The impact of phase to amplitude noise conversion for QPSK, 16-QAM, and 64-QAM coherent optical systems are investigated with electronically-compensated chromatic dispersion (CD). The electronic equalizer is shown to convert the phase noise from the local oscillator (LO) to amplitude noise, limiting the amount of CD that can ideally be compensated digitally. The simulation results demonstrate that the performance of coherent systems can significantly be degraded with digitally compensated CD and LO phase noise. The maximum tolerable LO linewidth is also investigated for the different modulation formats and found to become increasingly stringent for longer transmission distance and higher symbol rate.
© 2010 Optical Society of America
In coherent optical systems employing digital signal processing, the fiber chromatic dispersion (CD) can be compensated in the electrical domain using finite impulse response (FIR) filters without resorting to optical compensation techniques [1, 2]. A transmission system without dispersion compensation fiber is preferable due to its superior performance to nonlinearity [3, 4]. Recently, QSPK coherent experiments have successfully been demonstrated with electronically-compensated chromatic dispersion [5–7]. However, the performance of highspeed coherent systems can significantly be degraded with digital dispersion compensation [8, 9]. This is due to the fact that unlike an optical dispersion compensator, the electronic equalizer following the coherent receiver enhances the impairment from the local oscillator (LO) phase noise. With the LO phase noise converted into amplitude noise, this additional source of impairment can thus be expected to become a limiting factor as the number of taps in the FIR filter increases with large amount of CD and significant laser linewidths in a digital coherent system. The degradation in performance with digital filtering has analytically been presented in  for the QPSK modulation format. Numerical simulations were also performed in  to investigate the effect of electronically-compensated CD for a QPSK coherent system.
With the signal power degraded and inter-symbol interference intensified due to the phase noise induced impairment with digital compensation, the performance of a digital coherent receiver can significantly be affected. In this paper, we investigate the impact of phase to amplitude noise conversion for QPSK, 16-QAM, and 64-QAM coherent systems with digital dispersion compensation. To the authors’ knowledge, the degradation in performance for higher-order QAM with digital filtering has not yet been investigated in the literature. We show that the phase noise from the LO directly influences the amount of CD that can be compensated digitally for the different modulation formats. The maximum tolerable LO linewidth is also investigated assuming a penalty of 1 dB and was found to become increasingly stringent for longer transmission distance and higher symbol rate.
2. Dispersion compensation
Chromatic dispersion in an optical fiber can be modeled as an all-pass filter on the electric field of the lightwave, given by a complex transfer function in the frequency domain 
where λ is the wavelength, c is the speed of light, D is the dispersion coefficient of the fiber, and z is the transmission distance. Since the restored complex amplitude from a digital coherent receiver contains the information on the amplitude and phase of the distorted optical signal, the effect of chromatic dispersion can be compensated after detection with a transversal filter having an inverse transfer function of equation (1) in the time-domain .
A fractionally spaced equalizer can compensate for the chromatic dispersion when the signal is sampled above the Nyquist rate . Such a filter can effectively be realized by using the FIR filter shown in Fig. 1. The tap spacing is equal to the sampling interval, T/2, where T denotes the symbol period. The tap weights to compensate for CD accumulated through a transmission distance, z, are given as 
and the total number of taps, Ntaps, is computed using
where ⌊p⌋ means the largest integer not exceeding p.
3. Simulation results and discussions
Fig. 2 shows the simulation setup for the coherent system under investigation. At the transmitter side, the transmitted signal was bandwidth-limited by a pulse shaping function which was chosen to be the root-raised cosine with the roll-off factor, α = 1. In transmission systems without electronic equalizer to compensate for CD, it is often possible to lump the transmitter and the LO phase noise together as this approach does not influence the overall performance of the system. However, when digital filters are employed to compensate for a large amount of residual CD, the origin of the phase noise in the system was found to be critical on the performance of the coherent system. The signal was thus impaired first by the phase noise from the transmit laser as shown in Fig. 2. The phase noise was modeled as the Wiener process . The signal was then transmitted through a single-mode fiber having CD of 17 ps/nm/km with the response (1) and the amplifier noise, NASE, was also introduced. At the receiver end, an additional source of phase noise from the LO was included to emulate a realistic communication model. The signal was subsequently filtered by another root-raised cosine function to match the pulse shaping at the transmitter side for optimal performance. The signal was then sampled at twice the symbol rate and fed into the electronic filter described in Section II to compensate for CD from the fiber channel. Following the equalization, the signal was finally decimated to one sample per symbol for carrier phase estimation. The BER was calculated over 106 bits.
