Optics Communications 285 (2012) 3692–3696

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Optics Communications

0030-40

http://d

n Corr

E-m

journal homepage: www.elsevier.com/locate/optcom

Frequency estimation for optical coherent M-QAM system without removingmodulated data phase

Song Yu n, Yinwen Cao, Haijun Leng, Guohua Wu, Wanyi Gu

State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing 100876, PR China

a r t i c l e i n f o

Article history:

Received 17 January 2012

Received in revised form

15 May 2012

Accepted 16 May 2012Available online 31 May 2012

Keywords:

Coherent detection

Fast Fourier Transform (FFT)

M-ary quadrature amplitude modulation

(M-QAM)

18/$ - see front matter & 2012 Elsevier B.V. A

x.doi.org/10.1016/j.optcom.2012.05.027

esponding author.

ail address: yusong@bupt.edu.cn (S. Yu).

a b s t r a c t

For optical coherent M-ary quadrature amplitude modulation (M-QAM) system, the frequency offset

can be extracted directly by applying Fast Fourier Transform (FFT) to the signal’s argument, without

removing the modulated data phase. By categorizing the constellation points and rotating some

constellation points by p=4, this algorithm is robust to extract the frequency offset against the noise.

Numerical simulations of 16-QAM and 256-QAM coherent systems are presented to demonstrate this

algorithm.

& 2012 Elsevier B.V. All rights reserved.

1. Introduction

In optical coherent detection, digital signal processing (DSP)shows its competence for the impairment compensation of thelong-haul transmission [1–3]. Moreover, because of DSP, the stageof phase recovery can be transferred from hardware domain tosoftware domain [1]. Among different DSP-based phase estima-tions (PEs), considering laser linewidth tolerance and implemen-tation complexity, feed-forward approach [4–7] is preferred.However, this approach needs the frequency offset between thetransmit and the local oscillator (LO) laser to be small enough.Meanwhile, the frequency resolution of a typical tunable lasercannot satisfy the above requirement. Therefore, a frequencyestimator is necessary to ensure reliable performance of PE.

We have presented a frequency estimation algorithm forM-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 isextended 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 ofsymbols is introduced in this M-QAM scheme. In addition, it is toderive that specific constellation points should be rotated by p=4,to counteract the noise. Simulations of 10-Gbaud 16-QAM and256-QAM coherent systems demonstrate that the range of the

ll rights reserved.

frequency offset estimation can cover [�1.2 GHz, þ1.2 GHz],while the maximum estimation error is below 1 MHz.

2. Principle

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].

The nth received M-QAM symbol can be presented as

Sn ¼ An exp½jðyd,nþyl,nþ2pDfnTsÞ�þNn, ð1Þ

where An, yd,n represent modulated amplitude and phase ofM-QAM format, yl,n is the phase noise induced by laser linewidth(Df l), Ts is the symbol duration, Nn represents ASE noise, and Df

denotes the frequency offset.The argument of the signal is of the form

fd,n ¼ yd,nþ2pDfnTs�2pm

¼/fd,nSþy0d,n, ð2Þ

where m is an integer to ensure fd,n to fall into the range of½�p,þp�. 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/fd,nS and the residual y0d,n.

Fig. 1. The division of constellation for 16-QAM when 2pDfnTs is in range I. (a)–(d) denote r¼ 1;2,3;0, respectively.

Table 1The derivation of /mS for range I.

i m Possibility /mS

4k k 13=16 i=4þ3=16

kþ1 3/16

4kþ1 k 9=16 i=4þ3=16

kþ1 7/16

4kþ2 k 5=16 i=4þ3=16

kþ1 11/16

4kþ3 k 1=16 i=4þ3=16

kþ1 15/16

S. Yu et al. / Optics Communications 285 (2012) 3692–3696 3693

From Eq. (2), the average term is governed by

/fd,nS¼/yd,nSþ/2pDfnTsS�/2pmS¼ 2pDfnTs�2p/mS: ð3Þ

Without loss of generality, 16-QAM is demonstrated to illus-trate the algorithm. In order to ensure that m is only dependenton yd,n, 2pDfnTs is divided into three ranges, range I:ðp=4þ ip=2,p=2�aþ ip=2�; range II: ðp=2�aþ ip=2,p=2þaþ ip=2�;range III: ðp=2þaþ ip=2;3p=4þ ip=2�, where a¼ arctanð1=3Þ and i

is an integer which is denoted as 4kþr ðr¼ 0;1,2;3Þ.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 filled symbols). If the modulated data belongsto open symbols, m¼ kþ1. If not, m¼k. The explanation andanalysis will be given in detail later.

