Non-Data-Aided Symbol Rate Estimation of Linearly ... - IEEE Xplore

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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 2, FEBRUARY 2008

Non-Data-Aided Symbol Rate Estimation of Linearly Modulated Signals Carlos Mosquera, Member, IEEE, Sandro Scalise, Member, IEEE, and Roberto López-Valcarce, Member, IEEE

Abstract—The estimation of the symbol rate of a linearly modulated signal is addressed, with special focus on low signal-to-noise ratio (SNR) scenarios. This problem finds application in automatic modulation classification and signal monitoring. A maximum-likelihood (ML) approach is adopted to derive practical estimators, exploiting information on the cyclostationarity of the modulated signal as well as knowledge of the received signaling pulse shape. The structure of the ML estimator suggests a two-step estimation procedure, whereby an initial coarse search determines first a neighborhood from which a subsequent fine search yields the final symbol rate estimate. Links between the ML approach and previous results from the literature in symbol rate estimation are established as well. The proposed scheme is applicable even for small excess bandwidths, at the cost of a higher complexity with respect to simpler estimators known to fail under such conditions. Index Terms—Cyclostationarity, frequency estimation, maximum-likelihood (ML) estimation, non-data-aided, synchronization.

I. INTRODUCTION YMBOL rate estimation of a digital communication signal is an important task when performing passive signal analysis and automatic modulation classification. For example, symbol rates may differ among broadcasters for the same type of service, such as cable or satellite. In addition, signal sensing is becoming more essential in new applications needing knowledge (“cognition”) of the type of signals in the air [1]. In this paper, we present a maximum-likelihood (ML) approach to the problem of estimating the symbol rate of a linearly modulated signal. Due to its practical importance in the passive sensing scenario, we focus on the general case where all the synchronization parameters, including the mentioned baud rate, are unknown at the receiver. The information symbols, as well as the number of symbols in a frame, are also assumed unknown, so that the estimation process is totally blind. It will be shown how to exploit the structure of the received signal in order to anticipate the underlying symbol rate, without resorting to trial-and-error decoding for every possible rate.

S

Manuscript received December 29, 2006; revised July 12, 2007. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Zhi Tian. This work was partially supported by the EC-IST SatNEx project (IST-507052), SatNEx-II project (IST-27393), and the MEC project DIPSTICK (TEC2004-02551). C. Mosquera and R. López-Valcarce are with the Departamento de Teoría de la Señal y Comunicaciones, ETSE Telecomunicación, Universidad de Vigo, 36310 Vigo, Spain (e-mail: [email protected]; [email protected]). S. Scalise is with the DLR (German Aerospace Center), Institute for Communications and Navigation, 82230 Wessling, Germany (e-mail: [email protected]). Digital Object Identifier 10.1109/TSP.2007.907888

Blind rate estimation has been considered in the literature from different points of view. The use of a bank of filters has been proposed as an ad hoc approach, either matched to the different signaling pulse shapes [2], or unmatched [3]. Assuming a known number of symbols, [4] adopted an ML approach that also led to the use of a bank of matched filters. Alternatively, the fact that the received signal is cyclostationary with period , where is the symbol rate, can be exploited. A computationally simpler scheme not requiring the knowledge of the received pulse and avoiding the bank of front-end filters uses cyclic second-order statistics (SOS), with an asymptotic analysis of this scheme performed in [5]. Another important advantage of this estimate is its immunity to frequency offsets, unavoidable in practical scenarios due to oscillator mismatch and relative motion. On the other hand, performance of cyclic SOS-based methods degrades for small excess bandwidths. This is a consequence of the fact that, for the limiting case of a zero roll-off factor, the received signal becomes wide-sense stationary rather than cyclostationary. This fact led to the use of higher order statistics in [6], where it was proposed the use of a bank of samplers at the candidate symbol rates, followed by baud-spaced blind equalizers computed by minimizing some contrast at the equalizer output. The estimate is then given by the symbol rate providing the smallest value of the contrast. Although this approach is operative for small roll-off factors, proper operation requires that the signal-to-noise ratio (SNR) be sufficiently high. The use of powerful error-correcting codes in practice motivates the development of estimates applicable in low-SNR environments. In addition to lack of robustness with low excess bandwidths, cyclic SOS-based methods must address some practical issues. Essentially, these methods use the received signal to regenerate a spectral line that serves to identify the symbol rate or any other related parameter of interest [7]. The search for this spectral line entails quite a few practical problems, as its width collapses as the number of samples goes to infinity, and the “spectrum” in which this line is sought is plagued with spurious peaks. These effects make the search for the desired line a rather difficult task. We are interested in finding practical methods with good performance for low SNRs and/or low roll-off factors, being at the same time robust to frequency offsets. Our focus is on linearly modulated signals going through a frequency-flat channel. Our development will be guided by the ML criterion, which has proven to be a very useful tool in the rich field of synchronization and parameter estimation. The ML approach has served to derive practical synchronization schemes, and it has even been instrumental to give consistency to already known ad hoc methods. A general ML setting for low-SNR scenarios, which paved the way for future synchronization schemes, was introduced in the seminal paper [8]. Some evident links can be

