Design Criteria of Preamble Sequence for Multipath Fading
Design Criteria of Preamble Sequence for Multipath Fading Channels with Doppler Shift Tatsuro Higuchi and Yutaka Jitsumatsu† Depart. of Informatics, Kyushu University, Motooka 744, Nishi-ku, Fukuoka, 819-0395, Japan. Email: {higuchi, jitumatu}@me.inf.kyushu-u.ac.jp Abstract—The timing synchronization method proposed by Shmidl and Cox uses a reference block consisting of two identical parts, while the one proposed by Shi and Serpedin uses a reference block consisting of four parts with the third part being multiplied by −1. The accuracy of estimated delays of the latter method is higher than the former. In this paper, the number of division is generalized as an integer number M. Optimal M is investigated for single- and multi-path channels. We conclude that √ a recommended M value is approximately given by N/2 for both single- and muti-path channels, irrespectively of signal-to-noise ratio, where N is a length of the reference block.
I. Introduction Time and frequency synchronization is an important issue in Orthogonal Frequency Division Multiple Access (OFDMA) systems with a large Doppler shift [1]. Various methods of generating a preamble are already known (See [2] and references therein). A transmitted signal in an OFDMA system consists of a reference block and a data block; the former is a control signal and can be used for synchronization, while the latter is a payload. The Schmidl-Cox (S&C) method [3] in which a reference block consists of two identical parts is used as a coarse synchronization method in OFDMA systems. A receiver establishes its synchronization by finding the peak value of the auto-correlation between the received signal and the half-block delayed signal. Shi and Serpedin (S&S) [4] proposed an improved version of the S&C method, where a reference block consists of four identical parts, but the third part is multiplied by −1, i.e., the reference block is expressed by (B, B, −B, B). At the receiver, the sum of autocorrelation values for every pair of four parts is calculated. It is natural to ask 1) how many parts should we divide the reference block into for maximizing the synchronization performance in a multi-path environment with Doppler shift, and 2) if we divide a reference block into M parts, which parts should be multiplied by −1? We answer the former question in this paper, and will answer the second in a separate paper [8]. In this paper, effects of M and signal-to-noise ratio (SNR) on the sum of auto-correlation values are inves-
tigated. Firstly, we ignore the multi-path effect and assume that the channel impulse response (CIR) is a delta function. We analyze numerically how the peak value changes as M increases and as the SNR changes. Theoretical evaluation for this case is provided. Secondly, we take the multi-path effect into account. The power delay profile of CIR is assumed to decay exponentially and the maximum delay is denoted by LT . We analyze the effect of M and LT on the sum of auto-correlation values. Finally, numerical simulation results for various cases of M, LT and SNR are provided. II. Timing Synchronization A. Channel model A multi-path fading channel with a time delay and a Doppler shift can be modelled as follows: Consider a discrete-time and time-invariant system with impulse response (h0 , h1 , . . . hLT −1 ), where LT denotes the maximum delay. Let ε = N fD T s ∈ {0, 1, . . . , N − 1} be a normalized frequency offset, where fD is a Doppler frquency, T s is a sampling interval, and N is the size of a Discrete Fourier Transform (DFT) for an OFDMA that is equal to the length of a reference block. Then, the discrete-time received signal of the m-th time instance is expressed by L T −1 rm = e j2πεm/N h sm−θ− + wm , (1) =0
where sm is a transmitted signal, θ is a timing offset, and wm is a proper1 complex Gaussian noise with zero mean and variance σ2 . We assume that the filter coefficient h is a complexvalued and randomly generated with a power delay profile E[|h |2 ] = e−4/LT ,
(2)
where E[·] denotes the expectation. The real and imaginary parts of h follow, independently, Gaussian distri− 4 butions with mean zero and variance 12 e LT . 1 If Z is a proper complex Gaussian random variable, its real and imaginary parts are independent. †This work is supported by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Number 25820162.
correlation functions, defined by ˜ = Λk (θ)
M−k−1
di di+k
i=0
L−1
r¯n+θ+iL rn+θ+(i+k)L , ˜ ˜
(5)
n=0
where r¯m denotes the complex conjugate of rm .
