Reserach Review

平成２５年度研究報告書
平成２６年４月
東京大学
大学院工学系研究科
菊池和朗
A. 研究概要
(1) デジタルコヒーレント光受信器における信号処理技術
信号光と LO 光の周波数オフセットを除去するための新しい位相推定アルゴリズムを提案した。ナイ
キスト WDM システムにおいても，本方式を適用すれば，周波数オフセットを補償することによりペナ
ルティーなく波長チャンネルを分離できることが示された [b5]。また，ディジタルコヒーレント光受信
器の等化器出力を用いれば，波長チャンネル間にガードバンドがないため，測定が困難であったナイキ
スト WDM システムにおいても，OSNR の評価が可能であることを示した [a2, b2]。
さらに，バタフライ構成の FIR フィルタを実部と虚部に分離することにより，デジタルコヒーレント
光受信器の IQ ポート間の振幅誤差，位相誤差およびスキューを等化できることを示した [a4]。CMA を
修正することにより，QPSK 信号のキャリア位相推定を行うアルゴリズムに関しても検討を行った [b8]。
(2) 偏波変調方式
光信号のストークスベクトルは，光の絶対位相に依存せず，x および y 偏波間の位相差と振幅比で決
定される。したがって，ストークスベクトルを変調パラメータに用いれば，光信号の絶対位相に不感応
な光伝送システムを実現できる。このようなシステムでは，ストークス空間で効率的な多値化が可能と
なり，しかも多値信号を直接検波方式で復調することができる。一方で，適応等化が困難であるという
欠点はあるが，100 km 程度の大容量伝送には有効な技術であると考えられる。本研究では，ストークス
空間での多値化の方法，直接検波光受信器における復調アルゴリズムを提案し，2，4，8，16，32 相多
値信号の符号誤り率特性をシミュレーションにより解析した。特に 25 Gsymbol/s で 16 多重することに
より，比較的少ない計算量の信号処理で，100 Gbit/s を実現できることが明らかになった [a1, b1, c6]。
さらに派生技術として，偏波多重 QPSK 信号のストークスベクトルをサンプリング測定すれば，レー
ザ光の位相雑音の影響を排除して，コンステレーションダイアグラムをモニタできることを示した [b7]。
(3)IM·DD システムにおける偏波多重分離技術
本研究では，偏波多重・多値強度変調信号を復調する，新しい構成の直接検波光受信器を提案してい
る [a3, c2]。この受信器は，信号光の偏波状態を測定する光回路であるストークスアナライザとディジタ
ル信号処理回路からなる。測定されたストークスベクトルをディジタル信号処理することにより，信号
光の偏波変動を追尾しつつ，任意の多値強度変調信号に対して偏波多重分離を行うことができる。計算
機シミュレーションにより，本受信器を用いれば，25 Gbaud のシンボルレートにおいて，波長多重技
術や光学的な偏波コントローラを用いることなく，100 Gbit/s の伝送容量を実現できることが明らかに
なった。直接検波を用い，ディジタル信号処理の計算量も比較的小さい本受信器は，100G イーサーネッ
トへの応用に対して有効な技術と考えられる。
(4) 新しい 4 次元光変調フォーマット
コヒーレント光通信においては，2 次元自由度を持つ IQ 平面が 2 つの自由度を持つ偏波それぞれに
存在することから、合計 4 つの自由度を用いた変調が可能となる。近年，この 4 自由度のもとで信号点
配置を設計することにより，パワー効率が最大となる変調方式である PS-QPSK 変調方式が見出されて
いる。本研究では，4 次元空間での信号点配置の詳細な検討に基づき，DP-QPSK 方式と同一のパワー
効率でありながら周波数利用効率がより高い，新しい変調方式を提案した。次に，このような 4 次元空
間で変調された信号を復調するための一般的な信号処理アルゴリズムを提案する。さらに，信号空間を
4 次元以上に拡張することにより，周波数利用効率およびパワー効率の観点から，より優れた変調方式
が存在することを示した [b3, c4]。
さらに，4 次元空間で、DP-QPSK にセット分割を 3 回施すことによって生成された子信号集合を用
いた、8 状態を有する 4 次元トレリス符号化光変調方式 (4D-TCM) を提案した。最小自由距離を効率よ
く拡大できるため、非符号化 PS-QPSK に比べて周波数利用効率を維持したまま、より高感度となり得
ることを、シミュレーションにより示している [c1]。
(5) 光ファイバの非線形補償
光ファイバ伝送によって生じるカー効果は，信号の伝送可能距離を厳しく制限する。このため，長距
離伝送システムを構築するためには，カー効果によって生じた信号劣化を補償しなければならない。
本研究では，所望の光信号とその位相共役光を偏波多重して伝送することにより，ファイバの非線形
効果を効率的に低減する方法を提案した。ファイバ伝送路の全分散を付与したのち生成された位相共役
光は，伝送路を伝搬し，コヒーレント受信器で受信される。一方所望信号は，ファイバを伝送したのち，
コヒーレント受信器で分散補償される。この時，位相共役光の逆伝搬の原理によって，所望信号と位相
共役光における非線形歪みは逆相関を持つので，両者の和をとることにより非線形歪みがキャンセルさ
れる。計算機シミュレーションにより，この方法を用いると，20Gbit/s QPSK 信号を 1,000km 伝送し
た時，最適パワーは 7dB 向上することが示された [b4,c5]。
B. List of Publications
I. Journal Papers
[a1] K. Kikuchi and S. Kawakami, “Multi-level signaling in the Stokes space and its application to
large-capacity optical communications,” Optics Express, vol.22. no.7, pp.7374-7387, April 2014.
[a2] Md. S. Faruk, Y. Mori, and K. Kikuchi, “In-band estimation of optical signal-to-noise ratio from
equalized signals in digital coherent receivers,” IEEE Photonics J., vol.6, no.1, 7800109, Feb.,
2014.
[a3] K. Kikuchi, “Electronic polarization-division demultiplexing based on digital signal processing
in intensity-modulation direct-detection optical communication systems,” Optics Express vol.
22, no. 2, pp. 1971-1980, Jan. 2014.
[a4] Md. S. Faruk and K. Kikuchi, “Compensation for in-phase/quadrature imbalance in coherentreceiver front end for optical quadrature amplitude modulation,” IEEE Photonics J., vol.5, no.2,
7800110, April, 2013.
II. International Conference
[b1] K. Kikuchi and S. Kawakami, “16-ary Stokes-vector modulation enabling DSP-based direct
detection at 100 Gbit/s,” Optical Fiber Communications Conference (OFC 2014), San Francisco,
CA, USA, Th3K.6 (9-13 March 2014).
[b2] Md. S. Faruk, Y. Mori, and K. Kikuchi, “Estimation of OSNR for Nyquist-WDM transmission
systems using statistical moments of equalized signals in digital coherent receivers,” Optical
Fiber Communications Conference (OFC 2014), San Francisco, CA, USA, Th2A.29 (9-13 March
2014).
[b3] S. Ishimura and K. Kikuchi, “Multi-dimensional permutation modulation aiming at both high
spectral eﬃciency and high power eﬃciency,” Optical Fiber Communications Conference (OFC
2014), San Francisco, CA, USA, M3A.2 (9-13 March 2014).
[b4] H. Lu, Y. Mori, C. Han, and K. Kikuchi, “Novel polarization-diversity scheme based on mutual
phase conjugation for ﬁber-nonlinearity mitigation in ultra-long coherent optical transmission
systems,” European Conference on Optical Communication (ECOC2013), We.3.C.3, London,UK
(22-26 Sept. 2013).
[b5] Y. Mori, C. Han, H. Lu, and K. Kikuchi, “Wavelength demultiplexing of Nyquist WDM signals
under large frequency oﬀsets in digital coherent receivers,” European Conference on Optical
Communication (ECOC2013), Mo.4.C.6, London,UK (22-26 Sept. 2013).
[b6] K. Kikuchi, “Coherent optical communications: past, present and future (Plenary Talk),” The
Paciﬁc Rim Conference on Lasers and Electro-Optics (CLEO-PR 2013) & Optoelectronics and
Communications Conference (OECC 2013)/ Photonics in Switching (PS 2013), Kyoto, Japan
(30 June-4 July 2013).
[b7] K. Kikuchi and S. Y. Set, “Proposal of optical-sampling-based constellation monitor for DPQPSK signals,” The Paciﬁc Rim Conference on Lasers and Electro-Optics (CLEO-PR 2013) &
Optoelectronics and Communications Conference (OECC 2013)/ Photonics in Switching (PS
2013), TuR2-1, Kyoto, Japan (30 June-4 July 2013).
[b8] Md. S. Faruk and K. Kikuchi, “Blind equalization and carrier-phase recovery in QPSK coherent
optical receivers based on modiﬁed constant-modulus algorithm,” The Paciﬁc Rim Conference on
Lasers and Electro-Optics (CLEO-PR 2013) & Optoelectronics and Communications Conference
(OECC 2013)/ Photonics in Switching (PS 2013), ThR3-7, Kyoto, Japan (30 June-4 July 2013).
III. Domestic Conference
[c1] 石村昇太, 菊池和朗, “4 次元セット分割を適用した 8 状態トレリス符号化光変調方式の提案,” 電子
情報通信学会総合大会，B-10-46, 新潟大学, 2014 年 3 月 18 日-21 日.
[c2] 菊池和朗, “強度変調・直接検波光通信システムにおけるディジタル信号処理を用いた偏波多重分
離技術,” 電子情報通信学会総合大会，B-10-70, 新潟大学, 2014 年 3 月 18 日-21 日.
[c3] 菊池和朗, “ディジタルコヒーレント光通信システムの概要,” 電子情報通信学会ソイエティ大会，
BCI-1-1, 福岡工大, 2013 年 9 月 17 日-20 日.
[c4] 石村昇太, 菊池和朗, “コヒーレント光通信システムにおける多次元変復調方式の検討,” 信学技報,
vol.113, no.262, OCS2013-81, pp.159-164, 2013 年 10 月.
[c5] Hongbo Lu and Kazuro Kikuchi, “Pre-dispersed Mutual Phase-conjugation Scheme for Fibernonlinearity Mitigation in Coherent Optical Communication Systems – Proposal and Analysis
of Parallel Back-propagation Method,” 信学技報, vol.113, no.182, OCS2013-46, pp.55-60, 2013
年 8 月.
[c6] 菊池和朗, 川上彰二郎, “ストークス空間での多値化技術と大容量コヒーレント光伝送への応用,” 信
学技報, vol. 113, no. 156, OCS2013-28, pp. 49-54, 2013 年 7 月.
[c7] 菊池和朗, “［チュートリアル招待講演］ディジタルコヒーレント光通信の基礎,” 信学技報, vol.113,
no.156, OCS2013-29, pp.55-60, 2013 年 7 月.
[a1]
Multi-level signaling in the Stokes space
and its application to large-capacity
optical communications
Kazuro Kikuchi1,∗ and Shojiro Kawakami2
1
Department of Electrical Engineering and Information Systems, the University of Tokyo
7-3-1 Hongo, Bunkyo-Ku, Tokyo 113-8656, Japan
2 Photonic Lattice, Inc., ICR Building 2F, 6-6-3 Minami-Yoshinari, Aoba-Ku, Sendai, Miyagi
989-3204, Japan
∗ [email protected]
Abstract: The Stokes vector of an optical signal does not depend on its
absolute phase; therefore, we can construct the phase-insensitive optical
communication system, using the Stokes vector as a modulation parameter.
In such a system, multi-level optical signals can effectively be designed
in the three-dimensional Stokes space and demodulated either by direct
detection or by coherent detection, where low-complexity digital-signal
processing (DSP) is employed. Although this system has the disadvantage
that adaptive equalizers can hardly be implemented in the digital domain, it
is still an attractive solution to large-capacity (≥ 100 Gbit/s) and mediumor short-reach (≤ 100 km) transmission. In this paper, we discuss the
receiver configuration for the multi-level signal in the Stokes space and the
efficient DSP algorithm for demodulating such a signal. Simulation results
demonstrate that 2-, 4-, 8-, 16-, and 32-ary signals in the Stokes space have
good bit-error rate (BER) characteristics. Especially, the 16-ary signal at the
moderate symbol rate of 25 Gsymbol/s can reach the bit rate of 100 Gbit/s
even by using direct detection.
© 2014 Optical Society of America
OCIS codes: (060.2330) Fiber optics communications; (060.1660) Coherent communications;
(060.4080) Modulation.
References and links
1. E. Yamazaki, S. Yamanaka, Y. Kisaka, T. Nakagawa, K. Murata, E. Yoshida, T. Sakano, M. Tomizawa,
Y. Miyamoto, S. Matsuoka, J. Matsui, A. Shibayama, J. Abe, Y. Nakamura, H. Noguchi, K. Fukuchi, H. Onaka, K. Fukumitsu, K. Komaki, O. Takeuchi, Y. Sakamoto, H. Nakashima, T. Mizuochi, K. Kubo, Y. Miyata,
H. Nishimoto, S. Hirano, and K. Onohara, “Fast optical channel recovery in field demonstration of 100-Gbit/s
Ethernet over OTN using real-time DSP, ” Opt. Express 19, 13139-13184 (2011).
2. G. R. Welti and J. S. Lee, “Digital transmission with coherent four-dimensional modulation,” IEEE Trans. on
Information Theory IT-20, 497-502 (1974).
3. M. Karlsson and E. Agrell, “Which is the most power-efficient modulation format in optical links?,” Opt. Express
17, 10814-10819 (2009).
4. E. Agrell and M. Karlsson, “Power-efficient modulation formats in coherent transmission systems,” J. of Lightwave Technol. 27, 5115-5126 (2009).
5. K. Kikuchi, “Digital coherent optical communication systems: Fundamentals and future prospects,” IEICE Electron. Express 8, 1642-1662 (2011).
6. S. Benedetto and P. T. Poggiolini, “Multilevel polarization shift keying: Optimum receiver structure and performance evaluation,” IEEE Trans. on Comm. 42, 1174-1186 (1994).
#201666 - $15.00 USD
(C) 2014 OSA
Received 22 Nov 2013; revised 24 Jan 2014; accepted 4 Mar 2014; published 24 Mar 2014
7 April 2014 | Vol. 22, No. 7 | DOI:10.1364/OE.22.007374 | OPTICS EXPRESS 7374
7. S. Benedetto, A. Djupsj¨obacka, B. Lagerstr¨om, R. Paoletti, P. Poggiolini, and G. Mijic, “Multilevel polarization
modulation using a specifically designed LiNbO3 device,” IEEE Photon. Technol. Lett. 6,949-951 (1994).
8. S. Benedetto, R. Gaudino, and P. Poggiolini, “Direct detection of optical digital transmission based on polarization shift keying modulation,” IEEE J. Sel. Areas in Comm. 13, 531-541 (1995).
9. S. Betti, F. Curti, G. De Marchis, and E. Iannone, “Multilevel coherent optical system based on Stokes parameters
modulation,” J. Lightwave Technol. 8, 1127-1136 (1990).
10. S. Benedetto, R. Gaudino, and P. Poggiolini, “Polarization recovery in optical polarization shift-keying systems,”
IEEE Trans. on Comm. 45, 1269-1279 (1997).
11. C. Brosseau, Fundamentals of polarized light (John Wiley & Sons, Inc. 1998).
12. K. Kikuchi and S. Tsukamoto, “Evaluation of sensitivity of the digital coherent receiver,” J. of Lightwave Technol. 26, 1817-1822 (2008).
13. S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express 16, 804-817 (2008).
14. Y. Mori, C. Zhang, and K. Kikuchi, “Novel configuration of finite-impulse-response filters tolerant to carrierphase fluctuations in digital coherent optical receivers for higher-order quadrature amplitude modulation signals,”
Opt. Express 20, 26236-26251 (2012).
1.
Introduction
The complex amplitude of the optical electric field is represented by its in-phase (I) and quadrature (Q) components; thus, we have two degrees of freedom realizing quadrature amplitude
modulation (QAM) of a single-polarization optical carrier. On the other hand, the optical electric field has two degrees of freedom of polarizations. Now, the following question arises: How
can we utilize these four degrees of freedom for enhancing the performance of optical communication systems?
Figure 1 shows the genealogy of optical modulation formats studied so far. The dualpolarization (DP) 4-QAM (or quadrature phase-shift keying: QPSK) format has been commercialized at the single-carrier bit rate as high as 100 Gbit/s [1]. In such a scheme, IQ components of two polarizations are independently modulated in a binary manner. On the other
hand, coherent four-dimensional modulation was investigated in [2], where constellation points
were designed in the four-dimensional vector space. Applying such an idea to coherent optical
communication systems, we can constitute the four-dimensional vector space composed of IQ
vectors of two polarizations. Recently, the most power-efficient modulation format called the
polarization-switched QPSK (PS-QPSK) format has been found out in the four-dimensional
vector space [3, 4]. The receiver sensitivity of all of these modulation schemes depends on the
carrier phase, and the newly-developed digital coherent receiver can demodulate them relying
upon the digital phase-estimation technique [5].
Polarization
multiplexing
Polarization
modulation
Polarizationmultiplexed IQ
modulation
IQ
modulation
Fourdimensional
vector
modulation
Intensity
modulation
Stokes vector
modulation
Fig. 1. Genealogy of optical modulation formats. IQ modulation and four-dimensional
modulation are sensitive to the carrier phase. On the other hand, Stokes-vector modulation is independent of the absolute value of the carrier phase.
Meanwhile, optical communication systems based on multi-level modulation of the state of
polarization (SOP) were studied a long time ago. The SOP including the intensity is expressed
in terms of the Stokes vector S = [S1 , S2 , S3 ]T , and we can effectively design the multi-level
#201666 - $15.00 USD
(C) 2014 OSA
Received 22 Nov 2013; revised 24 Jan 2014; accepted 4 Mar 2014; published 24 Mar 2014
7 April 2014 | Vol. 22, No. 7 | DOI:10.1364/OE.22.007374 | OPTICS EXPRESS 7375
signal in the three-dimensional Stokes space [6]. The signal modulated in the Stokes space is
independent of the absolute phase of the carrier, but is dependent on the relative phase and the
amplitude ratio between the x- and y-polarization component. Such Stokes-vector modulation
is a natural extension of intensity modulation, which is immune to the carrier phase, adding two
degrees of freedom of SOP to one-dimensional intensity modulation as shown in Fig. 1.
Multi-level SOP modulation was demonstrated by using a LiNbO3 device [7]. In [8] and
[9], direct detection and coherent detection of the Stokes vector were studied, respectively, and
their bit-error rate (BER) performances were analyzed theoretically. In addition, the method for
tracking SOP fluctuations during propagation through a fiber was discussed in [10].
Following such preceding researches, this paper puts Stokes-vector modulation in the spotlight again with the help of state-of-the-art digital signal-processing (DSP) technologies, which
have been introduced in 100-Gbit/s digital coherent receivers. We develop an efficient DSP algorithm, which enables to track SOP fluctuations stemming from the random change in fiber
birefringence and demodulate multi-level Stokes-vector modulation signals. Our simulation results show that the DSP circuit implemented in direct-detection receivers or coherent receivers
can track such SOP fluctuations, using the decision-feedback control in a symbol-by-symbol
manner, and that we can obtain high system performance and stability owing to the fast tracking speed of the DSP circuit. In addition, our DSP algorithm can cope with any Stokes-vector
modulation format in a unified manner under severe conditions of carrier-phase fluctuations
as well as SOP fluctuations. In fact, we demonstrate that 2-, 4-, 8-, 16-, and 32-ary signals in
the Stokes space have good BER characteristics. The bit rate as high as 100 Gbit/s is obtained
especially when we employ the 16-ary signal at the moderate symbol rate of 25 Gsymbol/s
and the sampling rate of 25 GSa/s even in direct-detection systems. Although the Stokes-vector
modulation system has the disadvantage that adaptive equalizers can hardly be implemented in
the digital domain, it is a cost-effective solution to large-capacity (≥ 100 Gbit/s) and mediumor short-reach (≤ 100 km) transmission, where the time-varying effect from polarization-mode
dispersion (PMD) is ignored.
This paper is organized as follows: Section 2 reviews the definition of Stokes vectors, the
method of Stokes-vector modulation, and the method of Stokes-vector detection. In Sec. 3, we
propose a direct-detection receiver and a coherent receiver, which can demodulate the multilevel signal in the Stokes space using low-complexity DSP. In Sec. 4, after analyzing noise
characteristics of the Stokes vector, we perform intensive computer simulations on BERs of 2-,
4-, 8-, 16-, and 32-ary signals in the Stokes space. Section 5 concludes the paper.
2.
Review on methods of Stokes-vector modulation and detection
2.1. Definition of Stokes vectors
Let Ex and Ey respectively be the x-polarization and y-polarization component of the complex
amplitude of the signal electric field. The Jones vector of the signal is then expressed as
Ex
E=
(1)
Ey
and transformed into the Stokes vector S = [S1 , S2 , S3 ]T [11] as
2
S1 = |Ex |2 − Ey ,
S2 = 2Re (Ex∗ Ey )
= 2 |Ex | Ey cos δ ,
S3
=
(3)
2Im (Ex∗ Ey )
= 2 |Ex | Ey sin δ ,
#201666 - $15.00 USD
(C) 2014 OSA
(2)
(4)
Received 22 Nov 2013; revised 24 Jan 2014; accepted 4 Mar 2014; published 24 Mar 2014
7 April 2014 | Vol. 22, No. 7 | DOI:10.1364/OE.22.007374 | OPTICS EXPRESS 7376
where δ is the phase difference between y and x polarizations defined as
δ = arg (Ey /Ex ) .
(5)
The amplitude of the Stokes vector is equal to the signal intensity as shown by
q
S12 + S22 + S32
S0 =
2
= |Ex |2 + Ey .
(6)
The Stokes vector can also be obtained from light intensities as follows: Let Ix be the light
intensity of the x-polarization component of the signal, I45◦ that of the 45◦ linearly-polarized
component with respect to the x axis, and IR that of the right-circularly polarized component,
respectively. The Stokes parameters S1 , S2 , and S3 can be related to these intensities [11] as
S1
=
2Ix − S0 ,
(7)
S2
S3
=
=
2I − S0 ,
2IR − S0 .
(8)
(9)
45◦
From Eqs. (2)-(4) as well as Eqs. (7)-(9), we find that the Stokes vector is obtained by squarelaw detection of signal complex amplitudes and does not include the absolute phase of the
signal.
Stokes vectors are represented in the three-dimensional Stokes space. Especially when Stokes
vectors are normalized such that S0 = 1, they are distributed on the unit sphere called the
Poincar´e sphere, and each point on the sphere corresponds to SOP of the signal E.
Using a signal amplitude A given as
q
2
(10)
A = |Ex |2 + Ey .
we can express Eq.(1) as
θ

iφ 
2 E = Ae 
θ
i
δ
e sin
2

cos


.

(11)
In Eq. (11), φ is the absolute phase of the signal, and θ (0 ≤ θ ≤ π ) and δ (−π /2 ≤ δ ≤ π /2)
respectively determine the power ratio and the phase difference between y and x polarizations.
Then, using Eqs. (2)-(4), we obtain the Stokes vector corresponding to Eq. (11) as


cos θ
S = S0  sin θ cos δ  ,
(12)
sin θ sin δ
where S0 = A2 . The Stokes vector S given by Eq. (12) is illustrated in Fig. 2.
Since the Stokes vector does not include the absolute phase of the carrier φ , the demodulation
of Stokes-vector-modulated signals does not require the carrier estimation process similarly to
the demodulation of intensity-modulated signals. On the other hand, the Stokes vector can be
multi-level modulated in the three-dimensional Stokes space, whereas the intensity is modulated only one-dimensionally. This fact means that Stokes-vector modulation has much more
freedom to design multi-level signals than intensity modulation in spite of its phase-insensitive
nature.
#201666 - $15.00 USD
(C) 2014 OSA
Received 22 Nov 2013; revised 24 Jan 2014; accepted 4 Mar 2014; published 24 Mar 2014
7 April 2014 | Vol. 22, No. 7 | DOI:10.1364/OE.22.007374 | OPTICS EXPRESS 7377
S3
S0
δ
θ
S1
S2
Fig. 2. Illustration of the signal SOP and the intensity in the Stokes space.
2.2. Multi-level modulation of Stokes vectors
In the Stokes-vector modulation scheme, a symbol mapper at the transmitter maps a symbol
into a constellation point in the Stokes space shown by Fig. 2, using a prescribed mapping table.
Then, the desired Stokes vector is generated through modulation of S0 , θ , and δ in Eq. (12).
Following Eq. (11), we can construct the Stokes-vector modulator shown in Fig. 3 [7]. The
amplitude modulator (AM1) modulates the signal intensity S0 . The amplitude modulator (AM2)
is a Mach-Zehnder modulator which splits the intensity-modulated signal into two output ports
with an arbitrary splitting ratio by controlling θ . The phase modulator (PM) gives the phase
difference δ between the split signals. The linear polarization of one of the paths is rotated
by 90◦ by a half-wave plate (λ /2). These signals from the two paths have orthogonal linear
polarizations and are combined with a polarization-beam combiner (PBC).
θ
cos AM1
S0
PBC
2
λ/2
θ 2
PM
AM2
θ
sin δ
Fig. 3. Structure of the Stokes-vector modulator. AM: amplitude modulator, PM: phase
modulator, λ /2: half-wave plate, and PBC: polarization-beam combiner. Symbols k and ⊥
respectively show parallel and perpendicular linear polarizations.
For multi-level modulation of Stokes vectors, we should maximize the Euclidean distance
between nearest constellation points in the Stokes space under the condition that S0 is constant.
Figure 4 shows constellation maps for binary, quad, and octal Stokes-vector modulation [8]. In
the case of binary modulation, two constellation points are located at antipodal points. Constellation points in quad modulation are set at four vertices of a regular tetrahedron, whereas
those in octal modulation at eight vertices of a regular hexahedron. Hereafter, we call the unit
vector, which is directed from the origin to each constellation point in the Stokes space, the
basis Stokes vector. In addition to such multi-level SOP modulation, multi-level modulation of
S0 can increase the number of modulation levels.
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7 April 2014 | Vol. 22, No. 7 | DOI:10.1364/OE.22.007374 | OPTICS EXPRESS 7378
S3
S2
S1
S3
S2
S1
S3
S2
S1
Fig. 4. Constellation maps in the Stokes space. We show those of binary, quad, and octal
modulation. In addition, S0 can be modulated at the same time.
2.3. Detection of Stokes vectors
Figure 5 shows the schematic diagram of the Stokes analyzer [8, 11], which can determine the
Stokes vector based on direct detection. In the first branch, we measure the signal intensity
S0 with a photodiode (PD). Inserting a polarizer (0◦ Pol), whose transmission axis is the x
axis, in the second branch, we measure the intensity Ix . Using a polarizer (45◦ Pol), whose
transmission-axis is rotated by 45◦ with respect to the positive x axis, we detect the intensity
I45◦ in the third branch. With a quarter-wave plane (λ /4), whose fast axis is aligned to the x
axis, and a 45◦ -rotated polarizer (45◦ Pol), we measure the intensity IR in the fourth branch.
The Stokes vector is then calculated from Eqs. (6)-(9).
On the other hand, when we use a homodyne receiver comprising phase and polarization
diversities [5], the Stokes vector is obtained from measured complex amplitudes Ex and Ey by
using Eqs. (2)-(4).
PD
S0
PD
Ix
45 Pol
PD
I 45°
λ / 4 45° Pol
PD
IR
0° Pol
°
Fig. 5. Schematic diagram of the Stokes analyzer. From the four outputs from PDs, we can
determine the Stokes vector using Eqs. (6)-(9).
3.
Receivers for the demodulation of Stokes-vector-modulated signals
3.1. Direct-detection receiver
Figure 6 shows our proposed direct-detection-based receiver for the demodulation of Stokesvector-modulated signals. The incoming signal is pre-amplified by an Erbium-doped fiber amplifier (EDFA). After passing through an optical compensator for group-velocity dispersion
(GVD) of the fiber, the signal is filtered out to reduce amplified spontaneous emission (ASE)
from the EDFA. Then, the signal is incident on the Stokes analyzer given in Fig. 5.
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7 April 2014 | Vol. 22, No. 7 | DOI:10.1364/OE.22.007374 | OPTICS EXPRESS 7379
If the carrier-to-noise ratio (CNR) at the optical stage is determined from the average signal
power and the associated shot noise, the reduction of the CNR in each branch is as low as 1/NF
owing to optical pre-amplification before the power splitter, where NF is the noise figure of the
EDFA. Throughout this paper, however, we used the CNR value of the signal after detection;
therefore, its value is degraded by 1/NF from the optical CNR.
Four outputs from the Stokes analyzer are converted to digital data using four-channel
analog-to-digital converters (ADCs). The clock extracted from the second branch controls sampling instances of the ADCs. The sampling rate is thus one sample per symbol. The ADC outputs are sent to the Stokes-vector calculator, where the Stokes vector is determined by Eqs. (2)(4). Next, the output of the Stokes-vector calculator is sent to the following DSP circuit.
Optical
signal
EDFA
Optical GVD
compensator
Optical
filter
CLK
Stokes
analyzer
Discriminator
ADC
Stokes-vector
calculator
Decoded
symbol
Basis Stokesvector updater
Fig. 6. Direct-detection-based receiver for the demodulation of Stokes-vector-modulated
signals.
3.2. Coherent-detection receiver
Figure 7 shows the coherent-detection-based receiver for the demodulation of Stokes-vectormodulated signals. This coherent receiver is the homodyne receiver employing phase and polarization diversities. The optical CNR is degraded by 3 dB after detection [12]. If we introduce
an optical pre-amplifier having the noise figure of NF before the coherent receiver, the total
reduction of the CNR is 1/NF, which is the same as that in the direct-detection-based receiver
[12]. Hereafter, we use the CNR after coherent detection for describing the signal quality.
Four outputs from the coherent receiver are converted to digital data using four-channel
ADCs [5]. The clock for the ADCs can be extracted from the power of the x-polarization
component, which is obtained from the analog output of the coherent receiver. After fixed
equalization of accumulated GVD of the fiber and spectral filtering of the signal in the digital
domain, the Stokes vector S is calculated from Eqs. (2)-(4) and sent to the following DSP circuit. Note that in contrast to the direct-detection-based receiver shown in Fig. 6, the coherent
receiver enables fixed GVD compensation and signal filtering in the digital domain.
In principle, it is possible to introduce adaptive equalizers controlling the Jones vector into
the DSP circuit; actually, we can achieve adaptive equalization using finite-impulse-response
(FIR) filters in the butterfly-configuration adapted by the decision-directed least-mean-square
(LMS) algorithm [13]. However, the decision-feedback process involved in such adaptive
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7 April 2014 | Vol. 22, No. 7 | DOI:10.1364/OE.22.007374 | OPTICS EXPRESS 7380
equalization seriously suffers from carrier-phase fluctuations [14]. In such a case, computational complexity of the DSP circuit becomes larger to cope with this problem, and the advantage of the Stokes-vector-modulation scheme is greatly diminished.
Therefore, as far as we do not include adaptive equalizers in the receiver, the direct-detectionbased receiver is more cost-effective than the coherent-detection-based receiver. The advantage
of the coherent receiver is that fixed GVD compensation can be done in the digital domain;
on the other hand, the direct-detection receiver is more suitable for short-distance applications
and/or 1.3-µ m transmission systems, where optical GVD compensation is not necessary.
Coherent
Optical receiver
signal
ADC
Fixed GVD
compensator/
Filter
Stokes-vector
calculator
CLK
Discriminator
Decoded
symbol
Basis Stokesvector updater
Fig. 7. Coherent-detection-based receiver for the demodulation of Stokes-vector-modulated
signals.
3.3. DSP algorithm for symbol discrimination and polarization tracking
In both of the receivers described in 3.1 and 3.2, the calculated Stokes vector is processed by
the same DSP algorithm as shown below.
We demodulate the signal by comparing the measured Stokes vector with basis Stokes vectors. In order to prepare the basis Stokes vectors at the receiver, the training sequence with a
fixed SOP modulation pattern is transmitted through the fiber in advance. Assume that basis
Stokes vectors thus obtained at the receiver are given as v1 (k), v2 (k), · · ·, vn (k), where k denotes the number of samples and n the number of basis vectors. In binary, quad, and octal SOP
modulations, n are 2, 4, and 8, respectively. Then, after taking the inner product between the
received normalized Stokes vector S(k)/S0 and all of the n basis Stokes vectors, we find the
basis vector vm (k), which gives the maximum value of the inner products. The symbol is determined from the decided m-th basis Stokes vector, using the mapping table discussed in 2.2.
When the intensity is simultaneously modulated, the measured S0 is level-discriminated to find
the symbol.
The basis Stokes vectors are fluctuating slowly due to the change in fiber birefringence, even
if we have determined them using the training sequence; therefore, we need to track such fluctuations of the basis vectors. In the k-th sample, we have decided that the measured normalized
Stokes vector is the m-th basis vector. Then, the m-th basis vector in the next symbol duration
is updated by using the measured normalized Stokes vector as
vm (k + 1) =
v m (k) + µ e(k)
.
kvm (k) + µ e(k)k
(13)
In Eq. (13), µ is a step-size parameter, and the error signal e (k) given as
e (k) =
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S (k)
− vm (k)
S0 (k)
(14)
Received 22 Nov 2013; revised 24 Jan 2014; accepted 4 Mar 2014; published 24 Mar 2014
7 April 2014 | Vol. 22, No. 7 | DOI:10.1364/OE.22.007374 | OPTICS EXPRESS 7381
attempts to eliminate the deviation of the basis Stokes vector from its correct position. A smaller
value of µ improves the signal-to-noise ratio of vm (k) but reduces the SOP tracking speed;
therefore, we need to choose an optimum value of µ , depending on the SOP fluctuation speed.
It should be noted that we do not update the m-the basis Stokes vector and keep its value
until the next symbol where the m-the basis Stokes vector is selected again. However, since
the speed of SOP fluctuations is much slower than the symbol rate, such thinned-out operation
of the update process never degrades the BER performance. In addition, the decision-feedback
process included in Eq. (13) does not have any serious problem for circuit implementation
owing to the slow SOP fluctuation speed.
Compared with the digital coherent receiver for the DP-QPSK signal, Fig. 7 has much simpler structure, because DSP circuits for carrier phase estimation and adaptive equalization are
not included. On the other hand, since adaptive equalization cannot be implemented in our receiver, the Stokes-vector modulation format may be mainly applied to transmission systems
shorter than 100 km, where PMD is not significant.
4.
Receiver-sensitivity analyses of the Stokes-vector modulation scheme
4.1. Noise distribution in the Stokes space
The Stokes vector is obtained by square-law detection of the complex amplitude of the optical
electric field. Therefore, the noise distribution of the Stokes vector is broader than that of the
complex amplitude, which results in the degradation of the BER performance of the Stokesvector modulation scheme compared with the IQ modulation scheme. In 4.1, we compare the
noise distribution of the Stokes vector with that of the IQ vector through numerical calculations
in order to have comprehensive understanding of the BER performance in 4.2.
Adding white Gaussian noise to a unity signal amplitude with x polarization, we define the
Jones vector as
1 + nx
E=
.
(15)
ny
In Eq. (15), nx and ny are complex-valued Gaussian noises with x and y polarizations, respectively. The variance of the real part and that of the imaginary part of these noises are given as
σ 2 . Since CNR per polarization (CNR/pol) means the ratio between the average signal power
and the power of the single-polarization noise, it is given as
CNR/pol =
1
.
2σ 2
(16)
Figure 8 shows probability-density functions for the in-phase and quadrature components of
the complex amplitude of the signal. Blue curves are simulation results, whereas red ones are
Gaussian fits to them, when CNR/pol=7, 10, and 15 dB.
Next, we transform the Jones vector given by Eq. (15) into the Stokes vector, using Eqs. (2)(4), and calculate probability-density functions for Stokes parameters S1 , S2 , and S3 , when
CNR/pol=7, 10, and 15 dB. Blue curves are simulation results, whereas red ones are Gaussian
fits to them. These probability-density functions differ from Gaussian distributions and such
difference is remarkable when CNR/pol becomes lower. As shown by Figs. 8 and 9, probabilitydensity functions of the Stokes parameters are much broader than those of the IQ components
when CNR/pol is the same. This means that the receiver sensitivity in the Stokes-vector modulation scheme is generally lower than that in the IQ-modulation scheme. In addition, we find
that the probability-density function of the Stokes vector has a steeper slope when the S1 value
decreases along the radial direction of the Stokes vector. This fact implies that the receiver sensitivity in the Stokes-vector modulation scheme is dependent not only on the distance between
nearest constellation points but also on the configuration of constellation points.
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7 April 2014 | Vol. 22, No. 7 | DOI:10.1364/OE.22.007374 | OPTICS EXPRESS 7382
;<=>?@ABCDE
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Fig. 8. Probability-density functions for the in-phase and quadrature components of the
complex amplitude of the signal. Blue curves are simulation results, and red ones are Gaussian fits to them. Upper figures correspond to the in-phase component, whereas the lower
figures to the quadrature component. Values of CNR/pol are 7, 10, and 15 dB.
In fact, Fig. 10 shows the comparison of probability-density functions of Stokes vectors
among binary (black curve), quad (blue curve), and octal (red curve) modulation formats. In all
of the modulation formats, we assume that CNR/bit/pol is 13 dB and the noise-free amplitude
of the signal is unity. Note that CNR/pol defined in 4.1 is divided by the number of bits per
symbol to evaluate the receiver sensitivity. We calculate probability-density functions along the
line connecting two nearest constellation points, when the modulated SOP is in the positive side
of the line. The origin of the distance is the center of the segment of the line. The symbol error
occurs when the Stokes vector is distributed in the negative distance region. The binary and
quad formats have almost the same distribution where the slope is positive, indicating that both
formats have the very similar BER characteristics. On the other hand, the probability density
of the octal format is higher than those of the binary and quad formats, when the distance is
negative; hence, we can predict that the BER performance of the octal format is worse than
those of the binary and quad formats.
4.2. BER characteristics
Through computer simulations, we calculate BER characteristics of the binary, quad, and octal
SOP-modulation schemes as a function of the CNR/bit/pol. We also analyze 16-ary and 32-ary
Stokes-vector modulation schemes, where the intensity in the octal SOP-modulation scheme is
modulated in 2-level and 4-level manners, respectively. In the 16-ary modulation scheme, the
intensity S0 is modulated between I0 and 3I0, and the threshold of the level discrimination is set
at 2I0 to obtain the best BER performance. In the 32-ary modulation scheme, on the other hand,
the intensity modulation is performed among four levels, I0 , 2I0 , 4I0 , and 6I0. Thresholds for
the level discrimination are 1.5I0, 3I0 , and 5I0. Those parameters are determined so that the best
BER performance is obtained. In these modulation schemes accompanying the intensity modulation, we define CNR using the average signal power. Note that since we define CNR/bit/pol
after detection, BER characteristics are not dependent on the choice of receivers between Fig. 6
and Fig. 7.
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Received 22 Nov 2013; revised 24 Jan 2014; accepted 4 Mar 2014; published 24 Mar 2014
7 April 2014 | Vol. 22, No. 7 | DOI:10.1364/OE.22.007374 | OPTICS EXPRESS 7383

