final report 2018eeserver.ee.polyu.edu.hk/fyp/fyp_201718/ft/fyp_47/final...project id: fyp_47 signal...

THE HONG KONG POLYTECHNIC UNIVERSITY DEPARTMENT OF ELECTRICAL ENGINEERING

Project ID: FYP_47

Signal processing techniques for long-haul nonlinear optical fibre transmissions

by

LAM HO 15055846D

Final Report

Bachelor of Engineering* (Honors) in

Electrical Engineering*

Of

The Hong Kong Polytechnic University Supervisor: Dr. Alan Pak-tao LAU Date: 29/03/2018


Abstract

The optical fiber has been innovated form several years. It becomes a curial part of a data

transmission network because of its board bandwidth and supporting higher bit rate. On the other

hand, lower loss of optical fiber and optical amplifier innovation lead long-haul optical fiber

communication possible. A nonlinear distortion, however, raised up when the optical power

increase for long-distance transmission. In one signal channel, higher optical power will lead self-

phase modulation (SPM) while cross-phase modulation (XPM) and four-wave mixing (FWM) are

induced within the multichannel. The evolution of signal model by Nonlinear Schrodinger

equation (NLSE).

This project is proposed to implement the algorithm for Nonlinear Fourier Transform (NFT) and

Inverse Nonlinear Fourier Transform (INFT), which are both derivate from Lax Pair, to operate

on the signal (QPSK) then obverse its performance of relieving nonlinearity effect and noise along

a channel. The signal derivate form NFT is a soliton which mean its invariant in pulse shape. The

conventional approach, digital backpropagation, in a linear system model is to compensate the

nonlinearity distortion and dispersion then compare its simulated result against with those

derivated from NFT and INFT approach. The performance of two communication scheme will be

evaluated by Bit Error Ratio versus dBm. NFT and INFT has expected immunity of nonlinear,

linear effect and noise as well.

New communication scheme embedded NFT and INFT shows a better performance of data

transmission. Data in spectral also show less noise effect. This new way can explore the Nonlinear

Frequency Multiplexing which might be immunity of XPM and FWM. Furthermore, the

experiment is going to conduct after the submission of the final report because of time-limited;

then will include in the revised report.

Keywords: Nonlinear Fourier Transform (NFT), Inverse Fourier Transform (INFT), Soliton


Acknowledgement

I hereby thank my supervisor, Dr. Alan Lau, for his excellent academic leading during my final

year project. He gives care to him each FYP students. For my final year project, there are lots of

things, Nonlinear Fourier Transform, beyond my pieces of knowledge I had learned. Alan,

however, is willing to donate his countless time to educate and supervise me. He gives me lots of

inspiration not only this project but also my career path. I am again deeply indebted to him for

many helpful and bring me to this state-of-art lore and technology through this FYP. It's a great

research experience for me in a past year.

I also want to give gratitude to Alan's Ph.D. Student, Zhou Gai. I sometimes found him to ask the

questions about NFT when Alan is not in Hong Kong in a period. He also inspires me at time of

developing the algorithm.

Finally, I would like to thank PolyU EE department providing Final Year Project to me to have

that kind of research experiences.


TableofContents

INTRODUCTION 1

OBJECTIVE 3

BASICBACKGROUND 43.1LOSS 43.2CHROMATICSDISPERSION 43.3GROUP-VELOCITYDISPERSION 53.4KERRNONLINEARITY 53.5NONLINEARFEATURE 63.6NONLINEARSCHRODINGEREQUATION 73.7BASICCONCEPTOFAMPLIFIER 83.8LAXPAIRANDEVOLUTIONEQUATION 93.9COMMUNICATIONSCHEMEBASEDONNFT 103.10BASIESIGNALTRANSMISSIONCHANNEL 113.11BANDLIMITEDCHANNEL 14

LINEARSYSTEMMETHODOLOGY&RESULT 174.1.1MATHEMATICALEXPRESSIONOFSIGNALEVOLUTION: 174.1.2RESULTOFTHEPULSEPROPAGATIONUNDERNLSE: 194.2.1SIMULATIONOFDISPERSIONEFFECTINLONG-HAULTRANSMISSION: 204.2.2BERCURVEOFTHEDISPERSIONANDLOSSINTHE1600KMWITH80KMPERSPAN: 214.3.1ABLOCKDIAGRAMOFDISPERSIONANDNONLINEARITYIN80KM/SPANINTOTAL1600KM: 224.3.1BANALYSISOFOPTICALSIGNALINM-QAM 234.3.1CDBPINTHE1600KMWITH80KMPERSPAN: 244.3.2BERCURVEOFDBPCOMPENSATEINTHE1600KMWITH80KMPERSPAN: 254.3.3CONSTELLATIONDIAGRAMVARIOUSOFTHEDBM 26SUMMARY 32

NONLINEARSYSTEMMETHODOLOGY 335.1.1IMPLEMENTATIONOFNONLINEARFOURIERTRANSFORMINCOMMUNICATION: 335.1.2RESULTOFCOMPUTINGNFT 395.2.1IMPLEMENTATIONOFINVERSEOFNONLINEARFOURIERTRANSFORM: 415.2.2RESULTOFCOMPUTINGINFT 435.3.1SIMULATIONOFANEWCOMMUNICATIONSCHEME: 455.3.2RESULTOFCOMPUTINGNFT 47

CONCLUSION 50

DISCUSSION 51

REFERENCES 52


1

IntroductionAs the communication rate raise up following the population and various technology, the electrical

signal transmission system is no longer catch up the grow till optical fiber having a larger

bandwidth is innovated; and find it could support faster communication speed. The internet

company, for example, is used fiber optics to support Gb/s and Tb/s transmission instead of the

telephone line. On the order hand, the reliability measuring in BRE decrease with the speeding up

communication rate; and the fiber loss increase with the fiber distance. To overcome those

problems in the telecommunication system, the signal power should be enlarged. The relation

between bit-error-ratio (BRE) and the power or signal-to-noise ratio (SNR) would further discuss

the methodology. For an optical fiber, the amplified is added on the transmission part to increase

to a higher intensity and focusing light ray which is called Laser. The laser has higher intensity

hence a higher power light source. Raman amplification actually has been using commonly in

nowadays long-haul communication system [1]. In a long-distance optical fiber communication

system, a fiber link from the transmitter to the receiver is torn apart several spans; an amplifier is

installed between the span to compensate the loss along each span. The length of the span depends

on the total distance of the system. The 80km/span for 1600km is used to modelling at this article

simulation result.

The fiber loss and BER are on the other hand enhanced by high signal power in long-distance

communication; the fiber nonlinearity, however, occurs in that system. In 1973, the soliton-like-

pulse was suggested to propagate in an optical fiber as a result of an interaction between the

depressive and nonlinear effect [2]. The depressive and nonlinear effect in the communication

system then become a major issue to be the deal. Both depressive and nonlinear effect is included

into one of the nonlinear partial differential equation, Nonlinear Schrodinger equation (NLSE).

The depressive effect in fiber optics is consist of the chromatic depressive and group-velocity-

dispersion while we only deal with the self-phase modulation (SPM), the cross-phase modulation

(XPM) and four-wave mixing problem in the nonlinear effects. Nonlinear Schrodinger equation

(NLSE) has no analytical solution; it have to be solved by the numerical method, Slip-Step-Fourier

Method. The approach will be discussed in section 4.1.1. In 2004, Yousefi and Kschischang based

on the framework of nonlinear Fourier Transform (NFT) to proposed nonlinear frequency division

multiplexing (NFDM) mixing the soliton theory in the communication system to deal with the

problem [3] - [5].


2

As Nonlinear Fourier Transform (NFT) framework is developed completed, becomes an important

mathematical tool for frequency domain analysis and signaling processing in the nonlinear system

as same as the regular Fourier Transform in the linear system. With the NFT knowledge, deduce

an algorithm in the specially modified channel for nonlinearity.

