Periodic nonlinear Fourier transform for fiber-optic communications, Part II: Eigenvalue communication

: In this paper we propose the design of communication systems based on using periodic nonlinear Fourier transform (PNFT), following the introduction of the method in the Part I. We show that the famous “eigenvalue communication” idea [A. Hasegawa and T. Nyu, J. Lightwave Technol. 11 , 395 (1993)] can also be generalized for the PNFT application: In this case, the main spectrum attributed to the PNFT signal decomposition remains constant with the propagation down the optical ﬁber link. Therefore, the main PNFT spectrum can be encoded with data in the same way as soliton eigenvalues in the original proposal. The results are presented in terms of the bit-error rate (BER) values for different modulation techniques and different constellation sizes vs. the propagation distance, showing a good potential of the technique.


Introduction
In the Part I of this work we described the PNFT peculiarities, introduced the most important quantities (main and auxiliary spectra) resulting from the PNFT periodic signal decomposition, and explained the differences arising when dealing with the PNFT compared to its "ordinary" counterpart -the NFT attributed to burst-mode (or truncated at the time interval boundaries) signals.We also itemized the potential advantages of using the PNFT against the burst-mode NFT in the optical transmission.Although, as it was explained in Part I, the utilization of periodic signals can bring noticeable benefits in terms of the reduction of the processing complexity and the capability of control over the generated signal time-extent, much less attention has been paid to the PNFT applications due to a more complex mathematical background involved.In this regard we would like to mention a very resent paper of Wahls and Poor [1], where a quite exhaustive survey of the PNFT theory and numerical methods including the newest fast direct PNFT decomposition algorithms, was presented, perhaps aside from the computation of theta-functions (see subsection 4.2 of our Part I).In the first part, we also analyzed the available numerical tools for the direct PNFT computation, addressing the quality of the resulting main spectrum and the associated processing time consumption, and inferred that the best performance attributes to the Ablowitz-Ladik (AL) discretization algorithm.Thus, in this Part II, whenever we present any data for the PNFT spectrum, these are computed with the use of the AL routine described in Part I, see also [1].It is worth noting that the AL algorithm can be recast into the fast form using the fast polynomial arithmetic [1].
The main goal of Part II is to present possible design schemes for communication systems based on the PNFT processing and to show their potential.Here, we propose proof-of-concept designs of systems that function similarly to the burst-mode eigenvalue communication [2][3][4][5][6], but dealing with the periodically continued waveforms and utilizing the PNFT to retrieve the data encoded on the nonlinear spectrum (NS) at the receiver.In general, one can also design the PNFT-based communication system using the nonlinear inverse synthesis idea (NIS) [7][8][9]: within this approach one maps the encoded data onto the NS and then synthesises the respective signal in time domain (meaning the use of the inverse PNFT for the case considered) at the transmitter, submits the new synthesized signal into the fiber and extracts data from the received signal by performing the direct transformation (i.e. the direct PNFT) together with compensation of the evolution inside the nonlinear spectral domain.However, as was mentioned in our Part I, there is still currently a lack of a generic efficient approach for dealing with the inverse PNFT, and its computation brings about technical difficulties.Therefore, to simplify the task, instead of performing the inverse transformation we use signals with the analytically known NS and, hence, bypass the inverse PNFT step in the synthesis procedure.On the other hand, the efficient AL algorithm is used at the receiver to retrieve the encoded data from the main spectrum.We numerically evaluate the performance of our PNFT-based systems through direct counting of BER comparing the transmitted and received bit streams.The paper is organized as follows.In Section 2, we briefly describe the NS (main spectrum) calculation attributed to the PNFT, to make this Part II self-consistent.In Section 3, we introduce designs of the PNFT-based communication systems using three different signals with the known NS.These system are examined via the direct simulations of the noise-perturbed NLSE channel in Section 4: This section also contains the discussion of the emerging error sources.The results are summarised in the Conclusion section.

