U.S. patent application number 15/461718 was filed with the patent office on 2018-09-20 for fiber kerr nonlinear noise estimation.
This patent application is currently assigned to Ciena Corporation. The applicant listed for this patent is Maurice O'SULLIVAN, Michael Andrew REIMER, Qunbi ZHUGE. Invention is credited to Maurice O'SULLIVAN, Michael Andrew REIMER, Qunbi ZHUGE.
Application Number | 20180269968 15/461718 |
Document ID | / |
Family ID | 63519814 |
Filed Date | 2018-09-20 |
United States Patent
Application |
20180269968 |
Kind Code |
A1 |
ZHUGE; Qunbi ; et
al. |
September 20, 2018 |
Fiber Kerr Nonlinear Noise Estimation
Abstract
A method of fiber Kerr nonlinear noise estimation in an optical
transmission system comprises recovering received symbols from a
received signal, isolating a noise component of the received
signal, estimating coefficients of a matrix based on
cross-correlations between the isolated noise component and the
fields of a triplet of received symbols or training symbols or
estimated transmitted symbols, estimating doublet correlations of
the product or the quotient of the isolated noise component and the
field of a received symbol or of a training symbol or of an
estimated transmitted symbol, and estimating one or more parameters
related to nonlinear noise based on the estimated coefficients of
the matrix and based on the estimated doublet correlations.
Inventors: |
ZHUGE; Qunbi; (Kanata,
CA) ; REIMER; Michael Andrew; (Stittsville, CA)
; O'SULLIVAN; Maurice; (Ottawa, CA) |
|
Applicant: |
Name |
City |
State |
Country |
Type |
ZHUGE; Qunbi
REIMER; Michael Andrew
O'SULLIVAN; Maurice |
Kanata
Stittsville
Ottawa |
|
CA
CA
CA |
|
|
Assignee: |
Ciena Corporation
Hanover
MD
|
Family ID: |
63519814 |
Appl. No.: |
15/461718 |
Filed: |
March 17, 2017 |
Current U.S.
Class: |
1/1 |
Current CPC
Class: |
G06N 20/00 20190101;
H04B 10/07953 20130101; H04B 10/0775 20130101; G06N 3/08
20130101 |
International
Class: |
H04B 10/079 20060101
H04B010/079; H04B 10/2507 20060101 H04B010/2507; G06N 99/00
20060101 G06N099/00 |
Claims
1. A method of fiber Kerr nonlinear noise estimation in an optical
transmission system, the method comprising: recovering received
symbols from a received optical signal; isolating a noise component
of the received optical signal from the received symbols and from
either training symbols or estimated transmitted symbols determined
from the received symbols; estimating coefficients of a matrix
based on cross-correlations between the isolated noise component
and the fields of a triplet of received symbols or training symbols
or estimated transmitted symbols; estimating doublet correlations
of the product or the quotient of the isolated noise component and
the field of a received symbol or of a training symbol or of an
estimated transmitted symbol; and estimating one or more parameters
related to nonlinear noise based on the estimated coefficients of
the matrix and based on the estimated doublet correlations.
2. The method as recited in claim 1, wherein estimating the one or
more parameters related to nonlinear noise comprises: applying a
trained machine learning function to one or more inputs, the
trained machine learning function having as output the one or more
parameters related to nonlinear noise.
3. The method as recited in claim 2, wherein the one or more
parameters related to nonlinear noise include the intra-channel
nonlinear noise-to-signal ratio.
4. The method as recited in claim 2, wherein the one or more
parameters related to nonlinear noise include the total Kerr
nonlinear noise-to-signal ratio.
5. The method as recited in claim 2, wherein the one or more
parameters related to nonlinear noise include an optical
signal-to-noise ratio.
6. The method as recited in claim 2, wherein the one or more inputs
include the estimated coefficients of the matrix and/or results
obtained by pre-processing the estimated coefficients of the
matrix.
7. The method as recited in claim 2, wherein the one or more inputs
include the estimated doublet correlations and/or results obtained
by pre-processing the estimated doublet correlations.
8. The method as recited in claim 2, wherein the one or more inputs
include one or more attributes of a fiber link over which the
received optical signal was transmitted.
9. The method as recited in claim 1, wherein estimating the one or
more parameters related to nonlinear noise comprises: estimating an
intra-channel nonlinear noise-to-signal ratio from the estimated
coefficients of the matrix; estimating a total Kerr nonlinear
noise-to-signal ratio from the estimated doublet correlations; and
correcting the estimated intra-channel nonlinear noise-to-signal
ratio and the estimated total Kerr nonlinear noise-to-signal ratio
using calibrations based on one or more attributes of a fiber link
over which the received optical signal was transmitted.
10. The method as recited in claim 9, wherein estimating the total
Kerr nonlinear noise-to-signal ratio is based on intra-polarization
doublet correlations.
11. The method as recited in claim 9, wherein estimating the total
Kerr nonlinear noise-to-signal ratio is based on cross-polarization
doublet correlations.
12. The method as recited in claim 9, wherein estimating the
intra-channel nonlinear noise-to-signal ratio is based on
coefficients of intra-polarization matrices.
13. The method as recited in claim 9, wherein estimating the
intra-channel nonlinear noise-to-signal ratio is based on
coefficients of cross-polarization matrices.
14. The method as recited in claim 9, wherein the calibrations
involve polynomials that relate the one or more attributes of the
fiber link to an error in the estimated intra-channel nonlinear
noise-to-signal ratio and the estimated total Kerr nonlinear
noise-to-signal ratio, and coefficients of the polynomials have
been determined using fitting techniques.
15. A coherent optical receiver comprising: a digital signal
processor (DSP) implementing at least a carrier recovery module
operative to recover received symbols from a received signal, the
DSP implementing a decision block operative to output decisions
regarding the received symbols and from which estimated transmitted
symbols can be derived; components operative to isolate a noise
component of the received signal from the received symbols and from
either training symbols or the estimated transmitted symbols; and a
nonlinear noise-to-signal ratio (NSR) calculator operative to
estimate coefficients of a matrix based on cross-correlations
between the isolated noise component and the fields of a triplet of
received symbols or training symbols or estimated transmitted
symbols; estimate doublet correlations of the product or the
quotient of the isolated noise component and the field of a
received symbol or of a training symbol or of an estimated
transmitted symbol; and estimate one or more parameters related to
nonlinear noise based on the estimated coefficients of the matrix
and based on the estimated doublet correlations.
16. The coherent optical receiver as recited in claim 15, wherein
the nonlinear NSR calculator is operative to estimate the one or
more parameters related to nonlinear noise by applying a trained
machine learning function to one or more inputs, the trained
machine learning function having as output the one or more
parameters related to nonlinear noise.
17. The coherent optical receiver as recited in claim 16, wherein
the one or more inputs include the estimated coefficients of the
matrix and/or results obtained by pre-processing the estimated
coefficients of the matrix.
18. The coherent optical receiver as recited in claim 16, wherein
the one or more inputs include the estimated doublet correlations
and/or results obtained by pre-processing the estimated doublet
correlations.
19. The coherent optical receiver as recited in claim 15, wherein
the nonlinear NSR calculator is operative to estimate the one or
more parameters related to nonlinear noise by: estimating an
intra-channel nonlinear noise-to-signal ratio from the estimated
coefficients of the matrix; estimating a total Kerr nonlinear
noise-to-signal ratio from the estimated doublet correlations; and
correcting the estimated intra-channel nonlinear noise-to-signal
ratio and the estimated total Kerr nonlinear noise-to-signal ratio
using calibrations based on one or more attributes of a fiber link
over which the received signal was transmitted.
20. The coherent optical receiver as recited in claim 19, wherein
the calibrations involve polynomials that relate the one or more
attributes of the fiber link to an error in the estimated
intra-channel nonlinear noise-to-signal ratio and the estimated
total Kerr nonlinear noise-to-signal ratio, and coefficients of the
polynomials have been determined using fitting techniques.
Description
TECHNICAL FIELD
[0001] This document relates to the technical field of estimating
nonlinear characteristics of a channel in an optical fiber
communication system.
