An Initial Timing Offset Estimation Method for ... - Semantic Scholar
An Initial Timing Offset Estimation Method for OFDM Systems in Rayleigh Fading Channel Seung Duk Choi, Jung Min Choi, and Jae Hong Lee School of Electrical Engineering and INMC, Seoul National University Email: [email protected] Abstract - An initial timing offset estimation method for orthogonal frequency-division multiplexing (OFDM) systems is proposed. Conventional preamble-based synchronization methods result in performance degradation due to the time variation in the power delay profile of the channel. We propose a simple but efficient synchronization method using a specifically designed preamble. The performance of the proposed estimator is evaluated by computer simulations. The simulation results show that the proposed estimator has a significantly smaller mean squared error (MSE) than other estimators even in the fast varying channel.
Key words - orthogonal frequency-division multiplexing (OFDM), preamble, timing estimation, synchronization
I. INTRODUCTION
Orthogonal frequency-division multiplexing (OFDM) systems are very sensitive to the synchronization errors. By using the known preamble, several approaches have been proposed to estimate the timing and the frequency offset error either jointly or individually [1]-[5], [8]-[10]. The synchronization methods in [8] and [9] exploit the cyclic prefix of the OFDM symbol. Since the cyclic prefix is usually affected by preceding signals, the estimation result depends on the time variation in the power delay profile of the channel [12]. To avoid the channel dependent problem, the preamble-based synchronization methods are employed in [1]-[5], [10]. The algorithm of Schmidle and Cox (S&C) uses the preamble that consists of repetitive patterns [2]. At the receiver, the timing estimation is made by searching for the correlation peak between these patterns. The S&C method provides simple estimates for the timing. However, the correlation peak of the timing metric has plateau, which causes a large estimation variance in the timing estimation. To overcome the uncertainty in the timing metric, Minn proposes a special preamble which yields a unique peak in the timing metric at the correct starting point of the OFDM symbol [1], [3]. However, the estimation performance is still unsatisfactory since the side lobe in the timing metric results in large estimation variance [4]. To remove the side lobe in the Minn method, the proposed methods in [4] and [5] have the highly sharp timing metric, which yields a more accurate timing estimation. However, in the fast varying channel where the first channel path is not always dominant, the performance of those methods are degraded since they assumed the first arrived channel path is dominant. In this paper, a novel timing estimation method is proposed. To improve the estimation performance in the fast varying channel, channel path gains are exploited in a specifically designed windowing method. The proposed windowing method does not assume that a particular channel path is dominant since the window contains the channel gains regardless of their dominancy. By exploiting all path gains of the received signal which is above the threshold, the performance of the proposed This work was supported in part by the ITRC program and the Brain Korea 21 Project .
1-4244-0063-5/06/$2000 (c) 2006 IEEE
method is improved greatly with low complexity even in the fast varying channel. This paper is organized as follows. Section II describes the system and the existing method. In Section III, the proposed estimation method is presented. Simulation results and discussion are provided in Section IV. Finally, our conclusions are provided in Section V.
II. SYSTEM DESCRIPTION A. OFDM signal description The samples of the transmitted baseband OFDM signal are given by 1 N −1 j ( 2π / N ) dn , − N g ≤ d < N − 1 , (1) ∑ cn e N n=0 where cn is the complex valued information symbol, N is the number of subcarriers, and N g is the number of the samples in the guard interval. The sample at the receiver is given by x( d ) =
y (d ) = e xp( jφ )exp( j 2π dv / N )r ( d − nε ) + w1 (d ), (2) where
L −1
r (d ) = ∑ hm x( d − τ m ) + w1 ( d ),
(3)
m =0
φ is an arbitrary phase factor, v is the carrier frequency offset normalized by subcarrier spacing, nε is the timing offset, hm is the channel impulse response with path delay τ m , L is the length of channel memory, and w1 (d ) is the additive white Gaussian noise (AWGN). B. Timing synchronization The OFDM timing estimation is depicted in Fig. 1. As long as the starting point of the FFT window is within the guard interval, inter symbol interference (ISI) does not occur in the AWGN channel [8]. However, Due to the multipath propagation, a part of the guard interval is corrupted by preceding symbols. The estimated timing needs to be within the uncorrupted guard interval to prevent inter symbol interference (ISI) [8]. Consider the conventional preamble-based methods proposed in [1], [2], and [4]. Let us describe the timing estimation of the GR method under the multipath fading channel. The preamble of the GR method is given by PGR = [ AN / 2 AN / 2 ], (4) where AN / 2 represents the sequence of length N / 2 generated by the inverse fast Fourier transform (IFFT) of the constant amplitude zero autocorrelation (CAZAC) sequence. The sequence in the preamble satisfies the following condition xi = xi + N / 2 = C ⋅ eθi , i = 0,1,..., N / 2 − 1, (5) and || xk ||= C , k = 0,1,..., N − 1, (6) where C is the constant amplitude and θi is the phase of each sample of the sequence in the preamble.
(a) Cyclic prefix.
Ng
Fig. 1. OFDM timing estimation.
To improve the estimation performance at the receiver, each sample of the sequence in the preamble is multiplied by a PN sequence weighting factor. The sequence in the new preamble is given by xk' = sk xk , k = 0,1,..., N − 1, (7) where sk is the PN sequence weighting factor of the k -th sample of the preamble. The value of the PN sequence is +1 or −1 . The GR method finds the timing at the maximum point of the timing metric given by | P ( d ) |2 M GR (d ) = 1 2 , (8) R1 (d ) P1 (d ) =
where
N / 2 −1
∑ss
k k+N /2
k =0
R1 (d ) =
and
r * (d + k ) ⋅ r (d + k + N / 2)
N / 2 −1
∑ | r (d + m + N / 2) |
2
N
(b) Zero padding. Fig. 2. Two methods of inserting the guard interval.
.
