Discrete FRFT-Based Frame and Frequency Synchronization for Coherent Optical Systems

1  Abstract — A joint frame and carrier fr equency synchronization algorithm fo r coherent optical systems, based on the digital computation of the fractio nal Fourier transform (FRFT), is proposed. The algo rithm utilizes t he characteristics of energy c entralization of chirp signals in the FRFT domain, together with the time and phase shift properties of the FRFT. Chirp signals are used to construct a training sequence (TS), and fractional cross-correlation is e mp loyed t o define a detection metric for the TS , from w hich a set of equations can be obtained. Estimates o f both the tim ing offset and carrier frequ ency offset (CFO) a re obtained by solving these equations. T his TS is la ter employed i n a phase-dependent decision-directed le ast-mean square algorithm f or a daptive equalization . Simulation res ults of a 32 -Gbaud coherent polarization division multiplexed Ny quist system show that the proposed scheme has a wide CFO estimation range and accurate s ynchronization performance even in poor optical signal- to -noise ratio condition s. Index Terms — Chirp signals, coherent o ptical communication, fractional correlation, fractional Fourier Transform, fra me synchronization, frequency offset esti ma ti on, optical fiber communication, training sequence. I. I NTRODUCTI ON OHERENT op tical technology has been ac tively investigated in r ecent years a s a promising technique for next-generation high-capacity transport networks . Current state- of - the-art coheren t optical systems utilize digital si gnal processing (DSP) to compensate for various linea r impairments in optical transmission s uch as chromatic dispersion (CD) and polarizatio n-mode disper sion (P MD). In addition, these s ystems supp ort the use of a combinatio n of multi-level m odulation and polarization division multiplex ing (PDM) to increase the number o f transported bits. In digital coherent receivers, a static filter is usually employed for b ulk CD co mpensation, while a set o f ad aptive finite-impulse-respon se (FIR) filters are used in performing polarization de mu ltiplexing as w ell as to compensate for ti me- varying channel impairments such as P MD and the state o f polarization [1 ]. B lind tap adaptation algorithms like the constant-modulus algorithm (CM A) and t he multi-mod ulus algorithm (MMA) are commonly used to update t he tap coefficients of the adapti ve FIR filters. Ho wever, b oth CMA and MMA are disadvantaged by lo ng converge nce time and the singularity problem [ 2]. To avoid these p roblems, a training sequence (TS) -based phase-dependent decisio n- directed least- mean square (DD-LMS) al gorithm has been proposed [3 ]. T he DD - LMS algorithm requ ires accurate frame synchronization to identi fy the TS prior to adaptive equalization. In addition, the car rier frequency o ffset (CFO) has to be estimated and co mpensated for. For the fra me s ynchroniza tion, th e Schmidl and Cox’s algorithm [4] ca n be adap ted for coherent optical single-carrier systems as d emonstrated by Zhou i n [5]. Ho wever, as shown in [6], the Schmidl and Cox ’s al gorithm yields frame synchronization errors under poor optical signal - to -noise ra tio (OSNR) conditions. For the freq uency synchronization, most of the existing methods in t he literature dep end on using either the M -th power oper ation [7] o r a T S [5] to remove the modulated data phase . Notwithstanding, the M -th power operation is disadvanta ged b y large co mputational co mplexity , while the accurac y o f the Zhou’s T S -based algorithm [5] degrades in poo r OSNR conditions. A method which does n ot depend on removing t he modulated data phase h as b een proposed in [8 ]. Ho wever, this met hod is not modulatio n- format transparent, a nd has a small CFO esti mation range. In t his le tter, we propose an algorithm whic h utilizes fractional cross -correlation, together with the time and phase shift proper ties of the fractional Fourier tra nsform ( FRFT) , to carry out j oint frame and frequenc y synchronizatio n . Recentl y, the FRFT has also been prop osed for joint synchronization for coherent o ptical OFDM [9]. Ho wever, t he method in [ 9] , which utilize s only the FRFT time and phase shift properties , yields frame synchronization errors even in the absence of noise. In addition, this method has a CFO estimation range of ±4 GHz, and it needs t he Schmidl an d Cox ’s algorithm to compute the CFO. T he pr oposed scheme i s rob ust to a mplified spontaneous e mission (ASE) noi se, and has a wide CFO estimation range. The proposed technique is de monstrated b y means of simulatio ns in a 32-Gb aud 16-ary quadrat ure amplitude modulation (1 6-QAM) coherent PDM syste m. II. O PERATI ON P RINCIPLE The FRFT is a generalization o f the con ventional Fourier transform throu gh an angle parameter ϕ and an order parameter α [10 ] . For each value o f ϕ , t he  th -order FRFT rotates a time-do main signal counterclock wise by ϕ [1 1] . In