Decision-aided sampling frequency offset compensation for reduced-guard-interval coherent optical OFDM systems

We propose a decision-aided algorithm to compensate the sampling frequency offset (SFO) between the transmitter and receiver for reduced-guard-interval (RGI) coherent optical (CO) OFDM systems. In this paper, we first derive the cyclic prefix (CP) requirement for preventing OFDM symbols from SFO induced inter-symbol interference (ISI). Then we propose a new decision-aided SFO compensation (DA-SFOC) algorithm, which shows a high SFO tolerance and reduces the CP requirement. The performance of DA-SFOC is numerically investigated for various situations. Finally, the proposed algorithm is verified in a single channel 28 Gbaud polarization division multiplexing (PDM) RGI COOFDM experiment with QPSK, 8 QAM and 16 QAM modulation formats, respectively. Both numerical and experimental results show that the proposed DA-SFOC method is highly robust against the standard SFO in optical fiber transmission. ©2014 Optical Society of America OCIS codes: (060.1660) Coherent communications; (060.2330) Fiber optics communications. References and links 1. W. Shieh, X. Yi, Y. Ma, and Q. Yang, “Coherent optical OFDM: has its time come? [Invited],” J. Opt. Netw. 7(3), 234–255 (2008). 2. A. J. Lowery and L. B. Du, “Optical orthogonal division multiplexing for long haul optical communications: A review of the last five years,” Opt. Fiber Technol. 17(5), 421–438 (2011). 3. W. Shieh, “OFDM for flexible high-speed optical networks,” J. Lightwave Technol. 29(10), 1560–1577 (2011). 4. S. L. Jansen, I. Morita, T. C. Schenk, and H. Tanaka, “Long-haul transmission of 16×52.5 Gbits/s polarizationdivision-multiplexed OFDM enabled by MIMO processing (Invited),” J. Opt. Netw. 7(2), 173–182 (2008). 5. X. Liu, S. Chandrasekhar, B. Zhu, P. J. Winzer, A. H. Gnauck, and D. W. Peckham, “448-Gb/s reduced-guardinterval CO-OFDM transmission over 2000 km of ultra-large-area fiber and five 80-GHz-grid ROADMs,” J. Lightwave Technol. 29(4), 483–490 (2011). 6. Q. Zhuge, B. Chatelain, and D. V. Plant, “Comparison of intra-channel nonlinearity tolerance between reducedguard-interval CO-OFDM systems and nyquist single carrier systems,” in Proc. OFC’12, paper OTh1B.3 (2012). 7. M. Sliskovic, “Carrier and sampling frequency offset estimation and correction in multicarrier systems,” in IEEE Global Telecommunications Conference, 2001. GLOBECOM ’01 (IEEE, 2001), Vol.1, pp. 285–289. 8. Y. Chen, S. Adhikari, N. Hanik, and S. L. Jansen, “Pilot-aided sampling frequency offset compensation for coherent optical OFDM,” in Proc. OFC’12, paper OTh4C (2012). 9. X. Yi and K. Qiu, “Estimation and compensation of sample frequency offset in coherent optical OFDM systems,” Opt. Express 19(14), 13503–13508 (2011). 10. X. Liu and F. Buchali, “Intra-symbol frequency-domain averaging based channel estimation for coherent optical OFDM,” Opt. Express 16(26), 21944–21957 (2008). 11. S. K. Mitra, Digital Signal Processing: A Computer Based Approach, 4th ed. (McGraw Hill, 2011). 12. J. F. Kenney and E. S. Keeping, Mathematics of Statistics, 3rd ed. (Princeton, 1962). #221423 $15.00 USD Received 21 Aug 2014; revised 18 Oct 2014; accepted 24 Oct 2014; published 30 Oct 2014 (C) 2014 OSA 3 November 2014 | Vol. 22, No. 22 | DOI:10.1364/OE.22.027553 | OPTICS EXPRESS 27553 13. Q. Zhuge, M. Morsy-Osman, and D. V. Plant, “Low overhead intra-symbol carrier phase recovery for reducedguard-interval CO-OFDM,” J. Lightwave Technol. 31(8), 1158–1169 (2013). 14. W. Wang, Q. Zhuge, Y. Gao, M. Qiu, M. Morsy-Osman, M. Chagnon, X. Xu, and D. V. Plant, “Low overhead and nonlinear-tolerant adaptive zero-guard-interval CO-OFDM,” Opt. Express 22(15), 17810–17822 (2014). 15. M. Morsy-Osman, M. Chagnon, Q. Zhuge, X. Xu, and D. V. Plant, “Non-data-aided feedforward timing recovery for flexible transceivers employing PDM-MQAM modulations,” in Proc. OFC’14, paper W3B.4 (2014).


