Sergei K. Turitsyn

Aston Institute of Photonic Technologies, Aston University, UK

Nonlinear communication technologies

May 17, 2019

ONDM 2019 Conference and Summer School

Optical Network Design And Modelling

May 2019, Athens, Greece

Information

Library of signals Communication channel Received evolved signal Look-up table

receiveddataA big library of signals

Set ofparameters

Smaller dimension: amplitude, phase, frequency, position, etc.

*

Changing the parameters makes new signals

Classical communication theory

Classical linear channel Y=X+N: propagation is equivalent to adding noise

Y=X+N

Communication theory(Shannon, Kotel’nikov, Nyquist,…)

Linear communication channels Y= X + N

• Channel model P(Y|X) – Gaussian PDF

• Optimal signal distribution P(X) (Gaussian PDF) => Channel capacity

• Modulation and coding techniques to implement signals close to optimal

• Processing techniques for optimal signal (or its approximations)

Analogue-to-digital (sampling theorem):

B

0time

freq

uen

cy

n

nnBt

nBtxtx

)2(

)2(sin)(

TMTtotal T=1/(2B)

… or chBNB

)2/()2/(1 BNBT chs

chB

chB

chB

One of the key features of linear channels – their properties are not

changed with increase of signal power

Information theory:

Signal distribution (shaping/coding) in linear

additive white Gaussian noise channel

Probabilistic and/or geometrical shaping of )(xPx

)/(]/||exp[)( 2 SSxxPx

Optimal signal distribution

=> Shannon capacity

SNR [dB]

Some uniform signal

distributions are used in

practice, e.g. M - Quadrature Amplitude Modulation

Theory of linear communication channels is well developed and widely used

Optical fibre communications:different scales – different challenges

Trans-continental

(transoceanic, ultra-long-haul)

~6,000 km - 10,000 km

Long-haul systems

typically up to 5,000 km

also sub-bands of 1,000 km

and 3,000 km

Metro – typically up to 500 km

single or multimode fibres

Local Area Networks (LAN)

typically hundreds of metre

generally multimode fibres

Some physical effects that impact quality of signal transmission

accumulate with distance: they are more important in long-haul systems

Key physical effects in fibre links:

optical noise, dispersion, nonlinearity

Optical fibre channel:

more than just application Optical fibre is the most common

type of communication channels.

99% of global data traffic is

provided by fibre-optic networks.

Wireless channels deliver signal to

stations connected by fibre links.

By 2010, the length of installed

optical fibre is equivalent to the

distance from the Sun to the

planet Saturn.

Amazingly, underlying science of the

installed fibre-optic networks is far

from being fully understood.

Underlying science of installed

fibre-optic communication networks

What maximal amount of information can

be transmitted error-free through installed

fibre-optic communication channels ?

How well do we understand the existing

commercial optical networks?

Capacity: definition of terms

Information spectral density/Spectral efficiency (engineering) – number of

bits transmitted per second per Hz in a communication system

Channel capacity (also known as Shannon capacity) (information theory) -

the largest amount of information that can be reliably transmitted over a

given physical channel, e.g. channel capacity per bandwidth unit:

Several quantities have the same dimension [bits/s/Hz] – a possible

source of confusion with regards to fibre channel capacity

System capacity (common language) – the maximal information rate

that can be practically provided (e.g. to a customer) by a given system

(often is replaced by bandwidth, assuming the same meaning of both terms)

]/[~

][

sec]/[]sec//[ symbolbitsC

HzB

symbolsRHzbitsC s

Communication capacity crunch -

current technologies hitting the wall

- Compensation of nonlinear signal

distortions in coherent fibre-optic links.

- Parallel communications channels, e.g.

spatio-division multiplexing (SDM)

Current technologies use most of available degrees of freedom.

Spatial-division-multiplexing (SDM) is considered as an option, however,

it should be compared in cost and efficiency with use of multiple fibres.

Also SDM does not help to improve efficiency of use of installed fibre links.

