frequency synchronization scheme based on Radon–Wigner transform of LFM signals in CO-OFDM systems

Optics Communications 410 (2018) 744–750

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Optics Communications journal homepage: www.elsevier.com/locate/optcom

A novel joint timing/frequency synchronization scheme based on Radon–Wigner transform of LFM signals in CO-OFDM systems Jianfei Liu *, Ying Wei, Xiangye Zeng, Jia Lu, Shuangxi Zhang, Mengjun Wang School of Electronic and Information Engineering, Hebei University of Technology, Tianjin, China Key Laboratory of Electronic Materials and Devices of Tianjin, Hebei University of Technology, Tianjin, China

a r t i c l e

i n f o

Keywords: CO-OFDM Radon–Wigner transform Synchronization Linear frequency modulation signal

a b s t r a c t A joint timing and frequency synchronization method has been proposed for coherent optical orthogonal frequency-division multiplexing (CO-OFDM) system in this paper. The timing offset (TO), integer frequency offset (FO) and the fractional FO can be realized by only one training symbol, which consists of two linear frequency modulation (LFM) signals with opposite chirp rates. By detecting the peak of LFM signals after Radon– Wigner transform (RWT), the TO and the integer FO can be estimated at the same time, moreover, the fractional FO can be acquired correspondingly through the self-correlation characteristic of the same training symbol. Simulation results show that the proposed method can give a more accurate TO estimation than the existing methods, especially at poor OSNR conditions; for the FO estimation, both the fractional and the integer FO can be estimated through the proposed training symbol with no extra overhead, a more accurate estimation and a large FO estimation range of [−5 GHz, 5GHz] can be acquired. © 2017 Elsevier B.V. All rights reserved.

1. Introduction Orthogonal Frequency Division Multiplexing (OFDM) has attracted lots of interest due to its high spectral efficiency (SE) and robustness to fiber dispersion [1]. It can be realized either with direct detection [2] or with coherent optical detection [3,4]. Whereas direct detection optical OFDM is more suitable for cost effective short reach applications, the superior performance of coherent optical OFDM (CO-OFDM) makes it an excellent candidate for long haul transmission systems. Based on the CO-OFDM system, many research results show that this technology is robust against the channel impairments, such as chromatic dispersion (CD) [5] and polarization mode dispersion (PMD) [6,7]. However, the CO-OFDM system suffers from FO and phase noise, which make it more sensitive to synchronization errors than the single-carrier system [8,9]. So the synchronization at the receiver is an important step that must be performed in CO-OFDM systems. Compared with the schemes with pilot or cyclic prefix aided, using training symbol for synchronization can provide higher accuracy [10,11]. Most methods of timing and frequency estimation are based on exploiting the correlation property of periodical training symbols, like Schmidl algorithm [12] and the Minn algorithm [13]. But the traditional synchronization methods based on the correlation property need two steps to estimate the timing and frequency offsets individually. Ref. [14] proposed a joint frequency

offset estimation method based on training symbol and virtual subcarriers for CO-OFDM systems, which can estimate the integer FO and the fractional FO by one training symbol and virtual subcarriers. Recently a joint timing and frequency synchronization based on fractional Fourier transform (FrFT) has been proposed by Qi Liu [15], which can estimate the TO and integer FO simultaneously through two times FrFT. In addition, Ref. [16] proposed a novel joint synchronization algorithm based on FrFT encoded training symbols for PDM CO-OFDM systems. The TO and the integer FO estimation can be realized by only one time FrFT at the receiver [17,18], then the residual fractional FO can be estimated with another training symbol constructed by the classic Schmidl algorithm. But when the OSNR is very low, and there are more than one LFM signals exist in fractional Fourier domain, their spectrums will be shaded by the others [19,20], which will reduce the accuracy of the LFM signals’ detection. As a result, the accuracy of synchronization will be affected. Furthermore, the synchronization are realized totally by two training symbol in the joint synchronization method proposed in Ref. [16], which reduces the transmission efficiency. Radon–Wigner transform (RWT) has a superior performance on LFM signals detecting compared with FrFT [21,22]. In this paper a joint timing and frequency synchronization scheme based on RWT of LFM signals has been proposed in CO-OFDM systems by using only one training

* Correspondence to: Hebei University of Technology, School of Electronic and Information Engineering, 5340 Xiping Road, Tianjin, 300401, China.