(9) (10)
of | hm |2 without AWGN. Hence, applying the minimum variance unbiased estimator (MVUE) for the linear model of the system [11. pp. 90-94] in the fist term, (11) can be expressed as ^
P1 (d ) = C1 | hm |2 + * L−1 L −1 2 ' h' (m)a (k, m) + w (d ). (13) ' C s h ( m ) a ( k , m ) ∑ 0 1 k ∑ ∑ 2 ' k =0 ' mm≠=m'0 m =0 N / 2 −1 ' 1 C1 = C 2 ⋅ (14) where ( s (k )) 2 . ∑ 2 N k =0 Without interference term, the second term in (13), P1 (d ) gives estimates for the energy of the channel impulse response. N / 2−1
m =0
However, the performance of the GR method is degraded in the multipath fading channel with high Doppler frequency. The degradation of performance is described using the linear model of the received signal. Substituting (3) into (9) and rearranging yields P1 (d ) =
N / 2 −1
∑ (s
k
k =0
=
N / 2−1
N / 2 −1
∑
k =0
=
L−1
*
L−1
∑ s ∑h a (k, m)b (k, m)) ∑h a (k,m)b (k, m) + w (d) ' k
k =0
=
r (d + k ) ⋅ r (d + k + N / 2) )
' *
k =0
N / 2 −1
∑
k =0
m =0
0
'
m 1
1
m=0
2
* L −1 L −1 C 2 s k ' ∑ hm ' a 0 ( k , m ) ( ∑ hm ' a1 ( k , m ) + w 2 ( d ) m' = 0 m =0
N / 2 −1
∑
m 0
C 2s' k
L −1 | hm | 2 s ' ( d + k − m ) + ∑ m' =0 m = m'
L −1 C s k ∑ hm ' a0 ( k , m )) m =0 2 '
*
L −1 hm ' a1 ( k , m )) + w2 ( d ), ∑ m' = 0 m ≠ m'
(11)
where *
L −1 w 2 ( d ) = ∑ h ( m ) a 0 ( k , m )b0 ( k , m ) w ( d + k + N / 2 ) + w * ( d + k ) m' =0 L −1 ⋅ ∑ h ( m ) a1 ( k , m ) b1 ( k , m ) + w * ( d + k ) w ( d + k + N / 2), (1 2 ) m=0
sk sk + N / 2 = sk , sd +k −m = a0 (k , m), sd + k −m+ N / 2 = a1 (k , m), xd +k −m '
'
θd +k −m
= b0 ( k , m), xd +k −m + N / 2 = b1 (k , m), and h (m ) = h(m )e . The first term in (11) is the cross correlation between the transmitted PN sequence sk ' and the received sequence of sk ' in the multipath fading channel with the channel impulse response
B. Timing synchronization problem The performance of the GR method is degraded under the fast varying channel. Substituting (13) into (8), the estimated timing that maximize (8) is the starting point of the dominant channel path assuming the first arrived path is dominant. However, in the fast varying channel, the first channel tap is not always dominant with high probabilities, which result in degradation in the performance of the GR method. Simulation results also prove this. The performance degradation is explained in the case of the general preamble based estimation method If the preamble have the repetitive structure given by [ AN / 2 AN / 2 ], the first part and the second part in the preamble are equivalently affected by the multipath fading channel. However, In the GR method, the repetitive structure of the preamble in (4) is destroyed by multiplying the sequence in the preamble by a PN sequence weighting factor in (7). In that case, each part of the preamble is affected differently by the fading channel. Then, the timing metric that is based one the correlation or some kind of product summation between patterns in the preamble is also affected by the fading channel. Let Θ(d ) be the general output of the timing metric. Let Θm (d ), m = 0,..., L − 1 , be each multipath component. It is assumed that each path gain of the multipath component has the Rayleigh distribution and it is i.i.d. (independently and identically distributed). Then, Θ(d ) is given by L −1
Θ(d ) = Θ0 (d ) + Θ1 (d ) + .... + Θ L −1 (d ) = ∑ Θ m ( d ) .
(15)
m =0
In the view of the linear algebra, the timing that maximizes Θ(d ) is determined by the dominant component of Θm (d ), m = 0,..., L − 1 . In general, the maximum point of the timing metric is not necessarily the desired timing and it varies greatly in the fast varying channel.
1 N −1 (19) ∑ | r ( d + k − N / 2) |2 . 2 k =0 The proposed timing metric has peak at the correct symbol timing because of DN / 2 in the preamble [4]. It is robust to the frequency offset since each value from the product of N / 2 different pairs of samples in (18) has the equal phase and it is canceled out at the absolute operation in (17). Timing metrics of the estimators are shown in Fig. 3. : (a) shows the timing metrics of the conventional methods and S&C method. (b) shows the timing metric of the proposed method. In Fig. 3, we assumed that the correct timing is zero and there is no channel distortion. The Park and the GR methods have highly sharp timing metrics which make those methods outperform the S&C and the Minn methods. In the proposed method, the shape of the timing metric is the impulse which is ideal in the signal detection [11]. Furthermore, the small subpeak at the other positions of the correct timing in the Park and the GR methods is suppressed in the proposed method. R2 (d ) =
and
(a) Conventional methods and S&C method.
B. Timing metric in multipath fading channel By using the impulse of the proposed timing metric and the othorgonality of the CAZAC sequence [7], it is derived in the appendix that, in the multipath fading channel, P (d ) is approximated as 2 C h + w3 ( d , k ), if d = ti P( d ) ≈ 2 i (20) otherwise, w3 (d , k ),
(b) Proposed method. Fig. 3. Timing metrics of the estimators.
where
Hence, to make reliable estimation, the assumption that a particular channel path is dominant is inevitable for the conventional estimators since they do not use the repetitive patterns in the preamble. However, the assumption is not the case of the fast varying channel without line of sight which results in performance degradation.
w3 (d , k ) =
∑
w2 (d , k ) ,
(21)
k =0
L −1
w2 (d , k ) = w1 (d − k ) ∑ hm x( d − k − τ m' ) + w1 (d + k + 1) m' = 0
L −1
⋅ ∑ hm x (d − k − τ m ) + w1 ( d − k ) ⋅ w1 (d + k + 1),
III. PROPOSED ESTIMATION METHOD The proposed method is designed to be robust to the time variation of the fast varying channel and to improve the performance.
