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Mathematics of the Optical FiberCommunication:
Nonlinear Fourier Transforms and Information Theory
Mansoor I. Yousefi
Communications and Electronics DepartmentTélécom ParisTech, LTCI, Université Paris-Saclay, France
Horizon Maths 2017 - Mathématiques et réseaux
Télécom ParisTech, Paris, FranceNovember 30, 2017
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Thanks to:
Frank Kschischang (U. of Toronto)
Gerhard Kramer (TU Munich)
Hartmut Haffermann & Jan-Willem Goossens (Huawei Paris)
Xianhe Yangzhang (University College London)
also the orgonizers!
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Table of Contents
1 Introduction to Optical Networks
2 Nonlinear Fourier Transforms
3 Information Theory of Nonlinear Channels
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Wired Communication Networks
TX1
RX1
RX2
TX2 MUX
TX
Amplifier
span 1 span 2 span N
Fiber Amplifier MUX Fiber Amplifier MUX Fiber Amplifier
RX
(1) Multiple users with add-drop multiplexers (ADMs); (2) interference unknownto the user-of-interest (UOI); (3) network topology unknown
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Optical Fiber
‚ Adtantages: Low-loss („ 0.2 dB/km), huge bandwidth („5-10 THzbandwidth), all-optical prosseing (laser sources, amplifiers, detectors)
‚ Challenges: Refractive index is a function of frequency and intensity
npω, |q|2q “ n0 ` n1pω ´ ω0q ` n2pω ´ ω0q2 ` ¨ ¨ ¨loooooooooooooooooooooooomoooooooooooooooooooooooon
dispersion
` γ0|q|2loomoon
Kerr nonlinearity
` ¨ ¨ ¨
intuition
1 Dispersion: n depends on frequency
2 Kerr nonlinearity: the intensity of thesignal modifies the refractive index!
3 High reliability: Pe “ 10´15
4 High speed: 400 Gb/s
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Stochastic Nonlinear Schrödinger Equation
optical fiberqpt, 0qinput waveform
qpt,Lqoutput waveform
LPulse propagation in optical fibers can be modeled by thestochastic nonlinear Schrödinger (NLS) equation:
Bqpt, zqBz “ B2qpt, zq
Bt2loooomoooon
dispersion
˘ 2j |qpt, zq|2qpt, zqlooooooooomooooooooon
nonlinearity
` npt, zqloomoon
noise
qpt, zq is the signal, t is time, z is distanceDistributed white Gaussian noise` focusing regime, ´ defocusing regimeVectorial generalizations exit
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Fourier Analysis of the NLS EquationAssume a Fourier series with variable coefficients for qpt, zq at z ą 0
qpt, zq “N´1ÿ
k“0
qkpzqe j2πkWt .
Substituting (1) into the NLS equation
jBqkpzqBz “ ´4π2W 2k2qkpzq
looooooooomooooooooon
dispersion
` 2|qkpzq|2qkpzqlooooooomooooooon
SPM
` 4qkpzqÿ
`‰k
|q`pzq|2loooooooooomoooooooooon
XPM
` 2ÿ
`‰m`‰k
q`pzqq˚mpzqqk`m´`pzqloooooooooooooooomoooooooooooooooon
FWM
`nkpzq,
in which nk are the noise coordinates and where we have identified thedispersion, self-phase modulation (SPM), cross-phase modulation (XPM)and four-wave mixing (FWM) terms.
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Nonlinear Effects in Fibers
nonlinearitynoise
dispersion
, interactions
, deterministicsignal Ø signal
, inter-channel
, XPM , FWM
, intra-channel
, SPM , XPM , FWM
, stochastic
, signal Ø noise
, inter-channel
, intra-channel
, noise Ø noise
SPM & XPM= self- & cross- phase modulation; FWM = four-wave mixing.8
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Example of Signal Propagation
−15 −10 −5 0 5 10 150
1
2
3
4
t
|q(t)|
z = 0 km
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Example of Signal Propagation
−15 −10 −5 0 5 10 150
1
2
3
4
t
|q(t)|
z = 1000 km
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Example of Signal Propagation
−15 −10 −5 0 5 10 150
1
2
3
4
t
|q(t)|
z = 5000 km
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Example of Signal Propagation
−15 −10 −5 0 5 10 150
1
2
3
4
t
|q(t)|
z = 0 kmz = 1000 kmz = 5000 km
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The State-of-the-Art Approach
1 TX: Wavelength-division multiplexing (WDM)
inputoutputnoise
loomoon
COI
loomoon
out of band
loomoon
out of band f
W guard band
2 RX: Digital back-propagation (BP)
qpt,Lq “ KNLSpqpt, 0qq qpt, 0q “ K´1NLSpqpt,Lqq
qpt, 0q qpt, 0q
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Current Achievable Rates
0 10 20 30 40 50 602
4
6
8
10
12
14
log(
1+
SNR)
nonlinearityimpact
SNR [dB]
Ach
ieva
ble
rate
[bit
s/s/
Hz]
Linear AWGN channelsNonlinear fiber channel
Fiber nonlinearity places an upper limit on capacity
Nonlinear Shannon limit in fiber
Capacity crunch in fiber!
