## Abstract

We present a performance evaluation of a non-conventional approach to implement phase noise tolerant optical systems with multilevel modulation formats. The performance of normalized Viterbi-Viterbi carrier phase estimation (V-V CPE) is investigated in detail for circular m-level quadrature amplitude modulation (C-mQAM) signals. The intrinsic property of C-mQAM constellation points with a uniform phase separation allows a straightforward employment of V-V CPE without the need to adapt constellation. Compared with conventional feed-forward CPE for square QAM signals, the simulated results show an enhanced tolerance of linewidth symbol duration product (Δν*T _{s}*) at a low sensitivity penalty by using feed-forward CPE structure with C-mQAM. This scheme can be easily upgraded to higher order modulations without inducing considerable complexity.

© 2014 Optical Society of America

## 1. Introduction

The fast growing bandwidth demand has been driving recent research efforts on high-capacity and high-speed optical communication systems [1]. To increase capacity, the use of optical multi-level quadrature amplitude modulation (mQAM) – rather than the conventional binary modulation schemes [2,3] – has emerged as a promising solution. By using both phase and amplitude of optical fields, mQAM enhances the system spectral efficiency at no cost of optoelectronic bandwidth. The key technology which enables the demodulation of advanced modulation formats at high data rates is coherent detection with powerful digital signal processing (DSP) [4,5] instead of optical phase locking [6]. An important constraint in coherent systems is the phase noise induced by the signal laser and the free running local oscillator (LO) laser, which limits the laser linewidth tolerance and a cost-effective implementation. Nevertheless, phase noise can be compensated using digital carrier phase estimation (CPE) [7,8]. Several CPE algorithms have been proposed for conventional square QAM signals with both decision-directed feedback [9–12] and blind feed-forward structures [13–21]. In practice, it is desired to implement the digital CPE in a blind feed-forward manner for hardware efficiency and better phase noise tolerance [8,15]. For this reason, Viterbi-Viterbi (V-V) algorithm is widely adopted for CPE of phase shift keying (PSK) signals with uniform phase distribution [22]. However, adapting the V-V algorithm for CPE of square 16QAM signals is more complex due to intrinsic characteristics of square constellations i.e. non-uniform phase distribution. Different approaches have been proposed to address this problem, such as QPSK partitioning [13,14] and blind phase search (BPS) [15]. Due to the complexity of such techniques, which greatly increases with the modulation level, recent efforts have been focused on trying to reduce its complexity and improving accuracy using two stage [16–19] and/or pilot-aided carrier recovery [20,21]. Although some of these methods achieve a linewidth tolerance which allows implementation with some available distributed feedback (DFB) lasers, there is still a significant sensitivity penalty and a considerably reduced linewidth tolerance at higher modulation levels.

On the other hand, circular constellations, as opposed to square QAM, provide a more flexible constellation design which can improve performance and enhance phase noise tolerance [23,24]. The uniform phase distribution in circular mQAM (C-mQAM) enables a direct implementation of V-V CPE in an efficient, simple, feed-forward receiver without additional stages or complex hardware implementation. Additionally, due to the geometry of C-mQAM, this CPE scheme can be easily extended for high order modulation levels. Available multi-format transmitter structures can generate arbitrary modulations by only adapting the electrical driving signals of the optical modulators [25]. In this sense, C-mQAM transmission only increases the complexity in the electrical domain by adjusting the driving signals generated by the digital-to-analog converter (DAC), without introducing additional complexity in the optical domain. However, by using transmitter configurations with reduced optical complexity, like serial intensity and phase modulator (IM/PM), C-mQAM would require considerably less signal levels in the DAC than the equivalent mQAM format (4-/8-ary for C-16QAM and 3-/12-ary for 16QAM, 8-/16-ary for C-64QAM and 9-/52-ary for 64QAM) [25].

