1

Communication Using Eigenvalues of Higher
Multiplicity of the Nonlinear Fourier Transform

arXiv:1802.07456v2 [] 26 Feb 2018

Javier Garcı́a
Institute for Communications Engineering
Technical University of Munich
javier.garcia@tum.de
(This work has been submitted to the IEEE/OSA Journal of Lightwave Technology for possible publication.
Copyright may be transferred without notice, after which this version may no longer be accessible.)

Abstract—Eigenvalues of higher multiplicity of the Nonlinear
Fourier Transform (NFT) are considered for information transmission over fiber optic channels. The effects of phase, time or
frequency shifts on this generalized NFT are derived, as well
as an expression for the signal energy. These relations are used
to design transmit signals and numerical algorithms to compute
the direct and inverse NFTs, and to numerically demonstrate
communication using a soliton with one double eigenvalue.
Index Terms—Inverse Scattering Transform, Nonlinear
Fourier Transform, optical fiber, higher multiplicity eigenvalues,
spectral efficiency

I. I NTRODUCTION
Current optical transmission systems exhibit a peak in the
achievable rate due to the Kerr nonlinearity of the Nonlinear
Schrödinger Equation (NLSE) [1]. Several techniques have
been proposed to attempt to overcome this limit, of which
the Inverse Scattering Transform (IST) [2], or the Nonlinear
Fourier Transform (NFT) [3], has attracted considerable attention. Numerous algorithms have been developed to compute
the direct [4], [5] and inverse [5], [6], [7] NFT.
Information transmission using the NFT has been demonstrated both numerically and experimentally in several works,
such as [8], [9], [10]. For purely discrete spectrum modulation, the spectral efficiencies obtained so far are not very
high [11]. In this paper, eigenvalues of higher multiplicity in
the discrete spectrum are considered for communication. The
theory for these eigenvalues has been developed in [12], [13],
but its applications to communications have to the best of our
knowledge not been explored yet.
The paper is organized as follows. In Section II, we introduce the NLSE model. Section III briefly describes the NFT.
In Section IV, we explain the theory of higher multiplicity
eigenvalues from [12], [13], and we prove some properties
of this generalized NFT. In Section V, we show how to
compute the direct and inverse NFT with multiple eigenvalues.
Section VI numerically demonstrates information transmission
using a double eigenvalue, and Section VII concludes the
paper.
Date of current version February 27, 2018. J. Garcı́a was supported by the
German Research Foundation under Grant KR 3517/8-1.

II. S YSTEM

MODEL

Consider the slowly varying component Q(Z, T ) of an
electrical field propagating along an optical fiber, where Z
is distance and T is time. The field obeys the NLSE, which
is expressed as [14, Eq. (2.3.46)]:
β2 ∂ 2
∂
Q(Z, T ) + jγ |Q(Z, T )|2 Q(Z, T )
Q(Z, T ) = − j
∂Z
2 ∂T 2
+ N (Z, T )
(1)
where β2 is the group velocity dispersion (GVD) parameter,
and γ is the nonlinear coefficient. We neglect attenuation in (1)
because we assume that fiber loss is exactly compensated
by distributed Raman amplification. The noise term N (Z, T )
is the formal derivative of a band-limited Wiener process
W (Z, T ), i.e., we have
Z Z
p
N (Z ′ , T ) dZ ′ = Nase W (Z, T )
(2)
0

where Nase = αhνs KT is the distributed noise spectral density, α is the attenuation coefficient, h ≈ 6.626 ·
10−34 m2 kgs−1 is Planck’s constant, νs is the signal center
frequency, and KT is the phonon occupancy factor, which is
approximately 1.13 for Raman amplification [1]. Note that,
unlike [1], we do not include the distance in the definition of
Nase . The Wiener process W (Z, T ) may be defined as
⌊KZ⌋
1 X
Wk (T )
W (Z, T ) = lim √
K→∞
K k=1

(3)

where the Wk (T ) are independent and identically distributed
(i.i.d.) circularly symmetric complex Gaussian processes with
zero mean, bandwidth B, and autocorrelation
E [Wk (T )Wk∗ (T ′ )] = B sinc (B (T − T ′ ))

(4)

where sinc(x) , sin (πx) / (πx).
III. T HE N ONLINEAR F OURIER T RANSFORM
In this section, we briefly introduce the steps involved in
the NFT. For more detail, the reader is referred to [3].

