timedomainequalizationanddigitalback-propagation method

Research ArticleTime Domain Equalization and Digital Back-PropagationMethod-Based Receiver for Fiber Optic Communication Systems

Fazal Muhammad 1 Farman Ali 2 Usman Habib 3 Muhammad Usman4

Imran Khan 4 and Sunghwan Kim 5

1Department of Electrical Engineering City University of Science and Information Technology Peshawar 25000 Pakistan2Electrical Engineering Department Qurtuba University of Science and Information and Technology Dera Ismail Khan 29190KP Pakistan3School of Electrical Engineering Korea Advanced Institute of Science and Technology Daejeon 34141 Republic of Korea4Electrical Engineering Department University of Engineering Technology Mardan 23200 KP Pakistan5School of Electrical Engineering University of Ulsan Ulsan 44610 Republic of Korea

Correspondence should be addressed to Sunghwan Kim sungkimulsanackr

Received 23 November 2019 Revised 9 January 2020 Accepted 13 January 2020 Published 12 February 2020

Academic Editor Wonho Jhe

Copyright copy 2020 Fazal Muhammad et al is is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Fiber optic communication systems (FOCSs) have attained a lot of attention by revolutionizing the telecommunication industryand offering new possibilities with the technical advancements in state-of-the-art high speed digital electronics Advancedmodulation formats make use of the phase amplitude and polarization of the optical signals at the same time to provide highspectral efficiency as compared with 1 bitsHz for the intensity modulation direct detection system (IMDD) but are highly proneto transmission impairments us the effects that add up to the optical fiber impairments such as optical fiber chromaticdispersion (OFCD) polarizationmodel dispersion (PMD) and phase offset and noise (POaN) need to be addressed at the receiverside e development of components and algorithms to minimize these effects in next generation FOCSs with 100Gbps data rateand beyond with long-haul transmission is still a challenging issue In this paper digital signal processing- (DSP-) assisteddispersion and nonlinear compensation techniques are presented to compensate for physical layer impairments including OFCDPMD and POaNe simulations are performed considering Dual Polarization- (DP-) QPSK modulation format to achieve two-fold data rate to achieve spectral efficiency of 328 bitssHz by making use of the polarization diversity and system performance isinvestigated in terms of bit error rate (BER) constellation diagrams and quality factor (Q-factor) for different values of opticalsignal-to-noise ratio (OSNR) launch power (PL) and fiber length

1 Introduction

e introduction of telegraphy in 1830s [1] led to the de-velopment and design of many further telecommunicationtechnologiese increased demand for bandwidth and low-cost transmission over long distances led to the use of theoptical waves by 1970s to achieve ultralarge capacity and useoptical fiber as the transmission medium [2] e ad-vancement of fiber optic communication system (FOCS) isspread over various generations which was boosted by theadvent of Erbium-doped fiber amplifier (EDFA) in the 1980s

because it enabled long-haul transmission [3] e opticalamplification enabled increase in repeater spacing andwavelength division multiplexing (WDM) which resulted inultrahigh bit rates [4] and making the FOCSs a primarymean of information transportation on a global scale eultimate goal of a FOCS is to provide high throughputthrough the desired transmission range which requirescareful design considerations for the system configurationgenerally consists of a transmitter (Tx) optical fiber as thechannel and a receiver (Rx) Optical fiber is utilized in FOCSapplications in two basic forms from short range which use

HindawiInternational Journal of OpticsVolume 2020 Article ID 3146374 13 pageshttpsdoiorg10115520203146374

multi mode fiber (MMF) [5] to long-haul links employingsingle mode fiber (SMF) [6] To achieve high spectral effi-ciency (SE) which mainly represents the maximum theo-retical data throughput of a system and how efficiently thetotal available bandwidth is being used high order modu-lation formats can be used according to the available opticalsignal-to-noise ratio (OSNR) In a multichannel transmis-sion system such as wavelength division multiplexing(WDM) with constant transmit power the maximumachievable SE is an increasing function of the channelquantity and OSNR [7] To cope with the demand of multi-Gbs of data rates the channel impairments such as opticalfiber chromatic dispersion (OFCD) polarization modedispersion (PMD) and phase offset and noise (POaN) in theFOCS need to be treated ese effects are major bottleneckfor ultrahigh data rate systems such as 40Gbps and 100Gbpscapacity Passive Optical Networks (PONs) which are gen-erally supported by wavelength division multiplexing(WDM) and are operated for distances under 100 km [8]e situation is even worse for long-haul transmission evendeploying the coherent detection scheme [9 10] because ofcumulative dispersion and multiple bends throughout theoptical fiber link

In this paper the transmission impairments areaddressed in a Dense-WDM (DWDM) FOCS using con-ventional digital signal processing techniques [11] with aphase dithering technique and a modified version of thedigital back-propagation method We focus on long-haulFOCS using optical amplification with EDFA to compensatethe optical losses and RF amplification at the receiver end foroptical to electrical conversion loss e simulations areperformed by varying the system parameters such as lengthof the fiber input optical power and modulation scheme toquantify the performance in terms of OSNR bit error rate(BER) and quality factor (Q-factor) Moreover a constel-lation analyzer is used to show the performance of quad-rature modulation schemes before and after theimplementation of DSP compensation techniques C-band[12] with wavelength λ ranging from 152877 nm to156836 nm is used for simulation work for realization ofcommercially used equipment in a FOCS

11 Related Work For modulation schemes utilizing bothamplitude and phase of the optical carrier such as QPSK theaccumulated effects [13] pose a challenging situation at thereceiver end Recent work on a QPSK-based system [14] onmitigation of the channel effect shows the extend to whatchromatic dispersion and polarization mode dispersion candegrade the performance Even for coherent detection-basedsystems where advanced modulation formats are used [15]DSP assistance is required to achieve acceptable value ofBER As dispersion also causes power loss for the systemDSP techniques which provide power efficiency [16] canenable high data rates for such cases DP-QPSK has beenshown to provide high data rates with the use of dispersioncompensation fiber and DSP techniques [17] DSP tech-niques are a powerful tool to compensate optical fibertransmission impairments in single carrier systems as shown

by many researchers to achieve high capacity and error-freetransmission In [18] a frequency domain equalizer has beensuggested for FOCS to minimize PMD losses Some re-searchers [19] have shown an analysis of coupling behaviorsfor two linear polarized modes using DSP up to 10 kmtransmission range A demonstration of higher ordermodulation format data transmission with Raman amplifiersis performed with DSP techniques in [20] In [21] authorshave discussed a 4-channel FOCS over 26 km SMF in termsof DSP and digital filters but the implementation cost ofsuch filters is a real issue Similarly in [22] researchers usedmethodologies such as coherent optical orthogonal fre-quency multiplexing (COOFDM) for FOCS where theysuccessfully addressed the nonlinear issues but the trans-mission distance for an order of 100 km still needs to beinvestigated to analyze the cumulative channel effects forlong-haul FOCS In this work we focus on OFCD PMDand POaN impairments for long-haul transmission dis-tances using time domain equalization as compared withfrequency domain to support commercially used standardsusing TDM e use of DP-QPSK modulation format forhigh data rate and DSP-assisted receiver with the digitalback-propagation method to compensate nonlinear fibereffects has been shown in this work

12 Major Contributions Transmission over longer dis-tances using ultrahigh data rate systems is mainly carried outusing repeaters within the transmission spans For a het-erogeneous network or multi-Radio Access Technology(multi-RAT) the current optical fiber transmission tech-niques such as ethernet-over-fiber or digital-over-fiber re-quire an adaptive set of repeaters to support multiplestandards Large number of such repeaters will be a costlysolution and the requirement of regular maintenance andchecks is also an issue Moreover the additional noise in theregeneration or amplification steps will further deterioratethe performancee trade-off exists between the use of highdata rates which are mainly limited due to OFCD PMD andPOaN impairments for very long distances and overalldeployment cost of the complete FOCS In this work theprocess of adaptive equalization through advanced DSP anduse of Gaussian filters with the digital back-propagationtechnique to minimize PMD OFCD and POaN have onlybeen used at the receiver end to provide a cost-effectivesolution As the digital back-propagation technique has beenused in the previous work for mitigation of deterministicnon-linear effects a modified version is presented in thiswork with an adaptive equalizer to provide assistance againstthe randomly distributed PMD effect Similarly the phasedithering technique has been introduced at the transmitterend to assist the DSP receiver for mitigation of OFCD emain handouts of this paper are listed as follows

