Aris Moustakas, University of Athens

Unitary Matrices in Fiber Optical Communications: Applications

Aris Moustakas

A. Karadimitrakis (Athens) P. Vivo (KCL)

R. Couillet (Paris-Central) L. Sanguinetti (Pisa) A. Muller (Huawei)

H. Hafermann (Huawei)

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Introduction

• Need for high speed infrastructure Bandwidth (BW) hungry technologies emerging (~2 dB increase per year)

-Wired: IPTV, telepresence, online gaming, live streaming etc. - “Capacity- crunch” eminent

– Solutions so far

• Optical Networks (WDM-DWDM) have exploited many degrees of freedom (BW, available power, polarization diversity) except one: spatial.

– Soon the online devices will exceed the number of global population! – Use of optical Space Division Multiplexing (SDM) is compelling.

• Ideally: without changing the already infrastructure…

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Introduction

• What is Optical MIMO? Just like in wireless domain, in optical we can use N parallel transmission paths to greatly enhance the capacity of the system.

– First paper on Optical MIMO: H. R. Stuart, “Dispersive multiplexing in multimode optical fiber,” Science 289(5477), 281–283 (2000)

– Multi-Mode Fibers (MMF): Use multiple modes to carry information

– Multi-Core Fibers (MCF): Utilize different optical paths of different cores in the same fiber (within the same cladding)

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Introduction

Problem: Crosstalk phenomenon arises Due to : – Extensive fiber length – Bending of fiber – Limited area with multiple power distributions – Light beam scattering – Non-linearities

Crosstalking between adjacent cores in MCF Light beam scattering resulting in crosstalking in MMF

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Introduction

Problem: Crosstalk phenomenon • Two Approaches:

– Fight it

Section of 7-core optical fiber

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Introduction

Problem: Crosstalk phenomenon • Two Approaches:

– Take advantage of it

“Classic” MIMO techniques required but with some twists: – Low power constraints to avoid non-linear behavior. – Optical channel matrix just a subset of a unitary matrix.

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System Model

Consider a single-segment N-channel lossless optical fiber system: Nt ≤ N transmitting channels excited, Nr ≤ N receiving channels coherently. The 2N ×2N scattering matrix is Only t ( Haar-distributed t†t = tt† = IN) sub-matrix is of interest (r is ~0). Generally Nr , Nt < N: • Other channels may be used from different, parallel transceivers • Modelling of loss: additional energy lost during propagation

=

rTt

rttr

S (S=ST)

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System Model

Only t ( Haar-distributed t†t = tt† = IN) sub-matrix is of interest (r is ~0). Define Nt x Nr matrix U as • where P projection operator:

rt NT

N SPPU =

Nt

Nr

...

...

...=

NNN

N

tt

tt

1

111

t

=

0I

P t

t

NN

Nt

N-Nt

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System Model

Channel Equation: – Assume no differential delays between channels (frequency flat fading)the

mutual information is

• Gaussian noise z • Receiver knows the channel (pilot) • Transmitter does not know the channel • w.l.o.g. 𝑁𝑁𝑡𝑡 ≤ 𝑁𝑁𝑟𝑟

( ) ( )∑=

+=+=tN

kkNI

1

† 1logdetlog)( ρλρ UUIU

zUxy +=

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Information Metrics

– Outage Probability:

• Optimal: Assumes infinite codewords

What is the price of finite codelengths?

– Gallager error bound for M-length code:

( ) ( )[ ])(Prob)( UU NttNout IrNErNIrP −Θ=<=

++−< ≤≤ UUIU

†10 1

detlogmaxexp)(k

kkrNMErP tkerrρ

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Coulomb – Gas Analogy

• Joint probability distribution of eigenvalues of 𝐔𝐔†U

–

• Exponent is energy of point charges repelling logarithmically in the presence of external field

• Large N: charges coalesce to density (Dyson – Ben Arous/Maida )

– where

• Minimizing (convex) S0 w.r.t p(x) gives the “Marcenko-Pastur” distribution

( ) ( ) ( )

−+−+−∝ ∑∑∑

>Ν

mkmk

kktr

kk NNNP

tλλλλλλλ log2log1logexp,...,, 021

xxnxV

yxxpypdxdyxVxpdxS

eff

eff

log)1()1log()(

log)()()()(0

−−−−=

−−= ∫ ∫∫β

)1(2))((

)(xx

xbaxxpMP −

−−=

π

][02 pSNteP −∝

0210 ≥−−= NNNN

2

1)(1

,

+++±+

=β

ββn

nnba

tNNn 0=

t

r

NN

=β

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Outage Probability

• We need tails of distribution: Optical Comms operate at

– Fourier transform (or use large deviations arguments)

• k plays role of strength of logarithmic attraction/repulsion at

– k>0 shifts charge density to larger values (R>Rerg), k<0 to smaller ones (R<Rerg) • Minimizing S w.r.t p(x) gives the generalized Marcenko Pastur distribution • Equivalent to balancing forces on the charge located at x:

( )( )xkxVxV

rxxpdxkSS

effeff ρ

ρ

+−→

−+−→ ∫1log)()(

)1log()(0

10

−−= ρx

( ) ( )[ ])(Prob)( UU NttNout IrNErNIrP −Θ=<=

810−≈outp

∫ +−

−−

−=

− ρρβk

kxx

ndxxx

xpP1

11'

)'(2

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Generalized MP equation

• Use Tricomi theorem to calculate p(x)

• Obtain closed form expr. for energy S[p]

• E.g. β>1; n>0

– a, b, k calculated from

−+

−−−+

+−−

=abxbax

nxx

axxbxp 1

)1)(1()1()1(

)1(2))((

)( βρρπ

0)()( == apbp

( )xxpdxrb

aρ+= ∫ 1log)(

)(1 xpdxb

a∫=

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Generalized MP equation

• In general

• Phase transitions (a=0 to a>0) etc are third order – discontinuous – Relation to Tracy-Widom (?)

S0b Sab S01 Sa1 a=0 b<1 a>0 b<1 a=0 b=1 a>0 b=1

n=0; β=1 r<rc1 -- rc1 < r < rc2 r>rc2 n>0; β=1 r<rc3 r>rc3 -- -- n=0; β>1 -- r<rc4 -- r>rc4 n>0; β>1 -- all r -- --

)(''' crS

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Distribution Density of r

Finally – where is the variance at the peak of the distribution.

[ ]

erg

rSrSN

t veNrf

ergt

π2)(

)()(2 −−

≈

( )

++

+−+==

00

2

00

11411

log)(''

1baab

rSv

ergerg ρρ

ρρ

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Numerical Simulations (β > 1 and n0 > 0)

The LD approach demonstrates better behavior, following Monte Carlo.

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Numerical Simulations (β > 1 and n0 > 0)

The LD approach demonstrates better behavior, following Monte Carlo.

For small values of Nr, Nt and N0, the discrepancy is minimal

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Finite Block-Length Error Probability

• Here we need

– where

• Now k is bounded in [0,1] • Also when k<1

– otherwise no constraint

+++→

−

+++→ ∫

xk

kxVxV

rxk

xpdxkSS

effeff 11log)()(

11log)(0

ρα

ρα

++−< ∑≤≤

jjtkerr k

kkrNMErP λρ1

1logmaxexp)( 10U

++

+

++= ∫ x

xk

kxk

xpdxrb

a ρρρ

1111log)(

tΝΜ

=α

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Numerical Simulations (β > 1 and n0 = 0)

There are two types of phase transition points here: • rc1: At k=1 point – discontinuous

• rc2: Discontinuous

0 0.5 1 1.5 2 2.50

2

4

6

8

10

12

14

16

18

20Error Probability vs Rate; P=10; β=3; n=0

r

-log(

Per

r)/N t2

α=2

α=10

α=50

opt

( )1'' crS

( )2''' crS

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More Realistic Channel

• Chaotic cavity picture:

• Assuming very low backscattering at edges we obtain

– Deterministic (mode energy) – Random (complex Gaussian) – Diagonal loss matrix

( ) WWWWIS 12 −++ +−= ππ iHi

( ) trtr ic PΓGHPSPPU 10

−++ ++−== γ

0HG

Γ

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More Realistic Channel

• Mutual Information metric:

• I – Distribution (mean-variance) can be obtained using replica theory – I2 same with ρ=0 – also expressions for variance

( ) ( )[ ]rttr iiI PΓGHPPΓGHPIG 10

10detlog)( −+−+ −++++= γγρ

[ ]( )[ ] ( )22

01

21

)1)((detlog

)(

ptrNprtI

IIIE

−−−++++=

−=

γγγδρ H

G

( )

( )

( )

−++++++

=

−++++−

=

−+++++

=

20

20

0

20

)1)(()(1

)1)(()(1

)1)(()1(1

prttTr

Nt

prtpTr

Np

prtrTr

Nr

γγγδργδργ

γγγδργγ

γγγδργγ

H

HH

H

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Numerical Simulations

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Conclusions

• MIMO: promising idea in Optical Communications

• In this work:

– Simple model for Optical MIMO channel – Large Deviation Approach provides tails for MIMO mutual information – Method provides metric for outage throughput and finite blocklength error

• Many issues still open: – Channel modeling still at its infancy – Transmitter/Receiver Architectures – Multiple fiber segments – Nonlinearities – …

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Introduction

• Motivation: – Cap crunch, degrees of freedom

• Channel Model: – Unitary, random, lossy(reference for extra channel model) – Non-linearities, MDL

• Metrics: – Capacity Equation, dof explanation – Error exponent: Proof by Gallagher

• Eigenvalue picture: – Mapping to effective energy – minimize to maximize probability:

• MP result for most probable – Coulomb approach: effective density – Integral equation and solution – Examples of most probable densities (comparison between optimal and non-

optimal outage

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Introduction

• Phase transitions: – Second order and third order transitions in the Gallagher exponents

• Generalization to other approaches – Deterministic plus random channel – Lossy channel – Amplify and forward channel