IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 53, NO. 3, MAY 2004
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Continuous-Mode Frame Synchronization for Frequency-Selective Channels Yan Wang, Kai Shi, and Erchin Serpedin, Member, IEEE
Abstract—Although it has been studied extensively for additive white Gaussian noise and flat fading channels, the problem of continuous-mode frame synchronization over frequency-selective channels has received much less attention. In this paper, based on the maximum likelihood criterion, we derive a computationally efficient continuous-mode algorithm for joint frame synchronization, channel and frequency offset estimation for linear modulations transmitted through unknown frequency-selective channels affected by carrier-frequency offset. Computer simulations show that the proposed algorithms exhibit low implementation complexity and good performance. Index Terms—Channel estimation, frame synchronization, frequency offset, frequency-selective channel.
I. INTRODUCTION
I
N THE past decades, there has been much research into continuous-mode frame synchronization; most widely used methods concentrate on locating a fixed frame synchronization pattern or “sync word” inserted periodically into the continuous data stream [1], [4], [5], [8], [9], [11], [13], [15]. The optimum maximum likelihood (ML) rule for frame synchronization in additive white Gaussian noise (AWGN) channels with binary phase-shift keying (BPSK) signaling was originally proposed by Massey [9]. Nielsen subsequently reported that this ML rule and its high signal-to-noise ratio (SNR) approximation (high SNR ML rule) provided several decibels improvement over the well-known correlation rule [13]. Many years later, Liu and Tan extended these results to -ary phase-shift keying (PSK) modulations and corroborated the Nielsen’s conclusion [8]. Recently, based on the ML rule, frame-synchronization algorithms for flat fading channels were derived in [4] and [15]. Although it has been studied extensively for AWGN and flat fading channels [1], [4], [5], [8], [9], [13], [15], the problem of continuous-mode frame synchronization in the presence of frequency-selective channels has received much less attention. In [11], an ML-based frame synchronizer was derived assuming a binary pulse amplitude modulated system and known static dispersive intersymbol interference (ISI) channels. However, in many applications of interest, the channel is time varying and a priori unknown. This problem becomes further complicated in the presence of frequency offset, which is the case due to the fact that frame synchronization sometimes has to be achieved Manuscript received December 20, 2001; revised July 3, 2003 and February 4, 2004. This paper was presented in part at the Wireless Networking Symposium, University of Texas, Austin, 2003, and has been supported by the National Science Foundation Award CCR-0092901. The authors are with the Department of Electrical Engineering, Texas A&M University, College Station, TX 77843-3128 USA (e-mail:
[email protected]. edu;
[email protected];
[email protected]). Digital Object Identifier 10.1109/TVT.2004.827163
before carrier recovery is performed [1]. The objective of this paper is to contribute to filling this gap. Generally, in the presence of other unknown variables (e.g., channel coefficients and frequency offset), two possible approaches may be derived to estimate the frame boundary according to the ML criterion. One is the Bayesian approach, which consists of modeling other unknowns as random variables with certain probability density functions (pdfs) and computing the average of joint likelihood function with respect to their pdfs to produce the marginal likelihood of the frame boundary, from which the ML estimate of frame boundary can be obtained (see e.g., [1] and [4]). Another method aims at jointly estimating the frame boundary and other unknown variables [2], [6]. In this paper, following the latter approach, we propose a computationally efficient synchronization scheme for joint frame synchronization, channel and frequency offset estimation by exploiting the ML rule. Computer simulations show that the proposed algorithm exhibits low implementation complexity and good performance. It is interesting to note that the problem of frame synchronization over unknown frequency-selective channels is well covered for the scenario of asynchronous or spontaneous packet transmission (“one-shot” or burst-mode synchronization) [2], [3], [6], [7], where the sync word is prefixed to the data stream and is itself preceded by no signal or by a sequence of symbols to perform other synchronization tasks (a clock-recovery sequence or an unmodulated sequence for carrier estimation) [9], [10]. There are two essential differences between the methods dealing with the “one-shot” synchronization and those of continuous-mode frame synchronization. First, the observation sequence of one-shot synchronization is chosen long enough to contain the complete frame sync word [10], while for the latter, an -signal span of the received sequence, where denotes the frame length [9], is usually processed. Second, for the synchronization of spontaneous packet transmissions, it is always assumed that the position of the data packet is known up to an uncertainty in a finite interval that is centered about a coarse frame sync flag generated by the preceding automatic gain control unit [2], [6], [10]. This assumption guarantees that the frame sync word is contained entirely in the observation sequence and assumes its initial order. When tackling the problem of continuous-mode frame synchronization, it is generally assumed that the sync word is a priori equally likely to begin in any of the positions of the received sequence (see, e.g., [8] and [9]). After introducing the system model, an ML scheme for joint frame synchronization and channel estimation is developed. Then, an extension of the proposed ML scheme to
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Fig. 1. Frame synchronization model. a) Frame structure. b) Transmitted sequence. c) Received sequence with perfect sync. d) Practical observed sequence.
