Closed-Form Expressions for the Exact Cramér-Rao Bounds of Timing Recovery Estimators From BPSK, MSK and Square-QAM Transmissions

2474 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 6, JUNE 2011

Closed-Form Expressions for the Exact Cramér–RaoBounds of Timing Recovery Estimators From BPSK,

MSK and Square-QAM TransmissionsAhmed Masmoudi, Faouzi Bellili, Sofiène Affes, Senior Member, IEEE, and Alex Stéphenne, Senior Member, IEEE

Abstract—In this paper, we derive for the first time analyticalexpressions for the exact Cramér–Rao lower bounds (CRLB) forsymbol timing recovery of binary phase shift keying (BPSK), min-imum shift keying (MSK), and square QAM-modulated signals. Itis assumed that the transmitted data are completely unknown atthe receiver and that the shaping pulse verifies the first Nyquistcriterion. Moreover the carrier phase and frequency are consid-ered as unknown nuisance parameters. The time delay remainsconstant over the observation interval and the received signal iscorrupted by additive white Gaussian noise (AWGN). Our new ex-pressions prove that the achievable performance holds irrespectiveof the true time delay value. Moreover, they corroborate previousattempts to empirically compute the considered bounds therebyenabling their immediate evaluation.

Index Terms—Non-data-aided (NDA) estimation, QAM signals,stochastic Cramér–Rao lower bound (CRLB), symbol timing re-covery.

I. INTRODUCTION

I N modern communication systems, the received signal isusually sampled once per-symbol interval to recover the

transmitted information. But the unknown time delay, intro-duced by the channel, must be estimated a priori in order tosample the signal at the accurate sampling times. In this context,many time delay estimators have been developed to meet this re-quirement. These estimators can be mainly categorized into twomajor categories: data-aided (DA) and non-data-aided (NDA)estimators. In DA estimation, a priori known symbols are trans-mitted to assist the estimation process, although the transmis-sion of a known sequence has the drawback of limiting thewhole throughput of the system. Whereas, in the NDA mode,the required parameter is blindly estimated assuming the trans-mitted symbols to be completely unknown. In both cases, the

Manuscript received July 30, 2010; revised December 03, 2010 and February11, 2011; accepted February 23, 2011. Date of publication March 17, 2011; dateof current versionMay 18, 2011. The associate editor coordinating the review ofthis manuscript and approving it for publication was Prof. Alfred Hanssen. Thiswork was supported by a Canada Research Chair in Wireless Communicationsand a Discovery Accelerator Supplement from NSERC.A. Masmoudi, F. Bellili, and S. Affes are with the INRS-EMT, Montreal,

QC H5A 1K6, Canada (e-mail: [email protected]; [email protected];[email protected]).A. Stéphenne is with Huawei Technologies, Ottawa, ON K2K 3C9, Canada,

and also with the INRS-EMT, Montreal, QC H5A 1K6, Canada (e-mail:[email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TSP.2011.2128314

performance of an estimator affects the performance of the en-tire system. In the case of an unbiased estimation, the varianceof the timing error is usually used to evaluate the estimationaccuracy. The CRLB is a lower bound on the variance of anyunbiased estimator and is often used as a benchmark for theperformance evaluation of actual estimators [1], [2]. The com-putation of this bound has been previously tackled by many au-thors, under different simplifying assumptions. For instance, as-suming the transmitted data to be perfectly known and one canderive the DA CRLB. The modified CRLB (MCRLB), whichis also easy to derive, has been introduced in [3] and [4], butunfortunately it departs dramatically from the exact (stochastic)CRLB, especially at low signal-to-noise ratios (SNR).Actually, the time delay stochastic CRLBs of higher-order

