Nonlinear Fourier transform for optical data processing and transmission: advances and perspectives

Abstract

Fiber-optic communication systems are nowadays facing serious challenges due to the fast growing demand on capacity from various new applications and services. It is now well recognized that nonlinear effects limit the spectral efficiency and transmission reach of modern fiber-optic communications. Nonlinearity compensation is therefore widely believed to be of paramount importance for increasing the capacity of future optical networks. Recently, there has been steadily growing interest in the application of a powerful mathematical tool—the nonlinear Fourier transform (NFT)—in the development of fundamentally novel nonlinearity mitigation tools for fiber-optic channels. It has been recognized that, within this paradigm, the nonlinear crosstalk due to the Kerr effect is effectively absent, and fiber nonlinearity due to the Kerr effect can enter as a constructive element rather than a degrading factor. The novelty and the mathematical complexity of the NFT, the versatility of the proposed system designs, and the lack of a unified vision of an optimal NFT-type communication system, however, constitute significant difficulties for communication researchers. In this paper, we therefore survey the existing approaches in a common framework and review the progress in this area with a focus on practical implementation aspects. First, an overview of existing key algorithms for the efficacious computation of the direct and inverse NFT is given, and the issues of accuracy and numerical complexity are elucidated. We then describe different approaches for the utilization of the NFT in practical transmission schemes. After that we discuss the differences, advantages, and challenges of various recently emerged system designs employing the NFT, as well as the spectral efficiency estimates available up-to-date. With many practical implementation aspects still being open, our mini-review is aimed at helping researchers assess the perspectives, understand the bottlenecks, and envision the development paths in the upcoming NFT-based transmission technologies.

Full text

Nonlinear Fourier transform for optical data
processing and transmission: advances and
perspectives
SERGEI K. TURITSYN,1,2,*JAROSLAW E. PRILEPSKY,1SON THAI LE,3SANDER WAHLS,4LEONID L. FRUMIN,2,5
MORTEZA KAMALIAN,1AND STANISLAV A. DEREVYANKO6
1Aston Institute of Photonic Technologies, Aston University, Birmingham B4 7ET, UK
2Novosibirsk State University, Novosibirsk 630090, Russia
3Nokia Bell Labs, Stuttgart, Germany
4Delft Center for Systems and Control, Delft University of Technology, 2628 CD, Delft, The Netherlands
5Institute of Automation and Electrometry, Siberian Branch, Russian Academy of Sciences, Novosibirsk 630090, Russia
6Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev, Beer Sheva 84105, Israel
*Corresponding author: s.k.turitsyn@aston.ac.uk
Received 14 November 2016; revised 13 January 2017; accepted 15 January 2017 (Doc. ID 280880); published 28 February 2017
Fiber-optic communication systems are nowadays facing serious challenges due to the fast growing demand on capacity
from various new applications and services. It is now well recognized that nonlinear effects limit the spectral efficiency
and transmission reach of modern fiber-optic communications. Nonlinearity compensation is therefore widely believed
to be of paramount importance for increasing the capacity of future optical networks. Recently, there has been steadily
growing interest in the application of a powerful mathematical tool—the nonlinear Fourier transform (NFT)—in the
development of fundamentally novel nonlinearity mitigation tools for fiber-optic channels. It has been recognized that,
within this paradigm, the nonlinear crosstalk due to the Kerr effect is effectively absent, and fiber nonlinearity due to the
Kerr effect can enter as a constructive element rather than a degrading factor. The novelty and the mathematical com-
plexity of the NFT, the versatility of the proposed system designs, and the lack of a unified vision of an optimal NFT-type
communication system, however, constitute significant difficulties for communication researchers. In this paper, we
therefore survey the existing approaches in a common framework and review the progress in this area with a focus on
practical implementation aspects. First, an overview of existing key algorithms for the efficacious computation of the
direct and inverse NFT is given, and the issues of accuracy and numerical complexity are elucidated. We then describe
different approaches for the utilization of the NFT in practical transmission schemes. After that we discuss the
differences, advantages, and challenges of various recently emerged system designs employing the NFT, as well as
the spectral efficiency estimates available up-to-date. With many practical implementation aspects still being open,
our mini-review is aimed at helping researchers assess the perspectives, understand the bottlenecks, and envision
the development paths in the upcoming NFT-based transmission technologies. © 2017 Optical Society of America
OCIS codes: (060.1660) Coherent communications; (060.2330) Fiber optics communications; (070.4340) Nonlinear optical signal
processing; (290.3200) Inverse scattering.
https://doi.org/10.1364/OPTICA.4.000307
1. INTRODUCTION
The exponential surge in global data traffic driven by the skyrock-
eting proliferation of different bandwidth-hungry online services,
such as cloud computing, on-demand HD video streams, and on-
line business analytics, brings about escalating pressure on the
speed (capacity) and quality (bit error rate) characteristics of in-
formation flows interconnecting individual network participants
[1–5]. Optical fiber systems are the backbone of the global tele-
communication networks. It is hard to overstate the impact that
fiber communications have made on the economy, public ser-
vices, society, and almost all aspects of our lives. It is also well
recognized [3–12] that rapidly increasing data rates in the core
fiber communication systems are quickly approaching the limits
of current transmission technologies, many of which were origi-
nally developed for linear (e.g., radio) communication [13,14].
Optical fiber channels are very different from wireless and
other traditional linear channels. The main order effect here is
the signal attenuation due to fiber loss that is compensated by
optical amplifiers, e.g., erbium-doped amplifiers (EDFAs) or dis-
tributed Raman amplification (DRA) [1]. Optical amplification
adds amplified spontaneous emission (ASE) noise that mixes with
the signal during the transmission. In general, optical noise
2334-2536/17/030307-16 Journal © 2017 Optical Society of America
Review Article Vol. 4, No. 3 / March 2017 / Optica 307

together with dispersion and nonlinearity are the three key physical
effects having a major impact on signal transmission in optical
fiber links. The successful implementation of the “fifth genera-
tion”of optical transmission systems, operating with coherent de-
tection, wavelength division multiplexing (WDM), advanced
multilevel modulation formats, and digital signal processing
(DSP), has led to the possibility of channel rates exceeding
100 Gb/s [2,5,7]. The key to this breakthrough is the digital mit-
igation of the most important linear transmission impairments,
such as chromatic and polarization mode dispersion [1,2,15].
