We present an efficient channel estimation method for coherent optical OFDM (CO-OFDM) based on intra-symbol frequency-domain averaging (ISFA), and systematically study its robustness against transmission impairments such as optical noise, chromatic dispersion (CD), polarization-mode dispersion (PMD), polarization-dependent loss (PDL), and fiber nonlinearity. Numerical simulations are performed for a 112-Gb/s polarization-division multiplexed (PDM) CO-OFDM signal, and the ISFA-based channel estimation and the subsequent channel compensation are found to be highly robust against these transmission impairments in typical optical transport systems.
©2008 Optical Society of America
Orthogonal frequency-division multiplexing (OFDM) is a widely used modulation/multiplexing technology in wireless and data communications . With recent advances in high-speed CMOS technologies and optical modulation and detection technologies, optical OFDM at 40-Gb/s or even 100-Gb/s information rate becomes feasible [2–9]. Together with digital coherent detection, coherent optical OFDM (CO-OFDM) brings similar benefits such as high spectral efficiency and high receiver sensitivity as coherent single-carrier transmission [10–12]. A key feature of CO-OFDM is its capability to insert training symbols (TS’s) at the transmitter to facilitate channel estimation, which provides crucial information about the transmission channel and enables efficient digital compensation of optical transmission impairments such as chromatic dispersion (CD) and polarization-mode dispersion (PMD). In optical transmission, the accuracy of channel estimation is often limited by the presence of optical noise. To increase the accuracy of channel estimation, a time-domain averaging method that averages the channel matrices estimated by multiple TS’s for each frequency subcarrier was used [5,6,8]. Recently, we proposed the use of an intra-symbol frequency-domain averaging (ISFA) based method  where the averaging is over the estimated channel matrices for multiple adjacent frequency subcarriers in the same TS. This method offers the benefits of overhead reduction and reaction speed improvement. In this paper, we systematically study the robustness of the ISFA-based channel estimation against transmission impairments such as optical noise, CD, PMD, polarization-dependent loss (PDL), and fiber nonlinearity.
This paper is organized as follows. In Section 2, we describe the architecture of a 112-Gb/s polarization-division multiplexed (PDM) CO-OFDM system. Section 3 presents the concept of the ISFA-based channel estimation method and its performance in optical noise limited transmission. Section 4 discusses the impact of CD on the channel estimation. Section 5 shows the robustness of the channel estimation against PMD and the performance of digital PMD compensation. Section 6 discusses the impact of PDL, without and with PMD. Section 7 investigates the impact of fiber nonlinearity. Section 8 concludes this paper.
2. Architecture of a 112-Gb/s PDM CO-OFDM system
Figure 1 shows the schematic of a 112-Gb/s PDM CO-OFDM transmitter and receiver setup . The original 112-Gb/s data were first divided into x- and y-polarization branches, each of which was mapped onto 1280 frequency subcarriers with QPSK modulation, which, together with 16 pilot subcarriers, were transferred to the time domain by an IFFT of size 2048 with a filling ratio of ~63%. A cyclic prefix of length 512 was used to accommodate dispersion of up to ~20,000 ps/nm, resulting in an OFDM symbol size of 2560 time samples. The time-domain samples were then serialized and converted by two 56-GS/s DACs before driving two I/Q modulators. The modulated optical signals were combined by a polarization beam splitter (PBS) for polarization multiplexing. TS’s were inserted periodically into the OFDM symbol sequence after every 20 payload symbols. At the receiver, digital coherent detection with polarization diversity was used to sample the fields of two orthogonal components of the received optical signal. Symbol synchronization was then performed, and TS’s were extracted for channel estimation for each frequency subcarrier. To increase the accuracy of channel estimation in the presence of noise, previous authors [5,6,8] used a time-domain averaging algorithm that averages over multiple TS’s. Here, we used the intra-symbol frequency-domain averaging (ISFA) based method briefly described in  where the averaging was over the estimated channel matrices for multiple adjacent subcarriers in the same TS. ISFA offers the benefits of overhead reduction and reaction speed improvement. To save computational efforts, the channel estimation process can update the channel information at a speed that is much slower than the real-time data speed, but much faster than the speed of channel physical changes, which is usually in the order kilohertz. Other signal processing needed to recover the original data was similar to that described in [5,6,8]. Detailed descriptions of the channel estimation and compensation methods will be given in the following section. Figure 2 shows the allocations of the OFDM data subcarriers and pilot subcarriers. 16 pilot subcarriers were uniformly distributed among the data subcarriers. Table 1 summarizes key OFDM design parameters. A clipping ratio of 6 was chosen so that in DAC and ADC, the samples whose powers were more than 6 times of the mean signal power were clipped to 6 times the mean power.