2

By applying the following change of variables:
s
T02
1 |β2 |
T = T0 t,
Z=2
z,
Q(Z, T ) =
q(z, t)
|β2 |
T0
γ
E [N (Z, T )N ∗ (Z ′ , T ′ )] =

β22
E [n(z, t)n∗ (z ′ , t′ )]
2γT03

(5)

the NLSE (1) is normalized to
∂
∂2
q(z, t) = −j sign (β2 ) 2 q(z, t) + j2 |q(z, t)|2 q(z, t)
∂z
∂t
+ n(z, t)
(6)
and we choose β2 < 0 to focus on the case of anomalous
GVD [14, p. 131]. The parameter T0 can be freely chosen, and provides an additional degree of freedom for the
normalization. The nonlinear term |q|2 q causes inter-channel
interference.
The IST or NFT provides a domain in which the NLSE
channel is multiplicative, i.e., there exists a channel transfer
function H(z, λ) such that
Q̃(z, λ) = H(z, λ)Q̃(0, λ)

(7)

adjoint of the solutions. The four canonical eigenvectors
v 1 (t, λ), v 2 (t, λ), ṽ 1 (t, λ), and ṽ 2 (t, λ) satisfy
v 2 (t, λ) = a(λ)ṽ 1 (t, λ∗ ) + b(λ)v 1 (t, λ)
2

vz = M v

(9)

vt = P v

(10)

where a subscript indicates a derivative with respect to that
variable. For the NLSE (6), L, M and P are given by:
 ∂

−q
∂t
L(z, t) = j
(11)
∂
−q ∗ − ∂t


2
2jλ2 − j |q|
−2λq − jqt
(12)
M (z, t, λ) =
∗
∗
2λq − jqt
−2jλ2 + j |q|2


−jλ q
P (z, t, λ) =
.
(13)
−q ∗ jλ
The NFT is calculated by solving the Zakharov-Shabat system (10). In the following, we often drop the dependence on z
to simplify notation. Two solutions v 1 (t, λ) and v 2 (t, λ) that
are bounded in the upper complex half plane (λ ∈ C+ ) are
obtained using the boundary conditions
 
0 jλt
1
v (t, λ) →
e , t → +∞
(14a)
1
 
1 −jλt
v 2 (t, λ) →
e
, t → −∞.
(14b)
0
We define the adjoint of a vector v = (v1 , v2 )T as ṽ =
(v2∗ , −v1∗ )T . Two additional solutions ṽ 1 (t, λ∗ ) and ṽ 2 (t, λ∗ )
of (10) are calculated by solving vt (t, λ∗ ) = P (t, λ∗ )v(t, λ∗ )
using boundary conditions adjoint to (14), and taking the

∗

1

∗

∗

∗

1

ṽ (t, λ) = −b (λ )ṽ (t, λ ) + a (λ )v (t, λ)

(15a)
(15b)

where a(λ) and b(λ) do not depend on t. The NFT of the
signal q(z, t) is made up of two spectra:
b(λ)
• the continuous spectrum Qc (λ) = a(λ) , for λ ∈ R;

k)
, for the K
the discrete spectrum Qd (λk ) = ab(λ
λ (λk )
eigenvalues {λk ∈ C+ : a(λk ) = 0}
where aλ = da/dλ. To compute the NFT, the following
relations are useful

•

a(λ) = lim v12 (t, λ)ejλt

(16a)

b(λ) = lim v22 (t, λ)e−jλt .

(16b)

t→∞

t→∞

Given a signal q(z, t) propagating according to the
NLSE (6), its NFT evolves in z according to the following
multiplicative relations:
2

Qc (z, λ) = Qc (0, λ)e4jλ

where Q̃(z, λ) is the NFT of q(z, t). The NFT is based on the
existence of a Lax pair (L, M ) of operators that satisfies the
following condition:
∂L
= M L − LM.
(8)
∂z
As shown in [2, Section 1.4], the eigenvalues λ of L are
invariant in z, and the eigenvectors v of L satisfy:

∗

z

λk (z) = λk (0)

(17a)
(17b)

2

Qd (z, λk ) = Qd (0, λk )e4jλk z .
IV. E IGENVALUES OF

(17c)

