Non-Data-Aided Symbol Rate Estimation of Linearly Modulated Signals

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

Non-Data-Aided Symbol Rate Estimation ofLinearly 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 modu-lated signal is addressed, with special focus on low signal-to-noiseratio (SNR) scenarios. This problem finds application in automaticmodulation classification and signal monitoring. A maximum-like-lihood (ML) approach is adopted to derive practical estimators,exploiting information on the cyclostationarity of the modulatedsignal as well as knowledge of the received signaling pulse shape.The structure of the ML estimator suggests a two-step estimationprocedure, whereby an initial coarse search determines first aneighborhood from which a subsequent fine search yields thefinal symbol rate estimate. Links between the ML approach andprevious results from the literature in symbol rate estimation areestablished as well. The proposed scheme is applicable even forsmall excess bandwidths, at the cost of a higher complexity withrespect to simpler estimators known to fail under such conditions.

Index Terms—Cyclostationarity, frequency estimation, max-imum-likelihood (ML) estimation, non-data-aided, synchroniza-tion.

I. INTRODUCTION

SYMBOL rate estimation of a digital communication signalis an important task when performing passive signal anal-

ysis and automatic modulation classification. For example,symbol rates may differ among broadcasters for the sametype of service, such as cable or satellite. In addition, signalsensing is becoming more essential in new applications needingknowledge (“cognition”) of the type of signals in the air [1].

In this paper, we present a maximum-likelihood (ML) ap-proach to the problem of estimating the symbol rate of a lin-early modulated signal. Due to its practical importance in thepassive sensing scenario, we focus on the general case where allthe synchronization parameters, including the mentioned baudrate, are unknown at the receiver. The information symbols, aswell as the number of symbols in a frame, are also assumedunknown, so that the estimation process is totally blind. It willbe shown how to exploit the structure of the received signal inorder to anticipate the underlying symbol rate, without resortingto trial-and-error decoding for every possible rate.

Manuscript received December 29, 2006; revised July 12, 2007. The asso-ciate editor coordinating the review of this manuscript and approving it for pub-lication was Dr. Zhi Tian. This work was partially supported by the EC-ISTSatNEx project (IST-507052), SatNEx-II project (IST-27393), and the MECproject DIPSTICK (TEC2004-02551).

C. Mosquera and R. López-Valcarce are with the Departamento deTeoría de la Señal y Comunicaciones, ETSE Telecomunicación, Univer-sidad de Vigo, 36310 Vigo, Spain (e-mail: [email protected];[email protected]).

S. Scalise is with the DLR (German Aerospace Center), Institute forCommunications 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 literaturefrom different points of view. The use of a bank of filters hasbeen proposed as an ad hoc approach, either matched to the dif-ferent signaling pulse shapes [2], or unmatched [3]. Assuminga known number of symbols, [4] adopted an ML approach thatalso 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 compu-tationally simpler scheme not requiring the knowledge of thereceived pulse and avoiding the bank of front-end filters usescyclic second-order statistics (SOS), with an asymptotic anal-ysis of this scheme performed in [5]. Another important ad-vantage of this estimate is its immunity to frequency offsets,unavoidable in practical scenarios due to oscillator mismatchand relative motion. On the other hand, performance of cyclicSOS-based methods degrades for small excess bandwidths. Thisis a consequence of the fact that, for the limiting case of azero roll-off factor, the received signal becomes wide-sense sta-tionary rather than cyclostationary. This fact led to the use ofhigher order statistics in [6], where it was proposed the use ofa bank of samplers at the candidate symbol rates, followed bybaud-spaced blind equalizers computed by minimizing somecontrast at the equalizer output. The estimate is then given bythe 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 codesin practice motivates the development of estimates applicable inlow-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 regeneratea spectral line that serves to identify the symbol rate or any otherrelated parameter of interest [7]. The search for this spectral lineentails quite a few practical problems, as its width collapses asthe number of samples goes to infinity, and the “spectrum” inwhich this line is sought is plagued with spurious peaks. Theseeffects make the search for the desired line a rather difficult task.

