In this letter, we address the problem of pilot design

for Carrier Frequency Offset (CFO) and data detection in digital burst transmission systems. We consider a quasi-static flat-fading channel. We find that placing half of the pilot symbols at the beginning of the burst and the other half at the end of the burst is optimal for both CFO estimation and data detection. Our findings are based on the Cram´er-Rao bound and on empirical evaluations of the bit error rate for different pilot designs. The

equal-preamble-postamble pilot design is shown to provide a

significant gain in performance over the conventional preambleonly pilot design

Ghogho, M.

Ying, Y.Q.

English

White Rose Research Online

This is a repository copy of Optimal pilot placement for frequency offset estimation and data detection in burst transmission systems.White Rose Research Online URL for this paper:http://eprints.whiterose.ac.uk/691/Article:Ying, Y.Q. and Ghogho, M. (2005) Optimal pilot placement for frequency offset estimation and data detection in burst transmission systems. IEEE Communications Letters, 9 (6). pp.549-551. ISSN 1089-7798 https://doi.org/10.1109/LCOMM.2005.1437367eprints@whiterose.ac.ukhttps://eprints.whiterose.ac.uk/Reuse See Attached Takedown If you consider content in White Rose Research Online to be in breach of UK law, please notify us by emailing eprints@whiterose.ac.uk including the URL of the record and the reason for the withdrawal request. IEEE COMMUNICATIONS LETTERS, VOL. 9, NO. 6, JUNE 2005 549Optimal Pilot Placement forFrequency Offset Estimation andData Detection in Burst Transmission SystemsYeqiu Ying, Student Member, IEEE and Mounir Ghogho, Member, IEEEAbstract— In this letter, we address the problem of pilot designfor Carrier Frequency Offset (CFO) and data detection in digitalburst transmission systems. We consider a quasi-static flat-fadingchannel. We find that placing half of the pilot symbols at thebeginning of the burst and the other half at the end of the burstis optimal for both CFO estimation and data detection. Ourfindings are based on the Cramér-Rao bound and on empiricalevaluations of the bit error rate for different pilot designs. Theequal-preamble-postamble pilot design is shown to provide asignificant gain in performance over the conventional preamble-only pilot design.I. INTRODUCTIONP ILOT Symbol Assisted Transmission (PSAT) is a prac-tical technique in wireless communication systems. Pilotsymbols, which are usually time-division multiplexed with theinformation bearing symbols, are used for carrier recoveryand channel estimation. In a burst-mode transmission, theway the pilot symbols are placed in each burst often affectsthe performance of the system. Recent papers have derivedoptimal pilot designs for channel estimation and data detectionin the context of time/frequency-selective channels e.g. [1],[4]. The issue of optimal pilot design when there is a mismatchbetween the transmitter and receiver carrier frequencies hasreceived little attention in the literature. This frequency mis-match or Carrier Frequency Offset (CFO), which is causedby oscillator instability and Doppler effect in mobile commu-nications, needs to be accurately estimated and compensatedfor, particularly in the case of large size constellations (seee.g. [2], [3] and references therein). In [5], a white trainingsequence was shown to be asymptotically optimal for jointCFO and channel estimation using the worst-case asymptoticCRB. However, the pilot symbols were grouped into a singlecluster and optimal pilot placement was not investigated. In[6], for OFDM systems, the CFO was estimated using null-subcarriers and optimal placement of these null-subcarrierswas derived.In this letter, we consider a single carrier burst transmissionsystem and a quasi-static flat fading channel, i.e. the channelis a complex scalar which is fixed over each burst but changesacross the bursts. The system is also affected by a CFO whichmay vary across the bursts. This might occur in mobile satellitecommunications where the frequency mismatch can signifi-cantly vary over time. Here, we derive the pilot placementManuscript received October 