In the linear system, the frequency division multiplexed (FDM) is used for data transmission while

there is a nonlinear frequency division multiplexed (NFDM) as well for dealing with nonlinear

effect in the transmission channel. To derivate the new signal processing techniques or the

algorithm for the new nonlinear communication system based on Nonlinear Fourier Transform

(NFT) and that (NFDM) in order to mitigate the effect of fiber nonlinearity. Then, Import the

Algorithm into nonlinear transmission system based on the nonlinear frequency division

multiplexed (NFDW) and simulate the result from MATLAB, then compare the noise effect on

signals derived from nonlinear Fourier Transform (NFT) and regular Fourier Transform. In general,

to use the bit error ratio (BRE) versus the signal to noise (dBm) curve to investigate the reliability.

In the following section, we will find out how the signal evaluating according to the NLSE distort

such as chromatic dispersion and nonlinearity effect. Then, we will do some compensation such

as the chromatic compensation and digital backpropagation, to see if there is an enhancement in

BER. As chromatic dispersion and nonlinearity factor interplay each other, there is a particular

pulse shape keep unchanged propagating along the z-direction. This kind of pulse shape is called

soliton. This kind of solution can be found out from NFT and INFT. We, therefore, derivate an

algorithm for computing NFT and INFT then implement them into a communication scheme to

simulate the result. The result is expected to be better than using digital backpropagation


3

Objective

3.1 Study and Simulate the light pulse propagation with 80kmspan in 1600km total

3.2 Implement CD compensator in 3.1 model

3.3 Implement Digital Backpropagation in 3.1 model

3.4 Study NFT and INFT then derivate the corresponding algorithms

3.5 Implement the algorithms in 3.4 into a communication scheme


4

BasicBackground

3.1Loss In electrical signal, the significant loss is thermal power loss (I"R) due to the large current flowing

into some electronic devices. For optical signal, the fiber loss could be separated in three factors

which are Rayleigh scattering, material characterizes and the impurity. Rayleigh scattering, a

fundamental loss, will draw a fluctuation refractive index along the fiber so scatter the light in all

directions [6]. For the material characterizes, light intensity usually is absorbed by the glassy

material because it is not a perfectly transparent medium. Silicon glass, for example, absorb a few

light in the resonances wavelength region extending from 0.5 − 0.2*+ [6]. The impurity leading

the loss is from the fabrication. During the fabrication of fiber optics, the OH ion inside the fiber

has an oscillational absorption peak at about 2.7*+ [6]. For new generation fiber, it was found

there is loss below the 0.5dB level at the wavelength range near 1.4*+ [7]. From the year 2000,

the wavelength region for fiber-optic communication is set to 1.3 − 1.6*+ with minimum loss of

about 0.2 dB/km at around 1.55*+ [6].

Summing up all the fiber loss factor, the total loss of fiber can be determined by the attenuation

constant 1 with an equation given by

23 = 256789

23 is Output power 25 is Input power L is the fiber length

3.2ChromaticsDispersion Chromatics dispersion is a common phenomenon. As same as incident light ray into the prism, an

electromagnetic pulse with various frequency launched into the fiber at the same time will be

spread at an output. When the light entry to a medium, the refractive index depends on the

frequency of the electromagnetic wave. The refractive index can be approximated by Shellmeier

equation [8]

:" ; = 1 +=>;>"

;>" − ;"

?

>@A

where ; is optical frequency, => is the strength of jth resonance, ;> the resonance frequency


5

3.3Group-VelocityDispersion Group-Voltage BC is described an optical plus’s envelope propagation shown in figure 2.2.1.

Figure 2.2.1 optical pulse

On the order hands, the effect of fiber dispersion is determined by expanding the mode-propagation

constant D in a Taylor series near the frequency ;5 [6]

D ; = D5 + DA ; − ;5 +12 D" ; − ;5

" + ⋯

From this series, first order mode-propagation constant, DA, is equal to the reciprocal of BC in unit

of F/+ [1] while D" in unit of F/+" is the derivative of DA and could be interpret the dispersion of

the group velocity after passing through a distance of fiber This phenomenon is the Group-Velocity

Dispersion (GVD) with GVD parameter D". In general, the dispersion parameter D is defined [6]

H =IDAIJ = −

2KLJ" D" = −

JLI":IJ"

With both Dispersion effect, the output pulses would be broadened and overlap with each other to

lead a distortion signal to increase the BER.

3.4Kerrnonlinearity The fundamental nonlinear factor at the long-haul optical fiber transmission come from Kerr

effect. Nonlinear response can be interpreted that the additional applied electric field influence the

motion of the bound electrons of fiber medium molecule [6]. As the optical medium is a dielectric

material, there will be the electric dipole moment under the external electric field. Due to the

electric dipole, the polarization will be induced. The total polarization M can be written in a tensor

summation formula [6]:

M = N5(P A ∙ R + P " : RR + P T ⋮ RRR +⋯)

P > is jth order susceptibility


6

The total polarization in the formula go to nonlinearity as the jth larger than one. The linear

response come the first term,P A of which effect include the reflective index n and the attenuation

coefficient 1 [6].

Depending on the material in symmetric molecule, P " will disappear in some case; for instant,

WXY" [6].

3.5NonlinearFeature According to Kerr nonlinearity, there are three corresponding phenomena such as third-harmonic

generation, four-wave mixing, and nonlinear refraction [9]. In this project, we only concern about

four-wave mixing, and nonlinear refraction; third-harmonic generation will not discuss here.

Nonlinear refraction is shown the optical power Z "dependence of the refractive index, which can

be written as [1]

: ;, Z " = : ; + :" Z "

where : ; is the linear part given by Shellmeier equation

Two major nonlinear effects from this light intensity dependence of refractive index are known as

self-phase modulation (SPM) and cross-phase modulation (XPM). SPM is illustrated as self-

induced phase shift of the optical field along the its propagation inside the fibre; and its magnitude

can be obtained from [6]

\ = :]5^ = (: + :" Z ")]5^

where the nonlinear phase shift is :"]5^ Z "

Unlike SPM, XPM’s phase shift is induced by another fieldZ" with different frequency, direction

or state of polarization; and its given by [6]

is :"]5^( ZA " + 2 Z" ")

Four wave mixing is created by the third-order polarization term [6]. When all four field are

linearly polarized along the same axis of a birefringent fiber oscillating at four different frequency,

;A, ;", ;T, ;_ , the SPM and XPM effect will appear and there will be two type frequency

combination [6]. With four known different frequencies, there would be total 8 combinations.


7

3.6NonlinearSchrodingerequation It is known that there are the depression effect and fibre loss in the linear pulse propagation. Those

factors also exist in the nonlinear system; and there is a nonlinear term, γ A "A, to describe the

fibre nonlinearly. The pulse envelope can be approximated by the equation:

bc(d,e)bd

+ 8"f + ghi

"bicbei

− hijbkcbek

= Xl f "f + gmn

b( c ic)be

− opfb c i

be ____[1]

For the pulse width > 5ps, the parameter and are small enough to be ignored [6]. As the wavelength

we proposed in this project is not too close to the zero-dispersion, the 3rd order dispersion is also

quite small for the signal [6]. Finally, Nonlinear Schrodinger Equation (NLSE) which is proposed

one of the equations to interpret the pulse propagation in nonlinear system without noise is

simplifier in

X bc(d,e)bd

+ g8"f q, r − hi

"bisbei

+ l f q, r "f q, r = 0 ____(2)

where l is nonlinear parameter

In a true nonlinear system, the noise, :(q, r), should be added on the R.H.S of NLSE to replace

zero.


