# Frequency estimation for optical coherent M-QAM system without removing modulated data phase

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<ul><li><p>nt</p><p>W</p><p>ive</p><p>y q</p><p>by a</p><p>da</p><p>4, t</p><p>16-</p><p>igital signal processing (DSP)irment compensation of thever, bed froment DSdth tolpproacfrequenO) laseon ofent. T</p><p>symbols is introduced in this M-QAM scheme. In addition, it is to</p><p>to counteract the noise. Simulations of 10-Gbaud 16-QAM and</p><p>d Df</p><p>fd,n yd,n2pDfnTs2pm</p><p>Contents lists available at</p><p>els</p><p>Optics Comm</p><p>Optics Communications 285 (2012) 36923696/fd,nS and the residual y0d,n.E-mail address: yusong@bupt.edu.cn (S. Yu).256-QAM coherent systems demonstrate that the range of the where m is an integer to ensure fd,n to fall into the range ofp,p. For simplicity, laser phase noise and ASE noise areneglected. It is worth to mention that in Eq. (2) the randomvariable of fd,n has been decomposed into its average value</p><p>0030-4018/$ - see front matter & 2012 Elsevier B.V. All rights reserved.</p><p>http://dx.doi.org/10.1016/j.optcom.2012.05.027</p><p>n Corresponding author.derive that specic constellation points should be rotated by p=4, /fd,nSy0d,n, 2extended to M-ary quadrature amplitude modulation (M-QAM)format, which is more likely to be employed in the next genera-tion optical coherent communications system. Different from ourformer M-PSK scheme [8], partition according to the amplitude of</p><p>l,n</p><p>(Df l), Ts is the symbol duration, Nn represents ASE noise, andenotes the frequency offset.</p><p>The argument of the signal is of the formestimator is necessary to ensure reliable performance of PE.We have presented a frequency estimation algorithm for</p><p>M-ary phase-shift keying (M-PSK) optical coherent system [8].Compared with previous algorithm, the modulated data phase isnot required to be removed. In this paper, the algorithm is</p><p>The nth received M-QAM symbol can be presented as</p><p>Sn An expjyd,nyl,n2pDfnTsNn, 1</p><p>where An, yd,n represent modulated amplitude and phase ofM-QAM format, y is the phase noise induced by laser linewidthIn optical coherent detection, dshows its competence for the impalong-haul transmission [13]. Moreoof phase recovery can be transferresoftware domain [1]. Among differtions (PEs), considering laser linewitation complexity, feed-forward aHowever, this approach needs thetransmit and the local oscillator (LMeanwhile, the frequency resoluticannot satisfy the above requiremcause of DSP, the stagehardware domain to</p><p>P-based phase estima-erance and implemen-h [47] is preferred.cy offset between ther to be small enough.a typical tunable laserherefore, a frequency</p><p>2. Principle</p><p>For M-QAM coherent system, the modulated data phase israndomly chosen from the constellation states. Thus, the corre-sponding phase can be treated as a discrete noise. On the otherhand, the frequency offset is comparatively stable and its result-ing argument change would be periodical. Therefore, after Fouriertransform, the frequency offset can be extracted directly [8].1. Introduction frequency offset estimation can cover [1.2 GHz, 1.2 GHz],while the maximum estimation error is below 1 MHz.Frequency estimation for optical coheremodulated data phase</p><p>Song Yu n, Yinwen Cao, Haijun Leng, Guohua Wu,</p><p>State Key Laboratory of Information Photonics and Optical Communications, Beijing Un</p><p>a r t i c l e i n f o</p><p>Article history:</p><p>Received 17 January 2012</p><p>Received in revised form</p><p>15 May 2012</p><p>Accepted 16 May 2012Available online 31 May 2012</p><p>Keywords:</p><p>Coherent detection</p><p>Fast Fourier Transform (FFT)</p><p>M-ary quadrature amplitude modulation</p><p>(M-QAM)</p><p>a b s t r a c t</p><p>For optical coherent M-ar</p><p>can be extracted directly</p><p>removing the modulated</p><p>constellation points by p=Numerical simulations of</p><p>algorithm.