Random Matrix Theory

What is a Random Matrix? A matrix whose elements xi j are random

variables

x11 x12 · · · x1p

x21 x22 · · · x2p...

.... . .

...

xp1 xp1 · · · xpp

What is a random variable? A quantity whose value is random and to

which a probability distribution is assigned

RV Example (discrete): Number of siblings

x 0 1 2 3 . . .

P r 0.28 0.3 0.24 0.08 . . .

1

Random Matrix Theory

RV Example (continuous): Annual teaching income across US

Probability distribution function (pdf)of teachers’ salaries across US

40,000 100,000

x~

2

Random Matrix Theory

Natural tools which helps us explore relationships between RVs

RM Example: education (x1) and income (x2)

Inco

me

years education

This relationship can be express as a random matrix

var [x1] cov [x1, x2]

cov [x2, x1] var [x2]

3

Random Matrix Theory

Other applications:

Wireless communications

Nuclear physics

Finance

Genetics

. . .

4

Rare events in nonlinear lightwave systems

Elaine T. Spiller

SAMSI and Duke University

SAMSI/CRSC undergraduate workshop - May 21, 2007

5

Communications

Goal

send lots of information very quickly over very long distances

make almost no mistakes

Optical Communications

use light to represent information

transmit light (information) through fiber

Our goal

model this system mathematically

calculate the probability errors

6

Very brief history of optical fiber communications

Fiber Optic Communication: The transmission of information via a

signal comprised of light that travels through glass fiber which acts as a

wave guide.

made possible by laser (1960) and

low-loss, single-mode fiber (1970)

nonlinear Schrodinger equation exactly solvable by the inverse

scattering transform: Zakharov and Shabat, (1971)

NLS proposed for fibers by Hasegawa and Tappert, (1973)

experimental verification by Mollenauer, (1983)

made practical by advent of in-line fiber amplifiers (1988)

7

Optical fiber communication systems: large scale

BELGIUM

P OG E R M A N YNETHE

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CROATIAGemini South 2x6 (WDM) x 2.5Gbit/s

FLAG

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FRIC

A3

x56

0

CANUS-1

TAT-11 3 x 560

TAT-12 2 x 3 (WDM) x 5 Gbit/s

PTAT-1 3 + 1 x 420

Gemini North 2 x 6 (WDM) x 2.5 Gbit/s

AC-1

FLAG Atlantic-1 160 Gbit/s

TAT-9 2 + 1 x 560

Columbus-3 1 x 2 (WDM) x 2.5 Gbit/s

TAT-8 2 x 280

TAT-13 2 x 3 (WDM) x 5 Gbit/s

TAT-14

MA

C 2 x 4

(WD

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x 2.5 Gbit/s

MA

C

2 x

4 (W

DM

) x 2

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bit/s

CANTAT-3

2 + 1 x 2.5 Gbit/s

TAT-10 2 + 1 x 560

TAT-14AC-1

6

SEA-ME-WE 2

SEA-ME-WE 3

CA

RA

C

1 x

42

0

PT

AT

-1

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FLAG Atlantic-1 160 Gbit/s

Casablanca

Lisbon

St Hilaire de Riez

Sesimbra

Tetouan

Algiers Bizerte

CanaryIslands

Vestmannaeyjar

Faroes

Westerland

Bermuda

Azores

Medway Harbour

Manasquan

Pennant Point

Rhode IslandShirley

Long IslandNew York

Tuckerton

est Palm BeachHollywood

Ume

Dieppe

St Brieuc

HallstavikKarst

Norden/Grossheide

NorrtŠlje

System supplied by Alcatel

Circle denotes an underwater branching unit

Broken line indicates system under construction

Planned systems

Other manufacturersÕ systems

System supplied jointly byAlcatel and others

Unless otherwise stated, capacities show number of fibre pairs and bit rate in Mbit/ s

Optical Fibre Submarine Systems North Atlantic

[2002]

8

Propagation of optical pulses in fibers:From Maxwell’s equations to the nonlinear Schr odinger equation

fiber properties

-single mode fiber (only one transverse pulse shape)

-intensity-dependent index of refraction (nonlinearity)

-slowly varying envelope approx (quasi monochromatic)

-small, anomalous dispersion

x

y

z

NLS: iuz +1

2utt + |u|

2u = 0

Define signal in time (one pulse per “bit slot”), propagate in space

9

The first impairment is dispersion

∂u

∂z=i

2

∂2u

∂t2

2nd derivative makes pulse widen (disperse) with distance since

different frequencies have different group velocities

time dependent phase variations⇒“chirp” or instantaneous differences

in frequency across the pulse

10

The second impairment is nonlinearity

∂u

∂z= i |u|2u

Intensity-dependent phase rotation (as below)

−3

−2

−1

0

1

2

3

−2 0 20

0.2

0.4

0.6

0.8

1

−3

−2

−1

0

1

2

3

−2 0 20

0.2

0.4

0.6

0.8

1

11

Nonlinear dispersive waves: an example

dispersion

iuz + utt = 0

↓

pulse spreading

nonlinearity

iuz + |u|2u = 0

↓

self-steepening, shocks

dispersion+nonlinearity

iuz + utt + |u|2u = 0

↓

?