Fig. 1. Transmitted signals for Schmidl-Cox (upper) and ShiSerpedin (lower) methods
B. Schmidl-Cox and Shi-Serpedin methods A transmitted signal consists of a reference block and data block in an OFDMA system. A reference block, also known as a preamble, is utilized to establish time and synchronization. In the S&C method [3], a reference block consists of two identical parts, denoted by (B, B). The auto-correlation value between the received signal and its half-block delayed version is calculated. If the receiver finds a peak of autocorrelations, synchronization is declared. Otherwise, the auto-correlation value of the next timing is calculated. The S&C method has a drawback that the peak value exhibits a large plateau that greatly reduces the accuracy of the estimated delay [1]. In order to overcome this drawback, Shi-Serpedin used a reference block composed of four identical parts with the third part being multiplied by −1 (See Fig. 1). Then, the S&S method exhibits a smaller plateau than does the S&C method. This result shows that the synchronization performance of M = 4 is better than that of M = 2. Then, it is natural to ask: Can we get better performance by increasing the number of repetitive parts to more than four? This paper gives an answer to this question. In this paper, we consider a general case where the number of repetitive parts is M and each part is multiplied by +1 or −1. Let N be the length of a reference block. We denote the sequence of these ±1 by (d0 , d1 , . . . , d M−1 ), while the repetitive part B consists of X0 , X1 , . . . , XL−1 , where L = N/M. Then, the transmitted signal is expressed by sn+iL = di · Xn
(3)
for 0 ≤ i ≤ M − 1, 0 ≤ n ≤ L − 1. The timing metric is introduced by Shi and Serpedin for the case M = 4, which can be generalized as follows [5]: M−1 M−1 ˜ k=1 |Λk (θ)| ˜ = 2 Γ(θ) , (4) 1 ˜ M |Λ0 (θ)| is a binomial coefficient, θ˜ is a where M2 = M(M−1) 2 controlled parameter for estimating θ and {Λk } are auto-
Remark: The timing metric Eq.(5) is based on an autocorrelation of the received signal. On the other hand, Rick and Milstein [6] derived their optimal decision statistics based on a cross-correlation function between the received signal and a replica of the transmitted signal for spread-spectrum (SS) communications in Rayleigh and Rician fading channels. Doppler shift was not taken into account in [6]. A timing estimation method combining auto-correlation and crosscorrelation functions for OFDM systems with Doppler shift was proposed in [7]. We may obtain a better timing metric utilizing both SS and OFDM techniques. The parameter θ˜ that attains the maximum value of ˜ Γ(θ) is selected as an estimate of θ, i.e., ˜ θˆ = arg max Γ(θ). (6) θ˜
It may be worth noting that this estimate has bias for a multi-path channel, that is, the estimate is likely to be larger than the true timing offset because of the channel distortion. Let Δθ = E[θˆ − θ] (7) be the bias of the estimator. The bias becomes large as LT gets large. This topic will be discussed in the Section IV. III. Analysis of the timing metric for a single-path channel ˜ for a The expectation of the timing metric Γ(θ) single-path channel is analyzed in this section. The received signal in this case is given by rm = sm−θ + wm ,
sm = dm/L · Xm mod L ,
(8)
where x denotes the largest integer smaller than x. Let us consider the effect of noise on a peak value ˜ The function Γ(θ) ˜ is maximized by θ˜ = θ. of the Γ(θ). It is easy to find that Eq.(5) in this case is expressed by Λk (θ) =
M−k−1 L−1 i=0
(Xn + di wn+θ+iL )
n=0
× (Xn + di+k wn+θ+(i+k)L ),
(9)
where di ∈ {+1, −1} is a code for i-th part of a reference block. We evaluate the expectations of numerator and denominator of Eq.(4). First, the expectation of the
denominator in (4) is calculated as follows: 1 M E [Λ0 (θ)] M−1 L−1 1 = E 1 + 2Xn di [wn+iL ] + |wn+iL |2 M j=0 n=0
1.2
0.4 0.2 0 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 SNR(dB)
Fig. 2. A comparison between expectation of Γ(θ) obtained by simulations and their theoretical estimations for L = 8 and N = 32, 64, and 128.
1.2 1
M=4 M=16 M=64
0.8 0.6 0.4 0.2 0 0 10 20 30 40 50 60 70 80 90 100 110 120 130 ~ θ
˜ for a single-path Fig. 3. The average values of the timing metric Γ(θ) channel with M = 4, 16, and 64, where N = 64 and SNR is 30dB.
L2 (M−k) − 2V .
(13)
The theoretical and experimental values of Γ(θ) are illustrated in Fig. 2. The numerator and denominator of (4) are evaluated separately and therefore (13) is not the expectation of (4). However, the theoretical value coincides with a numerical simulation result for a high SNR. IV. Simulation Results We perform numerical simulations with the following parameters: • The length of reference block is N = 64 (Figs.2–9) and N = 60 (Figs.10–12). • The delay is fixed to be θ = 30, which is unknown to the receiver. • di , Xn ∈ {+1, −1} are randomly generated with equal probability.
1.2 The value of Timing Metric
=
0.6
The value of Timing Metric
k=1
However, this approximation does not hold for a moderate or a low SNR. We assume Λk (θ) (k = 1, . . . , M−1) are proper complex Gaussian random variables. Without loss of generality, assume θ = 0. The expectation and the variance of Λk (θ) are L(M − k) and 2V(M − k), where (12) 2V = L(2 · σ2 + σ4 ) (See the Appendix for the derivation). It is well-known that if Z is a proper complex Gaussian random variable with mean μ and variance 2σ2 , then U = |Z| fol 2 +L2 lows a Rice distribution pU (u) = Vu exp − u 2V I0 uL V , 2
μ πσ2 1 1 and that E[U] = 2 L 2 − 2σ2 , where L 2 (x) = x x e x/2 (1 − x)I0 − 2 − xI1 − 2 and Iα (·) is the modified Bessel function of the first kind with order α. Finally, we obtain M−1 M−1 k=1 E[|Λk (θ)|] 2
Γ(θ)
0.8
= L(1 + σ2 ), (10) where [·] denotes the real part. Next, evaluate the expectation of the numerator in (4). For a high SNR, where wi → 0 for all i, we have E[|Λk (θ)|] ≈ L(M − k). Thus, the expecatation of the numerator for a high SNR is evaluated as M−1 M−1 E[|Λk (θ)|] ≈ L. (11) 2
1 M E[Λ0 (θ)] M−1 M−1 πV(M−k) L 12 2 2 k=1 L(1 + σ2 )
experimental N=32 experimental N=64 experimental N=128 theoretical N=32 theoretical N=64 theoretical N=128
1
1
SNR=30(dB) SNR=0(dB) SNR=-30(dB)
0.8 0.6 0.4 0.2 0 0 10 20 30 40 50 60 70 80 90 100 110 120 130 ~ θ
˜ for a singleFig. 4. The average values of the timing metric Γ(θ) path channel with M = 4, where N = 64 and SNR is −30dB, 0dB, and 30dB.