As for range I, if r¼1, which is corresponding to Fig. 1(a), filledsymbols exhibit argument less than 3p=4 and 2pDfnTsAðp=4þ2kp,p=2�aþ2kp�. Therefore, for filled symbols, m¼k isrequired to keep fd,nA ½�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 filled symbols is 13=16. Therefore, /mS¼i=4þ3=16. Considering Eq. (3) and Table 1 together, if 2pDfnTs

is in range I, Eq. (3) can be rewritten as

/fd,nS¼ 2pDfnTs�3p=8�ip=2: ð4Þ

Similarly, for range II and range III, we have

/fd,nS¼ 2pDfnTs�p=2�ip=2; ð5Þ

or

/fd,nS¼ 2pDfnTs�5p=8�ip=2: ð6Þ

In order to express Eqs. (4)–(6) in a more general form, GðxÞ isintroduced

GðxÞ ¼�1, xry1,

0, y1oxry2,

1, x4y2:

8><>: ð7Þ

Therefore

/fd,nS¼ 2pDfnTs�pGð2pDfnTsÞ=8�y3�ip=2, ð8Þ

where y3 ¼ p=2.The trajectory of /fd,nS for 16-QAM (black solid line) is different

from QPSK (black dashed line), as shown in Fig. 2. The argumentchange induced by frequency offset (arg½expðj2pDfnTsÞ�) 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 ofarg½expðj2pDfnTsÞ�. Therefore, after applying FFT to /fd,nS, the firstharmonic frequency represents the absolute value of 4Df , as shownin Fig. 3.

However, compared with the spectrum of /fd,nS for QPSK and16-QAM in Fig. 3(a) and (b), the amplitude of the first harmonicfor 16-QAM is much lower, making it difficult to be extracted.

In order to increase the amplitude of first harmonic, firstly, 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

includes the rest rings. Specifically, CI is the middle ring and CII

includes the inner and outer rings. We rotate CI by y, as shown inFig. 4(a). Considering the symmetry of the constellation, rotationof ½0,p=2Þ is sufficient. The derivation of /mS is similar, and itsexpression is the same as Eq. (8). The only differences are y1, y2

and b, which is shown in Table 2.To facilitate the expression of the first harmonic F1, we use

continuous variable t to substitute nTs, yielding

F1 ¼1

T

Z t0 þT

t0

/fd,nSejo1t dt, ð9Þ

where T ¼ 1=ð4Df Þ and o1 ¼ 2p� 4Df . Here, we assume Df 40. Theintegral range is ðp=4;3p=4� and Eq. (9) can be further simplified

F1 ¼j

16ðe�j4y1þe�j4y2�2Þ: ð10Þ

The amplitude of F1 is

9F19¼ 18

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifficos½2ðy1�y2Þ�fcos½2ðy1�y2Þ��2 cos½2ðy1þy2Þ�gþ1

p: ð11Þ

It is worth to mention that all the constellation points best lineup when the CI points are rotated by y¼ p=4. And this can befurther confirmed by Figs. 3 and 4(b). It can be mathematically

T/4 T/2 3T/4 Tt

Pha

se (r

ad)

<φd,n,16−QAM><φd,n,QPSK>Arg[exp(j2πΔfnTs)]

π

π/4

0

−π/4

−π/2

3π/4

π/2

−3π/4

−π

period of <φd,n>

Fig. 2. /fd,nS and arg½expðj2pDfnTsÞ�, T ¼ 1=Df : The period of /fd,nS is T/4 for both 16-QAM and QPSK.

1 3 5 70

0.05

0.10

0.15

0.20

0.25

f/(4Δf)

|FFT

|

QPSK

1 3 5 70

0.05

0.10

0.15

0.20

0.25

f/(4Δf)

|FFT

|

16−QAM w/o Rotation

1 3 5 70

0.05

0.10

0.15

0.20

0.25

f/(4Δf)

|FFT

|

16−QAM w. Rotation

Fig. 3. Spectrum of /fd,nS. (a) /fd,nS for QPSK; (b) /fd,nS for 16-QAM before p=4 rotation; (c) /fd,nS for 16-QAM after p=4 rotation.

