1. Introduction

Recently, coherent optical orthogonal frequency division multiplexing (CO-OFDM) [1] has trigged world-wide research interests. It combines two powerful technologies, coherent optical detection and OFDM, to combat the fiber dispersion which has to be overcome for today’s high-speed optical fiber transmission systems, e.g., 10 Gb/s and beyond [2–5]. CO-OFDM is a multi-carrier transmission technique and has apparent difference to conventional single-carrier techniques. For instance, the latter may not use digital-to-analog/analog-to-digital converter (DAC/ADC), whereas CO-OFDM heavily relies on DAC at the transmitter and ADC at the receiver, because the OFDM modulation and demodulation are based on discrete-Fourier-transform (DFT) in digital signal processing (DSP). CO-OFDM also uses DSP for other critical functions, such as DFT window synchronization, frame synchronization, carrier frequency offset and channel estimation. These functions are important for the transmission performance and have been widely studied [2–5]. In contrast to single-carrier techniques, there is no explicit clock recovery circuit in CO-OFDM systems and therefore, the sample frequency synchronization can be considered as the clock recovery in conventional signal-carrier systems. Without the synchronization, the sample frequency offset (SFO) between the transmitter and receiver can destroy the orthogonality among the OFDM subcarriers [6,7]. Hence, the SFO must be estimated and compensated, especially for field optical OFDM transmissions where the transmitters and receivers are separated physically [8].

So far, there are only a few reports on this issue. One way is to do the re-sampling in the time domain [3], which requires interpolation of the sample points. Another way is to transmit a dedicated clock signal [9,10], which needs more circuits in the transmitter and receiver. On the other hand, the research in wireless transmissions shows that the SFO adds an additional phase shift proportional to the subcarrier index [6,7]. In this paper, we adapt the theory in [6,7] for optical transmissions and present a frequency-domain CO-OFDM transmission model in the presence of the SFO. We re-use the pilot subcarriers of the laser phase noise estimation to estimate and compensate for the SFO. We verify the method in experiment with negligible penalties. Except for some additional DSP computational resources, our method does not require additional hardware and overhead. With the estimated SFO, it is straightforward to feedback it to the receiver oscillator and thus the synchronization of the sample frequencies can be achieved. This synchronization not only improves the performance of CO-OFDM, but also provides a way to distribute the clock within the CO-OFDM transmission network.

2. Channel model in the presence of sample frequency offset

For simplicity, we mainly follow the terminologies in [11,12]. OFDM transmissions are usually based on a block structure. One OFDM block can be illustrated in the two dimensional time/frequency structure, which includes N_f OFDM symbols in time and N_SC subcarriers in frequency. The indices of OFDM symbol and subcarrier are i and k, respectively. k takes values from [–N_SC/2 + 1 N_SC/2]. The preamble is added at the beginning to facilitate frame synchronization, DFT window synchronization, and channel estimation.

It is found in [6] that the sample frequency offset between the DAC and ADC causes a phase shift ϕ which is proportional to the subcarrier index k,

where f_s and $\Delta f$represent the ideal sample frequency and the SFO, respectively. This phase shift is also proportional to the OFDM symbol index i. This is because we assume that the DFT window synchronization is performed on a per OFDM block basis. The beginning OFDM symbol has good DFT window synchronization, so the SFO-caused phase shift is very small. However, the phase shift is accumulated and is more significant for the rear OFDM symbols within one OFDM block. In the worst case, the information on one OFDM symbol with a large OFDM symbol index cannot be sampled correctly and may be totally lost [6].

Then we extend the frequency-domain channel model in [12] to include the SFO by inserting the phase shift term,

where $x_{ik}$ and $y_{ik}$are transmitted and received signal respectively, $h_k$is the channel response for the kth subcarrier, ${\Phi }_i$is the laser phase drift of the ith OFDM symbol, and $n_{ik}$ represents the white Gaussian noise. The difference between the laser phase shift and the SFO-caused shift is apparent. The latter depends on the subcarrier index but not the former.

