Joint Data Detection and Channel Tracking for OFDM ... - CiteSeerX

4230

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 9, SEPTEMBER 2008

Joint Data Detection and Channel Tracking for OFDM Systems With Phase Noise Stelios Stefanatos and Aggelos K. Katsaggelos, Fellow, IEEE

Abstract—This paper addresses the problem of data detection in orthogonal frequency division multiplexing (OFDM) systems operating under a time-varying multipath fading channel. Optimal detection in such a scenario is infeasible, which makes the introduction of approximations necessary. The typical joint data-channel estimators are decision directed, that is, assume perfect past data decisions. However, their performance is subject to error propagation phenomena. The variational Bayes method is employed here, which approximates the joint data and channel distribution as a separable one, greatly simplifying the problem. The data detection part of the resulting algorithm provides soft data estimates that are used for channel tracking. The channel itself is modeled as an autoregressive process allowing for a Kalman-like tracking algorithm. According to the developed algorithm, both data and channel estimates are exchanged and updated in an iterative manner. The performance of the proposed algorithm is evaluated by simulations. Furthermore, since OFDM is extremely sensitive to the presence of phase noise, the algorithm is extended to operate under severe phase noise conditions, with moderate performance degradation. Index Terms—Channel tracking, joint detection-estimation, orthogonal frequency division multiplexing (OFDM), phase noise, variational Bayes method.

I. INTRODUCTION RTHOGONAL frequency division multiplexing (OFDM) is now widely accepted as a modulation scheme that allows transmission of high data rates with reasonable complexity at the receiver. It has been accepted by many standards and is considered to be a strong candidate for future systems. As in every communication scheme, the channel is the single most crucial component that affects system performance, and therefore it has to be accurately known by the receiver. The most common method of channel estimation in standard OFDM systems involves the transmission of a pilot preamble at the beginning of the connection [13], that enables the use of one of the many proposed pilot-based channel estimation schemes (e.g., [25]). In environments where the channel varies significantly between consecutive OFDM symbols, the initial channel estimate

O

Manuscript received May 8, 2007; revised March 27, 2008. First published May 23, 2008; last published August 13, 2008 (projected). The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Manuel Davy. This work was supported in part by the project “EPEAEK II—PYTHAGORAS II—Support to Research Groups in Universities” and by the European Social Fund and Greek National Resources. The material in this paper was presented in part at the8th IEEE Workshop on Signal Processing Advances in Wireless Communications (SPAWC), Helsinki, Finland, June2007 . The authors are with the Department of Physics, National and Kapodistrian University of Athens, Athens, Greece. Digital Object Identifier 10.1109/TSP.2008.925968

becomes quickly outdated, thus, resulting in severe performance degradation [26]. A simple method to overcome this problem is to employ pilots during payload transmission dedicated to channel tracking. Many pilot-based channel estimation/tracking algorithms have been proposed (e.g., [9] and [14]). However, when the channel dynamics are considerable, the number of pilots required for reliable estimation needs to be large, thus reducing the bandwidth utilization. Furthermore, in current standards, the number of pilots during payload is too small to be of any practical use for channel estimation (these pilots are used for phase synchronization purposes). These observations suggest the use of more sophisticated channel tracking methods that do not rely on known symbols. Data-aided (i.e., nonpilot-based) channel tracking for OFDM systems has been addressed previously in, for example, [15] and [22]. In [22], a decision directed approach is employed, where channel estimation is based on the current data estimates that were obtained using the channel estimate corresponding to the previous OFDM symbol. In [15] this idea is extended by employing frequency and temporal filtering exploiting knowledge of the channel statistical properties, in order to obtain improved channel estimates. However, these approaches are suboptimal in terms of data detection, as they only address the channel estimation problem without considering data detection. The same argument also holds true for blind channel estimation methods (e.g., [26] and [34]). An optimal strategy would be to jointly estimate data and channel parameters [10], which is the topic of this paper. Unfortunately, the problem of joint estimation is known to be algorithmically infeasible, necessitating the introduction of approximations for complexity reduction. An algorithm for joint data and channel estimation in OFDM was proposed in [4], extending the results of [8], with the channel modeled as an autoregressive (AR) process. The algorithm provides optimal data and channel estimates, with the latter computed by a recursive (Kalman) filtering operation. However, to avoid infeasible data averaging computations, a decision directed approach is utilized, which is susceptible to error propagation phenomena, especially when multiamplitude, higher-order constellations are utilized, as is usually done in OFDM. In this paper, the variational Bayes method [31] is employed as an alternative method to obtain an approximate solution to the joint data and channel estimation problem. The variational Bayes method has been successfully applied in the last years in many scientific fields, such as physics [20], machine learning [3], and image processing [23], providing a computationally attractive alternative to Monte Carlo-based Bayesian inference methods [20] (see [33] for a recent paper that applies Monte

1053-587X/$25.00 © 2008 IEEE

STEFANATOS AND KATSAGGELOS: JOINT DATA DETECTION AND CHANNEL TRACKING

Carlo inference methods in a similar setting as the one considered in this paper). The resulting algorithm can be conceptually divided into a data detection part and a Kalman-based channel tracking part, which exchange information in an iterative manner. This operation results in improved performance, both in terms of data detection and channel estimation, compared to the straightforward decision directed approaches. The expectation-maximization (EM) algorithm of [1] will be shown to be a special case (resulting by appropriate modifications) of the proposed algorithm. Except for channel variations, OFDM is known to be very susceptible to phase impairments [29], namely frequency offset and phase noise. For the case of a constant frequency offset (a commonly used model) many estimators have been proposed that provide very good performance (e.g., [5] and [24]). This estimate can be used to deterministically compensate for the frequency offset effect. However, phase noise is a time varying process that needs to be estimated and compensated on a symbol-by-symbol basis. Compensation of the common phase error, either by using pilot subcarriers [27], [30], as is usually done in practice, or in a blind manner [33], is not optimal, and does not provide sufficient performance when the phase noise process is fast varying, as is the case with low cost RF oscillators. The proposed, joint data and channel estimator is therefore modified to allow for reliable system performance even in this case, i.e., when in addition to channel, phase noise also constitutes a major performance degradation parameter. It is noted that a phase noise compensation algorithm based on the variational Bayes method has been proposed in [16]. However, a perfect channel estimate is considered there. Therefore, the algorithm proposed in this paper can be viewed as a generalization of [16] for operating under a time varying channel. The paper is organized as follows. In Section II, the OFDM system model is described, along with the formulation of optimal data detection. In Section III, the problem of joint data and channel estimation is addressed, assuming negligible phase noise. Based on the variational Bayes method, an algorithm is derived, that is capable of providing soft data decisions. In Section IV, the algorithm is extended to handle the case of a non-negligible phase noise and the possibility of simplifying modifications is discussed in Section V. Performance evaluation of the proposed algorithms is done via simulations shown in Section VI. Finally, Section VII concludes the paper. Notation: Boldface lower (upper) case letters denote vectors (matrices). The operations of transposition and and , reHermitian transposition are denoted by is denoted by . The spectively. The th element of (unitary) DFT matrix is denoted by with elements , . denotes the diagonal matrix with diagonal elements , denotes the diagonal matrix with diagonal equal denotes the identity to the diagonal of , and matrix. The all-ones and all-zeros vectors are denoted by , , respectively. In some cases the matrix dimensions are explicitly shown as subscripts. Notation (or ) denotes the probability density function (pdf) of , or probability mass denotes function if is discrete. Specifically, the complex, circularly symmetric, multivariate Gaussian pdf of

