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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 27 — Dec. 17, 2012
  • pp: 29035–29062
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Estimating the Volterra series transfer function over coherent optical OFDM for efficient monitoring of the fiber channel nonlinearity

Gal Shulkind and Moshe Nazarathy  »View Author Affiliations


Optics Express, Vol. 20, Issue 27, pp. 29035-29062 (2012)
http://dx.doi.org/10.1364/OE.20.029035


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Abstract

We present an efficient method for system identification (nonlinear channel estimation) of third order nonlinear Volterra Series Transfer Function (VSTF) characterizing the four-wave-mixing nonlinear process over a coherent OFDM fiber link. Despite the seemingly large number of degrees of freedom in the VSTF (cubic in the number of frequency points) we identified a compressed VSTF representation which does not entail loss of information. Additional slightly lossy compression may be obtained by discarding very low power VSTF coefficients associated with regions of destructive interference in the FWM phased array effect. Based on this two-staged VSTF compressed representation, we develop a robust and efficient algorithm of nonlinear system identification (optical performance monitoring) estimating the VSTF by transmission of an extended training sequence over the OFDM link, performing just a matrix-vector multiplication at the receiver by a pseudo-inverse matrix which is pre-evaluated offline. For 512 (1024) frequency samples per channel, the VSTF measurement takes less than 1 (10) msec to complete with computational complexity of one real-valued multiply-add operation per time sample. Relative to a naïve exhaustive three-tone-test, our algorithm is far more tolerant of ASE additive noise and its acquisition time is orders of magnitude faster.

© 2012 OSA

1. Introduction

2. Volterra nonlinear formalism – third order fiber nonlinearity characterization

Our objective is fiber nonlinearity monitoring. The first step is to adopt a suitable mathematical representation for the fiber nonlinearity. This section provides the necessary background reviewing the VSTF, which is our target quantity for optical measurement.

The Volterra Series (VS) provides a general model for nonlinear transformations with memory. For our purposes it suffices to truncate the VS to third-order and adopt a discretized frequency-domain representation of the VS as elaborated in this section, which reviews the frequency-domain VS model from an optical physics vantage point, relating the mathematical model with FWM generation. Readers unfamiliar with frequency-domain VS nonlinear modeling should consult our review in Ch. 3 of [26

26. S. Kumar, Impact of Nonlinearities on Fiber Optic Communications, Ch. 3 (Springer, 2011).

] for a self-contained tutorial. Similar Volterra-based nonlinear models in slightly different notations have also appeared in [7

7. K. Peddanarappagari and M. Brandt-Pearce, “Volterra series transfer function of single-mode fibers,” J. Lightwave Technol. 15(12), 2232–2241 (1997). [CrossRef]

11

11. J. D. Reis and A. L. Teixeira, “Unveiling nonlinear effects in dense coherent optical WDM systems with Volterra series,” Opt. Express 18(8), 8660–8670 (2010). [CrossRef] [PubMed]

] and recently the number of papers approaching nonlinear compensation via VS-based tools rather than by SSF-based back-propagation is on the rise [17

17. L. Liu, L. Li, Y. Huang, K. Cui, Q. Xiong, F. N. Hauske, C. Xie, and Y. Cai, “Intrachannel nonlinearity compensation by inverse Volterra series transfer function,” J. Lightwave Technol. 30(3), 310–316 (2012). [CrossRef]

24

24. Z. Pan, C. Benoit, M. Chagnon, and D. V. Plant, “Volterra filtering for nonlinearity impairment mitigation in DP-16QAM and DP-QPSK fiber optic communication systems,” in OFC/NFOEC - Conference on Optical Fiber Communication and the National Fiber Optic Engineers Conference (2011).

]. This trend is consistent with the Volterra formalism being increasingly recognized as the proper tool for describing weak nonlinearities in all science and engineering areas and recently also in fiber-optic communication. The first application of the Volterra series to fiber transmission was presented in [7

7. K. Peddanarappagari and M. Brandt-Pearce, “Volterra series transfer function of single-mode fibers,” J. Lightwave Technol. 15(12), 2232–2241 (1997). [CrossRef]

].

We remark that even without the benefit of the VS mathematical framework, the theory of Kerr-induced fiber nonlinearity may be developed in the frequency domain in purely physical terms. The key is to identify the elementary mechanism of nonlinear generation as the interaction of frequency triplets via FWM. The overall nonlinear interference is obtained by summing up all the FWM elementary intermodulation contributions, each of which consists of a triple product of three complex amplitudes associated with three frequencies. Generally, each such triple product would be weighted differently in the overall nonlinear superposition. The VSTF is nothing but the complex weight attached to each frequency triplet, describing the FWM generation efficiency. Thus, the VSTF may be viewed as the extension of the conventional concept of linear transfer function (TF) (which in turn is the complex weight used to multiply the amplitude of single input frequency into a linear system by in order to obtain its output). For a nonlinear system, a full frequency domain characterization involves sweeping over all possible triplets of frequencies rather than over a single frequency. Hence, the FWM nonlinearity is fully determined by specifying the VSTF as a complex-valued function of three frequencies (which may be suitably sampled).

Finally, as this paper is devoted to proof of concept, for the sake of clarity, rather than using a vector Manakov system to handle both orthogonal polarizations we illustrate the key points for a single polarization, amounting to a treatment of scalar nonlinear Schro¨dinger equation (NLSE). Future work should explore extending the novel SID procedure derived here to a dual polarization context.

2.1 Volterra Series Transfer Function (VSTF)

Let three optical harmonic tones at freqs.νj,νk,νl generate a fourth FWM tone at freq. νi=νj+νkνl s.t. νjνi,νkνi.The rotating phasors describing the optical field complex envelopes (CE) of the three input tones are given by
Ej(t)=Ajej2πνjt,Ek(t)=Akej2πνkt,El(t)=Alej2πνlt
(1)
In elementary FWM analysis we seek the mixing product generated by the third-order ideal nonlinearity corresponding to a lumped FWM generation mechanism. The nonlinear optical field contribution from a differential length dzof nonlinear medium is given by [27

27. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express 16(20), 15777–15810 (2008). [CrossRef] [PubMed]

]:
ri;jkl(3)(t)=(jγdz)Ej(t)Ek(t)El*(t)
(2)
whereγis the fiber nonlinear coefficient.