frequency-selective channels affected by carrier frequency offset is presented. Finally, the performance of the proposed algorithms is demonstrated through computer simulations. II. SYSTEM MODEL We consider a linear modulation (e.g., PSK or QAM) transmitted through a slow time-varying frequency-selective are channel, whose coefficients1 assumed to remain constant over the duration of the observed represents the channel memory. The frame sequence and symbols, where the first of transmitted data consists of symbols form a fixed frame synchronization pattern followed by random [see Fig. 1(a) and (b)]. data symbols are zero-mean indepenWe assume that the data symbols dently and identically distributed (i.i.d.) with unit average and the training energy per symbol, i.e., are selected from the same set as that of data symsymbols bols , so that no restriction is made on the frame structure to prohibit the replication of the frame-synchronization pattern in the portion of random data [8], [13]. It is generally desirable to choose a sync word with good autocorrelation property satisfying the condition
which ensures the number of replications of the sync word amid random data to be minimized [8], [10], [13]. The transmitted signal is passed through the channel and is sampled at the symbol period. It is reasonable to assume and . The outputs of the channel corresponding to the th frame are modeled as [see Fig. 1(c)] (1) 1We use the superscripts position, respectively.
and
to denote transposition and conjugate trans-
where if if if Based on (1), the positions of , are defined as the frame boundaries in the channel outputs, where the first training symbol is involved in the first path of the channel. In the absence of a priori information, the received signal is a linear shift of the sequence with an arbitrary delay rather than itself and, hence, the frame boundaries may appear in any of the positions (i.e., the locamodulo ) with equal probability in an arbitrarily tion of the observed selected -signal span sequence [Fig. 1(d)] [4], [8], [9]. Therefore, the frame-synchronization problem that we pose resumes to estimating the index from the selected segment of channel-output observations. Defining the linear shift operator as , we can express the received segment as (2) where and the components are independent complex Gaussian random variables with . Note that when dealing with the zero-mean and variance problem of frame synchronization in AWGN and flat-fading channels, authors prefer to use (left) cyclic shift operator, , which is defined by instead of , since these two operations are statistically equivalent and the former makes the derivation more compact. However, this equivalence does not hold true in the case of frequency-selective channels. From (1), it is not difficult to and do not always involve the find that same set of unknown random data when varies; hence, they exhibit different statistical properties due to the memory of the channel. For simplicity, we will omit the dummy variable in the ensuing derivation.
WANG et al.: CONTINUOUS-MODE FRAME SYNCHRONIZATION FOR FREQUENCY-SELECTIVE CHANNELS
III. JOINT FRAME SYNCHRONIZATION AND CHANNEL ESTIMATION
Substituting (6) into (5), we can obtain the following estimator that is equivalent to (5):
From (2), the optimum maximum a posteriori (MAP) al, gorithm maximizes the posterior probability , which by the Bayes’ theorem [14, p. 84] , where stands for the becomes for all , the MAP algorithm repdf of . Since duces to the ML estimator, which maximizes
(3) where represents the set of unknown data involved in the . Note that the probability depends operation on the size of the symbol alphabet and the synchronization position . Moreover, the averaging over all possible data vectors of length varying with , whose complexity increases exponentially with the channel memory, is complicated enough to escape even an approximation and, hence, the optimal estimator (3) appears to be computationally prohibitive. To circumvent this difficulty, next we propose a suboptimal but computationally efficient algorithm, which does not necessitate the averaging over the unknown data. Exploiting (1), it is not difficult to observe that the subset can be expressed as
.. .