modulations were empirically computed in previous works.Their analytical expressions were tackled only for specificSNR regions, i.e., very low or very high-SNR values andthe derived bounds are referred to as ACRLBs (asymptoticCRLBs). In fact, in [5], the stochastic CRLB was tackled underthe low-SNR assumption and an analytical expression of theconsidered bound (ACRLB) was derived for arbitrary PSK,QAM, and PAM constellations. In this SNR region, the authorsof [5] approximated the likelihood function by a truncatedTaylor series expansion to obtain a relatively simple ACRLBexpression. An analytical expression was also introduced in[6] under the high-SNR assumption. This high-SNR ACRLBcoincides with the stochastic CRLB in this SNR region butunfortunately it cannot be used even for moderate (practical)SNR values. Another approach was later proposed in [7] and[8] to compute the NDA deterministic (or conditional) CRLBs,in which the symbols are considered as deterministic unknownparameters. Then the conditional CRLB is derived from thecompressed likelihood function in which standsfor the observed vector, is the parameter vector of interest(including the unknown time delay) and is the maximumlikelihood estimate of the transmitted symbols .However, it is widely known that the conditional CRLB does

not provide the actual performance limit (unconditional or sto-chastic CRLBs). In an other work, the stochastic CRLB wasempirically computed [9] assuming perfect phase and frequencysynchronization and a time-limited shaping pulse. Later in [10],its computation was tackled in the presence of unknown car-rier phase and frequency and pulses that are unlimited in time.Both [9] and [10] simplified the expression of the bounds butultimately resorted to empirical methods to evaluate the exactCRLB, without providing any closed-form expressions.

1053-587X/$26.00 © 2011 British Crown Copyright

MASMOUDI et al.: CLOSED-FORM EXPRESSIONS FOR THE EXACT CRAMÉR–RAO BOUNDS OF TIMING RECOVERY ESTIMATORS 2475

Motivated by these facts, in this work, we derive for the firsttime analytical expressions for the stochastic CRLBs of symboltiming recovery from BPSK,MSK and square QAM-modulatedsignals. We consider the general scenario as in [10] in whichthe carrier phase and frequency offsets are completely unknownat the receiver, and we show that this assumption does not ac-tually affect the performance of a time delay estimator fromperfectly frequency- and phase-synchronized received samples.The derivations assume an AWGN-corrupted received signaland a shaping pulse that verifies the first Nyquist criterion. Thelast assumption is verified in practice for most of the shapingpulses.This paper is organized as follows. In Section II, we introduce

the system model that will be used throughout this article. InSection III, we derive the analytical expression of the stochasticCRLB for any square QAM modulation. Then, in Section IV,we outline the derivation steps of the CRLB in the cases ofBPSK and MSK transmissions. Some graphical representationsare presented in Section V and, finally, some concluding re-marks are drawn out in Section VI.

II. SYSTEM MODEL

Consider a traditional communication system where thechannel delays the transmitted signal and a zero-mean proper1

AWGN, with an overall power , corrupts the received signal.In the case of imperfect frequency and phase synchronization,the received signal is expressed as

(1)

where is the time delay, is the channel distortion phase,is the carrier frequency offset and is the complex number

verifying . The parameters and are assumedto be deterministic but unknown. They can be gathered in thefollowing unknown parameter vector:

(2)

In (1), is a proper complex Gaussian white noise with in-dependent real and imaginary parts, each of variance , and

is the transmitted signal given by

(3)

with being the sequence of transmitted symbolsdrawn from a BPSK, an MSK or any square-QAM constella-tion and is the symbol duration. The transmitted symbolsare assumed to be statistically independent and equally likely,with normalized energy, i.e., . Finally, is asquare-root Nyquist shaping pulse function with unit-energywhich will be seen in Sections III and IV, as would be expected,to have an important impact on the CRLB and therefore on the

1A proper complex random process satisfies .

system’s performance. The Nyquist pulse obtained fromis defined as

(4)

and satisfies the first Nyquist criterion

for any integer (5)

Suppose that we are able to produce unbiased estimates, , ofthe vector from the received signal. Then the CRLB, whichverifies , is defined as [1], [2]

(6)

where is the Fisher informationmatrix (FIM)whose entriesare defined as

(7)

with being the log-likelihood function of the parametersto be estimated and are the elements of the unknownparameter vector .To begin with, we show in Appendix A that the problem of

time delay estimation is disjoint from the problem of carrierphase and frequency estimation. Indeed, we show that the FIMis block-diagonal structured as follows:

