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IEEE Transactions on Consumer Electronics, Vol. 48, No. 4. NOVEMBER 2002

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TRACKING OF TIME MISALIGNMENTS FOR OFDM SYSTEMS IN MULTIPATH FADING CHANNELS M. Julia Fernandez-Getino Garcia Univ. Carlos 111 de Madrid, Dpto. de Teoria de la Seiial y Corn., 28911 Madrid, Spain E-mail: rnjuliaOtsc.uc3m.e~ Jose M. Pdez-Borrallo Univ. Politecnica de Madrid, E.T.S.I.T., Dpto. SSR, 28040 Madrid, Spain

ABSTRACT In this paper, new time offset tracking techniques'for OFDM are proposed without devoting specific pilot symbols for synchronization. This can be attained since, in coherent OFDM, channel estimation is usually performed employing Pilot Symbol Assisted Modulation (PSAM), based on inserting known symbols in the time-frequency grid. We show bow this two dimensional embedded signalling can also he used for tracking time-frequency offsets. A two-stage procedure is proposed, focusing this work on the time correction step. The ML estimator has been derived, but though optimal, may not be practical due to its complexity. Hence, a frequency-domain low-complex estimator has been proposed, based on only the samples at pilot positions. Also, the stability of the generated estimate can be improved by averaging over a few number of OFDM symbols. This estimator has been proven to he efficient against time dispersive channels.

1. INTRODUCTION Orthogonal Frequency Division Multiplexing (OFDM) is receiving a growing interest for multipath wireless environments. This technique has been employed for terrestrial digital audio and video broadcasting in Europe, DAB [l] and DVB [Z] respectively, and several DAB systems for North America [3]. In addition, the three standards for wireless LANs, IEEE 802.11a (USA), ETSI HIPERLAN/:! (Europe) and ARIB MMAC (Japan), are all based on OFDM [4]; high-speed WLANs are going to play a major role in next-generation wireless systems. Also, joint OFDM and spread spectrum techniques, known as MultiCarrier-Spread-Spectrum (MC-SS) are gaining interest, since they provide robust communication capabilities [5]. Finally, in other wireless environments, like HF links, OFDM has been shown to outperform singlecarrier schemes [6]. In these wireless environments, OFDM has an interesting property of robustness against frequency selective channels, since parallel transmission through the narrow sub-channels experiences almost flat fading. However, synchronization is a critical problem in OFDM systems, since this technique is very sensitive to timing or fre quency offsets between the transmitter and the receiver

[7]. Firstly, time misalignments at the receiver mean an unknown symbol arrival time; sensitivity to a time offset is higher in Multi Carrier systems than in Single Carrier systems, and has been discussed in [SI, [9]. A false estimate in timing leads to Inter-Symbol Interference (ISI), which may disturb the orthogonality of the system; if considering a coherent system, a degradation of the channel estimate can also show up [lo]. On the other hand, a second problem is the mismatch of the oscillators; this generates a signal with an offset in the carrier frequency, which can cause a high bit error rate and degrade the performance of a symbol synchronizer. Regarding time synchronization, it is necessary at the receiver an initial acquisition step to determine the timing to correctly decode the data. This is usually a rough estimation which needs of a further fine timing mode for an accurate synchronization. Thus, time offset tracking must be carried out to both improve the coarse estimate of the acquisition stage, and also follow time misalignments caused by possible channel variations. Therefore, the design of an efficient and accurate symbol timing re covery algorithm is crucial in obtaining a satisfactory performance in OFDM systems. Many time synchronization algorithms for OFDM have been already proposed in the literature, with two main categories, namely Pilot-Symbol-Aided (PSA) and Non-Pilot-Symbol- Aided (NPSA) schemes: 0

NPSA schemes are based on the analysis of the received signal, taking into account the cyclostationarity properties due to the insertion of the Cyclic Prefix (CP) [ll], [12], [13], or the orthogonality condition [14]. In this case, the aid of pilot symbols specifically inserted for synchronization is not required but their reliability is limited since these methods are based on samples within the CP which are corrupted by the time dispersive channel. Regarding PSA algorithms, they are based on devoting specific training symbols for synchronization; they can acquire a more accurate estimation of time offsets but generate a loss in data rate. A distinction in two subcategories must be done. Firstly,