We are interested in finding practical methods with good per-formance for low SNRs and/or low roll-off factors, being atthe same time robust to frequency offsets. Our focus is on lin-early modulated signals going through a frequency-flat channel.Our development will be guided by the ML criterion, whichhas proven to be a very useful tool in the rich field of syn-chronization and parameter estimation. The ML approach hasserved to derive practical synchronization schemes, and it haseven been instrumental to give consistency to already knownad hoc methods. A general ML setting for low-SNR scenarios,which paved the way for future synchronization schemes, wasintroduced 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 665

drawn between several current practical schemes and the resultstherein. This will be also the case with the developments con-tained in this paper. It will be shown how the ML approachfor symbol rate estimation1 exploits the dependence of both theshaping 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 ina common formal framework, which allows to grasp some sub-tleties of the problem, unnoticed so far in the literature.

The structure of the paper is as follows. The problem is for-mally stated in Section II. Section III presents an ML approachbased on a low-SNR assumption. Practical aspects such assearch granularity and frequency offsets will be discussed inSections IV and V, respectively. The links between the pro-posed ML method and the cyclic SOS-based estimate analyzedin [5] will be also discussed in Section V. Numerical resultsare given in Section VI; finally, conclusions will be drawn inSection 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 trans-pose (Hermitian), respectively. The th element and the de-terminant of the matrix are denoted by and , re-spectively.

II. PROBLEM STATEMENT

The model of a linearly modulated signal after passingthrough a channel of complex gain can be written as

(1)

corresponding to transmitted symbols assumed zero-mean, independent and identically distributed (i.i.d.) with unitvariance . The signaling pulse is a square-root raised cosine (SRRC) pulse with roll-off factor

and baud rate , such that its Fourier transform satisfiesfor . The process is cir-

cularly symmetric additive white Gaussian noise (AWGN) withpower spectral density . and denote the carrier frequencyand phase offsets, respectively, whereas is the timing offset.Our focus is on the general case for which all the synchro-nization parameters as well as the symbol sequence

are unknown at the receiver. In addition, we do not pre-sume any knowledge on the noise power or the channel gain.The signal is sampled at a fixed rate , after being fil-tered by an analog low-pass filter with cutoff frequency .It is assumed that is small enough for all possible baud ratesand frequency offsets, so that the oversampled signal is free ofintersymbol interference and can be expressed as follows, for

:

(2)

1The term detection could also apply to the problem under study, as in manypractical settings the number of choices for the symbol rate are finite. Thus, thecontext should dictate whether detection or estimation is the appropriate term.

with the normalized frequency offset, andi.i.d. Gaussian circularly complex noise samples with variance

. The vector of the received samplescan be written as

(3)

The vector containsthe unknown information symbols, whereas the noise vector

is Gaussian distributed with covariance . Thenumber of symbols is considered as unknown.2 In fact, if wedefine the oversampling ratio , then for sufficientlylong records the number of symbols satisfies .Throughout the paper, we will use the terms symbol period,symbol rate and oversampling ratio as equivalent notations forthe same unknown. The matrix in (3) is param-eterized by , and it is given by

(4)

Note that for sufficiently large, the columns of becomeorthogonal, since

(5)

where is a raised cosine pulse withroll-off and baud rate , and therefore satisfies

. With no loss of generality,3 it will be as-sumed that for all , so thatfor sufficiently large. The following property of the matrices

will also be useful; the proof is given in Appendix I.Property 1: Let and

such that

(6)

and the corresponding matrices and. Then, for sufficiently large, the

matrix has orthonormalcolumns, i.e., .

III. MAXIMUM-LIKELIHOOD FORMULATION

Our goal is to obtain the ML estimator of the symbol periodgiven the received samples in . Under the Gaussian noise

assumption, the probability density function of conditioned tothe unknown parameters is given by

(7)

2In [4] the number of symbols is fixed and known; we assume that this param-eter is not available, as usually happens in practice. Note that if the receiver knewK , then a coarse estimate of T would be readily available as T̂ = NT =K .

