 Research Article
 Open Access
MaximumLikelihoodSemiblind Equalization of Doubly Selective Channels Using the EM Algorithm
 Gideon Kutz^{1}Email author and
 Dan Raphaeli^{1}
https://doi.org/10.1155/2010/709143
© Gideon Kutz and Dan Raphaeli. 2010
 Received: 5 August 2009
 Accepted: 9 June 2010
 Published: 1 July 2010
Abstract
Maximumlikelihood semiblind joint channel estimation and equalization for doubly selective channels and singlecarrier systems is proposed. We model the doubly selective channel as an FIR filter where each filter tap is modeled as a linear combination of basis functions. This channel description is then integrated in an iterative scheme based on the expectationmaximization (EM) principle that converges to the channel description vector estimation. We discuss the selection of the basis functions and compare various functions sets. To alleviate the problem of convergence to a local maximum, we propose an initialization scheme to the EM iterations based on a small number of pilot symbols. We further derive a pilot positioning scheme targeted to reduce the probability of convergence to a local maximum. Our pilot positioning analysis reveals that for high Doppler rates it is better to spread the pilots evenly throughout the data block (and not to group them) even for frequencyselective channels. The resulting equalization algorithm is shown to be superior over previously proposed equalization schemes and to perform in many cases close to the maximumlikelihood equalizer with perfect channel knowledge. Our proposed method is also suitable for coded systems and as a building block for Turbo equalization algorithms.
Keywords
 Orthogonal Frequency Division Multiplex
 Channel Estimation
 Expectation Maximization
 Recursive Little Square
 Pilot Symbol
1. Introduction
Next generation cellular communication systems are required to support high data rate transmissions for highly mobile users. These requirements may lead to doubly selective channels, that is, channels that experience both frequencyselective fading and timeselective fading. The frequency selectivity of the channel stems from the requirement to support higher data rates that necessitates the usage of larger bandwidth. Time selectivity arises because of the need to support users traveling at high velocities as well as the usage of higher carrier frequencies. It is therefore an important challenge to develop highperformance equalization schemes for doubly selective channels.
Doubly selective channels can rise both in singlecarrier systems and in Orthogonal Frequency Division Multiplexing (OFDM) systems. In singlecarrier systems, the doubly selective channel is modeled as a timevarying filter and introduces timevarying Inter Symbol Interference (ISI). In OFDM systems, the time selectivity of the channel destroys the orthogonality between subcarriers and introduces Inter Carrier Interference (ICI) while the frequency selectivity of the channel causes the ICI to be frequency varying. In this paper, we concentrate on singlecarrier systems only.
The problem of equalization for doubly selective channels has been extensively researched. Several methods for only trainingbasedequalization were proposed in [1, 2]. Semiblind equalization methods, that can benefit from both the training and data symbols, were proposed based on linear processing [3, 4] and DecisionFeedback Equalization (DFE) [5]. However, the performance of these equalization methods may not be satisfactory, especially when only one receiving antenna is present [6]. Moreover, the constant advance in processing power calls for more sophisticated equalization schemes that can increase network capacity.
Maximumlikelihood detection based on the Viterbi algorithm is a widely known technique for slowly fading channels. For higher Doppler rates, this method is not satisfactory due to the inherent delay in the Viterbi detector which causes the channel estimator part not to track the channel sufficiently fast. A partial remedy is offered by the Per Survivor Processing (PSP) approach, proposed originally in [7] and justified theoretically from the ExpectationMaximization (EM) principle in [8]. Using the PSP approach, the channel estimation is updated along each survivor path each symbol period using Least Mean Square (LMS) or Recursive Least Squares (RLS) [9]. However, for high Doppler rates, the performance is limited as these algorithms are not able to track fast fading channels [10]. Improved performance can be gained by using Kalman filtering [9, 11–13] but this approach requires the knowledge of the channel statistics which is normally not known a priori and its estimation will likely not be able to track fast fading. Lowcomplexity alternatives to the PSP were also proposed [11, 14].
