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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 5, MAY 2010 2771

Uplink Synchronization in OFDMASpectrum-Sharing Systems

Luca Sanguinetti, Member, IEEE, Michele Morelli, Senior Member, IEEE, and H. Vincent Poor, Fellow, IEEE

Abstract—Spectrum sharing employs dynamic allocation offrequency resources for a more efficient use of the radio spec-trum. Despite its appealing features, this technology inevitablycomplicates the synchronization task, which must be accom-plished in an environment that may involve interference. Thispaper considers the uplink of an orthogonal frequency-divisionmultiple-access (OFDMA)-based spectrum sharing system andprovides solutions for estimating the frequency and timing er-rors of multiple unsynchronized users. In doing so, we exploitsuitably designed training blocks and apply maximum likelihoodmethods after modeling the interference power on each subcarrieras a random variable with an inverse gamma distribution. Theresulting frequency estimator turns out to be the extension to amultiuser scenario of a scheme that has previously been proposedfor single-user spectrum-sharing systems. Timing recovery ismore challenging and leads to a complete search over a multidi-mensional domain. To overcome such a difficulty, two alternativeapproaches are proposed. The first one relies on a simplifyingassumption about the interference distribution in the frequencydomain, while the second scheme operates in an iterative fashionaccording to the expectation-maximization algorithm.

Index Terms—Expectation-maximization algorithm, frequencyestimation, narrowband interference, OFDMA, spectrum sharing,timing estimation.

I. INTRODUCTION

I N orthogonal frequency-division multiple-access(OFDMA), several users simultaneously transmit their

data by modulating exclusive sets of orthogonal subcarriers.This approach offers increased resistance to intra-cell inter-ference, while providing remarkable flexibility in resourcemanagement and simplified channel equalization. All thesefeatures justify the adoption of OFDMA as a physical layertechnique in emerging broadband wireless communications, in-cluding the IEEE 802.16e metropolitan area network (WMAN)standard [1].

The inherent flexibility in allocating power over distinctsubcarriers makes OFDMA a natural choice for the deployment

Manuscript received April 15, 2009; accepted January 13, 2010. Date of pub-lication January 29, 2010; current version published April 14, 2010. The as-sociate editor coordinating the review of this manuscript and approving it forpublication was Prof. Gerald Matz. This research was supported in part by theU. S. National Science Foundation under Grant CNS-09-05398.

L. Sanguinetti and M. Morelli are with the Department of Information En-gineering, University of Pisa, 56126 Pisa, Italy (e-mail: [email protected]; [email protected]).

H. V. Poor is with the Department of Electrical Engineering, Princeton Uni-versity, Princeton, NJ 08544 USA (e-mail: [email protected]).

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

Digital Object Identifier 10.1109/TSP.2010.2041867

of spectrum-sharing systems, where users opportunisticallyestablish a communication link by filling existing gaps inthe frequency spectrum [2]. This technique offers an effectivesolution to the radio spectrum shortage problem by allowing co-existence of different types of wireless services over a commonfrequency-band. A practical example in which spectrum sharingcan usefully be employed is the OFDM-based IEEE 802.11gwireless local area network (WLAN) standard [3], whichoperates in the same unlicensed 2.4 GHz band as the Bluetoothsystem [4]. In such an application, collision avoidance can beguaranteed by dynamically placing unmodulated subcarriersover the frequency band occupied by Bluetooth users. Spectrumsharing can also find application in future WiMAX femtocells,which represent a viable means to improve radio coverage inindoor environments [5]. In practice, a femtocell can be viewedas a simplified WiMAX base station (BS) providing broadbandwireless access to small building areas that cannot be reliablycovered by an outdoor BS due to the high penetration lossfrom walls. Since femtocells operate on the same frequencyband as macro BSs, co-channel interference becomes a primaryimpairment for such systems. A promising solution is based onthe use of an OFDMA spectrum sharing protocol, accordingto which femtocell users monitor the radio environment andinhibit transmission over subcarriers that have been interfered.

A fundamental weakness of OFDMA is its remarkablesensitivity to frequency and timing errors. A carrier frequencyoffset (CFO) between the receiver and transmitter destroysthe subcarrier orthogonality and causes interchannel inter-ference (ICI) as well as multiple-access interference (MAI).Timing errors produce interblock interference (IBI) and mustbe kept within a small fraction of the block duration for reli-able transmission. While timing and frequency estimation inthe OFDMA downlink can be accomplished with the samemethods employed for single-user OFDM, synchronization inthe OFDMA uplink is much more challenging. The reason isthat uplink signals arriving at the BS are normally affected bydifferent Doppler shifts and propagation delays, which resultinto user-dependent timing and frequency errors.