1 0.8 0.6 0.4 0.2 0
0 10 20 30 40 50 60 70 80 90 100 110 120 130 ~ θ (a) M = 4 The value of Timing Metric
1.2
LT =1 LT =5 LT =10 LT =20
1 0.8
1.2
LT =5 LT =10 LT =20
1 0.8 0.6 0.4 0.2 0 0
0.6 0.4 0.2 0 10 20 30 40 50 60 70 80 90 100 110 120 130 ~ θ
(b) M = 16
5 10 15 20 25 30 35 40 45 50 55 60 M
˜ versus M for a multi-path Fig. 6. The average value of maxθ˜ Γ(θ) channel with LT = 5, 10 and 20, where N = 60 and SNR is 30dB. 1.2
0
1
SNR=30(dB) SNR=0(dB) SNR=-10(dB)
0.8 0.6 0.4 0.2 0
1.2 The value of Timing Metric
The maximum value of Timing Metric
LT =1 LT =5 LT =10 LT =20
The value of Timing Metric
The value of Timing Metric
1.2
LT =1 LT =5 LT =10 LT =20
1 0.8
0 10 20 30 40 50 60 70 80 90 100 110 120 130 ~ θ
Fig. 7. The average values of the average values of timing metric ˜ for a multi-path channel with LT = 10, where N = 64 and SNRs Γ(θ) are −10, 0, and 30dB.
0.6 0.4 0.2 0 20 22 24 26 28 30 32 34 36 38 40 ~
θ
(c) M = 64
˜ for single-path Fig. 5. The average values of the timing metric Γ(θ) (LT = 1) and multi-path (LT = 5, 10, 20) channels, where N = 64 and SNR is 30dB.
1) Case 1 [Single-path Channel]: In Fig.3, simu˜ averaged over lation results for the timing metric Γ(θ) randomly generated di , Xn , and noise for M = 4, 16, and 64 are depicted. This figure shows that the timing metric becomes sharper by increasing M and that the support of peak is given by N/M approximately. ˜ when SNR is −30dB, 0dB, and Fig.4 shows Γ(θ) 30dB. This figure shows the peak value is decreased as SNR becomes lower. 2) Case 2 [Multi-path Channel]: Here, we give simulation results of the timing metric in a multi-path channel. Figures 5 (a) –(c) show the timing metric for M = 4, 16, and 64, where SNR is 30dB. For simplicity, Doppler frequency is assumed to be zero. These figures show that when M is small, the
plateau of the peak is enlarged as LT gets large. We observe that in Figs. 5 (a) and (b), the θˆ that attains the peak value is slightly larger than the actual delay. It is also shown that the plateau becomes smaller as M becomes large, while the value of the peak decreases. ˜ for In Fig. 6, the average values of maxθ˜ Γ(θ) 2 ≤ M ≤ 60 and LT = 5, 10, 20 are illustrated, where N = 60. The curve for LT = 1 is almost one for all M and is omitted. It is shown that the peak value is decreases as M increases. This figure shows that for LT = 20, the peak value for M = 20 is almost half of that for M = 2. Note also that for given M and LT , ˜ is always greater the average value of maximum Γ(θ) ˜ i.e., than the maximum value of the average of Γ(θ), ˜ ≤ E[maxθ˜ Γ(θ)] ˜ holds. Fig. 5 shows the maxθ˜ E[Γ(θ)] left hand side and Fig. 6 shows the right hand side. Fig. 7 shows the timing metric for a multi-path channel with additive noise, where M = 4. This figure shows that the peak value severely degrades for a low SNR environment. In Figs. 8 (a)–(c), results of Monte Carlo simulation for the variances of timing estimation errors, Var[θ] = E[(θ − (θˆ − Δθ))2 ], are shown, where Δθ is a bias of the estimator, defined by (7). The way how we estimate and compensate the bias of θ will be discussed
TABLE I.
Types of the shapes of the variance of estimated timing errors versus M, obtained by numerical simulation.
LT
N = 120 SNR=10dB SNR=15dB SNR=20dB
1 a a a
3 a a a
5 a a a
10 c b a
SNR=10dB SNR=15dB SNR=20dB
LT 1 a a a
5 a a a
7 c c b
2
V[estimated value of θ](sec )
8
3 a a a
6 4 2 0
LT
N = 240
10 c b b
5 a a a
7 b c c
10 c b c
SNR=20(dB), LT =1 SNR=20(dB), LT =3 SNR=20(dB), LT =5 SNR=20(dB), LT =7 SNR=20(dB), LT =10
2.5 2 1.5 1 0.5
10
20
30
40
50
60
0
5
10
M
2
V[estimated value of θ](sec )
2
20
(a) N = 120 3
SNR=10(dB), LT =1 SNR=10(dB), LT =5 SNR=10(dB), LT =10
8
15 M
(a) 5dB 10 V[estimated value of θ](sec )
3 a a a
0 0
6 4 2 0
SNR=20(dB), LT =1 SNR=20(dB), LT =3 SNR=20(dB), LT =5 SNR=20(dB), LT =7 SNR=20(dB), LT =10
2.5 2 1.5 1 0.5 0
0
10
20
30 M
40
50
60
0
5
10
10
3 2
V[estimated value of θ](sec )
2
20
25
30
35
(b) N = 180
SNR=20(dB), LT =1 SNR=20(dB), LT =5 SNR=20(dB), LT =10
8
15 M
(b) 10dB
V[estimated value of θ](sec )
1 a a a
SNR=10dB SNR=15dB SNR=20dB
3
SNR=5(dB), LT =1 SNR=5(dB), LT =5 SNR=5(dB), LT =10
2
V[estimated value of θ](sec )
10
7 a a a
N = 180
6 4 2 0
SNR=20(dB), LT =1 SNR=20(dB), LT =3 SNR=20(dB), LT =5 SNR=20(dB), LT =7 SNR=20(dB), LT =10
2.5 2 1.5 1 0.5 0
0
10
20
30 M
40
50
60
0
5
10
15
20
25 M
30
35
40
45
(c) 20dB
(c) N = 240
Fig. 8. The variance of timing estimation error for LT = 1, 5, 10, where N = 60.
