Post on 01-May-2018
Analytical approximations of the nonlinear Schrödinger
equation: applications to optical communications and
information theory
M. Secondini, D. Marsella, E. Forestieri
Nonlinear Evolution Equations and Linear Algebra
September 2%5, 2013, Cagliari, Italy
Supported in part by MIUR under the FIRB project COTONE
© 2013 Scuola Superiore Sant’Anna
Outline
• First, I’ll try to answer the following questions:
– What are typical problems concerning the nonlinear Schrödinger
equation (NLSE) in fiber%optic communication?
– Why the inverse scattering transform (IST) is not employed in this
case?
– Why the split�step Fourier method (SSFM) is widely employed?
– Are there other options?
• Then, I’ll introduce a few concepts from information theory and a
specific problem involving the NLSE.
• Finally, I’ll show how we can (approximately and partially) solve it
by using an approximate analytical solution of the NLSE.
© 2013 Scuola Superiore Sant’Anna
Digital communications in a nutshell
Typical problems
• Transmission techniques: how to map information bits on x(t) ?
• Detection strategies: how to retrieve information bits from y(t) ?
• Performance evaluation: what is the error probability ?
• Information theoretical limits: what is channel capacity (maximum
information rate for reliable transmission) ?
TX Channel RX…101101… …101101…x(t) y(t)
Modulated signals (e.g., the EM field)
What about fiber�optic communications?
• Propagation of the signal through the fiber%optic link is governed by
the nonlinear Schrödinger equation (NLSE)
© 2013 Scuola Superiore Sant’Anna
• Accounting for attenuation/amplification, group velocity dispersion, and
Kerr nonlinearity, in a reference frame moving with the signal group velocity,
the normalized low%pass complex equivalent signal propagates through the
fiber%optic channel according to the following equation:
Dispersion Nonlinearity
Attenuation/amplification
TX
Fiber spans
Amplifiers
RX
Fiber%optic channelInput signal Output signal
The propagation equation
•When g(z) is constant, the propagation equation can be rewritten (through
proper normalizations) in the following equivalent form
© 2013 Scuola Superiore Sant’Anna
Deterministic problem
Given the input signal at z=0 and
the evolution equationEvaluate the output signal at z=L
– Information bits mapped on
– Pulse shape p(t)
– Time duration: very long (K~billions), but can be properly truncated to a shorter length T
– Limited bandwidth: P(f)≃0 for |f|>B/2, with B of the order of 1/Ts
Typically:
• β2 and γ are constant or step%wise constant
• g(z): alternating exponential decay and concentrated amplification
• Input signal: pulse amplitude modulation (PAM)
t……
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Possible solutions
• Split%step Fourier method (SSFM)
• Inverse scattering transform (IST)
• Analytical approximations
– Perturbation methods (regular, logarithmic, …)
– Volterra series
– …
© 2013 Scuola Superiore Sant’Anna
Why SSFM is hard to beat in this case?
• It works in any practical system
• Its accuracy can be increased at will (by increasing computational
complexity)
• Among known numerical methods, it has the lowest complexity
(for a desired accuracy)
• Both theory and implementation are very simple
© 2013 Scuola Superiore Sant’Anna
• Propagation of a signal of duration T and bandwidth B through a fiber
of length L requires the computation of 2M FFT of size N, where
– N=BT is the total number of signal samples
– M=L/dz is the number of propagation steps, where dz is the step size
• The global error decreases quadratically with the step size dz
� Computational complexity (number of multiplications per output
sample) is
– M: trade off between accuracy and complexity
– N: propagation of a long signal performed by overlap & save, dividing it
into shorter overlapping blocks of duration T≥Tm, where Tm is channel
memory). Typically, N≈4πβ2LB2
SSFM computational complexity
© 2013 Scuola Superiore Sant’Anna
SSFM limitations
• It does not provide a closed�form expression of the output signal
• An even lower computational complexity is desirable
© 2013 Scuola Superiore Sant’Anna
Why IST is not employed in this case?
• The evolution equation is not integrable in the general case, but for specific attenuation/amplification profiles (e.g., g(z) constant)
• Computational complexity of direct and inverse scattering can be very high (and increases much faster than logN per sample)
• IST theory is quite involved for beginners.