HIGHER MULTIPLICITY

The relations (15) are consistent only for λ ∈ R ∪ {λk } ∪
{λ∗k }, because the boundary condition (14a) on ṽ 1 (t, λ∗ ) is
unbounded outside this region. As the eigenvalues λk come
in complex conjugate pairs, the spectrum at λ ∈ R ∪ {λk } is
enough to determine q(t) uniquely. However, to the best of our
knowledge, all the work on the NFT for optical communication
assumes that all the eigenvalues λk have multiplicity 1, i.e.,
the zeros of a(λ) are simple. There has been, however,
some work [12], [13] on the mathematical theory of higher
multiplicity eigenvalues, which we summarize in this section.
For a multiple zero λk of a(λ), we have aλ (λk ) = 0,
and the above definition of the discrete spectrum is not valid
anymore. If the multiplicity of the eigenvalue λk is Lk , we
need Lk constants qk0 , . . . , qk,(Lk −1) to determine the discrete
spectrum. In [13], these norming constants are calculated in
several intermediate steps.
• Dependency constants
The dependency constants γkℓ , k ∈ {1, . . . , K}, ℓ ∈
{0, . . . , Lk − 1} are defined by the following equation [13, Eq. (3.1)]:
m  
X
m
v 2,(m) (t, λk ) =
γk,(m−ℓ) v 1,(ℓ) (t, λk ) (18)
ℓ
ℓ=0

i,(m)

where v
denotes the m-th derivative of v i with
respect to λ. Taking the m-th derivative of (15), and using
a(ℓ) (t, λ) = 0 for ℓ ≤ Lk − 1, we have
γk,ℓ = b(ℓ) (λk ).
•

Generalized residues

(19)

3

The generalized residues tkℓ , k ∈ {1, . . . , K}, ℓ ∈
{1, . . . , Lk }, are the coefficients of the expansion of
1/a(λ) − 1 in inverse powers of λ − λk [13, Eq. (4.3)]:

tkLk
tk1
1
−1 =
+ O(1). (20)
+···+
L
k
a(λ)
(λ − λk )
(λ − λk )

They can be computed as
tkℓ
•

1
dLk −ℓ
=
lim
(Lk − ℓ)! λ→λk dλLk −ℓ

"

#
(λ − λk )Lk
.
a(λ)

LkX
−ℓ−1
m=0

b(m) (λk )
tk,(ℓ+m+1) .
m!

(22)

The generalization of the distance evolution equation (17c)
to the case Lk > 1 is given by [13, Eq. (4.9)]:


qk,(Lk −1) (z) · · · qk0 (z)


2
= qk,(Lk −1) (0) · · · qk0 (0) e−4jΛk z (23)

for all k ∈ {1, . . . , K}, where

−jλk
−1
0
···
 0
−jλ
−1
·
··
k

 ..
.
.
.
..
..
..
Λk =  .

 0
···
0 −jλk
0
0
···
0

0
0
..
.






 ∈ CLk ×Lk .

−1 
−jλk
(24)

NFT {q(t)} = (Qc (λ), {λk }, {qkℓ }).

(25)

A. Properties of the NFT with Higher Multiplicity Eigenvalues
We prove the following properties in Appendix A.
1) Phase shift:


NFT q(t)ejφ0 = (Qc (λ)e−jφ0 , {λk }, qkℓ e−jφ0 ).
(26)
2) Time shift: if q ′ (t) = q(t − t0 ) then
′
NFT {q ′ (t)} = (Q′c (λ), {λ′k } , {qkℓ
})

(27)

satisfies
Q′c (λ) = Qc (λ)e−2jλt0

(28a)

λ′k

(28b)
i

′
· · · qk0
= qk,(Lk −1)



COMPUTATION OF THE (I)NFT WITH
HIGHER MULTIPLICITY EIGENVALUES

Using the theory from Section IV, we extend the existing
numerical algorithms that compute the (I)NFT to include
multiple eigenvalues.
A. Direct NFT
Most algorithms that compute the direct NFT discretize the
Zakharot-Shabat system (10) to find a(λ) and b(λ) from (16).
Let u = (u1 , u2 )T , where u1 (t, λ) = v12 (t, λ)ejλt and
u2 (t, λ) = v22 (t, λ)e−jλt . Then from (10) we have


0
q(t)e2jλt
ut (t, λ) =
u(t, λ)
(32)
−q ∗ (t)e−2jλt
0
and from (16) we have
a(λ) = lim u1 (t, λ)

(33a)

b(λ) = lim u2 (t, λ).

