Capacity lower bounds of the noncentral Chi-channel with applications to soliton amplitude modulation

Citation for published version (APA): Shevchenko, N. A., Derevyanko, S. A., Prilepsky, J. E., Alvarado, A., Bayvel, P., & Turitsyn, S. K. (2018). Capacity lower bounds of the noncentral Chi-channel with applications to soliton amplitude modulation. IEEE Transactions on Communications, 66(7), 2978-2993. https://doi.org/10.1109/TCOMM.2018.2808286

DOI: 10.1109/TCOMM.2018.2808286

Document status and date: Published: 01/07/2018

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Download date: 30. Sep. 2021 2978 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 66, NO. 7, JULY 2018 Capacity Lower Bounds of the Noncentral Chi-Channel With Applications to Soliton Amplitude Modulation

Nikita A. Shevchenko, Stanislav A. Derevyanko, Jaroslaw E. Prilepsky, Alex Alvarado , Senior Member, IEEE, Polina Bayvel, and Sergei K. Turitsyn

Abstract— The channel law for amplitude-modulated solitons have undergone a long process of increasing engineering com- transmitted through a nonlinear optical fiber with ideal distrib- plexity and sophistication [1]Ð[3]. However, the key physical uted amplification and a receiver based on the nonlinear Fourier effects affecting the performance of these systems remain transform is a noncentral chi-distribution with 2n degrees of freedom, where n = 2andn = 3 correspond to the single- and largely the same. These are: attenuation, chromatic dispersion, dual-polarisation cases, respectively. In this paper, we study the fibre nonlinearity due to the optical Kerr effect, and optical capacity lower bounds of this channel under an average power noise. Although the bandwidth of optical fibre transmission constraint in bits per channel use. We develop an asymptotic systems is large, these systems are ultimately band-limited. semi-analytic approximation for a capacity lower bound for This bandwidth limitation combined with the ever-growing arbitrary n and a Rayleigh input distribution. It is shown that this lower bound grows logarithmically with signal-to-noise ratio demand for data rates is expected to result in a so-called (SNR), independently of the value of n. Numerical results for “capacity crunch” [4], which caps the rate increase of error- other continuous input distributions are also provided. A half- free data transmission [4]Ð[7]. Designing spectrally-efficient Gaussian input distribution is shown to give larger rates than a transmission systems is therefore a key challenge for future = , , Rayleigh input distribution for n 1 2 3. At an SNR of 25 dB, optical fibre transmission systems. the best lower bounds we developed are approximately 3.68 bit per channel use. The practically relevant case of amplitude shift- The channel model used in optical communication that keying (ASK) constellations is also numerically analyzed. For includes all three above-mentioned key effects for two states the same SNR of 25 dB, a 16-ASK constellation yields a rate of of polarisation is the so-called Manakov equation (ME) approximately 3.45 bit per channel use. [7, eq. (1.26)], [8, Sec. 10.3.1]. The ME describes the propa- Index Terms— Achievable information rates, channel capacity, gation of the optical field for systems employing polarisation mutual information, nonlinear optical fibres, nonlinear Fourier division multiplexing. The ME therefore generalises the popu- transform, optical solitons. lar scalar nonlinear Schrödinger equation (NSE) [6]Ð[9], used for single-polarisation systems. In both models, the evolution I. INTRODUCTION of the optical field along the fibre is represented by a nonlinear PTICAL fibre transmission systems carrying the over- partial differential equation with complex additive Gaussian Owhelming bulk of the world’s telecommunication traffic noise.1 The accumulated nonlinear interaction between the signal and the noise makes the analysis of the resulting channel Manuscript received April 1, 2017; revised September 8, 2017 and model a very difficult problem. As recently discussed in, e.g., January 31, 2018; accepted February 12, 2018. Date of publication February 20, 2018; date of current version July 13, 2018. Research supported [10, Sec. 1], [11], [12], exact channel capacity results for fibre by the Engineering and Physical Sciences Research Council (EPSRC) project optical systems are