Fig. 9. The variance of timing estimation error for LT = 1, 3, 5, 7, 10, where SNR is 20dB.
in a separate paper. The case LT = 1 implies a single path channel. The graph when SNR is 15dB is almost the same as that for 10dB and 20dB, and thus omitted here. These figures show that the variances of timing estimation errors are minimized by M = N = 60 for a single-path channel, but they are minimized by M = 5 and M = 6 for a multi-path channel with LT = 5 and LT = 10.
Simulation results for the variances of timing estimation errors for N = 120, 180, and 240 are shown in Fig. 9, where SNR is 20dB. Figures of the simulation results for 10dB and 15dB SNRs are omitted due to insufficient space. These curves versus M are categorized into three cases: (a) monotone decreasing, (b) first decreasing, taking minimum value at M = M ∗ , and then increasing, (c) it is difficult to judge (a) or (b).
TABLE II.
N M
60 5 or 6
Recommended values for M
120 6 or 8
180 9 or 10
240 10 or 12
The result of the categorization is shown in Table I. For the case (b), Var[θˆ − θ + Δθ] is minimized by M = M ∗ . On the other hand, for the case (a), it is minimized by M = N but there exists M = M0 such that Var[θˆ − θ + Δθ] takes almost the same value for M ≥ M0 . Since large M requires long computational time, M = M0 is recommended for the case (a) and (c). Finally, the recommended M values for N = 60, 120, 180 and 240 are summarized in Table II, where is M ∗ for case (b) and M0 for cases (a) and (c). M is independent of LT and Interestingly, the value M SNR if SNR is between 10dB to 30dB. From this table, we find that √ recommended M value is approximately given by N/2. V. Conclusion In this paper, we have considered an optimization of the performance of a timing synchronization method [5] which is a generalization of [3] and [4]. A reference block is divided into M repetitive parts, where each part is multiplied by a {+1, −1}-valued sequence. The timing ˜ was used to evaluate the synchronization metric, Γ(θ), ˜ performance. The effect of M as well as SNR on Γ(θ) was analyzed. We first considered a single-path channel for simplicity and then considered a multi-path channel. Single-path model was analyzed theoretically and multi-path model was investigated through numerical simulations. For a single-path channel with a high SNR, we observed that a plateau which is observed in the S&C method is decreased by increasing M and the peak is very sharp and the length of plateau is equal to a chip duration when M = N. The peak value of the timing metric decreases for a low SNR case. This phenomenon was analyzed theoretically. Then, we investigated the timing metric for a multipath channel. The number of resolvable paths for simulation were LT = 5, 10, 20 and the length of reference block was N = 64 or N = 60. For a high SNR, the length of plateau of timing metric increases as LT increases and it decreases as M increases. The peak value decreases as LT and M increases. In the presence of noise, the peak value √ decreases as SNR degrades. We conclude that M N/2 is recommended for both signle- and multi-path channels. References [1] M. Morelli, I. Scott, C.-C. J. Kuo, and M.-O. Pun, “Synchronization Techniques for Orthogonal Frequency Division Multiple Access (OFDMA): A Tutorial Review,” Proc. of the IEEE, Vol.95, pp.1394-1427, 2007.
[2] E. M. Silva C, F. J. Harris, G. J. Dolecek, “On Preamble Design for Timing and Frequency Synchronization of OFDM Systems over Rayleigh Fading Channels,” 18th Int. Conf. Digital Signal Processing, July, 2013. [3] 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. [4] K. Shi and E. Serpedin, “Coarse Frame and Carrier Synchronization of OFDM Systems: A New Metric and Comparison,” IEEE Trans Wireless Commun., Vol.3, No.4, pp.1271-1284, July 2004. [5] M. Ruan, M. C. Mark, and Z. Shi, “Training Symbol Based Coarse Timing Synchronization in OFDM Systems,” IEEE Tran. Wireless Commun., Vol.8, No.5, pp.2558-2569, 2009. [6] R. R. Rick and L. B. Milstein, “Optimal Decision Strategies for Acquisition of Spread-Spectrum Signals in FrequencySelective Fading Channel,” IEEE Tran. Commun., Vol. 46, No. 5, pp. 686–694, 1998. [7] A. B. Awoseyila, C. Kasparis, and B. G. Evans, “Improved Preamble-Aided Timing Estimation for OFDM Systems,” IEEE Tran. Commun. Lett., Vol.12, No.11, pp.825-827, 2008. [8] Y. Jitsumatsu, M. Hashiguchi, and T. Higuchi, “Optimal Sign Patterns for a Generalized Schmidl-Cox Method,” Proc. SEquences and Their Applications (SETA2014), Nov. 2014. (to be presented)
Appendix M−k−1 Zi,i+k , where Let Λk (θ) = i=0 L−1 Zi, j = (Xn + di wn+θ+iL )(Xn + d j wn+θ+ jL ).