N / 2 −1
(22)
m=0
C2 =
N / 2 −1
∑
x (ti − k − τ m ) x (ti + k + 1 − τ m ),
(23)
k =0
and
{ti : i = 0,1,", L − 1} .
(24)
A. Proposed timing metric based on a new preamble In order to get the more accurate timing estimation, a new time domain preamble is given by Ppro = [ C N / 2 DN / 2* ], (16)
From (20), the proposed timing metric has the impulses that are approximately proportional to the squared channel path gains. It is because C2 has impulse at the correct starting point of the channel path ti , i = 0,1," , L − 1, like in Fig. 3. (b).
where CN / 2 represents the sequence of the length N / 2 generated by the inverse fast Fourier transform (IFFT) of the constant amplitude zero autocorrelation (CAZAC) sequence and D*N / 2 is a complex conjugate of DN / 2 which samples of the
C. Proposed timing estimation method. The Block diagram of the proposed method is shown in Fig. 4. τ 0 that maximizes (17) is first obtained. The impulses which are from the timing metric are tresholded and combined in the moving summation. By using the windowing method, the time interval τ 1 between τ 0 and the starting point of the first channel path is estimated and then compensated in the block diagram. The thresholding is given by
sequence are time reversed version of CN / 2 . The cyclic prefix and the zero padding are used for the guard interval of the preamble [6]. The two methods are shown in Fig.2. In this paper, the zero padding is employed since the cyclic prefix introduces interference in timing estimation when the preamble does not contain the repetitive patterns. The proposed timing metric that uses the new preamble is given by | P ( d ) |2 M Pr o ( d ) = 2 2 , (17) R2 ( d ) where
P2 ( d ) =
N / 2 −1
∑ r (d − k ) ⋅ r (d + k + 1) k =0
(18)
^ ^ ^ M Pro (τ 0 − d + k ) = M Pro (τ 0 − d + k ), if M Pro (τ 0 − d + k ) > η otherwise, 0,
where η is the threshold. Then, the windowing method is given by ^
τ 1 = arg max{EP (d ) : d = 0,1, ", J }, d
(25)
(26)
Table 1. Simulation parameters Parameters Number of the subcarriers Number of FFT/IFFT points Guard interval length (samples) Window size S (samples) The distance J (samples) Threshold η Number of simulation runs Number of channel taps Channel tap spacing (samples) Ratio between first tap to last tap (dB) Fig. 4. Proposed synchronization method. S −1
where
^
EP (d ) = ∑ M Pro (τ 0 − d + k ) ,
(27)
k =0
where J is the maximum time interval between the first channel tab and the strongest one in the Rayleigh fading channel and S is the widow size. The window parameters, the time interval J and the window size S , are designed assuming that the channel has exponential power delay profile and the maximum delay spread is as large as the length of the cyclic prefix. J is designed to minimize the probability of meaningless summation in (27) since the squared channel tap gain with relatively long delay is below the threshold. As it can be seen in (27), the proposed window method is simply the moving summation of the thresholded values of (17). By using the proposed windowing method, the timing that maximizes the timing metric^is compensated. It is given by ^ ^ τ = τ 0 − τ1 . (28) The proposed timing estimation method has two desirable properties in the mutipath fading channel. First, the proposed method does not need the assumption that a particular channel path is dominant since the proposed window contains the squared channel paths regardless of their dominancy However, the conventional methods assume that the first arrived channel is dominant. Hence, the proposed method is more robust to the fast varying channel where the first channel path is not always strongest. Secondly, the proposed method combines the squared path gains of each channel in the timing estimation while the conventional method considers only the dominant path. It is interesting that (25) and (27) are expressed at the correct timing as follows C h 2 , if C2 h 2 k > η ' ⋅ C2 h 2 max C2 h 2 k = 2 k (29) otherwise, 0, s −1
E 'P ( d ) ≈ C2 ∑ hk 2 k =0
(30)
where η ' = 1/(SNR + 20) . It has the similar shape as the fine timing estimation method in [1]. Minn also proposed a scheme that exploits the channel tap gains in the fine timing estimation. However, the method requires too much complexity to be used in the initial timing estimation due to the ML channel estimation that is employed to obtain the channel tap gains. In the proposed method, the squared channel tap gains are obtained simply from peaks of the timing metric instead of the ML channel estimation. Hence, the complexity of the proposed estimator is now suitable for the initial timing estimation, but the performance is expected to be improved.
IV. SIMULATION RESULTS
A. simulation parameters
Value 2048 2048 256 105 81 η = M Pro (τ 0 ) /( SNR + 20) 100000 17 8 20
The Rayleigh fading channel and ISI channel is used in the simulation. The exponential power delay profile is assumed in all channels. The tap gains of the static ISI channel are fixed and the tap gain powers are the same as those of the Rayleigh fading channel. Table 1 shows simulation parameters.
B. Simulation results
Fig. 5 shows the MSE of the estimators versus SNR in the ISI channel. We can see that the Park and the GR methods outperform the S&C and the Minn methods. The improvement of the performance is due to the reduction of the side lobe of the timing metric. However, the performance is still unsatisfactory. To improve the performance, a windowing method is employed in the proposed scheme. The proposed scheme shows much smaller MSE than the Park and the GR methods in the whole SNR region. This is because the windowing method with the squared channel path gains yields a more accurate timing estimation. Fig. 6 shows the MSE of the estimators versus normalized Doppler frequency at SNR 10 dB in the Rayleigh fading channel. The conventional schemes which assume the first arrived channel path is dominant are degraded to have almost the same MSE as the S&C method at high normalized Doppler frequency. For the static ISI channel, the timing estimation with reference to the first channel path has some meaning since the first channel tap represents desired timing point. However, for the fast Rayleigh fading channel, all channel tap gains are time-varying, and missing the first tap with small gain result in performance degradation in the conventional method. However, the proposed method does not need the assumption that a particular channel path is dominant, which results in much smaller MSE than other estimators in whole Doppler frequency region. The question can be raised about applying the proposed method in (25),(26),(27), and (28) to the GR and the Park methods. Since they also has channel dependent component and the timing metrics have impulse-like shape, the improvement in the performance is expected by applying the proposed method. However, the GR method, as addressed in section II, have the interference term, the second term in (13), which keeps P1 (d ) from reliable estimation of channel tap energy. In Fig. 3 (a), both methods have small subpeak at the other position of correct timing even under noiseless condition assuming no channel distortion. Applying the proposed method, there was slight increase in the performance at high SNR. However, at low SNR, deciding the threshold factor η is impossible. Simulation result is not shown in this paper.