Introduction
Coherent optical orthogonal frequency division multiplexing (CO-OFDM) is regarded as a potential candidate for high capacity spectrally efficient optical transmission systems attributed to its inherent compact spectrum [1][2][3].In CO-OFDM, cyclic prefix (CP) is usually inserted between adjacent OFDM symbols to accommodate the inter-symbol interference (ISI) induced by linear transmission impairments such as chromatic dispersion (CD) and polarization mode dispersion (PMD) [3].However, the large CP overhead required to accommodate CD in long haul transmission sacrifices the high spectral efficiency (SE) of CO-OFDM.To maintain a moderate CP overhead (~25%), a large OFDM symbol size must be used [4].The reduced-guard-interval (RGI) CO-OFDM system was proposed to compensate CD with an overlapped frequency domain equalizer (OFDE) [5].This novel scheme enables the use of a small OFDM symbol size while maintaining a reduced CP overhead (≤5%).Moreover, a small OFDM symbol size (e.g., FFT size N DFT ) provides some additional benefits: (1) a reduced computational effort, (2) a reduced training symbol (TS) overhead, (3) a faster channel tracking speed, (4) a higher tolerance to laser phase noise, and (5) an potentially improved fiber nonlinearity tolerance [1][2][3][4][5][6].Therefore, RGI-CO-OFDM is a promising variation of CO-OFDM systems.
However, because of the reduced CP overhead, one major drawback of RGI-CO-OFDM systems is its high sensitivity to synchronization errors of the sampling frequencies between the digital to analog convertor (DAC) at the transmitter (Tx) and the analog to digital convertor (ADC) at the receiver (Rx).This sampling frequency offset (SFO) between the DAC and ADC destroys the orthogonality of OFDM subcarriers [3,7].Even a small amount of SFO, e.g.50~100 parts per million (ppm), can lead to a serious system performance degradation [8].
The typical method used in experiments to combat SFO is to manually synchronize the clock both at the Tx and Rx, or to use an external clock to drive DAC and ADC at the same time.However, both approaches are not applicable for real optical transmission systems with a standard 200 ppm SFO since the Tx and Rx are physically separated [8,9].In wireless communication, SFO is usually first estimated using training symbols and then compensated based on the digital time domain interpolation [7].However, the interpolation is computationally complex, and carriers close to Nyquist frequency might be strongly attenuated [7].The authors in [9] proposed a SFO estimation method for coherent optical transmission based on the re-use of pilot subcarriers (PS-SFOC).The SFO is then compensated either by multiplying additional frequency dependent phase correction term in the frequency domain or by the feedback of the estimated SFO to the receiver clock [9].This approach shows an ability to compensate up to 100 ppm of SFO and an ability to estimate up to 1000 ppm of SFO [9].As we will show in Section 3.1 and 3.2, we found the performance of the PS-SFOC is highly dependent on the CP overhead.However, for both methods, estimation errors are inevitable depending on the signal to noise power ratio, or equivalently the number of pilot symbols for one estimation [7,9].For PS-SFOC, 100 OFDM symbols are typically required to achieve a reliable SFO estimation [9].In reality, this requires a large buffer to store all the OFDM symbols before their successful demodulation and suffers from a large processing latency.In addition, the residual SFO from estimation error can cause a nonnegligible performance penalty.Moreover, the PS-SFOC approach is not very effective for time varying SFO without adding additional complexity.