E-mail address: [email protected] (J. Liu). https://doi.org/10.1016/j.optcom.2017.10.077 Received 7 August 2017; Received in revised form 27 October 2017; Accepted 29 October 2017 Available online 14 December 2017 0030-4018/© 2017 Elsevier B.V. All rights reserved.

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center frequency 𝑓0 can be described as 𝑧(𝑡) = exp[𝑗2𝜋(𝑓0 𝑡 + 𝜇𝑡2 )], and its Wigner–Ville distribution is ∞

𝑊𝑧 (𝑡, 𝑓 ) =

∫−∞

exp[𝑗2𝜋(𝑓0 + 𝜇𝑡)𝜏] ⋅ exp[−𝑗2𝜋]𝑑𝜏 = 𝛿[𝑓 − (𝑓0 + 𝜇𝑡)]

(4)

The Radon transform of it can be described as 𝐷z (𝜇, 𝛼) ∞

𝐷𝑧 (𝑢, 𝛼) =

∫

𝑊𝑧 (𝑡, 𝑓 )𝑑𝑣 = ∞

=

𝑊𝑧 (𝑡, 𝑓 )𝛿(𝑢 − 𝑢′ )𝑑𝑢𝑑𝑣

∞

∫−∞ ∫−∞

𝑊𝑧 (𝑡, 𝑓 )𝛿[sin 𝛼(𝑓 − 𝑓0 − 𝜇𝑡)]𝑑𝑓 𝑑𝑡 ∞

=

∞

∫−∞ ∫−∞

∞

1 𝑊 (𝑡, 𝑓 )𝛿[𝑓 − (𝑓0 + 𝜇𝑡)]𝑑𝑓 𝑑𝑡 |sin 𝛼| ∫−∞ ∫−∞ 𝑧

(5)

The expression indicates that if a certain LFM signal with parameters 𝜇 and 𝑓0 , the integral value is the maximum, the peak in its Radon– Wigner transform will appear at (𝜇, 𝑓0 ). When the parameters depart from 𝜇 and 𝑓0 , the value of RWT will decreases rapidly. The Wigner–Ville distribution of the LFM signal is a beeline, when transformed to Radon plane, it is an aiguille and the cross terms spread out, so it is easy to distinguish signals and cross terms apart on Radon plane. The linear integral of RWT can suppress the noise and cross terms well. In practical computation, the discrete RWT can be defined as

Fig. 1. Radon transform.

symbol, which takes advantage of the LFM signal’s concentration after Radon–Wigner transform. The TO and the integer FO can be estimated simultaneously by detecting the peak positions of the LFM signals in Radon–Wigner domain, then the fractional FO can be estimated with the same training symbol. The TO and FO estimation performance of the proposed method is evaluated on a CO-OFDM system at 112 Gbit/s by numerical simulations. Simulation results verify the effectiveness of the proposed synchronization scheme.

𝑅𝑊 𝑇 (𝑓 , 𝜇) =

∑∑ 𝑛

𝑧(𝑛 + 𝑘)𝑧∗ (𝑛 − 𝑘)𝑒−𝑗2𝑘(𝑓0 +𝜇𝑛)

(6)

𝑘

If 𝑧(𝑛) = 𝐴 exp[−𝑗𝜋𝜇𝑛2 ] is a LFM signal with length of 𝑁, we can calculate the maximum amplitude of its RWT to be 𝑁 2 𝐴2 /2. The discrete Fractional Fourier transform can be calculated through the algorithm proposed √ in Ref. [23], and the maximum amplitude of 𝑧(𝑛) after FrFT is |𝐴| 𝑁∕|sin 𝛼|1∕2 , where 𝜇 = tan 𝛼. To summarize, for a certain LFM signal, we can get a higher peak after RWT compared with FrFT. Fig. 2(a) and (b) show the RWT and FrFT of a LFM signal respectively. As it can be seen from the figures, the RWT of the LFM signal has a sharper peak compared with the FrFT of it, which means the RWT has a better performance in converging the LFM signal. Therefore, in the poor SNR condition, when the noise disturbs the signal seriously, the peak positions of LFM signals can be detected more accurately after RWT compared with FrFT. So we propose a joint TO and FO estimation method based on the RWT of LFM signals.