(14)
n=0
We assume Zi j follows a proper complex Gaussian distribution. The mean and the variance of the real part of Zi j are L and V, where V is defined by
2V = E |Zi j − L|2 L−1 = E {Xn (di wn+iL + d j wn+ jL ) n=0
2 + di d j wn+iL wn+ jL }
=
L−1 E |wn+iL |2 + |wn+ jL |2 + |wn+iL |2 |wn+ jL |2 n=0
(15) = L(2 · σ2 + σ4 ). Here, we have used the assumption that wn+iL and wn+ jL ( j i) are independent. The mean and the variance of the imaginary part of Zi j are 0 and V. The covarience between Zi,i+k and Z j, j+k for j i is found to be zero. Therefore, the mean and the variance of Λk (θ) are L(M− k) and 2V(M − k). Hence, we have (13).
I. Introduction Time and frequency synchronization is an important issue in Orthogonal Frequency Division Multiple Access (OFDMA) systems with a large Doppler shift [1]. Various methods of generating a preamble are already known (See [2] and references therein). A transmitted signal in an OFDMA system consists of a reference block and a data block; the former is a control signal and can be used for synchronization, while the latter is a payload. The Schmidl-Cox (S&C) method [3] in which a reference block consists of two identical parts is used as a coarse synchronization method in OFDMA systems. A receiver establishes its synchronization by finding the peak value of the auto-correlation between the received signal and the half-block delayed signal. Shi and Serpedin (S&S) [4] proposed an improved version of the S&C method, where a reference block consists of four identical parts, but the third part is multiplied by −1, i.e., the reference block is expressed by (B, B, −B, B). At the receiver, the sum of autocorrelation values for every pair of four parts is calculated. It is natural to ask 1) how many parts should we divide the reference block into for maximizing the synchronization performance in a multi-path environment with Doppler shift, and 2) if we divide a reference block into M parts, which parts should be multiplied by −1? We answer the former question in this paper, and will answer the second in a separate paper [8]. In this paper, effects of M and signal-to-noise ratio (SNR) on the sum of auto-correlation values are inves-
tigated. Firstly, we ignore the multi-path effect and assume that the channel impulse response (CIR) is a delta function. We analyze numerically how the peak value changes as M increases and as the SNR changes. Theoretical evaluation for this case is provided. Secondly, we take the multi-path effect into account. The power delay profile of CIR is assumed to decay exponentially and the maximum delay is denoted by LT . We analyze the effect of M and LT on the sum of auto-correlation values. Finally, numerical simulation results for various cases of M, LT and SNR are provided. II. Timing Synchronization A. Channel model A multi-path fading channel with a time delay and a Doppler shift can be modelled as follows: Consider a discrete-time and time-invariant system with impulse response (h0 , h1 , . . . hLT −1 ), where LT denotes the maximum delay. Let ε = N fD T s ∈ {0, 1, . . . , N − 1} be a normalized frequency offset, where fD is a Doppler frquency, T s is a sampling interval, and N is the size of a Discrete Fourier Transform (DFT) for an OFDMA that is equal to the length of a reference block. Then, the discrete-time received signal of the m-th time instance is expressed by L T −1 rm = e j2πεm/N h sm−θ− + wm , (1) =0
where sm is a transmitted signal, θ is a timing offset, and wm is a proper1 complex Gaussian noise with zero mean and variance σ2 . We assume that the filter coefficient h is a complexvalued and randomly generated with a power delay profile E[|h |2 ] = e−4/LT ,
(2)
where E[·] denotes the expectation. The real and imaginary parts of h follow, independently, Gaussian distri− 4 butions with mean zero and variance 12 e LT . 1 If Z is a proper complex Gaussian random variable, its real and imaginary parts are independent. †This work is supported by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Number 25820162.
correlation functions, defined by ˜ = Λk (θ)
M−k−1
di di+k
i=0
L−1
r¯n+θ+iL rn+θ+(i+k)L , ˜ ˜
(5)
n=0
where r¯m denotes the complex conjugate of rm .
Fig. 1. Transmitted signals for Schmidl-Cox (upper) and ShiSerpedin (lower) methods
B. Schmidl-Cox and Shi-Serpedin methods A transmitted signal consists of a reference block and data block in an OFDMA system. A reference block, also known as a preamble, is utilized to establish time and synchronization. In the S&C method [3], a reference block consists of two identical parts, denoted by (B, B). The auto-correlation value between the received signal and its half-block delayed version is calculated. If the receiver finds a peak of autocorrelations, synchronization is declared. Otherwise, the auto-correlation value of the next timing is calculated. The S&C method has a drawback that the peak value exhibits a large plateau that greatly reduces the accuracy of the estimated delay [1]. In order to overcome this drawback, Shi-Serpedin used a reference block composed of four identical parts with the third part being multiplied by −1 (See Fig. 1). Then, the S&S method exhibits a smaller plateau than does the S&C method. This result shows that the synchronization performance of M = 4 is better than that of M = 2. Then, it is natural to ask: Can we get better performance by increasing the number of repetitive parts to more than four? This paper gives an answer to this question. In this paper, we consider a general case where the number of repetitive parts is M and each part is multiplied by +1 or −1. Let N be the length of a reference block. We denote the sequence of these ±1 by (d0 , d1 , . . . , d M−1 ), while the repetitive part B consists of X0 , X1 , . . . , XL−1 , where L = N/M. Then, the transmitted signal is expressed by sn+iL = di · Xn