After the equalization of linear effects, noise and nonlinearity be-
come the principal factors deteriorating the performance of opti-
cal networks. Indeed, the Kerr nonlinear effect at high signal
powers leads to power-dependent nonlinear transmission signal
distortions in the fiber channel. In this sense, fiber nonlinearity
has a detrimental effect on the transmission of information, and,
thereby, serious worldwide efforts are aimed at the suppression or
compensation of nonlinear impairments. It was stressed in [6]
that, in contrast to linear channels [13], the spectral efficiency
of optical fiber WDM networks cannot increase indefinitely
and starts to decay at high signal powers due to the spectral chan-
nel crosstalk imposed by fiber nonlinearity. The nonlinear fiber
effects are behind the infamous “nonlinear capacity limit”prob-
lem [5,7,9,10,16].
In spite of the immense recent progress in optical communi-
cation technologies, the next step in the future systems’design has
appeared to be not so straightforward [17]. Space-division multi-
plexing (SDM) is considered by many engineers as a promising
direction in the evolution of optical transmission systems [18].
However, the SDM technology requires a considerable upgrade
in the infrastructure. The compensation of nonlinearity-induced
effects is a principal research and engineering challenge, and it is
likely to remain so in the future. A plethora of nonlinearity com-
pensation methods have been proposed, including digital back-
propagation (DBP) [19], digital [20] and optical [21,22] phase
conjugation (spectral inversion), and phase-conjugated twin
waves [23], to mention just a few important advances (see reviews
[17,24]). Note that in most of the compensation techniques, the
fiber nonlinearity is treated as an undesirable effect, and the
purpose of all of those methods is just to mitigate or suppress
its impact.
There is, however, an alternative and not yet widely popular
viewpoint: since fiber channels are inherently nonlinear, rather
than treating nonlinearity as a completely destructive feature,
it can be considered as an essential element in the design of fiber
communication systems. There is growing evidence of the neces-
sity of a novel paradigm and radically new approaches to coding,
transmission, and processing of information, which would take
into account the nonlinear properties of the optical fiber. In this
work, we describe one such recently resurrected approach: the
nonlinear Fourier transform (NFT). The NFT-based transmis-
sion method belongs to a conceptually different bevy of tech-
niques compared to those mentioned above [25]: here the
nonlinearity enters as an undetachable element of the processing
and transmission, defining the features of the system architecture
and its characteristics. The application of such paradigm-shifting
nonlinear methods means that some common “linear”method-
ology may need to be reconsidered or appended with a new mean-
ing. For instance, in addition to the usual notions of frequency,
spectral power, and bandwidth, one has to work with their
nonlinear analogues that can be drastically non-conventional,
but can serve as new well-defined and adjustable characteristics
of the optical signal in nonlinear systems. It will be convenient
further to distinguish between signal characteristics in the stan-
dard frequency domain and those in the so-called nonlinear
spectral domain. Note also that for the sake of clarity, within this
review we address only the single-mode and single-polarization
fiber transmission model, leaving aside the polarization degree
of freedom and specific peculiarities of multimode systems [5].
We would like to stress that the beauty of the mathematical
theory presented here is inevitably spoiled by the limits of
applicability of the master model—the integrable nonlinear
Schrödinger equation (NLSE)—for the description of signal
transmission in fiber links. The application of the NFT methods
is limited by deviations of the optical signal dynamics from the
NLSE channel model. Apart from the deviations due to periodic
variations of signal power caused by alternation of loss and gain in
practical systems (in that case the NLSE emanates as a leading
approximation within the path-averaged model), various other ef-
fects contribute to perturbations that are not accounted for by the
pure NLSE; e.g., higher-order dispersion [26–28], polarization
effects [27,29,30,15], the Raman effect [27,28,31], and the
acoustic effects (electrostriction) [32] all limit the validity of this
channel model. Consideration of the impact of these effects is
beyond the scope of this survey, which is focused on the NFT
techniques for the NLSE-based channel.
To assist reading of the paper, Supplement 1 contains a list of
acronyms used in our review.
2. PRINCIPLES OF INTEGRABILITY AND NFT
In physics and, notably, in photonics, many important phenom-
ena and the evolution of underlying systems can be modelled by
the NLSE [1,27,28,33–36]. In particular, the NLSE is a principal
master model governing the evolution of the slow-varying optical
field envelope qz;t(zwill further play the role of the distance
along the fiber while tis the time variable) along a single-mode
fiber,
i∂q
∂z1
2
∂2q
∂t2jqj2q0:(1)
Note that this is the NLSE in its normalized form. Here and in
what follows, the upper sign in Eq. (1)(“+“) corresponds to
anomalous fiber dispersion, while the lower one (“−“) refers to
the normal dispersion case. Formally, the NLSE (1) describes
the evolution of light in a lossless optical fiber under the effects
of dispersion and Kerr nonlinearity. Albeit all real fibers, certainly,
have losses, this model appears as a result of averaging over peri-
odic gain and loss variation, leading to effectively conservative sig-
nal evolution [27,28,33,34]. Close to ideal compensation of losses
along optical fiber is possible in specific schemes of the so-called
ultra-long fiber lasers DRA [37,38]. Such a quasi-lossless
transmission was demonstrated experimentally in [38,39].
The possibility to approximate signal evolution in practical
fiber channels by the NLSE gives a remarkable opportunity to
apply advanced mathematical techniques, developed in the
1970s, to optical communications. It was first shown in the semi-
nal work by Zakharov and Shabat [40] that Eq. (1) belongs to the
class of the so-called integrable nonlinear systems. The mathemati-
cal method, widely known in the physical and mathematical com-
munities as the inverse scattering transform (IST), can be applied
Review Article Vol. 4, No. 3 / March 2017 / Optica 308

to find the solution of integrable nonlinear equations. In this con-
text, one can think of integrability as an elegant transform of the
original nonlinear system into the so-called action-angle variables
corresponding to a set of uncoupled trivial evolutionary equa-
tions. Mathematically, this can be treated as an effective lineari-
zation of the nonlinear evolution. There exists a vast amount of
literature where the integrability notion is elucidated in great
detail; see, e.g., [28,35,36,40–44]. The integrability itself implies
a lot of consequences in both mathematical and physical contexts.
For example, the NLSE (1) possesses an infinite number of con-
served quantities: while the conservation of power, momentum,
and Hamiltonian for Eq. (1) is relatively obvious, the rest of the
conserved quantities are nontrivial. As we will show below, a lot of
fruitful ideas based on the conserved quantities have been
successfully implemented.