Contribuled Paper

Manuscript received August 13,2002

0098 3063/00 $10.00 e 2002 IEEE

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M. I. Femhdez-Getino Garcia and J. M. PAez-Borrallo: Tracking of Time Misalignments for OFDM Systems in Multipath Fading Channels

some schemes, e.g., (71, [15], [16], [17], employ a preamble or known training sequence to acquire the frame timing, using generally a correlation based algorithm. These preambles are typically PN sequences or duplicate structures; search for optimum preambles has been analyzed in (71, [16]. Secondly, other schemes use pilot-symbols inserted at certain subcarrier frequencies to attain frequency-domain correction [la]; a joint approach of this subcategory of PSA and NPSA based on CP, is discussed in [19]. Hybrid schemes, using both PSA and NPSA, have been proposed in [lo], [ZO], [21]. They carry out acquisition with a NPSA strategy, based on CP; tracking is however performed with a PSA scheme of the second subcategory. We present a novel time offset tracking scheme, overcoming the disadvantages of the two types of existing schemes already described. In the proposed method, specific pilot symbols for synchronization are not required and, therefore, transmission is not burdened with additional signalling and data rate is not reduced, as it happens with the second group of conventional methods, namely PSA techniques. Besides, these advantages mean no loss in performance; the proposed estimator performs as well as traditional methods, satisfying the requirements stated for a time estimator in wireless systems, as shown in simulation results. Our proposal lies on taking into account that coherent OFDM systems typically require the transmission of known symbols or pilot symbols spreaded out throughout the 2D time-frequency grid for channel estimation. This technique, known as Pilot-Symbol Assisted Modulation (PSAM), was first deeply analyzed for single-carrier systems [22]. When extending the idea of PSAM to multicarrier systems, named ZD-PSAM, pilot symbols are inserted in scattered time and frequency positions to estimate the channel [23], [24]. We show that these scattered symbols can also be employed to perform a twestage time and frequency synchronization, Fig. 1. At a first step, time offset tracking must be carried out. Then, in a second stage, frequency correction must be attained; the frequency tracking subblock has been already discussed by the authors in [25]. In this paper, we broadely analyze this new time tracking stage for OFDIh, where the 2D grid of pilots will then be used simultaneously for both channel estimation and synchronization. In section 11, the discrete-time signal model is described. In section 111, we derive the mozimvm lakelihood (ML) estimator for a symbol time offset in coherent OFDM. Then, a lower-complex frequency-domain algo rithm is presented in section N . Performance is evaluated in section V, and section VI discusses and concludes the paper.

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2. SIGNAL MODEL FOR SYNC ERRORS

The discrete-time signal model for a coherent OFDM system consisting of N sub-carriers considers that at certain OFDM-symbols, N p pilots are inserted; the set of time indexes which include pilots can be denoted as r. At the Ith OFDM-symbol, being I E r, we will denote as T the set of sub-carriers indexes where a pilot is transmitted. At time position 1 E r, the received time-domain signal, r[n], before the DFT operation is,

S(k) =

P ( k ) k E I-

{ X(k)

k @ I-

where H ( k ) is the channel's frequency response a t kth frequency and, P ( k ) and X ( k ) are respectively the pilotsymbol or complex data symbol transmitted in the kth subcarrier. Finally, w[n] denotes the complex additive white Gaussian noise process which affects the signal. The transmitted symbol can be then separated in two parts, one containing the ( N - N p ) data symbols, and the other containing the N p pilot-symbols. Then, the received signal can be rewritten as follows

r In1 = (P[I.

+ 2 [nl) + w [.I

(2)

where p[n] refers to the received pilots part (k E I")and the second part contains the data-symbols (k @ I-) and is denoted as z[n]. After the DFT, the received signal in frequency domain, R(i),i= 0, ...,N - 1 will be modelled as

c

1 N - l r[n]e-P?F R(i) = -

dR n=o

(3)

Under certain assumptions, substituting (1) into (3) we get to the complex multiplicative model given in (4), where W ( i )are the samples of the DFT performed over the additive Gaussian noise w[n].

