## Abstract

A joint timing offset (TO) and frequency offset (FO) estimation algorithm is proposed for polarization division multiplexing (PDM) coherent optical orthogonal frequency-division multiplexing (CO-OFDM) systems. It is realized by taking the advantage of the time-frequency property of the fractional Fourier transformation (FrFT) encoded training symbols. Compared with the classical Schmidl & Cox method, the proposed algorithm exhibits robust estimation result of timing offset with poor optical signal-to-noise ratio (OSNR) and nonlinear interference. For the frequency offset estimation, a quite large FO estimation ranges of [-5GHz + 5GHz] can be achieved. The mean normalized estimation error can be kept under 0.002 and the max normalized estimation error is no more than 0.008. The feasibility and effectiveness of the proposed joint estimation algorithm has been verified by experiments. The transmission performances with [-5GHz + 5GHz] FO are compared under the OSNR range from 14 to 27dB in a 106.8Gbit/s 16-ary quadrature amplitude modulation (16-QAM) PDM CO-OFDM transmission system. The proposed TO/FO estimation algorithm performs robustly and accurately without any induced BER degradations.

© 2016 Optical Society of America

## 1. Introduction

Coherent optical orthogonal frequency division multiplexing (CO-OFDM) communication system has been extensively investigated due to its advantages of high spectral efficiency and good tolerance of dispersion [1–3]. Moreover, a wider selection of modulation format and payload location in subcarriers makes it more flexible compared with other communication systems. On the other hand, polarization division multiplexing (PDM) is a technology that utilizes the polarization character of light to easily double the system capacity while keeping the signal bandwidth unchanged. Combining those benefits, PDM CO-OFDM communication technology is a promising candidate for the next generation optical long-haul transmission systems. However, the CO-OFDM system is very sensitive to the synchronization errors. The frequency offset (FO) between the local oscillator (LO) and the carrier laser source causes inter-carrier interference between subcarriers due to the loss of orthogonality [4]. The timing offset (TO) induces inter-symbol interference (ISI) and degrades the system performance [5]. Thus, estimation and correction of FO and TO for CO-OFDM has been investigated extensively in optical communication systems.

For the FO estimation (FOE), since the commercially-available lasers usually have a frequency accuracy with ± 2.5GHz over the lifetime, the practical requirement of the FOE algorithm will be satisfied as long as the estimation range is close to [-5GHz, + 5GHz] [6]. The classic Schmidl and Cox’s algorithm divides the frequency offset into the fraction and the integer part of the subcarrier spacing and estimates the two kinds of frequency offset respectively by using two training symbols (TSs) [7,8]. The estimation accuracy of fractional part of frequency offset is high, but the range is small. Though integral part of estimation in the frequency domain increases the estimation range, computational complexity and latency are also increased because FO compensation is performed in the time domain at the front end of the OFDM receiver. Another approach for FO estimation is based on a pilot subcarrier in the center of OFDM spectra [9,10]. By extracting the phase of the pilot, the phase impairments induced by the frequency offset can be estimated. However, the receiving filter cannot filter out the pilot subcarrier accurately without an additional coarse FO estimation method.

The Schmidl and Cox’s algorithm can also achieve TO estimation by using a training symbol with two identical halves. However, the timing metric of the algorithm has a plateau which results in a large timing offset estimation variance [8]. In order to improve the accuracy of the Schmidl and Cox’s algorithm, Minn et al [11] proposed a modified training symbol with four identical parts having specific sign changes for these parts. The timing metric has sharp peaks but the estimation result has a large variance in ISI channels. Some other algorithms have been used for TO estimation in CO-OFDM systems [12,13]. All of these methods rely on the correlation property of specially-designed training symbols. Correlation operation after sweeping the signal sample by sample is necessary in time domain.