In OFDM, the center frequencies (subcarriers) of the sub-channels fall on a regularly spaced frequency grid, νi=iΔν,i=1,2,...M,hence it is convenient to label all the discrete tones by their integer indices, i,setting a one-to-one correspondence between frequencies and their indices: νi=νj+νkνl=(j+kl)Δν.Substituting the three phasors (1) into (2) yields the nonlinear output field complex amplitude at the mixing frequency νi:
ri;jkl(3)(t)=(jγdz)AjAkAl*ej2π(νj+νkνl)t=Ri;jkl(3)ej2πνit
(3)
Ri;jkl(3)(jγdz)AjAkAl*
(4)
It was shown in [27

27. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express 16(20), 15777–15810 (2008). [CrossRef] [PubMed]

] that for a general amplified fiber link, accounting for the total distributed nonlinear distortions generated across the link, the factor jγdz in the elementary triple product of Eq. (4) is to be replaced by a more general complex scaling factor Hi;jkl, depending on the three input tones j,k,l (which in turn determine the output tone i=j+kl). This complex scaling factor Hi;jkl is defined as the Volterra Series Transfer Function (VSTF) of the 3rd order nonlinear system. The VSTF describes the amplitude attenuation or gain and the phase-shift experienced by the FWM mixing product excited by the three input tones. The VSTF is a generalization of the concept of linear Transfer Function (TF). Notice that for a specified output tone i, once the two input tones j,k are also given, the third input tone, l, becomes redundant, as it is uniquely determined as l=j+ki.We then discard this implied index, introducing the abbreviated VSTF notationHi;jkHi;j,k,j+ki, expressing the output FWM contribution due to the three tones as follows (j,k determine the third index l causing the mixing product to fall onto the specified i):
Ri;jk(3)=Hi;jkAjAkAj+ki*.
(5)
As detailed in [26

26. S. Kumar, Impact of Nonlinearities on Fiber Optic Communications, Ch. 3 (Springer, 2011).

], the CE of the output nonlinear signal Ri;jk(3)is specified at the input plane of the nonlinear system where the input tones CEs, Aj,Ak,Aj+ki,are measured. Thus, if we obtain an expression for the CE of the nonlinear field at some output plane, in order to extract the corresponding VSTF we ought to first (quasi-linearly) back-propagate the nonlinear generated field to the input plane and then express the resulting field as in Eq. (5).

When the input contains a multitude of tones, e.g. the multiple subcarriers in an OFDM signal, the mixing product contributions from all tone triplets must be coherently superposed. Let the input into the nonlinear system be given by a Fourier Series (FS), implying that it is either time-limited or periodic. Further assume that the input is approximately Band-Limited (BL) i.e. it may be expanded as a Finite FS (FFS) i.e. a FS with a finite number N of harmonics:
a(t)=k=1NAkej2πkΔνt;ΔνT1
(6)
The total nonlinear field at the i-th frequency is then
ri(3)(t)=j=1Nk=1NRi;jk(3)ej2πiΔνt;t[0,T]
(7)
with the summation over all index pairs in the domain [1,N]×[1,N]. We must allow for the possibility that given a target index i, then Ri;jk(3) may be null out for certain indices j,k, whenever l=j+kiis not a valid index falling in the range. Equivalently, given i, it suffices to restrict the summation to the set S[i][1,N]×[1,N]of “proper FWM triplets” [27

27. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express 16(20), 15777–15810 (2008). [CrossRef] [PubMed]

], namely index pairsj,ksubject to l=j+ki being a valid index, and excluding Self-Phase-Modulation (j = k = i) and Cross-Phase-Modulation (XPM) (j = i or k = i) contributions, which are separately treated:
S[i]={(j,k):1jN,1kN,1j+kiN,ji,ki}
(8)
The summation of Eq. (7), describing just the FWM contribution, then reduces to
riFWM(t)=[j,k]S[i]Ri;jk(3)ej2πiΔνt=[j,k]S[i]ej2πiΔνtHi;jkAjAkAj+ki*
(9)
The distinction between the two summations of Eqs. (7) and (9) is that the second one excludes terms for which the VSTF is zero. The total nonlinear field in-band (that is at the N sub-carriers of the given OFDM channel) is a superposition over all nonlinear contributions of all the i tones: rFWM(t)=i=1NriFWM(t).Substituting Eqs. (9) and (5) into the last equation yields a FS expansion of the nonlinear system output over the [0,T] interval:

rFWM(t)=i=1Nej2πiΔνt[j,k]S[i]Hi;jkAjAkAj+ki*=i=1NRiFWMej2πiΔνtRiFWM=[j,k]S[i]Hi;jkAjAkAj+ki*
(10)

2.2 VSTF of a general optically amplified dispersive fiber link with non-uniform parameters

2.3 Numeric validation of the analytic Volterra series based nonlinear fiber link description

In top curve of Fig. 1 we additionally apply nonlinear compensation (NLC) based on the analytic FWM generation model of Eq. (10), which predicts the nonlinear distortion complex amplitudes RiFWM,expressed as a summation of triple products of transmitted complex amplitudes weighted by the VSTF, Hi;jk.The FWM compensated samples for each subcarrier are now set to ρi=RiRiFWM,analytically calculating RiFWMas per Eq. (10) and subtracting it out, and the MER (Eq. (18)) is evaluated for the resulting FWM-compensated ρi.The very large improvement in MER attained relative to the FWM-uncompensated case is indicative of the high accuracy of the analytic FWM generation model of Eq. (10) and in particular verifies the analytic expressions for the VSTF. This simulation clearly validates our nonlinear propagation analytic model for OFDM as derived in [27

27. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express 16(20), 15777–15810 (2008). [CrossRef] [PubMed]

] and reviewed in this section.