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Fig. 9. Probability-density functions of Stokes parameters. Blue curves are simulation results for S1 , S2 , and S3 , whereas red ones are Gaussian fits to them. Values of CNR/pol are
7, 10, and 15 dB.
In the simulations, we first prepare the sequence of the Jones vector having a certain value
of CNR/bit/pol and the number of symbols N = 220 . The value of CNR/bit/pol is controlled by
the amount of Gaussian noise. We also assume that δ f · T = 10−2 . The parameter δ f denotes
the linewidth of the transmitter laser in the direct-detection case, whereas it denotes the sum
of linewidths of the transmitter laser and the local oscillator in the coherent-detection case.
The parameter T is the symbol duration. Supposing that the symbol rate is 10 Gsymbol/s,
the linewidth of the laser in the direct-detection system is as wide as 100 MHz, and that in
the coherent system is 50 MHz. Those values are much larger than the linewidth of common
distributed feedback (DFB) semiconductor lasers.
Next, we scramble the SOP of the signal to emulate the random fluctuation of fiber birefringence. Let the fiber birefringence be expressed by the Jones matrix as

 φr
− sin θ2r
ei 2 cos θ2r
.
(17)
J=
φr
e−i 2 cos θ2r
sin θ2r
The polar angle and the azimuthal angle of SOP randomly fluctuate on the Poincar´e sphere
in a symbol-by-symbol manner through φr and θr . Parameters φr and θr obey the following
equations:
φr (k + 1) = φr (k) + ∆φr (k) ,
θr (k + 1) = θr (k) + ∆θr (k) ,
(18)
(19)
where k is the number of symbols, and ∆φr (k) and ∆θr (k) are real-valued Gaussianq
noises.
In our simulations, we assume that their standard deviations are commonly given as σ02 =
2 × 10−3 [rad]. Since φr (k) and θr (k) are random-walk parameters, their variance σ (N)2 at the
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Received 22 Nov 2013; revised 24 Jan 2014; accepted 4 Mar 2014; published 24 Mar 2014
7 April 2014 | Vol. 22, No. 7 | DOI:10.1364/OE.22.007374 | OPTICS EXPRESS 7384
"
!
#$%&'(
)*&+
,-.&/
Fig. 10. Probability-density functions for Stokes vectors in binary (black curve), quad (blue
curve), and octal (red curve) SOP modulation formats along the line connecting two nearest
constellation points in each modulation format. The horizontal axis represents the distance
along the line where the origin is the center of the segment of the line. The signal amplitude
is unity, and CNR/bit/pol is 13 dB in all of the cases.
N-th symbol is written as
σ (N)2 = σ02 N [rad2 ] .
(20)
q
Since N = 220 , σ (N)2 ≃ 2 [rad]. In the 10-Gsymbol/s system, the total time span for 220
symbols is about 100 µ s, and the standard deviation of 2 rad in 100 µ s is much larger than SOP
fluctuations in real systems. Figure 11 shows symbol-by-symbol plots of SOP on the Poincar´e
sphere for 220 symbols, when the Jones matrix given by Eq. (17) is operated to the initial Jones
vector E = [1, 0]T . We find that SOP randomly fluctuates on the Poincar´e sphere starting from
S = [1, 0, 0]T .
Fig. 11. Fluctuations of SOP due to the change in fiber birefringence. SOP is plotted in a
symbol-by-symbol
manner for N = 220 symbols, when the initial Jones vector is E = [1, 0]T
q
and
σ02 = 2 × 10−3 [rad].
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Received 22 Nov 2013; revised 24 Jan 2014; accepted 4 Mar 2014; published 24 Mar 2014
7 April 2014 | Vol. 22, No. 7 | DOI:10.1364/OE.22.007374 | OPTICS EXPRESS 7385
Finally, the Jones vector including SOP fluctuations is converted to the Stokes vector. The
Stokes vector is demodulated by the algorithm given in 3.3, where the step-size parameter µ is
1/27 . Then, the number of bit errors is counted.
Figure 12 shows BERs for binary, quad, octal, 16-ary, and 32-ary modulation formats calculated as a function of CNR/bit/pol. We have confirmed that these curves are influenced neither
by laser linewidths nor by SOP fluctuations. For comparison, the BER curve for the DP-QPSK
format is also shown in Fig. 12, where laser linewidths and SOP fluctuations are neglected.
We find that the binary and quad formats have almost the same BER characteristics. On the
other hand, the receiver sensitivity of the octal format is worth than those of the binary and
quad formats by 2 dB at BER=10−5 . These results are in good agreement with the prediction
from probability-density functions of Stokes vectors, which was discussed in 4.1.
The 16-ary modulation format can reach 100 Gbit/s at the moderate symbol rate of 25 Gsymbol/s and the ADC sampling rate of 25 GSa/s even in direct-detection systems. The sensitivity
degradation from the binary modulation scheme is 3 dB, and that from the DP-QPSK format
is 7 dB at BER=10−5 . This format seems very attractive for medium- or short-reach (≤ 100
km) 100-Gbit/s transmission because of its low computational complexity of DSP, although the
receiver sensitivity is lower than the DP-QPSK format and adaptive equalization is difficult to
achieve.
41
MNOPQR
STOPQR
UVWPX
YZP[
\]^PQR
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43
L
K
J
I
H 46
G
F
E
45
40
0
12
10
32
789:;<=:>?@ ABCD
Fig. 12. BERs for binary, quad, octal, 16-ary, and 32-ary modulation formats calculated
as a function of CNR/bit/pol. The BER curve for the DP-QPSK format is also shown for
comparison.
5.
Conclusions
In this paper, we have discussed the receiver configuration for the multi-level signal in the
Stokes space and the efficient DSP algorithm for demodulating such a signal. Either directdetection receivers or coherent receivers can demodulate such a multi-level signal using lowcomplexity DSP. Intensive computer simulations show that 2-, 4-, 8-, 16-, and 32-ary signals
in the Stokes space have good bit-error rate (BER) characteristics. The bit rate as high as 100
Gbit/s is obtained especially when we employ the 16-ary signal in the Stokes space at the moderate symbol rate of 25 Gsymbol/s and ADC sampling rate of 25 GSa/s. Although the Stokesvector modulation system has the disadvantage that adaptive equalizers cannot be implemented
in the digital domain and its receiver sensitivity is lower than that of the IQ modulation sys#201666 - $15.00 USD
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7 April 2014 | Vol. 22, No. 7 | DOI:10.1364/OE.22.007374 | OPTICS EXPRESS 7386
tem, it is a cost-effective solution to large-capacity (≥ 100 Gbit/s) and medium- or short-reach
(≤ 100 km) transmission.
Acknowledgments
This work was supported in part by Grant-in-Aid for Scientific Research (A) (25249038), the
Ministry of Education, Culture, Sports, Science and Technology in Japan.
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[a2]
In-Band Estimation of Optical
Signal-to-Noise Ratio From
Equalized Signals in Digital
Coherent Receivers
Volume 6, Number 1, February 2014
Md. Saifuddin Faruk
Yojiro Mori
Kazuro Kikuchi, Fellow, IEEE
DOI: 10.1109/JPHOT.2014.2304557
1943-0655 Ó 2014 IEEE
IEEE Photonics Journal
In-Band Estimation of OSNR
In-Band Estimation of Optical
Signal-to-Noise Ratio From
Equalized Signals in Digital
Coherent Receivers
Md. Saifuddin Faruk, 1 Yojiro Mori, 2 and Kazuro Kikuchi,3 Fellow, IEEE
1
Department of Electrical and Electronic Engineering, Dhaka University of Engineering and
Technology, Gazipur 1700, Bangladesh
2
Department of Electrical Engineering and Computer Science, Nagoya University,
Nagoya 466-8550, Japan
3
Department of Electrical Engineering and Information Systems, The University of Tokyo,
Tokyo 113-0033, Japan
DOI: 10.1109/JPHOT.2014.2304557
1943-0655 Ó 2014 IEEE. Translations and content mining are permitted for academic research only.
Personal use is also permitted, but republication/redistribution requires IEEE permission.
See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
Manuscript received December 24, 2013; revised January 28, 2014; accepted January 29, 2014. Date
of publication February 4, 2014; date of current version February 12, 2014. Corresponding author:
Md. S. Faruk (e-mail: [email protected]).
Abstract: We propose a novel method of in-band estimation of optical signal-to-noise ratio
(OSNR) using a digital coherent receiver, where OSNR is determined from second- and
fourth-order statistical moments of equalized signals in any modulation format. Our
proposed method is especially important in recently-developed Nyquist wavelength-division
multiplexed (WDM) systems and/or reconfigurable optical-add/drop-multiplexed (ROADM)
networks, because in these systems and networks, we cannot apply the conventional OSNR
estimation method based on optical-spectrum measurements of the in-band signal and the
out-of-band noise. Effectiveness of the proposed method is validated with computer
simulations of Nyquist-WDM systems and ROADM networks using 25-Gbaud quadrature
phase-shift keying (QPSK) and 16 quadrature-amplitude modulation (16-QAM) formats.
Index Terms: Optical signal-to-noise ratio, digital coherent receivers, digital signal
processing, Nyquist-WDM, ROADM.
1. Introduction
1.1. General Background
In long-haul optical transmission systems and networks, erbium-doped fiber amplifiers (EDFAs)
compensate for loss of optical fibers for transmission and photonic-node components; then, the
optical signal is contaminated by amplified-spontaneous-emission (ASE) noise accompanied with
optical gain. The quality of the optical signal is generally assessed by means of optical signal-tonoise ratio (OSNR) which is defined as the ratio of the signal power and the ASE noise power in a
reference optical bandwidth. OSNR is one of the key parameters for optical performance monitoring,
which enables fault management of optical transmission systems and networks including fault
diagnosis and localization [1].
In a conventional wavelength-division-multiplexed (WDM) system employing guard bands
between WDM channels, the most common method of OSNR estimation is based on optical
Vol. 6, No. 1, February 2014
7800109
IEEE Photonics Journal
In-Band Estimation of OSNR
Fig. 1. Optical spectra in (a) conventional-WDM systems having guard bands, (b) Nyquist-WDM
systems without any guard band, and (c) ROADM networks. Blue curves show WDM signal spectra,
whereas red ones ASE-noise spectra. In (a), the spectral density of the in-band ASE noise is
determined from that of the out-of-band ASE noise. In (b), there is no guard band between WDM
channels. In (c), the out-of-band ASE noise is filtered out at each node, and the amount of the in-band
ASE noise depends on the wavelength path. Therefore, in (b) and (c), we cannot apply the conventional
approach for OSNR monitoring.
spectral analyses of out-of-band ASE noise between WDM channels [2]. Fig. 1(a) shows spectra of
WDM channels (blue curve) and ASE noise (red curve) in the conventional WDM system. We
measure power spectral densities in guard bands at i , where i is the central wavelength of
a target WDM channel and is generally set to the half of the WDM channel spacing. From such
power spectral densities of the out-of-band ASE noise, we estimate the power spectral density of
ASE noise in the signal bandwidth. Then, OSNR is obtained as the ratio of the signal power to the
in-band ASE noise power.
On the other hand, Fig. 1(b) shows spectra of WDM channels (blue curve) and ASE noise (red
curve) in Nyquist-WDM systems. We cannot measure the out-of-band ASE noise, since there is no
guard band between WDM channels. Fig. 1(c) shows those in reconfigurable optical-add/dropmultiplexed (ROADM) networks, where the optical signal is filtered by an add/drop optical filter at
every node. Therefore, out-of-band ASE noise is cut off, while the amount of in-band ASE noise
strongly depends on the network path that the wavelength channel travels through. Thus, the
conventional OSNR estimation method fails both in Nyquist-WDM systems and ROADM networks.
Therefore, a novel OSNR estimation method applicable to such systems and networks is strongly
desired.
1.2. Our Proposed OSNR Estimation Method
Recent advancement of digital coherent technologies enables optical performance monitoring by
digital signal processing (DSP) of the received signal [3]–[5]. As of the OSNR estimation in digital
coherent receivers, generally two approaches have been investigated so far: One is the data-aided
estimation [6], [7], and the other is the non-data-aided estimation [8]–[10]; however, the data-aided
OSNR estimation method has drawbacks that it requires the modification of transmitters and
reduces the spectral efficiency.
In our previous work [8], we proposed a non-data-aided OSNR estimation scheme based on
measuring the second- and fourth-order statistical moments of equalized signals in digital coherent
receivers. The performance of such OSNR estimation is inherently insensitive to the phase noise of
transmitter lasers and local oscillators, since it involves only the measurement of second- and
fourth-order moments. Also, it is not affected by linear fiber transmission impairments such as
chromatic dispersion (CD) and polarization-mode dispersion (PMD), because OSNR estimation is
done after adaptive equalization. However, the proposed method has been applicable only to the
quadrature phase-shift keying (QPSK) signal and its effectiveness has been demonstrated only in
the conventional WDM system shown in Fig. 1(a).
In this paper, maintaining the above-mentioned advantages, we extend our previous scheme so
that it can be applied to multi-level vector modulation formats such as the M-ary PSK (M-PSK)
format and the M-ary quadrature-amplitude-modulation (M-QAM) format where M 4. In the
newly-developed algorithm, we need to change only one constant parameter, that is, kurtosis of
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In-Band Estimation of OSNR
Fig. 2. Schematic of the coherent optical receiver and typical DSP blocks for data recovery. Adaptive
equalizer outputs are used for OSNR monitoring. (LO: local oscillator, PBS: polarization-beam splitter,
TIA: trans-impedance amplifier, ADC: analog-to-digital converter.)
prescribed signal constellation points, in order to select a particular modulation format. We perform
extensive computer simulations using 25-Gbaud QPSK and 16-QAM formats and verify the
effectiveness of the proposed method. Through such computer simulations, OSNR estimation in
Nyquist-WDM systems and ROADM networks is successfully demonstrated for the first time to the
best of our knowledge. OSNR is estimated with very good accuracy over a wide range for both of
the cases.
The organization of our paper is as follows: In Section 2, analyzing statistical properties of the
signal and ASE noise, we derive the formula for estimating OSNR from the equalized signal in
digital coherent optical receivers. To verify the effectiveness of the proposed scheme, we perform
numerical simulations using QPSK and 16-QAM signals in point-to-point Nyquist-WDM systems
(Section 3) and ROADM networks (Section 4). Finally, Section 5 concludes our paper.
2. Principle of Proposed OSNR Estimation Scheme
Fig. 2 shows the schematic of the coherent optical receiver employing phase and polarization
diversities and typical DSP stages for data recovery in polarization-division multiplexed
transmission systems [11]. For OSNR monitoring, we use the signal at the adaptive equalizer
output. The digital filters used in the equalizer can compensate for a large amount of linear fiber
transmission impairments without any notable penalty [12] and the signal at this stage is mainly
contaminated by ASE noise. Therefore, the output of the adaptive equalizer is the earliest stage of
the DSP chain where the OSNR estimation is done. In addition, DSP stages for frequency-offset
compensation and carrier-phase estimation can be placed after the OSNR estimation stage.
The output signal from the adaptive filter can be approximated as
pﬃﬃﬃﬃ
pﬃﬃﬃﬃ
yn C an e jn þ N wn
(1)
where an is the M-PSK or M-QAM symbol amplitude, C the signal-power scale factor, N the noisepower scale factor, wn the ASE noise, n the phase noise stemming from phase fluctuations of a
transmitter laser and a local oscillator, and n the number of samples. When we neglect the
polarization-dependent loss (PDL), the coherent optical channel has the all-pass nature; and thus,
the equalizer has an impulse response of an all-pass filter [13]. Under such a condition, we find that
the stochastic property of the ASE noise wn at the equalizer output is the same as that at the input [14].
Therefore, we can regard that the symbol amplitude an and the noise amplitude wn are stochasticallyindependent random variables with a zero-mean value.
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The second-order moment 2 of yn can be expressed as
2 ¼ E yn yn
pﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ ¼ CE an ejn an ejn þ CN E an e jn wn þ CN E an e jn wn þ NE wn wn
n
o pﬃﬃﬃﬃﬃﬃﬃﬃ n
o
¼ CE jan j2 þ CN E an e jn wn þ E an e jn wn þ NE jwn j2
(2)
where E fg represents the ensemble average and the superscript ðÞ denotes the complex
conjugate. Since the signal and the noise obey a mutually-independent complex-valued stochastic
process, we have
E an e jn wn ¼ 0
(3)
j
E an e n wn ¼ 0:
(4)
We also assume that the signal an and the noise wn are normalized to have an equal variance
given by
n
o
n
o
E jan j2 ¼ E jwn j2 ¼ :
(5)
Thus, we can rewrite Eq. (2) as
2 ¼ ðC þ NÞ
(6)
and the carrier-to-noise ratio (CNR) is expressed as
CNR ¼
C
:
N
On the other hand, the fourth-order moment 4 of yn can be written as
n
2 o
4 ¼ E yn yn
n
pﬃﬃﬃﬃﬃﬃﬃﬃ 2 o
¼ C 2 E an an
þ 2C CN E an an an ejn wn þ E an an an ejn wn
n
n
2 o
2 o þ CN E an e jn wn
þ 4E an an wn wn þ E an e jn wn
n
pﬃﬃﬃﬃﬃﬃﬃﬃ 2 o
þ 2N CN E wn wn an ejn wn þ E wn wn an e jn wn þ N 2 E wn wn
:
(7)
(8)
In Eq. (8), since
E fan g ¼ E an ¼ E fwn g ¼ E wn ¼ 0
(9)
we have
E an an an e jn wn ¼ 0
E wn wn an e jn wn ¼ 0
E an an an e jn wn ¼ 0
E wn wn an e jn wn ¼ 0:
(10)
(11)
(12)
(13)
Also note that the real part anI and the imaginary part anQ of an are uncorrelated in M-ary PSK and
M-ary QAM signals when M 4. Then, we find that
n
2 o
n
2 o
E an ejn wn
¼ E a2n E e jn wn
¼0
(14)
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In-Band Estimation of OSNR
because
E a2n ¼ E a2nI a2nQ þ 2janI anQ ¼ 0:
(15)
Similarly, we have
E
n
an ejn wn
2 o
¼E
n
wn ejn
2 o
E
n
an
2 o
¼ 0:
(16)
In addition, it is evident that
n
o
¼ E jan j4
(17)
n
n
o
2 o
wn wn
¼ E jwn j4
(18)
E
E
n
an an
2 o
n
o
E an an wn wn ¼ E jan j2 jwn j2 :
(19)
Taking all of these equations into consideration, we can simplify Eq. (8) as
n
o
n
o
n
o
4 ¼ C 2 E jan j4 þ 4CNE jan j2 jwn j2 þ N 2 E jwn j4
¼ ka 2 C 2 þ 4 2 CN þ kw 2 N 2
(20)
where
n
o
E jan j4
ka ¼ n
o2
E jan j2
(21)
n
o
E jwn j4
kw ¼ n
o2
E jwn j2
(22)
and
are kurtoses of the signal and the noise, respectively. The Gaussian distribution of ASE noise yields
kw ¼ 2.
Solving Eqs. (6) and (20), we obtain
sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
1 222 4
C¼
(23)
2 ka
8
sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ9
222 4 =
1<
2 :
N¼
:
2 ka ;
(24)
Therefore, determining 2 and 4 from experimental results and using Eqs. (7), (23) and (24), we
can estimate CNR as
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
222 4
CNR ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ :
(25)
2 2 ka 222 4
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In a practical system, we can calculate second- and fourth-order moments from a received data
block of L symbols as
2 L1
1X
jyn j2
L n¼0
(26)
4 L1
1X
jyn j4
L n¼0
(27)
and
respectively. As shown in Eqs. (26) and (27), measuring second- and fourth-order moments does
not include any effect of the phase noise and thus the proposed scheme operates phase
insensitively.
Equation (25) is a generalized equation to calculate CNR of any arbitrary modulation format. The
value of ka is dependent on the modulation format; for example, in the case of QPSK, we have
ka ¼ 1 since an 2 f1; 1; j; jg. Then, CNR is expressed as
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
222 4
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ :
CNRQPSK ¼
(28)
2 222 4
On the other hand, for the 16-QAM signal, since an 2 f1 i; 1 3i; 3 i; 3 3ig, ka ¼ 1:32.
Then, CNR is given as
qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
222 4
CNR16QAM ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ :
(29)
2 0:68 222 4
We can thus cope with any arbitrary modulation format just by changing the value of ka . When the
launched power is so low that fiber nonlinear effects can be neglected, the OSNR value in dB can
be estimated from the CNR value [15] as
Rs
OSNRdB ¼ 10log10 ðCNRÞ þ 10log10
(30)
Br
where Rs is the symbol rate and ðRs =Br Þ is a scaling factor adjusting the measured noise bandwidth
to the reference bandwidth Br . The bandwidth Br is usually set to 12.5 GHz, which is equivalent to
the 0.1-nm OSA resolution bandwidth.
3. Simulation Results for Nyquist-WDM Systems
In order to confirm the effectiveness of our proposed method for estimating OSNR in Nyquist-WDM
systems, we conduct computer simulations under the following conditions: Each wavelength
channel consists of a dual-polarization single-carrier 25-Gbaud QPSK signal (i.e., the
bit rate ¼ 100 Gbit/s) or a dual-polarization dual-subcarrier 25-Gbaud 16-QAM signal (i.e., the
bit rate ¼ 400 Gbit/s). At the transmitter, the spectrum of the signal on each carrier or subcarrier is
shaped by the Nyquist filter with a roll-off factor of zero. The channels thus obtained are aligned in
the frequency domain without any guard band. As system impairments, CD, the laser phase noise,
and the ASE noise are taken into account. Assuming that the ASE noise is white Gaussian noise,
we control OSNR by changing its average power. At the receiver side, the signal is detected by an
optical homodyne receiver and delivered to the DSP circuit as shown in Fig. 2. In the DSP circuit,
the signal on each carrier or subcarrier are separated by the Nyquist filter with a roll-off factor of
zero; then, the separated signal is two-fold oversampled and fed into a 21-tap half-symbol-spaced
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Fig. 3. OSNR-estimation results for the QPSK Nyquist-WDM system. (a) Estimated OSNR as a
function of the setup value of OSNR. (b) Estimation error as a function of the setup value of OSNR.
(ASE: amplified-spontaneous emission noise, PN: phase noise, and CD: chromatic dispersion).
Fig. 4. OSNR-estimation results for the 16-QAM Nyquist-WDM system. (a) Estimated OSNR as a
function of the setup value of OSNR. (b) Estimation error as a function of the setup value of OSNR.
(ASE: amplified-spontaneous emission noise, PN: phase noise, and CD: chromatic dispersion).
finite-impulse-response (FIR) filter, which is adapted by the constant-modulus algorithm (CMA) [16]
for the QPSK signal or the radial-directed equalization (RDE) algorithm [17] for the 16-QAM signal.
From the equalized signal, we estimate OSNR using Eqs. (28)–(30).
Fig. 3 shows OSNR-estimation results for the QPSK format. OSNR values estimated by our
proposed algorithm are plotted as a function of the setup value of OSNR. In Fig. 3(a), the red curve
is obtained when only the ASE noise is considered and CD and the phase noise are ignored. In the
calculation of the blue curve, the 3-dB linewidth f for the transmitter laser as well as the local
oscillator is 500 kHz and CD is ignored. The black curve is obtained when f ¼ 500 kHz and
CD ¼ 500 ps/nm. For all of the three cases, estimated OSNR values are very close to setup OSNR
values, proving that accurate OSNR estimation is done independently of the phase noise and CD.
As shown in Fig. 3(b), the maximum value of estimation error, which is defined as the deviation of
the estimated value from the setup value, is found below 0.2 dB over the OSNR estimation range
from 6 to 15 dB.
Fig. 4 shows estimation results of OSNR per subcarrier for the 16-QAM format. In Fig. 4(a), the
red curve is calculated when only ASE noise is included and CD and the phase noise are ignored.
The blue curve is obtained when f ¼ 150 kHz and CD ¼ 0. The black curve is obtained when
f ¼ 150 kHz and CD ¼ 500 ps/nm. Accurate estimation is found over a wide range of OSNR for all
of the three cases. As shown in Fig. 4(b), the maximum estimation error is found less than 0.6 dB.
It is important to note here that we consider ideal Nyquist-pulse generation by using a roll-off
factor of 0. However, if we increase the roll-off factor for pulse shaping in Nyquist-WDM systems, it
causes inter-channel crosstalk. The estimation error due to such crosstalk is determined by the rolloff factor and we can calibrate it out in estimating the OSNR.
Vol. 6, No. 1, February 2014
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In-Band Estimation of OSNR
Fig. 5. Estimated OSNR as a function of the number of nodes of a photonic network employing
ROADMs. In (a) and (b), QPSK and 16-QAM formats are used, respectively.
4. Simulation Results for ROADM Networks
We also confirm effectiveness of our proposed method in the simplest ROADM network as follows:
WDM signals for OSNR evaluation are the same as those in Section 3. A channel is added to the
network from a photonic node. The photonic node includes an optical filter whose transfer function
is given by the 4-th order super Gaussian function. The 10-dB bandwidth of the optical filter is set to
50 GHz and 75 GHz for the QPSK and 16-QAM systems, respectively. With such a filter, the
wavelength channel is filtered; thus, the out-of-band noise is cut off at the node. Then, the signal
passes through an optical-repeater span composed of a single-mode fiber (SMF) and an EDFA.
The span loss is 20 dB and fiber nonlinearity is neglected. The EDFA has the noise figure of 4 dB
and compensates for the loss of 20 dB. The power launched on the SMF is fixed at -6 dBm per
subcarrier and per polarization. Each node is connected every four optical-repeater spans. After
passing through multiple nodes, the signal is dropped and detected by a coherent optical homodyne
receiver and delivered to the DSP circuit. The configuration of the DSP circuit is the same as that in
Section 3.
To evaluate the effectiveness of the proposed scheme, we plot OSNR as a function of the
number of traversed nodes. Fig. 5(a) shows such results for the QPSK modulation format. The
estimated OSNR value is very close to the delivered OSNR, which can be calculated by the given
system parameters [18]. Fig. 5(b) shows OSNR per subcarrier for the 16-QAM modulation format.
We find that the estimated OSNR has good accuracy.
5. Conclusion
We have proposed a novel in-band OSNR-estimation technique, where OSNR is obtained from
statistical moments of the equalized signal in coherent optical receivers. This method is applicable
to Nyquist-WDM systems and ROADM networks, where we cannot apply the conventional OSNRestimation method using out-of-band noise measurements. Computer simulations show the very
good estimation performance in point-to-point Nyquist-WDM systems and ROADM networks
employing QPSK and 16-QAM formats.
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[4] Md. S. Faruk, Y. Mori, C. Zhang, K. Igarashi, and K. Kikuchi, BMulti-impairment monitoring from adaptive finite-impulseresponse filters in a digital coherent receiver,[ Opt. Exp., vol. 18, no. 26, pp. 26 929–26 936, Dec. 2010.
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[5] J. C. Geyer, C. R. S. Fludger, T. Duthel, C. Schulien, and B. Schmauss, BPerformance monitoring using coherent
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[6] F. Pittal, F. N. Hauske, Y. Ye, N. G. Gonzalez, and I. T. Monroy, BJoint PDL and in-band OSNR monitoring supported
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[8] M. S. Faruk and K. Kikuchi, BMonitoring of optical signal-to-noise ratio using statistical moments of adaptive-equalizer
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Vol. 6, No. 1, February 2014
7800109
[a3]
Electronic polarization-division
demultiplexing based on digital signal
processing in intensity-modulation
direct-detection optical communication
systems
Kazuro Kikuchi∗
Department of Electrical Engineering and Information Systems, the University of Tokyo
7-3-1 Hongo, Bunkyo-Ku, Tokyo 113-8656, Japan
∗ [email protected]
Abstract:
We propose a novel configuration of optical receivers for
intensity-modulation direct-detection (IM·DD) systems, which can cope
with dual-polarization (DP) optical signals electrically. Using a Stokes
analyzer and a newly-developed digital signal-processing (DSP) algorithm,
we can achieve polarization tracking and demultiplexing in the digital
domain after direct detection. Simulation results show that the power
penalty stemming from digital polarization manipulations is negligibly
small.
© 2014 Optical Society of America
OCIS codes: (060.2330) Fiber optics communications; (060.4080) Modulation.
References and links
1. E. Yamazaki, S. Yamanaka, Y. Kisaka, T. Nakagawa, K. Murata, E. Yoshida, T. Sakano, M. Tomizawa,
Y. Miyamoto, S. Matsuoka, J. Matsui, A. Shibayama, J. Abe, Y. Nakamura, H. Noguchi, K. Fukuchi, H. Onaka, K. Fukumitsu, K. Komaki, O. Takeuchi, Y. Sakamoto, H. Nakashima, T. Mizuochi, K. Kubo, Y. Miyata,
H. Nishimoto, S. Hirano, and K. Onohara, “Fast optical channel recovery in field demonstration of 100-Gbit/s
Ethernet over OTN using real-time DSP, ” Opt. Express 19, 13139–13184 (2011).
2. K. Kikuchi, “Performance analyses of polarization demultiplexing based on constant-modulus algorithm in digital coherent optical receivers,” Opt. Express 19, 9868–9880 (2011).
3. K. Kikuchi, “Digital coherent optical communication systems: Fundamentals and future prospects,” IEICE Electron. Express 8, 1642–1662 (2011).
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haul WDM transmission systems,” in 2004 OSA Technical Digest of Optical Fiber Communication Conference
(Optical Society of America, 2004), FI3.
1.
Introduction
The dual-polarization (DP) transmission scheme has been introduced into practical optical communication systems for the first time by using recently-developed digital coherent receivers [1].
Controlling the state of polarization (SOP) of the DP signal in the digital domain, such receivers
#202277 - $15.00 USD
(C) 2014 OSA
Received 29 Nov 2013; revised 11 Jan 2014; accepted 13 Jan 2014; published 23 Jan 2014
27 January 2014 | Vol. 22, No. 2 | DOI:10.1364/OE.22.001971 | OPTICS EXPRESS 1971
can demultiplex two polarization tributaries in an adaptive manner [2]. The efficient SOP control based on digital signal processing (DSP) is owing to the phase information of the DP signal,
which is obtained from coherent detection employing phase and polarization diversities [3].
On the other hand, it has been believed that in conventional intensity-modulation directdetection (IM·DD) systems, we cannot manipulate the signal SOP even by using DSP, because
the phase information of the DP signal is entirely lost after direct detection; therefore, in order
to demultiplex the DP signal, we need to rely on bulky and slow optical polarization controllers,
which prohibit practical implementation of DP-IM·DD systems.
Contrary to such a common belief, this paper proposes a novel configuration of directdetection receivers, which enables polarization-division demultiplexing the DP-IM signal in
the digital domain without using optical polarization controllers. Implementing the Stokes analyzer and low-complexity DSP in the receivers, we can achieve tracking of SOP fluctuations
and polarization-division demultiplexing of the DP-IM signal in the digital domain. Simulation
results show that the power penalty stemming from digital polarization manipulations is negligibly small even under very fast SOP fluctuations. This technique may be useful for 100-Gbit/s
short-reach optical transmission systems based on the IM·DD scheme, because the 50-GS/s
sampling rate of analog-to-digital converters (ADCs) is currently available [1] and the bit rate
can easily be doubled with our proposed method.
The organization of the paper is as follows: Section 2 discusses the SOP of the DP-IM signal.
Section 3 deals with the configuration of the proposed direct-detection receiver composed of the
Stokes analyzer. In Sec.4, we discuss the polarization-tracking and polarization-demultiplexing
algorithm used in the DSP circuit. Simulation results on the bit-error rate (BER) performance
of the proposed receiver is described in Sec.5, and the effectiveness of the proposed algorithm
is validated. Finally, Sec.6 concludes this paper.
2.
SOP of the DP signal
Figure 1 shows the configuration of the DP-IM transmitter. We assume that two independent
laser diodes, LD 1 and LD 2, are intensity-modulated with the same clock using either the
direct-modulation method or the external modulation method. The two signals are polarizationmultiplexed with a half-wave plate (λ /2) and a polarization beam combiner (PBC). The tributary 1 has the linear x polarization at the transmitter, whereas the tributary 2 does the linear y
polarization. The intensity of the lasers is modulated in a binary manner. In a low logic level,
the intensity of each tributary is zero. In the following analysis, we assume that the intensity of
each tributary in a high logic level is two so that the average intensity is normalized to unity
when both logic levels occur at the same probability of 1/2.
The total intensity of the DP-IM signal is classified into three cases shown in Table 1. In the
case (I), logic levels of both of the polarization tributaries are low and the total signal intensity
is zero. In the case (II), one tributary is in the high level, and the other in the low level; therefore,
the total intensity is two. In the case (III), both of the tributaries are in the high level, and the
total intensity is four.
Corresponding to these cases, the SOP of the DP-IM signal is classified as follows: In the
case (I), we have no signal. The SOP in the case (II) at the transmitter is determined either from
the linear x polarization ((II)(a)) or from the linear y polarization ((II)(b)). On the other hand,
in the case (III), the DP signal never has a fixed SOP, because phases of the two tributaries are
not correlated. Noting that intensities of x-polarization and y-polarization components of the DP
signal are the same, we find that S1 of the Stokes vector of the DP signal is zero, whereas S2 and
S3 are fluctuating at the speed of the laser linewidth under the condition that S22 + S32 = 4. Thus,
we have the relation of the three Stokes vectors shown in Fig. 2: Normalized Stokes vectors
S/S0 in cases (II)(a) and (II)(b) are pointed to opposite directions, and the normalized Stokes
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Pattern
generator 1
PBC
LD 1
Clock
~
λ
LD ２
2
Pattern
generator 2
Fig. 1. Configuration of the DP-IM transmitter. Intensities of two independent laser diodes
(LDs) are modulated either by direct modulation or by external modulation with the same
clock. The two signals are polarization-multiplexed by a half-wave plate (λ /2) and a polarization beam combiner (PBC).
Table 1. Classification of the state of the DP-IM signal based on the total peak intensity.
(I)
(a)
Logic level
tributary 1: Low
tributary 2: Low
Total peak intensity
0
(b)
(II)
tributary 1: High
tributary 2: Low
tributary 1: Low
tributary 2: High
2
(III)
tributary 1: High
tributary 2: High
4
vector in the case (III) is orthogonal to these vectors in spite of its fluctuations. This orthogonal
relation among the three normalized Stokes vectors is unchanged at the receiver although these
vectors walk around on the Poincar´e sphere due to fluctuations of fiber birefringence.
v ( n)
(II)(b)
(II)(a)
(III)
Fig. 2. Relation among normalized Stokes vectors in cases (II)(a), (II)(b), and (III). Normalized Stokes vectors in cases (II)(a) and (II)(b) are pointed to opposite directions, whereas
the normalized Stokes vector in the case (III) is orthogonal to these vectors. The vector
v(n) denotes the reference Stokes vector discussed in Sec. 4.
3.
Receiver configuration
Figure 3 shows the schematic diagram of our proposed receiver. The incoming DP-IM signal is
equally split into four branches after optical pre-amplification and optical filtering if necessary.
In the first branch, we measure the signal intensity It . Inserting a polarizer (0◦ Pol), whose
transmission axis is the x axis, we measure the intensity of the x-polarization component Ix in
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the second branch. Using a polarizer (45◦ Pol), whose transmission-axis is rotated by 45◦ with
respect to the positive x axis, we detect the intensity of the 45◦ linearly-polarized component
I45◦ in the third branch. With a quarter-wave plate (λ /4), whose fast axis is aligned to the x
axis, and a 45◦ -rotated polarizer (45◦ Pol), we measure IR , which is the intensity of the rightcircularly-polarized component, in the fourth branch. This configuration is known as the Stokes
analyzer, which determines Stokes parameters from It , Ix , I45◦ , and IR [4] as
S0
S1
S2
S3
=
It ,
(1)
=
=
2Ix − S0 ,
2I45◦ − S0 ,
(2)
(3)
=
2IR − S0 .
(4)
The four outputs of photodiodes (PDs) in Fig. 3 are converted to digital data using fourchannel ADCs. The clock (CLK) extracted from the first branch of the Stokes analyzer controls
sampling instances of the ADCs. The sampling rate is one sample/bit. The sampled data are
sent to the DSP circuit.
Stokes analyzer
PD
Input
signal
0° Pol
PD
45° Pol
PD
λ / 4 45° Pol
PD
CLK
ADC
DSP
Decoded
bits
x pol
y pol
Fig. 3. Receiver configuration for polarization-division demultiplexing the DP-IM signal.
The part surrounded by broken lines represents the Stokes analyzer. Four outputs from
photodiodes (PDs) are sent to the DSP circuit.
4.
DSP circuit
In the DSP circuit shown by Fig. 4, after calculations of Stokes parameters using Eqs. (1)-(4),
polarization tracking and demultiplexing are done by the algorithm given in the following.
Stokesvector
calculator
Intensity
discriminator
Stokes-vector
amplitude
discriminator
Bit aligner
Reference
Stokes-vector
updater
Fig. 4. DSP circuit for polarization-division demultiplexing the DP-IM signal. The intensity discriminator determines the case (I) shown in Table 1. Cases (II)(a), (II)(b), and (III)
in Table 1 are separated through discrimination of the Stokes-vector amplitude along the
reference Stokes-vector direction.
First, in the intensity discriminator, we separate the case (I) from cases (II) and (III) (see
Table 1) by intensity discrimination of the measured S0 (n) with the threshold Sth , where n
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denotes the number of samples. When S0 (n) ≤ Sth , both of the tributaries are decided to be in
the low level.
Next, in the Stokes-vector amplitude discriminator and the reference Stokes-vector updater,
we process the cases (II) and (III). Let the reference Stokes vector v(n) be a noise-free unit
vector expressing the SOP of the tributary 1 at the receiver (See Fig. 2). Note that the SOP for
the tributary 2 is given as −v(n) in such a case. Provided that v(n) is known, we can calculate
the inner product between the received normalized Stokes vector S(n)/S0 (n) and the reference
Stokes vector v(n) as
S(n)
· v(n) ,
(5)
u(n) =
S0 (n)
which means the normalized Stokes-vector amplitude along the direction of the reference
Stokes vector. Then, Fig. 2 shows that we can separate the three cases (II)(a), (II)(b), and (III),
discriminating the distribution of u(n) into three regions. Let discrimination thresholds for u(n)
be uth (> 0) and −uth . When u(n) ≥ uth , we decide that the measured sample belongs to the
tributary 1. In such a case, the tributary 1 is in the high level, whereas the tributary 2 is in the
low level. The reference Stokes vector is then updated as
S (n)
− v(n)
v (n) + μ
S (n)
,
0
(6)
v (n + 1) = v (n) + μ S (n) − v(n) S0 (n)
where μ is the step-size parameter. Equation (6) shows that the reference vector v(n) is modified
by using the error signal ε = S(n)/S0 (n) − v(n) and tracks the SOP of the tributary 1 even
when it fluctuates on the Poincar´e sphere due to the random change in fiber birefringence. A
smaller value of μ improves the signal-to-noise ratio of v(n) but reduces the SOP tracking
speed; therefore, we need to choose an optimum value of μ , depending on the SOP fluctuation
speed.
On the other hand, when u(n) ≤ −uth , we decide that the measured sample belongs to the
tributary 2. In such a case, the tributary 1 is in the low level, whereas the tributary 2 is in the
high level. Reversing the sign of the normalized Stokes vector in Eq. (6), we have the update