4231

and covariance matrix , whereas mean denotes the real Gaussian distribution. is the statistical , the cardinality of a set will expectation with respect to , denotes the Euclidean norm of be written as and the frequency domain representation of is denoted by . The sets of real and complex numbers are denoted as , and , respectively, while and are used to extract real and imaginary components, respectively.

II. SYSTEM MODEL AND OPTIMAL DATA DETECTION denote the -dimensional vector of data (and pilot, Let , is if any) symbols of the th OFDM symbol ( , , the time index) with elements where denotes a finite, two dimensional constellation, e.g., quadrature amplitude modulation (QAM). It is assumed that the data symbols are independent, identically distributed (i.i.d.), with unit variance. The symbols are loaded onto subcarriers , by an -point IDFT.1 The sampled received signal after dropping the cyclic prefix, can be shown to be [19] (1) is the time-varying channel impulse response, where assumed quasi-static for the duration of an OFDM symbol, is constructed from the first columns of , , with , is the represents the effect of phase noise, where real-valued, sampled phase noise sequence that affects the th represents complex additive OFDM symbol, and per dimenwhite Gaussian noise (AWGN) of variance sion. The phase noise model adopted in this paper is , where is the cyclic prefix length and is a scalar, Gaussian, wide sense stationary (WSS) AR(1) process [17], i.e., (2) is a white Gaussian process of unit variance, and where , are constants, specifying the phase noise time dynamics. [2] model will be assumed for the A th order AR AR channel statistics, i.e., (3) where is is complex white the channel state vector, Gaussian noise of unit variance and are constant matrices. It can be easily deduced from (3) that (in steady state). This of zero AR model can represent a WSS Gaussian process mean and arbitrary correlation using a sufficiently large order and proper choice of and [11]. 1It is assumed, here, that there are no virtual subcarriers. The presented results can be trivially applied to this case as well.

4232

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 9, SEPTEMBER 2008

According to the maximum a posteriori probability (MAP) is2 rule, the optimal, causal estimate of the symbol-vector

(4) denotes the set of indexed vectors with the Fourier transform of the time domain received signal. Employing the data pdf conditioned on rather than in (4) is done for algorithmic to is invertible, convenience (since the mapping from both sets contain the same information [7]). Unfortunately, the presence of unknown parameters (hidden variables) and makes exact computation of a difficult problem (this will be demonstrated in Section III). In what follows, approximate solution methods of (4) will be shown, initially for the case of ideal phase synchronization (i.e., negligible phase noise), where the channel is the only parameter that needs to be estimated (tracked), and later generalize to the case when phase noise can not be neglected. Note that the noise variance and the statistics of the channel and phase noise are assumed to be perfectly known at the receiver. This assumption is not restrictive as these parameters are usually slowly varying to allow a reliable estimate by training and/or tracking. where

III. DATA DETECTION AND CHANNEL TRACKING WITH PERFECT PHASE SYNCHRONIZATION For the case when phase noise is absent, the received th . The freOFDM symbol is given by (1) with quency-domain received signal can be written as (5)

Although is not known during payload transmission, (6) and (7) are utilized by the recursive joint data and channel estimators presented in the following. The algorithm described in Section III-A was proposed in [4] by extending the arguments of [8] for the OFDM case. It is repeated here for reference and comparison purposes with the second, novel algorithm developed in Section III-B, based on the use of the variational Bayes method. A. Joint Data and Channel Estimation Based on Kalman Filtering In order to provide a feasible data and channel estimation algorithm, it is assumed that is independent of . This is actually the case, when data are uncoded or each codeword is contained in only one OFDM symbol. Since the number of pilots per OFDM symbol during payload is too small to provide any useful information for channel estimation [26], it will be consists only of assumed in the following (w.l.o.g.) that data symbols. in (4) can be written as The required posterior pdf

(8) have been dropped [their exwhere the terms independent of plicit value is irrelevant since a maximization procedure is performed in (4)]. For the case of uncoded transmission , whereas for the case of coded transmission defines the code constraints [12]. In the following, uncoded data transmission will be assumed for simplicity. The first term inside the summation in (8) can be shown to be equal to [2]

where the data-dependent matrix

, with , is utilized in order to inin (5). Since the unitary DFT does troduce the state vector not change the statistical properties of the AWGN, the same noin both (1) and (5). tation is used for the noise vector Note that (3) and (5) constitute a pair of state and observation (measurement) equations. For the artificial case when the data are known, the optimal causal channel estimator (tracker) is the Kalman filter, specified by the time-measurement update equations [2], which are compactly presented here as operators , , respectively, i.e.

(9) where the recursively computed, as per (6) and (7), channel and corresponding error-covariance matrix estimate depend on the past data sequence being considchannel estimates must be computed, one ered, i.e., in (8). Even though the second term infor each possible side the summation in (8) can also be recursively computed as [8] (10)

(6) (7) is the conditional mean (minimum where given the observations up to time mean squared estimate) of , and is the corresponding error-covariance matrix. The recursions are initialized and . Of course, the channel by a proper choice of estimates depend on the symbol sequence , but this is not shown explicitly in notation for simplicity. 2w.l.o.g.

it is assumed that there are no pilot symbols included in x .

evaluation of (8) is still infeasible due to the extremely large number of possible sequences that grows with time. In order to resolve the dependency on past data, a decision directed approach can be employed. Specifically, the detected data can be treated as correct, so that evaluation vectors of (8) is done only for the possible sequences [21]. In [4] and [8], the choice of is considered, i.e., the receiver operates in a purely decision directed mode. This choice is actually the most appropriate one for the OFDM case in terms of complexity, as the cardinality of the data space is extremely large, for a moderate number of subcarriers and/or high-order constellations. In statistical terms, selecting

STEFANATOS AND KATSAGGELOS: JOINT DATA DETECTION AND CHANNEL TRACKING

is equivalent to setting (8), where

in , resulting in

4233

, from has been used. From (5), it follows that the first term in (14) equals:

(11) as given in (9), with and recursively com. puted, based on the detected sequence This approximation of can be used in (4) to obtain data estimates by direct maximization. However, the depenon [via in dency of the covariance matrix of (9)] results in a difficult maximization procedure when multiamplitude constellations are employed (e.g., 64-QAM) since all possible symbol-vectors have to be examined. Therefore, in order to further reduce complexity, this dependence is neglected, which is a reasonable approximation in steady state operation where the channel estimation error is small [21]. This approximation leads, finally, to a simple detector structure. Specifindependent per subcarrier detectors are employed, ically, based on the detection rule (12) , where is the for th element of the decision-directed channel estimate . Therefore, detection is accomplished in the same way as in the case of ideal channel knowledge, by employing the channel estimate (one-step prediction), recursively computed by (6) and (7) using the previously detected symbols. (Since the algorithm requires only the one-step prediction of the channel, the alternative form of the Kalman filter equations that propagate the one-step prediction could be used.) It is noted that the channel estimation scheme proposed in [15] leads to the same algorithm, when the channel is modeled as in (3). B. Joint Data and Channel Estimation Based on the Variational Bayes Method A novel algorithm for joint data and channel estimation will be derived in this section, based on the variational Bayes on-line inference method [31]. The main advantage of this algorithm is that it provides soft data decisions, resulting in better channel estimates. Furthermore, it can be generalized to handle the case of a non-negligible phase noise in a straightforward manner, as will be shown in Section IV. The algorithm’s derivation starts by noting that , required in (4), can also be written as (13) The integrand in (13) is proportional to

(15) However, exact calculation of the second term in (14), , is intractable, since it requires averaging over all possible previous vector-se[similarly to (8)]. One possible solution is to inquences voke the approximation using the previously detected symbols , which leads to the algorithm in Section III-A. However, as in every decision directed method, this approximation is susceptible to error propagation. Therefore, an alternative approximation method is used for the computation of the integrand in (13), namely the variational Bayes method [31]. The variational Bayes method, which can be viewed as a generalization of the EM algorithm, is most often utilized as a method that provides an approximation (lower bound) of to the normalization constant of the joint pdf vector variables [3], [20]. For that purpose by the separable pdf the method approximates . This approximation by itself can be exploited by many signal processing applications [31] and is the key for the derivation of the proposed algorithm. The , optimal set of (approximate) marginal distributions in the sense of minimizing the Kullback-Leibler (KL) distance and , is specified by the [7] between ) [3], [31] following system of equations (for

(16) up to a constant additive term,3 where . The solution of the above system of equations provides the opthat minimize the KL distance (in timal distributions case of more than one solutions, each of them must be checked in order to find the minimizer). Usually, a closed form solution is not available, so that the solution has to be found iteratively (with the risk of converging to a local minimum). Even with this approach, the solution of the system may still remain intractable, due to the cumbersome form of marginal (prior) and conditional distributions required in (16). However, when these distributions belong to the class of exponential distributions and are conjugate to each other, the solution of the system can be found in a straightforward manner [3]. and In the present context, by setting , the variational Bayes method can be used to approximate the integrand in (13) as (17)

(14) has been where the term dropped since it is independent of (uncoded data transmis, conditioned on sion is considered) and the independence of

The two distributions on the right-hand side (RHS) of (17) can be considered to be approximations of the true marginal dis, . Therefore, the integration in tributions 3For the rest of the paper, constant additive terms of no interest will be omitted from equations.

4234

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 9, SEPTEMBER 2008

(13) is avoided since the method directly provides for an approxneeded for use in (4). It therefore remains imation of and are actually calculated. to show how Note that, although seems to be of no interest for the data detection problem, it has to be calculated in parallel as both approximating marginal distributions with are coupled. is considered first for a fixed The evaluation of (to be specified later) with mean and covariance maand , respectively.4 Note that since is distrix should represent a discrete-valued pdf crete-valued, probabilities. From (14)–(16), it follows as specified by

It remains to specify be seen to be equal to (given

which, from (14)–(16) can )

(21) where with

,

, ,

.

can be regarded as a soft data estimate with Note that being a reliability indicator of the estimate ( , , required in (21), in general).5 As argued earlier, cannot be practically computed, and some approximation has to be invoked, which should also allow for a simple and easily . To this end, the following proposition computable results shows that imposing a Gaussian form for in a tractable evaluation of , and allows for a recur. sive, approximate computation of Proposition 1: Under the approximation (22)

(18) with , , and . It can be seen from (18) that actually separable, i.e., where

(23)

is

(24)

(19)

After computation of imated as

,

can be approx(25)

as in (20), shown at the bottom of the page, with where is the th diagonal element of . It is interesting (and fortunate) that the approximating marginal disturned out to be separable, given that no such tribution of (one expects that the elconstraint was placed on ements of are coupled under the true marginal distribu). Therefore, probabilities for each subcartion probabilirier symbol need to be computed, for a total of . Note also, that these probabilities depend ties, instead of only on the first and second moments of . 4For notational convenience the same notation has been used here as in (6) and (7). It is noted that, here, this notation represents approximate corresponding quantities.

with predictive mean and covariance matrix given by the Kalman time-update (6) and (7), using (23) and (24). Proof: See Appendix I. Equations (19), (20), and (22)–(25) specify the optimal, approximating, and marginal distributions. Note that the form of these distributions is invariant over the time index . However, the distributions are coupled and a closed form solution cannot be found. Therefore, an iterative scheme has to be used, that involves sequentially updating the pdfs as follows (see Fig. 1). 5x ^ should not be confused with the corresponding notation in Section III-A denoting hard decision detected symbols.

(20)

STEFANATOS AND KATSAGGELOS: JOINT DATA DETECTION AND CHANNEL TRACKING

able in cases where powerful channel encoders such as LDPC or turbo codes [20] are employed, because it simplifies the joint parameter-estimation/data-decoding problem. In general, the decoding (sum-product) algorithm must operate on the composite graph representing the joint pdf of received signal, hidden parameters (channel), and coding constraints [32]. However, utilization of an algorithm performing parameter estimation and soft symbol detection in an iterative manner as the one proposed in this paper allows for incorporation of the coding constraints in an efficient manner along the lines of the algorithm described, for example, in [18]. In terms of channel estimation, algorithm A employs a prefor data detection, whereas aldicted channel estimate

Fig. 1. Block diagram of the proposed algorithm in Section III-B.

