Post on 26-May-2020
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
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Da
ta
TX
Optical
Optical
Dig
ita
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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.