Channel model
We use as master model the NLSE with the account of the ASE noise (assuming an ideal Raman amplification) [11]: where q(t, z) is the slowly varying envelop of the electromagnetic field, β 2 < 0 is the chromatic dispersion parameter, γ is the nonlinearity (Kerr) coefficient, and n(t, z) is the circularly symmetric complex Gaussian noise process introduced by amplifiers.The latter is completely characterized by the autocorrelation function [11]: where α is the fiber loss coefficient, typically α ≈ 0.2 dB/km at the carrying wavelength λ 0 = 1.55 µm, h is the Planck constant, f s is the central frequency which we set to 193.41 THz, K T = 1.13 is the photon occupancy factor, and δ (•) is the Dirac delta-function.We note that the model in Eq. ( 1) can also be used as a leading approximation for the erbium-doped fiber amplifiers (EDFA) (see e.g.[9] and references therein).In this case, in the leading approximation one can use the averaged NLSE replacing γ with the effective nonlinearity coefficient where L span is the length of span.A similar approach can be used in the case of non-ideal Raman amplification [13,14].We then normalize Eq. (1) using the following standard redefinitions: where T s can be chosen as a free normalization parameter, and the respective z-scale becomes Z s = 2T 2 s /|β 2 |.The normalized NLSE reads: Note that the autocorrelation function intensity in Eq. ( 2) has to be renormalized as well according to the last expression in Eq. ( 4).Further we are going to work with periodic (in time variable) signals with the period T 0 , therefore, the additional condition to Eq. ( 5) is imposed: q(t, z) = q(t + T 0 , z).

Direct PNFT and main spectrum definition
Insofar as we described the PNFT details in our Part I, Section 4.1, here we reproduce just some basic elements that are directly necessary for our further analysis and communication system design.The PNFT spectrum of a periodic signal consists of two parts: main and auxiliary spectra.The NS attributed to the PNFT signal decomposition is calculated according to the the following procedure.The first step is to form the fundamental matrix of the periodic version of Zakharov-Shabat system (ZSS) [12]: by finding two linearly independent ZSS solutions via the eigenfunctions Φ = [φ 1 , φ 2 ] T with the conditions set at some point t 0 [10,17]: The monodromy matrix, M, is then defined as the value of the fundamental matrix at one period from the base point, M = [φ (t 0 + T,t 0 ; λ ), φ (t 0 + T,t 0 ; λ )] T .The main spectrum, M, is defined as the set of points in the complex plane at which the ZSS solutions are anti-periodic or periodic.This condition leads to the expressions defining the dependence of the main spectrum on the waveform parameters [10,17]: The important property of the main spectrum that one can utilize for the transmission purposes is that it remains invariant along the signal evolution within the unperturbed NLSE (for more information of the main spectrum role in the definition of the NLSE solution properties see Part I and [1,10,18]).

Eigenvalue communication
Using the PNFT it is possible to map data on the NS according to some deterministic one-to-one mapping rule, then synthesize the signal corresponding to this modulated NS in time domain, and launch the signal into the fiber; at the receiver side one uses the direct PNFT and retrieves the encoded data from the NS (see Fig. 1).Such a procedure can be understood as a variant of NIS [7][8][9] with the use of the PNFT.However, in spite of the fact that we know the evolution law inside the nonlinear spectral domain attributed to the PNFT, this evolution is nontrivial, in contrast to the one for the ordinary NFT, see Part I. To eliminate the evolution compensation step and simplify our system further, we can disregard the auxiliary spectrum and employ only the main spectrum part of the NS, of course, at the expense of losing some degrees of freedom available for modulation.This approach is the analogue of eigenvalue communication [2] or a variant of the nonlinear frequency division multiplexing [3].As the main spectrum part of the NS is invariant along the signal propagation (aside from the noise-induced corruptions), one automatically circumvents the compensation of NS evolution by means of employing the main spectrum only.