(8)

where is the CRLB of thetime delay parameter and is the (2 2) FIM pertainingto the joint estimation of and . Hence, we prove analyti-cally that we deal with two separable estimation problems; onone hand, time delay estimation and, on the other hand, carrierphase and frequency estimation. Actually, this conclusion hasbeen already made in [10] but the authors resorted to empiricalevaluations to find that the elements and ofthe FIM are almost equal to zero. Now, since the parameters aredecoupled, we only need to derive the first element of the globalFIM, in order to find the CRLB for time delay esti-mation under imperfect frequency and phase synchronization.Therefore, in the following, we consider the virtually derotatedreceived signal given by

(9)

where is also a proper AWGN withan overall power since the nuisance parameters are assumedto be deterministic.We mention that and return the mag-

nitude, real, imaginary and conjugate of any complex numberand is the statistical expectation. We also define the SNRof the system as .

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III. TIME DELAY CRLB FOR SQUAREQAM-MODULATED SIGNALS

In this section, we introduce the main contribution embodiedby this paper which consists in deriving closed-form expres-sions for the stochastic CRLBs of time delay estimation whenthe transmitted data are unknown and drawn from any -arysquare QAM-constellation (i.e., ).Before further development, it is important to emphasize that

an exact representation of requires an infinite-dimensionalvector representation . But let us consider the -dimensionaltruncated vectors and , representing the projection,over an orthonormal basis of dimensions, of and

, respectively. Then, the pdf of conditioned on the trans-mitted symbols and parameterized by is given by [4]

(10)

To derive the likelihood function which incorporates all the in-formation contained in , we should make tend to in-finity to get . However, convergence problems ap-pear. To overcome these problems, is divided by

to obtain

(11)and as tends to infinity, we obtain the conditional likelihoodfunction

(12)

To begin with, we note that since the transmitted symbolsare equally likely, then the desired likelihood function

of the derotated observation vector can be written as

(13)

where the expectation is performed with respect to the vector oftransmitted symbols and

(14)

It can be shown that (13) reduces simply to

(15)

where

(16)

in which is the constellation alphabet. Actually, the main dif-ficulty in deriving an analytical expression for the stochasticCRLB stems from the complexity of the log-likelihood func-tion. Therefore, we will manipulate the summation involved in(16). In fact, considering only square QAM-modulated signals,we are able, by exploiting the full symmetry of the constellation,to factorize which in turn linearizes the global log-likeli-hood function and ultimately linearizes all the derivations.Indeed, denoting by the subset of the alphabet points with

positive real and imaginary parts (i.e.,), the constellation alphabet is de-

composed as follows:

(17)

Note that is the inter-symbol distance derived under the as-sumption of a normalized-energy square QAM constellation asfollows:

(18)

Using (17), we rewrite (16) as

(19)

Now using the hyperbolic cosine function defined by, (19) reduces simply to

(20)

MASMOUDI et al.: CLOSED-FORM EXPRESSIONS FOR THE EXACT CRAMÉR–RAO BOUNDS OF TIMING RECOVERY ESTIMATORS 2477

Moreover, using the fact thatand noting that and

, we obtain

(21)

Recall that andhence the previous expression of is rewritten as

(22)

Then, splitting the two sums in (22), is factorized as fol-lows2:

(23)

where

(24)

(25)

and

(26)

Now, injecting the expression of in the likelihood func-tion of the received signal (15), we obtain

(27)

2Note that similar factorization was recently used to derive an analytical ex-pression for the NDA SNR estimation [11], [12].

Finally, the log-likelihood function of the received signal ex-pands to

(28)

Note from (28) that due to the factorization of in (19),the global log-likelihood function of interest in (28) involvesthe sum of two analogous terms. This reduces considerably thecomplexity of the stochastic CRLB derivation. In fact, the firstderivative of (28) with respect to is obtained as follows:

(29)

where is given by

(30)

Then, the first diagonal element of the FIM matrix is expressedas

(31)

where and are the derivatives of andwith respect to .Starting from (31), the derivation of involves the

evaluation of three expectations. However, it is easy to verifythat the first and the last expectations in the right-hand side of(31) are performed with respect to two random processes havingthe same statistical properties and they are therefore identicallyequal. Moreover, as shown in Appendix B, the second expecta-tion is equal to zero. Therefore, (31) reduces simply to