The synchronization issue in the OFDMA uplink has re-ceived much attention in recent years [6]–[14]. In the earlierapproaches, a subband carrier assignment scheme (CAS) wasassumed in which each user transmits over an exclusive set ofadjacent subcarriers. After performing user separation at the BSthrough a bank of bandpass filters, timing and frequency errorscan be estimated in a blind fashion by exploiting either theredundancy offered by the cyclic prefix (CP) [6] or by lookingfor the position of null (virtual) carriers [7]. Unfortunately,the subband CAS does not ensure full channel diversity as a

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2772 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 5, MAY 2010

deep fade might hit a substantial part of the user’s subband. Toovercome this problem, synchronization algorithms have beendevised for the interleaved CAS, in which channel diversityis guaranteed by providing maximum separation among thesubcarriers assigned to the same user. The solution proposed in[8] exploits the inherent structure of the interleaved OFDMAuplink to get CFO estimates by means of a multiple signalclassification approach. In an effort to reduce the computa-tional load, the estimation of signal parameters via rotationalinvariance technique is suggested in [9]. Particular attention hasrecently been devoted to the OFDMA uplink with generalizedCAS. This technology provides a multiuser diversity gain byallowing each terminal to select the best subcarriers (i.e., thoseexhibiting the smallest channel attenuations) that are currentlyavailable for data transmission. A method for estimating thetiming and frequency errors of a new user entering an OFDMAsystem with generalized CAS is discussed in [10] assumingthat all other users are already synchronized. A more generalsituation is considered in [11], in which the space-alternatinggeneralized expectation-maximization (SAGE) algorithm isemployed to deal with multiple asynchronous users. Alternatingprojection methods are also employed in [12] to reduce the jointmaximum likelihood (ML) estimation of the users’ CFOs andchannel responses to a series of more tractable one-dimensionaloptimization problems. A variant of [12] is illustrated in [13],in which MAI cancellation is performed both in the time andfrequency domains through a modified SAGE approach, whilean alternative low-complexity scheme providing CFO estimatesin closed form is derived in [14] by using a specially designedpilot block.

As mentioned previously, OFDMA is a natural choice formultiple-access spectrum-sharing systems thanks to its flexi-bility in resource management. One major challenge in theseapplications is that, in addition to ICI and MAI, the users’ sig-nals may also be plagued by narrowband jamming, which isexpected to substantially degrade the accuracy of conventionalsynchronization schemes. An efficient frequency estimation al-gorithm for a single-user orthogonal frequency-division multi-plexing (OFDM) system plagued by non-negligible narrowbandinterference (NBI) has recently been derived in [15] assumingthat the NBI is Gaussian distributed across the signal spectrumwith zero mean and unknown power [16].

In this work, we investigate the synchronization problemin OFDMA-based spectrum-sharing systems and present fre-quency and timing acquisition schemes that are robust to NBI.In doing so, we assume that each remote terminal transmitsa number of consecutive and identical training blocks at thebeginning of the uplink frame in which pilot symbols areorganized into small groups of physically adjacent subcarrierscalled tiles. In order to preserve the channel frequency diversity,the tiles are allocated to users according to a generalized CAS.At the BS, frequency and timing acquisition is achieved bymeans of a two-stage procedure that operates as follows. Inthe first stage, frequency recovery is accomplished using anML approach in which the NBI power over each subcarrier istreated as a nuisance parameter with an inverse gamma distri-bution. Interestingly, the resulting frequency estimator is the

extension to a multiuser uplink scenario of the scheme previ-ously proposed in [15] for single-user transmissions. However,while such an estimator is derived in [15] by means of heuristicreasoning, a rigorous ML approach is employed in this work. Inthe second stage, the timing offset of each user and the channelattenuation over the corresponding tiles are jointly estimatedthrough ML methods. Unfortunately, the exact solution of thisproblem turns out to be too complex for practical purposes asit involves a complete search over a multidimensional domain.To overcome this difficulty, two alternative approaches aresuggested. The first one relies on the simplifying assumptionthat the average NBI power does not vary over a tile, therebyproviding suboptimal performance in the event that such acondition is not fulfilled. The second scheme is based on theexpectation-maximization (EM) algorithm, which operatesin an iterative fashion and, under some regularity conditions,converges to the ML solution. To the best of the authors’ knowl-edge, timing synchronization in OFDMA spectrum-sharingtransmissions has never been addressed before. The proposedtiming recovery schemes provide an effective solution to thisproblem and, accordingly, they represent the main contributionof our work.

The remainder of the paper is organized as follows. The nextsection introduces the tile structure model of the consideredOFDMA uplink. The CFO estimation algorithm is derived inSection III, while timing recovery is discussed in Section IV.Simulation results are presented in Section V and some conclu-sions are drawn in Section VI.