BACKGROUND
[0002] For long-haul optical transmission, the link accumulated
optical noise consists of linear and nonlinear contributions. The
linear noise results from optical amplification, that is, amplified
spontaneous emission (ASE) noise. An important nonlinear noise
results from the Kerr effect in optical fiber. The ratio of linear
and nonlinear noise depends on the power of optical signals during
transmission. At small launch power, the link accumulated noise is
dominated by linear noise due to low optical signal-to-noise ratio
(OSNR). At high launch power, the fiber nonlinearities can
dominate. The highest SNR, which includes other power-independent
noise such as transceiver internal noise, is achieved at a launch
power where a fixed proportion of linear and nonlinear optical
noise is approximately 2 to 1.
[0003] Separate measurement of linear and nonlinear optical noise
provides a means of optical power optimization, capacity
estimation, and capacity maximization. However, it is not
straightforward to distinguish linear and nonlinear noise in the
time domain. Furthermore, direct OSNR measurement in the frequency
domain becomes impractical as the available bandwidth of optical
fibers is completely occupied by signals for higher spectral
efficiency.
[0004] It has been proposed in Zhenhua Dong, Alan Pak Tao Lau, and
Chao Lu, "OSNR monitoring for QPSK and 16-QAM systems in presence
of fiber nonlinearities for digital coherent receivers," Opt.
Express 20, 19520-19534 (2012) and in H. G. Choi, J. H. Chang, Hoon
Kim and Y. C. Chung, "Nonlinearity-tolerant OSNR estimation
technique for coherent optical systems," 2015 Optical Fiber
Communications Conference and Exhibition (OFC), Los Angeles,
Calif., 2015, Paper W4D.2 to use the correlation of amplitude noise
or symbol amplitude on received symbols to derive the nonlinear
noise. However, accuracy of results depends on net chromatic
dispersion (CD), fiber type, fiber length, inline CD compensation
and so forth.
SUMMARY
[0005] Estimating intra-channel Kerr nonlinear noise-to-signal
ratio based on received signals in coherent optical receivers is
described. Estimates of intra-channel Kerr nonlinear
noise-to-signal ratio are derived from measured triplet
coefficients in the received noise.
[0006] Estimating total Kerr nonlinear noise-to-signal ratio based
on received signals in coherent optical receivers is described.
Estimates of total Kerr nonlinear noise-to-signal ratio are derived
from the measured correlation of doublet in the received noise.
[0007] Calibrations are proposed to make the estimates sufficiently
accurate over link attributes such as, for example, net chromatic
dispersion, link length, and symbol error rate.
BRIEF DESCRIPTION OF THE DRAWINGS
[0008] FIG. 1 illustrates a method for characterizing a
channel;
[0009] FIG. 2 illustrates an example coherent receiver;
[0010] FIG. 3 illustrates a graph of indices for which C-matrix
coefficients are of interest;
[0011] FIG. 4 illustrates a method of determining and using
calibration information;
[0012] FIG. 5, FIG. 6, FIG. 7, and FIG. 8 illustrate fitting errors
in Signal to Noise Ratio (SNR) for various Quadrature Phase Shift
Keying (QPSK) modulation simulations regarding intra-channel Kerr
noise;
[0013] FIG. 9, FIG. 10, FIG. 11, FIG. 12, FIG. 13, and FIG. 14
illustrate fitting errors in Signal to Noise Ratio (SNR) for
various 16 Quadrature Amplitude Modulation (16QAM) simulations
regarding intra-channel Kerr noise;
[0014] FIG. 15, FIG. 16, FIG. 17, and FIG. 18 illustrate fitting
errors in Signal to Noise Ratio (SNR) for various Quadrature Phase
Shift Keying (QPSK) modulation simulations regarding total Kerr
noise;
[0015] FIG. 19, FIG. 20, FIG. 21, and FIG. 22 illustrate fitting
errors in Signal to Noise Ratio (SNR) for various 16 Quadrature
Amplitude Modulation (16QAM) simulations regarding total Kerr
noise;
[0016] FIG. 23 illustrates an example neural network structure
having two inputs, ten hidden nodes, and two outputs;
[0017] FIG. 24 illustrates another example neural network structure
having four inputs, ten hidden nodes, and two outputs;
[0018] FIG. 25 illustrates error histograms for training samples
for the example neural network structure illustrated in FIG.
23;
[0019] FIG. 26 illustrates error histograms for test samples for
the example neural network structure illustrated in FIG. 23;
[0020] FIG. 27 illustrates error histograms for training samples
for the example neural network structure illustrated in FIG. 24;
and
[0021] FIG. 28 illustrates error histograms for test samples for
the example neural network structure illustrated in FIG. 24.
DETAILED DESCRIPTION
[0022] Coherent detection gives access to optical fields after
transmission, and provides a new path for link monitoring. In
particular, the properties of Kerr nonlinear noise can be exploited
to allow separation of linear noise, and intra-channel and
inter-channel nonlinearities at fixed power provisioning.
Challenges include proposing a method to exploit the properties of
Kerr nonlinear noise based on the received signal, obtaining
sufficient measurement accuracy over the wide variety of link
applications services, and measuring nonlinear noise with ongoing
traffic (in service).
[0023] The Kerr effect is a third-order nonlinearity. Three fields
(separated in time, or separated in frequency) interact to produce
a fourth field. The Kerr effect is referred to as intra-channel
Kerr nonlinear noise in cases where the three interacting fields
are within a channel and the resulting fourth field is also within
that same channel. The Kerr effect is referred to as inter-channel
Kerr nonlinear noise in cases where the three interacting fields
are between different channels (that is, one interacting field is
from one channel and two interacting fields are from another
channel, or each interacting field is from a different channel) and
in cases where the three interacting fields are from one channel
and the resulting fourth field is in a different channel.
[0024] FIG. 1 illustrates a method 2 for characterizing a channel
and optionally, for determining nonlinear noise. At 4, a coherent
optical receiver isolates a noise component of a received signal.
As described in more detail below, the noise component may be
isolated from the received symbols and estimated transmitted
symbols, or from the received symbols and training symbols. At 6,
the coherent optical receiver determines C-matrices and doublet
correlations that characterize how the Kerr effect is manifested in
a channel. The C-matrices and doublet correlations, which are
described in more detail below, are determined using the nonlinear
noise component that was isolated at 4, and using received symbols
or training symbols or estimated transmitted symbols. As described
in further detail below, various techniques may be employed to
reduce the complexity of determining the C-matrices. Correction of
systematic error is also described. Optionally, at 8, the coherent
optical receiver uses the C-matrices and doublet correlations
determined at 6 to calculate an estimated intra-channel
noise-to-signal ratio (NSR) and to calculate a total nonlinear NSR.
Optionally, at 9, the coherent optical receiver derives a linear
NSR (also known as the optical NSR), for example, as the difference
between the isolated noise and the total nonlinear NSR.
[0025] Isolation of the noise component assumes that nonlinear
noise is a small perturbation of the otherwise linear transmit
signal:
A'=A+.DELTA.A (1)
where A=[A.sub.X, A.sub.Y].sup.T is the dual-polarization
transmitted signal, A'.sub.X=A'.sub.Y].sup.T is the
dual-polarization received signal, and .DELTA.A=[.DELTA.A.sub.X,
.DELTA.A.sub.Y].sup.T is the nonlinear noise component of the
received signal. A, A', and .DELTA.A are functions of time t and
distance z. Other noise, including linear noise and transceiver
internal noise, is ignored in equation (1).
[0026] Based on perturbation theory, the nonlinear Schrodinger
equation can be expressed as
.differential. .DELTA. A .differential. z - i .beta. 2 2
.differential. 2 .DELTA. A .differential. t 2 + .alpha. 2 .DELTA. A
= - i .gamma. A 2 A ( 2 ) ##EQU00001##
where .beta..sub.2 is a second order dispersion coefficient, and a
is an attenuation coefficient of the optical power.