3. ISFA-based channel estimation method
In OFDM, a large number of subcarriers are usually used so that the frequency-domain transfer function of a given transmission channel for each subcarrier can be regarded as constant or flat. The combined effect of CD, PMD and PDL on an optical OFDM signal can be generally expressed as
where the 2×1 vectors in the equation respectively represent the received and the transmitted OFDM signals for the k-th subcarrier each of which contains two orthogonal polarization components, and the 2×2 matrix is the channel matrix representing linear channel effects. To simplify the channel estimation, a pair of time-multiplexed TS’s across the two polarization branches, t1 and t2, is inserted into the OFDM symbol sequence at the transmitter . We can write t1 and t2, as
where tx and ty are two known symbols, preferably with low peak-to-average-power-ratios (PAPR). Note that the pair of TS’s need to be periodically inserted into the OFDM symbol sequence in order to capture dynamic channel behaviors.
Assuming that the two training symbols experience the same channel effect, the received training symbols are
The channel matrix can then be obtained as
To improve the accuracy of channel estimation in the presence of noise, the ISFA process is applied such that for each modulated subcarrier, its channel matrix is an average of the channel matrices estimated for itself and its multiple adjacent subcarriers according to Eq. (4). Typically, for subcarrier k, the averaging can be performed over subcarrier k and its m left neighbors and/or m right neighbors, or totally up to (2m+1) adjacent subcarriers. The improved channel matrix for subcarrier k’ after the ISFA process can be expressed as
where kmax and kmin are the maximum and minimum modulated subcarrier indexes, respectively. In Eq. (5), the elements of the estimated channel matrix for k outside [kmin, kmax] are not available and thus are set to zero in the averaging process.
Once the improved channel matrices for all the modulated subcarriers are obtained, they can be inverted and applied to the corresponding received subcarrier vectors in the payload symbols to obtain the original subcarrier vectors. This in effect realizes polarization demultiplexing and compensation of linear channel effects such as CD, PMD, and PDL.
Simulations were performed to illustrate the effect of the ISFA process. A 112-Gb/s PDM-OFDM signal was passed through a dispersive transmission link having a PMD with a mean differential group delay (<DGD>) of 100 ps. The original data were 211-1 pseudo random bit sequences. The loss of the transmission link was compensated by optical amplification and the optical signal to noise ratio (OSNR) of the received signal, defined with the common 0.1-nm noise bandwidth, was 15.5 dB. Figure 3 shows the first two channel matrix coefficients as a function of the modulated subcarrier index before and after the ISFA process with m=6. Evidently, the estimated channel coefficients without the ISFA process exhibit high-frequency fluctuations due to the presence of optical noise. With the ISFA, the noise-induced high-frequency fluctuations are removed.
To quantify the benefits of the ISFA process, we first performed simulations in the back-to- back case where the channel is only affected by optical noise. 20 OFDM symbols were simulated in order to be able to count ~100 errors at a bit error ratio (BER) of 10-3, a typical forward error correction (FEC) threshold. Figure 4 shows the back-to-back BER performance of the 112-Gb/s PDM-OFDM signal under different channel estimation (CE) conditions. Ideal CE refers to the case where the channel matrices are obtained in the absence of optical noise. The required OSNR for BER=10-3 is 15.2 dB in the ideal CE case. Direct CE refers to the case with noise but without the ISFA, the OSNR performance at BER=10-3 is ~5 dB worse. With the ISFA over 5 (m=2), 9 (m=4), and 13 (m=6) adjacent subcarriers, the channel estimation penalties with respect to the ideal CE case at BER=10-3 are significantly reduced to 1, 0.4, and 0.2 dB, respectively, showing substantial performance improvement achieved by the ISFA. Further increasing m in the ISFA process brings diminishing improvement in the BER performance and reduces the frequency resolution of the channel estimation, which may put a limit on the magnitude of frequency-dependent channel effects that the ISFA-based channel estimation can tolerate. In a recent experiment , the performance of the ISFA-based CE in linear transmission was found to be similar to that of the time-domain averaging based CE with the same averaging window size. The detailed comparison between these two CE approaches, however, is beyond the scope of this paper. We will investigate the impact of CD and PMD, which are frequency-dependent effects, on the ISFA-based channel estimation in the following two sections.