R(i) = S ( i ) H ( i )+ W ( i ) V i E {0,...,N - 1)

(4)

Considering that the transmitted signal s[n] is affected by additive Gaussian noise and a dispersive channel, time and frequency uncertainties can be described as,

r[n] = (s[n] @ h[n - u ] ) e j w + w[n]

(5)

The uncertainty in the arrival time is modelled as a delay in the channel impulse response S[n- u],where u is the integer-valued unknown arrival time of a symbol, 0 < u < N . The uncertainty in carrier frequency gives rise to a shift of all subcarriers; thus, in time domain, the received data is affected by e j w , where the normalized frequency offset is E = j e ( H z ) / A f ( H z ) ,with Af

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E

2D-PSAM

Figure 1: Two-stage correction of the normalized time offset, denoted here 0,and the normalized frequency offset, E , in the timefrequency grid by using 2D-PSAM channel information, primarily provided for channel estimation. Pilot symbols are marked in grey. the inter sub-carrier spacing and fE the frequency offset, given both in Hz. This paper is focused on the study of time offsets, so the frequency offset will be disregarded in the analysis. Two synchronization parameters are not taken into account in this work. Firstly, a carrier phase that may affect symbol error performance in a coherent scheme. Secondly, it has not also been considered the effect on BER performance of non-synchronized sampling. It will be assumed that such an offset is negligible.

2.1. S t u d y of time misalignments

At the receiver, after an initial acquisition step there is usually a h e timing mode for an accurate synchronization. The range of timing offsets that can be corrected in this tracking mode will depend on the length of the CP, L , and the length of the channel impulse response (c.i.r.), denoted with L h , both given in number of samples [19]. It must be fulfilled that the timing point is within the cyclic prefix, u 5 L,to guarantee that time-offset, in time domain drives to a phase offset in frequency domain [7] [lo], since larger time offsets should be corrected in the previous acquisition step. With this requirement, we can distinguish two situations; in the case v > ( L - Lh), the samples are affected by the channel impulse response of the previous OFDM-symbol, so it also suffers some ISI, apart from the phase rotation. If u 5 ( L - L h ) , Is1 is avoided. After this tracking step, the residual error must be smaller than half a sample, Iv - DI 5 0.5. In Fig. 2 is depicted an schematic of these requirements. As pointed out, as long as all the samples taken are within the length of the cyclically extended OFDM symbol, the data frame suffers a circular time shift of that number of samples. This drives in frequency-domain to a linear rotation in the phase of the FFT outputs by an amount of e j % u k , and then, suhcarriers corresponding to large values of kth sub-carrier index, may be seriously affected even by small symbol timing offsets [20]. In the following expressions (€4, the first column refers to timedomain and the second one is frequency-domain representation.

OFDM-lymbol lit11

OFDM-symbol (i)

OFDM-symbol (i-I)

time

n -

Lmghofai.r.

lime

Figure 2: A time offset within the grey area avoids IS1 and ICI.

+I

---t

T[n- V ] N

+

R(i) ~ , ( i= ) R(i)ejtui

(6)

Hence, the complex multiplicative model previously derived in (4) can be rewritten as (7), where subindex u in R,(i) denotes a time offset in the received signal and W(i) is a generalized complex gaussian noise. Using this notation, we will analyze next the effect on the signal of both, channel estimation and timing errors.

~ , ( i ) = R(i)ej%"i =

+

[ ~ ( i ) ~ ( wi ()i ) ] e j G V i

= S(i)H(i)ej%"'

+W(i)

(7)

2.2. Timing and channel estimation e r r o r s Timing and channel estimation errors can be considered jointly. At ith subcarrier, the estimated channel B(i) can be splitted into two parts; the channel's frequency' response H ( i ) and the estimation error AH(i). Hence, it can he written as: B(i) = H ( i ) A H ( i ) . At the receiver, the frequency-domain symbols can he equalized SNR > with a generic function G(i) which fulfils G(i) 3 H(i)-'. If we consider, as a first approach and to simplify the analysis, a Zero-Forcing scheme for the equalization, G ( i ) = H(i)-', the equalized frequency-domain symbols R.(i) are given by,