</p><p>journal homepage: www.M-QAM system without removing</p><p>anyi Gu</p><p>rsity of Posts and Telecommunications, Beijing 100876, PR China</p><p>uadrature amplitude modulation (M-QAM) system, the frequency offset</p><p>pplying Fast Fourier Transform (FFT) to the signals argument, without</p><p>ta phase. By categorizing the constellation points and rotating some</p><p>his algorithm is robust to extract the frequency offset against the noise.</p><p>QAM and 256-QAM coherent systems are presented to demonstrate this</p><p>& 2012 Elsevier B.V. All rights reserved.</p><p>SciVerse ScienceDirect</p><p>evier.com/locate/optcom</p><p>unications</p></li><li><p>pDfn</p><p>S. Yu et al. / Optics Communications 285 (2012) 36923696 3693From Eq. (2), the average term is governed by</p><p>Fig. 1. The division of constellation for 16-QAM when 2</p><p>Table 1The derivation of /mS for range I.</p><p>i m Possibility /mS</p><p>4k k 13=16 i=43=16k1 3/16</p><p>4k1 k 9=16 i=43=16k1 7/16</p><p>4k2 k 5=16 i=43=16k1 11/16</p><p>4k3 k 1=16 i=43=16k1 15/16/fd,nS/yd,nS/2pDfnTsS/2pmS 2pDfnTs2p/mS: 3Without loss of generality, 16-QAM is demonstrated to illus-</p><p>trate the algorithm. In order to ensure that m is only dependenton yd,n, 2pDfnTs is divided into three ranges, range I:p=4 ip=2,p=2a ip=2; range II: p=2a ip=2,p=2a ip=2;range III: p=2a ip=2;3p=4 ip=2, where a arctan1=3 and iis an integer which is denoted as 4kr r 0;1,2;3.</p><p>As for range I, all four cases are discussed in Fig. 1 and Table 1.Fig. 1 shows that the constellation is divided into two classes(open symbols and lled symbols). If the modulated data belongsto open symbols, m k1. If not, mk. The explanation andanalysis will be given in detail later.</p><p>As for range I, if r1, which is corresponding to Fig. 1(a), lledsymbols exhibit argument less than 3p=4 and 2pDfnTsAp=42kp,p=2a2kp. Therefore, for lled symbols, mk isrequired to keep fd,nA p,p and for the rest symbols (opensymbols), one additional 2p is required. The requirement fordifferent m for different modulate data accounts for the two setsof symbols in Fig. 1. The possibility of open symbols is 3=16, whilethe possibility of lled symbols is 13=16. Therefore, /mSi=43=16. Considering Eq. (3) and Table 1 together, if 2pDfnTsis in range I, Eq. (3) can be rewritten as</p><p>/fd,nS 2pDfnTs3p=8ip=2: 4Similarly, for range II and range III, we have</p><p>/fd,nS 2pDfnTsp=2ip=2; 5or</p><p>/fd,nS 2pDfnTs5p=8ip=2: 6In order to express Eqs. (4)(6) in a more general form, Gx isintroduced</p><p>Gx 1, xry1,0, y1oxry2,1, x4y2:</p><p>8>: 7</p><p>Therefore</p><p>/fd,nS 2pDfnTspG2pDfnTs=8y3ip=2, 8</p><p>where y3 p=2.The trajectory of /fd,nS for 16-QAM (black solid line) is different</p><p>from QPSK (black dashed line), as shown in Fig. 2. The argumentchange induced by frequency offset (argexpj2pDfnTs) is alsoplotted (gray solid line). It can be seen that the period of /fd,nSfor 16-QAM is the same as that of QPSK, a quarter ofargexpj2pDfnTs. Therefore, after applying FFT to /fd,nS, the rstharmonic frequency represents the absolute value of 4Df , as shownin Fig. 3.</p><p>However, compared with the spectrum of /fd,nS for QPSK and16-QAM in Fig. 3(a) and (b), the amplitude of the rst harmonicfor 16-QAM is much lower, making it difcult to be extracted.</p><p>In order to increase the amplitude of rst harmonic, rstly, wepartition different rings into two classes, CI and CII. In CI, the ringsconsist of constellation points that the shorter distance from axis Iand II is less than that from axis X and Y in Fig. 4(a), while CII</p><p>includes the rest rings. Specically, CI is the middle ring and CII</p><p>includes the inner and outer rings. We rotate CI by y, as shown inFig. 