a discrete-time polynomial model of single channel long-haul fiber-optic communication systems

A Discrete-Time Polynomial Model of SingleChannel Long-Haul Fiber-Optic Communication

Systems

Houbing Song and Maıte Brandt-Pearce

Charles L. Brown Department of Electrical and Computer EngineeringUniversity of Virginia, USA

song@virginia.edu, mb-p@virginia.edu

IEEE ICC2011Kyoto, JapanJune 7, 2011

Outline Introduction Model Development Model Validation Model Application: Constrained Coding Conclusion

Outline

1 IntroductionMotivationNonlinear Schrodinger Equation

2 Model DevelopmentStep 1: Extension of VSTF to Multispan Multipulse CaseStep 2: Simplification of Triple Integral to Simple IntegralStep 3: Conversion to Time DomainStep 4: Extension to include Photodetector

3 Model Validation

4 Model Application: Constrained CodingCoding SchemePerformance Evaluation

5 Conclusion

A Discrete-Time Polynomial Model of Single Channel Long-Haul Fiber-Optic Communication Systems:Houbing Song and Maıte Brandt-Pearce, University of Virginia 2/26

Outline

3 Model Validation

5 Conclusion

Motivation

Dense Wavelength Division Multiplexing (DWDM)

Physical impairments in DWDM systems

DispersionFiber nonlinearitiesNoise

Signal processing for optical communications requires a model2D: time and wavelength

Intrachannel impairmentsInterchannel impairments

Discrete-time: digital communications, digital signal processing(DSP)

Motivation

Previous Work: single span caseTriple integral

Volterra series transfer function (VSTF) method(Peddanarappagari and Brandt-Pearce, 1997)perturbation method (Ablowitz, 1998)

Single integral (Mecozzi, 2000; Essiambre, 2002) but ignoresfiber losschirpphotodetection

Our Work: a single channel multipulse multispan modelsimplifies the triple integral to a simple integral for Gaussianpulsescomputational advantage over the split-step Fourier (SSF)method with comparable accuracyeasily extendable to multichannel case and 2-polarization-modecasetakes into account the parameters ignored in the literature

Motivation

Nonlinear Schrodinger Equation

∂z= −α

2A− iβ2

∂t2+ iγ|A|2A

A = A(t, z): slowly varying complex envelope

t: propagation time

z : propagation distance

α: attenuation constant

β2: group-velocity dispersion (GVD) parameter

γ: nonlinear parameter

∂z= −α

2A− iβ2

∂t2+ iγ|A|2A

t: propagation time

∂z= −α

2A− iβ2

∂t2+ iγ|A|2A

t: propagation time

∂z= −α

2A− iβ2

∂t2+ iγ|A|2A

t: propagation time

∂z= −α

2A− iβ2

∂t2+ iγ|A|2A

t: propagation time

∂z= −α

2A− iβ2

∂t2+ iγ|A|2A

t: propagation time

Fiber Nonlinearities

Multipulse: A =∑K−1

k=0 Ak

K−1∑k=0

(∂Ak

∂z+α

2Ak + i

∂2Ak

∂t2) = iγ

K−1∑l ,m,n=0

AlA∗mAn

Time-matching location: (l −m + n)T

l = m = n: self phase modulation (SPM) ⇒ spectralbroadeningl = m 6= n or l 6= m = n: intrachannel cross phase modulation(IXPM) ⇒ timing jitterl 6= m 6= n or l = n 6= m: intrachannel four wave mixing(IFWM) ⇒ amplitude jitter and ghost pulse

k=0 Ak

K−1∑k=0

(∂Ak

∂z+α

2Ak + i

∂2Ak

∂t2) = iγ

K−1∑l ,m,n=0

AlA∗mAn

k=0 Ak

K−1∑k=0

(∂Ak

∂z+α

2Ak + i

∂2Ak

∂t2) = iγ

K−1∑l ,m,n=0

AlA∗mAn

k=0 Ak

K−1∑k=0

(∂Ak

∂z+α

2Ak + i

∂2Ak

∂t2) = iγ

K−1∑l ,m,n=0

AlA∗mAn

3rd-order VSTF model

A(ω, L)=H1(ω, L)A(ω, 0) +

∫ +∞

−∞

∫ +∞

−∞H3(ω1, ω2, ω−ω1+ω2, L)

