricalengineering (with highest distinction) from thePolytechnic Institute of Bucharest, Bucharest,Romania, in 1991. He received the specializationdegree in signal processing and transmission ofinformation from Supélec, Paris, France, in 1992, theM.Sc. degree from Georgia Institute of Technology,Atlanta, in 1992, and the Ph.D. degree from theUniversity of Virginia, Charlottesville, in December1998.

From 1993 to 1995, he was an Instructor with thePolytechnic Institute of Bucharest. In 1999, he joined the Wireless Commu-nications Laboratory, Texas A&M University, College Station, as an AssistantProfessor. His research interests lie in the areas of statistical signal processingand wireless communications.

Antoine Chevreuil was born in Caen, France, in1971. He received the M.Sc. and Ph.D. degrees fromEcole Nationale Supérieure des Télécommunica-tions, Paris, France, in 1994 and 1997, respectively.

In 1998, he was with the TelecommunicationsDepartment, the University of Louvain-la-Neuve,Louvain-la-Neuve, Belgium. He is currently anAssistant Professor with the CommunicationSystems Laboratory, University of Marne-la-Vallée,Marne-la-Vallée, France. His research interests arestatistical signal processing for communications,

system identification, and equalization.

Georgios B. Giannakis(F’96) received the Diplomain electrical engineering from the National TechnicalUniversity of Athens, Athens, Greece in 1981. FromSeptember 1982 to July 1986, he was with the Uni-versity of Southern California (USC), Los Angeles,where he received the M.Sc. degree in electrical engi-neering in 1983, the M.Sc. degree in mathematics in1986, and the Ph.D. degree in electrical engineeringin 1986.

After lecturing for one year at USC, he joinedthe University of Virginia (UVA), Charlottesville,

in 1987, where he became a Professor of electrical engineering in 1997,Graduate Committee Chair, and Director of the Communications, Controls,and Signal Processing Laboratory in 1998. Since January 1999, he has beenwith the University of Minnesota, Minneapolis, as a Professor of electricaland computer engineering. His general interests lie in the areas of signalprocessing and communications, estimation and detection theory, time-seriesanalysis, and system identification—subjects on which he published more than100 journal papers. Specific areas of expertise have included (poly)spectralanalysis, wavelets, cyclostationary, and non-Gaussian signal processing withapplications to sensor array and image processing. Current research focuseson transmitter and receiver diversity techniques for equalization of single- andmultiuser communication channels, mitigation of rapidly fading channels,compensation of nonlinear amplifier effects, redundant filterbank transceiversfor block transmissions, multicarrier, and wideband communication systems.

Dr. Giannakis was awarded the School of Engineering and Applied ScienceJunior Faculty Research Award from UVA in 1988 and the UVA-EE OutstandingFaculty Teaching Award in 1992. He received the IEEE Signal Processing So-ciety’s 1992 Paper Award in the Statistical Signal and Array Processing (SSAP)area and the 1999 Young Author Best Paper Award (with M. K. Tsatsanis). Heco-organized the 1993 IEEE Signal Processing Workshop on Higher-Order Sta-tistics, the 1996 IEEE Workshop on Statistical Signal and Array Processing,and the first IEEE Signal Processing Workshop on Wireless Communicationsin1997. He guest (co-)edited two special issues on high-order statistics (Interna-tional Journal of Adaptive Control and Signal Processingand the EURASIPjournalSignal Processing) and the January 1997 Special Issue on Signal Pro-cessing for Advanced Communications of the IEEE TRANSACTIONS ONSIGNAL

PROCESSING. He has served as an Associate Editor for the IEEE TRANSACTIONS

ON SIGNAL PROCESSINGand the IEEE SIGNAL PROCESSINGLETTERS, a Secre-tary of the Signal Processing Conference Board, a member of the SP Publica-tions Board, and a member and vice-chair of SSAP Technical Committee. Henow chairs the Signal Processing for Communications Technical Committee andserves as Editor-in-Chief of the IEEE SIGNAL PROCESSINGLETTERS. He is alsoa member of the IEEE Fellow Election Committee, the IMS, and the EuropeanAssociation for Signal Processing.

Philippe Loubaton (M’88) was born in 1958 inVillers Semeuse, France. He received the M.Sc. andPh.D. degrees from Ecole Nationale Supérieure desTélécommunications, Paris, France, in 1981 and1988, respectively.

From 1982 to 1986, he was a Member of TechnicalStaff with Thomson-CSF/RGS, where he worked indigital communications. From 1986 to 1988, he waswith the Institut National des Télécommunications asan Assistant Professor of electrical engineering. In1988, he joined the Signal Processing Department,

Ecole Nationale Supérieure des Télécommunications. Since 1995, he has beenProfessor of electrical engineering with Marne la Vallée University, Champssur Marne, France. His present research interests are in statistical signal pro-cessing, and digital communications with a special emphasis on blind equaliza-tion, multi-user communication systems and multi carrier modulations.

Dr. Loubaton is an associate editor for the IEEE TRANSACTIONS ONSIGNAL

PROCESSINGand is a member of the IEEE Signal Processing for Communica-tions technical committee.

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