Interplay of Probabilistic Shaping and the Blind Phase Search Algorithm

Accepted for publication: IEEE/OSA J ournal of Lightwav e T echnolo g y Interplay of Probabilistic Shaping and the Blind Phase Searc h Algo rithm Darli A. A. Mell o, Fabio A. Barbosa and Jacklyn D. Reis Abstract —Probabilistic shaping (PS) is a promising techniqu e to approach the Shannon l imit usin g typical constellation geome- tries. Howev er , th e impact of PS on the chain of signal processing algorithms of a coherent receiv er still needs further inv estigation. In this w ork we stud y the interplay of PS and phase recov ery using the blind p h ase search (BPS) algorithm, wh ich is widely used in optical communications systems. W e first in vestigate a supervised p h ase search (SPS) algorithm as a theoretical up per bound on t he BPS performance, assuming perfect decisions. It i s shown that PS influen ces the SPS algorithm, but its impact can be alleviated by moderate noise rejection w i ndow sizes. On the other hand , BPS is affected b y PS ev en for long windows because of correlated erro neous decisions in the phase recov ery scheme. The simulation results also show that t h e capacity-maximizing shaping i s near to th e BPS wo rst-case situation for squ are-QAM constellations, causing p otential implementation penalties. Index T erms —Coherent optical communications, phase r ecov - ery , probabilistic shaping. I . I N T R O D U C T I O N Probabilistic shaping (PS) is a digital transmission techniq ue by which constellation sym b ols are tr ansmitted with different a-prior i probab ilities. I n general, symbols with larger amp li- tudes are tran sm itted with lower prob abilities. PS maxim izes the mutual informa tio n (MI) ac h iev ed by the transmission scheme for a g i ven signal constellation a n d signal to no ise ratio (SNR) and allows , in certain conditions, to approach the Shannon limit. Alth ough PS has been known for decades [2], [3], its applica tio n on pra ctical systems is still in its infancy . Significant implemen tation advances have been recen tly p ro- posed by B ¨ ocherer et al. in [4]. In optical systems, the interest in PS has gained significant momentu m. T o our kn owledge, PS has bee n first add r essed in the context of optical commu nications by Beygi et al. in [ 5], wher e a ra te - adaptive coded modulation scheme with probab ilistic signal shaping has been proposed. The impact of rate-adap tive cod ed mod u lation with PS o n o ptical n etworking has been quan tified by Mello et a l. in [6]. Y ankov et al. have invest igated in [7] an implementatio n of PS for tur bo codes. The comb ination of PS with low-density parity-ch eck codes (LDPC) f or o ptical co mmunication s has been shown by Fehenberger et al. in [8]. Th e first experim e ntal demon stration of PS for o ptical comm unications has been accomp lish e d b y D. A. A. Mello and F . A. Barbosa are with the School of Electrical and Computer Engine ering, Univ ersit y of Campina s (Unicamp), Campinas, Bra zil. J. D. Reis is wit h Idea! E lectronic Systems, C ampinas, Brazil. At Unicamp, this work was supported by F APESP grants 2015/24341- 7 and 2015/24517-8. J. Reis was supported by CNPq grant 311871/2016- 0. W e would like to thank Omar Domingues for the constrained capacity calcu lation s. Part of this work appears in [ 1]. Buchali et al. in [9], for a 6 4-QAM mo dulated sign al. Since then, PS has been applied to different contexts, ran ging fro m transoceanic application s [1 0], [11] to unrep eatered o ptical transmission [12 ]. PS has already been demonstrated in a large set of experiments, but f ully super v ised equalizatio n and phase recovery , with co ntrolled cond itions, are largely u sed. One o f the first works to relate phase recovery and PS in more practical scen arios has been recently presented by Pilori et al. in [13]. Sup e rvised and p artially-super v ised pilo t- aided phase recovery were investigated. Su pervision using 2 % pilot overhead is applied to th e phase unwr apper to mitiga te cycle slips. The p ilot-aided sch eme achieved equiv alent perfo rmance as th e superv ised scheme at linear prop agation regimes, but exhibited some p e n alty in the p resence of no nlinear in terfer- ence. Ho we ver , the performan c e of phase recovery algorithm s was as sessed from a n end- to -end perspective and in particular configur ations. In [1], we have sh own th at PS can affect the p erforman ce of the blind ph ase search (BPS) algo rithm, which is widely used in optical c o mmunicatio