Iterative Frame Synchronization for

Frequency-Selective Channels

Ayman Abdel-SamadIC Design, MIMOS Berhad

Technology Park Malaysia, 57000 Kuala LumpurEmail: ayman.ahmed@mimos.my

Abstract- Iterative receiver techniques provide a powerful toolto enhance receiver performance and are gradually becoming thenorm in modern communication systems. Although they havebeen used in various receiver components like channel estimation,equalization and decoding, they have not been explicitly appliedto frame synchronization. In this paper, we propose an efficientiterative frame synchronization technique in which the hard-output, un-coded data at the output of the equalizer is utilizedto improve the accuracy of synchronization. Simulation resultsshow huge signal-to-noise-ratio (SNR) gains in terms of falseacquisition probability (FAP), mean square channel estimationerror (MSCEE) and un-coded bit error rate (BER) at allSNR values. The proposed approach can be easily applied tocontinuous or burst type frame synchronization under additivewhite Gaussian noise (AWGN) in addition to flat or frequency-selective channels.

I. INTRODUCTION

Frame synchronization is used to locate a fixed train-ing sequence that is inserted either periodically in a datastream in the case of continuous transmission [1], or atthe beginning/middle of a data packet in the case of bursttransmission [2]. It is also used in the process of locatingpilot symbols that are periodically inserted in data streams inpilot-symbol-assisted modulation (PSAM) [3]. The problem offrame synchronization received a lot of attention in the liter-ature starting from the derivation of the optimum maximumlikelihood (ML) rule for binary phase shift keying (BPSK)under AWGN channels [4]. Subsequently, various continuous-mode and burst type frame synchronizers have been derivedfor different modulation schemes under AWGN in addition toflat and frequency-selective fading channels (see [1], [2], [3]and references therein).

Since the introduction of Turbo codes [5], and promptedby the their unprecedented gains, the concept of iterativeprocessing has been extended to a variety of receiver functionssuch as equalization-decoding [6], [7], [8], which is knownas turbo equalization, and channel estimation [9]. Recentlyiterative processing was also extended to acquisition functionslike timing recovery [10] and estimation of carrier phase andfrequency or timing offsets [11]. The recent work, however,almost exclusively focuses on Turbo receivers and/or magneticrecording channels.The differences between continuous-mode and burst frame

synchronization pertain to the length of the observation periodin addition to the prior knowledge about when the frame starts.

0-7803-9206-X/05/$20.00 ©2005 IEEE

In continuous-mode, the length of the observation period isequal to the length of the whole frame which includes thelength of both the training and data. It is also assumed thatthe frame could start anywhere within the observation period.On the other hand, in the burst case the observation periodis shorter since it is only made long enough to include thetraining sequence while the frame is assumed to start aroundsome coarse estimate provided by automatic gain control(AGC) or an energy detector [2], [12]. Note that whereasin both continuous and burst modes the training sequenceis grouped together, in the case of PSAM it is actuallyinterspersed within the data stream in addition to being usuallychosen from a different symbol set than that of the data.

Generally, there are two main approaches for deriving theML frame synchronization rule. The first is the Bayesianapproach in which unknown variables, like channel coeffi-cients, are treated as random variables with known probabilitydensity functions (pdf's). The marginal likelihood function forframe synchronization is then obtained by averaging the jointlikelihood function over the pdf's of unknown variables. Theother approach is to jointly estimate the frame boundary alongwith other unknown variables. In this paper we follow the latterapproach and jointly estimate the frame boundary and thechannel coefficients. Other unknowns, like carrier frequencyoffset can be also estimated following the approach in [1],[12] and [13].The paper is organized as follows. The system model is

introduced in section II. This is followed, in section III, bydeveloping an ML-based joint frame synchronizer and channelestimator in addition to outlining the iterative frame synchro-nization process. Simulation results are given in section IVand finally conclusions are drawn in section V.

II. SYSTEM MODELWe consider the transmission of linearly modulated data

through an L tap frequency-selective channel given by h =

[h1,h2, . , hL]T, where L is the channel memory and [.]Tdenotes transposition. We assume throughout our formulationthat the channel is constant during the observation interval andthe case of a fast-fading channel is briefly discussed at the endof section III. The symbol-sampled received signal at time kis given by

L

rk =E Sk-i+l hi + nk

i=1(1)

539

where Sk's are the transmitted symbols and nk is additive,i.i.d. complex Gaussian noise with variance No. We assumefor simplicity that E{8sk12} - 1.

In order to simplify subsequent notation we reformulate thisequation into matrix notation. For this purpose, we define the(N-L + 1) x 1 vectors rk = [rk, rk+1,. , rk+N-L]T andnk = [nk,n+1, nk+N-L]T and finally the (N-L+ 1) xL matrix, Sk as

Sk Sk-1 ... Sk-L+1Sk=F Sk+1 Sk Sk-L+2 l

5k =. . (2)

Sk+N-L Sk+N-L-1 ... Sk+N-2L+lWe can now rewrite equation (1) as

rk = Skh + nk. (3)

Note that the symbols matrix, Sk contains N distinct symbolsand that, due to the channel memory of length L, every N -L + 1 received samples depend on N transmitted symbols.