A(ω1, 0)A∗(ω2, 0)A(ω − ω1 + ω2, 0)dω1dω2

H1(ω, L) = exp(−α2L + i

2ω2L),

H3(ω1, ω2, ω − ω1 + ω2, L)=iγ

4π2H1(ω, L)

0exp[−αz +

iβ2z(ω1 − ω)(ω1 − ω2)]dz

A(ω, z) : Fourier transform of A(t, z)H1(ω, L): linear transfer functionH3(ω1, ω2, ω − ω1 + ω2, L): nonlinear transfer function

A(ω, L)=H1(ω, L)A(ω, 0) +

∫ +∞

−∞

∫ +∞

−∞H3(ω1, ω2, ω−ω1+ω2, L)

A(ω1, 0)A∗(ω2, 0)A(ω − ω1 + ω2, 0)dω1dω2

H1(ω, L) = exp(−α2L + i

2ω2L),

H3(ω1, ω2, ω − ω1 + ω2, L)=iγ

4π2H1(ω, L)

0exp[−αz +

iβ2z(ω1 − ω)(ω1 − ω2)]dz

A(ω, L)=H1(ω, L)A(ω, 0) +

∫ +∞

−∞

∫ +∞

−∞H3(ω1, ω2, ω−ω1+ω2, L)

A(ω1, 0)A∗(ω2, 0)A(ω − ω1 + ω2, 0)dω1dω2

H1(ω, L) = exp(−α2L + i

2ω2L),

H3(ω1, ω2, ω − ω1 + ω2, L)=iγ

4π2H1(ω, L)

0exp[−αz +

iβ2z(ω1 − ω)(ω1 − ω2)]dz

Outline

3 Model Validation

5 Conclusion

Model Development

Laser Diode

Modulator Dispersion Compensator Amplifier Photo

DetectorDecision Device

Threshold

Sample at time

{ }∧

kb( )ts ( )tr ( )ty ( )qty

qTtq =

( )( )0,tA n ( )( )LtA n , ( ) ( )0,1 tA n+

n th span

Encoder

Figure: Schematic of a typical fiber-optic communication system

Assumptions:

Noiseless

No predetection optical filering

Input-output model: {ak} ⇒ y(tq)

Model Development

Laser Diode

Modulator Dispersion Compensator Amplifier Photo

DetectorDecision Device

Threshold

Sample at time

{ }∧

kb( )ts ( )tr ( )ty ( )qty

qTtq =

( )( )0,tA n ( )( )LtA n , ( ) ( )0,1 tA n+

n th span

Encoder

Figure: Schematic of a typical fiber-optic communication system

Assumptions:

Noiseless

No predetection optical filering

Input-output model: {ak} ⇒ y(tq)

Step 1: Extension of VSTF to Multispan Multipulse Case

Dispersion Compensation + Amplification:

H−11 (ω, L) = exp(

2L− i

2ω2L)

Multipulse:

S(ω) =K−1∑k=0

√2πT0 exp

[−ω

2− iωkT + iΦk

P0: launched peak power

T 20 =

1+iC , where T0 is pulse width and C is chirp parameter

T = 1/Rs , where Rs is symbol rate

H−11 (ω, L) = exp(

2L− i

2ω2L)

Multipulse:

S(ω) =K−1∑k=0

√2πT0 exp

[−ω

2− iωkT + iΦk

T 20 =

H−11 (ω, L) = exp(

2L− i

2ω2L)

Multipulse:

S(ω) =K−1∑k=0

√2πT0 exp

[−ω

2− iωkT + iΦk

T 20 =

Step 2: Triple Integral ⇒ Simple Integral

R(ω) =√

2πK−1∑k=0

0 T0 exp

(−ω

2− iωkT + iΦk

+ iNγ√

2πT 20 exp

(−3ω2T 2

)K−1∑l=0

K−1∑m=0

K−1∑n=0

ala∗man

0 exp[−iω(l −m + n)T + i(Φl − Φm + Φn)]

{−αz +

[2ωT 20 +i(l−m)T ][2ωT 2

0 +i(n−m)T ]3T 2

0 +iβ2z

3T 20 +

− i2β2z

[− (l − n)2T 2

3T 20 + β2

2z2/T 2

0 − i2β2z

Step 3: Conversion to Time Domain

r(t) =K−1∑k=0

[−(t − kT )2

+ iΦk

]+ iNγ

K−1∑l=0

K−1∑m=0

K−1∑n=0

ala∗manP

0 exp[i(Φl − Φm + Φn)]

)∫ L

0exp[−αz−K2(z , l , n)]

+(l−m)T}{

+(n−m)T}

T 20 +i3β2z

}K1(z)

wheretNL = t − (l −m + n)T

Step 3 (Con.)