8

3.7BasicconceptofAmplifier

An optical amplifier used to pump optically or electrically to obtain population inversion; The gain

depends not only on the frequency of the incident signal but also the intensity of the beam inside

the amplifier [1]. The amplification factor is given by

t ; = exp x ; ^

To compensated the loss, x ; in this project will be set the constant value a which is the fibre

loss coefficient. Comparing with the electrical signal, the noise (White noise) of optical signal is

induced by spontaneous-emission from the optical amplifier [1]. The spectral density of that noise

is almost constant and can be written as

Wyz B = t − 1 :yzℎB

Multiplying the bandwidth (BW) of the noise can obtain the noise variance (power):

|" = t − 1 :yzℎB ∙ =}

In the equation, :yz is named the spontaneous-emission factor which is given by

:yz =~"

~" − ~A

where ~Aand ~" are the atomic populations for the ground state and excited state.

In this project, we assume that the amplifier is kept in room temperature, therefore :yz is taken in

2 which is near the Raman-gain-peak.


9

3.8LaxpairandEvolutionEquation Lax pair is derivate by L(z) operator. It operates function into a Hilbert space which is extend to

infinity dimension. Similar with the finite dimension Matrix, “A”, there may be some eigenvalues

in L(z). The meaning of eigenvalue is that, the function or the vector will not change the shape and

direction respectively after operation of L(z). The final result just been scaled after L

transformation. In that case, we could rewrite L(z) in the form G(z)ΛG7A(z) . Λ is the

diagonalizable matrix containing the eigenvalue; G(z) is the eigenvector space formed from

corresponding eigenvalue. Since q also depend on z, L could write a form L(z) which mean L

operate q into a Hilbert space.

If the L(z) various smoothly respect to z, L(z) could be derivate with respect to z [13]

I^(q)Iq

= tÉt7A tΛt7A − tΛt7A tÉt7A

= Ñ q ^ q − ^ q Ñ q = [Ñ, ^]

where [Ñ, ^] is the Lax Pair

Compare with the nonlinear evolution equation of a function á, we see that the nonlinear evolution

equation could rewrite into

àâ

àä→

I^(q)Iq

Where L is defined by

å

ç

çr−é(r, q)

F(r, q) −ç

çr

While ^B = JB respect, we could obtain ^ − Jè B = 0. Derivate that equation by z and t, we

could an evolution equation (M-equation) of eigenfunction, B(r, q) and P-equation[13].

Bd = ÑB Be = 2B


10

P- operator are fixed as

å

ççr −é(r, q)

F(r, q) −ççr

With respective to NLSE, M and P operator are given [1]:

P= åç

çrá(r, q)

−á∗(r, q) − ç

çr

Ñ = 2åJ" − å á(r, q) " −2Já r, q − åáe(r, q)2Já∗ r, q − åáe∗(r, q) −2åJ" + å á(r, q) "

As the Lax equation,ë9(d)ëd

,shows a linear from, we could obtain the system block diagram [1]

3.9CommunicationSchemebasedonNFT According to above Lax convolution, the launching signal á(r, 0) derivate form NFT propagate in

time domain will give out a significant distortion such as inter symbol interference (ISI) but

formation in spectral data is invariant up to a complex-value scaled. The communication scheme

thus is proposal as below [13]

^d(á) = [Ñ(á), ^(á)] á(r, 0) á(r, ^)


11

3.10Basiesignaltransmissionchannel Even it is the nonlinear system related research project, is built up based on the linear system so

that some general technique and investigate approach are still workable.

A basic linear communication system can be formed by three components, which are transmission

part, receiver part, transmission medium and the White Gaussian Noise in zero variance. The

whole system can be shown in a simple block diagram:

In electrical signal, the noise comes from the thermal noise which relates to the random motion of

electrons while the noise of optical signal coming from the pumping in the amplifier. White

Gaussian Noise usually is accepted a noise model for a simulating communication system. “White”

means a uniform spectral noise power density; “Gaussian” describe noise added on each signal

point with the normal distribution probability.

As the additive noise will lead a distortion in the received signal, therefore the single processing

is involved in the communication to recovery the signal as much as possible. For MATLAB, the

main function, “find [x]”, is used for processing the signal with noise. The main concept of the

algorithm getting involved in the signal processing is detecting in which range each symbol/ pule

is, then assign it a certain value according to the detective result. Although the received signal is

cancelled out the noise result in a pulse shape as same as TX, the bit sequence error would occur

in the receiver.

Transmission Signal (TX)

F(r)

Received Signal F(r) +:íXF6(r)

noise(t)

Figure 3.10.1 Block diagram of simulation communication system


12

Figure 3.10.2a The transmitted signal with and without noise

Figure 3.10.2b The received signal with signal processing


13

From figure 3.1.2(a)(b), shows an example that the received signal, however, recovery to the prefer

square symbol, there are still some mistake symbols arise during the bit sequence. To represented

the probability of error bit in a digital signal, Bit Error Ratio (BER) is widely used. BRE also

illustrates how well the digital transmission system performs. BRE could be calculated by the

following equation:

=ìZ = ~î+ï6éíñ6ééíéïXrFoíróò:î+ï6éíñïXrF

If there is a lower BRE in the communication system, the system shows high reliability. Apart

from the noise, the power level of the signal also affects BRE and thus there is the relationship

between signal power level and the noise power level, Signal-to-Noise Ratio (SNR) in decibels

(dB). That relationship is governed by various theoretical formulas according to the different signal

transmitting way such as PAM, PSK, and 4-QAM etc. The simulating results also follow the rule

of the equations. The relationship between BRE and SNR with the simulated curve and theoretical

curve in 16-QAM with 10_ symbols is shown below:

Figure 3.10.3 Symbol error probability curve with 16-QAM


14

From the curves, it shows that lower BRE along with higher SNR. The relationship between SNR and BRE could show more clearly under the below complex plane.

Figure 3.10.4 16-QAM with 24dB

The green dots represent the unprocessed signal with noise while he black cross represent the TX signal. 3.11Bandlimitedchannel

In in a real linear system, there is another concern, band-limitation which means only allowing

some frequencies below the certain cut-off frequency pass through, in data transmission. The

bandlimited factor is due to the material, size and the geometry of the transmission medium even

if the glassy material in fiber optics. Butterworth filter, Fig 3.2.3, is an extremely flat magnitude

low-passed filter. That is the reason why we use it as modeling the bandlimited channel in that

project. By adding the Butterworth filter, the more realistic linear system could model in the as the

following diagram:


15

Because of the bandwidth of the medium, the signal passing through the channel will be distortion.

As the result, BRE at the receiver goes up in spite of higher SNR. Figure 3.11.2 is an example to

show the curves with lower cut-off frequency would no longer fitting into theoretical and the

simulating curve without filter.

Figure 3.11.3 Butterworth filter

Based on the study of the conventional signal transmission scheme above, it's clear to see that the

capacity of the electrical signal transmission network is limited by the bandlimited channel which

is an obstacle for the multi-channel and the long distance. Fortunately, optical fiber overcomes that

problem; also could achieve the higher data transmission rate. The story, however, is not the end.

When a lunching light pulse propagates confine a long-haul optical fiber, it will experience not

Transmission Signal (TX)

F(r) Received Signal

F(r) + :íXF6(r)

noise(t)

Butterworth

Figure 3.11.1 Block diagram of bandlimited communication system

Figure 3.11.2 (4-QAM with filter) Symbol error probability simulation


16

only the power loss and dispersion but also self-phase modulation, cross phase modulation and

four-wave mixing which come from the Kerr-nonlinearity we have discussed in the background

section.

The entire physics behaviour of the light pulse propagation can be summed up in a nonlinear partial

equation (evolution equation), Nonlinear Schrodinger equation, with 1+1 dimension. In the

following sections, it will be the focal point of this thesis. At first, a numerical method has been

introduced to solve the NLSE.