The power of V-V CPE combined with circular constellations inherent characteristics is an area of opportunity which has not been addressed previously. This unique approach would enable a cost-effective implementation of multi-level coherent systems by relaxing the strict laser linewidth requirements. In this article, we study the phase noise tolerance for C-mQAM transmissions along with normalized V-V CPE scheme, which includes an amplitude normalization stage on the received signal. Sensitivity penalties are used to estimate the phase noise tolerance, which are defined as the additional optical signal to noise ratio (OSNR) required to achieve symbol error rate (SER) of 10^{−3} compared to ideal (Δν*T _{s}* = 0) square 16QAM performance. For C-16QAM, a combined linewidth symbol duration product (Δν

*T*) of 1x10

_{s}^{−4}is tolerable at a minimum sensitivity penalty (~0.37 dB). Normalized V-V CPE was also tested for C-64QAM achieving a Δν

*T*of 3.6x10

_{s}^{−5}for a ~1 dB penalty and proving its simple extension for high order modulation formats. These results show that C-mQAM systems implemented with V-V CPE are a viable and potential alternative for phase noise tolerant high-speed optical transmissions.

## 2. Methodology

#### 2.1 Construction of circular 16QAM signal

Circular QAM constellations can be constructed with arbitrary phase distribution, number of amplitude circles, and number of symbols per circle. These parameters can be optimized and designed to increase robustness against phase noise or overall system performance [23,24]. The C-16QAM constellation considered in this work has 16 (4/4/4/4) symbols distributed in 4 amplitude circles (as opposed to conventional 16QAM which has 4/8/4 symbols distributed in 3 amplitude circles). Within each circle there is a constant phase separation of π/2 between symbols. In this way, C-16QAM has 16 symbols distributed along 8 possible phase positions and 4 amplitude levels. In this study, we construct a C-16QAM constellation in a way considering the influence of both additive and phase noise. The radii of the amplitude circles are defined so that the distance between neighboring symbols is close to the minimum distance between symbols in the inner circle. Following this rule, the radii for the 4 amplitude circles are chosen to be 1, $\sqrt{\text{2+}\sqrt{\text{3}}}$, $\text{1+}\sqrt{\text{2}}$, $\sqrt{\text{2+}\sqrt{\text{3}}}\text{+}\sqrt{\text{2}}$, respectively, as illustrated in Fig. 1(a), resulting a minimum distance of $\sqrt{2}$. High order circular constellations can be constructed in a similar way, e.g. C-64QAM with 64 symbols distributed along 8 amplitude levels and 16 phase positions.

To set as a reference, we firstly compare the performance of square 16QAM and C-16QAM at 28 GBaud in a back-to-back (B2B) implementation by simulations with VPItransmissionMaker^{TM} [26], where no or very low phase noise are applied to the signal in the form of setting the combined 3-dB linewidth of the transmitter and local oscillator (LO) lasers. It is noted that in this work the phase noise is modeled as Wiener process [26]. A total number of 2^{16} symbols (2^{18} bits) are transmitted for performance evaluation. Figure 1(b) shows the simulational comparison results of both cases without any phase recovery scheme. Under ideal (no phase noise) conditions, C-16QAM shows a slight penalty (< 0.3 dB) at SER = 10^{−3} compared to 16QAM. However, in the presence of low phase noise with laser 3-dB linewidth of 10 kHz (Δν*T _{s} =* 3.6 × 10

^{−7}), C-16QAM outperforms 16QAM due to its inherent phase noise tolerant property.

Since for C-16QAM the number of neighboring symbols for a given symbol is not equal or less than the number of bits per symbol, Gray mapping scheme is not possible. For this reason, a different mapping scheme is used and SER measurements are considered as they provide a general performance assessment of the constellation [23]. As indicated in Fig. 1(a), for C-16QAM the first symbol is assigned to the point in first circle with the lowest phase equal to 0. Subsequent mapping is assigned by increasing symbol number counterclockwise, from the lowest amplitude circle to the largest.