Introduce:

ISI coefficient (effect of pulse k on pulse q):

ρISIq,k = exp[−(q − k)2T 2/2T 2

]SPM, IXPM, IFWM coefficient:

ρSPM(IXPM,IFWM)l ,m,n = iγ

exp [−αz − K2(z , l , n)]

{−3(l −m)(n −m)T 2

T 20 + i3β2z

K1(z) =

√1 + i2β2z/T 2

0 + 3β22z

2/T 40

K2(z , l , n) =(l − n)2T 2

T 20 + i2β2z + 3β2

2z2/T 2

0A Discrete-Time Polynomial Model of Single Channel Long-Haul Fiber-Optic Communication Systems:Houbing Song and Maıte Brandt-Pearce, University of Virginia 14/26

Step 3 (Con.)

Introduce:

ISI coefficient (effect of pulse k on pulse q):

ρISIq,k = exp[−(q − k)2T 2/2T 2

]SPM, IXPM, IFWM coefficient:

ρSPM(IXPM,IFWM)l ,m,n = iγ

exp [−αz − K2(z , l , n)]

{−3(l −m)(n −m)T 2

T 20 + i3β2z

K1(z) =

√1 + i2β2z/T 2

0 + 3β22z

2/T 40

K2(z , l , n) =(l − n)2T 2

T 20 + i2β2z + 3β2

2z2/T 2

0A Discrete-Time Polynomial Model of Single Channel Long-Haul Fiber-Optic Communication Systems:Houbing Song and Maıte Brandt-Pearce, University of Virginia 14/26

Step 3 (Con.)

Discrete-time polynomial model:

r(tq) = aqP12

0 exp(iΦq) + P12

K−1∑k=0;k 6=q

akρISIq,k exp(iΦk)

+ N|aq|2aqP32

0 ρSPM exp(iΦq)

+ NP32

∑l=m 6=n;l 6=m=n

ala∗manρ

IXPMl ,m,n exp[i(Φl − Φm + Φn)]

+ NP32

∑l 6=m 6=n;l=n 6=m

ala∗manρ

IFWMl ,m,n exp[i(Φl − Φm + Φn)]

Advantages:

Reduce computational complexity

Isolate any individual physical impairment

Analyze impact of system, fiber, and pulse parametersA Discrete-Time Polynomial Model of Single Channel Long-Haul Fiber-Optic Communication Systems:Houbing Song and Maıte Brandt-Pearce, University of Virginia 15/26

Step 3 (Con.)

r(tq) = aqP12

0 exp(iΦq) + P12

K−1∑k=0;k 6=q

+ N|aq|2aqP32

0 ρSPM exp(iΦq)

+ NP32

∑l=m 6=n;l 6=m=n

ala∗manρ

+ NP32

∑l 6=m 6=n;l=n 6=m

ala∗manρ

Advantages:

Step 3 (Con.)

r(tq) = aqP12

0 exp(iΦq) + P12

K−1∑k=0;k 6=q

+ N|aq|2aqP32

0 ρSPM exp(iΦq)

+ NP32

∑l=m 6=n;l 6=m=n

ala∗manρ

+ NP32

∑l 6=m 6=n;l=n 6=m

ala∗manρ

Advantages:

Step 4: Extension to include Photodetector

y(tq) = |aq|2P0 + P0

∣∣∣∣∣∣K−1∑

k=0;k 6=q

∣∣∣∣∣∣2

+ 2P0Re

K−1∑k=0;k 6=q

ak(ρISIq,k)∗ exp[i(Φq − Φk)]

+ 2NP2

0Re{aq(ρSPM)∗

}+ N2|aq|6P3

∣∣∣ρSPM ∣∣∣2+ 2NP2

0Re{aq∑

ala∗man(ρIXPMl ,m,n )∗K3(l ,m, n)}

+ 2NP20Re{aq

∑ala∗man(ρIFWM

l ,m,n )∗K3(l ,m, n)}

+ N2P60

(∣∣∣∑ ala∗manρ

IXPMl ,m,n

∣∣∣2 +∣∣∣∑ ala

∗manρ

IFWMl ,m,n

∣∣∣2)A Discrete-Time Polynomial Model of Single Channel Long-Haul Fiber-Optic Communication Systems:Houbing Song and Maıte Brandt-Pearce, University of Virginia 16/26

