Range of Influence of Physical Impairments inWavelength-Division Multiplexed 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 GLOBECOM 2011Houston, Texas, USA

Wednesday, 7 December 2011

Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion

Outline

Introduction

2D Discrete-Time Model

Range of Influence

Conclusion

RoI of Physical Impairments in WDM Systems: Houbing Song and Maıte Brandt-Pearce, UVA 2/27

Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion

Motivation

Performance of long-haul WDM systems limited byPhysical Impairments

Fiber LossDispersionFiber Nonlinearity

Amplified Spontaneous Emission (ASE) Noise

Fiber Modeling: Prerequisite for development of physicalimpairment mitigation techniques

2D

Time: IntrachannelWavelength: Interchannel

Discrete-Time

Digital CommunicationsDigital Signal Processing (DSP)

RoI of Physical Impairments in WDM Systems: Houbing Song and Maıte Brandt-Pearce, UVA 3/27

Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion

Concept of Range of Influence (RoI)

1 ... 1 1 1 1... ... ...

1 ... 1 1 1 1... ... ...

1 ... 1 1 1 1... ... ...

1 ... 1 1 1 1... ... ...

1 ... 1 1 1 1... ... ...

Intrachannel RoI

Intrachannel RoI

Inte

rcha

nnel

RoI

Inte

rcha

nnel

R

oI

: : :

:::

: : :

: ::

Wav

elen

gth

Time

RoI of Physical Impairments in WDM Systems: Houbing Song and Maıte Brandt-Pearce, UVA 4/27

Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion

Significance

Signal Processing for Optical Communications

PredistortionEqualizationConstrained Coding......

RoI of Physical Impairments in WDM Systems: Houbing Song and Maıte Brandt-Pearce, UVA 5/27

Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion

Long-Haul WDM System

Laser 0

WDM

MUX

Dispersion

CompensatorAmplifier

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

n th span

Laser F-1

kFa 1−

ka0

( )ts F 1−

( )ts0

( )tr

Assumptions:

Chirped Gaussian pulses

Gaussian optical filters

No ASE noise

No predetection optical filtering

No photodetection

No postdetection electrical filtering

RoI of Physical Impairments in WDM Systems: Houbing Song and Maıte Brandt-Pearce, UVA 6/27

Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion

Input and Output

Input-output model: {afk} ⇒ {rf (tk )}

... 1 1 ...

... 1 ...

... 1 1 ...

... ... ...

Wav

elen

gth

Time

0

0

0

... ...

... ...

.........

[ ]fka

... ? ? ...

... ? ...

... ? ? ...

... ... ...

Wav

elen

gth

Time

?

?

?

... ...

... ...

.........

( )[ ] ?=kf trGiven input matrix Output matrix

RoI of Physical Impairments in WDM Systems: Houbing Song and Maıte Brandt-Pearce, UVA 7/27

Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion

Volterra Series Transfer Function (VSTF)

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

where

H1(ω, L) = exp(−α2L + i

β2

2ω2L),

H3(ω1, ω2, ω − ω1 + ω2, L)=iγ

4π2H1(ω, L)

∫ 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

RoI of Physical Impairments in WDM Systems: Houbing Song and Maıte Brandt-Pearce, UVA 8/27

Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion

Optical Equalization:

H−11 (ω, L) = exp(

α

2L− i

β2

2ω2L)

Input Signal:

S(ω) =√

2πF−1∑f =0

K−1∑k=0

afkAf Tf

exp

[−

(ω − f ∆)2T 2f

2− i(ω − f ∆)kTs + iΦfk

]

where

Af =√Pf , where Pf is launched peak power

T 2f =

T 20f

1+iCf, where T0f is pulse width, Cf is chirp parameter

Φfk : pulse phase

RoI of Physical Impairments in WDM Systems: Houbing Song and Maıte Brandt-Pearce, UVA 9/27

Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion

Model Development

Step 1: Extend VSTF to multispan multichannel multipulsecase to get R(ω)

Step 2: Simplify R(ω) from triple integral to simple integral

Step 3: Take inverse Fourier transform to get r(t)

Step 4: Sample r(t)

RoI of Physical Impairments in WDM Systems: Houbing Song and Maıte Brandt-Pearce, UVA 10/27

Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion

2D Model

r(t) =F−1∑f =0

K−1∑k=0

afkAf exp

[−(t − kTs)2

2T 2f

+ if ∆t + iΦfk

]

+ iNγF−1∑u=0

F−1∑v=0

F−1∑w=0

K−1∑l=0

K−1∑m=0

K−1∑n=0

aulavmawnAuAvAw TuTv Tw

× exp[i(Φul − Φvm + Φwn) + i∆Ts(ul − vm + wn)]

× E (t)

∫ L

0exp(−αz)J(t, z)dz

RoI of Physical Impairments in WDM Systems: Houbing Song and Maıte Brandt-Pearce, UVA 11/27

Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion

E(t) and J(t,z)

E (t) = exp

[−(u∆)2T 2

u

2− (v∆)2T 2

v

2− (w∆)2T 2

w

2

].