formula for v(n) given as
S (n)
− v(n)
v (n) + μ −
S0 (n)
,
(7)
v (n + 1) = v (n) + μ − S (n) − v(n) S0 (n)
where the error signal ε = −S(n)/S0 (n) − v(n) controls the reference Stokes vector.
When |u(n)| < uth , both of the tributaries are in the high level. We do not update the reference
Stokes vector, because the SOP is not fixed in such a case. It should be noted that in cases (I)
and (III), we do not update the reference Stokes vector and keep that defined in the nearest
preceding case of (II); however, since the fluctuation speed of the reference Stokes vector is
much slower than the bit rate, such thinned-out operation of the update process never degrades
the BER performance as shown in 5.3.
Although we have assumed that v(n) is known, the update process using Eqs. (6) and (7)
can start from an arbitrary reference vector in the blind mode. However, depending on the
initial choice of the reference Stokes vector, the tributaries 1 and 2 may be exchanged. After
a sufficient number of iteration with the proper choice of μ , the initial tracking process is
converged and we can find an accurate estimate for v(n) even under very fast SOP fluctuations.
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Thus, we can discriminate the four cases (I), (II)(a), (II)(b), and (III). Finally, we complete the
demodulation process, aligning bit sequences of the tributaries.
5.
5.1.
Simulation results
Simulation model
In computer simulations, we generate IM signals having binary random bit patterns for the two
polarization tributaries. The number of bits for each tributary is N = 220 . The Jones vector of
the DP signal at the transmitter is written as
Ein, x (n)
.
(8)
Ein (n) =
Ein, y (n)
Complex amplitudes of electric fields Ein, x (n) and Ein, y (n) are given as
Ein, x (n) = sx (n) exp [iφx (n)] + nx (n) ,
Ein, y (n) = sy (n) exp [iφy (n)] + ny (n) .
(9)
(10)
√
In these equations, sx (n) and sy (n) are signal amplitudes, which are either 2 (High level) or
0 (Low level) so that the average intensity of each polarization is unity. Parameters nx (n) and
ny (n) are complex-valued Gaussian noises. The variance of the real part of nx, y (n) and that of
the imaginary part of nx, y (n) are represented as σs2 . Then, the carrier-to-noise ratio (CNR) of
each polarization is expressed as
1
.
(11)
CNR/pol =
2σs2
The value of CNR/pol is controlled by the amount of Gaussian noise, while the average signal
intensity is kept at a unity for each tributary. We do not take the CNR reduction by branching
of the signal into account. This is valid for the optically pre-amplified signal [5].
Parameters φx (n) and φy (n) are phase noises of the lasers LD 1 and LD 2, respectively. We
express them as
φx (n + 1) = φx (n) + Δφx (n) ,
φy (n + 1) = φy (n) + Δφy (n) .
(12)
(13)
Parameters Δφx (n) and Δφy (n) are real-valued Gaussian noises and their variance σ p2 is given [6]
as
σ p2 = 2πδ f · T ,
(14)
where δ f is the 3-dB spectral width of the lasers and T the bit duration.
After suffering from the random change in fiber birefringence, the Jones vector of the DP
signal at the receiver is given as
Eout (n) = J(n)Ein (n) ,
(15)
where J(n) is the Jones matrix of the fiber for transmission. From Eout (n), Stokes parameters
of the received signal is obtained [4] as
2
S0 (n) = |Eout, x (n)|2 + Eout, y (n) ,
(16)
2
2 (17)
S1 (n) = |Eout, x (n)| − Eout, y (n) ,
(18)
S2 (n) = 2 |Eout, x (n)| Eout, y (n) cos [δ (n)] ,
(19)
S3 (n) = 2 |Eout, x (n)| Eout, y (n) sin [δ (n)] ,
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where δ (n) = arg [Eout, y (n)/Eout, x (n)]. Equations (16)-(19) are equivalent to Eqs. (1)-(4).
When we scramble the SOP of the signal to emulate random fluctuations of fiber birefringence, the Jones matrix is expressed as
⎤
⎡
− sin θr2(n)
exp i φr2(n) cos θr2(n)
⎦.
(20)
J(n) = ⎣
exp −i φr2(n) cos θr2(n)
sin θr2(n)
The polar angle and the azimuthal angle of the SOP randomly fluctuate on the Poincar´e sphere
in a bit-by-bit manner through φr (n) and θr (n). Parameters φr (n) and θr (n) including fluctuations obey the following equations:
φr (n + 1) = φr (n) + Δφr (n) ,
θr (n + 1) = θr (n) + Δθr (n) ,
(21)
(22)
where Δφr (n) and Δθr (n) are real-valued Gaussian noises having the variance given as
σ02 = AT .
(23)
The parameter A with the dimension of s−1 is a constant under a specific condition of the fiber
for transmission.
5.2.
Determination of discrimination thresholds
In 5.2, we determine the optimum threshold Sth for discriminating S0 (n) and uth for discriminating u(n) through computer simulations. Ignoring the laser phase noise, we assume that δ f = 0
in Eq. (14). The fluctuation of the received SOP is also neglected throughout 5.2; then, we
assume that J = 1 in Eq. (15).
Figure 5 shows the simulation result of the probability-density function of the intensity S0 (n)
when CNR/pol=10, 12, and 14 dB. The cases (I) and (II) are clearly separated, and we can
decide that both logic levels of the tributaries are low, when the measured intensity S0 is smaller
than the threshold Sth = 0.6 shown by the sold line. On the other hand, the discrimination ability
between (II) and (III) is so poor that the intensity discrimination shown by the broken line
cannot be applied to separate (II) and (III).
We can understand the noise distribution shown in Fig. 5 as follows [5]: When we consider
that the Gaussian noise originates from amplified spontaneous emission (ASE) of optical preamplifiers, the noise distribution in the case (I) is determined from the spontaneous-spontaneous
beat-noise process. On the other hand, since the signal-spontaneous beat noise is predominant
in the case (II), the probability distribution in the case (II) is broader than that in the case (I).
In the case (III), two stochastically independent signal-spontaneous beat noises for orthogonal
polarizations are added together; therefore, the variance of the noise in the case (III) is twice as
large as that in the case (II).
Figure 6 shows the simulation result on the probability-density function of the inner product
u(n) given by Eq. (5), when CNR/pol=10, 12, and 14 dB. Three peaks clearly appear in this
function and we can optimally discriminate (II)(a), (II)(b), and (III) when uth = 0.55 as shown
by solid lines.
5.3.
BER performance
In BER calculations, we scramble the SOP of the signal, assuming that the parameter A in
Eq. (23) is 105 [s−1 ]. The variance σ f (N)2 of φr (n) and θr (n) at the N-th bit is written as
σ f (N)2 = σ02 N = ANT .
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10
CNR/pol=10 dB
CNR/pol=12 dB
CNR/pol=14 dB
1
(I)
yit
0
sn10
ed
yti 10-1
ilb
ab -2
or10
P -3
10
0
(II)
2
(III)
4
Intensity
6
8
Fig. 5. Probability-density function of the intensity of the DP-IM signal. The cases (I)
and (II) are discriminated by the solid line; on the other hand, the intensity discrimination
shown by the broken line cannot be applied to separate (II) and (III).
10
CNR/pol=10 dB
CNR/pol=12 dB
CNR/pol=14 dB
1
yit
sn100
ed
yti 10-1
li
ba -2
bo10
r
P -3
(II)(b)
(III)
(II)(a)
10
-1 -0.5
0
0.5
1
Normalized Stokes-vector amplitude
Fig. 6. Probability-density function of u(n), which represents the normalized Stokes-vector
amplitude along the reference-vector direction. The three cases (II)(a), (II)(b), and (III) are
discriminated by solid lines.
Therefore, if we assume the 25-Gbit/s/pol system (T = 40 [ps]), the standard deviation is 2 rad
in a 40-μ s time span for N = 220 bits. This value is much larger than SOP fluctuations observed
in real systems [7]. The step-size parameter μ is set at 1/27 to track the SOP fluctuation most
accurately. We also include the effect of the laser linewidth δ f , assuming that δ f ·T = 1×10−3 ,
which means δ f = 25 MHz at the bit rate of 25 Gbit/s/pol.
Figure 7 shows the typical convergence property of the error magnitude ε controlling the
reference Stokes vector, where we use the moving average with the span of 21 samples. Within
1,000-sample periods, the SOP tracking process is stabilized; then bit errors are counted after
the convergence of the error magnitude.
Figure 8 shows BERs calculated as a function of CNR/pol. The red curve is the BER of each
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]
B
d[ 0
ed -10
uit
ng -20
a
m -30
ro
rr -40
E 0
500
1000
1500
Number of samples
2000
Fig. 7. Convergence property of the error magnitude updating the reference Stokes vector.
Since the SOP tracking process is stabilized within 1,000-sample periods, bit errors are
counted after the convergence of the error magnitude.
polarization tributary of the DP-IM signal demodulated with our proposed method. The black
curve represents the BER performance of the single-polarization (SP) IM signal for comparison.
In the DP-IM scheme, we find that both of the polarization tributaries have almost the same
BER characteristics and the power penalty from the SP-IM scheme is negligible. Thus, the
digital polarization-manipulation process does not generate any harmful effect even under very
fast SOP fluctuations.
Single pol
Dual pol x
Dual pol y
-2
)
R
E
B
(g-3
oL
-4
8
10
12
CNR/pol [dB]
14
Fig. 8. BERs as a function of CNR/pol for the DP-IM signal which is demodulated with
our proposed method. The BER curve of the single-polarization signal is also shown for
comparison.
The effect from chromatic dispersion of the link is also examined. We include the dispersion
value of β2 L/T 2 = 0.125, where β2 denotes the dispersion parameter and L the fiber length.
This value corresponds to a 10-km-long standard single-mode fiber (SMF) at the bit rate of 25
Gbit/s/pol and at the wavelength of 1.55 μ m. Red curves in Fig. 9 show BER characteristics
of the proposed DP-IM scheme with and without chromatic dispersion, whereas black curves
show those of the SP-IM signal. We find that the dispersion effect is severer in the proposed
DP-IM scheme than in the conventional SP-IM scheme; however, the difference in the receiversensitivity degradation due to chromatic dispersion is not so significant between the two cases.
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-2
)
R
E
B
(g-3
oL
-4
8
β 2 L / T 2 = 0.125
β2 L / T 2 = 0
∗
∗
Single pol
Dual pol x
Dual pol y
10
12
CNR/pol [dB]
14
Fig. 9. BERs as a function of CNR/pol for the DP-IM signal with and without chromatic
dispersion. BERs of the single-polarization signal are also shown for comparison.
6.
Conclusions
We have proposed a novel configuration of IM·DD receivers, which enables polarizationdivision demultiplexing in the digital domain after direct detection. Simulation results show
that the power penalty stemming from digital polarization-division demultiplexing is negligibly small even under very fast SOP fluctuations. The proposed method is useful for 100-Gbit/s
short-reach optical transmission systems based on the IM·DD scheme, because the bit rate can
easily be doubled.
Acknowledgments
This work was supported in part by Grant-in-Aid for Scientific Research (A) (25249038), the
Ministry of Education, Culture, Sports, Science and Technology in Japan.
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[a4]
Compensation for In-Phase/Quadrature
Imbalance in Coherent-Receiver Front End for
Optical Quadrature Amplitude Modulation
Volume 5, Number 2, April 2013
Md. Saifuddin Faruk
Kazuro Kikuchi, Fellow, IEEE
DOI: 10.1109/JPHOT.2013.2251872
1943-0655/$31.00 Ó2013 IEEE
IEEE Photonics Journal
Compensation for IQ Imbalance in Receivers
Compensation for In-Phase/Quadrature
Imbalance in Coherent-Receiver
Front End for Optical Quadrature
Amplitude Modulation
Md. Saifuddin Faruk 1 and Kazuro Kikuchi,2 Fellow, IEEE
1
Department of Electrical and Electronic Engineering, Dhaka University of Engineering and
Technology, Gazipur 1700, Bangladesh
Department of Electrical Engineering and Information Systems, The University of Tokyo,
Bunkyo-Ku, Tokyo 113-8656, Japan
2
DOI: 10.1109/JPHOT.2013.2251872
1943-0655/$31.00 Ó2013 IEEE
Manuscript received January 11, 2013; revised February 26, 2013; accepted March 5, 2013. Date of
publication March 13, 2013; date of current version March 22, 2013. Corresponding author: K. Kikuchi
(e-mail: [email protected]).
Abstract: We propose a novel method of compensation for imbalance between in-phase (I)
and quadrature (Q) channels in the front-end circuit of digital coherent optical receivers.
Adaptive finite-impulse-response (FIR) filters in the butterfly configuration, which are commonly used for signal equalization and polarization demultiplexing, are modified so as to
allow for adjustment of any imbalance between the IQ channels. IQ imbalances under
consideration include the gain mismatch, the phase mismatch, and the timing-delay skew.
Computer simulations for the dual-polarization quadrature-amplitude-modulation (QAM)
format up to an order of 256 show that such IQ imbalances can severely degrade the system
performance, especially for higher order QAM; however, using the proposed scheme, we
can compensate for them without any significant penalty over a wide range of imbalances.
Index Terms: Coherent optical receivers, in-phase/quadrature imbalance, digital signal
processing.
1. Introduction
The recent development of digital coherent optical receivers has brought the 100-Gbit/s dualpolarization quadrature phase-shift keying (QPSK) system into practical use [1]. For further increase in the bit rate of optical fiber transmission systems, quadrature amplitude modulation (QAM),
where both of the in-phase (I) and quadrature (Q) components of an optical carrier are modulated in
a multilevel manner, is the best candidate among various modulation formats [2]. However, when
higher order QAM formats are employed, their performances are seriously impaired by imperfections of the systems, such as phase noise of the transmitter laser and the local oscillator [3], fiber
nonlinearity [4], and imbalance between the IQ channels in the front end of coherent optical receivers [5]. In this paper, we focus on the impact of the imbalance between the IQ channels on QAM
signals. We propose a novel IQ-imbalance compensation scheme, where the conventional adaptive
finite-impulse-response (FIR) filters in the butterfly configuration are modified so as to allow for
adjustment of any imbalance between the IQ channels.
The front end of coherent receivers employing phase and polarization diversities converts complex amplitudes of the incoming dual-polarization optical signal into the electrical domain by means
of homodyne detection with a free-running local oscillator. It provides four outputs, namely, IQ
Vol. 5, No. 2, April 2013
7800110
IEEE Photonics Journal
Compensation for IQ Imbalance in Receivers
Fig. 1. Block diagram of the digital coherent receiver comprising phase and polarization diversities. LO:
local oscillator, PBS: polarization-beam splitter, TIA: transimpedance amplifier, ADC: analog-to-digital
converter, and DSP: digital signal-processing circuit.
components of the complex amplitudes for horizontal and vertical polarizations [6]. Fig. 1 shows the
configuration of the front end composed of polarization-beam splitters (PBSs), 90 optical hybrids,
balanced photodiodes, and transimpedance amplifiers (TIAs). The four outputs from TIAs are sent
to analog-to-digital converters (ADCs) followed by a digital signal-processing (DSP) circuit.
Imperfection in any of the 90 optical hybrids, balanced photodiodes, and TIAs in the front end
may introduce IQ imbalance stemming from the mismatch of the gain and/or the phase between the
IQ ports [7]. In addition, timing mismatch between the IQ ports may also be induced by the difference in the physical path length of the circuit trace, which is known as the IQ delay skew [8].
These IQ imbalances degrade the system performance severely if they are left uncompensated in
the DSP unit of the receiver.
Several methods of IQ-imbalance compensation in the digital domain have been reported so far
[9]–[11]. In [9], the Gram–Schmidt orthogonalization procedure (GSOP) is investigated for the
QPSK signal; however, when higher order QAM formats are employed, computational complexity is
increased and very high ADC resolution is required. In [10], the compensation is done by the
ellipse-correction method, which is neither applicable to higher order QAM signals nor effective
when the optical signal-to-noise ratio (OSNR) is low. The IQ-imbalance equalizer based on the
constant-modulus algorithm (CMA) is demonstrated for the QPSK signal in [11]; however, such an
approach is again not applicable to higher order QAM signals. It should be also noted that none of
the methods mentioned above can compensate for the IQ delay skew and that all of them need to
use dedicated DSP circuits for IQ-imbalance compensation.
On the other hand, in this paper, we propose a novel scheme, which can overcome the difficulties
of the previous schemes: First, our scheme can be applied to any modulation formats. Second, it
can compensate for the IQ gain mismatch, IQ phase mismatch, and IQ delay skew all at once.
Third, it can be implemented as a part of the conventional two-by-two butterfly-structured FIR filters,
which have been commonly used for signal equalization and polarization demultiplexing.
In fact, our proposed scheme modifies the configuration of the conventional adaptive FIR filters in
such a way that each of the complex-valued FIR filters is replaced with four real-valued FIR filters in
an inner two-by-two butterfly structure. Then, our scheme has two input ports for each polarization,
corresponding to the IQ outputs from the front end. Tap coefficients of the sixteen real-valued FIR
filters can be updated using any stochastic-gradient-decent-based adaptation algorithm such as the
decision-directed least-mean-square (DD-LMS) algorithm [12] and the CMA [13]. The initial convergence speed and the steady-state performance of the equalizer depend on the employed
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Compensation for IQ Imbalance in Receivers
adaptation algorithm. In this paper, we use the DD-LMS algorithm with the training mode. The
training mode ensures fast and reliable initial convergence and then switched to the decisiondirected mode. Such adaptation procedure provides the optimal steady-state performance for highorder QAM formats. By the proposed filter modifications, our scheme performs compensation for IQ
gain/phase mismatch and IQ delay skew simultaneously, along with other conventional tasks of
adaptive FIR filters such as sampling-phase adjustment [14], polarization demultiplexing [15], and
compensation for polarization-mode dispersion (PMD) [16].
With intensive computer simulations using the 10-Gsymbol/s dual-polarization 4-, 16-, 64-, and
256-QAM formats, we find that the system performance severely degrades if we use the conventional FIR-filter structure in the presence of IQ imbalance; however, our scheme can fully compensate
for IQ imbalance generated in the receiver front end over a wide range.
The organization of our paper is as follows: Section 2 presents the formulation of front-end IQ
imbalance based on the transfer-function matrix. In Section 3, we propose the novel scheme for IQimbalance compensation together with the tap-adaptation algorithm. Section 4 deals with intensive
computer simulations on higher order QAM signals, and we conclude our paper in Section 5.
2. Formulation of the Problem
Let rx I ðt Þ and rx Q ðt Þ respectively be received signals from the I and Q ports of the front-end circuit
for the x -polarization component, when IQ imbalances are not present. Similarly, let ry I ðt Þ and
ry Q ðt Þ be those for the y -polarization component. Then, the complex amplitude of the optical signal
is reconstructed as
rx ;y ðt Þ ¼ rx I;y I ðt Þ þ jrx Q;y Q ðt Þ:
(1)
First, we consider the case that the IQ phase mismatch is included in the front end. In such a
case, the I and Q axes in the complex plane are rotated by angles of x I and x Q , respectively, for
the x -polarization component. Similarly, those angles for the y -polarization component are denoted
as y I and y Q . IQ phase mismatches are given as x I x Q and y I y Q for x - and
y -polarization components, respectively, which represent the offset from the correct angle of 90 .
Using x I , x Q , y I , and y Q , the real and imaginary parts of the received complex amplitude
rxp;y ðt Þ are expressed as
rxpI;y I ðt Þ ¼ cosðx I;y I Þrx I;y I ðt Þ þ sinðx I;y I Þrx Q;y Q ðt Þ
rxpQ;y Q ðt Þ ¼ sinðx Q;y Q Þrx I;y I ðt Þ þ cosðx Q;y Q Þrx Q;y Q ðt Þ:
When we define the transfer matrix Px ;y stemming from the IQ phase mismatch as
"
#
sinðx I;y I Þ
cosðx I;y I Þ
Px ;y ¼
sinðx Q;y Q Þ cosðx Q;y Q Þ
equations (2) and (3) yield
h
iT
T
rxpI;y I ðt Þ; rxpQ;y Q ðt Þ ¼ Px ;y rx I;y I ðtÞ; rx Q;y Q ðt Þ
(2)
(3)
(4)
(5)
which expresses the mutual coupling of real and imaginary parts of the complex amplitude.
Second, we consider the case that only the IQ gain mismatch is involved in the front end. In such
a case, we can express the received complex amplitude as
rxg;y ðt Þ ¼ x I;y I rx I;y I ðt Þ þ jx I;y I rx Q;y Q ðt Þ:
(6)
In (6), x I and x Q are the gains of the I and Q ports, respectively, for the x -polarization component. Similarly, those values for the y -polarization component are y I and y Q . In case that
x I 6¼ x Q , the IQ gain mismatch exists in the x -polarization port and it does in the y -polarization
port when y I 6¼ y Q . The gain mismatching factor is defined as x I =x Q and x I =x Q for the
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x - and y - polarization components, respectively. Equations (1) and (6) yield the transfer matrix for
the IQ gain mismatch Gx ;y as
x I;y I
0
Gx ;y ¼
(7)
0
x Q;y Q
which leads to
h
iT
T
rxgI;y I ðt Þ; rxgQ;y Q ðt Þ ¼ Gx ;y rx I;y I ðt Þ; rx Q;y Q ðt Þ :
(8)
Third, we consider the case that only the IQ delay skew is involved between the I and Q ports of
the front end. When time delays in the four output ports of the front end are given as x I , x Q , y I ,
and y Q , the complex amplitude reconstructed from the front-end outputs can be expressed as
rxd;y ðt Þ ¼ rx I;y I ðt x I;y I Þ þ jrx Q;y Q ðt x Q;y Q Þ:
(9)
In case that x I 6¼ x Q , the IQ delay skew exists in the x -polarization component, whereas in case
that y I 6¼ y Q , it does in the y -polarization component. We transform (9) into the frequency
domain as
h
iT
T
RxdI;y I ð!Þ; RxdQ;y Q ð!Þ ¼ Dx ;y ð!Þ Rx I;y I ð!Þ; Rx Q;y Q ð!Þ
(10)
where ! is the angular frequency of the optical signal measured from the carrier frequency;
RxdI;y I ð!Þ, RxdQ;y Q ð!Þ, Rx I;y I ð!Þ, and Rx Q;y Q ð!Þ are Fourier transforms of rxdI;y I ðt Þ,
rxdI;y I ðt Þ, rx I;y I ðt Þ, and rx I;y I ðt Þ, respectively; and the transfer matrix Dx ;y ð!Þ is given as
expðj!x I;y I Þ
0
Dx ;y ð!Þ ¼
:
(11)
0
expðj!x Q;y Q Þ
On the other hand, frequency-domain expressions for Px ;y and Gx ;y are the same as Px ;y and Gx ;y ,
since they are time independent. Therefore, the overall transfer function expressing the three kinds
of IQ imbalances can be written in the frequency domain as
Qx ;y ð!Þ ¼ Px ;y Gx ;y Dx ;y ð!Þ:
(12)
Using (12), we find that the complex amplitude including the front-end IQ imbalances is measured as
h
iT
T
RxeI;y I ð!Þ; RxeQ;y Q ð!Þ ¼ Qx ;y ð!Þ Rx I;y I ð!Þ; Rx Q;y Q ð!Þ
(13)
in the frequency domain. To compensate for the IQ imbalances, we need to find the inverse
matrix Q1
x ;y ð!Þ, which eliminates the IQ phase mismatch, the IQ gain imbalance, and the IQ
delay skew, as shown by
h
iT
T
e
e
Rx I;y I ð!Þ; Rx Q;y Q ð!Þ ¼ Q1
ð!Þ
R
ð!Þ;
R
ð!Þ
:
(14)
x ;y
x I;y I
x Q;y Q
3. Proposed IQ Compensation Scheme
This section discusses how we can generate the inverse matrix Q1
x ;y ð!Þ using adaptive FIR filters.
We assume that the linear transfer-function matrix of the link is given as Hf ð!Þ in the absence of IQ
imbalances. It is a two-by-two matrix including polarization-mode coupling. Fig. 2(a) shows the
conventional adaptive FIR filters in the two-by-two butterfly structure. The discrete Fourier transform
(DFT) of tap-coefficient vectors in Fig. 2(a) is defined as
T
hij ðnÞ ¼ hij;0 ðnÞ; hij;1 ðnÞ; . . . ; hij;N ðnÞ
(15)
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Fig. 2. Adaptive FIR-filter configurations used in digital coherent receivers. (a) Conventional configuration with four complex-valued FIR filters. (b) Our proposed configuration with sixteen real-valued FIR
filters to enable IQ-imbalance compensation.
where ði; jÞ ¼ ðx ; y Þ, n denotes the index of the data sequence, and N is the tap length. As
discussed in [17], it can generate the inverse transfer function H1
f ð!Þ in an adaptive manner. Since
H1
f ð!Þ is a two-by-two matrix with four complex elements, four butterfly-structured complex-valued
FIR filters shown in Fig. 2(a) are capable of generating such matrix.
However, in order to compensate for IQ imbalances, tap-coefficient vectors hxx ðnÞ and hyx ðnÞ
should generate Q1
x ð!Þ because the input vector ux ðnÞ for them includes the IQ imbalances from
the x -polarization channel. Similarly, hxy ðnÞ and hyy ðnÞ should generate Q1
y ð!Þ to adjust the IQ
imbalances of the y -polarization channel that is included in their common input vector uy ðnÞ. Since
Q1
x ;y ð!Þ is a two-by-two matrix that contains four independent elements, it is evident that the
conventional FIR-filtering approach shown in Fig. 2(a) cannot generate Q1
x ;y ð!Þ.
On the other hand, Fig. 2(b) shows the proposed configuration, where each complex-valued FIR
filter is replaced by four real-valued FIR filters in an inner two-by-two butterfly structure. Our scheme
works on two inputs corresponding to real and imaginary parts of the input complex amplitude. In
the following, subscripts ðÞr and ðÞi correspond to the real and imaginary parts of a variable,
respectively. With an optimum tap-adaptation algorithm, such a configuration ensures that
"
#
hxx rr ðnÞ hxx ri ðnÞ
(16)
’ Q1
DFT
x ð!ÞH11 ð!Þ
hxxir ðnÞ hxxii ðnÞ
"
#
hxy rr ðnÞ hxy ri ðnÞ
(17)
DFT
’ Q1
y ð!ÞH12 ð!Þ
hxyir ðnÞ hxyii ðnÞ
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"
DFT
hyxir ðnÞ
"
DFT
hyx rr ðnÞ hyx ri ðnÞ
hyxii ðnÞ
hyy rr ðnÞ hyy ri ðnÞ
hyyir ðnÞ
#
hyyii ðnÞ
’ Q1
x ð!ÞH21 ð!Þ
(18)
’ Q1
y ð!ÞH22 ð!Þ
(19)
#
where four matrix elements of H1
f ð!Þ, i.e., H11 ð!Þ, H12 ð!Þ, H21 ð!Þ, and H22 ð!Þ, are transformed into
two-by-two matrices Hmn ð!Þ ððm; nÞ ¼ ð1; 2ÞÞ as
"
#
ReðHmn ð!ÞÞ ImðHmn ð!ÞÞ
Hmn ð!Þ ! Hmn ð!Þ ¼
:
(20)
ImðHmn ð!ÞÞ ReðHmn ð!ÞÞ
Equation (20) is the transfer matrix for implementing a complex multiplication with well-known
procedure of using four real multiplications. With such a modification, the matrix can be multiplied
with a column vector consisting of the real and imaginary parts of the complex amplitude.
For the proposed configuration, four input column vectors are given by
T
uðx ;y Þðr ;iÞ ðnÞ ¼ uðx ;y Þðr ;iÞ ðnÞ; uðx ;y Þðr ;iÞ ðn 1Þ; . . . ; uðx ;y Þðr ;iÞ ðn N 1Þ
(21)
and sixteen filter tap-coefficient vectors hpqab ðnÞ, where pq ¼ xx ; xy ; yx , or yy and ab ¼ rr ; ri; ir , or
ii are written as
T
hpqab ðnÞ ¼ hpqab ðnÞ; hpqab ðn 1Þ; . . . ; hpqab ðn N 1Þ :
(22)
The output signal vpq ðnÞ is expressed as
vxx ðnÞ ¼ hxx rr ðnÞux r ðnÞ þ hxx ri ðnÞux i ðnÞ þ j fhxx ir ðnÞux r ðnÞ þ hxx ii ðnÞux i ðnÞg
vxy ðnÞ ¼ hxy rr ðnÞuy r ðnÞ þ hxy ri ðnÞuy i ðnÞ þ j hxy ir ðnÞuy r ðnÞ þ hxy ii ðnÞuy i ðnÞ
vyx ðnÞ ¼ hyx rr ðnÞux r ðnÞ þ hyx ri ðnÞux i ðnÞ þ j hyx ir ðnÞux r ðnÞ þ hyx ii ðnÞux i ðnÞ
vyy ðnÞ ¼ hyy rr ðnÞuy r ðnÞ þ hyy ri ðnÞuy i ðnÞ þ j hyy ir ðnÞuy r ðnÞ þ hyy ii ðnÞuy i ðnÞ :
(23)
(24)
(25)
(26)
Then, the final outputs from the FIR-filter configuration, i.e., vx ðnÞ and vy ðnÞ, are computed as
vx ðnÞ ¼ vxx ðnÞ þ vxy ðnÞ
(27)
vy ðnÞ ¼ vyx ðnÞ þ vyy ðnÞ:
(28)
The error signal for updating the tap coefficients using the DD-LMS algorithm is calculated as
ex ;y ðnÞ ¼ dx ;y ðnÞ vx ;y ðnÞ
(29)
where dx ;y ðnÞ is either the training symbol in the training mode or the symbol decoded from vx ;y ðnÞ
in the tracking mode.
Finally, based on the DD-LMS algorithm, the filter tap coefficients are updated as
hpqrr ðn þ 1Þ ¼ hpqrr ðnÞ þ epr ðnÞuqr ðnÞ
(30)
hpqri ðn þ 1Þ ¼ hpqri ðnÞ þ epr ðnÞuqi ðnÞ
(31)
hpqir ðn þ 1Þ ¼ hpqir ðnÞ þ epi ðnÞuqr ðnÞ
(32)
hpqii ðn þ 1Þ ¼ hpqii ðnÞ þ epi ðnÞuqi ðnÞ
(33)
where is the step-size parameter, and ex r ;y r ðnÞ and ex i;y i ðnÞ are the real and imaginary parts
of ex ;y ðnÞ, respectively.
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Fig. 3. Eb =N0 penalty at BER of 103 as a function of the IQ phase mismatch when neither IQ gain
mismatch nor IQ delay skew is included. (a) 4-QAM. (b) 16-QAM. (c) 64-QAM. (d) 256-QAM. Red and
blue curves represent the penalty when the conventional configuration and the proposed configuration
are used, respectively.
4. Simulation Results and Discussions
To validate the proposed scheme, we conduct simulations on 10-Gsymbol/s dual-polarization 4-,
16-, 64-, and 256-QAM coherent systems. In the transmitter, the signal is band limited by a rootraised-cosine filter with a roll-off factor of 0.5. The signal is then impaired by phase noise of the
transmitter laser and passes through a 100-km-long standard single-mode fiber (SSMF), whose
transfer function consists of a chromatic-dispersion value of 1700 ps/nm and a Jones matrix for
fiber birefringence. Additive white Gaussian noise (AWGN) from a preamplifier and phase noise
from LO are then given to the signal. The 3-dB linewidths of lasers for the transmitter and LO are
500 kHz, 100 kHz, 10 kHz, and 1 kHz for 4-, 16-, 64-, and 256-QAM systems, respectively. To
obtain the optimal performance, the signal is subsequently filtered by another root-raised-cosine
filter to match the signal waveform shaped at the transmitter. After that, we intentionally introduce
IQ imbalances. Identical amounts of IQ imbalances are added to both of the x - and y -polarization
ports. The signal is then sampled at twice the symbol rate and fed into the adaptive FIR filters
configured as either in the conventional manner [see Fig. 2(a)] or in the proposed way [see
Fig. 2(b)]. The delay-tap spacing is T =2, where T is the symbol duration. Such FIR filters simultaneously perform clock recovery, equalization of linear impairments, polarization demultiplexing,
and IQ-imbalance compensation. The step-size parameter for the LMS algorithm is optimized so
that bit-error rate (BER) is minimized. Next, carrier phase estimation based on the decision-directed
LMS algorithm [18], symbol decoding, and BER calculations is done in this order. For the
performance evaluation of different modulation formats, we calculate the penalty for the energy-perbit-to-noise spectral-power-density ratio, i.e., Eb =N0 , at BER of 103 . In the following, results for the
x -polarization tributary are presented; however, similar results are found for the y -polarization
tributary.
Fig. 3 shows the Eb =N0 penalty for different IQ phase mismatches when neither IQ gain mismatch
nor IQ delay skew is included. In the simulation, we assume that x I ¼ y I ¼ 0 and vary the
rotation angle of the Q axis x Q ¼ y Q to generate the IQ phase mismatch. In the conventional
configuration, tolerances for the IQ phase mismatch are about 25 , 10 , 5 , and 2.5 for 4-, 16-, 64-,
and 256-QAM formats, respectively, when the Eb =N0 penalty is less than 2 dB. However, no
notable penalty is found over a wide range when the proposed scheme is used.
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Fig. 4. Eb =N0 penalty at BER of 103 as a function of the IQ gain mismatch when neither IQ phase
mismatch nor IQ delay skew is included. (a) 4-QAM. (b) 16-QAM. (c) 64-QAM. (d) 256-QAM. Red and
blue curves represent the penalty when the conventional configuration and the proposed configuration
are used, respectively.
Fig. 5. Eb =N0 penalty at BER of 103 as a function of the IQ delay skew when neither IQ phase nor IQ
gain mismatch is included. (a) 4-QAM. (b) 16-QAM. (c) 64-QAM. (d) 256-QAM. Red and blue curves
represent the penalty when the conventional configuration and the proposed configuration are used,
respectively.
The Eb =N0 penalty as a function of the IQ gain mismatch is shown in Fig. 4 when neither IQ
phase mismatch nor IQ delay skew is included. We set the gain of the I port as a reference such that
x I ¼ 1 and y I ¼ 1 and vary x Q and y Q to introduce the IQ gain mismatch. As far as the
Eb =N0 penalty is less than 2 dB, tolerances of the conventional configuration are about 4 dB,
1.75 dB, 0.75 dB, and 0.4 dB for 4-, 16-, 64-, and 256-QAM formats, respectively. However, with the
proposed scheme, no significant penalty is observed over a wide range of the IQ gain mismatch.
Next, we calculate the Eb =N0 penalty due to the IQ delay skew when neither IQ gain nor IQ phase
mismatch is involved. As shown in Fig. 5, we find that the conventional FIR-filtering scheme is very
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Fig. 6. Constellation diagrams of the recovered signal for (a) 4-QAM, (b) 16-QAM, (c) 64-QAM, and
(d) 256-QAM formats. Upper diagrams show constellation maps when the conventional FIR-filter configuration is used and any IQ imbalance is not included. Middle diagrams are those with the conventional
FIR-filter configuration when IQ imbalances given in Table 1 are involved. Lower diagrams are
constellation maps calculated with the proposed scheme when the same IQ imbalances are included.
TABLE 1
Values of IQ imbalances used in che Calculation of constellation maps shown in Fig. 6
sensitive to the IQ delay skew and the tolerance drastically decreases with the increase in the level
of modulation; however, the proposed method has a very wide range of tolerance to such IQ delay
skew. In fact, a quasi-continuous delay can be generated even by using T =2-spaced FIR filters, as
shown in [14]. Therefore, any amount of delay skew can be compensated for with T =2-spaced FIR
filters having a sufficient number of delay taps.
Finally, we investigate the combined effect of all of the IQ imbalances. Fig. 6 shows constellation
diagrams for different modulation formats. In the simulation, Eb =N0 for each polarization are 10 dB,
14 dB, 18 dB, and 23 dB for 4-, 16-, 64-, and 256-QAM formats, respectively. Upper diagrams show
constellation maps calculated with the conventional FIR-filter configuration when any IQ imbalances
are not included. Middle and lower diagrams are calculated with the conventional and proposed
schemes, respectively, when IQ imbalances listed in Table 1 are included. As shown in the middle
diagrams, the conventional approach fails to compensate for the IQ imbalances. However, the
lower constellations are as clear as the upper constellations, showing that the proposed method
can perfectly compensate for those IQ imbalances.
In our simulations, we do not include the effect of the limited ADC resolution. However, we
generally need ADCs with higher bit resolution as the increase in the IQ imbalance. This situation is
similar to other IQ-imbalance compensation methods.
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It is also important to mention the computational complexity of the proposed scheme compared
with that of the conventional FIR-filter scheme. The hardware-implementation complexity for a
multiplier is much higher than that of an adder. Hence, if we evaluate the computational complexity
in terms of the number of real multiplications, the conventional FIR-filter configuration and the
proposed configuration have the same computational cost. This is because a complex-number
multiplication in the DSP circuit is realized by using four real-number multiplications.
5. Conclusion
We have proposed a novel FIR-filter configuration, which can compensate for IQ imbalances
generated in the front-end circuit of coherent optical receivers. With intensive computer simulations,
we have evaluated the impact of IQ imbalances on the performance of dual-polarization 4-, 16-, 64-,
and 256-QAM systems and verified that the proposed scheme can effectively compensate for them.
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[b1]
Th3K.6.pdf
OFC 2014 © OSA 2014
16-ary Stokes-vector Modulation Enabling
DSP-based Direct Detection at 100 Gbit/s
Kazuro Kikuchi1,* and Shojiro Kawakami2
1
Department of Electrical Engineering and Information Systems, the University of Tokyo
7-3-1 Hongo, Bunkyo-Ku, Tokyo 113-8656, Japan
*[email protected]
2
Photonic Lattice, Inc., ICR Building 2F, 6-6-3 Minami-Yoshinari, Aoba-Ku, Sendai, Miyagi 989-3204, Japan
Abstract: Multi-level optical signals designed in the three-dimensional Stokes space can be
demodulated by a direct-detection receiver using low-complexity DSP. Simulation results
demonstrate that the 16-ary signal can achieve the bit rate of 100 Gbit/s.
OCIS codes: (060.2330) Fiber optics communications; (060.4080) Modulation.
1. Introduction
Stokes-vector modulation is a natural extension of intensity modulation, adding two degrees of freedom of the state
of polarization (SOP) to one-dimensional intensity modulation [1, 2]. We can modulate the three-dimensional
Stokes vector in a multi-level manner; on the other hand, such a signal can be demodulated with a direct-detection
receiver because the Stokes vector is independent of the absolute phase of the carrier.
This paper examines the practical impact of the Stokes-vector modulation scheme with the help of state-of-the-art
digital signal processing (DSP) technologies. Simulation results show that our proposed DSP circuit in the directdetection receiver can track fast fluctuations of SOP due to the random change in fiber birefringence; such SOP
tracking ability of the receiver enables stable demodulation of the multi-level Stokes-vector-modulated signal. The
bit rate as high as 100 Gbit/s is obtained when we employ the 16-ary Stokes-vector modulation format at the
moderate symbol rate of 25 Gsymbol/s and the sampling rate of 25 GSa/s. It is a cost-effective solution to largecapacity ( ≥ 100 Gbit/s) and short-reach (<100 km) transmission without relying upon costly coherent detection.
2. 16-ary Stokes-vector modulation
Desired Stokes vectors are generated by the Stokes-vector modulator shown in Fig.1. The amplitude modulator
(AM1) modulates the signal intensity S0 . The amplitude modulator (AM2) is a Mach-Zehnder modulator which
splits the intensity-modulated signal into two output ports with an arbitrary splitting ratio of tan(θ / 2). The phase
modulator (PM) gives the phase difference δ between the split signals. These signals from the two paths are
combined with a polarization-beam combiner (PBC) such that the signal in the upper path has x polarization and
the signal in the lower path y polarization. Using such a modulator, we can generate the Stokes vector given by
 cosθ 
S = S0 sin θ cos δ  .
 sin θ sin δ 
(1)
For multi-level modulation of Stokes vectors, we should maximize the Euclidean distance between nearest
constellation points in the Stokes space under the condition that S0 is constant. Figure 2 shows the constellation map
most suitable for octal SOP modulation, where transmitted constellation points are placed at eight vertices of a cube
in the Stokes space and corresponding Stokes vectors are given as
θ 
cos  
2
AM1
S0
S3
PBC
AM2
θ 
sin  
2
PM
eiδ
Fig.1. Structure of the Stokes-vector modulator. AM: amplitude modulator,
PM: phase modulator, and PBC: polarization-beam combiner.
S1
S2
Fig.2. Constellation map in the Stokes space which is
most suitable for octal SOP modulation.
Th3K.6.pdf
S tr =
OFC 2014 © OSA 2014
 ±1
1  
±1 .
3 
 ±1
(2)
In addition to the octal SOP modulation, binary modulation of S0 between I 0 and 3I 0 ( I 0 > 0) can increase the
modulation level up to 16.
3. Receiver configuration
Figure 3 shows the configuration of our proposed receiver. In Fig.3, the polarizer ( 0° Pol) has the x transmission
axis; the transmission axis of the polarizer ( 45° Pol) is rotated by 45° with respect to the positive x direction; the
fast axis of the quarter-wave plane ( λ / 4 ) is aligned to the x axis. The incoming signal is equally split into four
branches, where we measure the signal intensity S0 , the intensity of the x-polarization component I x , the intensity of
the 45° linearly-polarized component I 45 , and the intensity of the right-circularly-polarized component I R . These
°
four outputs from photodiodes (PDs) are converted to digital data using four-channel analog-to-digital converters
(ADCs). The clock (CLK) extracted from the first branch controls sampling instances of the ADCs at the rate of one
sample per symbol. Stokes vectors are then calculated using the following relation [3]:
 2 I x ( k ) − S0 ( k ) 