2. Principle 2.1. Radon–Wigner transform Radon transform is a line integration, which can rotate the axes by an angle 𝛼, then we can get a new rectangular coordinate system (𝑢, 𝑣). After that, integrate 𝑣 at different u values, as it is shown in Fig. 1, we can get the Radon transform. For any two-variables function 𝑧(𝑡, 𝑓 ), the Radon transform by an angle 𝛼 can be described as

2.3. The proposed method based on Radon–Wigner transform

∞

𝑅𝛼 (𝑢) =

∫−∞

𝑧(𝑢 cos 𝛼 − 𝑣 sin 𝛼, 𝑢 sin 𝛼 + 𝑣 cos 𝛼)𝑑𝑣

∞

=

The Radon–Wigner domain contains both the time and frequency domain information of the signals, when exists TO and FO, the peak position of the LFM signals after RWT will shift accordingly. By detecting the variation of the peak positions after the RWT, TO and integral part of FO can be derived simultaneously and then the fractional FO can be estimated according to the correlation of the proposed training symbol. When the signals have finished the joint synchronization, the FFT can be carried out correctly. Fig. 3 shows the block diagram of the proposed joint timing and frequency offset estimation algorithm. In order to estimate TO and FO at the same time, two LFM signals with different chirp rates are needed. In this paper, the proposed training symbol 𝑃 (𝑡) consists of two LFM signals with opposite chirp rates and opposite center frequencies, as Eqs. (7)–(9) show. If 𝑍1 (𝑡) and 𝑍2 (𝑡) are two LFM signals, the chirp rates of them are 𝑘 and −𝑘, and the center frequencies are 𝑓0 and −𝑓0 .

∞

∫−∞ ∫−∞

𝑧(𝑢 cos 𝛼 − 𝑣 sin 𝛼, 𝑢 sin 𝛼 + 𝑣 cos 𝛼) ⋅ 𝛿(𝑢 − 𝑢′ )𝑑𝑢𝑑𝑣

𝜋 𝜋 <𝛼< (1) 2 2 Radon–Wigner transform is the Radon transform of the Wigner–Ville distributed (WVD) time-frequency plane, and it can be defined as: − ∞ < 𝑢 < ∞, −

𝑅𝛼 (𝑢, 𝛼) ∞

=

∞

∫−∞ ∫−∞

𝑊 𝑉 𝐷(𝑢 cos 𝛼 − 𝑣 sin 𝛼, 𝑢 sin 𝛼 + 𝑣 cos 𝛼)𝛿(𝑢 − 𝑢′ )𝑑𝑢𝑑𝑣

(2)

If we use the intercept 𝑓0 and slope 𝜇 as parameters to denote a beeline, and 𝑓0 = 𝑢∕ sin 𝛼, 𝜇 = − cot 𝛼, then when we make the beeline integral along the 𝑓 = 𝑓0 + 𝜇𝑡, the integral path can be described with the parameter (𝜇, 𝑓0 ) as ∞

𝑅𝛼 (𝑢, 𝛼) =

∞

1 𝑊 (𝑡, 𝑓 ) ⋅ 𝛿[𝑓 − (𝑓0 + 𝜇𝑡)]𝑑𝑓 𝑑𝑡 |sin 𝛼| ∫−∞ ∫−∞ 𝑧

(7)

𝑃 (𝑡) = 𝑍1 (𝑡) + 𝑍2 (𝑡)

(3)

𝑍1 (𝑡) = exp[𝑗2𝜋(−𝑓0 𝑡 + 𝑘𝑡2 )] 2

𝑍2 (𝑡) = exp[𝑗2𝜋(𝑓0 𝑡 − 𝑘𝑡 )]

2.2. LFM signals with Radon–Wigner transform

(8) (9)

The OFDM frame is shown in Fig. 4. As shown in the figure, the training symbol we proposed is inserted in the beginning of the OFDM frame, which consists of two LFM signals with opposite chirp rates. The

A LFM signal is a waveform of which the frequency changes linearly according to the time instance. The LFM signal with chirp rate 𝜇 and 745

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Optics Communications 410 (2018) 744–750

Fig. 4. OFDM frame structure.