4. Impact of CD on the ISFA-based channel estimation
In the presence of large CD, there is a large phase variation across the subcarriers, particularly near the edges of the OFDM spectrum, and the ISFA process may cause inaccurate estimation of the channel matrices. As a design rule, it is desired that the CD-induced phase difference between the center subcarrier and the farthest subcarrier in the averaging process of the ISFA to be less than about 1 rad. After some derivations, it is found that the residual CD prior to the ISFA, denoted as DISFA, is desired to be limited such that
where DISFA is in units of ps/nm, ΔfOFDM(GHz) is the optical bandwidth of the OFDM signal in GHz, and ΔfISFA(GHz) is the optical frequency difference between the center subcarrier and the farthest subcarrier in the averaging process of the ISFA in GHz. For example, in our previous described 112-Gb/s PDM-OFDM system, we have ΔfOFDM(GHz) =56*(1296/2048)=35.4, and ΔfISFA(GHz)=56/2048*6=0.164 for m=6 in the ISFA. According to Eq. (6), we need | D | ~6850 ps/nm ISFA <. Figure 5 shows the phases of the OFDM subcarriers, estimated by the ISFA-based channel estimation process using m=6, when the OFDM signal experiences 6800-ps/nm and 21760-ps/nm dispersion. With CD=6800 ps/nm, condition (6) is satisfied, and the estimated phases accurately follow the quardratic curve (dashed line) that is calculated from the theory and shifted by a constant phase for easy comparison. This indicates that accurate channel estimation is obtained at the CD value. However, with CD=21780 ps/nm, condition (6) is severely violated, and the estimated phases for the edges subcarriers deviates from the theoretical curve, indicating inaccurate channel estimation for the edge subcarriers. The reason for the inaccurate channel estimation for the edges subcarriers is two folds. First, with the correct symbol synchronization, the CD-induced phase variations are larger towards the edges of the OFDM spectrum. Second, the averaging windows for the edges subcarriers are no longer symmetric, thereby making the averaging less effective.
To quantify the CD-induced penalty, we performed simulations to obtain the received signal Q factor, derived from the BER, as a function of the CD experienced by the signal prior to the ISFA process with m=6. Figure 6 shows the simulation results. When condition (6) is satisfied, i.e., | D | ~6850 ps/nm ISFA <, the CD-induced penalty is <0.2 dB. Note that 6850-ps/nm dispersion corresponds to about 400-km standard single-mode fiber (SSMF) worth of dispersion. To transmit the signal beyond 400 km in SSMF, one can satisfy condition (6) by choosing a smaller m in the ISFA process, using optical dispersion compensation in the fiber link, or performing a rough EDC prior to the ISFA process.
5. PMD estimation and compensation
PMD is a major transmission impairment for high-speed optical fiber transmission. PMD causes frequency-dependent phase change and polarization rotation. To ensure the accuracy of the ISFA-based channel estimation, the amount of DGD needs to be limited. As a rough design rule, it is desired to limit the DGD-induced phase difference within ΔfISFA to be much less than 2π, e.g., less than π/5, or
where DGD(ps) is the instantaneous DGD in ps. Using ΔfISFA(GHz)=0.164 for m=6, we have DGD(ps)<610 ps.
To quantify the PMD-induced penalty, we performed simulations to obtain the received signal Q factor, derived from the BER, as a function of the instantaneous DGD experienced by the signal prior to the ISFA process with m=6, as shown in Fig. 7. When condition (7) is satisfied, i.e., DGD(ps)<610 ps for the above case, the PMD-induced penalty is <0.2 dB. When condition (7) is violated, the PMD-induced penalty quickly takes off as DGD increases. PMD is a stochastic phenomenon, and the instantaneous DGD follows a Maxwellian distribution. For a given <DGD>, there is a probability of about 10-8 that the instantaneous DGD exceeds 4<DGD>. Thus, to limit the PMD-induced outage to be less than 10-8, it is desired to limit <DGD>, in the above example, to be <150 ps, which is easily achievable in most practical fiber transmission systems. This indicates that the ISFA-based channel estimation is highly robust against fiber PMD. In the following sections, we will use m=6 in the ISFA process.