S ( k )  2 I =
( k ) − S0 ( k ) ,
45°
 2 I R ( k ) − S0 ( k ) 


(3)
T
where k is the number of samples. Assume that initial SOPs given by Str = [ ±1 ±1 ±1] 3
/ are transformed to
reference Stokes vectors v i ( k ) (i = 1,,8) at the receiver through fiber birefringence. We demodulate the signal by
comparing the measured Stokes vector S ( k ) with the reference v i ( k ) as follows: First, we take inner products
between the received normalized Stokes vector S(k ) / S0 (k ) and all of the eight reference Stokes vectors v i ( k ) and
find the reference vector v n ( k ) which gives the maximum value of the inner products. Next, the measured S0 is leveldiscriminated in a binary manner. The measured Stokes vector is thus discriminated in the Stokes space and the
symbol is demodulated.
Incoming
optical
signal
CLK
PD
PD
Pol
Pol
ADC
PD
Pol
Intensity
discriminator
Stokes-vector
calculator
SOP
discriminator
Decoded
symbol
Reference
Stokes-vector
updater
PD
Fig.3. Direct-detection receiver for the demodulation of Stokes-vector-modulated signals.
The reference Stokes vectors are fluctuating slowly due to the change in fiber birefringence; therefore, we need
to track such fluctuations of the reference vectors. In the k-th sample, we have decided that the measured normalized
Stokes vector is the n-th reference vector. After such a decision, the n-th reference vector is updated by using the
measured normalized Stokes vector as
v n ( k +=
1)
S (k ) ,
{v ( k ) + µS ( k ) / S ( k )} / v ( k ) + µS ( k ) / n
0
n
0
(4)
where µ is a step-size parameter.
Since the speed of SOP fluctuations is slow and computational complexity of our DSP circuit is low enough, we
do not have any serious problem for DSP-circuit implementation. The disadvantage of our scheme is that adaptive
linear equalization is not implemented in the receiver; therefore, the Stokes-vector modulation format may be
mainly applied to transmission systems shorter than 100 km, where polarization-mode dispersion (PMD) is not
significant.
Th3K.6.pdf
OFC 2014 © OSA 2014
4. Bit-error rate characteristics
Through computer simulations, we calculate BER characteristics of the 16-ary Stokes-vector modulation scheme as
a function of the carrier-to-noise ratio (CNR) /bit/pol. The intensity S0 is modulated between I 0 and 3I 0 , and the
threshold of the level discrimination is adjusted to 2I 0 so that the best BER performance is obtained.
In the simulations, loading additive white Gaussian noise, we prepare the sequence of the Jones vector having a
certain value of CNR/bit/pol and the number of symbols N equals 220. We also assume that δ f ·T = 10−2 , where
δ f denotes the linewidth of the transmitter laser and T the symbol duration. Supposing that the symbol rate is 10
Gsymbol/s, the linewidth of the laser is as wide as 100 MHz, which is much larger than that of common distributedfeedback (DFB) semiconductor lasers. Next, we scramble the SOP of the signal to emulate the random fluctuation
of fiber birefringence. The azimuth and the altitude of the signal SOP represented on the Poincaré sphere are
randomly varied at the rate of 2 × 10−6 rad/symbol, which is much faster than that in the real system. Finally, the
Jones vector including fluctuations of the SOP and the carrier phase is converted to the Stokes vector. The Stokes
vector is demodulated by the algorithm given in Sec. 3, where the step-size parameter µ = 1 / 27. Finally, the number
of bit errors is counted.
The red curve in Fig. 4 shows BERs for the 16-ary Stokes-vector modulation format calculated as a function of
CNR/bit/pol. We have confirmed that this curve is influenced neither by laser linewidths nor by SOP fluctuations.
For comparison, the BER curve for the dual-polarization quadrature phase-shift keying (DP-QPSK) signal, which is
demodulated with a conventional digital coherent receiver [4], is shown by the black curve in Fig.4. In this
calculation, we ignore the laser linewidth and the SOP fluctuation. In addition, the blue curve shows BERs of the
single-polarization (SP) binary IM · DD scheme.
The 16-ary modulation format can achieve 100 Gbit/s at the moderate symbol rate of 25 Gsymbol/s and ADC
sampling rate of 25 GSa/s, although its receiver sensitivity is 7-dB lower than that of the DP-QPSK format at
BER= 10−5. Compared with the SP-IM · DD scheme, however, the receiver-sensitivity degradation is only 3 dB at
BER= 10−5 in spite of the number of modulation levels as high as 16. This modulation format is very attractive for
short-reach 100-Gbit/s transmission, because it can be demodulated by the direct-detection receiver with lowcomplexity DSP and has reasonably-high receiver sensitivity.
16-ary
DP-QPSK
SP-IM・DD
Log(BER)
-2
-3
-4
-5
6
8
10
12
14
16
18
CNR/bit/pol [dB]
Fig.4. BER characteristics of the 16-ary Stokes-vector modulation format calculated as a function of CNR/bit/pol.
BER curves for DP-QPSK and SP-IM signals are also shown for comparison.
5. Conclusions
Through intensive computer simulations, we have analyzed the receiver sensitivity of the 16-ary Stokes-vector
modulation format. Direct detection employing low-complexity DSP can cope with such a modulation format and
achieve the bit rate of 100 Gbit/s at the moderate symbol rate of 25 Gsymbol/s and ADC sampling rate of 25 GSa/s.
Therefore, our proposed receiver for large-capacity ( ≥ 100 Gbit/s) and short-reach (<100 km) transmission is a less
costly alternative to the digital coherent receiver.
References
[1] S. Betti, F. Curti, G. De Marchis, and E. Iannone, “Multilevel coherent optical system based on Stokes parameters modulation,” J. Lightwave
Technol. 8, 1127-1136 (1990).
[2] S. Benedetto and P. Poggiolini, “Performance evaluation of polarization shift keying modulation schemes,” Electron. Lett. 26, 256-258 (1990).
[3] C. Brosseau, Fundamentals of polarized light, John Wiley & Sons, Inc. (1998).
[4] K.Kikuchi, “Digital coherent optical communication systems: Fundamentals and future prospects,” IEICE Electronics Express 8, 1642-1662
(2011).
[b2]
Th2A.29.pdf
OFC 2014 © OSA 2014
Estimation of OSNR for Nyquist-WDM Transmission
Systems Using Statistical Moments of Equalized Signals in
Digital Coherent Receivers
Md. Saifuddin Faruk1,*, Yojiro Mori2, and Kazuro Kikuchi3
1
Department of Electrical and Electronic Engineering, Dhaka University of Engineering and Technology, Gazipur, Bangladesh
2
Departmrent of Electrical Engineering and Computer Science, Nagoya University, Japan
3
Department of Electrical Engineering and Information Systems, the University of Tokyo, Japan
*
[email protected]
Abstract: We propose a novel method of OSNR estimation in Nyquist-WDM transmission
systems based on the measurement of statistical moments of equalized signals in the digital
coherent receiver. Its effectiveness is verified with computer simulations.
OCIS codes: (060.2330) Fiber optic communication; (060.1660) Coherent communications; (060.2920) Homodyning.
1. Introduction
In order to increase the spectral efficiency of wavelength-division-multiplexed (WDM) transmission systems, one of
the most promising techniques is to use very tight spacing between WDM channels. If these channels are filtered out
with Nyquist filters having rectangular transmission windows, they can be placed in the frequency domain as close
as their symbol rate without any intersymbol interference (ISI). Such a multiplexing approach is called NyquistWDM [1]. The Nyquist-WDM technology is also considered as a promising method for generating super-channels
with very high bit rates.
In any WDM system, estimation of the optical signal-to-noise-ratio (OSNR) is very important because OSNR
clearly indicates channel health. In a conventional WDM system employing guard bands between WDM channels,
OSNR of the i-th channel at the center wavelength λi can be estimated by using an optical spectrum analyzer (OSA)
as follows: We measure noise powers in guard bands at λi±Δλ, where Δλ is generally the half of the WDM channel
spacing. From such out-of-band noise powers, we calculate the noise power in the signal bandwidth as shown in
Fig.1 (a). OSNR is then obtained as the ratio of the signal power to the in-band noise power. However, such out-ofband noise measurements are not possible in Nyquist-WDM systems, which have no guard bands as shown in Fig.
1(b). Therefore, the standard method of OSNR estimation using OSA fails in Nyquist-WDM systems.
λi
Δλ
λi
Power
Power
Interpolated noise
at λi
Δλ
Wavelength
Wavelength
(a)
(b)
Fig.1. WDM signal spectra in (a): conventional-WDM systems having guard bands and (b): Nyquist-WDM systems without any guard band.
Extending our previous work [2] in this paper, we derive a generalized formula for estimating OSNR from the
statistical moments of equalized signals in digital coherent receivers. Using this formula, we can estimate OSNR of
both of the M-ary phase-shift keying (M-PSK) signal and the M-ary quadrature-amplitude modulation (M-QAM)
signal under the Nyquist-WDM condition. The proposed method operates in a phase-noise insensitive manner, since
it involves only second- and fourth-order moments. Moreover, it is not affected by linear transmission impairments
such as chromatic dispersion (CD) and polarization-mode dispersion (PMD), because OSNR estimation is done after
adaptive filtering. The effectiveness of the proposed method is verified with computer simulations; estimation errors
in QPSK and 16-QAM Nyquist-WDM systems are found below 0.2 dB and 0.5 dB, respectively.
2. Principle of the proposed OSNR-estimation method
Assume that the channel power is modest so that the transmission system operates in the optically linear region and
that the tap length of the adaptive FIR filter implemented in the digital coherent receiver is sufficient enough to
compensate for all of the linear impairments. Then, the output signal from the adaptive filter can be approximated as
Th2A.29.pdf
OFC 2014 © OSA 2014
yn | Can e jTn N wnc * hn ,
(1)
where an is the M-PSK or M-QAM symbol amplitude, C the signal-power scale factor, N the noise-power scale
factor, w'n the amplified-spontaneous-emission (ASE) noise, θn the phase noise, * the convolution operator, and hn
the impulse response of the equalization filter. The symbol amplitude an and the noise amplitude w'n are
stochastically-independent random variables with a zero-mean value. When we neglect the polarization-dependent
loss, the coherent optical channel has the all-pass nature; and thus, hn has an impulse response of an all-pass filter
[3]. Under such a condition, we can write wnc * hn
(1) can be rewritten as
wn , where w'n is statistically equivalent to wn [4]. Therefore, Eq.
Can e jTn N wn .
yn
(2)
The second-order moment μ2 of yn can be expressed as
P2
E{ yn yn* } CE{ an } CN ( E{an e jTn wn* } E{an*e jTn wn }) NE{ wn },
2
2
(3)
where superscript (-)* denotes complex conjugate. Considering that the signal and the noise are mutuallyindependent random variables with zero means and variances of E{|an|2}= E{|wn|2}= v, we can rewrite Eq. (3) as
P2
v(C N ).
(4)
On the other hand, the fourth-order moment μ4 of yn can be written as
P4
E{( yn yn* )2 }
C 2 E{(an an* )2 } 2C CN ( E{an an* an e jTn wn* } E{an an* an*e jTn wn }) CN ( E{(an e jTn wn* ) 2 } 4 E{an an* wn wn* }
E{(an * e jTn wn )2 }) 2 N CN ( E{wn wn* an e jTn wn* } E{wn wn* an*e jTn wn }) N 2 E{(wn wn* )2 }.
ka v2C 2 4v2CN kwv2 N 2 ,
(5)
where ka=E{|an|4}/E{|an|2}2 and kw= E{|wn|4}/E{|wn|2}2 are kurtoses of the signal and the noise, respectively.
Solving Eq. (4) and (5), we obtain C

2P22 P4
1°
® P2 2 ka
v¯
°
1 2 P 22 P 4 and
,
N
2 ka
v
½
°
¾ . Therefore, determining P2 and P4 ,
°
¿
we can estimate the carrier-to-noise ratio (CNR) as
CNR
2P22 P4
^P
2 ka 2
`
2P22 P4 .
(6)
In a practical system, we can calculate second- and fourth-order moments from a block of L symbols as
1 L 1 2
1 L 1
4
P2 | ¦ yn and P 4 | ¦ yn , respectively.
Ln 0
Ln 0
Equation (6) is a generalized equation to calculate CNR of any arbitrary modulation format. The value of ka is
dependent on the modulation format; for example, in the case of QPSK, since an {1, 1, j , j} , we have ka=1 and
CNR is expressed as
CNRQPSK
2P22 P4
^P 2
`
2P22 P4 ,
(7)
which is similar to that derived in [2, 5]. On the other hand, for the 16-QAM signal, since
an {r1 r i, r1 r 3i, r3 r i, r3 r 3i} , ka =1.32 and CNR is given as
CNR16QAM
2P22 P4
^P
2
`
0.68 2P22 P4 .
(8)
The OSNR in dB can be estimated from the CNR value [6] as
OSNRdB
10log10 (CNR) 10log10 Rs / Br ,
(9)
Th2A.29.pdf
OFC 2014 © OSA 2014
where Rs is the symbol rate and (Rs/Br) is a scaling factor that adjusts the measured noise bandwidth to the reference
bandwidth Br. The bandwidth Br is usually set to 12.5 GHz, which is equivalent to the 0.1-nm OSA resolution
bandwidth.
3. Simulation results and discussions
In order to confirm effectiveness of our proposed OSNR-estimation method, we conduct computer simulations
under the following conditions: In the transmitter, spectra of QPSK or 16-QAM signals are shaped by the Nyquist
filter with a roll-off factor of zero. Baud rates are fixed at 25 GBaud for both of the QPSK system and the 16-QAM
system. Multiple wavelength channels are aligned in the frequency domain without any guard band. As system
impairments, CD, the laser phase noise, and the ASE noise are taken into account. We assume that ASE noise is a
white Gaussian noise and control OSNR by changing the average power of ASE noise. At the receiver side, after
wavelength demultiplexing with the Nyquist filter, the two-fold oversampled signal is fed into a 21-tap half-symbolspaced FIR filter, which is adapted by the constant-modulus algorithm (CMA) for the QPSK signal or the radialdirected equalization (RDE) algorithm for the 16-QAM signal. From the equalized signal, we estimate OSNR using
Eqs. (7)-(9).
Figure 2(a) shows OSNR estimation results for the QPSK modulation format. The OSNR estimated by our
proposed method is plotted as a function of the setup value of OSNR. The red curve is obtained when only ASE
noise is considered and CD and the phase noise are ignored. In the calculation of the blue curve, the 3-dB linewidth
δf for the transmitter laser as well as the local oscillator is 500 kHz and CD is ignored. The black curve is obtained
when δf =500 kHz and CD=500 ps/nm. For all of the three cases, the estimated OSNR value is almost the same as
the setup OSNR value, proving that accurate OSNR estimation is done independently of the phase noise and CD.
The maximum estimation error is found below 0.2 dB over the OSNR estimation range from 6 to 15 dB. Figure 2(b)
shows estimation results for the 16-QAM modulation format. The red curve is calculated when only ASE is included
and CD and the phase noise are ignored. The blue curve is obtained when δf =150 kHz and CD=0. The black curve is
obtained when δf =150 kHz and CD= 500 ps/nm. Accurate estimation is found over a wide range of OSNR for all of
the three cases with the maximum estimation error less than 0.5 dB.
22
Estimated OSNR [dB]
[ ] [dB]
Estimated OSNR
16
14
12
10
ASE
ASE+PN
ASE+PN+CD
8
6
6
8
10
12
14
Setup value of OSNR [dB]
(a)
16
20
18
16
ASE
ASE+PN
ASE+PN+CD
14
12
12
14
16
18
20
Setup value of OSNR [dB]
22
(b)
Fig. 2 Estimated OSNR versus setup values of OSNR in (a) QPSK systems and (b) 16-QAM systems. ASE: amplified-spontaneous emission
noise, PN: phase noise, and CD: chromatic dispersion.
4. Conclusions
Aiming at applications to Nyquist-WDM systems, we have proposed a novel OSNR estimation method based on the
measurement of statistical moments of the equalized signal in digital coherent receivers. The very good estimation
performance is verified in computer simulations of QPSK and 16-QAM Nyquist-WDM systems.
References
[1] G. Bosco, A. Carena, V. Curri, P. Poggiolini, and F. Forghieri, “Performance limits of Nyquist-WDM and CO-OFDM in high-speed PMQPSK systems,” IEEE Photon. Technol. Lett. 22, 1129–1131 (2010).
[2] M. S. Faruk and K. Kikuchi, “Monitoring of optical signal-to-noise ratio using statistical moments of adaptive-equalizer output in coherent
optical receivers,” Opto-Electronics and Communications Conference, 233–234 (2011).
[3] E. Ip and J. M. Kahn, "Digital equalization of chromatic dispersion and polarization mode dispersion," J. Lightwave Technol. 25, 2033-2043
(2007).
[4] G. Colavolpe, T. Foggi, E. Forestieri, and G. Prati, "Robust multilevel coherent optical systems with linear processing at the receiver," J.
Lightwave Technol. 27, 2357-2369 (2009).
[5] D. J. Ives, B. C. Thomsen, R. Maher, and Seb J. Savory, "Estimating OSNR of equalised QPSK signals," Opt. Express 19, B661-B666
(2011) .
[6] P. J. Winzer and R.-J. Essiambre, Optical Fiber Telecommunication V B (Academic press, 2008), Chap. 2.
[b3]
M3A.2.pdf
OFC 2014 © OSA 2014
Multi-dimensional Permutation Modulation Aiming at Both
High Spectral Efficiency and High Power Efficiency
Shota ISHIMURA and Kazuro KIKUCHI
Department of Electrical Engineering and Information Systems, the University of Tokyo
7-3-1 Hongo, Bunkyo-Ku, Tokyo 113-8656, Japan
[email protected]
Abstract: We analyze the performance of multi-dimensional permutation modulation formats.
With the increase in the dimension of modulation, their spectral efficiencies can approach the
Shannon limit even when their power efficiencies are kept high.
OCIS codes: (060.2330) Fiber optics communications; (060.4080) Modulation.
1. Introduction
With the recent development of digital coherent optical communication systems, the constellation diagram of
modulated optical signals can be designed in the multi-dimensional vector space. The optical electric field has two
degrees of freedom of the complex amplitude, that is, the in-phase and quadrature (IQ) components, and two degrees
of freedom of the state of polarization (SOP); thus, the four-dimensional vector space is constituted for a single
wavelength channel [1]; in addition, including k orthogonal wavelength channels, we can have the total number of
dimensions of 4k.
Permutation modulation is a general modulation format defined in such a 4k-dimensional vector space [2]. We
consider binary modulation where either +1 or -1 is allotted to each dimension. Then, we can express the
permutation modulation format as (n, m) ( n  m  1 ), which means that m elements of the n-dimensional vector are
either +1 or -1, and remaining n-m elements are zero. An extreme case where n=m is nothing but the dualpolarization (DP) wavelength-division-multiplexed (WDM) quadrature phase-shift-keying (QPSK) scheme. The
DP-QPSK WDM scheme is most commonly employed in current coherent optical communication systems to
enhance the spectral efficiency. On the other hand, the opposite extreme where m=1 is called the biorthogonal
modulation format, where the power efficiency is very high, while the spectral efficiency is reduced significantly [3,
4]. Especially in four-dimensional modulation (n=4), this scheme is equivalent to polarization-switched QPSK (PSQPSK) [1].
In the region between these two cases ( n  m  1 ), permutation modulation may create a novel modulation
format having both high spectral efficiency and high power efficiency within the limit determined from Shannon’s
theorem [3]; however, such a study has not been reported so far.
In this paper, we analyze spectral efficiency and power efficiency of permutation modulation formats in a
systematic manner. Through these analyses, we find out a novel four-dimensional modulation format that can
transmits 5.3 bits per symbol, keeping the power efficiency almost the same as that of the DP-QPSK format. We
also show that the relation between the spectral efficiency and the power efficiency can approach the Shannon limit
when we increase the number of dimensions of permutation modulation using multiple wavelength channels.
2. Spectral efficiency, power efficiency, and the Shannon limit between them
We first define the power efficiency as

2
d min
.
4 Eb
(1)
In Eq.(1), Eb is the average energy per bit, d min is the minimum Euclidean distance between the nearest
constellation points, and the squared distance of a constellation point from the origin is normalized so that it
represents the energy. Next, we define the spectral efficiency SE as the bit rate per polarization in a unit bandwidth:
C
,
(2)
SE 
WD / 2
where C denotes the bit rate, W the bandwidth of a single wavelength channel, and D the dimension of the system.
When we transmit a signal whose symbol duration is T, the Nyquist bandwidth is W=1/T. If the number of symbols
of a specific modulation format is M, C  log2 M / T  W log2 M . Then, Eq. (2) yield another expression for SE as
SE 
log 2 M
.
D/2
(3)
M3A.2.pdf
OFC 2014 © OSA 2014
On the other hand, Shannon’s law is given as
D
S
(4)
log 2 (1  ) ,
2
N
where S / N is the carrier-to-noise ratio of the single-wavelength and single-polarization optical electric field.
Equations (2) and (4) yield

E 
(5)
SE  log 2 1  SE b  ,
N0 

C
where Eb / N0 represents the energy per bit to noise power-spectral-density ratio. On the other hand, from Eq. (1),
we have
2
Eb 1 d min
.
(6)

N0  4 N0
The symbol-error rate (SER) is determined by
(7)
SER  erfc   / 2 ,
where


  dmin / 2 N0 .
(8)
If we assume SER=10-3,   2.18 . Substituting Eqs. (6) and (8) into Eq. (5), we have

SE 
(9)
SE  log 2 1   2
 .
 