The intercepts and slopes of the two signals can be acquired through searching the peak positions in the Radon–Wigner domain. Finally, TO and FO can be derived with the changes of the intercepts and the slopes. In order to analyze the relations between the peak positions and TO/FO, we carry out the following derivations. First of all, the two LFM signals can be described as 𝑓1 = −𝑓0 + 𝑘0 𝑡1

(10)

𝑓2 = 𝑓0 − 𝑘0 𝑡2

(11)

Assuming that 𝛿𝑡 and 𝛿𝑓 represent TO and FO respectively, the signals with TO and FO can be expressed as Eqs. (12) and (13). 𝑓1 = −𝑓0 + 𝑘0 (𝑡1 + 𝛿𝑡) + 𝛿𝑓

(12)

𝑓2 = 𝑓0 − 𝑘0 (𝑡2 + 𝛿𝑡) + 𝛿𝑓

(13)

That is, 𝑓1 = −𝑓0 + 𝛿𝑓 + 𝑘0 𝛿𝑡 + 𝑘0 𝑡1

(14)

𝑓2 = 𝑓0 + 𝛿𝑓 − 𝑘0 𝛿𝑡 − 𝑘0 𝑡2

(15)

And the center frequencies change into 𝑓01 and 𝑓02 at this time, while the chirp rates are still 𝑘0 and −𝑘0 .

Fig. 2. Transforms of LFM signal: (a) Radon-Wigner transform (b) Fractional Fourier transform.

training symbol is used to estimate the TO, integer FO and fractional FO. A cyclic prefix is also added to the OFDM frame. The LFM signal can be described by a linear function in the timefrequency plane, so the training symbol 𝑃 (𝑡) can be described as Fig. 5(a). When there exists FO/TO, the LFM signals in the Radon– Wigner domain will shift along the 𝑓 ∕𝑡 axis. As it shows in Fig. 5(a)–(d), 𝑓 and 𝑡 denote the frequency and time axes. The vertical and horizontal arrow are FO and TO respectively. When there is only FO or TO in the system, the signals will shift as Fig. 5(b) or Fig. 5(c) shows; and if there are both FO and TO, the signals will move as Fig. 5(d) shows. According to Fig. 5, when FO/TO exists, the intercepts 𝑓0 and −𝑓0 of the two LFM signals are varying, while the slopes −𝑘 and 𝑘 of them remain unchanged. Since the peak position of the LFM signal in the Radon–Wigner domain is determined by the intercept and the slope, the two peaks of training symbols in the Radon–Wigner domain will shift according to the TO and FO.

𝑓01 = −𝑓0 + 𝑘0 𝛿𝑡 + 𝛿𝑓

(16)

𝑓02 = 𝑓0 − 𝑘0 𝛿𝑡 + 𝛿𝑓

(17)

We can define the variations of the intercepts as 𝛿𝜌1 and 𝛿𝜌2 𝛿𝜌1 = 𝑓01 − (−𝑓0 ) = 𝛿𝑓 + 𝑘0 𝛿𝑡

(18)

𝛿𝜌2 = 𝑓02 − 𝑓0 = 𝛿𝑓 − 𝑘0 𝛿𝑡

(19)

Then, we can derive the TO and FO as follows 𝛿𝑓 = (𝛿𝜌1 + 𝛿𝜌2 )∕2 𝛿𝑡 = (𝛿𝜌1 − 𝛿𝜌2 )∕2𝑘0

(20) (21)

As long as we get the peak shift of the training symbol in Radon– Wigner domain, 𝛿𝜌1 and 𝛿𝜌2 are known at the same time, TO and the integer FO can be estimated according to Eqs. (20) and (21) simultaneously. Since the starting sample of the OFDM frame has been found already, the fractional part of FO can be easily estimated through the same training symbol.

Fig. 3. Block diagram of joint synchronization.