Central Question:Does fiber nonlinearity really place an upper limit on achievable spectralefficiency?
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Origin of the Capacity Limitation – 1Let T : H ÞÑ H be a linear map:
y “ T pxq ` n,
where x and y are input and output signals and n is noise.
Projecting signals onto an orthonormal basis tφkukPN:
x , y , n( “
8ÿ
k“1
xk , yk , nk(
φk , Thus:
yk “ xkxTφk , φky `ÿ
i‰k
xixTφi , φkylooooooomooooooon
linear interactions
`nk
However, if tφkptquk is the set of eigenvectors of T , then
yk “ λkxk ` nk
where λk is eigenvalue.12
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Origin of the Capacity Limitations – 2
Capacity crunch occurs if the basis used for communication is notcompatible with the channel.
Deterministic nonlinear effects are not a fundamental limitation. Itis the method of the communication causing the problem
After abstracting away non-essential aspects, current methods, inessence, modulate linear-algebraic modes
In nonlinear channels, this introduces interference and ISI
BP cannot remove the interference in a network scenario
Linearmultiplexing
Pulse trains
Polarization-division multiplexing
Time-division multiplexing
Space-division multiplexing
Orthogonal frequency-division multiplexing
Wavelength-division multiplexing
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Nonlinear frequency-division multiplexing (NFDM)
It was realized that the NLS equation supports nonlineareigenfunctions which have a crucial independence property, thekey to build a multiuser system
The tool necessary to reveal signal degrees of freedom is
Nonlinear Fourier Transform
Based on NFT, we constructed an NFDM, which can be viewed as ageneralization of OFDM to optical fiber
Exploiting the integrability, NFDM modulates non-interactingdegrees-of-freedom
Capacity of the NFDM in the deterministic model is infinite
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Table of Contents
1 Introduction to Optical Networks
2 Nonlinear Fourier Transforms
3 Information Theory of Nonlinear Channels
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Linear Convolutional Channel
Let Thpxq “ h ˙ x
yptq “ hptq˙ xptq ` nptq , 0 ď t ă T .
The eigenvectors and eigenvalues of Thpxq “ h ˙ x are
φkptq “ 1?T
expp´jkω0tq, ω0 “ 2πT,
and λk “ Fphptqqpkω0q.Fourier transform maps convolution into a multiplication operator
Y pωq “ Hpωq•X pωq ` Npωq , ω “ kω0.
1 Frequency ω is conserved in the channel
2 Channel is decomposed into parallel independent channels3 OFDM: information is encoded in spectral amplitudes X pωq
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General Waveform Channels
Instantaneous waveform channel
channelq0ptq qptq
qptq “ K pq0ptqq ` nptqEvolutionary channel. Here the signal evolves according to anevolution equation in 1+1 dimensions (time t, distance z)
BqBz “ K pqpt, zqq ` nptq
Examples: (qt :“ Btq)‚ K pqq “ j |q|2 (memoryless) ‚ K pqq “ ´jpqtt ` 2|q|2qq(NLS)‚ K pqq “ qtt (heat eq.) ‚ K pqq “ qttt ` 6qqt (KdV)
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Isospectral Flow
A Key IdeaWe seek an invariant under evolution (in the absence of noise). LetL be a linear differential operator (depending on qpt, zq). It may bepossible to find an L whose (eigenvalue) spectrum remains constant,even as q evolves (in z).
channel
Lpq0ptqq Lpqptqq
q0ptq qptq
0 ¨ ¨ ¨ z ¨ ¨ ¨ L
qpt, 0q ¨ ¨ ¨ qpt, zq ¨ ¨ ¨ qpt,Lq
Lpqpt, 0qq ¨ ¨ ¨ Lpqpt, zqq ¨ ¨ ¨ Lpqpt,Lqq
Constant Spectrum18
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Isospectral Families of Operators
If the eigenvalues of Lpzq do not depend on z , then we refer toLpzq as an isospectral family of operators.
Example:The operator L can be a matrix
Lpzq “ˆ
cospzq sinpzqsinpzq ´ cospzq
˙
, λ “ ˘1, Lpzq “ G pzqΛG´1pzq,
where Λ “ diagp1,´1q.
Compact self-adjoint operators can be diagonalized similarly, viaHilbert-Schmidt Spectral Theorem. Here, Λ is a multiplicationoperator.