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MOSQUERA et al.: NON-DATA-AIDED SYMBOL RATE ESTIMATION OF LINEARLY MODULATED SIGNALS

drawn between several current practical schemes and the results therein. This will be also the case with the developments contained in this paper. It will be shown how the ML approach for symbol rate estimation1 exploits the dependence of both the shaping pulse and its repetition rate with the symbol period , in terms of a cost function that can be neatly interpreted. Thus, several previous results from the literature will be integrated in a common formal framework, which allows to grasp some subtleties of the problem, unnoticed so far in the literature. The structure of the paper is as follows. The problem is formally stated in Section II. Section III presents an ML approach based on a low-SNR assumption. Practical aspects such as search granularity and frequency offsets will be discussed in Sections IV and V, respectively. The links between the proposed ML method and the cyclic SOS-based estimate analyzed in [5] will be also discussed in Section V. Numerical results are given in Section VI; finally, conclusions will be drawn in Section VII. Notation is as follows. Vectors and matrices are, respectively, denoted by lowercase and uppercase bold letters. Superscripts and denote the transpose and the conjugate transth element and the depose (Hermitian), respectively. The terminant of the matrix are denoted by and , respectively.

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with the normalized frequency offset, and i.i.d. Gaussian circularly complex noise samples with variance . The vector of the received samples can be written as (3) contains The vector the unknown information symbols, whereas the noise vector is Gaussian distributed with covariance . The number of symbols is considered as unknown.2 In fact, if we , then for sufficiently define the oversampling ratio long records the number of symbols satisfies . Throughout the paper, we will use the terms symbol period, symbol rate and oversampling ratio as equivalent notations for in (3) is paramthe same unknown. The matrix , and it is given by eterized by (4) Note that for sufficiently large, the columns of orthogonal, since

become

II. PROBLEM STATEMENT The model of a linearly modulated signal after passing through a channel of complex gain can be written as (1) corresponding to transmitted symbols assumed zeromean, independent and identically distributed (i.i.d.) with unit variance . The signaling pulse is a squareroot raised cosine (SRRC) pulse with roll-off factor and baud rate , such that its Fourier transform satisfies for . The process is circularly symmetric additive white Gaussian noise (AWGN) with power spectral density . and denote the carrier frequency and phase offsets, respectively, whereas is the timing offset. Our focus is on the general case for which all the synchronization parameters as well as the symbol sequence are unknown at the receiver. In addition, we do not presume any knowledge on the noise power or the channel gain. is sampled at a fixed rate , after being filThe signal tered by an analog low-pass filter with cutoff frequency . It is assumed that is small enough for all possible baud rates and frequency offsets, so that the oversampled signal is free of intersymbol interference and can be expressed as follows, for :

(5) is a raised cosine pulse with and baud rate , and therefore satisfies . With no loss of generality,3 it will be asfor all , so that sumed that for sufficiently large. The following property of the matrices will also be useful; the proof is given in Appendix I. and Property 1: Let such that where roll-off

(6) and the corresponding matrices . Then, for matrix columns, i.e., .

and sufficiently large, the has orthonormal

III. MAXIMUM-LIKELIHOOD FORMULATION Our goal is to obtain the ML estimator of the symbol period given the received samples in . Under the Gaussian noise assumption, the probability density function of conditioned to the unknown parameters is given by

(7) (2) 1The

term detection could also apply to the problem under study, as in many practical settings the number of choices for the symbol rate are finite. Thus, the context should dictate whether detection or estimation is the appropriate term.

2In [4] the number of symbols is fixed and known; we assume that this parameter is not available, as usually happens in practice. Note that if the receiver knew K , then a coarse estimate of T would be readily available as T N T =K . 3Note that any scaling of the transmitted pulse can be absorbed by the unknown channel gain h.

