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Data-Aided Maximum Likelihood Frequency

Synchronization for OFDM Systems

Yi-Ching Liao, Kwang-Cheng Chen

Institute of Communication Engineering, College of Electrical Engineering

National Taiwan University, Taipei, Taiwan, 10617, R.O.C.

E-mail: ycliao@fcom.ee.ntu.edu.tw, chenkc@cc.ee.ntu.edu.tw

Abstract— Distinct from conventional approaches, we apply a

generalized data-aided signal model which can jointly consider the cyclic prefix with the preamble/pilot to propose three data-aided frequency offset estimation algorithms based on maximum likelihood criterion. The proposed algorithms differ from each other in the processing domain and the observation space. They can not only systematically estimate both the integer part and the fractional part of the frequency offset, but also possess the maximum acquisition range of a discrete time system. The effectiveness of the proposed schemes is verified by mathematical analysis and computer simulation.

I. INTRODUCTION

In recent years, multicarrier transmission, in particular, orthogonal frequency division multiplexing (OFDM) has at-tracted much more attention for its possible application to the design of broadband wireless communication systems. Multicarrier transmission schemes are resistant to wireless impairments such as multipath fading and impulsive noise. Therefore, its most important application for future wireless multimedia is high speed data transmission over fading chan-nels.

In OFDM systems, the sensitivity to carrier frequency offset (CFO) is one of the major challenges for practical implementa-tion. CFO is mainly caused by Doppler shift, Doppler spread in fading channels and transmitter-receiver oscillator instabilities. Such an offset can be dozens of the subcarrier spacing and is usually classified as the integer part and the fractional part. The integer part is a multiple of the subcarrier spacing while the fractional part is confined to half the subcarrier spacing. Without properly compensation, the former results in a shift of the subcarrier indices and the latter produces inter-carrier interference [2].

Data-aided systems using pilot or preamble symbols are more suitable for packet oriented applications which require fast and reliable synchronization. Several data-aided schemes are proposed in the literature. The method proposed in [3] gives the maximum likelihood (ML) frequency estimator based on the observation of two consecutive and identical OFDM symbols. The estimation range is limited to half of the subcarrier spacing. In other words, an ambiguity of multiple subcarrier spacing exists even though this limit can be extended by shortening the training symbol duration at the cost of reduced estimation accuracy. A joint timing and frequency estimation is proposed in [4], where two training OFDM symbols are employed for both timing and frequency

offset estimation. In this scheme, the first symbol has two identical halves and serves to measure the fractional part. The second symbol serves to resolve the remaining ambiguity, however the estimation accuracy is still not satisfied.

On account of the significance of data-aided systems, it is substantial to investigate the data-aided estimation of CFO in OFDM systems. Data-aided frequency synchronization algo-rithms can be essentially classified by the processing domain (time or frequency), the observation space ( considering cyclic prefix or not) and the data-assistance (preamble-aided or pilot-aided). In this paper, different from conventional approaches, we apply ML criterion to propose a class of estimators based on a generalized data-aided signal model which can completely describe the foregoing affecting factors and only assume one general training symbol or pilot signal. The proposed schemes can effectively estimate both parts of CFO and have a wide acquisition range up to the whole OFDM transmission bandwidth.

In Section II, the generalized data-aided OFDM signal model is described. The class of data-aided estimators for the integer part are presented in Section III. And the estimation for the fractional part is given in Section IV. Performance evaluation via mathematical analysis and computet simulation are addressed in Section V. Finally, Section IV discusses and concludes this paper.

II. GENERALIZEDDATA-AIDEDOFDM SIGNALMODEL

We consider a generalized data-aided OFDM system using N -point inverse fast Fourier transform (IFFT) for modulation. Each OFDM symbol is composed of Nu complex symbols Xl,mwhere l denotes the OFDM symbol index and m denotes

the subcarrier index. Let D denote the set of indices for Nd

data-conveying subcarriers andP is the set of indices for Np

pilot subcarriers. Then, the set of indices of the Nu useful

subcarriersU can be defined as

U = D ∪ P (1)

and

Nu= (Nd+ Np) N. (2)

