IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 58, NO. 5, MAY 2010
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Uplink Synchronization in OFDMA Spectrum-Sharing Systems Luca Sanguinetti, Member, IEEE, Michele Morelli, Senior Member, IEEE, and H. Vincent Poor, Fellow, IEEE
Abstract—Spectrum sharing employs dynamic allocation of frequency resources for a more efficient use of the radio spectrum. Despite its appealing features, this technology inevitably complicates the synchronization task, which must be accomplished in an environment that may involve interference. This paper considers the uplink of an orthogonal frequency-division multiple-access (OFDMA)-based spectrum sharing system and provides solutions for estimating the frequency and timing errors of multiple unsynchronized users. In doing so, we exploit suitably designed training blocks and apply maximum likelihood methods after modeling the interference power on each subcarrier as a random variable with an inverse gamma distribution. The resulting frequency estimator turns out to be the extension to a multiuser scenario of a scheme that has previously been proposed for single-user spectrum-sharing systems. Timing recovery is more challenging and leads to a complete search over a multidimensional domain. To overcome such a difficulty, two alternative approaches are proposed. The first one relies on a simplifying assumption about the interference distribution in the frequency domain, while the second scheme operates in an iterative fashion according to the expectation-maximization algorithm. Index Terms—Expectation-maximization algorithm, frequency estimation, 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 interference, while providing remarkable flexibility in resource management and simplified channel equalization. All these features justify the adoption of OFDMA as a physical layer technique in emerging broadband wireless communications, including the IEEE 802.16e metropolitan area network (WMAN) standard [1]. The inherent flexibility in allocating power over distinct subcarriers makes OFDMA a natural choice for the deployment Manuscript received April 15, 2009; accepted January 13, 2010. Date of publication January 29, 2010; current version published April 14, 2010. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Gerald Matz. This research was supported in part by the U. S. National Science Foundation under Grant CNS-09-05398. L. Sanguinetti and M. Morelli are with the Department of Information Engineering, University of Pisa, 56126 Pisa, Italy (e-mail: luca.sanguinetti@iet. unipi.it;
[email protected]). H. V. Poor is with the Department of Electrical Engineering, Princeton University, Princeton, NJ 08544 USA (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TSP.2010.2041867
of spectrum-sharing systems, where users opportunistically establish a communication link by filling existing gaps in the frequency spectrum [2]. This technique offers an effective solution to the radio spectrum shortage problem by allowing coexistence of different types of wireless services over a common frequency-band. A practical example in which spectrum sharing can usefully be employed is the OFDM-based IEEE 802.11g wireless local area network (WLAN) standard [3], which operates in the same unlicensed 2.4 GHz band as the Bluetooth system [4]. In such an application, collision avoidance can be guaranteed by dynamically placing unmodulated subcarriers over the frequency band occupied by Bluetooth users. Spectrum sharing can also find application in future WiMAX femtocells, which represent a viable means to improve radio coverage in indoor environments [5]. In practice, a femtocell can be viewed as a simplified WiMAX base station (BS) providing broadband wireless access to small building areas that cannot be reliably covered by an outdoor BS due to the high penetration loss from walls. Since femtocells operate on the same frequency band as macro BSs, co-channel interference becomes a primary impairment for such systems. A promising solution is based on the use of an OFDMA spectrum sharing protocol, according to which femtocell users monitor the radio environment and inhibit transmission over subcarriers that have been interfered. A fundamental weakness of OFDMA is its remarkable sensitivity to frequency and timing errors. A carrier frequency offset (CFO) between the receiver and transmitter destroys the subcarrier orthogonality and causes interchannel interference (ICI) as well as multiple-access interference (MAI). Timing errors produce interblock interference (IBI) and must be kept within a small fraction of the block duration for reliable transmission. While timing and frequency estimation in the OFDMA downlink can be accomplished with the same methods employed for single-user OFDM, synchronization in the OFDMA uplink is much more challenging. The reason is that uplink signals arriving at the BS are normally affected by different Doppler shifts and propagation delays, which result into user-dependent timing and frequency errors. The synchronization issue in the OFDMA uplink has received much attention in recent years [6]–[14]. In the earlier approaches, a subband carrier assignment scheme (CAS) was assumed in which each user transmits over an exclusive set of adjacent subcarriers. After performing user separation at the BS through a bank of bandpass filters, timing and frequency errors can be estimated in a blind fashion by exploiting either the redundancy offered by the cyclic prefix (CP) [6] or by looking for the position of null (virtual) carriers [7]. Unfortunately, the subband CAS does not ensure full channel diversity as a
