## Abstract

The nonlinear transmission performance of quasi-Nyquist wavelength-division multiplexing (qN-WDM) and reduced guard interval orthogonal frequency-division multiplexing (RGI-OFDM) using polarization-division multiplexing quadrature phase-shift-keying (PDM-QPSK) and quadrature amplitude modulation (PDM-QAM-8 and PDM-QAM-16) with high information spectral densities have been compared for the first time, both by simulations and analytically. The results show that both systems are able to reach similar maximum transmission distances of approximately 6700km, 2600km and 1100km over standard single-mode fibre for the spectral efficiencies of 3.43 bits/s/Hz, 5.25 bits/s/Hz and 7 bits/s/Hz respectively.

© 2012 Optical Society of America

## 1. Introduction

To meet an expected tenfold increase in global Internet Protocol (IP) traffic over the next decade, it is envisaged that Terabit/s channel capacities with high spectral efficiencies of multiple bits/s/Hz will be required in order to maximize link throughput in core networks [1].

When scaling data channels to Tb/s rates, for a single channel, the limits of digital signal processing (DSP) are exceeded even with parallelization and the resulting baud rate in such systems is beyond the current limits of electronic circuits [2]. Due to this limitation two approaches, Nyquist wavelength-division multiplexing (N-WDM) and orthogonal frequency division multiplexing (OFDM), can both be used to combine multiple lower rate channels with high spectral efficiency to reach Tb/s per channel transmission capacity.

The first approach makes use of channels which operate at high baud-rates (e.g. >25 Gbaud) with Nyquist filtering to reduce the channel spacing to the symbol rate per carrier without incurring significant penalties from inter-channel crosstalk or intersymbol interference (ISI) [3]. Experimental demonstrations of the concept were shown in [4], in which transmission distances of greater than 8000 km were achieved for PDM-QPSK over pure silica core fiber. Quasi-Nyquist wavelength-division multiplexing (qN-WDM) transmission can be realized by relaxing the Nyquist condition and using a filter that approximates the ideal case with a channel spacing slightly larger than the symbol rate [5].

The alternative format, OFDM, has a very well defined narrow optical spectrum resulting from overlapping orthogonal subcarriers spaced at the modulation rate allowing signalling at or close to the Nyquist rate without the use of sharp cutoff filters. The subcarriers can be generated electrically with a reduced guard interval (RGI-OFDM) or optically with no guard interval (NGI-OFDM). Experimental verification of RGI-OFDM [6] and NGI-OFDM [7] using Ultra-Large-Area-Fibre, achieving transmission distances over 2000km (PDM-QAM-16) and 7200km (PDM-QPSK) respectively, shows that both are candidates for high spectral efficiency transmission.

In general, the achievable spectral efficiency is limited by optical fibre nonlinearities as the transmission distance reach increases [8–10]. However, little work has been done to directly compare the two formats in terms of their nonlinear performance over long distances and with channel spacings approaching the Nyquist limit. The only direct comparison between OFDM and N-WDM for long haul transmission performance, under the conditions of such densely spaced WDM channels is published in [3]. In this study, OFDM signalling with multi-Gbaud symbol rate is considered, requiring sampling rates of at least 4 samples per symbol. In our work, we consider the more practical implementation of optical OFDM signals with low baud rate subcarriers electrically multiplexed using an inverse fast Fourier transform / fast Fourier transform (IFFT/FFT) pair, which results in a near rectangular channel spectrum while allowing oversampling rates of 2 or less.

Recently, two closed-form expressions describing the nonlinear transmission performance of densely spaced coherent optical systems have been independently derived Nyquist-WDM formats [11] and OFDM [12]. However, to date, they have not been directly compared in terms of the assumptions used and the performance they predict, to yield the definitive answer on the optimum format for long-haul, high spectral efficiency transmission.

In this paper we describe for the first time a study directly comparing the long-haul transmission performance of PDM-QPSK, PDM-QAM-8 PDM-QAM-16 using qN-WDM and RGI-OFDM. The analysis is applied to standard-single mode fibre as the most widely deployed fibre type. We found that both systems have a similar nonlinear transmission performance and reach.

## 2. Analytical Expressions

As described above, two closed-form analytical expressions have been derived for WDM systems at the Nyquist limit based on the idea that fiber nonlinearities in such systems are to a good approximation represented as four wave mixing (FWM).

The expression for nonlinear noise power density *I*_{NL}, experimentally verified in [13], is given for N-WDM in [11] as:

*N*is the number of spans,

_{s}*γ*is the fiber non linear parameter,

*β*

_{2}is the group velocity dispersion parameter,

*L*

_{eff}is the nonlinear effective length,

*B*is the optical bandwidth, and

*I*

_{tx}is the optical signal power density. It is formally defined as

*I*

_{tx}=

*P*

_{tx}/Δ

*f*with

*P*

_{tx}being the optical power per channel and Δ

*f*the N-WDM channel spacing.

A similar expression has been derived by Shieh *et al* [12] for OFDM:

*h*being the noise enhancement factor given as

_{e}*α*being the fiber attenuation,

*ζ*representing the residual dispersion ratio after propagation and

*L*the length of the fibre span.