(33b)

t→∞

We write the NFT as

= λk
h
′
qk,(L
k −1)

k=0

V. N UMERICAL
(21)

Norming constants
The norming constants qkℓ , k ∈ {1, . . . , K}, ℓ ∈
{0, . . . , Lk − 1}, are given by [13, Eq. (4.1)]:
qkℓ = j ℓ

5) Parseval’s theorem:
Z ∞
Z


1 ∞
|q(t)|2 dt =
log 1 + |Qc (λ)|2 dλ
π −∞
−∞
K
X
+4
Lk ℑ {λk } .
(31)

· · · qk0 e2Λk t0 .
(28c)


3) Frequency shift:

NFT q(t)e−2jω0 t = (Qc (λ − ω0 ), {λk + ω0 }, {qkℓ }).
(29)
4) Time dilation: for T > 0
  n

  

λk
qkℓ o
t
1
= Qc (T λ),
,
.
q
NFT
T
T
T
T ℓ+1
(30)

t→∞

To compute the NFT of q(t), we discretize the time axis in the
interval t ∈ [t1 , t2 ]. Let tn = t1 + nǫ, qn = q(tn ), where n ∈
{0, . . . , N − 1}, N is the number of samples, and ǫ = (t2 −
t1 )/(N −1) is the step size. Similarly, let u[n] = u(t1 +nǫ, λ).
Starting at u[0] = (1, 0)T (see (14)), the following update step
is applied iteratively:
u[n + 1] = A[n]u[n], n ∈ {0, . . . , N − 2}

(34)

and we have a(λ) = u1 [N − 1] and b(λ) = u2 [N − 1]. The
kernel A[n] varies according to the discretization algorithm.
A few options are given in [4]. In this work, we consider the
trapezoidal kernel proposed in [5]:


cos (|qn | ǫ)
sin (|qn | ǫ) ej(θn +2λtn )
A[n] =
− sin (|qn | ǫ) e−j(θn +2λtn )
cos (|qn | ǫ)
(35)
where θn = arg qn . However, the following analysis is valid
for any kernel A[n]. To obtain the norming constants qkℓ , we
need to calculate higher order λ-derivatives of a(λ) and b(λ).
More specifically, from (22) we need the first Lk −1 derivatives
of b(λ). In the case of a(λ), we obtain an upper bound on the
order of the required derivatives.
Lemma 1. The value of tkℓ in (21) depends on λk only
through the functions a(m) (λk ) for m ∈ {Lk , . . . , 2Lk − ℓ}
Proof. See Appendix B.
For an eigenvalue of multiplicity Lk , we need to compute
the first 2Lk − 1 derivatives of u[N − 1]. We do this by setting
the initial conditions
 
0
u(m) [0] =
,
m ∈ {1, . . . , 2Lk − 1}
(36)
0

4

and applying the update steps
m  
X
m (r)
A [n]u(m−r) [n]
u(m) [n + 1] =
r
r=0

and we compute
(37)

where A(r) [n], the r-th order λ-derivative of A[n], is obtained
in closed form. For the trapezoidal kernel (35) we have

a(ℓ) (λk ) =

b(λk ) = R21 L11 + R22 L21
R21
L21
=
(R11 L11 + R12 L21 ) +
R11
R11
L21
R21
a(λk ) +
=
R11
R11


ej(θn +2λtn )
. (38)
0

Once we have the required values of a, b and their derivatives,
we use (22) and (21) to compute the norming constants.
In (21), the derivative is evaluated in closed form, and then
L’Hôpital’s rule is applied repetitively to obtain an expression
for tkℓ that depends only on nonzero derivatives of a. See (75)(77) in Appendix B for details. For Lk = 2, this gives
j2b(λk )
aλλ (λk )
2bλ (λk )
2 b(λk )aλλλ (λk )
=
−
aλλ (λk ) 3 aλλ (λk )2

qk1 =

(39a)

qk0

(39b)

and for Lk = 3 we have

(45)

To obtain b(ℓ) (λk ), note that

r

A(r) [n] = (2jtn ) sin (|qn |ǫ)

0
·
r+1 −j(θn +2λtn )
(−1)
e

ℓ  

X
ℓ
(m) (ℓ−m)
(m) (ℓ−m)
.
r2 l1
− r1 l2
m
m=0

(46)

where we used R22 = (1 + R12 R21 ) /R11 . The ℓ-th derivative
of the left summand in (46) is 0 for ℓ ≤ Lk − 1, because
a(ℓ) (λk ) = 0 for ℓ ≤ Lk − 1 . Therefore, we have
b(ℓ) (λk ) =

dℓ l2
dλℓ r2

(47)
λ=λk

which can be written in closed form using (72) below. Equations (47) and (45), together with (22) (or (39) or (40)), let us
compute the direct NFT with higher multiplicity eigenvalues
from the forward-backward method.
B. Inverse NFT