V. CONCLUSION
In this paper, an improved initial timing offset estimation method in OFDM systems is proposed. In order to improve the performance in the timing estimation, a specifically designed preamble is suggested. The timing metric based on a new preamble has the impulses that are proportional to the squared channel tap gains. Then, a more accurate timing estimation
[3] H. Minn, M. Zeng, and V. K. Bhargava, "On timing offset estimation for OFDM systems," IEEE Commun. Lett., vol. 4, no. 7, pp. 242-244, July 2000. [4] B. J. Park, H. S. Cheon, C. G. Kang, and D. S. Hong, “A novel timing estimation method for OFDM systems,” IEEE Commun. Lett., vol. 7, no. 5, pp. 239-241, May 2003. [5] G. Ren, Y. Chang, and H. Zhang, “Synchronization method based on a new constant envelop preamble for OFDM systems,” IEEE Trans. Broadcast., vol. 51, no. 1, pp. 139-143, Mar. 2005. [6] R. van Nee and R. Prasad, OFDM for Wireless Multimedia Communications. Artech House, 2000. [7] R. L. Frank and S. A. Zadoff, “Phase shift pulse codes with good periodic correlation properties,” IRE Trans. Inform. Theory, vol. 8, no. 6, pp. 381-382, Oct. 1962. [8] J-J van de Beek, M. sandell, and P. O. Borjesson, “ML estimation of time and frequency offset in OFDM systems,” IEEE Trans. Signal Processing, vol. 45, no. 7, pp. 1800-1805, July 1997. [9] M. speth, F. Classen, and H. Meyr, “ Frame synchronization of OFDM system over frequency selective fading channels,” in Proc. of VTC 1997-Spring, vol. 3, pp. 1807-1811, Phoenix, AZ, May 1997. [10] P. H. Moose, “A technique for orthogonal frequency division multiplexing frequency offset correction,” IEEE Trans Commun., vol. 42, no. 10, pp. 2908-2914, Oct. 1994. [11] Steven M. Kay. Fundamentals of Statistical Signal Processing: Estimation and detection Theory. Prentice-Hall, 1993. [12] C. Williams, M. A. Beach, and S. McLaughlin, "Robust OFDM timing synchronization," Electron. Lett., vol. 41, no. 13, pp. 751752, June 2005.
Fig. 5. MSE of the timing offset versus SNR in ISI channel.
Appendix Substituting (3) into (18) gives (31). The second term in (31) is approximately zero due to the orthogonal property of CAZAC sequence. Then, P (d ) is approximated as N /2−1 L−1 P(d ) ≈ ∑ ∑ h2m x(d − k −τ m )x(d + k +1 −τ m ) + w2 (d, k ) k=0 m=0 (m=m' )
Fig. 6. MSE of the timing offset versus normalized Doppler frequency under SNR 10 dB in Rayleigh fading channel.
is achieved by taking carefully designed window for the peaks of the timing metric. The windowing method is robust to the channel condition by combining the squared channel path gains of the received signal. Simulation results show that the performance of the proposed estimator is significantly improved in the ISI channel and the Rayleigh fading channel. Therefore, the proposed method is suitable for the initial timing estimation scheme in the mobile communication environment.
References [1] H. Minn, V. K. Bhargava, and K. B. Letaief, “A novel timing estimation method for OFDM systems,” IEEE Trans. Commun., vol. 2, no. 4, pp. 822-839, July 2003. [2] T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun., vol. 45, no. 12, pp. 1613-1621, Dec. 1997.
L−1
= ∑ h2 m m=0
( m= m' )
where
N / 2−1
∑ x( d − k − τ
m
) x( d + k + 1 − τ m ) +
k =0
N /2−1
∑ w (d , k ), 2
(32)
k =0
L −1
w2 (d , k ) = w1 (d − k ) ∑ hm x (d − k − τ m' ) + w1 (d + k + 1) m' = 0
L−1
⋅ ∑ hm x(d − k − τ m ) + w1 (d − k ) ⋅ w1 (d + k + 1).
(33)
m =0
Let the starting point of i -th channel path in time domain be {ti : i = 0,1," , L − 1} . Then, (32) is given by 2 C h + w3 (d , k ), if d = ti P( d ) ≈ 1 i otherwise, w3 (d , k ),
w3 (d , k ) =
where
N / 2−1
∑ w (d , k ) 2
(34)
(35) (36)
k =0
C1 =
and
N / 2−1
∑
x (ti − k − τ m ) x (ti + k + 1 − τ m ) .