3. Naïve VSTF monitoring using exhaustive 3-tone tests over coherent OFDM

In this section we consider an initial naïve approach to nonlinear monitoring making use of a coherent optical OFDM transceiver for VSTF system identification.

In Eq. (5) we characterized the VSTF as the coefficient Hi;jkto be applied to the triple product of complex amplitudes of three harmonic tones interacting through FWM, in order to obtain the CE of the mixing product, Ri;jkFWM=Hi;jkAjAkAj+ki*.This suggests that, in principle, the VSTF may be determined by probing the system with all possible triplets of tones – applying an exhaustive triple-tone test. Such nonlinear monitoring procedure may readily be realized if we set up coherent OFDM transmission in the particular channel over the fiber link. At the OFDM Tx we turn on three subcarriers at a time to be used as pilots for nonlinearity monitoring for each subcarrier, i, namely subcarriersj,kandl=j+ki.In each probe interval we transmit an OFDM symbol over the duration T, comprising just these three subcarrier frequencies. This has the effect of linearly exciting the same subcarriers frequencies in the Rx, however at subcarrier index i, (i.e., at the i-th port of the FFT output in the Rx), there appears a small nonlinear disturbanceRi;jkFWMdue to FWM mixing of the three frequencies. Assuming the probing subcarriers are sufficiently strong, the FWM disturbance will be above the noise. Alternatively, the signal-to-noise ratio (SNR) may be improved by having each triplet of pilot frequencies repeatedly transmitted. The samples collected at the i-th output of the FFT in the Rx in response to the repetitions of each triple-frequency training symbol should then be time-averaged. Then, assuming noise has been sufficiently suppressed relative to the signal, the particular value of the VSTF at the frequency triplet is simply obtained by H^i;jk=Ri;jkFWM¯/(AjAkAj+ki*),where Ri;jkFWM¯is the time-averaged value of multiple samples at the i-th FFT output, and the A-s represent the complex amplitudes of the transmitted training signal. In particular, it is convenient to set the CE of all transmitted tones to be real-valued, all of amplitude A, yieldingH^i;jk=Ri;jkFWM¯/A3.

While this method works in principle, its main shortcomings are an excessively long acquisition time and poor noise tolerance. Indeed, there are O(N3)OFDM VSTF coefficients to be evaluated (where N is the number of Analog to Digital Converter (ADC) samples in an OFDM symbol, i.e., the OFDM FFT size) thus the overall measurement time, expressed in ADC sample interval units, should beNO(N3)S=O(N4S),where S is the number averaging measurements, E.g., for N = 1024 and S = 100 (yielding 20 dB averaging gain), we would require O(1014) samples. In this particular example we would be able to measure the VSTF with a resolution ofΔν=BT/N,where BT is the OFDM signal aggregate bandwidth, e.g. forBT=25GHz,we would attain a resolution of Δν25MHz.

In the sequel we explore alternative VSTF monitoring approaches consisting sending more sophisticated training sequences (TS) rather than the simplistic three-tone OFDM symbols just considered. Such TS would contain more energy and improve SNR allowing to eliminate averaging and speed up the measurement, however the foreseen difficulty would be that the TS, now consisting of multiple (much more than three) pilot tones, would simultaneously excite superpositions of three-tone responses, which would fall on the same set of output frequencies and would be hard to separate out. Nevertheless, the challenges entailed in such an approach may be successfully addressed and a fast method of nonlinear channel identification be obtained on its basis, as will be explored below.

4. VSTF analytics revisited - compressed representations

In the remainder of this section we develop the mathematical properties of the compressed VSTF formulation, setting the mathematical background for developing the novel fast SID.

4.1 Compressed VSTF mathematical properties

Notice that Eq. (19) amounts to a many-to-one mapping from the pair of indices j,k to the index m (for a given target index, i). Certain indices, m, may each correspond to multiple (j,k) pairs as there may be multiple ways to factor m as a product of the j-i and k-i. In contrast, m-index values which are either prime or are expressible as products of two prime numbers are factorizable just in a single way as products of j-i and k-i. Moreover, there might be values of m which are not feasible (are undefined) as the j,k values of their integer factors fall outside S[i],for every 1iN. The inverse image of the many-to-one transformation of Eq. (19), i.e. the set of (j,k) pairs falling on target index i (uniquely defining FWM triplets) and satisfying the relation m=(ji)(ki),is denoted asSm[i](formally defined in Eq. (37)below),introducing the following sets of indices:
M˜={m|m=(ji)(ki):1iN,1jN,1kN,1j+kiN,ji,ki}Sm[i]={(j,k)|(ji)(ki)=m,1jN,1kN,1j+kiN,ji,ki}
(28)
where the indices defining Sm[i] satisfy 1iNand mM˜.

For points m which are not feasible, we take Sm[i]=, the empty set, with cardinality zero.

Comparing the definition of Sm[i] with that of S[i](Eq. (8)) it is apparent that the set of inverse images {Sm[i]}mM˜forms a disjoint partition of the aggregate setS[i](Eq. (8)), with each subset in the partition indexed by m, i.e., S[i]=mM˜Sm[i] and Sm[i]Sm[i]=ifmm.

4.2 Compressed representation of the FWM mixing products build-up

Based on the compressed representation of the VSTF we proceed to formulate a corresponding compressed representation of the FWM products build-up process, recasting Eq. (9) in the form:
riFWM(t)=(j,k)S[i]H(ji)(ki)AjAkAj+ki*ej2πiΔνt=mM˜Hm(j,k)Sm[i]AjAkAj+ki*ej2πiΔνt
(30)
where in the last expression we rearranged the summation as an outer sum over the inverse images of the mi:j,k mapping of Eq. (19), as well as an inner sum over the elements of each inverse image set of the mi:j,kmapping. While the two expressions in (30) are equivalent, we note that the inner sum in the second expression is VSTF-free, whereas the number of DOFs involved in the outer sum involving the compressed VSTF is substantially lower than the total number |S[i]| of terms in the first expression. Thus, in order to specify the overall FWM distortion across the entire OFDM symbol we need to invoke only |M˜|compressed VSTF coefficients, vs. the i=1N|S[i]|coefficients required for the FWM description in terms of the conventional VSTF.