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deep fade might hit a substantial part of the user’s subband. To overcome this problem, synchronization algorithms have been devised for the interleaved CAS, in which channel diversity is guaranteed by providing maximum separation among the subcarriers assigned to the same user. The solution proposed in [8] exploits the inherent structure of the interleaved OFDMA uplink to get CFO estimates by means of a multiple signal classification approach. In an effort to reduce the computational load, the estimation of signal parameters via rotational invariance technique is suggested in [9]. Particular attention has recently been devoted to the OFDMA uplink with generalized CAS. This technology provides a multiuser diversity gain by allowing each terminal to select the best subcarriers (i.e., those exhibiting the smallest channel attenuations) that are currently available for data transmission. A method for estimating the timing and frequency errors of a new user entering an OFDMA system with generalized CAS is discussed in [10] assuming that all other users are already synchronized. A more general situation is considered in [11], in which the space-alternating generalized expectation-maximization (SAGE) algorithm is employed to deal with multiple asynchronous users. Alternating projection methods are also employed in [12] to reduce the joint maximum likelihood (ML) estimation of the users’ CFOs and channel responses to a series of more tractable one-dimensional optimization problems. A variant of [12] is illustrated in [13], in which MAI cancellation is performed both in the time and frequency domains through a modified SAGE approach, while an alternative low-complexity scheme providing CFO estimates in closed form is derived in [14] by using a specially designed pilot block. As mentioned previously, OFDMA is a natural choice for multiple-access spectrum-sharing systems thanks to its flexibility in resource management. One major challenge in these applications is that, in addition to ICI and MAI, the users’ signals may also be plagued by narrowband jamming, which is expected to substantially degrade the accuracy of conventional synchronization schemes. An efficient frequency estimation algorithm for a single-user orthogonal frequency-division multiplexing (OFDM) system plagued by non-negligible narrowband interference (NBI) has recently been derived in [15] assuming that the NBI is Gaussian distributed across the signal spectrum with zero mean and unknown power [16]. In this work, we investigate the synchronization problem in OFDMA-based spectrum-sharing systems and present frequency and timing acquisition schemes that are robust to NBI. In doing so, we assume that each remote terminal transmits a number of consecutive and identical training blocks at the beginning of the uplink frame in which pilot symbols are organized into small groups of physically adjacent subcarriers called 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 by means of a two-stage procedure that operates as follows. In the first stage, frequency recovery is accomplished using an ML approach in which the NBI power over each subcarrier is treated as a nuisance parameter with an inverse gamma distribution. Interestingly, the resulting frequency estimator is the
extension to a multiuser uplink scenario of the scheme previously proposed in [15] for single-user transmissions. However, while such an estimator is derived in [15] by means of heuristic reasoning, a rigorous ML approach is employed in this work. In the second stage, the timing offset of each user and the channel attenuation over the corresponding tiles are jointly estimated through ML methods. Unfortunately, the exact solution of this problem turns out to be too complex for practical purposes as it involves a complete search over a multidimensional domain. To overcome this difficulty, two alternative approaches are suggested. The first one relies on the simplifying assumption that the average NBI power does not vary over a tile, thereby providing suboptimal performance in the event that such a condition is not fulfilled. The second scheme is based on the expectation-maximization (EM) algorithm, which operates in an iterative fashion and, under some regularity conditions, converges to the ML solution. To the best of the authors’ knowledge, timing synchronization in OFDMA spectrum-sharing transmissions has never been addressed before. The proposed timing recovery schemes provide an effective solution to this problem and, accordingly, they represent the main contribution of our work. The remainder of the paper is organized as follows. The next section introduces the tile structure model of the considered OFDMA uplink. The CFO estimation algorithm is derived in Section III, while timing recovery is discussed in Section IV. Simulation results are presented in Section V and some conclusions are drawn in Section VI. The notation adopted throughout the paper is defined as follows. Matrices and vectors are denoted by boldface letters, with being the identity matrix of order and the inverse of a matrix . We use , and for complex conjugation, transposition and Hermitian transposition, respectively. The norepresents the Euclidean norm of the enclosed vector, tation stands for the modulus and indicates the th entry of a matrix . II. SYSTEM DESCRIPTION AND SIGNAL MODEL A. System Description We consider the uplink of an OFDMA system employing subcarriers with index set and potentially affected by NBI. The transmission is organized into frames consecutive OFDMA blocks and each frame is preceded by in which the available subcarriers are grouped into synchronization subchannels and data subchannels. The former are employed by ranging subscriber stations (RSSs) that must complete their synchronization process, while the latter are assigned to data subscriber stations (DSSs) that entered the network at an earlier stage and have already achieved synchronization. We denote by the number of simultaneously active RSSs and assume that a given subchannel cannot be accessed by more than one RSS. Each subchannel is divided into subbands and a given subband comprises a set of adjacent subcarriers, which is called a tile. The subcarrier indexes of in the th the th tile subchannel are collected into a set . The