(37)
k =0
P( d ) =
N / 2−1
∑
r (d − k ) ⋅ r (d + k + 1)
k =0
=
N / 2−1 k =0
=
L−1
∑ ∑h x(d − k −τ
m
m=0
N /2−1 L−1
∑ ∑h k =0
m=0
( m=m' )
2 m
m
N /2−1 L−1 L−1 L−1 ) + w1(d − k ) ⋅ ∑ hm x(d + k +1 −τ m' ) + w1 (d + k + 1) = ∑ ∑hmx(d − k −τ m) ∑ hm' x(d + k +1 −τm' ) + w2 (d, k) m' =0 m' =0 k=0 m=0
x(d − k −τ m )x(d + k + 1 −τ m ) +
N /2−1 L−1 L−1
∑ ∑∑h h '
k=0 m =0 m=0
( m≠m' )
m m'
x(d − k −τ m ) x(d + k +1 −τ m' ) +
N /2−1
∑ w (d , k ) 2
k=0
(31)
Key words - orthogonal frequency-division multiplexing (OFDM), preamble, timing estimation, synchronization
I. INTRODUCTION
Orthogonal frequency-division multiplexing (OFDM) systems are very sensitive to the synchronization errors. By using the known preamble, several approaches have been proposed to estimate the timing and the frequency offset error either jointly or individually [1]-[5], [8]-[10]. The synchronization methods in [8] and [9] exploit the cyclic prefix of the OFDM symbol. Since the cyclic prefix is usually affected by preceding signals, the estimation result depends on the time variation in the power delay profile of the channel [12]. To avoid the channel dependent problem, the preamble-based synchronization methods are employed in [1]-[5], [10]. The algorithm of Schmidle and Cox (S&C) uses the preamble that consists of repetitive patterns [2]. At the receiver, the timing estimation is made by searching for the correlation peak between these patterns. The S&C method provides simple estimates for the timing. However, the correlation peak of the timing metric has plateau, which causes a large estimation variance in the timing estimation. To overcome the uncertainty in the timing metric, Minn proposes a special preamble which yields a unique peak in the timing metric at the correct starting point of the OFDM symbol [1], [3]. However, the estimation performance is still unsatisfactory since the side lobe in the timing metric results in large estimation variance [4]. To remove the side lobe in the Minn method, the proposed methods in [4] and [5] have the highly sharp timing metric, which yields a more accurate timing estimation. However, in the fast varying channel where the first channel path is not always dominant, the performance of those methods are degraded since they assumed the first arrived channel path is dominant. In this paper, a novel timing estimation method is proposed. To improve the estimation performance in the fast varying channel, channel path gains are exploited in a specifically designed windowing method. The proposed windowing method does not assume that a particular channel path is dominant since the window contains the channel gains regardless of their dominancy. By exploiting all path gains of the received signal which is above the threshold, the performance of the proposed This work was supported in part by the ITRC program and the Brain Korea 21 Project .
1-4244-0063-5/06/$2000 (c) 2006 IEEE
method is improved greatly with low complexity even in the fast varying channel. This paper is organized as follows. Section II describes the system and the existing method. In Section III, the proposed estimation method is presented. Simulation results and discussion are provided in Section IV. Finally, our conclusions are provided in Section V.
II. SYSTEM DESCRIPTION A. OFDM signal description The samples of the transmitted baseband OFDM signal are given by 1 N −1 j ( 2π / N ) dn , − N g ≤ d < N − 1 , (1) ∑ cn e N n=0 where cn is the complex valued information symbol, N is the number of subcarriers, and N g is the number of the samples in the guard interval. The sample at the receiver is given by x( d ) =
y (d ) = e xp( jφ )exp( j 2π dv / N )r ( d − nε ) + w1 (d ), (2) where
L −1
r (d ) = ∑ hm x( d − τ m ) + w1 ( d ),
(3)
m =0
φ is an arbitrary phase factor, v is the carrier frequency offset normalized by subcarrier spacing, nε is the timing offset, hm is the channel impulse response with path delay τ m , L is the length of channel memory, and w1 (d ) is the additive white Gaussian noise (AWGN). B. Timing synchronization The OFDM timing estimation is depicted in Fig. 1. As long as the starting point of the FFT window is within the guard interval, inter symbol interference (ISI) does not occur in the AWGN channel [8]. However, Due to the multipath propagation, a part of the guard interval is corrupted by preceding symbols. The estimated timing needs to be within the uncorrupted guard interval to prevent inter symbol interference (ISI) [8]. Consider the conventional preamble-based methods proposed in [1], [2], and [4]. Let us describe the timing estimation of the GR method under the multipath fading channel. The preamble of the GR method is given by PGR = [ AN / 2 AN / 2 ], (4) where AN / 2 represents the sequence of length N / 2 generated by the inverse fast Fourier transform (IFFT) of the constant amplitude zero autocorrelation (CAZAC) sequence. The sequence in the preamble satisfies the following condition xi = xi + N / 2 = C ⋅ eθi , i = 0,1,..., N / 2 − 1, (5) and || xk ||= C , k = 0,1,..., N − 1, (6) where C is the constant amplitude and θi is the phase of each sample of the sequence in the preamble.
(a) Cyclic prefix.
Ng
Fig. 1. OFDM timing estimation.
To improve the estimation performance at the receiver, each sample of the sequence in the preamble is multiplied by a PN sequence weighting factor. The sequence in the new preamble is given by xk' = sk xk , k = 0,1,..., N − 1, (7) where sk is the PN sequence weighting factor of the k -th sample of the preamble. The value of the PN sequence is +1 or −1 . The GR method finds the timing at the maximum point of the timing metric given by | P ( d ) |2 M GR (d ) = 1 2 , (8) R1 (d ) P1 (d ) =
where
N / 2 −1
∑ss
k k+N /2
k =0
R1 (d ) =
and
r * (d + k ) ⋅ r (d + k + N / 2)
N / 2 −1
∑ | r (d + m + N / 2) |
2
N
(b) Zero padding. Fig. 2. Two methods of inserting the guard interval.
.