14, 2004. The associate editor coordinatingthe review of this letter and approving it for publication was Dr. StellaBatalama.The authors are with the University of Leeds, LS2 9JT, UK. The work ofM. Ghogho was partly supported by the UK Royal Academy of Engineering.Digital Object Identifier 10.1109/LCOMM.2005.06007.that minimizes the Bit Error Rate (BER). We achieve this byminimizing the CRB on the CFO and channel estimations andthrough Monte-Carlo simulations.II. SIGNAL MODEL AND PRELIMINARIESConsider a single carrier burst transmission system oper-ating over a quasi-static flat fading channel. Assuming thattiming is perfectly synchronized, the received burst signal aftermatched filtering and sampling can be modelled as [2]x(n) = ρej(ωn+φ)s(n) + v(n) n ∈ [0, 1, . . . , N − 1] (1)where N is the burst length, ρ and φ are respectively themagnitude and phase of the channel, ω (−π < ω ≤ π) is thenormalized CFO, {s(n)} is the transmitted symbol sequencewhich encompasses both pilot and information-bearing (ordata) symbols, and v(n) is an additive noise with varianceσ2.In burst-by-burst processing, the CFO and the channel areestimated at each burst by making a number, say P , of the Ntransmitted symbols known to the receiver. Our objective is tofind the optimal placement of these pilot symbols in terms ofCFO and channel estimation accuracy and BER. In our pilotdesign, we make the following assumptions:(A1) The P pilot symbols have the same magnitude, as-sumed unity without loss of generality.(A2) v(n) is a circularly symmetric white Gaussian noise.Note that (A1) is usually made in PSAT systems, where thepilots are chosen to be either BPSK or QPSK constellations.However, this assumption could be relaxed by allowing the pi-lot symbols to have different magnitudes. The design problemin this case is outside the scope of this letter.Let P and D denote complementary subsets of {0, ..., N −1} which respectively represent the placements of the pilotand data symbols in a burst. In coherent PSAT systems, ω,φ and ρ are first estimated using {x(n), n ∈ P}; let theseestimates be denoted by ω̂, φ̂ and ρ̂. These estimates are thenused to compensate for the distortions due the CFO and thechannel. This provides a soft decision on the data symbols,which is given byŝ(n) = f(ρ̂)e−jψ̂nx(n), n ∈ D (2)where ψ̂n = ω̂n + φ̂ is an estimate of the overall phase tobe compensated for, ψn = ωn + φ, and f(ρ̂) = 1/ρ̂ in thecase of zero-forcing equalization and f(ρ̂) = ρ̂/(ρ̂2 + σ2)in the case of minimum mean square error equalization. Thedata detection is then performed on a symbol-by-symbol basisby minimizing the Euclidian distance between ŝ(n) and theconstellations points used in the symbol mapping stage of data1089-7798/05$20.00 c© 2005 IEEE550 IEEE COMMUNICATIONS LETTERS, VOL. 9, NO. 6, JUNE 2005generation. It is therefore imperative to obtain an accurateestimation of the ψn’s and ρ in order to achieve good detectionperformance.III. OPTIMAL PLACEMENT OF PILOT SYMBOLSThe criterion we use to derive the optimal pilot placement isbased on the CRB, which is an algorithm-independent measureof performance. Indeed, the CRB is a lower bound on thevariance of any unbiased estimator. The use of the CRB isalso motivated by the fact that the performance of maximumlikelihood (ML) estimation asymptotically achieves the CRB.A. Cramér-Rao BoundLet θ = [ρ, φ, ω]T be the parameter vector to be estimated.The CRB is obtained from the diagonal elements of the inverseof the Fisher Information Matrix (FIM), which is given by [7]F := −E{∂2L(xP |θ)∂θ∂θT}where E {·} denotes the statistical expectation operator andL(xP |θ) is the log-likelihood function of xP := {x(n), n ∈P} given θ. Under assumptions (A1)-(A2), L(xP |θ) is, afterremoving irrelevant terms, given byL(xP |θ) = −1σ2∑n∈P[−2ρℜ{x(n)s∗(n)e−j(ωn+φ)} + ρ2]where ℜ{·} is the real part operator. The FIM is found to beF =2ρ2σ2⎡⎢⎢⎣Pρ20 00 P∑n∈Pn0∑n∈Pn∑n∈Pn2⎤⎥⎥⎦Thus, the CRBs for ρ, φ and ω are given by:CRB(ρ) =σ22P(3)CRB(φ) =12 SNR∑n∈Pn2P∑n∈Pn2 −(∑n∈Pn)2 (4)CRB(ω) =12 SNRPP∑n∈Pn2 −(∑n∈Pn)2 (5)where SNR := ρ2σ2is the Signal-to-Noise Ratio (SNR).Using the inverse of the FIM, the CRBs for ψm, m ∈ D,are obtained asCRB(ψm) =12 SNRm2P − 2m∑n∈Pn +∑n∈Pn2P∑n∈Pn2 −(∑n∈Pn)2 , m ∈ D(6)B. Optimal Pilot PlacementSince the CRB for ρ is independent of P , only the CRBsfor {ψm, m ∈ D} are relevant in the pilot placement design.We therefore propose the following optimum pilot placementP∗ = arg minP∑m∈DCRB(ψm) (7)Preamble Data PostambleFig. 