(3)
for 0 ≤ i ≤ M − 1, 0 ≤ n ≤ L − 1. The timing metric is introduced by Shi and Serpedin for the case M = 4, which can be generalized as follows [5]: M−1 M−1 ˜ k=1 |Λk (θ)| ˜ = 2 Γ(θ) , (4) 1 ˜ M |Λ0 (θ)| is a binomial coefficient, θ˜ is a where M2 = M(M−1) 2 controlled parameter for estimating θ and {Λk } are auto-
Remark: The timing metric Eq.(5) is based on an autocorrelation of the received signal. On the other hand, Rick and Milstein [6] derived their optimal decision statistics based on a cross-correlation function between the received signal and a replica of the transmitted signal for spread-spectrum (SS) communications in Rayleigh and Rician fading channels. Doppler shift was not taken into account in [6]. A timing estimation method combining auto-correlation and crosscorrelation functions for OFDM systems with Doppler shift was proposed in [7]. We may obtain a better timing metric utilizing both SS and OFDM techniques. The parameter θ˜ that attains the maximum value of ˜ Γ(θ) is selected as an estimate of θ, i.e., ˜ θˆ = arg max Γ(θ). (6) θ˜
It may be worth noting that this estimate has bias for a multi-path channel, that is, the estimate is likely to be larger than the true timing offset because of the channel distortion. Let Δθ = E[θˆ − θ] (7) be the bias of the estimator. The bias becomes large as LT gets large. This topic will be discussed in the Section IV. III. Analysis of the timing metric for a single-path channel ˜ for a The expectation of the timing metric Γ(θ) single-path channel is analyzed in this section. The received signal in this case is given by rm = sm−θ + wm ,
sm = dm/L · Xm mod L ,
(8)
where x denotes the largest integer smaller than x. Let us consider the effect of noise on a peak value ˜ The function Γ(θ) ˜ is maximized by θ˜ = θ. of the Γ(θ). It is easy to find that Eq.(5) in this case is expressed by Λk (θ) =
M−k−1 L−1 i=0
(Xn + di wn+θ+iL )
n=0
× (Xn + di+k wn+θ+(i+k)L ),
(9)
where di ∈ {+1, −1} is a code for i-th part of a reference block. We evaluate the expectations of numerator and denominator of Eq.(4). First, the expectation of the
denominator in (4) is calculated as follows: 1 M E [Λ0 (θ)] M−1 L−1 1 = E 1 + 2Xn di [wn+iL ] + |wn+iL |2 M j=0 n=0
1.2
0.4 0.2 0 -30 -25 -20 -15 -10 -5 0 5 10 15 20 25 30 SNR(dB)
Fig. 2. A comparison between expectation of Γ(θ) obtained by simulations and their theoretical estimations for L = 8 and N = 32, 64, and 128.
1.2 1
M=4 M=16 M=64
0.8 0.6 0.4 0.2 0 0 10 20 30 40 50 60 70 80 90 100 110 120 130 ~ θ
˜ for a single-path Fig. 3. The average values of the timing metric Γ(θ) channel with M = 4, 16, and 64, where N = 64 and SNR is 30dB.
L2 (M−k) − 2V .
(13)
The theoretical and experimental values of Γ(θ) are illustrated in Fig. 2. The numerator and denominator of (4) are evaluated separately and therefore (13) is not the expectation of (4). However, the theoretical value coincides with a numerical simulation result for a high SNR. IV. Simulation Results We perform numerical simulations with the following parameters: • The length of reference block is N = 64 (Figs.2–9) and N = 60 (Figs.10–12). • The delay is fixed to be θ = 30, which is unknown to the receiver. • di , Xn ∈ {+1, −1} are randomly generated with equal probability.
1.2 The value of Timing Metric
=
0.6
The value of Timing Metric
k=1
However, this approximation does not hold for a moderate or a low SNR. We assume Λk (θ) (k = 1, . . . , M−1) are proper complex Gaussian random variables. Without loss of generality, assume θ = 0. The expectation and the variance of Λk (θ) are L(M − k) and 2V(M − k), where (12) 2V = L(2 · σ2 + σ4 ) (See the Appendix for the derivation). It is well-known that if Z is a proper complex Gaussian random variable with mean μ and variance 2σ2 , then U = |Z| fol 2 +L2 lows a Rice distribution pU (u) = Vu exp − u 2V I0 uL V , 2
μ πσ2 1 1 and that E[U] = 2 L 2 − 2σ2 , where L 2 (x) = x x e x/2 (1 − x)I0 − 2 − xI1 − 2 and Iα (·) is the modified Bessel function of the first kind with order α. Finally, we obtain M−1 M−1 k=1 E[|Λk (θ)|] 2
Γ(θ)
0.8
= L(1 + σ2 ), (10) where [·] denotes the real part. Next, evaluate the expectation of the numerator in (4). For a high SNR, where wi → 0 for all i, we have E[|Λk (θ)|] ≈ L(M − k). Thus, the expecatation of the numerator for a high SNR is evaluated as M−1 M−1 E[|Λk (θ)|] ≈ L. (11) 2
1 M E[Λ0 (θ)] M−1 M−1 πV(M−k) L 12 2 2 k=1 L(1 + σ2 )
experimental N=32 experimental N=64 experimental N=128 theoretical N=32 theoretical N=64 theoretical N=128
1
1
SNR=30(dB) SNR=0(dB) SNR=-30(dB)
0.8 0.6 0.4 0.2 0 0 10 20 30 40 50 60 70 80 90 100 110 120 130 ~ θ
˜ for a singleFig. 4. The average values of the timing metric Γ(θ) path channel with M = 4, where N = 64 and SNR is −30dB, 0dB, and 30dB.