−6b(λk )
qk2 =
(40a)
aλλλ (λk )
3aλλλλ (λk )
j6bλ (λk )
(40b)
− jb(λk )
qk1 =
aλλλ (λk )
2aλλλ (λk )2
6bλλ (λk )
3aλλλλ (λk )
qk0 =
− bλ (λk )
aλλλ (λk )
2aλλλ (λk )2
2
15aλλλλ (λk ) − 12aλλλ (λk )aλλλλλ (λk )
+ b(λk )
. (40c)
20aλλλ (λk )3
Forward-Backward Method: This technique was proposed
in [5] to improve numerical stability. We write (34) as


 
 
a(λ)
1
1
= A[N − 1] · · · A[1]A[0]
= RL
(41)
b(λ)
0
0
where R = A[N − 1] · · · A[n0 ] and L = A[n0 − 1] · · · A[0],
and n0 is chosen according to some criterion to minimize the
numerical error. The iterative procedure (34) is run forward
up to n0 − 1 to obtain
  

 
1
l1
L11
=L
(42)
=
0
l2
L21
T

and backward from r[N − 1] = (0, 1) down to r[n0 − 1]:

  
 
0
−R12
r1
.
(43)
=
= R−1
R11
1
r2
−1

The kernel A[n] is used to compute (43): for the trapezoidal
case this amounts to replacing ǫ with −ǫ in (35). Note that (43)
is valid only for kernels with unit determinant.
Using (38), we obtain r1 , r2 , l1 , l2 , and their derivatives up
to order 2Lk − ℓ. From (41) we have
a(λ) = R11 L11 + R12 L21

(44)

The inverse NFT can be computed using the generalized version of the Gelfand-Levitan-Marchenko equation (GLME) [7]:
K(t, y) − Ω∗ (t + y)
Z ∞
Z ∞
+
dx
ds K(t, s)Ω(s + x)∗ Ω(x + y) = 0. (48)
t

t

The kernel Ω(y) is given by [12]:
Z ∞
K LX
k −1
X
yℓ
1
Qc (λ)ejλy dλ +
qkℓ ejλk y . (49)
Ω(y) =
2π −∞
ℓ!
k=1 ℓ=0

The inverse NFT is then obtained as

q(t) = −2K(t, t).

(50)

A numerical procedure to solve the GLME (48) is given in [7,
Section 4.2]: it suffices to replace F (y) in the reference by
Ω(y) from (49).
When there is no continuous spectrum, a closed-form expression is given in [12] for the generalized K-solitons:
q(z, t)
H

= −2B H e−Λ
where

t

−1 −ΛH t+4j (ΛH )2 z

(I + M (z, t)N (t))



Λ1
0

Λ= .
 ..
0

e

0
Λ2
..
.

···
···
..
.

0
0
..
.

···

0

ΛK





,


CH
(51)

(52)

Λk is given by (24),
T
T
L ×1
B = B1 · · · BK , Bk = 0 · · · 0 1 ∈ {0, 1} k ,
(53)

5


· · · CK ,

Ck = qk,(Lk −1)


· · · qk0 ,
(54)

P
I is an identity matrix of size k Lk , and
Z ∞
H
H 2
2
M (z, t) =
e−Λ s+4j (Λ ) z C H Ce−Λs−4jΛ z ds (55)
Z ∞t
H
e−Λx BB H e−Λ x dx.
(56)
N (t) =
t

We compute the integrals (55) and (56) numerically, and then
apply (51) to obtain the inverse NFT of a purely discrete
spectrum with higher multiplicity eigenvalues.