[0027] Intra-Channel Kerr Nonlinear Noise
[0028] With dual-polarization signals (say, an X-polarization and a
Y-polarization), an approximate solution of the intra-channel Kerr
nonlinear noise component for each polarization can be derived
based on equation (2) as
.DELTA. A X [ k ] = m , n C X [ m , n ] A X [ k + m ] A X [ k + n ]
A X * [ k + m + n ] + m , n C XY [ m , n ] A X [ k + m ] A Y [ k +
n ] A Y * [ k + m + n ] ( 3 ) .DELTA. A Y [ k ] = m , n C Y [ m , n
] A Y [ k + m ] A Y [ k + n ] A Y * [ k + m + n ] + m , n C YX [ m
, n ] A Y [ k + m ] A X [ k + n ] A X * [ k + m + n ] ( 4 )
##EQU00002##
where .DELTA.A.sub.X[k] is the nonlinear noise component indexed by
k for the X-polarization, .DELTA.A.sub.Y[k] is the nonlinear noise
component indexed by k for the Y-polarization, A.sub.X[k] is the
transmitted symbol indexed by k for the X-polarization, A.sub.Y[k]
is the transmitted symbol indexed by k for the Y-polarization, and
the asterisk (*) denotes conjugation. A.sub.X or Y [k] is a sampled
version of A.sub.Xor Y in equations (1) and (2), and .DELTA.A.sub.X
or Y[k] is a sampled version of .DELTA.A.sub.x or y in equations
(1) and (2).
[0029] As mentioned above, in the Kerr effect, three fields
(separated in time, or separated in frequency) interact to produce
a fourth field. Thus the summations in equations (3) and (4)
involve triplets of symbols: a symbol indexed by k+m, a symbol
indexed by k+n, and a symbol indexed by k+m+n. The triplet is
represented by the shorthand (k,m,n,m+n).
[0030] In the time domain, the three interacting fields may be
represented with A[k]=A(k.DELTA.t) where .DELTA.t is the sample
duration, and the corresponding C-matrices may be in the time
domain.
[0031] In the frequency domain, the three interacting fields may be
represented with A[k]= (k.DELTA.f), where (f) is the Fourier
transform of A(t) and .DELTA.f is the frequency spacing, and the
corresponding C-matrices may be in the frequency domain.
[0032] In the symbol domain, the three interacting fields may be
represented with with A[k]=A(kT) where T denotes the symbol
duration, and the corresponding C-matrices may be in the symbol
domain.
[0033] The indices m and n can take on negative values, zero values
and positive values. For example, the sequence A.sub.X[-2],
A.sub.X[-1], A.sub.X[0], A.sub.X[1], A.sub.X[2] represents five
consecutively transmitted symbols for the X polarization. The
contribution of the nonlinear intra-channel interaction between
A.sub.X[-1], A.sub.X[1], and A.sub.X[2] to the intra-channel Kerr
nonlinear noise for the X polarization, .DELTA.A.sub.X[0], is given
by C.sub.X[-1,2]A.sub.X[-1]A.sub.X[2]A.sub.X[1].
[0034] The coefficients C.sub.X[m, C.sub.Y[m, C.sub.XY[m, n], and
C.sub.YX[m, n] appearing in equations (3) and (4) form four
separate C-matrices C.sub.X, C.sub.Y, C.sub.XY, and C.sub.YX,
respectively, that characterize how the Kerr effect is manifested
in the channel over which the symbols have been transmitted and
received. That is, the C-matrices C.sub.x, C.sub.y, C.sub.xy, and
C.sub.yx characterize the fourth field that is produced by the Kerr
effect. In the absence of polarization effects, the C-matrices
C.sub.X, C.sub.Y, C.sub.XY, and C.sub.YX are identical to each
other.
[0035] Assuming the power of symbols is normalized, the C-matrices
can be estimated by evaluating the cross-correlation between the
intra-channel noise component and the fields of each (k,m,n,m+n)
triplet, and taking into account the correlation between the
triplets.
[0036] The C-matrices C.sub.X[m, n] and C.sub.XY[m, n] can be
evaluated as follows:
C'.sub.X[m,
n]=E[.DELTA.A.sub.X[k]A*.sub.X[k+m]A*.sub.X[k+n]A.sub.X[k+m+n]]
(5)
C'.sub.XY[m,
n]=E[.DELTA.A.sub.X[k]A*.sub.X[k+m]A*.sub.Y[k+n]A.sub.Y[k+m+n]]
(6)
Q.sub.X[k, m, n]=A.sub.X[k+m]A.sub.X[k+n]A*.sub.X[k+m+n] (7)
Q.sub.XY[k, m, n]=A.sub.X[k+m]A.sub.Y[k+n]A*.sub.Y[k+m+n] (8)
{right arrow over (C)}.sub.X=vec(C'.sub.X[m, n], C'.sub.XY, [m, n])
(9)
{right arrow over (Q)}.sub.X=vec(Q.sub.X[k, m, n], Q.sub.XY[k, m,
n]) (10
R.sub.X=E[{right arrow over (Q)}.sub.X{right arrow over
(Q)}.sub.X.sup.H] (11)
{right arrow over (C)}.sub.X={right arrow over
(C)}.sub.Xinv(R.sub.X) (12)
(C.sub.x[m, n], C.sub.XY[m, n])=vec.sup.-1({right arrow over
(C)}.sub.X) (13)
where vec() is a function that converts the matrices to a single
vector with a certain order of all elements, vec.sup.-1() is the
inverse operation of vec(), and inv() is inverse of a matrix.
[0037] The C-matrices C.sub.Y[m, n] and C.sub.YX[m, n] can be
evaluated as follows:
C'.sub.Y[m, n]=E[.DELTA.A.sub.Y[k]A*.sub.Y, [k+m]A*.sub.Y,
[k+n]A.sub.Y[k+m+n]] (14)
C'.sub.YX[m,
n]=E[.DELTA.A.sub.Y[k]A*.sub.Y[k+m]A*.sub.X[k+n]A.sub.X[k+m+n]]
(15)
Q.sub.Y[k, m, n]=A.sub.Y[k+m]A.sub.Y[k+n]A*.sub.Y[k+m+n] (16)
Q.sub.YX[k, m, n]=A.sub.Y[k+m] A.sub.X[k+n]A*.sub.X[k+m+n] (17)
{right arrow over (C)}.sub.Y=vec(C'.sub.Y[m, n], C'.sub.YX[m, n])
(18)
{right arrow over (Q)}.sub.Y=vec(Q.sub.Y[k, m, n], Q.sub.YX[k,m,
n]) (19)
R.sub.Y=E[{right arrow over (Q)}.sub.Y{right arrow over
(Q)}.sub.Y.sup.H] (20)
{right arrow over (C)}.sub.Y={right arrow over
(C)}'.sub.Yinv(R.sub.Y) (21)
(C.sub.Y[m, n], C.sub.YX[m, n])=vec.sup.-1({right arrow over
(C)}.sub.Y) (22)
where vec() is a function that convert the matrices to a single
vector with a certain order of all elements, vec.sup.-1() is the
inverse operation of vec(), and inv() is inverse of a matrix.
[0038] In equations (5), (6), (11), (14), (15) and (20), E is an
expectation, which is a weighted average. In equations (3) through
(22), the transmitted symbols can be replaced by the received
symbols.
[0039] Note that C.sub.X[m, 0], C.sub.X[0, n], C.sub.XY[M, 0],
C.sub.Y[m, 0], C.sub.Y[0, n], and C.sub.YX[M, 0] cannot be
evaluated in this manner due to correlations between the various
triplet terms. For example,
C.sub.X[1,0]A.sub.X[k]|A.sub.X[k+1]|.sup.2 is correlated with
C.sub.X[2,0]A.sub.X[k]|A.sub.X[k+2]|.sup.2 and it is not possible
to separate C.sub.X[1,0] and C.sub.X[2,0] in the proposed
calculations. Therefore, the coefficients C.sub.XY[0, n] and
C.sub.YX[0, n] are used to approximate these 6 coefficients. The
coefficients C.sub.X[0,0], C.sub.Y[0,0], C.sub.XY[0,0], and
C.sub.YX[0,0] cannot be evaluated.