17

LinearsystemMethodology&Result

4.1.1MathematicalExpressionofSignalEvolution: Nonlinear Schrodinger equation (NLSE) involves describing the signal growth versus fiber length

in the long-haul transmission system. It's an evolution equation which can express the derivate of

function respecting with one domain to another domain derivate. To model and predicate the signal

waveform at the receiver after the propagation along the fiber from the transmitter, that partial

equation must be solved. As solving a nonlinear partial differential equation in an analytical

solution is complicated, the numerical method, Split-Step Fourier Method (SSFM), was

introduced. Another advantage of SSFM is to reduce the computational power of the computer.

SSFM is a member of the family of pseudo-spectral methods which were used to solve the time-

dependent nonlinear partial differential equation [7]. In this project, I shall illustrate the

implementation of SSFM with MATLAB in algorithm 1.

With this method, NLSE will be separated into time-independent linear and nonlinear operators,

then is solved in "nonlinear step" and "linear" step individually. For convenience, the linear

operator is taken in Fourier domain to calculate. Mathematically, the solution will be demonstrated

(in following few step) by below equation:

Consider the Nonlinear term first only

çô

çq= ål ô "ô

ô(r, q5, + ℎ) = ô(r, q5,)exp(ål ô r, q5,"ℎ)

Then, use this solution be an input to the linear part

çô(r, q5,)exp(ål ô r, q5,"ℎ)

çq= −

åD"2

ç2ô(r, q5,)exp(ål ô r, q5,"ℎ)

çr2

Final, take both side Fourier Transform to get the follow result:

ö7A ö A(q5,t)6Pú(Xl f "ℎ) 6Pú XD"2 ;"ℎ −

12 ℎ

The dominant error, however, is found from that equation; therefore, the SSFM can be improved

by adding a different procedure also speed up the computational power [6]. The final expression


18

programmed in MATLAB is:

f(^, r) ≈ A 0, t exp(−~ℎ2) 6Pú Hℎ exp ~ℎ

û

?@A

exp(~ℎ2)______[6]

where His linear operator, ~ = is the nonlinear operator.

On the other hand, Fast Fourier Transform (FFT) is used to realize a Fourier Transform in digital.

MATLAB already have the functions, which are fft(), ifft() and fftshift(), in the library to deal with

the Fourier Transform and inverse Fourier Transform. More detail will be show in below

algorithm.

Algorithm 1: Split Step Fourier Method

Input: the signal ô(r), time vector o, Fibre distance ^ and the no_of step ℎ

Output: renew signal ô(r)

N=9† %no of step

; ← o;

ô r = ô(r)6Pú(−Xl ô " †");

for X=1 to N

ô r = ô(r)6Pú(Xl ô "ℎ);

Wú6Lréî+ = öóFr_öíîéX6é_oéó:Fñéí+(ô r );

Wú6Lréî+ = Wú6Lréî+ ∗ 6Pú X hi";"ℎ − 8

"ℎ ;

ô r = è:B6éF6_öóFr_öíîéX6é_oéó:Fñéí+(Wú6Lréî+);

end

ô r = ô(r)6Pú(−Xl ô " †");


19

4.1.2ResultofthepulsepropagationunderNLSE:

Figure 3.1.1 Pulse under dispersion effect only

Figure 3.1.2 Pulse under dispersion and nonlinear effect

The above simulation result shows that dispersion will lead the spreading of the shape while nonlinear effect will lead compressing of the shape.


20

4.2.1SimulationofDispersioneffectinlong-haultransmission: The single-mode fiber, in fact, is commonly used in long-haul transmission due to its board

bandwidth. One of the benefits of single-mode transmission is no chromatic dispersion (DA). Since

the wavelength of the light, the carrier of the signal, moreover is around 1.55um (193THz) which

is far away from the zero-dispersion, the third order dispersion (DT ) will also not occur.

Corresponding to the modelling of this project, we only consider the second order dispersion which

is the group velocity dispersion (D").The final equation in each step "h", therefore can be rewritten

in the form:

A(L, t) ≈ö7A ö A(q5,t) 6Pú XD"2 ;"^ −

12 ^

On the other hand, the optical fiber in the long-haul transmission will be divided into several spans

by adding the Raman amplifier between the span to compensate the loss. The noise 2c£§, however,

will be induced after the amplification from Raman amplifier; that noise can be model by Gaussian

Whit noise. As the average power of the noise will vanish, 2c£§ can seem as a variance of the

Gaussian Whit noise with mean zero. The noise will be added to the amplified signal in time

domain at each span. A simulation model of 1600km with 80km per span is shown below:

where H(w) is 6Pú X hi";"^ − 8

"^ , G is 6Pú 8

"^

The algorithm to simulated dispersion effect in long-haul transmission is also implied SSFM in a loop which is given in the following Algorithm 2 but without the nonlinear factor.

With the compensator: 6Pú −X hi";"^

TX

+ + RX

~A(r) ~•7A(r)

G G G

~•(r)

H(w)

H(w)

TX

+ + compensator

~A(r) ~•7A(r)

G G G

~•(r)

H(w)

H(w)

RX


21

4.2.2BERcurveofthedispersionandlossinthe1600kmwith80kmperspan:

Figure 3.2.1 Result of Dispersion effect with and without Compensator in QPSK

Figure 3.2.2 Result of Dispersion compensation with QPSK, 16QAM, 64QAM

From the above result, it's shown that compensation at the receiver improves the performance of

the system. Even if there is the compensator at the receiver end, higher power is necessary for

delivery the signal in the higher level of QAM to reduce BER as the dispersion effect at high power

level is quite small.


22

4.3.1aBlockdiagramofdispersionandnonlinearityin80km/spanintotal1600km:

Seeing the pulse duration larger than 5ps, the derivate with respecting to time in nonlinear term

become so small [6]. In that scenario, the nonlinear operator in NLSE is aa only. The evolution for

each step “h” can refer to eq (2). With the addition of nonlinear effect, a simulation model of

1600km signal evolution is shown below:

Algorithm 2: ~ − Fúó:in ^ distance

Input: the pulse train ô(r), time vector o, span distance ^ and the no of step ℎ, no of span :

Output: renew pulse train ô(r)

N=9† %no of step

; ← o;

for X=1 to n

Algorithm 1: SSFM loop

ô r = ô r ∗ t; %amplifier

ô r = ô r + :íXF6;

end

TX + + RX

~A(r) ~•7A(r)

G G G

~•(r)

Wg•(q, r) 6Pú ßX

DA2 ;"ℎ −

12 ℎ

®

exp(Xl|f|"ℎ)

W™´e(q + ℎ, r) ö{} x ö7A{}


23

4.3.1bAnalysisofopticalsignalinM-QAM

The SNR in long-haul optical fibre transmission system can be calculated in

¨≠Æ"Ø∞£∞±≤≥¥

where the 2g• is the input power of the pulse, ~c is the number of the optical Amplifier. The factor

“2” represents two orthogonally polarizing fibre mode [1].

In contrast to the electrical signal, the performance of the optical fiber communication system will

compare the Bit Error Ratio (BER) with either 2g• or dBm. dBm value is given by

I=+ = 10log2g•1+}

The noise variance in fact will be increased after across the amplifier due to a specified noise figure of that kind of amplifier; therefore, it’s better to use dBm rather than SNR in long-haul optical fiber network.