1

0
20
0
−20

t

0

0.2

0.6

0.4

0.8

z

Figure 1. Propagation of a double soliton

C. Example: Double Soliton
A soliton q(z, t) with a 2nd order eigenvalue at λ = ξ + jη,
with norming constants q11 and q10 , can be derived in closed
form from (51). The result is
h(z, t)
q(z, t) =
f (z, t)

(57)

where
n
2
2
h(z, t) = −j4ηe−j arg q11 e−j2ξt e−j4(ξ −η )z
i
h
2
∗
q10
e−X − |q11 | (2ηt + 8η (ξ + jη) z + 2) − ηq11
h
io
2
∗
+eX |q11 | (2ηt + 8η (ξ − jη) z) + ηq11 q10
(58)
2

f (z, t) = |q11 | [cosh (2X) + 1]

2

+ 2 |q10 η + q11 (2ηt + 8η (ξ + jη) z + 1)|

(59)

and
X = 2ηt + 8ηξz − log

|q11 |
.
4η 2

(60)

We refer to this soliton as a double soliton. The evolution of
the norming constants (23) reduces to
2

q11 (z) = q11 (0)e4jλ

z

q10 (z) = (q10 (0) + 8λzq11 (0)) e

(61a)
4jλ2 z

.

(61b)

Note from (61b) that a soliton with higher multiplicity eigenvalues does not exhibit periodic (breathing) behavior in z.
Only the first norming constant qk,(Lk −1) evolves periodically
if the eigenvalue is purely imaginary. In particular, the ratio
|q10 |/|q11 | is unbounded with distance, which causes the unbounded pulse broadening that can be inferred from Figure 1.
We remark that the following closed form expression for
the center time of the double soliton seems to be valid based
on our experiments, although we have not found a proof:
R∞


2
1
|q11 |
−∞ t |q(t)|
.
(62)
R∞
2 = 2η log
4η 2
|q(t)|
−∞

2

|q(z, t)|

C = C1

VI. I NFORMATION

TRANSMISSION USING HIGHER
MULTIPLICITY EIGENVALUES

We simulated a communications system with the parameters
given in Table I. Three different launch signals were compared:
• a double-soliton with an eigenvalue at λ = 1.25i,
• a 2-soliton with eigenvalues λ1 = 1.5i and λ2 = 1i,
• and a 1-soliton with an eigenvalue at λ0 = 2.5i.
From (31), the three signals have the same pulse energy in the
normalized domain. The 1-soliton uses multi-ring modulation
on q0 = Qd (λ0 ) with 32 rings and 128 phases per ring.
The 2-soliton has the two spectral amplitudes q1 = Qd (λ1 )
and q2 = Qd (λ2 ), while the double-soliton system has the
two norming constants q11 and q10 . Both the 2-soliton and
the double-soliton have 4 rings and 16 phases per spectral
amplitude. With this design, the three launch signals can
transmit up to 12 bits per channel use. The amplitudes of the
rings have been heuristically optimized to obtain a small timebandwidth product (TBP) of 10.5 sec·Hz that is approximately
the same for the three signals. The ring amplitudes for the 1soliton are given by

|q0 | ∈ 0.088754 · 1.6142k : k ∈ {0, . . . , 31} .
(63)
The ring amplitudes for the two-soliton and double-soliton
are given in Tables II and III, respectively. The phases are
uniformly spaced in [0, 2π), starting at 0 for q0 and q11 , at
π/128 for q2 , and at π/16 for q10 . The optimal criterion
for choosing ring amplitudes has not yet been found, but
expressions such as (62) suggest that geometric progressions
are better suited than arithmetic progressions.
The free normalization parameter T0 in (5) was used to
obtain the desired powers. This means that the pulse duration
and bandwidth are different for each power value. This is
unavoidable in soliton systems. The TBP of 10.5 is, however,
constant for all values of power.
Propagation according to (1) was simulated using the splitstep Fourier method. In all systems, the transmitter used
closed-form expressions to generate the solitons, and the
receiver used forward-backward computation with the trapezoidal kernel to obtain the norming constants. Equalization
was performed by inverting (23). The mutual information
of the transmitted and received symbols was measured and

6

Table III
R ING AMPLITUDES FOR THE DOUBLE - SOLITON SYSTEM

Table I
S IMULATION PARAMETERS
Symbol
β2
γ
z
Nase

|q11 |
|q10 |

Value
−21.667 ps2 /km
1.2578 W−1 km−1
4000 km
6.4893 · 10−24 Ws/m

Table II
R ING AMPLITUDES FOR THE 2- SOLITON SYSTEM
|q1 |(λ = 1.5i)
|q2 |(λ = 1i)

2.5355
0.2662

2.8364
1.0211

3.1730
3.9173

3.5496
15.0283

normalized by the TBP to obtain the spectral efficiency. In the
2-soliton and double-soliton systems, the mutual information
was computed jointly for the two spectral amplitudes,
(T X) (T X) (RX) (RX)
I(q1
, q2
; q1
, q2
)
(T X)