In this paper, we propose a decision-aided SFO compensation (DA-SFOC) method for RGI-CO-OFDM, which is suitable for directly compensating the standard SFO or for compensating residual SFO due to estimation errors with other estimation methods.Unlike the previous estimation and compensation based two step approaches, the DA-SFOC does not require the SFO estimation, and hence does not suffer from SFO estimation errors.The proposed method is also able to relax the SFO induced CP requirement with small additional complexity.This paper is organized as follows.In Section 2, we review and discuss the effects of SFO on OFDM signals, and present the concept of the DA-SFOC algorithm.In Section 3, we numerically study the performance of PS-SFOC and DA-SFOC for various conditions.In Section 4, the proposed DA-SFOC approach is verified in a single channel 28 Gbaud PDM RGI-CO-OFDM experiment for different modulation formats including QPSK, 8-QAM and 16-QAM.Both numerical and experimental results demonstrate that the standard SFO (200 ppm) for optical transmission system can be compensated with a negligible performance penalty.Section 5 finally concludes the paper.

Effect of SFO on the received OFDM signals
Adopting the analysis of [7,9], the received OFDM signal in the presence of SFO and laser phase noise can be described as follows: where ki s and ki r are the transmitted and received ith information symbol on the kth subcarrier in one frame, h k is the channel transfer function for the kth subcarrier, i φ is the common phase error (CPE) originated from laser phase noise, ki η is the sum of the laser phase noise and SFO induced inter-carrier interference, ki n is channel additive white Gaussian noise (AWGN), and s ε is the normalized SFO, which is defined as where f Δ is the sampling frequency offset between the DAC and ADC, and s f is the sampling frequency at the transmitter.k takes values from [-Nsc/2 + 1, Nsc/2] where Nsc is the total number of subcarriers.And i takes values from [1, N] with N being the frame length (FL).
According to Eq. ( 1), SFO has two detrimental effects on the received OFDM signals: (1) a time and frequency dependent phase shift of the transmitted symbols, and (2) the intercarrier interference whose variance increases with the absolute subcarrier indices and the SFO values [7].A third effect caused by SFO can be observed from Fig. 1.As the number of samples accumulates, sampling points belong to one OFDM symbol window would finally appear in another OFDM symbol window, inducing the so called inter-symbol interference (ISI) [3].
Since DFT window synchronization is usually performed on a frame basis with the aid of training symbols at the beginning of the frame, the OFDM symbols in the front of the frame normally have good DFT window synchronizations.However, the dislocation of DFT window becomes more and more severe for the rear OFDM symbols and the ISI induced penalty is also increasing.

Cyclic prefix requirement for preventing SFO induced ISI
Cyclic prefix is known to be able to eliminate all ISI in OFDM systems [3].Therefore, we reuse CP to prevent the ISI induced penalty caused by SFO.Since the receiver side sampling frequency r f might be either larger or smaller than s f , it is necessary to append CP at both ends of one OFDM symbol.In order to completely eliminate ISI, the maximum number of samples that appear in a wrong OFDM symbol period, i.e., the symbol in the last DFT window of one OFDM frame, should not exceed the single side cyclic prefix length, i.e.
( ) where N DFT is the DFT window size, N CP is the total number of CP for one OFDM symbol, FL is the frame length, and | | denotes the absolute value operation.After some simple derivations, it is found that the total desirable number of CP is Figure 2 shows the required CP overhead (N CP /N DFT ) against the SFO induced ISI in the system.Although CP overhead increases linearly with the SFO, a large SFO is rarely encountered in practice, or could be compensated by a rough estimation and feedback to the Rx clock.Therefore, only the standard SFO or residual SFO needs to be considered.Assuming a common frame length of 100 OFDM symbols, only ~4% CP (e.g. 6 CPs for an OFDM symbol size of 128) is enough for combating a standard 200 ppm SFO.However, in the following section we will show a way, combining with the DA-SFOC method, to reduce the required amount of CP induced by SFO to only 2 samples per OFDM symbol despite the OFDM symbol size and the SFO value.