.. .
..
.
.. .
(4)
and is affected by the sync word only and not by the random data. Define the (right) cyclic shift of the observed signals as , where and of , namely choose a subwindow of length . Under the assumption that is the correct position of frame boundaries, can be expressed in terms of the frame-synchronization pattern , as in (4). A reduced-complexity ML-based estimator of that exploits the information provided by can be obtained by maximizing the likelihood function
or, equivalently, the log-likelihood function (5) For a fixed [6], [12])
, the ML estimate of
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is given by (see, e.g., [2],
(6)
(7) identity matrix and denotes the projection matrix. In summary, the proposed frame synchronization algorithm is as follows. Step 1) Select an arbitrary-length signal segment of the received signal. , choose the subwindow of Step 2) For each and compute the metric . observation Step 3) Find a value such that the corresponding metric achieves the the maximum among the metrics (7). . Step 4) The channel estimate is given by (6) with (i.e., ), is Note that if square matrix and nonsingular, then one can see that an . In this condition, is always 0 and the estimator (7) is meaningless [12]. Therefore, the length of sync pattern has . Physically, this means that the first to be chosen as symbols of the sync pattern are guard symbols that prevent the remaining sync symbols from being affected by random data uncorrupted sync symbols are required to and at least unknown parameters ( , ). estimate the The proposed synchronizer (7) is based on one frame length of channel observations. For certain applications where the constraint on the processing delay is not stringent, we may improve the performance of (7) by using multiple frames of channel observations to estimate the index . One method is to make individual estimates based on single-frame observations successive frames and then to use a majority decision for rule that decides the estimate of on the majority of these independent estimates (e.g., [8]). Another approach, which we successive frames will present here, is to jointly exploit the of observations to obtain a single estimate of . Assuming that the channel coefficients remain constant frames of channel observations and during the consecutive following the procedure used to derive (7), one can obtain the -frame based synchronizer, which takes a similar expression to (7) as where is the
(8) is the where matrix and
identity
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IV. SYNCHRONIZATION IN THE PRESENCE OF FREQUENCY OFFSET We now consider that there is a residual frequency offset in the received signal. Hence, the channel model (2) becomes
(9) where stands for the unknown frequency offset normalized to the symbol rate. Defining diag with chosen as the subof and adopting window the procedure presented in the last section, we can obtain the following ML-based estimator for : Fig. 2.
Similar to (6), the ML estimate of
FAP versus SNR with fixed
N.
(10) now takes the expression (11)
and the estimates of
and
can be obtained by maximizing (12)
To proceed, we derive an estimate of as a function of can be expressed in the form that
. Note
Re
(13) where Re denotes the real part of the enclosed quantity, stands is the -entry of . for the complex conjugation, and The second term of the right-hand side (RHS) of (13) is independent of and, based on the definition of , the first term or of the RHS of (13) can be rewritten for as
Fig. 3. MSCEE versus SNR with fixed
N.
When is in the range , defining , lengthy and tedious algebra manipulations lead to the following expressions for the first term of the RHS of (13):
where (15) (16)
if if if
where
,
(17)
if if if
. (14)
if if
,
(18)
WANG et al.: CONTINUOUS-MODE FRAME SYNCHRONIZATION FOR FREQUENCY-SELECTIVE CHANNELS
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Fig. 4. Improvement of FAP versus SNR with multiple-frame synchronization.
(19) Obviously, for each fixed
,
can be estimated by
(20) is equal to 0 or given by (18), depending on the where value of . It is easy to find that the estimator (20) can be efficiently implemented by fast Fourier transform (FFT) methods (see, e.g., [6] and [12]) and the estimation range of is , the maximum range that can be expected for any frequency offset estimator operating on baud rate samples [12]. Finally, plugging (20) back into (12), the ML estimate of can be obtained as
(21) The proposed algorithm in the presence of frequency offset can be summarized as follows. Step 1) Select an arbitrary-length signal segment of the received signal. , choose the subwindow of Step 2) For each observation • if or , comaccording to (14) and set pute the term ; and • otherwise, compute the terms according to (15)–(19). Step 3 For each , estimate according to (20) and compute the metric based on (21).