(9) (10)
of | hm |2 without AWGN. Hence, applying the minimum variance unbiased estimator (MVUE) for the linear model of the system [11. pp. 90-94] in the fist term, (11) can be expressed as ^
P1 (d ) = C1 | hm |2 + * L−1 L −1 2 ' h' (m)a (k, m) + w (d ). (13) ' C s h ( m ) a ( k , m ) ∑ 0 1 k ∑ ∑ 2 ' k =0 ' mm≠=m'0 m =0 N / 2 −1 ' 1 C1 = C 2 ⋅ (14) where ( s (k )) 2 . ∑ 2 N k =0 Without interference term, the second term in (13), P1 (d ) gives estimates for the energy of the channel impulse response. N / 2−1
m =0
However, the performance of the GR method is degraded in the multipath fading channel with high Doppler frequency. The degradation of performance is described using the linear model of the received signal. Substituting (3) into (9) and rearranging yields P1 (d ) =
N / 2 −1
∑ (s
k
k =0
=
N / 2−1
N / 2 −1
∑
k =0
=
L−1
*
L−1
∑ s ∑h a (k, m)b (k, m)) ∑h a (k,m)b (k, m) + w (d) ' k
k =0
=
r (d + k ) ⋅ r (d + k + N / 2) )
' *
k =0
N / 2 −1
∑
k =0
m =0
0
'
m 1
1
m=0
2
* L −1 L −1 C 2 s k ' ∑ hm ' a 0 ( k , m ) ( ∑ hm ' a1 ( k , m ) + w 2 ( d ) m' = 0 m =0
N / 2 −1
∑
m 0
C 2s' k
L −1 | hm | 2 s ' ( d + k − m ) + ∑ m' =0 m = m'
L −1 C s k ∑ hm ' a0 ( k , m )) m =0 2 '
*
L −1 hm ' a1 ( k , m )) + w2 ( d ), ∑ m' = 0 m ≠ m'
(11)
where *
L −1 w 2 ( d ) = ∑ h ( m ) a 0 ( k , m )b0 ( k , m ) w ( d + k + N / 2 ) + w * ( d + k ) m' =0 L −1 ⋅ ∑ h ( m ) a1 ( k , m ) b1 ( k , m ) + w * ( d + k ) w ( d + k + N / 2), (1 2 ) m=0
sk sk + N / 2 = sk , sd +k −m = a0 (k , m), sd + k −m+ N / 2 = a1 (k , m), xd +k −m '
'
θd +k −m
= b0 ( k , m), xd +k −m + N / 2 = b1 (k , m), and h (m ) = h(m )e . The first term in (11) is the cross correlation between the transmitted PN sequence sk ' and the received sequence of sk ' in the multipath fading channel with the channel impulse response
B. Timing synchronization problem The performance of the GR method is degraded under the fast varying channel. Substituting (13) into (8), the estimated timing that maximize (8) is the starting point of the dominant channel path assuming the first arrived path is dominant. However, in the fast varying channel, the first channel tap is not always dominant with high probabilities, which result in degradation in the performance of the GR method. Simulation results also prove this. The performance degradation is explained in the case of the general preamble based estimation method If the preamble have the repetitive structure given by [ AN / 2 AN / 2 ], the first part and the second part in the preamble are equivalently affected by the multipath fading channel. However, In the GR method, the repetitive structure of the preamble in (4) is destroyed by multiplying the sequence in the preamble by a PN sequence weighting factor in (7). In that case, each part of the preamble is affected differently by the fading channel. Then, the timing metric that is based one the correlation or some kind of product summation between patterns in the preamble is also affected by the fading channel. Let Θ(d ) be the general output of the timing metric. Let Θm (d ), m = 0,..., L − 1 , be each multipath component. It is assumed that each path gain of the multipath component has the Rayleigh distribution and it is i.i.d. (independently and identically distributed). Then, Θ(d ) is given by L −1
Θ(d ) = Θ0 (d ) + Θ1 (d ) + .... + Θ L −1 (d ) = ∑ Θ m ( d ) .
(15)
m =0
In the view of the linear algebra, the timing that maximizes Θ(d ) is determined by the dominant component of Θm (d ), m = 0,..., L − 1 . In general, the maximum point of the timing metric is not necessarily the desired timing and it varies greatly in the fast varying channel.
1 N −1 (19) ∑ | r ( d + k − N / 2) |2 . 2 k =0 The proposed timing metric has peak at the correct symbol timing because of DN / 2 in the preamble [4]. It is robust to the frequency offset since each value from the product of N / 2 different pairs of samples in (18) has the equal phase and it is canceled out at the absolute operation in (17). Timing metrics of the estimators are shown in Fig. 3. : (a) shows the timing metrics of the conventional methods and S&C method. (b) shows the timing metric of the proposed method. In Fig. 3, we assumed that the correct timing is zero and there is no channel distortion. The Park and the GR methods have highly sharp timing metrics which make those methods outperform the S&C and the Minn methods. In the proposed method, the shape of the timing metric is the impulse which is ideal in the signal detection [11]. Furthermore, the small subpeak at the other positions of the correct timing in the Park and the GR methods is suppressed in the proposed method. R2 (d ) =
and
(a) Conventional methods and S&C method.
B. Timing metric in multipath fading channel By using the impulse of the proposed timing metric and the othorgonality of the CAZAC sequence [7], it is derived in the appendix that, in the multipath fading channel, P (d ) is approximated as 2 C h + w3 ( d , k ), if d = ti P( d ) ≈ 2 i (20) otherwise, w3 (d , k ),
(b) Proposed method. Fig. 3. Timing metrics of the estimators.
where
Hence, to make reliable estimation, the assumption that a particular channel path is dominant is inevitable for the conventional estimators since they do not use the repetitive patterns in the preamble. However, the assumption is not the case of the fast varying channel without line of sight which results in performance degradation.
w3 (d , k ) =
∑
w2 (d , k ) ,
(21)
k =0
L −1
w2 (d , k ) = w1 (d − k ) ∑ hm x( d − k − τ m' ) + w1 (d + k + 1) m' = 0
L −1
⋅ ∑ hm x (d − k − τ m ) + w1 ( d − k ) ⋅ w1 (d + k + 1),
III. PROPOSED ESTIMATION METHOD The proposed method is designed to be robust to the time variation of the fast varying channel and to improve the performance.
N / 2 −1
(22)
m=0
C2 =
N / 2 −1
∑
x (ti − k − τ m ) x (ti + k + 1 − τ m ),
(23)
k =0
and
{ti : i = 0,1,", L − 1} .