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Spectrum of Bounded Linear Operators
Spectrum of an operator is defined as
σpLq “ tλ ˇˇ L´ λI is not invertibleu
Example: Linear Schrödingeroperator
Lpqpt, zqq “ ´ B2
Bt2 ` qpt, zq. <pxq
=pxqspectrum of L
Classification: Spectrum can be discrete (like matrices),continuous, residual, essential, etc.
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The Lax Equation
We have Lpzq “ G pzqΛG´1pzq, where Λ does not depend on z .Assuming that Lpzq varies smoothly with z , we can form
dLpzqdz
“ G 1ΛG´1 ` GΛ`´G´1G 1G´1˘
“ G 1G´1loomoon
Mpzq
`
GΛG´1˘
loooomoooon
Lpzq
´ `
GΛG´1˘
loooomoooon
Lpzq
G 1G´1loomoon
Mpzq
“ MpzqLpzq ´ LpzqMpzq “ rM, Ls , (1)
where rM, Ls ∆“ ML´ LM is the commutator bracket. In otherwords, every diagonalizable isospectral operator Lpzq satisfies thedifferential equation (1).The converse is also true.
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Lax Pairs
LemmaLet Lpzq be a diagonalizable family of operators. Then Lpzq is anisospectral family if and only if it satisfies
dL
dz“ rM, Ls, (2)
for some operator M, where rM, Ls “ ML´ LM.
DefinitionThe operators L and M satisfying (2) are calleda Lax Pair (after Peter D. Lax, who introducedthe concept [1968]).
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Integrable System
Let L and M be operators (depending on qpt, zq).BLBz “ rM, Ls ðñ Bq
Bz “ K pqqoperator form signal form
Example [KdV]: Let qpt, zq be a real-valued function and choose
L “ B2t ` q3, M “ 4B3
t ` qt ` qBt .
Then:BLBz “ rM, Ls ðñ qz “ qttt ` qqt .
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NLS Equation
For the normalized nonlinear Schödinger equation
jqz “ qtt ` 2|q|2q,Zakharov and Shabat (1972) found a Lax pair:
L “ j
¨
˚
˝
BBt ´qpt, zq
´q˚pt, zq ´ BBt
˛
‹
‚
,
M “ˆ
2jλ2 ´ j |qpt, zq|2 ´2λqpt, zq ´ jqtpt, zq2λq˚pt, zq ´ jq˚t pt, zq ´2jλ2 ` j |qpt, zq|2
˙
.
As qpt, zq evolves according to the NLS equation, the spectrum ofL is preserved.
Thus the NLS equation is indeed generated by a Lax pair!
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Nonlinear Fourier Transform. Summary-1
L “ j
¨
˚
˝
BBt ´qptq
´q˚ptq ´ BBt
˛
‹
‚
Generalized frequencies: eigenvalues λ of L
Nonlinear Fourier coefficients: a, b where
V pλq “ˆ
ab
˙
is a normalized eigenvector of L
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Nonlinear Fourier Transform. Summary-2
The Zakharov-Shabat operator has two types of spectra:
A discrete (or point) spectrum which occurs in C` andcorresponds to solitons
A continuous spectrum, which in general includes the wholereal line R
<(λ)
=(λ)
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Nonlinear Fourier Transform. Summary-3
−1.5 −1 −0.5 0 0.5 1 1.5
A
t
q(t)
−4 −2 0 2 4−5
0
5
0
1
spectrum of L
<λ=λ
∣ ∣ ∣NFT(q)(λ)∣ ∣ ∣
ba′
ba
0
0.5
1
1.5
2
2.5
-30 -20 -10 0 10 20 30
|q(λ)|
λ
-10123456
-1 -0.5 0 0.5 1
ℑ(λ)
ℜ(λ)
0
0.5
1
1.5
2
2.5
-40 -30 -20 -10 0 10 20 30 40
|q(λ)|
λ
-10123456
-1 -0.5 0 0.5 1
ℑ(λ)
ℜ(λ)
0
0.5
1
1.5
2
2.5
-40 -30 -20 -10 0 10 20 30 40|q(
λ)|
λ
-10123456
-1 -0.5 0 0.5 1
ℑ(λ)
ℜ(λ)
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Properties of the NFT
1 The NFT shares some of properties of the Fourier transform (FT)
2 FT is a special case of the NFT if ||q||L1 ! 1
3
Linear: yptq “ hptq˙ xptq ÐÑ Y pωq “ HpωqX pωqIntegrable: yptq “ xptq˙ pL,M;Lq ÐÑ NFTpyqpλq “ Hpλ,LqNFTpxqpλq
where Hpλ,Lq “ e´4jλ2L is the channel filter. The generalizedfrequencies are invariant in the channel.
4 When there are a finite number of parameters, the solutions can beexpressed via theta functions.Let K be an N ˆ N complex matrix with =pKq ą 0. The Riemanntheta function is defined by
θpt|Kq “ÿ
mPZN
exp´
2πjpmT t` 12mTKmq
¯
, t P CN .