Step 4 Find a value such that the corresponding metric achieves the maximum among the metrics. . Step 5 Frequency offset is obtained by (20) with and Step 6 Channel estimate is given by (11) with . Exact theoretical analysis of frame-synchronization algorithms does not generally appear to be tractable for the decision rules other than the correlation rule, even for AWGN channels [8], [15]. Therefore, next we resort to computer simulations to evaluate the performance of the proposed synchronizers. V. SIMULATION RESULTS In computer simulations, the false acquisition probabilities ) and the mean square channel-estimation (FAP, i.e., error (MSCEE) of the proposed joint frame-synchronization and channel-estimation algorithms are evaluated. In the presence of frequency offset, the mean square error ) is investigated, (MSE) of (i.e., too. All experiments are performed assuming 100 000 Monte are taken from Carlo trials, the transmitted symbols is generated as a QPSK constellation, the additive noise , and SNR is defined as white Gaussian noise with variance . The frequency-selective channel coefficients are modeled as i.i.d. complex Gaussian random . variables with zero mean and variance Experiment 1—Performance of the proposed synchronizer , with fixed length : Fixing the total length of frame we plot the FAP and MSCEE of the synchronizer (7) versus SNR in Figs. 2 and 3, respectively, where two different channel and are assumed. The frame synchroorders nization patterns used are the seven-symbol Barker sequence , the 13-symbol Neuman-Hofman sequence , and the midamble adopted in the GSM system, i.e., with length
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Fig. 5. Improvement of MSCEE versus SNR with multiple-frame synchronization.
Fig. 6. (a) FAP, (b) MSCEE, and (c) MSE(f^ ) versus SNR in the presence of f .
WANG et al.: CONTINUOUS-MODE FRAME SYNCHRONIZATION FOR FREQUENCY-SELECTIVE CHANNELS
The results presented in Figs. 2 and 3 show that the performance of the proposed algorithm deteriorates when channel memory increases or the length of sync words decreases. The failure in the case of channel memory is due of (7) with is not satisfied in this to the fact that the condition of scenario. Varying the length , additional simulations show that the performance of the proposed synchronizer is not sensitive to the frame length, especially at medium and high SNRs, which is a pleasing property in the sense that we can increase the length of useful data sequence to obtain a high transmission efficiency. Experiment 2—Multiple-frame synchronization: In Figs. 4 and 5, we compare the performance of multiple-frame and ) with that of the synchronizer (8) ( single-frame-based algorithm (7), as well as the method of frames, single-frame majority-decision (SFMD) with , , and . assuming the parameters One can find that the estimator (8) can provide considerable improvement over the methods of (7) and SFMD. Experiment 3—Performance of the proposed synchronizer in , , the presence of frequency offset: Fixing , Fig. 6(a)–(c) illustrates the FAP, MSCEE, and and of the proposed algorithm in the presence and absence of frequency offset, respectively. In each simulation run, the fre, quency offset is selected randomly from the interval assuming a uniform distribution. Two values of the upper-bound , (large frequency offset) and (small frequency offset) are used. Fig. 6(a) illustrates that the proposed frame-acquisition algorithm is quite robust to frequency offsets, while Fig. 6(b) and (c) shows that the proposed channel and frequency offset estimators exhibit almost the same performance in the presence of large and small frequency offsets. VI. CONCLUSION In this paper, we have proposed an ML synchronizer for joint frame, channel and frequency offset estimation for continuous-mode linearly modulated transmissions through frequency-selective channels affected by carrier frequency offset. The proposed algorithms are computationally efficient, robust to frequency offsets, do not necessitate detection of the unknown data symbols, and exhibit good performance. REFERENCES [1] Z. Y. Choi and Y. H. Lee, “Frame synchronization in the presence of frequency offset,” IEEE Trans. Commun., vol. 50, pp. 1062–1065, July 2002. [2] S. A. Fechtel and H. Meyr, “Fast frame synchronization, frequency offset estimation and channel acquisition for spontaneous transmission over unknown frequency-selective radio channels,” in Proc. PIMRC’93, 1993, pp. 229–233. [3] , “Improved frame synchronization for spontaneous packet transmission over frequency-selective radio channels,” in Proc. PIMRC’94, 1994, pp. 353–357. [4] J. A. Gansman, M. P. Fitz, and J. V. Krogmeier, “Optimum and suboptimum frame synchronization for pilot-symbol-assisted modulation,” IEEE Trans. Commun., vol. 45, pp. 1327–1337, Oct. 1997. [5] C. N. Georghiades, “Chapter 19: synchronization,” in The Communications Handbook, J. D. Gibsson, Ed. Boca Raton, FL: CRC, 2002. [6] Y. Koo and Y. H. Lee, “A joint maximum likelihood approach to frame synchronization in presence of frequency offset,” in Proc. ICC’02, vol. 3, 2002, pp. 1546–1550.