Information theory and

optical communications

R.J. Essiambre, G. Kramer,

P. J. Winzer, G. J. Foschini,

B. Goebel, “Capacity limits of

optical fiber networks,”

JLT, 28, 662, 2010.

Capacity scales

slowly with SNR

– approximately

+3.3 bit/s/Hz per

10 dB of SNR at

high SNR

Nonlinear Shannon limit

Nonlinear refractive index n=no +n2 I(t); I(t) – light intensity, n2>0

Need for the next generation of disruptive

optical communication technologies

Courtesy of P. Winzer

Experimentally achieved record spectral efficiency (in two polarizations)

versus transmission distance (markers), shown with the nonlinear

Shannon limit estimate on SSMF (dashed).

How important is fibre nonlinearity

from the practical view point? Effect of signal power on the capacity (through SNR) under

logarithm, . However, it defines what you can do per

channel (and is multiplied by the factor of parallelism M).

To compare cost/bit in deployment of parallel fibres with products

based on nonlinear technology, nonlinearity mitigation should first move

from research (where it is now) to technology and to products.

SDM and any parallel fibre channels are, actually, not instead of

solving nonlinear transmission problems by messing with SNR.

Nonlinearity mitigation techniques are complimentary to SDM.

As long as the fibre channel models will have

nonlinear terms, nonlinear technologies will be

important for fibre-optic systems.

)(log 2 SNR

Nonlinearity in fibre-optic

communications

Optical nonlinearities to a large

extent determined the evolution

of communications systems.

Andrew R. Chraplyvy, The Second Era in the Age of

Optical Fiber Communications, at “Light: Enabling the

Global Internet Era” Symposium-ECOC 2015.

Evolution of an optical signal in a cascaded fibre transmission line is

governed by a combined action of attenuation, group velocity dispersion

nonlinearity and polarization effects.

Sources of signal degradation

in fibre transmission line:

(a) fibre loss

(b) dispersion

(c) nonlinear effects

(d) polarization effects

Signal power attenuation can be compensated using the optical fibre amplifiers.

However: signal recovering in an amplifier is not complete, because amplified

spontaneous emission noise is added to the signal.

The fundamental limitations come from noise and nonlinearity

Principal solutions:

(a) optical amplifiers (but they add noise)

(b) dispersion compensation

(c) mixing with noise - only partial

compensation is possible

(d) linear effects can be compensated; for

nonlinear effects due to mixing noise and

nonlinearity, only partial mitigation is

possible

Basic physical effects in optical fibre systems

100 Mm

Physical effects in optical fibre systems have a range of scales

Light wavelength1 m

10 m

100 m

1 mm

10 mm

100 mm

1 m

100 m

10 m

1 km

100 km

10 km

1 Mm

10 Mm

Core diameter

Pulse

durations

Polarization

beat length

Attenuation

length

Nonlinear

length

Fibre correlation

length

Dispersion

length

trans-Pacific

trans-Atlantic

Average model

approximations

Slowly varying

envelope

approximation

Maxwell’s equations

terrestrial

links

In practical telecom applications, slowly varying envelope approximations averaged

over fast random changes of birefringence parameters or gain/loss, are typically used

Modeling of optical links: enormous

range of length scales in fibre systems

Nonlinear fibre channel

)(2 z

)(

)(2

0

2

zA

zn

eff

),(22

)()(

2

)( 2

2

2

2 tzzg

zit

zi

z

Nonlinear parameter: Gain/loss:

Simplified model that captures all key propagation effects: gain/loss,

dispersion, nonlinearity and noise.

Dispersion:

)(zg

…plus polarisation, higher order dispersion terms, but this simple model is good.

 

h(z,t)Accumulated amplifier noise:

)'()'()()','(),( * zzttzghntztz sp

Signal power evolution in optical fibre

due to gain and loss (over one span)

Raman amplification can make signal variations along the

fibre line more smooth, improving system performance.