• Initialization: , Set • For iteration : update: — 1) Update

.

according to (19) and (20) using

and . In order to avoid numerical instabilities the value of is , where is a small constant. replaced by 2) Compute , — update: Compute

and

4235

. from (23) and (24) using

, . • At the end of the final iteration: , (i.e., set 1) Set ). 2) Compute , from (6) and (7). 3) Obtain the (approximate) MAP data esth OFDM symbol as timates of the , . for This scheme is repeated for every OFDM symbol, thus specifying a joint, causal data detection and channel tracking algorithm. A natural criterion for the algorithm’s convergence is the normalized difference of the data pdf among consecutive iterations, i.e.

that is (apgorithm B employs a filtered channel estimate proximately) generated and updated by the iterative procedure. We argue that employing a filtered, instead of predicted, channel estimate is beneficial. It is noted, however, that the channel estimates provided by the two algorithms can not be directly associated. The channel prediction provided by algorithm A is optimal in the case when past decisions are correct, but in practice this estimate is subject to error propagation phenomena. On the other hand, the channel estimate provided by algorithm B is only an approximation of the optimal filtered estimate, even when perfect decisions are employed, since its derivation is based on the approximation (17) and the iterative algorithm employed may converge to a local minimum. Another difference between the two algorithms is the incorporation of reliability information. Specifically, in algorithm B, data detection [(19)–(20)] utilizes a correction term depending , while channel on the channel error-covariance matrix estimation [(23)–(24)] utilizes which provides reliability information about the data estimates. This type of information exchange is typical in applications of the variational Bayes method. On the other hand, algorithm A does not provide any sort of reliability information that could be employed for better performance. Further comparisons between the two algorithms are made in Section VI based on simulation results.

(26)

IV. DATA DETECTION AND CHANNEL TRACKING WITH SEVERE PHASE NOISE VARIATION

is the vector representation of the data pdf evaluated at iteration , and is a small constant (ideally equal to the numerical precision). A fixed number of iterations per OFDM symbol could have been used, although clearly in this case the solution’s convergence is not guaranteed.

When the level of phase noise is considerable, e.g., when high frequency, low-cost RF oscillators are employed, the estimation algorithms in Section III present significant performance degradation, since they ignore the presence of phase noise. It is therefore necessary to estimate/compensate the phase noise effect in order to achieve reliable performance. The standard method toward this goal is to employ a few pilot subcarriers per OFDM symbol, that are used to estimate the mean phase .6 The MPR rotation (MPR), defined as , is then used to derotate the received symbol, estimate, [27]. For small varying phase noise, this i.e., compute method almost completely cancels the phase noise effect, as the residual inter-carrier interference (ICI) is negligible. However,

where

C. Comparison of the Algorithms Both algorithms described in Sections III-A and III-B (which henceforth are referred to as algorithms A and B, respectively) provide approximate solutions for the causal data detection problem. As a byproduct a channel estimate is also available. However, several differences can be identified between them, which are described next. As far as data detection are concerned, algorithm B provides soft symbol decisions instead of hard ones, as in the case with algorithm A, along with a reliability measure. This feature is valu-

6The MPR is also defined as    (n)=N . However, even for  fast varying phase noise, both definitions are equivalent in the sense u exp j  .

f

g



4236

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 9, SEPTEMBER 2008

under large phase noise dynamics, this method results in a performance error floor due to the uncompensated ICI [27], thus, implying the need for a more sophisticated phase noise compensation method [16]. For this purpose, algorithm B can be extended to handle the presence of phase noise. Since the algorithm is based on the Bayesian framework, it is necessary to specify a prior . It was stated in Section II that phase distribution for noise is modeled as a scalar AR(1), WSS Gaussian process. , , are correlated, which could be This implies that taken into account for a more accurate estimation, i.e., perform phase tracking along the incoming OFDM symbols, similarly to channel estimation. However, when expressing the scalar AR(1) phase process in vector form, similarly to (3), which is convenient for the derivation of the phase tracking algorithm, has very small it turns out that the corresponding matrix coefficients when phase noise is moderately-to-fast varying, provides only little inforindicating that knowledge of . Therefore, and in order to keep the complexity mation for low, is treated as an independent per OFDM symbol vector-process, specified by an invariant over the symbol index , Gaussian pdf , of zero mean and covariance matrix , known to the receiver. Having specified the phase statistical model, the algorithm can now be derived. The variational Bayes method is invoked, similarly to Section III-B, as a method to calculate (27) where now, in contrast with (13), the presence of phase noise has been taken into account. The approximation to be utilized is (28) is written Note that the approximate marginal distribution of instead of the freas conditioned on the time-domain signal in order to emphasize the fact that quency-domain signal estimation of , a time-domain quantity itself, is more conve. niently performed by processing , Adopting (16) for the present context, with , , the approximate marginal distributions in (28) can be found in a similar manner to Section II, with the results summarized in the following proposition. Proposition 2: and given 1) Under the (linear) approximation , , the phase distribution is obtained by (29)

with (30) (31) and

where , with and 2) For given for suming distribution is obtained by

. as in (29), and as, the data

(32)

with as in (33), shown at the bottom , with of the page, where , is the Fourier transform of the derotated time-domain received OFDM symbol, and , , with , the mean and covariance matrix of , respectively. 3) Under the approximation , given and as in (29), the channel distribution is obtained by

(34) with

(35) (36) where ements

is a diagonal matrix with diagonal el, , , with

, Proof: See Appendix II.

.

(33)

STEFANATOS AND KATSAGGELOS: JOINT DATA DETECTION AND CHANNEL TRACKING

In Appendix III, it is shown that and , required to completely specify the approximate marginal phase pdf are as follows: (37)

(38) where denotes Hadamard (element-wise) matrix product. , which is Note that the linear approximation reasonable even for relatively fast varying phase noise, is only employed in the first part of Proposition 2 in order to obtain a ; Gaussian phase distribution. The approximation is employed in the second part of Proposition 2 in order to obtain a separable data distribution. This can be inevitably leads to some information loss, as viewed as the estimation error variance of the th element of the phase vector. Hopefully, under normal operational conditions of the algorithm, the phase estimate is good enough so that the loss is small. The results of proposition 2 suggest the following iterative algorithm for the joint data-channel-phase estimation for the th OFDM symbol: • Initialization: ,

1) Set

.

2) Obtain an initial phase estimate (to be discussed in the following) : • For iteration — update: according to (32) and (33) using 1) Update ,

and

. In order to avoid numerical instabilities the is replaced by , where is a small value of constant. 2) Compute , . update: — 1) Compute

,

according to (37) and (38)

using

,

,

,

.