^=

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The likelihood function (7) is of extended use and has been widely studied for synchronization purposes [9]. Before maxwith regard to (or, equivalently, ), the reimizing maining unknown parameters must be taken into account. The unkown symbols are usually dealt with by resorting to with different assumptions, given that the expectation of regard to cannot be obtained in closed form except for very particular cases. A good overview is presented in [10]: lowSNR, high-SNR, and Gaussian symbols approximations (this last one is known as the Gaussian maximum-likelihood (GML) method) provide solutions applicable in different scenarios. Although in practice the symbols do not follow a Gaussian distribution, the GML method usually yields meaningful estimators; moreover, it converges to the low-SNR and high-SNR sogoes to infinity and zero, respeclutions as the noise power tively. The reason lies in the fact that for low SNR, the actual symbol distribution becomes irrelevant from an ML perspective, whereas for high SNR, the ML criterion leads to a decision-directed scheme whereby the symbols in are to be estimated. In between these two extreme cases, the GML method provides an estimator specifically tuned to the SNR operating point. We therefore apply the GML method to the problem at hand. Assuming that is Gaussian with zero mean and covariance , the expectation of (7) with respect to is given by

(8) After computing the integral (using the fact that ), taking the logarithm, and dropping irrelevant constants, the log-likelihood function (LLF) is found to be

(9) As expected, the LLF obtained with the GML method depends on the unknown . Note that the dependence of the LLF on is not only via the last term as a function of , but also through the unknown parameter . Hence, the GML estimator is not in general the result of maximizing with respect to unless the SNR is sufficiently high, as discussed next. A. High-SNR Case When , the LLF (9) does boil down to . This would also be the result of applying the conditional maximum-likelihood (CML) criterion [10], which maximizes the likelihood function (7) by using the ML estimate of the unknown symbols . However, the uncertainty on the symbol period introduces an important peculiarity: namely, that the same sequence of received samples can be synthesized with different symbol rates, by choosing appropriately the sequence of symbols. Let us illustrate this fact with an example. Consider a simple noiseless

Fig. 1. CML function kG ( )r k for a noiseless received signal with = 0:2 and N = 6, computed as detailed in (17). There is no frequency offset.

case with no frequency, phase or timing offsets, and a unity gain channel, so that the received analog signal is given by . Now let be another symbol in). Then, it is terval such that (6) is satisfied (with possible to sample at a rate without aliasing. Moreover, it is easily seen that the SRRC pulse can be used as interpolation filter to reconstruct from these samples, i.e., , where accounts for the proper scaling. In this way, the analog signal can be interpreted as being generated by the “information symbols” with symbol rate . Thus, the fact that different combinations of candidate information symbols and candidate symbol rates yield the same observation will result in a CML criterion with multiple maxima of equal height. Indeed, if (noiseless case), then for all such that (6) holds, Property 1 applies and . Fig. 1 shows an example (with ) illustrating this undesirable “plateau” effect in the CML criterion: although a sharp peak is seen at the true symbol interval , this peak level is also attained for all such that . As the roll-off factor decreases, this plateau moves closer to the desired peak, until for the prominent peak disappears. A possible way out of this ambiguity accounts for the non-Gaussianity of the information symbols: in order to minimize , substitute by the hard decisions instead of , where is the projection operator onto the constellation employed. The performance of this decision-directed approach, however, will degrade quickly as the SNR drops. In consequence, we turn our attention to low-SNR approximations of the LLF. B. Low-SNR Case As has been said, the motivation to focus on the low SNR regime is twofold: to avoid the ambiguities affecting the CML method in the high SNR case and to improve the performance of estimators known to fail in noisy environments such as the


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decision-directed scheme or those mentioned in the Introduction (Section I). Let us start by maximizing the LLF (9) with respect to the , we can approximate (9) as noise power. For

(10) Maximizing this with respect to noise variance estimate:

, we obtain the following

(11) which implicitly assumes that the only contribution to the observations is the noise. On the other hand, if the LLF (9) is max, one obtains imized with regard to

(12)

Fig. 2. Low-SNR approximation (13) of the LLF computed as detailed in (17) for a noiseless received signal with = 0:2 and N = 6. There is no frequency offset.

Substituting (11) and (12) in the LLF (9), and using the relation linking the total number of symbols and the oversampling ratio, one finally obtains

Note that the term in (13) computes the energy at the output of the matched filter sampled at the symbol rate

(14)

(13) It is interesting to note that this low-SNR approximation of the LLF turns out to be applicable for high SNR as well, as illustrated by Fig. 2. The same noiseless environment as that used for generating the CML cost shown in Fig. 1 was considered. The plateau effect observed in the CML criterion is not present in this case. This is readily checked if one assumes (noiseless case) and evaluates (13) at those satisfying (6): given that in view of Property 1 and that , in that region the LLF (13) be, which goes to zero comes as . C. Nuisance Synchronization Parameters At this point, the effect of the unknown symbols has been averaged out of the likelihood function, whereas the channel gain and noise power have been handled as deterministic unknowns (and consequently estimated). However, the low-SNR LLF (13) still depends on the unknown synchronization parameters contained in . In our context, the only parameter of interest is the symbol rate, whereas the remaining variables are considered as nuisance parameters. The phase does not play any role in (13), but we still have to deal with the normalized frequency and timing offsets, and , before we can obtain the estimate of the symbol period .