#### 2.2 Principle of normalized V-V carrier phase recovery

The V-V algorithm, which has been described and studied in detail [27], raises a received complex signal to the *M*-th power, taking advantage of the geometric properties of a constellation in order to remove modulation. A received symbol can be represented as a complex signal${x}_{k}={r}_{k}{e}^{j{\phi}_{k}}$ by its amplitude and phase components. The phase offset for symbol ${x}_{k}$is estimated by averaging a total block size${N}_{block}=2N+1$considering the preceding and following *N* symbols of ${x}_{k}$. After raising the signal to the *M*-th power, summation, argument calculation and phase unwrapping are performed, and an average phase offset for the block is estimated and applied to ${x}_{k}$. It is noted that we perform amplitude normalization prior to V-V CPE, which aims to reduce the ambiguity in phase estimation due to additive noise and different amplitude levels in the QAM signal. Thus, the overall estimator can be defined by [13]:

For the described C-16QAM constellation, the optimum *M* value is 8, since there are 8 different phase positions in this constellation. Implementation for C-64QAM is possible using exactly the same procedure and only changing *M* to 16 to consider additional phase positions.

## 3. Simulation results and discussion

A C-16QAM coherent system with B2B implementation transmitting 2^{16} symbols at 28 GBaud rate was simulated with VPItransmissionMaker^{TM}. Normalized V-V CPE was implemented using MATLAB / VPI cosimulation [28]. SER was measured using Monte-Carlo approach with direct error counting.

The V-V block size range for a successful phase recovery is limited by the signal OSNR and laser linewidth. Figure 2 shows an overview of relationship between block size, signal linewidth and OSNR. A maximum linewidth tolerance was determined if the constellation was recovered for an OSNR penalty lower than 1 dB when compared to ideal (Δν*T _{s}* = 0) performance at a target SER of 10

^{−3}. It is observed that increasing the block size is necessary at low OSNR to cancel the effects of additive noise in phase estimation (i.e. the minimum block size is determined by the OSNR). On the other hand, using a too large block size leads to wrong phase estimation and a decrease of linewidth tolerance (i.e. the maximum block size is eventually determined by the linewidth). For all OSNR cases the optimum block size is near the minimum block size. As expected, the maximum linewidth tolerance increases with signal OSNR. At high OSNR of 32 dB, the linewidth tolerance is even close to 1x10

^{−3}and can achieve an error-free transmission.

Based on the block size analysis done to consider an optimum block size, SER performance of C-16QAM with normalized V-V CPE was analyzed for different values of Δν*T _{s},* as shown in Fig. 3. OSNR penalties are obtained by comparing the performance of C-16QAM to the ideal (Δν

*T*= 0) curve of square 16QAM at a target SER of 10

_{s}^{−3}. Most of the sensitivity penalty (~0.28 dB) is related to the use of the circular constellation which can be seen by comparing the ideal curves. Despite a decrease in linewidth tolerance at low OSNR, the penalties are almost negligible for Δν

*T*< 1.4x10

_{s}^{−4}when compared to the performance of ideal C-16QAM. Furthermore, a linewidth of Δν

*T*= 1.8x10

_{s}^{−4}is still tolerable at a penalty lower than 1 dB, which slightly outperforms the widely adopted BPS scheme with max tolerance of Δν

*T*of 1.4x10

_{s}^{−4}for square 16-QAM [15]. This can be seen more clearly in Fig. 4(a) which summarizes the linewidth tolerance for the simulated system. Examples of C-16QAM constellation before and after CPE are also shown in Fig. 4(b) to visualize its performance.

C-64QAM transmissions were also tested considering an optimum block size and the same simulation parameters previously described. The linewidth tolerance for C-64QAM with V-V CPE and the penalties compared to square 64QAM performance were estimated at a SER of 10^{−3}. As with C-16QAM, most of the penalty (~0.8 dB) comes from use of the circular constellation which is illustrated in Fig. 5. A linewidth of Δν*T _{s}* = 3.6x10

^{−5}, which corresponds to 1 MHz 3-dB linewidth in a 28 GBaud system, is tolerable at a penalty close to 1 dB. The same normalized V-V CPE is implemented by only changing the

*M*-th power to 16 due to phase positions of constellation. These results confirm the straightforward adaptation when upgrading to higher modulation orders in a C-QAM system with this phase recovery scheme.