Despite being derived for two different signal formats, the analytical expressions are quite similar, and differ in two main aspects only. The first main difference between them is that in *I*_{NL,2} a multi-span phase array effect is accounted for. This effect describes the build-up of nonlinearities due to interference between them over several spans. The second main difference addresses dispersion compensation. The expression for *I*_{NL,1} is only applicable to the case of systems with no in-line dispersion compensation whereas the Shieh approach can be used for an arbitrary in-line dispersion compensation ratio.

The nonlinear noise power density *I*_{NL,i} of Eq. (1) or Eq. (2) can be expressed in a more concise form with the definition of nonlinear characteristic power density *I*_{o,i} as follows:

Fibre nonlinearities, modelled as FWM, induce random fluctuations in the channel of interest which is exponentially related to the intensity of the neighboring WDM channels. The signal power in presence of the nonlinear interference can be expressed as [8]

*i*is either 1 or 2, depending on whether Eq. (1) or Eq. (2) is to be used.

*I*

_{ASE}is the ASE noise power density per polarization defined as: where

*G*is the gain per amplifier,

*F*the noise figure and

*h*·

*ν*is the photon energy defined as the product of Planck’s constant

*h*and the frequency of the propagating light,

*ν*.

The signal-to-noise ratio (SNR) of the transmission system can be expressed as:

Using the SNR definition of Eq. (10), an analytical reach curve for a qN-WDM or OFDM transmission system can be determined numerically by searching for the maximum *N _{s}* achievable before the optical signal to noise ratio (OSNR) that corresponds to the target bit error rate BER is exceeded. The SNR of Eq. (10) is the symbol SNR evaluated at the target bit error rate (BER) for a particular modulation format and has to account for the implementation penalty of the transmitter and receiver.

## 3. Simulation setup

The system diagrams for qN-WDM and RGI-OFDM used in this work are shown in fig. (1). At the transmitter a 2^{15} deBruijn sequence is generated, duplicated, de-correlated and then mapped to either QPSK, QAM-8 or QAM-16. All modulation formats were differentially encoded to prevent cycle slips. The RGI-OFDM signal consists of 128 sub-carriers all of which carry data to maximize the spectral efficiency and a cyclic prefix of 4% for QPSK, 2% for QAM-8 and QAM-16 was added respectively, to accommodate the ISI induced by transmitter bandwidth limitations and fiber polarization mode dispersion (PMD). OFDM symbols suffer from high peak to average power ratios (PAPR), which is not the case with qN-WDM, and to reduce the PAPR the signals were clipped prior to quantization. The clipping ratio for each modulation format is given in Table (1) along with the digital-to-analog converter (DAC) resolution used in both systems. The DAC resolution for each modulation format was chosen to limit the back-to-back sensitivity implementation penalty to within 1dB of that achievable for a single channel. In practice this limit was set by the RGI-OFDM system which was more sensitive to the quantization and clipping noise than the qN-WDM system. The in-phase and quadrature components of the qN-WDM signal are electrically shaped using a root raised cosine (RRC) finite impulse response filter (FIR), with a roll-off factor of equal to the RGI-OFDM cyclic prefix overhead for each modulation format, which generates an approximately rectangular spectrum. The electrical bandwidth in the transmitter and receiver was modeled using a 5th-order Bessel filter with an optimized bandwidth of 0.7*R _{s}* and both DACs and ADCs were assumed to operate at 56GS/s. A pre-emphasis filter was used to compensate for the electrical filter roll-off on the edge sub-carriers for RGI-OFDM. The electrical signals for both systems drive an IQ-Mach Zehnder modulator (MZM) operating in the linear region which is modulating a continuous wave (CW) laser at 1550nm with a linewidth of 100kHz. The overall bit rate per channel for PDM-QPSK was 112Gb/s, 168Gb/s for PDM-QAM-8 and 224Gb/s for PDM-QAM-16 respectively, which includes 12% overhead for forward error correction and Ethernet. The channel spacing for both qN-WDM and RGI-OFDM PDM-QPSK was 28.875GHz giving a spectral efficiency of 3.43 bits/s/Hz. For PDM-QAM-8 and PDM-QAM-16 a 28.4375GHz channel spacing was used giving a spectral efficiency of 5.25 bits/s/Hz and 7 bits/s/Hz respectively. The difference in channel spacing between QPSK and QAM-8 and QAM-16 results from the differing lengths of the cyclic prefixes, needed to accommodate ISI due to fibre PMD which increases with the transmission distance.

Signal propagation over single-mode fiber spans is modeled with the split-step Fourier method, at a resolution of 32 SpS, including the effect of chromatic dispersion, dispersion slope, first-order polarization mode dispersion and Kerr effect. The erbium-doped fiber amplifiers (EDFA) noise figure was 4.5dB and all noise was added in line to model the interaction between ASE noise and fibre nonlinearity. Table (2) summarizes the fibre parameters used throughout the simulations. The state of polarization (SOP) for all channels was the same when they were multiplexed together at the transmitter and we assumed that randomizing the SOP of each channel, in relations to the others, would not have any noticeable effect.