2) Compute

and

Compute , . • Final iteration: 1) Set

Termination of the iterations can be accomplished using the criterion of (26). based on For the initial phase estimate one could set the phase prior. However, employing this estimate for data estiiteration results in severe degradation, mation in the first since there is no phase compensation achieved. Since a good initialization is essential in every iterative algorithm with nonunique fixed point, a better initial phase estimate has to be employed. Improved phase estimate initialization can be achieved using the MPR estimate provided by setting denotes the by a pilot aided estimator (e.g., [27]), where argument of a complex number. Employing this estimate compensates for the MPR, thus providing improved data estimates which are, however, still quite unreliable due to the presence of . In other ICI in the phase compensated symbol words, the approximations concerning the phase error-covariance matrix employed in the second part of proposition 2 do not hold. Unfortunately, due to these approximations, no reliability measure for the phase estimate is passed to the data estimator, leading to overconfident data estimates. In order to avoid this phenomenon a small modification can be applied to the data estimation, only for the first iteration, by replacing the value of in (33) by , where is the variance of the ICI noise “seen” at the th subcarrier [27]. The resulting data estimates are less confident as they are obtained in a more noisy environment. Assuming can that the MPR estimate is perfect, the value of be easily calculated when the channel is known by averaging over the symbols and the phase noise statistics [27, eq. 8]. In this case, the only channel information is the one provided by the algorithm which for simplicity is assumed to be perfect in calculation. Another approach would be to obtain the by averaging over a channel independent value for iterations the channel statistics. For the subsequent the algorithm is executed with no modifications. Incorporating as a phase estimate reliability measure in the following iterations might lead to better performance but unfortunately its calculation is not simple. However, the phase estimate provided in the following iterations is expected to be much closer to the true phase noise so that the approximations of proposition 2 are accurate and there is no need to calculate the residual ICI effect.

according to (30) and (31) using

and update:

—

4237

V. MODIFICATIONS OF THE PROPOSED ALGORITHM

.

and

from (35) and (36) using ,

,

and

(i.e., set

). 2) Compute , from (6) and (7). 3) Obtain the (approximate) MAP data estimates th OFDM symbol as of the for .

The proposed algorithm can be employed as the baseline for deriving algorithms aiming at lower complexity. Some possible modifications that could be applied are as follows. Obtain the final data decisions by quantizing the values rather than maximizing . Constrain the approximate marginal data pdf to have a multivariate complex Gaussian form, i.e., and perform data detection by quantization of . Constrain the data pdf as , a degenerate distribution, where are hard data decisions.

4238

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 9, SEPTEMBER 2008

Constrain the phase pdf as , (MPR a degenerate distribution, where compensation only). In [1], an EM-based algorithm for joint data detection and channel tracking was proposed (without considering the phase noise problem). This algorithm can be obtained by setting in (20) and (33) and employing modification M1) for final data decisions. Note that setting in the context of the proposed algorithm is not justifiable is nonetheless computed by the Kalman tracker, since and incorporating it into the data pdf results in insignificant complexity increase. However, under the EM-context, the possibility of employing information of the channel estimate for data detection is lost. In [16], where the problem of data detection under severe phase noise in a static (known) channel was considered, modification M2) was employed for the data pdf as a means to simplify the data pdf update step, i.e., an update of the mean and covariance matrix is required. However, this constraint on the pdf form is suboptimal and the computational complexity advantage is not clear, especially for large number of subcarriers, since large matrix inversions are required in [16]. The algorithm resulting by employing modification M3) could be viewed as an extension of the algorithm A derived in Section III-A for the case of significant phase noise. Clearly, many other modifications are possible, that can provide a fine trade off between performance and complexity. Various algorithms based on modifications M1)–M4) will be compared in Section V in terms of performance. VI. SIMULATION RESULTS The performance of the proposed algorithms is investigated subcarriers by simulations. An OFDM system of is considered, with 64-QAM modulation. The discrete-time channel model employed is the typical multipath Rayleigh fading channel, of length smaller than the cyclic prefix (here set equal to 16 samples) whose taps are independent from each other, having a power delay profile of channel model A used in [26, Table I], truncated to 16 samples. The channel is assumed to , and with be time-varying, of average energy each tap independently fading according to the Jakes’ model, with normalized Doppler frequency , where is the Doppler frequency (in Hz) and is the system’s sampling period (in sec). This Doppler frequency corresponds to a relative average speed of 30 m/s for an OFDM system with 20–MHz bandwidth and a carrier frequency of 5 GHz. It is noted that for the considered operational signal-to-noise (SNR) range, the assumed Doppler frequency is the largest possible such as the ICI introduced by the channel variations during an OFDM symbol interval can be safely assumed to be negligible [6], i.e., the channel obeys accurately enough the AR model of (3). This is essentially a worst-case study, and one can be assured that for milder channel variations performance would be at least the same or better than the one shown here. Furthermore, in case of unknown channel statistics it is typical to match the algorithm parameters to the worst case scenario that the system is expected to encounter. This approach, although

Fig. 2. Symbol/bit error rate performance of the algorithms A and B for different AR channel models.

not providing the same performance as with point optimal adjustment of the algorithm, makes the system robust to unknown channel conditions as long as they are “milder” [15]. Frame-based data transmission is assumed, where each frame consists of 100 consecutive (payload) OFDM symbols. As done in practice, a preamble is introduced in every frame for channel estimation purposes. In order to investigate the steady state performance of the tracking algorithms, perfect initial channel estimation is assumed (a reliable channel estimation scheme in the presence of phase noise was recently proposed in [17]). It is assumed that the receiver has knowledge of the exact channel and phase statistics. The results shown in the following were obtained by averaging over 1000 independently generated frames and corresponding channel realizations. 1) Negligible Phase Noise: Fig. 2 shows the performance of algorithms A and B in the case of negligible phase noise, in terms of the uncoded symbol error rate (SER) and bit error rate (BER) over , where is the average received energy per bit. Information bits are Gray mapped to the 64-QAM constellation. No pilot subcarriers were employed (i.e., all subcarriers are loaded with information symbols). The SER under ideal channel knowledge is also shown as a performance lower bound. The channel is modeled by either an AR(1) or an AR(2) and in (3). The AR(2) process, with proper selection of model represents more accurately the channel dynamics but results in a more computationally complex algorithm, since the is larger. The stopping rule of (26) was utidimension of lized for algorithm B with and the maximum number of iterations was set to 20 (results on the number of iterations are discussed later). In order to avoid numerical instabilities of algorithm B at high SNR, a fixed value of was employed in all cases, corresponding to an artificial noise level of about 33 dB. In terms of SER, it can be seen that algorithm B outpervalues when an AR(1) channel forms algorithm A for all model is adopted. Specifically, under an AR(1) channel model, algorithm A exhibits an error floor which can be attributed to error propagation, whereas algorithm B closely follows the ideal