(1) Channel impairments like OFCD PMD and POaNare investigated for long-haul FOCS transmission toquantify the capacity and transmission range

(2) A simple low-complexity receiver design for theFOCS is proposed involving several DSP steps as

2 International Journal of Optics

offline processing in MATLAB after the terminationof transmission

(3) Simulation results are performed-based on mathe-matical model of the proposed set up Optisystemsimulation software is used to conduct simulationwork and results are obtained by varying inde-pendent variables such as length of fiber and inputoptical power to achieve performance parameterssuch as Q-factor BER and OSNR

(4) e simulation results of the designed FOCS arecompared with the use of conventional signal de-tection schemes at the receiver as compared with thetime domain equalization DSP assistance anddigital back-propagation technique e results de-clare that the proposed framework mitigates theeffects of OFCD PMD and POaN impairments andproves to be several times efficient than the currentFOCS

(5) To provide high SE an advance modulation DP-QPSKmethod is implemented at the transmitter sideof the proposed FOCS to provide polarization di-versity along with the phase diversity

13 Organization and Notation of Paper e remainingportions of this paper are arranged as follows Section 2discusses the framework of the proposed model and Section3 consists of the analytical model with mathematical modelrepresentation Section 4 includes results and discussion ofthe proposed model Conclusions from the presented workare in Section 5 Table 1 presented the notations used in thepaper for the convenience of the readers

2 Framework of the Proposed Model

In Section 1 the background of FOCS and literature revieware presented Moreover as the FOCS acts as the backboneof the current communication networks the optimization ofthe FOCS framework is necessary against channel impair-ments such as OFCD PMD and POaN Figure 1 shows theblock diagram of the proposed system e system model iscomposed of various components and the parametric valuesof the components are varied to optimize the performance ofthe overall system e data are transmitted through acontinuous wave (CW) laser which modulates the user dataA pseudorandom binary sequence (PRBS) generator is usedto generate data followed by a DP-QPSK modulator and aRF amplifier to adjust the gain e laser beam is split intotwo polarization orientations using a Polarization BeamSplitter (PBS) A set of identical MachndashZehnder Modulators(MZMs) is used to modulate the two optical carriers fromthe PBS which are afterwards combined and transmittedthrough 500 km length of SMF To compensate the fiberattenuation an erbium-doped fiber amplifier (EDFA) [23] isinstalled after each 100 km of fiber range After transmissionover optical fiber the transmitted optical signals arereconverted to electrical form using a DP-QPSK receiverwhich consists of a Laser LO for coherent detection a PBS a90 degree optical hybrid to mix the signals and four

photodiodes for balanced photodetection en theresulting electrical signals are proceeded to a low pass RFfilter electrical amplifier and a DSP unit to recover the datafrom the OFCD PMD and POaN effects reshold de-tection and parallel to serial converter are utilized by theBER test set to analyze the performance of the system at theRx side

e conventional receiver for the coherent detectionscheme performs DC blocking compensation of In-phaseQuadrature (IQ) imbalance timing recovery and fre-quencyphase offset compensation e effect of highamplitude DC component on the data transmission is thusremoved and the signal is compensated for the IQ im-balance effects which are caused due to the imperfectionbetween the amplitude and phase of I and Q componentse symbols are synchronized using timing recovery whichmainly estimates the sampling frequency and phase Fre-quency and phase offset between the received signal and thelocal oscillator at the receiver are common problems incoherent systems which can be compensated by utilizingpilot tones or preamble frames as a training sequence eproposed receiver uses DSP block from the Opitsystemand two additional DSP steps are performed usingMATLAB script e first one is the use of the phasedithering technique with a digital filter to reduce thenonlinear distortion effects by introducing a ditheringsignal at a high RF frequency so that the noise shape in thefrequency domain changes to shift high noise level towardsthe higher frequencies e second step is the use of thedigital back-propagation technique to further improve theperformance of the system against cumulative chromaticdispersion which will be explained later in mathematicalmodel of the proposed system Also an adaptive polari-zation controller has been used to estimate the mismatchbetween the two polarization modes of the received signaland to perform its correction

21 Optical Fiber Channel Impairments In this paperOFCD PMD and POaN impairments are analyzed andmitigated for the proposed FOCS model e effect of theseimpairments in a long-haul FOCS are explained as follows

Table 1 Notations and their descriptions used for the proposedmodel

Description NotationWavelength λAttenuation of the fiber αTransmitter TxReceiver RxPhase offset θSignal amplitude ζs(t)

Signal carrier frequency ψt

Responsivity κStokes parameter τFiber attenuation αDispersion slope ΨSpeed of light cIntensity of noise ξIN

International Journal of Optics 3

211 Optical Fiber Chromatic Dispersion Broadening of theoptical signals happens due to the OFCD as light pulsetravels in SMF at several speeds due to multiple factors [24]e transmission quality and hence the data rate is reduceddue to this critical factor as shown in Figure 2 us thiseffect needs to be treated for higher transmission speeds Inour proposed model we compensate the OFCD by usingdigital filters in the time domain due to its several ad-vantages over the frequency domain implementation [25]for improved selection of data signals With the help of theseprocedures the proposed framework performs better interms of BER OSNR and Q-factor

212 Polarization Mode Dispersion Use of polarizationdiversity doubles the transmission rate as different data canbe sent over the two propagationmodes of light which traveltogether inside the fiber But this technique is mainly af-fected by the PMD e two modes of light are supposed topropagate with same velocity inside SMF but this is not thecase in practical systems A small birefringence is generatedwhen the light source is encountered inside the fiber [26]is phenomenon becomes valuable in case of high speedtransmissions utilizing polarization diversity As a resultdelay shift occurs among transmittedmodes as mentioned inFigure 3 Just like OFCD the PMD distorts the optical pulseinside SMF and thus limits the transmission rates in long-haul FOCSus to enable long-haul and high-speed FOCSmitigating measures have to be adopted

213 Phase Offset and Noise e FOCS performance isimpaired by another key limiting factor known as POaN Ithappens when several wavelengths travel over an SMF withsame phase angle but reaches at the other end with a phaseoffset [27] due to the reflections and refraction from core andcladding As a result the transmitted optical pulses facephase offset and related phase noise [28]

22 DP-QPSK Compared with Other Modulation FormatsAmplitude phase shift keying (ASK) also known as on offkeying (OOK) [29] is a well-known modulation scheme inFOCS due to its simplicity but is less efficient in terms ofpower and is very susceptible to noise interference OOK is adigital modulation technique in which analog carrier ismodulated by a digital signal e intensity of the carrier

light is varied to represent binary data inputs while fre-quency and phase of the carrier signal remain constant eother mostly used modulation formats are frequency shiftkeying (FSK) and phase shift keying (PSK) which modulatethe frequency and phase of the carrier with respect to theinput binary data signal respectively [30] e FSK mod-ulation technique uses two different carrier frequencies torepresent binary 1 and binary 0 and thus uses largerbandwidth compared with other modulation techniquesPSK is a digital modulation technique which changes phaseof the analog carrier to represent digital binary data [31] In atwo-level PSK scheme a difference of 180 phase shift is usedbetween binary 1 and binary 0 Another modulation formatcalled Differential PSK (DPSK) provides better performancethan the rest in which the information is encoded in thephase between two successive bits Quadrature PSK is a formof advanced modulation formats in which the data infor-mation modulates both amplitude and phase of the signalthus providing improved energy efficiency and SE Hencethe DP-QPSK is proposed to be used in this work to increasethe data rate by making use of the polarization diversityalong with the QPSK modulation formate block diagramof DP-QPSK is shown in Figure 4 which consists of a po-larization beam splitter and two different QPSK datastreams are modulated over the X and Y polarized opticalcarrier signals

23 Proposed Digital Signal Processing Unit e phaselocking OFCD treatment and polarization adjustment arethe main challenges in the traditional FOCSs using advancedmodulation schemes which are addressed in the proposedDSP unit whose framework is shown in Figure 5 In additionthe phase dithering technique is applied at the transmitterend for noise shaping and to reduce the effects of nonlineardistortions In FOCS the electrical signals output from thephotodetector are transformed into discrete signals usinganalogue to digital converters (ADCs) to be processed bythe DSP unit [32] e DSP module also provides timedomain equalization for channel impairments and is able todetect DP-QPSK with superior receiver sensitivity and timerecovery is section summarizes the proposed frameworkphysical impairments SE and the techniques used formitigating channel effects and DSP steps e followingSection 3 involves an analytical approach for the proposedmodel