Fast timevarying channel estimation might be achieved using the basis expansion (BE) model [15]. In this method, the channel's time behavior is modeled as a linear combination of basis functions. Basis functions can be polynomials [15], oversampled complex exponentials [2, 16], discrete prolate spheroidal sequences [17], and KarhunenLoeve decomposition of the fading correlation matrix [10]. Several receiver structures were proposed based on the combination of the BE with Viterbi algorithm variants like PSP [10], algorithm [14], and minimum survivor sequence [6, 18]. One common drawback of these methods is that the channel estimation part uses only hard decisions and does not weight the probability of different hypothesis for the symbol sequences. Moreover, all of these methods provide only hard decisions outputs which make them unsuitable for coded systems.
In order to enable the channel estimation part to benefit from soft decisions, several MAPbased algorithms combined with recursive, RLSbased, channel estimation were proposed [19–22]. The combination of MAP decoding and maximum likelihood channel estimation can be justified using the EM principle. This leads to an iterative detection and channel estimation algorithm based on the BaumWelch (BW) algorithm, proposed in [23] and modified for reduced complexity in [24] for nontimevarying channels (See [25] and references therein for more nontimevarying semiblind equalization methods). Adaptation for doubly selective channels is found in [26] based on incorporation of LMS and RLS in the algorithm.
Iterative MAP detection combined with polynomial BE was proposed in [27]. Unfortunately, this method cannot be directly extended to higher order BE models required in high mobility environments because the choice of polynomial expansion creates numerical difficulties for higher BE models. Furthermore, the equalization and channel estimation in [27] are done in a twostep ad hoc approach which is not a true EM (see Appendix A) and exhibits degraded performance in our simulations.
Finally, Turbo equalization schemes, encompassing iterative detection and decoding, were proposed based on RLS/LMS channel estimation [22, 26] and BE channel estimation [28]. The latter method employs a lowcomplexity approximation to the MAP algorithm for the detection part. It requires, however, that the channel statistics is fully known a priori.
In this paper, we present a novel method for semiblind MLbased joint channel estimation and equalization for doubly selective channels. The method is based on an adaptation of the EMbased algorithm for doubly selective channels by incorporating a BE model of the channel in the EM iterations. Using the BE method, we can simultaneously use long blocks thereby enhancing the performance in noisy environments without compromising the ability to track the channel because of the usage of sufficiently highorder BE to model the channel time variations. The proposed method is shown to have superior performance over previously proposed methods with the same block size and number of pilots in the block. Alternatively, it requires a lower number of pilots to achieve the same performance thereby enabling more bandwidth for the information. In addition, it is shown to have good performance for relatively small blocks, which is important if low latency in the communication system is required. The proposed algorithm outputs are the loglikelihood ratios (LLRs) of the transmitted bits, making it ideally suited for coded systems and also suitable as a building block for Turbo equalization algorithm that iterates between detection and decoding stages to improve the performance further. We treat the case of uncorrelated channels paths which is the worst case in terms of number of required BE functions. In Appendix B, we discuss the generalization to correlated paths.
Another contribution of the paper is the determination of a pilot positioning scheme that improves the equalizer's performance. In the context of our proposed algorithm, the main purpose of the pilots is the enablement of sufficient quality initialization of the EM iterations so that the probability of convergence to a local maximum is minimized. To that end, we propose an initialization scheme based on a small number of pilots and find the optimal pilot positioning such that the initial channel parameters guess is as close as possible to the channel parameters obtained assuming perfect knowledge of transmitted symbols (this is the channel estimation expected at the end of the BW iterations). It is shown that the pilot positioning depends on the channel's Doppler. For high Doppler rates, our results indicate that spreading the pilot symbols evenly throughout the block leads to the best initial channel guess. This result is surprising as it is different from previous results where the optimal positioning scheme was found to be spreading of groups of pilots whose length depended on the channels delay spread [29, 30]. These previous results, however, were obtained using different criteria and channel model. More importantly, the analysis in these papers was restricted and did not consider pilot groups shorter than the channel's delay spread as done in this paper. Therefore, these previous results do no contradict with our new result.