To further verify the performance of the PMD estimation and the PMD compensation (PMDC) capability of the ISFA-based OFDM system, numerical simulations with many different realizations of fiber link PMD were performed. For each realization, the fiber link was divided into 6,400 birefringent sections each having a random orientation for its principal state of polarization axes (PSP’s) and a DGD of 0.3125 ps, resulting in a mean link DGD of 25 ps. Note that higher-order PMD effects, in addition to the first-order PMD effect, were naturally included in the channel emulation. Figure 8 shows the magnitudes of the channel matrix coefficients of all the modulated OFDM subcarriers, estimated after the ISFA, for 20 different PMD realizations. Note that the spectral bandwidth covered by the OFDM subcarriers is about 35.4 GHz. If only the first-order PMD is considered, then a DGD of 25 ps would cause a 40-GHz periodicity in the frequency dependence of the matrix coefficients. With the inclusion of higher-order PMD effects, the frequency-dependence of the matrix coefficients becomes irregular, as shown in Fig. 8. Notice also that in the absence of PDL, the channel matrix is a unitary matrix with the following properties
As shown in Fig. 8, the above relationships are precisely followed, indicating the robustness of the ISFA-based channel estimation method.
To better visualize the frequency dependences of the channel coefficients, Fig. 9 shows the contour plots of the estimated coefficients |a(k)|2 and |b(k)|2 for 50 different PMD realizations with a <DGD> of 25 ps. The characteristic 40-GHz periodicity in the frequency dependence of the matrix coefficients is apparent. The complementary relationship between |a(k)|2 and |b(k)|2 is also apparent. Figure 10 shows the contour plots of the estimated coefficients |a(k)|2 and |b(k)|2 with a <DGD> of 100 ps. With the 4-fold increase in <DGD> as compared to Fig. 9, the characteristic periodicity in Fig. 10 is about 4 times reduced, as expected.
With the channel matrix coefficients being available after the ISFA-based channel estimation, PMDC can be performed to compensate for the PMD effect, as described earlier. To quantify the performance of the PMDC, we calculate the BER of the recovered data at the OFDM receiver through direct error counting for 50 different fiber PMD realizations. Optical noise was added along the link so that the received OSNR was 16.5 dB. The signal launch power was 4 dBm, for which the single-channel nonlinear penalty is about 1 dB . Figure 11 shows the distribution of the Q factors (derived from the BER) of the received 112-Gb/s OFDM signal with 25-ps and 100-ps <DGD>. For each of the two cases, the mean additional penalty due to PMD is virtually zero, and the standard deviation of the signal Q factor is as small as ~0.2 dB, indicating the effectiveness of the ISFA-based channel estimation process and the subsequent PMDC process.
6. Impact of PDL
PDL is another major transmission impairment in long-haul optically amplified transmission systems. PDL causes one polarization component of a signal to suffer more loss than its orthogonal counterpart. Even when the total power of the signal is maintained, e.g., by optical amplification, the effective OSNR’s of the two signal components are unequal, thereby resulting in an OSNR penalty. Unlike CD and PMD which could be fully compensated for in digital coherent receiver, full compensation of PDL can not be realized from a fundamental standpoint due to the loss of OSNR .