DP-QPSKTx

DP-QPSKRx

SMF 100km

EDFA

Loop N = 5

Optical filter

BERtester

resholddetector

Low passGaussian

filter

DSP usingMATLAB

Constellationviewer

RFspectrum

Figure 1 DP-QPSK-based long-haul FOCS framework with DSP unit


Optical pulse input

Delay shi due to polarizationmode dispersion

Figure 3 Delay Shift between the two traveling modes in SMF

X pol l data

MZM

MZM 90deg

90deg

MZM

MZM

X pol Q data

Y Pol Q data

Y Pol l dataCWlaser

Opticalcoupler

DP-QPSKoutputPol

beamcombiner

5050

Figure 4 System architecture for a typical DP-QPSK modulator

Subs

eque

ntbl

ocks

Dig

ital b

ack

prop

agat

ing

met

hod

Tim

e dom

ain

digi

tal fi

lter

Freq

pha

se o

ffset

Tim

ing

reco

very

DC

bloc

king

and

QI c

ompe

nsat

ion

ADC

DSP

Figure 5 Basic structure of the digital signal processing unit

Spectral width ofinput optical signal

Spectrum disturbed due to opticalfiber chromatic dispersion

λ λ

Single mode fiber

Figure 2 Optical fiber chromatic dispersion in single mode fiber


3 Analytical Model

is paper highlights the OFCD PMD and POaN im-pairments in FOCSs and proposes a DSP-assisted DP-QPSKsystem to mitigate these effectse abovementioned sectiondiscussed the proposed framework for the mentioned im-pairments while in this section a detailed analytical model ispresented In an FOCS a complexmodulated signal containsinformation in its phase as well as amplitude [33] which canbe represented as

Es(t) ζs(t)9j(ψt+forallt)

(1)

where Es(t) is a complex modulated signal ζs(t) representsthe amplitude component of the signal the carrier frequencyof the signal is denoted by ψt and the parameter forallt meansthe time-dependent phase Optical field associated with localoscillator (LO) at the receiver for coherent detection can bewritten as

ELO(t) ζLO(t)9j ψLOt+forallLOt( ) (2)

where ζLO(t) denotes amplitude of the LO pulse ψLOt is LOcarrier frequency and LO is the time-dependent phaserepresented by forallLOt Both ELO(t) and Es(t) fields are as-sumed to be identically polarized Moreover we investigatein this section balanced photodetection quadrature anddual polarized coherent detection in FOCSs For all thesecoherent detection in FOCS low and thermal noises areignored in this theoretical analysis e Es(t) and ELO(t)

input signals to PD are given as

EPD 12

radic1 j

j 1⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

Es(t)

Es(t)

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ (3)

where j minus 1

radic and the output of PD is written as

E(PD)o κ Es(t) + jELO

111386811138681113868111386811138681113868111386811138682 (4)

where κ is responsivity which is equal to 1 e power of thepropagated optical signal is given as

pi(t) px + p(t)sin[Δθt + Δβ] (5)

where pi(t) and p(t) are defined as follows

p(t) 2ζLO(t)ζs(t) (6)

pi(t) ζ2LO(t) + ζ2s (t) (7)

In equation (5) Δθ and Δβ explain frequency offset andphase offset respectively ese two parameters denote thetransmitted signal information e OSNR of the trans-mitted signal can also be achieved by balanced PD outputsof which are presented from Equations (5)ndash(7)

E+11(t) px + p(t)sin[Δθt + Δβ] + ξIN (8)

Eminus11(t) px + p(t)sin[Δθt + Δβ] + ξIN (9)

Here ξIN explains the intensity noise and it is consid-ered identical for ELO(t) and Es(t) us from equations (8)and (9) the outcome of balanced PD [34] is written as

E11(t) E+11(t) minus E

minus11(t) (10)

From these analytical analyses it is concluded thatbalanced PD enhances the quality of FOCS in terms of BERand OSNR However advance modulation formats needquadrature detection to transmit a reliable signal over FOCSe quadrature scheme detects both amplitude and phaseinformation e outcomes of a transmitted signal in termsof quadrature detection is defined from Equations (4) (8)and (9) which are given as

EQud

E+11

Eminus11

E+22

Eminus22

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Es(t) + ELO1113868111386811138681113868

11138681113868111386811138682

Es(t) minus ELO1113868111386811138681113868

11138681113868111386811138682

Es(t) + jELO1113868111386811138681113868

11138681113868111386811138682

Es(t) + jELO1113868111386811138681113868

11138681113868111386811138682

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(11)

After 90deg quadrature hybrid the outputs [35] are given asfollows

EN(t) 2p(t)cos[Δθt + Δβ] (12)

EM(t) 2p(t)sin[Δθt + Δβ] (13)

whereas the complex parameter quadrature detector is de-fined as

ER(t) 2p(t)9[jΔθt+Δβ]

(14)

ese equations show that the quadrature detectionscheme is efficient but requires accurate polarization set ofvalues In order to fulfill the 100Gbps data rate requirementdual polarization detection is a promising technique as itdoubles the transmission capacity for a quadrature detec-tion-based system e outputs of orthogonally polarizedsignals a

rarr and brarr

[36] are addressed as

EPOL ararr

E+11

Eminus11

E+22

Eminus22

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ararr

Es(t) + ELO1113868111386811138681113868

11138681113868111386811138682

ararr

Es(t) minus ELO1113868111386811138681113868

11138681113868111386811138682

ararr

Es(t) + jELO1113868111386811138681113868

11138681113868111386811138682

ararr

Es(t) + jELO1113868111386811138681113868

11138681113868111386811138682

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(15)

EPOL b

rarr

E+11

Eminus11

E+22

Eminus22

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

brarr

Es(t) + ELO1113868111386811138681113868

11138681113868111386811138682

brarr

Es(t) minus ELO1113868111386811138681113868

11138681113868111386811138682

brarr

Es(t) + jELO1113868111386811138681113868

11138681113868111386811138682

brarr

Es(t) + jELO1113868111386811138681113868

11138681113868111386811138682

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)

Equations (1)ndash(16) discuss the transmission opticalsignal reliability and the performance of dual polarizationdetection compared with single balance and quadraturedetection of signal In order to investigate the losses inside anoptical fiber medium Equation (17) explains the attenua-tion α losses inside the fiber [37] which is given as follows

α(l) 10l

εoutεin

1113888 1113889 (17)


where εout and εin present output and input power values ofoptical fiber and l is the fiber length Similarly OFCD effectis mentioned in Equations (18) and (19) given as

Ω 2πc

χ2ϑ2 (18)

Ψ 4πc

χ3ϑ2 +

πc

χϑ31113888 1113889 (19)

Here ϑ2 and ϑ3 mean second order and third orderOFCD coefficients respectively Ω is group velocity dis-persion Ψ denotes dispersion slope c represents speed oflight and wavelength of the pulse [38 39] is represented byχ In addition the Stokes vector in terms of the two or-thogonal polarization states a

rarr and brarr can be represented as

X

τ0τ1τ2τ3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

E2wararr + E2

u brarr

E2w ararr + E2

u brarr

EwararrE

brarr cos Φ

wararr minus Φ

ubrarr

111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113874 1113875

EwararrE

ubrarr sin Φ

wararr minus Φ

ubrarr

111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113874 1113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)

where Ew ararr and E

u brarr denote optical fields Φ

ubrarr and Φ

w ararr

define optical field phase and τ0 to τ3 represent stokesparameters ese orthogonally polarized fields transmittingthrough SMF generate different group velocities

e digital filter for the compensation of dispersion canbe implemented as Finite Impulse Response (FIR) with filtertap weights given as

ck

jcT2

Dλ2l

1113971

exp minus jn2πcT2

Dλ2l1113888 1113889 (21)

where c is defined by the limits [N2]le cle [N2] D is thedispersion coefficient λ defines the wavelength of the opticalwave l represents fiber length and T is the sampling time Inorder to discuss back-propagation let us consider a step sizeequal to fiber span of channel which is written as