3Note that any scaling of the transmitted pulse can be absorbed by the un-known channel gain h.

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

The likelihood function (7) is of extended use and has beenwidely studied for synchronization purposes [9]. Before max-imizing with regard to (or, equivalently, ), the re-maining unknown parameters must be taken into account.

The unkown symbols are usually dealt with by resorting todifferent assumptions, given that the expectation of withregard to cannot be obtained in closed form except for veryparticular cases. A good overview is presented in [10]: low-SNR, high-SNR, and Gaussian symbols approximations (thislast one is known as the Gaussian maximum-likelihood (GML)method) provide solutions applicable in different scenarios. Al-though in practice the symbols do not follow a Gaussian dis-tribution, the GML method usually yields meaningful estima-tors; moreover, it converges to the low-SNR and high-SNR so-lutions as the noise power goes to infinity and zero, respec-tively. The reason lies in the fact that for low SNR, the actualsymbol distribution becomes irrelevant from an ML perspective,whereas for high SNR, the ML criterion leads to a decision-di-rected scheme whereby the symbols in are to be estimated. Inbetween these two extreme cases, the GML method provides anestimator 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 dependson the unknown . Note that the dependence ofthe LLF on is not only via the last term as a function of , butalso through the unknown parameter . Hence, the GML es-timator is not in general the result of maximizingwith respect to unless the SNR is sufficiently high, as dis-cussed next.

A. High-SNR Case

When , the LLF (9) does boil down to .This would also be the result of applying the conditional max-imum-likelihood (CML) criterion [10], which maximizes thelikelihood function (7) by using the ML estimate of the unknownsymbols .

However, the uncertainty on the symbol period introducesan important peculiarity: namely, that the same sequence of re-ceived samples can be synthesized with different symbol rates,by choosing appropriately the sequence of symbols. Let us il-lustrate this fact with an example. Consider a simple noiseless

Fig. 1. CML function kGGG ( )rrrk 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 unitygain channel, so that the received analog signal is given by

. Now let be another symbol in-terval such that (6) is satisfied (with ). Then, it ispossible to sample at a rate without aliasing. More-over, it is easily seen that the SRRC pulse can be usedas interpolation filter to reconstruct from these samples,i.e., , where ac-counts for the proper scaling. In this way, the analog signal canbe interpreted as being generated by the “information symbols”

with symbol rate .Thus, the fact that different combinations of candidate infor-

mation symbols and candidate symbol rates yield the same ob-servation will result in a CML criterion with mul-tiple maxima of equal height. Indeed, if (noise-less case), then for all such that (6) holds, Property 1 appliesand . Fig. 1 shows anexample (with ) illustrating this undesirable “plateau”effect in the CML criterion: although a sharp peak is seen at thetrue symbol interval , this peak level is also attained forall such that . As the roll-off factor decreases,this plateau moves closer to the desired peak, until forthe prominent peak disappears.

A possible way out of this ambiguity accounts for thenon-Gaussianity of the information symbols: in order tominimize , substitute by the hard decisions

instead of , where isthe projection operator onto the constellation employed. Theperformance of this decision-directed approach, however, willdegrade quickly as the SNR drops. In consequence, we turn ourattention to low-SNR approximations of the LLF.

B. Low-SNR Case

As has been said, the motivation to focus on the low SNRregime is twofold: to avoid the ambiguities affecting the CMLmethod in the high SNR case and to improve the performanceof estimators known to fail in noisy environments such as the

MOSQUERA et al.: NON-DATA-AIDED SYMBOL RATE ESTIMATION OF LINEARLY MODULATED SIGNALS 667

decision-directed scheme or those mentioned in the Introduction(Section I).