In a nutshell, NFTs can be used to solve initial value problems
for a special class of nonlinear evolutionary equations. In fiber op-
tics, where the signal evolution occurs along the fiber, the initial
conditions correspond to the time-domain waveform at the trans-
mitter. Similar to conventional Fourier transform (FT), initial con-
ditions of the integrable nonlinear equations (such as the NLSE or
a Manakov system [45], corresponding to the integrable two-com-
ponent NLSE generalization) can be decomposed into (nonlinear)
spectral data. For the NLSE this is done through the solution of a
linear scattering problem known as the Zakharov–Shabat problem
(ZSP) [40,41]. Any solution of the NLSE can be presented as the
evolution of nonlinear spectral data that evolve effectively in a lin-
ear manner. The inverse transform, namely the recovery of the
space-time-domain field distribution from the known nonlinear
modes, is classically implemented through solution of the
Gelfand–Levitan–Marchenko equation (GLME). Altogether, this
means that such a spectral transform can be interpreted as the di-
rect nonlinear analog of the FT [35,40,41]: similarly to the FT
transforming the dispersion of a linear propagation to a phase ro-
tation in frequency space, the NFT recasts both the nonlinearity
and dispersion of the NLSE into a simple decoupled evolution of
nonlinear spectral data inside the NFT domain (see Fig. 1); the
latter plays the role of Fourier spectrum for nonlinear problems.
By performing the NFT of a given profile qz; t, we segregate
two distinct components, the dispersive nonlinear radiation and
the non-dispersive solitons, although either of these two can be
absent for some specific profiles. For normal dispersion, the in-
puts localized in time cannot nucleate solitons. For the dispersive
part of the nonlinear spectrum, the NLSE evolution produces ex-
actly the linear phase rotation of spectral components as we have
for linear systems. For the anomalous dispersion, the solitons, as-
sociated with the complex “nonlinear frequencies”(eigenvalues),
in addition to the rotation of soliton phases, can involve either the
motion as a whole or a more nontrivial beating dynamics of
bound states—the so-called multisoliton breathers [28,33],
although inside the NFT domain the solitonic degrees of freedom
remain decoupled. Note that NFT methods are much richer, more
flexible, and more versatile with respect to the system design and
performance compared to just soliton-based techniques, studied
previously in much detail [27,28,33]. In the NFT methods deal-
ing with the discrete part of the nonlinear spectrum (solitonic
eigenvalues), the information carriers are not the fundamental sol-
itons themselves but the NFT parameters (nonlinear spectral data)
attributed to a multisolution pulse. In this sense, the traditional
soliton-based transmission emerges as the simplest (and not nec-
essarily optimal) subclass of the NFT methods. The NFT com-
munications are, to some extent, the extension of not only the
soliton-based approach but also the coherent communication idea
itself: while for the latter both the signal’s amplitude and phase are
used for modulation, the NFT approach goes further and employs
the nonlinear characteristics of the signal.
By using the NFT all effects caused by the fiber Kerr nonlin-
earity can be described as a trivial change of the phase in the non-
linear spectrum. This paves the way to fundamentally novel
nonlinear techniques for compensation of the effects of chromatic
dispersion and fiber nonlinearity. In 1993 Hasegawa and Nyu
[42] (see also Chapter 4 of [28]) proposed the truly innovative
idea of eigenvalue communications based on the exploitation of
discrete eigenvalues (corresponding to solitons) emerging in
the NFT signal decomposition to encode and transmit informa-
tion [40–42]. This approach potentially solves the problem of
nonlinear crosstalk that is one of the major challenges in optical
WDM networks. The concept of Hasegawa and Nyu has recently
been resurrected with various modifications and further exten-
sions [25,44,46–79], including the new direction employing
the modulation of continuous nonlinear spectrum [51–
60,62,63,79], first experiments using transmission and processing
of discrete eigenvalues [64–67] and continuous spectrum [58,59],
NFT-based DBP [50,75], and, most recently, polarization divi-
sion multiplexing with the NFT [74]. The transition from the
space-time domain into the nonlinear spectral domain and back
is achieved by performing the NFT operations. Generally, there
exists the straightforward interrelation not only in the ideology of
FT and NFT methods but also between the linear and NFT spec-
tra [35]: in the low-power limit one can prove the asymptotic
equivalence of the linear FT and NFT [41]. However, in spite
of the similarity, the explicit form of the NFT operations is much
more mathematically involved as compared to the simple profile
convolution with exponentials for the usual FT. Thus, the com-
plexity of the NFT operations and the “change of notions”often
bring about difficulties for the communication engineers. The
purpose of our survey is to demonstrate without going deep into
mathematical details how the NFT method and various integra-
bility features can be employed for the sake of efficient optical
transmission, also summing up the existing numerical tools that
can be employed for the computation of the NFT.
This paper is organized as follows. First, in Section 3,we
describe the NLSE model for realistic optical fiber systems. In
Fig. 1. Exemplary NF spectrum (anomalous dispersion case), contain-
ing solitons (discrete eigenvalues) and continuous nonlinear spectrum
(depicted on the real axis ξ).
Review Article Vol. 4, No. 3 / March 2017 / Optica 309

Section 4we introduce minimally required notations for the NFT
operations, including the periodic NFT variant. Then, in
Section 5, we overview the existing numerical methods for the
calculation of the direct and inverse NFT operations.
Subsection 5.C is focused on fast NFT algorithms. Then, in
Section 6, we directly address different NFT-based transmission
methods, also presenting some new results and generalizations.
After that, in Section 7, we overview recent results with regard
to the efficiency of NFT-based optical transmission mehods.
The paper ends with Section 8, the conclusion, where we also
outline some NFT perspectives and further development
directions.
