Publication in the conference proceedings of EUSIPCO, Florence, Italy, 200

Boucheret, Marie-Laure

Dervin, Mathieu

Tourneret, J.-Y.

English

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A COMPARISON OF SOFT AND HARD DECISION-DIRECTED FEEDFORWARDPHASE ESTIMATORSMathieu Dervin, Marie-Laure Boucheret, Jean-Yves TourneretIRIT / ENSEEIHT2, rue Charles Camichel, BP 7122, 31071 Toulouse Cedex 7, FrancePhone : (+33 5) 61 58 80 17, Fax : (+33 5) 61 58 83 06, Email : dervin@enseeiht.frABSTRACTThis paper is devoted to the comparison of hard and soft de-cision directed feedforward phase estimators based on themaximum likelihood principle. The particular structure ofthese estimators is taken into account to derive a new lowerbound on the estimation variance for a constant phase error.Equivalent noise bandwidths for decision directed estimatorsare also studied. They allow us to characterize the estima-tor behaviour for the tracking of a time varying phase error.Simulation results are finally proposed, comparing the rootmean square errors of the different estimators, for a constantphase error and for a time varying phase error respectively.1. INTRODUCTIONFeedforward carrier recovery is usually preferred to feedbacktechniques for burst transmission systems. Indeed, contraryto the classical phase locked loop, a feedforward structureprovides phase estimates without any acquisition delay. Be-sides, several phase estimation strategies are discussed inthe literature when the transmitted data are totally unknownby the receiver [1]. These strategies are commonly dividedinto non data aided (NDA) and decision directed (DD) tech-niques. The recent success of solutions where channel de-coding and phase estimation are embedded [2] highlights theinterest of DD approaches. In this case, the phase estima-tion relies on the symbol detection, possibly more reliablebecause of the decoding process. This study focuses on feed-forward DD phase recovery algorithms.The DD phase estimation can either involve hard or softdecisions from the receiver. We propose here to compareboth approaches it absence of channel decoding. The rel-evance of the decisions relies on a preliminary phase pre-correction step. To ensure this pre-correction, a feedbackcomponent is introduced into the estimation structure. Asa consequence, the DD estimators cannot be studied as theclassical non data aided feedforward estimators.This paper derives a specic lower bound for the vari-ances of the hard- and soft-decision directed estimators inpresence of constant carrier phase error. Moreover, equiva-lent noise bandwidths are dened and evaluated for the pro-posed schemes, allowing a better understanding of the esti-mators behaviour when the phase error is not constant. Theperformances of the hard and soft DD estimators are nallycompared for a constant and a time varying carrier phase er-ror. All simulation results presented in this paper have beenobtained with QPSK modulation.The paper is organized as follows: the proposed phase re-covery schemes are described in Section 2. Section 3 derivesthe modied Cramer Rao lower bound for the DD phase es-timation variance. Section 4 characterizes the estimators interms of equivalent noise bandwidths. The estimator perfor-mances are nally compared through simulation results inSection 5. Conclusions are reported in Section 6.2. PROBLEM FORMULATION2.1 Decision directed phase estimationThis paper addresses the problem of feedforward phase re-covery. A random sequence of N complex QPSK symbolsd = (d0,d1, . . . ,dN−1) ∈ {±1± j}N is sent through a Gaus-sian channel. The transmitted signal is affected by a phaseerror ϕ . The possible phase variations are assumed to beslow: ϕ is assumed to be constant on duration NTs, whereTs denotes the symbol duration. Assuming perfect timingrecovery in a classical baseband model - including Nyquistpulse shaping and