ns s ystems. Th e per formance of the algorith m was e valuated by simulations. In this paper, we extend the resu lts of [1], and pr ovide a detailed analysis on the inte r play of PS and BPS. Su p ervised ph ase search (SPS), a phase recovery algor ith m with th e same architecture of BPS, but with perf ect decisions, is in vestigated by analytical deriv ations an d Monte Carlo simulations. T his config u ration is used to derive an u pper bound on the BPS p erforma n ce. BPS is only studied by simu lations, as th e ana ly tical mo deling becomes overly complex beca u se of the decision pro cess. As in [1], the in vestigated phase recovery algorithms are first assessed by the mean squar e error (MSE) of a co n stant phase shift estimated over a g i ven ob servation window , in a way to allow an analytical tr eatment of the prob lem. Sub sequently , we in vestigate the impact of phase recovery on the mutual informa tio n ( M I) of a chan nel with Wiener pha se noise. Here, ev entual cycle slips are cir cumvented by a superv ised phase unwrapp er . The remainde r o f this pape r is divided as follows. Section II details the system mo d el, inclu ding the PS techniq u e and the BPS an d SPS alg orithms. Sectio n II I presents the simulation setup an d resu lts. Lastly , Section IV co ncludes the pap er . I I . S Y S T E M MO D E L A. Pr obab ilistic shaping (PS) Probabilistic sh a ping is usually implemented by applying the Maxwell-Bolzma n n distrib ution to the a-prior i prob abilities Copyri ght ( c) 2018 IEEE . Personal use of this m ateri al is pe rmitted . Ho we ver , permissi on to use t his material for any other purposes must be obtained from t he IEEE by sending a request to p ubs-permissions@i eee.or g. Accepted for pu blication: IEE E /OSA Journ al of Lightwave T echn ology (a) optimum (b) 64-QAM 64-QAM 256-QAM 256-QAM 256-QAM 128-QAM 64-QAM 32-QAM 16-QAM Shannon Limit PS-256-QAM PS-64-QAM Fig. 1. (a) MI for typical M-QAM formats. Dashed lines: uniform constellati ons. Solid lines: probabilistic ally shaped constellati ons. The dotted lines indicate the interv al of in terest for PS-64 -QAM and PS-256-QAM. (b) Optimum values of λ for PS-64-QAM a nd PS-256-QAM. The dot ted lines indicate th e range of λ s that corresponds t o the interv al of interest shown in Fig. 1(a). Note t hat λ = 0 corresponds to a uniform constellat ion. P m of symbols s m of th e transm itted constellation [2]: P m = e − λ | s m | 2 P M k =1 e − λ | s k | 2 (1) where λ is the shaping p arameter and M is the con stellation size. The cho ice of λ must be made caref u lly , as the optimum value varies acco rding to the signal p ower , m odulation format and SNR. Fig . 1(a) s hows th e MI fo r typical mo dulation formats with uniform ( dashed line) and sh aped (solid lin e ) constellations. For the sake of clarity , we fo cus in this pape r on the PS-6 4-QAM and PS-256- QAM fo rmats, but the analysis can also be easily extended to oth e r scheme s. Fig. 1(a) help s to un d erstand the range of SNRs for which shap ing should be applied for a s pecific modulation for m at. For PS -64-QAM , for example, PS should no t be app lied for SNRs high er than 22 dB, as unifo rm and shap ed con stellations achieve the same MI. On the o th er hand, PS s hould not be deployed with an SNR b elow 12 dB, as PS-32- Q A M ach iev es equiv a- lent perf ormance causin g a potentially lower imp lementation penalty . A n analogou s analysis can be carried o u t for the PS-256-QAM form a t, for which the SNR interval of interest ranges fro m 17 dB to 27 dB. Fig. 1( b ) shows the op tim um λ para m eter f or th e PS-64 -QAM a n d PS-2 56-QAM for mats, with in-ph ase and quadrature compo nents ha ving amplitudes 1 ± (2 i + 1) , i = 0 , 1 , ..., √ M / 2 − 1 . Th e figure allows to infer th e ra n ge of λ for which PS should b e imp lemented, namely , from 0 to 0 .05 for PS-6 4-QAM a n d fr om 0 to 0 .015 for PS-256-QAM. Note that, λ = 0 correspo nds to a u n iform constellation. B. Supervised and blind phase searc h algorithms ( SPS and BPS) Let the i th constellation symbol s i be transmitted over a complex ad d iti ve white Gau ssian noise (A WGN) chann el. The phase no ise associated with the tran smitter and lo cal oscillator 1 Note that the choice of λ dep ends on the sig nal power . lasers is expressed by a multiplicative factor e j θ n , so that the received symbol r i is given by: r i = s i e j θ n + n ′ i (2) where the co mplex Ga u ssian noise term n ′ i has zero mean and variance 2 σ 2 n . W e d e fin e the signa l