(32)

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First, we consider the case where , and we show inAppendix C that and are statistically independent.This results in

(33)

These two expectations involved in the right-hand side of (33)are easily evaluated as follows:

(34)

(35)

where and are the first and second derivative of ,respectively. We simplify (34) by changing byand we obtain the following result:

(36)where

(37)

(38)

We now consider the case where . The intersymbol inter-ference results in a statistical dependence between andthe first derivatives and (likewise for and thefirst derivatives and ). Thus, using a standard probo-bility approach to derive the expectations involved in (32), wefirst average by conditioning on and , then averagethe resulting expression with respect to these two random vari-ables. To that end, consider the expectation of andconditioned on and

(39)

(40)

Using (39) and (40), it follows that

(41)

and we obtain

(42)

where the last equality follows from the statistical independenceof and and

(43)

Finally, gathering all these results, we obtain the analytical ex-pression of the stochastic CRLB for symbol timing estimation.From square QAM-modulated signals in the presence of carrierphase and frequency offsets as follows:

(44)

Note that for large values of , one can use the following ac-curate approximation [10]:

(45)

It is worth mentioning that the new analytical expression in(44) allows the immediate evaluation of time delay stochasticCRLBs, contrarily to the empirical approaches presented in [9]and [10], and this is made possible for any square QAMmodula-tion order. Second, the shaping pulse is involved only viaand , and is separate from the factors resulting

MASMOUDI et al.: CLOSED-FORM EXPRESSIONS FOR THE EXACT CRAMÉR–RAO BOUNDS OF TIMING RECOVERY ESTIMATORS 2479

from the modulation order. Moreover, to the best of our knowl-edge, we show here for the first time, through our new analyticalexpression, that the true value of the time delay parameter doesnot affect the actual achievable performance as intuitively ex-pected, i.e., the variance of the estimation error holds irrespec-tive of the time delay value to be estimated.

IV. CRLB FOR BPSK AND MSK MODULATED SIGNALS

In this section, we consider the BPSK and MSK modula-tions. In BPSK transmissions, the data symbols take values in

with equal probabilities. In MSK transmissions, thesymbols are defined as where is a sequence ofBPSK symbols and is the original value drawn from the set

. For these two transmission schemes, the keyderivation steps of the NDACRLBwill be briefly outlined in thefollowing. All derivation details can be found in Appendix D.First, the likelihood function of interest based on the received

signal is:

(46)

where is equal to 1 and for BPSK and MSK, respec-tively. Therefore, we show that the useful log-likelihood func-tion of is given by

(47)

Note that is defined in (9). After some algebraic manipu-lations, detailed in Appendix D, it turns out that the analyticalexpression of the stochastic CRLB for time delay estimation isthe same for BPSK and MSK modulations, and it is given by:

(48)

where is defined as

(49)

V. GRAPHICAL REPRESENTATIONS

In this section, we provide graphical representations of thetime delay CRLBs and the CRLB/MCRLB ratio for differentmodulation orders. First, we mention that the even integrand

functions and involved in(44) and (49), respectively, decrease rapidly as increases.Therefore, the integrals over can be accurately ap-proximated by a finite integral over an interval and theRiemann integrationmethod can be adequately used. In our sim-ulations, we note that and a summation step of 0.5provided accurate values for the infinite integral.First, we plot in Fig. 1 the CRLBs for different modulation

orders and compare them to the ones previously obtained em-pirically in [10]. We see a good agreement between the two ap-proaches thereby validating the developments above. Then, weconfirm through Fig. 2 that, at low SNR values, the MCRLB isa looser bound compared to the exact CRLB. Indeed, this figuredepicts the CRLB/MCRLB ratio as a function of the SNR. Thisratio quantifies the performance degradation that arises fromrandomizing the transmitted data and it approaches 1 at highSNR values. Hence, in this SNR region, the MCRLB can beused as a benchmark to evaluate the performance of unbiasedtime delay estimators instead of the exact CRLB, since it iseasier to evaluate. However, the gap between the two boundsbecomes important as soon as the SNR drops below 7 dB, evenfor QPSK-modulated signals, where the stochastic CRLB quan-tifies the actual performance limit. Moreover, we consider inthis figure two values of the roll-off factor, 0.2 and 1, in order toillustrate the effect of the roll-off factor on timing estimation.Clearly, timing estimation is less accurate at a lower roll-offfactor (larger intersymbol interference). Moreover, we see fromFig. 3 that the different CRLBs tend to ultimately coincide withthe MCRLB as long as the SNR gets increases. This actually,in the high SNR region, the achievable performance of NDAestimation of the signal time delay is equivalent to the one ob-tained when the received symbols are perfectly known since inthis SNR range the MCRLB coincides with the DA CRLB. Inthe specific case where is time limited to the symbol dura-tion, the corresponding CRLB follows directly from the generalexpression in (44) by taking for all