K3(l ,m, n) = exp

{− [q − (l −m + n)]2T 2

6(T 20 )∗

}exp{i [Φq − (Φl − Φm + Φn)]}

Mapping: binary input vector ⇒ sampled photodetectoroutput vector

intrachannel interference (ICI): ICI{bk},q

K3(l ,m, n) = exp

{− [q − (l −m + n)]2T 2

6(T 20 )∗

}exp{i [Φq − (Φl − Φm + Φn)]}

Mapping: binary input vector ⇒ sampled photodetectoroutput vector

intrachannel interference (ICI): ICI{bk},q

Outline

3 Model Validation

5 Conclusion

Model Validation (Model vs SSF simulation)

Normalized squared deviation (NSD):

NSD(N) =

∫ KT0 |rModel(t)− rSSF (t)|2dt∫ KT

0 |rSSF (t)|2dt

2 4 6 8 10 12 14 16

x 10−4

Number of Spans

P0=1 mW; OOKP0=3 mW; OOKP0=10 mW; OOKP0=1 mW; DBPSKP0=3 mW; DBPSKP0=10 mW; DBPSK

Outline

3 Model Validation

5 Conclusion

Coding Scheme

Using a constrained code to avoid bit patterns that will most likelybe detected incorrectly

Metric (Bit pattern): ICI{bk} =∑K−1

q=0 |ICI{bk},q|Scheme:

Rank the bit patterns in order of increasing ICIMap every K-bit pattern with bad ICI metric to one of K+1-bitpatterns with good ICI metric

Coding Scheme

Evaluation Method

For a given proportion of bit patterns constrained,

Calculate Q factor before coding: Qbefore = µ1−µ0σ1+σ0

whereµi = E [y(tq)|aq = i ], i = 0, 1

σ2i = E{[y(tq)− µi ]2|aq = i}, i = 0, 1

Constrained coding

Calculate Q factor after coding: Qafter

Q factor improvement: Qimprovement(dB) = 20log10QafterQbefore

Evaluation Method

σ2i = E{[y(tq)− µi ]2|aq = i}, i = 0, 1

Constrained coding

Evaluation Method

σ2i = E{[y(tq)− µi ]2|aq = i}, i = 0, 1

Constrained coding

Evaluation Method

σ2i = E{[y(tq)− µi ]2|aq = i}, i = 0, 1

Constrained coding

Evaluation Result

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.56

Proportion of Sequences Constrained

OOK; 1 mW

OOK; 3 mW

OOK; 10 mW

DBPSK; 1 mW

DBPSK; 3 mW

DBPSK; 10 mW

OOK; 1 mW

OOK; 3 mW

OOK; 10 mW

DBPSK; 1 mW

DBPSK; 3 mW

DBPSK; 10 mW

Encoded

Uncoded

Figure: Q factor improvementA Discrete-Time Polynomial Model of Single Channel Long-Haul Fiber-Optic Communication Systems:Houbing Song and Maıte Brandt-Pearce, University of Virginia 23/26

Outline

3 Model Validation

5 Conclusion

Conclusion

Development, validation and application of a discrete-timetime model of single channel multipulse multispan systemswith periodic amplification and dispersion management2D model

”Range of Influence of Physical Impairments in WDMSystems”, GLOBECOM 2011 (Under Review)”A Two-Dimensional Discrete-Time Model of PhysicalImpairments in WDM Systems”, The IEEE/OSA Journal ofLightwave Technology (JLT) (Submitted)

Extending to the general case, including ASE noise andpostdetection electrical filteringPotential Applications

multichannel signal processing for intersymbol and interchannelinterference mitigationmultiuser coding, multichannel detection and path-diversity forall-optical networksconstrained coding for WDM systems

Conclusion

Thank You

a discrete-time polynomial model of single channel long-haul fiber-optic communication systems

time domainstep

digital communications

optical communications

mate brandtpearcecharles

extension of vstf

simple integralstep

multispan multipulse

brown department of

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