to no ise ratio (SNR) as SNR = P s / 2 σ 2 n , where P s = E {| s i | 2 } . Phase recovery algorithm s resort to the fact that θ n varies slowly over tim e, in such a way that it is appro ximately co n stant over N symb ols. In prac tice, the s ize of N also depends o n th e symb ol rate and on the linewidth o f transmitter and local o scillator lasers. The B PS alg orithm estimates the ph ase no ise rotation θ n as the ang le that minimizes the sum of squared d istances b e tween N adjacent symbols s i , r o tated b y a test phase θ r , and their respective estimates ˆ s i . I n this section we assume an infinite number of test phases and do no t delve into resolution issues. In m athematical terms, estimate ˆ θ n is o btained as: ˆ θ n = min θ r J ( θ r ) (3) where the cost fun ction J ( θ r ) is g i ven by: J ( θ r ) = N X i =1 | e − j θ r ( s i e j θ n + n ′ i ) − ˆ s i | 2 (4) = N X i =1 | s i e j ( θ n − θ r ) − ˆ s i + n i | 2 (5) T erm n i is a rotated Gaussian pro cess of the same mean and variance o f n ′ i . It is also possible to write J ( θ r ) as a function of th e symbol erro r e i = s i − ˆ s i : J ( θ r ) = N X i =1 | s i e j ( θ n − θ r ) − s i + e i + n i | 2 (6) In ord e r to provide an analy tica l insight to the problem , we first inv estigate an ideal algorithm called SPS, which follows the same steps of BPS, except for the fact that the algorithm is not a ffected by erron eous d e cisions. In practical Copyri ght ( c) 2018 IEEE . Personal use of this m ateri al is pe rmitted . Ho we ver , permissi on to use t his material for any other purposes must be obtained from t he IEEE by sending a request to p ubs-permissions@i eee.or g. Accepted for pu blication: IEE E /OSA Journ al of Lightwave T echn ology implementatio ns, SPS can be dep loyed in bursts to per iodically refresh BPS. In SPS e i = 0 and the an alytical modelin g is simplified . It can be shown th at the MSE of SPS in the estimation of θ n can be approximated by (see Appendix A fo r the complete d eriv ation): MSE SPS ( N ) = E { ( θ n − ˆ θ n ) 2 } ≈ E    " P N i =1 ( n (1) i ) | s i | P N i =1 | s i | 2 # 2    (7) where n (1) i is the noise com ponent in the direction o f th e subtraction of s i and its ro ta ted version s i e j ( θ n − θ r ) . The computation of (7) is n o t trivial for intermediate values of N , but the extre m e cases offer interesting insigh ts. Setting N = 1 gives: MSE SPS (1) ≈ E    " n (1) i | s i | # 2    = σ 2 n M X m =1 1 | s m | 2 P m (8) Clearly , fo r small windows the SPS perfo rmance depe nds not only on th e SNR, but a lso on th e a-pr iori probab ility dis- tribution of tr a n smitted symbols. I n comm u nications systems with PS implemented by the Maxwell-Bolzm ann distrib ution, it can be shown, by differentiating (8) with respect to λ and setting th e result equal to zero, that the MSE is maxim ized by the following condition:  E {| s i | 4 } − 2 E {| s i | 2 } 2  E (     1 s i     2 ) + E {| s i | 2 } = 0 (9) In the deriv ation of (9), it s hould be noted that σ 2 n = P s / (2 SNR ) , wh ere P s = E { | s i | 2 } , also depe n ds on λ . By inspection o f ( 9) one can observe that the SPS perfo rmance is affected by se veral mom ents of | s i | a n d 1 / | s i | , including the fourth centra l mom e n t of | s i | . For M-QAM con stellations, where E { s 2 i } = 0 , its possible to rewrite (9) in term s o f its Kurtosis, g i ven by K s = E { | s i | 4 } − 2 E 2 {| s i | 2 } − | E { s 2 i }| 2 . Thus, the MSE is max imized when: K s = − E {| s i | 2 } E n | 1 /s i | 2 o (10) On the other h and, sup posing a large value of N , c a lled h ere N L , the Law of Large Numbe rs can be inv oked to assume th at, in the observ ation windo w , N L P m symbols of type s m occur . Thus, MSE SPS ( N L ) be c o mes: MSE SPS ( N L ) ≈ σ 2 n N L P M m =1 | s m | 2 P m N 2 L ( P M m =1 | s m | 2 P m ) 2 (11) = σ 2 n N L P s = 1 2 N L SNR − 1 (12) Thus, for la rge window sizes, the SPS perf o rmance depend s on the SNR, but is weakly affected by th e transmitted constel- lation. This can b e explained by the sums o f N L indepen d ent and iden tica lly distributed random variables in (7), allowing us to in v oke the Cen tr al L imit Theo rem. In BPS, e i 6 = 0 , and th e a nalytical modeling becomes challengin g becau se e i depend s on n i , s i and θ r . Theref ore, the analysis o f BPS is carried out b y simulatio n. I I I . S I M U L AT I O N S E T U P A N D R E S U LT S A. MSE performance W e assume that sh a ping changes the a-p riori proba b ility of transmitted symbo