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SANGUINETTI et al.: UPLINK SYNCHRONIZATION IN OFDMA SPECTRUM-SHARING SYSTEMS
only constraint in the selection of is that the tiles cannot for overlap in the frequency domain, i.e., or . The th subchannel is thus composed of subcarriers with indexes belonging to . The uplink signal transmitted by the th RSS propagates through a multipath channel with impulse response , where depends on the 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 synchrothe timing nized with the local references. We denote by error of the th RSS expressed in unit of sampling intervals , is the frequency offset normalized by the subcarrier while spacing. In practice, each RSS achieves coarse timing and frequency synchronization through a dedicated downlink control channel before starting the uplink transmission. This way, frequency errors in the uplink are only due to Doppler shifts and/or downlink estimation errors and, consequently, they are . Timing errors are related normally smaller than to the distances of the RSSs from the BS and their maximum roughly corresponds to the round trip propagation value delay for a user located at the cell boundary [10]. This amounts , where is the cell radius. In to putting our study, we assume that the cyclic prefix (CP) is long enough to accommodate both the channel delay spread and timing errors. This results in a quasi-synchronous scenario in which no IBI is present at the input of the receive discrete-Fourier transform (DFT) device [12]. Although such a solution can be adopted during the synchronization stage, the CP of data blocks should be made just greater than the channel length in order to minimize unnecessary overhead. It follows that accurate timing information must be acquired during the uplink synchronization time-slot in order to align all active users with the BS time scale and avoid IBI over the data section of the frame.
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where
is the channel frequency response over the while
th subcarrier,
(3) accounts for ICI generated by subcarriers of the same tile. Following [16], we model as a circularly symmetric complex Gaussian random variable with zero mean and variance , where is the thermal noise power, while accounts for NBI plus CFO-induced MAI and depends on if the DFT output is the subcarrier index . Clearly, is asfree of NBI and MAI. It is worth observing that sumed to be independent of the block index , which amounts to saying that the interference power does not change over a synchronization time-slot. This assumption is reasonable as long is sufficiently small. To facilitate the discusas the value of sion, in all subsequent derivations the quantities are assumed to be statistically independent for different values of , and . While this assumption is reasonable when applied to thermal noise, some correlation is expected between NBI terms over closely spaced subcarriers. On the other hand, since such correlation could be exploited to increase the robustness of the system against NBI, the independence assumption may be viewed as a means for describing a worst-case scenario. Our goal is the estimation of the unknown parameters and based on the observations for , and . III. MAXIMUM-LIKELIHOOD FREQUENCY OFFSET ESTIMATION
B. Signal Model Without loss of generality, in all subsequent derivations we concentrate on the th synchronization subchannel and omit the subscript for notational simplicity. Furthermore, we denote by the DFT output over the th subcarrier of the th tile during the th OFDMA block. Since DSSs are already aligned . to the BS references, their signals do not contribute to In contrast, the presence of uncompensated CFOs destroys the subcarrier orthogonality of the RSSs and gives rise to ICI and as MAI. Hence, we may write (1) is the length of the cyclically extended where denotes the signal component of the OFDMA block, RSS allocated over the considered subchannel and, finally, is a disturbance term collecting the contribution of is given by thermal noise, NBI and MAI. The quantity [17]
We begin by investigating the CFO estimation problem. For th DFT output across the this purpose, we arrange the synchronization slot into an -dimensional vector and rewrite (1) as (4) where collects the phase shifts induced by the CFO and is a Gaussian vector with zero mean and covariance matrix . are exploited to obtain the joint ML Vectors and , with estimate of . Denoting by the set of unknown parameters, the probability density function conditioned on a trial value is (pdf) of given by
(2)
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(5)
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and depends on the variances , which are generally unknown. A possible way to deal with this problem is suggested in [15], where the variances are treated as nuisance by deterministic parameters and estimated together with applying ML methods. Here, we follow an alternative approach are modeled as statistically independent in which random variables distributed according to an inverse gamma distribution. The latter is expressed by its pdf:
TABLE I COMPUTATIONAL LOAD
(6) where is a design parameter. The main advantage of the inverse gamma distribution is that it provides closed-form analysis in many otherwise intractable mathematical problems. For this reason, it is adopted in the radar signal processing literature to statistically describe the sea clutter (see, for example, [18] and [19]) and in wireless communications to model the inter-cell interference in severely fading channels [20]. Its use is also suggested in textbooks on estimation theory as a prior distribution for the noise power (see, for example, [21, pp. 329 and 355]). Under the above framework, the marginal log-likeliis obtained by averaging the hood function (LLF) for right-hand-side of (5) with respect to (7) where