24

4.3.1cDBPinthe1600kmwith80kmperspan:

One of the compensated techniques which are used to compensate the both group velocity

dispersion and nonlinearities is Digital Backpropagation (DBP) [11]. It’s commonly used in

nowadays for digital signal compensation. The approach of DBP is inputting the receiver signal

through a virtual fiber with reverse sign of D" and l in NLSE. As the loss will be compensated

by the amplifier, 1 is not necessary taken in negative sign in value. Some of the researchers

have extended the Digital Backpropagation to Optical Backpropagation which is needed to

install optical phase conjugation equipment and highly nonlinear fiber [12]. For the current

progress, I have simulated the Digital backpropagation to compensated dispersion and

nonlinearities.

Algorithm 3: Digital Backpropagation

Input: ô(r) from algorithm 2, time vector o, span distance ^ and the no of step ℎ,

no of span :

Output: renew pulse train ô(r)

N=9† %no of step

; ← o;

algorithm 2 with reveres sign of parameter D", 1ó:Il

TX G G RX G TX G G

Actual fiber Virtual fiber (DBP)

Figure 3.2.1.1 Imaginary diagram of DBP


25

4.3.2BERcurveofDBPcompensateinthe1600kmwith80kmperspan:

The BER curve of 120G/s is reduce greatly after adding DBP in front of the receiver, it performs

much better than dispersion compensation only. The lunch power of signal however is limited in

a few mW. In the above plotting graph, it shows three schemes which are doing CD compensation,

doing the backpropagation and doing nothing in the receiver to compare. Intuitively, we expect

that the blue curve should represent the minimum probability of error bit among three of them; but

the result doesn't achieve our expectation.

The BER curve of only compensate with chromatic dispersion in the receiver cross over the blue

one, then override it to reach almost 100% probability of error bits or symbols. We see there is an

obvious flat line within a certain dBm range; however, it doesn't happen on the digital-

backpropagation BER curve. To understand why that particular issue occurs, I will introduce sets

of signal distributions corresponding to various fixes power on the complex plane in the following

few pages. Starting from -5dBm, we can see there is a slight self-rotation of QPSK in the signal

distribution with chromatic compensation only while the clearest plots are from -1dBm to 5dBm.

In -1dBm, we have seen that there is an about 90-degree shift of the whole set of the received

signal after CD compensation; but no happen after digital-backpropagation. Digital-

backpropagation is one of the methods to compensate the nonlinear part and the dispersion part at

the same time; on the other hand, a factor,exp(Xl f "q), from SSFM exactly interpret a phase shift

on the complex plane from a geometry view. In fact, it is the self-phase modulation we mentioned

form the chapter 2, background.

Figure 4.3.2a BER curve with 2G/s Baud rate Figure 4.3.2b BER curve with 120G/s Baud rate


26

4.3.3ConstellationDiagramvariousofthedBm

Power = 12.6mW


27

Power = 3.16mW


28

Power = 1.26mW


29

Power = 7.94e-4W


30

Power = 3.16e-4W


31

Power = 7.94e-5W


32

Summary From the pulse propagation simulation, we see that the pulse propagating in a long-haul system

will appear the spreading of dispersion and the compressing of nonlinear effect at the same time.

With the consideration of only dispersion effect in long-haul signal transmission, a linear system's

performance is improved by a compensator added in front of the receiver. The improved BER

curve shows that high power need for high-level M-QAM such as 16QAM and 64QAM to reduce

the probability of symbol error. The laser used to transfer the signal, however, is within mW as

high energy leaser will induce heat as a result of damage to the fiber optics and the equipment. It

is the limitation of the compensation. The Digital Backpropagation (DBP) compensating both

nonlinear effect and the dispersion of an optical signal, on the other hand, decrease more BER than

just adding dispersion compensator before the receiver. In contrast, the launched power is limited

to a few mW to obtain a low probability of symbol error after the addition of DBP. As the linear

compensation approach have no longer enhance the capability of the signal transmission. The new

way, nonlinear Fourier transform, has been proposed to deal with the long-haul optical fiber

transmission.

As the simulation shows that the dispersion will lead spreading of the signal while nonlinear effect

will lead the compression, it's possible to find a function which could balance the broaden and

denseness to keep the pulse shape unchanged. This function is named in soliton. In the following

section, I will briefly introduce Nonlinear Fourier Transform (NFT) and Inverse Nonlinear Fourier

Transform (INFT) first; also realize algorithmic to compute the signal/ distributional function. I

then imply those new algorithmic into the span model I set up in previous chapter to observer the

overall performance compare with digital back-propagation. In contrast to the linear system, signal

is stored in frequency domain to transmit after INFT; it also be compatible on both nonlinear effect

and dispersion rather than overcome them [13]. In final, an experiment is conducted to demonstrate

this new strategy; also compare this actual result to simulation result.


33

NonlinearsystemMethodology

5.1.1ImplementationofNonlinearFourierTransformincommunication:

Nonlinear Fourier Transform actually is the inverse scattering transform which we have briefly

introduced in the background section [13]. This method makes use of the Lax Pair to solve some

particular differential equation. Comparing with a linear system, Lax pair solves the PDE in a

soliton function rather than an impulse response. In the previous chapter, we have already derivate

not only the Lax Pair but also two linear equations from an operator, L. For the case of the NLS

equation, s t, z and r t, z are – q∗(t, z) and q t, z respectively. P, M and L are therefore defined

[13] in

2 = −åJ á r, q – á∗(r, q) åJ Ñ = 2åJ" − å á(r, q) " −2Já r, q − åáe(r, q)

2Já∗ r, q − åáe∗(r, q) −2åJ" + å á(r, q) "

The P-equation are given by

Be =−åJ á r, q

– á∗(r, q) åJ B where á r, q is a time invariant signal distribution vanishing at r → ∞, B is canonical eigenvector

in Zº space.

The nonlinear Fourier Transform only refer to P-equation while M-equation, which will be

discussion in section 5.3.1, is related to the propagation of the NFT coefficient along z direction.

To understand the work of the Nonlinear Fourier Transform, one can visualize that a measure

function, canonical eigenvector, is applied evolves forward in time to interact with the signal from

r = −∞ to r = +∞. The results coefficient of the canonical eigenvector at r = +∞.

To obtain the canonical eigenvector, B, with r → ∞, á r, q and – á∗(r, q) could be assumed both 0 in P operator.

Be =−åJ 00 åJ B

The final solutions bounded in ℑ J ≥ 0 are solved in

BA(+∞, J) → 01 exp(åJr) B"(−∞, J) → 1

0 exp(−åJr)

The solution of á r, q supporting within [rA, r"] can be calculated in follow approach:

Be = 2B2 −∞, J → B = exp 2 r − rA ∙ B2 −∞, J ……… (1)

On the other hand, let 2 = fHf7Aøℎ6é6HXFóIXóxí:óòÑóréXP

Be = fHf7AB"


34

f7ABe = Hf7AB"

let f7AB = ;, f7ABr = ;r

f−1Be = Hf−1B2→ ;e = H;

let the eigenvalue in H be ∆, and involved eq (1)

exp 2 r − rA ∙ B2 −∞, J = f;>è ∙ Lí:Fró:r

exp 2 r − rA ∙ B2 −∞, J = f;>f7A ∙ B2 −∞, J

exp 2 r − rA = f6Pú(H(r − rA))f7A

Before further develop a numerical method for computing NFT, consider implying the above P-

equation’s in a rectangular pulse at q = 0:

á r = f, r ∈ [rA,r"]0, P ≥ 0

Thus, the solution [13] is given by

B" +∞, J = exp 2 r2 − r1 ∙ B" −∞, J

=cos ∆ r2 − r1 −

åJΔ sin(∆(r2 − r1))

fΔ sin(∆(r2 − r1))

−f∗

Δ sin(∆(r2 − r1)) cos ∆ r2 − r1 +åJΔ sin(∆(r2 − r1))

where exp 2 r" − rA is scattering Matrix “S” which correspond to the inverse scattering

transform and ∆= J" + f ".