ls, but keeps th eir loc ation in the comp lex plane in ± (2 i + 1 ) , i = 0 , 1 , ..., √ M / 2 − 1 . Th is a ssum ption has a direct in fluence in the choice of λ , as it dep ends on th e constellation amp litudes (althoug h P m depend s on ly on the SNR). I n practice, the absolu te values of signal and additive n oise powers are meaningless for the phase recovery algorithm , as only the SNR dictates the transcei ver pe rfor- mance. Monte Carlo simulations w e r e carried out considering 2 19 symbols. An ar b itrary constant rotation o f π / 6 r ad was applied to the symbo ls, to represent a constant phase noise in a giv en window . Thus, the larger the window size, th e better the perfo rmance of the estimation alg o rithm. In pra ctical applications, the optim um window size depends on the system operating cond itions, suc h a s the op tica l signal to noise ratio (OSNR), laser linewidth, and symbo l rate. In this section, the size of the window was varied to simulate these different operating co nditions witho ut enterin g into system issues. As the phase deviation i s kept constant through out the simulation in the first quad rant, ther e is n o need to implem e nt a ph ase unwrapp er after BPS. A WGN was ad ded to the generated signals to g uarantee a co nstant SNR, independently of the amount of shapin g applied to th e con stellation. T o circumvent resolution issues, 900 test phases ar e used in th e SPS and BPS algorithm s. Fig. 2 shows the MSE of θ n for SPS. The solid lines indica te analytical pre dictions, while the sy m bols corresp o nd to the results produc ed by Monte Carlo simulations. T he results for SNR = 30 dB and SNR = 3 5 dB were include d as a h igh- SNR ref erence. Figs. 2( a) and 2(b) show the results f o r the PS-64-QAM fo rmat an d N = 1 and N = 100 , respec tively . The analytica l appr oximation for N = 1 exhib its a g o od agreemen t with the simu la tio ns, with increasing ac c uracy fo r higher SNRs. At N = 10 0 the m o del accuracy is preserved ev en at lower SNRs. The sam e beh a vior is ob served fo r the PS-256-QAM format in Figs. 2(c) and 2(d). As pr e d icted by the an a lytical mo del, for N = 1 the MSE can increa se as a result of shaping co mpared with the u niform distribution. T here are two main pro cesses th at explain the shape in Figs. 2(a) and 2(c). T o un derstand them, let us on c e again assume that in the shaping pro cess th e po sition of the constellation sym bols is retained , but its freque n cy is altered. In the first process, an in creasing λ red u ces th e occur rence of large amplitu d e symbols, imp airing the BPS per formanc e . This o c curs because p hase deviations are mo re easily detected in large am plitude symbo ls. In th e second process, shaping reduces the sig n al power and, to main tain th e SNR constant, the additive noise power is also downscaled, helping the estimation process. The do minance of the first process for low λ values, an d of the second process for high λ values, explains the e xistence of a maximum in the MSE curves. This depend ence o f the SPS perform a nce on shaping can be easily alleviated by longer noise rejection windows, for which the MSE is practically ind ependen t on th e mo dulation format. Th e Copyri ght ( c) 2018 IEEE . Personal use of this m ateri al is pe rmitted . Ho we ver , permissi on to use t his material for any other purposes must be obtained from t he IEEE by sending a request to p ubs-permissions@i eee.or g. Accepted for pu blication: IEE E /OSA Journ al of Lightwave T echn ology 0 0.05 0.1 0.15 0.2 10 -3 10 -2 10 -1 MSE [rad 2 ] SNR = 30 dB SNR = 22 dB SNR = 17 dB SNR = 12 dB max 0 0.05 0.1 0.15 0.2 10 -5 10 -4 SNR = 30 dB SNR = 22 dB SNR = 17 dB SNR = 12 dB (a) (b) (c) (d) 0 0.01 0.02 0.03 0.04 0.05 0.06 10 -3 10 -2 10 -1 MSE [rad 2 ] max SNR = 17 dB SNR = 22 dB SNR = 27 dB SNR = 35 dB 0 0.01 0.02 0.03 0.04 0.05 0.06 10 -6 10 -5 10 -4 MSE [rad 2 ] SNR = 17 dB SNR = 22 dB SNR = 27 dB SNR = 35 dB MSE [rad 2 ] Fig. 2. MSE for SPS wi th PS-64-QAM and (a) N = 1 and (b) N = 100. MSE for PS-256-QAM with (c) N = 1 and (d) N = 100. The solid lines indicate analyt ical results obtained by MSE SPS (1) and MSE SPS ( N L = 100 ), while the symbols were generated by Monte Carlo simulations. The transmission channel include s A WGN and a constant phase shift of π / 6 . The curves indicate that moderate noise rejection windo ws are suf ficie nt to make th e SPS performance indepen dent on PS. The vertic al dotted line s in figure s (a) and (c) indic ate λ max , ca lcula