[0040] The intra-channel nonlinear noise-to-signal ratio (NSR),
denoted in this document as NL-NSR.sub.INTRA,X or Y, may be
calculated as:
NL-NSR.sub.INTRA,X.apprxeq.E[{right arrow over
(C)}.sub.X.sup.H{right arrow over (Q)}.sub.X{right arrow over
(Q)}.sub.X.sup.H{right arrow over (C)}.sub.X]={right arrow over
(C)}.sub.X.sup.HE[({right arrow over (Q)}.sub.X{right arrow over
(Q)}.sub.X.sup.H]{right arrow over (C)}.sub.X={right arrow over
(C)}.sub.X.sup.HR.sub.X{right arrow over (C)}.sub.X (23)
NL-NSR.sub.INTRA,Y.noteq.E[{right arrow over (C)}.sub.Y.sup.H{right
arrow over (Q)}.sub.Y{right arrow over (Q)}.sub.Y.sup.H{right arrow
over (C)}.sub.Y]={right arrow over (C)}.sub.Y.sup.HE[{right arrow
over (Q)}.sub.Y{right arrow over (Q)}.sub.Y.sup.H]{right arrow over
(C)}.sub.Y={right arrow over (C)}.sub.Y.sup.HR.sub.Y{right arrow
over (C)}.sub.Y (24)
In equations (23) and (24), E is an expectation, which is a
weighted average.
[0041] Total Kerr Nonlinear Noise
[0042] With dual-polarization signals (say, an X-polarization and a
Y-polarization), the total Kerr nonlinear noise component--which
includes both the intra-channel Kerr nonlinear noise component and
the inter-channel Kerr nonlinear noise--can be approximated by:
.DELTA. A X [ k ] .apprxeq. n = - N N A X [ k + n ] { m C SPM [ m ,
n ] ( A X [ k + m ] A X * [ k + m + n ] + A Y [ k + m ] A Y * [ k +
m + n ] ) + m C XPM [ m , n ] ( 2 B X [ k + m ] B X * [ k + m + n ]
+ B Y [ k + m ] B Y * [ k + m + n ] ) } + n = - N N A Y [ k + n ] m
C XPM [ m , n ] B X [ k + m ] B Y * [ k + m + n ] ( 25 ) .DELTA. A
Y [ k ] .apprxeq. n = - N N A Y [ k + n ] { m C SPM [ m , n ] ( A Y
[ k + m ] A Y * [ k + m + n ] + A X [ k + m ] A X * [ k + m + n ] )
+ m C XPM [ m , n ] ( 2 B Y [ k + m ] B Y * [ k + m + n ] + B X [ k
+ m ] B X * [ k + m + n ] ) } + n = - N N A X [ k + n ] m C XPM [ m
, n ] B Y [ k + m ] B X * [ k + m + n ] ( 26 ) ##EQU00003##
where .DELTA.A.sub.X[k] is the total nonlinear noise component
indexed by k for the X-polarization in the channel,
.DELTA.A.sub.Y[k] is the total nonlinear noise component indexed by
k for the Y-polarization in the channel, A.sub.X[k] is the
transmitted symbol indexed by k for the X-polarization in the
channel, A.sub.y[k] is the transmitted symbol indexed by k for the
Y-polarization in the channel, B .sub.x[k] is the transmitted
symbol indexed by k for the X-polarization in a different channel,
B.sub.y[k] is the transmitted symbol indexed by k for the
Y-polarization in the different channel, and the asterisk (*)
denotes conjugation.
[0043] For each polarization, the following doublets are correlated
over k: [0044] .SIGMA..sub.mC.sup.SPM[m,
n](A.sub.X[k+m]A*.sub.X[k+m+n]+A.sub.Y[k+m]A*.sub.Y[k+m+n]) in
equation (25); [0045] .SIGMA..sub.mC.sup.XPM[m,
n](2B.sub.X[k+m]B*.sub.X[k+m+n]+B.sub.Y[k+m]B*.sub.Y[k+m+n]) in
equation (25); [0046] .SIGMA..sub.mC.sup.XPM[m,
n]B.sub.X[k+m]B*.sub.Y[k+m+n] in equation (25);
[0047] and [0048] .SIGMA..sub.mC.sup.SPM[m,
n](A.sub.Y[k+m]A*.sub.Y[k+m+n]+A.sub.X[k+m]A*.sub.X[k+m+n]) in
equation (26); [0049] .SIGMA..sub.mC.sup.XPM[m,
n](2B.sub.Y[k+m][k+m+n]+B.sub.X[k+m]B*.sub.X[k+m+n]) in equation
(26); [0050] .SIGMA..sub.mC.sup.XPM[m,
n]B.sub.Y[k+m]B*.sub.Y[k+m+n] in equation (26).
[0051] Moreover, the doublet across two polarizations is partially
correlated. These correlations can be exploited to estimate the
total Kerr nonlinearities even though the symbols of the channels B
are not accessible.
[0052] In equations (25) and (26), one can focus on the
intra-polarization correlation (the C-matrix C.sup.SPM) or on the
cross-polarization correlation (the C-matrix C.sup.XPM) or on both
correlations. In general, the correlations of the doublet can be
substituted by the expressions .DELTA.A.sub.Y[k]/A.sub.X[k+n] and
.DELTA.A.sub.X[k]/A.sub.Y[k+n], respectively. These correlations
can be used to estimate the total Kerr nonlinear noise-to-signal
ratio. For example, assuming transmitted symbols are uncorrelated
and normalized, the following is an example formulation for an
approximation of the total Kerr nonlinear noise-to-signal ratio,
denoted in this document as NL-NSR.sub.TOTAL:
NL - NSR TOTAL .apprxeq. n = - N N p ( n ) ( 27 ) p ( 0 ) = E [
.DELTA. A X [ k ] A Y * [ k ] .DELTA. A Y [ k ] A X * [ k ] ] ( 28
) p ( n ) = E [ .DELTA. A X [ k ] A X * [ k + n ] .DELTA. A Y [ k ]
A Y * [ k + n ] ] , for n .noteq. 0 ( 29 ) ##EQU00004##
where N denotes the symbol delay in the estimation, which is
related to estimation accuracy and time. For a link with a long
memory, that is, with high net chromatic dispersion, a large N
should be used in principle. In equations (28) and (29), E is an
expectation, which is a weighted average.
[0053] The following is a summary of doublet correlations that can
be derived based on equation (25) and equation (26). These doublet
correlations can be used to estimate the total Kerr nonlinear
noise-to-signal ratio. The correlations between transmitted symbols
should also be included, if they exist. Any one or more of the
doublet correlations and the symbol correlations can be used as the
input to machine learning functions (e.g. a neural network) to
obtain the estimate of the nonlinear noise-to-signal ratio or other
related system parameters.
[0054] Intra-Polarization Doublet Correlation for the
X-Polarization
p 1 X ( n , i ) = E [ .DELTA. A X [ k ] A X [ k + n ] .DELTA. A X [
k + n + i ] A X [ k + i ] ] ( 30 ) p 2 X ( n , i ) = E [ .DELTA. A
X [ k ] A X * [ k + n ] .DELTA. A X [ k + n + i ] A X * [ k + i ] ]
( 31 ) p 3 X ( n , i ) = E [ .DELTA. A X [ k ] A X [ k + n ]
.DELTA. A X * [ k + i ] A X * [ k + n + i ] ] , i .noteq. 0 ( 32 )
p 4 X ( n , i ) = E [ .DELTA. A X [ k ] A X * [ k + n ] .DELTA. A X
* [ k + i ] A X [ k + n + i ] ] , i .noteq. 0 ( 33 )
##EQU00005##
[0055] Intra-Polarization Doublet Correlation for the
Y-Polarization
p 1 Y ( n , i ) = E [ .DELTA. A Y [ k ] A Y [ k + n ] .DELTA. A Y [
k + n + i ] A Y [ k + i ] ] ( 34 ) p 2 Y ( n , i ) = E [ .DELTA. A
Y [ k ] A X * [ k + n ] .DELTA. A Y [ k + n + i ] A Y * [ k + i ] ]
( 35 ) p 3 Y ( n , i ) = E [ .DELTA. A Y [ k ] A Y [ k + n ]
.DELTA. A Y * [ k + i ] A Y * [ k + n + i ] ] , i .noteq. 0 ( 36 )
p 4 Y ( n , i ) = E [ .DELTA. A Y [ k ] A Y * [ k + n ] .DELTA. A Y
* [ k + i ] A Y [ k + n + i ] ] , i .noteq. 0 ( 37 )
##EQU00006##
[0056] Cross-Polarization Doublet Correlation
p 5 ( n , i ) = E [ .DELTA. A X [ k ] A X [ k + n ] .DELTA. A Y * [
k + i ] A Y * [ k + n + i ] ] ( 38 ) p 6 ( n , i ) = E [ .DELTA. A
X [ k ] A X * [ k + n ] .DELTA. A Y * [ k + i ] A Y [ k + n + i ] ]
( 39 ) p 7 ( n , i ) = E [ .DELTA. A X [ k ] A X [ k + n ] .DELTA.