The fractional Fourier transformation (FrFT) is a generalization form of the Fourier transformation (FT) and has been utilized to represent the signals on an orthonormal basis formed by chirps. And, the FrFT can induce rotations in various time-frequency transforms, including the Wigner distribution and the short-time Fourier transform, further enhancing its interpretation as a rotation operator [14]. Thus, it can be used to analyze the signal both in the time and frequency domains. Thanks to its unique properties, the FrFT has been used in multiple applications such as solving differential equations [15], quantum mechanics [15], optical information processing [16] and signal processing [17–19]. It has advantages in dealing with linear frequency modulated (LFM) or chirped signals and has been deployed for detecting and estimating LFM signal’s characteristics [20]. A blind CD estimation method for coherent optical communication system based on FrFT has also been proposed and demonstrated recently [21].

By further exploiting its properties on manipulating the signal in time-frequency transforms, we proposed a novel joint TO and FO estimation/correction algorithm based on FrFT encoded training symbols for CO-OFDM system. Although the general principle and preliminary simulations for a single polarization CO-OFDM system have been presented in [22], in this paper, the detailed operation principle, robustness analysis and comprehensive simulations are conducted with systematic long-distance transmission experimental verifications for PDM CO-OFDM systems. We take advantage of the time-frequency property of the FrFT encoded TSs instead of the correlation property to achieve fast and accurate FO and TO estimation. The unique structure of TSs in time-frequency space ensures robust and accurate estimations of TO and FO simultaneously. The TO estimation performance of our proposed algorithm is better than that using Schmidl and Cox’s method at poor signal to noise ratio (SNR) conditions. The range of FO estimation is close to ± 5GHz. The performance of our proposed algorithm was also tested by a 3-tone 106.8 Gb/s 16QAM PDM CO-OFDM based 480 km standard single mode fiber (SSMF) transmission system. The transmission performances with [-5GHz + 5GHz] FO are compared under the OSNR range from 14 to 27dB, which shows the feasibility of proposed TO/FO estimation method. The rest of the paper is organized as follows: in Section 2, the operation principle of proposed algorithm is introduced. In Section 3, simulations are conducted to prove the robustness of this method. In Section 4, we performed experiments to confirm the method’s feasibility in long-distance optical fiber transmission systems. Finally, conclusions are drawn in Section 5.

## 2. Operation principle

#### 2.1 Brief introduction of FrFT

The FrFT can induce rotations in Wigner domain [23]. The rotation angle$\alpha $corresponds to the rotation angle caused by the *p* order FrFT with the relationship as follows:

Because of the rotation of the Wigner domain, some signals like linear frequency modulated (LFM) or chirped signals can be dealt with in specific fractional domain instead of just either in time domain or in frequency domain. In some specific fractional domains, LFM or chirped signals may present unique properties that will be helpful for signal processing. There are several digital FrFT (DFrFT) computation algorithms. One of them, which is proposed by H. M. Ozaktas et al is used in our work. This quick DFrFT algorithm computes the fractional transform in O(NlogN) time, where N is the time-bandwidth product of the signal, similar to the FFT algorithm. Thus, the computation of the fractional transform does not sacrifice the computation efficiency compared with the ordinary Fourier transform [24].

#### 2.2 Design of the training symbol

The FrFT was shown to induce order-dependent rotations in time-frequency transforms, which can be interpreted by the Wigner distribution or the short-time Fourier transform [25]. After a P order of FrFT, a direct current (DC) signal’s frequency, which corresponds to a fixed value 0, will vary linearly along with the time due to an$\text{\theta}=\text{P}\cdot \pi \text{/2}$angle rotation of time-frequency distribution as Fig. 1(a) shows. We call this kind of signal as the FrFT-DC signal, which is similar to LFM signal, like Eq. (5).

S(t) is a DC signal,*FrFT*denotes the operation of FrFT, P is the order of FrFT and$\text{T(}t\text{,P)}$ stands for the P order FrFT-DC signal in time domain. For this kind of FrFT-DC signal, we can find a Q order fractional domain where the energy of signal can be gathered to the utmost extent, see in Fig. 1(b), where

*u*is the time axis after FrFT [25]. The order Q and P satisfy:

After Q order FrFT, the FrFT-DC signal will have an energy peak. And the position of the energy peak is related to both of the TO and FO, which will be discussed in Section 2.3. In order to estimate the TO and FO at the same time respectively, two FrFT-DC signals with different orders are needed. In this paper, the proposed training symbol consists of two FrFT-DC signals with opposite orders of P and -P, like Eq. (7).

