## Estimating the Volterra series transfer function over coherent optical OFDM for efficient monitoring of the fiber channel nonlinearity |

Optics Express, Vol. 20, Issue 27, pp. 29035-29062 (2012)

http://dx.doi.org/10.1364/OE.20.029035

Acrobat PDF (1438 KB)

### 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

*Orthogonal Frequency Division Multiplexing*(OFDM) transmission have been conducted [1

1. A. Bononi, P. Serena, N. Rossi, E. Grellier, and F. Vacondio, “Modeling nonlinearity in coherent transmissions with dominant intrachannel-four-wave-mixing,” Opt. Express **20**(7), 7777–7791 (2012). [CrossRef] [PubMed]

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]

*nonlinear compensation*(NLC) methods have been considered. Three prominent classes of NLC methodologies are digital back-propagation (DBP) [12

12. E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. **26**(20), 3416–3425 (2008). [CrossRef]

14. D. Rafique, M. Mussolin, M. Forzati, J. Mårtensson, M. N. Chugtai, and A. D. Ellis, “Compensation of intra-channel nonlinear fibre impairments using simplified digital back-propagation algorithm,” Opt. Express **19**(10), 9453–9460 (2011). [CrossRef] [PubMed]

16. L. B. Y. Du and A. J. Lowery, “Pilot-based XPM nonlinearity compensator for CO-OFDM systems,” Opt. Express **19**(26), B862–B867 (2011). [CrossRef] [PubMed]

*Volterra Series*(VS) based methods [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]

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. 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]

*nonlinear*(NL) mitigation approaches just arbitrarily assumed a-priori knowledge of the nonlinear parameters characterizing the link, either by means of an unspecified measurement process, or from a theoretical calculation. In particular, to the best of our knowledge, all VS-based mitigation methods have heretofore assumed analytic formulas for either the time-domain

*Volterra Series Kernel*(VSK) or for the corresponding frequency domain

*Volterra Series Transfer Function*(VSTF). In practice, the exact link configuration must be estimated in terms of its nonlinear and dispersion parameters, amplifier gains and fiber span optical lengths, which may discontinuously jump to new values due to protection switching or network reconfiguration over different paths. A key question which remains unanswered in all NLC proposals is ‘how the receiver ‘knows’ the amount of nonlinearity to be compensated in the link. Upon connection initialization, an optical monitoring procedure would be required in order to estimate the nonlinear channel, measuring and inferring either the distributed profile of the nonlinear parameter

*system identification*(SID) algorithm for efficient estimation of the VSTF of the fiber link, and determine its noise robustness, latency and computational complexity. Section 6 treats the additional slightly lossy compression attainable by prioritized discarding of low-power and low-multiplicity VSTF samples. In section 7 we present simulations validating the proposed SID methods vs.

*Split-Step-Fourier*(SSF) numerical modeling. Section 8 concludes the paper.

## 2. Volterra nonlinear formalism – third order fiber nonlinearity characterization

*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] 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. 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]

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]

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

*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).

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]

1. A. Bononi, P. Serena, N. Rossi, E. Grellier, and F. Vacondio, “Modeling nonlinearity in coherent transmissions with dominant intrachannel-four-wave-mixing,” Opt. Express **20**(7), 7777–7791 (2012). [CrossRef] [PubMed]

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. Express **20**(2), 1022–1032 (2012). [CrossRef] [PubMed]

*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)

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]

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]

*j,k,l*(which in turn determine the output tone

*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

*j,k*determine the third index

*l*causing the mixing product to fall onto the specified

*i*):As detailed in [26], the CE of the output nonlinear signal

*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:The total nonlinear field at the

*i*-th frequency is thenwith the summation over all index pairs in the domain

*i*, then

*i*, it suffices to restrict the summation to the set

**16**(20), 15777–15810 (2008). [CrossRef] [PubMed]

*Self-Phase-Modulation*(

*j*=

*k*=

*i*) and

*Cross-Phase-Modulation*(XPM) (

*j*=

*i*or

*k*=

*i*) contributions, which are separately treated:The summation of Eq. (7), describing just the FWM contribution, then reduces toThe 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:

*T*] interval:

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

**16**(20), 15777–15810 (2008). [CrossRef] [PubMed]

*L*, with the spans not necessarily identical, with arbitrary z-varying dispersion and nonlinear fiber parameters

*z*= 0) to a point

*z*is given by

*chromatic dispersion*(CD) is given by Eq. (3.70) of [26]:This general form is readily reduced to specific expressions for successively specialized cases. In particular, if the GVD parameter is constant along the link,