general, we ca n relate ϕ and α as follo ws [10]:       󰇛 󰇜 Discrete FRFT - Based Frame and Frequency Synchronization for Coherent Optical Systems Oluyem i Omomukuyo, Shu Z hang, Octav ia Dobre, Ramachandran Venkatesan, and Tel ex M. N. Ngat ched C 2 The  th -order FRFT of a signal  󰇛 󰇜 can be defined as [12 ]:     󰇛 󰇜    󰇛  󰇜   󰇛    󰇜     󰇛󰇜   󰇛    󰇜         󰇣 󰇡             󰇢󰇤  󰇛 󰇜 where   is the F RF T op erator associated with angle ϕ , and   󰇛    󰇜 is the trans form kernel, defined in (3) for values of ϕ that are not multiples of π . T here are several discr ete computational algorit hms for the FRFT , but in the pro posed scheme, we make use o f t he algorithm in [10 ] b ecause of its computational efficienc y  󰇛  󰇜  for an N -length signal . A. Training S equence Design The TS used to perform the j oint synchronization is obtained from t wo d iscrete-ti me linear chirp signals with different chirp r ates. For simplicity, we consider a finite- duration d iscrete-time linear chirp with zero initial phase and a center frequency of 0 Hz , which can be expressed as:  󰇟  󰇠   󰇟  󰇛      󰇜 󰇠        󰇛 󰇜 where  is the c hirp rate,  is the sampling period, and   is the number of discrete samples. The optimum angle ,   , at which the FRFT of the c hirp signal yie lds an impulse is [13]:                  󰇛󰇜 In designing t he TS, two differ ent values o f   are selected , and the co rresponding values of the chirp rate s are obtained from (5). These chirp rates are then used in (4) to construct t he actual chirp signals. Since the constellation p oints o f the c hirp signals lie i n a unit circle, t he chirp signals are “sliced” and converted into 4-Q AM symbols using t he method in [14 ]. At the receiver, the chirp signals are detected by per forming the fractio nal cro ss-correlatio n of t he recei ved T S and the transmitted one. This operatio n yields two im pulses who se peaks would shift by di fferent a mounts dep ending on the values of the frame offset a nd CFO. B. Joint F rame and Frequen cy Synchronization For each polarization, t he received s ymbols are divided i nto  b locks, each of length   . To detect chirp signal 1, for each block, we define a d etection metric   󰇛󰇜 , o btained from the fractional cro ss -correlation [1 1 ] of the block with the original transmitted chirp signal 1 as follo ws:   󰇛󰇜  󰇻    󰇣   󰆓   󰇛 󰇜   󰆓   󰇛󰇜 󰇤󰇻   󰇛󰇜 where   󰆒       , and     is the op timal angle for chirp signal 1,   󰇛󰇜    󰇛      󰇜 ,   repr esents the discrete r eceived time-domain samples,         is the b lock index ,          , 󰇛 󰇜 are the discrete samples corr esponding to t he transm itted c hirp sig nal 1, and * is the complex conjugatio n o peratio n. For each block  , the FRFT sample index,    , where   󰇛󰇜 has its peak value i s:       󰇟   󰇛   󰇛󰇜 󰇜 󰇠  󰇛󰇜 We select the specific b lock   at which the m aximum value of the detection metric is obtained using the follo wing rule:      󰇟   󰇛   󰇛   󰇜 󰇜 󰇠  󰇛󰇜 The peak shift for chirp signal 1 ,   , is then obtained as:               󰇛󰇜 where     is the value of    co rresponding to block   . The peak shift for chirp signal 2,   , is obtained in a similar manner. A time shift  and phase shift  of a signal in the time do main correspond to shifts of    and     in the FR FT do main , respectivel y [12 ]. W e can then constru ct the followin g set of equation s which governs the pea k shifts:                              󰇛  󰇜 The solution of (10) is :                                           󰇛  󰇜 The frame offset estimate,   , and the CFO e stimate,   , are:      󰇛  󰇜       󰇛  󰇜            󰇛  󰇜 where  󰇛  󰇜 r ounds to wards the nearest integer , and   is the s ymbol rate . As shown in (11) an d (13), t he frequency resolution o f the CFO estimation , w ould depend on      ,   , and   . It can also be ded uced from (10) -(13) that the CFO estimatio n range ,   , of the propo sed algorithm is :     󰇩    󰇡        󰇢   󰇡       󰇢      󰇪      (1 4) 3 Fig. 1. Simulation setup. C1: chirp signal 1. C2: chirp signal 2. Tr: Training symbols (in serted every 1000 data symbols). PRBS: pseudo-random b inary sequence. TS: training sequence. RRC: root raised-cosine. DAC: digital- to -analog converter. IQ: in-phase/quadrature phase. PBS: polarization beam splitter. PBC: polarization beam coupler. WDM: wavelength division multiplexing. EDFA : Erb ium-doped fiber amplifier. SSMF: standard single-mode fiber. OBPF: optical band-pass filter. ADC: analog- to -digital converter . CD: chromatic dispersion. LMS: least-mean squa re. CPR: carrier phase recove ry. Inse t (a): Frame structure. I nset (b): Metric for bo th chirp sig nals using (6) for a 100-symbol f rame offset and a 3-GHz CFO. III. S IMULA TION S ETUP AN D R ESULTS To investigate the performance of the pro posed scheme, a model of a 32-Gbaud coherent P DM Nyquist system, whose schematic is depicted in Fig. 1, is built using VPI TransmissionMaker. Five channels are simulated with a channel spacing of 32 GHz, and the per formance is asses sed on the central channel. The DSP at the transmitter and receiver is perfor