3. GENERALIZED NLSE MODEL OF OPTICAL
FIBER
The principal master model for the electrical field qz; tevolu-
tion inside a single-mode optical fiber with the account of am-
plification can be written as a generalized NLSE (GNLSE)
[1,27,28]:
i∂q
∂z
−
β2
2
∂2q
∂t2γjqj2qigzqηz; t;(2)
where zis the distance (in kilometers) along the fiber, and tis the
time (in picosconds) in the frame co-moving with the velocity of
the envelope. The parameter β2(in ps2∕km) is the characteristic
of chromatic dispersion that is negative for the anomalous
dispersion (the most important practical case) or β2>0for
the normal dispersion (jβ2jcan vary from 5ps
2∕km to
60 ps2∕km at the typical operating wavelength of 1550 nm);
further, for a standard single-mode fiber we assume β2
−22 ps2∕km.γis the nonlinear Kerr coefficient, typically γ
1.27 W−1km−1. The function gzcharacterizes thegain–loss
profile of a particular amplification scheme. For the quasi-lossless
DRA scheme, the function gz≡0[38,39], resulting in the
lossless NLSE perturbed by an additive white Gaussian noise
(AWGN) term ηz; t(having zero mean). The latter is
completely characterized by the ASE spectral power density D:
Eηz;t¯
ηz0;t0  2Dδt−t0δz−z0;(3)
where the overbar means the complex conjugate, E·is the ex-
pectation value, and δ·is the Dirac delta-function. In the case of
ideal DRA we have 2Dhν0KTα, where αis the fiber loss co-
efficient, typically α≈0.2 dB∕km at the carrying wavelength
λ01.55 μm;KTis the temperature-dependent factor (related
to the phonon-occupancy factor) that characterizes the Raman
pump providing the distributed gain; KTis typically in the range
from 1.1 to 1.2; and ν0is the carrying frequency of the signal
corresponding to λ0:ν0193.55 THz. Taking these typical
values of parameters, one estimates the order of characteristic
noise intensity per complex signal component (polarization),
per unit of propagation length and per unit of bandwidth, to
be D∼10−21 J∕km; for KT1.13 we have: D≈3.3 ·10−21 J∕
km. Such an idealized form of optical channel [the lossless
integrable NLSE (1) weakly perturbed by AWGN] suits NFT
applications [52] best as it is close to the integrable NLSE (1).
However, the NFT method can still be successfully applied to
the EDFA (lumped) [53] or non-ideal DRA [54,55] cases. For the
EDFA we have gz−α∕2in between point-action (lumped)
amplifiers, but the signal is boosted to the initial power level after
each span of length Za. For the DRA scheme we have a more
complicated non-flat profile of gz, e.g., that corresponding
to the open-cavity random distributed feedback laser-based am-
plification as it provides the best performance among various
other Raman amplification schemes [80], where the gain profile
recurs periodically after each span of the length Za. Now, by using
the path-averaged approach [27,28,34,53–55] one can introduce
the new field variable as ˜qz;tqz; tG1∕2z, where
Gzexp2RZa
0gzdz, and this substitution recasts Eq. (2)
into the lossless NLSE for ˜qz; twith the z-dependent
factor Gznear the nonlinear term. In the leading order with
respect to Za∕Zd, with Zdbeing the dispersion length
[ZdW2jβ2j−1, where Wis the signal’s bandwidth], the dis-
tance-dependent nonlinearity coefficient can be approximated
with the averaged value ˜γγZ−1
aRZa
0Gzdz, such that we ar-
rive at the lossless path-averaged (LPA) NLSE written for ˜qz;t
with constant coefficients with ˜γin place of the original γfrom
Eq. (2); for EDFA system ˜γγGa−1∕lnGawith Ga
exp−αZa. In general, the applicability limits of the LPA
NLSE model depend on the signal power, bandwidth, and trans-
mission distance. The accuracy of the LPA NLSE for optical links
with EDFA was investigated in [53] for a link distance of
2000 km, signal powers up to 8 dBm, and bandwidths up to
80 GHz. It was found that the LPA NLSE model can be applied
with a normalized mean square error below −20 dBm when the
signal power is below 3 dBm, almost independently of the signal
bandwidth. The LPA model was found to work under more re-
laxed requirements with non-ideal RDA [55] as this amplification
scheme provides a lower gain variation along the link, depending
on the specific RDA scheme. The applicability limits of this
model for the EDFA case with regards to NFT applications were
presented in [53], and for the RDA scheme in [55]. The noise
term is assumed to possess the same properties as we have for
the ideal RDA case; i.e., it is the circular AWGN with only a dif-
ferent expression for the intensity
˜
D∼10−21 J∕km. For the
EDFA system 2
˜
Dnsphν0G−1
a−1∕Za, where nsp ≈1is the
spontaneous emission factor [33].
Having recast our GNLSE (2) to the approximate LPA NLSE
form with the distance-independent coefficients or using the ideal
RDA model, we introduce the normalizations
t∕Ts→t; z∕Zs→z; q∕ffiffiffiffiffi
P0
p→q; (4)
with P0γZs−1(or the same with the ˜γfor EDFA or non-ideal
DRA and resulting LPA NLSE), ZsT2
s∕jβ2j, and we finally
have the standard NLSE model, Eq. (1), but with the AWGN
term in the r.h.s. In Eq. (4) any of three parameters, Ts,Zs,
or P0, can be taken for the normalization, but then the remaining
two have to be properly adjusted: Tscan be, e.g., the extent of our
symbol, or setting it to be the reciprocal bandwidths, TsW−1,
our normalized distance unit becomes the dispersive length men-
tioned above; in soliton-related problems Tsis often set as an
individual soliton full width at half maximum (FWHM). The
noise intensity has to be normalized in accordance with
Eq. (4): DZsP0Ts−1→D. We also omit tildes in ˜q,˜γ, and
˜
Dfurther, assuming that Eq. (1) refers to a simplified description
pertaining to a particular amplification scheme. Of course, the
results for the NFT application for non-ideal DRA or EDFA
schemes are expected to show slightly worse performance as com-
pared to the ideal DRA [53–55], though the higher-order correc-
tions with respect to Za∕Zdmay also be taken into account by
using, e.g., the guiding center approximation [28].
Review Article Vol. 4, No. 3 / March 2017 / Optica 310

4. EXPLICIT FORM OF NFT OPERATIONS
In this section, the direct (forward) and inverse NFT (INFT) are
introduced. The NFT considers the signal qz; tat a fixed loca-
tion zz0and returns the corresponding NFT spectrum. The
INFT reverses this process; i.e., given a NFT spectrum it returns
the corresponding signal qz0;t. Since only the main features can
be outlined here, the reader is referred to [28,35,40–47] for fur-
ther details. The section ends with some properties of the periodic
NFT (PNFT).
A. Direct NFT
The direct NFT is computed from specific (auxiliary) solutions
v1;2t;ζv1;2t;ζ;z0to the ZSP [40]
dv1
dt qz0;tv2−iζv1;dv2
dt ¯qz0;tv1iζv2(5)
for different values of the complex parameter ζξiη, which
will play the role of a nonlinear analog of frequency. The signal
qz;tacts as a potential. The upper and lower signs correspond
to the anomalous and normal dispersion according to Eq. (1).