ted analytical ly by (9). Note that λ = 0 correspo nds to a u niform constel lation . figures for N = 1 a lso sho w th e λ parame ter value which maximizes the MSE SPS ( λ max ) calculated by ( 9). Figs. 3(a) and 3(b) show the MSE as a fun ction of λ , for BPS ev aluated with PS-64 - QAM at SNR = 12 dB and with PS-256-QAM at SNR = 17 dB, respectively . The ho rizontal dashed lines show th e analytical pred ictions for SPS with large N , obtain ed by ( 12 ), for N = 1 00 and N = 500. Clearly , BPS is af fected by a third process at lo w SNRs, wh ich is directly influen c ed by the two processes described for the SPS. It is the generation of decision erro rs in the estimation o f the transmitted symbol. Lon g er noise reje c tion wind ows redu ce the MSE, but the filterin g gains dep end strong ly on λ . For example, for the 64- QAM format without shaping , increasing N fro m 30 to 100 p roduces a 10-fo ld reduction on the MSE. On the oth er hand, this gain is strongly red uced if the system operates at λ = 0 .05. A similar trend can be observed f o r the PS-256-QAM mo dulation at λ = 0.015. It is interestin g to note that, for b oth PS-64-QAM an d PS-256-QAM formats, the maxim um MSE is achieved near λ optimum . Figs. 3(e ) an d 3(f) show the MSE f o r BPS ev aluated with PS-64-Q AM at SNR = 22 d B a nd with PS-256-QAM at SNR = 2 7 dB, respectively , which are the highest SNR v alues fo r which shaping should be applied. Interesting ly , for both cond itions, in most cases the MSE r emains con stant or decreases with λ , in dicating that PS can improve the BPS per f ormance. Figs. 3 (c) and 3(d) are in termediate ca ses, where the MSE is evaluated with PS-64-QAM a t SNR = 17 dB and with PS- 256-QAM at SNR = 22 d B. In all observed cases, λ optimum approa c h es t he worst-case con d ition for BPS. This effect was not pr e sent in th e SPS analysis, f or which a modera te no ise rejection wind ow was enou gh to mitigate the im pact of PS on th e MSE . T h erefore, we conjecture th a t the ca pacity- maximizing shapin g is near to the th e worst-case con dition for the decision proc e ss in sid e the BPS algorithm . T o ev aluate this trend, we simu late BPS with a window N = 10 , and find λ max for each SNR. The obtained λ max is co mpared with λ optimum . Previous analyses in th is paper have fo cused on th e PS-64-QAM and PS-2 56-QAM formats. Here, in order to increase the compr ehensiveness of th e results, we also evaluate the PS-3 2-QAM f ormat – built by pruning the previously defined PS-64-QAM constellation – and the PS-16 -QAM format, with a m plitudes ± (2 i + 1) , i = 0 , 1 . Th e results are Copyri ght ( c) 2018 IEEE . Personal use of this m ateri al is pe rmitted . Ho we ver , permissi on to use t his material for any other purposes must be obtained from t he IEEE by sending a request to p ubs-permissions@i eee.or g. Accepted for pu blication: IEE E /OSA Journ al of Lightwave T echn ology (a) (b) (c) (d) (e) (f) 0 0.01 0.02 0.03 0.04 0.05 0.06 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 MSE [rad 2 ] optimum N = 10 N = 30 N = 100 N = 500 0 0.01 0.02 0.03 0.04 0.05 0.06 10 -6 10 -4 10 -2 10 0 MSE [rad 2 ] optimum N = 10 N = 30 N = 100 N = 500 0 0.01 0.02 0.03 0.04 0.05 0.06 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 MSE [rad 2 ] N = 10 N = 30 N = 100 N = 500 optimum 0 0.05 0.1 0.15 0.2 10 -4 10 -3 10 -2 10 -1 10 0 MSE [rad 2 ] N = 10 N = 30 N = 100 N = 500 optimum 0 0.05 0.1 0.15 0.2 10 -4 10 -3 10 -2 10 -1 MSE [rad 2 ] N = 10 N = 30 N = 100 N = 500 optimum 0 0.05 0.1 0.15 0.2 10 -5 10 -4 10 -3 10 -2 MSE [rad 2 ] optimum N = 10 N = 30 N = 100 N = 500 Fig. 3. MSE for BPS wit h N = 1 0, 30, 100, and 500, e v aluat ed with PS-64-QAM a t (a) SNR = 12 dB, and (c) SNR = 17 dB , and (e) SNR = 2 2 dB; a nd e v aluat ed with PS-256-QAM at (b) SNR = 17 dB, (d) SNR = 22 dB, a nd (f) SNR = 27 dB. The transmission channel includes A WGN and a constant phase shift of π / 6 . The dotted lines indicat e λ optimum for the correspondi ng configuration. The dashed lines indicate the SPS predictions for large N , for N = 100 and N = 500 , give n by (12). Note that λ = 0 corresponds to a uniform constella tion. Copyri ght ( c) 2018 IEEE . Personal use of this m ateri al is pe rmitted . Ho we ver , permissi on to use t his material for any other purposes must be obtained from t he IEEE by sending a request to p ubs-permissions@i eee.or g. Accepted for pu blication: IEE E /OSA Journ al of Lightwave T echn ology (c) (d) (a) (b) 12 13 14 15 16 17 18 1 20 21 22 SNR [dB] 0 0.01 