4(a). Considering the symmetry of the constellation, rotationof 0,p=2 is sufcient. The derivation of /mS is similar, and its</p><p>Ts is in range I. (a)(d) denote r 1;2,3;0, respectively.expression is the same as Eq. (8). The only differences are y1, y2and b, which is shown in Table 2.</p><p>To facilitate the expression of the rst harmonic F1, we usecontinuous variable t to substitute nTs, yielding</p><p>F1 1</p><p>T</p><p>Z t0 Tt0</p><p>/fd,nSejo1t dt, 9</p><p>where T 1=4Df ando1 2p 4Df . Here, we assume Df40. Theintegral range is p=4;3p=4 and Eq. (9) can be further simplied</p><p>F1 j</p><p>16ej4y1 ej4y22: 10</p><p>The amplitude of F1 is</p><p>9F19 18cos2y1y2fcos2y1y22 cos2y1y2g1</p><p>p: 11</p><p>It is worth to mention that all the constellation points best lineup when the CI points are rotated by y p=4. And this can befurther conrmed by Figs. 3 and 4(b). It can be mathematically</p></li><li><p>S. Yu et al. / Optics Communications 285 (2012) 369236963694</p><p>3/4calculated from Eq. (10) that the increase in 9F19 is 0.777, which isconsistent with Fig. 3.</p><p>Fig. 5(a) is presented to illustrate the modied frequencyestimator. Firstly, partition the rings including CI by the amplitude;</p><p>T/4 T/2</p><p>Pha</p><p>se (r</p><p>ad)</p><p>/4</p><p>0</p><p>/4</p><p>/2</p><p>/2</p><p>3/4</p><p>period of </p><p>Fig. 2. /fd,nS and argexpj2pDfnTs, T 1=Df : The pe</p><p>1 3 5 70</p><p>0.05</p><p>0.10</p><p>0.15</p><p>0.20</p><p>0.25</p><p>f/(4f)</p><p>|FFT</p><p>|</p><p>QPSK</p><p>1 30</p><p>0.05</p><p>0.10</p><p>0.15</p><p>0.20</p><p>0.25</p><p>f</p><p>|FFT</p><p>|</p><p>16QAM w</p><p>Fig. 3. Spectrum of /fd,nS. (a) /fd,nS for QPSK; (b) /fd,nS for 16-QA</p><p>0.09</p><p>0.10</p><p>0.11</p><p>0.12</p><p>0.13</p><p>0.14</p><p>0.15</p><p>0.16</p><p>|F1|</p><p>Y</p><p>X</p><p>III</p><p>Fig. 4. (a) Rotation of CI by y. Black symbols: CI before rotation. Gray s</p><p>then, rotate these rings by p=4; next, obtain the argument ofall the symbols; after that, use FFT to get the correspondingfrequency of the rst harmonic frequency; nally, divide it by 4 toget 9Df 9.</p><p>3T/4 Tt</p><p>Arg[exp(j2fnTs)]</p><p>riod of /fd,nS is T/4 for both 16-QAM and QPSK.</p><p>5 7/(4f)</p><p>/o Rotation</p><p>1 3 5 70</p><p>0.05</p><p>0.10</p><p>0.15</p><p>0.20</p><p>0.25</p><p>f/(4f)</p><p>|FFT</p><p>|</p><p>16QAM w. Rotation</p><p>M before p=4 rotation; (c) /fd,nS for 16-QAM after p=4 rotation.</p><p>0 /8 /4 3/8 /2</p><p>0.09</p><p>0.16</p><p>ymbols: CI after rotation. Open symbols: CII. (b) The curve of 9F19.</p></li><li><p>Another frequency estimator is necessary to decide the sign ofthe frequency offset. The integrated structure of the algorithm isillustrated in Fig. 5(b). Here, four steps are required to get Df :</p><p>1. Use frequency estimator to get absolute value of frequencyoffset, denoted as 9Df 19.</p><p>2. Assume Df 140, then, S0n Sn expj2p9Df 19nTs.3. Repeat step 1 on S0n once again and get 9Df 29.4. Compare 9Df 29 with 9Df 19. If 9Df 2949Df 19, it means Df 1o0,</p><p>then, S00n S0n expj4p9Df 19nTs; and if not, it means Df 140.Then, output S00n S0n directly.</p><p>For frequency estimation of M-PSK, rings partition and p=4rotation are omitted. Meanwhile, the divisor 4 is replaced by M.</p><p>3. Simulation results</p><p>A 10-Gbaud 16-QAM and 256-QAM transmission system withsingle polarization is simulated to demonstrate the proposedalgorithm. Here, 215 symbols are processed, laser linewidth is100 kHz, with only ASE noise included.</p><p>Fig. 6 shows the effect of p=4 rotation in 16-QAM and 256-QAM. Here, the frequency offset is 0.5 GHz. Without p=4 rotation,there is a chance that the rst harmonic corresponding tofrequency offset is buried in the noise (the left part of Fig. 6).