where denotes the output of a receive filter matched to the pulse corresponding to the symbol period , including frequency offset correction

(15) In (14), samples are taken at the rate , thus spanning the whole observation time window. This is in contrast with the ML approach in [4], where the number of information symbols in the observation time window was assumed to be known a priori, resulting in an LLF that only included the first term in (13). In addition, no frequency offset was considered in [4] and the timing offset was assumed to follow a uniform distribution. The timing offset can be extracted for a given symbol period using the received signal second-order cyclostationarity at the output of the corresponding matched filter [9]. ML estimation of is detailed in Appendix II, and turns out to yield the so-called Oerder and Meyr estimator [11] associated to the candidate oversampling ratio (16)

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Substituting (16) in obtained (see Appendix II):

, the following expression is

(17) (18) where the vector given by

and the diagonal matrix

are, respectively,

(19) (20) The first term in (17) and (18) measures the energy at the output of the filter matched to the corresponding signaling pulse. The second term measures the spectral correlation, with the goal of exploiting the cyclostationarity of the information signal; it can be regarded as a matched filter followed by a square-law nonlinearity and a bandpass filter at the cyclic frequency rad, thus resembling well-known results from classical synchronization theory [8]. For roll-off factors close to zero, this spectral correlation term will be very small, and the only useful second-order information is found in the shape of the received pulse. Any frequency mismatch in (1) which is not correspondingly compensated before matched filtering will degrade the performance of the ML symbol rate estimator. If some kind of a priori knowledge is available about the distribution of the frequency error, it can be exploited by considering as a nuisance random variable that could be averaged out. This Bayesian approach turns out to be useful in some settings (see, e.g., [12]) but cannot be adopted here due to the structure of the LLF under consideration. Alternatively, the normalized frequency offset can be considered as an unknown deterministic parameter. Unfortunately, attempting to maximize the LLF (13) with respect to does not yield a closed-form estimate. Some hints as to how to handle this issue will be provided in Section V. IV. PROPOSED ESTIMATOR

^ ) from (22) and its corresponding coarse apFig. 3. Complete LLF `(r jT; proximation (24), for a realization of N =2000 samples. SNR = 0 dB, = 0:25. The maximum of the LLF is at the true value N = 5.

The vector is composed by the samples at the output of the filter matched to the pulse corresponding to the symbol period , after frequency correction with [see (15) and (19)] (23) in an illustraFig. 3 plots the log-likelihood function tive case without frequency offset. The spectral correlation term is responsible for the narrow peak at the true parameter value as well as for all the spurious peaks. However, as the number of samples goes to infinity, the width of the desired “spectral line” goes to zero. This poses a problem in practical settings and makes it necessary to have some cautions, as anticipated in [5]. On the other hand, the smooth curve is the result obtained when this spectral correlation term is left out in the computation of (18). In view of these considerations, we propose a two-step strategy for the estimation of the symbol period: 1) an initial search using only the energy at the output of the matched filters, that is, seeking the maximum of (24)

Assume that a suitable frequency offset estimate is available. The proposed estimator is summarized next. After inserting (18) in (13) and leaving out, one has (21) with

(22)

is a smooth function of , the number of as matched filters to use in this initial search need not be too large; 2) a refined search around the coarse estimate of the previous stage; if the possible symbol rates are known, a simple evaluation of the LLF in (22) at each of them within a predetermined range around the coarse estimate will suffice. A two-step approach was also suggested in [5], therein referred to as coarse-search and fine-search, to handle the problems associated to the location of the global maximum of a specified cost function, namely the cyclic-correlation cost


that will be discussed in Section V. The approach derived from the ML criterion, however, is different in that a smoothed approximate LLF [given by (24)] is used in the coarse search, rather than the complete LLF (22), which is used in the finesearch step only. Notably, if the cyclostationary content of the signal is of relevance (that is, if the roll-off factor of the signaling pulse is not too low), it turns out that there is no significant difference in the results obtained at the fine-search step when the “pruned” LLF (25) is maximized instead of the complete LLF from (22). This will be illustrated in the simulation results of Section VI. in order to obtain the fine estimate of Maximizing the symbol rate is in correspondence with the ML estimation of the timing offset (16): the ML estimate of the symbol rate is the one providing a matched filter output whose squared magnitude rad, has the largest spectral content at the frequency whereas the phase of that content yields the ML estimate of the timing offset. The next section addresses the issue of frequency offset estimation and draws some links between the ML symbol rate estimate and the cyclic-correlation-based (CCB) estimator [5].