ΓSMF(forall) eminus jΩlSMFforall2 (22)

Γfilter(forall) eminus jΩlfilterforall2 (23)

BnlSMF BcSMFleff SMF (24)

Bnlfilter Bcfilterleff filtergeminus αSMFlSMF (25)

In equations (22) to (25) B is optimization coefficientand equal to (minus cleff(Ns) + 12) Ns is total number ofspans Γ is the fiber span g is the gain of the EDFA leff is theeffective length c denotes nonlinear coefficient and lSMFdefines total length of SMF e range of B is varyingamong 0 and 1 and at B 1 it is considered as pure back-propagation is back-propagation presents better per-formance at high power values Moreover the system isefficiency maintained by treating OFCD because it addsnoise to walk off from pulse erefore overcome ondispersion is needed To deduce the computational re-quirements and implementing multispan the simulationmodel is calculated by using BER which is working basedon following equation

BER Q2φ

1113968 (26)

Here φ defines OSNR per bit n denotes the number ofreceived bits with errors and the function of q is calculatedas

010001

1E ndash 51E ndash 71E ndash 9

1E ndash 111E ndash 131E ndash 15BE

R

1E ndash 171E ndash 191E ndash 211E ndash 231E ndash 251E ndash 27

0 20 40 60 80 100Data rate (Gbps)

DP-QPSK w qualified DSPDP-QPSK w 5km amp ndash40 psnmkm DCFDP-QPSK w 5km amp ndash45 psnmkm DCFDP-QPSK wo qualified DSP

Figure 6 Comparison of BER performance with qualified DSP with different DCF magnitudes and without qualified DSP and DCF fordifferent data rates


Q(n) 12π

1113946infin

1exp minus n

221113872 1113873dn (27)

4 Results and Discussion

e full width at a half maximum (FWHM) bandwidth of305GHz is used for the modulated signal with a roll-offfactor of 082 for the pulse shaping filter e spectral ef-ficiency of the system (defined by system bit rate per BW of

the modulated signal) for the presented system thus be-comes 328 bitssHze performance analysis is performedby first using a conventional receiver for DP-QPSK mod-ulation which only performs DC blocking IQ compensa-tion and carrierphase recoveryen dispersionmitigationthrough a Dispersion Compensation Fiber (DCF) is appliedto analyze the effect of dispersion Finally the performance iscompared with the presented system with a qualified re-ceiver consisting of time domain digital filter-based dis-persion compensation and digital back propagation-based

28

26

24

22

20

18OSN

R16

14

12

100 20 40 60 80 100

Data rate (Gbps)


Figure 7 OSNR for different data rates

01

0001

1E ndash 5

1E ndash 7

1E ndash 9

1E ndash 11

1E ndash 13

BER

100 200 300Fiber length (km)

400 500


Figure 8 Comparison of BER performance with qualified DSP with different DCF magnitudes and without qualified DSP and DCF fordifferent fiber lengths


nonlinear treatment Figure 6 shows the BER results byvarying the data rate of the system for different configu-rations and verifies the effectiveness of the qualified receiverin the presence of dispersion and nonlinearities Figure 7shows the OSNR results for the same configurations withdifferent data rates which suggest that use of a qualifiedreceiver can allow the use of high order modulation schemesAs the BER and OSNR are related to each other the resultsfor OSNR are only shown here for one set of values to verifythe results e reference BER of 10minus 4 has been used in forthe OSNR results

Figure 8 shows the performance analysis for differentlengths of transmission fiber ranging from 100 km to500 km e results show that using DP-QPSK withqualified DSP module can increase the long-haul range upto 500 km e complete set of results depict that channelimpairments become more severe after 300 km of trans-mission distance where BER exceeds the maximumthreshold for the system even using DCF For systemsbased on coherent detection one of the major factors thatlimit the performance is the LO frequency offset As thetransmitter and receiver are located far away from each

70

65

60

55

50Q-fa

ctor

45

40

3515490 15495 15500 15505

Reference wavelength (nm)15510 15515 15520


Figure 9 Q-factor comparison for different frequency offset

001

1E ndash 4

1E ndash 6

1E ndash 8

1E ndash 10

1E ndash 12

BER

1E ndash 14

1E ndash 16

1E ndash 18

1E ndash 20ndash40 ndash35 ndash30 ndash25

Recieved power (dBm)ndash20 ndash15


Figure 10 Comparison of BER performance with qualified DSP with different DCF magnitudes and without qualified DSP and DCF fordifferent received power levels


001

1E ndash 5

1E ndash 9

1E ndash 3BER

1E ndash 17

1E ndash 21

1E ndash 2504 05 06 07 08

PMD (pskm2)09


Figure 11 BER for different polarization dispersion values

(a) (b)

Figure 12 Continued


other the LO laser used at the receiver end may notperfectly match with the incoming signal and down-conversion of the received signal generates errors in thedata transmission Figure 9 shows the Q-factor analysis forcases where the LO laser has an offset from the receivedsignal e results shows the highest Q-factor when the LOlaser matches exactly to the transmitter laser wavelength of1552 nm To analyze the performance of the system interms of receiver sensitivity Figure 10 shows the BERperformance for system configurations with and withoutthe qualified DSP module for different optical power levelsat the photodiode To achieve BER of 10minus 4 the results showthat power penalty is 35 dB for the DP-QPSK system withconventional DSP receiver which shows the significance ofthe proposed DSP techniques in FOCS Finally as thesystem employs polarization-based diversity it is impor-tant to investigate the effect of Polarization Mode Dis-persion (PMD) for the simulated system Figure 11 showsthe effect of PMD on BER of the described system fordifferent values of PMD e initial results with the pre-sented qualified receiver were not good enough to toleratePMD so an adaptive polarization controller is added to thequalified receiver to mitigate the PMD effects e pro-posed model outcomes are also compared in terms of

constellation diagrams Figures 12(a) to 12(d) explain thatthe constellation of the proposed model is much improvedby using the qualified receiver than DCF or conventionalDSP receiver for the DP-QPSK-based FOCS Table 2presents a comparison of performance for the presentedwork with two other relevant work [17 18] based on DP-QPSK modulation-supported FOCS Table 2 mentions theimprovement in receiver sensitivity (as shown in Figure 10)with a very low BER as compared with the recently pre-sented work on DP-QPSK e improvement in receiversensitivity shows reduced noise level at the receiver endwhich is mainly due to the noise shaping performed in thiswork as advanced DSP processing e channel noise isdistributed to the higher frequency region with an insertionof dithering signal to ensure low noise level for the DP-QPSK signal

5 Conclusion

is paper presents a DP-QPSK-based FOCS for long-haultransmission and advanced DSP-assisted receiver forcompensation of different types of cumulative opticaltransmission impairments including chromatic dispersionpolarization mode dispersion and nonlinear effects Instead

(c) (d)

Figure 12 Constellation diagram of (a) DP-QPSKwith qualified DSP (b) DP-QPSK with 5 km and minus 45 psnmkm (c) DP-QPSK with 5 kmand minus 40 psnmkm and (d) DP-QPSK with a conventional DSP receiver

Table 2 Performance comparison of the proposed receiver with the existing literature on DP-QPSK-based FOCS

Description Kakati and Arya [17] Zheng et al [18] Proposed receiverReciever sensitivity minus 34 dBm Not available minus 40 dBmAlleviation of PMD Not considered Yes YesBER at 100Gbs data rate 20 times 10minus 3 1 times 10minus 4 1 times 10minus 4

Considered distance for DP-QPSK 7200 km 20 km 500 km


of using costly hardware components such as DCF andadaptive repeaters throughout the transmission span theproposed DSP module implementation at the baseband levelon the receiver side provides a low-cost and feasible ap-proach for a long-haul FOCS e analysis depicts that theuse of coherent detection scheme provides improved sen-sitivity for the receiver which is affected by the noise levelwhich is treated using the phase dithering technique fornoise shapinge simulation results show that the proposedDSP-assisted approach compensates the transmission im-pairments to support high data rates and spectral efficiencyand thus can be used for long-haul FOCSs

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work (S2666095) was supported by project for Coop-erative RampD between Industry Academy and ResearchInstitute funded Korea Ministry of SMEs and Startups in 20