</p><p>Fig. 7 shows absolute estimation errors after frequency esti-</p><p>10-Gbaud 16-QAM and 256-QAM coherent systems demonstratethat the estimator could accurately estimate the frequency offset</p><p>Table 2</p><p>y1, y2 and b in different ranges of y.</p><p>y y1 y2 b</p><p>0,a p=2ay ip=2 p=2ay ip=2 p=2a,p=4a p=2ay ip=2 p=2ay ip=2 5p=8p=4a,p=4a ay ip=2 p=2ay ip=2 p=2p=4a,p=2a ya ip=2 ya ip=2 3p=8p=2a,p=2 ya ip=2 ya i=2p p=2</p><p>Fig. 5. Block diagram of the modied algorithm. (a) Frequency estimator employ-ing partition and rotation; (b) the integrated structure of the algorithm.</p><p>0.04</p><p>0.06</p><p>|</p><p>0.12</p><p>0 1 2 3 4 50</p><p>0.02</p><p>0.04</p><p>0.06FFT w/o rotation</p><p>Frequency (GHz)</p><p>|FFT</p><p>|</p><p>Frequency (GHz)256 QAM</p><p>16 QAM</p><p>S. Yu et al. / Optics Communications 285 (2012) 36923696 36950 1 2 3 4 50</p><p>Frequency (GHz)0.02|FF</p><p>TFig. 6. FFT on the argument of the signal with and without pFrequency (GHz)</p><p>0 1 2 3 4 50</p><p>0.04</p><p>0.08</p><p>|FFT</p><p>|</p><p>2GHzup to 1.2 GHz. Meanwhile, the estimation error remains below0.2 MHz, signicantly satisfying the requirement of the subse-quent phase estimation.</p><p>0 1 2 3 4 50</p><p>0.04</p><p>0.08</p><p>0.12FFT w. rotation</p><p>|FFT</p><p>|</p><p>2GHzmation. The range of frequency offset is [1.2 GHz, 1.2 GHz]. Itcan be seen that the mean estimation error is about 0.1 MHz, andthat the maximum is still below 1 MHz.</p><p>4. Conclusion</p><p>A frequency offset estimator for coherent M-QAM system isproposed in this paper based on selective rotation and the fastFourier transform. The proposed scheme need not remove theimpact of the modulated data phase, which can be modeled as adiscrete noise. The rotation of p=4 for specic constellation pointsrenders this scheme more robust against noise. Simulations of=4 rotation. Frequency offset is set to be 0.5 GHz.</p></li><li><p>Acknowledgments</p><p>This work was supported by Key Project of Chinese NationalPrograms for Fundamental Research and Development No.2012CB315605), National Science Foundation (61072054,61008049 and 60932004), Program for New Century ExcellentTalents in University (NCET-10-0243), and Fundamental Research</p><p>[3] H. Kato, Optics Communications 8 ((4) August) (1973) 378.[4] R. Noe, Journal of Lightwave Technology 23 ((2) February) (2005) 802.[5] E. Ip, J.M. Kahn, Journal of Lightwave Technology 25 ((9) September) (2007)</p><p>2675.[6] T. Pfau, S. Hoffmann, R. Noe, Journal of Lightwave Technology 27 ((8) April)</p><p>(2009) 989.[7] I. Fatadin, D. Ives, S.J. Savory, Photonics Technology Letters 22 ((9) May) (2010)</p><p>631.[8] Y. Cao, S. Yu, J. Shen, W. Gu, Y. Ji, Photonics Technology Letters 22 ((10) May)</p><p>(2010) 691.</p><p>1.2 1.0 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1.0 1.20</p><p>0.2</p><p>0.4</p><p>0.6</p><p>0.8</p><p>Frequency offset (GHz)</p><p>Est</p><p>imat</p><p>ion </p><p>erro</p><p>r (M</p><p>Hz)</p><p>Maximum (16QAM)</p><p>Maximum (256QAM)</p><p>Mean (16QAM)</p><p>Mean (256QAM)</p><p>Fig. 7. Absolute values of frequency estimation error under [1.2 GHz,1.2 GHz] (1000 simulations for each frequency offset).</p><p>S. Yu et al. / Optics Communications 285 (2012) 369236963696[1] G. Li, Advances in Optics and Photonics 1 (February) (2009) 279.[2] Y. Gao, F. Zhang, L. Dou, Z. Chen, A. Xu, Optics Communications 282 ((5) 1</p><p>March) (2009) 992.Funds for the Central Universities.</p><p>References</p><p>Frequency estimation for optical coherent M-QAM system without removing modulated data phaseIntroductionPrincipleSimulation resultsConclusionAcknowledgmentsReferences</p></li></ul>