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is especially well suited to scenarios where multipath or frequency-selective fading may distort the pulse shape. On the other hand, its main drawback is the degradation in performance exhibited for low roll-off applications: for baseband bandwidths Hz, the peak of at the true value of the close to oversampling ratio decreases, becoming zero when the excess bandwidth is null. Note that both the CCB cost (28) and the “pruned” LLF (25) provide a measure of the spectral correlation content at the rad, but before and after the matched cyclic frequency filter, respectively. Moreover, it is possible to establish an insightful connection between both estimators, showing that if any potential uncertainties in the received pulse shape and the frequency offset are properly addressed, then the ML estimator maximizing (25) will reduce to the CCB estimator. The key to this link is the so-called cyclic correlation [13] of the received , at time lag and for cycle , denoted by signal and defined as

(29) The natural estimate of is given by

from the finite sample record

V. FREQUENCY-OFFSET ESTIMATION The last step needed in order for the proposed ML estimate to be usable is frequency offset correction. For this purpose, the family of methods presented in [13] is especially appealing, since they work with the oversampled signal and can be tailored to different needs. For example, one may consider the estimate (26) which is computationally simple and, as the numerical simulations in Section VI will confirm, results in good performance of our symbol rate estimate. The frequency-offset estimate (26) is analyzed in detail in [9]. It is worth mentioning that symbol rate estimates robust to frequency offsets are available from the literature. In particular, the CCB estimator is a well-known scheme exploiting signal cyclostationarity, which was thoroughly analyzed in [5]. The CCB estimate of the oversampling ratio is given by (27) (28) included in The number of correlation terms has an important influence on the variance of the estimate, as shown in [5]; nevertheless, increasing beyond a certain value (which depends on the particular scenario) does not provide any significant gain. It is easily seen that is independent of the received signal frequency offset, which is clearly a desirable property. Moreover, the CCB estimate does not make any assumptions about the particular shape of the received pulse, and thus it

(30) (for the summation limits in (30) must be for suitably modified). This estimate is asymptotically unbiased and mean-square-sense consistent [13]. It can be used as a starting point in the derivation of timing and frequency offsets, since after substituting in (29) the expression of the received signal (2) one finds that

(31) It is shown in Appendix III how the cyclic correlation estimate (30) can be used in the computation of the “pruned” LLF to implicitly account for lack of knowledge about the frequency offset and the pulse shape, resulting in the cost yielding the CCB estimator. Hence, two possibilities exist in order to account for a frequency offset when using the proposed ML estimate: explicit estimation of using, e.g., (26), or implicit estimation using the cyclic correlation (30), in which case the ML estimate reduces to the CCB estimate. This latter estimate does not exploit the pulse shape, which is known in frequency flat channels, and it is outperformed by the ML method, as the results in Section VI will confirm. A detailed comparison of both methods in frequency-selective channels would depend on the particular type of channel and is out of the scope of this paper. In this regard, let us say that symbol rate estimation in the presence of an unknown frequency-selective channel has been addressed in [6], based on a bank of samplers at the candidate symbol rates followed by blind

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TABLE I MEAN ESTIMATED VALUE OF THE COARSE ESTIMATE AFTER 10 REALIZATIONS. N = 5; = 0:25

baud-spaced equalizers minimizing some suitable contrast; the symbol rate estimate is then taken as that providing the smallest value of this contrast at its equalizer’s output. This approach, however, suffers from an important degradation for low SNR, especially considering that locating the minimum requires short signal records due to the fact that as the number of observations increases, the width of this minimum tends to zero. On the other hand, this width does not collapse for excess bandwidths going to zero, in contrast with the CCB method [5]. VI. NUMERICAL RESULTS In the simulations conducted, the estimators were tested for quadrature phase-shift-keying (QPSK) signals with SNR 5 to 5 dB. The true value of the oversamranging from , and the sampling period was pling ratio was set at normalized as , so that all the numerical results apply indistinguishable to the symbol period and the oversampling and realizations ratio. For each set of parameters, were used in the averaging for the coarse and the fine search, respectively. Fine searches were conducted on a fine grid of oversampling ratios in the range . Except where noted, the frequency offset was randomly and uniformly chosen in the interval . First, we assessed the performance of the coarse estimate resulting from the maximization of in (24). It turns out that this coarse estimate presents a bias, which depends on the SNR, the number of samples , the roll-off factor, and the true oversampling ratio. For example, Table I shows the mean value of this coarse estimate; note how the bias reduces for higher SNR and longer records. Bias analysis is difficult due to the logarithm in the last term of (24). Fig. 4 shows the coarse search performance, measured as the probability of making an initial estimate within 10% of the true oversampling ratio. For a given setting, once established the desired probability and the tolerance of this initial acquisition, the number of necessary samples can be determined provided that curves as those shown in Fig. 4 are known for the entire range of possible symbol rates. Next, we compared the performance obtained by using the complete LLF from (22) and the “pruned” LLF from (25) in the fine-search step. Fig. 5 shows the in both cases, for Mean Square Error (MSE) different roll-off factors. No frequency offset was present in this example. It is seen that both schemes behave similarly except with small roll-off values, in which case the complete LLF clearly outperforms the “pruned” LLF. In addition, for , we must highlight the following two facts. • The “pruned” LLF performs even better than the complete LLF for some SNR values. The reason for this is that the maximum of is biased away from

