Frame Synchronization for OFDMUsing Cyclic Prefix and Pilot Carriers

Stephen G. Wilson, Rui Shang, Ruixuan HeDepartment of Electrical and Computer Engineering

University of VirginiaCharlottesville, VA 22904

Abstract—We present an optimal frame timing estimator forOFDM signals that utilizes both cyclic prefix (CP) and pilot-carrier-aided channel estimation. Combining the periodic natureof the embedded pilot signal information with the correlationpresent in the cyclic prefix allows improved frame timing esti-mates relative to use of either one. Following a derivation of theML estimator, a processor diagram is provided, and simulationresults compare the performance of the new synchronizer withthat of Sandell et al [1] that employs only the CP structure. Inthe present paper, we assume negligible frequency offset due toDoppler and/or oscillator frequency error.

I. INTRODUCTION

OFDM represents a popular physical layer access techniquefor a variety of wireless (802.11g, LTE, DAB) as well aswired (DSL) scenarios. The primary attraction of OFDM is theability to efficiently equalize for frequency-selective fading ona channel, relative to the case of single-carrier transmissionwith the same bandwidth, [2]. Information is delivered inOFDM frames, or symbols, and it is important to be able tocorrectly parse the received sample stream into frames priorto FFT processing and channel equalization in the frequencydomain. This the frame sync problem.

While special synchronization symbols can be inserted intoa sequence of frames to provide frame timing, these representoverhead and loss of spectral efficiency. So techniques basedon the intrinsic OFDM signal format are of interest. Mostnotable of these is the procedure of Sandell et al [1], whoutilize the repetition of the complex time domain signal presentin cyclic prefixing to develop a frame synchronizer (as wellas frequency offset estimator). The technique is relativelysimple and can be implemented as a streaming processor ona sequence of OFDM frames. It is also a non-coherent, orphase-blind, synchronizer, as required.

In this paper, we utilize additional information residing inthe form of known pilot subcarriers, typically spaced uni-formly across the frequency band. These subcarriers provide ameans of estimating the channel frequency response at thesefrequencies, and interpolation provides channel estimates atall subcarrier frequencies. These are then used to compensateeach of the FFT outputs in the frequency domain. Pilots alsorepresent overhead, but the technique is standard, and there islittle alternative. The IFFT of these pilot signals is a complexsequence in time which is known except for time delay, and

Work supported by Laulima Systems under National Spec-trum Consortium contract NSC-16-0140-002

the pilot signal alone provides a means of establishing framesync, via matched filtering the received data against the pilotsignal template. However, the pilot signal power is a smallfraction of the total power, and estimator variance may becorrespondingly too large.

So, we formulate the problem of ML estimation of theframe boundary in a sequence of frames, using both the CPand pilot signal structure. The resulting processing is morecomplex than that derived from use of the pilot signal alone, orthe CP structure alone, but feasible for DSP/FPGA hardware,and will perform better than either simpler frame estimator.Our estimator collapses to simpler structures if one choosesto ignore CP structure, or to ignore the pilot information.

II. PROBLEM FORMULATION

We adopt the standard OFDM signal format, with N subcar-riers per OFDM symbol. P of these subchannel symbols areknown pilot symbols, with P typically being a small fraction ofN . Usually some of the remaining N−P carriers are zeroed outto provide interchannel guard bands in frequency, and oftenthe subcarrier corresponding to zero frequency (DC) is alsonulled out.

Following the IFFT at the transmitter, producing N complextime domain samples per frame, a cyclic prefix (CP) of lengthL is appended at the beginning of every frame. L shouldbe longer than the anticipated channel impulse response inequal units, so that the physical channel’s linear convolutionaction can be equated with a circular convolution by discardingthe cyclic prefix segment at the receiver. Hence transform-domain multiplication of the channel can be undone by simplecomplex gain correction versus frequency. For good efficiencywe want N >> L and N >> P.

We denote the time domain sequence corresponding to asequence of frames, following CP insertion, by xn. This signalis the superposition of a signal µn which is the IFFT of onlythe pilots, and the contribution of the information-carryingdata subcarriers, dn, which is unknown but containing the CPrepetition in each frame:

xn = dn + µn (1)

The pilot sequence is periodic in time with period N + L,assuming the pilot symbols are fixed over frames. (BeforeCP extension the pilot signal has period N .) At the receiver,

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we measure a delayed version of the signal xn, together withadditive noise, denoted rn1.