is an -dimensional vector collecting the variances for and This operation is accomplished in Appendix A and leads to the following objective function:
provides ambiguous estimates unless the CFO belongs to the . interval It is worth pointing out that the MLFE is equivalent to the modified ML estimator (MMLE) presented in [15]. However, while the MMLE is heuristically introduced in [15] just to improve the performance of the joint ML estimator of and , the MLFE is derived here through a rigorous ML approach in which the regularization parameter appears naturally in the as a consequence of the inverse gamma frequency metric distribution adopted for . The computational load of MLFE can be assessed in terms of the number of required floating point operations (flops). In needs doing so, we observe that computing flops for each pair , while flops are required to for any set . Hence, denoting by evaluate the number of candidate CFO values, the overall complexity of MLFE approximately amounts to flops, as shown in the first row of Table I. IV. ESTIMATION OF THE TIMING DELAY
achieves its yields
After CFO recovery, the BS must acquire information about the timing delay . For this purpose, we first compensate the DFT output for the phase shift induced by the CFO. This leads to the quan, where is provided tities , from (1) we by MLFE. Assuming for simplicity that obtain
(9)
(13)
and substituting this result back into (8) leads to the ML frequency offset estimator (MLFE):
where is statistically equivalent to . To proceed further, we assume that the channel response is nearly flat over a tile so that we may with an average reasonably replace the quantities frequency response
(8) The ML estimate of is the location where global maximum. Maximizing with respect to
(10) where
is the frequency metric (14) (11) Under the above assumption, from (2) it follows that be approximated as
with
can (15)
(12)
where
and we have defined
As is periodic in with period , its maxima occur from each other. This means that MLFE at a distance of Authorized licensed use limited to: UNIVERSITA PISA S ANNA. Downloaded on April 19,2010 at 12:11:40 UTC from IEEE Xplore. Restrictions apply.
(16)
SANGUINETTI et al.: UPLINK SYNCHRONIZATION IN OFDMA SPECTRUM-SHARING SYSTEMS
with yields
as given in (3). Finally, substituting (15) into (13)
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In this hypothesis, the conditional pdf in (18) reduces to
(17) depends on From the above equation, it follows that and, consequently, it can be exploited for timing recovery. Unfortunately, this operation is complicated by the presence of the and . The apnuisance vectors proach we follow here aims at jointly estimating the parameter while still considering the variances set as statistically independent random variables with an inverse gamma distribution. For notational conciseness, in the ensuing with , derivations the quantities and are collected into a single -dimensional vector . A. Maximum-Likelihood Estimation Given the unknown parameters , the frequency-rotated DFT outputs expressed in (17) are statistically indepenand varident Gaussian random variables with mean ances . Hence, the pdf of conditioned on and takes the form
(21) where are modeled as statistically independent random variables with an inverse gamma distribution. Paralleling the steps of the previous subsection, we compute the with respect to the variances and expectation of take the logarithm of the resulting function. This provides the marginal LLF for in the form
(22) has the favorable property that each Compared to (19), logarithmic term in (22) can be independently maximized with respect to . In doing so, we obtain (23) is obtained by averaging where served blocks, i.e.,
over the ob-
(24) (18) Substituting (23) into (22) and maximizing over
produces
Averaging the right-hand-side of (18) with respect to the pdf of and taking the logarithm of the resulting function, yields the following marginal LLF:
(25) where (26)
(19) aside from some irrelevant terms and factors independent of . is the location where achieves The ML estimate of its global maximum. Unfortunately, the maximization with recannot be accomplished in closed-form and needs spect to a multidimensional grid-search. To overcome this difficulty, in the following two alternative approaches are proposed by which timing estimates can be obtained with affordable complexity. B. Suboptimal Maximum-Likelihood Estimation Assume that the quantities do not vary significantly with and can reasonably be replaced by their arithmetic mean over a tile: (20)
and (27) is the signal energy over the th tile collected across the observed blocks. We point out that the estimator (25) provides the true ML timing estimate under the condition that the variances are independent of , otherwise it operates in a mismatched mode. For this reason, in the sequel we refer to (25) as the approximate ML timing estimator (AMLTE). The computational load of AMLTE is assessed as follows. are available, evaluating Assuming that the observations for and requires flops. Moreover, from (16) we see that computing for needs flops for any given
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. Hence, the overall complexity involved in the computation of starting from the quantities amounts to flops, where approximately is the number of candidate values . Evaluating requires flops, while the energies additional flops are required to compute the summation in (25). The above results lead to the overall complexity of AMLTE as summarized in the second line of Table I.