Signals with known NS
As a simple example we select a two-phase signal with the main spectrum consisting of one complex conjugate pair of single points, λ 1 and λ * 1 , and two nearly degenerate (solution points in Eq. ( 8) with multiplicity of 2 as opposed to nondegenerate points with multiplicity 1) pairs, λ 2 , λ * 2 , λ 3 and λ * 3 .For the case when λ 2 approaches λ 3 , the periodic signal resulting from such an NS can be written as [21]: where the parameters in Eq. ( 9) are expressed through the main spectrum eigenvalues λ 1 and λ 3 as follows with the assumption that Reλ 1 = −Reλ 3 → 0 and Imλ 1 > Imλ 3 .The selection of eigenvalues by using the rules in Eqs.(10) ensures that each λ i belongs to the main spectrum.Now we can construct the signal by using Eq. ( 9) at z = 0 for different values of λ 1 , and use the invariant value of λ 1 to encode and transmit our data.The value of λ 3 (and λ 2 ) is selected with respect to the chosen period of the signal T 0 by means of the relation: , such that the resulting signals have the same period.Hence, the only data carrier parameter is the imaginary part of λ 1 which is drawn from a one dimensional constellation (like the one in Fig. 8 in Section 4 below).Fig. 2 depicts an example of such a signal and the corresponding main spectrum.Since the two-phase signal of Eq. ( 9) is periodic, it can be recast as a Fourier series representation.The leading term of this series is by far larger than the succeeding one; the latter, in addition, can be made even smaller by decreasing the imaginary part of the eigenvalue.In this case, the bandwidth of different signals related to various values of eigenvalue remains almost constant.Thus, the important advantage of the periodic signals based on Eq. ( 9) is that it can be used to generate a set of encoded signals with similar characteristics, i.e. the similar bandwidth and time domain occupation (period).
In the case of the two-phase signal of Eq. ( 9) with two eigenvalues (located in the complex half-plane corresponding to positive imaginary parts, C + ) since we intend to use only one of the points in the main spectrum, the area in which we can settle that point is limited by the positions of remaining points.So, the location of eigenvalues is confined, and hence, the power of different signals from the generated set varies insignificantly.For the example used in this work, the signal power varies between −1 dBm and 0 dBm for the smallest and largest eigenvalue respectively, set for signals with time duration of 0.42 ns giving the data rate 2.4 GSym/s.For the first time this signal was used in a PNFT-based communication in [22].Now we have an alternative for the inverse transformation which provides us with adjustable period and one free point in the main spectrum, but the choices of power are limited and the main spectrum is restricted to a part of the imaginary axis.Because of the latter fact we can deal with the one dimensional constellations only.Note that the solution in Eq. ( 9) can also be obtained by means of a one-step Bäcklund transformation [23].

Perturbed plane wave
In the previous case, the main spectrum was confined to a part of the imaginary axis and we could use only one-dimensional constellations.To proceed one step forward for having twodimensional constellations without calculating algebraic-geometric loop integrals [18,19], one can use the analytical formulas given in [10,19] derived for a special spectrum corresponding to a perturbed plane wave.As opposed to the two-phase signal of Eq. ( 9), the separation between points in these pairs could be controlled, and this provides us with more degrees of freedom to design data-carrying eigenvalues forming our constellation.A finite-gap periodic solution of NLSE can be constructed by carrying out the steps explained in Part I, subsection 4.2.For doing that one needs to find the Riemann spectrum, and this procedure is generally rather challenging.However, for the special cases of a perturbed plane wave the situation is much simpler [10].The main spectrum of a plane wave q(t) = A = const.in a time window T 0 (i.e. the PNFT period is T 0 ) consists of a simple (nondegenerate) point, λ = A i, and some double (degenerate) points on the real and imaginary axes, with their location given by: Then, by adding a weak perturbation in the form of cosine waves to the initial constant waveform we can split up the degenerate points into the pairs of simple points (see Fig. 3).The main spectrum of the new waveform is characterized by the parameters: i) λ = A i, ii) the vector of g degenerate points of the plane wave λ λ λ in Eq. ( 11), and iii) ε ε ε, the vector with g elements determining the separation of the split points.For this main spectrum we can calculate the Riemann spectrum using the following formulas [10,19] (see subsection 4.2. of Part I for the definition of Riemann spectrum parameters): where the vector of Riemann sheet indices, σ σ σ , consists of ±1s and is the element-wise multiplication.In fact, for small modulation amplitudes ε ε ε, in the first order in |ε j | we have where a j are some phases, k j are the elements of k k k from Eqs. ( 12) independent of ε ε ε.Hence, if in a communication system based on this kind of signals our data mapped on the values of ε ε ε, in this approximation all transmitted signals share the same bandwidth.Fig. 4 shows an example of such a signal where a degenerate point is split into two nondegenerate ones; for this figure we used ε = 0.045 + 0.045i and T 0 = 3.55.For this type of signals one can encode the data on the main spectrum by picking the complex values of the separations, ε j , between the simple split eigenvalues.Since the analytical formulas of Eqs.(12) are derived in the first order of approximation in ε j , we can control the modulation of the main spectrum only in the areas close to initial degenerate points, where the contribution of terms ∼ |ε| 2 to these expressions are small.By picking the set of separations ε j as complex values taken from an arbitrary constellation, we get the main spectrums in the form depicted in the right panel of Fig. 3, where one of the nondegenerate points and its mirrored one (with respect to a degenerate point of the plane wave, a blue dot in the left panel of Fig. 3) form the main spectrum.Due to the noise-induced corruptions along the transmission, the received main spectrum becomes smeared.Thus, in order to minimise level of errors, it is necessary to have a sufficiently large minimum distance between the constellation points.However increasing the values of separations |ε j | one simultaneously decreases the accuracy of Eqs.(12)    This is one of the current limitations of this approach in achieving high spectral efficiency communication system.Since the error in constructing the signal based on the chosen main spectrum of the form in Fig. 3 is deterministic, it is possible to equalize it at the receiver: as long as the deformation is not too large, one can replace the constellation points with the main spectrum of the initial signal.This new constellation will play the role of reference in the decision making stage of demodulation.Another solution to this problem is to use pilot symbols to equalize the received points at the receiver.