Fig. 4. Coarse search results. The probability of acquiring the correct symbol rate within an error margin of 10% is plotted against the SNR and for different sample sizes. The true oversampling ratio is N = 5, and the roll-off factor = 0:25 is assumed to be known.

Fig. 5. Fine search performance of the Maximum Likelihood estimator (22) and its simplified version (25). There is no frequency offset. N = 3000.

the true value of the symbol interval, due to the previously mentioned bias in the coarse LLF from (24). • The slope of the MSE increases sharply for SNR below 2 dB. This so-called “outliers effect” [14] is also found in harmonic retrieval methods in which a narrow spectral line must be sought: for low SNR and/or short signal records, the variability within the search range of the location of


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channel model differs from that considered in the analysis. Thus, Fig. 6(a) shows the performance of the ML estimator using a SRRC roll-off factor of 0.2, when the received signaling pulse corresponds to a SRRC pulse with roll-off factor of 0.35. This roll-off mismatch could be the case, for example, when estimating the symbol rate in a satellite broadcasting scenario under the emerging standard DVB-S2 [16]. In addition to a number of possible symbol rates, the SRRC pulse of DVB-S2 signals can have any of three roll-off factors: . Even under this roll-off mismatch, the performance of the ML estimator is better than that of the CCB method. Optimality is also lost under channel linear distortions, since the SRRC pulse is no longer matched to the received pulse. Fig. 6(b) depicts the degradation stemming from this fact for a single-echo channel of the form . The performance of both ML and CCB degrades, in the CCB case due exclusively to the lower cyclostationarity content of the received signal. As a general remark, we can say that such cyclostationary content will determine the performance of the symbol rate estimation. This statement applies also to frequency-selective channels, for which the received distorted pulse must be considered. Although suboptimum, the proposed ML method can be still applied under unknown channel distortion: the filtering operation involved in (25) will help to fight noise and guarantee a minimum performance even for low excess bandwidth.4 Finally, let us say that pulses with higher roll-off factors than those shown, although less common in practice, would improve the performance of both methods. VII. CONCLUSION

Fig. 6. Fine search MSE results versus SNR: performance of the ML estimate (25) and the CCB estimate (27). For low SNR all curves converge to an MSE value corresponding to a uniform distribution in the search range [4.5,5.5]. (a) = 0:35; N = 3000. = 0:2 for the mismatched case. (b) = 0:2; N = 3000. The time response of the channel with echo is given by (t) + 0:25 (t T ).

0

the spectral peak increases, and eventually this location becomes a uniform random variable. Thus, the MSE apfor low SNR in all the plotted curves. proaches The “outliers effect” is difficult to analyze and renders the family of classical Cramér–Rao bounds of little value for the range of relevance of the outliers. Some results about the computation of the outliers’ probability as well as some alternative bounds for the MSE can be found in [14] and [15] for some specific harmonic retrieval problems. In order to assess the fine search performance, we have evaluated both the ML estimate (25) and the CCB estimate (27). For the latter, the parameter in (27) yielding best performance was chosen at each case. As shown in Fig. 6 for two different cases, the new proposed estimator outperforms the CCB estimator for a given number of processed samples, at the cost of a higher complexity. These figures depict also the degradation when the

A maximum-likelihood approach to the symbol rate estimation problem has been addressed. With low-SNR scenarios in mind, new results have been obtained which clearly show the different kinds of second-order information that can be exploited for symbol rate estimation, namely, the shape of the received pulse together with the cyclostationarity of the information signal. As a result, a practical two-step strategy has been devised, which avoids the drawbacks related to spurious peaks akin to those found in harmonic retrieval problems, by means of a first coarse search. The expression of the fine estimate handles the same function as the Oerder and Meyr estimator, the well-known timing offset ML estimator for low SNR. An initial precorrection for the frequency error makes this method suitable to practical scenarios. The proposed ML estimator has been compared with a well-known cyclic-correlation-based suboptimal estimator, whose main virtues are its simplicity and immunity to frequency offsets. It turns out that this suboptimal estimator can be derived from the ML estimator by a certain means of jointly dealing with received pulse and frequency offset uncertainties. Performance has been shown to improve notably for low SNRs and low roll-off factors, which makes the new method especially well suited for square-root raised cosine signals employing powerful error-correcting codes. The case of frequency-selective channels can be considered as an interesting line of research to pursue: both the proposed and the CCB estimators are suboptimum, although for the low-SNR and low-cyclostationary content of the received signal, the 4On the other side, the coarse search will suffer a more severe degradation, since its good performance is strictly due to the knowledge of the pulse shape for the different symbol rates.