III. JOINT FRAME SYNCHRONIZATION AND CHANNELESTIMATION

In this section we derive the joint likelihood function forframe synchronization and channel estimation. Let Sm denotethe matrix of the known Nt training symbols {t1,, - tN, }arranged according to equation (2) with N = Nt. Note thatby restricting the size of rk to (N - L + 1) x 1 and underthe assumption that k is the correct starting position of thetraining sequence, rk only depends on the training sequence.Thus, the conditional pdf of rk does not depend on unknownrandom data. This results in a substantially less complex MLapproach since we do not have to average the conditional pdfover all possible unknown data.The conditional pdf of rk assuming h and m (the correct

starting position) is given by

e- 1 (rk-S,,h)71(rk-S,hP(rklh,m) = e (r,No )h * (4)

where [.]X denotes conjugate transposition. Because the train-ing sequence is assumed equally likely to start anywhere inthe search interval, it is straightforward to show that the MLestimate, min, of where the training sequence starts1 is given by

m = arg max - (rk - Smh) (rk - Smh). (5)k

Since the channel h is unknown, we use the ML channelestimate, assuming that the training sequence starts at k, whichis given by

h(k) = (SHSm) S"rk. (6)Finally, the ML synchronization is obtained as

n = arg max rH (P-I) rk (7)k

k

'Note that we actually synchronize on the Lth symbol of the N knownsymbols in Sk, in this case Sk.

where P = Sm (SHSm)' SX is a projection matrix.The formulation so far is general and accommodates both

burst and continuous-mode frame synchronization. Indeed,the only difference between this metric and that of [1] forcontinuous-mode is that the elements of rk are obtained hereby linear shifts of the received data while they are obtained bycircular shifts in the case of continuous-mode synchronization,due to the periodicity of occurrence of the training sequence.

It is also worth mentioning that the computation of thismetric involves only multiplications and additions since thematrix P is known before hand and thus the matrix inversionneed not be done in real-time. Although we did not accountfor the presence of carrier frequency offset, it could be easilyincorporated but will involve performing an FFT for each kin the search range as explained in [1], [12], [13].

A. Iterative synchronizationIn this section we outline the steps involved in per-

forming iterative synchronization. After obtaining the ini-tial synchronization, mii(°), using equation (7) the data isequalized. We then form the new matrix S(), using equa-tion (2) with N = Nt + Nd, with the Nt training sym-bols appended by the Nd estimated data symbols d(, i.e.{t1,..., tNt, d(°),... d()}, in scenarios where the trainingsequence precedes the data (preamble). If the training sequencefalls in the middle of the data (midamble) then we haveN = Nd, + Nt + Nd2 and the matrix S() is formed using thesequence {d( )

I d , t1 ... I tNt Id, I ... Id( . UsingS(°) in place of Sm in equation (7), we can obtain mn(1) andso on. The recursion could be written as

n('+-)= argmax (p(i) - I rkk r

(8)

where p(i) - S t) (S -t)S(-$)) S(-)R and i = 1,2,....During that process, new and improved channel and dataestimates h(i) and d(i) are obtained.

Contrary to the case of training based synchronization in (7),the matrix P(i) involves data that is not know before handand thus the inversion of the matrix (S(i)HS()) has to bedone in real-time. Fortunately, the matrix size is only L xL and thus the complexity of the matrix inversion is only0(L3). The matrix could be inverted using available efficientimplementations of the Cholesky decomposition [14]. Notethat the ideal values of 7hnj) for i > 0 are not equal to mn(°).

Although it is a common practice in frame synchroniza-tion to assume that the channel is constant throughout theobservation interval, this assumption only holds true in slow-fading channels. For fast-fading channels, which is not pursuedhere, it is possible to adopt the approach proposed in [9] foriterative channel estimation. In this approach, the data burst isdivided into blocks over which the channel is assumed to beapproximately constant. The choice of a block size involvesa tradeoff between two types of errors. If the block is tooshort, this will result in a noisy channel estimate. On theother hand, if the block is too long, the channel undergoes

540

a large change during that block. Note that specificationsof wireless communication systems are designed in order toensure adequate performance in specific fading environments.Thus, it is fair to assume that in general the channel wouldnot change much during at least the length of the trainingsequence. Indeed, it was found out in [9] that using a blocksize equal in length to the training sequence offers a goodcompromise under fading conditions specified by the GSMstandard.

Having divided the data burst into blocks, we can computethe synchronization metric for each block and use the multi-frame majority rule in [1]. The summation rules in [2] and [15]can also be used although they will be suboptimal since thechannel realizations in different blocks are not independent.