_{1}and Q

_{2}of these two fractional domains are related to the original order of FrFT-DC signal, in Eq. (8).

#### 2.3 Principle of joint TO and FO estimation

Figure 3(a) represents the time-frequency distribution of our proposed training symbol. According to Fig. 2(b) and Fig. 3(a), we can find that the energy peak of the signal in Q_{1} or Q_{2} order fractional domain is at the middle of the range of training symbol without any TO and FO. According to Eq. (1), the rotation angle is related to the order P.

However, when there are some TO or FO occurs in training symbols, the peaks of the Q_{1} and Q_{2} order signals will shift. As we can see in Figs. 3(b)–3(d), *f* and *t* denote the frequency and time axis. The vertical and horizontal green arrow mean the FO and TO respectively. The position’s shift of the signal along with the green arrow represent the signal’s change after FO or TO. The purple arrow means the peak shift after Q_{1} and Q_{2} FrFT. When there is only FO or TO, the peaks’ shift in Q_{1} and Q_{2} are the opposite or same way respectively. The detail relationship between the peaks’ shift and the offsets can be obtained by simple geometrical analysis.

In order to analyze the shift quantitatively, we use the sample number to represent the offset of frequency and time as in Eq. (10):

where$\Delta F$and$\Delta T$are the frequency and time offset respectively,$dt$and$df$are the sampling interval in the time and frequency domain after digital sampling,$\Delta {N}_{f}$and$\Delta {N}_{t}$are the samples number of frequency offset and time offset respectively.$\Delta {Q}_{1}$and$\Delta {Q}_{\text{2}}$are sample number of peak shift in Q_{1}and Q

_{2}fractional domain respectively.

We can get the relationship between the peak shifts and the offsets of frequency and time by geometrical analysis in Fig. 4. Notice that$\Delta {N}_{f}$,$\Delta {N}_{t}$,$\Delta {Q}_{1}$and$\Delta {Q}_{\text{2}}$are negative numbers when the shift direction is opposite to the forward direction of the axis representing *f*, *t*, or *u*. Therefore, the peak shift can be represented by the offset sample number, as Eq. (11) shows.

_{1}and ΔQ

_{2}of the training symbols in two fractional domains by peak searching, the TO and FO can be estimated according to Eq. (12), combining the Eq. (9) and Eq. (11). According to the Eq. (12), the range and accuracy of TO and FO estimation are related to the order P. For the purpose of joint TO/FO estimation, the estimated range of FO is associated with the worst TO condition, which equals to the half of the training symbol’s length as shown in Fig. 5(a).

In this condition, the TO ($\Delta {N}_{t}\text{=N/2}$) and FO ($\Delta {N}_{f}$) satisfy the Eq. (13).

*F*is the sample rate of the signal. Figure 5(b) shows the relationship between the largest estimated FO and the order P when the

_{sam}*F*is 12Gsamples/s as we set in simulations and experiments. A larger order P, which means a lager rotation angel$\text{\theta}$, gives a better resolution for$\Delta {N}_{t}$and leads to a better TO estimation result. Instead, the FO estimation result will be better and have a larger range with a smaller order P. Since there are cyclical prefixes (CPs) to tolerate the TO estimation error, a relative small P should be set to make sure the good FO estimation accuracy in a larger range. According to the Eq. (14) and Fig. 5(b), in order to achieve the accurate FO estimation with a range of [-5GHz + 5GHz], the order must be smaller than 0.115. Considering the slight frequency jitter of the laser, we set the P to 0.1 to ensure reliable FO estimations when FO is set to ± 5GHz in experiments. Once the range of the FO estimation can be guaranteed, a smaller P will induce the degradation of the TO estimation because of the coarser resolution according to the Eq. (12). Therefore, considering the range of the FO estimation and the accuracy of the TO estimation, P is set to 0.1 in our simulation and experiments.

_{sam}According to the Eq. (10) and Eq. (12), the minimum estimated FO value *FO _{min}* of our proposed training symbol can be obtained when the absolute value of ΔQ

_{1}+ ΔQ

_{2}equals to 1.