Under the assumption that qz0;tdecays at least exponentially
for t→∞, specific solutions (the so-called Jost functions)
ϕ1;2t;ζand ψ1;2t;ζto the ZSP can be obtained from the
boundary conditions:
ϕ1t;ζe−iζto1;ϕ2t;ζo1for t→−∞;
(6)
ψ1t;ζo1;ψ2t;ζeiζto1for t→∞:
(7)
In practical realization of the transmission schemes, the pulse qt
is truncated and we operate in the so-called burst mode [52]; see
Fig. 2. The above pairs of functions solve the ZSP, and all these
different solutions are linearly dependent:
ϕ1ϕ2aζ ˜
ψ1
˜
ψ2bζ ψ1ψ2;(8)
˜
ϕ1
˜
ϕ2−˜aζ ψ1ψ2
˜
bζ ˜
ψ1
˜
ψ2:(9)
The functions aζand bζare known as the Jost scattering
coefficients. They serve as the basis on which the NFT spectrum
is defined. Due to the boundary conditions, we have
aζ lim
t→∞
ϕ1t;ζeiζt;bζ lim
t→∞
ϕ2t;ζe−iζt:(10)
Another important property of the Jost scattering coefficients is
that they satisfy jaξj2jbξj21for all real ξ, where the
upper and lower signs refer to those in Eqs. (1) and (5).
The NFT spectrum of the signal qz0;tconsists of two parts.
The first part is given by either the left or the right reflection
coefficient (RC), respectively:
lξ¯
bξ∕aξ;rξbξ∕aξ;ξ∈R:(11)
The second part of the NFT spectrum consists of the discrete
eigenvalues ζnξniηn, which are the eigenvalues of the
ZSP with a positive imaginary part η>0, and their associated
left or right norming constants (also often referred to as spectral
amplitudes), which are defined by the residue of lζ(or rζ)at
the point ζn:
lnbζna0ζn−1;r
nbζn∕a0ζn;(12)
where the prime designates the derivative with respect to ζ.We
therefore have four real parameters defining each solitary degree of
freedom. The complete (left or right) NFT spectrum of the signal
qz0;tis given by
Σlflξ;ζn;lnN
n1g;Σrfrξ;ζn;rnN
n1g;(13)
where Nis the total number of solitons in the signal; an exem-
plary NF spectrum is shown in Fig. 1. The NF spectrum char-
acterizes the signal qz0;tcompletely and can be used to recover
the corresponding time-domain signal given that it vanishes suf-
ficiently fast for jtj→∞. Note that in the normal dispersion
case, the signal cannot have solitonic components and either lξ
or rξis sufficient to uniquely recover the corresponding profile
qz0;t. The zdependence of the NF spectrum, Σl;rz, is given
by the following expressions. The eigenvalues ζnare independent
on z. For the remaining quantities, we have
lξ;zlξ;z0e−2iξ2z−z0;l
nξ;zlnξ;z0e−2iζ2
nz−z0;
rξ;zrξ;z0e2iξ2z−z0;r
nξ;zrnξ;z0e2iζ2
nz−z0:
(14)
Finally, we remark that the solitons disappear and the NFT re-
duces to conventional FT when the signal power becomes small.
Any rescaled signal qϵtϵqtsatisfies [35,41]
ϵ−1¯rϵξ;ϵ−1lϵξ→−qωjω−2ξwhen ϵ→0;(15)
where qωR∞
−∞qte−iωtdt. Also note that, in optics, the ZSP
(5) also appears widely in the field of Bragg grating synthesis
[1,81–83], where the functions v1;2play the role of slowly varying
coupled mode amplitudes: the anomalous dispersion [the upper
sign in Eqs. (5) and (16)] corresponds to the coupling of co-
propagating waves, while the normal dispersion (the lower sign)
refers to counter-propagating modes.
B. INFT Operation (Left Set of Scattering Data)
The INFT maps the scattering data Σl;r onto the field qt. This is
classically achieved via the GLME for the unknown functions
K1;2t;t0[28,35,40,41,44]. The GLME, written in terms of
the left scattering data, reads
¯
K1τ;τ0Zτ
−∞
dyLτ0yK2τ;y0;
¯
K2τ;τ0Lττ0Zτ
−∞
dyLτ0yK1τ;y0
(16)
for τ>τ0, where the upper and lower signs correspond to upper
and lower ones in Eqs. (1) and (5). In the realistic applications,
where the operations are performed on a finite interval of τ, say
0<τ<T, we have a finite region for the change of τ0,τ0<jτj.
For the anomalous dispersion [the sign “−“in Eqs. (16)] the
quantity Lτcan contain contributions from both solitonic
(discrete) and radiation (continuous) spectrum parts, Lτ
LsolτLrad τ, where
Lsolτ−iX
n
lne−iζnτ;L
radτ 1
2πZ∞
−∞
dξlξe−iξτ
(17)
and we have assumed that all discrete eigenvalues have a multi-
plicity one. The “nonlinear time”variable τis thus Fourier
Review Article Vol. 4, No. 3 / March 2017 / Optica 311

conjugated to the “nonlinear frequency”ξ, so that one can start
not from the ξdomain but immediately from the functions given
by Eq. (17) in the τdomain. In this paper, we have chosen to
work with the left reflection coefficient, l, corresponding to
the GLME inversion around −∞. The reason for such choice
is that (as we shall see below) this is a common convention in
the fiber Bragg grating reconstruction problems from which
we borrow most of our INFT numerical algorithms. Having
solved the GLME (16) for K1;2τ;τ0, the sought solution in
the space-time domain is recovered as qt−2¯
K2t;t. For
the soliton-free case we have Lsolτ0, and the only quantity
participating in Eq. (16) is the FT of RC lξ:Lτ≡Lradτ.
When one is interested in the solution qz0;tat some distance
zz0, the quantity lξin Eq. (17) is replaced with lz0;ξ.
The resulting solution of the GLME (16) becomes a function
of z0:K1;2z0;τ;τ0.
C. Periodic NFT
The usual NFT operations assume that the optical signal, qz0;t,
decays as t→∞. So the ordinary NFT assumes that we have a
burst-mode transmission, Fig. 2; i.e., at each zz0the signal
duration and the processing region coincide. However, in com-
munication applications it is often more convenient to work with
periodic signals for the processing of a data stream: the periodicity
assumption in our notations is expressed as qz0;tqz0;t
Tpfor the period Tp. Thus, the PNFT may be considered as a
natural choice for the replacement of linear (say, FFT-based)
processing elements. The PNFT was recently introduced within
the circle of available solutions for the nonlinear signal processing
in [84,85]. Basically, the PNFT offers the same possibilities for
the communication system design and concepts (with the use
of the periodically continued signals) as the NFT does for the
vanishing signals by adding a cyclic prefix extension instead of
zero-padded wings for ordinary NFT (Fig. 2). Together with this,
the periodicity assumption can bring about some other benefits:
(i) Only a finite part of a periodic signal (one period) represents
the whole signal, so we do not have to process the entire interval
accounting for the dispersion-induced memory, as it occurs for
the ordinary NFT; see Fig. 2. Because of this, one can have
a considerable processing speed-up when using PNFT.