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What about analytical approximations?
• There are several:
– Regular perturbation (RP), equivalent to Volterra series.
– Logarithmic perturbation (LP), more accurate.
• Typically, they don’t provide significant computational gain with
respect to SSFM for the solution of the deterministic problem
• However, they provide some physical insight and are a valuable
tool for the analysis and design of fiber%optic systems.
� After recalling the RP and LP approximation, I’ll introduce a
modified LP approximation, with a simpler 1st%order term but similar
accuracy, and show its application to a specific problem.
© 2013 Scuola Superiore Sant’Anna
• The NLSE solution is expanded in power series in γ (nonlinearity is a small
additive perturbation of the linear signal)
…
• By substituting it into the NLSE, we obtain:
linear solution
…
1st�order perturbation
is the impulse response in linear regime
Regular perturbation (RP)
© 2013 Scuola Superiore Sant’Anna
• We have proposed the following alternative expansion in γ (nonlinearity is a small
multiplicative perturbation of the linear signal)
LP terms are simply related to RP terms:
same complexity, higher accuracy!
• By substituting it into the NLSE, we obtain:
…
1st�order perturbation
linear solution
…
Logarithmic perturbation (LP)
© 2013 Scuola Superiore Sant’Anna
A simplified (linearized) propagation equation
• Linear Schrödinger equation with time and space varying potential
• Preserves the L2 norm as the original equation.
• u0 evaluated analytically
• Good approximation to the original equation
• Equivalent to the original equation when considering a 1st%order
approximated solution based on RP or LP.
Potential |u|2 of the NLS is replaced by the approximation |u0|2
is the linearly propagated signal (for γ=0)
© 2013 Scuola Superiore Sant’Anna
• Represent input signal through its Fourier transform U(f)
• Solve the propagation equation separately for each frequency component by
using a 1st%order LP approximation
• Recombine all the components at the output (exploit linearity of the equation)
Frequency%resolved logarithmic perturbation
perturbation term
nonlinear interaction efficiency
(for g(z)=1 and β2=const.)
© 2013 Scuola Superiore Sant’Anna
WDM systems and amplifier noise
In this case, we have a stochastic problem:
• Optical amplifiers introduce random noise
• In wavelength division multiplexing (WDM), unknown random signals
from other TX%RX pairs share the same channel and may interfere with
the observed channel (because of nonlinearity).
TXiRXi
TX1
Fiber spans
AmplifiersTXN RXN
…
RX1
…
……
Fiber%optic channel
© 2013 Scuola Superiore Sant’Anna
• NLS equation modified to include cross�phase modulation (XPM) from other
copropagating signals and optical noise:
– Noise ni from i%th amplifier is added at z=zi
– ni: independent complex zero%mean white Gaussian processes with PSD Ni
Typically:
• u(0,t) and w(0,t): PAM signals (at different center frequencies) with i.i.d. symbols
• F(z,t): white Gaussian noise (space and time uncorrelated)
evaluate some output statistic
e.g.,
Given the evolution equation and
the statistical properties of
u(0,t), w(0,t), and F(z,t)
Stochastic problem
© 2013 Scuola Superiore Sant’Anna
Solution methods
• SSFM and Monte Carlo method
• Analytical Approximations and Analysis
• What about inverse scattering transform?
© 2013 Scuola Superiore Sant’Anna
A problem from information theory: channel capacity
Channel capacity
• What is the maximum rate at which information can be reliably transmitted
through the channel?
– Maximum: any modulation, encoding, and decoding is possible
– Reliably: with arbitrarily low error probability
� It depends on channel characteristics. Analytical solutions are available for
a few classes of channels.
TX Channel RX…101101… …101101…x(t) y(t)
Fiber�optic channel
In this case, propagation through the channel is governed by the NLSE and
the problem has not yet been completely solved.
© 2013 Scuola Superiore Sant’Anna
• Mutual information between xxxxN and yyyyN
• Average mutual information rate for ergodic processes
• Capacity: supremum of the mutual information rate taken over all possible
input distributions satisfying some constraints
Mutual information and capacity
ChannelxxxxN yyyyN
Noisy channel coding theorem (Shannon, 1948)
There exist channel codes that make it possible to achieve reliable
communication if the transmission rate is R<C. If R>C, it is not possible to
achieve reliable communication with any code.