References

[1] G P Agrawal Fiber-Optic Communication Systems JohnWiley amp Sons Hoboken NJ USA 3rd edition 2002

[2] R Ramaswami and K SivarajanOptical Networks A PracticalPerspective Wiley Hoboken NJ USA Second edition 2002

[3] F Ali Y Khan and S S QureshiNon-Linear Long-Haul HighCapacity Fiber Optics Communication Lap Lambert Aca-demic Publishing Saarbrucken Germany 2019

[4] J G Proakis Digital Communications McGraw-Hill NewYork NY USA 1995

[5] D Sadot ldquoPushing optical fiber communications to theShannon limit advanced modulation formats and digitalsignal processingrdquo in Proceedings of the 18th InternationalConference on Transparent Optical Networks (ICTON)pp 1ndash3 Trento Italy 2016

[6] D Sadot G Dorman A Gorshtein E Sonkin and O VidalldquoSingle channel 112Gbitsec PAM4 at 56Gbaud with digitalsignal processing for data centers applicationsrdquo Optics Ex-press vol 23 no 2 pp 991ndash997 2015

[7] X Zhou and L Nelson ldquoAdvanced DSP for 400 Gbs andbeyond optical networksrdquo Journal of Lightwave Technologyvol 32 no 16 pp 2716ndash2725 2014

[8] G Agrawal ldquoFiber-optic communication systemsrdquo WileySeries inMicrowave and Optical Engineering Wiley HobokenNJ USA 4th edition 2012

[9] A J Lowery and B Corcoran ldquoNanosecond-latency IMDDDSB short-haul to coherentSSB long-haul converterrdquo Journalof Lightwave Technology vol 37 no 20 pp 5333ndash5339 2019

[10] Y Loussouarn M Song E Pincemin G MillerA Gibbemeyer and Mikkelsen ldquo100 Gbps coherent digitalCFP interface for short reach regional and ultra long-hauloptical communicationsrdquo in Proceedings of the IEEE Euro-pean Conf on Optical Communication (ECOC) pp 1ndash3Valencia Spain September 2015

[11] J Li J Du L Ma et al ldquoCoupling analysis of non-circular-symmetric modes and design of orientation-insensitive few-mode fiber couplersrdquo Optics Communications vol 383 no 3pp 42ndash49 2017

[12] D Soma S Beppu Y Wakayama et al ldquo257-Tbits weaklycoupled 10-mode C+L-bandWDM transmissionrdquo Journal ofLightwave Technology vol 36 no 6 pp 1375ndash1381 2018

[13] R Dar M Feder A Mecozzi and M Shtaif ldquoAccumulationof nonlinear interference noise in fiber-optic systemsrdquo OpticsExpress vol 22 no 12 pp 14199ndash14211 2014

[14] A N Ermolaev G P Krishpents V V Davydov andM G Vysoczkiy ldquoCompensation of chromatic and polari-zation mode dispersion in fiber-optic communication lines inmicrowave signals transmittionrdquo Journal of Physics vol 741no 1 Article ID 012171 2016

[15] A Nespola ldquoExperimental demonstration of fiber nonline-arity mitigation in a WDM multi-subcarrier coherent opticalsystemrdquo in Proceedings of the IEEE European Conference onOptical Communication (ECOC) pp 1ndash3 Valencia SpainSeptember 2015

[16] O Ishida K Takei and E Yamazaki ldquoPower efficient DSPimplementation for 100G-and-beyond multi-haul coherentfiber-optic communicationsrdquo in Proceedings of the IEEEOptical Fiber Communications Conference and Exhibition(OFC) pp 1ndash3 Anaheim CA USA March 2016

[17] D Kakati and S C Arya ldquoPerformance of grey-coded IQM-based optical modulation formats on high-speed long-hauloptical communication linkrdquo IET Communications vol 13no 18 pp 2904ndash2912 2019

[18] Z Zheng N Cui H Xu et al ldquoWindow-split structuredfrequency domain Kalman equalization scheme for largePMD and ultra-fast RSOP in an optical coherent PDM-QPSKsystemrdquo Optics Express vol 26 no 6 pp 7211ndash7226 2018

[19] M Morsy-Osman M Sowailem E El-Fiky et al ldquoDSP-freersquocoherent-litersquo transceiver for next generation single wave-length optical intra-datacenter interconnectsrdquo Optics Expressvol 26 no 7 pp 8890ndash8903 2018

[20] S W Pelouch ldquoRaman amplification an enabling technologyfor long-haul coherent transmission systemsrdquo Journal ofLightwave Technology vol 34 no 1 pp 6ndash19 2015

[21] E Al-Rawachy R P Giddings and J Tang ldquoExperimentaldemonstration of a real-time digital filter multiple access PONwith low complexity DSP-based interference cancellationrdquoJournal of Lightwave Technology vol 37 no 17 pp 4315ndash4329 2019

[22] E Giacoumidis Y Lin J Wei I Aldaya A Tsokanos andL P Barry ldquoHarnessing machine learning for fiber-inducednonlinearity mitigation in long-haul coherent opticalOFDMrdquo Future Internet vol 11 no 1 pp 1ndash20 2018

[23] Y-W Chen S Shen Q Zhou et al ldquoA reliable OFDM-basedMMW mobile fronthaul with DSP-aided sub-band spreadingand time-confined windowingrdquo Journal of Lightwave Tech-nology vol 37 no 13 pp 3236ndash3243 2019

[24] F Ali S Ahmad F Muhammad Z H Abbas U Habib andS Kim ldquoAdaptive equalization for dispersion mitigation inmulti-channel optical communication networksrdquo Electronicsvol 8 no 11 p 1364 2019

[25] H Zhang H G Batshon C R Davidson D G Foursa andA Pilipetskii ldquoMulti-dimensional coded modulation in long-haul fiber optic transmissionrdquo Journal of Lightwave Tech-nology vol 33 no 13 pp 2876ndash2883 2015

[26] N Stojanovic and X Changsong ldquoAn efficient method for skewestimation and compensation in coherent receiversrdquo IEEEPhotonics Technology Letters vol 28 no 4 pp 489ndash492 2016


[27] K Benyahya C Simonneau A Ghazisaeidi et al ldquoMulti-terabit transmission over OM2 multimode fiber with wave-length and mode group multiplexing and direct detectionrdquoJournal of Lightwave Technology vol 36 no 2 pp 355ndash3602018

[28] Q Zhuge and X Chen ldquoPrefacerdquo Optics Communicationsvol 409 pp 1ndash136 2018

[29] D Maharana and R Rout ldquoA 4 channel WDM based hybridoptical FiberFSO communication system using DP QPSKmodulation for bit rate of 100112 Gbsrdquo Int Journal ofEngineering Research and Technology vol 8 no 6 2019

[30] H M Obaid and H Shahid ldquoAchieving high gain using Er-Ybcodoped waveguidefiber optical parametric hybrid amplifierfor dense wavelength division multiplexed systemrdquo OpticalEngineering vol 57 no 5 2018

[31] A Niaz F Qamar K Islam A Shahzad R Shahzadi andM Ali ldquoPerformance analysis and comparison of QPSK andDP-QPSK based optical fiber communication systemsrdquo ITEEJournal vol 7 no 3 pp 34ndash39 2018

[32] E Tipsuwannakul J Li M Karlsson and P A AndreksonldquoPerformance comparisons of DP-16qam and duobinary-shaped DP-QPSK for optical systems with 41 bitshz spectralefficiencyrdquo Journal of Lightwave Technology vol 30 no 14pp 2307ndash2314 2012

[33] F I El-Naha ldquoCoherent quadrature phase shift keying opticalcommunication systemsrdquo Optoelectronics Letters vol 14no 5 pp 372ndash375 2018

[34] J M Kahn and D A B Miller ldquoCommunications expands itsspacerdquo Nature Photonics vol 11 no 1 pp 5ndash8 2017

[35] J K Perin A Shastri and J M Kahn ldquoDesign of low-powerDSP-free coherent receivers for data center linksrdquo Journal ofLightwave Technology vol 35 no 21 pp 4650ndash4662 2017

[36] X Miao M Bi Y Fu L Li and W Hu ldquoExperimental studyof NRZ duobinary and PAM-4 in O-band DML-based 100 g-EPONrdquo IEEE Photonics Technology Letters vol 29 no 17pp 1490ndash1493 2017