)(zg

𝑃(𝑧) = 𝑃 0 exp[ 𝑔 𝑧 − 𝛼]

Gain and loss distribution

affects nonlinear effects and SNR

Signal power evolution:

Minimal nonlinear distortion

for a given SNR or maximum

SNR for a given nonlinearity

level would be achieved in

lossless fibre span:

𝑔 𝑧 − 𝛼 = 0

Creating quasi-lossless fibre spans using

second-order distributed Raman amplification

Distributed Raman fibre amplification provide a gain medium for signal in

optical fibre over distances on the scale of 100 km and more.

Design of quasi-lossless fibre span includes:

• Use of two bi-directional primary pumps around 1365 nm.

• Two fibre Bragg gratings at 1455 nm form a cavity for 1455 nm radiation.

• Cascaded second-order Raman effect creates a gain medium for signal

propagating at 1555 nm

Transmission FibreAmplified

Spontaneous

Emissions

Amplified

Spontaneous

Emissions

pump wave output

Weak Signal

AmplifiedSignal

pump wave input

Distributed Raman amplification for quasi-lossless spans

Quasi-lossless fibre span using second-

order distributed Raman amplification

Primary pumps at 1365 nm generate radiation at 1455 nm.

Due to symmetry, field spatial

distribution is uniform at 1455 nm.

Signal at 1550 nm is amplified

by the evenly distributed power

at 1455 nm leading to quasi-

lossless amplification:

)(zg

Continuous compensation of attenuation along fibre span is feasible

TX

Span #1 Span #2 Span #N

RX

Average

power

Averaging Power

or

Path-averaged fibre channel model:

effectively lossless fibre propagation

),(||2

1 2

2

2

tzNAAt

A

z

Ai

]exp[ LG

Nonlinear properties define optimal performance in fibre links.

At low powers impact of fibre nonlinearity is small, channel is effectively linear

Nonlinear fibre channels: What is changed

compared to linear one and why would we care?

Of course, you are still welcome to use low power.

Nonlinear fibre channels require nonlinear technologies

Performance degradation

due to nonlinear cross-talk

Propagation effects are

more complex than in linear

channels

R.J. Essiambre, G. Kramer, P. J. Winzer,

G. J. Foschini and B. Goebel, JLT, 2010.

Nonlinear fibre channel:

Continuous input, continuous output

),(||2

1 2

2

2

tznqqt

q

z

qi

A simplified master model with additive noise n(z,t):

),()0,( Lztqyztqx ?)|( xyP

),(2

12

2

tznt

q

z

qi

In the low signal power limit we have the classical linear additive white

Gaussian noise channel:

)()0,(),(])(

2[

0 22

NeqLqL

dzzi

Nonlinear channel requires nonlinear communication technologies

Fourier transform helps to describe signal evolution in linear channel

OFC workshop presentation template

Fourier transform

Any practical signal q(t) can be presented through

its spectral (Fourier) components (spectral data) .

Spectral modes are orthogonal

and propagate without interaction

with each other in a linear dispersive

channel => basis for WDM.

Effects of dispersion can be

compensated in the spectral domain

by digital signal processing.

Fast Fourier transform algorithms make possible fast processing

)(~ q

Science: Fourier transform (published in 1822, submitted in 1807)

Technology: Fast FT – (published in 1965, also by Gauss in 1805)

Backward FT

Spectral components evolve independently of each other:

( , 0)q t z ( , )q t z L

How to make

evolution trivial?

Forw

ard

FT

Backw

ard

FT

Signal propagation:

linear dispersive medium

Fourier transform for fast digital signal processing in coherent communications

Forward FT

02

)(2

2

2

t

qzi

z

q

]2

)(exp[)0,(),(2

02

L

dzziqLq

)0,( zq ),( Lzq

dtetqqti

)()(

deqtqti

)(2

1)(

Time

domain

Frequency

domain

Amplitude

Fourier transform basics

]4

exp[2

]exp[22

2

2

tied

t

Not always a clear simplification. In general, pulse can be presented by few parameters.