med in MATLAB. T wo i ndependent pseudo -random binary sequences are generated for the two polarizati on branches. For each p olarization, an identical T S, comprising two chirp signals with different chirp rates, is p laced at the beginning of each frame to be transmitted to achie ve the j oint synchronization. An additio nal 2 4 training symbols are inserted ever y 1000 transm itted data symbols to track t he dynamic channel b ehaviors. T he frame struct ure is sho wn in inset (a) o f Fig. 1. The symbols are upsampled to 2 samples/symbol, and digitally shaped usin g a 73 -tap root raised-cosine (RRC) filter with a roll - of f factor of 0.13. For each channel, the electrical signals fro m eac h polarization br anch are fed to digital- to -a nalog co nverters, and then used to drive two null -biased I/Q modulators. The o ptical source to the I/Q modulators is a continuous wave laser with a linewidth of 1 00 kHz . T he multiplexed P DM optical signal is launched into a transmission link consisting of 10 s pans o f standard single -mode fiber , with 80 km a nd a 16-dB gain erbium-doped fiber amplifier per span. A t the receiver, the central channel is selected using a 0 .4-n m o ptical band -pass filter, a nd coherentl y detecte d with a po larization -diversi ty optical hybrid. A laser with a linewidth of 100 kHz is used as the local o scillator. T he coherently-detected signal is sampled by the analo g- to -digital converters , and then processed by the matched RRC filters. An overlap ped frequency-domain equalizer is used for CD compensatio n. After CD compensation and downsampling to 1 sample/symbol, joint synchronization is carried out using the p roposed algor ithm . The processing o f the algorithm is ca rried out indep endently for each polarization. T he TS is th en used f or polarization demultiplexing using t he phas e-dependent D D-LMS algorithm [3]. T he DD-LMS al gorithm is also used to estimate the carrier phase and fo r residual CFO compensatio n. For all simulation results, unless other wise mentioned, the TS length is 1024, the frame offset i s 100 symbols, the CFO is 3 GHz, and the OSN R is 1 0 dB. In additio n, 1 000 trial run s have bee n per formed for each assessment. It is clear fro m ( 11) and (14 ) that the performance of the propo sed scheme depend s on the selectio n of app ropriate values o f the angle p arameters   and   in the d esign of the TS. For the perfor mance assessment, we have selected       . Inset (b) of Fig. 1 shows that the metric for both ch irp signals is i mpulse-shaped. Fig. 2 sho w s the fra me synchronization performance as a function of   . It is observed that the timing esti mation error is minimum aro und       . Consequentl y, we have carried out the simulations u sing this value of   . Fig. 3 s hows the i mpact of a variation of the T S length on the frame a nd freque ncy synchronization p erformance. It is o bserved that for a T S length of 10 24, no timing estimation erro rs ar e o bserved, an d the CFO estimation error is ~7 MHz. The fram e synchronization perfor mance is more robust than the frequency synchronizatio n performance to further reduction in the TS len gth. Wit h the above simulation p ara meters,   , as obtained using (14), is ~  16 .3 GHz. Fig . 4 sho ws that the proposed algor ithm can comfortably estimate C FOs as high as  5 GHz, with a maximum CFO estimation error of ~11 MHz obtained. In Fig. 5 , the fra me and frequency synchronization performance of the p roposed algorithm is co mpared to the T S- based Schmidl- Cox’s [4] and the Zhou’s [5] algorithms, respectively, in t he presence of varying levels of the OSN R. The proposed algorithm de monstrates s uperior robustness to ASE noise than bo th algorithms. 4 Fig. 2. Frame sy nchronizatio n perfo rmance as a functio n of   . Fig. 3. Frame and frequency synchronizat ion perfor mance as a function of the TS length. Fig. 4. Mean of estimated CF O and mean of CFO estimat ion error a s a function of the a ctual CF O. IV. C ONCLUSION A novel j oint frame and frequ ency synchr onization scheme based on the FRFT h as b een pro posed for co herent op tical PDM systems. T he proposed sche me, which utilizes fractional cross-correlation , has been s hown to be robust to ASE noise, with a wide CFO e stimation ran ge, greater than half the symbol rate. T he FRFT angle p arameter can b e varied in t he design of the TS in the sc heme to increase the accurac y of t he offset estimation. R EFERENCES [1] S. J. Savory , “Digital coherent optical receive rs: algorithms and subsystems,” IEEE J . Sel. Topics Quantum El ectron ., vol. 16, no. 5, pp. 1164-1179, Sep.-Oc t. 2010. [2] K. Kikuchi, “Perfo rmance analyses of polarization demultiplex ing based on constant- modulus algorithm in digital coherent receivers,” Opt. Exp ., vol. 19, no. 10, p p. 9868-9880, May 2011. Fig. 5 . Synchronization performance in the presence of ASE noise. (a) Frame synchronization. (b) Fre quency sy nchronization. [3] Y. Mori, C. Zhang, an d K . 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