A Y [ k + n + i ] A Y [ k + i ] ] ( 40 ) p 8 ( n , i ) = E [
.DELTA. A X [ k ] A X * [ k + n ] .DELTA. A Y [ k + n + i ] A Y * [
k + i ] ] ( 41 ) p 9 ( n , i ) = E [ .DELTA. A X [ k ] A Y [ k + n
] .DELTA. A Y [ k + n + i ] A X [ k + i ] ] ( 42 ) p 10 ( n , i ) =
E [ .DELTA. A X [ k ] A Y * [ k + n ] .DELTA. A Y [ k + n + i ] A X
* [ k + i ] ] ( 43 ) p 11 X ( n , i ) = E [ .DELTA. A X [ k ] A Y [
k + n ] .DELTA. A X * [ k + i ] A Y * [ k + n + i ] ] , i .noteq. 0
( 44 ) p 12 X ( n , i ) = E [ .DELTA. A X [ k ] A Y * [ k + n ]
.DELTA. A X * [ k + i ] A Y [ k + n + i ] ] , i .noteq. 0 ( 45 ) p
11 Y ( n , i ) = E [ .DELTA. A Y [ k ] A X [ k + n ] .DELTA. A Y *
[ k + i ] A X * [ k + n + i ] ] , i .noteq. 0 ( 46 ) p 12 Y ( n , i
) = E [ .DELTA. A Y [ k ] A X * [ k + n ] .DELTA. A Y * [ k + i ] A
X [ k + n + i ] ] , i .noteq. 0 ( 47 ) ##EQU00007##
[0057] In equations (30) through (47), E is an expectation, which
is a weighted average. In equations (30) through (47), the
transmitted symbols can be replaced by the received symbols.
[0058] FIG. 2 illustrates an example coherent optical receiver 10.
A polarizing beam splitter 12 is operative to split a received
optical signal 14 into orthogonally-polarized components 16, 18. An
optical hybrid 20 is operative to process the
orthogonally-polarized components 16, 18 with respect to a
reference optical signal 22 produced by a laser 24. Photodetectors
26 are operative to convert the output of the optical hybrid 20 to
electrical signals corresponding to the in-phase (I) and quadrature
(Q) components on both polarizations (say, an X-polarization and a
Y-polarization). Amplifiers 28 are operative to amplify the
electrical signals, and analog-to-digital converters (ADCs) 30 are
operative to sample the amplified electrical signals.
[0059] An application specific integrated circuit (ASIC) 32
comprises components to process the output of the ADCs 30 to
recover the data that was transmitted. The ASIC 32 comprises a
digital signal processor (DSP) 34.
[0060] The DSP 34 is operative to process the output of the ADCs 30
to perform dispersion compensation, polarization compensation,
clock recovery, carrier recovery, symbol decoding, and so
forth.
[0061] A carrier recovery module 36 implemented by the DSP 34 is
operative to track and remove the phase noise from both the
transmit laser (not shown) and the laser 24 so as to produce
digital representations 42 of analog received symbols (also
referred to as "received waveforms") for the X polarization and Y
polarization. Each received symbol, and hence its digital
representation, is composed of the true transmitted symbol and a
noise component. The true transmitted symbol is the symbol
modulated onto a polarized component of an optical carrier at a
transmitter, the modulated optical carrier being sent over a
channel to the coherent optical receiver 10, where it is received
as the received optical signal 14.
[0062] A decision block 38 implemented by the DSP 34 is operative
to output decisions 44 regarding the received symbols for the X
polarization and Y polarization. In cases where the decision block
38 implements a hard decision, each decision 44 output by the
decision block 38 is a set of bits representing an estimated
transmitted symbol. The estimated transmitted symbol includes some
error with respect to the true transmitted symbol. Over time, there
might be more than 10% of the estimated transmitted symbols that
are erroneous, that is, not identical to the true transmitted
symbol. In cases where the decision block 38 implements a soft
decision, the most significant bit (MSB) of each soft decision
output by the decision block 38 is part of the string of bits that
is converted to the estimated transmitted symbol.
[0063] The example coherent optical receiver 10 differs from
conventional coherent optical receivers in that it comprises
additional functionality to enable calculation of nonlinear
noise-to-signal ratio. The remaining discussion of FIG. 2 describes
this additional functionality as being implemented in the ASIC 32.
In an alternative implementation, this additional functionality or
a portion thereof is implemented in a low-speed processor 48 that
is operatively coupled to the ASIC 32. The low-speed processor 48
is illustrated as an optional component of the example coherent
optical receiver 10.
[0064] To enable calculation of nonlinear noise-to-signal ratio,
the ASIC 32 may comprise a bit-to-symbol converter 50, subtraction
elements 52, 54, a delay element 56, and a nonlinear
noise-to-signal ratio (NSR) calculator 60. The subtraction element
52 is for the X-polarization, and the subtraction element 54 is for
the Y-polarization. The nonlinear NSR calculator 60 has access to
calibration information 61, as described in further detail below.
The nonlinear NSR calculator 60 may also have access to training
symbols 63, as described in further detail below. The calibration
information 60 and, if present, the training symbols 63, may be
stored in a memory 59. The nonlinear NSR calculator 60 may be
implemented in software (also stored in the memory 59) or in
firmware or in hardware or in any combination thereof.
[0065] The bit-to-symbol converter 50 is operative to convert a set
of bits into a digital representation 62 of an analog pristine
symbol with no noise. The bit-to-symbol converter 50 receives, for
each polarization, an input 64 that is the set of bits representing
an estimated transmitted symbol (because the set of bits are a hard
decision or the MSBs of a soft decision), and the digital
representation 62 is of an analog pristine symbol identical to the
estimated transmitted symbol with no noise.
[0066] Each of the subtraction elements 52, 54 is operative to
subtract the digital representation 62 of the analog pristine
symbol with no noise from a delayed version 66 of the digital
representation 42 of the analog received symbol. The delay element
56 is operative to delay the digital representation 42 of the
analog received symbol to yield the delayed version 66, so that the
subtraction element 52, 54 effects a comparison of the analog
received symbol corresponding to a particular instant in time to an
estimated transmitted symbol from the same instant in time. The
output of the subtraction element 52, 54 is a digital
representation 68 of a noise component in the analog received
symbol.
[0067] The subtraction by the subtraction element 52 can be
represented as .DELTA.A.sub.X[k]=R.sub.X[k]-A.sub.X[k], and the
subtraction by the subtraction element 54 can be represented as
.DELTA.A.sub.Y[k]=R.sub.Y[k]-A.sub.Y[k], where R.sub.X[k] and
R.sub.Y[k] are the digital representations of the received symbol
indexed by k for the X-polarization and for the Y-polarization,
respectively.