Phase recovery with decision aided SFO compensation
With the aid of CP eliminating SFO induced ISI, we can compensate SFO induced phase shifts using the following decision-aided method.The first step of the phase recovery procedure in the presence of SFO is to use the pilot subcarrier aided CPE estimation and compensation (CPEC) technique as is done in the conventional OFDM phase recovery process [1].Since SFO and laser phase noise both introduce a phase shift to the transmitted information symbol, it is important to make sure that the SFO does not influence the CPEC process [9].As shown in Eq. ( 1), the SFO induced phase shifts of positive subcarriers are equal in magnitude but opposite in sign to those of the negative subcarriers with the same absolute indices (e.g.index of 10 and −10).In order to separate the CPEC and SFOC stage, pilot subcarriers can be arranged in a symmetric way, i.e. a pilot with a positive index should have a counterpart with the same negative index, such that the SFO induced phase shifts of different subcarriers are averaged out in the CPE estimation and therefore do not interfere with the CPE estimation process [9].After CPEC, the SFO induced phase shift of the ith symbol on the kth subcarrier, ki SFO φ , is obtained by ) where r CPE denotes the received symbol after channel equalization and CPE correction, max ' k and min ' k in Eq. ( 5) are the maximum and minimum modulated subcarrier indices, * denotes the phase conjugation, ki s  is the decision of the transmitted information symbol ki s , and N TS is the number of training symbols at the beginning of a frame.In Eq. ( 5), intra-symbol frequency domain averaging (ISFA) with an averaging length 2m + 1 is used to improve the accuracy of the estimation in the presence of noise [10].The ISFA is performed for positive and negative subcarriers separately due to the SFO induced frequency miscorrelation between positive and negative subcarriers.A detail study of the ISFA algorithm can be found in [10].The term i/(i-1) in Eq. ( 6) is a revising factor for using information on past symbols.The process described by Eq. ( 5) and Eq. ( 6) can be initiated using training symbols inserted at the beginning of a frame originally used for channel estimation (CE) purposes, or simply by setting 1 0 k SFO φ ⋅ = , which is a valid approximation from Eq. (1).The procedure can then be propagated until the end of the frame.It is notable that since DA-SFOC operates on a symbolby-symbol basis, and OFDM symbol rate is much lower than the actual transmitted bit rate, the implementation of DA-SFOC does not require buffers and very high-speed power consuming electronics.Moreover, since the compensation is performed on a symbol-bysymbol basis, the DA-SFOC is robust against time varying SFO.

Method of relaxing the CP overhead requirement
In this sub-section, we provide one optional way to relax the above mentioned CP requirement to only 2 samples per symbol regardless of the SFO values and the OFDM symbol size.According to digital signal processing theory, circular time-shift is equivalent to a linear phase shift in the frequency domain, and the relationship is given by [11] where g is a time domain digital signal, n is the time domain sample index, n 0 is the time shift of integer samples, G is the DFT transform of g, k is frequency domain sample index, N is the DFT size, and <> N denotes modulo-N operation [11].From Fig. 1, because of the appended CP, the SFO induced sampling point dislocation for one OFDM symbol can be treated as a circular time shift as long as the amount of shifting does not exceed the CP length, and if we ignore the slight differences in the time shifting amount of each sample within one DFT window.This means that by monitoring the linear phase shift in the frequency domain, we can know the sampling point dislocation in the time domain.For example, when the total accumulated shift equals to one integer sample, we should observe a linear phase shift equal in the frequency domain, where we set 0 1 n = in Eq. ( 7).Therefore, if there is a stringent CP overhead requirement for the system, we only need 2 CP samples appended at each end of one OFDM symbol to cover the fractional part of the sampling point shift.And through monitoring the phase shift on each subcarrier of the OFDM symbol, we can move the entire DFT window of one symbol forward or backward by 1 sample when the monitored frequency phase shift gets close to , making sure the SFO induced sample dislocation does not exceed the current OFDM symbol window.
DA-SFOC provides a convenient way to monitor the above mentioned linear phase in the frequency domain.From Eq. ( 5) and Eq. ( 6), ki SFO φ is essentially the additional phase shift of the transmitted ith symbol on the kth subcarrier.Therefore, we can build the following simple linear regression without interception model to derive an estimation of the slope of the linear phase shift in Eq. ( 7) from ki SFO φ , assuming all the samples in one OFDM symbol window have the same shifting amount.The model is given by [12] 2 The slope can be estimated with the least square based method using information on several subcarriers [12].Once the slope s is calculated, the time domain shift amount n can be derived from 2 . If for ith OFDM symbol n = 1, the (i + 1)th DFT window should be shifted forward by 1 sample, and vice versa.And the phase compensation process described by Eq.