At the receiver side, the laser phase noise is mixed with the SFO-caused phase shift. To separate the two effects, we have two assumptions: (i) the laser phase noise is a white Gaussian frequency noise, so that $E_i\{{\Phi }_i-{\Phi }_{i-1}\}=0$, where $E_i$ stands for the average over i;(ii) the channel response and the SFO are semi-static, i.e., almost no change for tens of OFDM symbols. Then, with these assumptions, we can estimate the SFO as

where $\arg (\cdot )$ stands for the angle. In the above estimation, the white Gaussian frequency noise from the laser has been averaged out following our first assumption and the subcarrier-index-dependent frequency offset estimation is from the SFO. Having estimated the SFO, we may compensate for the SFO and then continue the laser phase noise estimation. However, we can arrange the pilot subcarriers in a symmetrical way, e.g., with the paired-up negative and positive subcarrier indices, so that we can skip the SFO compensation to reduce some DSP computation and use the same method in [12] to estimate the laser phase noise as,

where the SFO phase noise has been averaged out because of the symmetrical arrangement of the pilot subcarrier indices, referring to Eq. (1). Equation (5) also means that the laser phase noise estimation can be performed independently of the SFO effect, if the pilot subcarriers are distributed properly. Note that k in Eq. (5) takes values from a set of pilot subcarriers, typically including eight ones.

After the compensation of the channel response, the estimated signal, denoted as${\overline{x}}_{ik}$, is,

where ${\overline{h}}_k^{*}$ is the conjugate of the estimated channel response. Finally, the constellation can be constructed and decided.

Note that the SFO causes not only the additional phase shift in Eq. (1), but also the amplitude reduction and the inter-carrier interference (ICI) due to the loss of orthogonality [6,7]. When the SFO becomes large, the latter will affect the system performance, even with the compensation method in Eq. (6). However, in a real application, we can use the result of Eq. (4) as a monitor and feedback the monitoring result to the receiver oscillator, so the sample frequency synchronization between the transmitter and receiver is achieved. Consequently, this synchronization can reduce the SFO-caused ICI. Due to the experimental limitation, we only discuss the estimation and compensation of the SFO in the ensuing experiments.

3. Experimental setup

The experimental setup is similar to that in [11,13] with some modified parameters. The time-domain waveform of the OFDM signal is firstly generated with a Matlab program, including mapping 2¹⁵-1 PRBS into corresponding 36 subcarriers with 16-QAM encoding within multiple OFDM symbols, which are subsequently converted into the time domain using IDFT, and inserted with cyclic prefix (CP). The number of OFDM symbols in one OFDM block is 50. The length of IDFT is 128 and the CP is 1/17 of one OFDM symbol. Note that our choice of 36 subcarriers is to accommodate our heterodyne detection receiver with a relatively small bandwidth. Eight pilot-subcarriers are added for the SFO and laser phase noise estimation and therefore, the SFO compensation does not require additional pilot subcarriers. The waveform is loaded into an Arbitrary Waveform Generator (AWG) as DAC with a sample frequency of 10 Gs/s. The AWG outputs the complex-valued waveform using two ports. A dual MZM modulator configured as an IQ modulator up-converts the RF OFDM signal to the optical domain. The so-generated OFDM waveform carries 10.6-Gb/s data.

The generated optical OFDM signal passes through an EDFA, 66-km standard single-mode fiber (SSMF), an optical attenuator, another EDFA, a band-pass filter, and a polarization controller (PC) to align the polarization states of the optical signals, and is received with a local laser (LO) for heterodyne coherent optical detection with a pair of balanced photodiodes. The wavelengths of the transmitter and receiver LO lasers both are approximately 1551 nm and the laser linewidths 100 kHz. The RF signal is then input into a Tektronix Time Domain-sampling Scope (TDS) as ADC and acquired at 25 Gs/s. The digitized RF traces are uploaded into a computer for off-line signal processing [2,11,13]. The SFO estimation and compensation method by Eqs. (3) and (6) is inserted in the signal processing. Between the AWG and TDS, there is a clock connection, which can be connected or disconnected according to the experimental requirement.

4. Experimental results and discussions

The first part experiment shows the effect of the SFO. The AWG and TDS are synchronized to a 10-MHz reference clock and the sample frequency of the AWG is intentionally changed to emulate the SFO. For example, we can set the sample frequency 10.001 GHz to emulate 1-MHz SFO. Note that our choice of the AWG instead of the TDS is due to the configuration limitation of the latter. We use electrical SNR at the receiver to estimate the performance and the calculation method has been reported in [11]. Without using the SFO compensation, Fig. 1 illustrates the SNR degradation across the subcarriers in an electrical back-to-back setup, i.e., the AWG outputs are connected to the TDS directly. The SFO values are relative to 10 Gs/s. When there is the SFO, the SNR of the 0-index subcarrier has minimum change. However, the SNR degradation is more significant for the larger index. This can be understood from Eq. (1), where the SFO-caused phase shift depends on the subcarrier index. Without the SFO compensation, this SFO phase shift can degrade the performance significantly. The positive and negative SFO with the same magnitude show similar SNR degradation across the subcarriers. Figure 1 shows ±1-MHz SFO can totally destroy the OFDM signal, but ±0.01-MHz SFO causes little degradation, which means that CO-OFDM can tolerate the SFO within certain values.