The following setup follows closely that of Sandell, withminor difference due to our incorporation of a known meansignal. Figure 1 depicts a time line of M(N + L) samples atthe receiver, which encompasses M − 1 complete frames plusfragments of an earlier frame and the next frame. Samples areindexed 0 to M(N + L) − 1. No is the unknown index of thebeginning of the first complete frame, where the beginning offrame is the first sample of the CP. Successive frames begin atNo + N + L, No + 2(N + L), .... In Figure 1, we have identifiedthe CP segments as Im, as well as the segments at the end ofthe frame to which the CP is identical, denoted I

′

m.

Fig. 1. Record of M(N+L) samples

We seek the maximum likelihood estimate for the parameterNo, and require a probabilistic model for the observations,conditioned upon No. For our initial development, we assumethat the known sequence µn arrives without phase rotation.In practice there will be an unknown phase rotation due topropagation delay and other physical effects, but incorporationof unknown phase is reserved until after development of thesynchronizer for zero phase angle, for notational simplicity.

The samples rn are modeled as Gaussian random variables,owing to a central limit theorem argument when N is large.The marginal p.d.f. for rn is Gaussian with mean µn−No andthe variance of each sample is σ2

d+ σ2

n , where σ2d

is the datasignal power and σ2

n is the complex noise power. For later usewe define SNR = σ2

d/σ2

n . Except for pairs of samples residingin the CP interval, we assume rn’s are also independent r.v.’s,also a result attached to FFT’s of large size.

Because of CP repetition, we have that rn and rn+N , for n ∈Im are strongly correlated, and only decorrelated because ofadditive noise. Such pairs have a bivariate Gaussian p.d.f. withmean values µn−No and µn+N−No respectively. The covariancematrix for this pair of samples is

K =[σ2d+ σ2

n σ2d

σ2d

σ2d+ σ2

n

](2)

Consequently, again following the notation of Sandell, we havethe log likelihood for the observation, conditioned on someoffset No, is

Λ(No) = log( ∏k∈I1

f2(rk, rk+N |No) ·∏k∈I2

f2(rk, rk+N |No)·

· · ·∏

k∈IM−1

f2(rk, rk+N |No) ·∏

other k

f1(rk |No)

)(3)

1In our derivation we ignore dispersive channel effects due to multipath.

= logM−1∏m=1

( ∏k∈I1

f2(rk+m(N+L), rk+N+m(N+L) |No)

f1(rk+m(N+L) |No) · f1(rk+N+m(N+L) |No)

·∏

other k

f1(rk |No)

)(4)

where f1(.) is a one-dimensional p.d.f for a complex Gaussianr.v. and f2(.) is a bivariate Gaussian p.d.f.

After substitution of the Gaussian p.d.f.’s, taking the loga-rithm, and expanding the quadratic form in f2(.), as well aseliminating additive constant terms in the log likelihood thatare independent of No we find that the ML estimator is

N̂0 = argmaxM−1∑m=1

{No+L−1∑k=No

{2Re

(rk+m(N+L)r∗k+N+m(N+L)

+ µk+m(N+L)−Noµ∗k+N+m(N+L)−No

− rk+m(N+L)µ∗k+N+m(N+L)−No

− rk+N+m(N+L)µ∗k+m(N+L)−No

)− ρ

[|rk+m(N+L) |2 + |rk+N+m(N+L) |2

+ |µk+m(N+L)−No|2 + |µk+N+m(N+L)−No

|2

− 2Re(rk+m(N+L)µ∗k+m(N+L)−No

+ rk+N+m(N+L)µ∗k+N+m(N+L)−No

)]}+ C

No+N+L−1∑k=No

2Re(rk µ∗k−No)

}(5)

where C =σ4

n+2σ2nσ

2d

σ2d(σ2

n+σ2d)= 1+2 SNR

SNR2+SNR, and ρ =

σ2d

σ2n+σ

2d

. ρ isapproximately 1 for large SNR.

The metric Λ(No) to be maximized over No involves sumsover the entire record and sums over the portions corre-sponding to the CP intervals. Several of these terms involvecorrelation of the received samples with the pilot sequence.Others involve sums of products such as rkr∗

k+Nas in [1].