(24)
A. Proposed timing metric based on a new preamble In order to get the more accurate timing estimation, a new time domain preamble is given by Ppro = [ C N / 2 DN / 2* ], (16)
From (20), the proposed timing metric has the impulses that are approximately proportional to the squared channel path gains. It is because C2 has impulse at the correct starting point of the channel path ti , i = 0,1," , L − 1, like in Fig. 3. (b).
where CN / 2 represents the sequence of the length N / 2 generated by the inverse fast Fourier transform (IFFT) of the constant amplitude zero autocorrelation (CAZAC) sequence and D*N / 2 is a complex conjugate of DN / 2 which samples of the
C. Proposed timing estimation method. The Block diagram of the proposed method is shown in Fig. 4. τ 0 that maximizes (17) is first obtained. The impulses which are from the timing metric are tresholded and combined in the moving summation. By using the windowing method, the time interval τ 1 between τ 0 and the starting point of the first channel path is estimated and then compensated in the block diagram. The thresholding is given by
sequence are time reversed version of CN / 2 . The cyclic prefix and the zero padding are used for the guard interval of the preamble [6]. The two methods are shown in Fig.2. In this paper, the zero padding is employed since the cyclic prefix introduces interference in timing estimation when the preamble does not contain the repetitive patterns. The proposed timing metric that uses the new preamble is given by | P ( d ) |2 M Pr o ( d ) = 2 2 , (17) R2 ( d ) where
P2 ( d ) =
N / 2 −1
∑ r (d − k ) ⋅ r (d + k + 1) k =0
(18)
^ ^ ^ M Pro (τ 0 − d + k ) = M Pro (τ 0 − d + k ), if M Pro (τ 0 − d + k ) > η otherwise, 0,
where η is the threshold. Then, the windowing method is given by ^
τ 1 = arg max{EP (d ) : d = 0,1, ", J }, d
(25)
(26)
Table 1. Simulation parameters Parameters Number of the subcarriers Number of FFT/IFFT points Guard interval length (samples) Window size S (samples) The distance J (samples) Threshold η Number of simulation runs Number of channel taps Channel tap spacing (samples) Ratio between first tap to last tap (dB) Fig. 4. Proposed synchronization method. S −1
where
^
EP (d ) = ∑ M Pro (τ 0 − d + k ) ,
(27)
k =0
where J is the maximum time interval between the first channel tab and the strongest one in the Rayleigh fading channel and S is the widow size. The window parameters, the time interval J and the window size S , are designed assuming that the channel has exponential power delay profile and the maximum delay spread is as large as the length of the cyclic prefix. J is designed to minimize the probability of meaningless summation in (27) since the squared channel tap gain with relatively long delay is below the threshold. As it can be seen in (27), the proposed window method is simply the moving summation of the thresholded values of (17). By using the proposed windowing method, the timing that maximizes the timing metric^is compensated. It is given by ^ ^ τ = τ 0 − τ1 . (28) The proposed timing estimation method has two desirable properties in the mutipath fading channel. First, the proposed method does not need the assumption that a particular channel path is dominant since the proposed window contains the squared channel paths regardless of their dominancy However, the conventional methods assume that the first arrived channel is dominant. Hence, the proposed method is more robust to the fast varying channel where the first channel path is not always strongest. Secondly, the proposed method combines the squared path gains of each channel in the timing estimation while the conventional method considers only the dominant path. It is interesting that (25) and (27) are expressed at the correct timing as follows C h 2 , if C2 h 2 k > η ' ⋅ C2 h 2 max C2 h 2 k = 2 k (29) otherwise, 0, s −1
E 'P ( d ) ≈ C2 ∑ hk 2 k =0
(30)
where η ' = 1/(SNR + 20) . It has the similar shape as the fine timing estimation method in [1]. Minn also proposed a scheme that exploits the channel tap gains in the fine timing estimation. However, the method requires too much complexity to be used in the initial timing estimation due to the ML channel estimation that is employed to obtain the channel tap gains. In the proposed method, the squared channel tap gains are obtained simply from peaks of the timing metric instead of the ML channel estimation. Hence, the complexity of the proposed estimator is now suitable for the initial timing estimation, but the performance is expected to be improved.
IV. SIMULATION RESULTS
A. simulation parameters
Value 2048 2048 256 105 81 η = M Pro (τ 0 ) /( SNR + 20) 100000 17 8 20
The Rayleigh fading channel and ISI channel is used in the simulation. The exponential power delay profile is assumed in all channels. The tap gains of the static ISI channel are fixed and the tap gain powers are the same as those of the Rayleigh fading channel. Table 1 shows simulation parameters.
B. Simulation results
Fig. 5 shows the MSE of the estimators versus SNR in the ISI channel. We can see that the Park and the GR methods outperform the S&C and the Minn methods. The improvement of the performance is due to the reduction of the side lobe of the timing metric. However, the performance is still unsatisfactory. To improve the performance, a windowing method is employed in the proposed scheme. The proposed scheme shows much smaller MSE than the Park and the GR methods in the whole SNR region. This is because the windowing method with the squared channel path gains yields a more accurate timing estimation. Fig. 6 shows the MSE of the estimators versus normalized Doppler frequency at SNR 10 dB in the Rayleigh fading channel. The conventional schemes which assume the first arrived channel path is dominant are degraded to have almost the same MSE as the S&C method at high normalized Doppler frequency. For the static ISI channel, the timing estimation with reference to the first channel path has some meaning since the first channel tap represents desired timing point. However, for the fast Rayleigh fading channel, all channel tap gains are time-varying, and missing the first tap with small gain result in performance degradation in the conventional method. However, the proposed method does not need the assumption that a particular channel path is dominant, which results in much smaller MSE than other estimators in whole Doppler frequency region. The question can be raised about applying the proposed method in (25),(26),(27), and (28) to the GR and the Park methods. Since they also has channel dependent component and the timing metrics have impulse-like shape, the improvement in the performance is expected by applying the proposed method. However, the GR method, as addressed in section II, have the interference term, the second term in (13), which keeps P1 (d ) from reliable estimation of channel tap energy. In Fig. 3 (a), both methods have small subpeak at the other position of correct timing even under noiseless condition assuming no channel distortion. Applying the proposed method, there was slight increase in the performance at high SNR. However, at low SNR, deciding the threshold factor η is impossible. Simulation result is not shown in this paper.