## 4. Conclusions

In this paper we investigated the performance of a non-conventional approach of combining circular QAM constellations with a normalized V-V CPE scheme for high capacity coherent transmissions. The geometrical characteristics of the circular constellation are exploited to implement the described method in a feed-forward blind receiver with low complexity. Results show that, at a target SER of 10^{−3}, a linewidth symbol duration product (Δν*T _{s}*) of 1x10

^{−4}is tolerable at a minimum sensitivity penalty (~0.37 dB). For a ~1 dB penalty, Δν

*T*of 1.8x10

_{s}^{−4}and 3.6x10

^{−5}are tolerable for C-16QAM and C-64QAM, respectively. The achieved linewidth tolerance enables implementation with available cost-effective DFB lasers with the same baud rates. The low sensitivity penalty and simple extension to high order modulation formats represents an enhanced performance over CPE proposed for square mQAM systems. Results show that C-mQAM transmissions with normalized V-V CPE is a viable alternative to be applied in high-capacity, low complexity, phase noise tolerant optical systems. Further investigation on this scheme should include the consideration of long-distant fiber transmission induced impairments.

## Acknowledgments

Support from FP7-PEOPLE-2012-IAPP (project GRIFFON, No. 324391) is acknowledged.

## References and links

**1. **M. Seimetz, “High spectral efficiency phase and quadrature amplitude modulation for optical fiber transmission - configurations, trends, and reach,” in Proceedings of ECOC 2009, paper 8.4.3 (2009).

**2. **I. Garrett and G. Jacobsen, “Theory for optical heterodyne narrow-deviation FSK receivers with delay demodulation,” J. Lightwave Technol. **6**(9), 1415–1423 (1988). [CrossRef]

**3. **G. Jacobsen, “Performance of DPSK and CPFSK systems with significant post-detection filtering,” J. Lightwave Technol. **11**(10), 1622–1631 (1993). [CrossRef]

**4. **S. J. Savory, “Digital Coherent Optical Receivers: Algorithms and Subsystems,” IEEE J. Sel. Top. Quantum Electron. **16**(5), 1164–1179 (2010). [CrossRef]

**5. **T. Xu, G. Jacobsen, S. Popov, J. Li, K. Wang, and A. T. Friberg, “Normalized LMS digital filter for chromatic dispersion equalization in 112-Gbit/s PDM-QPSK coherent optical transmission system,” Opt. Commun. **283**(6), 963–967 (2010). [CrossRef]

**6. **F. Mogensen, G. Jacobsen, and H. Olesen, “Light intensity pulsations in an injection locked semiconductor laser,” Opt. Quantum Electron. **16**(2), 183–186 (1984). [CrossRef]

**7. **T. Xu, G. Jacobsen, S. Popov, J. Li, A. T. Friberg, and Y. Zhang, “Analytical estimation of phase noise influence in coherent transmission system with digital dispersion equalization,” Opt. Express **19**(8), 7756–7768 (2011). [CrossRef] [PubMed]

**8. **T. Pfau, “Carrier Recovery Algorithms and Real-time DSP Implementation for Coherent Receivers,” in *Proceedings of OFC 2014* (OSA, 2014), paper W4K.1. [CrossRef]

**9. **G. Jacobsen, T. Xu, S. Popov, J. Li, A. T. Friberg, and Y. Zhang, “Receiver implemented RF pilot tone phase noise mitigation in coherent optical nPSK and nQAM systems,” Opt. Express **19**(15), 14487–14494 (2011). [CrossRef] [PubMed]

**10. **N. K. Jablon, “Joint blind equalization, carrier recovery and timing recovery for high-order QAM signal constellations,” IEEE Trans. Signal Process. **40**(6), 1383–1398 (1992). [CrossRef]

**11. **P. Winzer, A. H. Gnauck, C. R. Doerr, M. Magarini, and L. L. Buhl, “Spectrally Efficient Long-Haul Optical Networking Using 112-Gb/s Polarization-Multiplexed 16-QAM,” J. Lightwave Technol. **28**(4), 547–556 (2010). [CrossRef]

**12. **I. Fatadin, D. Ives, and S. J. Savory, “Blind Equalization and Carrier Phase Recovery in a 16-QAM Optical Coherent System,” J. Lightwave Technol. **27**(15), 3042–3049 (2009). [CrossRef]

**13. **M. Seimetz, “Laser linewidth limitations for optical systems with high order modulation employing feed forward digital carrier phase estimation,” in *Proceedings of OFC/NFOEC 2008* (OSA, 2008), paper OTuM2. [CrossRef]