matched ltering - the received samplesy = (y0, . . . ,yN−1) can be expressed as:yk = dke jϕ +nk, (1)where the complex noise samples n = (nk)k=0,...,N−1 can bedecomposed into independent Gaussian real and imaginarycomponents with zero means and variances σ 2.The log-likelihood function for the joint phase and sym-bol estimation problem can be expressed as follows (up to anadditive constant)[1]:Λ(y|d,ϕ) = −12σ 2N−1∑k=0∣∣yk−dke− jϕ ∣∣2 . (2)In a DD approach, (hard or soft) decisions dk are taken fromthe received samples yk in order to obtain a more tractablephase log-likelihood expression:ΛDD(y| d,ϕ)=−12σ 2N−1∑k=0∣∣yk− dke− jϕ ∣∣2 . (3)The maximization of the log-likelihood function (3) leads tothe following feedforward estimator [1]:ϕDD = arg(N−1∑k=0yk dk∗), (4)where ∗ denotes complex conjugation.Remark: when the decisions dk are taken directly from thereceived samples, their relevance depends on the phase14th European Signal Processing Conference (EUSIPCO 2006), Florence, Italy, September 4-8, 2006, copyright by EURASIPEstimationDecisionDirectedPhaseykyke−jϕˆϕˆDDDecisionsdˆky¯ke−j(...)Figure 1: DD phase estimation with data pre-correction.error ϕ , which has to be avoided. It is therefore prefer-able to pre-correct the signal phase before making the de-cisions. The phase estimated from a block of data is clas-sically used to pre-correct the samples of the next block[3, 4], as shown on Fig. 1. The initial pre-correction stepis based on some non data aided algorithm: the Viterbi& Viterbi estimator [5] is considered in our simulations.The pre-corrected samples are denoted as flyk , yke− jϕ¯ ,where flϕ is the phase estimate obtained from the previousblock of samples.2.2 DD algorithmsWe propose to compare the following algorithms for the es-timation of a constant phase error:1. The Hard Decision Directed (HDD) estimator ϕHDD [3]given by (4) wheredk = sign(ℜ( flyk))+ jsign(ℑ( flyk)) , (5)and ℜ( flyk), ℑ( flyk) denote the real and imaginary parts offlyk, respectively,2. The Soft Decision Directed (SDD) estimator ϕSDD givenby the same expression, except that the hard decisions dkare replaced by soft decisions δk:δk , ∑s=±1± jP [dk = s| flyk]s. (6)The probabilities P [dk = s| flyk] are classically computedas [1]:P [dk = s| flyk] = λ exp(−|s− flyk|22σ 2), (7)withλ = 1∑s=±1± j exp(− |s−y¯k|22σ 2) . (8)3. LOWER BOUND ON THE ESTIMATIONVARIANCESection 3.1 shows that the phase pre-correction step (men-tioned in 2.1) provides a priori information about the un-known parameter ϕ . Based on this remark, Section 3.2 de-rives a lower bound on the variance of the SDD/HDD phaseestimators when the phase error is constant.EstimationDecisionDirectedPhaseykyky¯kDecisionsdˆk/δkϕˆHDD or ϕˆSDD( Hard or soft)Hard / Softe−jϕ¯ϕ¯ ∼ (ϕ, σ20)Figure 2: Phase estimation model including a priori informa-tion .3.1 Phase estimation with a priori informationThe feedback pre-correction step illustrated on Fig. 1 intro-duces additional information about the phase to be estimated.Indeed, the phase estimate from the previous block (denotedas flϕ) provides additional information to the current data sam-ples. We assume that the distribution of flϕ is Gaussian withmean ϕ and variance σ 20 :flϕ ∼N(ϕ ,σ 20). (9)Under this assumption, we propose the equivalent model de-picted on Fig. 2: each phase estimate ϕSDD or ϕHDD is ob-tained from N received samples (y0, . . . ,yN−1) and flϕ .Remarks:• The Gaussian assumption for flϕ has been motivated bymany simulations for both HDD and SDD schemes,• The expectation of flϕ is equal to ϕ , reecting the unbias-ness of HDD and SDD estimators,• The looped structures proposed in this paper ensure smallvariations of the variance σ 20 in consecutive blocks (insteady state regime), inducing σ 20 ' σ 2ϕ , where σ 2ϕ is thephase jitter,• It is interesting to note that flϕ does not depend on y (sincethis estimate is issued from a former data block). Asa