1 0.8 0.6 0.4 0.2 0
0 10 20 30 40 50 60 70 80 90 100 110 120 130 ~ θ (a) M = 4 The value of Timing Metric
1.2
LT =1 LT =5 LT =10 LT =20
1 0.8
1.2
LT =5 LT =10 LT =20
1 0.8 0.6 0.4 0.2 0 0
0.6 0.4 0.2 0 10 20 30 40 50 60 70 80 90 100 110 120 130 ~ θ
(b) M = 16
5 10 15 20 25 30 35 40 45 50 55 60 M
˜ versus M for a multi-path Fig. 6. The average value of maxθ˜ Γ(θ) channel with LT = 5, 10 and 20, where N = 60 and SNR is 30dB. 1.2
0
1
SNR=30(dB) SNR=0(dB) SNR=-10(dB)
0.8 0.6 0.4 0.2 0
1.2 The value of Timing Metric
The maximum value of Timing Metric
LT =1 LT =5 LT =10 LT =20
The value of Timing Metric
The value of Timing Metric
1.2
LT =1 LT =5 LT =10 LT =20
1 0.8
0 10 20 30 40 50 60 70 80 90 100 110 120 130 ~ θ
Fig. 7. The average values of the average values of timing metric ˜ for a multi-path channel with LT = 10, where N = 64 and SNRs Γ(θ) are −10, 0, and 30dB.
0.6 0.4 0.2 0 20 22 24 26 28 30 32 34 36 38 40 ~
θ
(c) M = 64
˜ for single-path Fig. 5. The average values of the timing metric Γ(θ) (LT = 1) and multi-path (LT = 5, 10, 20) channels, where N = 64 and SNR is 30dB.
1) Case 1 [Single-path Channel]: In Fig.3, simu˜ averaged over lation results for the timing metric Γ(θ) randomly generated di , Xn , and noise for M = 4, 16, and 64 are depicted. This figure shows that the timing metric becomes sharper by increasing M and that the support of peak is given by N/M approximately. ˜ when SNR is −30dB, 0dB, and Fig.4 shows Γ(θ) 30dB. This figure shows the peak value is decreased as SNR becomes lower. 2) Case 2 [Multi-path Channel]: Here, we give simulation results of the timing metric in a multi-path channel. Figures 5 (a) –(c) show the timing metric for M = 4, 16, and 64, where SNR is 30dB. For simplicity, Doppler frequency is assumed to be zero. These figures show that when M is small, the
plateau of the peak is enlarged as LT gets large. We observe that in Figs. 5 (a) and (b), the θˆ that attains the peak value is slightly larger than the actual delay. It is also shown that the plateau becomes smaller as M becomes large, while the value of the peak decreases. ˜ for In Fig. 6, the average values of maxθ˜ Γ(θ) 2 ≤ M ≤ 60 and LT = 5, 10, 20 are illustrated, where N = 60. The curve for LT = 1 is almost one for all M and is omitted. It is shown that the peak value is decreases as M increases. This figure shows that for LT = 20, the peak value for M = 20 is almost half of that for M = 2. Note also that for given M and LT , ˜ is always greater the average value of maximum Γ(θ) ˜ i.e., than the maximum value of the average of Γ(θ), ˜ ≤ E[maxθ˜ Γ(θ)] ˜ holds. Fig. 5 shows the maxθ˜ E[Γ(θ)] left hand side and Fig. 6 shows the right hand side. Fig. 7 shows the timing metric for a multi-path channel with additive noise, where M = 4. This figure shows that the peak value severely degrades for a low SNR environment. In Figs. 8 (a)–(c), results of Monte Carlo simulation for the variances of timing estimation errors, Var[θ] = E[(θ − (θˆ − Δθ))2 ], are shown, where Δθ is a bias of the estimator, defined by (7). The way how we estimate and compensate the bias of θ will be discussed
TABLE I.
Types of the shapes of the variance of estimated timing errors versus M, obtained by numerical simulation.
LT
N = 120 SNR=10dB SNR=15dB SNR=20dB
1 a a a
3 a a a
5 a a a
10 c b a
SNR=10dB SNR=15dB SNR=20dB
LT 1 a a a
5 a a a
7 c c b
2
V[estimated value of θ](sec )
8
3 a a a
6 4 2 0
LT
N = 240
10 c b b
5 a a a
7 b c c
10 c b c
SNR=20(dB), LT =1 SNR=20(dB), LT =3 SNR=20(dB), LT =5 SNR=20(dB), LT =7 SNR=20(dB), LT =10
2.5 2 1.5 1 0.5
10
20
30
40
50
60
0
5
10
M
2
V[estimated value of θ](sec )
2
20
(a) N = 120 3
SNR=10(dB), LT =1 SNR=10(dB), LT =5 SNR=10(dB), LT =10
8
15 M
(a) 5dB 10 V[estimated value of θ](sec )
3 a a a
0 0
6 4 2 0
SNR=20(dB), LT =1 SNR=20(dB), LT =3 SNR=20(dB), LT =5 SNR=20(dB), LT =7 SNR=20(dB), LT =10
2.5 2 1.5 1 0.5 0
0
10
20
30 M
40
50
60
0
5
10
10
3 2
V[estimated value of θ](sec )
2
20
25
30
35
(b) N = 180
SNR=20(dB), LT =1 SNR=20(dB), LT =5 SNR=20(dB), LT =10
8
15 M
(b) 10dB
V[estimated value of θ](sec )
1 a a a
SNR=10dB SNR=15dB SNR=20dB
3
SNR=5(dB), LT =1 SNR=5(dB), LT =5 SNR=5(dB), LT =10
2
V[estimated value of θ](sec )
10
7 a a a
N = 180
6 4 2 0
SNR=20(dB), LT =1 SNR=20(dB), LT =3 SNR=20(dB), LT =5 SNR=20(dB), LT =7 SNR=20(dB), LT =10
2.5 2 1.5 1 0.5 0
0
10
20
30 M
40
50
60
0
5
10
15
20
25 M
30
35
40
45
(c) 20dB
(c) N = 240
Fig. 8. The variance of timing estimation error for LT = 1, 5, 10, where N = 60.