The output of the IFFT has a duration of T seconds which is equivalent to N samples. A cyclic prefix of duration Tg

seconds or Ng samples longer than the channel impulse

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(ISI). The resulting training signal is of duration Ts= T + Tg

seconds or equivalently, Ns = N + Ng samples. And the

transmitted baseband complex signal can be represented by s(t) = 1 N  l=−∞  m∈U Xl,mej2π(m/T )(t−Tg−lTs)g(t−lTs) (3)

where g(t) is the rectangular pulse given by g(t) =



1, t∈ [0, Ts)

0, otherwise. (4)

The transmitted signal can be separated into two parts and modelled as s(t) = d(t) + p(t), (5) where d(t) = 1 N  l=−∞  m∈D Xl,mej2π( m T)(t−Tg−lTs)g(t− lTs), (6) p(t) = 1 N  l=−∞  m∈P Xl,mej2π( m T)(t−Tg−lTs)g(t− lT s). (7)

Here, we assume that {Xl,m} belong to some constellation

with zero mean and two kinds of average power σX2 = {E{|Xl,m|2}|m ∈ D}, and σP2 = {E{|Xl,m|2}|m ∈ P}.

When D is a null set, this model can also characterize the preamble signal.

In the following, we assume a zero mean additive white Gaussian noise n(t) and a frequency offset ∆f . At the receiver, timing recovery is assumed to be accomplished and the received signal sampled at tk= kT /N is

r(kT N ) = d( kT N )e j2π∆fkTN + p(kT N )e j2π∆fkTN + n(kT N ). (8) In order to simplify the interpretation and terminology, we use r(k) to represent r(kTN ) hereafter. In addition,  is used to represent the frequency offset normalized to the subcarrier spacing. Therefore, the received signal can be rearranged as

r(k) = d(k)ej2πkN + p(k)ej2πkN + n(k) (9)

The signal-to-noise ratio is defined as η= (σ 2d+ σ2p)/σ2nwith σd2 = E {|d(k)|2}, σp2 = E {|p(k)|2} and σ2n = E{|n(k)|2}. Let rl,k denotes the kth sample of the lth received OFDM

symbol. Then, the lth received OFDM symbol can be repre-sented by

˜

rl= [rl,0, rl,1,· · · , rl,Ns−1]

T

, (10)

after removing the guard interval, the lth received OFDM symbol is represented by rl=  rl,Ng, rl,Ng+1,· · · , rl,Ns−1 T . (11)

Taking FFT to rl, the lth frequency domain decision vector

Rl= [Rl,0, Rl,1,· · · , Rl,N−1]T (12)

is obtained for further inner-receiver processing like equaliza-tion and detecequaliza-tion. With these vector variables defined, we are ready to develop the maximum-likelihood estimator.

III. DA-ML FREQUENCYESTIMATION: INTEGERPART

Here, we consider the data-aided maximum likelihood es-timation for the integer part of the CFO in OFDM systems. Taking the processing domain and the observation space into account, there are three possible schemes to approach this problem, they can be categorized as

❖ Time domain approach with cyclic prefix. ❖ Time domain approach without cyclic prefix. ❖ Frequency domain approach.

The first two approaches differ from each other in the ob-servation space while the third one diverges from others in the processing domain. We shall systematically derive the synchronization algorithms based on ML criterion with the assist of preamble or pilot signal.

A. Stochastic Characteristics ofr(k)

As in [5], we can simplify the statistical characteristics of r(k) to derive a tractable estimator. First, with the assumption of sufficiently large number of data-conveying subcarriers on which the modulating symbols are uncorrelated, the cen-tral limit theorem can be applied to model d(k) as a zero mean complex Gaussian random variable with variance σ2d= NdσX2/N . It is evident that{d(k)} are independent with each

other provided that they are not cyclic prefix pair. Since d(k) and n(k) are both zero-mean complex Gaussian distributed and p(k) is a deterministic signal which is known at the receiver, we can claim that r(k) is also complex Gaussian distributed with time-varying mean p(k)ej2πkN and variance

σ2d + σ2n. Besides, {r(k)} are independent with each other provided that they are not cyclic prefix pair.