[0068] In the implementation where the input 64 to the
bit-to-symbol converter 50 is the set of bits representing an
estimated transmitted symbol, A.sub.X[k] and A.sub.Y[k] are the
digital representations of the estimated transmitted symbol indexed
by k for the X-polarization and for the Y-polarization,
respectively, and .DELTA.A.sub.X[k] and .DELTA.A.sub.X[k] are
digital representations of the estimated noise component of the
analog received symbol indexed by k for the X-polarization and for
the Y-polarization, respectively.
[0069] The nonlinear NSR calculator 60 is operative to perform
nonlinear noise-to-signal ratio calculations based on its input.
The nonlinear NSR calculator 60 receives as input, for the
X-polarization and for the Y-polarization, the digital
representations 68 of the noise component in the analog received
symbol. The nonlinear NSR calculator 60 may also receive as input,
for the X-polarization and for the Y-polarization, either the
digital representation 58 of the analog pristine symbol (the
estimated transmitted symbol) with no noise or the delayed version
66 of the digital representation 42 of the analog received
symbol.
[0070] The nonlinear NSR calculator 60 is operative to provide a
measure or estimate of intra-channel Kerr nonlinear noise-to-signal
ratio and a measure or estimate of total Kerr nonlinear
noise-to-signal ratio. The intra-channel Kerr nonlinear
noise-to-signal ratio is derived from measured triplet coefficients
in the received noise. The total Kerr nonlinear noise-to-signal
ratio is derived from doublet correlations in the received
noise.
[0071] With respect to the intra-channel Kerr nonlinear
noise-to-signal ratio, the nonlinear NSR calculator 60 may evaluate
the C-matrices C.sub.X and C.sub.XY according to equations (5) to
(13), and the C-matrices C.sub.Y and C.sub.YX according to
equations (14) to (22). In some implementations, the nonlinear NSR
calculator 60 may calculate NL-NSR.sub.INTRA,X or Y according to
equations (23) and (24).
[0072] With respect to the total Kerr nonlinear noise-to-signal
ratio, the nonlinear NSR calculator 60 may evaluate
NL-NSR.sub.TOTAL according to equations (27) to (29).
[0073] The previous discussion shows how the noise component is
isolated from the received symbols and the estimated transmitted
symbols. In an alternative implementation, the noise component is
isolated from the received symbols and training symbols, where the
training symbols 63 are substituted for the digital representation
62 of the analog pristine symbol with no noise.
[0074] The previous discussion shows how coefficients of the
C-matrices are evaluated from the isolated noise component and the
fields of a triplet of received symbols or estimated transmitted
symbols. In an alternative implementation, the training symbols 63
are substituted for the received symbols or for the estimated
transmitted symbols in the calculations.
[0075] The previous discussion shows how doublet correlations are
evaluated from the product or quotient of the isolated noise
component and the field of a received symbol or of an estimated
transmitted symbol. In an alternative implementation, the training
symbols 63 are substituted for the received symbols or for the
estimated transmitted symbols in the calculations.
[0076] Implementation Considerations
[0077] Various techniques, alone or in combination, may be employed
by the nonlinear NSR calculator 60, to increase the accuracy of the
calculations, or to reduce the complexity of the calculations, or
to reduce the number of calculations. Such techniques include
exploitation of symmetry, limiting the range of values for the
indices m and n over which the coefficients are evaluated,
interpolation, truncation, and using machine learning techniques to
extract information from fewer inputs.
[0078] Reduced-complexity implementations for determining
C-matrices are proposed. For example, inter-channel noise may be
ignored. In an alternative example, when considering inter-channel
noise, only nearest neighbor channels may be included, and other
channels may be ignored.
[0079] Symmetry
[0080] The extent of the Kerr effect is related to the memory of
the channel, and the values for the indices m and n that are
considered in the summation relate to the memory accounted for in
the calculations. As mentioned above, the indices m and n in
equations (5) to (22) can take on negative values, zero values and
positive values. Symmetry may be exploited, to reduce the number of
coefficients that are evaluated, or to increase accuracy by
averaging over symmetric coefficients, or both. For example, the
C-matrices C.sub.X, C.sub.Y, C.sub.XY, and C.sub.YX could be
calculated only for positive values of the indices m and n, relying
on symmetry of the quadrants to determine the coefficients for the
other quadrants. In another example, averaging could be evaluated
as follows, where C could be any one of the C-matrices C.sub.X,
C.sub.Y, C.sub.XY, and C.sub.YX:
C'[m, n]=1/4(C[m, n]+C[-m, n]+C [m, -n]+C[m, n]), for m>0,
n>0 (48)
C'[0, n=1/2(C [0, n]+C[0, n]), for n>0 (49)
[0081] For the C-matrices C.sub.XY and C.sub.YX a further symmetry
can be exploited to increase accuracy:
C''[m, n]=1/2(C'[m, n]+C'[n, m]), for m>0,n>0 (50)
[0082] Limited Range of Values for Indices
[0083] The extent of the Kerr effect is related to the memory of
the channel. The memory of the channel is determined by factors
such as dispersion, link length, and signal bandwidth. When the
symbols and coefficients are indexed by absolute time units, then
as the baud increases the number of symbols to be included in the
summations has to increase, because the number of symbols in a
particular time span has increased. As the baud increases, the
spectral extent of the channel increases, and through dispersion,
the amount of spread in time increases. So the memory of the
channel, when indexed or measured in absolute time units, goes
roughly as the baud squared.
[0084] In some cases, it may be sufficient to consider m and n
values in the range of -10 through +10. In other cases, it may be
prudent to consider m and n values in the range of -100 through
+100, or in the range of -200 through +200.
[0085] FIG. 3 illustrates a Cartesian graph with axes labeled m and
n. An interior area 70 indicates the values of m and n for which
the C-matrix coefficients are of interest. That is, one can ignore
C-matrix coefficients C[m,n] for values of m,n that are not within
the interior area 70, but are within the remaining areas 72.
[0086] If a channel employs frequency division multiplexing (FDM),
then although the channel as a whole may have a wide spectral
extent, the effective baud that has to be contended with within a
given C-matrix has been reduced, and separate C-matrices may be
determined on a per-division basis.
[0087] Interpolation
[0088] In many cases, more than ten thousand C-matrix coefficients
are required for each the C-matrices C.sub.X, C.sub.Y, C.sub.XY,
and C.sub.YX to obtain a relatively accurate estimation of
NL-NSR.sub.INTRA,X or Y. However, computation of so many C-matrix
coefficients might take too long, based on hardware that is
currently available.
[0089] For example, assuming the triplets are uncorrelated,
NL-NSR.sub.INTRA,X or Y can be calculated based on the power of the
coefficients in the C-matrices. For each index m we can define
P X [ m ] = 8 n > 0 , m .noteq. n C X [ m , n ] 2 + 4 C X [ m ,
m ] 2 + 4 n > 0 C XY [ m , n ] 2 , m > 0 ( 51 ) P X [ 0 ] = 2
n > 0 C XY [ n , 0 ] 2 ( 52 ) P Y [ m ] = 8 n > 0 , m .noteq.
n C Y [ m , n ] 2 + 4 C Y [ m , m ] 2 + 4 n > 0 C YX [ m , n ] 2
, m > 0 ( 53 ) P Y [ 0 ] = 2 n > 0 C YX [ n , 0 ] 2 ( 54 ) NL
- SNR X [ m ] = m .gtoreq. 0 P X [ m ] ( 55 ) NL - SNR Y [ m ] = m
.gtoreq. 0 P Y [ m ] ( 56 ) ##EQU00008##
[0090] In one example, the quantity P.sub.X [m] (equations (51) and
(52)) or the quantity P.sub.Y[m] (equations (53) and (54)) may be
evaluated for the index m belonging to the set {0, 1, 2, 3, 4, 6,
9, 19, 29, 49, 69, 99, 129, 159, 199}. The values of the C-matrix
coefficients are expected to be very small when the index m is
large. Therefore more precision is used for small values of the
index m than for large values of the index m. When m=0 the
summation is from n=1 to n=150. When m=1, the summation is from n=1
to n=100. When m=2, the summation is from n=1 to n=90. When m=3,
the summation is from n=1 to n=80. When m=4, the summation is from
n=1 to n=70. When m=6, the summation is from n=1 to n=60. And when
m is equal to any other value in the set above, the summation is
from n=1 to n=50. The values of the quantity P.sub.X[m] or the
quantity P.sub.Y[m] for other values of the index m .di-elect cons.