**14. **I. Fatadin, D. Ives, and S. J. Savory, “Laser Linewidth Tolerance for 16-QAM Coherent Optical Systems Using QPSK Partitioning,” IEEE Photon. Technol. Lett. **22**(9), 631–633 (2010). [CrossRef]

**15. **T. Pfau, S. Hoffmann, and R. Noé, “Hardware-efficient coherent digital receiver concept with feedforward carrier recovery for M-QAM constellations,” J. Lightwave Technol. **27**(8), 989–999 (2009). [CrossRef]

**16. **X. Zhou, “An Improved Feed-Forward Carrier Recovery Algorithm for Coherent Receivers with M -QAM Modulation Format,” IEEE Photon. Technol. Lett. **22**(14), 1051–1053 (2010). [CrossRef]

**17. **J. Li, L. Li, Z. Tao, T. Hoshida, and J. C. Rasmussen, “Laser-Linewidth-Tolerant Feed-Forward Carrier Phase Estimator with Reduced Complexity for QAM,” J. Lightwave Technol. **29**(16), 2358–2364 (2011). [CrossRef]

**18. **K. P. Zhong, J. H. Ke, Y. Gao, and J. C. Cartledge, “Linewidth-Tolerant and Low-Complexity Two-Stage Carrier Phase Estimation Based on Modified QPSK Partitioning for Dual-Polarization 16-QAM Systems,” J. Lightwave Technol. **31**(1), 50–57 (2013). [CrossRef]

**19. **C. Spatharakis, N. Argyris, S. Dris, and H. Avramopoulos, “Frequency offset estimation and carrier phase recovery for high-order QAM constellations using the Viterbi-Viterbi monomial estimator,” in Proceedings of IEEE Conference on Communication Systems, Networks & Digital Signal Processing (IEEE, 2014), pp. 781–786. [CrossRef]

**20. **M. Magarini, L. Barletta, A. Spalvieri, F. Vacondio, T. Pfau, M. Pepe, M. Bertolini, and G. Gavioli, “Pilot-Symbols-Aided Carrier-Phase Recovery for 100-G PM-QPSK Digital Coherent Receivers,” IEEE Photon. Technol. Lett. **24**(9), 739–741 (2012). [CrossRef]

**21. **F. Zhang, Y. Li, J. Wu, W. Li, X. Hong, and J. Lin, “Improved Pilot-Aided Optical Carrier Phase Recovery for Coherent M –QAM,” IEEE Photon. Technol. Lett. **24**(18), 1577–1580 (2012). [CrossRef]

**22. **D. Ly-Gagnon, S. Tsukamoto, K. Katoh, and K. Kikuchi, “Coherent detection of optical quadrature phase-shift keying signals with carrier phase estimation,” J. Lightwave Technol. **24**(1), 12–21 (2006). [CrossRef]

**23. **T. Pfau, L. Xiang, and S. Chandrasekhar, “Optimization of 16-ary Quadrature Amplitude Modulation constellations for phase noise impaired channels,” in Proceedings of ECOC2011, Tu.3.A.6. [CrossRef]

**24. **S. Tharranetharan, M. Saranraj, S. Sathyaram, and V. R. Herath, “A performance comparison of nonlinear phase noise tolerant constellation diagrams,” in Proceedings of IEEE Conference on Industrial and Information Systems (IEEE, 2011), pp. 439–442. [CrossRef]

**25. **M. Seimetz, “Multi-format transmitters for coherent optical M-PSK and M-QAM transmission,” in Proceedings of IEEE Conference on Transparent Optical Networks (IEEE, 2005), pp. 225–229. [CrossRef]

**26. **VPItransmissionMaker Optical Systems 9.1, VPIphotonics GmbH, Berlin, Germany, www.vpiphotonics.com/.

**27. **A. J. Viterbi and A. M. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Trans. Inf. Theory **29**(4), 543–551 (1983). [CrossRef]

**28. **MATLAB 8.3 (R2014a), The MathWorks Inc., Natick, Massachusetts, USA, www.mathworks.se/.