0.02 0.03 0.04 0.05 optimum max fit 5 6 7 8 9 10 11 12 13 14 15 SNR [dB] 0 0.02 0.04 0.06 0         optimum max fit   10 11 12 13 14 15 16   SNR [dB] 0 0.02 0.04 0.06   optimum max fit S ! " d# $ %& '( )* 20 21 22 23 24 25 26 2+ Fig. 4. Theoret ical λ optimum for N = 10 (solid lines) a nd simulated λ max (marke rs) fo r (a) PS-16-QAM, (b) PS-32-QAM, (c) PS-64- QAM, and (d) PS-256- QAM. The transmission channel includes A WGN and a co nstant phase shift of π/ 6 . The dashed lines cor respond to fitted v alues of λ max ( λ fit ). shown in Fig. 4. It is observed that, for th e 16 /64/256- QAM constellations, λ max is in the vicinity o f λ optimum in the full range o f SNRs, indicating that BPS can cause implementatio n problem s in systems with probabilistic shaping. Ho wev er , this effect is not ob served fo r PS-32-QAM (see Fig. 4(b)). The se results sug gest tha t PS may im p air the BPS perfo rmance for square-QAM constellations, but this behavior may chan ge for other constellation g eometries. B. MI performan ce In th e previous section, we observed that the MSE o f the estimated phase dep ends o n the shapin g parameter λ , and th at the worst-ca se MSE is ach ie ved in the vicinity o f λ optimum for square-QAM constellations. I n this section we assess the imp act of this effect on the MI of a channel with W iener phase n oise. The simulations ar e perfo rmed with 2 17 symbols. The BPS is implemente d with 60 test p hases. If left uncomp ensated, the occurren ce of cycle slips in simu lations with p hase noise would distur b the estimation of the M I, which in this paper is b ased on th e m e thod used in [1 2]. For this reason, we app ly a supe rvised cycle slip comp ensation me th od that ro tates every symb ol at the outpu t of BPS by m ultiples of π / 2 to minimize th e E uclidean d istan ce to the corr esponding transmitted sym b ol. Fig. 5 e valuates the ch annel MI under the sam e no ise le vels and wind ow sizes as in Fig. 3. Th e dashed lines i ndicate the MI obtain ed numerically f or an A WGN chann el, and the crosses show si mulation results used to validate th e simu- lation setup. The simulated W iener pha se noise correspon ds to ∆ ν = 2 0 0 kHz a n d a symbol rate of 50 GBd. It can be observed that errors in the phase estimatio n pr ocess cau se a significant impact on the cha nnel MI. For the lowest SNRs (Figs. 5(a) and 5(b)) , in mo st cases the MI achieved withou t shaping ( λ = 0) is higher than th at obtained with shaping. That is, instead of incr easing the MI, th e shaping cau ses the inverse effect of actu ally d ecreasing the ch annel MI. For example, this problem is clearly observed in Fig. 5(a) for the 64-QAM for mat. The maximum theor etical MI for the ch annel without shaping ( d ashed line for λ = 0 ) is reached for N = 500, and alm o st reach ed for N = 1 00. The MI for N = 100 exhibits a sudd en drop in λ = λ optimum , making tr ansmission uninteresting in this situation. Althoug h the effect is milder f or N = 500, the M I reached at λ = λ optimum is still lower than that ob tained in the u niform case ( λ = 0 ). Th is situatio n is alleviated for intermed ia te SNR values ( Fig s. 5(c) an d 5 (d)), Copyri ght ( c) 2018 IEEE . Personal use of this m ateri al is pe rmitted . Ho we ver , permissi on to use t his material for any other purposes must be obtained from t he IEEE by sending a request to p ubs-permissions@i eee.or g. Accepted for pu blication: IEE E /OSA Journ al of Lightwave T echn ology Theory 0 0.05 0.1 0.15 0.2 2 2.5 3 3.5 4 MI [bit/symbol] 0 0.05 0.1 0.15 0.2 3 3.5 4 4.5 5 5.5 MI [bit/symbol] 0 0.05 0.1 0.15 0.2 3.5 4 4.5 5 5.5 6 MI [bit/symbol] 0 0.01 0.02 0.03 0.04 0.05 0.06 2 2.5 3 3.5 4 4.5 5 5.5 6 MI [bit/symbol] optimum optimum 0 0.01 0.02 0.03 0.04 0.05 0.06 3 4 5 6 7 MI [bit/symbol] 0 0.01 0.02 0.03 0.04 0.05 0.06 5 5.5 6 6.5 7 7.5 8 MI [bit/symbol] (f) (e) (c) (d) (a) (b) Theory N = 500 N = 100 N = 30 N = 10 Theory N = 500 N = 100 N = 30 N = 10 Theory N = 500 N = 100 N = 30 N = 10 Theory N = 500 N = 100 N = 30 N = 10 Theory N = 500 N = 100 N = 30 N = 10 Theory N = 500 N = 100 N = 30 N = 10 optimum optimum optimum optimum Fig. 5. MI e v alua ted with BPS assuming N = 10 , 30, 100, and 500, for PS-64-QAM at (a ) SNR = 12 dB, (c) SNR = 17 dB, and (e) SNR = 22 dB; and for PS-256-QAM a t (b) SNR = 17 dB, (d) SNR = 22 dB, and (f) SNR = 27 dB. The simul ations in clude A WGN and Wie ner ph ase noise c orrespondin g to ∆ ν = 200 kHz and a symbol rate of 50 GBd. Note that λ = 0 corresponds to a uniform c onstell ation. Copyri