[0,199] may be obtained by simple linear interpolation. This
example involves approximately four thousand C-matrix coefficients
for each of the C-matrices C.sub.X, C.sub.Y, C.sub.XY, and
C.sub.YX, compared to the ten thousand C-matrix coefficients
mentioned above.
[0091] Calibration
[0092] There are errors inherent in some of the calculations
performed by the nonlinear NSR calculator 60. The errors may be due
to one or more of the following factors: the model being an
approximation of the actual nonlinearity, the use of estimated
transmitted symbols rather than true transmitted symbols to
determine the C-matrices, the use of techniques to reduce
computation complexity, the use of interpolation, and
simplifications in the equations (for example, focusing on the
cross-polarization correlation (the C-matrix C.sup.XPM) in
equations (25) and (26)). Calibration techniques are employed by
the nonlinear NSR calculator 60 to reduce the errors and improve
accuracy.
[0093] FIG. 4 illustrates a method 80 for developing and using
calibration information. Calibration information 61 used by the
nonlinear NSR calculator 60 is an example of calibration to be
developed and used according to the method 80.
[0094] An application is defined as a particular configuration of
an optical communication system, characterized by fiber type,
distance between transmitter and receiver, line chromatic
dispersion compensation (CDC) ratio, number of wavelength-division
multiplexing (WDM) channels, and the launch power of the
signal.
[0095] For example, the following applications were considered, for
56.8 Gbaud signals with 62.5 GHz spacing:
TABLE-US-00001 TABLE 1 Applications for systematic error study Line
CDC Power (dBm) Power (dBm) Fiber Type Distance (km) ratio WDM CHs
0% CDC 90% CDC NDSF 320, 640, 1200, 0%, 90% 1, 3, 5, 7 -1.5, 0.5,
2.5 -4, -2, 0 2000, 3200 TWC 320, 640, 1200, 0%, 90% 1, 3, 5, 7
-4.5, -2.5, -0.5 -6, -4, -2 2000, 3200 ELEAF 320, 640, 1200, 0%,
90% 1, 3, 5, 7 -3, -1, 1 -5, -3, -1 2000, 3200 NDSF & 320, 640,
1200, 0%, 90% 1, 3, 5, 7 -3, -1, 1 -5, -3, -1 TWC 2000, 3200 NDSF
& 320, 640, 1200, 0%, 90% 1, 3, 5, 7 -2.5, -0.5, 1.5 -4.5,
-2.5, -0.5 ELEAF 2000, 3200 ELEAF & 320, 640, 1200, 0%, 90% 1,
3, 5, 7 -4, -2, 0 -5.5, -3.5, -1.5 TWC 2000, 3200
[0096] Split-step Fourier method (SSFM) simulations are performed
at 82. Each simulation of a particular application (see Table 1
above) involves known transmitted symbols, known nonlinear
inter-channel Kerr noise and known nonlinear intra-channel Kerr
noise. A coherent optical receiver that implements the method 2
described with respect to FIG. 1 and FIG. 2 is simulated. The
simulated coherent optical receiver isolates a noise component of a
received signal or symbol, calculates C-matrices using the isolated
noise component and either an estimated transmitted symbol or the
known transmitted symbol, and, optionally, calculates a nonlinear
contribution to the total NSR. The calculated nonlinear
contribution to the total NSR can be compared to the known noise
used in the simulation.
[0097] By performing the simulations over different applications
(see Table 1 above), one can develop at 84 a series of
relationships between attributes of the link (such as, for example,
net chromatic dispersion along the link, the link length or
distance, fiber type, span length, WDM configuration, etc.) and how
the calculated nonlinear contribution to the total NSR differs from
the known noise used in the simulation. The calibration information
61 is based on the series of relationships.
[0098] In one implementation, the relationships are polynomials,
developing the relationships involves determining coefficients of
the polynomials using known fitting techniques, and the calibration
information 61 includes the polynomials, their determined
coefficients, and one or more attributes of the link. In another
implementation, the relationships are developed using neural
networks, and the calibration information 61 includes a trained
neural network and optionally one or more attributes of the
link.
[0099] At 86, the nonlinear NSR calculator 60 uses the calibration
information 61 based on the series of relationships developed at 84
to calibrate or correct the calculation of the C-matrices (or to
calibrate or correct the calculation of the nonlinear contribution
to total NSR).
[0100] Calibration of Intra-Channel Kerr Nonlinear Noise-to-Aignal
Ratio Using Polynomials
[0101] QPSK (Quadrature Phase Shift Keying) Examples
[0102] Simulations were performed for applications with QPSK
modulation. Using net chromatic dispersion only to fit the results
of the simulations using a second order polynomial regression
as
.epsilon.=c.sub.0+cx+c.sub.2x.sup.2 (57)
where .epsilon. is the offset in dB, x=log.sub.10 (net chromatic
dispersion in ps/nm), and the coefficients obtained were
c.sub.0=7.029, c.sub.1=3.343, and c.sub.2=0.472 with a root mean
square error (RMSE) of 0.3008 dB. The fitting error in SNR (dB) is
illustrated in FIG. 5.
[0103] Using both net chromatic dispersion and link length to fit
the results as
.epsilon.=c.sub.0+c.sub.1x+c.sub.2y+c.sub.3x.sup.2+c.sub.4xy+c.sub.5y.su-
p.2 (58)
where .epsilon. is the offset in dB, x=log.sub.10 (net chromatic
dispersion in ps/nm), y=[net chromatic dispersion in ps/nm]/[link
length in km], and the coefficients obtained were c.sub.0=6.438,
c.sub.1=2.695, c.sub.2=0.08515, c.sub.3=0.294, c.sub.4=0.03263, and
c.sub.5=0.009783 with a root mean square error (RMSE) of 0.2255 dB.
The fitting error in SNR (dB) is illustrated in FIG. 6.
[0104] Even with reduced complexity, calibration information can be
determined. For example, evaluating the quantity P.sub.X or Y [0]
given in equation (52) or (54) above with a half-width of 10
symbols yields the following fitting results with net chromatic
dispersion and link length: c.sub.0=32.0806, c.sub.1=19.8526,
c.sub.2=1.1470, c.sub.3=3.4696, c.sub.4=0.2033, and c.sub.5=0.0124
with a root mean square error (RMSE) of 0.5026 dB. The fitting
error in SNR (dB) is illustrated in FIG. 7.
[0105] Similarly, evaluating the quantity P.sub.X or Y [m] given in
equation (51) or (53) above for m=1 with a half-width of 20 symbols
yields the following fitting results with net chromatic dispersion
and link length: c.sub.0=14.5311, c.sub.1=4.4661, c.sub.2=0.6968,
c.sub.3=0.3787, c.sub.4=0.0399, and c.sub.5=0.0352 with a root mean
square error (RMSE) of 0.5478 dB. The fitting error in SNR (dB) is
illustrated in FIG. 8.
[0106] 16QAM (16 Quadrature Amplitude Modulation) Examples
[0107] Simulations were performed for applications with 16QAM
modulation. Using net chromatic dispersion only to fit the results
of the simulations using a second order polynomial regression as in
equation (57) above, the coefficients obtained were c.sub.o=21.93,
c.sub.1=9.112, and c.sub.2=1.055 with a root mean square error
(RMSE) of 0.2214 dB. The fitting error in SNR (dB) is illustrated
in FIG. 9.
[0108] Using both net chromatic dispersion and link length to fit
the results as in equation (58) above, the coefficients obtained
were c.sub.0=19.6769, c.sub.1=7.3748, c.sub.2=0.2014,
c.sub.3=0.7243, c.sub.4=0.0865, and c.sub.5=0.0070 with a root mean
square error (RMSE) of 0.1971 dB. The fitting error in SNR (dB) is
illustrated in FIG. 10.