Let us start by maximizing the LLF (9) with respect to thenoise power. For , we can approximate (9) as

(10)

Maximizing this with respect to , we obtain the followingnoise variance estimate:

(11)

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

(12)

Substituting (11) and (12) in the LLF (9), and using the relationlinking the total number of symbols and the over-

sampling ratio, one finally obtains

(13)

It is interesting to note that this low-SNR approximation ofthe LLF turns out to be applicable for high SNR as well, asillustrated by Fig. 2. The same noiseless environment as thatused for generating the CML cost shown in Fig. 1 was con-sidered. The plateau effect observed in the CML criterion isnot present in this case. This is readily checked if one assumes

(noiseless case) and evaluates (13) at thosesatisfying (6): given that in view of Prop-erty 1 and that , in that region the LLF (13) be-comes , which goes to zeroas .

C. Nuisance Synchronization Parameters

At this point, the effect of the unknown symbols has beenaveraged out of the likelihood function, whereas the channelgain and noise power have been handled as determin-istic unknowns (and consequently estimated). However, thelow-SNR LLF (13) still depends on the unknown synchro-nization parameters contained in . In our context, the onlyparameter of interest is the symbol rate, whereas the remainingvariables are considered as nuisance parameters. The phasedoes not play any role in (13), but we still have to deal with thenormalized frequency and timing offsets, and , before wecan obtain the estimate of the symbol period .

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

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

(14)

where denotes the output of a receive filter matchedto the pulse corresponding to the symbol period , includingfrequency offset correction

(15)

In (14), samples are taken at the rate , thusspanning the whole observation time window. This is in con-trast with the ML approach in [4], where the number of infor-mation symbols in the observation time window was assumedto be known a priori, resulting in an LLF that only included thefirst term in (13). In addition, no frequency offset was consid-ered in [4] and the timing offset was assumed to follow a uni-form distribution.

The timing offset can be extracted for a given symbol pe-riod using the received signal second-order cyclostationarity atthe output of the corresponding matched filter [9]. ML estima-tion of is detailed in Appendix II, and turns out to yield theso-called Oerder and Meyr estimator [11] associated to the can-didate oversampling ratio

(16)

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

Substituting (16) in , the following expression isobtained (see Appendix II):

(17)

(18)

where the vector and the diagonal matrix are, respectively,given by

(19)

(20)

The first term in (17) and (18) measures the energy at theoutput of the filter matched to the corresponding signalingpulse. The second term measures the spectral correlation, withthe goal of exploiting the cyclostationarity of the informationsignal; it can be regarded as a matched filter followed by asquare-law nonlinearity and a bandpass filter at the cyclicfrequency rad, thus resembling well-known results fromclassical synchronization theory [8]. For roll-off factors closeto zero, this spectral correlation term will be very small, andthe only useful second-order information is found in the shapeof the received pulse.

Any frequency mismatch in (1) which is not correspondinglycompensated before matched filtering will degrade the perfor-mance of the ML symbol rate estimator. If some kind of a prioriknowledge is available about the distribution of the frequencyerror, it can be exploited by considering as a nuisance randomvariable that could be averaged out. This Bayesian approachturns out to be useful in some settings (see, e.g., [12]) but cannotbe adopted here due to the structure of the LLF under consid-eration. Alternatively, the normalized frequency offset canbe considered as an unknown deterministic parameter. Unfor-tunately, attempting to maximize the LLF (13) with respect todoes not yield a closed-form estimate. Some hints as to how tohandle this issue will be provided in Section V.

IV. PROPOSED ESTIMATOR

Assume that a suitable frequency offset estimate isavailable. The proposed estimator is summarized next. Afterinserting (18) in (13) and leaving out, one has

(21)

with

(22)

Fig. 3. Complete LLF `(rrrjT; ̂) from (22) and its corresponding coarse ap-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 thefilter matched to the pulse corresponding to the symbol period

, after frequency correction with [see (15) and (19)]

(23)

Fig. 3 plots the log-likelihood function in an illustra-tive case without frequency offset. The spectral correlation term

is responsible for the narrow peak at the true param-eter value as well as for all the spurious peaks. However, asthe number of samples goes to infinity, the width of the desired“spectral line” goes to zero. This poses a problem in practicalsettings and makes it necessary to have some cautions, as antic-ipated in [5]. On the other hand, the smooth curve is the resultobtained when this spectral correlation term is left out in thecomputation of (18).