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ML-based fine search is expected to perform better provided that its extra complexity can be afforded.

APPENDIX I PROOF OF PROPERTY 1 The

element of

is given by

. This is also the case in conventional synchronization for a known and fixed symbol period. We will follow a similar reasoning to that employed in [9] or [12]. The details of the , which will allow to estimate the interpretation of optimum timing offset for a given symbol period, are included next, giving their relevance in the subsequent derivation of the symbol period estimator. It can be easily seen that the diagonals of the matrix show a periodic behavior related to the cyclostationarity of the linearly modulated signal. For large , the element of this matrix is given by

(32) and . Recall that if two signals are band-limited to Hz, then their crosscorrelation can be written in terms of the sampled signals as where

(33) for any

(37) The summation in (37) is lowing Fourier series [12]:

-periodic in

and admits the fol-

. Applying this fact to (32), one finds that

(38)

(34) has Fourier trans. Now if (6) holds, then since is constant in the support of . The scaling can be determined imposed from the fact that the normalization on implies that , . Therefore, since , and where form

where

(39) We have not included the terms in the Fourier series expansion as in the case (38), which null out for a band-limited pulse under consideration. , one Using expression (38) for the elements of obtains that, for large

(35) large enough, the Using (35), and assuming both element of is seen to be given by

(40) (36) where the last equality follows from (33), since (6) implies that is band-limited to Hz. Therefore, as was to be proved.

Next, we show the algebraic manipulations which lead to a likelihood function easier to interpret. Using (39), the term in (40) corresponding to can be written as

APPENDIX II TIMING OFFSET ESTIMATION If the timing offset is regarded as a deterministic unknown in (13), it can be extracted after maximizing with regard to . Setting , this translates into ; that is, ML estimation of the timing with regard to offset amounts to maximizing

(41)

(42)


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where denotes the output of the matched filter corresponding to the signaling pulse and normalized frequency offset , as given by (15). The remaining terms in (40) are given by

(47) where the summation limits have been ommited for clarity; for to . large , all summations can be assumed to run from Now, note from (31) that

(48) (43)

This allows to write (47) in terms of the cyclic correlation :

where the first equality can be readily checked noting that is the conjugate of the correand exchanged and that sponding expression with . For the last equality, we have used (39). Using (42) and (43), one has from (40) that

(49) Independence of with respect to the received shaping pulse and the frequency offset can be achieved if is subfrom (30), so that stituted in (49) by its estimate

(44) The timing offset can be extracted in explicit form, since the above expression can be readily maximized with regard to . This estimator turns out to be the Oerder and Meyr estimator [11]:

(50) which after being substituted in (49) leads to

(45) As mentioned in [12], this timing offset estimate is optimal for baseband transmission or perfectly known frequency offset . (51) APPENDIX III RELATION BETWEEN THE ML AND CCB ESTIMATORS from (25) for the fine estimaThe “pruned” LLF tion of the symbol period or, equivalently, the oversampling ratio , can be written as follows, after discarding irrelevant constants:

The CCB function in (28) is an approximation to (51) obtained by including only the terms running from to . The appropriate value for the summation limit is usually some tens as detailed in [5]. REFERENCES

(46)

[1] S. Haykin, “Cognitive radio: Brain-empowered wireless communications,” IEEE J. Sel. Areas Commun., vol. 23, no. 2, pp. 201–220, Feb. 2005. [2] J. Y. Lee, Y. M. Chung, and S. U. Lee, “On a timing recovery technique for a variable symbol rate signal,” in Proc. IEEE Vehicular Technology Conf., Phoenix, AZ, May 1997, pp. 1724–1728. [3] Z. Yu, Y. Q. Shi, and W. Su, “Symbol-rate estimation based on filter bank,” in Proc. IEEE Int. Symp. Circuits Syst., Kobe, Japan, May 2005, pp. 1437–1440. [4] H. Wymeersch and M. Moeneclaey, “Blind symbol rate detection for low-complexity multi-rate receivers,” in Proc. Vehicular Technology Conf. (VTC), Stockholm, Sweden, May 2005, pp. 1171–1175.