[37] S Zhu X Wu J Liu and C Guo ldquoStokes-space modulationformat identification for coherent optical receivers utilizingimproved hierarchical clustering algorithmrdquo in Proceedings ofthe Opto-Electronics and Communications Conf (OECC) andPhotonics Global Conference (PGC) pp 1ndash3 Singapore July2017

[38] Y Dong E Al-Rawachy R P Giddings W Jin D Nessetand J M Tang ldquoMultiple channel interference cancellation ofdigital filter multiple access PONsrdquo Journal of LightwaveTechnology vol 35 no 1 pp 34ndash44 2017

[39] S Jawla and R K Singh ldquoPhase-shift modulation formats inoptical communication systemrdquo International Journal ofAdvancements in Research and Technology vol 2 no 11pp 72ndash76 2013


multi mode fiber (MMF) [5] to long-haul links employingsingle mode fiber (SMF) [6] To achieve high spectral effi-ciency (SE) which mainly represents the maximum theo-retical data throughput of a system and how efficiently thetotal available bandwidth is being used high order modu-lation formats can be used according to the available opticalsignal-to-noise ratio (OSNR) In a multichannel transmis-sion system such as wavelength division multiplexing(WDM) with constant transmit power the maximumachievable SE is an increasing function of the channelquantity and OSNR [7] To cope with the demand of multi-Gbs of data rates the channel impairments such as opticalfiber chromatic dispersion (OFCD) polarization modedispersion (PMD) and phase offset and noise (POaN) in theFOCS need to be treated ese effects are major bottleneckfor ultrahigh data rate systems such as 40Gbps and 100Gbpscapacity Passive Optical Networks (PONs) which are gen-erally supported by wavelength division multiplexing(WDM) and are operated for distances under 100 km [8]e situation is even worse for long-haul transmission evendeploying the coherent detection scheme [9 10] because ofcumulative dispersion and multiple bends throughout theoptical fiber link

In this paper the transmission impairments areaddressed in a Dense-WDM (DWDM) FOCS using con-ventional digital signal processing techniques [11] with aphase dithering technique and a modified version of thedigital back-propagation method We focus on long-haulFOCS using optical amplification with EDFA to compensatethe optical losses and RF amplification at the receiver end foroptical to electrical conversion loss e simulations areperformed by varying the system parameters such as lengthof the fiber input optical power and modulation scheme toquantify the performance in terms of OSNR bit error rate(BER) and quality factor (Q-factor) Moreover a constel-lation analyzer is used to show the performance of quad-rature modulation schemes before and after theimplementation of DSP compensation techniques C-band[12] with wavelength λ ranging from 152877 nm to156836 nm is used for simulation work for realization ofcommercially used equipment in a FOCS

11 Related Work For modulation schemes utilizing bothamplitude and phase of the optical carrier such as QPSK theaccumulated effects [13] pose a challenging situation at thereceiver end Recent work on a QPSK-based system [14] onmitigation of the channel effect shows the extend to whatchromatic dispersion and polarization mode dispersion candegrade the performance Even for coherent detection-basedsystems where advanced modulation formats are used [15]DSP assistance is required to achieve acceptable value ofBER As dispersion also causes power loss for the systemDSP techniques which provide power efficiency [16] canenable high data rates for such cases DP-QPSK has beenshown to provide high data rates with the use of dispersioncompensation fiber and DSP techniques [17] DSP tech-niques are a powerful tool to compensate optical fibertransmission impairments in single carrier systems as shown

by many researchers to achieve high capacity and error-freetransmission In [18] a frequency domain equalizer has beensuggested for FOCS to minimize PMD losses Some re-searchers [19] have shown an analysis of coupling behaviorsfor two linear polarized modes using DSP up to 10 kmtransmission range A demonstration of higher ordermodulation format data transmission with Raman amplifiersis performed with DSP techniques in [20] In [21] authorshave discussed a 4-channel FOCS over 26 km SMF in termsof DSP and digital filters but the implementation cost ofsuch filters is a real issue Similarly in [22] researchers usedmethodologies such as coherent optical orthogonal fre-quency multiplexing (COOFDM) for FOCS where theysuccessfully addressed the nonlinear issues but the trans-mission distance for an order of 100 km still needs to beinvestigated to analyze the cumulative channel effects forlong-haul FOCS In this work we focus on OFCD PMDand POaN impairments for long-haul transmission dis-tances using time domain equalization as compared withfrequency domain to support commercially used standardsusing TDM e use of DP-QPSK modulation format forhigh data rate and DSP-assisted receiver with the digitalback-propagation method to compensate nonlinear fibereffects has been shown in this work

12 Major Contributions Transmission over longer dis-tances using ultrahigh data rate systems is mainly carried outusing repeaters within the transmission spans For a het-erogeneous network or multi-Radio Access Technology(multi-RAT) the current optical fiber transmission tech-niques such as ethernet-over-fiber or digital-over-fiber re-quire an adaptive set of repeaters to support multiplestandards Large number of such repeaters will be a costlysolution and the requirement of regular maintenance andchecks is also an issue Moreover the additional noise in theregeneration or amplification steps will further deterioratethe performancee trade-off exists between the use of highdata rates which are mainly limited due to OFCD PMD andPOaN impairments for very long distances and overalldeployment cost of the complete FOCS In this work theprocess of adaptive equalization through advanced DSP anduse of Gaussian filters with the digital back-propagationtechnique to minimize PMD OFCD and POaN have onlybeen used at the receiver end to provide a cost-effectivesolution As the digital back-propagation technique has beenused in the previous work for mitigation of deterministicnon-linear effects a modified version is presented in thiswork with an adaptive equalizer to provide assistance againstthe randomly distributed PMD effect Similarly the phasedithering technique has been introduced at the transmitterend to assist the DSP receiver for mitigation of OFCD emain handouts of this paper are listed as follows

(1) Channel impairments like OFCD PMD and POaNare investigated for long-haul FOCS transmission toquantify the capacity and transmission range

(2) A simple low-complexity receiver design for theFOCS is proposed involving several DSP steps as























DP-QPSKTx

DP-QPSKRx

SMF 100km

EDFA

Loop N = 5

Optical filter

BERtester

resholddetector

Low passGaussian

filter

DSP usingMATLAB

Constellationviewer

RFspectrum



Optical pulse input



X pol l data

MZM

MZM 90deg

90deg

MZM

MZM

X pol Q data

Y Pol Q data

Y Pol l dataCWlaser

Opticalcoupler

DP-QPSKoutputPol

beamcombiner

5050


Subs

eque

ntbl

ocks

Dig

ital b

ack

prop

agat

ing

met

hod

Tim

e dom

ain

digi

tal fi

lter

Freq

pha

se o

ffset

Tim

ing

reco

very

DC

bloc

king

and

QI c

ompe

nsat

ion

ADC

DSP




λ λ

Single mode fiber



3 Analytical Model



(1)





EPD 12

radic1 j

j 1⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

Es(t)

Es(t)

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ (3)

where j minus 1



111386811138681113868111386811138681113868111386811138682 (4)





pi(t) ζ2LO(t) + ζ2s (t) (7)






minus11(t) (10)


EQud

E+11

Eminus11

E+22

Eminus22

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Es(t) + ELO1113868111386811138681113868

11138681113868111386811138682

Es(t) minus ELO1113868111386811138681113868

11138681113868111386811138682

Es(t) + jELO1113868111386811138681113868

11138681113868111386811138682

Es(t) + jELO1113868111386811138681113868

11138681113868111386811138682

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(11)






(14)


rarr and brarr


EPOL ararr

E+11

Eminus11

E+22

Eminus22

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ararr

Es(t) + ELO1113868111386811138681113868

11138681113868111386811138682

ararr

Es(t) minus ELO1113868111386811138681113868

11138681113868111386811138682

ararr

Es(t) + jELO1113868111386811138681113868

11138681113868111386811138682

ararr

Es(t) + jELO1113868111386811138681113868

11138681113868111386811138682

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(15)

EPOL b

rarr

E+11

Eminus11

E+22

Eminus22

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

brarr

Es(t) + ELO1113868111386811138681113868

11138681113868111386811138682

brarr

Es(t) minus ELO1113868111386811138681113868

11138681113868111386811138682

brarr

Es(t) + jELO1113868111386811138681113868

11138681113868111386811138682

brarr

Es(t) + jELO1113868111386811138681113868

11138681113868111386811138682

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)