In view of these considerations, we propose a two-stepstrategy for the estimation of the symbol period:

1) an initial search using only the energy at the output of thematched filters, that is, seeking the maximum of

(24)

as is a smooth function of , the number ofmatched filters to use in this initial search need not be toolarge;

2) a refined search around the coarse estimate of the previousstage; if the possible symbol rates are known, a simpleevaluation of the LLF in (22) at each of themwithin a predetermined range around the coarse estimatewill suffice.

A two-step approach was also suggested in [5], therein re-ferred to as coarse-search and fine-search, to handle the prob-lems associated to the location of the global maximum of a spec-ified cost function, namely the cyclic-correlation cost

MOSQUERA et al.: NON-DATA-AIDED SYMBOL RATE ESTIMATION OF LINEARLY MODULATED SIGNALS 669

that will be discussed in Section V. The approach derived fromthe ML criterion, however, is different in that a smoothed ap-proximate LLF [given by (24)] is used in the coarse search,rather than the complete LLF (22), which is used in the fine-search step only.

Notably, if the cyclostationary content of the signal is of rel-evance (that is, if the roll-off factor of the signaling pulse is nottoo low), it turns out that there is no significant difference in theresults 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.

Maximizing in order to obtain the fine estimate ofthe symbol rate is in correspondence with the ML estimation ofthe timing offset (16): the ML estimate of the symbol rate is theone providing a matched filter output whose squared magnitudehas the largest spectral content at the frequency rad,whereas the phase of that content yields the ML estimate ofthe timing offset. The next section addresses the issue of fre-quency offset estimation and draws some links between the MLsymbol rate estimate and the cyclic-correlation-based (CCB)estimator [5].

V. FREQUENCY-OFFSET ESTIMATION

The last step needed in order for the proposed ML estimateto 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 tailoredto different needs. For example, one may consider the estimate

(26)

which is computationally simple and, as the numerical simula-tions in Section VI will confirm, results in good performance ofour symbol rate estimate. The frequency-offset estimate (26) isanalyzed in detail in [9].

It is worth mentioning that symbol rate estimates robust tofrequency offsets are available from the literature. In particular,the CCB estimator is a well-known scheme exploiting signalcyclostationarity, which was thoroughly analyzed in [5]. TheCCB estimate of the oversampling ratio is given by

(27)

(28)

The number of correlation terms included inhas an important influence on the variance of the estimate, asshown in [5]; nevertheless, increasing beyond a certain value(which depends on the particular scenario) does not provide anysignificant gain.

It is easily seen that is independent of the receivedsignal frequency offset, which is clearly a desirable property.Moreover, the CCB estimate does not make any assumptionsabout the particular shape of the received pulse, and thus it

is especially well suited to scenarios where multipath or fre-quency-selective fading may distort the pulse shape. On theother hand, its main drawback is the degradation in performanceexhibited for low roll-off applications: for baseband bandwidthsclose to Hz, the peak of at the true value of theoversampling ratio decreases, becoming zero when the excessbandwidth is null.

Note that both the CCB cost (28) and the “pruned” LLF (25)provide a measure of the spectral correlation content at thecyclic frequency rad, but before and after the matchedfilter, respectively. Moreover, it is possible to establish aninsightful connection between both estimators, showing that ifany potential uncertainties in the received pulse shape and thefrequency offset are properly addressed, then the ML estimatormaximizing (25) will reduce to the CCB estimator. The key tothis link is the so-called cyclic correlation [13] of the receivedsignal , at time lag and for cycle , denoted by

and defined as

(29)

The natural estimate of from the finite sample recordis given by

(30)

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

(31)

It is shown in Appendix III how the cyclic correlation esti-mate (30) can be used in the computation of the “pruned” LLF

to implicitly account for lack of knowledge about thefrequency offset and the pulse shape, resulting in the costyielding the CCB estimator.