*F*is the sample rate of the signal. N is the sample number of proposed training symbol TS1, which equals to the IFFT size. Then, we can get the relationship between the

_{sam}*FO*and the subcarrier frequency spacing

_{min}*f*.

_{sc}*FO*is smaller than

_{min}*f*, which proves that our proposed idea can deal with the fractional part of FO with a coarse estimation. Besides, the FO estimation range of the Schmidl and Cox’s algorithm is limited to ±

_{sc}*f*. In order to provide a FO estimation with high accuracy, another training symbol constructed by the classic Schmidl and Cox’s algorithm is adopted to achieve the residual FO estimation after coarse estimation of proposed training symbol.

_{sc}## 3. Simulation results and discussions

#### 3.1 Simulation setup and OFDM frame structure

In order to investigate the performance of the proposed algorithm, we built a simulation system in MATLAB and VPI Transmission Maker 9.1 environment. The numerical simulation model of PDM CO-OFDM transmission system is shown in Fig. 6(a). In the OFDM transmitter, The OFDM baseband signals are generated by a DSP seen in Fig. 6(b). The IFFT size is 512. A pseudo-random bit sequence (PRBS) with a length of 2^{15}-1 as the transmitted data stream is mapped into 16QAM and filled into 232 payload subcarriers and 24 pilot subcarriers are used for phase estimation. After IFFT, 64-sample CPs is added in the OFDM symbol, resulting in an OFDM symbol size of 576. The OFDM frame structure is shown in Fig. 6(d). Eleven training symbols are subsequently inserted at the beginning of each OFDM frame. TS1 is our proposed training symbol for OFDM TO and FO estimation. TS2 is the training symbols constructed by the classic Schmidl and Cox’s algorithm, which is used for estimation of fractional part of frequency offset. In our simulation and experiment, TS2 is for the residual FO estimation. TS3, TS4, TS5 and the NULL symbols inserted in training symbols are used for channel estimation. Then, we minimize the impact that the 128th data symbol of Y-pol has on the TS1of X-pol by adding a NULL symbol in the end of the OFDM frame. At last, each OFDM frame contains 139 OFDM symbols. The sampling rate of AWG is 12Gsamples/s. According to Eq. (14), in order to satisfy the FO estimation’s range of [-5GHz, + 5GHz], the order P of TS1 is set to 0.1. Therefore, the 16QAM PDM CO-OFDM baseband signal is obtained with a 35.6Gbit/s (≈2 × 4 × 12GSam/s × 232 ÷ 576 × 128 ÷ 139) data rate and 23.4375MHz (≈12G ÷ 512) subcarrier frequency spacing *f _{sc}*.

After modulation, the optical signals are divided into two arbitrary states of polarization via a polarization beam splitter (PBS). Then the Y polarization signal is delayed by one OFDM symbol period (48 ns) to emulate the polarization multiplexing. Fiber link composes of single model fiber (SMF), erbium doped fiber amplifier (EDFA) and optical filter. Here, optical signal-to-noise ratio (OSNR) can be adjusted by changing the noise figure of EDFA. After fiber propagation, the received signals are divided into two arbitrary states of polarization via a PBS. Then, both states of polarization are mapped from the optical field into four electrical signals by utilizing the passive quadrature hybrid with balanced detectors. Next, the electrical signals were digitized by ADCs at 50Gsamples/s and stored for DSP using MATLAB, as shown in Fig. 6(c).

#### 3.2 Detail process and simulation results

The joint TO and FO estimation is first accomplished by scanning the signal to find the rough place of training symbols. First, we scan the signal once a symbol length and they are called temp symbols, as denoted by the green frames in Fig. 7(a). Second, for each temp symbol, we transform it to Q_{1} and Q_{2} fractional domains by Q_{1} and Q_{2} order FrFT. Only these temp symbols which cover the most part of TS1 can have peaks in Q_{1} and Q_{2} fractional domains, shown in Fig. 7(b). Therefore, we can estimate the coarse place of every TS1. Third, after we find the coarse place of every TS1, the specific temp symbols’ peaks appear in Q_{1} and Q_{2} fractional domains, just like Fig. 7(c). For Q_{1} and Q_{2} fractional domains, we can get the peak’s position, respectively. Last, the peak’s shifts ΔQ_{1} and ΔQ_{2} equal the peak’s position minus the middle position of the signal in Q_{1} and Q_{2} fractional domains. The precise TO estimation can be achieved by Eq. (12). To cope with the residual part of frequency offset, we further adapt the Schmidl and Cox’s algorithm to estimate the fractional frequency offset accurately. Because the starting sample of OFDM signal have been found already, it is much simpler to estimate the fractional frequency offset by using Schmidl and Cox’s method. After that, the residual frequency offset (RFO) is compensated as demonstrated in [26] without any extra overhead.