Pulse description:

• temporal position

• amplitude

• pulse width

• frequency

• phase

Time

domain

“Nonlinear

frequency

domain”

Im λ

Can pulse be presented betterin some other (orthogonal) basis?

It is not only about presentation in some new basis, but also about capability of new presentation to facilitate analysis

Time

domain

Nonlinear

frequency

domain

Im λ

Can complex pulse be presented better (compared to FT) in some appropriate basis?

22

2

1| | 0

2

q qi q q

z t

Nonlinear spectral components

evolve independent of each other

and in a trivial (linear-like) manner

( , 0)q t z ( , )q t z L

NLSE channel

is integrable - IST

(Zakharov, Shabat, 1972)

Forw

ard

NFT

Backw

ard

NFT

Signal propagation:

nonlinear Schrödinger equation (NLSE)

Nonlinear Fourier Transform (NFT)for nonlinear fibre channels

Forward nonlinear

Fourier transform

(Zakharov-Shabat

spectral problem)

Backward nonlinear

Fourier transform

(Gelfand-Levitan

Marchenko equation)

},...,1,,);({ Nkr kk

Time domain

Nonlinear spectral domain

)0,( z ),( Lz i

LiLi

kkkk erLreLL k22

22)0,(),(,)0()(),0()(

discrete eigenvalues

correspond to solitons

Signal, e.g. q(t,z=o)=q(t) at z=0

Inverse Scattering Transform (IST)also known as

Nonlinear Fourier Transform (NFT)

Nonlinear spectrum is related to the kernel

viutqdt

dv

vtquidt

du

)(

)(

*

ti

k

k

ti keierdzt

)(2

1)0,(

},...,1,,);({ Nkr kk

kλ- plane

Re

Forward NFT: Zakharov-Shabat spectral problem

for a given q(t) compute the nonlinear spectrum:

Discrete nonlinear spectrum

Continuous nonlinear spectrum

Discrete spectrum

Continuous spectrum

Forward NFT:

},...,1,,);({)( Nkrtq kk

Complex

frequency

Real

frequency

Discrete

eigenvalue

Discrete

spectrum

Continuous

spectrum

Forward nonlinear Fourier transform

q(t)

Continuous

spectrum

Discrete eigenvalue position

Discrete spectrum parameters

Real

frequency

Nonlinear spectrum: single discrete point

Four free parameters:

real and imaginary parts of lambda, amplitude and phase

,,Im,Re 11 A

iAe 11)(

Soliton

amplitude

Soliton

velocity

Pulse

positionDiscrete

eigenvalue

Discrete

spectrum

Signal

phase

Link between discrete eigenvalue parameters and pulse characteristics

Single eigenvalue corresponds to optical soliton pulse

)](cosh[

]2/)(exp[),(

22

tztA

zAitiiAztq

amplitude phase frequency

time position

Modulation/coding is possible over four soliton parameters

Nonlinear Spectrum – invariant during

transmission => no nonlinear cross talks

Nonlinear Spectrum – invariant during propagation

We define the nonlinear spectrum (NS) function as:

The NS converges to the ordinary FT of q(0,t) in the linear limit

Rich opportunities for fundamentally new coding, transmission,

in-line control, detection and signal processing techniques.

Making nonlinearity friend, not enemy.

Advantage: Nonlinear spectrum (NS) for the NLSE channel evolves trivially and is

invariant => no nonlinear cross-talks that limit performance of fibre channels when

linear approaches are used. Information coded in NS is available at any point of network.

OFC workshop presentation template

Why might nonlinear Fourier transformbe interesting for the NLSE channel/systems?

Any practical signal q(t) can be presented through its nonlinear spectral

components (nonlinear spectral data) Σ(ξ).

Nonlinear spectral modes are orthogonal and propagate without

interaction with each other in the NLSE channel.