00.09

0.10

0.11

0.12

0.13

0.14

0.15

0.16

π/8 π/4 3π/8 π/2θ

0.09

θ

|F1|

0.16

Y

X

III

Fig. 4. (a) Rotation of CI by y. Black symbols: CI before rotation. Gray symbols: CI after rotation. Open symbols: CII. (b) The curve of 9F19.

S. Yu et al. / Optics Communications 285 (2012) 3692–36963694

calculated from Eq. (10) that the increase in 9F19 is 0.777, which isconsistent with Fig. 3.

Fig. 5(a) is presented to illustrate the modified frequencyestimator. Firstly, partition the rings including CI by the amplitude;

then, rotate these rings by p=4; next, obtain the argument ofall the symbols; after that, use FFT to get the correspondingfrequency of the first harmonic frequency; finally, divide it by 4 toget 9Df 9.

S. Yu et al. / Optics Communications 285 (2012) 3692–3696 3695

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 :

1.

Tab

y1,

y

½0

½a½p½p½p

Figing

Use frequency estimator to get absolute value of frequencyoffset, denoted as 9Df 19.

le 2

y2 and b in different ranges of y.

y1 y2 b

,aÞ p=2�aþyþ ip=2 p=2þaþyþ ip=2 p=2

,p=4�aÞ p=2�aþyþ ip=2 p=2þaþyþ ip=2 5p=8

=4�a,p=4þaÞ aþyþ ip=2 p=2�aþyþ ip=2 p=2

=4þa,p=2�aÞ y�aþ ip=2 yþaþ ip=2 3p=8

=2�a,p=2Þ y�aþ ip=2 yþaþ i=2p p=2

. 5. Block diagram of the modified algorithm. (a) Frequency estimator employ-

partition and rotation; (b) the integrated structure of the algorithm.

0 1 2 3 4 50

0.02

0.04

0.06

Frequency (GHz)

|FFT

|

0 1 2 3 4 50

0.02

0.04

0.06FFT w/o rotation

Frequency (GHz)

|FFT

|

256 QAM

16 QAM

Fig. 6. FFT on the argument of the signal with and withou

2.

|FFT

||F

FT|

t p

Assume Df 140, then, S0n ¼ Sn expð�j2p9Df 19nTsÞ.

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,

then, S00n ¼ S0n expðj4p9Df 19nTsÞ; and if not, it means Df 140.Then, output S00n ¼ S0n directly.

For frequency estimation of M-PSK, rings partition and p=4rotation are omitted. Meanwhile, the divisor 4 is replaced by M.

3. Simulation results

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.

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 first harmonic corresponding tofrequency offset is buried in the noise (the left part of Fig. 6).

Fig. 7 shows absolute estimation errors after frequency esti-mation. The range of frequency offset is [�1.2 GHz, þ1.2 GHz]. Itcan be seen that the mean estimation error is about 0.1 MHz, andthat the maximum is still below 1 MHz.

4. Conclusion

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 specific constellation pointsrenders this scheme more robust against noise. Simulations of10-Gbaud 16-QAM and 256-QAM coherent systems demonstratethat the estimator could accurately estimate the frequency offsetup to 1.2 GHz. Meanwhile, the estimation error remains below0.2 MHz, significantly satisfying the requirement of the subse-quent phase estimation.

0 1 2 3 4 50

0.04

0.08

0.12

Frequency (GHz)

0 1 2 3 4 50

0.04

0.08

0.12FFT w. rotation

Frequency (GHz)

2GHz

2GHz

=4 rotation. Frequency offset is set to be 0.5 GHz.

−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

0.2

0.4

0.6

0.8

Frequency offset (GHz)

Est

imat

ion

erro

r (M

Hz)

Maximum (16−QAM)

Maximum (256−QAM)

Mean (16−QAM)

Mean (256−QAM)

Fig. 7. Absolute values of frequency estimation error under [�1.2 GHz,þ1.2 GHz] (1000 simulations for each frequency offset).

S. Yu et al. / Optics Communications 285 (2012) 3692–36963696

Acknowledgments

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 ResearchFunds for the Central Universities.

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