scarce, and many aspects related to this UNLOC (EP/J017582/1), by the Netherlands Organisation for Scientific problem remain open. Research (NWO) via the VIDI Grant ICONIC (project number 15685), and a UCL Graduate Research Scholarship (GRS). The associate editor Until recently, the common belief among some researchers coordinating the review of this paper and approving it for publication was in the field of optical communication was that nonlinearity H. Wymeersch. (Corresponding author: Alex Alvarado.) was always a nuisance that necessarily degrades the system N. A. Shevchenko and P. Bayvel are with the Optical Networks Group, Department of Electronic and Electrical Engineering, University College Lon- performance. This led to the assumption that the capacity of don, London WC1E 7JE, U.K. (e-mail: [email protected]). the optical channel had a peaky behaviour when plotted as a S. A. Derevyanko is with the Department of Electrical and Computer function of the transmit power.2 Partially motivated by the idea Engineering, Ben-Gurion University of the Negev, Beersheba 84105, Israel (e-mail: [email protected]). of improving the data rates in optical fibre links, a multitude J. E. Prilepsky and S. K. Turitsyn are with the Aston Institute of Pho- of nonlinearity compensation methods have been proposed tonic Technologies, Aston University, Birmingham B4 7ET, U.K. (e-mail: (see, e.g., [16]Ð[21]), each resulting in different discrete-time [email protected]). A. Alvarado is with the Signal Processing Systems Group, Department channel models. Recently, a paradigm-shifting approach for of Electrical Engineering, Eindhoven University of Technology, Eindhoven 5600 MB, The Netherlands (e-mail: [email protected]). 1The precise mathematical expressions for both channel models are given Color versions of one or more of the figures in this paper are available in Sec. II-A. online at http://ieeexplore.ieee.org. 2However, nondecaying bounds can be found in the literature, e.g., in [10] Digital Object Identifier 10.1109/TCOMM.2018.2808286 and [13] (lower bounds) and [14] and [15] (upper bounds). 0090-6778 © 2018 IEEE. Translations and content mining are permitted for academic research only. Personal use is also permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. SHEVCHENKO et al.: CAPACITY LOWER BOUNDS OF THE NONCENTRAL CHI-CHANNEL 2979 overcoming the effects of nonlinearity has been receiving equivalent to a well-known time-domain amplitude-modulated increased attention. This approach relies on the fact that both soliton transmission system.4 In this paper, we consider this the ME and NSE in the absence of losses and noise are exactly simple set-up, where a single eigenvalue is transmitted in integrable [22], [23]. every time slot. The obtained results are applicable not only to One of the consequences of integrability is that the signal classical soliton communication systems, but also to the novel evolution can be represented using nonlinear normal modes. area of the eigenvalue communications. While the pulse propagation in the ME and NSE is nonlin- Although the set-up we consider in this paper is one ear, the evolution of these nonlinear modes in the so-called of the simplest ones, its channel capacity is still unknown. nonlinear spectral domain is essentially linear [24], [25]. The Furthermore, the only results available in the literature [29], decomposition of the waveform into the nonlinear modes [42], [43], [46]Ð[49] are exclusively for the NSE, leaving (and the reciprocal operation) is often referred to as non- the ME completely unexplored. In particular, previous results linear Fourier transform (NFT), due to its similarity with include those by Meron et al. [48], who recognised that mutual the application of the conventional Fourier decomposition in information (MI) in a nonlinear integrable channel can (and linear systems [26].3 The linear propagation of the nonlinear should) be evaluated through the statistics of the nonlinear modes implies that the nonlinear cross-talk in the NFT domain spectrum, i.e., via the channel defined in the NFT domain. is theoretically absent, an idea exploited in the so-called Using a Gaussian scalar model for the amplitude evolution nonlinear frequency division multiplexing [24], [27]. In this with in-line noise, a lower bound on the MI and capacity of method, the nonlinear interference can be