To assess the performance of the CO-OFDM system in the presence of PDL, we performed numerical simulations with 10,000 different realizations of fiber PDL. For each realization, the fiber link was divided into 64 birefringent sections each having a random orientation for its PSP’s and a PDL of 0.5 dB, leading to a mean link PDL, denoted as <PDL>, of 4 ps. Optical noise was added after every 4 sections to emulate an optically amplified transmission link. The received OSNR for the 0-dB and 4-dB <PDL> cases were 15.5 dB and 16.5 dB, respectively, which correspond to baseline Q factors of 9.8 dB and 10.8 dB. The effect of fiber PMD was also taken into consideration. Figure 12 shows the distributions of the received signal Q factors (derived from BER) after transmission over a fiber link with 10,000 different fiber realizations for <PDL>=0 dB and 4 dB. There are three main observations. First, without PDL, PMD is fully compensated with virtually zero penalty. The fluctuations of the signal Q factors are small and are independent of <DGD> for the <DGD> values under investigation. Second, with 4-dB PDL, the mean PDL-induced penalty is about 0.5 dB. At 10-3 probability, the PDL-induced penalty, in the absence of PMD, is ~1.4 dB, which is smaller than that in conventional single-polarization direct-detection systems . Thirdly, the presence of PMD not only does not cause additional penalty as does in conventional systems, but reduces the variance of the PDL-induced Q factor fluctuation and reduces the worst-case PDL penalties. Similar effect was also reported for a single-polarization CO-OFDM system by W. Shieh . Thus the ISFA-based channel estimation and compensation scheme is robust against PDL and the combined effect of PDL and PMD.
7. Impact of fiber nonlinear effects
It is known from early studies that CO-OFDM is highly susceptible to fiber nonlinearity . In this section, we investigate the nonlinear tolerance (NLT) of the 112-Gb/s PDM-OFDM signal in both single-channel and wavelength-division multiplexed (WDM) transmission when the ISFA-based channel estimation is applied.
7.1 Single-channel nonlinear effect
To assess the NLT, a transmission link consisting of 16 optically amplified 80-km fiber spans was used. The fiber span loss was 20 dB, and the fiber nonlinear coefficient was 1.22/W/km. Random rotation of the polarization fields during transmission was taken into consideration. ASE noise was added at each optical amplifier such that the received OSNR was 16.5 dB, which was 1 dB above what was required for BER=10-3 in the back-to-back case. We performed simulations for three cases. In the first case, we assumed that the fiber has zero dispersion and the signal is singly polarized (no PDM) in order to compare the simulation results with the analytical predictions made by Lowery et al. . In the second case, we assumed zero-dispersion fiber and PDM signal to assess the impact of PDM. In the third case, we assumed realistic fiber dispersion and PDM signal.
Figure 13(a) shows the Q factor of the recovered signal, derived from the calculated BER, as a function of the signal power launched into each span (Pin) for the above three cases. A commonly used dimensionless parameter to represent the magnitude of the fiber nonlinearity is the total nonlinear phase shift experienced by a signal, ΦNL, which is defined as ∫z=Lz=0 γ(z)P(z)dz, where γ(z) and P(z) are respectively, the fiber nonlinear coefficient and the signal power at location z, and L is the total length of the transmission link. Figure 13(b) shows the Q factor of the recovered signal as a function of Φ NL for the above three cases. The Q factor was derived both from the BER and from the variance of the symbols in the signal constellation . The Q factors derived from the BER are similar to those derived from the symbol variance, indicating that the single-channel nonlinear distortions in this system are approximately Gaussian distributed . At 3-dB Q penalty, the signal power is -0.5 dBm, corresponding to an effective total nonlinear phase shift of 0.27 rad, which is in good agreement with that predicted by Eqs. (7) and (12) in . With PDM, the NLT in the dispersion-less transmission at 1-dB penalty is improved by ~2 dB. We attribute this partially to the reduced probability of having large signal peak powers, since to have a large signal peak power after PDM, the powers of both polarization components have to be large at the same time location. With the consideration of fiber dispersion, the NLT at 1-dB penalty is further improved by 5 dB. We attribute this to dispersion-induced broadening of high-power peaks in the OFDM signal waveform and consequent removal of persistent high-power peaks. This could also be explained by dispersion-induced phase mismatching in the FWM interactions among subcarriers. Analytical explanation of this effect was recently reported by M. Nazarathy et al. . This effect was also reported in a simulated 26.3-Gb/s single-polarization CO-OFDM system . We point out that for the same fiber link, the dispersion-induced improvement in the NLT of an OFDM signal is larger for higher symbol rates, at which the dispersion effect is stronger .