J(t, z) =

exp

{{u∆T 2

u +v∆T 2v +w∆T 2

w +A1+B1+i [t−(l−m+n)Ts ]}2

2(T 2u +T 2

v +T 2w +2A2)

}√[(

T 2u + T 2

v

)(T 2

v + T 2w

)−(T 2

v + iβ2z)2]

× exp(A0 + B0 + C )√T 2

u + T 2v + T 2

w + 2A2

.

RoI of Physical Impairments in WDM Systems: Houbing Song and Maıte Brandt-Pearce, UVA 12/27

Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion

Impairment Coefficients

ISI

ρISIk,k

= exp

[−(k − k)2T 2

s

2T 2f

].

Intrachannel

ρintraf ,k,l ,m,n = iγE intra

l ,m,n

∫ L

0exp(−αz)J intra

l ,m,ndz .

intra = SPM, IXPM, IFWM

Interchannel

ρinterf ,k,u,v ,w = iγE inter

u,v ,w

∫ L

0exp(−αz)J inter

u,v ,wdz .

inter = XPM,FWM

RoI of Physical Impairments in WDM Systems: Houbing Song and Maıte Brandt-Pearce, UVA 13/27

Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion

Intrachannel Example

Table: Index Triplets [lmn] for a Triple Pulse Case

Nonlinearity Time Location l −m + n-2 -1 0 1 2 3 4

SPM 000 111 222

IXPM 011 001 002022 221 112110 100 200220 122 211

IFWM 020 010 121 012 101 102 202021 210 201120 212

RoI of Physical Impairments in WDM Systems: Houbing Song and Maıte Brandt-Pearce, UVA 14/27

Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion

Intrachannel Coefficients Example

Table: Intrachannel Coefficients for a Triple Pulse Case

NL Time Location l −m + n-2 -1 0 1 2 3 4

SPM 0.0212 0.0212 0.0212

IXPM 0.0096 0.0096 0.00520.0052 0.0096 0.00960.0096 0.0096 0.00520.0052 0.0096 0.0096

IFWM 0.0003 0.0028 0.0028 0.0028 0.0028 0.0012 0.00030.0012 0.0028 0.00120.0012 0.0028

RoI of Physical Impairments in WDM Systems: Houbing Song and Maıte Brandt-Pearce, UVA 15/27

Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion

Interchannel Example

Table: Index Triplets [uvw] for a Triple Channel Case

Nonlinearity Frequency Location u − v + w-2 -1 0 1 2 3 4

XPM 011 001 002022 221 112110 100 200220 122 211

FWM 020 010 121 012 101 102 202021 210 201120 212

RoI of Physical Impairments in WDM Systems: Houbing Song and Maıte Brandt-Pearce, UVA 16/27

Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion

Interchannel Coefficients Example

Table: Interchannel Coefficients for a Triple Channel Case

NL Frequency Location u − v + w-2 -1 0 1 2 3 4

XPM 0.0207 0.0207 0.01910.0191 0.0207 0.02070.0207 0.0207 0.01910.0191 0.0207 0.0207

FWM 0.0187 0.0206 0.0206 0.0197 0.0206 0.0194 0.01870.0194 0.0197 0.01940.0194 0.0206

RoI of Physical Impairments in WDM Systems: Houbing Song and Maıte Brandt-Pearce, UVA 17/27

Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion

2D Discrete-time Model:

rf (kTs) = afkAf e(if ∆kTs +iΦfk ) +

K−1∑k=0;k 6=k

af kAf e(if ∆kTs +iΦf k )ρISI

k,k

+ Na3fkA

3f T

3f e

(if ∆kTs +iΦfk )ρSPM

+ NK−1∑

l ,m,n=0

aflafmafnA3f T

3f e

(if ∆kTs )

× e [i(Φfl−Φfm+Φfn)][ρIXPM

f ,k,l ,m,n + ρIFWMf ,k,l ,m,n

]+ N

F−1∑u,v ,w=0

aukavkawkAuAvAw TuTv Twe(if ∆kTs )

× e [i(Φuk−Φvk +Φwk )][ρXPM

f ,k,u,v ,w + ρFWMf ,k,u,v ,w

]+

F−1∑f =0;f 6=f

K−1∑k=0

af kAf e(if ∆kTs +iΦf k )e

[− (k−k)2T 2

s2T 2

f

]