The notation adopted throughout the paper is defined as fol-lows. Matrices and vectors are denoted by boldface letters, with

being the identity matrix of order and the inverse ofa matrix . We use , and for complex conjugation,transposition and Hermitian transposition, respectively. The no-tation represents the Euclidean norm of the enclosed vector,

stands for the modulus and indicates the th entryof a matrix .

II. SYSTEM DESCRIPTION AND SIGNAL MODEL

A. System Description

We consider the uplink of an OFDMA system employingsubcarriers with index set and potentiallyaffected by NBI. The transmission is organized into framesand each frame is preceded by consecutive OFDMA blocksin which the available subcarriers are grouped into synchro-nization subchannels and data subchannels. The former areemployed by ranging subscriber stations (RSSs) that mustcomplete their synchronization process, while the latter areassigned to data subscriber stations (DSSs) that entered thenetwork at an earlier stage and have already achieved synchro-nization. We denote by the number of simultaneously activeRSSs and assume that a given subchannel cannot be accessedby more than one RSS. Each subchannel is divided intosubbands and a given subband comprises a set of adjacentsubcarriers, which is called a tile. The subcarrier indexes ofthe th tile in the thsubchannel are collected into a set . The


SANGUINETTI et al.: UPLINK SYNCHRONIZATION IN OFDMA SPECTRUM-SHARING SYSTEMS 2773

only constraint in the selection of is that the tiles cannotoverlap in the frequency domain, i.e., for

or . The th subchannel is thus composed ofsubcarriers with indexes belonging to .

The uplink signal transmitted by the th RSS propa-gates through a multipath channel with impulse response

, where depends onthe maximum expected channel delay spread. We assume that

does not vary significantly over a synchronization time-slot.Ranging signals arriving at the BS are typically not synchro-nized with the local references. We denote by the timingerror of the th RSS expressed in unit of sampling intervals ,while is the frequency offset normalized by the subcarrierspacing. In practice, each RSS achieves coarse timing andfrequency synchronization through a dedicated downlink con-trol channel before starting the uplink transmission. This way,frequency errors in the uplink are only due to Doppler shiftsand/or downlink estimation errors and, consequently, they arenormally smaller than . Timing errors are relatedto the distances of the RSSs from the BS and their maximumvalue roughly corresponds to the round trip propagationdelay for a user located at the cell boundary [10]. This amountsto putting , where is the cell radius. Inour study, we assume that the cyclic prefix (CP) is long enoughto accommodate both the channel delay spread and timingerrors. This results in a quasi-synchronous scenario in whichno IBI is present at the input of the receive discrete-Fouriertransform (DFT) device [12]. Although such a solution can beadopted during the synchronization stage, the CP of data blocksshould be made just greater than the channel length in order tominimize unnecessary overhead. It follows that accurate timinginformation must be acquired during the uplink synchronizationtime-slot in order to align all active users with the BS time scaleand avoid IBI over the data section of the frame.

B. Signal Model

Without loss of generality, in all subsequent derivations weconcentrate on the th synchronization subchannel and omit thesubscript for notational simplicity. Furthermore, we denote by

the DFT output over the th subcarrier of the th tileduring the th OFDMA block. Since DSSs are already alignedto the BS references, their signals do not contribute to .In contrast, the presence of uncompensated CFOs destroys thesubcarrier orthogonality of the RSSs and gives rise to ICI andMAI. Hence, we may write as

(1)

where is the length of the cyclically extendedOFDMA block, denotes the signal component of theRSS allocated over the considered subchannel and, finally,

is a disturbance term collecting the contribution ofthermal noise, NBI and MAI. The quantity is given by[17]

(2)

where

is the channel frequency response over the th subcarrier,while

(3)

accounts for ICI generated by subcarriers of the same tile. Fol-lowing [16], we model as a circularly symmetric com-plex Gaussian random variable with zero mean and variance

, where is the thermal noise power, whileaccounts for NBI plus CFO-induced MAI and depends on

the subcarrier index . Clearly, if the DFT output isfree of NBI and MAI. It is worth observing that is as-sumed to be independent of the block index , which amountsto saying that the interference power does not change over a syn-chronization time-slot. This assumption is reasonable as longas the value of is sufficiently small. To facilitate the discus-sion, in all subsequent derivations the quantitiesare assumed to be statistically independent for different valuesof , and . While this assumption is reasonable when ap-plied to thermal noise, some correlation is expected betweenNBI terms over closely spaced subcarriers. On the other hand,since such correlation could be exploited to increase the robust-ness of the system against NBI, the independence assumptionmay be viewed as a means for describing a worst-case scenario.

Our goal is the estimation of the unknown parameters andbased on the observations for ,

and .