is a biased estimate of the maximum of
at the th iteration. For a given , with respect to is located at
(32) where
is given in (24) and
C. EM-Based Timing Estimation One inherent drawback of AMLTE is that it is derived under the assumption that the NBI power does not vary over a tile, which is reasonable only for small values of . Such constraint can be relaxed by following an alternative approach based on the EM algorithm. As is well-known, in the EM formulation the observed measurements are replaced with so-called complete data, from which the original measurements can be obtained through a many-to-one mapping [22]. Then, the EM algorithm iteratively alternates between an E-step, calculating the expectation of the LLF of the complete data, and an M-step, maximizing 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, while is still the unknown parameter vector. Hence, during the th iteration the EM algorithm proceeds as follows: is computed: E-step—The function
(33) Next, inserting respect to yields
back into (30) and maximizing with
(34) where (35) The channel estimate is eventually obtained from (32) in the following form: (36)
(28) and are conditional pdfs, denotes the statistical expectation over the pdf of and is the current estimate of . with respect to is M-step—The maximum of found. This produces the updated estimate where
(29) In order to evaluate the expectation on the right-hand-side of (28), we still assume that the entries of are statistically independent and follow the inverse gamma distribution in (6). Then, in Appendix B we show that, after omitting irrelevant factors can be equivalently and additive terms, the function replaced by
(30) where
(31)
Inspection of (31)–(36) reveals the rationale behind the EM algorithm. Specifically, from (34)–(36) we see that a new estimate is computed at the th iteration by exploiting an estimate of the interference power obtained from the previous step. is then employed in (31) to obtain , which will Vector th iteration and so forth. Clearly, an initial be used in the estimate of , say , is required to start the iterative procedure. One possible solution is based on the signal model (1) and takes the form (37) is obtained as in (9) where is provided by (10), while after replacing by . In the sequel, we refer to (34) as the EM-based timing estimator (EMTE). In assessing its computational complexity, from (32) and (33) we observe that, before starting with the iterations, it is convenient to precompute the quantities and for any set , which approximately requires flops. Then, evaluating the variances in (31) for each pair starting from needs flops, while additional flops are required in (32) for all canto get the channel estimates . Finally, flops are didate values in (35) starting from needed to evaluate the timing metric and . The overall complexity of EMTE
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SANGUINETTI et al.: UPLINK SYNCHRONIZATION IN OFDMA SPECTRUM-SHARING SYSTEMS
is summarized in the third row of Table I, where number of employed iterations.