1. Optimal pilot placement design.Solving the above optimization problem analytically seemsuntractable. Hence, we resort to computer analysis. By car-rying out an exhaustive evaluation of the design criterionover all possible pilot placements, P∗ is found to consistof a preamble (i.e. a cluster of symbols at the beginning ofthe burst) and a postamble (i.e. a cluster of symbols at theend of the burst) of equal size; see Fig. 1. If P is odd,either the preamble or the postamble can have one moresymbol than the other without affecting the design criterion.Interestingly, the conventional preamble-only design gives theworst performance, as illustrated in Figs. 2 and 3 (details aregiven in Section IV). The following results have also beenobtained by computer analysis:• The equal-preamble-postamble (EPP) pilot design alsominimizes the CRB for ω.• The pilot placement that minimizes the CRB for φ is alsoa preamble-postamble placement. However, the size ofthe postamble is either one or two symbols. Neverthless,the CRB for φ when the preamble and postamble are ofequal size is quite close to the minimum of CRB(φ).• The EPP pilot design also minimizes the CRB(ψm)’sindividually.We now make the following relevant comments.1) It is worth recalling that in the absence of CFO, the CRBfor φ is independent of P . Therefore, it is the presence of anunknown CFO that makes the CRB of φ dependent on thepilot placement.2) Let am and bm denote the amplitude and angle of thedata symbol s(m). Thus, from (1), the soft detection outputin eq. (2) can be expressed as:ŝ(m) = f(ρ̂)ρamej(bm+∆ψm) + ṽ(m), m ∈ Dwhere ∆ψm = ψm − ψ̂m and ṽ(m) = f(ρ̂)v(m). Sincethe estimation performance for ρ is independent of P , it isreasonable to expect that the probability of detection errorfor each data symbol, s(m), is minimized when the meansquare of ∆ψm is also minimized, which is obtained with theEPP pilot design. Although this statement lacks a rigorouslytheoretical proof, it is well supported by simulation results(see Section IV.)3) After some algebra, the CRB for ω in eq. (5) can bere-expressed asCRB(ω) =1SNRPP∑i=1P∑j=1;j =i(ni − nj)2where ni, i = 1, ..., P , denote the time indexes of the pilotsymbols. This expression shows that CRB(ω) decreases withthe sum of square differences (ni−nj)2. Using this expression,we can analytically prove for P = 2, 3, 4 that the EPPdesign minimizes CRB(ω). For P ≥ 5, the analytical proofbecomes tedious. As to why the EPP design is best in termsof CFO estimation, we present the following explanation. Forsimplicity, assume that P = 2 and that the pilot symbolsYING and GHOGHO: OPTIMAL PILOT PLACEMENT FOR FREQUENCY OFFSET ESTIMATION AND DATA DETECTION 5510 5 10 15 20 25 3010−610−510−410−310−210−1100101102103Number of pilot symbols in postambleMSEN=148, P=26, SNR=3dBCRB(ω)MLE(ω)CRB(h)MLE(h)CRB(ψ)MLE(ψ)Fig. 2. MSEs and CRBs for different preamble-postamble designs.0 5 10 15 20 25 3010−510−410−310−210−1N=148, P=26Number of pilot symbols in postambleBERQPSK 12dB16−QAM 18dB64−QAM 23dBFig. 3. Bit error rate for different preamble-postamble designs.are both equal to unity. The received pilot symbols are thenx(ni) = ρ exp(jωni + φ) + v(ni), i = 1, 2, where n1and n2 can take any distinct values from {0, ..., N − 1}.The maximum likelihood estimate of ω can be shown to beω̂ = arg{x∗(n1)x(n2)}/(n2 − n1) where arg{.