W = ó(J) −ï∗(J∗)ï(J) ó∗(J∗)

øℎXò6ó J ó:Iï J rℎ6:í:òX:6óéñíîéX6éLí6ññXLX6:ríïróX:6Iï∆ñíòòíøXI6:rXrX6F 13

ó J = lime→»

BA"6>Jr ï J = lime→»

B""67>Jr

The elementary properties Nonlinear Fourier Transform is almost the same, unless layer-peeling

property which I would like to introduce detail here since the algorithm computing NFT is mainly

depended on this property. Consider the signals áA r, q + á" r, q +áT r, q +…. +á• r, q

r

... ...... ...

áA(r, q) á"(r, q)

, á•(r, q)

r•7A rA r" rT r•

B"(−∞, J) B"(+∞, J)


35

With above diagram, the layer-peeling property could be write in a Mathematical form as:

B" +∞, J = exp 2 r] − r]−1"

…@•

∙ B" −∞, J

B" +∞, J = W:−(]−1)•

…@A

∙ B" −∞, J

If n=2

W"∗A =ó"(J) −ï"∗(J∗)ï"(J) ó"∗(J∗)

∙ óA(J) −ïA∗(J∗)ïA(J) óA∗(J∗)

W"∗A =óA J ó" J − ïA(J)ï"∗(J∗) −ó" J ïA∗ J∗ − óA∗(J∗)ï"∗(J∗)óA J ï" J + ïA(J)ó"∗(J∗) ó"∗(J∗)óA∗(J∗) − ï" J ïA∗ J∗

Thus

B" +∞, J =ó1 J ó2 J − ï1(J)ï2∗(J∗) −ó2 J ï1

∗ J∗ − ó1∗(J∗)ï2

∗(J∗)ó1 J ï2 J + ï1(J)ó2∗(J

∗) ó2∗(J∗)ó1∗(J

∗) − ï2 J ï1∗ J∗

∙ 10 exp(−åJr)

B" +∞, J = ó1 J ó2 J − ï1(J)ï2∗(J∗)ó1 J ï2 J + ï1(J)ó2∗(J

∗)∙ exp(åJr)

where ó"∗A J 6áîXBò6:rrí[óA J ó" J − ïA J ï"∗ J∗ ]exp(åJ(r" − rA))

ï"∗A J 6áîXBò6:rrí[óA J ï" J + ïA(J)ó"∗(J∗)]exp(−åJ(r" + rA))

After ó J and ï J are well defined, the spectrum amplitude could be calculated by ºÀ º

. ó J ,

however, is an analytic function which might have the roots,J>@A,",T,…•,on ℂŒ so that ºÀ º

will be

lead to be undefined. In that particular case, the spectrum amplitude could be derivate from by

ºœ–— “œ–“ ºœ”‘,i,k…Æ

. More precisely, signal in Nonlinear Fourier Domain are form by continuous

spectrum ºÀ º

with J ∈ ℝ and discrete spectrum ºœ–— “œ–“ ºœ”‘,i,k…Æ

with J ∈ ℂŒ

Unless layer-peeling, two others existed numerical analysis method should also be included to

work together. They are Euler's Methods and Trapezoidal integration.

With a given ó J and ï J equation of a rectangular pulse and layer-peeling formula above, there

is an idea to compute an arbitrary pulse shape numerically. Firstly, we could cut an arbitrary pulse


36

shape in the number of rectangular pulses with different amplitude with sufficient step size, ∆t.

The sketch map of this idea is shown as:

Then, calculate each narrow pulse's Nonlinear Fourier Transform's coefficient. In final, use layer

peeling iteration to add them up. The related equations of coefficient are given below [13]:

ó ] + 1 = ó ] P ] − ï ] ∆ ]

ï ] + 1 = ó ] ∆ ] + ï ] P ]

óÉ ] + 1 = óÉ ] P ] + ó ] PÉ ] − (ïÉ ] ∆ ] + ó ] ∆É ] )

ïÉ ] + 1 = ó ] ∆ ] + ï ] ∆É ] + ïÉ ] P ] + ï[]]PÉ[]]

With initial iteration ó 0 = 1, ï 0 = óÉ 0 = ïÉ 0 = 0

P ] = cos H÷ −åJH sin(H÷)

∆ ] = −á∗[]]H sin(H÷)

PÉ ] =åJ"÷H" cos H÷ −

å + J÷H −

åJ"

HT sin(H÷)

∆É ] =á∗[]]JHT H÷cos H÷ − sin(H÷)

PÉ ] = −åJ"÷H" cos H÷ −

−å + J÷H +

åJ"

HT sin(H÷)

∆É ] =á[]]JHT H÷cos H÷ − sin(H÷)

whereH = J" + á[]] ", ÷ = o 2 − o[1]

r

... ...


37

Seeing that there is a derivate of the óÉ ] , the Euler's Method or Newton-Raphson Method is such

as a good tool to search for the roots (discrete) eigenvalues. The iteration scheme of this method

is

J…ŒA = J… − 1…ï J…óÉ J…

where 1… is keep always 1 in below algorithm.

In order to make sure that all of the eigenvalue are found, the trace formula for n=1,2,3 is involved

[13]. n=1,2,3 respected to the Energy, Momentum and Hamiltonian respectively. In data,

communication, there is no momentum and Hamiltonian so that only n=1 should be consider only.

According to the energy conservation (n=1), sum of energy in time domain should be equal to the

sum of it in discrete and continuous spectral domain:

Z = Zëgy◊ + Z◊™•e

In data communication, energy

Zëgy◊ = 4 ÿØ>@A (J>) Z◊™•e =

AŸ

J…7Alog(1 + á(J) ")IJ»7» Z = á(r) "Ir»

7»

With Calculation of finite integration numerically, Trapezoidal integration is the best scheme to

approach to the solution approximately. The general form of the Trapezoidal rule is:

è =ℎ2 x P5 + 2 x Pg

•7A

g@A

+ x P•


38

Algorithm 4: Nonlinear Fourier Transform

Input: Signalô[1rí~], Time: o[1rí~] Set J form a final interval [−:, :] E_t=Trapiz(ô[1rí~],) %Calculate ô[1rí~] energy %Calculate continuous spectrum a=1, b=0; for ii=1→ N calculate P XX, J and ∆ XX, J then update ó XX + 1, J ï XX + 1, J

end

E_c=Trapiz(b[]/a[]) %Calculate continuous spectral energy

Calculate whether there are discrete spectral

Error=E_t-E_c

a=1, b=a’=b’=0; If Error >N

Random drawn J>> ∈ ℂŒ and do while loop else not do while loop;

While norm (Error) >N

for ii=1→ N Calculate P åå , ∆ åå , PÉ åå ∆É åå PÉ åå ∆É åå then update ó åå , ï åå , óÉ åå ïÉ åå end If J>> not ∈ ℂŒ

Random drawn J ∈ ℂŒ else do J>>ŒA = J>> − ºœœÀ⁄ ºœœ

end

if ºœœÀ⁄ ºœœ

< ‹, Error = Error-4*Imag(J>>) %update Error and check if there is any roots

Final_roots[jj]=J>> end

á(J>>) ºœœÀ⁄ ºœœ

%calculate the discrete spectrum


39

5.1.2ResultofComputingNFT Take a rectangular pulse with r ∈ [0.5,0.5] and the amplitude are 1 and 6 respectively as examples

From the simulation result, we can see that the continuous spectrum of which J in real show as

same as a sinc function derivate from ordinary Fourier Transform when the amplitude is equal to

1 or sufficient small. There is also no discrete spectrum as J ∈ ℂŒ almost vanish. If ∆=

J" + f " andf ≪ 0, the continuous spectrum could write in [13]

á J = −f∗o6Pú −åJoÉ FX:L 2oñ

where J = 2Kñ

While f larger than 1 in the rectangular pulse, the continuous spectrum is no longer to hold on a

sinc function shape as the ∆ also depend on the amplitude. As the same time, we observe that there

are existing discrete spectrum as J ∈ ℂŒ which are 5.3776i and 3.0014i has been find.