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Launching a Pulse (revisited)
0
0.5
1
1.5
2
2.5
3
3.5
4
-15 -10 -5 0 5 10 15
|q(t)|
t
z=0 kmz=1000 kmz=5000 km
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Launching a Pulse (revisited)
0
1
2
3
4
5
6
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
|q(λ)|
λ
-1
-0.5
0
0.5
1
1.5
2
-1 -0.5 0 0.5 1
ℑ(λ)
ℜ(λ)
z=0 km
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Launching a Pulse (revisited)
0
1
2
3
4
5
6
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
|q(λ)|
λ
-1
-0.5
0
0.5
1
1.5
2
-1 -0.5 0 0.5 1
ℑ(λ)
ℜ(λ)
z=1000 km
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Launching a Pulse (revisited)
0
1
2
3
4
5
6
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
|q(λ)|
λ
-1
-0.5
0
0.5
1
1.5
2
-1 -0.5 0 0.5 1
ℑ(λ)
ℜ(λ)
z=5000 km
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NFDM vs WDMW “ 60 GHz, z “ 2000 km, 15 users, one symbol per user, defocusing.
´10 0 100
0.5
1
λ
|NFTpλq|
input output
NFDM
´6 ´4 ´2 0 2 4 60
1
2
f
|qpfq|
input output
WDM
−25 −20 −15 −10 −5 00
2
4
6
8
10
12
14
P [dBm]
Achievable
rate
[bits/2D
]
AWGNNFDMWDM
´1.42 ´0.71 0 0.71 1.42
¨1.41
´0.71
0
0.71
¨1.41P “ ´0.33 dBmWDM
<s r?mWs
=sr?
mWs
´1.42 ´0.71 0 0.71 1.42
¨1.41
´0.71
0
0.71
¨1.41P “ ´0.33 dBmNFDM
<s r?mWs
=sr?
mWs
Focusing regime, vectorial models, experiments, robustness toperturbations, ... 30
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Table of Contents
1 Introduction to Optical Networks
2 Nonlinear Fourier Transforms
3 Information Theory of Nonlinear Channels
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Shannon’s Formula for Linear Channels
The capacity of a channel pY |X py |xqC “ sup
pX pxq
I pX ;Y q
The mutual information is defined as
I pX ;Y q “ hpY q ´ hpY |X qwhere
hpX q “ ´ż
pxpX q logppX pxqqdx .
For a linear channels
C “ logp1` SNRq, bits/s/Hz
The capacity of optical fiber is unknown, for about 50 years. Evenppy |xq is unknown!
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Upper Bound
0 10 20 30 40 502
4
6
8
10
12
14
log(
1+
SNR)
nonlinearityimpact
?
SNR [dB]
Ach
ieva
ble
rate
[bit
s/s/
Hz]
upper boundmodified lower boundlower bound
TheoremConsider the discrete-time periodic model Cn ÞÑ Cn. We have
CpPq ď logp1` SNRq.
The proof combines:
Energy and entropy conservation
Shannon’s entropy power inequality33
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Proof: Invariant Measures for PDEs
Lemma (Volume Preservation in NLS)
Let Ω “ p`2, E , µq be a measure space, where `2 ∆“
qn | ř |qk |2 ă 8(
and
µpAq “ volpAq “ż
A
˜
nź
k“1
dqkdq˚k
¸
, @A P E ,
is the Lebesgue measure. Transformation Tz underlying the NLS equation,as a dynamical system on Ω, is measure-preserving. That is to say
µpTzpAqq “ µpAq, @A P E .
Application 1: Theorem. The flow of Tz is entropy preserving!Application 2: There are invariant measures. Gibbs measure:
dµx “ 1Z
exp
#
´α´
mÿ
i“1
|qi |4 ´ |qi ´ qi´1|2¯
+
mź
i“1
dqi χ||q||ď1
where α ą 0, Z is the partition function, and χS is the indicator function.34
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Asymptotic Capacity
TheoremDiscrete-time periodic model Cn ÞÑ Cn:
CpPq “ 1nlogplogPq ` c ,
where c∆“ cpn,Pq ă 8.
With n signal DOFs, n ´ 1 DOFsare asymptotically lost to signal-noise interactions.
−20 −15 −10 −5 04
6
8
10
12
14
signal-signalinteractions
signal-noiseinteractionslog
P +c
P [dBm]
Achievable
rate
[bits/2D
]
AWGNNFDMWDM
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Conclusions
We showed examples where advanced mathematics help make progress inlong-standing engineering problems.
Nonlinear Fourier transforms could be used for data transmission
The growth of the capacity is too small compared to the linearchannel
C “ 1nlogplogPq ` c
36