consequence, the joint log-likelihood of (y, flϕ) can bewritten:Λ(y, flϕ |d,ϕ) =N−1∑k=0lnp [yk|dk,ϕ ]+ln p( flϕ) . (10)3.2 Modified Cramer-Rao lower boundDetermining a lower bound on the variance of a phase es-timator is a classical problem in synchronization [1]. Themodied Cramer Rao lower bound (MCRB) is usually pre-ferred to the true Cramer Rao bound (CRB) for its simplicity[6]. In absence of prior information regarding ϕ , the MCRBis dened as:MCRB(ϕ) , −1E[∂ 2∂ ϕ2 (Λ(y|d,ϕ))] , (11)yielding the well known result [6]:MCRB(ϕ) = 12N (Es/N0), (12)14th European Signal Processing Conference (EUSIPCO 2006), Florence, Italy, September 4-8, 2006, copyright by EURASIPwhere Es/N0 denotes the signal to noise ratio (SNR) on thetransmission channel.This section takes advantage of the additional informa-tion p( flϕ) dened in (9) (coming from a previous block ofsamples) to derive a new MCRB on the variance of the DDphase estimators:MCRB(ϕ) , −1E[∂ 2∂ ϕ2 (Λ(y, flϕ |d,ϕ))] . (13)Combining (10) and (13), straightforward computationsyield:MCRB(ϕ) =(1MCRB(ϕ) +1σ20)−1, (14)Remarks:• Eq. (14) clearly shows that the proposed bound is smallerthan the classical MCRB. This means that the knowledgeof an a priori information flϕ improves the phase estima-tion,• When σ 20 tends to zero, MCRB(ϕ) tends to zero as well.This makes sense, since flϕ is itself an estimator of ϕ , withvariance σ 20 ,• When σ 20  MCRB(ϕ), the benet brought by flϕ is neg-ligible and we recover the classical bound:MCRB (ϕ)'MCRB(ϕ) , (15)• It is important to note that MCRB(ϕ) dened in (12) isno longer a bound for the phase estimation variance inpresence of feedback pre-correction.4. EQUIVALENT NOISE BANDWIDTHSSection 3 derived a lower bound for the variance of phaseestimation errors. This bound is applicable when the carrierphase is constant. When the phase to estimate is not con-stant, we propose to resort to the equivalent noise bandwidthparameter in order to characterize the estimator behaviour.It should indeed be noted that the consecutive phase esti-mates are correlated, due to:• The ltering operation (4), and• The phase pre-correction step discussed in Sect. 2.1.Considering the problem in the frequency domain, it is in-teresting to compare the power spectral density (PSD) of thephase estimates for both HDD and SDD schemes. For in-stance, Fig. 3 shows the normalized PSD of the estimates(obtained by simulations with a constant phase error) whenEs/N0 = 0dB (SNR) and N = 32 (number of samples perblock): the PSD of the HDD estimates appears to be largerthan the PSD of the SDD estimates.As for classical feedback algorithms, we propose to com-pare the HDD/SDD estimation schemes in terms of equiva-lent noise bandwidth [7]. This parameter, denoted as Bl , isdened as the bandwidth of an equivalent rectangular PSDfunction with the same maximum value A0, such that A0Blequals the average power of the phase estimates, as shownon Fig. 4.A theoretical expression for the HDD estimator’s equiva-lent noise bandwidth was analytically derived in [3]. A simi-lar result is much more difcult to obtain when soft decisionsFigure 3: PSD of phase estimates for Es/N0 = 0dB and N =32.                                                                                                                                                                                ~~           ~~           ~~           ~~           ~~           ~~           ~~           ~~           ~~           ~~                                                                                                                         fBlA0Rs2NPSD of the estimates∫ Rs/2N0 psd(f)df = A0BlFigure 4: Denition of the equivalent noise bandwidth.are used in the decision process. Thus, we resort to simula-tions in order to evaluate the noise bandwidth of the SDDestimator. More precisely, the values of Bl are deduced fromthe PSD of the estimates when the phase error is constantand normalized by the symbol rate 1Ts . They appear to be in-versely proportional to the estimation block size N. Fig. 5shows the product BlTsN for both HDD and SDD estimationstructures, as a function of the channel SNR Es/N0. It is im-portant to mention the following points:• It is noticeable that BlTs tends to 12N for high SNRs, asfor classical data aided estimators. This was predictable,since at high SNRs, the (soft or hard) decisions are likelyto be reliable. Consequently, both HDD and SDD algo-rithms behave like data aided estimators.• When the SNR decreases and for a given value of N, theSDD noise bandwidth decreases faster than that of theHDD scheme.14th European Signal Processing Conference (EUSIPCO 2006), Florence, Italy, September 4-8, 2006, copyright by EURASIPFigure 5: Equivalent noise bandwidths for the SDD andHDD estimators (obtained by simulation).Finally, we recall that a small equivalent noise bandwidthis favorable to the mitigation of the Gaussian noise effectsin the phase estimation. On the other hand, a large noisebandwidth is more suited to the tracking of a time varyingphase error. A trade-off on this parameter is thus necessary,in order to optimize the estimation performance.5. SIMULATION RESULTSIn this section, the HDD and SDD estimator performance arecompared in terms of root mean square error (RMSE). Sim-ulations with a constant phase error are analysed in section5.1. A time-varying phase error is then considered in section5.2, including a DVB-S2 compliant phase noise model.5.1 Constant phase errorFig. 6 shows the root mean square errors (RMSEs) for thephase estimates versus the signal to noise ratio Es/N0. Thecircle (resp. square) curves corresponds to the the SDD (resp.HDD) estimates. The parameters for this example are N =32 and ϕ = 0◦ (without loss of generality). The MCRBsare computed from (14), where σ 20 has been estimated fromMonte Carlo runs:σ20 '1KK−1∑k=0( ϕk−ϕ)2 , (16)and ϕk are consecutive phase estimates.For a given estimation block size N, the SDD estima-tor outperforms the HDD estimator, especially at low SNRs.For instance, in order to guaranty a RMSE < 8◦ (withN = 32 QPSK symbols per block), the SDD estimator re-quires Es/N0 >−3dB, whereas the HDD estimator requiresEs/N0 > 1dB. A gain of 4dB is thus provided at these SNRlevels.We recall that at low SNR and for a xed value of N, theSDD estimator noise bandwidth is smaller than the HDD es-timator noise bandwidth (cf. Fig. 5), which is more favorablewhen the phase error is constant. However, in most applica-tions, the receiver has to estimate and correct a time vary-ing phase error. The next section studies the performance ofHDD and SDD estimators in the presence of phase noise.Figure 6: RMSEs and MCRBs for HDD and SDD algorithms(Constant phase error, N=32 , QPSK modulation).H2(z)H1(z)ϕphn(k)wk ∼ N(0, 1)Figure 7: Phase noise model .5.2 Time varying phase errorWe propose to evaluate the HDD and SDD estimator perfor-mance in presence of phase noise. The proposed model forphase noise is decribed in Sect. 5.2.1. The estimator perfor-mances are presented in Sect. 5.2.2.5.2.1 Phase noise modelThe proposed phase noise model is based on a white Gaus-sian sequence, sent through two parallel recursive lters H1and H2, as depicted on Fig. 7. With the transfer functionsH1 (z) and H2 (z) given in [8], this model complies with theaggregate mask suggested in [9] for the evaluation of carrierrecovery in DVB-S2 receivers, as shown on Fig. 8.With the notation dened of Fig. 7, the phase error ϕ (k)affecting the k-th transmitted sample can be expressed as:ϕ (k) = ϕ0 +ϕphn (k) , (17)where ϕ0 is the phase error affecting the sample y0.5.2.2 Estimation performanceA trade-off on the noise bandwidth allows to minimize thephase jitter, as explained in section 4. The RMSE is plottedon Fig. 9 for HDD and SDD estimators, when the estimationblock