Spectral efficiency (bits/s/Hz)

Parameter
Dispersion coefficient
Nonlinear coefficient
Fiber length
Noise spectral density

5.3785
34.3750

6.5708
45.0440

7.2627
51.5625

1-soliton
2-soliton
Double soliton

1

0.5

0
−40

−30

−20

(64)

−10

0

P (dBm)

(RX)

where qk
refers to the transmitted symbols and qk
refers to the received symbols after equalization. The effective
number of transmitted symbols was 614440.
Figure 2 shows the spectral efficiency for the three systems.
The double soliton performs better than the 1-soliton, but
worse than the 2-soliton. This is expected: the higher order
derivatives of a(λ) in (39) lead to a loss of accuracy. However,
the experiment demonstrates that the generalized NFT with
multiple zeros can be used to transmit information.
VII. C ONCLUSION
We started from the theory in [12], [13], and proved some
properties of the generalized NFT that are useful for communications. We designed and implemented algorithms to compute
the generalized NFT, and we numerically demonstrated the
potential of higher multiplicity eigenvalues for information
transmission. With this, we extend the class of signals that
admit an NFT, and we show that there are additional degrees
of freedom for NFT-based optical communications systems.
There are several directions for future work. For example,
an extension of the Darboux algorithm to the generalized NFT
would speed up the computation of the INFT. More insight into
the duration, bandwidth and robustness to noise of multiple
eigenvalue signals would be useful.

Figure 2. Spectral efficiency of three solitonic signals

we have
L

(λ − λk ) k
a(λ)
dLk −ℓ
1
tkLk (λ).
tkℓ (λ) =
(Lk − ℓ)! dλLk −ℓ

tkLk (λ) =

OF THE PROPERTIES OF THE

NFT

t′kℓ (λ) = e−jλt0

In the following, all primed variables (a′ ) refer to the
spectral functions of the shifted signal q ′ (t).
1) Phase shift: replacing q with qejφ0 in (32), we have
′
a (λ) = a(λ) and b′ (λ) = b(λ)e−jφ0 . The property then
follows from (17a) and (22).
2) Time shift: the change of variable t → t − t0 in (10)
proves that a′ (λ) = a(λ)ejλt0 , and b′ (λ) = b(λ)e−jλt0 . The
expressions (28a) and (28b) follow immediately. Defining
#
"
L
1
dLk −ℓ (λ − λk ) k
tkℓ (λ) =
(65)
(Lk − ℓ)! dλLk −ℓ
a(λ)

LX
k −ℓ
i=0

(66b)

1
i
(−jt0 ) tk,ℓ+i (λ).
i!

(67)

From b′ = be−jλt0 we have
′ (ℓ)

(b )

(λ) = e

−jλt0

ℓ  
X
ℓ

i

i=0

(−jt0 )ℓ−i b(i) (λ).

(68)

Using (67) and (68) in (22), we have
′
qkℓ

ℓ −2jλk t0

=j e

LkX
−ℓ−1

m  
1 X m (i)
m−i
b · (−jt0 )
m! i=0 i

m=0
Lk −ℓ−m−1
X

WITH HIGHER

MULTIPLICITY EIGENVALUES

(66a)

Using a′ = aejλt0 , we have t′kLk (λ) = tkLk (λ)e−jλt0 and,
from (66b),

A PPENDIX A
P ROOF

5.9449
39.3496

r=0

1
r
(−jt0 ) tk,ℓ+m+r+1 . (69)
r!

The change of variable m = u + i − r yields
′
qkℓ

=

e

−2jλk t0

LkX
−ℓ−1
u=0

Lk −ℓ−u−1
X
i=0

=

e−2jλk t0

j

ℓ+u

i!

which is the same as (28c).

u
X
r=0

1
r! (u − r)!

!

b(i) tk,ℓ+u+i+1

LkX
−ℓ−1
u=0

(−t0 )

u

1
(−2t0 )u qk,ℓ+u
u!