Pilot positioning was discussed in [31–33] and conditions for MMSE optimality of both the pilot sequence and positioning were derived. The resulting sequences and positioning are, however, less attractive for practical implementations. This is because most of them require that the pilots and data overlap in time which complicated the receiver structure. The only optimal scheme proposed in [32, 33] with nonoverlapping pilots and data requires a pilot pattern that results in very high peak to average transmission which is not desirable in practical communications systems. In contrast, we optimize the pilot positioning given a predefined pilot sequence (in this paper we use, as an example, Barker sequence). This allows us to derive optimal pilot positioning for a given pilot sequence that meets some other constrains (e.g., constant envelope signals, low peaktoaverage ratio, etc.).
The rest of the paper is organized as follows. In Section 2, we present the system model and introduce the BE model. In Section 3 we present our proposed method for semiblind joint channel estimation and equalization for doubly selective channels. Section 4 discussed the BE functions set selection. Our results regarding optimal pilot placement are presented in Section 5. Section 6 presents our simulation results and conclusions are drawn in Section 7. Partial results of this work were introduced in a conference paper [34].
2. Problem Formulation
2.1. System Model
where is the received vector (observation vector) and represents a branch (transition) on the trellis formed by the channel's memory [23]. There are possible branches at each time sample . Each possible branch is denoted by the row vector where and . Finally, is an Additive White Gaussian Noise (AWGN) sequence with zero mean and an unknown variance .
where is the communication system carrier frequency, is the user's velocity, is the speed of light, and is the time of one symbol.
 (A1)
Information symbols, channel realization, and noise samples are statistically independent.
 (A2)
The channel's paths are statistically independent (uncorrelated scattering [35]).
2.2. Basis Expansion Model
where .
3. The BaumWelch Algorithm for Equalization of Doubly Selective Channels
In this section, we present our new algorithm for semiblind maximumlikelihood joint channel estimation and equalization for doubly selective channels. We treat channels with uncorrelated paths as this is the worst case in term of number of required basis functions in the BE description. In Appendix B, we extend the algorithm for channels with correlated paths.
3.1. Algorithm for Blind Equalization
The values of are efficiently computed using the BahlCockeJelinekRaviv (BCJR) algorithm [36]. For nontimevarying channels, we have and (11) reduces to equation (11) in [23]. Our algorithm may also be extended to channels with correlated paths. In this case, the number of BE parameters used for describing all channel's paths may be reduced. In that sense, uncorrelated channels paths may be considered as the worst case. The correlated case is discussed in Appendix B.
3.2. Adaptation to the Semiblind Case
Adaptation of the above algorithm to the semiblind case where we have some known pilot symbols is straightforward. The only required change is in the computation of using the BCJR algorithm. We modify the branch metrics so that all transitions that are not consistent with known pilots are assigned zero probability. This ensures that transition probabilities are calculated with the a priori information about the pilots.
3.3. Initialization of the Algorithm
Optimization of the EM objective function (8) is a nonlinear process that may converge to a local maximum. It is therefore important to calculate good initial guess for the channel parameters so that the probability of convergence to a local maximum is minimized. We suggest using the available pilot symbols for finding initial channel parameters using the following method. First, we run the BCJR algorithm where the branch metrics are initialized without any initial channel guess by assigning zero a priori probability to all transitions that are not consistent with the known pilots and equal (nonzero) a priori probability to all transitions that are consistent with the pilots. This initialization of the branch metrics represents our best a priori knowledge about the transitions probability in the trellis. From the BCJR algorithm we find and then use them in (11) to obtain an initial guess for the channel BE parameters. More details on this initialization method can be found in Appendix C.
An important feature of this initialization scheme is that all observations that have some content of pilots in them are taken into account including those with mixed pilot and data contributions. This is in contrast to most other pilot based estimations that take into account observations based on pilots only [30]. This fact turns out to be significant when we discuss how to position the pilot symbols in the block in Section 5.
It should be noted that the above algorithm requires initial synchronization stage to ensure that all major channels taps fall within the searched multipath window. This synchronization stage can be done at a much lower rate than channel estimation update, as the channel tap positions typically drift at a much slower rate compared to the fading rate, and therefore its complexity is negligible. The synchronizations stage is outside the scope of this paper and we assume perfect synchronization throughout the paper.
3.4. Computational Complexity
Computational complexity summary.
Equation  Stage  MAC operations 

Update channel estimate 


(11) 



 

 

 
Update noise variance 


(12) 


4. Selection of the Basis Functions
The functions in (14) model the channel time behavior as polynomial in time. This choice of basis functions may be regarded as a generalization of the channel description in [27] where it was suggested to use first and secondorder polynomials to model the channel time variations.