Next, we evaluate the efficiency improvement attained by switching from conventional to compressed VSTF description by evaluating |M˜|vs. i=1N|S[i]|.

The cardinality of the set S[i] was derived in [27

27. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express 16(20), 15777–15810 (2008). [CrossRef] [PubMed]

]:
|S[i]|=(N25N+2)/2+(N+1)ii2
(31)
An expression for the total number of triplets with FWM falling in-band within the OFDM spectrum, e.g. as used in the simple-minded three-tone testing of section 3, is given by
i=1N|S[i]|=N(N25N+2)/2+(N+1)12(N+1)N16N(N+1)(2N+1)=23N(N1)(N2)<23N3
(32)
seen to scale as O(N3).This precise total number of triplets, as required in principle for the VSTF description, is tabulated in Table 1

Table 1. Number of Coefficients – H vs. H’

table-icon
View This Table
, which further presents, for each N, the corresponding sizes of the set M˜of m-indices which are seen to be much smaller. As the setM˜contains the products of two integer factors (ji),(ki) each ranging from –(N-1) to (N-1), it approximately scales asO(N2)which is factor of N less than the order of i=1N|S[i]|.Actually,O(N2)is a loose upper bound.

In the case of interest of N being a power-of-two, we have for the maximum value included in the M˜set the expression:
mmax=maxs.t.1j,k,i,j+kiN(ji)(ki)=N2(N21)N24.
(33)
Symmetrically, the minimum m-index is given bymmin=mmax.The M˜set is properly included in the integer range{mmax,mmax+1,...,mmax1,mmax},as there are non-feasible values of m, as already indicated. Thus, we have the upper bound|M˜|2mmaxN2/2. In fact 0.5N2 is a loose upper bound: the ratio |M˜|/N2is tabulated in Table 1 and is seen to fall well under 0.5, essentially falling in the 0.2...0.3 range. For N128,which is the range of OFDM sizes of interest, the table indicates a tighter upper bound |M˜|<14N2.

This approximately quantifies the initial compression achievable with our ‘fast SID’ approach. The precise compression factor is presented in the rightmost column of Table 1 which presents the compression ratioi=1N|S[i]|/|M˜|.An approximation for this ratio is
i=1N|S[i]|/|M˜|23N3/(14N2)=83N.
(34)
This is the ‘compression factor’ describing the complexity reduction by virtue of using the compressed VSTF. Finally, consider the total FWM power at an in-band frequency indexed by i. Assuming uncoded OFDM n-ary PSK transmission with constellation PT/Nejα2π/n,α=0,1,...,n1 (as will be used for our training sequences), we follow the approach of [27

27. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express 16(20), 15777–15810 (2008). [CrossRef] [PubMed]

] to express the total FWM power falling on the i-th subcarrier in terms of the individual triplets powers.
[riFWM(t)]2N32PT3(j,k)S[i]|H(ji)(ki)|2=N32PT3mM˜|Sm[i]||Hm|2
(35)
In the last expression we introduced the compressed m-index notation, realizing that all triplets corresponding to a given m-index share the same|Hm|2,hence their power contribution is weighted by the factors |Sm[i]| which were referred to as the multiplicities of the m-indices. The expression of Eq. (35) for the FWM power was described in [27

27. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express 16(20), 15777–15810 (2008). [CrossRef] [PubMed]

] - approximately akin to having individual triplets incoherently superpose in power, except that the triplets j,k,j+kiand k,j,j+ki(related by transposing j,k) are indistinguishable in their respective coherent field contributions, hence re-enforce coherently, yielding a factor of two in Eq. (35). The approximation stems from neglecting the ‘degenerate’ triplets of the form j,j,2ji.Summing the nonlinear interference power (Eq. (35)) over all sub-carriers yields
i|[riFWM(t)]|22N3PT3mM˜|Sm||Hm|2
(36)
This expression will be used to further compress the VSTF SID process with slight loss of information.

5. Efficient nonlinear system identification of the fiber link

At the end of section 3 we proposed to explore the potential for SID nonlinear performance monitoring (measurement of the FWM VSTF) by transmitting training sequences which are ‘rich’ in frequency content and more energetic, as opposed to the initial naïve three-tone-test approach which was based on training symbols containing just three pilot subcarriers.

Resorting to compressed VSTF formalism of the last section, we reformulate the nonlinear channel model of Eq. (30) as follows:
riFWM(t)=RiFWMej2πiΔνt;RiFWMmM˜Hm(j,k)Sm[i]AjAkAj+ki*=mM˜HmAi,mFWM
(37)
where in the last expression we introduced a compact notation, Ai,mFWM,for the aggregate FWM distortion onto the i-th subcarrier, due to all triplets corresponding to the m-th subset, Sm[i]:
Ai,mFWM(j,k)Sm[i]AjAkAj+ki*
(38)
Notice that Ai,mFWMdepends on both indices i and m, since out of the triplets falling on i (belonging to the setS[i]) we must single out a subset Sm[i] of triplets satisfying the requirement (ji)(ki)=m.The modified formulation of Eq. (37), with Ai,mFWM given by Eq. (38) describes a simple linear transformation underlying our proposed efficient SID method, to be introduced next.

5.1 SID procedure using arbitrary training sequences formulated as least-squares problem

We proceed to formulate the SID problem with general training sequences{Ai}i=1N,as estimation of the compressed VSTF, Hm based on measuring the subcarrier complex amplitudes, RiFWM,under transmission of arbitrary training sequences. We transmit a TS consisting of a succession of OFDM symbols, each having all its N sub-carriers generally non-zero. The received complex amplitudes are labeled as Ri[t],wherei{1,2,...,N}is the subcarrier frequency index (i-th output of the Rx FFT), and tT˜ indices the transmitted OFDM symbol in the TS (the ‘training symbol’). Here T˜+is a set of training symbol discrete-time indices, assumed for simplicity of notation to consist of contiguous integers. We also allow for the possibility of repetition transmissions, in order to enable averaging the additive noise for SNR improvement, however the averaging process is to be modeled outside the context of the set T˜of TS indices, i.e. the training symbols labeled by tT˜ are independent rather than being repeated.

