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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)
Aris Moustakas, University of Athens 2
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…
Aris Moustakas, University of Athens 3
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)
Aris Moustakas, University of Athens 4
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
Aris Moustakas, University of Athens 5
Introduction
Problem: Crosstalk phenomenon • Two Approaches:
– Fight it
Section of 7-core optical fiber
Aris Moustakas, University of Athens 6
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.
Aris Moustakas, University of Athens 7
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)
Aris Moustakas, University of Athens 8
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
Aris Moustakas, University of Athens 9
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 +=
Aris Moustakas, University of Athens 10
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
Aris Moustakas, University of Athens 11
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
Aris Moustakas, University of Athens 12
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
Aris Moustakas, University of Athens 13
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∫=
Aris Moustakas, University of Athens 14
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
Aris Moustakas, University of Athens 15
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
Aris Moustakas, University of Athens 16
Numerical Simulations (β > 1 and n0 > 0)
The LD approach demonstrates better behavior, following Monte Carlo.
Aris Moustakas, University of Athens 17
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
Aris Moustakas, University of Athens 18
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
Aris Moustakas, University of Athens 19
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
Aris Moustakas, University of Athens 20
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
Aris Moustakas, University of Athens 21
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
Aris Moustakas, University of Athens 22
Numerical Simulations
Aris Moustakas, University of Athens 23
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 – …
Aris Moustakas, University of Athens 24
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
Aris Moustakas, University of Athens 25
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