Out of all eigenvectors, the vectors that correspond to the largest eigenvalues are selected as the basis set. The target function in (15) is suitable for flat fading channel. For frequencyselective channel, the mean square error will be simply the sum of the mean square errors of the individual paths and therefore the same solution is optimal for multipath channels. We note that a similar argument is given in [38].
An obvious alternative to the equalization approach we propose in this paper is to divide the data block into small subblocks such that the channel can be considered approximately constant within a subblock period and then equalize each subblock separately using the BaumWelch algorithm for nontimevarying channels [23]. Interestingly, this subblock scheme can be considered as an instance of the BE approach if we choose basis functions for subblocks where the th basis function is equal to one in the symbols time that correspond to the th subblock and zero elsewhere, that is , where is a vector on ones. To justify our approach, we would like to compare it to this subblock approach.
5. Placement of Pilot Symbols
5.1. Pilot Positioning Problem Formulation
where is a vector of the pilot positions in the block. The maximum function in (23) and expectation in (24) are taken with respect to the data symbols, noise, and channel realizations. Using these criteria, it might be possible to optimize both the pilots positions and the pilot patterns. We, however, select known pilot patterns (e.g., Barker sequences) so that we keep constant envelope signals and optimize the positioning for this given pilot pattern. The usage of these two criteria is detailed in the next sections. Interestingly, both criteria lead to the same positioning scheme for high Doppler rates.
5.2. Worst Case Analysis for Flat Fading Channels
where is the vector of eigenvalues of the matrix . The second equality follows from the fact that for flat fading channels and (Assume that is eigenvalue of , that is, , define , then and . Matrices and have therefore the same eigenvalues).
The matrix is a deterministic function of the BE functions, block size, pilot positioning, and the pilot pattern (sequence). It is therefore possible to find the best positioning scheme for the desired block size, BE model, and pilot sequence with a computer search. For simplicity, we limit the search for patterns in which the pilots are grouped in groups of length and these groups are spread throughout the block as evenly as possible. This means that the pilot positioning we find with this limited search is only optimal amongst all positioning with evenly spaced pilot clusters. However, all previous works on pilot positioning arrived at positioning schemes that are consistent with this structure. It turns out that the best positioning scheme is obtained with for all tested block sizes. It is interesting to note that this result is identical to the result in [29] which was obtained using different channel model and criterion.
5.3. Mean Case Analysis for Frequency Selective and Frequency Flat Channels
The optimal positioning strategy is shown to be dependent on the Doppler rate and the number of pilots in the block. As can be seen from Figure 2, for low Doppler rates it is better to use group of pilots as also indicated by [29, 30] (although the difference is not very significant, at least for short delay spreads). For high Doppler rates and a small number of pilots, however, it turns out that using leads to much better results. This is because there is a tradeoff between accurate estimation of the multipath at specific points in time (that is better achieved by grouping the pilots) and tracking the channel time variations (that is better achieved by spreading the pilots throughout the block). Our results indicate that for high velocities using leads to a lower metric value as this means better ability to track time variations. Note that this result is obtained for severe ISI channel with three equal energy paths (and similar result was obtained for channel with 5 equal energy paths). We have also simulated channels with less severe ISI (that is, decaying power profiles), and the advantage of using was even larger, as could be expected. The switching point (Doppler rate beyond which it is advantageous to use ) is dependent mainly on the percentage of pilots in the block. For larger number of pilots, the switching point will occur at higher Doppler rate. The reason is that for large number of pilots there will be sufficient number of groups in the block to allow tracking of path time variations even when the group size is kept , so both multipath profile and time variations could be estimated accurately. We, however, are interested in the smallest number of pilots that enables good performance, and in these conditions, is advantageous even for moderate Doppler rates (see Figure 2). This conclusion is somewhat surprising as it is different from previous conclusions in [29, 30]. However, these previous works used different channel models and performance criteria. Moreover, both works considered only pilot groups equal to [29] or [30] or longer, to facilitate their analysis.