(5) and Eq. ( 6) should be reset.Because any fractional movement of the DFT window can be compensated by Eq. ( 5) and Eq. ( 6), the estimation does not have to be highly accurate.In fact, all we need to do is to ensure that the entire DFT window stays within one OFDM symbol period.In the simulation and experiment we found that only 4 subcarriers (can be either data or pilot subcarriers) are enough for each estimation, and the requirement of n = 1 can be relaxed to n > = 0.9 to give some error margin without inducing any additional penalties.

Simulation conditions
Extensive simulations were conducted to investigate the performance of DA-SFOC. Figure 3 depicts the transmitter and receiver digital signal processing (DSP) diagram.At the Tx, a pseudo random binary sequence (PRBS) was mapped on to 110 subcarriers, which, together with one pair of symmetrically scattered pilot subcarriers and one unfilled DC subcarrier, were transferred to the time domain by an IFFT of size 128.Then a pair of correlated dual polarization (CDP) TSs was inserted for every 100 payload symbols for CE, synchronization and SFOC initialization purposes.The DAC was operating at 32 GS/s, resulting in a raw data rate of ~112 Gb/s.The laser line width was set to 100 kHz for both Tx and Rx.At the receiver, CD was first compensated with OFDE.Then frame synchronization was performed by detecting the peak of the cross correlation between the TS and the received signal.After CP removal, serial to parallel and FFT operation, the signal was transformed back to the frequency domain, and channel estimation and 1-tap frequency domain equalization were performed.The phase recovery procedure was the same as described in Section 2. The ISFA parameter in the SFOC process was m = 3, if not otherwise stated.To reduce the performance penalty due to insufficient pilot subcarriers, a maximum likelihood phase recovery (MLPR) was performed before symbol decisions to refine the phase estimation process [13].After symbol demodulation and parallel to serial, bit error rate (BER) was finally calculated via direct error counting.

Simulation results of DA-SFOC and PS-SFOC
Figure 4 shows the optical signal to noise ratio (OSNR) penalty at BER = 3.8E-3 versus SFO for a 28 Gbaud RGI PDM-QPSK-OFDM system.The result of PS-SFOC is also included for comparison [9].In [9], the OFDM System violated the CP condition described by Eq. ( 4), and it did not perform DFT window synchronization on a frame basis.Therefore, the PS-SFOC in [9] can only compensate for ~100 ppm of SFO.We satisfied both rules in this simulation.In addition, the performance of PS-SFOC depends both on the estimation and compensation stages, and the estimation stage is actually not very reliable [9].Therefore, in order to eliminate the influence of the estimation error, that is, to only evaluate the performance of the correction algorithm, we assumed perfect SFO estimation for PS-SFOC.Moreover, three CP cases are compared for two algorithms: w/o CP refers to zero CP condition, fixed 6 samples CP (4.69%), and with CP refers to CP calculated from the corresponding SFO based on Eq. ( 4).For DA-SFOC, 2 CP combined with the DFT window shifting mechanism is also included for comparison.4) is severely violated for large SFO, and the performance degrades significantly due to the ISI induced penalties.Second, for a ≤ 200 ppm SFO, there is no performance difference between the fixed 6 CP and full CP conditions.However, for an SFO larger than 200 ppm, the performance difference becomes gradually obvious since 6 samples CP is borderline to mitigate 200 ppm of SFO induced ISI, as indicated by Eq. ( 4).Third, even with full CP condition, there is an increasing OSNR penalty when the SFO gets larger, indicating that the ICI induced penalty becomes the dominant reason for performance degradation.Fourth, DA-SFOC performs similar to PS-SFOC with ideal estimation.However, in the experiment, the PS-SFOC method showed 5% to 300% estimation error depending on the SFO value in the system [9].In addition, we found the estimation accuracy of PS-SFOC also relates to the system OSNR, the number of pilot subcarriers, indices of the pilot subcarriers, and the system buffer length.The optimum performance could only be achieved with very careful system design.On the other hand, since DA-SFOC does not require an estimation stage, the above performance is the final result.Allowing for a 1 dB OSNR penalty, the maximum tolerable SFO for QPSK format is 800 ppm.