*Fourier Transform*(FT) of the product

*j*,

*k*deviate from the target index

*i*. The FT specific value yielding the VSTF is evaluated at a spatial frequency equal to

**16**(20), 15777–15810 (2008). [CrossRef] [PubMed]

*array-factor*, which is akin to the one arising in the radiation pattern from an antenna array. It is this array-factor which accounts to the nonlinear tolerance advantage of dispersion-uncompensated links over dispersion-compensated ones. We quote here the result for the VSTF in that case

**16**(20), 15777–15810 (2008). [CrossRef] [PubMed]

*z*-varying and slowly time-varying profile of these (non)linear parameters is extremely challenging to measure [28

28. H. W. Hatton and M. Nishimura, “Temperature dependence of chromatic dispersion in single mode fibers,” J. Lightwave Technol. **4**(10), 1552–1555 (1986). [CrossRef]

30. H. Onaka, K. Otsuka, H. Miyata, and T. Chikama, “Measuring the longitudinal distribution of four-wave mixing efficiency in dispersion-shifted fibers,” Photon. Technol. Lett. **6**(12), 1454–1456 (1994). [CrossRef]

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

*nonlinear compensation*(NLC) based on the analytic FWM generation model of Eq. (10), which predicts the nonlinear distortion complex amplitudes

**16**(20), 15777–15810 (2008). [CrossRef] [PubMed]

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

*i*, namely subcarriers

*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 disturbance

*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

*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*, yielding

*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 be

*N*= 1024 and

*S*= 100 (yielding 20 dB averaging gain), we would require

*N*= 128 rather than

*N*= 1024 OFDM FFT size, the measurement time would be reduced by a factor of

*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

**16**(20), 15777–15810 (2008). [CrossRef] [PubMed]

*i,j,k*) and the following function,the sole argument of which is a single integer index,

*m*. Equation (20) is henceforth referred to as ‘compressed VSTF’. It is the composition of the two mappings that yields the VSTF:Remarkably, the compressed VSTF

*i*). This is indicative of potential substantial reduction in the number of degrees of freedom (DOFs) required for nonlinear SID: The compressed VSTF number of DOFs will be seen to grow quadratically, as

*N*. As for the coefficient in the

*m*-values happens to include “gaps”, i.e., there may be non-consecutive values of

*m*, corresponding to missing

*m*indices for which no combination of valid

*i,j,k*subcarrier indices may be found.

*m*(corresponding to certain triplets), whenever the array factor generates destructive interference. Additional functional symmetry properties of

*m*:The substantially reduced number of DOFs (mainly because of the composition of mappings of Eqs. (21), but also further compressed by discarding very low power or low multiplicity values) will be seen to lead to a significantly faster

*System Identification*(SID) procedure as derived in the next section.

### 4.1 Compressed VSTF mathematical properties

*m*-index notation, our previous results for the VSTF of various links. Notice that the following relation always holds:with

*m*-index, the compressed VSTF magnitude for a multi-span link with identical spans, as described by Eq. (25). The figure also shows the constituent multiplicative factors, namely the single-span VSTF and the array factor as per the respective equations Eqs. (26),(27). The array factor is seen to modulate the slowly and smoothly decaying profile of the single-span VSTF, superimposing a spiky periodic structure.

*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

*j,k*) pairs falling on target index

*i*(uniquely defining FWM triplets) and satisfying the relation

*m*which are not feasible, we take

*m*, i.e.,

*m*-index corresponding to all the subcarriers:The cardinality of

*m,*referred to as ‘

*m-index multiplicity’*is plotted in Fig. 2(b), featuring a large variation. Thus, whereas all

**16**(20), 15777–15810 (2008). [CrossRef] [PubMed]

*i*as an incoherent summation of the powers of the individual FWM intermodulation products falling on the target frequency indexed

*i*.

### 4.2 Compressed representation of the FWM mixing products build-up

*S*[

*i*] was derived in [27

**16**(20), 15777–15810 (2008). [CrossRef] [PubMed]

*N*, the corresponding sizes of the set

*m*-indices which are seen to be much smaller. As the set

*N*-1) to (

*N*-1), it approximately scales as

*N*less than the order of

*N*being a power-of-two, we have for the maximum value included in the

*m*-index is given by

*m*, as already indicated. Thus, we have the upper bound

## 5. Efficient nonlinear system identification of the fiber link

*i*-th subcarrier, due to all triplets corresponding to the

*m*-th subset,

*i*and

*m,*since out of the triplets falling on

*i*(belonging to the set

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

*N*sub-carriers generally non-zero. The received complex amplitudes are labeled as

*i*-th output of the Rx FFT), and

**16**(20), 15777–15810 (2008). [CrossRef] [PubMed]

*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:If

*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:where the constituent vectors and matrix are defined as follows:

*t*. The mean square error now becomes

**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**

*A**pseudoinverse*(PI)

*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

*Recursive Least Squares*(RLS).