6. Simulation Results
6.1. Performance of the Proposed Equalization Scheme
where . This number was tested numerically to enable good accuracy description of the channel with the BE complex exponents and polynomial functions (below 1% error). This is also the number of basis functions used in [30]. For the selection of eigenvectors as the functions set, we could have decreased this number slightly.
 (i)
Maximum Likelihood equalization using perfect channel knowledge.
 (ii)
Maximum likelihood equalization with channel estimation based only on the pilots. This is identical to the first iteration of the proposed algorithm.
 (iii)
Timevarying BW algorithm with BE based on complex exponential functions (13) (BWBEexp).
 (iv)
Timevarying BW algorithm with BE based on complex exponential functions (13) and initial channel guess identical to the true channel (BWBEexp & perfect init.). The difference between the error curve of this simulation and the previous one will indicate if we have an issue of convergence to a local maximum.
 (v)
Timevarying BW algorithm with BE based on polynomial functions (14) (BWBEpoly). This might be considered a significant improvement of [27].
 (vi)
Timevarying BW algorithm with BE based on optimal basis functions (BWBEeig).
 (vii)
Nontimevarying BW algorithm based on dividing the data blocks into shorter blocks in which channel is assumed to be constant (BWSB). This is essentially the method of [23].
 (viii)
The BWRLS method in [26] (called APPSDDRLS in [26]). This method was initialized using the same initialization scheme we used for the BWBE methods. After the parameters of the BE are found, the actual channel responsed estimate is computed for every time instance. Finally, the BCJR algorithm uses this estimate to calculate the transitions probabilities which are the starting point for the BWRLS in [26].
 (ix)
 (x)
Iterative Viterbibased equalization with BEbased channel estimation. Reduced complexity variants of this algorithm appeared in [14, 18].
7. Conclusion
A novel method for maximum likelihood semiblind joint channel estimation and equalization for doubly selective channels is proposed. This method is based on expectation maximization (EM) principle combined with a BE model of the doubly selective channel. It is shown that the main drawback of the proposed approach is its possible convergence to local maxima. To alleviate this problem, we propose an initialization scheme to the equalization based on a small number of pilot symbols. We discuss the selection of basis functions set and find the optimal basis set. We further derive a pilot positioning scheme targeted to reduce the probability of convergence to a local maximum. The resulting algorithm is shown to be significantly superior over previously proposed equalization schemes and to perform in many cases close to an ML equalizer with perfect channel knowledge.
Appendices
A. Comparison of the proposed method to [27]
In this appendix, we highlight the differences between the algorithm proposed in [27] and the algorithm presented in this paper. For simplicity, we discuss only the nontimevarying case as the generalization to timevarying case is straightforward.
Comparing (A.5) to (A.1), we see that the two methods are not equivalent. Moreover, the method in [27] converges to the true BW only when is constant for all values of . This is likely to happen for long blocks. However, for short block it is not necessarily true and in this case the approach presented in [27] leads to a suboptimal solution.
B. Extension to correlated channel's paths
C. Proof of the proposed BW initialization method
By combining the results from (C.6) and (C.7), it can be verified that (21) is identical to (B.6) for this particular choice of branch metrics.
Declarations
Acknowledgment
Parts of the paper were presented at ISCCSP2008 [34].