Nonlinear spectrum is fully defined at the receiver and is not changed

(apart from trivial phase change) during evolution in the NLSE channel.

Both effects of dispersion and nonlinearity (Kerr effect) can be

compensated in the nonlinear spectral domain.

All nonlinear effects due to fibre Kerr effect (SPM, XPM, FWM) can be

compensated in the nonlinear spectral domain.

No nonlinear cross-talk limiting modern

fibre-optic communications techniques

History of the nonlinear

Fourier transform (NFT)

V. E. Zakharov and A. B. Shabat, Soviet Physics-JETP, 34, 62–69 (1972).

M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, “The inverse

scattering transform - Fourier analysis for nonlinear problems,” Stud. Appl.

Math. 53, 249–315 (1974).

A.S. Fokas and I. M. Gelfand. "Integrability of linear and nonlinear evolution

equations and the associated nonlinear Fourier transforms," Letters in

Mathematical Physics, 32, 189-210 (1994).

A.R. Osborne, "The inverse scattering transform: tools for the nonlinear

Fourier analysis and filtering of ocean surface waves," Chaos, Solitons &

Fractals, 5, 2623-2637 (1995).

In the context of the NLS equation the NFT was developed in 70s:

At a large time scale, history just started

OFC workshop presentation template

Nonlinear Fourier transform:recent re-emergence and progress

This idea (“eigenvalue communication”) was first proposed by A. Hasegawa

and T. Nyu in 1993, and resurrected recently in different versions by many

groups (Aston, Toronto/Telecom Paris Tech, Nokia Bell Labs, Delft/Princeton,

Athens, DTU, Novosibirsk University, Hong Kong Poly, Pisa, Osaka, Ben Gurion

Uni, TUM, Stuttgart, Huawei, Brescia, UCL, Orange Lab, Edinburgh, Kiel,…)

Nonlinear Frequency-Division Multiplexing

NFT as digital backward propagation technique

Ultra-fast NFT algorithms

Experiments on NFT transmission

Periodic NFT-based optical communications

New schemes of coding, modulation and detection using NFT

Polarisation-division-multiplexing with NFT…

History of NFT communications just started

Designs of NFT-based transmission systems

Coding/transmission in the NFT domain

Information is encoded in the

time domain, but the detection

stage involves NFT operations

and decision is made using

nonlinear spectrum data

The information is

encoded directly onto the

nonlinear spectrum

The NFT is used to cancel

the nonlinear distortions

at the receiver – NFT-DBP

INF

T

Dig

ita

l

Da

ta

TX

Optical

Optical

Dig

ita

l

NF

T

Phase

shift

Dig

ita

l

Da

ta

RX

Dig

ita

l

Da

ta

TX

Optical

Optical

Dig

ital

NF

T

Phase

sh

ift

Dig

ita

l

Da

ta

RX

Dig

ita

l

Da

ta

TX

Optical

Optical

Dig

ita

l

NF

T

Phase

shift

Dig

ita

l

Da

ta

RX

INF

T

NFT-Digital Backward Propagation

Hybrid NFT scheme

Compatible with standard digital signal processing

-10 -8 -6 -4 -2 02.5

3.5

4.5

5.5

Power (dBm)

Q-f

acto

r (d

B)

~ 5 dB

BER ~ 2.7e-7

OFDM

NFDM, cont inous part

z=1464 km

BER ~ 7.3e-7

• Modulation of both parts

• OFDM on continuous part

• 2 8PSK eigenvalues

• Data rate up to 65.16 Gb/s

NFT in experiments. Nokia Bell Labs

In 2017 - first experiments showing that nonlinear

techniques can outperform linear ones

50 70 90 110 130 150

Gross data rate (Gb/s)

10-3

10-2

10-1

BE

R

OFDM, before SD-FEC

NFDM, before SD-FEC

NFDM, after SD-FEC, 'error free'

(a)

N = 64 N = 132 N = 154 N = 176 N = 198 N = 222(c) (d) (e) (f) (g) (h)

(b)

60 80 100 120 140 160 180 200 220 240

Number of nonlinear subcarriers

40

60

80

100

120

140

160

Dat

a ra

te (

Gb

/s)

Gross data rate

Net data rate after SD-FEC, BER <1e-6

Achivable rate [Mutual information]

Achivable rate with BICM [GMI]

Record data rate of 400Gbps (SE~ 7.2bits/s/Hz)

have been demonstrated. ECOC 2018 We4F.1

NFT seems to be too mathematical.