+

M.J. Femindez-Getino Garcia and J. M. PAez-Borrallo: Tracking of Time Misalignments for OFDM Systems in Multipath Fading Channels

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where L denotes the real part of a complex expression and (,)* denotes complex conjugate. Consequently, the frequency-domain ML estimation for time correction becomes,

R,(i) = R,(i)G(i) = RV(i)fi(i)-' =

[ S ( i ) H ( i ) e j s u+ i W(i)]fi(i)-'

=

[S(i)(I?(i) - A H ( i ) ) e J % Y +w(i)]fi(i)%) i

The final expression is given by (9), where the second term produces misplacement of the constellation points, more significant for outer symbols [7]. The third term is the equalized gaussian noise, which still keeps its statistical properties.

3. M A X I M U M LIKELIHOOD ESTIMATION Based on the OFDM signal model already given, the ML estimator is derived by maximizing the joint probability of the received signal given a certain time offset u. AWGN is assumed when deriving the ML estimator, and conventional assumptions are made in the postdemodulation noise. The log-likelihood function will be taken as the score function and the ML estimation algorithm consists of maximizing this log-likelihood function, A(u) (10). Hence, the ML estimate is given by the solution of a / a u ( A ( u ) ) = 0. A(u) =log(Pr{R(.)(u}) = l o g f ( R ( . ) I u ) '

(10)

In the frequency domain, the received signal at pilot Vi E T,is given by R(i) = P ( i ) H ( i ) e j s U i W ( i ) ,denoting [i] a generalized additive Gaussian prccess which includes the data part. Let's us denote as e,(.) the following expression, that we call the reference signal:

positions,

w

+,(u) = P ( i ) H ( i ) e j s Y i

3.1. J o i n t P S A M and decision-directed

In previous section, it has been derived the ML estimator when only pilots are taken into account, and hence, non-pilot subcarriers were excluded. However, a decision on the data can he taken and the D obtained at data sub-carriers can be incorporated to the estimation procedure. Therefore, it will be analyzed a joint ZD-PSAM and Decision-Directed (DD) scheme; in this case, the signal model considers Vi E {O, . .. ,N - 1). The reference signal will be then, * ; ( u ) = i?(i)H(i)ej%vi

(15) and now we require an estimation of the data-symbols, as shown in (M), and all the factors in that expression are either known or can he estimated.

S(i)=

(

P(i) k E T X(i) k

I-

Therefore, the estimated timing offset can be found by maximizing,

+

(11)

where all the factors are known, excepting u to be estimated. Considering that the noise samples W [ i ]are Gaussian i.i.d, then we have (12). As we already stated, maximizing joint gaussian statistics with a quadratic exponent provides the guarantee of local convexity with respect to Bi(u).

Hence, developing (10) we have for the Gaussian case that,

4.

FINE TIMING ALGORITHM

In general, the complexity of salving the ML estimator is too high and we need to look at more computationally efficient algorithms, as presented in this section. In this algorithm, we will just take into account the pilot positions, i E T,so the signal model is given by ~ ( i=) P ( i ) H ( i ) e - j +

+~

( i )

(18)

In the following, the Gaussian noise will be omitted to simplify the analysis. Assuming a frequency-selective channel, we may have a good initial channel estimate, obtained at the acquisition stage jointly with coarse timing estimation. So, for a first analyticaj approach, we will assume perfect channel knowledge, H ( i ) = H ( i ) ,Vi E T. Then, the equalized received symbols R'(i)at pilot positions will be

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Let's construct a function T ( i ) ,Vi E T,in the following manner: T ( i ) = R'(i)P*(i) = p(i)p*(i)e-j%uY" -

I~(~)I~~-J+Fw

(20)

Since the phase of T ( i )does not depend on the values

la

of the pilots, it can be considered that lP(i)I = K for the

0.8

pilot positions, and we can then write T ( i )= K 2 e - j a U i . If we take the phase argument of this expression we have, 2lr

L(T(i))= -j-vz N

.

(21)

0.2

so we can calculate partial estimates a t every frequency position V i E T,as follows.