denotes the
D. Refinement of the Timing Estimates In order to avoid IBI over the data section of the frame, the timing error must be confined within the interval , where is the CP length adopted during the payload period. A simple way to counteract the insurgence of IBI is suggested in [23], in which the estimate is pre-advanced so as to move the mean value of the resulting timing . This amounts to considerror toward the middle point of , with ering a refined estimate . Extensive simulations indicate that closely approaches the mean channel delay spread, which can reasonably . This yields be approximated by (38) where is provided by either AMLTE or EMTE and is still integer-valued. assumed to be even such that
is
V. SIMULATION RESULTS A. System Parameters The performance of the proposed synchronization schemes has been assessed by Monte Carlo simulations in a IEEE 802.16, while the based OFDMA system. The DFT size is sampling period is ns. We assume that subcarriers are available for transmission. This means that 64 null (virtual) subcarriers are placed at both edges of the signal spectrum to avoid aliasing problem at the receiver. A ranging OFDMA blocks comprising both time-slot consists of synchronization and data subchannels. Any subchannel is ditiles. The latter contain subcarriers vided into and are uniformly spaced over the signal spectrum at a dissubcarriers. We assume that 48 subchantance of nels are employed for data transmission, while the remaining 8 subchannels are reserved for synchronization and are used by terminals that intend to establish a communication link . Their with the BS. The channel responses have order entries are modeled as independent and circularly symmetric Gaussian random variables with zero-mean and an exponential power delay profile, i.e., (39) where is chosen such that . Channels of different users are statistically independent of each other. We consider a cell radius of 1.5 km, corresponding to a maximum sampling periods. In order to propagation delay of accommodate both the channel delay spread and the propagation delays, each ranging block is preceded by a CP of length . The normalized CFOs and timing errors are randomly generated at each simulation run. Specifically, the CFOs with uniform distribution, while vary in the interval with equal timing errors are taken from the set
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. 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 interis inversely polation. Since the width of the main lobe of proportional to , the candidate CFO values during the coarse . search are spaced by Without loss of generality, we concentrate on a specific synchronization subchannel and provide results only for the corresponding user. In addition to background noise with , the uplink signal is plagued by NBI, which is variance introduced in the frequency domain after passing the received samples through the DFT unit. This is done by adding statistically independent zero-mean Gaussian terms of variance to jammed subcarriers. Unless otherwise stated, we a set of throughout simulations. Although this approach set does not take into account the spectral leakage induced by the DFT windowing effect, it has the advantage of facilitating the generation of NBI. On the other hand, since the proposed algorithms are able to estimate and mitigate the interference power on a subcarrier basis no matter what the source of interference is, we expect that their accuracy is only marginally affected by the leakage phenomenon. Two different scenarios are envisaged: S1) the jammed subcarriers are randomly distributed in the subchannel; S2) the jammed subcarriers are contiguous and occupy an entire tile. with The signal-to-noise ratio (SNR) is defined as being the average power of the received signal component, while the signal-to-interference ratio (SIR) . over the jammed subcarriers is The accuracy of the frequency estimates is measured in terms of 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. In our simulations, a timing error is declared whenever the refined produces IBI during the data transmission period. estimate As mentioned previously, such situation occurs if lies , where is the outside the interval CP length over the data section of the frame.
B. Performance Evaluation 1) Frequency Estimation: We begin by assessing the impact of the parameter on the performance of MLFE. Fig. 1 illustrates the MSEE of the frequency estimates versus as obtained in the S1 scenario. The SNR is fixed at 10 dB, while the SIR over the jammed subcarriers is either 5 or 0 dB. The curves are qualitatively similar to those reported in [15] and provide represents a good guidelines for the design of . Since choice irrespective of the SIR level, in the sequel such a value is adopted for MLFE. The accuracy of MLFE is depicted in Fig. 2 in terms of MSEE versus SNR. The results are obtained in the S1 scenario with 0 dB. Comparisons are made with an alternative CFO recovery scheme that does not take NBI into consideration. This is into (5) for achieved by putting and , with being an unknown deterministic parameter. The resulting frequency estimator is called
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Fig. 1. Accuracy of MLFE versus and or 0 dB.
in the S1 scenario with
dB,
Fig. 2. Accuracy of MLFE versus SNR in the S1 scenario with and .
0 dB
MLFE for an interference-free scenario (MLFE-IFS) and looks for the maximum of the following metric:
(40) is defined in (12). The Cramér–Rao bound for where a single-user OFDM spectrum-sharing system operating in the presence of NBI is also shown as a benchmark [15]. Inspection of Fig. 2 indicates that MLFE is robust against NBI and its accuracy is only 1 dB from the bound. Conversely, MLFE-IFS is severely affected by NBI and exhibits an irreducible floor at large SNR values. Results obtained in the S2 scenario are not reported as they are virtually the same as those illustrated in Figs. 1 and 2. This fact can be explained by observing that MLFE is derived by assuming statistically independent interference powers over different frequency bins and, accordingly, its performance is not affected by the position of the jammed subcarriers across the signal spectrum. 2) Timing Estimation: Fig. 3 illustrates the impact of parameter on the performance of AMLTE and EMTE in terms of with 10 dB. The jammed subcarriers are distributed according to the S1 scenario with a SIR level of either 5 or 0 dB. The number of iterations with EMTE is taken large enough so that the process can converge to a steady-state solution. Interestadopted by MLFE provides ingly, we see that the value nearly optimal performance also for timing estimation, and it is therefore adopted in all subsequent simulations. Another important design parameter is the number of iteraneeded by EMTE to achieve convergence. Fig. 4 shows tions as obtained in the S1 scenario. the accuracy of EMTE versus The SNR is 5, 10, or 15 dB, while the SIR is fixed at 0 dB. We see that the estimator converges in approximately three itera. This tions and no significant gains are obtained with means that EMTE can be stopped after completion of the third iteration.