This equation expresses the relation between the spectral efficiency and the power efficiency in the Shannon limit.
By using Eq. (9), SE in the Shannon limit is plotted by a solid curve in Figs.1 and 2 as a function of 1/  , which
is the inverse of the power efficiency and means the receiver-sensitivity penalty. In addition, SE of M-ary
quadrature-amplitude-modulation (QAM) formats is plotted as a function of 1/  for comparison. We find that
multi-level modulation formats such as higher-order QAM approaches the Shannon limit only when 1/  is much
larger than 0 dB; on the other hand, in the power-efficient region where 1/   0 dB, multi-level modulation
formats can improve the spectral efficiency toward the Shannon limit as will be discussed in Sections 3 and 4.
3. Four-dimensional permutation modulation
The four-dimensional vector space is constituted by two polarizations and IQ components of the optical electric field,
using a single wavelength channel. Applying permutation modulation in this vector space, we have four sets of
vectors C1=(4,1), C2=(4,2), C3=(4,3), and C4=(4,4) shown in Table 1. All of the vector elements are obtained by
possible permutations of rows of vectors in these sets. The sets, C1, C2, C3, and C4, have the characteristics
summarized in Table 2. The power efficiency and the spectral efficiency are calculated from Eqs. (1) and (3),
respectively.
Table 1. Sets of vectors for four-dimensional permutation modulation.
C1=(4,1)
( 1,0,0,0)
( 1, 1,0,0)
( 1, 1, 1,0)
( 1, 1, 1, 1)
Table 2. Performance of four-dimensional permutation modulation.
C1
Number of symbols M
Minimum distance d min
Spectral efficiency SE [bit/s/Hz/pol]
Power efficiency  [dB]
C3
8
24
32
16
2
40
1.5
1.76
2.3
0.59
2.5
-0.79
2
0
2.7
0.1
Modulation formats C1, C2, and C4 have already been studied and called PS-QPSK [1], POL-QAM [5], and DPQPSK, respectively. The modulation format C3 has not been investigated yet; it has the spectral efficiency higher
than DP-QPSK (5 bits/symbol), but its power efficiency is lower than that of DP-QPSK. On the other hand,
combining C3 with C1 (C1 C3), we can improve both spectral efficiency and power efficiency of C3 as shown in
M3A.2.pdf
OFC 2014 © OSA 2014
Table 2, because C1 C3 maintains the minimum distance 2 of C1 and C3. This format can transmit 5.3 bits per
symbol, keeping the power efficiency as high as 0.1 dB, which is slightly higher than that of DP-QPSK. Such
characteristics are even better than those of the 32-SP-QAM format recently investigated in [6]. In Fig.1, spectral
efficiencies of these four-dimensional permutation modulation formats are plotted by red dots as a function of
sensitivity penalties. C4 and DP-4-QAM (DP-QPSK) have the same characteristics.
Note that we can demodulate such four-dimensional modulation formats in a systematic manner, using the
decision-directed least-mean-square (LMS) algorithm implemented in digital coherent receivers [7].
4. Higher-dimensional permutation modulation
In this section, the modulation dimension is extended more than four by using multiple wavelength channels. As an
example, we consider the twelve-dimension permutation modulation format Cm  12, m  , which employs three
wavelength channels. Similarly to four-dimensional modulation, we can calculate the spectral efficiency and the
power efficiency of twelve-dimension permutation modulation formats. The SE-versus- 1/  plot is shown in Fig.2
by red dots, where C12= (12, 12) is omitted because its performance is the same as that of (4, 4) (DP-QPSK).
With the increase in the number of dimension from four to twelve, we find that the SE-versus- 1/  curve
approaches the Shannon limit closely. Especially, C6, C7, and C8 can transmit more than 5 bits per symbol, while
their power efficiencies are better than that of DP-QPSK. Such increase in the spectral efficiency without reduction
of the power efficiency is realized owing to multi-dimensional modulation.
Using the OFDM transmitter and receiver [4], we can cope with modulation and demodulation of higherdimensional modulation formats, where multiple wavelengths are employed.
10
Shannon limit
256
64
4
16
C1C3
C3
C2
2
C1
4
1
-5
Spectral
Spectralefficiency
Efficiency [bit/s/Hz/pol]
[bit/s/Hz/pol]
Spectral
Spectralefficiency
Efficiency[bit/s/Hz/pol]
[bit/s/Hz/pol]
10
M-ary
M-aryQAM
QAM
dimensions
44 dimension
0
5
10
15
Sensitivity penalty 1/ γ [dB]
Fig.1. Spectral efficiency as a function of sensitivity penalty for fourdimensional permutation modulation. The Shannon limit and the result
for M-ary QAM are also shown for comparison.
Shannon limit
256
64
4
CC
C C7 8 9C10
C5 6
C
C4
C3
2
C2
16
11
4
1
M-ary QAM
M-ary
QAM
4 dimensions
dimension
12dimensions
12
dimension
C1
-5
0
5
10
15
Sensitivity penalty 1/ γ [dB]
Fig.2. Spectral efficiency as a function of sensitivity penalty for
twelve-dimensional permutation modulation. The Shannon limit
and results for M-ary QAM and four-dimensional permutation
modulation are also shown for comparison.
5. Conclusions
We have analyzed multi-dimensional permutation modulation and found a novel four-dimensional modulation
format transmitting 5.3 bits per symbol with the power efficiency which is almost the same as that of DP-QPSK. It
is also shown that with the increase in the number of dimensions, the SE-versus-1/γcurve can approach the
Shannon limit in the power-efficient region.
References
[1] M. Karlsson and E. Agrell, “Which is the most power-efficient modulation format in optical links?” Optics Express 17, 10814-10819 (2009).
[2] I. Jacobs, “Comparison of M-ary modulation systems,” Bell System Technical Journal 46, 843-864 (1967).
[3] J. G. Proakis, Digital Communications, 4th ed. (McGraw Hill, 2001).
[4] K. Kikuchi and M. Osaki, “Highly-sensitive coherent optical detection of M-ary frequency-shift keying signal,” Optics Express 19, B32-B39
(2011).
[5] H. Bülow, “Polarization QAM (POL-QAM) for coherent detection schemes,” Optical Fiber Communication Conference (OFC2009), OWG2
(2009).
[6] L. Coelho and N. Hanik, “Global optimizationof fiber-optic communication systems using four-dimensional modulation formats,” European
Conference on Optical Communication (ECOC2011), Mo.2.B.4 (2011).
[7] K. Kikuchi, “Digital coherent optical communication systems: Fundamentals and future prospects,” IEICE Electronics Express 8, pp.16421662 (2011).
[b4]
We.3.C.3.pdf
Novel Polarization-diversity Scheme Based on Mutual Phase
Conjugation for Fiber-nonlinearity Mitigation in Ultra-long
Coherent Optical Transmission Systems
Hongbo Lu, Yojiro Mori*, Changyo Han**, and Kazuro Kikuchi
Department of Electrical Engineering and Information Systems, The University of Tokyo
7-3-1 Hongo, Bunkyo-Ku, Tokyo 113-8656, Japan, [email protected]
*Currently with Nagoya University, Nagoya, Japan
**Currently with Electronics and Telecommunications Research Institute (ETRI), Daejeon, Korea
Abstract We propose a novel method of fiber-nonlinearity mitigation for ultra-long optical transmission
systems, which employs two polarization modes mutually phase-conjugated. Simulations and
experiments show that in the nonlinear region, this diversity method significantly outperforms both of
the single- and dual-polarization schemes at the same bit rate.
Introduction
In long-haul optical fiber transmission systems,
nonlinear (NL) waveform distortion stemming
from self-phase modulation (SPM) ultimately
limits the system performance. Current
researches have been concentrated on NLmitigation algorithms such as the back1
propagation (BP) method , where NL distortion
is compensated for through digital signal
processing (DSP) at the receiver. However, too
complicated DSP implementation makes the BP
method infeasible for practical usage.
For practical nonlinearity mitigation, the
diversity scheme2 is an attractive solution; the
signal power is dissipated into multiple replicas,
which are detected by multiple coherent
receivers. In this scheme, the signal-to-noise
ratio (SNR) can be restored by coherently
processing multiple received signal replicas,
while the dissipation of the original signal power
into several replicas leads to the reduction of the
signal power of each replica; therefore, SPMinduced waveform distortion during fiber
transmission might be alleviated.
Although space, wavelength, and polarization
can be applied as dimensions of diversity, the
polarization diversity is the best candidate for
coherent optical communications, because a
single polarization-diversity coherent receiver
can process the signals. However, simple
transmission of two identical replicas by
polarization-division multiplexing (PDM) has the
performance same as single-polarization (SP)
transmission with the same total signal power. In
order to reallocate the signal power effectively,
we propose a novel polarization-diversity
scheme where the signals on two polarization
modes are mutually phase-conjugated (MPC).
Compared with the SP scheme at the same
bit rate, our proposal can always improve NL
tolerance by 3 dB and further obtain the extra
NL mitigation effect depending on the
transmission distance.
Through extensive computer simulations and
experiments, the advantage of our proposed
method is verified in 20-Gbit/s QPSK
transmission systems. Our experimental results
show about 4-dB improvement of NL tolerance
by the proposed polarization-diversity scheme
based on MPC. Such nonlinearity mitigation
extends the current limit of the transmission
distance over twice, while computational cost at
the receiver is almost the same as the
conventional scheme.
System configuration of the MPC method
The process of modulation and demodulation in
the proposed scheme is shown as follows.
At the transmitter, a pair of optical IQ
modulators (IQMs) is used to generate a desired
signal and its phase-conjugated replica, which
are transmitted simultaneously by PDM. When
complex amplitudes of horizontal and vertical
, and
polarization modes are denoted as
, ,
respectively,
the
y-polarization
component in our scheme is given as the phaseconjugation of the x-polarization component at
0,
0, .
the input (z=0), that is,
At the receiver (z=L), the DSP circuit performs
chromatic-dispersion
(CD)
compensation,
polarization demultiplexing, and carrier recovery
in a conventional manner3. Only the operation of
,
,
, /2 is added at
the final stage of DSP, where
, is the
,
signal after carrier recovery.
Principle of nonlinearity mitigation in our
proposed scheme
First of all, we should note that the MPC scheme
manages to keep its linear SNR equal to that of
the SP scheme. Since we use two orthogonal
polarization modes to transmit the original signal
and its MPC replica, the amount of ASE noise is
We.3.C.3.pdf
Simulations
We conduct extensive simulations of the 20Gbit/s QPSK system to examine our MPC
scheme in comparison with SP and DP
schemes. Each span of the 1,000-km-long link
consists of a 100-km standard single-mode fiber
(SMF) and an EDFA with 4-dB noise figure,
which fully compensates for the span loss. The
loss coefficient, the CD parameter, and the
nonlinear coefficient of the SMF are 0.2 dB/km,
17 ps/nm/km, and 1.5 /W/km, respectively. We
ignore linewidths of the transmitter laser and the
local oscillator (LO) and assume perfect balance
of the two IQMs to solely examine the influence
of the NL effect. For evaluation of the signal
quality, we calculate the variance σ of the
constellation-point distribution of received
signals, where the carrier amplitude is
normalized to unity.
Figure1 shows σ in the 1,000-km 20-Gbit/s
QPSK system as a function of the lauched
signal power. Blue, black, and red curves
correspond to SP, DP, and MPC schemes,
respectively. We find that the NL tolerance of
the MPC scheme is improved by 5 dB and 4 dB
compared with SP and DP ones, respectively.
0.1
2
the linear SNR is maintained by the MPC
scheme.
On the other hand, the NL mitigation benefits
from the joint effort of two mechanisms
described in the following.
First, in our scheme, cross phase modulation
(XPM) between the two polarization modes is
reduced compared with SPM in the
corresponding SP transmission system. In the
MPC scheme, the signal power is dissipated into
the two polarization modes, which are
decorrelated through MPC. In such a case, XPM
between the two decorrelated polarization
modes is half the amount of SPM in the SP
transmission system as far as the total power
and the bit rate are the same; thus, we can
expect 3-dB improvement of NL tolerance with
our scheme independently of the symbol rate,
the transmission distance, and the modulation
format.
Second, we can have an extra NL mitigation
effect stemming from mutual phase conjugation.
The x-polarization mode obeys the two-mode
4
nonlinear Schrödinger equation given as
| |
, (1)
where ɑ, β2, and γ represent the loss coefficient,
the CD parameter, and the nonlinearity
coefficient, respectively. On the other hand, the
complex conjugate of the y-polarization
component obeys
| |
, (2)
Note the opposite sign of the NL terms in Eqs.
,
(1) and (2). The operation of
,
, /2 may cancel the NL terms
in Eqs. (1) and (2) since
0,
0, at the
input end.
However, signs of dispersion terms in Eqs. (1)
and (2) are opposite; then, the NL cancellation
effect becomes imperfect, because the evolution
, is not the same as that of
, . As
of
a result, this extra NL mitigation works best in
the first several spans of the link where
correlation of NL phase rotations between the
two mutually phase-conjugated signals is strong.
As the signal experiences severer CD after longdistance transmission, the extra NL mitigation
effect gradually decreases. Taking these two NL
mitigation effects into account, we can enjoy
more than 3-dB improvement of NL tolerance.
As an alternative approach to transmit a
signal at the same bit rate, we can choose the
dual-polarization (DP) scheme, where two
independent tributaries at the half symbol rate
are transmitted. However, the NL reduction is
less than 3 dB because of the lower symbol rate.
Moreover, we cannot achieve the NL
cancellation effect; therefore, the improvement
of NL tolerance is smaller than 3 dB.
Variance σ
twice as large as that in the SP scheme. The
, where is the
noise power is given as 2
required signal bandwidth equal to the symbol
is the noise spectral density.
rate and
Meanwhile,
the
signal
power
remains
2
, where
and
are
unchanged:
respectively the power in the SP scheme and
that of one polarization tributary in the MPC
scheme. At the coherent receiver of the MPC
scheme, the correlation between the two MPC
signals is restored after CD compensation;
therefore, adding the two MPC signals in the
digital domain coherently, we can increase the
restored signal power by four times. Therefore,
the SNR level of the MPC scheme is given
, showing that
as
SP QPSK
DP QPSK
MPC QPSK
0.01
-15
-10
-5
0
5
Launched Power [dBm]
Fig.1: Simulation results of σ in the 1,000-km 20Gbit/s QPSK system calculated as a function of the
lauched power for SP, DP and MPC schemes.
We.3.C.3.pdf
Solid blue and red curves in Fig. 2 illustrate σ
calculated as a function of the transmission
distance in the 20-Gbit/s QPSK system for SP
and MPC schemes, respectively. The launched
signal power is set to 0 dBm. The variance σ in
the MPC scheme at 1,000 km is smaller than
that in the SP scheme at 500 km. This
improvement is mainly owing to the 3-dB power
reduction for each tributary in the MPC scheme.
We also confirm that σ increases as a slightlysuperlinear function of the transmission distance,
in line with the previous work5. Therefore, the
limit of the transmission distance is generally
extended over twice by the MPC scheme.
0.08
SP QPSK simulation
SP QPSK experiment
MPC QPSK simulation
MPC QPSK experiment
Variance σ2
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0
400
500
600
700
800
900
and clock-phase recovery; and carrier-phase
estimation. The signal was differentially decoded.
Figure 3 shows σ as a function of the
launched signal power for SP (blue), DP (black),
and MPC (red) schemes. In the NL region, the
MPC scheme increases NL power tolerance by
4 dB and 3 dB against SP and DP schems,
respectively. Small disagreement from the
simulation result (Fig.1) may be due to
imbalance of the two IQMs.
Dashed blue and red curves in Fig. 2 show
experimental results of σ as a function of the
transmission distance for SP and MPC schemes,
respectively. The launched signal power was 0
dBm and the transmitted signal was received
every 100-km span until the total distance
reached 1,000 km. The experiment results are in
good agreement with those of simulations,
which confirms that our proposal can extend the
limit of transmission distance over twice. Inset
constellations are measured at 1,000 km for SP
and MPC schemes, clearly showing the NL
mitigation effect of the MPC scheme.
1000
Transmission Distance [km]
Experiments
We experimentally verify the performance of our
proposal in comparison with SP and DP
schemes, using the 1,000-km 20-Gbit/s QPSK
system. A distributed-feedback laser diode
(DFB-LD) with a 100-kHz linewidth was used as
a transmitter laser whose center wavelength
was 1,552 nm. A differentially-encoded NRZQPSK signal was generated by a LiNbO3 optical
IQM which was driven by a 10-GSymbol/s
arbitrary waveform generator with 29-1 PRBS. In
the DP scheme, PDM was conducted by the
split-delay-combine method; in the MPC scheme,
the MPC signals were generated and calibrated
by two IQMs. The fiber link was the same as the
one used in the simulations. However, the actual
loss of each 100-km SMF span was 23 dB
including the splicing loss, which was
compensated for by an EDFA.
At the receiver, the signal was detected by a
homodyne phase- and polarization-diversity
coherent receiver. The LO was a DFB-LD with a
100-kHz linewidth and a 10-dBm output power.
The received signal was asynchronously
sampled by 20-GS/s ADCs with 8-bit resolution.
Data of 1-M samples were processed offline by
DSP including fixed CD compensation; 255-tap
FIR-filter-based equalization adapted by the DD6
LMS algorithm for polarization demultiplexing
0.1
Variance σ2
Fig. 2: Simulation and experimental results of σ
in the 20-Gbit/s QPSK system as a function of the
transmission distance for SP and MPC schemes.
The launched power is 0 dBm.
SP QPSK
DP QPSK
MPC QPSK
0.01
-15
-10
-5
0
Launched Power [dBm]
5
Fig.3: Experimental results of σ in the 1,000-km 20Gbit/s QPSK system measuered as a function of the
lauched power for SP, DP, and MPC schemes.
Conclusions
We have proposed the polarization-diversity
scheme based on MPC for NL mitigation in
coherent optical transmission systems. With
effective signal-power allocation on the two
polarization modes, NL signal distortion is
significantly
alleviated.
Simulations
and
experiments show that at the same bit rate, the
MPC
scheme
has better
transmission
characteristics than SP and DP schemes
References
[1] E.Ip, J. Lightwave Technol. 28, 939 (2010).
[2] D.Tse et al., Fundamentals of Wireless
Communication, Cambridge University Press
(2005).
[3] S.Savory, Opt. Express 16, 804 (2008).
[4] S.G.Evangelides Jr. et al., J. Lightwave
Technol. 10, 28 (1992).
[5] A.Carena et al., J. Lightwave Technol. 30,
1524, (2012)
[6] Y.Mori et al., Opt. Express 20, 26236 (2012).
[b5]
Mo.4.C.6.pdf
Wavelength Demultiplexing of Nyquist WDM Signals under
Large Frequency Offsets in Digital Coherent Receivers
Yojiro Mori*, Changyo Han**, Hongbo Lu, and Kazuro Kikuchi
Department of Electrical Engineering and Information Systems, The University of Tokyo
7-3-1 Hongo, Bunkyo-Ku, Tokyo 113-8656, Japan
*Currently with Nagoya University, Nagoya, Japan, [email protected]
**Currently with Electronics and Telecommunications Research Institute (ETRI), Daejeon, Korea
Abstract We propose a novel wavelength-demultiplexing algorithm applicable to Nyquist WDM
systems. The proposed scheme features the ability of perfect wavelength demultiplexing even under
the frequency offset half as large as the symbol rate. Through simulations and experiments,
effectiveness of our scheme is demonstrated in 16-QAM Nyquist WDM systems.
Introduction
The Nyquist wavelength-division-multiplexed
(WDM) system can achieve the ultimate spectral
efficiency
without
inducing
intersymbol
interference and inter-channel crosstalk1,2. In
such a system, wavelength demultiplexing is
generally performed by non-adaptive digital
filters having sharp cutoff characteristics.
However, when the received signal includes the
frequency offset between the transmitter laser
and the local oscillator, non-adaptive filters cut
off a part of the target-channel spectrum and
take up a part of the adjacent-channel spectrum.
As a result, the demultiplexed signal is
deteriorated both by intersymbol interference
and inter-channel crosstalk.
On the other hand, adaptive filters can
demultiplex WDM channels even in the
presence of the frequency offset3. In this case,
adaptive filters must consist of long-delay taps
to realize steep cutoff characteristics. However,
the convergence property of such filters is
disrupted by adjacent channels contained in the
target channel, especially when the received
signal includes the large frequency offset.
In order to overcome these problems, we
propose a novel wavelength-demultiplexing
scheme for Nyquist WDM systems, in which the
adaptive filter and the non-adaptive filter are
combined so as to achieve wavelength
demultiplexing under the large frequency offset.
The proposed scheme undergoes two
processes: First, in the training mode, adaptive
finite-impulse-response (FIR) filters with shortdelay taps roughly perform wavelength
demultiplexing in addition to polarization
recovery. After that, the phase estimator
estimates the frequency offset. Next, in the
tracking mode, wavelength demultiplexing is
done in the frequency domain by the Nyquist
filter with a roll-off factor of 0. The passband of
the filter is shifted by the offset frequency
obtained from the phase estimator in the training
mode. With such a scheme, we can
simultaneously realize the fast tap-adaptation in
the training mode and the sharp attenuation
slope of the wavelength-demultiplexing filter
even under the frequency offset.
Through computer simulations on 3-ch
Nyquist
WDM
dual-polarization
16-ary
quadrature-amplitude-modulation (QAM) signals,
we confirm that the proposed scheme enables
perfect wavelength demultiplexing of the Nyquist
WDM signal under the frequency offset half as
large as the symbol rate. Such theoretical
predictions are validated by demodulation
experiments at 10 Gsymbol/s.
Principle of operation
Figure 1 illustrates the block diagram of the
proposed demodulator for Nyquist WDM signals.
In
the
training
mode,
wavelength
demultiplexing and polarization recovery are
simultaneously conducted by two-by-two
butterfly-structured adaptive FIR filters with
short-delay taps. The adaptive FIR filters need
not achieve perfect wavelength demultiplexing
at this time, but the signal must be recovered
accurately enough to estimate the frequency
offset. After the frequency offset ∆fest is
estimated by the phase estimator, the
adaptation mode is switched to the tracking
mode.
In the tracking mode, after the discrete Fourier
transform (DFT) of the blocked-signal sequence,
wavelength demultiplexing is performed in the
frequency domain by the Nyquist filter with a
roll-off factor of 0, whose passband is shifted by
the estimated offset frequency ∆fest. Then, the
two-by-two butterfly-structured adaptive FIR
filters perform polarization recovery. Finally,
phase fluctuations induced by the laser phase
noise and the frequency offset are eliminated by
the phase estimator. We repeat these processes
Mo.4.C.6.pdf
in a block-by-block manner.
The fluctuation speed of the frequency offset
is usually much lower than the symbol rate. Let
f l be the maximum frequency of such
fluctuation. In addition, Lb denotes the block
length, Lt the insertion interval of the training
signal, and B the symbol rate. The requirement
for these parameters in our proposed scheme is
1 / f l ≥ Lt / B ≥ Lb / B. When this relation is
satisfied, the frequency offset is almost constant
within the block and the block-by-block
processing is effective. On the other hand, the
training
sequence
should
be
inserted
periodically so that we can track the change in
the frequency offset; however, it is not
necessary to add the training sequence to each
block as far as the above relation is maintained.
For tap adaptation of FIR filters, we employ
the decision-directed least-mean-square (DDLMS) algorithm with the training mode. This is
because blind algorithms such as the constantmodulus algorithm (CMA) may filter out a
channel next to the target channel when the
frequency offset is too large. On the other hand,
the DD-LMS algorithm never fails to select the
target channel owing to the use of the training
sequence.
Wavelength
demultiplexing
& polarization Phase-frequency
recovery
recovery
Training
mode
Input
Switch
Δfest
Nyquist
Tracking
filters
mode Wavelength
demultiplexing
Adaptive
FIR filters
estimators
Adaptive
FIR filters
Phase Output
estimators
Phase
Polarization Phase-frequency
recovery
recovery
FIR filters is 16. Each DFT block is composed of
212 sampled data. To estimate the frequency
offset, we employ the dual-stage phase
estimator based on the DD-LMS algorithm4,5.
Figure 2 shows BER characteristics at Eb/N0 =
9 dB as a function of the frequency offset
normalized to the symbol rate. Red, green, and
blue curves represent BERs obtained by the
proposed scheme, the non-adaptive-filter-based
scheme, and the adaptive-FIR-filter-based
scheme, respectively.
From Fig. 2, we find that the non-adaptivefilter-based
scheme
provides
the
best
performance when the frequency offset is
negligible; however, its BER performance
drastically degrades with increase in the
frequency offset, since a part of the targetchannel spectrum is outside the filter bandwidth.
On the other hand, the wavelengthdemultiplexing property of the adaptive-FIRfilter-based scheme is entirely independent of
the frequency offset, but BERs are worse than
those of our proposed scheme. This is because
adaptive FIR filters with short-delay taps cannot
form sharp cutoff characteristics, and crosstalk
from adjacent channels is inevitable. Although
the increase in the number of taps improves the
BER performance, we have found that the
adaptation speed of the long-delay-tap filter
seriously decreases under the large frequency
offset because of crosstalk from adjacent
channels.
In contrast, the proposed scheme can achieve
the best BER performance independent of the
frequency offset. The allowable frequency offset
is half of the symbol rate, which is determined
from the bandwidth of the anti-aliasing filter.
0
Simulations
In order to evaluate the frequency-offset
tolerance of the proposed scheme, we conduct
computer simulations on 3-ch Nyquist WDM
dual-polarization 16-QAM systems.
We calculate bit-error ratios (BERs) using 217
symbols per polarization, where 212 symbols are
utilized as the training signal. The frequency
offset is assumed to be constant within the
calculated symbol-sequence interval. The laser
phase noise is neglected. Before twofold
oversampling, the double-sided signal spectrum
is shaped by an anti-aliasing filter whose
bandwidth is twice as large as the symbol rate.
The sampled sequence is then processed by the
proposed scheme. The tap length of adaptive
log10(BER)
Fig. 1: Block diagram of the proposed wavelengthdemultiplexing scheme in conjunction with
polarization recovery and phase/frequency recovery.
-1
-2
Proposed scheme
Non-adaptive filter
Adaptive filter
-3
-0.4
-0.2
0
0.2
0.4
Frequency offset / Symbol rate
Fig. 2: Calculated frequency-offset tolerance.
Experiments
We measure the BER performance of the
proposed scheme in a 3-ch Nyquist WDM 10-
Mo.4.C.6.pdf
the proposed scheme can achieve perfect
wavelength demultiplexing under the frequency
offset as far as the whole signal spectrum is
processed.
log10(BER)
0
-1
-2
Proposed scheme
Non-adaptive filter
Adaptive filter
-3
-5
-4
-3
-2
-1
0
1
2
3
4
5
Frequency offset [GHz]
Fig. 3: Measured frequency-offset tolerance.
5
Measured frequency offset [GHz]
Gsymbol/s dual-polarization 16-QAM system.
At the transmitter, we used three distributedfeedback laser diodes (DFB-LDs). Linewidths of
DFB-LDs were around 100 kHz. The wavelength
of the center channel was 1552 nm. The center
channel and its adjacent channels were
modulated
independently
with
two
IQ
modulators (IQMs). The IQMs were driven by an
arbitrary-waveform generator (AWG), which
generated 4-level Nyquist-shaped IQ signals.
Polarization multiplexing was conducted in the
split-delay-combine manner. The 3-ch Nyquist
WDM 10-Gsymbol/s dual-polarization 16-QAM
signal was thus obtained and incident on the
receiver.
In front of the receiver, the total optical power
of three WDM channels was fixed at -25 dBm
with a variable optical attenuator. The signal
was then pre-amplified by an erbium-doped fiber
amplifier and detected by a phase- and
polarization-diversity optical receiver, where
another DFB-LD having the characteristics
same as the transmitter lasers was used as a
local oscillator. The frequency offset ∆f between
the center-channel transmitter laser and the
local oscillator was controlled by monitoring the
beat frequency with a radio-frequency spectrum
analyzer. Output signals from the receiver were
filtered out by anti-aliasing filters having the 3dB cutoff frequency of 7 GHz and sampled at 20
Gsample/s by a 4-ch analog-to-digital converter.
Finally, the digitized signal was processed
offline by the proposed demodulation algorithm.
The tap length of adaptive FIR filters, the block
size for DFT, and the length of the training
sequence were the same as those used in the
simulation.
Figure 3 shows BER performances of the
center channel as a function of the frequency
offset. Definitions of colors of curves are the
same as those in Fig. 2. We find that the BER
performance is gradually degraded even with
the proposed scheme when |∆f| > 3 GHz. This is
because the signal spectrum is partially cut off
by the anti-aliasing filter when the frequency
offset is larger than 3 GHz. Except for this point,
experimental results are in good agreement with
simulations.
In Fig. 4, diamonds represent actually
measured frequency offsets against frequency
offsets estimated from the phase estimator. As a
reference, the black line shows the desired state
where the estimated frequency offset and the
measured one are identical. We can observe
that the frequency offset is properly estimated in
the training mode using short-delay-tap FIR
filters.
From these results, we have confirmed that
4
Experimental results
Desired state
3
2
1
0
-1
-2
-3
-4
-5
-5
-4
-3
-2
-1
0
1
2
3
4
5
Estimated frequency offset [GHz]
Fig. 4: Measured frequency offsets against estimated
frequency offsets.
Conclusions
We have proposed a novel wavelengthdemultiplexing scheme for Nyquist WDM signals.
Computer simulations on 3-ch Nyquist WDM
dual-polarization 16-QAM signals demonstrate
that the proposed scheme can perform
wavelength demultiplexing under the influence
of large frequency offsets. Such theoretical
predictions are validated by demodulation
experiments at 10 Gsymbol/s.
References
[1] G.Bosco et al., J. Lightwave Technol. 29, 53
(2011).
[2] R.Schmogrow et al., Opt. Express 20, 317
(2012).
[3] J.Wang et al., Proc. OFC’13, OTu2I.3 (2013).
[4] Y.Mori et al., Opt. Express 20, 26236 (2012).
[5] Y.Mori et al., Proc. OFC’13, OTu3I.7 (2013).
[b7]
2013 18th OptoElectronics and Communications Conference held jointly with 2013 International Conference on Photonics in Switching
(OECC/PS)
TuR2-1
Proposal of Optical-Sampling-Based
Constellation Monitor for DP-QPSK Signals
Kazuro Kikuchi1,2 and Sze Y. Set2
1 Department of Electrical Engineering and Information Systems, The University of Tokyo
7-3-1 Hongo, Bunkyo-Ku, Tokyo 113-8656, Japan
2 Alnair Labs Corporation, Westside Gotanda, 6-2-7 Nishi-Gotanda, Shinagawa-Ku, Tokyo 141-0031, Japan
Abstract
We propose a novel linear optical-sampling system for
monitoring the constellation diagram of DP-QPSK signals.
A polarization-constellation in the Stokes space is obtained
with phase-noise-free optical-sampling measurements and
converted to an IQ-constellation diagram.
I. INTRODUCTION
Optical sampling systems can measure the temporal
intensity waveform of high bit-rate optical signals with
low-speed electronics; therefore, they have been
commercialized as cost-effective solutions to optical
performance monitoring in intensity-modulation/directdetection (IMDD) systems [1].
On the other hand, the recent development of digital
coherent optical receivers has brought the 100-Gbit/s
dual-polarization quadrature phase-shift keying (DPQPSK) system into practical use [2]. In such system, the
constellation diagram is a good measure of the signal
quality, and the constellation monitor with low cost and
complexity is strongly desired.
The linear optical-sampling system was proposed for
this purpose [3]. Such system utilizes the sampling pulse
train for the local oscillator (LO) in the homodyne phasediversity receiver. The constellation diagram can be
drawn through digital signal processing (DSP) of the
receiver outputs. However, the laser phase noise induces
the phase diffusion of constellation points within a long
sampling-time interval, which disenables the accurate
measurement of the constellation diagram.
The one-symbol-delayed dual-channel linear opticalsampling system was also proposed to observe the
constellation diagram of differential QPSK (DQPSK)
signals [4]. To avoid the phase-diffusion problem, it
measures the phase difference between two adjacent
symbols by using two homodyne phase-diversity
receivers in parallel; however, the system configuration is
much more complex than the standard linear sampling
system and its application is limited to DQPSK signals.
In this paper, we propose a novel linear opticalsampling system, which enables the measurement of the
constellation diagram of DP-QPSK signals with a lowspeed standard coherent receiver comprising phase and
polarization diversities [5]. The polarization-constellation
diagram, which is not influenced by the laser phase noise,
is measured in the Stokes space and mapped on a twodimensional IQ plane through DSP. Computer