Several feedforward carrier phase recovery algorithms [11–14] have recently been proposed for the different modulation formats investigated in this paper. To minimise the penalty incurred from the carrier phase recovery, we employed a decision-directed approach  where we fed back the ideal decisions based on the apriori knowledge of the transmitted data, tk, to demodulate the received signals, rk, giving the maximum likelihood estimate of the phase, , as
where N is the block size in the simulation. The penalty above the theoretical sensitivity at a BER of 10−3 of the phase estimation algorithm is shown in Fig. 3 for QPSK, 16-QAM, and 64-QAM, where Δv is the combined linewidths for the transmitter laser and the LO. The optimum block size in (4) was found to be 5 < N < 20 for all the modulation modulations.
The performance of the different modulation formats was investigated next with digital filtering in the presence of laser phase noise from both the transmitter and the LO. The linewidth per laser was fixed to 5 MHz, 1 MHz, and 100 kHz for QPSK, 16-QAM, and 64-QAM, respectively. It should be noted that feedforward carrier phase recovery algorithms have also been investigated at these levels of phase noise in coherent optical systems for the different modulation formats. It is therefore of importance to assess the performance of a digital coherent receiver with these levels of phase noise and electronically-compensated CD. The linewidths of the transmitter laser and the LO were assumed to be the same in the simulation. This would be the case in a coherent optical system where the same laser technology is used at both the transmitter and the receiver.
Fig. 4 shows that the LO phase noise has a significant impact on transmission systems employing digital filters, in particular, for large amount of chromatic dispersion. For all the investigated modulation formats, the performance was significantly degraded at higher transmission speeds. As such, LO phase noise to amplitude noise conversion with digital filtering will be a limiting factor for higher speed coherent systems. No such degradation in performance was observed with digital filtering when the combined linewidths were introduced at the transmitter only. This is due to the fact that any arbitrary amount of CD can ideally be compensated provided sufficient number of taps are used in the equalizer as given by (4). The phase noise that originates from the transmit laser does not degrade the performance of a digitally compensated filter as the phase noise is fed through both the fiber channel and the equalizer to reverse the effect of CD. On the other hand, adding the combined transmitter and LO phase noise at the receiver may provide a pessimistic estimation on the performance of a transmission system as the phase noise from the LO is converted into amplitude noise by the electronic equalizer in contrast to that from the transmit laser. The distribution of phase noise is thus critical in a coherent system employing electronic equalizer and cannot simply be lumped together, for example, in simulation where only digital compensation of dispersion is assumed. Therefore the analysis of phase noise requirements should be performed by considering the large amount of CD that may be present in non-dispersion compensated systems. Table 1 shows the penalty with digital filtering for a transmission distance of 2000 km at 56 Gbaud for the different modulation formats.
Finally, the maximum tolerable LO linewidth to achieve a BER of 10−3 was investigated assuming a penalty of 1 dB for the different modulation formats. The LO and the transmitter laser linewidths were assumed to be the same. Fig. 5 shows that the constraint on the LO linewidth became increasingly stringent for longer transmission distance and higher symbol rate. This in contrast to a system without digital compensation where the required linewidth is generally relaxed at a higher symbol rate [15, 16]. Phase to amplitude noise conversion will thus have a significant impact on coherent systems with electronically-compensated CD imposing even tighter requirement on the LO linewidth for the higher modulation formats such as 16- and 64-QAM.
Coherent optical systems can technically rely on the advances of powerful digital signal processing to completely compensate for CD in the electrical domain. However, unlike the optical dispersion compensator, an electronic equalizer converts the LO phase noise into amplitude noise, limiting the amount of CD that can ideally be compensated digitally. We have demonstrated that for uncompensated links, the constraint on the LO linewidth becomes increasingly stringent with longer transmission reach and higher symbol rate for all the investigated modulation formats. Electronically-compensated CD will impose tighter requirement on the maximum tolerable LO linewidth for higher-order modulation formats.
S. J. Savory gratefully acknowledges financial support through EPSRC project EP/G066159/1
References and links
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