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[7] U. Lambrette, J. Horstmannshoff, and H. Meyr, “Techniques for frame synchronization on unknown frequency selective channels,” in Proc. VTC’97, vol. 2, 1997, pp. 1059–1063. [8] G. L. Liu and H. H. Tan, “Frame synchronization for Gaussian channels,” IEEE Trans. Commun., vol. COM-35, pp. 818–829, Aug. 1987. [9] J. L. Massey, “Optimum frame synchronization,” IEEE Trans. Commun, vol. COM-20, pp. 115–119, Apr. 1972. [10] R. Mehlan and H. Meyr, “Optimum frame synchronization for asynchronous packet transmission,” in Proc. ICC’93, vol. 2, 1993, pp. 826–830. [11] B. H. Moon and S. S. Soliman, “ML frame synchronization for the Gaussian channel with ISI,” in Proc. ICC’91, 1991, pp. 1698–1702. [12] M. Morelli and U. Mengali, “Carrier-frequency estimation for transmissions over selective channels,” IEEE Trans. Commun., vol. 48, pp. 1580–1589, Sept. 2000. [13] P. T. Nielsen, “Some optimum and suboptimum frame synchronizers for binary data in Gaussian noise,” IEEE Trans. Commun., vol. COM-21, pp. 770–772, June 1973. [14] A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. New York: McGraw-Hill, 1991. [15] P. Robertson, “Maximum likelihood frame synchronization for flat fading channels,” in Proc. ICC’92, 1992, pp. 1426–1430.
Yan Wang received the B.S. degree from Peking University, Beijing, China, in 1996, the M.Sc. degree from Beijing University of Posts and Telecommunications (BUPT) in 1999, and the Ph.D. degree from Texas A&M University, College Station, in December 2003. He is currently an Intern with Nokia, Dallas, TX. His research interests are in the area of signal processing for communications systems.
Kai Shi received the B.S. degree from Nanjing University, Nanjing, China, in 1998, and the M.Sc. degree from Southeast University, National Communications Research Laboratory, China, in 2001. He is now working toward the Ph.D. degree at Texas A&M University, College Station. Since January 2002, he has been a Research Assistant with the Department of Electrical Engineering, Wireless Communications Laboratory, Texas A&M University. His research interests are in the areas of synchronization and equalization of ultrawide-band systems, OFDM transmissions, and PRML channels, and design of turbo/LDPC codes.
Erchin Serpedin (S’96–M’99) received the Dipl. degree in electrical engineering (with highest distinction) from the Polytechnic Institute of Bucharest, Bucharest, Romania, in 1991, the specialization degree in signal processing and transmission of information from Ecole Superiéure D’Electricité, Paris, France, in 1992, the M.Sc. degree from Georgia Institute of Technology, Atlanta, in 1992, and the Ph.D. degree in electrical engineering from the University of Virginia, Charlottesville, in January 1999. From 1993 to 1995, he was an Instructor with the Polytechnic Institute of Bucharest and from January to June 1999, a Lecturer at the University of Virginia. In July 1999, he joined the Wireless Communications Laboratory, Texas A&M University, College Station, as an Assistant Professor. His research interests are the areas of statistical signal processing and wireless communications. Dr. Serpedin has received the National Science Foundation Career Award in 2001 and is currently an Associate Editor for the IEEE COMMUNICATIONS LETTERS and the IEEE SIGNAL PROCESSING LETTERS.