greatly suppressed a single-soliton transmission system was presented. The case by assigning users different ranges in the nonlinear spectrum, of two and more solitons per one time slot was also analysed, instead of multiplexing them using the conventional Fourier where data rate gains of the continuous soliton modulation domain. versus an on-off-keying (OOK) system were also shown. A bit- Integrability (and the general ideas based around NFT) has error rate analysis for the case of two interacting solitons also lead to several nonlinearity compensation, transmission has been presented in [50]. The derivations presented there, and coding schemes [28]Ð[38]. These can be seen as a gener- however, cannot be used straightforwardly for information alisation of soliton-based communications [8], [9], [39, Ch. 5], theoretic analysis. Yousefi and Kschischang [29] addressed which follow the pioneering work by Hasegawa and Nyu [40], the question of achievable spectral efficiency for single- and and where only the discrete eigenvalues were used for com- multi-eigenvalue transmission systems using a Gaussian model munication. The development of efficient and numerically for the nonlinear spectrum evolution. Some results on the stable algorithms has also attracted a lot of attention [41]. continuous spectrum modulation were also presented. Later Furthermore, there have also been a number of experimental in [42], the spectral efficiency of a multi-eigenvalue transmis- demonstrations and assessments for different NFT-based sys- sion system was studied in more detail. In [43], the same tems [33]Ð[38]. However, for systems governed by the ME, problem was studied by considering the correlation functions the only results available come from the recent theoretical of the spectral data obtained in the quasi-classical limit of work of Maruta and Matsuda [32]. large number of eigenvalues. Achievable information rates Two nonlinear spectra (types of nonlinear modes) exist for multi-eigenvalue transmission systems utilising all four in the NSE and the ME. The first one is the so-called degrees of freedom of each scalar soliton in NSE were continuous spectrum, which is the exact nonlinear analogue analytically obtained in [46]. These results were obtained of the familiar linear FT, inasmuch as its evolution in an within the framework of a Gaussian noise model provided optical fibre is exactly equivalent to that of the linear spectrum in [29] and [47] (non-Gaussian models have been presented under the action of the chromatic dispersion and the energy in [51] and [52]) and assuming a continuous uniform input contained in the continuous spectrum is related to that in distribution subject to peak power constraints. The spectral the time domain by a modified Parseval equality [26], [31]. efficiency for the NFT continuous spectrum modulation was The unique feature of the NFT is, however, that apart from considered in [53]Ð[55]. Periodic NFT methods have been the continuous spectrum, it can support a set of discrete recently investigated in [56]. eigenvalues (the nondispersive part of the solution). In the In [49], we used a non-Gaussian model for the evolution of time domain, these eigenvalues correspond to stable localised a single soliton amplitude and the NSE. Our results showed multi-soliton waveforms immune to both dispersion and non- that a lower bound for the capacity per channel use of such linearity [8]. The spectral efficiency of the multiple-eigenvalue a model grows unbounded with the effective signal-to-noise encoding schemes is an area actively explored at the moment ratio (SNR). In this paper, we generalise and extend our [29], [42], [43]. Multi-soliton transmission has also received results in [49] to the ME. To this end, we use perturbation- increased attention in recent years, see, e.g., [44] and [45] based channel laws for soliton amplitudes previously reported and references therein. Finding the capacity of the multi- in [51] and [52] (for the NSE) and [57] (for the ME). Both eigenvalue-based systems in the presence of in-line noise that channel laws are a noncentral chi (χ) distribution with 2n breaks integrability still remains an open research problem. degrees of freedom, where n = 2andn = 3 correspond to If only a single eigenvalue per time slot is used, the problem is the NSE and ME, respectively. Motivated by the similarity of