simulations of the proposed scheme show that the
measured variance of the Gaussian distribution of
constellation points is about 3-dB larger than that of each
polarization
tributary.
However,
the
accurate
Copyright ©2013 IEICE
constellation measurement becomes possible without
depending on the laser phase noise as well as the
sampling rate.
II. PROPOSAL OF A NOVEL CONSTELLATION
MONITOR FOR DP-QPSK SIGNALS
A. System Configuration
Figure 1 shows the configuration of the proposed
system. Let the symbol duration be T. The incoming DPQPSK signal is filtered out and its clock is extracted. The
sampling-pulse generator is operated at the rate of 1/(nT),
where n is a sufficiently large integer, and the sampling
phase is adjusted so that the sampling instance is at the
center of the symbol duration. Such phase-locking
process for LO can also be done in the digital domain by
using the asynchronous optical-sampling pulse train [6].
Let the complex amplitude from the x-polarization port
of the coherent receiver be Ex(k) and that from the ypolarization port be Ey(k), where k denotes the number of
samples. Such sampled data are converted to the digital
domain with a four-channel analog-to-digital converter
(ADC) and sent to the DSP circuit consisting of the
Stokes vector generator and the two-dimensional mapper.
S3
DP-QPSK
Signal
Optical
Filter
Clock frequency
=1/(nT)
S1
Coherent
Receiver
ADC
Q
S2
Stokes
Vector
I
Two-dimensional
Mapping
Sampling Pulse
Generator
Fig.1. Configuration of the proposed linear optica-sampling system.
B. Principle of Operation
The state of polarization (SOP) of the DP-QPSK signal
is determined from the phase difference between the two
polarization tributaries. As shown in Fig.2, the SOP of
the DP-QPSK signal is each of linear polarizations (+45°,
-45°) and circular polarizations (right, left), depending on
such phase difference. Therefore, four constellation
points are generated in the Stokes space from four
constellation points of each polarization tributary in the
IQ plane.
Due to randomly-fluctuating fiber birefringence, the
plane S, on which the center of each polarization
constellation point is placed, is rotating threedimensionally; however, the positional relation among
2013 18th OptoElectronics and Communications Conference held jointly with 2013 International Conference on Photonics in Switching
(OECC/PS)
S3
Phase difference between State of polarization
y- and x-polarizations
c
S1
a
d
0
π/2
π
3π/2
S2
(a)
a: Linear (+45°)
b: Circular (Right)
c: Linear (-45°)
d: Circular (Left)
In phase
S
Fig.2. Relation between the Stokes vector and the phase difference of
the two polarization tributaries of the DP-QPSK signal.
(b)
Quadrature
b
larger in our proposed system than in the conventional
system. However, it should be noted that the performance
of our system depends upon none of the laser phase noise,
the symbol rate, and the sampling rate; therefore, our
proposed system can employ an extremely low sampling
rate in spite of considerable laser phase noise.
Quadrature
the four centers is unchanged. Therefore, when the threedimensional constellation points are mapped on the plane
S, we have the two-dimensional constellation diagram.
As shown in simulations given in Sec. III, we find that
the distribution of such constellation points is almost
Gaussian and its variance is about 3-dB larger than that of
each polarization tributary.
The calculation procedure for the constellation diagram
is as follows: Three-dimensional coordinates of the
Stokes vector are given from Ex(k) and Ey(k) as
2
(1)
S2 ( k ) = 2 Ex ( k ) E y ( k ) cos (θ ( k ) ) ,
(2)
S3 ( k ) = 2 Ex ( k ) E y ( k ) sin (θ ( k ) ) ,
(3)
where θ ( k ) = arg ( E y ( k ) / Ex ( k ) ) . Then, four centers of
constellation points are obtained, and the two
dimensional constellation diagram is displayed on the
plane S that is constituted from the four centers. The
measurement time interval for such constellation
monitoring should be shorter than the fluctuation period
of SOP.
III. SIMULATION RESULTS
Figure 3(a) shows the simulation result of our proposed
scheme. The carrier-to-noise ratio (CNR) is fixed at 15
dB at the output of the optical filter. The number of
samples is 216. The constellation diagram (left) is entirely
independent of the laser phase noise, the symbol rate, and
the sampling rate. The right figure shows the histogram
of the amplitude of constellation points in the in-phase
direction. The distribution is almost Gaussian.
On the other hand, the constellation diagram and the
distribution of constellation points of one polarization
tributary are shown in Fig. 3(b), which is obtained by the
conventional linear sampling system. We assume that
δfT=10-4, where δf is the 3-dB linewidth of the laser,
n=10, and the averaging sample number for phase
estimation is 23. For example, when the symbol rate is 25
Gbaud, the sampling rate is as high as 2.5 GS/s and the
laser linewidth is 2.5 MHz. We find that constellation
points are diffusing due to the phase noise of lasers (left
figure), and the distribution of constellation points is
broadened accordingly (right figure).
Only when the laser phase noise is ignored, we can
have the ideal constellation-point distribution even by the
conventional linear sampling system as shown in Fig.
3(c). Comparing Fig. 3(a) with Fig. 3(c), we find that the
variance of the distribution of constellation points is 3-dB
Copyright ©2013 IEICE
(c)
Quadrature
2
S1 ( k ) = Ex ( k ) − E y ( k ) ,
In phase
In phase
Fig.3. Constellation diagrams (left) and distributions of the normalized
amplitude of constellation points in the in-phase direction (right). (a):
our proposed scheme, (b): conventional scheme with laser phase noise,
and (c): conventional scheme without laser phase noise.
IV. CONCLUSION
We have proposed a novel linear optical-sampling
system, which enables the measurement of the
constellation diagram of DP-QPSK signals with a lowspeed coherent receiver. Mapping the polarizationconstellation diagram on a two-dimensional IQ plane, we
have the system performance independent of the laser
phase noise, the symbol rate, and the sampling rate. Thus,
the proposed system can provide us with a cost-effective
solution to constellation monitoring of DP-QPSK signals.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
http://www.alnair-labs.com/
E. Yamazaki et al., “Fast optical channel recovery in field
demonstration of 100-Gbit/s Ethernet over OTN using real-time
DSP,'' Opt. Exp., vol.19, pp.13139-13184, Jul. 2011.
C. Dorrer, C.R. Doerr, and I. Kang, “Measurement of eye diagram
and constellation diagram of optical sources using linear optics
and waveguide technology,” J. Lightwave Technol., vol.23,
pp.178-186, Jan. 2005.
K. Okamoto and F. Ito, “Ultrafast measurement of optical DPSK
signals using 1-symbol delayed dual-channel linear optical
sampling,” IEEE Photon. Technol. Lett., vol.20, pp.948-950, 2008.
K. Kikuchi, “Coherent optical communications: Historical
perspectives and future directions,” Chapter 2 in High Spectral
Density Optical Communication Technologies, edited by M.
Nakazawa, K. Kikuchi, and T. Miyazaki, Springer, 2010.
T. Kiatchanog, K. Igarashi, T. Tanemura, D. Wang, K. Katoh, and
K. Kikuchi, “Real-time all-optical waveform sampling using a
free-running passively mode-locked fiber laser as the sampling
pulse source,” Optical Fiber Communication Conference (OFC
2006), OWN1, Anaheim, CA, March 5-10, 2006.
[b8]
2013 18th OptoElectronics and Communications Conference held jointly with 2013 International Conference on Photonics in Switching
(OECC/PS)
ThR3-7
Blind Equalization and Carrier-Phase Recovery
in QPSK Coherent Optical Receivers Based on
Modified Constant-Modulus Algorithm
Md. Saifuddin Faruk* and Kazuro Kikuchi**
* Department of Electrical and Electronic Engineering, Dhaka University of Engineering and Technology, Gazipur-1700, Bangladesh
** Department of Electrical Engineering and Information Systems, The University of Tokyo, 7-3-1 Hongo, Bunkyo-Ku,
Tokyo 113-8656, Japan
Abstract
We propose and experimentally verify a fully-blind
scheme for equalization and carrier-phase recovery in
QPSK coherent optical receivers, which is based on the
modified CMA. Its computational complexity is much less
than the conventional approach.
I. INTRODUCTION
Digital coherent receivers allow compensation for
linear transmission impairments and carrier-phase
recovery in the digital domain. In such receivers
employing blind algorithms for digital signal processing
(DSP), data are generally recovered in the following way
[1]: First, analog-to-digital converters (ADCs) sample
received signals at the rate twice as high as the symbol
rate. Next, finite-impulse-response (FIR) filters, where
filter-tap weights are adapted every two samples through
the constant-modulus algorithm (CMA), are operating as
equalizers for linear transmission impairments. Finally,
M-th power carrier-phase recovery is done prior to
symbol decoding of M-ary PSK signals. However, such
process requires high computational cost due to several
nonlinear operations such as taking the M-th power,
taking the argument, and unwrapping the phase.
A multi-tap finite-impulse-response (FIR) filter
adapted by the modified CMA with lower computational
cost was investigated for simultaneous equalization and
carrier recovery in wireless communication systems [2].
However, such approach is not suitable for optical
communication systems, because a large phase noise
destroys phase correlation between symbols at filterdelay-time intervals.
On the other hand, in this paper, we propose and
experimentally verify a novel equalization and carrierrecovery scheme for the quadrature phase-shift-keying
(QPSK) signal. We use a multi-tap FIR filter for blind
equalization and a one-tap phase rotator for carrier-phase
recovery, both of which are adapted by the modified
CMA. The computation cost of such approach is low
because no nonlinear operation is required. In addition, it
works phase-insensitively unlike the conventional
modified-CMA-based scheme in [2] because we separate
phase-estimation functionality from the equalizer.
A similar configuration using the decision-directed
least-mean-square (DD-LMS) algorithm was proposed in
[3]. However, the DD-LMS algorithm introduces large
feedback delay from the decision circuit during tap
adaptation and also fails to provide reliable blind startup.
On the other hand, the proposed algorithm can be
implemented in the full-blind mode.
Copyright ©2013 IEICE
II. PROPOSED SCHEME
In this section, superscripts (•)T and (•)* are transpose
and conjugate operators, respectively, and subscripts (•)R
and (•)I of a variable denote real and imaginary parts of
the variable. For simplicity, we consider the singlepolarization system; however, the proposed scheme can
easily be extended to the butterfly configuration of the
equalizer applicable to the dual-polarization system.
The conventional CMA-based equalizer followed by
M-th power carrier-phase recovery is shown in Fig. 1(a).
On the other hand, we propose a multi-tap FIR filter for
equalization and a one-tap phase rotator for carrier phaserecovery, where the modified CMA updates their tap
coefficients, as shown in Fig. 1(b).
Let h(n) be a tap-coefficient column vector of the
equalizer, c(n) a tap coefficient of the phase rotator, and
u(n) a column vector of the FIR-filter input, where n is
the sampling index. Then, the output of the equalizer is
given by u ′(n) = h(n)T u(n) and the output of the phase
rotator is v ( n) = c( n)u ′( n). The tap coefficient is updated
by using the stochastic gradient algorithm as
(1)
c(n + 1) = c(n) + μ ec (n)u ′* (n),
where µc is the step-size parameter and ec(n) is the error
signal. Differently from the conventional CMA, the error
signal in the modified CMA is computed from real and
imaginary parts of the output signal as
{
(
ec (n) = vR (n) RR − vR (n)
where
RR,I
is
a
4
2
)} + j {v (n) ( R − v (n) )} ,
prescribed
I
I
constant
2
(2)
given
as
I
2
RR,I = E{ aR,I (n) } E{ aR,I (n) }, a(n) is the transmitted symbol,
and E the expectation operator. In the conventional CMA,
4
2
R = E{ a(n) } E{ a(n) }.
The modified CMA moves the real part of the output
on lines of x = ± RR and the imaginary part on lines of
y = ± RI . Thus, it can control the amplitude and phase of
the output to proper values, and carrier phase recovery is
accomplished. Next, tap-coefficients of the equalizer is
updated as
(3)
h( n + 1) = h( n) + μ h eh ( n)u* (n),
where µh is the step-size parameter and the error signal is
computed by multiplying ec(n) with the estimated phase
as eh ( n) = ec ( n){ c( n) c( n)}. Such an approach eliminates
the phase noise involved in u*(n) of Eq. (3), leading to
phase-noise insensitive adaptation of the multi-tap filter
with the modified CMA.
2013 18th OptoElectronics and Communications Conference held jointly with 2013 International Conference on Photonics in Switching
(OECC/PS)
u/(n)
h(n)
CMA
Error
e (n) calculation
R
(a)
u(n)
u/(n)
h(n)
Modified
CMA
Modified
CMA
c( n )
e h (n)
v (n)
c (n)
c( n )
Error
calculation
ec (n)
RI,RQ
(b)
Fig. 1. Schematic of the blind-equalization and carrier-phase recovery
unit. (a): Conventional approach consisting of CMA-based equalization
and M-th power carrier phase recovery. (b): Proposed approach of
modified CMA-based blind equalization and carrier phase recovery.
III. SIMULATIONS
We conduct numerical simulations to show that our
modified-CMA-based scheme can work in a phaseinsensitive manner because functions of the equalizer and
the phase estimator are separated. In the simulation, the
phase fluctuation of the restored complex amplitude is
assumed to have the Gaussian distribution with a variance
σ 2 = 4πδ f T , where δf is the 3-dB linewidth of the
transmitting laser as well as the local oscillator (LO) and
T is the symbol duration. We consider a 10-GSymbol/s
QPSK signal that is differentially encoded to avoid cycle
slips. The modified-CMA-based symbol-spaced FIR filter
is used for signal equalization. For all of the cases, stepsize parameters are optimized so that BER is minimized.
As shown in Fig. 2(a), when the conventional
modified-CMA scheme is applied, the 21-tap filter causes
2-dB SNR penalty for δf =150 kHz; on the other hand,
when δf =1 MHz, the performance is very bad even for
the 9-tap filter as shown in Fig. 2(b). In contrast, the
proposed scheme has the performance similar to the onetap phase rotator even with the 21-tap FIR filter, proving
the phase-insensitive operation of our proposed scheme.
0
Theory
Modified CMA (1-tap)
Modified CMA (9-taps)
Modified CMA (21-taps)
Proposed scheme (21-taps)
log10 (BER)
-1
-3
4
6
8
Eb/N0 [dB]
10
12
(a)
log10 (BER)
-1
-2
-4
2
Theory
Modified CMA (1-tap)
Modified CMA (9-taps)
Modified CMA (21-taps)
Proposed scheme (21-taps)
4
6
8
Eb/N0 [dB]
10
12
(b)
Fig. 2. BER characteristics as a function of Eb/N0 for different filter
orders. (a): δf =150 kHz and (b): δf =1MHz.
Copyright ©2013 IEICE
-1.5
Proposed scheme
Conventional approach
-2
-2.5
-3
-3.5
-4
-4.5
-48
-46
-44
-42
-40
Received power [dBm]
Fig. 3. BER characteristics as a function of the received power for the
conventional approach and the proposed scheme.
2
2
1
1
0
0
-1
-1
-2
-2
-1
0
1
2
-2
-2
-1
(a)
0
1
2
(b)
Fig. 4. Constellation diagram after (a) the equalizer and (b) the carrierphase recovery in the proposed scheme.
We have proposed a modified-CMA-based
equalization and carrier-phase-recovery scheme for
QPSK coherent optical receivers. Effectiveness of such
scheme is verified through numerical simulations and 10Gsymbol/s QPSK transmission experiments.
REFERENCES
0
-3
To prove the principle of operation of our proposed
scheme, we conduct 10-Gsysmbol/s single-polarization
QPSK unrepeated transmission over 200 km. The
transmitter laser and the local oscillator have 3-dB
linewidths of 150 kHz. The accumulated chromatic
dispersion is about 3,200 ps/nm. The signal is received by
a phase-diversity homodyne receiver where the
polarization of the incoming signal is controlled
manually. The output of the receiver is sample at 20
Gsample/s and stored for offline DSP. In the DSP circuit,
equalization and carrier-phase recovery are done by either
the conventional method (Fig. 1(a)) or the proposed
scheme (Fig. 1(b)). Symbols are then decoded and BER
is estimated.
As shown in Fig. 3, the proposed scheme has the
performance similar to that of the conventional approach;
however, it has lower computational cost. The rotating
constellation at the output of the multi-tap equalizer
confirms that it operates phase-insensitively (Fig. 4(a)).
On the other hand, the clear QPSK constellation at the
output of the phase rotator (Fig. 4(b)) ensures correct
phase-noise estimation.
V. CONCLUSIONS
-2
-4
2
IV. EXPERIMENTS
v (n)
M-th power
CPE
log10 (BER)
u(n)
[1] K. Kikuchi, “Digital coherent optical communication
systems: fundamental and future prospect,” IEICE
Electronics Express, vol. 20, pp. 1642-1662, 2011.
[2] K. N. Oh and Y. O. Chin, “Modified constant modulus
algorithm: Blind equalization and carrier phase recovery
algorithm,” Proc. ICC, vol. 1, pp. 498 -502, 1995.
[3] Y. Mori, C. Zhang, and K. Kikuchi, “Novel configuration
of finite-impulse-response filters tolerant to carrier-phase
fluctuations in digital coherent optical receivers for higherorder quadrature amplitude modulation signals,” Opt.
Express, vol. 20, pp. 26236–26251, 2012.
[c1]
4 次元セット分割を適用した 8 状態トレリス符号化光変調方式の提案
8-State Trellis-Coded Optical Modulation Scheme with 4-Dimensional Set Partitioning
石村昇太
Shota Ishimura
菊池和朗
Kazuro Kikuchi
東京大学大学院工学研究科電気系工学専攻
Department of Electrical Engineering and Information Systems, Graduate School of Engineering, the University of Tokyo
1. はじめに
近年、4 次元の信号空間を用いることにより従来以上に
感度を高めた光変調方式として、PS-QPSK が注目されてい
る[1]。PS-QPSK のシンボル点は、DP-QPSK のシンボル点
から 4 次元空間中で再隣接点を間引くことにより得られ、
効率よく最小符号間距離を拡大することができる。この分
割手法はセット分割法と呼ばれ、トレリス符号化変調
(Trellis coded modulation: TCM)と共に考案されたものであ
る [2] 。トレリス符号化変調とは、所要帯域を増加させる
ことなく電力利用効率を高める技術であり、電力・帯域制
限の大きい衛星通信などに用いられる。
本稿では、4 次元空間で、DP-QPSK にセット分割を 3 回
施すことによって生成された子信号集合を用いた、8 状態
を有する 4 次元トレリス符号化光変調方式(4D-TCM)を提
案する。最小自由距離を効率よく拡大できるため、非符号
化 PS-QPSK に比べて周波数利用効率を維持したまま、よ
り高感度となり得ることを、シミュレーションにより示す。
2. 符号化器構成及び符号化利得の漸近値
図 1 に符号化器構成を示す。3 ビットが入力され、その
うちの 2 ビットが拘束長 3、符号化率 2/3 の畳み込み符号
器により符号化される。また図 2 に、4 次元 DP-QPSK のセ
ット分割法とそれぞれの集合の最小距離を示す。符号化さ
れていないビットに対しては平行パス間での距離が最大に
なるように、S0 から S7 内の信号点を割り当てる。また各
ステートから分岐する枝と各ステートで合流する枝の間に、
次に大きい最小距離を有する PS-QPSK1、2 を割り当てる。
この構成の最小自由距離を、図 3 をもとにして考察する。
トレリス線図の最近接パスは、図 3 の赤で示した 2 つのパ
スとなる。この時の最小自由距離を  2free とすると
 2free  d{S 0 , S1}2  d{S 0 , S 2 }2  4  4  8
図 2: 4 次元セット分割法
図 3: 8 状態トレリス線図
方式全体の最小自由距離は、
2
d 2free  min[ 2free ,  min
]8
(3)
となり、平行パスによる制限を受けない。
この最小自由距離を用いると、DP-QPSK を基準とした
パワー効率は 4.77dB となる。 PS-QPSK のパワー効率
1.76dB と比較して高感度になっていることがわかる。
3. 計算機シミュレーション
本提案手法の性能を評価するため、計算機シミュレーシ
ョンを行った。信号にガウス雑音を付加し、ディジタルコ
ヒーレント光受信器では、ビタビアルゴリズムを用いて復
号を行う。図 4 で、本手法と非符号化 QPSK、PS-QPSK の
ビット誤り率(Bit-error rate: BER)を比較している。提案手
法は BER=10 3 において、QPSK に対し 3dB、PS-QPSK に
対し 2dB の符号化利得が得られていることがわかる。
4. まとめ
本稿では 4 次元セット分割を用いたトレリス符号化変調
を用いることで、PS-QPSK に対し周波数利用効率を落とす
ことなく 2dB 以上の符号化利得を得られることを示した。
(1)
となる。一方で、平行パスの存在も考慮すると、この最小
2
距離  min
は S0 から S7 内の信号点を割り当てているので、
d 22 と等しくなり、
2
 min
 d 22  8
(2)
となることがわかる。したがってこのトレリス符号化変調
図 1: 畳み込み符号器構成
図 4: 4D-TCM-DP-QPSK の BER 特性
参考文献
[1] M. Karlsson et al., Optics Express 17, 10814-10819 (2009).
[2] G. Ungerboeck, IEEE Trans. Inf. Theory 28, 55 – 67 (1982).
[c2]
強度変調・直接検波光通信システムにおけるディジタル信号処理
を用いた偏波多重分離技術
Electronic Polarization-division Demultiplexing based on Digital Signal Processing in Intensity-modulation
Direct-detection Optical Communication Systems
菊池和朗
Kazuro Kikuchi
東京大学大学院工学系研究科電気系工学専攻
Department of Electrical Engineering and Information Systems, Graduate School of Engineering, the University of Tokyo
まえがき
偏波多重技術はディジタルコヒーレント光ファイバ通
信システムに導入され，QPSK 変調方式とあわせて一波
長あたり 100 Gbit/s の伝送速度が達成された [1]。ディジ
タルコヒーレント光受信器を用いれば，偏波多重信号の
位相情報を抽出できるので，ディジタル領域での偏波制
御が可能となる [2]。一方，強度変調・直接検波 (IM·DD)
方式では，位相情報が失われるため，受信後のディジタ
ル信号処理 (DSP) による偏波制御は不可能であると考
えられてきた。本論文はこの常識を覆す IM·DD 方式に
おける DSP による偏波多重分離技術を提案し，シミュ
レーションによりその有効性を示す。
1
受信器の構成と動作原理
図 1 は，偏波多重 IM 信号に対する受信器の構成を示
す。光回路はストークスアナライザと呼ばれる構成であ
る [3]。４つの分枝の出力は AD 変換され，図 2 に示す
DSP 回路に送られる。偏波多重 IM 信号のストークスパ
ラメータ S0,1,2,3 を求めた後，偏波状態の追尾と偏波多
重分離が行われる。
2
Stokes analyzer
PD
Input
signal
図1
0° Pol
PD
45° Pol
PD
λ / 4 45° Pol
PD
CLK
ADC
DSP
Decoded
bits
x pol
y pol
ル v を更新すれば，このベクトルは光ファイバの複屈折
の変動に起因する偏波変動を追尾する。次に v と各規格
化ストークスベクトルとの内積をとることにより，状態
(II)(a)，(II)(b)，(III) を弁別する。
(I)
(II)
(a) 偏波 1:
論理レベル偏波 1: 低
偏波 2:
偏波 2: 低 (b) 偏波 1:
偏波 2:
ピーク強度
0
2
表 1 強度による偏波多重 IM 信号の分類。
シミュレーション結果
シミュレーションにより，2 で述べた動作原理の検証
を行った。偏波多重 IM 信号は，位相相関のない２台の
独立なレーザから生成される。レーザのスベクトル幅
δf · T = 1 × 10−3 とした。ここで T はビット間隔であ
る。光ファイバの複屈折によって偏波変動はランダムに
変化する。偏波状態は，ポアンカレ球上での角度変化が
10µs で約 1 rad となるように，高速でスクランブルし
た。各偏波 220 ビットの信号列について，2 で述べたア
ルゴリズムを用いて，符号誤り率 (BER) を求めた。図
3 は，偏波あたりのキャリア対雑音比 (CNR) の関数と
して求められた BER 特性を示す。単一偏波の場合に対
して，提案する方式を用いた偏波多重分離に伴うパワー
ペナルティーは極めて小さいことがわかる。
3
偏波多重分離のための受信器構成。
Stokesvector
calculator
Intensity
discriminator
Stokes-vector
amplitude
discriminator
Bit aligner
-4
8
偏波追尾および偏波多重分離のための DSP 回路。
偏波多重信号の全光強度は，偏波トリビュータリの論
理レベルによって，表 1 のように分類できる。ここで
は，各偏波成分の平均強度を 1 に規格化している。ま
ず，全光強度を閾値弁別することにより， (I) の状態と
(II)，(III) を分離する。状態 (II)(a) の規格化ストークス
ベクトル Sa /S0 と (II)(b) の規格化ストークスベクトル
Sb /S0 は反対方向を向く。また，(III) の規格化ストーク
スベクトル Sc /S0 は，これらに直交する。したがって，
Sa /S0 と −Sb /S0 に対して平均操作を行って基準ベクト
Single pol
Dual pol x
Dual pol y
-2
)
R
E
B
(g-3
oL
Reference
Stokes-vector
updater
図2
(III)
高
低偏波 1: 高
低偏波 2: 高
高
4
10
12
CNR/pol [dB]
14
図 3 CNR/pol の関数として求められた BER 特性。
むすび
IM·DD 方式においても，DSP により偏波多重分離が
可能なことを示した。低価格性が要求される短距離大容
量伝送には有効な方式と考えられる。
4
参考文献
[1] E. Yamazaki et al., Optics Express 19, 13139-13184 (2011).
[2] K. Kikuchi, Optics Express 19, 9868-9880 (2011).
[3] C. Brosseau, Fundamentals of polarized light, John Wiley
& Sons, Inc. (1998).
[c3]
ディジタルコヒーレント光通信システムの概要
Overview of Digital Coherent Optical Communication Systems
菊池和朗
Kazuro Kikuchi
東京大学大学院工学系研究科電気系工学専攻
Department of Electrical Engineering and Information Systems, Graduate School of Engineering, The University of Tokyo
1
はじめに
インターネットを流れる情報トラフィックは，年率 1.4
倍の増加率で伸び続けている。光ファイバ通信システム
は，このような膨大なデータを流通させるグローバル
ネットワークのための基盤技術として，大きく発展して
きた。しかし現在に至るまで，ほとんどの光伝送システ
ムは，強度変調 (Intensity modulation: IM)・直接検波
(Direct detection: DD) 方式と呼ばれる極めて単純な方
式を用いてきた。
一方，無線通信の歴史を振り返ると，この IM・DD
方式は，100 年以上も前のマルコーニの時代における通
信方式に対応する。無線通信技術はその後，アームスト
ロングによるスーパーヘテロダイン技術の発明を経て，
電磁波の位相情報を最大限に活用して情報を伝送する
方式が定着し，現在の興隆をみるに至っている。光通信
技術もさらなる発展を目指すには，無線技術と同様な
位相情報の活用が不可欠と考えられる。このような期待
に応える技術が，コヒーレント光通信技術である。この
方式では，受信端に用意された CW 局部発振光（Local
oscillator: LO）と信号光との干渉によって信号光の電
界複素振幅を測定し，信号光の復調を行う。この技術は
1980 年代に一時精力的に研究されたが [1]，技術的困難
さから，その後 15 年以上にわたって忘れ去られていた。
しかし，従来のコヒーレント光通信技術と高速ディジタ
ル信号処理 (Digital signal processing: DSP) を組み合わ
せたディジタルコヒーレント光通信方式が 2005 年に提
案され [2]，現在，100 Gbit/s 光伝送を可能にする唯一
の技術として，世界各国で活発に研究開発が進められて
いる。
本講演では，コヒーレント光通信技術の開発の歴史を
簡単に振り返ったのち，新世代のディジタルコヒーレン
ト光通信技術の概要について解説する。
2
コヒーレント光ファイバ通信の歴史
コヒーレント光受信器は，ショット雑音限界の高い受
信感度を持つ。十分に強度の大きい LO を用いれば，受
信器出力が受信器回路雑音に打ち勝つほど大きくできる
ため，ショット雑音限界まで受信感度を改善できる。高い
受信感度を活用すれば再生中継器間隔を延伸できるので，
1980 年代には長距離コヒーレント光伝送システムの研
究が各国で競って行われた。しかし 1990 年代に入ると，
エルビウム添加光ファイバ増幅器 (Erbium-doped fiber
amplifier: EDFA) と波長分割多重 (Wavelength-division
multiplexing: WDM) 技術の導入により，IM・DD 方式
の光ファイバ通信技術が新たな発展段階に入った。EDFA
を用いて多中継する長距離伝送システムでは，受信光パ
ワーは一定に維持されるので，コヒーレント光受信器の
持つ高受信感度の重要性が薄れた。またコヒーレント受
信器では，信号の位相や偏波の揺らぎに適応的に対処す
る必要があり，この問題の解決が容易ではなかった。こ
のため，コヒーレント光通信の研究開発は，その後 15
年以上にわたり中断されることになった。
1990 年代には，EDFA と WDM 技術に基づく IM・
DD 方式による光伝送システムの大容量化が急速に進ん
だ。この結果，EDFA 帯域を用いて伝送できる情報量の
限界が見え始め，2000 年以降，有限な EDFA 帯域を効
率よく利用するための多値光変調技術が注目を集め始め
た。2005 年には，位相ダイバーシティ・ホモダイン受信
器と高速ディジタル信号処理との組み合わせにより，従
来のコヒーレント受信器の技術的困難性を解決するディ
ジタルコヒーレント光受信器という新しい概念が登場し
た [2]。このようなディジタルコヒーレント受信器では，
受信後のディジタル信号処理により，電気領域で位相雑
音を除去し位相変調を抽出するので，いかなる多値変調
にも対応できる。また，偏波揺らぎもディジタル信号処
理によって補償できるほか，電気段でファイバの群速度
分散や偏波分散を適応的に等化できる。
このような優れた特長のために，ディジタルコヒーレ
ント光通信の研究開発は，近年，世界各国で急速に進ん
でいる。2008 年にはカナダの Nortel 社（現 Ciena 社）
により，本方式を用いて 50 Gbit/s で動作する偏波多重
QPSK 用の ASIC が開発された [3]。さらに 2012 年には，
我が国の国家プロジェクトによって，100 Gbit/s で動作
する ASIC が初めて開発された [4]。モジュール/システ
ムベンダー各社によって，この LSI を用いた 100 Gbit/s
偏波多重 4 相位相変調 (DP-QPSK) 信号用のディジタル
コヒーレント光受信器が商用化され，100 Gbit/s 伝送シ
ステムへの導入が進んでいる。
3
ディジタルコヒーレント光受信器の構成
図 1 に，ディジタルコヒーレント光受信器の構成を示
す。コヒーレント光受信器では，信号光と LO とのビー
トにより電気的な出力を得る。ビート信号は光電界複素
振幅の情報を完全に保持しているので，このビート信号
からディジタル信号処理を用いて，信号光の振幅のみな
らず位相変調をも抽出することができる。図 1 で用いら
れている光回路は，位相ダイバーシティ・ホモダイン検
波回路と呼ばれる。信号光周波数と LO 光周波数は，ほ
ぼ等しくなるように制御される。また，ここでは信号光
の偏波は，LO 光の偏波と一致していると仮定している。
LO1 と LO2 には，90◦ 光ハイブリッドにより 90◦ の位相
差が与えられる。LO の位相を基準として，信号光と LO1
とのビートは信号のコサイン成分を，信号光と LO2 と
のビートは信号光のサイン成分を与える。ダブルバラン
ス型フォトダイオードで測定されたこれら 2 つのビート
信号は，AD 変換器 (Analog-to-digital converter: ADC)
を経て DSP 回路に送られる。次にディジタル領域にお
いて，位相雑音を除去して信号光電界複素振幅が再生さ
れる。この方式では LO を信号光に位相同期する必要が
ないので，フリーランニング状態にある比較的線幅の大
きい半導体レーザを，送信用光源および LO として用い
ることができるようになった。
90° optical hybrid
E1
I1
E2
E4
I2
I4
E3
I3
II
LO1
Optical signal
π/2
LO2
LO
図1
IQ
位相ダイバーシティ・ホモダイン受信器の構成。
図 1 のホモダイン検波回路では，受信感度が信号光の
偏波に依存するという問題があるが，この偏波依存性を
解決する手段として，図 2 に示す偏波ダイバーシティ技
術が開発されている。信号光の垂直および水平偏波を，
同一の LO を用いてそれぞれ別々に位相ダイバーシティ・
ホモダイン検波することにより，各偏波成分に対して光
複素振幅が得られる。これらをディジタル領域で処理す
ることにより，偏波多重分離や偏波分散補償を行うこと
が可能となる。この受信器は，時変的な偏波変動に対し
ても十分に対応できる。位相・偏波ダイバーシティ・ホモ
ダイン検波光回路に関しては，石英系平面光回路 (Planar
lightwave circuit: PLC) を用いたハイブリッド光集積回
路や InP を用いたモノリシック光集積回路の開発が進め
られている。
EEs ,inx, x
Es
PBS
°
90
Optical
Hybrid
E1
E3
ELO , x
IIPDIx1
E2
IIPDQx
2
E4
ELO
PBS
Eins ,,yy
ELO , y
E5
°
90
Optical
Hybrid
IIPDIy3
E6
E7
IIPDQy
4
信号に変換される。2 倍オーバーサンプルすなわちシン
ボルレートの 2 倍のサンプリングレートでサンプルし，
エリアシングを除去するためのアナログフィルタへの要
求を軽くすることが一般的である。
AD 変換後の DSP には，図 3 に示すように，WDM
チャンネル選択，固定波長分散補償，適応等化，キャリ
ア位相推定，シンボル識別などが含まれる。このように
ディジタルコヒーレント光受信器は，多値信号を復調で
きるだけでなく，ディジタル領域で種々の信号処理を行
うことができるという特長がある。特に適応等化部は，
偏波多重分離，偏波分散補償，残留波長分散補償，タイ
ミングジッタ補償，光/電気回路の帯域制限の補償など，
多くの機能を含む。クロック抽出の方法は種々提案され
ているが，図 3 では，分散補償後にクロック周波数を
抽出して ADC のサンプリングレートを制御している。
この場合，クロック位相の制御は適応等化回路で行われ
る。このような適応等化は，有限インパルス応答フィル
タ (Finite-impulse-response filter: FIR filter) のタップ
係数を適応的に更新することにより実現される。キャリ
ア位相推定部では，位相雑音や信号光と LO 光の間の周
波数オフセットが除去される。
ASIC の低消費電力化には，計算量の軽い DSP アル
ゴリズムの開発が重要である。また，光ファイバの非線
形効果の等化なども今後解決すべき課題として残されて
いる。
CLK
X -pol.
Y -pol.
EX
4-Ch
ADC
EY
Carrier
phase
estimator
図3
WDM
channel
selector
Ex ,in
Fixed
equalizer
Adaptive
equalizer
E y ,in
x-pol.
Decoder
Symbol
y -pol.
シンボルを復号するための DSP 回路の概要。
むすび
コヒーレント光受信器にディジタル技術を導入するこ
とにより，安定なホモダイン検波が実現され，光の複素
振幅の情報を完全に抽出することが可能になった。この
ようなディジタルコヒーレント受信器を用いれば，任意
の多値光変調に対応できるだけでなく，電気領域での分
散補償，偏波制御などの新しい機能が実現できる。光集
積デバイス，ASIC，DSP アルゴリズムの一体となった
開発によって，本技術の更なる発展を期待したい。
5
E8
図 2 位相・偏波ダイバーシティ・ホモダイン受信器の
構成。
4
ディジタル信号処理の概要
位相・偏波ダイバーシティ・ホモダイン受信器の 4 つ
の出力端子からは，3 で述べたように，2 つの偏波状態に
対する光複素振幅の情報が得られる。高速の 4 チャンネ
ル AD 変換器を用いて，これら 4 つの出力はディジタル
参考文献
[1] T. Okoshi and K. Kikuchi, Coherent Optical Communication Systems, KTK/Kluwer, 1988.
[2] S. Tsukamoto et al., OFC 2005, Anaheim, CA, USA,
PDP29, March 2005.
[3] H. Sun et al., Opt. Express, vol.16, no.2, pp.873-879,
Jan. 2008.
[4] E. Yamazaki et al., Opt. Express, vol.19, no.14,
pp.13139-13184, July 2011.
[c4]
社団法人電子情報通信学会
THE INSTITUTE OF ELECTRONICS,
INFORMATION AND COMMUNICATION ENGINEERS
信学技報
TECHNICAL REPORT OF IEICE.
コヒーレント光通信システムにおける多次元変復調方式の検討
石村昇太†
菊池和朗†
† 東京大学大学院工学系研究科電気系工学専攻〒 113–8656 東京都文京区本郷 7–3–1
E-mail: †{ishimura,kikuchi}@ginjo.t.u-tokyo.ac.jp
あらまし
コヒーレント光通信においては，2 次元自由度を持つ IQ 平面が 2 つの自由度を持つ偏波それぞれに存在
することから、合計 4 つの自由度を用いた変調が可能となる。近年，この 4 自由度のもとで信号点配置を設計するこ
とにより，パワー効率が最大となる変調方式である PS-QPSK 変調方式が見出されている。本論文では，4 次元空間
での信号点配置の詳細な検討に基づき，DP-QPSK 方式と同一のパワー効率でありながら周波数利用効率がより高い，
新しい変調方式を提案する。次に，このような 4 次元空間で変調された信号を復調するための一般的な信号処理アル
ゴリズムを提案する。さらに，信号空間を 4 次元以上に拡張することにより，周波数利用効率およびパワー効率の観
点から，より優れた変調方式が存在することを示す。
キーワード
コヒーレント光通信、シャノン限界、PS-QPSK、多次元変復調方式
Multi-dimensional modulation and demodulation schemes
in coherent optical communication systems
Shota ISHIMURA† and Kazuro KIKUCHI†
† Department of Electrical Engineering and Information Systems, University of Tokyo
7–3–1 Hongo, Bunkyo-ku, Tokyo 113–8656, Japan
E-mail: †{ishimura,kikuchi}@ginjo.t.u-tokyo.ac.jp
Abstract In newly-developed coherent optical communication systems, we can employ four-dimensional optical
modulation including two-dimensional freedom of polarizations. The most power-eﬃcient modulation format named
PS-QPSK has recently been discovered in such four-dimensional space. This paper proposes another novel four-dimensional modulation format, which is more spectrally-eﬃcient than the DP-QPSK format and has the power
eﬃciency same as the DP-QPSK format. Next, we propose a general demodulation algorithm applicable to any
four-dimensional modulation formats. Finally, we show that with the increase in the dimension of modulation larger
than four, both of the power eﬃciency and the spectral eﬃciency can be improved further.
Key words Coherent optical communication, Shannon limit, PS-QPSK, Multi-dimensional modulation and demodulation formats
1. はじめに
コヒーレント光通信の一つの大きな利点として、直交した二
せることで 2bit 送信して、合計 3bit を送信できるというもの
で、Polarization Switshed-QPSK(PS-QPSK) と名付けられて
いる。この PS-QPSK を用いることにより、パワー効率は IQ
つの偏波状態を利用できることがあげられる。一つの偏波が複
平面上のみで変調を考えた際に最大であった BPSK よりも、
素振幅の実部・虚部という二つの自由度を持つことから、もう
1.76[dB] 改善されることがわかっている。しかし周波数利用効
一つの直交した偏波も合わせて考えれば、合計 4 つの自由度が
率という観点からは Dual Polarization-QPSK(DP-QPSK) に
存在するといえる。この 4 自由度をもとに変調を行うことで、
劣る。このように、一般にパワー効率と周波数利用効率にはト
従来の 2 次元の自由度を持つ IQ 平面上での変調方式よりもパ
レードオフの関係があり、PS-QPSK は周波数利用効率を犠牲
ワー効率が優れたものが存在することが報告されている。[1][2]
にしてパワー効率を高めた変調方式であると言える。
この変調方式は、直交した二つの偏波のうちどちらの偏波を用
本論文では、まずパワー効率と周波数利用効率との関係につ
いるかで 1bit 送信し、さらにその選択した偏波に QPSK をの
いて、シャノン限界から導かれるパワー効率と周波数利用効の
—1—
限界値について定量的に評価する。また 4 次元空間において、
DP-QPSK と同一のパワー効率でありながら周波数利用効率が
より高い、新しい変調方式を提案し、これが DP-QPSK に比べ
てシャノン限界に近づき得るものであることを確認する。また
4 次元空間での復調法についても、変調方式によらない一般的
なアルゴリズムを提案する。さらに４次元空間だけでなく次元
数を拡張することより、周波数利用効率およびパワー効率の観
点から、より優れた変調方式が存在し、それらがよりシャノン
限界に近づき得るものであることを示す。
2. パワー効率-周波数利用効率限界
図 1 シャノン限界と M-PSK、M-QAM のパワー効率周・周波数利
種々の変調方式は「パワー効率」と「周波数利用効率」とい
用効率の関係
う二つの尺度によって定量的に評価することができる。パワー
効率とは以下のように定義される。
γ=
という関係式が得られる。ところで (1) のパワー効率の定義式
d2min
(1)
4Eb
ただし Eb = Es / log2 M は 1 ビット当たりの送信パワー、dmin
の分子分母を N0 で割り、(6) を代入することで
γ=
d2min SE
4N0 2SE − 1
(7)
はシンボルとシンボルの最小距離を表している。つまり、パ
という式が得られる。この式はシャノン限界を組み込んだ、パ
ワー効率とは最小距離の二乗に比例し、パワーに反比例してい
ワー効率と周波数利用効率の関係の限界を示すものと考えるこ
ることから、より信号間隔が離れていると高くなり、また送信
とができる。
パワーが少なくても高くなることがわかる。BPSK、QPSK に
M-PSK および M-QAM がこの限界にどの程度接近しうるの
おいてはパワー効率はいずれも 1 となる。また偏波多重を行っ
か考察する。M-PSK 及び M-QAM のパワー効率と周波数利用
た DP-QPSK でも 1 となる。一般に多重化された変調信号の
効率の関係は以下のようになる。
パワー効率は、多重化を行っていないものと一致する。なぜな
ら、(1) の定義から、最小信号間距離は多重化には無関係であ
り、さらに Es は多重した分だけ増加するが、その分送信でき
るビット数も同じ比率で増加するため、Eb としても変化はな
(8)
(9)
(8)(9) と (7) の限界式の関係を図 1 に示す。(7) では
√
√
N0 ) が 10−3 となる dmin /2 N0 の値を用いてい
いからである。
1
erfc(dmin /2
2
周波数利用効率は以下のように定義される。
SE =
γM −P SK = sin2 (π/2SE )SE
3SE
γM −QAM =
2(2SE − 1)
る。ここで横軸はパワー効率の逆数である Sensitivity penalty
log2 M
N/2
(2)
ただし M は送信シンボル数、N は次元数である。これは単位
周波数あたり、単位偏波あたりのビット数を表している。
ここでパワー効率と周波数利用効率の関係について考えてみ
としていて、右にいくほどパワー効率は下がる。縦軸は周波
数利用効率である。このように M-PSK 及び M-QAM は、パ
ワー効率を低下させることでシャノン限界には近づき得るが、
パワー効率が高いところではシャノン限界から離れてしまうこ
とがわかる。
る。そのためまずシャノン・ハートレーの定理を以下に示す。
C=
3. 4 次元ベクトル集合を用いた変調方式の考察
(
S)
D
W log2 1 +
2
N
(3)
ここではベクトルの要素として、2 値（+1 と-1）のみを考え
ただし D は次元数、W は使用帯域である。この式を変形して
いくと
のベクトル集合を考えてみる。C1 は 4 つの自由度のうち一つ
(
C
Eb C )
= log2 1 +
W D/2
N0 W D/2
C
W D/2
る。この 2 値を用いることで、４次元空間で表 1 のような４つ
(4)
のみを選択し、そこに+1 か-1 を入れる。４つから 1 つを選び
出し、さらに+1 か-1 のどちらを入れるかで、全 8 パターン存
とは単位周波数あたり、単
在する。C2 は４つの自由度のうち 2 つを選択して用いる。こ
位偏波あたりのビット数を表しており、これは周波数利用効率
のため 4 つの自由度のうち、どの２つの自由度を用いるかの選
なる表式が得られる。ここで
択と、選んだ２つの自由度に+1 か-1 のどちらを入れるかで全
そのものである。以降これを SE と置くことにすると、
(
)
Eb
SE = log2 1 +
SE
N0
(5)
24 パターンの集合が存在する。同様に C3 は 32 パターン、C4
表 1 4 次元ベクトル集合
となる。これを Eb /N0 について解くことで
Eb
2SE − 1
=
N0
SE
C1
(6)
C2
C3
C4
(±1,0,0,0) (±1,±1,0,0) (±1,±1,±1,0) (±1,±1,±1,±1)
—2—
表 2 4 次元ベクトル集合のパワー効率および周波数利用効率
周波数利用効率
C1
C2
1.5
2.3
パワー効率 [dB] 1.76 0.59
C3
C4
2.5
2
-0.79
0
は 16 パターンの集合が存在する。C4 は DP-QPSK と等価で
ある。このようにして定めたベクトル集合を表 1 に示す。
これらのベクトル集合による変調方式のパワー効率と周波数
利用効率を、(1) 及び (2) に基づいてそれぞれ求めると表 2 の
ようになる。ここで C1 は PS-QPSK と等価であり、確かにこ
れら４つのベクトル集合の中で最もパワー効率が高いことがわ
図 2 DP-QPSK、C5′ のシンボル誤り率 (union bound)
かる。
今度はこれらベクトル集合を組み合わせてできる新たなベク
トル集合について考えてみる。ここで、いま一度 C1 ∼C4 の集
合数と最小信号間距離の関係についてまとめると、
表 3 4 次元ベクトル集合の集合数と最小信号間距離
C1
C3
C4
8
24 32
√ √ √
最小信号間距離
2
2
2
C2
16
集合数
2
となる。二つの集合を組み合わせるにあたって気をつけなけれ
ばいけないことは、それぞれの集合と最小信号間距離の関係で
図3
C1 から C5′ の周波数利用効率とパワー効率の関係
ある。具体的には、一つ目の集合の最小信号間距離、二つ目の
集合の最小信号間距離、この二つの集合間の最小信号間距離の
三つを考えなければならない。仮にそれぞれの集合の最小信
号間距離が大きいものを組み合わせて新しい集合を作ったとし
ても、二つの集合間の最小信号間距離が小さければ、全体のパ
ワー効率は低下する。これは誤り率の特性は、基本的には最小
信号間距離に支配され、とくに SN 比が高いところでは、誤り
のほとんどは最小信号間距離間でしか起きないからである。こ
のようなことに注意すると、もっとも効率のよい組み合わせは、
C1 と C3 であることに気付く。なぜなら C1 の最小信号間距離
と C3 の最小信号間距離、およびこれらふたつの集合の最小信
号間距離全てが等しいためである。以降この集合を C5 とする。
しかし、C5 は全ベクトル数が 40 と半端であるため、32 にす
るよう C3 から任意の 8 個を捨て去って新しい集合
C5′
を作る。
この C5′ についてパワー効率を考えてみる。
bound は近似的手法ではあるが、誤り率が 10−3 以下では理論
値とほとんど一致することが知られている。図 2 に DP-QPSK
と C5′ のシンボル誤り率を示す。理論的にはパワー効率が等し
いことから両者は一致するはずである。しかし図 2 では両者に
はわずかに差がある。この差は Eb /N0 が高い部分では徐々に
小さくなっていくことがわかる。ここで注意しなければならな
いことは、パワー効率はあくまで‘ 漸近的な ’値であって、い
つ漸近するかということについては全く述べていない。つまり、
この場合には確かに C5′ のパワー効率は１となり、DP-QPSK
と等しいのだが、この結果を見る限り完全に一致していると言
えるまで漸近するには、高い SN 比を必要としてしまう。
以上より C5′ は、誤り率が 10−4 付近では DP-QPSK に比べ
てパワー効率は 0.5dB 程度劣化してしまうが、それでも DP-
QPSK より偏波多重系にて 1bit 多く送信できる変調方式であ
まず送信する際必要なパワーは、１シンボルあたり平均して
√
(1 × 8+3 × 24)/32 で 2.5 となる。最小信号間距離間は 2、送
信ビット数は 5 であることから、パワー効率は 1 となる。つま
ることが確認できた。また図 3 では、C1 ∼C5′ のパワー効率
限界と周波数利用効率をシャノン限界と比較したものを示して
いる。
り、理論的には DP-QPSK と同じパワー効率でさらに 1 ビット
多く送信できることになる。これを確かめるため、union bound
を用いて、この妥当性を論ずることとする。union bound とは
シンボル誤り率を求めるための近似的手法であり、以下のよう
に表わされる。
M M
( d )
1 ∑∑ 1
ij
SER <
erfc √
(i =
| j)
= M
2
2
N0
i=1 j=1
4. 4 次元空間での復調法
4 次元空間での変調方式については前節で述べたとおりであ
る。しかし４次元信号を受信して復調するアルゴリズムは変調方
式によってバラバラで統一的に扱われてはいない。PS-QPSK の
(10)
ここでＭはシンボル数、d は符号間距離を表している。union
復調アルゴリズムはいくつか提案されており、[3] はストークス
パラメータを用いて偏波状態の判定を行っているが、PS-QPSK
のみに適用できるもので一般性はない。[4] は直交行列を用い
て 4 次元空間上で回転を行い、判定を行いやすくしているがこ
—3—
表 4 8 次元ベクトル集合
C1
(±1,0,0,0,0,0,0,0)
C2
(±1,±1,0,0,0,0,0,0)
C3
(±1,±1,±1,0,0,0,0,0)
C4
(±1,±1,±1,±1,0,0,0,0)
C5
(±1,±1,±1,±1,±1,0,0,0)
C6
(±1,±1,±1,±1,±1,±1,0,0)
C7
(±1,±1,±1,±1,±1,±1,±1,0)
表 5 8 次元ベクトル集合のパワー効率および周波数利用効率の関係
C3
C4
1.7
2.2
2.53
パワー効率 [dB] 3.01 2.30
1.64
1
周波数利用効率
図 4 4 次元ベクトル判定アルゴリズムを組み込んだ DD-LMS の構成
C1
C2
1
C5
C6
C7
2.7
2.7
2.5
0.33 -0.45 -1.48
れも PS-QPSK にのみ適用できるもので一般性はない。そこで
一般的な 4 次元空間における復調アルゴリズムを提案する。
まず、二つの直交した偏波状態により二つの複素電界を出力
として得る、偏波多重系にて受信を行ったとする。これをス
トークスパラメータに変換せず以下のようにそれぞれの複素電
界の実部・虚部をとり 4 次元ベクトルを生成する。