Second, in OFDM systems employing cyclic prefix, r(k) and r(k + N ) (k ∈ [lNs, lNs + Ng − 1] for some l) are

correlated since d(k) = d(k + N ). This correlation must be taken into account in the time domain approach with cyclic prefix. To simplify the derivation of the likelihood function, we shall find out the conditional complex Gaussian PDF of r(k) and the conditional joint complex Gaussian PDF of r(k) and r(k + N ) given .

1) Conditional Complex Gaussian PDF of r(k): From the preceding discussion, it is straightforward to obtain the conditional complex Gaussian PDF of r(k) as

f (r(k)|) = 1 π (σd2+ σn2)e   r(k)−p(k)ej 2πkN    2 σ2d +σ2n . (13)

2) Conditional Joint Complex Gaussian PDF of the Cyclic Prefix Pair: r(k) and r(k + N ): Here, we assume that r(k) and r(k+N ) belong to the same OFDM symbol and are cyclic prefix pair, this condition implies that s(k) = s(k + N ). Let [r(k), r(k + N )]T be denoted byx, of which the mean vector

and the covariance matrix are

µ =  p(k)ej2πk N p(k + N )ej2π(k+N)N  (14)

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f (r(k), r(k + N )|) = e

−|r(k)−p(k)ej 2πkN |2−2ρ{[r(k)−p(k)ej 2πkN ][r(k+N)−p(k+N)ej 2π(k+N)N ]∗ej2π}+|r(k+N)−p(k+N)ej2π(k+N)N |2

(σ2d +σ2n )(1−ρ2) π2d2+ σn2)2(1− ρ2) (17) Λlcp() =− Ng−1 k=0  rl,k− αlpl,kej 2πk N  2 Ng−1 k=0  rl,k+N− αlpl,k+Nej 2π(k+N) N  2 − (1 − ρ2)N−1 k=Ng  rl,k− αlpl,kej 2πk N  2 + Ng−1 k=0 2ρ  rl,k− αlpl,kej 2πk N  rl,k+N− αlpl,k+Nej 2π(k+N) N ej2π (19) and Cx= σ2d+ σ2n σd2e−j2π σd2ej2π σd2+ σn2 . (15)

Then, the conditional joint complex Gaussian PDF of r(k) and r(k + N ) can be expressed as

f (r(k), r(k + N )|) = e−(x−µ)

HC−1

x (x−µ)

π2det (Cx) . (16) To simplify the derivation, we define the correlation coeffi-cient between r(k) and r(k +N ) to be ρ. It is easy to find that ρ = σd2/(σd2+ σn2) and the signal-to-noise ratio is related to this correlation coefficient by ρ = σ2dη/[(1 + σ2d)η + σp2]. After some algebraic manipulations, the explicit form of (16) can be obtained in (17). To avoid misunderstanding, we emphasize here that (16) and (17) only hold true for some k pertaining to cyclic prefix. Based on the complex Gaussian PDF obtained, we can now derive the DA-ML estimators.

B. Time Domain Approach with Cyclic Prefix

With the help of cyclic prefix in time domain, the DA-ML estimation should be based on the observation of ˜rl. Hence,

the conditional PDF of ˜rl given  can be written as f (˜rl|) = N g−1 k=0 f (rl,k, rl,k+N|) N−1 k=Ng f (rl,k|) . (18)

With some algebraic manipulations and defining pl,k  p(lNs+k), the log-likelihood function corresponding to ˜rlcan

be obtained in (19), where α ej2πNsN stands for the phase

shift produced by the normalized frequency difference  and the time difference NsT /N . Thus, the time domain

cyclic-prefix-assisted DA-ML frequency estimation corresponding to the lth OFDM symbol shall be

ˆ

lcp= arg max



Λlcp(). (20)

C. Time Domain Approach without Cyclic Prefix

Without the help of cyclic prefix in time domain, the DA-ML estimation should be based on the observation of

rl. The conditional PDF of rl given  can be written as

f (rl|) =

N−1

k=0 f (rl,k+Ng|). After some algebraic

manip-ulations, the log-likelihood function corresponding to rl can

be derived as Λlt() = N−1  k=0 rl,k+Ngα−lβ∗p∗l,k+Nge−j 2πk N  , (21) where β  ej2πNNg stands for the phase shift produced by

the normalized frequency difference  and the time difference NgT /N . Thus, the time domain DA-ML frequency estimation

corresponding to the lth OFDM symbol shall be ˆ

lt= arg max



Λlt() . (22)