Hence, two possibilities exist in order to account for a fre-quency offset when using the proposed ML estimate: explicitestimation of using, e.g., (26), or implicit estimation using thecyclic correlation (30), in which case the ML estimate reducesto the CCB estimate. This latter estimate does not exploit thepulse shape, which is known in frequency flat channels, and itis outperformed by the ML method, as the results in Section VIwill confirm.

A detailed comparison of both methods in frequency-selec-tive channels would depend on the particular type of channeland is out of the scope of this paper. In this regard, let us saythat symbol rate estimation in the presence of an unknown fre-quency-selective channel has been addressed in [6], based on abank of samplers at the candidate symbol rates followed by blind

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

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

baud-spaced equalizers minimizing some suitable contrast; thesymbol rate estimate is then taken as that providing the smallestvalue 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 shortsignal records due to the fact that as the number of observationsincreases, the width of this minimum tends to zero. On the otherhand, this width does not collapse for excess bandwidths goingto zero, in contrast with the CCB method [5].

VI. NUMERICAL RESULTS

In the simulations conducted, the estimators were testedfor quadrature phase-shift-keying (QPSK) signals with SNRranging from 5 to 5 dB. The true value of the oversam-pling ratio was set at , and the sampling period wasnormalized as , so that all the numerical results applyindistinguishable to the symbol period and the oversamplingratio. For each set of parameters, and realizationswere used in the averaging for the coarse and the fine search,respectively. Fine searches were conducted on a fine grid ofoversampling ratios in the range . Except wherenoted, the frequency offset was randomly and uniformlychosen in the interval .

First, we assessed the performance of the coarse estimate re-sulting from the maximization of in (24). It turns outthat this coarse estimate presents a bias, which depends on theSNR, the number of samples , the roll-off factor, and the trueoversampling ratio. For example, Table I shows the mean valueof this coarse estimate; note how the bias reduces for higherSNR and longer records. Bias analysis is difficult due to the log-arithm in the last term of (24).

Fig. 4 shows the coarse search performance, measured as theprobability of making an initial estimate within 10% of the trueoversampling ratio. For a given setting, once established the de-sired probability and the tolerance of this initial acquisition, thenumber of necessary samples can be determined provided thatcurves as those shown in Fig. 4 are known for the entire rangeof possible symbol rates.

Next, we compared the performance obtained by usingthe complete LLF from (22) and the “pruned” LLF

from (25) in the fine-search step. Fig. 5 shows theMean Square Error (MSE) in both cases, fordifferent roll-off factors. No frequency offset was present inthis example. It is seen that both schemes behave similarlyexcept with small roll-off values, in which case the completeLLF 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 reasonfor this is that the maximum of is biased away from

Fig. 4. Coarse search results. The probability of acquiring the correct symbolrate within an error margin of 10% is plotted against the SNR and for differentsample 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 previouslymentioned bias in the coarse LLF from (24).

• The slope of the MSE increases sharply for SNR below2 dB. This so-called “outliers effect” [14] is also found inharmonic retrieval methods in which a narrow spectral linemust be sought: for low SNR and/or short signal records,the variability within the search range of the location of

MOSQUERA et al.: NON-DATA-AIDED SYMBOL RATE ESTIMATION OF LINEARLY MODULATED SIGNALS 671

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

the spectral peak increases, and eventually this locationbecomes a uniform random variable. Thus, the MSE ap-proaches for low SNR in all the plotted curves.

The “outliers effect” is difficult to analyze and renders thefamily of classical Cramér–Rao bounds of little value for therange of relevance of the outliers. Some results about the com-putation of the outliers’ probability as well as some alternativebounds for the MSE can be found in [14] and [15] for somespecific harmonic retrieval problems.