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[5] P. Ciblat, P. Loubaton, E. Serpedin, and G. B. Giannakis, “Asymptotic analysis of blind cyclic correlation-based symbol-rate estimators,” IEEE Trans. Inf. Theory, vol. 48, no. 7, pp. 1922–1934, Jul. 2002. [6] S. Houcke, A. Chevreuil, and P. Loubaton, “Blind equalization-case of an unknown symbol period,” IEEE Trans. Signal Process., vol. 51, no. 3, pp. 781–793, Mar. 2003. [7] A. Dandawate and G. B. Giannakis, “Statistical tests for presence of cyclostationarity,” IEEE Trans. Signal Process., vol. 42, no. 9, pp. 2355–2369, Sep. 1994. [8] W. A. Gardner, “The role of spectral correlation in design and performance analysis of synchronizers,” IEEE Trans. Commun., vol. COM-34, no. 11, pp. 1089–1095, Nov. 1986. [9] H. Meyr, M. Moeneclaey, and S. A. Fechtel, Digital Communication Receivers: Synchronization, Channel Estimation and Signal Processing. New York: Wiley, 1998. [10] J. Villares and G. Vazquez, “Second-order parameter estimation,” IEEE Trans. Signal Process., vol. 53, no. 7, pp. 2408–2420, Jul. 2005. [11] M. Oerder and H. Meyr, “Digital filter and square timing recovery,” IEEE Trans. Commun., vol. 36, no. 5, pp. 605–612, May 1988. [12] J. A. López-Salcedo and G. Vazquez, “Asymptotic equivalence between the unconditional maximum likelihood and the square-law nonlinearity symbol timing estimation,” IEEE Trans. Signal Process., vol. 54, no. 1, pp. 244–257, Jan. 2006. [13] F. Gini and G. B. Giannakis, “Frequency offset and symbol timing recovery in flat-fading channels: A cyclostationary approach,” IEEE Trans. Commun., vol. 46, no. 3, pp. 400–411, Mar. 1998. [14] P. Ciblat and M. Ghogho, “Blind NLLS carrier-frequency offset estimation for QAM, PSK and PAM modulations: Performance at low SNR,” IEEE Trans. Commun., vol. 54, no. 10, pp. 1725–1730, Oct. 2006. [15] P. Ciblat, M. Ghogho, P. Forster, and P. Larzabal, “Harmonic retrieval in the presence of non-circular Gaussian multiplicative noise: Performance bounds,” Signal Process., vol. 85, pp. 737–749, 2005. [16] Digital Video Broadcasting (DVB); Second Generation Framing Structure, Channel Coding and Model Systems for Broadcasting, Interactive Services, News Gathering and Other Broadband Satellite Applications, ETSI EN 302 307 VI. 1.2, 2006. Carlos Mosquera (S’93–M’98) was born in Vigo, Spain, in 1969. He received the undergraduate degree in telecommunication engineering from the Universidad de Vigo, Vigo, Spain, and subsequently the M.S. degree from Stanford University, Stanford, CA, in 1994 and the Ph.D. degree from the Universidad de Vigo in 1998. In 1999, he spent six months with the European Space Agency at ESTEC, The Netherlands. He is currently an Associate Professor at the Universidad de Vigo, where his research and teaching focus on the area of signal processing applied to communications. Dr. Mosquera participates in the European Satellite Communications Network of Excellence and has served as member of several international technical program commmittees.

Sandro Scalise (S’00–M’06) was born in Utrecht, The Netherlands, in April 1973. He received the Electrical Engineering degree (in telecommunications) with honors from the University of Ferrara, Italy, in 1999 and the Ph.D. degree from the University of Vigo, Vigo, Spain, in 2007. Since 2001, he has been with the Institute for Communications and Navigation, DLR (German Aerospace Center), Germany. Since October 2004, he has been leading the Mobile Satellite Systems Group. His research activity deals with forward error correction and synchronization schemes for mobile satellite applications, land mobile satellite channel modeling, and link performance evaluation. He was the main contributor and editor of the chapter “Satellite Channel Impairments” within the book Digital Satellite Communications (New York: Springer, 2007). Dr. Scalise has been Co-Chairman of the Advanced Satellite Mobile System Conference and Chairman of the R&D Working Group of ISI (Integral SatCom Initative) European Technology Platform.

Roberto López-Valcarce (M’01) received the Telecommunications Engineer degree from the Universidad de Vigo, Vigo, Spain, in 1995 and the M.S. and Ph.D. degrees in electrical engineering from the University of Iowa, Iowa City, in 1998 and 2000, respectively. During 1996, he was with Intelsis, Santiago, Spain. Since 2001, he has been with the Signal Theory and Communications Department at the Universidad de Vigo, where he is currently an Associate Professor. His research interests lie in the area of signal processing applied to communications. Dr. López-Valcarce was the recipient of a 2005 Best Paper Award of the IEEE Signal Processing Society.