STEFANATOS AND KATSAGGELOS: JOINT DATA DETECTION AND CHANNEL TRACKING

curve. For a SER, algorithm B is about 3.5 dB better than algorithm A and about 1.3 dB away from the lower bound. As expected, employing the more accurate AR(2) channel model results in improved performance. The error floor of algorithm A is now completely eliminated and performance is very close to the lower bound. However, algorithm B still performs better values up to 30 dB. For a than algorithm A for SER with an AR(2) channel model, algorithm B is about 0.5 dB better than algorithm A and 0.45 dB away from the lower bound. For larger SNR (smaller SER) both algorithms achieve essentially the same performance. The performance difference between AR(1) and AR(2) modeling with algorithm B is about 1–1.5 dB for the high SNR range. The performance gain of the AR(2) comes, however, with the cost of complexity as the diis doubled. On the other hand, an AR(2) channel mension of modeling seems mandatory for algorithm A, as performance with an AR(1) model is far from optimal. This can be seen as a disadvantage, as an AR(2) model requires more precise information about channel statistics compared with an AR(1) model, which might be difficult to obtain in practice. An interesting phenomenon is observed about the BER performance. For the AR(1) model, the conclusions made above about the performance of the algorithms remain essentially the same, with the exception that at very low SNR values algorithm A performs better. The situation changes when an AR(2) model is employed. In this case, algorithm A performs better than algorithm B by about 1 dB for high SNR values. This behavior can be interpreted by the fact that both algorithms are derived for optimal symbol decisions. When Gray demapping is applied on these decisions, the BER performance is not necessarily expected to follow the corresponding SER performance. Fig. 3 depicts the performance of both algorithms in terms of channel tracking (for clarity purposes, curve fitting was employed in this figure only). The channel estimation mean , is utisquared error (MSE), defined as lized here as the performance measure. In order to have a fair comparison, a filtered channel estimate was generated for algorithm A, after the data decisions of the current OFDM symbol had been obtained. The MSE in the artificial case when the data are known is also shown as the optimal channel tracking performance (for a given AR channel model). As expected, since an AR(2) process better models the channel, it results in a lower minimum MSE. When an AR(1) model is employed for algorithm A the channel estimate MSE is far from optimal, whereas, algorithm B exhibits much better performance, approaching the optimal solution for higher SNR values. For an AR(2) channel model, there is a dramatic improvement for algorithm A, reaching the optimal performance at high SNR values, since the data decisions are essentially perfect as far as the channel estimation scheme is concerned. However, for smaller SNR values, the channel tracking performance of algorithm A still remains worse compared to algorithm B with an AR(1) channel model, due to error propagation. Algorithm B also shows an improved performance with an AR(2) channel model, although the gain is much less pronounced. An interesting observation is that the MSE for algorithm B does not reach the optimal MSE with increasing SNR. This is in line with the comments

4239

Fig. 3. Channel estimate MSE.

Fig. 4. SER performance of algorithm B in the presence of phase noise.

made in Section III-C about the channel estimate provided by algorithm B not being the truly optimal estimate, even with (close to) perfect past data decisions. 2) Severe Phase Noise: The performance of the modified version of algorithm B, derived in Section IV, in the case of a nonnegligible phase noise is considered, with the phase noise modand . eled as in (2), with parameters These values were selected such as the simple MPR compensation results in a large error floor in terms of SER performance, and can be considered as representing a worst case phase noise dynamics. Since a pilot-aided MPR estimate is required in order , 4 equispaced pilot subcarriers with ento set ergy equal to the data subcarriers energy are inserted into every OFDM symbol. The SNR loss due to pilot insertion is not taken into account in the figures. The values of and were the same as in the negligible phase noise case. Fig. 4 depicts performance under both an AR(1) and AR(2) channel model. The curves labeled “MPR comp.” correspond to the case where the only phase estimate used by the algorithm is the MPR estimate provided by the pilot-aided estimator, i.e., the phase estimate is not updated

4240

at each iteration, rather the algorithm sets for all . This straightforward modification of algorithm B, presents an error floor due to the presence of the uncompensated ICI for both channel models, although an AR(1) model shows better perfor, performance mance at the low SNR region. For a SER of is about 7.5 dB away from the lower bound, thus justifying the characterization of phase noise as “severe.” The performance is drastically improved by updating the phase estimate at each iteration, thus compensating for the ICI (“ICI comp.” curves). As can be seen, with an AR(1) model, the performance degradation compared to the lower bound is about , and the error floor exhibited by the 1 dB for a SER of “MPR comp.” approach is significantly reduced (although not eliminated). For an AR(2) model, performance improvement is achieved only for high SNR values, although not considerable. Furthermore, the AR(1) channel modeling outperforms the AR(2) modeling in the range of practical SNR value. Therefore, one can conclude that AR(2) modeling is more sensitive to the presence of phase noise compared to AR(1) modeling, which is another argument that justifies selecting an AR(1) model for the algorithm’s operation. A possible reason for the error floor depicted by the AR(1) model and the SNR gap depicted by the AR(2) model is the phase ambiguity between the channel and phase noise, as can also be seen from (1). The algorithm may associate a rotation due to phase noise with the channel or the other way around. However, if this is done randomly along the various OFDM symbols the channel’s assumed AR model on which the Kalman filter is based is no longer valid, making channel tracking less accurate. 3) Number of Iterations: Fig. 5 shows the average number of iterations performed by algorithm B for an AR(1) and AR(2) channel model, for both negligible and severe phase noise conditions. In the practical SNR range, say 25–35 dB, the average number of iterations does not exceed the value of 6. The number of iterations increases for lower SNR values and when phase noise is present. Note that there were simulation runs, where the maximum number of iterations (20) was reached, especially in the low SNR region. It can be concluded that both AR(1) and AR(2) models result in a similar average number of iterations, with the exception of the case AR(2) -negligible phase noise, which is not, however, considerably smaller. In a practical communication system the number of iterations is usually fixed. Simulations with the number of iterations fixed to 6 (not shown here) showed that the performance degradation is tolerable. 4) Performance of Modified Versions of the Algorithm: Fig. 6 depicts the SER performance of various modifications of the proposed algorithm for a fixed SNR of 40 dB and two channel cases: time varying channel of the same statistics as in the previous simulations and static (known) channel, where in the latter case no channel tracking is required. The performance is dethat determines picted for various values of the parameter fixed the level of variation of the phase noise process, with to a value of 0.99 as in the previous simulations (similar results are obtained for other values of ). The legend of the figure corresponds to the following modifications of the original algorithm.

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 9, SEPTEMBER 2008

Fig. 5. Average number of iterations for AR(1)-AR(2) channel model with/ without phase noise.

Fig. 6. Performance of the proposed algorithm and its modified versions for various channel and phase noise statistics.