(ii) When using an ordinary NFT, in particular, within nonlinear
synthesis [51–53], it is difficult to control the time duration of
the resulting wave-shapes. Using the PNFT, where signals have a
finite “meaningful”time duration (the PNFT period), we can
attain more control over the time-domain profiles. (iii) For the
PNFT, the encoding schemes can be, to some extent, based
on the encoders of currently used communication systems, as
the PNFT shares a cyclic-prefix profile extension idea.
(iv) Producing periodic solutions of the NLSE could be generally
done using Riemann theta functions that can be seen as the multi-
dimensional generalization of the FT, such that some properties
of linear modulation can still be kept within the PNFT paradigm.
(v) By using periodically extended signals we can have a continu-
ous stream of data without sudden droppings of power, thus
reducing the peak-to-average power ratio (PAPR), in contrast
to the burst mode with the ordinary NFT usage. For the sake
of completeness we briefly describe below some basic elements
of PNFT (see [85–89] for examples of such a communication
system).
1. Direct PNFT
Similarly to the case of ordinary NFT, in the periodic problem we
have two parts of nonlinear spectrum associated with a general
periodic time-domain profile: the constant main spectrum, which
serves as an analog of soliton eigenvalues, and the dynamical aux-
iliary spectrum. In contrast to the ordinary NFT, here both parts
of the spectrum consist of discrete points and there is no continu-
ous component. To define the scattering data we now have to deal
with the solutions (the so-called Bloch solutions) of ZSP (5) with
a periodic potential, qz0;tqz0;t Tp, subject to condi-
tions φt0;t0;ζ1;0Tand ˜
φt0;t0;ζ0;1T, where t0is
an arbitrary base point. The so-called 2×2fundamental matrix is
defined through the Bloch ZSP solutions as Φt; t0;ζ
φt;t0;ζ;˜
φt;t0;ζ. Evaluating this fundamental matrix at
tt0Tp, one gets the monodromy matrix, Mt0;ζ
Φt0Tp;t0;ζ. The monodromy matrix plays a crucial role
in the Floquet theory, which deals with differential systems with
periodic structure. At the endpoints of stable bands, the Bloch
solutions are (anti-)periodic and the values of parameter ζ
corresponding to these endpoints, i.e. the main spectrum, M, can
be defined through the Floquet discriminant Δζ
1∕2Tr Mt0;ζas [84]
Mζm



Δζm1;dΔ
dζ


ζζm
≠0:(18)
Ag-band (g-gap) periodic solution of NLSE is the solution in
which there are only 2gelements in M[90,91]. The important
property of the main spectrum is that it remains invariant during
the pulse evolution along the zdirection. The definition of the
auxiliary spectrum, μiz; t, is given in Supplement 1.
2. Inverse PNFT
The inverse PNFT is the procedure for getting the (periodic in
time) profile qz;tstarting from given main and (evolved) aux-
iliary nonlinear spectrum parts. There are several methods to
construct finite-gap periodic solutions of the NLSE; see [84].
One can use the theta-function representation [92,93]
qz;tq0;0ΘW−jτ
ΘWjτeik0z−iω0t;(19)
where k0and ω0are some constants obtained from the nonlinear
spectrum, and the Riemann theta function, ΘWjτ, is defined
as [93]
Fig. 2. Burst mode for the window in vanishing signal processing
(ordinary NFT) and the processing window for the periodic signal with
cyclic extension (PNFT).
Review Article Vol. 4, No. 3 / March 2017 / Optica 312

ΘWjτ X
m∈Zg
exp2πimTWπimTτm:(20)
Here mis a g-dimensional vector with integer elements, and
Wπkzωtδ∕2is a vector calculated from the non-
linear spectrum. The set fk;ω;δ;τgis called the Riemann spec-
trum, and τis the Riemann (period) matrix [93]; their particular
values can be, again, obtained from the full set of nonlinear spec-
tral data. Thus, within the representation [Eq. (19)] the inverse
PNFT procedure can be reformulated as the problem of finding
the Riemann spectrum from the given nonlinear spectrum.
Although there is still currently a lack of a generic approach for
how to deal with the inverse PNFT, there are several software
packages allowing one to construct the profile in the time domain
using the periodic spectral data. For finding the Riemann spec-
trum, there are some packages and codes embedded in Maple,
Sage, and Mathematica [94–96]. For the second stage, which
is to construct the Riemann theta functions (20) using the
Riemann spectrum, in addition to the symbolic implementations
[97,98], some “hyper-fast”algorithms for the numerical
reconstruction of special classes of signals were proposed [93].
5. NUMERICAL ALGORITHMS FOR THE NFT
In this section, we overview existing numerical methods for the
forward (5) and inverse (16) NFT, paying particular attention for
the methods that have already been tested for the transmission
purposes. The goal of the forward NFT is to calculate the non-
linear spectrum Σz0(13) from the given space-time-domain
profile qz0;t. The INFT method must provide the time-
domain waveform starting from given Σ.
The signal qz0;tis in practice only known at the specific
points in time due to sampling operations, which means that
for the forward NFT the nonlinear spectrum has to be approxi-
mated based on the samples
qmqz0;T1εm−1;m1;…;M;
where T1is close enough to −∞such that the boundary condi-
tion in Eq. (6) is approximately satisfied for tT1and T2>T1
is sufficiently close to ∞such that Eq. (10) is approximately
satisfied, respectively. The parameter εT2−T1∕M−1
denotes the sampling interval.
The methods are classified according to how their numerical
complexity (in terms of floating point operations, flops) and the
accuracy of the result change as the number of sample points M
increases.
A. Algorithms for Direct NFT
Numerous algorithms for computing the NFT have been de-
scribed in the literature. The two most well-known are probably
the methods of Ablowitz–Ladik (AL) and Boffetta–Osborne
(BO). We will first describe these two methods and then briefly
list other approaches. More details can be found in [46,84,99].
