## Abstract

We analytically study received waveform asymmetries induced by chromatic dispersion (CD) for signed CD monitoring in differential quadrature phase-shift keying (DQPSK) systems and show that the asymmetries are results of differential detection and the ±*π*/4 phase shifters used in conventional DQPSK receivers. The theoretical insights developed help explain various published results on signed CD monitoring based on waveform asymmetries and allow us to further propose signed CD monitoring for differential eight phase-shift keying (D8PSK) systems without any modification to the receiver. Simulation results also show that the CD-induced waveform asymmetric features are preserved in presence of self-phase modulation (SPM) and polarization-mode dispersion (PMD).

© 2011 Optical Society of America

## 1. Introduction

Long-haul fiber-optic communication systems are currently moving towards differentially-coherent as well as coherent detection where the information is encoded in the phase of the optical carrier [1]. In particular, modulation formats such as polarization-multiplexed differential quadrature phase-shift keying (PM-DQPSK) are believed to be the standard for next-generation systems operating at 100 Gb/s and are currently under intensive research. Signal transmission in optical fibers is subjected to impairments such as chromatic dispersion (CD) and polarization-mode dispersion (PMD) which vary with temperature [2], component aging, fiber plant maintenance and dynamic routing in optical networks. Therefore, monitoring of the channel parameters such as CD is required for their appropriate compensation using optical or electronic techniques. Consequently, CD monitoring is a vital part of optical networks [3], especially for next-generation systems in which the tolerance to residual CD of the link decreases dramatically with increasing symbol rate.

Traditionally, CD can be monitored by phase-modulating the optical carrier [4] or inserting tones at the transmitter [5–8] and measuring the power of a CD-induced RF tone at the receiver. Alternatively, such RF tone can be obtained from clock tones for pulse shapes such as Non Return-to-Zero (NRZ)-, Return-to-Zero (RZ)- and Carrier-Suppressed (CS)-RZ On-Off Keying (OOK) [9]. CD monitoring techniques for coherent or differentially-coherent systems have also been studied [10–16]. However, the majority of the proposed techniques are not able to distinguish between the sign of CD which is vital for residual CD compensation. Sandel [17], Noé [18] and Takushima [19] studied signed CD monitoring by amplitude modulating a CW laser and measuring the signal arrival time or frequency modulating a CW laser and measuring received power fluctuations. However, complicated phase-lock loops are required. Liu et al. [20] added an additional dispersion element at the receiver to induce asymmetric dependence of the power ratio between two received RF tones on the sign of CD. On the other hand, Kawakami [21] proposed the use of asymmetric waveform distortions as a simple signed CD monitoring technique for RZ-DQPSK systems and our previous works [22, 23] also exploited such asymmetric features for signed CD monitoring for DQPSK and DPSK systems. However, no in-depth study on the nature of this asymmetry has been undertaken and the applicability and limitations of signed CD monitoring using waveform asymmetry are mostly unknown.

In this paper, we analytically study received waveform asymmetries induced by CD for DQPSK signals and show that such asymmetries are results of differential detection and the presence of ±*π*/4 phase shifters in conventional DQPSK demodulators. The theoretical insights developed thus explain the origins of waveform asymmetry obtained from DPSK receivers with an imperfectly tuned phase shifter [23]. In addition, the underlying principles can be similarly exploited for D8PSK systems without any modifications to the receiver. We study the use of delay-tap plots to exploit this asymmetry for signed CD monitoring through simulations and note that many other techniques can be used for exploiting the asymmetry. Furthermore, simulation results show that CD-induced waveform asymmetries are preserved in long-haul systems with SPM and PMD.

## 2. Waveform asymmetry for DQPSK systems in presence of chromatic dispersion

Consider a fiber-optic transmission system with transmitted signal

where*x*∈ {1, −1,

_{k}*j*,−

*j*} denotes QPSK symbols that are independent and identically distributed (i.i.d.) random variables,

*T*is the symbol period and

*b*(

*t*) is the pulse shape centered at

*t*= 0 which can be NRZ, RZ or CS-RZ. Polarization, multi-channel effects and self-phase modulation (SPM) are neglected for simplicity. In addition, noise from optical amplifiers and other components are also excluded in our analysis to highlight the origins of waveform asymmetry caused by CD. We note that the analytical derivations in this paper equally applies to general D

*M*PSK systems.