RoI of Physical Impairments in WDM Systems: Houbing Song and Maıte Brandt-Pearce, UVA 18/27

Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion

Advantages

Mapping: binary input matrix ⇒ sampled output matrix

Greatly reduced computational complexity

Arbitrarily isolate any individual physical impairment

Strong analytic capacity

Effects of system parameters: F ,∆,T , L,NEffects of pulse parameters: K ,Af ,T0f ,Cf ,Φfk

Effects of fiber parameters: α, β2, γ

RoI of Physical Impairments in WDM Systems: Houbing Song and Maıte Brandt-Pearce, UVA 19/27

Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion

Range of Influence (RoI) Definitions

Definition: the number of adjacent symbols/channels causing asignificant effect (smaller than some tolerance)

D ISI (k) =∑k+k

k=k+1

∣∣∣ρISIk,k

∣∣∣D IXPM(IFWM)(k) =

∑k+kl ,m,n=k+1

∣∣∣ρIXPM(IFWM)f ,k,l ,m,n

∣∣∣DXPM(FWM)(f ) =

∑f +fu,v ,w=f +1

∣∣∣ρXPM(FWM)f ,k,u,v ,w

∣∣∣

RoI of Physical Impairments in WDM Systems: Houbing Song and Maıte Brandt-Pearce, UVA 20/27

Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion

Range of Influence (RoI) of ISI

0 1 2 3 4 5 6 7 8 9 1010

−3

10−2

10−1

100

Number of Symbols

Cum

ulat

ive

ISI D

egra

datio

n

Rs=40 Gs/s

Rs=100 Gs/s

RoI(40)=RoI(100)=1

Figure: Computation of cumulative degradation due to ISI for SMF fiberoperating at 1.55 µm for various symbol rates

RoI of Physical Impairments in WDM Systems: Houbing Song and Maıte Brandt-Pearce, UVA 21/27

Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion

Range of Influence (RoI) of IXPM

0 20 40 60 80 100 120 140 160 180 20010

−3

10−2

10−1

100

Number of Symbols

Cum

ulat

ive

IXP

M D

egra

datio

n (m

W−1

ps−3

)

Rs=40 Gs/s

Rs=100 Gs/s

RoI(100)=185

RoI(40)=19

Figure: Computation of cumulative degradation due to IXPM for SMFfiber operating at 1.55 µm for various symbol rates

RoI of Physical Impairments in WDM Systems: Houbing Song and Maıte Brandt-Pearce, UVA 22/27

Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion

Range of Influence (RoI) of IFWM

0 50 100 150 200 250 300 350 40010

−3

10−2

10−1

100

101

Number of Symbols

Cum

ulat

ive

IFW

M D

egra

datio

n (m

W− 1p

s− 3)

Rs=40 Gs/s

Rs=100 Gs/sRoI(40)=65

RoI(100)=300

Figure: Computation of cumulative degradation due to IFWM for SMFfiber operating at 1.55 µm for various symbol rates

RoI of Physical Impairments in WDM Systems: Houbing Song and Maıte Brandt-Pearce, UVA 23/27

Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion

Range of Influence (RoI) of XPM

0 2 4 6 8 10 12 14 16 18 2010

−4

10−3

10−2

Number of Channels

Cum

ulat

ive

XP

M D

egra

datio

n (m

W−1

ps−3

)

∆=50 GHz

∆=100 GHz

RoI(50)=3

RoI(100)=2

Figure: Computation of cumulative degradation due to XPM for SMFfiber operating at 1.55 µm for various channel spacings

RoI of Physical Impairments in WDM Systems: Houbing Song and Maıte Brandt-Pearce, UVA 24/27

Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion

Range of Influence (RoI) of FWM

0 2 4 6 8 10 12 14 16 18 2010

−5

10−4

10−3

10−2

Number of Channels

Cum

ulat

ive

FW

M D

egra

datio

n (m

W−1

ps−3

)

∆=50 GHz

∆=100 GHz

RoI(50)=4

RoI(100)=3

Figure: Computation of cumulative degradation due to FWM for SMFfiber operating at 1.55 µm for various channel spacings

RoI of Physical Impairments in WDM Systems: Houbing Song and Maıte Brandt-Pearce, UVA 25/27

Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion

Conclusion

Development of a 2D discrete-time model of physicalimpairments in long-haul WDM systems

Determination of Range of Influence (RoI) of physicalimpairments

Potential foundation of signal processing for opticalcommunications

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

RoI of Physical Impairments in WDM Systems: Houbing Song and Maıte Brandt-Pearce, UVA 26/27

Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion

Thank You

RoI of Physical Impairments in WDM Systems: Houbing Song and Maıte Brandt-Pearce, UVA 27/27