In order to assess the fine search performance, we have eval-uated both the ML estimate (25) and the CCB estimate (27). Forthe latter, the parameter in (27) yielding best performance waschosen at each case. As shown in Fig. 6 for two different cases,the new proposed estimator outperforms the CCB estimator fora given number of processed samples, at the cost of a highercomplexity. These figures depict also the degradation when the

channel model differs from that considered in the analysis. Thus,Fig. 6(a) shows the performance of the ML estimator using aSRRC roll-off factor of 0.2, when the received signaling pulsecorresponds to a SRRC pulse with roll-off factor of 0.35. Thisroll-off mismatch could be the case, for example, when esti-mating the symbol rate in a satellite broadcasting scenario underthe emerging standard DVB-S2 [16]. In addition to a number ofpossible symbol rates, the SRRC pulse of DVB-S2 signals canhave any of three roll-off factors: . Even underthis roll-off mismatch, the performance of the ML estimator isbetter than that of the CCB method. Optimality is also lost underchannel linear distortions, since the SRRC pulse is no longermatched to the received pulse. Fig. 6(b) depicts the degradationstemming from this fact for a single-echo channel of the form

. The performance of both ML and CCB de-grades, in the CCB case due exclusively to the lower cyclosta-tionarity content of the received signal. As a general remark, wecan say that such cyclostationary content will determine the per-formance of the symbol rate estimation. This statement appliesalso to frequency-selective channels, for which the received dis-torted pulse must be considered. Although suboptimum, the pro-posed ML method can be still applied under unknown channeldistortion: the filtering operation involved in (25) will help tofight noise and guarantee a minimum performance even for lowexcess bandwidth.4 Finally, let us say that pulses with higherroll-off factors than those shown, although less common in prac-tice, would improve the performance of both methods.

VII. CONCLUSION

A maximum-likelihood approach to the symbol rate esti-mation problem has been addressed. With low-SNR scenariosin mind, new results have been obtained which clearly showthe different kinds of second-order information that can beexploited for symbol rate estimation, namely, the shape of thereceived pulse together with the cyclostationarity of the infor-mation signal. As a result, a practical two-step strategy has beendevised, which avoids the drawbacks related to spurious peaksakin to those found in harmonic retrieval problems, by meansof a first coarse search. The expression of the fine estimatehandles the same function as the Oerder and Meyr estimator,the well-known timing offset ML estimator for low SNR. Aninitial precorrection for the frequency error makes this methodsuitable to practical scenarios. The proposed ML estimator hasbeen compared with a well-known cyclic-correlation-basedsuboptimal estimator, whose main virtues are its simplicity andimmunity to frequency offsets. It turns out that this suboptimalestimator can be derived from the ML estimator by a certainmeans of jointly dealing with received pulse and frequencyoffset uncertainties. Performance has been shown to improvenotably for low SNRs and low roll-off factors, which makesthe new method especially well suited for square-root raisedcosine signals employing powerful error-correcting codes. Thecase of frequency-selective channels can be considered as aninteresting line of research to pursue: both the proposed and theCCB estimators are suboptimum, although for the low-SNRand 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 shapefor the different symbol rates.

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

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

APPENDIX IPROOF OF PROPERTY 1

The element of is given by

(32)

where and . Recall that if two signalsare band-limited to Hz, then their crosscorre-

lation can be written in terms of the sampled signals as

(33)

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

(34)

where has Fourier trans-form . Now if (6) holds, then

since is constant inthe support of . The scaling can be determinedfrom the fact that the normalization imposedon implies that ,since . Therefore,

, and

(35)

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

(36)

where the last equality follows from (33), since (6) implies thatis band-limited to Hz. Therefore,

as was to be proved.

APPENDIX IITIMING OFFSET ESTIMATION

If the timing offset is regarded as a deterministic unknownin (13), it can be extracted after maximizing withregard to . Setting , this translates into

; that is, ML estimation of the timingoffset amounts to maximizing with regard to

. This is also the case in conventional synchronization fora known and fixed symbol period. We will follow a similarreasoning to that employed in [9] or [12]. The details of theinterpretation of , which will allow to estimate theoptimum timing offset for a given symbol period, are includednext, giving their relevance in the subsequent derivation of thesymbol period estimator.

It can be easily seen that the diagonals of the matrixshow a periodic behavior related to the cyclo-

stationarity of the linearly modulated signal. For large , theelement of this matrix is given by

(37)

The summation in (37) is -periodic in and admits the fol-lowing Fourier series [12]:

(38)

where

(39)

We have not included the terms in the Fourier series expansion(38), which null out for a band-limited pulse as in the caseunder consideration.