IV. SIMULATION RESULTSIn this section we demonstrate the performance of the

proposed algorithm using simulations. We consider BPSKmodulation with the SNR defined as 101og10(l/No). Thefrequency-selective channel coefficients are zero-mean, i.i.d.complex Gaussian random variables and we use L = 4.The channel h is normalized such that llhll = 1 in orderto suppress time-selective fading and diversity effects therebyonly focusing on ISI effects [16]. We use a Max-Log-MAPequalizer and assume a GSM-like packet structure in whichthe training sequence is in the middle of data. The GSM burststructure has 26 training symbols preceded and followed by58 symbols of data. However, instead of having a 26-symboltraining sequence, we use a 16-symbol training sequence inthe middle of 126 data symbols, thus maintaining the samepacket length. By doing this, we demonstrate how iterativeframe synchronization can improve the performance whileusing less training overhead. All simulations are done using100,000 Monte Carlo runs.

Fig. 1 shows the FAP (rm(i) im). It shows the FAP whenonly the training sequence is used for synchronization (iter-ation 0) and when iterative synchronization is used (iteration1-3) in addition to the reference case in which the transmitteddata is perfectly known and used for synchronization. It isclear from this figure that iterative synchronization achieves ahuge SNR gain, even at low SNR values. It also shows that atreasonable SNR values we easily approach the FAP accuracyachievable if we perfectly knew all the data transmitted.

Fig. 2 shows the MSCEE, th(i)-h12 . Note that the MSCEEis hugely dependent on achieving correct synchronizationwhich is evident from the two cases of estimated and perfectsynchronization when using only the training and all data. Itis again obvious that iterative synchronization results in hugeSNR gains and that we approach the case of perfect dataknowledge at moderate SNR.

In Fig. 3 we examine the probabilities of correction anderror regarding FAP. The probability of correction, P,(i -*

j), corresponds to the event in which we make a wrongsynchronization in iteration i that gets corrected in iterationj > i while the probability of error, Pe(i -* j), correspondsto us making a correct synchronization in iteration i but

........ Iteration 0... ..... . . . . . ......... ......... ....... ......

.i:yIteration I.. Iteration 2

-- Iteration 3.~~~~~~--9 Perfectdata knowledge

...

... ...\ . ...... 0 .. ....... .. ' s.,>. \ . ...... .. ..... . . . . . . . . . . . ...

.. ......... ..... ..... ...... .. . .. . . .

... .... ......

.............. ..... .D.......

........

10o-3

n I

False Acquisition Probability

u I S3 4 5 6 7SNR (dB)

Fig. 1. False Acquisition Probability versus SNR

end up making a wrong one in iteration j > i. The figureshows that the probabilities of correction PC are orders ofmagnitude higher that probabilities of error Pe and that Pe'sfall sharply with SNR which is expected. It also shows thatPC(O -* 1), P,(0 -- 2) and P,(O -÷ 3) increase with SNRwhile PC(1 -* 2), PC(1 -- 3) and PC(2 -+ 3) start fallingat higher SNR. The decline in PC at higher SNR is due tothe fact that the probability of a wrong synchronization afteriterations 1 and 2 is very low and that in this case furtheriterations are less likely to correct them. This highlights whathas been apparent in Figs. 1 and 2 which is that not much isgained in terms of FAP or MSCEE after the first or seconditerations.

Finally, Fig. 4 shows the BER at the output of the equalizerand it is obvious that, even with a 40% reduction in the lengthof the training sequence, BER performance close to that ofideal synchronization and channel knowledge is achieved withonly one iteration. It is again clear that subsequent iterationsprovide little gain. Note that the SNR gain in BER, whilesubstantial, is less than that for FAP and MSCEE. This isbecause the dependence of BER on FAP and MSCEE is highlynonlinear.

V. CONCLUSIONSIn this paper, we proposed an efficient iterative frame

synchronization technique in which we used the hard-output,un-coded data at the output of the equalizer to obtain a new andimproved synchronization. We verified through simulationsthat huge SNR gains are obtained for FAP, MSCEE andun-coded BER with performance very close to the case ofperfect synchronization and channel knowledge. The proposedapproach can be applied to burst or continuous-mode framesynchronization under AWGN and flat or frequency-selectivefading channels. Using this approach, a considerable reductionin the training overhead can be easily achieved with the

541

ino

(LLI

8 9 1011

Mean Square Channel Estimation Error10.. ..

-*-iteration-Iteration

Iteration 2-Iteration 3

-0-E Pertect sync, trainingEstimated sync, perfect data

100 ~~~~~~~~~~~~~~~~~~~~---Pertect sync, pertect data

10

Fig. 2. Mean square channel estimation error versus SNR

id'

1 0-''

-2-,Nz22a.

10o-

o-i

Probabilities ot correctiona and errors

----

Nip~~~~~~

....3.

0 1 2 3 4 5 6 7 8 9

SNR (dB)

Fig. 3. Probabilities of correction and error versus SNR

possibility of achieving even higher gains when using coded

data or when combining this approach with turbo equalization

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IVSNA (dB)