Can we just ignore it and use conventional

communication signals/formats/methods in

nonlinear fibre channel ?

NFT methods and conventional systems

What happens with “conventional”

signal in nonlinear (NLSE) channel ?

Conventional signals in nonlinear fibre

(NLSE) channel

Conventional signals in nonlinear

(NLSE) channel

Consider conventional multi-subcarrier/multi-channel optical signal

Here K is a number of symbols, M is a number of spectral channels

(or subcarriers in the case of OFDM signal, is digital data for k-th symbol

in the n-th spectral channel. Consider return-to-zero carrier pulses and solve

direct NFT for a single symbol to determine nonlinear spectrum of this symbol.

𝑐𝑛𝑘

As present information (random process), we study statistics of nonlinear

spectrum of a signal q(t). Nonlinear spectrum is invariant during transmission.𝑐𝑛𝑘

Standard orthogonal frequency division multiplexing (OFDM)

signal, M – number of subcarriers, T – symbol duration

Put this q(t) signal (potential) in the Zakharov-Shabat spectral

problem (direct NFT) and compute nonlinear spectrum

Probability of discrete eigenvalues

occurrence in OFDM symbol

f(t) – rectangular pulse over interval T=10 ns

Probability of soliton occurrences

in OFDM symbol, M=128;QPSK/16QAM

Probability of occurrence

of discrete eigenvalues

(corresponding to solitons)

in the conventional OFDM

signal sent to nonlinear fibre

(NLSE) channel

Power level at which

all OFDM signal have

soliton content

At relatively low power level conventional OFDM signal

can have nonlinear (soliton) content

Optical OFDM signal transmission

Optimal power is around -15 dBm

At the maximum power there is a soliton component

Surprises of nonlinear fibre channels:solitons occurrence in conventional signals

“Good Heavens! For more than forty years I have

been speaking prose without knowing it.”Moliere, Le Bourgeois Gentilhomme

The conventional information-bearing orthogonal

frequency-division multiplexing input optical signal has

soliton component statistically created by the random process

corresponding to the information content.

At signal powers optimal for transmission, OFDM symbol

incorporates multiple solitons with high probability.

Conventional signals developed for linear channels and used

in optical fibre links at optimal transmission power have

soliton component

Can capacity of some communication

channel be higher than the Shannon

capacity of the linear white

Gaussian noise channel?

Shannon capacity of

nonlinear channels

𝐶 = 𝐵 𝑙𝑜𝑔2(1 +𝑆

𝑁)

Shannon capacity

)|( xyP conditional probability – characteristic of a communication channel

)(xPx distribution of input signal (e.g. defines modulation)

SxxPDx x 2||)(

xy

)1(log~

2N

SC

For the additive white Gaussian noise (AWGN) channel

(e.g. complex input signal x, with ):

with power constraint

Mutual information – no optimization over )(xPx

])|()(

)|([log)|()(max

~2

)(

xyPxDxP

xyPxyPxDxDyPC

X

XxPX

)/(]/||exp[)|( 2 NNxyxyP

Fibre channel capacity

Each physical communication channel has its own capacity, and thus,

different channels may have different Shannon capacities.

Different physical channels can be designed in optical communications by, e.g.,

• changing the transmission fibre

• modifying Tx or Rx in a way that changes channel conditional probability

• adding nonlinear (e.g. regenerative) elements in the channel affecting

signal and/or noise propagation

The question is not just : What is the capacity of a fibre channel?, but rather:

Which of (the infinite number of) fibre channels has the highest capacity?