Fig. 3.
versus in the S1 scenario with 5 or 0 dB.
10 dB,
and
Figs. 5 and 6 compare the performance of AMLTE, EMTE and a third scheme denoted as the ad hoc timing estimator (AHTE). The latter ignores the possible presence of interference in the considered subchannel and is heuristically derived by assuming that the CFO is small in (16) can reasonably be approxienough such that . Substituting this result into (17) yields mated by from which, neglecting the disturbance term and bearing in mind (24) and (38), a timing estimate can be obtained in closed-form as
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SANGUINETTI et al.: UPLINK SYNCHRONIZATION IN OFDMA SPECTRUM-SHARING SYSTEMS
Fig. 4. .
of EMTE versus
Fig. 5.
in the S1 scenario with
versus SNR in the S1 scenario with
0 dB and
0 dB and
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Fig. 6.
versus SNR in the S2 scenario with
0 dB and
.
Fig. 7.
versus SIR in the S1 scenario with
10 dB and
.
.
with being the integer value closest to . The S1 scenario is considered in Fig. 5, while Fig. 6 reports the results obtained in the S2 scenario. In both cases, the SIR is fixed at 0 dB. Inspection of Fig. 5 reveals that EMTE provides excellent results and largely outperforms AMLTE, which is plagued by an irreducible floor. The reason for this behavior is that AMLTE relies on the assumption that the NBI power is the same over a tile and, accordingly, it operates in a mismatched mode under the S1 scenario. The situation is different in Fig. 6, where the NBI is actually concentrated on a single tile. In this case, AMLTE exis practically the same as hibits improved accuracy and its that obtained with EMTE. Note that the latter assumes statistically independent interference power over different subcarriers and, accordingly, it performs similarly in both the S1 and S2
scenarios. As for AHTE, it does not take NBI into consideration and exhibits poor accuracy. Fig. 7 illustrates the performance of the timing estimators 10 dB. As versus the SIR in the S1 scenario and with expected, EMTE provides the best performance thanks to its reincreases by markable robustness against NBI. In particular, less than a factor of two when the SIR passes from 0 to 10 dB. Larger degradations occur with AMLTE, even though it remains considerably better than AHTE. is shown in Fig. 8 The impact of the CFO magnitude on for 10 dB and varying in the range [0, 0.4]. We still consider the S1 scenario with a SIR level of 0 dB. We see that AHTE performs poorly as the CFO increases, while the performance of AMLTE and EMTE depends weakly on . The reason is that the latter schemes have been derived from (17) which naturally accounts for the CFO-induced ICI, while the signal model
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Fig. 8.
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versus
in the S1 scenario with
10 dB and
0 dB.
employed by AHTE does not take the ICI term into consideration. All previous results have been obtained under the assumption that only 4 subcarriers are plagued by interference. Fig. 9 illustrates the performance of the timing estimators as a function of of jammed subcarriers in the S1 scenario. The the number . As exSNR is fixed at 10 dB while the SIR is 0 dB and increases with for all investigated schemes. Furpected, thermore, we observe that AMLTE achieves the same accuracy approaches 16. A possible explanation is that of EMTE as all subcarriers in the considered subchannel are when plagued by interference and, accordingly, there is no difference between the S1 and S2 scenarios. In such a case, it was found in Fig. 6 that both AMLTE and EMTE have the same accuracy and outperform AHTE. 3) Computational Complexity: We now compare the processing load of the investigated schemes in the assumed simcandidate ulation model. We observe that a total of CFO values are needed to cover a frequency uncertainty range of [ 0.4, 0.4]. Hence, from Table I it follows that approximately 8,660 flops are required by MLFE for each unsynchronized user. A slight complexity reduction is possible with MLE-IFS, which needs 8160 flops. Computing the timing estimates by means of AMLTE and EMTE involves 32 600 and 64 200 flops, respecin closed form with only 224 tively, while AHTE provides flops. This means that the improved accuracy of AMLTE and EMTE comes at the price of a substantial computational burden as compared to AHTE.