Re[Exout ]

 Im[Exout ]
E=

 Re[Eyout ]






(11)
Im[Eyout ]
この出力 4 次元ベクトルに対して、あらかじめ保存しておいた
基準となる 4 次元ベクトルとの二乗ユークリッド距離を総当り
的に求め、その距離が最小のものを判定ベクトルとする。この
ようにあらかじめ基準となる 4 次元ベクトル集合をすべて保
存しておけば、どのような 4 次元変調であっても最適受信が行
えることがわかる。この判定法を組み込んだ Decision-directed
least-mean-square algorithm (DD-LMS) の構成を図 4 に示す。
タップ係数の更新は通常の偏波多重時のバタフライ構成アルゴ
リズムと同様であるが、判定時のみ２つの複素振幅を４次元ベ
クトルに変換する。そして、最小自乗距離判定により最尤の 4
次元ベクトルを決定した後、これを再び２つの複素振幅に戻す。
図 5 8 次元ベクトル集合のパワー効率および周波数利用効率の関係
ではさらに 8、12、16 と次元を拡大していくとどうなるであろ
うか。以下 4 次元と同様にベクトル集合を考え、8、12、16 次
元における変調方式のパワー効率と周波数利用効率について検
討する。
まず 8 次元において表 4 のような 8 つのベクトル集合を考考
える。ただし 8 つのすべての自由度が ±1 で埋まっているもの
は DP-QPSK を多重したに過ぎず、パワー効率・周波数利用効
率ともに DP-QPSK あるいは QPSK と完全に等しくなるため
考察には入れない。以下いずれの次元においてもすべての自由
度が ±1 で埋まっているものは省く。これらのベクトル集合
このように判定法のみを 4 次元化することで、通常の偏波多重
系における DD-LMS と大きく異ならない構成となっている。
このようなアルゴリズムを用いる大きなメリットとして、4 次
元ベクトル判定によって一般化されたことにより、変調方式に
大きく依存しない受信が可能となり、変調方式の切り替えにも
表 6 12 次元ベクトル集合
C1
(±1,0,0,…………,0,0,0)
C2
(±1,±1,0,…………,0,0,0)
C3
.
..
(±1,±1,±1,…………,0,0,0)
.
..
対応できるようになることががあげられる。例えば、PS-QPSK
C10
(±1,±1,±1,…………,±1,0,0)
と DP-QPSK を状況によって使い分けながら通信を行う場合
C11
(±1,±1,±1,…………,±1,±1,0)
を仮定する。基準となる 4 次元ベクトルを PS-QPSK(8 個) と
DP-QPSK(16 個) の 2 パターン保存しておき、使用したい方
式のどちらかの基準ベクトルをメモリーから呼び出して、最小
自乗判定を行うことにより、容易に変調方式の切り替えに対応
した受信が実現できる。
5. 多次元空間での変調方式の検討
表 7 12 次元ベクトル集合のパワー効率および周波数利用効率
C1
C2
C3
C4
0.76
1.34
1.79
2.15
2.43 2.64 2.77
パワー効率 [dB] 3.59
1.64 1.20 0.74
周波数利用効率
3.03
2.52
2.08
C8
C9
C10
C11
2.82
2.79
2.67
2.43
C5
C6
C7
0.21 -0.31 -0.96 -1.80
以上の議論はすべて 4 次元空間におけるものであった。それ
—4—
図 6 12 次元ベクトル集合のパワー効率および周波数利用効率
表 8 16 次元ベクトル集合
図 7 16 次元ベクトル集合のパワー効率および周波数利用効率
表 10
各ベクトル集合のパワー効率および周波数利用効率 (8,12,16
C1
(±1,0,0,………………,0,0,0)
C2
(±1,±1,0,………………,0,0,0)
C3
.
..
(±1,±1,±1,………………,0,0,0)
.
..
C14
(±1,±1,±1,………………,±1,0,0)
周波数利用効率
2.5
C15
(±1,±1,±1,………………,±1,±1,0)
パワー効率 [dB]
1
次元）
8 次元
C4
表 9 16 次元ベクトル集合のパワー効率および周波数利用効率
C1
C3
C5
C7
2
2.5
1.5
2
2.5
2.08 1.20 3.05 2.32 1.65
16 次元において C3 , C5 , C7 を考えてみる。これらをシンボルが
C5
C6
C7
C8
0.62 1.11 1.51
1.85
2.13
2.37
2.56
2.70
パワー効率 [dB] 3.97 3.46 3.05
2.67
2.32
1.98
1.65
1.30
パワー効率と周波数利用効率の関係を表 10 に示す。この中でも
C10
C11
C12
C13
C14
C15
特に優れたものは、16 次元の C5 , C7 である。C5 は DP-QPSK
2.81 2.87 2.88
2.85
2.76
2.61
2.37
C8
C9
C3
16 次元
C6
C4
周波数利用効率
C2
12 次元
C4
0.93 0.59 0.20 -0.21 -0.70 -1.27 -1.98
2 のべき乗になるよう、あまりのものを破棄してできた集合の
と同じ周波数利用効率でパワー効率が PS-QPSK を超えるとい
うもので、C7 は DP-QPSK よりも周波数効率が 0.5 高く、パ
ワー効率も PS-QPSK とほぼ等しい。
の周波数利用効率とパワー効率の関係を表 5 と図 5 に示す。
次に 12 次元において考察を行う。今までと同様にベクトル
そこでこの 16 次元変調方式を OFDM と組み合わせた、多
次元コヒーレント光通信としての送信器及び受信器構成の図 8、
集合を表 6 のように定義する。これらのベクトル集合の周波数
図 9 に示す。これは、4 波長ごとに変調・復調を行うことを意
利用効率とパワー効率の関係を表 7 と図 6 に示す。
味している。
（1 波長あたり 4 つの自由度が存在するので、4 波
同様に 16 次元においてのベクトル集合を表 8 に、これらの
周波数利用効率とパワー効率の関係を表 9 と図 7 に示す。
長を用いれば、16 次元変復調が行えることになる。）
例えば、16 次元でのベクトル集合 C5 を考えてみる。この変
このように次元を拡張していくことで、ベクトル集合がとり
調方式により、4 波長にて全 16 ビットが送信される。16 ビッ
得るパワー効率と周波数利用効率の関係は、よりシャノン限界
トによりシンボルは 216 = 65536 パターン存在することにな
に接近していくことがわかる。
る。この 65536 パターンすべてのビット列とシンボルとの関係
6. 実用的な多次元変調方式の検討
前章でも見てきたように、8,12,16 次元においては DP-QPSK
と比較して、パワー効率・周波数利用効率どちらの観点からも
すぐれている変調方式があることがわかった。しかし、これら
はいずれも周波数利用効率が中途半端な値となっている（つま
り送信ビット数が整数でない）。したがって実際の方式として用
いるためには、送信ビット数を整数、つまり全シンボルパター
ンを 2 のべき乗にしなければならない。そこで、周波数利用効
率ができる限り 0.5 の倍数に近いものを選び、いくつかのシン
ボルパターンを破棄することで送信ビットを整数にすることを
考える。そこで、周波数利用効率が 0.5 の倍数にできるだけ近
いものとして、8 次元において C4 、12 次元において C4 , C6 、
図 8 16 次元コヒーレント光通信 OFDM 送信器構成の概略図
—5—
文
図 9 16 次元コヒーレント光通信 OFDM 受信器構成の概略図
献
[1] M. Karlsson and E. Agrell, “ Which is the most powereﬃcient modulation format in optical links?, ”Opt. Express,
vol. 17, pp.10814-10819, 2009.
[2] E. Agrell, and M. Karlsson, “ Power-eﬃcient modulation
formats in coherent transmission systems, ” J. Lightwave.
Technol., vol.27, pp.5115 - 5126, 2009.
[3] P. Poggiolini, G. Bosco1, A. Carena, V. Curri, and F.
Forghieri,“ Performance evaluation of coherent WDM PSQPSK (HEXA) accounting for non-linear fiber propagation
eﬀects, ” Opt. Express, vol. 18, pp.11360-11371, 2010.
[4] P. Johannisson, M. Sjdin, M. Karlsson, H. Wymeersch, E.
Agrell and P. A. Andrekson,“ Modified constant modulus
algorithm for polarization-switched QPSK, ” Opt. Express,
vol. 19, pp.7734-7741, 2011.
を記憶させておき、任意のビット列に対してシンボルを生成・
伝送した後、受信されたシンボルに対して再び 65536 パターン
との最小自乗距離を求めることは計算量的に現実的ではない。
そこで、送信されるシンボルの正負の判断を先に行うことで、
パターン数の削減を目指す。具体的には C5 には 16 個の自由度
のうち、5 個の自由度のみを用いて信号の送信を行うが、この
5 個の値が正か負かで 5 ビット分は計算量が削減される。つま
り、総当り的に行う計算量は 211 = 2048 となり、ある程度現
実的なものになる。
また OFDM との併用のように周波数を次元の資源として用
いるのではなく、時間軸を次元の資源として用いる方法もある。
例えば、一つの波長内には 4 つの自由度が存在するのだから、
信号の受信を 4 回行ってその 4 回全てで一つの 16 次元ベクト
ルを生成する、あるいは 1 つの自由度のみを用いて 16 回の受
信を行った後、その 16 回全てで一つの 16 次元ベクトルを生成
する、といったように行えば多次元変調方式に対応できる。
以上のような手法を用いることで多次元変復調が実現可能と
なる。
7. まとめ
本論文では、シャノン・ハートレーの定理を組み込んだ、周
波数利用効率とパワー効率の限界値に関する表式を導いた。こ
の式は、使用する次元には無関係であり、より次元を拡張して
いくことで、パワー効率を高く保ったままこの限界値に接近で
きると考えられる。一般に PSK や QAM 等のような 2 次元変
調方式では、低いパワー効率時にはこの限界には近づくことは
できない。
（パワー効率を犠牲にして多値化を行えば限界に近づ
くことは可能である。）そこでまず 4 次元空間にて一般的な復
調法とともに、DP-QPSK よりもこの限界に近づき得る新たな
変調方式を提案した。また多次元空間にて、さらにこの限界に
近づき得る変調方式とそれを実現するための手法を提案した。
以上の提案は、従来の DP-QPSK の多重化技術に比べて、よ
り少ないパワーで同じ情報量を送信できる、さらにはより少な
いパワーでより多くの情報を送信できることを示しており、通
信の効率化に貢献しう得るものであると考えられる。しかし。
そのためにはハードウェア的なさらなる検討も必要である。
—6—
一般社団法人電子情報通信学会
THE INSTITUTE OF ELECTRONICS,
INFORMATION AND COMMUNICATION ENGINEERS
[c5]
信学技報
IEICE Technical Report
送信端における分散制御と位相共役を用いた
光ファイバ非線形効果の補償
－並列逆伝搬法の提案と解析－
陸
弘波
菊池
和朗
東京大学大学院工学系研究科電気系工学専攻
東京都文京区本郷 7-3-1
E-mail: {hongbolu, kikuchi}@ginjo.t.u-tokyo.ac.jp
あらまし本論文では，所望の光信号とその位相共役光を偏波多重して伝送することにより，ファイバの非線形
効果を効率的に低減する方法を提案する。ファイバ伝送路の全分散を付与したのち生成された位相共役光は，伝送
路を伝搬し，コヒーレント受信器で受信される。一方所望信号は，ファイバを伝送したのち，コヒーレント受信器
で分散補償される。この時，位相共役光の逆伝搬の原理によって，所望信号と位相共役光における非線形歪みは逆
相関を持つので，両者の和をとることにより非線形歪みがキャンセルされる。計算機シミュレーションにより，こ
の方法を用いると，20Gbit/s QPSK 信号を 1,000km 伝送した時，最適パワーは 7dB 向上することが示された。
キーワードコヒーレント光通信，非線形光学，位相共役
Pre-dispersed Mutual Phase-conjugation Scheme for Fiber-nonlinearity
Mitigation in Coherent Optical Communication Systems
－Proposal and Analysis of Parallel Back-propagation Method－
Hongbo LU and Kazuro KIKUCHI
Department of Electrical Engineering and Information Systems, The University of Tokyo
7-3-1 Hongo, Bunkyo-Ku, Tokyo 113-8656, Japan
E-mail: {hongbolu, kikuchi}@ginjo.t.u-tokyo.ac.jp
Abstract We propose and analyze an effective and practical method for mitigating signal distortion caused by fiber
nonlinearity. In the proposed method, an original signal and a pre-dispersed phase-conjugated signal are simultaneously
transmitted through the fiber on two polarization modes. The nonlinear distortion is generated in a back-propagation manner
on the pre-dispersed phase-conjugated signal and cancels the nonlinear distortion on the original signal transmitted in parallel.
Through extensive simulations, we demonstrate that the optimal power of the quadrature phase-shift keying (QPSK) signal is
improved by over 7 dB in a 1,000-km-long dispersion-unmanaged link using standard single-mode fibers (SMFs) only.
Keyword Coherent Optical Communications, Nonlinear Optics, Phase Conjugation
The optical back-propagation method utilizes either the
1. Introduction
In ultra-long haul optical fiber communication systems,
nonlinear optical device [1] or the coherent optical-
the nonlinear (NL) distortion is generated from the
electrical-optical (O-E-O) convertor [2] at the middle
interaction of chromatic dispersion (CD) with self-phase
position of the transmission link for generating the phase-
modulation (SPM) induced by the Kerr effect, seriously
conjugated signal. While the phase-conjugated signal
limiting
NL
travels through the second hal f of the link, the NL
we
employ
distortion generated in the first half of the link is undone
quadrature-amplitude-modulation
(QAM)
in the back-propagation manner; however, the requirement
the
distortion
higher-order
is
transmission
more
characteristics.
significant
when
Such
formats.
The back-propagation method is very effective for
mitigating the NL distortion, and several schemes have
been proposed so far.
that the phase conjugator should be placed at the midpoint
of the link limits the system flexibility.
In [3, 4], the digital back-propagation (DBP) method
was
proposed.
In
this
method,
the
NL-distortion
This article is a technical report without peer review, and its polished and/or extended version may be published elsewhere.
Copyright ©20●●
by IEICE
mitigation is done in the digital-signal-processing (DSP)
On the contrary, our method generates the NL mitigation
circuit in the coherent optical receiver,
where the
pattern on the pre-dispersed phase-conjugated signal in
back-propagated
the real link, whereas the NL distortion is generated on the
through the fiber. However, computational complexity for
original signal transmitted in parallel; thus, we can cancel
such
out the NL distortion just by adding these two signals at
phase-conjugated
DSP
signal
is
too
is
virtually
large
to
obtain
satisfactory
nonlinearity-cancelling results. Although several research
works
[5,
6]
proposed
simplified
DBP
the receiver.
algorithms,
The principle of generation of the NL mitigation pattern
computational complexity for DSP is still far from the
follows
practical level enabling real-time implementation in
back-propagation scheme and can achieve the satisfactory
ultra-long haul transmission systems.
NL-mitigation effect. This method is also ready for
On the other hand, we have proposed a polarization for
NL
communication
mitigation
systems
[7].
in
coherent
This
method
mechanism
similar
to
the
conventional
real-time implementation because it does not require any
diversity scheme using mutually phase-conjugated (MPC)
signals
the
large modification of the conventional coherent receiver.
optical
Extensive computer simulations are performed on
provides
20-Gbit/s quadrature phase-shift keying (QPSK) signals
intrinsic 3-dB improvement of the NL power tolerance in
transmitted
comparison with the single-polarization (SP) scheme and
unmanaged link consisting only of standard single-mode
the dual-polarization (DP) scheme at the same bit rate
fibers (SMFs). We find that the optimal signal power is
without increasing computational cost of the conventional
improved by more than 7 dB with our proposed method.
scheme.
through
a
1,000-km-long
dispersion-
The organization of this paper is as follows: In Sec.2,
To further improve the NL mitigation effect achieved in
we introduce the system setup of the pre -dispersed MPC
[7], we propose the use of pre-dispersed MPC signals in
system based on a polarization-diversity scheme. In Sec.3,
the
the principle of parallel back-propagation is explained.
above-mentioned
polarization-diversity
scheme.
Although a similar scheme has recently been demonstrated
Section
in [8], our paper provides us with the design guideline of
mitigation effect. Finally, Sec.5 concludes this work.
the
pre-dispersed
MPC
scheme
based
on
which
we
hereafter
simulation
results
on
the
NL
2. System setup
Our proposed method is based on the novel principle for
mitigation,
discusses
clearer
understanding of the principle of operation．
NL
4
call
Figure 1 shows the system setup of the pre-dispersed
parallel
MPC scheme.
back-propagation (PBP). All of the previous digital
Two binary sequences are transmitted using the QPSK
back-propagation schemes perform the NL mitigation
format
process after receiving the nonlinearly-distorted signal.
sequences constitute an IQ-vector sequence, in other word,
in
the
x-polarization
Signal generator
Pre-dispersion
DL
Conjugation
Optical line
Electrical line
IQM
Laser
PBC
EDFA
IQM
×N
LO
Polarization
diversity 90
degree
hybrid
Chromatic
dispersion
compensation
ADC
Pol-demux&
adaptive
equalizer
DSP
Carrier
recovery
Signal
enhancement
(Ex+Ey*)/2
Fig.1 System setup for the pre-dispersed MPC scheme.
channel.
Such
binary
a
complex-number
sequence
constellation points.
expressing
QPSK
We then obtain the IQ-vector
𝜕𝐸𝑥∗
𝜕(−𝑧)
sequence for the y-polarization channel, pre-dispersing
∗
𝜕𝐸𝑦
and phase-conjugating the IQ-vector sequence for the
𝜕(−𝑧)
1
1 𝜕2
2
2 𝜕𝑡 2
= ( 𝛼 + 𝑗𝛽2
1
1 𝜕2
2
2 𝜕𝑡 2
= ( 𝛼 + 𝑗𝛽2
2
8
) 𝐸𝑥∗ − 𝑗𝛾 (|𝐸𝑥 |2 + |𝐸𝑦 | ) 𝐸𝑥∗ , (3)
9
2
8
) 𝐸𝑦∗ − 𝑗𝛾 (|𝐸𝑥 |2 + |𝐸𝑦 | ) 𝐸𝑦∗ . (4)
9
x-polarization channel. However, the IQ-vectors for both
of the channels can be pre-dispersed with different
amounts of CD, and the required condition for such
We assume that the influence of the power distribution
along the link is ignored ( 
0 ) and that the system
pre-dispersion is that the amount of CD given to the
length is L. In such a case, re-transmitting Ex* ( L, t ) and
phase-conjugated channel should be larger than that to the
E*y ( L, t ) in the backward direction from the output at z  L ,
desired channel by 𝐷𝐿, where D is the dispersion value of
we obtain
the link and L the length of the link. As far as this
This undo process is called back propagation and can
condition is satisfied, any amount of pre-dispersion can
Ex* (0, t ) and E*y (0, t ) at the input end z  0 .
A pair of optical
restore the initial waveform at the input end.
In the midway phase-conjugation method shown in
IQ modulators (IQMs) is used to generate a desired optical
Fig.2 (a), the transmitted signal is phase-conjugated at the
signal
are
middle point of the whole transmission link either
simultaneously using polarization-division
optically or electrically. The phase-conjugated signal is
realize the same NL mitigation effect.
and
transmitted
the
phase-conjugated
signal,
which
then transmitted through the second half of the link
multiplexing (PDM).
At the receiving end (z=L), the coherent receiver
instead of the actual back propagation. I f the property of
employing phase and polarization diversities detects the
the second half of the link is identical with that of the first
transmitted
half
signals.
The
DSP
circuit
performs
of
the
link,
dispersive/nonlinear
impairments
chromatic-dispersion compensation; adaptive equalization
generated along the first half of the link are undone along
including polarization demultiplexing, polarization-mode
the second half of the link in the back-propagation manner.
dispersion compensation, and clock recovery; and carrier
However, the need for the phase conjugator at the middle
recovery in a conventional manner. Let 𝐸𝑐𝑥 (𝐿, 𝑡) and
𝐸𝑐𝑦 (𝐿, 𝑡) represent output signals after such conventional
point strongly prohibits its practical application.
signal processing. Then, only the additional operation in
principle is adopted. The back-propagation through the
our scheme is 𝐸𝑜𝑢𝑡 (𝐿, 𝑡) = {𝐸𝑐𝑥 (𝐿, 𝑡) +
∗ (𝐿, 𝑡)}/2
𝐸𝑐𝑦
at the
In the digital back-propagation method, the same
fiber is virtually carried out by DSP. Physical phase
conjugation
final stage of DSP.
at
the
midpoint
is
not
necessary,
but
computational complexity for the virtual back-propagation
3. Principle of operations of the pre-dispersed
is so large that the real-time operation of DSP is very
difficult to achieve.
MPC scheme
3.1 Two-mode nonlinear Schrödinger equations
The transmission characteristics of PDM signals obey
the two-mode nonlinear Schrödinger equation s [9] as
3.3 Principle of the parallel back-propagation method
Instead of the conventional back-propagation scheme
mentioned
𝜕𝐸𝑥
𝜕𝑧
𝜕𝐸𝑦
𝜕𝑧
1 𝜕2
2 2 𝜕𝑡 2
1
= (− 𝛼 + 𝑗𝛽
2
1
1 𝜕2
2 2 𝜕𝑡 2
= (− 𝛼 + 𝑗𝛽
2
8
2
) 𝐸𝑥 − 𝑗𝛾 (|𝐸𝑥 |2 + |𝐸𝑦 | ) 𝐸𝑥 , (1)
9
8
2
) 𝐸𝑦 − 𝑗𝛾 (|𝐸𝑥 |2 + |𝐸𝑦 | ) 𝐸𝑦 . (2)
9
In these equations, complex amplitudes of horizontal and
vertical polarization modes are denoted as 𝐸𝑥 (𝑧, 𝑡) and
𝐸𝑦 (𝑧, 𝑡), respectively. Parameters 𝛼 , 𝛽2 , and 𝛾 represent
the
attenuation
coefficient,
the
chromatic-dispersion
parameter, and the nonlinearity coefficient, respectively.
in
3.2,
we
introduce
the
parallel
back-propagation scheme shown in Fig.2 (b).
The generation of the NL-mitigation pattern of the PBP
method follows the principle of the conventional back
propagation, but it is executed in a parallel manner. Thus,
we need neither the second half of the link in the midway
phase-conjugation
calculation
process
scheme
in
the
nor
the
digital
time-consuming
back-propagation
scheme.
In order to elucidate the principle of the PBP scheme,
we apply the perturbation method to Eqs. (1)-(4). For clear
3.2 Principle of the conventional back-propagation
method
Complex conjugation of Eqs. (1) and (2) yields
analyses of NL distortion, we assume the system is free
from noise and ignore the loss/gain distribution of the
system.
Fig.2 Schematic diagram of accumulation of the CD- and NL-distortion as a function of the transmission distance (a) in
the midway phase-conjugation scheme and (b) in the pre-dispersed MPC scheme.
The
original
𝐸𝑖𝑛
signal
is
transmitted
on
the
𝑥-polarization mode. The waveform of the received signal
is conceptually composed of two parts as
𝐸𝑥−
𝑥
On the other hand, the original signal is transmitted on
the orthogonal polarization mode. After compensation for
= [𝐸𝑖𝑛 ] = 𝐷(𝐸𝑖𝑛 ) + (𝐸𝑖𝑛 ),
where 𝐷(𝐸𝑖𝑛 ) and
the perfect compensation in a back-propagation manner.
(5)
(𝐸𝑖𝑛 ) represent linearly-dispersive
CD, the recovered signal becomes
𝐸𝑥−𝑐 = 𝐷−1 [𝐷(𝐸𝑖𝑛 ) + (𝐸𝑖𝑛 )]= 𝐸𝑖𝑛 + 𝐷−1 [ (𝐸𝑖𝑛 )].
(10)
term and nonlinearly-distorted term, respectively, and T
Note that nonlinear terms in the two parallel channels are
denotes the total transmission operator.
anti-correlated; therefore, adding these two signal together,
In the midway phase-conjugation system, the signal is
we obtain
phase-conjugated at the midpoint and re-transmitted
𝐸
through the same link. Then, the output signal returns to
=
∗
𝐸𝑥−𝑐𝑑+𝐸𝑜𝑢𝑡−𝑚𝑝𝑐
2
= 𝐸𝑖𝑛 .
(11)
the phase-conjugated input signal in the back-propagation
manner. This process is written as
𝐸𝑜𝑢𝑡−
𝑖
𝑦
The NL-distortion term is thus cancelled out with each
∗
∗
∗
∗
= [𝐸𝑥−
𝑥 ] = [𝐷(𝐸𝑖𝑛 ) + (𝐸𝑖𝑛 ) ] = 𝐸𝑖𝑛 . (6)
In the case of the PBP method shown in Fig.2 (b), we
∗
cannot transmit 𝐸𝑥−
𝑥
= 𝐷(𝐸𝑖𝑛 )∗ + (𝐸𝑖𝑛 )∗ , because such
waveform is unknown at the transmitter.
other and the original signal is recovered .
The above analysis describes the NL distortion from
SPM in the original signal and the phase-conjugated
replica. In the PDM scheme, we need to take cross -phase
Instead, we can
modulation (XPM) between two polarization modes into
which is pre-dispersed phase-conjugated
account; however, this analysis is still applicable in such a
signal. At the receiving end, the waveform of the received
case. This is because XPM-induced phase rotations on
phase-conjugated signal is obtained as
both signals are the same and XPM-induced distortions are
transmit 𝐷(𝐸𝑖𝑛
)∗ ,
𝐸𝑜𝑢𝑡−
𝑝𝑐
= [𝐷(𝐸𝑖𝑛 )∗ ].
(7)
Then, Eqs.(6) and (7) yield
𝐸𝑜𝑢𝑡−
𝑝𝑐
𝑝𝑐
In the above analysis, the amount of dispersion DL,
which is the total amount of dispersion of the link, is
∗
= 𝐸𝑖𝑛
− [ (𝐸𝑖𝑛 )∗ ].
(8)
If we ignore the second-order nonlinear term, we have
𝐸𝑜𝑢𝑡−
anti-correlated at the output ports.
∗
≃ 𝐸𝑖𝑛
− 𝐷[ (𝐸𝑖𝑛 )∗ ].
given to the phase-conjugated channel at the transmitter.
However,
if
the
amount
of
CD
given
to
the
phase-conjugated channel is larger than that given to the
(8)
desired signal by 𝐷𝐿, any amount of pre-dispersion can
realize the same NL mitigation effect.
Taking the operation of conjugation of the received signal
in the phase-conjugated channel, Eq. (8) yields
∗
𝐸𝑜𝑢𝑡−
𝑝𝑐
= 𝐸𝑖𝑛 −
𝐷−1 [
(𝐸𝑖𝑛 )].
4. Simulation results and discussions
(9)
We conduct extensive simulations of the 20 -Gbit/s
QPSK
system
to
examine
the
performance
of
The second term in Eq. (9) is generated because we
NL-distortion compensation in the pre-dispersed MPC
transmit 𝐷(𝐸𝑖𝑛 )∗ instead of 𝐷(𝐸𝑖𝑛 )∗ + (𝐸𝑖𝑛 )∗ that realizes
scheme.
The system setup is the same as that illustrated in Fig. 1.
35
Each repeater span consists of a 100-km-long SMF and an
erbium-doped fiber amplifier (EDFA) with a 4-dB noise
total length of the system is 1,000 km (ten spans). In some
simulations, for the pure examination of the effect of
NL-distortion mitigation, we ignore ASE noise from
EDFAs.
The loss coefficient, the CD parameter, and the
nonlinear coefficient of SMF are 0.2 dB/km, 17 ps/nm/km,
and 1.5 /W/km, respectively.
Laser linewidths for both
30
Compensation Ratio [dB]
figure. The EDFA fully compensates for the span loss. The
25
20
15
10
transmitter and local oscillator (LO) are ignored.
5
We define the following three parameters for the
evaluation of the system performance: 1) the variance σ
Virtual loss/gain distribution
Actual loss/gain distribution
2
0
-6
-4
-2
Launched Signal Power [dBm]
of the constellation-point distribution of the signal after
nonlinearity mitigation, where the carrier amplitude is
normalized to unity; 2) the compensation ratio for the
examination of the SPM mitigation effect, which is the
ratio between variances σ 2 of the signal E add after the
summation in Eq. (11) and the signal E x- cd before the
summation; 3) the NL power tolerance, which is the
0
Fig. 3 Simulation results of the compensation ratio in
a 1,000-km 20-Gbit/s pre-dispersed MPC QPSK
system as a function of the launched power. The red
curve corresponds to the virtual BP link and the blue
one to the actual link.
25
difference in the launched signal power giving a certain
2
variance σ between the SP and MPC schemes.
link (red) and the actual one (blue). In the virtual BP link,
the link for the phase-conjugated y-polarization channel
has the reversed loss coefficient of -0.2 dB/km as well as
the reversed gain of EDFAs, whereas the link for the
x-polarization
channel
has
the
normal
loss/gain
distribution. On the other hand, the actual link has the
normal loss/gain distribution for both of the channels. The
Compensation Ratio [dB]
Figure 3 shows the compensation ratio of the virtual BP
20
15
10
ASE noise is neglected in both cases. As shown in Fig.3,
5
in the signal power range from -7 dBm to 0 dBm, the
compensation ratio of the virtual BP link is over 20 dB and
around 5-dB better than that of the actual link. The
-4
enhancement of the compensation ratio owing to the
virtual power map suitable for the phase conjugated
channel proves that the generation of the NL mitigation
pattern
follows
the
mechanism
similar
to
the
back-propagation scheme. Note that the back-propagation
method works ideally when the sign of the loss/gain
coefficient is reversed [1].
Next, we calculate the compensation ratio in the real
-3
-1
-2
0
1
2
3
Amount of Pre-dispersion [ps/nm/km]
4 4
x10
Fig.4 Simulation results of th e compensation ratio in a
1,000-km 20-Gbit/s pre-dispersed MPC QPSK system
as a function of the amount of pre -dispersion. The
launched power is fixed at 0 dBm
that to the original signal by 𝐷𝐿. In Fig.4, we find that the
compensation ratio of the proposed method slightly
link as a function of the amount of pre-distortion. The
fluctuates
launched power is fixed at 0 dBm and the ASE noise is
pre-dispersion on the MPC signals is varied in a wide
around
17
dB
as
the
amount
of
the
neglected. The original signal is pre-dispersed with CD,
range.
whose amount ranges from −2𝐷𝐿 to 2𝐷𝐿. The amount of
any amount of CD, whereas in [8], the amount of
CD given to the phase-conjugated replica is larger than
pre-dispersion is restricted to that of a half of the link.
It should be noted that in our analysis, we can use
obtained in real-time coherent receivers because this
method
requires
only
small
modifications
to
the
conventional coherent receiver.
10-1
Variance 
References
[1] C. Lorattanasane and K. Kikuchi, “Design theory of
long-distance optical transmission systems using
midway optical phase conjugation,” J. Lightwave
Technol., vol.15, 948-955 (1997).
10-2
SP scheme
Pre-dispersed MPC scheme
10-3
-15
-10
-5
0
5
10
Launched Signal Power [dBm]
Fig.5 Simulation results of σ2 in the 1,000-km 20-Gbit/s
QPSK system as a function of the launched power for
the SP (blue) and the pre-dispersed MPC scheme (red).
Figure 5 shows variances σ2 of the pre-dispersed
MPC system (red curve) and the SP scheme (blue curve) as
a function of the launched signal power. The amount of
pre-dispersion on the MPC signals is set to 𝐷𝐿/2 because
such a value induces the least SPM distortion on the
transmitted signal [10]. In terms of the variance
σ2 ,
the
optimal power is increased to over 3 dBm in the proposed
scheme, while that of the SP scheme is -4 dBm. The
optimal signal power is improved by 7 dB
Figure 5 also shows that the NL power tolerance of our
system is over 10 dB when σ2 = 0.02, achieving the 6-dB
improvement in comparison with our previous work [7].
5. Conclusions
We have analyzed the principle of operation of the
pre-dispersed MPC scheme for NL mitigation and shown
its effectiveness through intensive computer simulations.
As
an
extension
of
our
previous
work
[7],
the
pre-dispersed MPC scheme provides a powerful SPM
mitigation
effect
using
the
NL
mitigation
pattern
generated by the PBP principle. Our analyses show that
the
same
SPM
mitigation
effect
can
be
obtained
independently of the amounts of pre-dispersion, which
introduces more flexibility to the system design. Th rough
extensive simulations, we find the optimal signal power is
improved by 7 dB and the NL power tolerance when
σ2 = 0.02 is over 10 dB in a 1,000-km 20-Gbit/s QPSK
system.
Such highly-effective NL-mitigation result could be
[2] X. Chen, X. Liu, S. Chandrasekhar, B. Zhu, and R.
Tkach, “Experimental demonstration of fiber
nonlinearity
mitigation
using
digital
phase
conjugation,”
Optical
Fiber
Communication
Conference/National
Fiber
Optic
Engineers
Conference (OFC/NFOEC 2012), paper OTh3C.1,
March 6-8, 2012, Los Angeles, CA, USA.
[3] E. Ip and J. M. Kahn, “Compensation of dispersion
and
nonlinear
impairments
using
digital
backpropagation,” J. Lightwave Technol., vol.26,
3416-3425 (2008).
[4] L. Li, Z. Tao, L. Dou, W. Yan, S. Oda, T. Tanimura, T.
Hoshida, and J. C. Rasmussen, “Implementation
efficient nonlinear equalizer based on correlated
digital
backpropagation,”
Optical
Fiber
Communication Conference/National Fiber Optic
Engineers Conference (OFC/NFOEC 2011), paper
OWW3, March 6-10, 2011, Los Angeles, CA, USA.
[5] J. Leibrich and W. Rosenkranz, “Efficient numerical
simulation of multichannel WDM transmission
systems limited by XPM,” IEEE Photon. Technol.
Lett., vol.15, 395–397 (2003).
[6] E. Mateo, T. Inoue, F. Yaman, S. Zhang, D. Qian, T.
Wang, Y. Inada, T. Ogata, and Y. Aoki,
“Intra-channel XPM compensation for single -stage
backward-propagation,” Optical Fiber Communication Conference/National Fiber Optic Engineers
Conference (OFC/NFOEC 2013), paper OTh3G.4,
March 17-21, 2013, Anaheim, CA, USA.
[7] H. Lu, Y. Mori, C. Han, and K. Kikuchi, “Novel
polarization-diversity scheme based on mutual phase
conjugation for fiber-nonlinearity mitigation in
ultra-long coherent optical transmission systems,”
European Conference and Exhibition on Optical
Communication (ECOC 2013), paper We3C.3, Sept.
22-26, 2013, London, UK.
[8] X. Liu, A. R. Chraplyvy, P. J. Winzer, R. W. Tkach ,
and S. Chandrasekhar “Phase-conjugated twin waves
for communication beyond the Kerr nonlinearity
limit,” Nature Photonics, vol.7, 560–568 (2013).
[9] S.G. Evangelides, L.F. Mollenauer, J.P. Gordon, and
N. Bergano, “Polarization multiplexing with solitons,”
J. Lightwave Technol., vol.10, 28-35 (1992).
[10] S. Fujisawa, D. Ogasahara, E. de Gabory, T. Ito, and
K. Fukuchi, “Mitigation of intra-channel nonlinear
distortions based on PAPR reduction with CD
pre-compensation in real-time 50Gbps PM-QPSK
transmission,” Opto-Electronics
and
Communications Conference (OECC 2012), paper 5B1-1, July
2-6, 2012, Busan, Korea.
[c6]
社団法人電子情報通信学会
THE INSTITUTE OF ELECTRONICS,
INFORMATION AND COMMUNICATION ENGINEERS
信学技報
TECHNICAL REPORT OF IEICE.
ストークス空間での多値化技術と大容量コヒーレント光伝送への応用
菊池和朗†
川上彰二郎††
† 東京大学大学院工学系研究科電気系工学専攻
〒 113-8656 東京都文京区本郷 7-3-1
†† (株) フォトニックラティス
〒 989-3204 宮城県仙台市青葉区南吉成 6-6-3 ICR ビル 2F
E-mail: †[email protected], ††[email protected]
あらまし
光信号のストークスベクトルは，光の絶対位相に依存せず，x および y 偏波間の位相差と振幅比で決定さ
れる。したがって，ストークスベクトルを変調パラメータに用いれば，光信号の絶対位相に不感応なコヒーレント光
伝送システムを実現できる。このようなシステムでは，ストークス空間で効率的な多値化が可能となり，しかも多値
信号を復調するためのディジタル信号処理が極めて軽いという特長がある。一方で，適応等化が困難であるという欠
点はあるが，100 km 程度の大容量伝送には有効な技術であると考えられる。本論文では，ストークス空間での多値化
の方法，ディジタルコヒーレント光受信器における復調アルゴリズムについて述べたのち，2，4，8，16，32 相多値
信号の符号誤り率特性のシミュレーション結果を示す。特に 25 Gsymbol/s で 16 多重することにより，比較的少ない
計算量の信号処理で，100 Gbit/s を実現できることが明らかになった。
キーワード
コヒーレント光通信，偏波変調，ディジタル信号処理
Multi-level Signaling in the Stokes Space and Its Application to
Large-capacity Coherent Optical Transmission
Kazuro KIKUCHI† and Shojiro KAWAKAMI††
† Department of Electrical Engineering and Information Systems, Graduate School of Engineering
The University of Tokyo
7-3-1 Hongo, Bunkyo-Ku, Tokyo 113-8656, Japan
†† Photonic Lattice, Inc.
ICR Building 2F, 6-6-3 Minami-Yoshinari, Aoba-Ku, Sendai, Miyagi 989-3204, Japan
E-mail: †[email protected], ††[email protected]
Abstract The Stokes vector of an optical signal does not depend on its absolute phase but is dependent on the
relative phase and the amplitude ratio between the x- and y-polarization component; therefore, we can construct
the coherent system whose performance is not dependent on the absolute phase of the signal, using the Stokes
vector as the modulation parameter. In such system, multi-level-coded optical signals can eﬀectively be designed
in the Stokes space, whereas computational complexity for demodulating the signals is low enough. Although such
system has the disadvantage that adaptive equalizers are hardly to be implemented, it is an attractive solution to
large-capacity (>
= 100 Gbit/s) and short-reach (< 100 km) transmission. In this paper, we discuss how we design
the multi-level signal in the Stokes space and how we demodulate such signal using the the digital signal processing
(DSP) circuit in the coherent receiver. Simulation results demonstrate that 2-, 4-, 8-, 16-, and 32-ary signals in
the Stokes space have good bit-error rate (BER) characteristics. The transmission capacity of 100 Gbit/s is obtained through low-complexity DSP when we employ the 16-ary signal in the Stokes space at the symbol rate of 25
Gsymbol/s.
Key words Coherent optical communications, Polarization modulation, Digital signal processing
—1—
偏波多重偏波変調
1. はじめに
図 1 は，光信号の IQ 成分，偏波状態，光強度を用いた各種
の光変調方式をまとめたものである。
ディジタルコヒーレント光受信器の実用化によって，光電界
偏波多重
IQ変調
複素振幅を IQ 平面上で多値変調し，偏波軸上で多重化するシス
を処理する ASIC(Application-speciﬁc integrated circuit) の開
発により，1 波長あたり，100 Gbit/s の伝送容量が達成されて
いる [2]。
一方，2 つの偏波成分の間に位相相関が保たれているとき，偏
クトルと呼ぶ。ジョーンズベクトルは次式により，ストークス
ベクトル S に変換される。
S = [S1 , S2 , S3 ]T
この時，IQ 軸と偏波軸から構成される 4 次元空間におけるベク
S1 = |Ex | − |Ey |
トルを用いて符号を設計することが可能となり [3]，偏波スイッ
2Re (Ex∗ Ey )
式などの新しい変調方式が提案されている [4], [5]。このような
4 次元変調方式は，偏波多重方式に比べて符号化の自由度が格
段に増すため，新しい変調方式を探索する上で重要なアプロー
ストークス
ベクトル変調
位相不感応型
図 1 光信号の IQ 成分，偏波状態，光強度を用いた各種光変調方式。
波軸は IQ 軸と同等な変調の直交軸として用いることができる。
チ 4 相位相変調 (Polarization-switched QPSK: PS-QPSK) 方
変調強度変調
次元ベクトル
変調
4
位相感応型
テムの構築が可能となった [1]。偏波多重４相位相変調（Dual-
polarization Quadrature phase-shift keying: DP-QPSK) 信号
IQ
2
S2 =
(2)
2
(3)
= 2 |Ex | |Ey | cos δ
(4)
S3 = 2Im (Ex∗ Ey )
= 2 |Ex | |Ey | sin δ
(5)
チとなると考えられる。この変調方式も光信号の絶対位相に感
応するため，ディジタルコヒーレント光受信器を用いたキャリ
ア位相推定技術によって，初めて実現が可能となった。
ここで δ は
δ = arg (Ey /Ex )
(6)
これに対して，偏波状態を多値化する光変調方式も古くから
研究されてきた [6], [7]。偏波状態はストークス (Stokes) パラ
であり，x，y 偏波間の位相差を示す。また，ストークスベクト
メータ S1 ，S2 ，S3 を用いて記述することができる [8]。ストー
ルの大きさは
クスベクトル S = [S1 , S2 , S3 ]T で構成される 3 次元ベクトル空
S0 = |Ex |2 + |Ey |2
(7)
間（以後，ストークス空間と呼ぶ）において，強度変調と偏波変
調を含む自由度の高い多値信号設計ができるが，これまでディ
である。ストークスベクトルは，光電界複素振幅とその複素共
ジタルコヒーレント光通信への適用は検討されていなかった。
役の積をとること（すなわち二乗検波）によって得られること
ストークス空間で定義された多値変調信号は光信号の絶対位相
がわかる。このため，ストークスベクトルは，光電界の絶対位
に依存しないので，ディジタルコヒーレント光受信器を用いて
相を含まない。
受信した場合，ディジタル信号処理 (Digital signal processing:
ストークスベクトルが形成する空間はストークス空間と呼ば
DSP) に関わる計算量が大幅に低減できることが期待される。
れ，その座標は，光信号 E の強度を含む偏波状態に１対 1 に
ストークスベクトルは，受信器出力を 2 乗検波することによっ
対応する。特に S0 = 1 とした場合，ベクトル S の覆う単位球
て得られるので，光電界複素振幅の適応等化 [9] が困難である
面はポアンカレ (Poincar´
e) 球と呼ばれ，球面上の各点が偏波状
という欠点はあるが，100 km 程度の短距離の大容量伝送には
態を表す。
実数 A を
有効な技術であると考えられる。
本論文では，3 次元ストークス空間での多値信号の設計方法
√
A=
|Ex |2 + |Ey |2
(8)
について述べたのち，多値信号を復調するための DSP アルゴ
リズムについて論ずる。次に，2, 4, 8 相多値偏波変調, 16, 32
相多値強度・偏波変調信号の符号誤り率特性について検討する。
2. ジョーンズベクトルとストークスベクトルの
関係
光信号の電界複素振幅を，x 偏波成分 Ex と y 偏波成分 Ey
を用いて
[
E=
Ey
Ex = Aeiϕ cos
( )
θ
2
Ey = Aei(ϕ+δ) sin
(9)
( )
θ
2
(10)
と表すことができる。したがって式 (1) は，次のように変形さ
れる。

]
Ex
で定義すると，
( )

θ
iϕ 
2
( ) 
E = Ae
θ
eiδ sin
2
cos
(1)
とベクトル表示する時，このベクトルをジョーンズ (Jones) ベ
(11)
ϕ は光信号の絶対位相である。x 偏波と y 偏波との間のパワー
—2—
θ 

2
分岐比および位相差は，θ (0 <
=θ<
= π) および δ (−π <
= δ < π)
を用いて決定され，この時，偏波状態も θ，δ により一意に定
まる。
AM1
AM2
式 (2)-(5) より，式 (11) に対応するストークスベクトルは