3In mathematics and physics literature, the name inverse scattering trans- 4Since the imaginary part of a single discrete eigenvalue is proportional to form method for the NFT is more commonly used. the soliton amplitude. 2980 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 66, NO. 7, JULY 2018 the channel models mentioned above, in this paper we study when addressing the ME. In the case of a single polarisation asymptotic lower bound approximations on the capacity (in state, the propagation equation above reduces to the lossless bit per channel use) of a general noncentral chi channel with generalised scalar NSE [6], [9] an arbitrary (even) number of degrees of freedom. To the best β 2 2 õE − Eττ + γ |E| E = N(, τ). (2) of our knowledge, this has not been previously reported in the 2 literature. Similar models, however, do appear in the study of In this paper we consider the case of anomalous dispersion noise-driven coupled nonlinear oscillators [58]. (β2 < 0), i.e., the focusing case. In this case, both the ME The first contribution of this paper is to numerically obtain in (1) and the NSE in (2) permit bright soliton solutions lower bounds for the channel capacity for three continuous (“particle-like waves”), which will be discussed in more detail input distributions, as well as for amplitude shift-keying (ASK) in Sec. II-B. constellations with discrete number of constellation points. It is customary to re-scale (1) to dimensionless units. For all the continuous inputs, the lower bounds are shown We shall use the following normalisation: The power will to be nondecreasing functions of the SNR under an average be measured in units of P0 = 1 mW since it is a typical power constraint. The second contribution of this paper is to power level used in optical communications. The√ normalised provide an asymptotic closed-form expression for the MI of (dimensionless) field then becomes q = E/ P0.Forthe the noncentral chi-channel with an arbitrary (even) number of distance and time, we define the dimensionless variables z degrees of freedom. This asymptotic expression shows that the and t as z = /0 and t = τ/τ0,where MI grows unbounded and at the same rate, independently of   |β | the number of degrees of freedom.  = (γ ∗ )−1,τ=  |β |= 2 . 0 P0 0 0 2 ∗ (3) γ P0 II. CONTINUOUS-TIME CHANNEL MODEL For the scalar case (2), we use the same normalisation but we A. The Propagation Equations replace γ ∗ by γ . Then, the resulting ME reads The propagation of light in optical fibres in the presence of 1 õ q + q +q, q¯ q = n(z, t), (4) amplified spontaneous emission (ASE) noise can be described z 2 tt by a stochastic partial differential equation which captures while the NSE becomes the effects of chromatic dispersion, nonlinear polarisation 1 2 mode dispersion, optical Kerr effect, and the generation of õqz + qtt + |q| q = n(z, t). (5) 2 ASE noise from the optical amplification process. Throughout ( , ) =[ ( , ), ( , )] this paper we assume that the fibre loss is continuously The ASE noise n z t n1 z t n2 z t in (4) is a (, τ) compensated along the fibre by means of (ideal) distributed normalised version of N , and is assumed to have the following correlation properties Raman amplification (DRA) [59], [60]. In this work we   consider the propagation of a slowly varying 2-component E [ni (z, t)] = E ni (z, t) n j (z , t ) = 0, 2       envelope E(, τ) =[E1(, τ), E2(, τ)]∈C over a nonlinear E ni (z, t) n¯ j (z , t ) = D δij δ z − z δ t − t , (6) birefringent optical fibre, where τ and  represent time and propagation distance, respectively. Our model also includes the with i, j ∈{1, 2}, with δij being a Kronecker symbol, E [·] δ (·) 2-component ASE noise N(, τ) =[N1(, τ), N2(, τ)] due is the mathematical expectation operator, and is the to the DRA. We also assume a uniform change of polarised Dirac delta function. The correlation properties (6) mean that state on the Poincaré sphere [61]. each noise component ni (z, t) is assumed to be a zero-mean, The resulting lossless ME is then given by [7, eq. (1.26)], independent, white circular Gaussian noise. The scalar case [8, Sec. 10.3.1], [57], [62]5 follows by considering a single noise component only. β γ The noise intensity D in (6) is (in dimensionless units) 2 8 ¯ õ E − Eττ + E, E E = N(, τ), (1)  σ 2 2 9 = σ 2 0 =  0 , D 0 (7) where the retarded time τ is measured in the reference P0τ0 γ ∗ |β | 3 2 P0 frame moving with the optical pulse average group veloc- σ 2 ity, E ≡ E(, τ) represents the slowly varying 2-component where 0 is the spectral density of the noise, with real world / ( · ) σ 2 envelope of electric field, β2 is the group velocity dispersion units [W km Hz ]. For ideal DRA, this 0 can be expressed coefficient characterising the chromatic dispersion, and γ is through the optical fibre and transmission system parameters σ 2 = α · ν α the fibre nonlinearity coefficient. The pre-factor 8/9in(1) as follows: 0 fibre KT h 0,where fibre is the fibre comes from the averaging of the fast polarisation rotation attenuation coefficient, hν0 is the average photon energy, KT [8, Sec. 10.3.1], [61]. For simplicity we will further work is a temperature-dependent phonon occupancy factor [6]. with the effective averaged nonlinear coefficient γ ∗  8γ/9 From now on, all the quantities in this paper are in normalised units unless specified otherwise. Furthermore, 5Throughout this paper, vectors are denoted by boldface symbols x = we define the continuous time channel as the one defined by [x1, x2, x3, ...], while scalars are denoted by nonboldface symbols. The scalar product is denoted by · , ·, and over-bar denotes complex conjugation. The the normalised ME and the NSE. This is shown schematically 2 2 2 Euclidean norm is denoted by x  |x1| + |x2| + .... The partial in the inner part of Fig. 1, where the transmitted and received derivatives in the partial differential equations are expressed as subscripts, e.g., waveforms are x(t) ≡ q(0, t) and y(t) ≡ q(Z, t), respectively, ∂ ∂2 √  E ττ  E  − E ∂ , E ∂τ2 , etc. The imaginary unit is denoted by õ 1. where Z is the propagation distance. SHEVCHENKO et al.: CAPACITY LOWER BOUNDS OF THE NONCENTRAL CHI-CHANNEL 2981