### 5.2 Exploiting the anti-hermitian symmetry of the compressed VSTF

*u*,

*l*indicate upper and lower blocks.

*L,R*):The optimization problem of Eq. (45), applied to

**A**-matrix is now

**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

### 5.3 Training sequences

*N*subcarriers having the complex amplitudes

*i*and

*t*indices) with elements drawn from some distribution, e.g.

### 5.4 Noise analysis

*three-tone-test*(3TT) SID described in section 3. The PI-SID amounts to a matrix multiplication by

*n*-PSK training sequences, which are our preferred option, we have

**A**-matrix is then expressed as

*Singular-Value-Decomposition*(SVD) of the FWM-normalized matrix (assumed to have full-column-rank

*i*, in response to a “3-tone” OFDM training symbol with amplitudes

*j,k,j + k-i*and zero elsewhere. Now, writing

*N*= 64 and 16-PSK TS (evaluated for the symmetrized A matrix of section 5.2). It is apparent that doubling the measurement time

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

*m*-indices corresponding to higher frequencies) or preferably by smart prioritization of the coefficients according to their power and multiplicity.

*m*-indices,

**A**matrix.

*lossy compression quality*(LCQ) may be quantified in terms of the fraction of the power of the participating triplets: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.

*m*is approximately proportional to

*m*. other coefficients may contribute only an insignificant portion of the nonlinear distortion power due to their small multiplicity

### 6.1 Lossy compression of the VSTF by band-limitation

*m*-index multiplicity, and the single span VSTF decrease with increasing

*m*, a simple approach towards discarding of

*m*-indices is to exclude all

*m*-values beyond a certain distance from the origin,

*m*= 0, i.e. select the target

*m*-indices as the following ‘reduced’ subset of the full set,

*m*-index represents a sampled spatial frequency we may refer to this set as low-pass-filtered or band-limited (BL) around DC. Figure 4(a) plots the LCQ (Eq. (59)) incurred in this simple lossy compression procedure. The LCQ falls off monotonically with increasing size

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

*m*-indices of the sequence of summand terms

*m*-indices. Denoting the sorted indices by

*m*), and further denoting the ordered list of indices by

*i*may then be approximated by truncating the power-sorted list,

*m*-indices or any other permutation of these indices. Figure 4(b) plots the LCQ (Eq. (59)) incurred in this compression method. It is apparent that the sorted policy is superior to the band-limited one in its performance vs. complexity tradeoff; E.g., for an LCQ of −0.4dB (the horizontal red line), 350 coefficients are retained, as compared with the 958 coefficients required for the band-limited compression of the previous subsection.

### 6.3 Complexity of the proposed nonlinear system identification procedure

*N*elements of

*N*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.

*multiply-accumulates*(MAC)) is given bywhere

*N*, as detailed in Fig. 5 (assuming the sorted prioritization of section 6.2).

*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

*N*= 128,256,512 the SID measurement time is less than 1 msec, whereas for

*N*= 1024 it steeply rises to 10 msec.

*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

### 7.1 Simulation setup

*N*subcarriers akin to that described in sub-section 2.4. At the Tx we launch a uniformly distributed white 16-PSK pseudo random training sequence. At the Rx we compensate for CD and SPM/XPM, providing

*N*subcarrier measurements

*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

*N*= 32 subcarriers. Figure 7(a) presents SID performance vs. the number of VSTF coefficients

*N*= 32 the full group of

*m*-indices,

*N*= 256. It is apparent that for sufficient acquisition time the nonlinear estimation results are substantially improved.

## Conclusions

*N*= 512,1024 points per 25 GHz channel, the proposed SID algorithm is fast (1,10 msec respectively) and the nonlinear optical performance monitoring complexity is negligible relative to the overall OFDM Rx complexity.

## Acknowledgments

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28. | H. W. Hatton and M. Nishimura, “Temperature dependence of chromatic dispersion in single mode fibers,” J. Lightwave Technol. |

29. | G. Ishikawa and H. Ooi, “Demonstration of automatic dispersion equalization in 40 Gbit/s OTDM transmission,” in European Conference of Optical Communication (ECOC) (1998), 519–520. |

30. | H. Onaka, K. Otsuka, H. Miyata, and T. Chikama, “Measuring the longitudinal distribution of four-wave mixing efficiency in dispersion-shifted fibers,” Photon. Technol. Lett. |

31. | S. Haykin, |

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 |

**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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