A channel with CD can be modeled as a linear and time-invariant (LTI) system with transfer function

where*β*

_{2}is the group velocity dispersion (GVD) parameter and

*L*is the length of the fiber. In this case, the corresponding impulse response

*h*(

*t*) is given by

*g*(

*t*), the received signal is given by where

*p*(

*t*) =

*b*(

*t*) *

*g*(

*t*) denotes the signal pulse shape with optical band-pass and electrical low-pass filtering. For DQPSK systems,

*E*(

_{r}*t*) will be mixed with a delayed version of itself plus a ±

*π*/4 phase shift using a demodulator shown in Fig. 1. The delayed signal is given by In this configuration, the output of the balanced receiver is given by

*y*(

_{I}*t*) for the rest of the paper and note that subsequent results will be similar to those from

*y*(

_{Q}*t*). Now,

**E**[

*y*(

_{I}*t*)] is periodic with period

*T*, we will just focus on the symbol centered at

*t*= 0 and consider the term

*z*(

*t*) =

*A*(

*t*)

*A*(

^{*}*t – T*) +

*A*(

*t*+

*T*)

*A*(

^{*}*t*). Now,

*p*(

*t*) is real for practical pulse shapes and filters. In this case, let

*τ*

_{3}= −

*τ*

_{1}and

*τ*

_{4}= −

*τ*

_{2},

*w*

_{2}(−

*t*) = −

*w*

_{1}(

*t*) and therefore

*t*and hence asymmetric about

*t*= 0. On the other hand, it can be shown that ℜ𝔢{

*z*(

*t*)} is an even function of

*t*and will not result in any waveform asymmetry. Since the sum of an symmetric and asymmetric waveform is still asymmetric, we have just shown that the expected value of the received waveform

**E**[

*y*(

_{I}*t*)] of a DQPSK demodulator exhibits an asymmetry about the center of each bit. Furthermore, ℑ𝔪{

*z*(

*t*)} and ℜ𝔢{

*z*(

*t*)} is also an odd and even function of

*β*

_{2}respectively. Therefore, any waveform asymmetric feature for a given

*β*

_{2}will be reversed about the center of the bit when the sign of

*β*

_{2}is changed. Such asymmetry reversal is exactly the feature that allows one to distinguish the sign of CD from the received waveform. To illustrate the analytical insights with an example, consider the transmission of a pulse train with Gaussian pulse shape Neglecting the effect of optical and electrical filters for simplicity, it can be shown that

*z*(

*t*)} is an odd function of

*t*and

*β*

_{2}and is shown in Fig. 2 for a 20G Sym/s system with

*T*= 15 ps and ± 136 ps/nm of accumulated CD. It is clear from Eq. (18) that when the sign of

_{o}*β*

_{2}changes, the direction of asymmetry also changes, enabling one to distinguish the sign of CD from the received waveform. As

*y*(

_{I}*t*) is an ergodic process, the ensemble average

**E**[

*y*(

_{I}*t*)] will be equal to the time-averaged waveform of one particular random realization of

*y*(

_{I}*t*) and therefore the CD-induced asymmetric features can be extracted from an eye diagram.

Note that **E**[*y _{I}*(

*t*)] is non-zero only because of the second line of Eq. (9) i.e. the product of each individual pulse with the conjugate of its delayed self. Consequently, CD-induced waveform asymmetries should only be present for differentially-coherent phase-modulated systems and not present for coherent modulation formats such as BPSK, QPSK or 16-Quadrature Amplitude Modulation(QAM). In addition, the asymmetry is due to the presence of the term ${\sum}_{k=-\infty}^{\infty}\Im \U0001d52a\left\{z\left(t-kT\right)\right\}$ which is brought about by the ±

*π*/4 phase shifters used in a typical DQPSK demodulator. The two above insights together explain the waveform asymmetries reported in [21,22] for DQPSK systems. Furthermore, they also explain how an intentional phase mismatch of

*π*/4 in one arm of a DPSK receiver enables signed CD monitoring for DPSK systems [23]. In addition, Fig. 3 shows the received eye diagrams for 40 Gb/s (20 GSym/s) NRZ-, RZ- and CS-RZ-DQPSK systems with an accumulated dispersion of ±136 ps/nm obtained from simulations. Noises from lasers, optical amplifiers and receiver circuitries are neglected to illustrate the CD-induced waveform asymmetries. From the figure, it can be seen that asymmetries are present for all pulse shapes. This is in agreement with expectations as the analytical derivations above only assume the pulse shape

*p*(

*t*) to be real and symmetric about its center. Therefore, the waveform asymmetric features should be present for all types of pulse shapes commonly used in communication systems.