ght ( c) 2018 IEEE . Personal use of this m ateri al is pe rmitted . Ho we ver , permissi on to use t his material for any other purposes must be obtained from t he IEEE by sending a request to p ubs-permissions@i eee.or g. Accepted for pu blication: IEE E /OSA Journ al of Lightwave T echn ology (c) (a) (b) (d) N = 30 N = 100 N = 500 N = 10 N = 30 N = 100 N = 500 N = 10 N = 30 N = 100 N = 500 N = 10 N = 30 N = 100 N = 500 N = 10 Fig. 6. MI as a function of SNR ev aluat ed with BPS assuming N = 10, 30, 100, and 500, for (a) 64-QAM, (b) 256-QAM, (c) PS-64-QAM, and (d) PS-256-QAM. The simulations include A WGN and W iener phase noise corresponding to ∆ ν = 200 kHz and a symbol rate of 50 GBd. The shaping paramete r λ optimum is used for a ll SNRs. for which N = 100 is sufficient to pr actically eliminate the impact o f phase recovery on system performa nce. For h igh SNR values (Fig s. 5(e ) an d 5(f) ), N = 30 is sufficient to guaran tee a suitable op eration for b oth modu lation fo rmats, howe ver , in this case, the shaping parameter is very lo w , and the constellation practically do es not h ave shaping. It is interesting to note that, for the PS-256 -QAM fo rmat and SNR = 27 d B (Fig. 5f), the c urve for N = 500 exhibits significant penalties because the noise rejection window is excessi vely lo ng for the g i ven balance of additive noise and phase noise. Possible SNR p enalties due to ph ase recovery in shaped transmissions can be ob served in Fig. 6, which sho ws the MI versus SNR performan ce for u niform (top) and shaped (bottom) cases fo r both both 64-QAM (left) an d 2 56-QAM (right) for mats. For th e uniform case, setting N = 100 is enoug h to provide n egligible imp lementation penalties for a wide rang e o f SNRs. On the othe r hand, the shaped case exhibits steep drop s, a n d even fo r long noise r ejection windows the simulated cu rves detach from the theo r etical ones at moderate SNR values, eliminating the exp e cted SNR shapin g gains. W e also evaluated the MI a s a function of the noise rejection window len g th fo r dif ferent values of SNR and las er linewidths. Th e results f or the 64 - QAM form at ar e shown in Figs. 7(a) ( SNR = 12 dB) and 7(b) (SNR = 22 d B). At SNR = 12 dB and unif o rm transmission the add iti ve n o ise is do minant, and little depen dence of the bit error ra te on N is observed, provided that the wind ow is lo nger than ap proxim a tely 20 0 symbols. Unde r th ese condition s, incre a sin g the window size (e.g. up to 500) does not result in system degradatio n , but increases the complexity and p ower consu m ption of the al- gorithm. Th e per formanc e with prob abilistic shap ing is con- siderably p oorer . Her e, fo r a ∆ ν = 200 kHz, only N = 45 0 ensures a pe r forman c e equ iv alent to the u niform c ase, and fo r ∆ ν = 2 MHz th e perform ance of the un iform case is never reached. For SNR = 22 dB the shaped and uniform cases coincide, as the shaping parameter is very low . A minimum window of appr o ximately 20 symbo ls is sufficient to e nsure adequate perf ormance for b oth cases. Howe ver , using larger windows im p airs the phase recovery p rocess an d co nsequently degrades the MI. The per formanc e for the 256 -QAM f ormat is shown in Fig s. 7(c) (SNR = 17 dB) and 7(d) (SNR = 27 dB). For SNR = 17 dB without shaping, a filterin g windo w o f Copyri ght ( c) 2018 IEEE . Personal use of this m ateri al is pe rmitted . Ho we ver , permissi on to use t his material for any other purposes must be obtained from t he IEEE by sending a request to p ubs-permissions@i eee.or g. Accepted for pu blication: IEE E /OSA Journ al of Lightwave T echn ology (c) (a) (b) (d) 0 100 200 300 400 500 600 N 1 2 3 4 5 6 MI [bit/symbol] 0 100 200 300 400 500 600 N 1.5 2 2.5 3 3.5 4 4.5 MI [bit/symbol] 0 100 200 300 400 500 600 N 2.5 3 3.5 4 4.5 5 5.5 6 MI [bit/symbol] 0 100 200 300 400 500 600 N 2 3 4 5 6 7 8 MI [bit/symbol] Fig. 7. MI as a function of N for uniform and proba bilisti cally shaped 64-QAM at (a) SNR = 12 dB and (b) SNR = 22 dB; and for uniform and probabilistic ally shaped 256-QAM at (c) SNR = 17 dB and (d) SNR = 27 dB. The si mulations incl ude A WGN a nd W iene r phase noise corresponding to a symbol rate of 50 GBd and ∆ ν = 200 kHz or ∆ ν = 2 MHz. approx imately 1 00 symb ols is en ough to a c hiev e a relatively high MI. Again, PS strongly impairs the system p erforma nce. For b oth ∆ ν = 200 k Hz a n d ∆ ν = 2 MHz, the perform ance obtained by the unif o rm con stellation is never achieved. For SNR = 27 dB the shap ed