12

Some invariances of the NLS equation

NLS: iuz +1

2utt + |u|

2u = 0 is invariant to

scale (amplitude) changes

phase rotations

“timing” (position) shifts

Galilean (velocity) shifts

Example: phase Since NLS is invariant to phase rotations if us is a

solution to NLS, then use iθ is also a solution to NLS. Let’s check

i∂(use

iθ)

∂z+1

2

∂2(useiθ)

∂t2+use

iθu∗s e−iθuse

iθ = (i∂us∂z+1

2

∂2us∂t2+|us |

2us)eiθ = 0

13

Solitons and solitary waves

“Solitons” = solitary waves that interact “elastically”.

us(t, z) = A sech[S(t, z)] exp[iϕ(t, z)]

S(t, z) = A (t − T −Ωz) ,

ϕ(t, z) = Ω t − 12(Ω2 − A2)z + φ .

Soliton parameters:

A↔ amplitude

T ↔ timing

Ω↔ frequency shift

φ↔ phase

red: Reusblue: Imusblack: |us |

Any combination of parameters results in a valid solution

How do perturbations changes this solution?

14

Noise in nonlinear lightwave systems

TX RX

fiber fiber fiber fiber fiber

amplifier amplifier amplifier

Signal amplification by stimulated emission is always accompanied

by spontaneous emission of incoherent photons⇒ noise

iuz +1

2utt + |u|

2u =Namp∑

k=1

nk(t)δ(z − kza)

That is, u(kz+a , t) = u(kz−a , t) + nk(t), where nk(t) is i.i.d. Gaussian

〈nk(t)〉 = 0,

〈nk(t)n∗j (t′)〉 = σ2δjkδ(t − t

′) .

noise is high dimensional (128xNamp)

Amplified spontaneous emission (ASE) noise produces jitter in soliton

parameters (A, T,Ω,Φ)

15

Amplification and noise

The light signal, u, is attenuated and amplified periodically

0 1000

1

|u|

0 100 0 100 0 100 0 100

· · ·

· · ·

0 50 1000

1

|u|

Amplifier noise is approximately Gaussian, but when coupled with

nonlinearity resulting statistics may not be

Strong noise-induced signal distortion results in errors

16

Amplitude -shift keying (ASK)

0 1 0 0 1 0 1 1

logical one ‘1’ represented by a pulse of light

logical zero ‘0’ represented by space (no optical energy)

What do we detect at the receiver?

Pulse energy (note E = E(A))

17

Differential phase -shift keying (DPSK)

Time

Am

plitu

de

Time

Am

plitu

de

π phase difference→ ‘0’ no phase difference→ ‘1’

encode information on relative phase of adjacent pulses

more robust than ASK when faced with noise and power fluctuations

But signal is amplified to compensate for loss⇒ amplification noise

how does noise + propagation change ‘0’ to ‘1’ (or vise-versa)?

18

How do we decode the phase information?

Optical signal at

detector of bit pattern

11100010

Detector:

V =A∗1A∗2 cos(ϕ

∗1 − ϕ

∗2)

−1

0

1

−1

0

1

0

1

0

1

0 100 200 300 400 500 600 700 800−1

0

1

time (ps)

signal

delayed

signal

|sum|2

voltage

=|sum|2

-|diff|2

|diff|2

To decode: delay one bit period, interfere constructively and

destructively, take intensity of each, take difference

Errors: ’1’ sent but ’0’ detected or vice-versa

Threshold between a ’1’ and ’0’ is zero voltage

19

Noise-induced errors

Predicting error rates:

Error rates must be small; e.g., 10−9 or 10−12

Simulating rare events with Monte Carlo is prohibitively expensive

Understanding why errors occur:

What combination of impairments produces errors?

If one knows how errors arise, they can potentially be corrected

One way to see errors: use importance sampling to bias simulations

toward these events

Crucial to understand most likely noise realizations that

lead to desired events

Perturbation theory to study noise-induced signal degradation

20

Aside on MC simulation

Calculating a probability amounts to calculating an integral

Numerical integration methods: trapezoid, Simpson, MC, . . .

∞∫

−∞

f (x)dx =1

M

M∑

k=1

f (xk) (1)

−5 0 5 10 15 20 25 300

2

4

6

8

21

Perturbation theory for NLS solitons

NLS+generic perturbation:

iuz +1

2utt + |u|

2u = N(t, z) .

Look for perturbative solution:

u(t, z) = us(t, z) + v(t, z) ,

with us= solution of the unperturbed problem

us = A sech(A[t − T − Ωz ]) exp(i [Ωt +1

2(A2 −Ω2)z + φ]) ,

Then v(t, z) solves the linearized NLS equation:

L v = N(t, z) ,

with L = linearized NLS operator:

L v = ivz +1

2vtt + 2|us |

2v + u2s v∗ .