: The algorithm derived in Section IV without any modifications. - : Employs modification M1) (see Section V) and ignores the information of the channel estimate (needed in the varying channel case) for the data pdf update. The algorithm can be viewed as an extension of the algorithm in [1] for the case of severe phase noise. - : Employs modification M2). The algorithm can be viewed as an extension of the algorithm in [16] for the case of a time-varying channel. The equations for the data update step (mean and covariance matrix) are not shown here for brevity. These can be obtained by a generalization of the methodology in [16]. - : Employs modification M3). The algorithm can be viewed as an extension of algorithm A derived in Section III-A for the case of severe phase noise. : Employs modifications M3) and M4). This algorithm is the simplest in terms of complexity

STEFANATOS AND KATSAGGELOS: JOINT DATA DETECTION AND CHANNEL TRACKING

and more related to the approach used by current OFDM standards, i.e., conventional decision directed channel tracking and MPR compensation. and are assumed Note that in all cases both values of known at the receiver as well as the channel statistics. The channel is modeled by an AR(1) process for tracking purposes and a maximum of six iterations was performed in all instances of the algorithm expect in the conventional case where one iteration was employed (in order to obtain a filtered channel estimate). The performance under the scenario of known channel and no phase noise is also shown as a performance bound. It can be seen that for the static channel case, the conventional approach has sufficient performance for a limited range values. On the other hand the proposed algorithm, along of with its modified versions, manages to increase the system’s values. The proposed-M1 algorithm robustness for larger performs very close to the original version of the proposed algorithm whereas the proposed-M2 and proposed-M3 algovalues. For rithms show an increased sensitivity for larger the varying channel case the conventional algorithm shows a significant degradation compared to the performance bound even for values of which did not affect performance in the static channel case. Clearly, this effect is attributed to inaccurate channel tracking and error propagation phenomena. Employing the proposed algorithm keeps performance close to the one experienced in the static channel case, thus making the system robust to channel variations. The proposed-M1 algorithm has practically the same performance, whereas the proposed-M2 and proposed-M3 algorithms have considerable performance loss. Interestingly, approximation M3) seems to be more robust to channel variations compared to approximation M2), even though the latter provides soft data estimates (but suboptimal nonetheless). These results indicate the ability of the proposed algorithm to perform well for a wide range of phase noise and channel variations, with the various modifications available providing a tradeoff between performance and complexity.

4241

APPENDIX I PROOF OF PROPOSITION 1 in Substituting the Gaussian approximation of with a quadratic expo(21), results in a exponential , i.e., a complex Gaussian pdf, whose nent with respect to mean is the solution of (39) given by (23), and whose covariance matrix is the solution of (40) given by (24). Assuming that is a good approximation of , (25) follows from (41) APPENDIX II PROOF OF PROPOSITION 2 For notational simplicity, the conditioning variable will be . Using the rule of (16), the optimal approxomitted from is given by imate marginal distribution

(42) where proximation

is the prior of . Using the apin (1) it can be shown that

(43)

VII. CONCLUSION A new algorithm for joint data detection and channel tracking was proposed based on the variational Bayes method. The algorithm’s advantage is that it provides (approximately optimal) soft data estimates, that result in improved channel estimation which in turn leads to enhanced data detection performance. The case of a nonnegligible phase noise, which is very important in OFDM systems, was also addressed by appropriately modifying the algorithm. Simulations showed that, in terms of performance in a nonphase-noise-corrupted environment, the proposed algorithm outperforms the straightforward, hard decision-directed data and channel estimation algorithm, both in terms of symbol error rate and channel estimate mean squared error. Furthermore, performance degradation under the presence of severe phase noise was shown to be only moderate, as the algorithm successfully managed to compensate for the most part of the ICI effect.

Substituting into (42) and after taking expectations, it follows that is Gaussian with mean and covariance matrix given in (30) and (31), respectively. is The optimal approximate marginal distribution given by (44) Using (1) and after some straightforward algebra

(45)

4242

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 9, SEPTEMBER 2008

where . Using the results for the characteristic function of a Gaussian pdf [28], it follows that , . Since, in gen, , the elements of are coueral, probabilities must be pled under (45), which means that . Howcalculated for each of the possible configurations of ever, assuming that ; , which is reasonable for normal operational conditions of the algorithm, becomes the last two terms in (45) cancel each other and separable, given by (32) and (33). is The optimal approximate marginal distribution given as

(46) where the assumption has been used, and is a diagonal matrix with diagonal ele. It is clear from (46) ments is complex Gaussian with its mean and covariance that matrix as given by (35) and (36), respectively. APPENDIX III PROOF OF (37) AND (38) Equation (37) follows readily from the definition of plying the linearity of the expectation argument, written as

. Apcan be

(47) The inner expectation of (47) equals . Using the identity for any vector and matrix , that can be easily verified by direct substitution, (38) follows. REFERENCES [1] T. Y. Al-Naffouri, A. Bahai, and A. Paulraj, “An EM-based OFDM receiver for time-variant channels,” in Proc. IEEE Globecom ’02, pp. 589–593. [2] B. D. O. Anderson and J. B. Moore, Optimal Filtering. Englewood Cliffs, NJ: Prentice-Hall, 1979. [3] M. Beal, “Variational algorithms for approximate Bayesian inference,” Ph.D., The Gatsby Computational Neuroscience Unit, Univ. College London, London, U.K., 2003. [4] S. B. Bulumulla, S. A. Kassam, and S. S. Venkatesh, “A systematic approach to detecting OFDM signals in a fading channel,” IEEE Trans. Commun., vol. 48, pp. 725–728, May 2000.