On the other hand, the accuracy of algorithm of solving NFT is really depend on the sample point

we take. Take the rectangular pulse of A=6 as an example again. The continuous spectrum of

sample point = 32 is not as same as the one of sample point = 64. For A=6 case, 128 sample point

is fair enough.


40

(a) (b)

(c)

Figure 4.3 Continuous and Discrete Spectrum of a rectangular pulse at A=6 with sample point (a)32 (b)64 (c)128


41

5.2.1ImplementationofInverseofNonlinearFourierTransform: In the section, inverse Fourier transform will be discussed. It is placed at the transmitter along the

communication scheme to encode the information form spectral into the time domain. There are

lots of work developing the maths to do the inverse Nonlinear Fourier Transform. Some would

characterize the signal form both continuous spectral and discrete, such as Riemann-Hilbert

Method. Some would only use either discrete or continuous spectral to reshape the signal. In this

project, we will introduce the method used discrete spectrum only. It based on Darboux Transform.

With only one soliton, the signal actually could be characterized by a sech(t) function. That one-

soliton solution is given by the following equation [13]:

á r = −å;exp(−å1r)sech(;(r ± r5))

where 1 = º7º∗

_, r5 =

‡·‚ „‰

m

If there are N-solitons (N-eigenvalues), this formula is no longer be true. That is why the general

method, Darboux Transform, is used. It is the best way to recover the function with only discrete

spectrum till nowadays because it comes up with lower truncation error and faster. The main reason

is that N-eigenvalues have already been assigned in transmitter.

The idea of Darboux box is obtained by the two main equation as shown below [13]:

î r, *; á = *è − WΓW7A B r, *; á …… (1)

á = á − "> º∗7º Êi∗Ê‘

Ê‘ iŒ Êi i ……(2)

where WXFóFúóL6ñéí+6Iï∆rℎ6]:íø:FíòîrXí:\, ΓisadiagonalmatrixwithλandJ∗

The eigenvector B refer to the input discrete spectrum and the eigenvalue in ℂŒ. The iteration loop

is according the number of the eigenvalue. Alternatively, above equation (2) could be simplified

by dividing numerator and denominator part by either \""and \A ". á then could update directly

by founding the ratio of the solution, Î. The equation (2), therefore, could be rewrite in:

á = á −

2å J∗ − J \"∗\A\A "

1 + \" "

\A "

á = á −2å J∗ − J Î∗

1 + Î 2


42

The algorithm using Darboux Transform is derivate by Vahid Aref [14]; the algorithm is shown below:

Algorithm 4: INFT From Darboux Transform


43

5.2.2ResultofComputingINFT Based on above algorithm, there are the simulation result of 1-solitonic pulse,J = 1åó:I á = 2å,

in figure (a). We can clearly to see that the sech function, which is continuous in time domain,

have only one discrete spectrum in nonlinear spectral domain. In Contrast, the choosing of

eigenvalue and the value of its corresponding discrete spectral must be careful. According Zëgy◊ =

4 ÿØ>@A (J>), the amplitude of the discrete spectrum is 2 times of its correspond imaginary part

of eigenvalue. If the amplitude of the discrete spectrum is not exactly the 2 times of the imaginary

part of eigenvalue. The time shift of the sech will be induced. The time shift value directly refer to

the value of the amplitude of the discrete spectrum. In spectral domain, it can be illustrate that the

discrete spectrum times a factor 6Pú(ln f) while this factor is corresponding to time-shift value.

This transform is the time shift properties of the Ordinary Fourier Transform. Figure (b) shows 1-

solitonic pulse with = 1åó:I á = 10å, there is forward time shift. If the amplitude of the discrete

spectrum is smaller than 2 times of its correspond imaginary part of eigenvalue, it will do the left-

hand-side shift in time domain. This particular issue is curial in doing amplitude modulation in the

data transmission system. We cannot directly amplifier the soliton while we should module the

eigenvalue when doing amplitude modulation. In the next section that there is a simulation of

QPSK by using NFT and INFT communication scheme, you will see that dBm various should be

done by changing the eigenvalue, Otherwise, the transmission pulse train derivate by INFT would

be distorted in higher order M-QAM.

(a) (b)

Figure 4.4 1-solitonic pulse J = 1åó:IIXFLé6r6Fú6Lréóòó+úòXrîI6 = (a)2j (b)10j


44

The simulation in next chapter, we only consider 1-solitonic pulse. When there is N-solitonic pulse,

the above rule is no longer be true. We, however, will not give more detail on it. Below are the

figures of a pulse contain two eigenvalues.

(a)

(b)

Figure 4.3 2-solitonic pulse J = 1å, 2åó:IIXFLé6r6Fú6Lréóòó+úòXrîI6 = (a)2,4j (b)6j,-12j respectively


45

5.3.1Simulationofanewcommunicationscheme: To simulate this new communication scheme, the evolution of the eigenvector in Nonlinear Fourier

Transform domain show understands first. It can be obtained by M-equation mentioned before

section. To do that, the eigenvector B" r, J, ; q should be scale up first by exp(−2åJ"q) to change

into î" r, J, ; q . By change variable, M-equation is transform in [13]:

îd"(∞, J; q) =0 00 −4åJ"q î"(∞, J; q)

The evolution of ó J, q and ï J, q are then obtained

ó J, q = limr→∞

îA"(r, J; 0) exp åJr = ó J, 0

ï J, q = limr→∞

î""(r, J; 0) exp −4åJ2q exp −åJr = ï J, 0 exp −4åJ2q

As a result, the Nonlinear Fourier Transform spectral either in continuous and discrete is scaled by the

factor exp −4åJ"q . From the above solutions, we found that ó J is invariant along z. The eigenvalue,

J>, thus is invariant along a distance. In NFT perspective, we find that the operation of the Lax

convolution could be rewrite in a simple multiplication by exp −4åJ"q which is corresponding to

the particular J channel [1].

The input output channel model then is simplified in a below block diagram:

where Z J ,Z J> are noise in the spectral domain

H J ,H J> (exp −4åJ"q , exp −4åJ>"q )could be treated as a filter which cause the phase shift

of the signal. Theoretical, we should multiply exp 4åJ"q to compensate the phase shift filter. The

channel however is not ideally same as the filter factor in practical; thus we will launch a (J) pulse

represent (J) channel to obtain the real phase shift factor.

As different channels are with different eigenvalue ( J ), the nonlinear frequency division

multiplexing (NFDM) could be realized [1].

Now, we are going to simulate transmitting a QPSK basing on NFT and INFT in one channel with

one soliton modulation to see the performance of this new communication scheme. In practical,

H(J),HÓJ>Ô x XÒ(J),XÚÓJ>Ô YÒ(J),YÚÓJ>Ô

ZÙ(J),ZıÓJ>Ô


46

INFT + +

~A(r) ~•7A(r)

G G G

~•(r) 6±

ˆ≠i ,6±

kˆ≠i

NFT

RX

á̃,J 6Pú ßX

DA2 ;"ℎ −

12 ℎ

®

exp(Xl|f|"ℎ)

W™´e(q + ℎ, r) ö{} x ö7A{}

we cannot use above block diagram to model the pulse evolution because of two reasons. One is the loss that the Lax convolution have included in, the solution is not exactly as same as the solution we solve from

NFT. Even there are the distributed amplifier to compensate the loss, the amplifier is a linear

operator which could not directly adding to the nonlinear operator, Lax convolution. Thus, we

have to combine Split-Step Fourier Method used to solve the NLSE with INFT and NFT to

simulation. Firstly, we have to assign the eigenvalue(s) and a corresponding discrete spectrum.