[0109] Even with reduced complexity, calibration information can be
determined. For example, evaluating the quantity P.sub.X or Y [0]
given in equation (52) or (54) above with a half-width of 10
symbols yields the following fitting results with net chromatic
dispersion and link length: c.sub.0=38.9976, c.sub.1=21.1308,
c.sub.2=0.7726, c.sub.3=3.4388, c.sub.4=0.1241, and c.sub.5=0.0102
with a root mean square error (RMSE) of 0.4008 dB. The fitting
error in SNR (dB) is illustrated in FIG. 11.
[0110] Similarly, evaluating the quantity P.sub.X or Y [m] given in
equation (51) or (53) above for m=1 with a half-width of 20 symbols
yields the following fitting results with net chromatic dispersion
and link length: c.sub.0=17.4486, c.sub.1=3.3473, c.sub.2=0.2427,
c.sub.3=0.0016, c.sub.4=0.1418, and c.sub.5=0.0338 with a root mean
square error (RMSE) of 0.5731 dB. The fitting error in SNR (dB) is
illustrated in FIG. 12.
[0111] Using the symbol error as the calibration attribute, one
could consider:
NL-NSR'.sub.INTRA,X or Y=NL-NSR.sub.INTRA, X or Yf(CD.sub.net,
D.sub.avg, SER,NL-NSR.sub.INTRA,X or Y) (59)
here NL-NSR.sub.INTRA,X or Y is the calibrated estimation, f() is
the calibration relationship for application-dependent systematic
errors as well as symbol error rate (SER)-induced errors,
CD.sub.net is the net chromatic dispersion, and D.sub.avg is the
average dispersion.
[0112] Assuming the implementation noise is 16.8 dB and the total
SNR is 11.5 dB (3.4% BER threshold for 16QAM), if the nonlinear
noise is a third of the total other noise, then the nonlinear SNR
is 17.8 dB. In the 60 applications in Table 1, applications with
nonlinear SNR in the range of 1421 dB were chosen, which results in
30 applications. In the following results, a second order
polynomial function with four variables was used to fit the
simulation results. Evaluating the quantity P.sub.X or Y [0] given
in equation (52) or (54) above with a half-width of 10 symbols
yielded an RMSE of 0.2475 dB and the fitting error in SNR is
illustrated in FIG. 13. Evaluating the quantity P.sub.x or Y [m]
given in equation (51) or (53) above for m=1 with a half-width of
20 symbols yielded an RMSE of 0.4240 dB and the fitting error in
SNR is illustrated in FIG. 14. Because of the reduced number of
applications, the RMSE value is actually reduced compared to the
other 16QAM results discussed above in this document. For the
purpose of comparison, the RMSE is 0.2203 dB and 0.3454 dB
respectively with these 30 applications. Therefore, the extra
standard deviation caused by SER calibration is very small in the
intra-channel nonlinear estimation.
[0113] Calibration of Total Kerr Nonlinear Noise-to-Signal Ratio
Using Polynomials
[0114] Simulations involving the applications listed in Table 1 and
evaluating the quantities given in equations (27), (28) and (29)
above for N=10 were performed.
[0115] A second order two-variable polynomial model was used to fit
the simulated data as
.epsilon.=c.sub.0+c.sub.1x+c.sub.2y+c.sub.3x.sup.2+c.sub.4xy+c.sub.5y.su-
p.2 (60)
where .epsilon. is the offset in dB, x=log.sub.10 (net chromatic
dispersion in ps/nm), and y=[net chromatic dispersion in
ps/nm]/[link length in km].
[0116] Fitting results for WDM channels 1, 3, 5 and 7 for QPSK are
illustrated in FIG. 15, FIG. 16, FIG. 17 and FIG. 18, respectively.
The coefficients obtained are summarized in Table 2.
TABLE-US-00002 TABLE 2 Fitting results with net chromatic
dispersion and link length for different WDM cases for QPSK RMS
Fitting coefficients (dB) Rsquare WDM ch 1 = [-0.2924, -0.9895,
0.0625, 0.2055, 0.1510 0.9613 0.0464, -0.0114] WDM ch 3 = [7.0640,
-4.2868, 0.2040, 0.6243, 0.2504 0.8908 0.0540, -0.0223] WDM ch 5 =
[7.9336, -4.6839, 0.2533, 0.6915, 0.2618 0.8770 0.0399, -0.0226]
WDM ch 7 = [10.5651, -5.5514, 0.1515, 0.7590, 0.2788 0.8229 0.0611,
-0.0218]
[0117] Fitting results for WDM channels 1, 3, 5 and 7 for 16QAM are
illustrated in FIG. 19, FIG. 20, FIG. 21 and FIG. 22, respectively.
The coefficients obtained are summarized in Table
TABLE-US-00003 TABLE 3 Fitting results with net chromatic
dispersion and link length for different WDM cases for QPSK RMS
Fitting coefficients (dB) Rsquare WDM ch 1 = [5.5340, 4.9668,
0.1698, 0.8156, 0.1591 0.9585 0.0000, -0.0075] WDM ch 3 = [7.4041,
-4.9358, 0.2256, 0.7388, 0.2375 0.9120 0.0413, -0.0203] WDM ch 5 =
[8.6977, -5.5329, 0.2575, 0.8064, 0.2515 0.9054 0.0461, -0.0233]
WDM ch 7 = [8.7642, 5.4589, 0.3049, 0.8062, 0.2716 0.8750 0.0234,
-0.0211]
[0118] Calibration of Intra-Channel Kerr Nonlinear Noise-to-Signal
Ratiio And Total Kerr Nonlinear Noise-to-Signal Ratio Using Neural
Network
[0119] A neural network implementation is used to improve the
accuracy of the determination of intra-channel Kerr nonlinear
noise-to-signal ratio and of total Kerr nonlinear noise-to-signal
ratio by training the neural network with data sets of known inputs
and known outputs (the outputs being related to the intra-channel
Kerr nonlinearity and to the total Kerr nonlinearity), and then
applying the trained neural network to actual inputs. For example,
neural network structures having two outputs may be used, with one
output related to the intra-channel Kerr nonlinearity (for example,
NL-NSR.sub.INTRA,X or Y) and the other output related to the total
Kerr nonlinearity (for example, NL-NSR.sub.TOTAL). The inputs may
include the coefficients of the C-matrices C.sub.X, C.sub.Y,
C.sub.XY, and C.sub.YX, and doublet correlations (for example, any
one or any combination of the doublet correlations of equations
(30) through (47)). The inputs may also include the results
obtained by pre-processing the C-matrices and the one or more
doublet correlations. The inputs may also include one or more
attributes of the link, for example, one or more of net chromatic
dispersion, link length, fiber type, span length, WDM
configuration, and the like.
[0120] FIG. 23 illustrates an example neural network structure
having two inputs, a single hidden layer with ten nodes, and two
outputs. FIG. 24 illustrates another example neural network
structure having four inputs, a single hidden layer with ten nodes,
and two outputs. In both examples, the two outputs are
intra-channel Kerr nonlinearity and total Kerr nonlinearity, and
two of the inputs are C-matrix measurements and doublet
correlation. In the example neural network structure having four
inputs, the additional two inputs are net chromatic dispersion and
link length.
[0121] To train the neural network, and then to demonstrate the
performance of the trained neural network, the applications listed
above in Table 1 were considered, for 56.8 Gbaud signals with 62.5
GHz spacing, using dual polarization 16 quadrature amplitude
modulation (DP-16QAM), 60 links, 70% training and 30% testing.
[0122] Error histograms for training samples and test samples are
illustrated in FIG. 25 and FIG. 26, respectively, for the example
neural network structure having two inputs illustrated in FIG. 23.
With two inputs, the training mean square error (MSE) was 0.84 Db
and the test MSE was 0.94 dB. Error histograms for training samples
and test samples are illustrated in FIG. 27 and FIG. 28,
respectively, for the example neural network structure having four
inputs illustrated in FIG. 24. With four inputs, the training mean
square error (MSE) was 0.075 Db and the test MSE was 0.079 dB.
* * * * *