.I

-0.21

Therefore, the estimate at every Ith stage is given by the averaged estimate,

4 8 ,

-1

where Np denotes the number of elements of the T set. For the purpose of illustration, if we have a system with N = 32, and a pilot every 8-th subcarrier, that means that T = {0,8,16,24}. Hence, T ( i ) ,V i E T and if v = 1, is given by: T ( 0 ) = 1, T(8) = e d ? , T(16) = e d T , T(24) = e - j t = e j * . In a realistic analysis, without assuming perfect channel knowledge, we must include a distortion effect, which can be modelled as a multiplicative factor I k l ( i ) = H ( i ) - ' A H ' ( i ) , see Appendix. In this case, the equalized signal will be:

4 8

-06 -04

R

e

o

,

,

4 2

0

02

01

06

O

,

011

i

I

Figure 3: Effect of timing errors in OFDM with QPSK signal constellation, when v = 1. To analyze time synchronization, the channel effects to be taken into account are AWGN and a time dispersive effect which produces ISI. The frequency-selective channel considered is a 4-tap impulse response and also, a certain remaining frequency offset E 5 0.5. Results are shown in Figs. 4-6, including batch estimation depending on the number T of OFDM-symbols averaged in time dimension. In previous sections, an analytical approach has been derived when only considering frequency dimension for estimation; thus, information is extracted from one OFDM-symbol. However, we can extend the estimator over T OFDM-symbols. A straightforward way of improving performance is averaging the partial estimates, sequentially obtained at different OFDM-symbols, and stored in a T-length memory shifted at every new estimation. This averaging approach is suboptimal, but do not imply any additional delay as the optimal one, and only requires a latency at initialization step. A t every stage, t = 1,.. . , T , a new Ct is obtained from the ML estimator, Ct = maw, At(v). Then, the average ML estimate ija is the mean of the T estimated values, and then

+ Et=, Figure 4 shows the error of estimated time offset, and

C, = and L(T(i))= -$J - j g v i , which reduces to the previous expression with a certain level of phase distortion. This effect will be analyzed through the simulations, that show how this algorithm performs successfully at low Signal-to-Noise-Ratio (SNR) in dispersive channels. I

T

-

vt.

it can be observed that if considering T N 2 - 3, the error of the estimation is already negligible. Mean-SquaredError fMSE) curve is dotted in Fia. 5: it must be noticed that the signal-to-noise ratio considered is significantly low. Finally, we have also considered the case where a -

I

M. J . Femindez-Getino Garcia and J. M. PBez-Borrallo: Tracking of Time Misalignments for OFDM Systems in Multipath Fading Channels

987

remaining frequency offset is present when performing the time correction, see Fig. 6; in this case, it can he observed how the estimation error is below 0.5 even in the presence of a considerable frequency misalignment E = 0.3. The time offset estimator shows an error standard deviation of less than 0.5 samples, which satisfies the requirements previously discussed, even at low values of the signal-to-noise-ratio. Also, in Fig. 7, an histogram of the estimation is presented when Y = 1; the mean of the estimation is E { t } = 1.0361, and the variance is given by E{($- Y ) ~ = } 0.0669' = 4.5 x which is quite low. These results confirm the feasibility of these synchronization schemes for time correction in practical OFDM systems. Figure 4: Error of averaged time offset estimation as a function of T for S N R = 0 dB.

,I

It

a'a' 0,

N.. Of OFDM-rymbolr sVerassd

10

Figure 7: Histogram of the time estimation D, when v = 1 sample, for 100 trials and S N R = 10 dB.

Figure 5: MSE of time estimation as a function of T ; S N R = 0 dB.

-0.05

i

6. CONCLUSIONS In this paper, we have shown how by using ZD-PSAM in coherent OFDM we can accomplish simultaneously both channel estimation and synchronization in a tracking mode; in this work, it has been hroadely analyzed the time-tracking step. The advantages of the proposed methods for time correction compared to traditional ones are: Specific pilot symbols for synchronization are not required, so the data rate is not reduced.

Figure 6: Error of averaged time offset estimation as a function of T for S N R = 0 dB. Also, a frequency offset is present, with a value of E = 0.3.

The range of time offsets that can he tracked is upper bounded by the length of the cyclic prefix. This time span allows to cover any possible time misalignment due to channel variations or a mismatch between transmitter and receiver. 0

It makes more efficient use of the pilots introduced primarily for channel estimation.