(50)

Note from (50) that the resulting CRLB becomes the product oftwo separate terms; one depending on the shaping pulse func-tion and the other on the signal modulation. This special boundis plotted in Fig. 4.We see again a good agreement in this specialcase between the CRLBs obtained from our analytical expres-sion in (50) and their empirical counterparts plotted in Fig. 1 of[9]. This particular expression still finds applications in manyconventional systems and in the emerging impulse radio tech-nology [13], [14] where, precisely, synchronization stands todayas a very challenging issue.

VI. CONCLUSION

In this paper, we derived, for the first time, analytical expres-sions of the Cramér–Rao lower bound for symbol timing esti-mation in the cases of BPSK, MSK and square-QAM modula-

2480 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 6, JUNE 2011

Fig. 1. Compression between the empirical CRLB and the analytical expres-sion in (44) for different modulation orders using and a raised-cosinepulse with roll-off factor of 0.2.

Fig. 2. CRLB/MCRLB ratio versus SNR for different modulation orders usingand a raised-cosine pulse with roll-off factor of 0.2 and 1.

tions.We considered the stochastic CRLBwhere the transmitteddata are unknown and randomly drawn. The carrier phase andfrequency offsets are also supposed to be unknown (nuisanceparameters). We showed that the knowledge of the phase andfrequency does not bring any additional information to the timedelay estimation problem and that the latter is decoupled fromthe joint estimation of the carrier frequency and phase offsets.Moreover, our analytical expressions for the CRLBs underlinethe fact that these bounds do not depend on the time delay value,which used to be stated only intuitively. We confirmed also thatthe modified CRLB is a valid approximation of the exact CRLBin the high SNR region and that it can be used as a benchmarksince it is easier to evaluate. Furthermore, the derived analyticalexpressions corroborate previous works that empirically com-puted the stochastic CRLBs via Monte Carlo simulations, and

Fig. 3. CRLB versus SNR for different modulation orders using anda raised-cosine pulse with roll-off factor of 0.2.

Fig. 4. CRLB/MCRLB ratio versus SNR for different modulations and a time-limited shaping pulse.

hence provide a useful tool for a quick and easy evaluation ofthe CRLBs with BPSK, MSK and square-QAM modulations.

APPENDIX APROOF OF THE BLOCK-DIAGONAL STRUCTURE OF THE FIM

To show that and are decoupled, we con-sider the actual received signal instead of the virtually dero-tated signal . Then we follow the same derivation steps from(13) to (28) to retrieve the log-likelihood function parameterizedby as follows:

(51)

MASMOUDI et al.: CLOSED-FORM EXPRESSIONS FOR THE EXACT CRAMÉR–RAO BOUNDS OF TIMING RECOVERY ESTIMATORS 2481

The first derivatives of this function with respect to the th ele-ment of , and are, respectively, given by

(52)

and

(53)

where is defined in (30). Then we average asin (31) to obtain the following result:

(54)

In order to simplify the calculations, without loss of generality,we consider . To begin with, we first differentiatewith respect to and we obtain

(55)

is a function of the imaginary part of the transmittedsymbols and the derotated noise, which are mutually indepen-dent from the real part of the transmitted symbols and the dero-tated noise. As a result, is independent from

and . This allows us to split the expectations in(54):

(56)

Noting that the last expectation is equal to zero, it follows imme-diately that is also equal to zero. Thus, we show analyt-ically that the two parameters and are decoupled. The samemanipulations are used to prove that and are also decoupled.Therefore, the FIM is block-diagonal structured as given by (8).