In addition, we also show the result of the DA-SFOC method combined with the DFT window shifting mechanism described in Section 2.4 (indicated as the case of "DA with 2 CP + Shifting" in the figure).In the simulation only 2 samples of CP were appended for each OFDM symbol to cover the fractional part of the DFT window shift.For each OFDM symbol, the least square based method with only four subcarriers with large indices were used for monitoring.The integer part was then compensated by shifting the DFT window in the appropriate direction when the monitored n > 0.9 or n < −0.9.In Fig. 4, we can see that the performance of the method is similar to DA-SFOC with full CP conditions.In fact this was expected, because both adding additional CP and moving the DFT window are essentially two ways to provide the circular time shift condition from a mathematical point of view.
It was found that higher order QAM modulation is more sensitive to SFO induced penalties [8].Simulations were conducted to further investigate the SFO tolerance of the proposed algorithm at higher order QAM constellations.Figure 5 shows the OSNR penalty at BER = 3.8E-3 versus SFO for 8 QAM and 16 QAM, respectively.Similar conclusions can be observed as was the case for the QPSK format.The maximum tolerable SFO decreases to 600 ppm for 8 QAM and ~400 ppm for 16 QAM with a 1 dB OSNR penalty at a full CP condition.However, at the standard 200 ppm SFO, both constellations can be compensated with a negligible OSNR penalty (~0.1 dB for 8 QAM and ~0.3 dB for 16 QAM).

Scalability on different FFT sizes
Due to the loss of orthogonality, SFO also causes ICI among different OFDM subcarriers.It was found in [7] that SFO induced ICI increases with the absolute subcarrier index, which means that a larger OFDM symbol size suffers more from the SFO induced penalty.Therefore we conducted simulations to investigate the proposed algorithm against the OFDM symbol size.It is known that a larger OFDM symbol size also suffers more from laser phase noise for the same reason [3].To rule out the impact of laser phase noise, we turned off the laser linewidth in the simulation.In addition, frame length is also adjusted according to the OFDM symbol size to maintain a fixed TS sample separation.In order to take advantage of the efficiently of the radix-2 FFT algorithm, only OFDM symbol sizes that are the power of 2 are considered (64, 128, 256 and 512 with a fixed filling ratio of 87.5%).Figure 6 illustrates the OSNR penalty versus the OFDM symbol size.As shown in the figure, there is an additional OSNR penalty for both the smallest (64) and the largest OFDM symbol (512) sizes.The former one is caused by the ISFA process.Since SFO introduces a frequency dependent phase shift, which can be treated as a type of frequency miscorrelation, certain numbers of the edge subcarriers, depending on the ISFA length, suffer from an additional penalty due to asymmetrical averaging [14].For OFDM symbols with a smaller size, the number of impaired edge subcarriers will have a stronger impact on the OSNR penalty than when using a larger OFDM symbol size.On the other hand, for OFDM signals with a symbol size of 512, the ICI induced penalty increases significantly for large SFO values, achieving 1 dB OSNR penalty for 200 ppm of SFO.Fortunately, for the RGI CO-OFDM system, the symbol size is not restricted by the accumulated CD and can be designed based on other requirements.Former studies found that OFDM systems with a smaller symbol size are more robust to laser phase noise and fiber nonlinearity, but these systems have more side lobe power leakage and thus are more sensitive to narrow filtering effects (when the FFT size < 100) [5,6].Therefore, a symbol size of 128 samples would be a suitable choice for optical transmission considering the overall conditions.