III. MAXIMUM-LIKELIHOOD FREQUENCY OFFSET ESTIMATION

We begin by investigating the CFO estimation problem. Forthis purpose, we arrange the th DFT output across thesynchronization slot into an -dimensional vector

and rewrite (1) as

(4)

where col-lects the phase shifts induced by the CFO and

is a Gaussianvector with zero mean and covariance matrix .Vectors are exploited to obtain the joint MLestimate of and , with

. Denoting bythe set of unknown parameters, the probability density function(pdf) of conditioned on a trial value isgiven by

(5)



and depends on the variances , which are generallyunknown. A possible way to deal with this problem is sug-gested in [15], where the variances are treated as nuisancedeterministic parameters and estimated together with byapplying ML methods. Here, we follow an alternative approachin which are modeled as statistically independentrandom variables distributed according to an inverse gammadistribution. The latter is expressed by its pdf:

(6)

where is a design parameter. The main advantage of the in-verse gamma distribution is that it provides closed-form analysisin many otherwise intractable mathematical problems. For thisreason, it is adopted in the radar signal processing literature tostatistically describe the sea clutter (see, for example, [18] and[19]) and in wireless communications to model the inter-cell in-terference in severely fading channels [20]. Its use is also sug-gested in textbooks on estimation theory as a prior distributionfor the noise power (see, for example, [21, pp. 329 and 355]).

Under the above framework, the marginal log-likeli-hood function (LLF) for is obtained by averaging theright-hand-side of (5) with respect to

(7)

where is an -dimensional vector collecting the variancesfor and This opera-

tion is accomplished in Appendix A and leads to the followingobjective function:

(8)

The ML estimate of is the location where achieves itsglobal maximum. Maximizing with respect to yields

(9)

and substituting this result back into (8) leads to the ML fre-quency offset estimator (MLFE):

(10)

where is the frequency metric

(11)

with

(12)

As is periodic in with period , its maxima occurat a distance of from each other. This means that MLFE

TABLE ICOMPUTATIONAL LOAD

provides ambiguous estimates unless the CFO belongs to theinterval .

It is worth pointing out that the MLFE is equivalent to themodified ML estimator (MMLE) presented in [15]. However,while the MMLE is heuristically introduced in [15] just to im-prove the performance of the joint ML estimator of and ,the MLFE is derived here through a rigorous ML approach inwhich the regularization parameter appears naturally in thefrequency metric as a consequence of the inverse gammadistribution adopted for .

The computational load of MLFE can be assessed in termsof the number of required floating point operations (flops). Indoing so, we observe that computing needsflops for each pair , while flops are required toevaluate for any set . Hence, denoting bythe number of candidate CFO values, the overall complexity ofMLFE approximately amounts toflops, as shown in the first row of Table I.

IV. ESTIMATION OF THE TIMING DELAY

After CFO recovery, the BS must acquire informa-tion about the timing delay . For this purpose, wefirst compensate the DFT output for the phase shift

induced by the CFO. This leads to the quan-tities , where is providedby MLFE. Assuming for simplicity that , from (1) weobtain

(13)

where is statisticallyequivalent to . To proceed further, we assume thatthe channel response is nearly flat over a tile so that we mayreasonably replace the quantities with an averagefrequency response

(14)

Under the above assumption, from (2) it follows that canbe approximated as

(15)

where and we have defined

(16)



with as given in (3). Finally, substituting (15) into (13)yields

(17)

From the above equation, it follows that depends onand, consequently, it can be exploited for timing recovery. Un-fortunately, this operation is complicated by the presence of thenuisance vectors and . The ap-proach we follow here aims at jointly estimating the parameterset while still considering the variancesas statistically independent random variables with an inversegamma distribution. For notational conciseness, in the ensuingderivations the quantities with ,

and are collected into a single-dimensional vector .

A. Maximum-Likelihood Estimation

Given the unknown parameters , the frequency-rotated DFToutputs expressed in (17) are statistically indepen-dent Gaussian random variables with mean and vari-ances . Hence, the pdf of conditioned on and

takes the form

(18)

Averaging the right-hand-side of (18) with respect to the pdf ofand taking the logarithm of the resulting function, yields the

following marginal LLF:

(19)

aside from some irrelevant terms and factors independent of .The ML estimate of is the location where achievesits global maximum. Unfortunately, the maximization with re-spect to cannot be accomplished in closed-form and needsa multidimensional grid-search. To overcome this difficulty, inthe following two alternative approaches are proposed by whichtiming estimates can be obtained with affordable complexity.

B. Suboptimal Maximum-Likelihood Estimation

Assume that the quantities donot vary significantly with and can reasonably be replaced bytheir arithmetic mean over a tile:

(20)

In this hypothesis, the conditional pdf in (18) reduces to

(21)

where are modeled as statistically independentrandom variables with an inverse gamma distribution. Paral-leling the steps of the previous subsection, we compute theexpectation of with respect to the variances andtake the logarithm of the resulting function. This provides themarginal LLF for in the form

(22)Compared to (19), has the favorable property that eachlogarithmic term in (22) can be independently maximized withrespect to . In doing so, we obtain

(23)

where is obtained by averaging over the ob-served blocks, i.e.,

(24)

Substituting (23) into (22) and maximizing over produces

(25)

where

(26)

and

(27)

is the signal energy over the th tile collected across the ob-served blocks. We point out that the estimator (25) provides thetrue ML timing estimate under the condition that the variances

are independent of , otherwise it operates in a mis-matched mode. For this reason, in the sequel we refer to (25) asthe approximate ML timing estimator (AMLTE).