APPENDIX BPROOF OF

In the following, we briefly show that. By definition,

depends on the real part of , while involvesthe imaginary part of , which are statistically independent.It follows that and are independent. The samearguments hold to show the statistical independence ofand . Then, it immediately follows that

(57)

And since and are statistically independent (seeAppendix C), each with mean zero, we obtain

(58)

APPENDIX C

A. PDFS OF AND

In this Appendix, we establish the joint pdf ofand defined, respectively, in (24) and (25). Tothat end, we define the proper complex random variable

. It can be easily seen thatand that .

Using the same algebraic manipulations from (16) through (23),we establish the pdf of as follows:

(59)

where

(60)

(61)

Note that the factorization of the joint pdf ofand to their elementary pdfs confirms that these are

two independent random variables.

B. PROOF OF STATISTICAL INDEPENDENCE OFAND

First, note that, can be written as

(62)

2482 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 6, JUNE 2011

where

(63)

Therefore, is given by

(64)

In addition, and are independent since the noiseand the transmitted symbols are independent. Recall alsothat (the maximum of is located at 0). Then,

and are also independent.Moreover, and are obtained by a linear transformation ofthe Gaussian process . Hence, they are also Gaussianprocesses. Then, since the cross-correlation of and isequal to zero, as shown below:

(65)

then, and are actually two uncorrelated Gaussianrandom processes and therefore they are independent. Thus,

and are independent.

APPENDIX DDERIVATION OF THE ANALYTICAL EXPRESSIONS FOR THECRLBS IN CASE OF BPSK AND MSK MODULATIONS

Starting from the expression of the log-likelihood functiongiven in (47), we will consider the two cases of BPSK andMSK separately. Starting with BPSK-modulated signals, weshow that the log-likelihood function in (47) reduces to

(66)

where

(67)

Then, the first derivative of the log-likelihood function with re-spect to the time delay parameter, , is given by

(68)

where denotes the first derivative of with respectto . It is easy to see that

(69)

Now injecting (68) in (7), we obtain

(70)

Note that (70) is similar to (32) (obtained in the case of squareQAM modulations). Thus, for the same reasons, it is more con-venient to separate the cases when and . Moreover, itcan be shown that the pdf of is given by

(71)

Thus, it can be shown that, after some manipulations, the expec-tations involved in (70) reduce to

(72)

(73)

(74)

MASMOUDI et al.: CLOSED-FORM EXPRESSIONS FOR THE EXACT CRAMÉR–RAO BOUNDS OF TIMING RECOVERY ESTIMATORS 2483

Finally, we obtain the closed-form expression for the stochasticCRLB of BPSK-modulated signals as follows:

(75)

where is defined in (49). Now, consider a MSK-modulatedsignal. In order to find the derivative of (47) with respect to thetime delay , we need to separate the cases where is real orimaginary. To do so, we assume, without loss of generality, thatis an even number (i.e., ) and . Using these

assumptions, the log-likelihood function can be written as

(76)

where

(77)

(78)

Then, the first derivative of (76) with respect to is given by

(79)

with and being the derivatives of andwith respect to , respectively. Then, the first diagonal

element of the FIM matrix is expressed as

(80)

Note that (80) is equivalent to (31). Then for the same reasons,reduces simply to

(81)

which is similar to (70) in the case of BPSK modulation. Thus,we obtain the same expression for the stochastic CRLB in caseof MSK and BPSK transmissions as given by (75).

REFERENCES

[1] C. R. Rao, “Information and accuracy attainable in the estimation ofstatistical parameters,” Bull. Calcutta Math. Soc., vol. 37, pp. 81–91,1945.

[2] S. M. Kay, Fundamentals of Statistical Signal Processing: EstimationTheory. Upper Saddle River, NJ: Prentice-Hall, 1993.

[3] A. N. D’Andrea, U. Mengali, and R. Reggiannini, “The modifiedCramer-Rao bound and its applications to synchronization problems,”IEEE Trans. Commun., vol. 24, pp. 1391–1399, Feb.–Apr. 1994.

[4] U. Mengali and A. N. D’Andrea, Synchronization Techniques for Dig-ital Receivers. New York: Plenum, 1997.