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the frequency offset 𝜀 is not zero, there will be a phase rotation angle ̂ 𝜑 = 𝜋𝜀 caused by the frequency offset. Under this circumstances, 𝑝(𝑑) can be expressed as: ∑

𝑁∕2−1

𝑃 (𝑑) =

𝑟∗ (𝑑 + 𝑛) ⋅ 𝑟(𝑑 + 𝑛 + 𝑁∕2)

𝑛=0 𝑁∕2−1

=

∑

𝑠∗ (𝑑 + 𝑛) ⋅ 𝑠(𝑑 + 𝑛 + 𝑁∕2) exp[−𝑗2𝜋𝜀(𝑑 + 𝑛)∕𝑁]

𝑛=0

⋅ exp[𝑗2𝜋𝜀(𝑑 + 𝑛 + 𝑁∕2)∕𝑁] ∑

𝑁.∕2−1

= exp[𝑗𝜋𝜀]

(a)

|𝑠(𝑑 + 𝑛)|2

(24)

𝑛=0

where the range of 𝜑 is (−𝜋, 𝜋), we can acquire the fractional part of frequency offset 𝜀𝑓 by Eq. (25). ̂ 𝜀𝑓 = 𝑎𝑟𝑔𝑙𝑒(𝑃 (𝑑))∕𝜋

(25)

In a word, the fractional frequency offset 𝜀𝑓 can be estimated in a range of (−1, 1) through the same training symbol by Eq. (25). 3. Simulation and analysis 3.1. CO-OFDM system

(b)

Fig. 6 shows the structure diagram of CO-OFDM transmission system. It is composed of five parts including radio frequency (RF) OFDM transmitter, optical I/O modulator, optical transmission link, coherent detector and RF OFDM receiver. The data stream is transformed to multiple low-speed data streams after serial–parallel conversion, and then it is mapped. The training symbol is inserted in the data streams for synchronization. The lowspeed data streams with particular training symbol are modulated to the sub-carriers by an IFFT. Finally, after the parallel–serial conversion, adding a cyclic prefix (CP), D/A conversion processing, a complete OFDM signal is generated. The OFDM signal is modulated to optical domain from electric domain through MZM modulator, and is coupled to the fiber channel then transmitted. The corresponding inverse process is completed in the receiver. To evaluate our synchronization scheme, the CO-OFDM system in this paper is built by Virtual Photonics Inc. (VPI) software, and the corresponding simulation parameters are set as follows. The data rate is 112 Gbit/s. The number of subcarriers is 1024. QPSK modulation is adopted. The length of CP is 128. There is a 100 km single-mode fiber (SMF) with 0.2×10−3 dB/m attenuation, and the nonlinearity coefficient is 2.6 × 10−20 m2 /W. The core area is 80 μm2 . In order to get the TO estimation range of half of the training symbol length, like it shows in Fig. 7. The chirp rate and the center frequency should satisfy Eq. (26).

f

t

0

(c)

(d) Fig. 5. (a) LFM signals without offset (b) with only FO (c) with only TO (d) with both TO and FO.

𝑁∕2∗ 𝑑𝑡 = 𝑓0 ∕𝑘

The 𝑓0 and 𝑘 represent the chirp rate and the center frequency respectively, 𝑁 is the sample number of training symbol and 𝑑𝑡 is the sampling rate. Thus, if we choose different 𝑓0 , the 𝑘 is determined accordingly. As it is shown in Eqs. (13) and (14), the signal with larger 𝑘 will move more obviously when the TO is the same, so a larger 𝑘 is better for TO estimation. Fig. 8 depicts the MSE of TO estimation under different normalized center frequencies in 5 dB OSNR, the MSE is 0 when the absolute value of the normalized center frequency is larger than 0.3. As it can be seen from Eq. (20), the accuracy of FO estimation is only determined by the variation of center frequency instead of the value of the center frequency. In order to ensure the estimation accuracy, the normalized center frequency is 0.35 in our simulation, that is the normalized frequency is changing from −0.35 to 0.35 linearly while the sample number of the training symbol from 0 to 𝑁.