We, however, note that our review does not, of course, cover all
existing possibilities for the NFT operations implementation
(e.g., in recent work [100] a bi-direction algorithm for the calcu-
lation of soliton norming constants was described), and we rather
concentrate on the methods that have already found their way
into optical transmission studies, although there are some new
methods that have yet to be tested; see, e.g., [101].
1. Boffetta–Osborne Transfer Method
The general idea is to approximate the Jost scattering functions
aζand bζusing Eq. (10). Therefore, Boffetta and Osborne
[102] assumed that the signal qz0;tis piecewise constant,
i.e., qz0;tqmconst:for t∈T1m−0.5ε;T1
m0.5ε, and solved the ZSP [Eq. (5)] in closed form under
that assumption. For each interval T1m−0.5ε;T1
m0.5ε, one has
ϕ1T1m0.5ε;ζ
ϕ2T1m0.5ε;ζTmϕ1T1m−0.5ε;ζ
ϕ2T1m−0.5ε;ζ;
(21)
where Tmζexpmε−iζqm
¯qmiζ:(22)
Here, expm·denotes the matrix exponential. Taking the boun-
dary conditions (6) and (10) into account, one finds that
aζ
bζ≈diageiζT20.5ε;e
−iζT20.5εTMζ×…
×T1ζe−iζT1−0.5ε:(23)
This approximation can be used straight-away to evaluate the RC
Eq. (11). In order to locate the discrete eigenvalues ζn, Boffetta
and Osborne proposed to apply Newton’s method to aζ. A non-
linear version of Parseval’s relation can be used to check whether
all discrete eigenvalues have been found [102]. The complexity
for evaluating Eq. (23) in a straightforward way is OM. The
total complexity of a search method to find the discrete eigenval-
ues is therefore OkiterNguessesM, where kiter is the average num-
ber of iterations per initial guess and Nguesses is the number of
initial guesses used. The complexity of evaluating the RC
Eq. (11) on a grid of Mnonlinear frequencies is OM2.
The BO method has a second-order approximation accuracy;
i.e., for any fixed ζζ0, the distance between the numerical
approximations of aζ0and bζ0and their true values is of
the order OM−2[102,103]. Note that the hidden constant
in the big-O notation depends on ζ. For the BO method, the
hidden constant was found to be ∼jζj−1for large ζin [103].
The BO method was used in the works [49,51–53] for the cal-
culation of continuous nonlinear spectrum for the nonlinear in-
verse synthesis scheme (see Subsection 5.B below). It also
demonstrated good results in the calculation of the perturbed dy-
namics of solitonic eigenvalues [103–105]. The calculation of
norming constants, requiring a0ζn, is described in [102,103].
In [103], the BO method was compared to the direct fourth-order
Runge–Kutta integration of the ZSP (5), where for the latter
method the hidden constant in the big-Onotation was found
to be ∼jζj4. It was concluded that, generally, the BO method
is more convenient especially when the wide range of ζvalues
is addressed.
2. Ablowitz–Ladik Discretization Method (Normalized)
The AL discretization [106,107] is another method widely used
for the NFT-based transmission [62–67]. It corresponds to the
approximation of the NLSE by a discrete integrable problem.
The method also takes the form in Eq. (23), but with
Tmζ 1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1ε2jqmj2
pZεqm
ε¯qmZ−1;Zeiζε:(24)
Review Article Vol. 4, No. 3 / March 2017 / Optica 313

For the location of zeros of aζ, one can again apply a search
routine. It was shown [46] that the AL algorithm can produce
some small spurious solitonic eigenvalues, which, however, can
be readily sorted out. In Ref. [56] the AL method was compared
with the BO method for realistic NFT-based transmission param-
eters, and it turned out that the AL method demonstrates slightly
better performance when applied to the calculation of continuous
spectrum (RC). The accuracy of the AL algorithm is of the second
order as for the BO scheme. This was, e.g., shown in [108] for a
variant of the AL algorithm that is commonly used in fiber Bragg
grating design. The relation of the scheme in [108] to the AL
algorithm below is elaborated in [109]. Apparently in contrast
to this, Boffetta and Osborne had observed in [102,110] that
the AL discretization achieves only first-order accuracy, when
the discrete eigenvalues computed by the AL and BO methods
were compared to exact analytical values. The AL discretization
that was investigated in [110], however, was an early version
[106], in which the coordinate transform Z1−iζε was used
instead of the now common transform Zeiζε that was given
later in [107]. In various numerical experiments that were reported
in [46], the errors of the AL and BO schemes decrease at similar
rates w.r.t. M. The same inference was confirmed in the study [56],
related to the true NFT-based transmission profiles.
3. Fourier Collocation Method
The Fourier collocation method has been used by the Osaka
group and co-authors [68–74]. Within this method the ZSP sol-
ution components v1;2are expanded in the Fourier series and the
ZSP itself is reformulated as an eigenvalue problem in the Fourier
space [43,46]. However, this method is inconvenient for the com-
putation of the continuous nonlinear spectrum and soliton norm-
ing constants, and it has been used only for eigenvalue
communication where the solitonic discrete eigenvalues them-
selves are adopted as information carriers. Another drawback
of this method is its numerical complexity: the method requires
the diagonalization of the non-Hermitial (for anomalous
dispersion) matrix, where the number of required flops is OM3.
4. Direct Toeplitz Inner Bordering Method
A new efficacious algorithm for the computation of continuous
nonlinear spectrum using the Toeplitz matrix transformations was
proposed in [111]. In the numerical example considered in [111],
this method outperforms the BO method in terms of speed and
accuracy; it has an error level of OM−2and the number of flops
OM2. This method is based on the reversion of the Toeplitz
matrix-based INFT algorithm; we provide the corresponding
INFT in the next subsection and description of the method in
Supplement 1. However, when dealing with the direct Toeplitz
inner bordering (TIB) method, one has to keep in mind that it
recovers the kernel of the GLME Lτthat in general includes
both discrete (solitonic) and continuous spectral components,
Eq. (17), simultaneously.
B. Numerical Methods for the INFT
The methods for numerical INFT computation were largely stud-
ied within the Bragg gratings’synthesis and characterization. Here,
using the traditional “matrix-inversion”terminology, we name the
INFT methods requiring OM2operation as “fast”and those
with lower complexity as “superfast.”Almost all INFT approaches
are based on the numerical solution of the GLME (16). After the
discretization, one aims at determining functions K1;2τ;τ0on the
grid of M×Mpoints. Note that the straightforward path there
based on the solution of Mnested linear matrix equations takes
OM4flops and is therefore unproductive.