V. CONCLUSION
In this paper, an improved initial timing offset estimation method in OFDM systems is proposed. In order to improve the performance in the timing estimation, a specifically designed preamble is suggested. The timing metric based on a new preamble has the impulses that are proportional to the squared channel tap gains. Then, a more accurate timing estimation
[3] H. Minn, M. Zeng, and V. K. Bhargava, "On timing offset estimation for OFDM systems," IEEE Commun. Lett., vol. 4, no. 7, pp. 242-244, July 2000. [4] B. J. Park, H. S. Cheon, C. G. Kang, and D. S. Hong, “A novel timing estimation method for OFDM systems,” IEEE Commun. Lett., vol. 7, no. 5, pp. 239-241, May 2003. [5] G. Ren, Y. Chang, and H. Zhang, “Synchronization method based on a new constant envelop preamble for OFDM systems,” IEEE Trans. Broadcast., vol. 51, no. 1, pp. 139-143, Mar. 2005. [6] R. van Nee and R. Prasad, OFDM for Wireless Multimedia Communications. Artech House, 2000. [7] R. L. Frank and S. A. Zadoff, “Phase shift pulse codes with good periodic correlation properties,” IRE Trans. Inform. Theory, vol. 8, no. 6, pp. 381-382, Oct. 1962. [8] J-J van de Beek, M. sandell, and P. O. Borjesson, “ML estimation of time and frequency offset in OFDM systems,” IEEE Trans. Signal Processing, vol. 45, no. 7, pp. 1800-1805, July 1997. [9] M. speth, F. Classen, and H. Meyr, “ Frame synchronization of OFDM system over frequency selective fading channels,” in Proc. of VTC 1997-Spring, vol. 3, pp. 1807-1811, Phoenix, AZ, May 1997. [10] P. H. Moose, “A technique for orthogonal frequency division multiplexing frequency offset correction,” IEEE Trans Commun., vol. 42, no. 10, pp. 2908-2914, Oct. 1994. [11] Steven M. Kay. Fundamentals of Statistical Signal Processing: Estimation and detection Theory. Prentice-Hall, 1993. [12] C. Williams, M. A. Beach, and S. McLaughlin, "Robust OFDM timing synchronization," Electron. Lett., vol. 41, no. 13, pp. 751752, June 2005.
Fig. 5. MSE of the timing offset versus SNR in ISI channel.
Appendix Substituting (3) into (18) gives (31). The second term in (31) is approximately zero due to the orthogonal property of CAZAC sequence. Then, P (d ) is approximated as N /2−1 L−1 P(d ) ≈ ∑ ∑ h2m x(d − k −τ m )x(d + k +1 −τ m ) + w2 (d, k ) k=0 m=0 (m=m' )
Fig. 6. MSE of the timing offset versus normalized Doppler frequency under SNR 10 dB in Rayleigh fading channel.
is achieved by taking carefully designed window for the peaks of the timing metric. The windowing method is robust to the channel condition by combining the squared channel path gains of the received signal. Simulation results show that the performance of the proposed estimator is significantly improved in the ISI channel and the Rayleigh fading channel. Therefore, the proposed method is suitable for the initial timing estimation scheme in the mobile communication environment.
References [1] H. Minn, V. K. Bhargava, and K. B. Letaief, “A novel timing estimation method for OFDM systems,” IEEE Trans. Commun., vol. 2, no. 4, pp. 822-839, July 2003. [2] T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun., vol. 45, no. 12, pp. 1613-1621, Dec. 1997.
L−1
= ∑ h2 m m=0
( m= m' )
where
N / 2−1
∑ x( d − k − τ
m
) x( d + k + 1 − τ m ) +
k =0
N /2−1
∑ w (d , k ), 2
(32)
k =0
L −1
w2 (d , k ) = w1 (d − k ) ∑ hm x (d − k − τ m' ) + w1 (d + k + 1) m' = 0
L−1
⋅ ∑ hm x(d − k − τ m ) + w1 (d − k ) ⋅ w1 (d + k + 1).
(33)
m =0
Let the starting point of i -th channel path in time domain be {ti : i = 0,1," , L − 1} . Then, (32) is given by 2 C h + w3 (d , k ), if d = ti P( d ) ≈ 1 i otherwise, w3 (d , k ),
w3 (d , k ) =
where
N / 2−1
∑ w (d , k ) 2
(34)
(35) (36)
k =0
C1 =
and
N / 2−1
∑
x (ti − k − τ m ) x (ti + k + 1 − τ m ) .
(37)
k =0
P( d ) =
N / 2−1
∑
r (d − k ) ⋅ r (d + k + 1)
k =0
=
N / 2−1 k =0
=
L−1
∑ ∑h x(d − k −τ
m
m=0
N /2−1 L−1
∑ ∑h k =0
m=0
( m=m' )
2 m
m
N /2−1 L−1 L−1 L−1 ) + w1(d − k ) ⋅ ∑ hm x(d + k +1 −τ m' ) + w1 (d + k + 1) = ∑ ∑hmx(d − k −τ m) ∑ hm' x(d + k +1 −τm' ) + w2 (d, k) m' =0 m' =0 k=0 m=0
x(d − k −τ m )x(d + k + 1 −τ m ) +
N /2−1 L−1 L−1
∑ ∑∑h h '
k=0 m =0 m=0
( m≠m' )
m m'
x(d − k −τ m ) x(d + k +1 −τ m' ) +
N /2−1
∑ w (d , k ) 2
k=0
(31)