In practical systems, the OSNR increases proportionally with the increase of signal power, and due to the takeoff of the nonlinear penalty at high signal powers, there is an optimal power at which the signal Q factor is optimized. Figure 14 shows the simulated signal Q factor as a function of signal launch power Pin while the received OSNR increases proportionally with Pin and equals 15.5 dB when Pin=2 dBm. The Q factors were derived both by the BER and by the variance in the recovered symbol constellation. The Q factors derived by the BER are in good agreement with those derived by the variance, indicating that the distribution of the SPM-induced signal distortions are Gaussian-like . The optimal Q factor is obtained when Pin=5 dBm, which corresponds to a total nonlinear phase shift of ~0.95 rad. While the optimal total nonlinear phase shift appears to be much larger than those reported in recent experiments at lower symbol rates [8,9], it is similar to that predicted for single-carrier coherent signal (ΦNL~1 rad) without considering the dispersion effect by Gordon and Mollenauer . This indicates the robustness of the ISFA-based channel estimation in the presence of single-channel fiber nonlinearity.
In most installed long-haul transport systems, in-line optical dispersion management is applied. Usually, optical dispersion compensating fiber (DCF) is inserted in the optical amplifier that follows each transmission fiber span to compensate for the CD of the span. It is thus important to assess the NLT of CO-OFDM in dispersion-managed systems. We performed simulations using the same transmission fiber spans as before, i.e., 16×80-km SSMF spans. For simplicity, the fiber nonlinear effects in the DCF were neglected. This is a reasonable assumption since the signal power launched into the DCF can be much lower than that launched into the transmission fiber and the DCF length is generally many times shorter than the transmission fiber length. Figure 15 shows the Q factor as a function of the signal power for optically dispersion-managed transmission with three different values of residual dispersion per span (RDPS), 400 ps/nm, 100 ps/nm, and 25 ps/nm. The results for dispersion-less (D=0) and dispersion-unmanaged (RDPS=1360 ps/nm) transmissions are also plotted for comparison. Evidently, the NLT decreases as the RDPS decreases. This is reasonable since larger RDPS leads to large overall dispersion excursion that helps to suppress the nonlinear interactions among the OFDM subcarriers as discussed previously. These results are inline with those found in a numerical study of a 26.3-Gb/s single-polarization OFDM system . There is thus a need to improve the NLT of CO-OFDM in dispersion-managed transmission.
7.2 Preliminary study on WDM nonlinear effect
It is important to assess the NLT in WDM environment where inter-channel cross-phase modulation (XPM) degrades signal quality. Since the XPM depends on the data patterns of the WDM channels and the relative timing and polarization alignments among these channels , extensive simulations are needed. Here, we present preliminary results on the NLT of the 112-Gb/s PDM-OFDM signal in a 7-channel 50-GHz spaced WDM transmission over 16×80-km SSMF spans. The OSNR after transmission was assumed to increase with Pin and equal 18.5 dB at Pin=1 dBm. Figure 16 shows the optical spectrum of the WDM signal after the transmission. Figure 17 shows the Q factor (derived from simulated BER) of the center wavelength channel as a function of signal launch power. The optimal signal launch power per wavelength channel is ~0 dBm, at which the nonlinear penalty is ~1 dB. Compared to the single-channel case, the NLT is substantially reduced (by ~4 dB). However, in terms of the total power launched into the fiber link, the WDM case allows ~4.5 dB higher power than the single-channel case for a 1-dB penalty. The simulated NLT compares favorably with recent experimental results obtained without using the ISFA-based channel estimation [8,9]. In future studies, it is desired to further improve the NLT of CO-OFDM in WDM transmission.
We have presented an efficient channel estimation method for CO-OFDM based on ISFA, and systematically investigated its robustness against transmission impairments such as optical noise, CD, PMD, and PDL through numerical simulations. Preliminary study on the impact of fiber nonlinear effects in both single-channel and WDM transmission has also been presented. It is found that the ISFA-based channel estimation and the subsequent channel compensation are highly robust against the above transmission impairments in typical optical transport systems. It is also suggested that future studies to further improve the nonlinear tolerance of CO-OFDM in dispersion-managed transmission and in WDM transmission are desired.
The authors wish to thank G. Charlet, Y. K. Chen, R. Dischler, R. Essiambre, and R. C. Giles for valuable discussions, and A. R. Chraplyvy, R. W. Tkach, and G. Veith for support.
References and links
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