and unifo rm cases coincide, as the shaping p arameter is very low . In this case, ag a in, N = 2 0 is enoug h to ach ie ve the expected theor etical MI. I V . C O N C L U S I O N The inter play of PS an d the BPS algo rithm is in vestigated analytically and by simulation. W e start by ana lyzing the perfor mance o f an SPS algorithm, which has the same ar- chitecture of BPS, except for the decision pro cess, which is assumed perfec t. W e pr ovide an an a lytical expression fo r the MSE o f SPS, which exhibits a go od agr e ement with simula- tions. T he results demo nstrate that PS affects the perfor mance of SPS at short noise rejection windows, b ut this impact is easily mitigated at wind ows of moderate sizes. At large windows, the SPS MSE is independen t on the modulation format and, thu s, insensitiv e to PS. The BPS algo r ithm, howe ver , reveals a strong dependen ce on PS, even for lon g noise re jection win d ows. Given the d ifferences in behavior of SPS and BPS, we infer that the decisions made inside the BPS algorith m are affected by shaping. For this reaso n, ev en long noise rejection windows may provid e on ly m odest gains to th e algor ith m perfor mance. It is also o bserved th at the worst shaping cond ition for th e BPS algorithm is near to the capacity -maximizing oper ation poin t for squar e-QAM constellations. Finally , simulations of the MI of a c hannel with W iener p hase noise show that the PS impact on BPS can affect the overall system perfo rmance, specially at low SNRs. I n this co n dition, the MI degradation caused by BPS can exceed potential cap a city gains exp ected by PS. This effect can be e ventually mitigated by e xtremely long noise rejection windows, which may increase com p lexity an d requir e lo w linewidth lasers. These findings sugg est the need for a lter nativ e phase recovery algorithms to be dep loyed in proba b ilistically- shaped tran smissions. A C K N O W L E G E M E N T W e would like to thank the editor and the anonymous revie wers for their essential contributions to imp rove the quality of th e pape r . Copyri ght ( c) 2018 IEEE . Personal use of this m ateri al is pe rmitted . Ho we ver , permissi on to use t his material for any other purposes must be obtained from t he IEEE by sending a request to p ubs-permissions@i eee.or g. Accepted for pu blication: IEE E /OSA Journ al of Lightwave T echn ology A P P E N D I X A. Derivation of the MSE fo r SP S A ge ometric an a ly sis of th e pr oblem enable s us to rewrite (6) as: J ( θ r ) = N X i =1 "  2 | s i | sin  θ n − θ r 2  + n (1) i  2 + ( n (2) i ) 2 # (13) where n (1) i is the noise com ponent in the direction o f th e subtraction of s i and its r otated version s i e j ( θ n − θ r ) , and n (2) i is the p erpendic u lar compon ent. Both n (1) i and n (2) i are zero mean real Gaussian pr ocesses with variance σ 2 n each. W e find ˆ θ n by differentiating J ( θ r ) with r e spect to θ r : dJ ( θ r ) dθ r = (14) N X i =1 − 2  2 | s i | sin  θ n − θ r 2  + n (1) i  | s i | cos  θ n − θ r 2  Setting the deriv ati ve equal to zero, and supposing a small θ n − θ r , yields: N X i =1 2 | s i | sin θ n − ˆ θ n 2 ! + n (1) i ! | s i | ≈ 0 (15) N X i =1 2 | s i | 2 sin θ n − ˆ θ n 2 ! + N X i =1 ( n (1) i ) | s i | ≈ 0 (16) sin θ n − ˆ θ n 2 ! ≈ − 1 2 P N i =1 ( n (1) i ) | s i | P N i =1 | s i | 2 (17) Approx imating sin( x ) ≈ x , gives: θ n − ˆ θ n 2 ≈ − 1 2 P N i =1 ( n (1) i ) | s i | P N i =1 | s i | 2 (18) ˆ θ n ≈ θ n + P N i =1 ( n (1) i ) | s i | P N i =1 | s i | 2 (19) Finally , th e mean squared error (MSE) in th e estimation of θ n can be g i ven as: MSE SPS ( N ) = E { ( θ n − ˆ θ n ) 2 } = E    " P N i =1 ( n (1) i ) | s i | P N i =1 | s i | 2 # 2    (20) R E F E R E N C E S [1] F . A. Barbosa, J. D. Reis, and D. A. A. 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Hanik, “Experimen tal compa rison of probabil istic shaping methods for unrepe ated fiber transmission, ” Jou rnal of Light- wave T echnol ogy , vol . 35, no. 22, pp . 4871–4879, Nov 2017. [13] D. Pilori, L. Bertign ono, A. Nespola, F . Forghie ri, and G. Bosco, “Comparison of probabilisti cally shaped 64QAM with lower cardina lity uniform c onstell ations in long-hau l optical systems, ” J ournal of Light- wave T echnol ogy , vol . 36, no. 2, pp. 501–509, Jan 2018. Copyri ght ( c) 2018 IEEE . Personal use of this m ateri al is pe rmitted . Ho we ver , permissi on to use t his material for any other purposes must be obtained from t he IEEE by sending a request to p ubs-permissions@i eee.or g.