Moreover, because we have µi = µi+N, i = 0, 1, ...L − 1 by thecyclic prefix property, two pairs of the partial sums in the loglikelihood expansion of (5) are identical, except for a scalefactor ρ. Also, the terms involving only µk samples may besimplified to |µk−No − µk−N−No |

2, which is zero by the samereasoning. The contribution to the log likelihood for a singleframe of length N + L then becomes

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No+N+L−1∑l=No+N

{2Re

(rlr∗l−N

)− ρ

(|rl−N |2 + |rl |2

)}︸ ︷︷ ︸

1©

+ 2Re

{(ρ − 1)

No+N+L−1∑l=No+N

rl−N µ∗l−No︸ ︷︷ ︸2©

+ (ρ − 1)No+N+L−1∑l=No+N

r∗l µl−N−No︸ ︷︷ ︸3©

+ C∑

all l in f rame

rlµ∗l−No︸ ︷︷ ︸4©

}(6)

The outer sum in (5) represents accumulation over Mframes.

III. STREAMING IMPLEMENTATION

The above formulation is a batch estimation approach, butfor practical reasons a streaming processor is attractive. Weassume still that the unknown channel phase is 0 degrees.

By expressing correlations as outputs of filters whose im-pulse responses are matched to the correlating waveform, wemay derive a processor in Figure 2 that sequentially producesΛ(No). It is important to time-align the various contributionsto the total metric. We assert this is the optimum framesynchronizer, but note for large SNR, one may want to invokethe approximation ρ ' 1, in which case this diagram simplifiesby eliminating two of the matched filter channels.

Fig. 2. Diagram of log likelihood computer

Notice the frame summation at the righthand end of thisdiagram. The matched filtering performs computation over asingle frame, and these calculations for a given hypothesiscan then be summed over M − 1 complete frames to improveperformance.

We now provide details on the matched filters in Fig 2.Consider term 4©. We define z4(No) =

∑No+N+L−1l=No

rlµ∗l−No.

This sequence can be produced by matched filter below inFigure 3.

Fig. 3. Diagram of the matched filter with length N+L

Similarly, we need two additional (shorter) matched filtersproducing terms 2©, 3©:

z2(No) =

No+N+L−1∑l=No+N

rl−N µ∗l−No(7)

z3(No) =

No+N+L−1∑l=No+N

r∗l µl−N−No (8)

whose implementations are shown in Figure 4:

Fig. 4. Two matched filters with length L

These matched filters define the corresponding boxes inFigure 2.

A. Unknown-Phase Processor

Though our derivation has assumed the mean sequencearrives with zero phase rotation, and only an unknown delayexists, the practical reality is that the mean signal will beµn−No e jθ , where θ is modeled as uniform on [0, 2π). Thoughnot developed here, we can find the likelihood function underunknown phase conditions by averaging a known-phase resultwith respect to θ, as done in any treatment of noncoherentdetection in digital communication, e.g [3]. For the contribu-tion from the pilot signal, we can show that the noncoherentstatistic is obtained by processing as in Figure 2 with zerophase rotation on µn and simultaneously with π/2 rotationon µn, summing over appropriate indexes as in Figure 2,then squaring and summing, followed by square-rooting. Thecontribution based only on the CP in Figure 2 becomes anoncoherent detector if the "real part of" operation is replacedby absolute value, agreeing with [1].

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One has the options of doing frame accumulation either afteror before the squaring and summing operation, equivalent topost- or pre-detection combining. The issue of frequency offsetplays into this decision. In our study we adopt accumulationprior to squaring/summing.

B. Complexity

Before presenting to results, we give a rough evaluationof the complexity of our proposed method, considering realmultiplications per input sample.

To simplify evaluation, assume ρ = 1 when high SNR isthe case, so branches 2© and 3© in Fig. 2. will be eliminated,leaving a matched filter with length N + L in branch 4©,which implies N + L complex multiplications, or 4(N + L)real multiplications. Note when taking only cyclic prefix intoaccount with diagram collapsing into Sandell-only branch1©, there are three absolute value operations (when taking

non-coherent results, the 2Re(·) turns into 2| · |) and onecomplex multiplication giving in total 10 real multiplications.In other words, our approach gives 4(N + L) additional realmultiplications per input sample comparing to CP-only metric.

IV. RESULTS

We have simulated a case with N = 128, P = 15, L = 20,with four-frame integration, on an ideal channel with additivenoise at SNR = σ2

d/σ2

n = 3 dB. Figure 5 shows a typicaloutput for the noncoherent detector for several consecutiveframes, clearly showing periodic peaks spaced by frame inter-vals in time (N + L = 148).