[5] H. Steendam and M. Moeneclaey, “Low-SNR limit of the Cramer-Raobound for estimating the time delay of a PSK, QAM, or PAM wave-form,” IEEE Commun. Lett., vol. 5, pp. 31–33, Jan. 2001.

[6] M. Moeneclaey, “On the true and the modified Cramer-Rao bounds forthe estimation of a scalar parameter in the presence of nuisance param-eters,” IEEE Trans. Commun., vol. 46, pp. 1536–1544, Nov. 1998.

[7] J. Riba, J. Sala, and G. Vàzquez, “Conditional maximum likelihoodtiming recovery: Estimators and bounds,” IEEE Trans. Signal Process.,vol. 49, no. 4, pp. 835–850, Apr. 2001.

[8] G. Vàzquez and J. Riba, “Non-data-aided digital synchronization,” inSignal Processing Advances in Wireless Communications. Engle-wood Cliffs, NJ: Prentice-Hall, 2000, vol. 2, ch. 9.

[9] I. Bergel and A. J. Weiss, “Cramér-Rao bound on timing recovery oflinearly modulated signals with no ISI,” IEEE Trans. Commun., vol.51, pp. 134–641, Apr. 2003.

[10] N. Noels, H. Wymeersch, H. Steendam, and M. Moeneclaey,“True Cramér-Rao bound for timing recovery from a bandlimitedlinearly modulated waveform with unknown carrier phase andfrequency,” IEEE Trans. Commun., vol. 52, pp. 473–383, Mar.2004.

[11] F. Bellili, A. Stéphenne, and S. Affes, “Cramér-Rao bounds for NDASNR estimates of square QAM modulated signals,” presented at theProc. IEEEWireless Commun. Netw. Conf. (WCNC), Budapest, Hun-gary, Apr. 5–8, 2009.

[12] F. Bellili, A. Stéphenne, and S. Affes, “Cramér-Rao lower boundsfor non-data-aided SNR estimates of square QAM modulatedtransmissions,” IEEE Trans. Commun., vol. 58, pp. 3211–3218,Nov. 2010.

[13] C. J. Le Martret and G. B. Giannakis, “All-digital PAM impulse radiofor multiple access through frequency-selective multipath,” in Proc.IEEE GLOBECOM, 2000, pp. 77–81.

[14] M. Z. Win and R. A. Scholtz, “Impulse radio: How it works,” IEEECommun. Lett., vol. 2, no. 2, pp. 36–38, Feb. 1998.

2484 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 59, NO. 6, JUNE 2011

Ahmed Masmoudi was born in Ariana, Tunisia,on February 10, 1987. He received the Diplômed’Ingénieur degree in telecommunication fromthe Ecole Supérieure des Communications deTunis-Sup’Com (Higher School of Communicationof Tunis), Tunisia, in 2010. Since September 2010,he is working toward the M.Sc. degree in the InstitutNational de la Recherche Scientifique (INRS),Montréal, QC, CanadaHis research activities include signal processing

and parameters estimation for wireless communica-tion.Mr. Masmoudi is the recipient of the National grant of excellence from the

Tunisian Government.

Faouzi Bellili was born in Sbeitla, Kasserine,Tunisia, on June 16, 1983. He received the Diplômed’Ingénieur degree in signals and systems (withHons.) from the Tunisia Polytechnic School in2007 and the M.Sc. degree, with exceptional grade,at the Institut National de la Recherche Scien-tifique-Energie, Matériaux, et Télécommunications(INRS-EMT), Université du Québec, Montréal, QC,Canada, in 2009. He is currently working towards thePh.D. degree at the INRS-EMT. His research focuseson statistical signal processing and array processing

with an emphasis on parameters estimation for wireless communications.During hisM.Sc. studies, he has authored/coauthored six international journal

papers and more than ten international conference papers.Mr. Bellili was selected by the INRS as its candidate for the 2009–2010 com-

petition of the very prestigious Vanier Canada Graduate Scholarships program.He also received the Academic Gold Medal of the Governor General of Canadafor the year 2009–2010 and the Excellence Grant of the Director General ofINRS for the year 2009–2010. He also received the award of the best M.Sc.thesis of INRS-EMT for the year 2009–2010 and twice—for both the M.Sc. andPh.D. programs—the National Grant of Excellence from the Tunisian Govern-ment. He was also rewarded in 2011 the Merit Scholarship for Foreign Studentsfrom the Ministère de l’Éducation, du Loisir et du Sport (MELS) of Québec,Canada. He serves regularly as a reviewer for many international scientific jour-nals and conferences.