D. Frequency Domain Approach

In frequency domain, the DA-ML estimation should be derived based onRl. From similar derivation, the likelihood

function in frequency domain corresponds toRl is f (Rl|) = N−1 n=0 1 N πσn2e −|Rl,n−αlβPl,n()|2N σ2 n (23) where Pl,n()  DFT  pl,k+Ngej2π k N  with k, n {0, 1, · · · , N −1}. Please notice that when  happens to be an integer, Pl,n() is exactly the pilot symbol on the ((n−))N th

subcarrier of the lth OFDM symbol, where the notation ((n))N

denotes (n modulo N ). After some algebraic manipulations, the log-likelihood function can be obtain as

Λlf() = N−1 k=0 Rl,kα−lβ∗Pl,k∗ ()  . (24)

And the frequency domain DA-ML estimator shall be ˆ

lf = arg max



Λlf(). (25)

Comparing (21) and (24), we can find that they are essentially equivalent to each other.

E. DA-ML Frequency Acquisition Using Preamble

When it comes to ”one-shot” synchronization, a preamble signal is usually suggested. The proposed algorithms are explicitly of pilot-aided scenario, however, it is straightforward

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to degenerate them to obtain their preamble versions. Here, we discuss two kinds of preamble, one is the ”mixed preamble” which contains both training and data symbols in frequency domain. The other is the ”pure preamble” which is purely composed of training symbols.

1) Mixed Preamble: The signal model of the mixed pream-ble is exactly consistent with the generalized data-aided OFDM system model described in Section II. Assume only one OFDM symbol is used, we can set l equals to 0 to get the mixed-preamble-aided algorithms.

2) Pure Preamble: Since the d(k) in (9) is absent, the stochastic property (specifically the variance) of r(k) is changed. We also assume only one OFDM symbol is em-ployed, then set l equals to 0. With similar derivation, it can be shown that both the time domain approach without cyclic prefix and the frequency approach have the same results with (21) and (24), wherein p0,kand P0,k() now stand for the pure preamble signal in time and frequency domain respectively.

As for the time domain approach with cyclic prefix, we need to make some crucial modifications. Compared with other two alternatives, this approach relies on the correlation between cyclic prefix pairs to provide extra information. However, this correlation results solely from d(k). When d(k) vanishes, σ2d and ρ become zero and each sample in one OFDM symbol become independent with each other. Under this condition, the two time domain approaches have the same form of log-likelihood function, except that considering the cyclic prefix gives a larger observation space. Briefly, when a pure preamble is adopted, the log-likelihood function of the time domain approach with cyclic prefix becomes

Λ0t() = N s−1  k=0 r(k)p∗(k)e−j2πkN  . (26) F. Acquisition Range

The acquisition range is vital to a frequency estimator, and the conventional schemes generally suffer from small acquisition range within half to several subcarrier spacing. To examine the acquisition range of the three proposed ap-proaches, the log-likelihood functions are depicted in Fig-ure 1. The frequency domain approach is omitted due to its equivalence to the time domain approach without cyclic prefix. From these waveforms, we can find that the log-likelihood functions exhibit periodicity with the same period N (normalized frequency). Therefore, the acquisition range of the proposed schemes is [−N/2, N/2). Please note that for a discrete-time OFDM system with Nyquist sampling rate of 1/T , the recognizable frequency range is exactly [−N/2, N/2), accordingly, the proposed frequency acquisition schemes possess the maximum acquisition range which is sufficient for any discrete-time OFDM system.

IV. DA-ML FREQUENCYESTIMATION: FRACTIONALPART

In [1], the phase of the log-likelihood function before taking real-part is used to estimate the frequency offset, this motivates us to examine the corresponding phases of the proposed

−20 −15 −10 −5 0 5 10 15 20 −20 −10 0 10 20 30 40 50

Normalized Freqency Offset Candidate ε

Λ

)

Time Domain Approach with Cyclic Prefix, N=16, Ng=4 Pure Preamble Mixed Preamble, Np=8

(a) Time domain approach with cyclic prefix. −20 −15 −10 −5 0 5 10 15 20 −15 −10 −5 0 5 10 15 20 25 30 35

Normalized Freqency Offset Candidate ε

Λ

)

Time Domain Approach without Cyclic Prefix, N=16 Pure Preamble Mixed Preamble, Np=8

(b) Time domain approach without cyclic prefix.