0 50 100 150 200 250 300 350 400 450 500

0

100

200

300

400

500

600

700

Fig. 5. Four-frame integration with CP and µ, SNR = 3 dB

For comparison, we show in Figure 6, similar results for theSandell synchronizer under the same conditions on the samescale, which employs only the CP structure.

We observe that the combined test statistic has a taller peakvalue, relative to the baseline and its fluctuation due to randomdata and additive noise. Moreover, the contribution of the pilotsignal to the test statistic is very sharp in time, owing to the

0 50 100 150 200 250 300 350 400 450 500

0

100

200

300

400

500

600

700

Fig. 6. Four-frame integration with only CP, SNR = 3 dB

narrow autocorrelation properties of the pilot signal. This pro-vides much better localization of the beginning of the OFDMframe than the Sandell procedure does. Synchronization withthe method described here is robust for SNR’s much lower thansuitable for data communication, so that the synchronizer doesnot become the weak link in the receiver processing chain.

0 50 100 150 200 250 300 350 400 450 500-200

-150

-100

-50

0

50

100

150

200

CP only

CP and Mu

Fig. 7. Four-frame integration with only CP and with CP and µ at 13dB

In Figure 7 where SNR = 13 dB, using CP and µ still givesus strong sharper peaks than using CP only. However, usingonly CP can reduce complexity at such high SNR.

Figure 8 presents a two-ray multipath case, where differ-ential delay is 5 µs and SNR is 13 dB, we still get betterequally-spaced sharp peaks than those in Figure 9 which onlyuses CP. We also see that using CP and µ can help us identifythe number of multipath components; note two sharp peaks inFigure 8 and the main peak is higher than the second peak in

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0 50 100 150 200 250 300 350 400 450 500

-200

-150

-100

-50

0

50

100

150

Fig. 8. Four-frame integration with CP and µ, two-ray multipath, SNR = 13dB

each frame. Using CP only is more likely to produce wrongpeaks since the peaks become flatter when there’s multipath,compared to using CP and µ together in Figure 8.

0 50 100 150 200 250 300 350 400 450 500

-200

-150

-100

-50

0

50

100

150

Fig. 9. Four-frame integration with CP only , two-ray multipath, SNR = 13dB

Finally, we present simulation result of another parameterset with N ′ = 256, P′ = 15, L ′ = 20. With four-frameintegration and additive noise at SNR = 13dB, we observein Figure 10 that with both CP and signal µ utlized, framedetection gives much higher peaks spaced by N ′ + L ′ = 276samples, comparing to CP-only metric.

V. CONCLUSIONS

An optimal frame synchronizer for OFDM has been de-veloped that utilizes the traditional CP structure as well asthe known information in the pilot signal contribution to the

0 100 200 300 400 500 600 700 800-200

-150

-100

-50

0

50

100

150

200

CP only

CP and Mu

Fig. 10. N = 256, four-frame integration with CP only and with CP and µat 13dB

transmitted signal. Gaussian approximation of the receiveddata has been invoked. The structure of the synchronizer,though complicated, is a sensible combination of contributionsfrom both the CP and the pilot signal, and the entire detectorstatistic can be computed by a bank of three filters matched toportions of the pilot signal. The noncoherent processor, neededin practice, roughly doubles the processor computation, as thenumber of matched filters doubles. This detector collapses tothe processor in [1] in the case that pilots are either absent,or are ignored in frame detection.

Testing with the algorithm to date shows that it is quiterobust to multipath on the channel. Under further study areextensions to the work to incorporate frequency offset, as donein Sandell for example, where a joint estimation of timing andfrequency offset is presented.

REFERENCES

[1] Sandell, M., J van de Beek, and P. Borjesson, "Timing and Frequency Syn-chronization in OFDM Systems Using the Cyclic Prefix," InternationalSymposium on Synchronization, pp. 16-19, Essen, Germany, December1995.

[2] L.Hanzo, T.Keller "OFDM and MC-CDMA: A Primer", Chichester: JohnWiley, 2006.

[3] S.G. Wilson, "Digital Modulation and Coding", Pearson-Prentice Hall,1996.

Effort sponsored by the U.S. Government under Other Transaction numberW15QKN-15-9-1004 between the NSC, and the Government. The US Gov-ernment is authorized to reproduce and distribute reprints for Governmentalpurposes notwithstanding any copyright notation thereon.

The views and conclusions contained herein are those of the authors andshould not be interpreted as necessarily representing the official policies orendorsements, either expressed or implied, of the U.S. Government.

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