Sofiène Affes (S’94–M’95–SM’04) received theDiplôme d’Ingénieur degree in electrical engineeringin 1992, and the Ph.D. degree (with hons.) in signalprocessing in 1995, both from the Ecole NationaleSupérieure des Télécommunications (ENST), Paris,France.He has since been with INRS-EMT, University of

Quebec, Montreal, QC, Canada, as a Research Asso-ciate from 1995 until 1997, then as an Assistant Pro-fessor until 2000. Currently, he is an Associate Pro-fessor in the Wireless Communications Group. His

research interests are in wireless communications, statistical signal and arrayprocessing, adaptive space-time processing, and MIMO. From 1998 to 2002,he has been leading the radio design and signal processing activities of theBell/Nortel/NSERC Industrial Research Chair in Personal Communications atINRS-EMT, Montreal, QC, Canada. Since 2004, he has been actively involvedin major projects in wireless communication of PROMPT (Partnerships for Re-search on Microelectronics, Photonics and Telecommunications).Prof. Affes was the corecipient of the 2002 Prize for Research Excellence

of INRS. He currently holds a Canada Research Chair in Wireless Communi-cations and a Discovery Accelerator Supplement Award from NSERC (Nat-ural Sciences & Engineering Research Council of Canada). In 2006, he servedas a General Co-Chair of the IEEE Vehicular Technology Conference (VTC)2006—Fall, Montreal, QC, Canada. In 2008, he received from the IEEE Vehic-ular Technology Society the IEEEVTCChair RecognitionAward for exemplarycontributions to the success of IEEE VTC. He currently serves as a member ofthe Editorial Board of the IEEE TRANSACTIONS ON SIGNAL PROCESSING, theIEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, and the Wiley Journalon Wireless Communications and Mobile Computing.

Alex Stéphenne (S’94–M’95–SM’04) was born inQuebec, Canada, on May 8, 1969. He received theB.Eng. degree in electrical engineering from McGillUniversity, Montreal, QC, Canada, in 1992, and theM.Sc. degree and Ph.D. degree in telecommunica-tions from INRS-Télécommunications, Université duQuébec, Montreal, QC, Canada, in 1994 and 2000,respectively.In 1999, he joined SITA, Inc., Montreal, QC,

Canada, where he worked on the design of remotemanagement strategies for the computer systems of

airline companies. In 2000, he became a DSP Design Specialist for Dataradio,Inc., Montreal, a company specializing in the design and manufacturing ofadvanced wireless data products and systems for mission critical applications.In January 2001, he joined Ericsson and worked for over two years in Sweden,where he was responsible for the design of baseband algorithms for WCDMAcommercial base station receivers. From June 2003 to December 2008, he wasstill working for Ericsson, but was based in Montreal, where he was a researcherfocusing on issues related to the physical layer of wireless communicationsystems. Since 2004, he has also been an Adjunct Professor at INRS, wherehe has been continuously supervising the research activities of multiple stu-dents. His current research interests include Coordinated Multi-Point (CoMP)transmission and reception, Inter-Cell Interference Coordination (ICIC) andmitigation techniques in Heterogeneous Networks (HetNets), wireless channelmodeling/characterization/estimation, statistical signal processing, arrayprocessing, and adaptive filtering for wireless telecomDr. Stéphenne is a member of the organizing committee and a Co-Chair of

the Technical Program Committee (TPC) for the 2012—Fall IEEE VehicularTechnology Conference (VTC’12—Fall) in Quebec City, Canada. He has servedas a Co-Chair for the Multiple Antenna Systems and Space-Time Processingtrack for VTC’08—Fall in Calgary, Canada, and as a Co-Chair of the TPC forVTC’06—Fall in Montreal, Canada.