The computational load of AMLTE is assessed as follows.Assuming that the observations are available, evaluating

for and requiresflops. Moreover, from (16) we see that computing

for needs flops for any given



. Hence, the overall complexity involved in the computation ofstarting from the quantities amounts to

approximately flops, whereis the number of candidate values . Evaluatingthe energies requires flops, whileadditional flops are required to compute the summation in (25).The above results lead to the overall complexity of AMLTE assummarized in the second line of Table I.

C. EM-Based Timing Estimation

One inherent drawback of AMLTE is that it is derived underthe assumption that the NBI power does not vary over a tile,which is reasonable only for small values of . Such constraintcan be relaxed by following an alternative approach based onthe EM algorithm. As is well-known, in the EM formulation theobserved measurements are replaced with so-called completedata, from which the original measurements can be obtainedthrough a many-to-one mapping [22]. Then, the EM algorithmiteratively alternates between an E-step, calculating the expec-tation of the LLF of the complete data, and an M-step, maxi-mizing this expectation with respect to the unknown parameters.In order to fit the assumptions of the method, in what follows

is viewed as the complete data set, whileis still the unknown parameter vector. Hence, during the th it-eration the EM algorithm proceeds as follows:

E-step—The function is computed:

(28)

where and are conditional pdfs,denotes the statistical expectation over the pdf of

and is the current estimate of .M-step—The maximum of with respect to isfound. This produces the updated estimate

(29)

In order to evaluate the expectation on the right-hand-side of(28), we still assume that the entries of are statistically inde-pendent and follow the inverse gamma distribution in (6). Then,in Appendix B we show that, after omitting irrelevant factorsand additive terms, the function can be equivalentlyreplaced by

(30)

where

(31)

is a biased estimate of at the th iteration. For a given ,the maximum of with respect to is located at

(32)

where is given in (24) and

(33)

Next, inserting back into (30) and maximizing withrespect to yields

(34)

where

(35)

The channel estimate is eventually obtained from (32) in thefollowing form:

(36)

Inspection of (31)–(36) reveals the rationale behind the EM al-gorithm. Specifically, from (34)–(36) we see that a new estimate

is computed at the th iteration by exploiting an estimateof the interference power obtained from the previous step.

Vector is then employed in (31) to obtain , which willbe used in the th iteration and so forth. Clearly, an initialestimate of , say , is required to start the iterative proce-dure. One possible solution is based on the signal model (1) andtakes the form

(37)

where is provided by (10), while is obtained as in (9)after replacing by .

In the sequel, we refer to (34) as the EM-based timing esti-mator (EMTE). In assessing its computational complexity, from(32) and (33) we observe that, before starting with the iterations,it is convenient to precompute the quantitiesand for any set , which approximately re-quires flops. Then, evaluating the variances

in (31) for each pair starting from needsflops, while additional flops are required

to get the channel estimates in (32) for all can-didate values . Finally, flops areneeded to evaluate the timing metric in (35) starting from

and . The overall complexity of EMTE



is summarized in the third row of Table I, where denotes thenumber of employed iterations.

D. Refinement of the Timing Estimates

In order to avoid IBI over the data section of the frame, thetiming error must be confined within the interval

, where is the CP length adopted duringthe payload period. A simple way to counteract the insurgenceof IBI is suggested in [23], in which the estimate is pre-ad-vanced so as to move the mean value of the resulting timingerror toward the middle point of . This amounts to consid-ering a refined estimate , with

. Extensive simulations indicate that closely ap-proaches the mean channel delay spread, which can reasonablybe approximated by . This yields

(38)

where is provided by either AMLTE or EMTE and isassumed to be even such that is still integer-valued.