Modulated CW signal
One of the signals with known main spectrum is the plane wave q(t) = Ae iµt with period T 0 , µ = kπ/T 0 with integer k.The main spectrum of this signal is a shifted version (for −µ/2 along real axis) of the main spectrum of a plane wave with amplitude A and period T 0 , i.e. of Eq. (11).Since µ and A determine the real and imaginary part of the points in the main spectrum respectively, one can make up a QAM constellation and map data on it to have a two dimensional constellation.Like the other above-mentioned signals, the requirement to have distinguishable points in the main spectrum entails the limited extent of variation for the imaginary part of points in main spectrum.

Simulation results
Now we assess the performance of eigenvalue communication system based on signal waveforms described in Section 3. Our criteria to quantify the performance of each system is the bit error rate (BER) value which is directly counted by the comparison of the transmitted and received bit sequences.
For the initial set of simulations we evaluate the accuracy of the whole procedure of mapping data on the NS and their retrieving via the direct PNFT in a back-to-back (B2B) scenario.The left panel of Fig. 5 shows the B2B error for a two-phase and a modulated CW signals for different powers.For these two signals the constellation is formed on an imaginary area between 1.2i and 1.5i.These constraints are imposed to make the points of the main spectrum to be distinguishable from symbol to symbol: since the main spectrum for each symbol comprises more than one point (among which only one bears our data), so we have to be capable to distinguish between them.This figure shows the improvement in the B2B procedure performance with the increase in the number of samples and the decrease of power, both effects are basically due to the numerical AL PNFT routine functioning.The performance for the modulated CW is better signal than that for the two-phase one.Since for the perturbed plane wave we are going to equalize the received signal, i.e. to use the distorted main spectrum of the constructed signal at the transmitter as the reference constellation at the receiver, the B2B performance does not correctly characterize the error, and thus it is not depicted in our figure.From Fig. 5 (left), we see that setting the number of samples (per period) to be ≈ 500 should be sufficient for an accurate system functioning to avoid a noticeable contribution from the PNFT numerical error at the receiver.
In our fiber simulations in Eq. ( 1) we use the typical parameters β 2 = −21.67ps 2 /km and γ = 1.27W −1 km −1 .For each eigenvalue communication system we pick a point from the corresponding constellation, construct the signal having a desirable characteristics, and then launch it into the fiber.At the receiver, the main spectrum is calculated by performing the direct PNFT, the data is extracted and the resulting system BER is evaluated.To simulate the signal propagation down the fiber we use the split-step Fourier (SSF) routine setting a small enough step in z to ensure the adequate resolution.Since the SSF uses FFT to perform the linear filtering part of the evolution, the periodicity assumption is automatically fulfilled and, therefore, sending a single signal implies that we effectively have the exact periodicity of our pattern, or, in other words, we effectively have the infinite duration of our cyclic prefix (CP).Although a very large

Fig. 1 .
Fig.1.General design of the NFT-based communication system concept.The data stream is mapped onto the NS (here -on the main spectrum) and then the signal is constructed from this main spectrum.At the receiver side, performing the direct transform, the original data are retrieved.

Fig. 2 .
Fig.2.The two-phase signal in Eq. (9) with λ 1 = 1.2 i (see the explanations in the text) and the time period T 0 = 3 defining the remaining values λ 2,3 (left).The corresponding main spectrum is shown in the right panel.

Fig. 3 .
Fig. 3. Main spectrum of an unperturbed plane wave with T 0 = 4 and A = 2 (left), and a perturbed one with cosine waves with amplitudes ε 1 and ε 2 .

Fig. 5 .
Fig. 5. Left: error vs. the number of samples in the B2B scenario for modulated CW and two-phase signals.Right: BER vs. distance in a single-symbol transmission with the constellation size K = 64 for modulated CW, two-phase and perturbed plane wave signals having −0.55 dBm power.