S = S0  sin θ cos δ 
eiδ
−π ≤ δ < π
0 ≤θ ≤π
(12)
sin θ sin δ
PM
θ 
sin  
2

cos θ
PBC
cos 
図 3 ストークスベクトルの変調法。AM:振幅変調器，PM:位相変調
器，PBC:偏波ビームコンバイナ。
と表される。ただし S0 = A2 である。図 2 に，式 (12) で示さ
れるベクトル S を表示している。通常の 3 次元極座標表示と
は，パラメータ θ，δ の定義が異なることに注意されたい。
2 相の場合には，2 つの直交する偏波状態，すなわちポアンカ
レ球上の２つの対蹠点を信号点とすれば良い。図 4 では，右旋
S3
円偏波と左旋円偏波の場合を示している。4 相の場合には正四
面体の頂点に，8 相の場合には正六面体の頂点に，それぞれ信
S0
号点を配置するのが最適であることは明らかである。しかし 8
δ
S1
θ
値を超える多値数に対しては，頂点の数が 8 を超える正多面体
は存在しないので，計算機解析によって最適な信号点配置を求
める必要がある。また，偏波状態を維持したまま光強度 S0 ，す
S2
なわち原点から信号点までの距離を多値変調すれば，さらに多
値数を上げることができる。
図 2 ストークス空間における偏波状態の表示。θ および δ は式 (12)
S3
Binary
を参照。
ストークスベクトルは，光信号の絶対位相 ϕ を含まない。こ
のため，ストークスベクトルを変調パラメータとして用いた時，
S2
S1
その復調操作にはキャリア位相推定を必要としない。したがっ
Quad
てストークスベクトル変調方式は，強度変調方式の拡張と考え
Octal
S3
S3
ることもできる。ただし強度変調方式は，変調空間が１次元で
あるのに対し，ストークスベクトル変調方式は，3 次元空間で
の変調が可能である。この事実は，ストークスベクトル変調方
式では，強度変調方式に比べて，多値変調を行うための設計の
S2
S1
S1
S2
自由度が大きいことを示唆している。
図 4 ストークス空間におけるコンステレーションマップ。2 相 (Bi-
3. ストークスベクトルの変調
nary)，4 相 (Quad)，8 相 (Octal) の例を示す。偏波状態と同
時に S0 ，すなわち原点から信号点までの距離を変調することが
式 (12) における S0 ，θ，δ を変調パラメータとして，ストー
可能である。
クスベクトルを変調することができる。図 3 に変調器の構成を
示す。変調器 AM(Amplitude modulator)1 は光強度 S0 を変調
ワー分岐するための，マッハ・ツェンダー (Mach Zehnder) 変
4. ディジタルコヒーレント光受信器によるス
トークスベクトルの検出
調器である。PM(Phase modulator) は分岐された信号間に位
図 5 は，多値強度・偏波変調信号を復調するための DSP
相差を与えるための位相変調器である。これら 2 つのパスの信
回路の構成である。位相・偏波ダイバーシティーホモダイン
号光は，偏波ビームコンバイナ (Polarization beam combiner:
光受信器からの 4 つの出力は，アナログ・ディジタル変換器
PBC) を用いて，一方が x 偏波，他方が y 偏波となるように合
(Analog-to-digital converter: ADC) によりディジタル信号に
波される。
変換される。この時，クロックは外部回路で抽出され，抽出さ
する。変調器 AM2 は，この強度変調光を 2 つの出力ポートにパ
ストークス空間で偏波状態に対する多値変調を行う場合，パ
れたクロックによって ADC におけるサンプリングが行われる。
ワー一定 (すなわち S0 一定) の条件のもとで，信号点間のユー
したがってサンプリングレートは 1 sample/symbol である。次
クリッド距離が最大となるような信号点配置をとることが望ま
に，半固定的な分散補償，スペクトルフィルタリングの後に，
しい。図 4 は，2 相 (Binary)，4 相 (Quad)，8 相 (Octal) の信
式 (1)-(5) を用いてストークスベクトル S が計算される。
号点配置の例である。これらは，3 次元ストークス空間におけ
るコンステレーションマップ (Constellation map) と呼ばれる。
受信されたストークスベクトル S を復号するアルゴリズム
は，以下の通りである。
—3—
（ 1 ）トレーニング信号の導入
あらかじめ伝送されたトレーニング信号に基づき，ストー
5. ストークスベクトル変調方式の受信感度
クス空間に多値偏波変調を表現するノルム１の基準ベクトル
5. 1 ストークス空間における雑音分布
v 0 (k)，v 1 (k), · · ·，v n−1 (k) を作る。k はサンプル番号であ
ストークスベクトルは，光電界複素振幅の積をとる操作で求
る。トレーニング信号を用いないブラインドモードでの動作は
められる。したがってその雑音分布は，光電界複素振幅の雑音
原理的には可能であるが，多値数が高いと，基準ベクトルの更
分布よりも大きな広がりを持ち，符号誤り率 (Bit-error rate:
新アルゴリズム ((3) 参照) の収束が遅くなる。
BER) 特性を劣化させる。本節では，ストークス空間における
雑音分布を解析する。
（ 2 ）受信信号の偏波状態の決定
受信したストークスベクトルと各基準ベクトルとの内積をと
まず，一定の光電界複素振幅に複素ガウス雑音を重畳し，任
意のキャリア対雑音比 (Carrier-to-noise ratio: CNR) を実現す
り，内積の最大値を探索する。すなわち
る。すなわちジョーンズベクトルを
[
I (k) = max(S (k) · v 0 (k) , S (k) · v 1 (k) , · · · ,
S (k) · v n−1 (k))
(13)
E=
]
1 + nx
(15)
ny
を求め，内積の最大値を与える基準ベクトルを復号された偏波
とする。ここで nx および ny は複素ガウス雑音である。光増幅
状態とする。光強度が偏波状態と同時に変調されている場合に
器の使用を前提としているので，x，y 両偏波成分に雑音を仮
は，さらに S0 を閾値弁別して強度変調成分をも復号する。
定している。これらの雑音の実部および虚部の分散を σ 2 とす
ると，信号の x 偏波成分の CNR は
（ 3 ）基準ベクトルの更新
光ファイバの複屈折変動により，(1) で求めた基準ベクトル
は時間的に変動する。i 番目の基準ベクトルが復号されるごと
CNR =
1
2σ 2
(16)
となる [10]。
に，この基準ベクトルは以下の式で更新される。
図 6 は，x 偏波成分の雑音分布のシミュレーション結果であ
v i (k + 1) = {v i (k) + µS(k)} / |v i (k) + µS(k)|
(14)
µ はステップサイズパラメータである。ただし偏波変動の速度
は高々10 kHz 程度であるので，必ずしもシンボルごとに基準
ベクトルを更新する必要はない。このため，実装上問題となる
図 5 におけるフィードバック遅延が，DSP 性能を劣化させる
ことはない。
以上のように，ストークスベクトルの復調アルゴリズムには，
キャリア位相推定，偏波分離などの計算量の多い操作が含まれ
ない。また，基準ベクトルの更新も，低速な偏波変動に対応で
きる程度の速度で良い。これらの特長は，上記アルゴリズムの
実装を極めて容易にすると考えられる。その反面，上記のアル
ゴリズムでは適応的な信号等化を行うことができない。これは，
ジョーンズベクトルとは異なり，ストークスベクトルは信号の
絶対位相を含まないためである。このため，ストークスベクト
ルを用いた光伝送は，100 km 程度の距離に限定されると思わ
れる。
る。220 サンプルに対して，同相成分 (In-phase) および直交成
分 (Quadrature) の振幅分布を，CNR=7, 10, 15 dB について
示している。青線がシミュレーション結果，赤線が青線に対し
てガウス曲線をフィッティングしたものである。
次に，式 (15) で与えられたジョーンズベクトルを，式 (2)-(5)
を用いてストークスべクトルに変換し，ストークスパラメータ
S1 ，S2 ，S3 の振幅分布を計算する。図 7 に，CNR が 7，10，
15 dB の時の計算結果を示す。青線はシミュレーション結果，
赤線は青線にフィッティングされたガウス分布を示す。
ストークスパラメータは，2 乗検波により得られるので，そ
の振幅分布はガウス分布とは異なる。そのずれは CNR が低い
ほど顕著である。また，動径方向（図 7 では S1 軸方向) の分布
は，ガウス分布とは大きく異なることがわかる。このようにス
トークスベクトルは対応するジョーンズベクトルに比べて，振
幅分布の広がりが大きく，ストークス空間での閾値弁別により
受信感度が劣化することが予想される。
5. 2 符号誤り率特性
Optical
signal
Coherent
receiver
ADC
Fixed GVD
compensator/
Filter
Stokes-vector
calculator
CLK
図 4 に示すコンステレーションマップを持つ 2，4，8 相偏波
変調信号の BER 特性を，4 章のアルゴリズムを用いて計算し
た。さらに 8 相偏波変調と 2 相，4 相強度変調を組み合わせた
Discriminator
Decoded
symbol
Basis Stokesvector updater
16 相，32 相強度・偏波変調についても同様の計算を行った。
16 相変調では，光強度を 2 値，すなわち S0 ，3S0 に変調した。
強度弁別の閾値は 2S0 とした。32 相変調では，光強度を 4 値，
すなわち S0 ，2S0 ，4S0 ，6S0 に変調した。強度弁別の閾値は
1.5S0 ，3S0 ，5S0 である。また，シンボル数 N は 220 (≃ 106 )
図5
多値強度・偏波変調信号を復調するための DSP 回路の構成。
であり，低速な DSP を模擬するためにステップサイズパラメー
タは µ = 1/27 とした。
送信レーザと局発用レーザのスペクトル幅の合計を δf ，シ
—4—
CNR=7dB
CNR=10dB
CNR=15dB
I
Q
図 6 IQ 平面における雑音分布。青線はシミュレーション結果，赤線は青線にフィッティングさ
れたガウス分布を示す。CNR は，7，10，15 dB であり，図の上段は同相（In-phase) 成
分，下段は直交 (Quadrature) 成分である。
CNR=7dB
CNR=10dB
CNR=15dB
S1
S2
S3
図 7 ストークス空間における雑音分布。青線はシミュレーション結果，赤線は青線にフィッティ
ングされたガウス分布を示す。CNR は，7，10，15 dB であり，図の上段は S1 , 中段は
S2 ，下段は S3 の分布を示す。
ンボル間隔を T とした時，δf · T = 10−2 を仮定した。すなわ
∆ϕr (k) と ∆θr (k) は実数のガウス雑音であり，これらの標準
ちシンボルレートが 10 Gsymbol/s では，レーザのスペクトル
偏差を
幅は 50 MHz である。この値は，通常の DFB レーザの線幅の
ムウォークするので，N 番目のシンボルでの分散 σ (N )2 は
10 倍程度である。
一方，光ファイバの複屈折変動を模擬するために，偏波状態
に対しても高速のスクランブルをかけた。ファイバの複屈折を
表現するジョーンズ行列を
[
J=
e
ϕ
i 2r
( )
cos θ2r
( θr )
sin
2
( ) ]
− sin θ2r
( θr )
ϕ
−i r
e
2
cos
√
σ02 = 2 × 10−3 rad とした。ϕr (k) と θr (k) はランダ
σ (N )2 = σ02 N
(rad2 )
で与えられる。N = 220 では，
(20)
√
σ (N )2 ≃ 2 rad である。
10 Gsymbol/s では測定時間が 100 µs 程度なので，この測定時
間内での標準偏差 2 rad という値は，10 kHz 以下といわれる
(17)
2
ファイバの通常の偏波変動の上限に近い値である。図 8 は，式
(17) のジョーンズ行列に E = [1, 0]T を作用させた時の偏波状
とする。ϕr と θr は
態の変動を，シンボル毎にポアンカレ球上にプロットした例で
ϕr (k + 1) = ϕr (k) + ∆ϕr (k)
(18)
ある。S = [1, 0, 0]T からスタートして，偏波状態がランダムに
θr (k + 1) = θr (k) + ∆θr (k)
(19)
変化することがわかる。
に従い，シンボル毎に変化させた。k はシンボル番号である。
レーザの位相変動，ファイバの偏波変動のもとで，偏波，ビッ
—5—
6. むすび
本論文では，3 次元ストークス空間での強度・偏波多値信号の
設計方法について述べた。次に，このような多値信号を，ディ
ジタルコヒーレント光受信器を用いて復調するための DSP ア
ルゴリズムについて論じた。2, 4, 8 相多値偏波変調, 16, 32
相多値強度・偏波変調信号の符号誤り率特性をシミュレーショ
ンにより検討し，16 相多値変調に対する多値化によるペナル
ティーは，BER=10−5 において，3 dB 程度にとどまることを
示した。この変調方式を用いると，DP-QPSK 方式と同様にシ
ンボルレート 25 Gsymbol/s において 100 Gbit/s の伝送速度
が得られる。しかし，DP-QPSK 方式に比べて，ディジタルコ
ヒーレント光受信器における信号処理のための計算量が極めて
図 8 光ファイバの複屈折変動による出力偏波状態の変化。E = [1, 0]T
少ない。信号等化が困難なこと，DP-QPSK に比べて受信感度
の入力に対する出力の偏波状態を，シンボル毎にプロットして
が劣化するなどの欠点はあるものの，100 km 以下の短距離伝
いる。
送には有効な方式であると考えられる。
トあたりの CNR の関数として求めた BER を，図 9 に示す。
この BER 曲線は，レーザスペクトル線幅，ファイバの偏波
変動の影響を全く受けていないことを確認している。また，
DP-QPSK の BER 曲線も比較のために示している。16 相多
値変調は，8 相偏波変調と 2 相強度変調の組み合わせで実現
しているが，多値化によるペナルティーは BER=10−5 におい
て，3 dB 程度にとどまっている。この変調方式は，シンボル
レート 25 Gsymbol/s で 100 Gbit/s の伝送速度が得られるが，
100 Gbit/s DP-QPSK 変調方式に対するペナルティーは 7 dB
程度である。信号等化が困難なこと，DP-QPSK に比べて受信
感度が劣化するなどの欠点はあるものの，信号処理のための計
算量は極めて少ないため，100 km 以下の短距離の伝送には有
効な方式であると考えられる。
-1
32-ary
16-ary
octal
quad
binary
DP-QPSK
-2
)
R
E
B
(g -3
o
L
-4
-5
5
10
15
CNR/bit/pol [dB]
20
図 9 2，4，8 相多値偏波変調，16，32 相多値強度・偏波変調信号の
BER 特性。DP-QPSK 信号の BER 特性も，比較のために示し
ている。
謝辞
本研究の一部は，文部科学省科学研究費補助金基盤研究
(A) (研究代表者: 菊池，課題番号: 22246046) の支援を受けて
実施された。
文
献
[1] K. Kikuchi, “Digital coherent optical communication systems: Fundamentals and future prospects,” IEICE Electronics Express, vol.8, no.20, pp.1642-1662, Oct. 2011.
[2] E. Yamazaki, S. Yamanaka, Y. Kisaka, T. Nakagawa,
K. Murata, E. Yoshida, T. Sakano, M. Tomizawa,
Y. Miyamoto, S. Matsuoka, J. Matsui, A. Shibayama,
J. Abe, Y. Nakamura, H. Noguchi, K. Fukuchi, H. Onaka, K. Fukumitsu, K. Komaki, O. Takeuchi, Y. Sakamoto,
H. Nakashima, T. Mizuochi, K. Kubo, Y. Miyata, H. Nishimoto, S. Hirano, and K. Onohara, “Fast optical channel
recovery in ﬁeld demonstration of 100-Gbit/s Ethernet over
OTN using real-time DSP, ” Optics Express, vol.19, no.14,
pp.13139-13184, July 2011.
[3] G. R. Welti and J. S. Lee, “Digital transmission with coherent four-dimensional modulation,” IEEE Trans. on Information Theory, vol.IT-20, no.4, pp.497-502, July 1974.
[4] M. Karlsson and E. Agrell, “Which is the most powereﬃcient modulation format in optical links?,” Optics Express, vol.17, no.13, pp.10814-10819, June 2009.
[5] E. Agrell and M. Karlsson, “Power-eﬃcient modulation formats in coherent transmission systems,” J. of Lightwave
Technol., vol.27, no.22, pp.5115-5126, Nov. 2009.
[6] S. Betti, F. Curti, G. De Marchis, and E. Iannone, “Multilevel coherent optical system based on Stokes parameters
modulation,” J. Lightwave Technol., vol.8, no.7, pp.11271136, July 1990.
[7] S. Benedetto and P. Poggiolini, “Performance evaluation
of polarization shift keying modulation schemes,” Electron.
Lett., vol.26., no.4, pp.256-258, Feb. 1990.
[8] C. Brosseau, Fundamentals of polarized light, John Wiley
& Sons, Inc., 1998.
[9] 菊池和朗, “ディジタルコヒーレント光受信器における適応等化技
術,” 電子情報通信学会論文誌 (B), vol.J96-B, no.3, pp.212-219,
2013 年 3 月.
[10] 斉藤洋一, ディジタル無線通信の変復調，電子情報通信学会，
1996 年.
—6—
[c7]
社団法人電子情報通信学会
THE INSTITUTE OF ELECTRONICS,
INFORMATION AND COMMUNICATION ENGINEERS
信学技報
TECHNICAL REPORT OF IEICE.
[チュートリアル招待講演] ディジタルコヒーレント光通信の基礎
菊池和朗†
† 東京大学大学院工学系研究科電気系工学専攻
〒 113-8656 東京都文京区本郷 7-3-1
E-mail: †[email protected]
あらまし
超高速ディジタル信号処理技術とコヒーレント光技術の融合が，近年，“ディジタルコヒーレント光通信”
と呼ばれる新技術を生み出した。この技術によれば，光の位相や偏波の情報が安定に抽出できるので，多値光変調や
偏波多重を用いることが可能となる。偏波多重 4 相位相変調を導入することによって，商用システムにおいて一波長
チャンネルあたり 100 Gbit/s の伝送速度が実現されている。さらに，適応ディジタルフィルタを用いれば，光ファイ
バの群速度分散や偏波分散などに起因する線形な信号の歪みを，電気段でほぼ完全に等化することができる。本講演
では，このようなディジタルコヒーレント光通信システムを構成する光回路，ディジタル信号処理の概要について，
基礎から説き起こす。
キーワード
コヒーレント光通信，ディジタル信号処理
[Tutorial Invited Lecture] Fundamentals of Digital Coherent Optical
Communication Systems
Kazuro KIKUCHI†
† Department of Electrical Engineering and Information Systems, Graduate School of Engineering
The University of Tokyo
7-3-1 Hongo, Bunkyo-Ku, Tokyo 113-8656, Japan
E-mail: †[email protected]
Abstract The new technology called “Digital Coherent Optical Communications ” has recently emerged, combining ultrafast digital signal processing and coherent optical technologies. Such technology enables multi-level optical
modulation and polarization-division multiplexing, because we can achieve carrier-phase recovery and polarization
alignment in a stable manner in the digital domain; thus, we can obtain the single-channel transmission capacity
as high as 100 Gbit/s, introducing the dual-polarization quadrature phase-shift keying (DP-QPSK) modulation
format. Moreover, we can compensate for linear impairments stemming from group-velocity dispersion and polarization-mode dispersion of fibers, using adaptive equalizers implemented in the receiver. In this tutorial lecture,
we talk about the fundamentals of digital coherent optical communications including optical circuits and digital
signal-processing units in the receiver.
Key words Coherent optical communications, Digital signal processing
1. はじめに
して光強度を “on” にし，符号 “0” に対しては光強度を “oﬀ”
にして変調を行う。一方受信端では，図 1 のように，フォトダ
インターネットを流れる情報トラフィックは，年率 1.4 倍の
イオードで光電界を二乗検波することにより光強度情報を得
増加率で伸び続けている。光ファイバ通信システムは，このよ
る。この方式は，強度変調 (Intensity modulation: IM)・直接
うな膨大なデータを流通させるグローバルネットワークのため
検波 (Direct detection: DD) 方式と呼ばれる。光ファイバや半
の基盤技術として，大きく発展してきた。しかし現在に至るま
導体レーザなどの光デバイスの優秀さにより，このような単
で，ほとんどの光伝送システムは，極めて単純な通信の原理を
純な方式でも１波長あたり 10 Gbit/s，複数の波長を用いれば
用いている。すなわち，送信端ではディジタル符号 “1” に対
1 Tbit/s 以上の伝送容量が可能になった。しかし現在，大容量
—1—
コンテンツの配信を目的とするブロードバンドアクセスの普及
リア位相雑音，ωIF は中間角周波数で，信号光と LO 光の角周
により，光ファイバ通信システムには，さらなる超高速・大容
波数差を表す。R は受信器の感度である。ωIF が変調帯域より
量の信号伝送能力が要求されている。光強度のみを利用する従
ずっと大きい場合をヘテロダイン検波， ωIF が 0 とみなせる
来技術の延長によってこれらの要求に対応することは，もはや
場合をホモダイン検波という。いずれの場合にも，このビート
困難になりつつある。
信号から，信号光の振幅のみならず位相変調をも抽出すること
Data
0 1 1 0 10
ができる。ただし，位相を正しく復調するには，位相変調を位
PD
Photocurrent
相雑音から分離する必要がある。無線通信で用いられる RF 信
Optical
signal
号と比べると，一般にレーザ光の位相雑音はかなり大きいので，
Square-law
detection
Intensity
modulation
位相変調と雑音を分離するプロセスには技術的困難が伴う。
位相を検出できるだけでなく，コヒーレント光受信器はショッ
図 1 IM・DD 方式における受信器構成の概念図。
ト雑音限界の高受信感度を持つ。式 (1) に示すように，十分に
強度の大きい LO を用いれば，受信器出力が受信器回路雑音に
一方，無線通信の歴史を振り返ると，この IM・DD 方式は，
100 年以上も前のマルコーニの時代における通信方式に対応す
る。無線通信技術はその後，真空管を用いた正弦波発振器の開
発，アームストロングによるスーパーヘテロダイン技術の発明
を経て，電磁波の位相情報を最大限に活用して情報を伝送する
方式が定着し，現在の興隆をみるに至っている。
位相情報の活用が不可欠と考えられる。このような期待に応え
る技術が，コヒーレント光通信技術である。この方式では，図
2 のように受信端に別途用意された連続波局部発振光（Local
oscillator: LO）と信号光との干渉によって信号光の電界複素
振幅を測定し，信号光の復調を行う。この技術は，1980 年代
に一時精力的に研究されたが [1]，その後 15 年にわたって忘れ
去られていた。しかし，従来のコヒーレント光通信技術と高速
ディジタル信号処理 (Digital signal processing: DSP) を組み
合わせたディジタルコヒーレント光通信方式が 2005 年に提案
され [2], [3]，現在，100 Gbit/s 光伝送を可能にする唯一の技術
として，世界各国で活発に研究開発が進められている。
延伸できるので，1980 年代には長距離コヒーレント光伝送シ
ステムの研究が各国で競って行われ, 1990 年にはフィールドテ
ストが行われる段階まで開発が進んだ [4]。
doped ﬁber ampliﬁer: EDFA) の研究開発が急進展し，1990
年代前半には早くも実用化された。さらに波長分割多重
(Wavelength-division multiplexing: WDM) 技術の導入によ
り，IM・DD 方式の光ファイバ通信技術が新たな発展段階に
入った。EDFA を用いて多中継する長距離伝送システムでは，
受信光パワーは一定に維持されるので，コヒーレント光受信器
の持つ高受信感度の重要性が薄れた。当時コヒーレント光通信
には，高い受信感度以外にその構成の複雑さに見合うだけの
利点が見いだせなかったので，その後の研究が停滞することと
なった。またコヒーレント受信器では，信号の位相や偏波の変
動に適応的に対処する必要があり，この問題の解決が容易では
なかったことも実用化を阻害する原因であった。これらの理由
本講演では，コヒーレント光通信技術の開発の歴史を簡単に
振り返ったのち，新世代のディジタルコヒーレント光通信技術
の特長と，この技術が将来の光通信システムにもたらす可能性
について解説する。
Optical
signal
度を改善できる。高い受信感度を活用すれば再生中継器間隔を
しかしその後，エルビウム添加光ファイバ増幅器 (Erbium-
光通信技術もさらなる発展を目指すには，無線技術と同様な
で，コヒーレント光通信の研究開発は，その後 15 年にわたり
中断されることになった。
1990 年代には，インターネットの普及と相俟って，EDFA と
WDM 技術を用いた IM・DD 方式による光伝送システムの大
Interference
Data
打ち勝つほど大きくできるため，ショット雑音限界まで受信感
0 11 0 10
PD
容量化が急速に進んだ。この結果，EDFA 帯域を用いて伝送で
Photocurrent
きる情報量の限界が見え始め，2000 年以降，有限な EDFA 帯
域を効率よく利用するための多値光変調技術が，注目を集め始
Linear
detection
Phase
modulation
めた。多値位相変調 (Phase-shift keying: PSK) や多値直交振
LO
幅変調 (Quadrature amplitude modulation: QAM) などの新
しい変調フォーマットを用いれば，1 シンボルあたり複数ビッ
図 2 コヒーレント光通信方式における受信器構成の概念図。
トの情報を送れるので，シャノン限界 [5] に迫る周波数利用効
率を実現できる。
2. コヒーレント光ファイバ通信の歴史
多値光変調信号を復調する方法として，まず光遅延検波の研
図 2 に示すコヒーレント光受信器では，信号光と LO との
ビートにより次のような電気的出力が得られる。
I (t) = 2R
√
Ps (t) PLO cos {ωIF t + θs (t) + θn (t)}
究開発が進展した。これは，あるシンボルと１つ前のシンボル
との位相比較を行う検波方式で，振幅変化を伴わない多値 PSK
信号の復調に特に有効である。2002 年には 4 相位相変調・光
(1)
遅延検波 (Diﬀerential Quadrature PSK: DQPSK) 方式の実
験が報告された [6]。その後，波長多重 DQPSK 伝送方式に関
ここで Ps (t) は信号光パワー，PLO は LO 光パワー，θs (t) は
信号光位相変調，θn (t) は LO の位相を基準とした信号光のキャ
する研究開発が進展し，現在ではビットレート 40 Gbit/s(シン
ボルレート 20 Gsymbol/s) において，実用に至っている。光
—2—
遅延検波器は位相を抽出できるという意味で，広義のコヒーレ
converter: ADC) および DSP 回路に送られる。次にディジタ
ント受信器ではあるが，信号光同士の掛け算（二乗検波）を用
ル領域において，信号光電界複素振幅が再生される。
いているためにコヒーレント光通信方式の全ての利点を引き出
90° optical hybrid
すことはできない。これに対して 2005 年に，位相ダイバーシ
E1
I1
E2
E4
I2
I4
E3
I3
ティ・ホモダイン受信器 [7](イントラダイン受信器とも呼ばれ
る [8]) と高速ディジタル信号処理との組み合わせにより，従来
II
LO1
のコヒーレント受信器の技術的困難性を解決するディジタルコ
Optical signal
π/2
LO2
ヒーレント光受信器という新しい概念が登場した [2]。このよう
なディジタルコヒーレント受信器の利点は，以下の点にある。
LO
IQ
（ 1 ）受信後のディジタル信号処理により，電気領域で位相
雑音を除去し位相変調を抽出することが可能であるので，安定
図 3 位相ダイバーシティ・ホモダイン受信器の構成。
性の高いホモダイン光受信器が構築できる。
（ 2 ）どのような変調フォーマットにも対応することができ
るので，光周波数帯域利用効率を高めることが可能となる。
（ 3 ）偏波揺らぎをディジタル信号処理によって補償できる
しかし信号光と LO の位相差は，位相雑音によってランダム
に変化するので，位相変調を抽出するにはこの位相雑音を除
去する必要がある。従来のホモダイン受信器では，光位相同期
ループ (Optical phase-locked loop: OPLL) を用いてこの機能
ので，偏波多重が可能となる。
（ 4 ）コヒーレント光受信器は線形な受信器なので，信号光
を実現してきた。すなわち，位相雑音を打ち消すように LO の
の位相情報が検波後も保持される。このため光ファイバの群速
光周波数をフィードバック制御する。しかし位相雑音を有効に
度分散 (Group-velocity dispersion: GVD) や偏波モード分散
除去するためには，OPLL 帯域が信号光のスペクトル線幅より
(Polarization-mode dispersion: PMD) も，ディジタル信号処
十分に大きい必要があり，光回路を介したフィードバックルー
理技術を駆使して電気領域で補償することが可能となる。
プを用いた場合，この条件を満足させることは一般には容易で
このような優れた特長のために，ディジタルコヒーレント光
ない。また，位相変調と位相雑音の分離も容易ではなかった。
通信の研究開発は，近年，世界各国で急速に進んでいる。2008
一方，ディジタルコヒーレント受信器では，検波後のディジタ
年にはカナダの Nortel 社（現 Ciena 社）により，本方式を用い
ル領域での高速な信号処理により位相雑音を推定し，位相変調
て 50 Gbit/s で動作する偏波多重 QPSK 用の専用 LSI が開発
と位相雑音を分離することができる。このとき LO を信号光に
された [9]。さらに 2012 年には，我が国の国家プロジェクトに
位相同期する必要がないので，フリーランニング状態にある比
よって，100 Gbit/s で動作する LSI が初めて開発された [10]。
較的線幅の大きい半導体レーザを，送信用光源および LO とし
モジュール/システムベンダー各社によって，この LSI を用いた
て用いることができるようになった。
100 Gbit/s 偏波多重 4 相位相変調 (DP-QPSK) 信号用のディ
図 3 のホモダイン検波回路では，受信感度が信号光の偏波
ジタルコヒーレント光受信器が商用化され，100 Gbit/s 伝送シ
に依存するという問題があるが，この偏波依存性を解決する手
ステムへの導入が進んでいる。
段として，図 4 に示す偏波ダイバーシティ技術が開発されてい
る [11]。信号光の垂直および水平偏波を，同一の LO を用いて
3. ディジタルコヒーレント光受信器の構成
それぞれ別々に位相ダイバーシティ・ホモダイン検波すること
図 3 に，ディジタルコヒーレント光受信器の構成を示す。コ
により，各偏波成分に対して光複素振幅が得られる。これらを
ヒーレント光受信器では，信号光と LO とのビートにより電気
ディジタル領域で処理することにより，偏波多重分離や偏波分
的な出力を得る。ビート信号は光電界複素振幅の情報を完全に
散補償を行うことが可能となる。この受信器は，時変的な偏波
保持しているので，このビート信号からディジタル信号処理を
変動に対しても十分に対応できる。位相・偏波ダイバーシティ・
用いて，信号光の振幅のみならず位相変調をも抽出することが
ホモダイン検波光回路に関しては，石英系平面光回路 (Planar
できる。図 3 で用いられている光回路は，位相ダイバーシティ・
lightwave circuit: PLC) を用いたハイブリッド光集積回路や
ホモダイン検波回路またはイントラダイン検波回路と呼ばれ
InP を用いたモノリシック光集積回路の開発が進められている。
る。信号光周波数と LO 光周波数は，ほぼ等しくなるように制
御される。また，ここでは信号光の偏波は，LO 光の偏波と一
EEs ,inx, x
◦
致していると仮定している。LO1 と LO2 には，90 光ハイブ
Es
リッドにより 90◦ の位相差が与えられる。式 (1) に示すように，
PBS
E1
E3
ELO , x
IIPDIx1
E2
IIPDQx
2
E4
LO の位相を基準として信号光と LO1 とのビートは信号のコ
ELO
サイン成分（複素振幅の実部）を，信号光と LO2 とのビート
は信号光のサイン成分（複素振幅の虚部）を与える。コサイン
PBS
Eins ,,yy
ELO , y
成分は In-phase (I) 成分，サイン成分は Quadrature (Q) 成分
E5
°
90
Optical
Hybrid
IIPDIy3
E6
E7
IIPDQy
4
E8
とも呼ばれる。ダブルバランス型フォトダイオードで測定され
たこれら 2 つのビート信号は，AD 変換器 (Analog-to-digital
°
90
Optical
Hybrid
図4
位相・偏波ダイバーシティ・ホモダイン受信器の構成。
—3—
4. コヒーレント受信器におけるディジタル信号
処理
信号の線形歪を補償する原理について述べる。
送信された複素振幅のフーリエ変換を E in (ω)
[Ein,x (ω) , Ein,y (ω)]
T
=
とする。ここで，Ein,x (ω) および
4. 1 信号処理の概要
Ein,y (ω) は，偏波多重信号の x 偏波成分複素振幅のフーリ
位相・偏波ダイバーシティ・ホモダイン受信器の 4 つの出力
エ変換および y 偏波成分複素振幅のフーリエ変換を示す。また，
端子からは，3. で述べたように，2 つの偏波状態に対する光複
T は転置行列をとることを意味する。このとき送信端から受信
素振幅 (cos 成分および sin 成分) の情報が得られる。高速の 4
端に至る光伝送システムが線形であれば，伝達関数行列 H (ω)
チャンネル AD 変換器を用いて，これら 4 つの出力はディジタ
を用いて，受信器出力のフーリエ変換は
ル信号に変換される。符号間干渉を起こさないという条件下で，
[
]
Er,x (ω)
シンボルレート 1/T （T はシンボル間隔）で変調された光信号
Er,y (ω)
が占める最小の帯域は, 光領域で Bo = 1/T である。このとき，
[
]
Ein,x (ω)
= H (ω)
(2)
Ein,y (ω)
位相ダイバーシティ・ホモダイン受信器の IQ ポートからの電
で与えられる。ここで ω は光搬送波角周波数の中心周波数から
気出力は，帯域 Be = 1/(2T ) を占める。したがって，この信
のずれを示す。送信信号波形は伝達関数の周波数依存性により
号に対する最小のサンプリングレート（ナイキストレート）は
歪むとともに，行列の非対角成分により x 偏波成分と y 偏波成
R = 1/T である。しかし，ナイキストレートのサンプリングで
分が混合する。信号の等化とは，次式のように，H (ω) の逆伝
は，エリアシングを避けるために，AD 変換を行う前に受信器
達関数を生成して，受信信号から送信された複素振幅を再生す
の出力を帯域 Be = 1/(2T )，ロールオフ=0 のナイキストフィ
る操作に他ならない。
ルタで低域フィルタする必要がある。アナログ領域でのこのよ
うな急峻なフィルタリングは困難であるため，2 倍オーバーサ
[
]
Ein,x (ω)
=H
Ein,y (ω)
]
[
−1
Er,x (ω)
(ω)
(3)
Er,y (ω)
ンプルすなわちナイキストレートの 2 倍のサンプリングレート
R = 2/T でサンプルし，アナログフィルタへの要求を軽くする
伝達関数行列 H (ω) は，種々の原因で生じる。まず，光伝送
路の特性を考えよう。光ファイバ伝送路を伝送される光パワー
ことが一般的である。
AD 変換後の DSP には，図 5 に示すように，WDM チャン
が十分小さい時，伝送路の特性は線形であるので，それは伝達関
ネル選択，固定波長分散補償，適応等化，キャリア位相推定，
数行列で表現できる。波長分散のスカラー伝達関数 D (ω)，偏波
シンボル識別などが含まれる。
モード分散 (Polarization-mode dispersion: PMD) を表す 2 × 2
このようにディジタルコヒーレント光受信器は，多値信号を
の伝達関数行列 U (ω)，偏波依存損失 (Polarization-dependent
復調できるだけでなく，ディジタル領域で種々の信号処理を行
loss: PDL) を表す 2 × 2 の行列 K, 光ファイバの複屈折を表す
うことができるという特長がある。特に適応等化部は，偏波多
2 × 2 の Jones 行列 T を用いて，伝送路の伝達関数は
重分離，偏波分散補償，残留波長分散補償，タイミングジッタ
H o (ω) = D (ω) U (ω) KT
(4)
補償，光/電気回路の帯域制限の補償など，多くの機能を含む。
クロック抽出の方法は種々提案されているが，図 5 では，分散
補償後にクロック周波数を抽出して ADC のサンプリングレー
トを制御している。この場合，クロック位相の制御は適応等化
回路で行われる。キャリア位相推定部では，位相雑音の除去や
と表される。
伝達関数 D (ω) は，
(
D(ω) = exp −j
ω 2 β2 z
2
)
(5)
信号光と LO 光の間の周波数オフセットの除去が行われる。LSI
である。β2 は二次波長分散パラメータ，z は伝搬距離である。
の低消費電力化には，計算量の軽い DSP アルゴリズムの開発
経路の切り替えがなければ波長分散の時間変動は小さいので，
が重要である。また，光ファイバの非線形効果の等化なども今
計算量を低減するために半固定的な補償を行ったのち，残留し
後解決すべき課題として残されている。
た分散を適応等化することが多い。
PMD 行列 U (ω) はユニタリであり，
CLK
X -pol.
Y -pol.
EX
4-Ch
ADC
EY
Carrier
phase
estimator
WDM
channel
selector
Adaptive
equalizer
E y ,in
x-pol.
Decoder

Ex ,in
Fixed
equalizer
Symbol
y -pol.

U (ω) = R−1
1
(
exp j
ω∆τ
2
)

(
0
exp −j
0
ω∆τ
2
)  R1
(6)
で与えられる。行列 R1 は，2 つの固有偏波状態 (Principal
state of polarization: PSP) を x および y 偏波に変換するユ
ニタリ行列，∆τ は PSP 間の遅延時間差（Diﬀerential group
図 5 シンボルを復号するための DSP 回路の概要。
4. 2 信号等化の原理
ディジタルコヒーレント受信器における DSP の中で，適応
等化は最も重要な機能である。本節では，適応等化回路により
delay: DGD) である。行列 K は次式のエルミート行列で定義
される。
K=
R−1
2
[ √
]
Γmax
0
√
0
Γmin
R2
(7)
—4—
ここで R2 は，PDL に対する固有偏波モードを x および y 偏
E y (n) = [Ey (n) , Ey (n − 1) , · · · , Ey (n − k)]T
(10)
波に変換するユニタリ行列，Γmax と Γmin は，これらの固有
モードに対するパワー透過率を示す。T は，ファイバの複屈折
を示す周波数に依存しない 2 × 2 のユニタリ行列で，Jones 行
列と呼ばれる。これらの偏波に起因する伝達関数は，時間的に
変動するので，伝送特性を安定化させるには，適応等化を行う
ここで Ex (n)，Ey (n) は，適応等化器への x および y ポート
入力，n はサンプルインデックスである。次に FIR タップ係数
ベクトル hp (n) を
hp (n) = [hp,0 (n), hp,1 (n), · · · , hp,k (n)]T
(11)
ことが不可欠である。
伝送路中に挿入される光フィルタ，受信器回路の電気フィル
タなどの特性も，スカラー伝達関数 De (ω) で表現できる。こ
れらの伝達関数が未知であっても，適応等化を行うことによっ
て，その周波数特性を補償することができる。
と定義する。するとフィルタ出力は次式となる。
EX (n) = hxx (n)T E x (n) + hxy (n)T E y (n)
T
T
EY (n) = hyx (n) E x (n) + hyy (n) E y (n)
(12)
(13)
また，光ファイバ伝送中にタイミングジッタが生じることに
kT /m が系のインパルス応答より十分長ければ，4. 4 で述べる
より，AD 変換器のサンプリング位相と信号クロックの位相と
適応等化アルゴリズムを用いて，タップ係数ベクトルを収束さ
の間にずれが生じる。この効果は，サンプリングされた波形の
せることができる。このタップ係数ベクトルを，離散フーリエ
時間シフトと見なされるので，時間シフトを表すスカラー伝達
変換 (Discrete Fourier transform: DFT) すれば，逆伝達関数
関数 Dt (ω) = exp (jωτj ) を用いて記述できる。ここで τj はサ
行列 H −1 (ω) に近似的に一致する。
ンプリング点の最適な時刻からのずれを示し，時間的に変動す
4. 4 適応等化アルゴリズム
る量である。すなわち図 5 のように，クロック周波数さえ抽出
次に，FIR フィルタを適応制御するアルゴリズムについて検
されていれば，クロック位相は適応等化回路で制御できる。
討する。これまで QPSK 信号の等化には，計算量の軽さとブラ
インド等化が可能なことから，定包絡アルゴリズム (Constant
したがって系の全伝達関数行列は
H (ω) = Dt (ω) De (ω) H o (ω)
(8)
で与えられる。このように線形系では，伝送された光信号に歪
modulus algorithm: CMA) が好んで用いられている。しかし
このアルゴリズムでは，偏波多重分離する際，2 つのポートが
同一偏波に収束する特異点問題を完全に解決することはできな
を与える要素は全て，式 (8) のように一つの伝達関数行列にま
い。一方，判定指向最小二乗誤差アルゴリズム (Decision-driven
とめられる。この逆関数行列は，4. 3 で述べるように FIR フィ
least-mean square algorithm: DD-LMS algorithm) において，
ルタで表現され，4. 4 に示すアルゴリズムを用いて，適応的に
トレーニング信号を用いれば，2 つのポートが同一偏波に収束
生成できる。
する問題は避けられる。
4. 3 FIR フィルタによる適応等化
逆伝達関数行列 H
−1
DD-LMS アルゴリズムによれば，タップ係数は次式にした
(ω) の各行列要素は，FIR フィルタで
実現することができる。図 6 に FIR フィルタの 2 × 2 バタフ
ライ構成を示す。各行列要素 hp (p = xx, xy, yx, yy) が，FIR
がって更新される [12]。
hxx (n + 1) = hxx (n) + µeX (n)E x (n)∗
∗
(14)
フィルタにより構成される。図 7 は，（xx) 要素の FIR フィル
hxy (n + 1) = hxy (n) + µeX (n)E y (n)
(15)
タの構成を示す。シンボル間隔が T ，ADC 変換器のオーバサン
eX (n) = dX (n) − EX (n) (16)
プリングレートが m であるとき，タップ間の遅延時間を T /m
hyx (n + 1) = hyx (n) + µeY (n)E x (n)∗
としている。また，タップ段数は k である。
E x ( n)
h xx
+
E X (n)
+
EY ( n )
∗
hyy (n + 1) = hyy (n) + µeY (n)E y (n)
eY (n) = dY (n) − EY (n)
(17)
(18)
(19)
h xy
ここで，µ はステップサイズパラメータ，eX (n)，eY (n) は誤
h yx
E y ( n)
h yy
差信号である。dX (n)，dY (n) は, トレーニングモードではト
レーニングシンボルを，トラッキングモードではデコードされ
図 6 適応等化のための FIR フィルタのバタフライ構成。
c
Ex ( n)
hCxx0,0
T /m
hxx ,1
T /m
hxx ,k −1
たシンボルを表す。トレーニングモードでタップ係数を十分収
束させた後に，トラッキングモードに切り替える。2 倍オーバ
T /m
サンプリングの場合には，上記のタップ係数の更新は，2 サン
hxx ,k
プルに 1 回行われる。
EX ( n )
一方 CMA では，複素振幅の絶対値が一定になるようにタッ
I (nT )
プ係数を更新する [13]。すなわち，タップ更新は，誤差信号と
図 7 FIR フィルタの構成。行列要素 (xx) の例を示す。
して以下の式を用いて行われる。
[
]
[
]
2
FIR フィルタへの入力列ベクトルを，次のように定義する。
eX (n) = 1 − |EX (n)|2 EX (n)
(20)
E x (n) = [Ex (n) , Ex (n − 1) , · · · , Ex (n − k)]T
eY (n) = 1 − |EY (n)|
(21)
(9)
EY (n)
—5—
無歪の QPSK 信号は，シンボルの識別判定を行うサンプル点
高性能化を遂げた無線通信技術と同様に，コヒーレント光通信
では電界振幅が一定であるので，このアルゴリズムが有効であ
とディジタル信号処理の融合は必然的な技術の流れと考えられ
る。また，偏波多重信号を受信したときにも，2 つの偏波成分
る。今後の更なる発展を期待したい。
が混合した時に誤差信号が生じるので，CMA が偏波多重分離
に有効であることが証明されている [14]。しかし，送信した偏
波多重信号がどちらの出力ポートにあらわれるかは制御できず，
また，2 つの出力ポートに同一の偏波が収束する可能性も完全
には排除できないという欠点もある。この誤差信号の大きさは
キャリア位相を含まないので，図 5 の適応等化部とキャリア位
相推定部とを，独立に動作させることができる。これに対して
DD-LMS アルゴリズムでは，誤差信号の大きさはキャリア位
相に依存するため，適応等化部とキャリア位相推定部が干渉し
ないような DSP の設計が必要となる [15]。
4. 5 キャリア位相推定
位相雑音推定を行ないこれを除去するための信号処理につい
て, QPSK 信号を例にとって検討する。
図 8 はディジタル信号処理を用いた位相雑音除去の原理を
示す。適応等化を行った後，各偏波の複素振幅について１シン
ボルあたり１サンプルのデータが得られる。まず，再生した複
素振幅 Es (i) ∝ exp {jθs (i) + θn (i)} を 4 乗し，位相変調を取
り除く。４乗演算 Es (i)4 ∝ exp {4jθn (i)} により位相角は 4 倍
されるが，このとき π/2 の整数倍である 4 相位相変調成分は
除去される。次に，SN 比を高めるために，t = (i − k)T から
t = (i + k)T にわたる連続した 2k + 1 サンプルの Es (i)4 を足
し合わせる。位相雑音の推定値はこの複素数の位相角を 4 で
割ることによって，
[
1
θe (i) = arg
4
k
∑
]
4
Es (i + j)
(22)
j=−k
のように得られる。
複素振幅 Es (i) の位相からこの θe を引くと位相雑音を除去
できるので，位相変調は４シンボルに弁別することができる。
ただしこの方法では，シンボルに π/2 だけの位相回転の曖昧さ
が残るので，これを排除するために，信号はあらかじめ差動符
号化しておく必要がある。
Q
Q
×
4 θn
θn
I
4-th power
図8
Q
I
I
−θn
ディジタル信号処理を用いた位相雑音除去の原理。
5. むすび
コヒーレント光受信器にディジタル技術を導入することによ
り，安定なホモダイン検波が実現され，光の複素振幅の情報を
完全に抽出することが可能になった。このようなディジタルコ
ヒーレント受信器を用いれば，任意の多値光変調に対応できる
だけでなく，電気領域での分散補償，偏波制御などの新しい機
文
献
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能が実現できる。1990 年代にディジタル化によって飛躍的な
—6—