Authors’ Affiliations
References
 Rousseaux O, Leus G, Moonen M: Estimation and equalization of doubly selective channels using known symbol padding. IEEE Transactions on Signal Processing 2006, 54(3):979989.View ArticleGoogle Scholar
 Giannakis GB, Tepedelenlioǧlu C: Basis expansion models and diversity techniques for blind identification and equalization of timevarying channels. Proceedings of the IEEE 1998, 86(10):19691986. 10.1109/5.720248View ArticleGoogle Scholar
 Leus G, Barhumi I, Rousseaux O, Moonen M: Direct semiblind design of serial linear equalizers for doublyselective channels. Proceedings of the IEEE International Conference on Communications, June 2004, Paris, France 5: 26262630.Google Scholar
 Colonnese S, Panci G, Rinauro S, Scarano G: Semiblind bussgang equalization for sparse channels. IEEE Transactions on Signal Processing 2009, 57(12):49464952.MathSciNetView ArticleGoogle Scholar
 Leus G: Semiblind channel estimation for rapidly timevarying channels. Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP '05), March 2005, Philadelphia, Pa, USA 3: 773776.Google Scholar
 Barhumi I, Moonen M: MLSE and MAP equalization for transmission over doubly selective channels. IEEE Transactions on Vehicular Technology 2009, 58(8):41204128.View ArticleGoogle Scholar
 Raheli R, Polydoros A, Tzou C: Persurvivor processing: a general approach to MLSE in uncertain environments. IEEE Transactions on Communications 1995, 43(2–4):354364.View ArticleGoogle Scholar
 ZamiriJafarian H, Pasupathy S: Adaptive MLSDE using the EM algorithm. IEEE Transactions on Communications 1999, 47(8):11811193. 10.1109/26.780454View ArticleMATHGoogle Scholar
 Kubo H, Murakami K, Fujino T: Adaptive maximumlikelihood sequence estimator for fast timevarying intersymbol interference channels. IEEE Transactions on Communications 1994, 42(2–4):18721880.View ArticleGoogle Scholar
 Borah DK, Hart BD: Receiver structures for timevarying frequencyselective fading channels. IEEE Journal on Selected Areas in Communications 1999, 17(11):18631875. 10.1109/49.806817View ArticleGoogle Scholar
 Dai Q, Shwedyk E: Detection of bandlimited signals over frequency selective Rayleigh fading channels. IEEE Transactions on Communications 1994, 42(2–4):941950.Google Scholar
 Iltis RA: A Bayesian maximumlikelihood sequence estimation algorithm for a priori unknown channels and symbol timing. IEEE Journal on Selected Areas in Communications 1992, 10(3):579588. 10.1109/49.127780View ArticleGoogle Scholar
 Hart BD, Taylor DP: Maximumlikelihood synchronization, equalization, and sequence estimation for unknown timevarying frequencyselective rician channels. IEEE Transactions on Communications 1998, 46(2):211221. 10.1109/26.659480View ArticleGoogle Scholar
 Hwang SJ, Schniter P: Nearoptimal noncoherent sequence detection for doubly dispersive channels. Proceedings of the Asilomar Conference on Signals, Systems, and Computers (ACSSC '06), November 2006, Pacific Grove, Calif, USA 134138.Google Scholar
 Borah DK, Hart BD: Frequencyselective fading channel estimation with a polynomial timevarying channel model. IEEE Transactions on Communications 1999, 47(6):862873. 10.1109/26.771343View ArticleGoogle Scholar
 Thomas TA, Vook FW: Multiuser frequencydomain channel identification, interference suppression, and equalization for timevarying broadband wireless communications. Proceedings of the IEEE Sensor Array and Multichannel Signal Processing, March 2000, Boston, Mass, USA 444448.Google Scholar
 Zemen T, Mecklenbräuker CF: Timevariant channel estimation using discrete prolate spheroidal sequences. IEEE Transactions on Signal Processing 2005, 53(9):35973607.MathSciNetView ArticleGoogle Scholar
 ElMahdy AES: Adaptive channel estimation and equalization for rapidly mobile communication channels. IEEE Transactions on Communications 2004, 52(7):11261135. 10.1109/TCOMM.2004.831391View ArticleGoogle Scholar
 Baccarelli E, Cusani R: Combined channel estimation and data detection using soft statistics for frequencyselective fastfading digital links. IEEE Transactions on Communications 1998, 46(4):424427. 10.1109/26.664292View ArticleGoogle Scholar
 Hart BD, Pasupathy S: Innovationsbased MAP detection for timevarying frequencyselective channels. IEEE Transactions on Communications 2000, 48(9):15071519. 10.1109/26.870018View ArticleGoogle Scholar