[5] Y.-S. Choi, P. J. Voltz, and F. A. Cassara, “ML estimation of carrier frequency offset for multicarrier signals in Rayleigh fading channels,” IEEE Trans. Veh. Technol., vol. 50, pp. 644–655, Mar. 2001. [6] Y.-S. Choi, P. J. Voltz, and F. A. Cassara, “On channel estimation and detection for multicarrier signals in fast and selective Rayleigh fading channels,” IEEE Trans. Commun., vol. 49, pp. 1375–1387, Aug. 2001. [7] T. M. Cover and J. A. Thomas, Elements of Information Theory. New York: Wiley-Interscience, 1991. [8] R. Haeb and H. Meyr, “A systematic approach to carrier recovery and detection of digitally phase modulated signals on fading channels,” IEEE Trans. Commun., vol. 37, pp. 748–754, Jul. 1989. [9] P. Hoeher, S. Kaiser, and P. Robertson, “Two-dimensional pilot-symbol aided channel estimation by wiener filtering,” in Proc. 1997 IEEE Int. Conf. Acoust., Speech, Signal Process., Munich, Germany, Apr. 1997, pp. 1845–1848. [10] C. K. Kaleh and R. Vallet, “Joint parameter estimation and symbol detection for linear or nonlinear unknown channels,” IEEE Trans. Commun., vol. 42, pp. 2406–2413, Jul. 1994. [11] C. Komninakis, C. Fragouli, A. H. Sayed, and R. D. Wessel, “Multiinput multioutput fading channel tracking and equalization using Kalman estimation,” IEEE Trans. Signal Process., vol. 50, pp. 1065–1076, May 2002. [12] F. R. Kschischang, B. J. Frey, and H. Loeliger, “Factor graphs and the sum-product algorithm,” IEEE Trans. Inf. Theory, vol. 47, pp. 498–519, Feb. 2001. [13] E. G. Larsson, G. Liu, J. Li, and G. B. Giannakis, “Joint symbol timing and channel estimation for OFDM based WLANs,” IEEE Commun. Lett., vol. 5, pp. 325–327, Aug. 2001. [14] Y. (G.) Li, “Pilot-symbol—aided channel estimation for OFDM in wireless systems,” IEEE Trans. Veh. Technol., vol. 49, pp. 1207–1215, Jul. 2000. [15] Y. (G.) Li , L. J. Cimini, and N. R. Sollenberger, “Robust chanel estimation for OFDM systems with rapid dispresive fading channels,” IEEE Trans. Commun., vol. 46, pp. 902–914, Jul. 1998. [16] D. D. Lin and T. J. Lim, “The variational inference approach to joint data detection and phase noise estimation in OFDM,” IEEE Trans. Signal Process., vol. 55, pp. 1862–1874, May 2007. [17] D. D. Lin, R. A. Pacheco, T. J. Lim, and D. Hatzinakos, “Joint estimation of channel response, frequency offset and phase noise in OFDM,” IEEE Trans. Signal Process., vol. 54, pp. 3542–3554, Sep. 2006. [18] B. Lu, X. Wang, and K. R. Narayanan, “LDPC-based space-time coded OFDM systems over correlated fading channels: Performance analysis and receiver design,” IEEE Trans. Commun., vol. 50, pp. 74–88, Jan. 2002. [19] X. Ma, C. Tepedelenlioglu, G. B. Giannakis, and S. Barbarossa, “Non-data-aided carrier offset estimators for OFDM with null subcarriers: Identifiability, algorithms and performance,” IEEE J. Sel. Areas Commun., vol. 19, pp. 2504–2515, Dec. 2001. [20] D. J. C. MacKay, Information Theory, Inference, and Learning Algorithms. Cambridge, U.K.: Cambridge Univ. Press, 2002. [21] H. Meyr, M. Moeneclaey, and S. A. Fetchel, Digital Communication Receivers: Synchronization, Channel Estimation, and Signal Processing. New York: Wiley-Interscience, 1997. [22] V. Mignone and A. Morello, “CD3-OFDM: A novel demodulation scheme for fixed and mobile receivers,” IEEE Trans. Commun., vol. 44, pp. 1144–1151, Sep. 1996. [23] R. Molina, J. Mateos, and A. K. Katsaggelos, “Blind deconvolution using a variational approach to parameter, image, and blur estimation,” IEEE Trans. Image Process., vol. 15, pp. 3715–3727, Dec. 2006. [24] P. H. Moose, “A technique for orthogonal frequency division multiplexing frequency offset correction,” IEEE Trans. Commun., vol. 42, pp. 2908–2914, Oct. 1994. [25] M. Morelli and U. Mengali, “Comparison of pilot-aided channel estimation methods for OFDM systems,” IEEE Trans. Signal Process., vol. 49, pp. 3065–3073, Dec. 2001. [26] B. Muquet, M. d. Courville, and P. Duhamel, “Subspace-based blind and semi-blind channel estimation for OFDM systems,” IEEE Trans. Signal Process., vol. 50, pp. 1699–1712, Jul. 2002. [27] K. Nikitopoulos and A. Polydoros, “Phase-impairment effects and compensation algorithms for OFDM systems,” IEEE Trans. Commun., vol. 53, pp. 698–707, Apr. 2005. [28] A. Papoulis and S. U. Pillai, Probability, Random Variables and Stochastic Processes, 4th ed. New York: McGraw-Hill, 2002. [29] T. Pollet, M. v. Bladel, and M. Moeneclaey, “BER sensitivity of OFDM systems to carrier frequency offset and wiener phase noise,” IEEE Trans. Commun., vol. 43, pp. 191–193, Feb./Mar./Apr. 1995. [30] P. Robertson and S. Kaiser, “Analysis of the effects of phase noise in OFDM systems,” in Proc. IEEE Int. Conf. Commun., (ICC95), Seattle, WA, Jun. 1995, pp. 1652–1657.

STEFANATOS AND KATSAGGELOS: JOINT DATA DETECTION AND CHANNEL TRACKING

[31] V. Smidl and A. Quinn, The Variational Bayes Method in Signal Processing. Heidelberg, Germany: Springer-Verlag, 2006. [32] A. P. Worthen and W. E. Stark, “Unified design of iterative receivers using factor graphs,” IEEE Trans. Inform. Theory, vol. 47, pp. 843–849, Feb. 2001. [33] D. Yee and J. P. Reilly, “A blind sequential Monte Carlo detector for OFDM systems in the presence of phase noise, multipath fading, and channel order uncertainty,” IEEE Trans. Signal Process., vol. 55, pp. 4581–4598, Sep. 2007. [34] S. Zhou and G. B. Giannakis, “Finite-alphabet based channel estimation for OFDM and related multicarrier systems,” IEEE Trans. Commun., vol. 49, pp. 1402–1414, Aug. 2001.

Stelios Stefanatos received the B.S. degree in physics and the M.S. degree in communications engineering from the National Kapodistrian University of Athens (NKUA), Athens, Greece, in 2004 and 2006, respectively. Since 2005, he has been working as a Research Associate for NKUA and the Institute of Accelerating Systems and Applications (IASA). His research interests are in the area of statistical signal processing for wireless communications.

4243

Aggelos K. Katsaggelos (F’98) received the Diploma degree in electrical and mechanical engineering from the Aristotelian University of Thessaloniki, Thessaloniki, Greece, in 1979, and the M.S. and Ph.D. degrees in electrical engineering from the Georgia Institute of Technology, Atlanta, in 1981 and 1985, respectively. In 1985, he joined the Department of Electrical and Computer Engineering at Northwestern University. He was the holder of the Ameritech Chair of Information Technology from 1997 to 2003. He is also the Director of the Motorola Center for Communications and a member of the Academic Affiliate Staff, Department of Medicine, Evanston Hospital. He is the editor of Digital Image Restoration (Springer-Verlag, 1991), coauthor of Rate-Distortion Based Video Compression (Kluwer, 1997), co-editor of Recovery Techniques for Image and Video Compression and Transmission (Kluwer, 1998), and coauthor of SuperResolution for Images and Video (Claypool, 2007), and Joint Source-Channel Video Transmission (Claypool, 2007), and the co-inventor of 12 patents. Dr. Katsaggelos has served the IEEE and other professional societies in many capacities; he was, for example, Editor-in-Chief of the IEEE Signal Processing Magazine (1997–2002), a member of the Board of Governors of the IEEE Signal Processing Society (1999–2001), and a member of the Publication Board of the IEEE PROCEEDINGS (2003–2007). He is the recipient of the IEEE Third Millennium Medal (2000), the IEEE Signal Processing Society Meritorious Service Award (2001), an IEEE Signal Processing Society Best Paper Award (2001), an IEEE International Conference on Multimedia and Expo Paper Award (2006), and an IEEE International Conference on Image Processing Paper Award (2007). He is a Distinguished Lecturer of the IEEE Signal Processing Society (2007–2008).