For the Schmidl and Cox’s algorism, correlation operation after scanning the signal sample by sample is necessary in time domain. For the proposed method, we do not have to scan samples one by one, but only scan once a symbol length and two FrFTs are needed for each scanning step. The two kind of operation after each scanning steps, one correlation operation for Schmidl and Cox’s algorism and two FrFTs for proposed method, have similar calculation load. But, the proposed method needs much fewer scanning steps than the Schmidl and Cox’s algorism. For example, if the OFDM frame contains 128 OFDM symbols and the size of the OFDM symbol is 576 (512 FFT size and 64 CPs), 73728 (128 × 578) scanning steps are needed for the Schmidl and Cox’s algorism while only 128 scanning steps are needed for the proposed methods. Therefore, the proposed method reduces the computation complexity significantly compared with the Schmidl and Cox’s algorism.

At the receiver, a timing offset is modeled as a delay in the received signal. Figure 8(a) shows the measured timing metric of Schmidl & Cox algorithm with 5GHz frequency offset in different OSNR condition. The correct timing instant is 288 ( = 576/2) in the figure. The peak position of the timing metric means the estimated timing instant. As can be easily found, the timing metric of Schmidl & Cox algorithm is no longer valid at a degraded OSNR condition of 10dB with a relatively large TO estimation error. Figure 8(b) shows the peaks shift of proposed training symbol in Q_{1} and Q_{2} fractional domain after finding the rough place of training symbols in different OSNR condition with 5GHz frequency offset. Based on the correlation operation, the timing metric of the classical Schmidl and Cox’s method has a plateau which results in a large timing offset estimation variance especially in a low OSNR condition in Fig. 8(a). Based on the time-frequency property of the proposed training symbol, the energy peak in fractional domain is not sensitive to the noise because the energy of the training symbol will be centralized while the noise is decentralized. Therefore, we can still find the sharp energy peaks at low OSNR scenarios in Fig. 8(b). Comparing the performance in Figs. 8(a) and 8(b), the highly stable peak position of proposed method implies a better TO estimation in a poor OSNR condition.

In order to compare the estimation accuracy of Schmidl & Cox algorithm and that of our proposed method, more than 100 sets of data are run for each OSNR condition. For Schmidl & Cox algorithm, the peak position of the timing metric is the estimation result. The result of proposed method can be calculated by the peaks’ shift according to Eq. (12). Figure 9(a) depicts estimation error of two methods in different OSNR conditions. Note that we didn’t set any FO for Fig. 9(a) to compare the timing estimation performance individually. The estimation error of our proposed method stays stable and equals to 1 sample. This 1 sample error is not due to the channel noise but caused by rounding the result of Eq. (12) to integer. The estimation result of Schmidl & Cox algorithm is sensitive to the noise especially in poor OSNR condition. For evaluating the FO estimation performance of the proposed algorithm, we add FO to the received OFDM signal arbitrarily and compare it with the estimated one. Figure 9(b) shows the absolute values of the normalized FOE error (with respect to *f _{sc}*) as a function of frequency offset by using the proposed FOE algorithm under a poor OSNR condition (OSNR = 15dB). To verify the estimation results statistically, more than 100 sets of data are simulated for each measurement point. Considering the estimation range and the accuracy, Fig. 9(b) shows the satisfactory FO estimation results by combining the large range estimation of TS1 and accurate estimation of TS2. Results indicate that the mean normalized estimation error can be kept under 0.002 and the max estimation error is no more than 0.008 with the estimation range from −5GHz to 5GHz, which can basically guarantee the requirement of the practical application.