22

Linearized NLS operator

L is non-self-adjoint linear operator

uΦ, uT , uA, uΩ eigenmodes and generalized eigenmodes.

uΦ = i uo u

T = A tanh(At) u

ouΩ = i t u

o uA = (1 − At tanh(At)) u

o/A

key points

The linear modes are associated to invariances of the NLS equation

They each correspond to changes in the soliton parameters

Each value of the soliton parameters yields an unperturbed solution,

so the system does not resist perturbations along these directions.

23

Soliton perturbation theory cont.

us = A sech(A[t − T −Ωz ]) exp(i [Ωt +1

2(A2 −Ω2)z + φ])

Invariances of the NLS equation suggest that it cannot resist

parameter jitter

Build up of parameter changes⇒ large pulse distortion

perturbation induced change to a soliton solution, u = us + v , given by

v = ∆AuA +∆ΩuΩ +∆TuT +∆φuφ + R(t, z)

where uK are modes of linearized NLS corresponding to parameter K

Linear modes link perturbations to parameter changes

24

Importance sampling: a very simple example

Experiment: 100 coin flips

Question: What is the probability of 70 or more heads?

Answer: 2.4× 10−13

But, how do we simulate this directly?

Solution: Use an unfair coin!

Optimal: Use a coin that gives heads 70% of the time.

Correct for biased coin by using likelihood ratio:

If on a flip one gets heads, multiply by 0.5/0.7If on a flip one gets tails, multiply by 0.5/0.3

This corrects statistics: get results for a fair coin

But, 10 orders of magnitude simulation speedup

25

IS application: calculating rare events in nonlinear commu nicationsystems

goals: predict error rates and understand why errors occur

recall: error rates must be small < 10−10

know: how noise changes soliton parameters⇒ linear modes

idea: use linear modes to bias simulations toward rare events of interest

but:Crucial to understand most likely noise realizations that lead to desiredevents, need to be careful

26

Optimal biasing for DMNLS

Consider a noise-induced perturbation, v , to the solution us

v = ∆TuT +∆ΩuΩ +∆φuφ +∆λuλ + R

Bias noise with function of time b(t)

n(t) white, Gaussian i.i.d., v(t) = n(t) + b(t)

Maximize probability of hitting, on average, a desired parameter

change, ∆K, (K ∈ T,Ω, φ, λ)

Solution: b(t) ∝ ∆K uK(t)

27

Importance-sampled Monte-Carlo for NLS

Pulse without and with biasing by the amplitude mode

50 100 150 200 2500

0.2

0.4

0.6

0.8

1u

time (ps)

+

50 100 150 200 250

−0.5

0

0.5

u

time (ps)

=

50 100 150 200 2500

0.2

0.4

0.6

0.8

1u

time (ps)

50 100 150 200 2500

0.2

0.4

0.6

0.8

1u

time (ps)

+

50 100 150 200 250

−0.5

0

0.5

u

time (ps)

=

50 100 150 200 2500

0.2

0.4

0.6

0.8

1u

time (ps)

correct with likelihood ratio⇒ unbiased statistics of rare events

how much should be bias noise at each amplifier?

28

Samples following targeted path

0 500 1000 1500 2000 2500 3000 3500 4000

0.4

0.6

0.8

1

1.2

1.4

1.6

distance, km

ener

gy, a

.u.

Targeted output energy

correct statistics with likelihood ratio⇒ unbiased statistics of rare

events

29

Importance sampled Monte-Carlo algorithm

1. Generate a signal

2. Propagate signal to next amplifier (split-step Fourier)

3. Fit soliton to noisy signal

4. Using analytic formulas, compute eigenmodes of the NLS equation

linearized about the soliton

5. Generate zero-mean normal random variables

6. Bias r.v. with the optimal combo of linearized modes

7. Update the likelihood ratio, p(X)/p∗(X)

8. Repeat steps 2-7 until end of transmission line

9. Detect signal

10. Update appropriate bin with likelihood ratio

30

IS-MC results: energy pdf

0.5 0.75 1 1.25

10−10

10−8

10−6

10−4

10−2

100

energy (normalized by back−to−back energy)

prob

abili

ty d

ensi

ty

black dots - 50,000 MC samples

black curve - Gaussian fit to MC simulations

pink curve - 42,000 IS-MC samples (DMNLS)

31

Typical biased parameter paths

Parameter paths from a few biased trials – anti-correlated

0 0.2 0.4 0.6 0.8 10.5

1

1.5

transmission distance

A1

A2

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

transmission distance

∆φ

32

Full simulation: voltage PDFs for DPSK

−2 −1 0 1 210

−15

10−10

10−5

100

prob

abili

ty(b)

(a)(a)

(b)

107 unbiased

BER = (4.9±0.4)× 10−10

−2 −1 0 1 20

0.20.40.6

voltage (a.u.)

coefficient ofvariation

PDFs clearly not Gaussian.

Blue and red curves are contributions to the overall PDFs from

correlated (large prob) and anticorrelated (small prob) events.

33

Rare events in nonlinear lightwave systems

Conclusions

Demonstration of importance sampled Monte-Carlo simulations of

rare events in soliton-based optical communication systems

PDFs can be sampled way down into the tails

Perturbation theory to guide importance sampling

Examples: pdfs of energy and voltage

This method can be used as a practical tool to analyze the

performance of nonlinear optical systems

34