The subcarrier complex amplitudes Ri[t] received in training mode are logged after CD equalization and after SPM/XPM compensation (a deterministic counter-rotation of all subcarrier complex amplitudes to correct for SPM/XPM [27

27. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express 16(20), 15777–15810 (2008). [CrossRef] [PubMed]

]). Assuming perfect CD equalization and SPM/XPM compensation, and neglecting noise-signal interaction, the received signal would be equal to the transmitted signalAi[t],plus the FWM nonlinear distortion terms, plus additive noises:
Ri[t]=Ai[t]+RiFWM[t]+ni[t]=Ai[t]+mM˜Ai,mFWM[t]Hm+ni[t]
(39)
or equivalently,
mM˜Ai,mFWM[t]Hm+ni[t]=δRi[t]Ri[t]Ai[t],i=1,2,...,N,tT˜
(40)
Given the measured perturbationsδRi[t],to which we refer as ‘observations’ (comprising the nonlinear FWM distortions plus noise), and given the known Ai,mFWM[t]coefficients (these are expressible in terms of the TS {Ai[t]}i=1Nas in Eq. (38)), we wish to find an optimal estimate for Hm,which we denote H^m,such that mM˜Ai,mFWM[t]H^m be as close in Euclidean distance as possible to the observations δRi[t].Formulated this way, the problem reduces to a least-squares (LS) one (under additive white gaussian noise this is equivalent to a maximum-likelihood estimator), expressed as a minimization of the mean-squared error:
{H^m}mM˜=argmin{Hm}mM˜tT˜i=1N|δRi[t]mM˜Ai,mFWM[t]Hm|2
(41)
If T˜contained a single index, denoted byt0,then the outer sum over t would be discarded, and the Mean Square Error (MSE) would be expressible in terms of a summation over the subcarriers, i, represented in vectorial notation as a squared norm:
MSE=i=1N|δRi[t0]mM˜Ai,mFWM[t0]Hm[t0]|2=δR[t0]A[t0]H2
(42)
where the constituent vectors and matrix are defined as follows:
δR[t0][δR1[t0],δR2[t0],...,δRN[t0]]T;H[Hm1,Hm2,...,Hm|M˜|]T[AN×|M˜|[t0]]i,m=Ai,mFWM[t0](j,k)Sm[i]Aj[t0]Ak[t0]Aj+ki*[t0]
(43)
The formulation above describes the transmission of a single OFDM training symbol. Further introducing the TS index, tT˜,now allowed to range over multiple values, is a matter of extending the vectors and the matrix by vertically juxtaposing additional vector and matrix blocks, as labeled by the TS index, t. The mean square error now becomes MSE=δRAH2 where the |M˜|×1 column vector H is defined just as in Eq. (43), whereas the observation vector, δR,and theAmatrix are now defined in terms of the following vertical concatenations of blocks:
δRN|T˜|×1[δR[t0]T,δR[t0+1]T,...,δR[t0+|T˜|1]T]TAN|T˜|×|M˜|=[A[t0]T,A[t0+1]T,...,A[t0+|T˜|1]T]T
(44)
with the various vector and matrix blocks defined in terms of their own individual components just as in Eq. (43). Conceptually, we may set up a system of N|T˜| equations in |M˜| unknowns, expressed in matrix form asAH=δR,then attempt to solve for H.We reiterate that theAmatrix contains as elements sums of FWM triple products of the form of Eq. (38), whereas the unknown vector H is the compressed VSTF. If we selected a sufficiently large number |T˜| of training sequences, ensuring that N|T˜|>|M˜|,then we would have more equations N|T˜| than there are unknowns, Hm1,Hm2,...,Hm|M˜|.In this case the system would be over-determined (whenever A has full column rank, which it typically does) and there would be no precise solution. However, we may obtain an optimal ‘pseudo-solution’ by selecting H such thatAHbe closest in Euclidean distance to the observed δRvector. Formally, Eq. (41) is rewritten as
H^=argminHδRAH2
(45)
The optimal LS solution H^is in principle obtained by projecting δRonto the column space of theAmatrix. The optimal solution may be simply expressed if Amatrix has a full column rank [31

31. S. Haykin, Adaptive Filter Theory (Prentice Hall, 2002).

] in which case the projection amounts to a linear transformation corresponding to the following pseudoinverse (PI) |M˜|×N|T˜|matrix:
A(AA)1A
(46)
The optimal estimate of the compressed VSTF is simply obtained by left-multiplying the observation vector δRof length N|T˜|by the PI matrix (which is compatible in dimensions):
H^=AδR
(47)
Remarkably, the PI matrix, A,is independent of the received data but is rather solely determined by the transmitted training sequences {Ai[t]}i=1N,according to Eq. (38). Fortunately then, the PI may be evaluated offline from the training sequences, which are specified in advance. The remaining real-time task of the Rx is to perform the matrix-vector multiplication. Thus, the estimation of the |M˜|-element column vector H^of VSTF samples amounts to a linear processing task consisting of multiplying the accumulated observations vector δR (which contains N|T˜|samples), by this fixed pre-evaluated |M˜|×N|T˜| PI matrix. This is more efficient than the nonlinear SID method [32

32. S. W. Nam, S. B. Kim, and E. J. Powers, “On the identification of a third-order Volterra nonlinear system using a frequency-domain block RLS adaptive algorithm,” in Acoustics, Speech, and Signal Processing ICASSP-90, 2407–2410 (1990).

], wherein the matrix inversion must be dynamically evaluated using the far more complex adaptive Recursive Least Squares (RLS).