Then, we will modulate the phase of discrete spectrum, á let say4å by ±zg"ó:I ± Tzg

" to obtain

QPSK. For the doing amplitude modulate to vary dBm, we could change the eigenvalue(s) to

change the power as Zëgy◊ = 4 ÿØ>@A (J>). After that, we will do INFT to transform á in time

domain to obtain the soliton(s) which is shown an example of only on soliton; then transmit it into

80km per span with 1600km in total through Split-Step Fourier Method.

The simulation block diagram is shown below:


47

5.3.2ResultofComputingNFT Let take the eigenvalue 1å an example, the pulse train after doing the phase modulation then INFT

is given by following:

Regards of the loss and noise in an ideal channel, the signal in the receive shows invariant

The channel, however, is not ideal. The fiber itself has the loss; that why we have to design a

distribution amplification along the distance. The amplifier on the other hand, would produce the

noise after the amplification. The signal in the receiver thus may not exactly be as same as figure

(a) but just slight distorted. Each pulse in the pulse trains can still be separated.

Figure 4.5 1-soilton pulse train in transmitter

(a)

(b) (c)

Figure 4.5 Reciever siganl with (a)loss and amplifier (b) loss, amplifier and noise


48

Now, we will simulate 1-solitionic pulse modulating on QPSK to investigate its BER versus dBm

in 1600km with 80km span. At that time, I set the baud rate 2G/s and 100 symbols in total because

the computational power of my laptop. The BER curve versus dBm result is show below:

In the simulation, it’s clearly to see that the error bit is almost zeros from dBm -10 to 7 while the

corresponding J are from 0.0441å to 2.2101å.

Comparing with the curve in figure 4.3.2a, it is shown that the performance of transmitting data

NFT and INFT communication scheme is better than the conventional linear compensation method

which are CD chromatics and Digital Backpropagation. The spectral information is not only to

compatible with both nonlinear effect and dispersion effect but also robust to the noise. To more

specify justify how well this method is, we show three kind dBm constellation diagram to see the

spectral signal distribution. From dBm -6 to -1, it’s clearly to see that the signal could be easy to

separate. The noise is also relative small. dBm=5 siganl distribution show a stranger Patten on the

complex plane but the information can still be encoded.


49

5 dBm

-1 dBm

dBm -6


50

Conclusion In long-haul optical fibre transmission, the high power optical pulse experience the chromatic

dispersion and nonlinear effect both at the same time when propagating along the z-direction.

There are several ways to describe these physics pheromones, but Nonlinear Schrodinger equation

(NLSE) nowadays is the most common one to use. NLSE have found that it can be solved by Split-

Step Fourier Method. As SSFM is a numerical method, the simulation result shows more accuracy

along with decreasing step size.

After doing the CD compensation, the received signal in higher dBm doesn't seem better. In the

signal distribution, constellation diagram, we found the whole signal distribution do the self-phase

rotation on the complex plane. This action is called self-phase modulation caused by the factor

exp(Xl f "q) . To deal with this problem, we review the SSFM then find that digital

backpropagation has a possibility to compensate the interplay of nonlinear and chromatic

dispersion effect. As simulation result, the signal doing the digital backpropagation perform much

better, however, lower BER is confined within few dBm ranges.

Until Professor Frank R. Kschishang find there is a relation between the Lax pair and the Nonlinear

Schrodinger equation (NLSE). Lax Pair this concept is developed by the mathematician, Professor

Peter D. Lax. It can be helpful for solving some nonlinear differential equation. Nonlinear Fourier

Transform thus is derivated from Lax Pair. Review to NLSE, it is known that there is the dispersion

term and nonlinear term. From the geometry view of the forward propagating pulse, dispersion

term will spread the pulse width while the nonlinear term does the compression effect on the pulse.

If there is a particular pulse which keeps the shape invariant undergo the interplay between

compression and spreading, this pulse can be used to do the data transmission. This invariant is

known as the soliton. To find the soliton, Nonlinear Fourier Transform is in use. J form the

positive complex plane is obtained from NFT. These J are formed the discrete spectra. These

discrete spectra are corresponding to a soliton in the time domain.

In practical, these solitons cannot be exactly invariant along propagation in the z-direction because

of the action of noise and amplifiers. We, however, can still see the pulse shape then the

information can be encoded form that. From simulation result the BER curve versus with dBm,

we could see BER almost zeros from dBm -10 to 8. It is shown that NFT can be implied into the

communication scheme; it is doing better than linear system approach.


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Discussion I'm going to prepare an experiment for QPSK transmission with NFT and INFT. The objective of

this experiment is to see if the simulated result from MATLAB fit the experimental data. If the

difference between these two data is too large, there might be something wrong with the simulating

algorithm. This algorithm will be modified. The experiment and simulation result will be written

in the revised report and documents.

From the previous sections, we found that Lax pair has no considering the loss term. Although the

amplifier seems compensates the loss only, it will affect the output shape of the soliton. To do this,

some researcher will try to find a suitable scaler factor then multiply the launching soliton directly

for a particular fiber link. This scaling process, however, is not always true in every case. It,

therefore, be better to develop another Lax pair which could derivate into NLSE with the loss term.

With that particular Lax pair, we then imply Nonlinear Fourier Transform to find out the solution.

On the other hand, there is a no noise model with more than one eigenvalue soliton. It will be one

of the research trades for developing the Nonlinear Frequency Multiplexing communication

scheme. Finally, a faster NFT algorithm has to be developed. One of reason is that the algorithm

using layer-peeling and searching method will be slow down when there are lots of eigenvalues.

The second reason is that we have to cut the pulse train then do the NFT by this algorithm one by

one. This part is really time consuming if there are more than 10^3 symbols.


52

References [1] C. Headley and G. P. Agrawal, Eds., Raman Amplification in Fiber Optical Communication

Systems (Academic Press, Boston, 2005). [2] A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 142 (1973).

[3] M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, part I: Mathematical tools,” IEEE Trans. Inf. Theory, vol. 60, no. 7, pp. 4312–4328, Jul. 2014. [4] M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, part II: Numerical methods,” IEEE Trans. Inf. Theory, vol. 60, no. 7, pp. 4329–4345, Jul. 2014. [5] M. I. Yousefi and F. R. Kschischang, “Information transmission using the nonlinear Fourier transform, part III: Spectrum modulation,” IEEE Trans. Inf. Theory, vol. 60, no. 7 , pp. 4346–4369, Jul. 2014. [6] G. P. Agrawal, 4th Eds., Nonlinear Fiber Optics (Academic Press, San Diego, 2007). [7] G. A. Thomas, B. L. Shraiman, P. F. Glodis, and M. J. Stephan, Nature 404, 262 (2000). [8] D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, New York, 1982), Chaps. 8 and 12. [9] Y. R. Shen, Principles of Nonlinear Optics (Wiley, New York, 1984). [10]P.U. Suarez, “An introduction of the Split Step Fourier Method using MATLAB,”, 18 Jau.

2013 [11]A. Kalander and W. Wang., Time-Domain Digital Back Progration for Optical

Communication in 28nm FD-SOI (Chalmers University of Technology and University of Gothenburg, Gothenburg, Sweden 2017)

[12] J. Shao, S. Kumar and X. Liang, “Ideal optical backpropagation of scalar NLSE using

dispersion-decreasing fibers for WDM transmission,” OPTICS EXPRESS, vol. 21, no. 23, Nov 2013

[13] A.I. Yousefi., “Information Transmission Using the Nonlinear Fourier Transform” (University of Toronto 2013)

[14] V. Aref “Control and Detection of Discrete of Spectral Amplitude in Nonlinear Fourier

Spectrum (Nokia Bell Labs, Stuttgart, Gemany)

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