 Davis LM, Collings IB, Hoeher P: Joint MAP equalization and channel estimation for frequencyselective and frequencyflat fastfading channels. IEEE Transactions on Communications 2001, 49(12):21062114. 10.1109/26.974257MATHView ArticleGoogle Scholar
 Otnes R, Tüchler M: Iterative channel estimation for turbo equalization of timevarying frequencyselective channels. IEEE Transactions on Wireless Communications 2004, 3(6):19181923. 10.1109/TWC.2004.837421View ArticleGoogle Scholar
 Kaleh GK, Vallet R: Joint parameter estimation and symbol detection for linear or nonlinear unknown channels. IEEE Transactions on Communications 1994, 42(7):24062413. 10.1109/26.297849MATHView ArticleGoogle Scholar
 Choi J: A joint channel estimation and detection for frequencydomain equalization using an approximate EM algorithm. Signal Processing 2004, 84(5):865880. 10.1016/j.sigpro.2004.01.005MATHView ArticleGoogle Scholar
 Aktas E: Belief propagation with Gaussian priors for pilotassisted communication over fading ISI channels. IEEE Transactions on Wireless Communications 2009, 8(4):20562066.MathSciNetView ArticleGoogle Scholar
 Nissilä M, Pasupathy S: Adaptive Bayesian and EMbased detectors for frequencyselective fading channels. IEEE Transactions on Communications 2003, 51(8):13251336. 10.1109/TCOMM.2003.815050View ArticleGoogle Scholar
 AntónHaro C, Fonollosa JAR, Faulí C, Fonollosa JR: On the inclusion of channel's time dependence in a hidden Markov model for blind channel estimation. IEEE Transactions on Vehicular Technology 2001, 50(3):867873. 10.1109/25.933319View ArticleGoogle Scholar
 Hwang SJ, Schniter P: Fast noncoherent decoding of block transmissions over doubly dispersive channels. Proceedings of the 41st Asilomar Conference on Signals, Systems and Computers (ACSSC '07), November 2007, Pacific Grove, Calif, USA 10051009.Google Scholar
 Adireddy S, Tong L: Optimal placement of known symbols for slowly varying frequencyselective channels. IEEE Transactions on Wireless Communications 2005, 4(4):12921296.View ArticleGoogle Scholar
 Ma X, Giannakis GB, Ohno S: Optimal training for block transmissions over doubly selective wireless fading channels. IEEE Transactions on Signal Processing 2003, 51(5):13511366. 10.1109/TSP.2003.810304MathSciNetView ArticleGoogle Scholar
 Kannu AP, Schniter P: MSEoptimal training for linear timevarying channels. Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP '05), March 2005, Philadelphia, Pa, USA 789792.Google Scholar
 Kannu AP, Schniter P: Design and analysis of MMSE pilotaided cyclicprefixed block transmissions for doubly selective channels. IEEE Transactions on Signal Processing 2008, 56(3):11481160.MathSciNetView ArticleGoogle Scholar
 Whitworth T, Ghogho M, McLernon D: Optimized training and basis expansion model parameters for doublyselective channel estimation. IEEE Transactions on Wireless Communications 2009, 8(3):14901498.View ArticleGoogle Scholar
 Kutz G, Raphaeli D: Maximum likelihood semiblind equalization of doubly selective channels. Proceedings of the 3rd International Symposium on Communications, Control, and Signal Processing (ISCCSP '08), March 2008, St. Julians, Malta 15801584.Google Scholar
 Sklar B: Rayleigh fading channels in mobile digital communication systems, part I: characterization. IEEE Communications Magazine 1997, 35(7):90100. 10.1109/35.601747View ArticleGoogle Scholar
 Bahl LR, Cocke J, Jelinek F, Raviv J: Optimal decoding of linear codes for minimizing symbol error rate. IEEE Transactions on Information Theory 1974, 20(2):284287.MATHMathSciNetView ArticleGoogle Scholar
 Tugnait JK, He S: Doublyselective channel estimation using datadependent superimposed training and exponential basis models. IEEE Transactions on Wireless Communications 2007, 6(11):38773883.View ArticleGoogle Scholar
 Teo KAD, Ohno S: Optimal MMSE finite parameter model for doublyselective channels. Proceedings of the IEEE Global Telecommunications Conference (GLOBECOM '05), December 2005, St. Louis, Mo, USA 6: 35033507.Google Scholar
 Dong M, Tong L, Sadler BM: Optimal insertion of pilot symbols for transmissions over timevarying flat fading channels. ACSP TR010302 Cornell University; January 2003.Google Scholar
 Li Y, Guan YL: Modified Jakes' model for simulating multiple uncorrelated fading waveforms. Proceedings of the IEEE International Conference on Communications (ICC '00), June 2000 4649.Google Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work are properly cited.