Experimental setup and results
A single channel 28 Gbaud PDM RGI-CO-OFDM experiment was conducted to verify the proposed algorithm.Figure 7 shows the experimental setup.The 28 Gbaud signal was generated with exactly the same DSP procedure as described in the simulation section.In order to maintain an appropriate CP overhead, fixed 8 samples of CP, half at the beginning and half at the end of an OFDM symbol, are used to combat for the overall ISI including residual CD, PMD, SFO, and all the narrow filtering effects, leading to a 6.25% CP overhead.The OFDM samples were stored in the memory of two field-programmable gate array (FPGA) boards driving two 32 Gs/s DAC with 6 bits resolution.An optical in-phase (I) and quadrature (Q) modulator was employed for the electrical to optical conversion.A PDM signal was formed using the PDM emulator with a delay of 2176 samples in order to fully decorrelate the two polarizations.The signal was amplified by a booster and was then launched into a re-circulating loop, which consisted of 4 spans each having 80 km standard single mode fiber (SSMF) and an EDFA with 5 dB noise figure.At the receiver, the signal out of the loop was filtered, amplified and noise loaded before coherent detection.Two real-time oscilloscopes operating at 80 Gs/s with a 33 GHz analog bandwidth were used to digitize the signal.Offline processing began with the front end correction and resampling the signal to the transmitted sampling rate.The rest of the procedures are the same as in Fig. 3.The SFO innate in our system was only ~30 kHz (about 1 ppm), and was too small to cause an observable degradation within one OFDM frame.Therefore, the data was resampled depending on the SFO value under which the algorithm is desired to be tested, as the way used in [15].
Figure 8 shows the experimental results for the system with a QPSK modulation format.As shown in Fig. 8(a), without SFO compensation, the received signal constellations are completely contaminated by the standard 200 ppm SFO, causing a BER around 0.5.Figure 8(b) shows the BER versus the received OSNR after applying the DA-SFOC method.At a BER threshold of 3.8E-3, the tolerable SFO is ~600 ppm with a 6.25% CP overhead, assuming 1 dB OSNR penalty.We can also observe that when SFO becomes larger than the tolerable SFO predicted by Eq. ( 4), i.e.SFO = 800 ppm in Fig. 8(b), there is a sudden increase in the performance penalty because of insufficient CP length and thus an increased ISI penalty.However, after applying the DFT window shifting mechanism at SFO = 800 ppm, the BER was significantly reduced, indicating the ISI penalty due to insufficient CP was eliminated (performances of DA-SFOC + DFT window shift at other SFO values were similar to the performances of DA-SFOC because of sufficient CP length, as discussed in Section 3.2, and were not showed in the figure for clarity).With the proposed DA-SFO compensation method, the signal constellations are successfully recovered at 200 ppm with a negligible penalty, as shown in Fig. 8(c  Figure 9 and 10 show the experimental results for 8 QAM and 16 QAM modulation formats, respectively.Similar to the QPSK format, a standard 200 ppm SFO seriously degrades the system performance.However, clean constellation points were recovered successfully with the proposed DA-SFOC algorithm.Due to the typically high sensitivity of the higher order QAM symbols to the SFO impairment, the maximum tolerable SFO drops to 400 ppm and 300 ppm for 8-QAM and 16-QAM, respectively, which matches the simulation results well.Nevertheless, the DA-SFOC method is able to completely compensate for the standard 200 ppm SFO with a negligible penalty.

Conclusions
We have presented a decision aided sampling frequency offset compensation method for RGI CO-OFDM systems and studied its performance for various conditions.Both numerical and experimental results show that the proposed method is robust against standard SFO in optical transmission systems.The benefits of our work are fivefold: (1) unlike previous estimation and compensation based approaches, the method does not suffer from estimation errors.(2) A compatible mechanism to reduce the SFO induced CP overhead is proposed.(3) Because of the decision directed mode, the proposed algorithm can be used in time varying SFO situations automatically.(4) The proposed algorithm performs OFDM symbol recovery on a symbol by symbol basis, which does not require buffers and high speed power consuming electronics.(5) DA-SFOC does not require any additional pilots, and can be used to directly compensate standard SFO or can be used to compensate any residual SFO in other SFO estimation and compensation based OFDM systems.

Fig. 1 .
Fig. 1.Sample dislocations due to SFO. Green dots: correct sample positions.Black arrows: actual sample positions of OFDM symbol 1 because of SFO.Blue arrows: actual sample positions of OFDM symbol 2 because of SFO.DFT: discrete-Fourier-transform