Fig. 1. Waveforms of the log-likelihood functions of the proposed acquisition schemes. (the exact frequency offset=0, SNR=10 dB)

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −4 −3 −2 −1 0 1 2 3 4

Normalized Freqency Offset Candidate ε

Phase of

Γ

)

Time Domain Approach with Cyclic Prefix, N=64, Ng=16 Pure Preamble Mixed Preamble, Np=8 Mixed Preamble, Np=4

(a) Time domain approach with cyclic prefix. −2 −1.5 −1 −0.5 0 0.5 1 1.5 2 −4 −3 −2 −1 0 1 2 3 4

Normalized Freqency Offset Candidate ε

Phase of

Γ

)

Time Domain Approach without Cyclic Prefix, N=64 Pure Preamble Mixed Preamble, Np=8 Mixed Preamble, Np=4

(b) Time domain approach without cyclic prefix.

Fig. 2. Phases of the log-likelihood functions before taking real part of the proposed acquisition schemes. (the exact frequency offset=0, SNR=10 dB)

algorithms. Prior to this, we define {Γlt()} = Λlt() and {Γl

cp()} = Λlcp(). Their phases are illustrated in Figure 2.

We can observe that there exhibit a linear range if the fre-quency offset falls within the range of [−1/2, 1/2]. Provided that the linear range is acquired, we can use either a linear interpolation or a number control oscillator (NCO) to track the fractional frequency offset. Three kinds of scenario which can simultaneously compensate both the integer part and the fractional part of the frequency offset are thus suggested as follows:

The proposed acquisition schemes followed by a linear interpolator to give an one-shot estimation or open-loop tracking.

The proposed acquisition schemes followed by a NCO to give a close-loop tracking.

The proposed acquisition schemes followed by a linear interpolator which feedback to a NCO to give a close-loop tracking.

The illustration of the third scenario is shown in Figure 3. V. PERFORMANCEEVALUATION

In this section, we evaluate the performance of the proposed acquisition and tracking schemes via computer simulation and mathematical analysis. As for acquisition schemes, the channel is assumed to affect the signal by a CFO with discrete value and an additive white Gaussian noise. The acquisition resolution is set to 1 (normalized frequency). And the channel

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¯ ¼  ¯ ½  ¯  ½  º º º Maximum Select Linear Interpolation Feed-Forward Acquisition r(k) to FFT ¾      ¾ ¡ ¾  ¯ ¯¯ ½ Close-Loop Tracking NCO ¯¯ ½  Open-Loop Tracking

Fig. 3. The third scenario of the proposed tracking schemes.

assumption in tracking schemes remains the same except that the CFO has a continuous value within the range of [-1/4,1/4]. And the open-loop tracking by linear interpolation, i.e., the first scenario in Section IV is employed. The QPSK modulation is adopted for both data and pilot symbols, N and N u are set to be 64 and Np is generally set to be 16 if not

specifically defined. Then, we simulate the mean-squared error (MSE) of the proposed acquisition and tracking schemes and each simulation point uses 107 OFDM symbols. Again, the performance evaluation of the frequency domain approach is omitted here due to its equivalence.

The comparison of the MSE of the time domain approaches using pure preamble with different cyclic prefix length is illustrated in Figure 4-(a). From Figure 4-(a), we can find that an incremental cyclic prefix length of 8 samples gives about an 0.5 dB gain.

The comparison of the MSE of the proposed acquisition schemes using different preambles are illustrated in Figure 4-(b). From Figure 4-(b), the acquisition schemes using pure preamble outperform the same schemes using mixed preamble with Np = 32 about 8 dB. The performance differences

between the acquisition schemes using mixed preamble are about 4 dB. This result is reasonable and conform to our expectation.

The validness of the simulation results in Figure 4-(a) and Figure 4-(b) are justified by Figure 5-(a) in which the results of computer simulation conform to that of mathematical analysis. The selected algorithm is the time domain acquisition using pure preamble with N = 4. The detail derivation is omitted here due to the lack of space.