## 3. Signed CD monitoring for D8PSK systems using delay-tap sampling

The theoretical insights developed above indicate that CD-induced waveform asymmetry will be present as long as the modulation format is differentially-coherent and ±*π*/4 phase shifters are used in the receiver structures. In fact, Eqs. (10) and (11) actually suggest that the term
${\sum}_{k=-\infty}^{\infty}\Im \U0001d52a\left\{z\left(t-kT\right)\right\}$, which causes the asymmetry, will be present as long as the phase shift of the 3-dB coupler is not 0. Consequently, for a D8PSK system with proposed receiver structures [24] such as the one shown in Fig. 4, one should be able to exploit the waveform asymmetric features in similar manners for signed CD monitoring.

In particular, one of the balanced-detector output can be ‘tapped’ for signed CD monitoring without modifying the receiver structure. Fig. 5 shows the eye diagrams corresponding to the output signal *y*_{3}_{π}_{/8}(*t*) of a 60 Gb/s (20 GSym/s) RZ-D8PSK system with −136 ps/nm, 0 ps/nm and 136 ps/nm of CD, indicating the presence of CD-induced waveform asymmetry. Similar to our previous studies, one can generate delay-tap plots from *y*_{3π/8} (*t*) or *y*_{−π/8} (*t*), evaluate the distances of sample pairs *d*_{1}, *d*_{2} furthest away on both sides of the diagonal *D* and calculate the distance ratio *d*_{1}/*d*_{2} for signed CD monitoring [22, 23]. Some delay-tap plots generated from *y*_{3π/8} (*t*) are shown in Fig. 6 and the distance ratio *d*_{1}/*d*_{2} vs. CD is shown in Fig. 7. It can be seen that the distance ratio corresponding to *y*_{−π/8}(*t*) is not as sensitive to CD compared with that from *y*_{3π/8} (*t*). This is in agreement with theoretical predictions as the coefficient of
${\sum}_{k=-\infty}^{\infty}\Im \U0001d52a\left\{z\left(t-kT\right)\right\}$ in Eqs. (10) and (11) will become large when the phase shift of the 3-dB coupler deviates away from 0. Moreover, in addition to delay-tap plots, we emphasize that many other techniques such as amplitude histograms and artificial neural networks can also be used to extract the CD-induced waveform asymmetric features for signed CD monitoring.

## 4. CD-induced waveform asymmetry in presence of SPM and PMD

In long-haul systems, signal transmission will also be affected by SPM as well as PMD and the performance of any CD monitoring technique maybe affected [25, 26]. To investigate the applicability of the signed CD monitoring technique for long-haul systems, Fig. 8 shows the distance ratio as a function of residual CD obtained from simulations for a 40 Gb/s RZ-DQPSK system with length 500 km and 1000 km. The transmission link consists of multiple spans of fiber with span length around 60 km and dispersion compensating fibers at each span such that the accumulated residual CD is equally distributed across the spans. The coarse-step model with 1-km step size is used for the simulation and the birefringence is random across different sections. The birefringence beat length is 15 m. The distance ratio corresponding to a link without SPM and PMD is also shown as reference. The input power, fiber nonlinear coefficient *γ* and the PMD coefficient is 0 dBm, 1.2/*W* ·km and
$0.1\text{ps}/\sqrt{\text{km}}$ respectively. From the figure, it can be seen that the CD-induced asymmetric features are still present for systems with SPM and PMD. Although the mapping between the distance ratio and residual CD will generally be different than the case without SPM and PMD and may need to be calibrated for each system setup, such mappings will still be able to distinguish between positive and negative CD. Consequently, the proposed signed CD monitoring technique will be applicable in realistic long-haul systems with high data rates in which residual CD compensation plays an important role in the overall performance of the transmission link.

## 5. Conclusions

In this paper, we analytically studied signed CD monitoring using waveform asymmetries for DQPSK systems and showed that such asymmetries are results of differential detection and ±*π*/4 phase shifters present in conventional DQPSK receivers. The theoretical insights developed help elucidate the operating principles behind other signed CD monitoring techniques based on waveform asymmetries reported in the literature and naturally suggest that similar methods can be used for D8PSK systems without modifications to the receiver. Signed CD monitoring using delay-tap plots is investigated for D8PSK systems through simulations and results are in agreement with theoretical predictions. In addition, the asymmetric features are shown to be preserved in presence of SPM and PMD and therefore the proposed technique can be used for signed CD monitoring in practical long-haul systems.

## Acknowledgments

The authors would like to acknowledge the support of the Hong Kong Government General Research Fund (GRF) under project number PolyU 519910 and the Open Project of State Key Laboratory of Advanced Optical Communication Systems and Networks ( 2008SH04) of the People’s Republic of China.

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