At the receiver the incoming signals are mixed with an optical local oscillator with a linewidth of 100kHz and we assumed a zero frequency offset between the Tx and Rx lasers. Each polarization is then detected separately by a phase diversity receiver using a 90° optical hybrid and balanced photodiodes. The optical front ends for both qN-WDM and RGI-OFDM receivers were assumed to be ideal. The I and Q components of each detected polarization were resampled at 2 SpS and then quantized using the same resolution as the transmitter DAC. In the qN-WDM receiver a matched RRC-FIR filter is applied prior to CD compensation. A frequency domain equalizer using the overlap and add method was used to fully compensate the fiber dispersion for both transmission systems. For OFDM the cyclic prefix was removed prior to the demodulation of the sub-carriers using an FFT. Dynamic-channel equalization, in the qN-WDM receiver, for PMD and any residual dispersion was performed using four FIR filters (7 taps long) in a butterfly configuration with the tap weights adaptively updated using the least-mean-square constant-modulus-algorithm (LMS-CMA) for QPSK and a radially directed equalizer for QAM-8 and QAM-16 [14]. In the RGI-OFDM system training symbols were used to estimate the channel and a Zero Forcing Equalizer used to compensate for PMD and any narrowband filtering distortions to the OFDM subcarriers. We assume that after the initial training period, channel estimation, for OFDM, can then be updated on a symbol by symbol basis or by insertion of periodic training sequences, with loss in spectral efficiency of < 0.5%. Carrier phase estimation (CPE) was performed for both systems using the Viterbi and Viterbi phase estimation algorithm for QPSK and QAM-8 with a QPSK partitioning scheme for QAM-16 [15]. The number of taps used in the Viterbi and Viterbi phase estimation window was 24 for both schemes. These algorithms were chosen to maximize the spectral efficiency of the OFDM system. After CPE the symbols were decoded and the bit error rate (BER) was measured by counting the number of errors. The number of errors counted was greater than 100 for a target BER of 10^{−3}.

## 4. Simulation results on maximum transmission reach for qN-WDM and RGI-OFDM

To investigate the long distance transmission performance of both formats a 9 channel WDM transmission system was simulated and the maximum reach, for a BER of 10^{−3} evaluated for different channel launch powers, swept from −8dBm to 3dBm, in increments of 1dB. All the results obtained were plotted for the central channel. The back-to-back sensitivity for both systems using 9 channels was measured and is plotted in fig. (2). It can be seen from fig. (3) that the optimum launch power to achieve the maximum transmission distance was −2dBm for both systems and all modulation formats. As the minimum WDM channel spacing for error free transmission with the RGI-OFDM system was higher than that achievable with qN-WDM due to the inclusion of the cyclic prefix, to ensure a fair comparison we used the minimum RGI-OFDM channel spacing for both system simulations.

The maximum transmission distance versus channel launch power for both qN-WDM and RGI-OFDM, and each modulation format is plotted in fig. (3). Results obtained with both closed-form expressions discussed in section (2) were also plotted. Figure (4) shows the maximum transmission distance achieved at the optimum launch power of −2dBm, versus the spectral efficiency. Very good agreement between the simulation results and the analytical curves can be clearly seen. Note that the expressions are for the Nyquist condition for spectral efficiencies of 4, 6 and 8 bits/s/Hz. The maximum transmission distances for qN-WDM and RGI-OFDM are the same for PDM-QPSK and PDM-QAM-8 while qN-WDM performs slightly better than RGI-OFDM in terms of the maximum reach achieved for PDM-QAM-16. The shorter reach for QAM-8 and QAM-16, at low launch powers, than that predicated by the analytical curves can be attributed to some noise enhancement from the equalization of the signals at a lower received OSNR.

The difference between the analytical curves can be explained by the noise enhancement factor *h _{e}* in Eq. (2). which takes into account that the noise due to fibre nonlinearities is not independent of the nonlinearities in previous spans because the nonlinearities constructively build up while the signal is propagating over several spans. We can clearly see from both figs. (3) and (4) that the analytical expression for OFDM slightly underestimates the maximum transmission reach at a given spectral efficiency.

## 5. Conclusion

A study to directly compare the two alternative approaches to achieving very high spectral efficiency WDM transmission, quasi-Nyquist-WDM pulse shaping and Reduced Guard Interval OFDM for PDM-QPSK, PDM-QAM-8 and PDM-QAM-16 is described for the first time. The nonlinear transmission performance has been compared at near baud rate channel spacing, both numerically and analytically. The results show that both systems achieve similar maximum transmission distance of approximately 6700km, 2600km and 1100km over standard single-mode fibre for the same spectral efficiency of 3.43 bits/s/Hz, 5.25 bits/s/Hz and 7 bits/s/Hz respectively.

It is known that OFDM and single carrier systems, such as qN-WDM, have the same nonlinear performance, both in presence and in absence of neighboring WDM channels over non-dispersion managed links, under the assumption of large channel separations [16,17]. We have shown that for a similar back-to-back receiver sensitivity for both formats this is also the case approaching the Nyquist limit.

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