(70)

7

3) Frequency shift: using the change of variable λ → λ−ω0
in (32), we have a′ (λ) = a(λ − ω0 ) and b′ (λ) = b(λ − ω0 ),
from which the property follows.
4) Time dilation: the change of variable t → t/T in (32)
proves that a′ (λ) = a(T λ) and b′ (λ) = b(T λ). Using this
in (21) and changing λ → T λ shows that t′kℓ = tkℓ /T ℓ.
Together with (22), this proves (30).
5) Parseval’s theorem: this is a particular case (n = 0) of
the more general trace formula:
Z


1 ∞
(2jλ)n log 1 + |Qc (λ)|2 dλ
Cn =
π −∞
K
X

4
n
(71)
(2j)
+
Lk ℑ λn+1
k
n+1
k=0

where Cn are the constants of motion, of which C0 is the
signal energy (31). The proof for simple eigenvalues is given
in [2, Sec. 1.6]. The result is extended to multiple eigenvalues
by allowing several ζm in [2, Eq. (1.6.18)] to be equal.
A PPENDIX B
P ROOF OF L EMMA 1

Applying Faà di Bruno’s formula [15, pp. 43-44] to 1/a(λ),
and then the generalized product rule to c(λ)·(1/a(λ)), for an
arbitrary c(λ), we can write a quotient rule for higher order
derivatives:
" 
n
X
n (n−m)
dn c
c
=
dλn a m=0 m

m
X
(−1)|p| m!
|p|! Y  (i) pi 
. (72)
a
p1 !1!p1 · · · pm !m!pm a|p|+1 i=1
p∈P(m)

Recall that a(i) denotes an i-th order derivative. Here, P(m)
denotes the set of partitions p of m:
m
X

p = [p1 , · · · , pm ] ,
and |p| =
we have

Pm

i=1

ipi = m,

i=1

pi ∈ N ∪ {0} (73)

pi is the cardinality of p. Using (72) in (21)
tkℓ = lim

λ→λk

g(λ)
a(λ)Lk −ℓ+1

(74)

where
LX
k −ℓ

"

Lk − ℓ
Lk !
(λ − λk )ℓ+m
g(λ) =
(ℓ
+
m)!
m
m=0
·

X

p∈P(m)


m 

Y
p
i
(−1) m!
a(i)  .
|p|!aLk −ℓ−|p|
p1 !1!p1 · · · pm !m!pm
i=1
|p|

(75)

Note that a has a zero of order Lk , and therefore a(m) (λk ) =
0 for m ∈ {0, . . . , Lk − 1}. To compute tkℓ , we repeatedly
apply L’Hôpital’s rule until the numerator and the denominator
become nonzero in the limit:
tkℓ =

g (r) (λk )
.
[dr a(λ)Lk −ℓ+1 /dλr ]|λ=λk

(76)

The number r of times we need to differentiate is equal to the
order of the zero in the denominator:
r = Lk (Lk − ℓ + 1) .

(77)

The summands in g (r) (λ) are of the form
m 
pi
Y
d(Lk −ℓ+1)Lk
ℓ+m Lk −ℓ−|p|
(i)
a
(λ
−
λ
)
a
k
dλ(Lk −ℓ+1)Lk
i=1
(78)
where s is an index, and Ks is a constant independent of λ. If
we apply the generalized product rule to (78), we see that the
only summands that will become nonzero for λ = λk are those
that apply ℓ + m differentiations on the factor (λ − λk )ℓ+m .
This leaves (Lk − ℓ + 1)Lk − ℓ − m differentiations for the
other factors. Note that the derivative of a term such as
m 
pi
Y
Lk −ℓ−|p|
(79)
a(i)
a

gs (λ) = Ks

i=1

is a sum of terms
P of the same form. Each new term has the
same amount i pi of a-factors as the original (an a-factor
here refers to a or one
P of its derivatives), and where the number
of differentiations i ipi in the a-factors is increased by 1. We
conclude that gs (λ) is made up of summands that contain
P
• Lk − ℓ − |p| +
− ℓ a-factors that have
i pi = LkP
• (Lk − ℓ + 1)Lk − ℓ − m +
i ipi = (Lk − ℓ + 1)Lk − ℓ
differentiations.
Only the summands where all the a-factors are differentiated at
least Lk times will be nonzero. We want to know the highestorder derivative of a that appears in a nonzero term. The worst
case occurs when we have Lk − ℓ − 1 a-factors with an Lk -th
order derivative. The remaining a-factor must have a derivative
of order [(Lk − ℓ + 1)Lk − ℓ] − [Lk (Lk − ℓ − 1)] = 2Lk − ℓ.
ACKNOWLEDGMENT
The author wishes to thank Prof. G. Kramer and B. Leible
for useful comments and proofreading the paper.
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