How to create communication channels with the highest capacity?

Nonlinear channels with capacity higher than

linear additive white Gaussian noise channel

Can we have the Shannon capacity above the

capacity of the linear AWGN channel ?

The answer is - yes, but we have to design and use different (from AWGN)

channels, e.g. insert additional nonlinear elements.

Three stages new idea

passes through:

First, they say: This can

not be.

Second stage : May be in can

be. Sometimes. Somehow.

But this is neither important

or practical.

And finally: It has long

been known. This is trivial.

Capacity of different channels can be different

2) 𝑦2 = 𝑇 𝑥 + 𝜉2

3) 𝑦3 = 𝑇(𝑥 + 𝜉3)+𝜉3

4) 𝑦4= 𝑇 𝑥 +𝜉4

Tx Rx

Tx Rx

Tx Rx

1) 𝑦1= 𝑥 + 𝜉1 Tx Rx - linear AWGN channel

- Nonlinear element

- AWGN

𝑪𝟏 ≤ 𝑪𝟑

𝐶1 ≥ 𝐶2𝐶1 ≥ 𝐶4

- optimal pre-processing is implied in the Shannon capacity definition

< |𝜉1|2 >=< |𝜉2|

2 >=2 < |𝜉3|2 > = < |𝜉4|

2 >

Consider four different channels with

the same accumulated noise

What can we say about capacities of these four channels? kC

- optimal post-processing is implied in the Shannon capacity definition: no

processing of 𝑦, deterministic or random, can increase the information that 𝑦contains about 𝑥

The capacity of (3) can be larger than the capacity of (1):

< |𝑥| 2 >≤ 𝑆

and the same signal power

GN (N)

Capacity can be increased by (nonlinear) squeezing of the noise.

Nonlinear communication channels with

Shannon capacity above the capacity of

the corresponding linear AWGN channel

Long dashed red - 50 nonlinear elements

Dashed blue – 20 nonlinear elements

Dotted green – 10 nonlinear elements

The suppression of noise keeps the signal within the corresponding

domains => increased capacity.

Increase of capacity in the system with 10, 20 and 50 nonlinear noise

squeezing elements compared to the capacity of the linear AWGN channel

Tx Rx

Nonlinear communication channels with

Shannon capacity above the capacity of

the corresponding linear AWGN channel

Nonlinear communication channels with

capacity above the linear Shannon limit

There is a wide spread misconception that non-linearity only limits

the capacity of communication channels.

In general, nonlinearity in transmission channels may be both - destructive,

leading to nonlinear impairments during transmission; and - constructive, e.g.

assisting nonlinear control of signal transmission and providing an effective

suppression of noise.

Although it may sound counterintuitive, the nonlinearity, however, might

improve performance beyond the limitations of linear channels.

Capacity of communication channels with in-line nonlinear noise filtering

(regeneration) elements, under certain conditions, can be higher than the

Shannon capacity for the corresponding linear white Gaussian noise channel.

A cross-section of a solutionat an arbitrary distance

Parameters with known evolution in fibre

Towards nonlinear communication theory

Information(random data)

Set ofparameters

Z-dependent setof parameters

Nonlinear spectrum is a set of numberswhich parametrise solutions to the nonlinear channel with trivial dependence on z

Modulation and coding in nonlinear channels can/should

take into account solutions of a master model in (t,z)

Solutions to differential equations are definedby boundary conditions

San Diego, March, 2019

Nonlinear fibre channels require

nonlinear communication techniques

Understanding and mastering of nonlinear communication

systems will lead to new breakthroughs in technology.

Nonlinear communication theory.

Application of techniques specific for the nonlinear

fibre channels like, NFT, is not about choice between

traditional (linear) signal coding/modulation and new

approaches.

The choice is between understanding or ignoring

nonlinear nature of the channel.