V. SIMULATION RESULTS

A. System Parameters

The performance of the proposed synchronization schemeshas been assessed by Monte Carlo simulations in a IEEE 802.16-based OFDMA system. The DFT size is , while thesampling period is ns. We assume thatsubcarriers are available for transmission. This means that 64null (virtual) subcarriers are placed at both edges of the signalspectrum to avoid aliasing problem at the receiver. A rangingtime-slot consists of OFDMA blocks comprising bothsynchronization and data subchannels. Any subchannel is di-vided into tiles. The latter contain subcarriersand are uniformly spaced over the signal spectrum at a dis-tance of subcarriers. We assume that 48 subchan-nels are employed for data transmission, while the remaining 8subchannels are reserved for synchronization and are used by

terminals that intend to establish a communication linkwith the BS. The channel responses have order . Theirentries are modeled as independent and circularly symmetricGaussian random variables with zero-mean and an exponentialpower delay profile, i.e.,

(39)

where is chosen such that . Channels ofdifferent users are statistically independent of each other. Weconsider a cell radius of 1.5 km, corresponding to a maximumpropagation delay of sampling periods. In order toaccommodate both the channel delay spread and the propaga-tion delays, each ranging block is preceded by a CP of length

. The normalized CFOs and timing errors are ran-domly generated at each simulation run. Specifically, the CFOsvary in the interval with uniform distribution, whiletiming errors are taken from the set with equal

a priori probabilities. Unless otherwise stated, we set .As in [15], the maximum of the MLFE frequency metric in (11)is sought through a coarse search followed by a parabolic inter-polation. Since the width of the main lobe of is inverselyproportional to , the candidate CFO values during the coarsesearch are spaced by .

Without loss of generality, we concentrate on a specificsynchronization subchannel and provide results only for thecorresponding user. In addition to background noise withvariance , the uplink signal is plagued by NBI, which isintroduced in the frequency domain after passing the receivedsamples through the DFT unit. This is done by adding statisti-cally independent zero-mean Gaussian terms of variance toa set of jammed subcarriers. Unless otherwise stated, weset throughout simulations. Although this approachdoes not take into account the spectral leakage induced by theDFT windowing effect, it has the advantage of facilitating thegeneration of NBI. On the other hand, since the proposed algo-rithms are able to estimate and mitigate the interference poweron a subcarrier basis no matter what the source of interferenceis, we expect that their accuracy is only marginally affected bythe leakage phenomenon.

Two different scenarios are envisaged: S1) the jammedsubcarriers are randomly distributed in the subchannel; S2) thejammed subcarriers are contiguous and occupy an entire tile.The signal-to-noise ratio (SNR) is defined as with

being the average power of the receivedsignal component, while the signal-to-interference ratio (SIR)over the jammed subcarriers is .

The accuracy of the frequency estimates is measured in termsof mean square estimation error (MSEE), which is defined as

. The probability of making a timing error, say ,is used as a performance indicator for the timing estimators. Inour simulations, a timing error is declared whenever the refinedestimate produces IBI during the data transmission period.As mentioned previously, such situation occurs if liesoutside the interval , where is theCP length over the data section of the frame.

B. Performance Evaluation

1) Frequency Estimation: We begin by assessing the impactof the parameter on the performance of MLFE. Fig. 1 illus-trates the MSEE of the frequency estimates versus as obtainedin the S1 scenario. The SNR is fixed at 10 dB, while the SIRover the jammed subcarriers is either 5 or 0 dB. The curvesare qualitatively similar to those reported in [15] and provideguidelines for the design of . Since represents a goodchoice irrespective of the SIR level, in the sequel such a valueis adopted for MLFE.

The accuracy of MLFE is depicted in Fig. 2 in terms of MSEEversus SNR. The results are obtained in the S1 scenario with

0 dB. Comparisons are made with an alternative CFO re-covery scheme that does not take NBI into consideration. This isachieved by putting into (5) forand , with being an unknown deter-ministic parameter. The resulting frequency estimator is called



Fig. 1. Accuracy of MLFE versus in the S1 scenario with dB,and or 0 dB.

MLFE for an interference-free scenario (MLFE-IFS) and looksfor the maximum of the following metric:

(40)

where is defined in (12). The Cramér–Rao bound fora single-user OFDM spectrum-sharing system operating in thepresence of NBI is also shown as a benchmark [15]. Inspectionof Fig. 2 indicates that MLFE is robust against NBI and its accu-racy is only 1 dB from the bound. Conversely, MLFE-IFS is se-verely affected by NBI and exhibits an irreducible floor at largeSNR values. Results obtained in the S2 scenario are not reportedas they are virtually the same as those illustrated in Figs. 1 and2. This fact can be explained by observing that MLFE is de-rived by assuming statistically independent interference powersover different frequency bins and, accordingly, its performanceis not affected by the position of the jammed subcarriers acrossthe signal spectrum.

2) Timing Estimation: Fig. 3 illustrates the impact of param-eter on the performance of AMLTE and EMTE in terms ofwith 10 dB. The jammed subcarriers are distributed ac-cording to the S1 scenario with a SIR level of either 5 or 0 dB.The number of iterations with EMTE is taken large enough sothat the process can converge to a steady-state solution. Interest-ingly, we see that the value adopted by MLFE providesnearly optimal performance also for timing estimation, and it istherefore adopted in all subsequent simulations.