α(l) 10l

εoutεin

1113888 1113889 (17)



Ω 2πc

χ2ϑ2 (18)

Ψ 4πc

χ3ϑ2 +

πc

χϑ31113888 1113889 (19)



X

τ0τ1τ2τ3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

E2wararr + E2

u brarr

E2w ararr + E2

u brarr

EwararrE

brarr cos Φ

wararr minus Φ

ubrarr

111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113874 1113875

EwararrE

ubrarr sin Φ

wararr minus Φ

ubrarr

111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113874 1113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)



ubrarr and Φ

w ararr



ck

jcT2

Dλ2l

1113971

exp minus jn2πcT2

Dλ2l1113888 1113889 (21)







BER Q2φ

1113968 (26)


010001



R


0 20 40 60 80 100Data rate (Gbps)




Q(n) 12π

1113946infin

1exp minus n

221113872 1113873dn (27)




28

26

24

22

20

18OSN

R16

14

12

100 20 40 60 80 100

Data rate (Gbps)



01

0001

1E ndash 5

1E ndash 7

1E ndash 9

1E ndash 11

1E ndash 13

BER


400 500






70

65

60

55

50Q-fa

ctor

45

40

3515490 15495 15500 15505




001

1E ndash 4

1E ndash 6

1E ndash 8

1E ndash 10

1E ndash 12

BER

1E ndash 14

1E ndash 16

1E ndash 18






001

1E ndash 5

1E ndash 9

1E ndash 3BER

1E ndash 17

1E ndash 21

1E ndash 2504 05 06 07 08

PMD (pskm2)09



(a) (b)

Figure 12 Continued




5 Conclusion


(c) (d)







Data Availability




Acknowledgments


References































































DP-QPSKTx

DP-QPSKRx

SMF 100km

EDFA

Loop N = 5

Optical filter

BERtester

resholddetector

Low passGaussian

filter

DSP usingMATLAB

Constellationviewer

RFspectrum



Optical pulse input



X pol l data

MZM

MZM 90deg

90deg

MZM

MZM

X pol Q data

Y Pol Q data

Y Pol l dataCWlaser

Opticalcoupler

DP-QPSKoutputPol

beamcombiner

5050


Subs

eque

ntbl

ocks

Dig

ital b

ack

prop

agat

ing

met

hod

Tim

e dom

ain

digi

tal fi

lter

Freq

pha

se o

ffset

Tim

ing

reco

very

DC

bloc

king

and

QI c

ompe

nsat

ion

ADC

DSP




λ λ

Single mode fiber



3 Analytical Model



(1)





EPD 12

radic1 j

j 1⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

Es(t)

Es(t)

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ (3)

where j minus 1



111386811138681113868111386811138681113868111386811138682 (4)





pi(t) ζ2LO(t) + ζ2s (t) (7)






minus11(t) (10)


EQud

E+11

Eminus11

E+22

Eminus22

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Es(t) + ELO1113868111386811138681113868

11138681113868111386811138682

Es(t) minus ELO1113868111386811138681113868

11138681113868111386811138682

Es(t) + jELO1113868111386811138681113868

11138681113868111386811138682

Es(t) + jELO1113868111386811138681113868

11138681113868111386811138682

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(11)






(14)


rarr and brarr


EPOL ararr

E+11

Eminus11

E+22

Eminus22

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ararr

Es(t) + ELO1113868111386811138681113868

11138681113868111386811138682

ararr

Es(t) minus ELO1113868111386811138681113868

11138681113868111386811138682

ararr

Es(t) + jELO1113868111386811138681113868

11138681113868111386811138682

ararr

Es(t) + jELO1113868111386811138681113868

11138681113868111386811138682

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(15)

EPOL b

rarr

E+11

Eminus11

E+22

Eminus22

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

brarr

Es(t) + ELO1113868111386811138681113868

11138681113868111386811138682

brarr

Es(t) minus ELO1113868111386811138681113868

11138681113868111386811138682

brarr

Es(t) + jELO1113868111386811138681113868

11138681113868111386811138682

brarr

Es(t) + jELO1113868111386811138681113868

11138681113868111386811138682

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)


α(l) 10l

εoutεin

1113888 1113889 (17)



Ω 2πc

χ2ϑ2 (18)

Ψ 4πc

χ3ϑ2 +

πc

χϑ31113888 1113889 (19)



X

τ0τ1τ2τ3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

E2wararr + E2

u brarr

E2w ararr + E2

u brarr

EwararrE

brarr cos Φ

wararr minus Φ

ubrarr

111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113874 1113875

EwararrE

ubrarr sin Φ

wararr minus Φ

ubrarr

111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113874 1113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)



ubrarr and Φ

w ararr



ck

jcT2

Dλ2l

1113971

exp minus jn2πcT2

Dλ2l1113888 1113889 (21)







BER Q2φ

1113968 (26)


010001



R


0 20 40 60 80 100Data rate (Gbps)




Q(n) 12π

1113946infin

1exp minus n

221113872 1113873dn (27)




28

26

24

22

20

18OSN

R16

14

12

100 20 40 60 80 100

Data rate (Gbps)



01

0001

1E ndash 5

1E ndash 7

1E ndash 9

1E ndash 11

1E ndash 13

BER


400 500






70

65

60

55

50Q-fa

ctor

45

40

3515490 15495 15500 15505




001

1E ndash 4

1E ndash 6

1E ndash 8

1E ndash 10

1E ndash 12

BER

1E ndash 14

1E ndash 16

1E ndash 18






001

1E ndash 5

1E ndash 9

1E ndash 3BER

1E ndash 17

1E ndash 21

1E ndash 2504 05 06 07 08

PMD (pskm2)09



(a) (b)

Figure 12 Continued




5 Conclusion


(c) (d)







Data Availability




Acknowledgments


References










































Optical pulse input



X pol l data

MZM

MZM 90deg

90deg

MZM

MZM

X pol Q data

Y Pol Q data

Y Pol l dataCWlaser

Opticalcoupler

DP-QPSKoutputPol

beamcombiner

5050


Subs

eque

ntbl

ocks

Dig

ital b

ack

prop

agat

ing

met

hod

Tim

e dom

ain

digi

tal fi

lter

Freq

pha

se o

ffset

Tim

ing

reco

very

DC

bloc

king

and

QI c

ompe

nsat

ion

ADC

DSP




λ λ

Single mode fiber



3 Analytical Model



(1)





EPD 12

radic1 j

j 1⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

Es(t)

Es(t)

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ (3)

where j minus 1



111386811138681113868111386811138681113868111386811138682 (4)





pi(t) ζ2LO(t) + ζ2s (t) (7)






minus11(t) (10)


EQud

E+11

Eminus11

E+22

Eminus22

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Es(t) + ELO1113868111386811138681113868

11138681113868111386811138682

Es(t) minus ELO1113868111386811138681113868

11138681113868111386811138682

Es(t) + jELO1113868111386811138681113868

11138681113868111386811138682

Es(t) + jELO1113868111386811138681113868

11138681113868111386811138682

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(11)






(14)


rarr and brarr


EPOL ararr

E+11

Eminus11

E+22

Eminus22

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ararr

Es(t) + ELO1113868111386811138681113868

11138681113868111386811138682

ararr

Es(t) minus ELO1113868111386811138681113868

11138681113868111386811138682

ararr

Es(t) + jELO1113868111386811138681113868

11138681113868111386811138682

ararr

Es(t) + jELO1113868111386811138681113868

11138681113868111386811138682

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(15)

EPOL b

rarr

E+11

Eminus11

E+22

Eminus22

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

brarr

Es(t) + ELO1113868111386811138681113868

11138681113868111386811138682

brarr

Es(t) minus ELO1113868111386811138681113868

11138681113868111386811138682

brarr

Es(t) + jELO1113868111386811138681113868

11138681113868111386811138682

brarr

Es(t) + jELO1113868111386811138681113868

11138681113868111386811138682

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)


α(l) 10l

εoutεin

1113888 1113889 (17)



Ω 2πc

χ2ϑ2 (18)

Ψ 4πc

χ3ϑ2 +

πc

χϑ31113888 1113889 (19)