The comparison of the MSE of the tracking schemes are illustrated in Figure 5-(b). We can observe that when pure preamble is used, the performance is insensitive to the decay of SNR. Because when SNR> 0, the acquisition result sent to linear interpolation is extremely accurate, hence, the perfor-mance is dominated by the limit of linear interpolation itself. When mixed preamble is used, the performance is sensitive to the decay of SNR. The effectiveness of the tracking schemes without cyclic prefix is proportional to the size of Np while

that with cyclic prefix is inversely proportional to Np. The

main reason is because when cyclic prefix is considered, the linearity of phase response in Figure 2-(a) will be destroyed

−20 −18 −16 −14 −12 −10 −8 −6 −4 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 101 102

103 Comparison of ML−DA frequency acquisition schemes using pure preamble

Mean Squared Error

SNR ML MLcp G=8 MLcp G=16 MLcp G=24 N=64, QPSK 107 OFDM symbols

(a) Time domain approaches using pure preamble. −10 −8 −6 −4 −2 0 2 4 6 8 10 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 101 102 103 SNR (dB)

Mean Squared Error

Comparison of ML−DA Frequency Acquisition Schemes Using Preamble

T,pure T,mixed,Np=32 T,mixed,Np=16 T,mixed,Np=8 T,mixed,Np=4 Tcp,pure Tcp,mixed,Np=32 Tcp,mixed,Np=16 Tcp,mixed,Np=8 Tcp,mixed,Np=4

(b) Time domain approaches using Preamble.

Fig. 4. Comparison of the MSE of the acquisition schemes using preamble.

−10 −8 −6 −4 −2 0 2 4 6 8 10 10−10 10−9 10−8 10−7 10−6 10−5 10−4 10−3 10−2 10−1 100 SNR

Mean Squared Error

Performance Evaluation of OFDM ML−DA Frequency Synchronization Analysis Simulation

N=4,L=107 ,Ps=1

(a) Acquisition: time domain ap-proach without cyclic prefix using pure preamble. 0 2 4 6 8 10 12 14 16 18 20 10−3 10−2 10−1 SNR (dB)

Mean Squared Error (1/T

2)

Mean Squared Error of the Linear Interpolation Estimator Np=2 N p=2,cp Np=4 Np=4,cp Np=8 N p=8,cp Np=64 Np=64,cp

(b) Tracking: time domain approaches using preamble.

Fig. 5. Comparison of the MSE of the acquisition and tracking schemes.

when Np grows. And this loss of linearity dominate the

performance.

VI. CONCLUSION

In this paper, three kinds of data-aided maximum likelihood frequency estimation algorithm based on generalized data-aided signal model are proposed. The main differences of them lies in the processing domain and the observation space. The proposed algorithms have a wide acquisition range and can systematically estimate both the integer part and the fractional of the frequency offset. Finally, We justify the effectiveness of the proposed acquisition and tracking schemes by simulation and analysis. The acquisition schemes using pure preamble are shown to be quite accurate even with a negative signal-to-noise ratio while the open loop tracking schemes have moderate performance and fairly low complexity.

REFERENCES

[1] J. J. Van de Beek, M. Sandell, and P. O. Borjesson, ”ML estimation of time and frequency in OFDM systems,” IEEE Trans. Signal Processing, vol. 45, pp. 1800-1805, July 1997.

[2] T. Pollet and M. Moeneclaey, “Synchronizability of OFDM signals,” in

Proc. Globecom’95, vol.3, pp. 2054-2058, Singapore Nov. 1995.

[3] P. H. Moose, “A technique for orthogonal frequency division multiplexing frequency offset correction,” IEEE Trans. Commun., vol. 42, pp. 2908-2914, Oct. 1994

[4] T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchro-nizaiton for OFDM,” IEEE Trans. Commun., vol. 45, pp. 1613-1621, Dec. 1997.

[5] D. Landstrom, S. K. Wilson, J.J˙van de Beek, P. Odling, P. O. Borjesson, “Symbol time offset estimation in coherent OFDM systems,” in Proc.

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數據

Fig. 2. Phases of the log-likelihood functions before taking real part of the proposed acquisition schemes
Fig. 4. Comparison of the MSE of the acquisition schemes using preamble.

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