Another important design parameter is the number of itera-tions needed by EMTE to achieve convergence. Fig. 4 showsthe accuracy of EMTE versus as obtained in the S1 scenario.The SNR is 5, 10, or 15 dB, while the SIR is fixed at 0 dB. Wesee that the estimator converges in approximately three itera-tions and no significant gains are obtained with . Thismeans that EMTE can be stopped after completion of the thirditeration.

Fig. 2. Accuracy of MLFE versus SNR in the S1 scenario with 0 dBand .

Fig. 3. versus in the S1 scenario with 10 dB, and5 or 0 dB.

Figs. 5 and 6 compare the performance of AMLTE,EMTE and a third scheme denoted as the ad hoc timingestimator (AHTE). The latter ignores the possible pres-ence of interference in the considered subchannel and isheuristically derived by assuming that the CFO is smallenough such that in (16) can reasonably be approxi-mated by . Substituting this result into (17) yields

from which, neglectingthe disturbance term and bearing in mind (24) and(38), a timing estimate can be obtained in closed-form as

(41)



Fig. 4. of EMTE versus in the S1 scenario with 0 dB and.

Fig. 5. versus SNR in the S1 scenario with 0 dB and .

with being the integer value closest to . The S1 sce-nario is considered in Fig. 5, while Fig. 6 reports the results ob-tained in the S2 scenario. In both cases, the SIR is fixed at 0 dB.Inspection of Fig. 5 reveals that EMTE provides excellent re-sults and largely outperforms AMLTE, which is plagued by anirreducible floor. The reason for this behavior is that AMLTE re-lies on the assumption that the NBI power is the same over a tileand, accordingly, it operates in a mismatched mode under the S1scenario. The situation is different in Fig. 6, where the NBI isactually concentrated on a single tile. In this case, AMLTE ex-hibits improved accuracy and its is practically the same asthat obtained with EMTE. Note that the latter assumes statisti-cally independent interference power over different subcarriersand, accordingly, it performs similarly in both the S1 and S2

Fig. 6. versus SNR in the S2 scenario with 0 dB and .

Fig. 7. versus SIR in the S1 scenario with 10 dB and .

scenarios. As for AHTE, it does not take NBI into considera-tion and exhibits poor accuracy.

Fig. 7 illustrates the performance of the timing estimatorsversus the SIR in the S1 scenario and with 10 dB. Asexpected, EMTE provides the best performance thanks to its re-markable robustness against NBI. In particular, increases byless than a factor of two when the SIR passes from 0 to 10 dB.Larger degradations occur with AMLTE, even though it remainsconsiderably better than AHTE.

The impact of the CFO magnitude on is shown in Fig. 8for 10 dB and varying in the range [0, 0.4]. We stillconsider the S1 scenario with a SIR level of 0 dB. We see thatAHTE performs poorly as the CFO increases, while the perfor-mance of AMLTE and EMTE depends weakly on . The reasonis that the latter schemes have been derived from (17) which nat-urally accounts for the CFO-induced ICI, while the signal model



Fig. 8. versus in the S1 scenario with 10 dB and 0 dB.

employed by AHTE does not take the ICI term into considera-tion.

All previous results have been obtained under the assumptionthat only 4 subcarriers are plagued by interference. Fig. 9 illus-trates the performance of the timing estimators as a function ofthe number of jammed subcarriers in the S1 scenario. TheSNR is fixed at 10 dB while the SIR is 0 dB and . As ex-pected, increases with for all investigated schemes. Fur-thermore, we observe that AMLTE achieves the same accuracyof EMTE as approaches 16. A possible explanation is thatwhen all subcarriers in the considered subchannel areplagued by interference and, accordingly, there is no differencebetween the S1 and S2 scenarios. In such a case, it was foundin Fig. 6 that both AMLTE and EMTE have the same accuracyand outperform AHTE.

3) Computational Complexity: We now compare the pro-cessing load of the investigated schemes in the assumed sim-ulation model. We observe that a total of candidateCFO values are needed to cover a frequency uncertainty rangeof [ 0.4, 0.4]. Hence, from Table I it follows that approximately8,660 flops are required by MLFE for each unsynchronized user.A slight complexity reduction is possible with MLE-IFS, whichneeds 8160 flops. Computing the timing estimates by means ofAMLTE and EMTE involves 32 600 and 64 200 flops, respec-tively, while AHTE provides in closed form with only 224flops. This means that the improved accuracy of AMLTE andEMTE comes at the price of a substantial computational burdenas compared to AHTE.

VI. CONCLUSION

We have presented synchronization algorithms for the uplinkof an OFDMA-based spectrum sharing network, in whichtraining blocks with identical tile structure are employed tojointly estimate the frequency and timing errors of multiple un-synchronized users. The proposed schemes have been derivedby applying ML estimation methods to a scenario characterized

Fig. 9. versus in the S1 scenario with dB, dBand .

by the presence of NBI. We have also investigated suboptimalapproaches that avoid computationally demanding searchesover multidimensional domains. Comparisons have been madewith conventional algorithms that do not take NBI into con-sideration. Computer simulations indicate that the proposedmethods are inherently robust to NBI and can effectively beemployed in a spectrum-sharing uplink system. The price forthis improved interference rejection capability is a remarkableincrease of the computational burden, especially when thetiming recovery task is addressed.