5.2 Exploiting the anti-hermitian symmetry of the compressed VSTF

In the last subsection we have shown how to select an optimalH^as a solution to the optimization problem of Eq. (45). We have previously shown that the compressed VSTF features anti-hermitian symmetry (Eq. (22)), which is equivalent to the real and imaginary parts satisfying:
H'Re[m]=H'Re[m];H'Im[m]=H'Im[m]
(48)
In a noisy environment we may just approximately retrieve an anti-hermitian solution to the optimization problem, as formulated in the last sub-section. Thus, we ought to constrain our solutions to satisfy Eq. (48), restricting the vector Hof unknowns to assume the form,
H=[HuHl]=[Hure+iHuimHlre+iHlim]=[Hure+iHuimHure+iHuim]
(49)
satisfying anti-hermitian symmetry, with subscripts u,l indicate upper and lower blocks.

Without loss of generality we then decompose the δRvector into real and imaginary parts,
δR=δRre+iδRim
(50)
and further separate the A matrix into left and right blocks (denoted by the subscripts L,R):
A=[ALre+iALimARre+iARim]
(51)
The optimization problem of Eq. (45), applied to Hconstrained to the form of Eq. (49), and further using the decompositions of Eqs. (50),(51), yields after separation to real and imaginary parts the formulation:
[H^ureH^uim]=argmin[Hure,Huim]T[δRreδRim][ALreARreALimARimALimARimALre+ARre][HureHuim]2
(52)
Once, H^ure,H^uimare determined, Eq. (49) is used to assemble a full H^solution satisfying anti-hermiticity. The modified optimization problem of Eq. (52) has twice the number of equations (the modified A-matrix is now 2N|T˜|×|M˜| rather than N|T˜|×|M˜|and the modified PI matrix is now dimensioned|M˜|×2N|T˜|) albeit with real-valued rather than complex-valued unknowns.

We have thus gained in two respects. First, for a given number of transmitted OFDM symbols, we collect twice as many equations each with the same number of unknowns. Indeed, the threshold condition for a unique valid optimal least-squares solution is having a full column rank A-matrix, for which it is necessary that the number of equations be equal or greater than the number of unknowns– we have thus halved the number of OFDM symbols necessary to obtain a solution and subsequently halved the acquisition time. Secondly, as each complex multiplier in the original A-matrix formulation amounts to three real multipliers, it is apparent that the anti-hermitian-constrained optimization provides a complexity reduction factor of 3/2 relative to the original unconstrained optimization problem, as the new Ais now real-valued rather than complex-valued.

5.3 Training sequences

As for the training sequences to be used for best performance, as we probe the fiber nonlinearity, it is advantageous to excite the system with as strong signals as possible, within the regime of validity of the 3rd order Volterra model formalism. Under a transmit-power-limited constraint, Ppeak,it is thus worthwhile to set all the components of the training sequence at an equal maximum power level. Our proposed training sequence then consists of OFDM symbols with their N subcarriers having the complex amplitudes Ppeakejθi[t],where the phases sequence θi[t] is white (over both the i and t indices) with elements drawn from some distribution, e.g. θi[t]~Unif[0,2π],or more practically from a QPSK constellation: θi[t]PT/Nejα2π/n,α=0,1,...,n1 It is sufficient to store one realization of a training sequence in the OFDM Tx and offline evaluate the corresponding PI matrix to be applied in the Rx processing.

5.4 Noise analysis

Let us derive the noise performance of the proposed PI-based system identification (PI-SID) procedure and compare it with that of the three-tone-test (3TT) SID described in section 3. The PI-SID amounts to a matrix multiplication by A of the noisy observation vector δR=δR+n,where n denotes the vector of additive noises accompanying the components of the observed vector δR.Propagating signal and noise through the PI we have:
AδR=A(δR+n)=AδR+An=H+nPI-SID;nPI-SIDAn
(53)
At this point let us clarify the dependence of the SID procedure on the amplitudes of the TS elements. Let the subcarrier complex amplitudes of the TS symbols be all selected from a TS constellation with average powerPo,and let AoPo be the RMS radius of the transmission constellation. Let us introduce unity-modulus-normalized versions of the complex-valued transmitted subcarriers: Ai=Ai/Ao.In the special case of n-PSK training sequences, which are our preferred option, we have |Ai|=1,i.e. the only DOFs in TS design are the phases.

The FWM-matrix elements are then expressed as:
Ai,mFWMAo3(j,k)Sm[i]AjAkAj+ki*=Ao3Ai,mFWMwhereAi,mFWM(j,k)Sm[i]AjAkAj+ki*
(54)
The overall A-matrix is then expressed as A=Ao3A,whereAis defined as the normalized FWM-matrix with elements [A]i,m=Ai,mFWMgiven in Eq. (38). SubstitutingA=Ao3Ainto Eq. (46) relates the un-normalized and normalized PIs as follows: A=Ao3A.

The total power of the output noise vector,nPI-SID,is then given by
nPI-SID2=mM˜|n˜mPI-SID|2=Tr{(nPI-SID)(nPI-SID)}=Tr{Ann(A)}=σn˜2Tr{A(A)}=σn˜2Tr{(AA)1}=Ao6σn˜2Tr{(AA)1}
(55)
where autocorrelation matrix, nn=σn˜2I of the white noise vector was expressed as a scaled unity matrix.