Using expression (38) for the elements of , oneobtains that, for large

(40)

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

(41)

(42)

MOSQUERA et al.: NON-DATA-AIDED SYMBOL RATE ESTIMATION OF LINEARLY MODULATED SIGNALS 673

where denotes the output of the matched filter corre-sponding to the signaling pulse and normalized frequencyoffset , as given by (15). The remaining terms in (40) are givenby

(43)

where the first equality can be readily checked noting thatis the conjugate of the corre-

sponding expression with and exchanged and that. For the last equality, we have

used (39). Using (42) and (43), one has from (40) that

(44)

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

(45)

As mentioned in [12], this timing offset estimate is optimal forbaseband transmission or perfectly known frequency offset .

APPENDIX IIIRELATION BETWEEN THE ML AND CCB ESTIMATORS

The “pruned” LLF from (25) for the fine estima-tion of the symbol period or, equivalently, the oversamplingratio , can be written as follows, after discarding irrelevantconstants:

(46)

(47)

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

(48)

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

(49)

Independence of with respect to the received shaping pulseand the frequency offset can be achieved if is sub-stituted in (49) by its estimate from (30), so that

(50)

which after being substituted in (49) leads to

(51)

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 usuallysome tens as detailed in [5].

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[4] H. Wymeersch and M. Moeneclaey, “Blind symbol rate detection forlow-complexity multi-rate receivers,” in Proc. Vehicular TechnologyConf. (VTC), Stockholm, Sweden, May 2005, pp. 1171–1175.

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[7] A. Dandawate and G. B. Giannakis, “Statistical tests for presence ofcyclostationarity,” 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 per-formance analysis of synchronizers,” IEEE Trans. Commun., vol.COM-34, no. 11, pp. 1089–1095, Nov. 1986.

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[16] Digital Video Broadcasting (DVB); Second Generation Framing Struc-ture, Channel Coding and Model Systems for Broadcasting, InteractiveServices, 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 de-gree in telecommunication engineering from the Uni-versidad de Vigo, Vigo, Spain, and subsequently theM.S. degree from Stanford University, Stanford, CA,in 1994 and the Ph.D. degree from the Universidadde Vigo in 1998.

In 1999, he spent six months with the EuropeanSpace Agency at ESTEC, The Netherlands. He is cur-rently an Associate Professor at the Universidad deVigo, where his research and teaching focus on the

area of signal processing applied to communications.Dr. Mosquera participates in the European Satellite Communications Net-

work of Excellence and has served as member of several international technicalprogram commmittees.

Sandro Scalise (S’00–M’06) was born in Utrecht,The Netherlands, in April 1973. He received theElectrical Engineering degree (in telecommunica-tions) with honors from the University of Ferrara,Italy, in 1999 and the Ph.D. degree from the Univer-sity of Vigo, Vigo, Spain, in 2007.

Since 2001, he has been with the Institute forCommunications and Navigation, DLR (GermanAerospace Center), Germany. Since October 2004,he has been leading the Mobile Satellite SystemsGroup. His research activity deals with forward error

correction and synchronization schemes for mobile satellite applications, landmobile satellite channel modeling, and link performance evaluation. He wasthe 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 SystemConference and Chairman of the R&D Working Group of ISI (Integral SatComInitative) European Technology Platform.

Roberto López-Valcarce (M’01) received theTelecommunications Engineer degree from theUniversidad de Vigo, Vigo, Spain, in 1995 and theM.S. and Ph.D. degrees in electrical engineeringfrom the University of Iowa, Iowa City, in 1998 and2000, respectively.

During 1996, he was with Intelsis, Santiago, Spain.Since 2001, he has been with the Signal Theory andCommunications Department at the Universidad deVigo, where he is currently an Associate Professor.His research interests lie in the area of signal pro-

cessing applied to communications.Dr. López-Valcarce was the recipient of a 2005 Best Paper Award of the IEEE

Signal Processing Society.