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s

These advantages mean no loss in performance. The convergence of the proposed estimator is as good as in traditional schemes, satisfying the requirements stated for wireless systems; after this tracking step, the residual error must be smaller than half a sample, Iu -PI 5 0.5.

Different approaches of this proposal have been analyzed in this paper and frequency domain estimators have been addressed. Maximum likelihood estimation in frequency-domain has been derived, considering also a joint approach based on pilots and decisions taken over the received data. A fine timing low-complex algorithm in frequency domain is proposed, which performs successfully at low Signal-to-Noise-Ratio. Also, t o improve performance further, the estimator can be extended by averaging over a few OFDM-symbols. Simulations have shown that this algorithm performs successfully in timedispersive environments suffering IS1 from previous symbol and also affected by a certain frequency offset. These results satisfy the accuracy requirements in practical applications.

7. APPENDIX In this Appendix, we analyze the distortion effect in channel estimation. At ith subcarrier, the estimated channel k(i)can be splitted into two parts; the channel's frequency response H ( i ) at that subcarrier, and the corresponding estimation error A H ( i ) , whatever the method of estimation is. Therefore, it can be modelled as an additive effect: H(i) = H ( i ) + A H ( i ) . In this expression, if we take the inverse, (26)

where we can operate on the second term ( H ( i ) + A H ( i ) ) - ' to write, 1

+

1

1

1

-

H(i) - H(i) + H ( i ) + A H ( i )

-

-1

H(i)

H(i) +

+ A H ( i )- H ( i

+

H(i)(H(i) A H ( i ) h 7 )

yielding that expression in Eq. (28) equals t o the expres sion in Eq. (29) 1

[l] Radio broadcasting systems; Digital Audio Broadcast-

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(k(i))-'= ( H ( i )+ A H ( i ) ) - '

H(i) AH(i)

8. REFERENCES

AH(i)

H(i) + H ( i ) ( H ( i )+ A H ( i ) )

A),

(28)

and denoting A H ' ( i ) = (1 + we can write finally, (l?(i))-' = ( H ( i ) ) - ' A H ' ( i ) (30) Thus, the channel distortion effect can be modelled as a multiplicative factor.

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engineering, both from the Polytechnic University of Madrid, Spain, in 1996 and 2001, respectively. Currently, she is with the Department of Signal Theory and Communications of Carlos 111 University of Madrid, Spain, as an Assistant Professor since 2001. She was on leave during 1998 at Bell Laboratories, Murray Hill, NJ, working on channel coding design for satellite broadcasting systems. She also visited Lund University, Sweden, during two periods in 1999 and 2000. Her research interests include multicarrier communications, coding and signal processing for wireless systems. In 1998, she received the best 'Master Thesis' award from the Professional Association of Telecommunication Engineers of Spain, and in 1999, she was awarded the 'Student Paper Award' at the IEEE 10th International Symposium on Personal, Indoor and Mobile Radio Communications, PIMRC.

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"An Analysis of Two-Dimensional Pilot-Symbol Assisted Modulation for OFDM". Proceedings of the IEEE Int. Conf. on Universal Personal Communications, ICUPC'97, San Diego, USA, October 12-16, 1997. [24] P. Hiiher, S. Kaiser and P. Robertson, "Two-Dimensional

Pilot-Symbol-Aided Channel Estimation by Wiener Filtering". Proceedings of IEEE Int. Conf. on Acoustics, Speech and Sig. Proc., ICASSP'97, Vol3, pp. 1845-1848, Munich, Germany. April 21-24 1997.

JosC M. Pdez-Borrallo was born in Huelva, Soain. in 1957. He received the Telecommunications Eneineer and Dr. Engineer degrees from the Universidad Politknica de Madrid, Spain, in 1982 and 1987 respectively. Rom 1982 to 1987, he was employed in AEG-Telefuuken and Ensa, where he was involved with the development and training of COMINT and ELINT systems. Since 1987 he has been a faculty member in the ETSI Telecomunicadn at the Universidad Polithica de Madrid, first as Associate Professor until 1998, and then as Full Professor, His main research activities are in the area of signal processing with special emphasis in the fields of communications and audio.

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