Let us now introduce the Singular-Value-Decomposition (SVD) of the FWM-normalized matrix (assumed to have full-column-rank |M˜|): A=UDVwhere UN|T˜|×N|T˜|,V|M˜|×|M˜| are unitary, and DN|T˜|×|M˜|=diag[λ1,λ2,...,λ|M˜|] is a rectangular (“portrait”N|T˜|>|M˜|) matrix containing the square-roots of the singular values along its diagonal, zero elsewhere. The average noise power in the reconstructed compressed VSTF vector, H (averaged over all its |M˜|components) is then obtained by dividing the total noise power (Eq. (55)) by|M˜|:
σ¯n˜PI-SID21|M˜|mM˜|n˜mPI-SID|2=Ao6σn˜21|M˜|Tr{(VDDV)1}=Ao6σn˜21|M˜|Tr{V(DD)1V}=Ao6σn˜21|M˜|Tr{(DD)1}=Ao6σn˜21|M˜|i=1|M˜|λi1
(56)
The term A6σn˜2in the last expression coincides with the noise power accompanying each of the components of the reconstructed VSTF using the naïve three-tone-test of section 3. Indeed, the 3TT-SID amounts in the current notation to H^i;jk=δRi;jk/Ao3where δRi;jkis the FWM distortion plus the additive noise in subcarrier i, in response to a “3-tone” OFDM training symbol with amplitudes Aofor subcarriers j,k,j + k-i and zero elsewhere. Now, writing δRi;jk=δRi;jk+n˜i3TT-SID,the fluctuations in the reconstructed VSTF are given by:
n˜i3TT-SIDH^i;jkH^i;jk=n˜i/Ao3;σn˜i3TT-SID2|n˜i3TT-SID|2=|n˜i|2/Ao6=Ao6σn˜2
(57)
Dividing Eq. (56) by the last equation we obtain the figure of merit describing the reduction in noise power attained upon using the PI-SID rather than the naïve 3TT-SID method:
σ¯n˜PI-SID2/σn˜i3TT-SID2=1|M˜|i=1|M˜|λi1
(58)
The noise reduction figure of merit of the PI-SID relative to the 3TT-SID has compactly been expressed as the arithmetic average of the inverses of the singular values of the normalized FWM-matrix A,the elements of which are given by Eq. (54).

Figure 3
Fig. 3 Noise reduction figure-of-merit for N = 64 OFDM subcarriers and 16-PSK constellation.
plots the noise reduction figure-of-merit (Eq. (58)) vs. the compressed|M˜|(as described in section 6) and parameterized by T|T˜|for N = 64 and 16-PSK TS (evaluated for the symmetrized A matrix of section 5.2). It is apparent that doubling the measurement time T˜results in a noise reductions larger than 3dB, indicating that this technique is preferred to simple averaging of the received noisy samples.

6. Extra ‘lossy’ compression by band-limitation and prioritized discarding of m-indices

In principle, in order to fully characterize the VSTF over all triplets we ought to evaluate Hm (Eq. (30)) over all feasible indices,mM˜.However, this task might be time and computationally demanding, especially when higher frequency resolution is sought. It turns out that the computational and acquisition time demands may be substantially relaxed while sacrificing a very small controlled amount of estimation accuracy, by introducing additional ‘lossy’ compression of the VSTF, either applying band-limitation (discarding m-indices corresponding to higher frequencies) or preferably by smart prioritization of the coefficients according to their power and multiplicity.

In either case the idea is to restrict the domain of the estimated VSTF by excluding the triplets corresponding to smaller nonlinear power contributions. Thus, the VSTF is only evaluated over a ‘relevant’ subset M˜M˜of the full domain of m-indices, M˜.The compressed VSTF is set to zero outside the restricted domain, with the expectation that this restriction have negligible or little impact on the evaluation of the nonlinear distortions. This is akin to video or image compression where low spatial transform coefficients are simply set to zero.

We note that the methods developed in section 5 apply to the new 'lossy-compressed' VSTF estimation problem by simply replacing M˜byM˜,which amounts to eliminating corresponding elements of Hand columns of the A matrix.

The lossy compression quality (LCQ) may be quantified in terms of the fraction of the power of the participating triplets:
LCQ(M˜)=mM˜|Sm||Hm|2/mM˜|Sm||Hm|2
(59)
Operationally, it may be preferable to assess the LCQ in terms of its impact on the quality of nonlinear compensation, once the evaluated 'lossy-compressed’ VSTF is plugged into an idealized NLC.

Equation (36) indicates that not all Hmcoefficients contribute on equal footing to the overall nonlinear distortion. The nonlinear power contribution associated with index m is approximately proportional to|Sm||Hm|2.Many coefficients are weighed very low in terms of their squared absolute value |Hm|2as they either fall between the array factor's side lobes, or as a result of the diminishing single span VSTF with increasing m. other coefficients may contribute only an insignificant portion of the nonlinear distortion power due to their small multiplicity|Sm|,resulting in nonlinear contribution from a small number of triplets. The various lossy compression strategies differ by their selection of the reduced subset, M˜.

6.1 Lossy compression of the VSTF by band-limitation

6.2 Sorted prioritization of the VSTF coefficients according to power and multiplicity

The compressed representation of Eq. (62) should also be instrumental in optimizing complexity-performance tradeoffs in active nonlinear compensation of OFDM by means of a frequency-domain Volterra equalizer, optimizing NLC performance for a given number of VSTF coefficients, however this topic is outside the scope of the current paper, which is dedicated to VSTF system identification.

6.3 Complexity of the proposed nonlinear system identification procedure

The proposed PI-SID procedure amounts to a matrix-vector multiplication via the PI matrix of size|M˜|×2N|T˜|.Fortunately, the processing may be performed ‘on-the-fly’. It is not necessary to collect the full δRobservations vector (of length2N|T˜|) prior to commencing the matrix-vector multiplication. Rather, matrix multiplication partial results may be gradually accumulated as soon as the responses due to each TS arrive. To this end, the PI matrix A(in its anti-hermitian symmetrized version), is viewed as being horizontally partitioned into sub-blocks of |M˜|×2N dimension (there are |T˜| such sub-blocks). For each arriving sub-block of 2N elements of δRwe proceed to multiply the received sub-block by the corresponding sub-block of the Amatrix, yielding a partial result consisting of an |M˜| elements column vector, which is element-by-element accumulated in a running sum, yielding the overall |M˜| elements estimated vector, H.This practically eliminates latency, as the multiplication of the δRvector may commence right after the negligible delay entailed in receiving its first 2N elements, rather than waiting for the entire vector to be received. This indicates that the SID procedure completion time is practically equal to the SID computation time.