© 2013 Scuola Superiore Sant’Anna
Waveform channels
• x(t) and y(t) are expanded into a complete set of orthonormal functions
and represented by the corresponding coefficients
• Shannon’s sampling theorem: a band%limited signal x(t) of bandwidth
B/2 can be represented through its samples taken at rate B
Channelx(t) y(t)
ChannelxxxxN yyyyN
© 2013 Scuola Superiore Sant’Anna
Additive white Gaussian noise (AWGN) channel
Shannon’s capacity
n(t): white Gaussian noise with power spectral density N
Channelx(t) y(t)
Channelxk yk
nk: i.i.d. Gaussian variables with variance NB
Memoryless channel:
Achieved when the input samples xkare i.i.d. Gaussian variables
© 2013 Scuola Superiore Sant’Anna
• Processing theorem:
� Same channel capacity as the AWGN channel
Fiber%optic channel: linear regime (γ=0)
Channelx(t) y(t)
n(t): white Gaussian noise
with power spectral density
AWGN channel!
© 2013 Scuola Superiore Sant’Anna
Fiber%optic channel: XPM+AWGN
~ compensated by processingapproximated as AWGN
Channelx(t) y(t)
approximated by linear propagation
FRLP solution:
XPM term
nonlinear interaction efficiency
(for g(z)=1 and β2=const.)
© 2013 Scuola Superiore Sant’Anna
Fiber%optic channel: XPM+AWGN
Channelxk yk
According to the FRLP model: linear channel with random time%varying
coefficients plus AWGN
� channel coefficients:
We cannot evaluate the exact channel statistics, but we can easily evaluate the input%output covariance matrix for xk and yk
© 2013 Scuola Superiore Sant’Anna
A capacity lower bound: achievable information rate
• Achievable information rate
• Mismatched decoding theorem: it is achievable on the real channel
when using the optimum detector for the auxiliary channel
• Corollary: it is a lower bound to the actual information rate and to
channel capacity
Real channelxxxxN yyyyN
Aux. channelxxxxN yyyyN
© 2013 Scuola Superiore Sant’Anna
Fiber%optic channel: AIR
(for g(z)=1, β2=const.,
Nc interfering channels)
stand. 4th moment (kurtosis)
power
interfering signal w(t)
TXiRXi
TX1Fiber spans
AmplifiersTXN RXN
…
RX1
…
……
x(t) y(t)
• Input distribution: i.i.d. Gaussian
• Auxiliary channel: AWGN channel with same covariance matrix as real channel
© 2013 Scuola Superiore Sant’Anna
Conclusions
• Analytical (approximated) solution of the i.v.p. for the NLSE in fiber%
optic systems are required for:
– Reduce computational complexity w.r.t. SSFM
– Evaluate output statistics
• Solutions must possibly account for attenuation, variable coefficients,
forcing term (noise)
� We have shown an approximated solution and its application to an
information theory problem concerning the fiber%optic channel.
Things to do
• Find numerical methods with lower computational complexity
• Find tighter bounds to channel capacity by
– Using more refined auxiliary channel models
– Looking for the optimum input distribution
© 2013 Scuola Superiore Sant’Anna
References
Part of this work is based on results published in
• E. Forestieri, M. Secondini, “Solving the nonlinear Schrödinger equation,” in Optical
Communication Theory and Techniques, E. Forestieri, Ed. New York: Springer,
2005, pp. 3–11.
• M. Secondini, E. Forestieri, “The nonlinear Schrödinger equation in fiber%optic
systems”, Rivista di Matematica dell’Università di Parma, vol. 8, 2008, pp. 69–97
• M. Secondini, E. Forestieri, “Analytical fiber%optic channel model in the presence of
cross%phase modulation,” IEEE Photonics Technology Letters, vol. 24, no. 22, pp.
2016 –2019, nov.15, 2012.
• M. Secondini, E. Forestieri, G. Prati, “Achievable Information Rate in Nonlinear
WDM Fiber%Optic Systems with Arbitrary Modulation Formats and Dispersion
Maps”, submitted to Journal of Lightwave Technology.
See references therein for a complete bibliography.