 

Fig. 1 SNR degradation across the subcarriers, caused by the SFO in an electrical back-to-back setup. The dash lines are for the positive SFO and the solid lines for the negative SFO. The inset constellations are for two subcarriers with indices of −10 and 0, respectively.

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The next investigation is based on an optical back-to-back experiment, i.e., without the SSMF fiber in Fig. 2 . Figure 3 shows the phase evolution of the 8 pilot subcarriers with −1-MHz SFO. The indices of the 8 pilot sub-carriers are [-20 −19 −18 −17 17 18 19 20], corresponding to the curves in Fig. 3, from the bottom to top in order. They are grouped into two, symmetrically distributed and intentionally chosen with larger indices. This selection of the indices follows the same reasoning for Eq. (5), so the laser phase noise estimation can be performed independently of the SFO effect. The phase evolution curves in Fig. 3 are the summation of two effects: the laser phase noise and the SFO-caused phase shift. The phase evolution curves of different subcarriers diverge over time and this divergence is caused by the SFO, explained in Eq. (1). At the same time, the curves share the same fluctuation over time, which is the laser phase noise. The curves also serve as an additional illustration for the estimation methods in Eqs. (3)–(5). Figure 4 demonstrates the effectiveness of the SFO compensation. After the compensation of −1 MHz SFO, the SNR distribution across all the subcarriers is almost flat. The dip in the central subcarriers is from the optical carrier caused dc-leakage.

 

Fig. 2 Experimental setup.DMZ: dual Mach-Zehnder modulator, EDFA: erbium-doped fiber amplifier, PC: polarization controller.

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Fig. 3 Phase evolution of the 8 pilot subcarriers without the SFO compensation. The SFO is −1 MHz.

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Fig. 4 SNR distribution with or without the SFO compensation in an optical back-to-back setup. The SFO is −1 MHz.

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In the following experiment, we add the 66-km SSMF fiber and run the AWG and the TDS without the clock synchronization by the 10-MHz reference clock. We use the similar method to set the SFO as in the electrical back-to-back measurement. The monitoring values are calculated from the average of 100 OFDM blocks, i.e., 100 times estimations. Table 1 shows the SFO monitoring results. The estimated SFO can track the polarity and the technique works well for ±10 and ±1 MHz SFO. Note that 10-MHz SFO relative to 10 Gs/s can be considered as 1000-ppm jitter, so our technique can track large values of SFO. However, the monitoring errors, in terms of percentage, become large for small SFO less than 0.1 MHz. Therefore, this technique may control the SFO within 0.1 MHz, or 10 ppm. On the other hand, the small SFO will not degrade the system performance as shown in Fig. 1, even the SFO monitoring fails. Table 1 also shows the limited accuracy of the SFO monitoring, which may be improved through the longer time average of more times of SFO estimation. Another possibility is to optimize the algorithm to correct some systematic errors.

Table 1. SFO monitoring results relative to 10 Gs/s.

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Figure 5 shows the BER measurement results for ±1 MHZ SFO after the 66-km transmission. With the SFO compensation, there is no apparent BER penalty when there is ±1-MHZ SFO. Therefore, by using our method, we expect the field systems can operate well, even with certain sample frequency offset between DAC and ADC.

 

Fig. 5 BER results with the SFO compensation after the optical transmission. The inset constellation is with −1-MHz SFO.

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5. Conclusion

Coherent optical OFDM systems employ DAC at the transmitters and ADC at the receivers and the sample frequencies of ADC and DAC need to be synchronized. This paper has introduced a channel model including the SFO. We have illustrated the effect of SFO, which causes a phase shift proportional to the subcarrier index. We have also verified the monitoring and compensation of the SFO and found no apparent penalty even when there is a large SFO in the experimental system. This method is based on DSP and does not require additional hardware and overhead, except for some additional DSP computational resources. The benefits of our work are two folds: (1) the transmitter and the receiver in CO-OFDM can work asynchronously within certain extent; (2) by feeding the monitoring result back to the oscillator, the sample frequency of the transmitter and receiver can be synchronized and therefore, the clock can be distributed in the CO-OFDM network.