The complexity of evaluation is defined as the MAC rate, namely the number of MACs per sample, obtained by dividing Eq. (64) by the number of ADC samples, SSID=TSID/Ts, transmitted during the SID procedure duration TSID,where Rs=Ts1is the ADC sampling rate in the OFDM transmitter:
cMAMAC/SSID10flossy2|M˜|2/(TSIDRs)
(65)
Evidently, the complexity is inversely proportional to the time TSIDallocated for the multiplication by the PI matrix. Inspecting Eq. (65), it is apparent that the complexity rapidly rises as the fourth power of the number of subcarriers (frequency resolution), N. If we insist on keeping the SID procedure complexity down to a low level of just 1 MAC per unit time, then the measurement time must accordingly be scaled to rise as N4:
cMA(TSID)=1TSID(N)=10flossy2(N)|M˜(N)|2/(Rs).
(66)
The last relation is plotted in Fig. 6
Fig. 6 SID procedure computation time vs. number of sub-carriers, N, such that the complexity rate is precisely one MAC per sample. Here we assumed Rs = 25 GS/s (25 GBaud OFDM), and the same fiber parameters as in Fig. 5.
, parameterized by the LCQ threshold.

Figure 6 indicates that for the specific channel under test and for N = 128,256,512 the SID measurement time is less than 1 msec, whereas for N = 1024 it steeply rises to 10 msec.

We note that the fiber nonlinearity is quite a stable effect, thus the nonlinear SID estimate, rapidly obtained in just 1msec (10 msec for N = 1024) should be applicable for an extended period of time – as suitable for protection switching reconfiguration scenarios.

7. Simulations of operational performance of the proposed SID

The estimated VSTF is plugged into an ideal NLC whereby a ‘genie’ makes available to the NLC the transmitted symbols, which are then distorted via a ‘synthetic fiber link’ modeled in terms of the SID-estimated VSTF. The synthesized distortion is subtracted off the actually ‘measured’ distortion of the uncompensated link. The residual FWM distortion per subcarrier is represented by the variance of the nonlinear interference in each sub-carrier, expressed as the MER of the noisy constellation per sub-carrier, which is used as the criterion of merit for the SID performance.

7.1 Simulation setup

The quality of the estimated VSTF is tested at the NLC output in terms of the variance of the nonlinear interference per subcarrier, expressed as the perturbed constellation modulation error ratio (MER). We compare the MER performance for the uncompensated received signal, with post-NLC performance loading our identified VSTF into the NLC, and also with the post-NLC performance obtained with an analytically calculated VSTF according to Eq. (12), which results are used for reference.

7.2 SID simulated performance

For N = 32 the full group of m-indices, M˜,includes 282 coefficients. As M˜approaches this limit we get improved results, until finally inclusion of additional coefficients provides diminishing improvement in approximating the nonlinear distortion. Upon increasing the acquisition time the results approach the optimal selection, which is a band-limited truncation of the analytically calculated Hm.

Conclusions

Future research directions include extending the acquired insights into the compressed representations of the VSTF beyond nonlinear optical monitoring in order to improve active nonlinear compensation techniques. Another important direction relegated to future work is to possibly use the estimated VSTF in order to extract the spatial profile of the nonlinear parameter, γ(z)along the fiber link.

Acknowledgments

This work was supported in part by the Israeli Science Foundation (ISF) and by the Chief Scientist Office of the Israeli Ministry of Industry, Trade and Labor within the ‘Tera Santa’ consortium.

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A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol. 30(10), 1524–1539 (2012). [CrossRef]

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J. K. Fischer, C.-A. Bunge, and K. Petermann, “Equivalent single-span model for dispersion- managed fiber-optic transmission systems,” J. Lightwave Technol. 27(16), 3425–3432 (2009). [CrossRef]

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F. Vacondio, O. Rival, C. Simonneau, E. Grellier, A. Bononi, L. Lorcy, J.-C. Antona, and S. Bigo, “On nonlinear distortions of highly dispersive optical coherent systems,” Opt. Express 20(2), 1022–1032 (2012). [CrossRef] [PubMed]

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OCIS Codes
(060.1660) Fiber optics and optical communications : Coherent communications
(060.2270) Fiber optics and optical communications : Fiber characterization

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: August 15, 2012
Revised Manuscript: October 21, 2012
Manuscript Accepted: December 3, 2012
Published: December 14, 2012

Citation
Gal Shulkind and Moshe Nazarathy, "Estimating the Volterra series transfer function over coherent optical OFDM for efficient monitoring of the fiber channel nonlinearity," Opt. Express 20, 29035-29062 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-27-29035


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References

  1. A. Bononi, P. Serena, N. Rossi, E. Grellier, and F. Vacondio, “Modeling nonlinearity in coherent transmissions with dominant intrachannel-four-wave-mixing,” Opt. Express20(7), 7777–7791 (2012). [CrossRef] [PubMed]
  2. G. Bosco, P. Poggiolini, A. Carena, V. Curri, and F. Forghieri, “Analytical results on channel capacity in uncompensated optical links with coherent detection,” Opt. Express19(26), B440–B449 (2011). [CrossRef] [PubMed]
  3. A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol.30(10), 1524–1539 (2012). [CrossRef]
  4. X. Chen and W. Shieh, “Closed-form expressions for nonlinear transmission performance of densely spaced coherent optical OFDM systems,” Opt. Express18(18), 19039–19054 (2010). [CrossRef] [PubMed]
  5. J. K. Fischer, C.-A. Bunge, and K. Petermann, “Equivalent single-span model for dispersion- managed fiber-optic transmission systems,” J. Lightwave Technol.27(16), 3425–3432 (2009). [CrossRef]
  6. F. Vacondio, O. Rival, C. Simonneau, E. Grellier, A. Bononi, L. Lorcy, J.-C. Antona, and S. Bigo, “On nonlinear distortions of highly dispersive optical coherent systems,” Opt. Express20(2), 1022–1032 (2012). [CrossRef] [PubMed]
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