IEEE GLOBECOM 2011, Houston, TX, December 7, 2011

1. Range of Influence of Physical Impairments in
Wavelength-Division Multiplexed Systems
Houbing Song and Ma¨ e Brandt-Pearce
ıt´
Charles L. Brown Department of Electrical and Computer Engineering
University of Virginia, USA
song@virginia.edu, mb-p@virginia.edu
IEEE GLOBECOM 2011
Houston, Texas, USA
Wednesday, 7 December 2011

2. 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 Ma¨ e Brandt-Pearce, UVA
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3. Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion
Motivation
Performance of long-haul WDM systems limited by
Physical Impairments
Fiber Loss
Dispersion
Fiber Nonlinearity
Amplified Spontaneous Emission (ASE) Noise
Fiber Modeling: Prerequisite for development of physical
impairment mitigation techniques
2D
Time: Intrachannel
Wavelength: Interchannel
Discrete-Time
Digital Communications
Digital Signal Processing (DSP)
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4. Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion
Concept of Range of Influence (RoI)
Time
Intrachannel Intrachannel
RoI RoI
1 ... 1 ... 1 ... 1 ... 1
: : :
1 ... 1 ... 1 ... 1 ... 1
Interchannel Interchannel
RoI
: : :
1 ... 1 ... 1 ... 1 ... 1
RoI
: : :
Wavelength
1 ... 1 ... 1 ... 1 ... 1
: : :
1 ... 1 ... 1 ... 1 ... 1
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5. Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion
Significance
Signal Processing for Optical Communications
Predistortion
Equalization
Constrained Coding
......
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6. Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion
Long-Haul WDM System
a0k n th span
Laser 0
s 0 (t )
WDM s (t ) A (n ) (t , 0 ) A (n ) (t , L ) Dispersion A (n + 1 ) (t , 0 ) r (t )
Amplifier
MUX Compensator
s F − 1 (t )
Laser F-1
a F −1 k
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 Ma¨ e Brandt-Pearce, UVA
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7. Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion
Input and Output
Input-output model: {afk } ⇒ {rf (tk )}
Time Time
... ... ... ... ... ... ... ... ... ...
... 1 0 1 ... ... ? ? ? ...
Wavelength
Wavelength
... 0 1 ... ... ? ? ...
... 1 0 1 ... ... ? ? ? ...
... ... ... ... ... ... ... ... ... ...
Given input matrix [a ]
fk Output matrix [r (t )] = ?
f k
RoI of Physical Impairments in WDM Systems: Houbing Song and Ma¨ e Brandt-Pearce, UVA
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8. 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)dω1 dω2
where 1st order Volterra Kernel
α β2
H1 (ω, L) = exp(− L + i ω 2 L),
2 2
L
iγ
H3 (ω1 , ω2 , ω − ω1 + ω2 , L)= H1 (ω, L) exp[−αz +
4π 2 0
iβ2 z(ω1 − ω)(ω1 − ω2 )]dz
3rd order Volterra Kernel
A(ω, z) : Fourier transform of A(t, z)
H1 (ω, L): linear transfer function
H3 (ω1 , ω2 , ω − ω1 + ω2 , L): nonlinear transfer function
RoI of Physical Impairments in WDM Systems: Houbing Song and Ma¨ e Brandt-Pearce, UVA
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9. Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion
Optical Equalization: Amplifier Dispersion compensation
−1 α β2
H1 (ω, L) = exp( L − i ω 2 L)
2 2
Input Signal:
Multipulse
F −1 K −1
√
S(ω) = 2π afk Af Tf
f =0 k=0
Multichannel
(ω − f ∆)2 Tf2
exp − − i(ω − f ∆)kTs + iΦfk
2
where
Chirped Gaussian Pulse
√
Af = Pf , where Pf is launched peak power
2
T0f
Tf2 = 1+iCf , where T0f is pulse width, Cf is chirp parameter
Φfk : pulse phase
RoI of Physical Impairments in WDM Systems: Houbing Song and Ma¨ e Brandt-Pearce, UVA
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10. Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion
Model Development
Step 1: Extend VSTF to multispan multichannel multipulse
case 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 Ma¨ e Brandt-Pearce, UVA
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11. Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion
2D Model
F −1 K −1
(t − kTs )2
r (t) = afk Af exp − + if ∆t + iΦfk
f =0 k=0 2Tf2
F −1 F −1 F −1 K −1 K −1 K −1
+ iNγ aul avm awn Au Av Aw Tu Tv Tw
u=0 v =0 w =0 l=0 m=0 n=0
× exp[i(Φul − Φvm + Φwn ) + i∆Ts (ul − vm + wn)]
L
× E (t) exp(−αz)J(t, z)dz
0
Simple Integral
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12. Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion
E(t) and J(t,z)
(u∆)2 Tu
2 (v ∆)2 Tv
2 (w ∆)2 Tw
2
E (t) = exp − − − .