size (which determines the noise bandwidth) is setto its optimal value. We observe that the gain over theSNR provided by the soft decisions approaches 1 dB for−2dB < Es/N0 < 1dB. As a conclusion, the SDD estimatorremains more performant than the HDD estimator for thetracking of phase noise.14th European Signal Processing Conference (EUSIPCO 2006), Florence, Italy, September 4-8, 2006, copyright by EURASIPFigure 8: PSD of the synthetized phase noise and target mask.Figure 9: RMSE in presence of phase noise, with the HDDand SDD estimators and for the optimal estimation blocksizes (QPSK modulation).Remark:At low signal to noise ratio, the optimal block sizes (al-lowing to minimize the MMSEs in presence of phase noise)are greater than the reference block size N = 32 (consideredin sect. 5.1). This explains why in such case the estimatorsseem to perform better in presence of phase noise (Fig. 6)than with a constant phase error (Fig. 9).6. CONCLUSIONThe decision directed feedforward estimators require a signalphase pre-correction step. To ensure this pre-correction, thephase estimate obtained from a block of samples is used tocorrect the samples of the next block. In other words, a feed-back component is introduced into the classical feedforwardestimation structure.This paper derived the modied Cramer-Rao lowerbound for these estimators, assuming that the carrier phaseerror is constant. Besides, equivalent noise bandwidths wereevaluated for both HDD and SDD schemes by means ofsimulations, for different SNR levels. The estimator per-formances obtained for constant and time-varying phase er-rors were also presented. The main conclusion was that theSDD estimator outperformed the HDD estimator in both sit-uations.REFERENCES[1] H. Meyr, M. Moeneclaey, and S.A. Fechtel. DigitalCommunication Receivers: Synchronization, Channelestimation and Signal Processing. J. Wiley and Sons,1998.[2] V. Lottici and M. Luise. Embedding carrier phaserecovery into iterative decoding of turbo-coded linearmodulations. IEEE Transactions on Communications,52(4):661669, April 2004.[3] R. de Gaudenzi. Performance analysis of decision-directed maximum-likelihood phase estimators for M-PSK modulated signals. IEEE Transactions on Commu-nications, 43(12):30903100, December 1995.[4] W. G. Cowley. Phase and frequency estimation for PSKpackets: bounds and algorithms. IEEE Transactions onCommunications, 44(1):2628, January 1996.[5] A. J. Viterbi and A. M. Viterbi. Nonlinear estimation ofPSK-modulated carrier phase with application to burstdigital transmission. IEEE Transactions on InformationTheory, 29(4):543551, July 1983.[6] A.N. D’Andrea, U. Mengali, and R. Reggiannini. Themodied cramer-rao bound and its application to syn-chronization problems. IEEE Transactions on Commu-nications, 42(4):13911399, April 1994.[7] D.R. White. The noise bandwidth of sampled data sys-tems. IEEE Transactions on Instrumentation and Mea-surement, 38(6):10361043, December 1989.[8] L. Benvenuti, L. Giugno, V. Lottici, and M. Luise. Code-aware carrier phase noise compensation on turbo-codedspectrally-efcient high-order modulations. In Proceed-ings of the ESA 8th International Workshop on SignalProcessing for Space Communications (SPSC), pages177184, September 2003.[9] Digital Video Broadcasting (DVB) - Second GenerationFraming Structure, Channel Coding and ModulationSystems for Broadcasting, Interactive Services, NewsGathering and other Broadband Satellite Applications -Ref. ETSI EN 302 307 v1.1.1, January 2004.14th European Signal Processing Conference (EUSIPCO 2006), Florence, Italy, September 4-8, 2006, copyright by EURASIP

A comparison of soft and hard decision-directed feedforward phase estimators

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A comparison of soft and hard decision-directed feedforward phase estimators

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