} denotes theargument (or angle) operator. The variance of the numerator isindependent of n1 and n2. Thus, the variance of ω̂ is inverselyproportional to (n2 − n1)2. This variance is minimum whenn1 = 0 and n2 = N − 1, which is the EPP design. It ishowever difficult to extend this reasoning to the case whereP > 2.4) The EPP pilot design guarantees a full acquisition rangeprovided P > 2. Indeed, with P = 3, either the preamble orthe postamble consists of two consecutive pilot symbols.5) Using the EPP pilot design and assuming P even, weobtain the following close-form CRB expressionsCRB(ω) =12 SNR12P [P 2 − 3NP + (3N2 − 1)]CRB(φ) =12 SNRP 2 − 3NP + 6N2 − 6N + 2P [P 2 − 3NP + (3N2 − 1)]IV. SIMULATION RESULTSHere, we illustrate the superiority of the proposed EPPdesign over the conventional preamble-only design using theCRB, the empirical MSE of the maximum likelihood (ML)estimates and the BER. In the evaluation of the latter, we usedQPSK, 16-QAM and 64-QAM. The amplitude ρ was set tounity and the phase φ was randomly generated using a uniformdistribution over (−π, π]. The CFO was set to ω = 0.2π. Theburst size was N = 148 and the total number of pilot symbolswas P = 26. We evaluate the CRB for ω, h := ρ exp(jφ), andthe sum of the CRBs for the ψm’s, m ∈ D, versus differentpreamble-postamble designs. Figs. 2 and 3 display the resultsversus the number of pilot symbols in the postamble. It can beseen that the EPP design provides a significant improvementover the preamble-only design in terms of both estimationaccuracy and data detection performance. It is however worthpointing out that if the preamble and postamble are of unequalsize, the performance is close to that of the EPP designprovided that the preamble or the postamble consists of morethan two pilots. We have also run simulations (not shown herebecause of space limitation) to compare the performance of theEPP design with that of the equally spaced pilot (ESP) design(i.e. the pilots are uniformly scattered across the block), whichis traditionally used for channel estimation [4]. Although theEPP design outperforms the ESP design, the difference inperformance is not as significant as that between the EPPdesign and the preamble-only design. However, it is worthpointing out that unlike the EPP design the ESP design doesnot provide full frequency acquisition range.V. CONCLUSIONSIn burst-mode transmission where the carrier frequency-offset needs to be estimated, we have proven that by split-ting the pilot sequence into a preamble and a postamble ofequal size, a significant gain in system performance over thepreamble-only pilot design can be obtained. Although the gainin performance over the equally spaced pilot design is notsignificant, the proposed design has the advantage of providingfull frequency acquisition range. The proposed pilot designmay be particularly useful when the frequency-offset mayvary across the bursts as might be the case for some mobilecommunication systems where a significant Doppler effect ispresent. The disadvantage of the preamble-postamble schemeis its unsuitability for real-time burst processing. Future workwill address the case of frequency selective channels.REFERENCES[1] S. Adireddy, L. Tong, and H. Viswanathan, “Optimal placement oftraining for frequency-selective block-fading channels,” IEEE Trans.Info. Theory, vol. 48, pp. 2338-2352, Aug. 2002.[2] U. Mengali and M. Morelli, “Data-aided frequency estimation for burstdigital transmission,” IEEE Trans. Commun., vol. 45, pp. 23-25, Jan.1997.[3] J. Li, G. Liu, and G. B. Giannakis, “Carrier frequency offset estimationfor OFDM-based WLANs,” IEEE Signal Processing Lett., vol. 8, pp.80-82, Mar. 2001.[4] M. Dong, B. M. Sadler, and L. Tong, “Pilot assisted wireless trans-missions: general model, design criteria, and signal processing,” IEEESignal Processing Mag., vol. 21, pp. 12-25, Nov. 2004.[5] P. Stoica and O. Besson, “Training sequence design for frequency offsetand frequency-selective channel estimation,” IEEE Trans. Commun., vol.51, pp. 1910-1917, Nov. 2003.[6] M. Ghogho, A. Swami, and G. Giannakis, “Opimized null-subcarrierselection for CFO estimation in OFDM over frequency-selective fadingchannels,” in Proc. Globecom-2001, vol. 1, pp. 25-29, Nov. 2001.[7] S. Kay, Fundamentals of Statistical Signal Processing, Vol I - EstimationTheory. Prentice Hall PTR, 1993.

Optimal pilot placement for frequency offset estimation and data detection in burst transmission systems

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Optimal pilot placement for frequency offset estimation and data detection in burst transmission systems

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