2 2 2
2 2 2
{u∆Tu +v ∆Tv +w ∆Tw +A1 +B1 +i[t−(l−m+n)Ts ]}2
exp
2(Tu +Tv +Tw +2A2 )
2 2 2
J(t, z) =
2
2 2
Tu + Tv 2 2 2
Tv + Tw − Tv + iβ2 z
exp(A0 + B0 + C )
× .
2 2 2
Tu + Tv + Tw + 2A2
RoI of Physical Impairments in WDM Systems: Houbing Song and Ma¨ e Brandt-Pearce, UVA
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13. Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion
Impairment Coefficients
ISI
ˆ
(k − k)2 Ts2
ρISIˆ = exp −
k,k
.
2Tf2
Intrachannel
L
ρintra intra
f ,k,l,m,n = iγEl,m,n
intra
exp(−αz)Jl,m,n dz.
0
intra = SPM, IXPM, IFWM
Interchannel
L
ρinter ,w
f ,k,u,v = inter
iγEu,v ,w inter
exp(−αz)Ju,v ,w dz.
0
inter = XPM, FWM
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14. 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 011 001 002
022 221 112
110 100 200
220 122 211
IFWM 020 010 121 012 101 102 202
021 210 201
120 212
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15. 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 0.0052
0.0052 0.0096 0.0096
0.0096 0.0096 0.0052
0.0052 0.0096 0.0096
IFWM 0.0003 0.0028 0.0028 0.0028 0.0028 0.0012 0.0003
0.0012 0.0028 0.0012
0.0012 0.0028
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16. 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 011 001 002
022 221 112
110 100 200
220 122 211
FWM 020 010 121 012 101 102 202
021 210 201
120 212
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17. 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 0.0191
0.0191 0.0207 0.0207
0.0207 0.0207 0.0191
0.0191 0.0207 0.0207
FWM 0.0187 0.0206 0.0206 0.0197 0.0206 0.0194 0.0187
0.0194 0.0197 0.0194
0.0194 0.0206
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18. Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion
2D Discrete-time Model:
K −1
rf (kTs ) = afk Af e (if ∆kTs +iΦfk ) + af k Af e (if ∆kTs +iΦf k ) ρISIˆ
ˆ
ˆ
k,k
ˆ ˆ
k=0;k=k
+ Nafk A3 Tf3 e (if ∆kTs +iΦfk ) ρSPM
3
f
K −1
+ N afl afm afn A3 Tf3 e (if ∆kTs )
f
l,m,n=0
× e [i(Φfl −Φfm +Φfn )] ρIXPM + ρIFWM
f ,k,l,m,n f ,k,l,m,n
F −1
+ N auk avk awk Au Av Aw Tu Tv Tw e (if ∆kTs )
u,v ,w =0
[i(Φuk −Φvk +Φwk )]
× e ρXPM ,w + ρFWM ,w
f ,k,u,v f ,k,u,v
F −1 K −1 (k−k)2 Ts
ˆ 2
−
(if ∆kTs +iΦf k ) 2T 2
+ af k Af e
ˆˆ ˆ
ˆˆ
e ˆ
f
ˆ ˆ ˆ
f =0;f =f k=0
RoI of Physical Impairments in WDM Systems: Houbing Song and Ma¨ e Brandt-Pearce, UVA
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19. 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, N
Effects 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 Ma¨ e Brandt-Pearce, UVA
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20. Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion
Range of Influence (RoI) Definitions
Definition: the number of adjacent symbols/channels causing a
significant effect (smaller than some tolerance)
¯
k+k
¯
D ISI (k) = ρISIˆ
ˆ
k=k+1 k,k
k+k¯ IXPM(IFWM)
¯
D IXPM(IFWM) (k) = ρf ,k,l,m,n
l,m,n=k+1
¯
f +f XPM(FWM)
¯
D XPM(FWM) (f ) = ρf ,k,u,v ,w