APPENDIX A

In this Appendix, we highlight the major steps leading to themarginal LLF for as given in (8). We begin by averaging theright-hand-side of (5) with respect to . This yields

(42)

where

(43)

is the a priori pdf of with support . Substi-tuting (5) and (43) into (42) and letting , pro-duces

(44)with . Using the identity

(45)



from (44) we obtain the LLF in the form

(46)Finally, omitting irrelevant terms independent of , we mayequivalently replace the LLF with the objective functionshown in (8).

APPENDIX B

In this Appendix, we compute the function definedin (28), which can be rewritten as

(47)

with as given in (43) and . From (18), weobserve that

(48)and

(49)

where we have used the following notation:

(50)

and

(51)

Substituting (43), (48), and (49) into (47) and letting, yields

(52)

with and , while

(53)

and

(54)

Using the identity (45), we rewrite (53) into the equivalent form

(55)

where the quantities and are independent of . Substi-tuting (55) into (52) yields

(56)

which coincides with in (30) after skipping the addi-tive terms and the irrelevant factor .

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Luca Sanguinetti (S’04–M’06) received the LaureaTelecommunications Engineer degree (cum laude)and the Ph.D. degree in information engineeringfrom the University of Pisa, Italy, in 2002 and 2005,respectively.

Since 2005, he has been with the Departmentof Information Engineering of the University ofPisa. In 2004, he was a visiting Ph.D. student atthe German Aerospace Center (DLR), Oberpfaffen-hofen, Germany. During the period June 2007–2008,he was a Postdoctoral Associate in the Department

of Electrical Engineering at Princeton University, Princeton, NJ. His expertiseand general interests span the areas of communications and signal processing,estimation, and detection theory. Current research topics focus on channelestimation, equalization, and synchronization in multicarrier systems withunknown intereference, linear and nonlinear prefiltering for interferencemitigation in multiuser environments.

Michele Morelli (SM’07) received the Laurea (cumlaude) degree in electrical engineering and thePremio di Laurea SIP degree from the Universityof Pisa, Italy, in 1991 and 1992 respectively, andthe Ph.D. degree in electrical engineering fromthe Department of Information Engineering of theUniversity of Pisa.

In September 1996, he joined the CentroStudi Metodi e Dispositivi per Radiotrasmissioni(CSMDR) of the Italian National Research Council(CNR), Pisa, Italy, where he held the position of

Research Assistant. Since 2001, he has been with the Department of Informa-tion Engineering of the University of Pisa, where he is currently an AssociateProfessor of Telecommunications. His research interests are in wireless com-munication theory, with emphasis on synchronization algorithms and channelestimation in multiple-access communication systems.

Prof. Morelli was a corecipient of the VTC 2006 (Fall) Best StudentPaper Award and is currently serving as an Associate Editor for the IEEETRANSACTIONS ON WIRELESS COMMUNICATIONS.

H. Vincent Poor (S’72–M’77–SM’82–F’87) re-ceived the Ph.D. degree in electrical engineeringand computer science from Princeton University,Princeton, NJ, in 1977.

From 1977 until 1990, he was on the faculty of theUniversity of Illinois at Urbana-Champaign. Since1990, he has been on the faculty at Princeton, wherehe is the Michael Henry Strater University Professorof Electrical Engineering and Dean of the Schoolof Engineering and Applied Science. His researchinterests are in the areas of stochastic analysis,

statistical signal processing, and information theory, and their applications inwireless networks and related fields. Among his publications in these areas arethe recent books Quickest Detection (Cambridge University Press, 2009) andInformation Theoretic Security (Now Publishers, 2009).

Dr. Poor is a member of the National Academy of Engineering, a Fellow ofthe American Academy of Arts and Sciences, and an International Fellow ofthe Royal Academy of Engineering (U.K.). He is also a Fellow of the Instituteof Mathematical Statistics, the Optical Society of America, and other organiza-tions. In 1990, he served as President of the IEEE Information Theory Society,and in 2004–2007 he served as the Editor-in-Chief of the IEEE TRANSACTIONSON INFORMATION THEORY. He is the recipient of the 2005 IEEE EducationMedal. Recent recognition of his work includes the 2007 Technical Achieve-ment Award of the IEEE Signal Processing Society, the 2008 Aaron D. WynerDistinguished Service Award of the IEEE Information Theory Society, and the2009 Edwin Howard Armstrong Achievement Award of the IEEE Communica-tions Society.


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