Based on timing synchronization, we can find the correct starting ̂ The correlation of the received signal 𝑟(𝑛) and its delay version sample 𝑑. ̂ is: with a delay of 𝐿 = 𝑁∕2 samples at the correct starting point 𝑝(𝑑) ∑

𝑁∕2−1

̂ = 𝑃 (𝑑)

𝑟∗ (𝑑̂ + 𝑛) ⋅ 𝑟(𝑑̂ + 𝑛 + 𝑁∕2)

(22)

𝑛=0

where the 𝑟(𝑛) is the received training symbol, which consists of two LFM signals with opposite chirp rates and center frequencies, and it can be described as: 𝑟(𝑛) = exp[𝑗2𝜋(𝑓0 𝑛 − 𝜇𝑛2 )] + exp[𝑗2𝜋(−𝑓0 𝑛 + 𝜇𝑛2 )]

(26)

(23)

𝑟∗ (𝑛),

For 𝑟(𝑛) = we know that 𝑟(𝑛) is an integer. So, when no ̂ 𝑛) and 𝑟(𝑑̂ + 𝑛 + fractional frequency offset exists in the system, 𝑟∗ (𝑑 + ̂ is an integer as well. But when 𝑁∕2) are both integers. As a result, 𝑝(𝑑) 747

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Fig. 6. The structure of CO-OFDM transmission system.

Fig. 7. Training symbol with largest TO.

Fig. 9. MSE of TO estimation.

The performance of the proposed algorithm is studied in terms of the mean squared error (MSE), and the results are compared with the Schmidl algorithm, Minn algorithm, algorithm proposed in Ref. [14] and the synchronization algorithm based on FrFT. Each point in the figures is the averaged result after 200 runs. 3.2. Timing offset estimation performance Fig. 9 shows the MSE of timing estimation errors versus OSNR of different algorithms, and the FO in the system is 1 GHz. The TO of the proposed algorithm can be estimated by Eq. (21). As can be seen from the figure, when the OSNR is as low as 0 dB, the MSE of the proposed algorithm is 0.04, which is approximately zero, when the OSNR is up to 4dB, the MSE is exactly zero, which means it can estimate the TO correctly. The algorithm based on FrFT has one sample estimation error, when the OSNR is above 6 dB, but the TO estimation performance of it becomes bad when the OSNR get lower. We can see that the Schmidl algorithm keeps a large MSE of timing offset estimation throughout the various OSNR values, which is affected by the platform of the timing metric. The Minn algorithm reduces the uncertainty and has a smaller MSE, but it is not accurate enough. Obviously the proposed method outperforms others methods even at poor OSNR condition.

Fig. 8. MSE of TO estimation under different center frequency.

When detecting the training symbol at the receiver, we do not need to scan the sample signal one by one at first. To finish the TO and the integer FO estimation, we only need to scan the sample for once in every training symbol length of the received signal. When the most part of the training symbol has been acquired, we will find the peaks in the Radon–Wigner domain. So the maximum value of the TO will be half of the training symbol length. After the estimation of the TO and the integer FO, we can find the starting point of the received signal, and then the fractional FO can be estimated by scanning the samples one by one like it shows in Schmidl algorithm and other methods using the correlation characteristic of the training symbol. Therefore, the total calculation steps of the proposed method is same with the FrFT method, which needs fewer steps than Schmidl’s method.

3.3. Frequency offset estimation performance In order to evaluate the FO estimation performance of the proposed method, we add FO to the OFDM signals and compare it with the estimated one. Figs. 10 and 11 show the FO estimation performance 748

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of these methods are almost independent of the actual value of the frequency offset. Because of the excellent performance of RWT in LFM signals detecting, the method we proposed has the best FO estimation performance compared with the other two methods, and the MSE of our new method can be kept under 0.002 when the OSNR is 2 dB. In order to investigate the performance of the CO-OFDM system with the proposed synchronization method, we compare the constellations of QPSK signals without/with the synchronization method based on RWT. Fig. 12(a) shows the constellations of the QPSK signals without synchronization when the OSNR is 10 dB, affected by the TO and FO in the system, we cannot receive the signals accurately. Since the algorithm can estimated the TO and FO in the system, the constellations of the signals after the proposed synchronization method are divided into four points clearly as shown in Fig. 12(b). In a word, the new joint TO and FO estimation method can improve the system performance effectively even if in a poor OSNR condition through only one training symbol. 3.4. Computation complexity Fig. 10. MSE of FO estimation error versus OSNR.