## 4. Experimental verifications and discussions

The setup utilized for the practical PDM CO-OFDM transmission experiment is illustrated in Fig. 10. The OFDM transmitter and receiver and the frame structure are the same with those in our simulation shown in Fig. 6. After modulation, three optical OFDM tones are generated by driving an optical intensity modulator with 7.031GHz RF tone, which extends the data rate to 106.8Gbit/s ( = 3 **×** 35.6Gbit/s). The EDFA1 and VOA1 are used for changing the input power after polarization multiplexing. The optical channel is a 480km distributed Raman amplified fiber link including six 80km spans. After fiber transmission, VOA2 and EDFA2 are used for changing the OSNR. The OSNR can be adjusted by changing the power of optical power through the VOA2. Then the middle tone of OFDM signal is filtered by the TOF. We change the gain of EDFA3 to keep the same received optical power for every OSNR conditions. The OSNR of the fiber link is monitored by OSA at a 0.02 nm resolution.

First of all, we compare the transmission performance with and without our proposed FO estimation method for different input power. The OSNR is also monitored at the same time as shown in Fig. 11(a). For Fig. 11, the frequency of the local oscillator (LO) is experimentally set to be the same as the frequency of the carrier laser source. However, the FO is inevitable due to the random frequency jitter of the laser and its actually value changes from time to time. The Fig. 11(b) shows one example of experimentally estimated FO with different input power. We found that the FO is always larger than the subcarrier frequency spacing *f _{sc}* (23.4375MHz), which will cause the totally distorted signal and the approximately 0.5 BER without FO estimation and compensation. With our proposed FO estimation, the signal is demodulated successfully and the best input power is −6dBm. After that, the BER increases when the input power gets lager, which is mainly caused by fiber nonlinear interference. The results show that our method is effective for different input power even if there are obvious nonlinear distortions when the input power is large. Figure 12 shows the measured constellations of the 16QAM signals. By comparing Fig. 12 (a) with Fig. 12 (b), our proposed method can estimate and correct the FO to recover the constellation. Figure 12(c) shows the feasibility of proposed method in a poor 14dB OSNR condition.

Figure 13(a) shows the system’s BER performance when the input power is set to the best value −6dBm. More specifically, the BER versus OSNR with −5GHz, + 5GHz and without FO are compared under the OSNR range from 14 to 27dB. We can find that they have almost the same performance according to the BER curves. It means that the BER degradation is mainly caused by the worse OSNR rather than the FO estimation error. Our proposed algorithm is still feasible and stable at poor OSNR conditions. Besides, in order to verify the estimation range, different FO, which ranges from −5GHz to 5GHz, are set for 18, 22 and 26dB OSNR conditions. The BER performances are shown in Fig. 13(b). The BER of different FO are almost the same for one given OSNR condition. When the FO is too large, there are a little bit of BER increases, which can be attributed to the non-flat frequency responses rather than the FO estimation error. Experimental results confirm that the proposed algorithm had a [-5GHz + 5GHz] estimation range, satisfying almost all practical application requirements.

## 5. Conclusions

A joint TO and FO estimation method based on FrFT encoded training symbols for PDM CO- OFDM system is proposed and evaluated numerically and experimentally. Unlike other estimation methods, our proposed method takes advantage of the time-frequency property of the FrFT encoded TSs instead of the correlation property to achieve fast and accurate FO and TO estimation. We do not have to scan samples one by one to calculate the correlation, but only scan once a symbol length to analyze the time-frequency property with a quick DFrFT algorithm, which reduces the computation complexity significantly. According to the result of our simulations, our method has more stable estimation results compared with the classical Schmidl & Cox algorithm in a poor OSNR condition. For FO estimation, combining with a simple residual FO estimation method, the final normalized estimation error is less than 0.002. Through a 480km 106.8Gbit/s PDM CO-OFDM transmission experiment, the proposed method is proved to be robust against noise and nonlinear interference with a [-5GHz + 5GHz] FO estimation range.

## Funding

National Natural Science Foundation of China (NSFC) (Grant No. 61331010), and the Program for New Century Excellent Talents in University (NCET-13-0235)

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