Fig. 9. The variance of timing estimation error for LT = 1, 3, 5, 7, 10, where SNR is 20dB.
in a separate paper. The case LT = 1 implies a single path channel. The graph when SNR is 15dB is almost the same as that for 10dB and 20dB, and thus omitted here. These figures show that the variances of timing estimation errors are minimized by M = N = 60 for a single-path channel, but they are minimized by M = 5 and M = 6 for a multi-path channel with LT = 5 and LT = 10.
Simulation results for the variances of timing estimation errors for N = 120, 180, and 240 are shown in Fig. 9, where SNR is 20dB. Figures of the simulation results for 10dB and 15dB SNRs are omitted due to insufficient space. These curves versus M are categorized into three cases: (a) monotone decreasing, (b) first decreasing, taking minimum value at M = M ∗ , and then increasing, (c) it is difficult to judge (a) or (b).
TABLE II.
N M
60 5 or 6
Recommended values for M
120 6 or 8
180 9 or 10
240 10 or 12
The result of the categorization is shown in Table I. For the case (b), Var[θˆ − θ + Δθ] is minimized by M = M ∗ . On the other hand, for the case (a), it is minimized by M = N but there exists M = M0 such that Var[θˆ − θ + Δθ] takes almost the same value for M ≥ M0 . Since large M requires long computational time, M = M0 is recommended for the case (a) and (c). Finally, the recommended M values for N = 60, 120, 180 and 240 are summarized in Table II, where is M ∗ for case (b) and M0 for cases (a) and (c). M is independent of LT and Interestingly, the value M SNR if SNR is between 10dB to 30dB. From this table, we find that √ recommended M value is approximately given by N/2. V. Conclusion In this paper, we have considered an optimization of the performance of a timing synchronization method [5] which is a generalization of [3] and [4]. A reference block is divided into M repetitive parts, where each part is multiplied by a {+1, −1}-valued sequence. The timing ˜ was used to evaluate the synchronization metric, Γ(θ), ˜ performance. The effect of M as well as SNR on Γ(θ) was analyzed. We first considered a single-path channel for simplicity and then considered a multi-path channel. Single-path model was analyzed theoretically and multi-path model was investigated through numerical simulations. For a single-path channel with a high SNR, we observed that a plateau which is observed in the S&C method is decreased by increasing M and the peak is very sharp and the length of plateau is equal to a chip duration when M = N. The peak value of the timing metric decreases for a low SNR case. This phenomenon was analyzed theoretically. Then, we investigated the timing metric for a multipath channel. The number of resolvable paths for simulation were LT = 5, 10, 20 and the length of reference block was N = 64 or N = 60. For a high SNR, the length of plateau of timing metric increases as LT increases and it decreases as M increases. The peak value decreases as LT and M increases. In the presence of noise, the peak value √ decreases as SNR degrades. We conclude that M N/2 is recommended for both signle- and multi-path channels. References [1] M. Morelli, I. Scott, C.-C. J. Kuo, and M.-O. Pun, “Synchronization Techniques for Orthogonal Frequency Division Multiple Access (OFDMA): A Tutorial Review,” Proc. of the IEEE, Vol.95, pp.1394-1427, 2007.
[2] E. M. Silva C, F. J. Harris, G. J. Dolecek, “On Preamble Design for Timing and Frequency Synchronization of OFDM Systems over Rayleigh Fading Channels,” 18th Int. Conf. Digital Signal Processing, July, 2013. [3] 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. [4] K. Shi and E. Serpedin, “Coarse Frame and Carrier Synchronization of OFDM Systems: A New Metric and Comparison,” IEEE Trans Wireless Commun., Vol.3, No.4, pp.1271-1284, July 2004. [5] M. Ruan, M. C. Mark, and Z. Shi, “Training Symbol Based Coarse Timing Synchronization in OFDM Systems,” IEEE Tran. Wireless Commun., Vol.8, No.5, pp.2558-2569, 2009. [6] R. R. Rick and L. B. Milstein, “Optimal Decision Strategies for Acquisition of Spread-Spectrum Signals in FrequencySelective Fading Channel,” IEEE Tran. Commun., Vol. 46, No. 5, pp. 686–694, 1998. [7] A. B. Awoseyila, C. Kasparis, and B. G. Evans, “Improved Preamble-Aided Timing Estimation for OFDM Systems,” IEEE Tran. Commun. Lett., Vol.12, No.11, pp.825-827, 2008. [8] Y. Jitsumatsu, M. Hashiguchi, and T. Higuchi, “Optimal Sign Patterns for a Generalized Schmidl-Cox Method,” Proc. SEquences and Their Applications (SETA2014), Nov. 2014. (to be presented)
Appendix M−k−1 Zi,i+k , where Let Λk (θ) = i=0 L−1 Zi, j = (Xn + di wn+θ+iL )(Xn + d j wn+θ+ jL ).
(14)
n=0
We assume Zi j follows a proper complex Gaussian distribution. The mean and the variance of the real part of Zi j are L and V, where V is defined by
2V = E |Zi j − L|2 L−1 = E {Xn (di wn+iL + d j wn+ jL ) n=0
2 + di d j wn+iL wn+ jL }
=
L−1 E |wn+iL |2 + |wn+ jL |2 + |wn+iL |2 |wn+ jL |2 n=0
(15) = L(2 · σ2 + σ4 ). Here, we have used the assumption that wn+iL and wn+ jL ( j i) are independent. The mean and the variance of the imaginary part of Zi j are 0 and V. The covarience between Zi,i+k and Z j, j+k for j i is found to be zero. Therefore, the mean and the variance of Λk (θ) are L(M− k) and 2V(M − k). Hence, we have (13).