Computation complexity of the proposed method should be taken into account because it has an effect on the speed of calculation and decreases system performance ultimately. In our proposed synchronization scheme, the complexity mainly lies on the Radon–Wigner transform. In our practical computation, we do not need two steps to calculate the Radon–Wigner transform as it shown in Eqs. (4) and (5). There are several fast RWT computation algorithms has been proposed. The algorithm proposed by John C. Wood and Daniel T. Barry is used in our work [24]. The RWT can be calculated through dechirping. Computationally, dechirping is O(2𝑁 log 2𝑁) instead of O(𝑁 3 ) for direct projection, and the computation is dominated by the fast Fourier transform calculation. The computation complexity can be reduced efficiently with this algorithm compared with the direct computation of RWT. In other joint synchronization method such as Ref. [16], there are two times FrFT in the method to realize the joint TO/FO estimation, and the computation complexity of digital FrFT is O(𝑁 log 𝑁), so the total computation complexity is O(2NlogN) time. Though the computation complexity of the algorithm we proposed is a little higher than the algorithm based on FrFT, there are only one training symbol in our algorithm while two in Ref. [16] to estimate the TO, integer FO and the fractional FO. In a word, the little sacrifice of computation complexity can increase the transmission efficiency and accuracy obviously in our method.

Fig. 11. MSE of FO estimation error versus different FO.

of the proposed method, the method based on FrFT and the method proposed in Ref. [14] with different OSNR and different FO. Fig. 10 shows the MSE of FO estimation performance of the three algorithms versus OSNR with 1 and 5 GHz FO. We can see from the figure, all of the three synchronization methods can estimate the FO with MSE below 10−2 with OSNR ranges from 0 to 14 dB. The algorithm proposed in this paper which utilizes only one training symbol to estimate the fractional FO and integer FO has a relatively lower MSE compared with the other algorithms. However, the FrFT method needs two training symbols to finish the FO estimation and the method proposed in Ref. [14] needs the training symbol and virtual subcarriers to get the whole FO estimation results. Overall, the proposed algorithm is superior to the existing methods for its lower training symbol cost and accuracy in FO estimation. Fig. 11 depicts the MSE of the three algorithms in a poor OSNR condition of 2 dB. We set different FO in the system, and the linewidth of laser and oscillator are 100 kHz. Since the frequency accuracy of commercially-available lasers are usually ±2.5 GHz over the lifetime, the estimation range of [−5 GHz, +5GHz] can satisfy the practical requirement of the FO estimation algorithms. It can be seen from the figure that when the FO ranges from −5 to 5 GHz, the MSE of the three methods all stay stable, which means that the performance

4. Conclusions A novel joint timing and frequency synchronization scheme based on Radon–Wigner transform of LFM signals in CO-OFDM systems is proposed in this paper. The symbol timing offset, integer frequency offset and the fractional frequency offset can be estimated through only one training symbol. The proposed method takes advantage of the convergence characteristic of LFM signals in Radon–Wigner domain. Through the training symbol we proposed, the TO and integer FO can be estimated simultaneously by detecting peak positions in the Radon–Wigner domain, and then the fractional FO can be acquired through the self-correlation characteristic. The new method based on RWT can overcome the shortage of classical synchronization algorithms in efficiency and accuracy. Simulation results show that, compared with the existing algorithms, the new method based on RWT can estimate the TO and FO more accurately versus different OSNR conditions. When the frequency offset ranges from −5 to 5 GHz, the MSE of the FO estimation of the proposed method can be kept about 10−3 and stays stable, which is more accurate than the other algorithms in a poor OSNR condition of 2 dB. Overall, the method proposed in this paper can not only reduce the training symbol cost but also improve accuracy of synchronization. 749

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Fig. 12. Constellations of QPSK signals (a) without synchronization (b) with the proposed synchronization method.

Acknowledgments

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