The earlier approaches utilize iterative methods of matrix inver-
sion with the computational complexity in the order of OkiM3,
where ki<Mis a number of iterations. As an example, we men-
tion the group of methods with the GLME kernel parametrization
[112,113]. A similar method was also employed recently for optical
transmission tasks [60]. The main drawbacks of these algorithms
are the problem of choosing an initial approximation and high
computational complexity. A more advanced family of algorithms
is based on the layer peeling (LP) method. This class of methods is
built on the representation of the RC attributed to a particular pro-
file qtthrough the sequence of individual actions of Mpoint
reflectors [81,82,114]. The LP algorithms are comparatively fast
and require about OM2flops. Some of them provide an error
that is globally proportional to M−2. Conventional algorithms
based on LP show numerical instabilities with exponential ampli-
fication of noise when the reflection coefficients (participating in
the LP step) contain noise [115]. This means that there exists no
signal such that forward scattering with Msamples can result in the
desired reflection coefficient [108,116], or, in other words, when
one has independent noisy additions to the scattering data them-
selves. Physically, when some profile corrupted by noise in the
space-time domain was converted into the NF spectrum, and then
this spectrum was used for the superfast LP algorithm considered
further, we observed that the instability in all numerical examples
was absent; see Supplement 1. For the properties of space-time
noise conversion into the NF domain, see Refs. [117–120]. In
some transmission systems this limitation has recently been cir-
cumvented in some first algorithms where the reflection coefficient
is ensured to be realizable by construction [75,109,121]. On the
other hand, this instability can reveal itself when one synthesizes a
profile starting from some randomly encoded spectral data in the
NFD. This question requires further analysis.
The LP with improved accuracy [83], known as an integral LP,
has some issues with the overall efficiency, as it requires many more
arithmetic operations. The drawback of these algorithms is the ac-
cumulation of computational errors during calculation and the re-
sulting decrease in their accuracy when enlarging the qtextent.
Another interesting group of algorithms is based on recasting
the GLME as the system of partial differential equations
[122,123] (see also [124] for the comparison of such algorithms),
including also the “leap-frog”algorithm [125]. The numerical
complexity is OM2flops; however, the error there is only of
the first order, OM−1{[123], Fig. 2(b)}.
In [126] another algorithm was proposed, based on a different
computational approach, whose error was proportional to M−2.
This algorithm is “slow,”requiring OM3flops, and addresses
only the case of normal dispersion. However, the important fea-
ture there is that it introduced the very idea of the bordering pro-
cedure itself. Later, a more efficient algorithm that has a M−2error
and at the same time uses OM2flops was described in
[111,127]. The algorithm exploits the Toeplitz symmetry of dis-
cretized GLME using TIB, similar to technique for common
Toeplitz matrices [128,129]. As the TIB was successfully used
in a number of transmission-related works [51–55], we provide
here more details on the TIB method. First, we change the
variables in Eq. (16)as
Review Article Vol. 4, No. 3 / March 2017 / Optica 314

uτ;τ0K1τ;τ−τ0;wτ;τ0¯
K2τ;τ0−τ:(25)
In new notations, explicitly assuming the finite extent of qt,
0≤t≤T, after the complex conjugation of the first of
GLME, we get
uτ;yZ2τ
y
¯
Lτ0−ywτ;τ0dτ00;
wτ;τ0Zτ0
0
Lτ0−yuτ;ydyLτ00;(26)
0≤y; τ0<2τ,0≤τ≤T. The sought solution in the time do-
main now reads as qt2wt;2t−0. The GLME form (26)
allows one to obtain the Toeplitz-type problem after the discre-
tization and to use the fast Toeplitz matrix-inversion algorithms
[128,129] for the recovery of qt. Further details of the
TIB-based INFT are given in Supplement 1.
At the end we mention a recent work on the INFT methods by
Civelli et al. [130]: the authors introduced yet another INFT first-
order solution algorithm based on iterated convolutions with the
GLME kernel using the FFT, which demonstrated the better per-
formance in terms of accuracy and time consumption than the
first-order TIB [52] and the Nyström conjugate gradient method
[131]. However, the last approach has not been tested so far on
transmission-related problems. Note that NFT can be formulated
in terms of the so-called Riemann–Hilbert problem (see, e.g., [43]
and references therein), and numerical solution of the NFT can
be implemented using this approach [132].
C. Superfast NFT Algorithms
It has recently been observed that the AL method (and others) for
computing the NFT can be significantly sped up, leading to a
superfast NFT analogous to the celebrated FFT [84,99]. We
illustrate how to deal with fast NFTs using AL discretization.
The matrix Tmζin Eq. (24) can be written as
TmζSmZ∕dmZ;Zeiλε;(27)
where SmZand dmZare polynomials with respect to Z:
SmZ 1
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1ε2jqmj2
pZ2εqmZ
−ε¯qmZ1;d
mZZ:
(28)
Consequently, with SZSMZ×…×S1Zand
dZdMZ×…×d1Z, Eq. (23) can be written as
aζ
bζ≈diageiζT20.5ε;e
−iζT20.5εSZ
dZe−iζT1−0.5ε:
(29)
Since the SmZand dmZare polynomials with degrees at most
two, both SZand dZare again polynomials whose degrees are
upper bounded by 2M. The superfast NFT exploits this observa-
tion and proceeds in two steps. First, the monomial expansions of
the polynomials SZand dZhave to be computed. That is, the
unique matrices Skand scalars dkneed to be found such that
SZ X
2M−1
k0
SkZk;dZ X
2M−1
k0
dkZk:(30)
One first needs a fast method to compute the monomial expan-
sions. A naive implementation, e.g., to compute the expansion
of SZ, would proceed as follows:
SZSMZSM−1ZSM−2Z   S1Z…:(31)
However, this leads to a OM2or even OM3runtime, depend-
ing on how the product of polynomials is found. In order to get a
superfast NFT algorithm, a divide-and-conquer strategy is used
instead. One starts with the elementary polynomials SmZ,
m0;…;M −1, partitions them into pairs, and computes the
products of these pairs. The products are again partitioned into
pairs, and then multiplied. This process is iterated until only
one product is left, which will be SZ. It turns out that this
algorithm finds the monomial expansion Sk,k0;…;M −1
using only OMlog2Mflops given that polynomial products
are computed with the FFT. The pseudocode for this algorithm
is provided in Supplement 1.
The second step of the superfast NFT now applies algorithms
for fast polynomial arithmetic in order