X

τ0τ1τ2τ3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

E2wararr + E2

u brarr

E2w ararr + E2

u brarr

EwararrE

brarr cos Φ

wararr minus Φ

ubrarr

111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113874 1113875

EwararrE

ubrarr sin Φ

wararr minus Φ

ubrarr

111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113874 1113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)



ubrarr and Φ

w ararr



ck

jcT2

Dλ2l

1113971

exp minus jn2πcT2

Dλ2l1113888 1113889 (21)







BER Q2φ

1113968 (26)


010001



R


0 20 40 60 80 100Data rate (Gbps)




Q(n) 12π

1113946infin

1exp minus n

221113872 1113873dn (27)




28

26

24

22

20

18OSN

R16

14

12

100 20 40 60 80 100

Data rate (Gbps)



01

0001

1E ndash 5

1E ndash 7

1E ndash 9

1E ndash 11

1E ndash 13

BER


400 500






70

65

60

55

50Q-fa

ctor

45

40

3515490 15495 15500 15505




001

1E ndash 4

1E ndash 6

1E ndash 8

1E ndash 10

1E ndash 12

BER

1E ndash 14

1E ndash 16

1E ndash 18






001

1E ndash 5

1E ndash 9

1E ndash 3BER

1E ndash 17

1E ndash 21

1E ndash 2504 05 06 07 08

PMD (pskm2)09



(a) (b)

Figure 12 Continued




5 Conclusion


(c) (d)







Data Availability




Acknowledgments


References










































3 Analytical Model



(1)





EPD 12

radic1 j

j 1⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦

Es(t)

Es(t)

⎡⎢⎢⎢⎣ ⎤⎥⎥⎥⎦ (3)

where j minus 1



111386811138681113868111386811138681113868111386811138682 (4)





pi(t) ζ2LO(t) + ζ2s (t) (7)






minus11(t) (10)


EQud

E+11

Eminus11

E+22

Eminus22

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Es(t) + ELO1113868111386811138681113868

11138681113868111386811138682

Es(t) minus ELO1113868111386811138681113868

11138681113868111386811138682

Es(t) + jELO1113868111386811138681113868

11138681113868111386811138682

Es(t) + jELO1113868111386811138681113868

11138681113868111386811138682

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(11)






(14)


rarr and brarr


EPOL ararr

E+11

Eminus11

E+22

Eminus22

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

ararr

Es(t) + ELO1113868111386811138681113868

11138681113868111386811138682

ararr

Es(t) minus ELO1113868111386811138681113868

11138681113868111386811138682

ararr

Es(t) + jELO1113868111386811138681113868

11138681113868111386811138682

ararr

Es(t) + jELO1113868111386811138681113868

11138681113868111386811138682

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(15)

EPOL b

rarr

E+11

Eminus11

E+22

Eminus22

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

brarr

Es(t) + ELO1113868111386811138681113868

11138681113868111386811138682

brarr

Es(t) minus ELO1113868111386811138681113868

11138681113868111386811138682

brarr

Es(t) + jELO1113868111386811138681113868

11138681113868111386811138682

brarr

Es(t) + jELO1113868111386811138681113868

11138681113868111386811138682

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)


α(l) 10l

εoutεin

1113888 1113889 (17)



Ω 2πc

χ2ϑ2 (18)

Ψ 4πc

χ3ϑ2 +

πc

χϑ31113888 1113889 (19)



X

τ0τ1τ2τ3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

E2wararr + E2

u brarr

E2w ararr + E2

u brarr

EwararrE

brarr cos Φ

wararr minus Φ

ubrarr

111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113874 1113875

EwararrE

ubrarr sin Φ

wararr minus Φ

ubrarr

111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113874 1113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)



ubrarr and Φ

w ararr



ck

jcT2

Dλ2l

1113971

exp minus jn2πcT2

Dλ2l1113888 1113889 (21)







BER Q2φ

1113968 (26)


010001



R


0 20 40 60 80 100Data rate (Gbps)




Q(n) 12π

1113946infin

1exp minus n

221113872 1113873dn (27)




28

26

24

22

20

18OSN

R16

14

12

100 20 40 60 80 100

Data rate (Gbps)



01

0001

1E ndash 5

1E ndash 7

1E ndash 9

1E ndash 11

1E ndash 13

BER


400 500






70

65

60

55

50Q-fa

ctor

45

40

3515490 15495 15500 15505




001

1E ndash 4

1E ndash 6

1E ndash 8

1E ndash 10

1E ndash 12

BER

1E ndash 14

1E ndash 16

1E ndash 18






001

1E ndash 5

1E ndash 9

1E ndash 3BER

1E ndash 17

1E ndash 21

1E ndash 2504 05 06 07 08

PMD (pskm2)09



(a) (b)

Figure 12 Continued




5 Conclusion


(c) (d)







Data Availability




Acknowledgments


References











































Ω 2πc

χ2ϑ2 (18)

Ψ 4πc

χ3ϑ2 +

πc

χϑ31113888 1113889 (19)



X

τ0τ1τ2τ3

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

E2wararr + E2

u brarr

E2w ararr + E2

u brarr

EwararrE

brarr cos Φ

wararr minus Φ

ubrarr

111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113874 1113875

EwararrE

ubrarr sin Φ

wararr minus Φ

ubrarr

111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113874 1113875

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(20)



ubrarr and Φ

w ararr



ck

jcT2

Dλ2l

1113971

exp minus jn2πcT2

Dλ2l1113888 1113889 (21)







BER Q2φ

1113968 (26)


010001



R


0 20 40 60 80 100Data rate (Gbps)




Q(n) 12π

1113946infin

1exp minus n

221113872 1113873dn (27)




28

26

24

22

20

18OSN

R16

14

12

100 20 40 60 80 100

Data rate (Gbps)



01

0001

1E ndash 5

1E ndash 7

1E ndash 9

1E ndash 11

1E ndash 13

BER


400 500






70

65

60

55

50Q-fa

ctor

45

40

3515490 15495 15500 15505




001

1E ndash 4

1E ndash 6

1E ndash 8

1E ndash 10

1E ndash 12

BER

1E ndash 14

1E ndash 16

1E ndash 18






001

1E ndash 5

1E ndash 9

1E ndash 3BER

1E ndash 17

1E ndash 21

1E ndash 2504 05 06 07 08

PMD (pskm2)09



(a) (b)

Figure 12 Continued




5 Conclusion


(c) (d)







Data Availability




Acknowledgments


References










































Q(n) 12π

1113946infin

1exp minus n

221113872 1113873dn (27)




28

26

24

22

20

18OSN

R16

14

12

100 20 40 60 80 100

Data rate (Gbps)



01

0001

1E ndash 5

1E ndash 7

1E ndash 9

1E ndash 11

1E ndash 13

BER


400 500






70

65

60

55

50Q-fa

ctor

45

40

3515490 15495 15500 15505




001

1E ndash 4

1E ndash 6

1E ndash 8

1E ndash 10

1E ndash 12

BER

1E ndash 14

1E ndash 16

1E ndash 18






001

1E ndash 5

1E ndash 9

1E ndash 3BER

1E ndash 17

1E ndash 21

1E ndash 2504 05 06 07 08

PMD (pskm2)09



(a) (b)

Figure 12 Continued




5 Conclusion


(c) (d)







Data Availability




Acknowledgments


References












































70

65

60

55

50Q-fa

ctor

45

40

3515490 15495 15500 15505




001

1E ndash 4

1E ndash 6

1E ndash 8

1E ndash 10

1E ndash 12

BER

1E ndash 14

1E ndash 16

1E ndash 18






001

1E ndash 5

1E ndash 9

1E ndash 3BER

1E ndash 17

1E ndash 21

1E ndash 2504 05 06 07 08

PMD (pskm2)09



(a) (b)

Figure 12 Continued




5 Conclusion


(c) (d)







Data Availability




Acknowledgments


References










































001

1E ndash 5

1E ndash 9

1E ndash 3BER

1E ndash 17

1E ndash 21

1E ndash 2504 05 06 07 08

PMD (pskm2)09



(a) (b)

Figure 12 Continued




5 Conclusion


(c) (d)







Data Availability




Acknowledgments


References












































5 Conclusion


(c) (d)







Data Availability




Acknowledgments


References











































Data Availability




Acknowledgments


References










































timedomainequalizationanddigitalback-propagation method

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