u,v ,w =f +1
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21. Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion
Range of Influence (RoI) of ISI
0
10
Rs=40 Gs/s
Rs=100 Gs/s
Cumulative ISI Degradation
−1
10
RoI(40)=RoI(100)=1
−2
10
−3
10
0 1 2 3 4 5 6 7 8 9 10
Number of Symbols
Figure: Computation of cumulative degradation due to ISI for SMF fiber
operating at 1.55 µm for various symbol rates
RoI of Physical Impairments in WDM Systems: Houbing Song and Ma¨ e Brandt-Pearce, UVA
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22. Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion
Range of Influence (RoI) of IXPM
0
10
Cumulative IXPM Degradation (mW−1ps−3)
Rs=40 Gs/s
−1
10 Rs=100 Gs/s
RoI(100)=185
−2
10
RoI(40)=19
−3
10
0 20 40 60 80 100 120 140 160 180 200
Number of Symbols
Figure: Computation of cumulative degradation due to IXPM for SMF
fiber operating at 1.55 µm for various symbol rates
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23. Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion
Range of Influence (RoI) of IFWM
1
10
Cumulative IFWM Degradation (mW−1ps−3)
0
10
RoI(100)=300
−1
Rs=40 Gs/s
10
Rs=100 Gs/s
RoI(40)=65
−2
10
−3
10
0 50 100 150 200 250 300 350 400
Number of Symbols
Figure: Computation of cumulative degradation due to IFWM for SMF
fiber operating at 1.55 µm for various symbol rates
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24. Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion
Range of Influence (RoI) of XPM
−2
10
Cumulative XPM Degradation (mW−1ps−3)
∆=50 GHz
∆=100 GHz
RoI(50)=3
−3
10
RoI(100)=2
−4
10
0 2 4 6 8 10 12 14 16 18 20
Number of Channels
Figure: Computation of cumulative degradation due to XPM for SMF
fiber operating at 1.55 µm for various channel spacings
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25. Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion
Range of Influence (RoI) of FWM
−2
10
Cumulative FWM Degradation (mW−1ps−3)
∆=50 GHz
∆=100 GHz
−3
10
RoI(50)=4
−4
10
−5
10
RoI(100)=3
0 2 4 6 8 10 12 14 16 18 20
Number of Channels
Figure: Computation of cumulative degradation due to FWM for SMF
fiber operating at 1.55 µm for various channel spacings
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26. Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion
Conclusion
Development of a 2D discrete-time model of physical
impairments in long-haul WDM systems
Determination of Range of Influence (RoI) of physical
impairments
Potential foundation of signal processing for optical
communications
multichannel signal processing for intersymbol and interchannel
interference mitigation
multiuser coding, multichannel detection and path-diversity for
all-optical networks
constrained coding for WDM systems
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27. Outline Introduction 2D Discrete-Time Model Range of Influence Conclusion
Thank You
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