Fig. 9. and
versus .
in the S1 scenario with
dB,
dB
by the presence of NBI. We have also investigated suboptimal approaches that avoid computationally demanding searches over multidimensional domains. Comparisons have been made with conventional algorithms that do not take NBI into consideration. Computer simulations indicate that the proposed methods are inherently robust to NBI and can effectively be employed in a spectrum-sharing uplink system. The price for this improved interference rejection capability is a remarkable increase of the computational burden, especially when the timing recovery task is addressed. APPENDIX A In this Appendix, we highlight the major steps leading to the marginal LLF for as given in (8). We begin by averaging the right-hand-side of (5) with respect to . This yields (42) where (43) is the a priori pdf of with support tuting (5) and (43) into (42) and letting duces
. Substi, pro-
VI. CONCLUSION We have presented synchronization algorithms for the uplink of an OFDMA-based spectrum sharing network, in which training blocks with identical tile structure are employed to jointly estimate the frequency and timing errors of multiple unsynchronized users. The proposed schemes have been derived by applying ML estimation methods to a scenario characterized
(44) with
. Using the identity
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(45)
SANGUINETTI et al.: UPLINK SYNCHRONIZATION IN OFDMA SPECTRUM-SHARING SYSTEMS
from (44) we obtain the LLF in the form
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where the quantities and tuting (55) into (52) yields
(46) Finally, omitting irrelevant terms independent of , we may equivalently replace the LLF with the objective function shown in (8).
are independent of
. Substi-
(56) which coincides with in (30) after skipping the addiand the irrelevant factor . tive terms
APPENDIX B defined
In this Appendix, we compute the function in (28), which can be rewritten as
(47) with as given in (43) and observe that
. From (18), we
(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)
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Luca Sanguinetti (S’04–M’06) received the Laurea Telecommunications Engineer degree (cum laude) and the Ph.D. degree in information engineering from the University of Pisa, Italy, in 2002 and 2005, respectively. Since 2005, he has been with the Department of Information Engineering of the University of Pisa. In 2004, he was a visiting Ph.D. student at the German Aerospace Center (DLR), Oberpfaffenhofen, Germany. During the period June 2007–2008, he was a Postdoctoral Associate in the Department of Electrical Engineering at Princeton University, Princeton, NJ. His expertise and general interests span the areas of communications and signal processing, estimation, and detection theory. Current research topics focus on channel estimation, equalization, and synchronization in multicarrier systems with unknown intereference, linear and nonlinear prefiltering for interference mitigation in multiuser environments.
Michele Morelli (SM’07) received the Laurea (cum laude) degree in electrical engineering and the Premio di Laurea SIP degree from the University of Pisa, Italy, in 1991 and 1992 respectively, and the Ph.D. degree in electrical engineering from the Department of Information Engineering of the University of Pisa. In September 1996, he joined the Centro Studi 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 Information Engineering of the University of Pisa, where he is currently an Associate Professor of Telecommunications. His research interests are in wireless communication theory, with emphasis on synchronization algorithms and channel estimation in multiple-access communication systems. Prof. Morelli was a corecipient of the VTC 2006 (Fall) Best Student Paper Award and is currently serving as an Associate Editor for the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS.
H. Vincent Poor (S’72–M’77–SM’82–F’87) received the Ph.D. degree in electrical engineering and computer science from Princeton University, Princeton, NJ, in 1977. From 1977 until 1990, he was on the faculty of the University of Illinois at Urbana-Champaign. Since 1990, he has been on the faculty at Princeton, where he is the Michael Henry Strater University Professor of Electrical Engineering and Dean of the School of Engineering and Applied Science. His research interests are in the areas of stochastic analysis, statistical signal processing, and information theory, and their applications in wireless networks and related fields. Among his publications in these areas are the recent books Quickest Detection (Cambridge University Press, 2009) and Information Theoretic Security (Now Publishers, 2009). Dr. Poor is a member of the National Academy of Engineering, a Fellow of the American Academy of Arts and Sciences, and an International Fellow of the Royal Academy of Engineering (U.K.). He is also a Fellow of the Institute of Mathematical Statistics, the Optical Society of America, and other organizations. 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 TRANSACTIONS ON INFORMATION THEORY. He is the recipient of the 2005 IEEE Education Medal. Recent recognition of his work includes the 2007 Technical Achievement Award of the IEEE Signal Processing Society, the 2008 Aaron D. Wyner Distinguished Service Award of the IEEE Information Theory Society, and the 2009 Edwin Howard Armstrong Achievement Award of the IEEE Communications Society.
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