Fig. 1. System model under consideration. The symbols X =[X1, X2, X3,...] are converted to amplitudes, and then mapped to a waveform x(t). The noisy received waveform y(t) is obtained by propagating x(t) in (4). The forward NFT processes the waveform y(t) symbol-by-symbol, and gives a soft estimate of the transmitted symbols Y =[Y1, Y2, Y3,...].

B. Fundamental Soliton Solutions As shown by (10) and (12), the solitons in (8) and (11) only It is known that the noiseless (n(z, t) = 0) ME (4) pos- acquire a phase rotation after propagation. When the noise is sesses a special class of solutions, the so-called fundamental not zero, however, these solutions will change. This will be bright solitons.6 In general, the Manakov fundamental soliton discussed in detail in the following section. is fully characterised by 6 parameters [57] (4 in the NSE case): frequency (also having the meaning of velocity in III. DISCRETE-TIME CHANNEL MODEL some physical applications), phase, phase mismatch, centre- A. Amplitude-Modulated Solitons: One and of-mass position, polarisation angle, and amplitude (the latter Two Polarisations is inversely proportional to the width of the soliton). In this We consider a continuous-time input signal x(t) = paper we consider amplitude-modulated solitons, and thus, [x1(t), x2(t)] of the form no information is carried by the other 5 parameters. The initial ∞ values of these 5 parameters can therefore be set to arbitrary x(t) = sk(t), (13) values. In this paper, all of them have been set to zero. For k=1 the initial frequency, this can be further motivated to avoid where s (t) =[s , (t), s , (t)] and k is the discrete-time deterministic pulse walk-offs. As for the initial phase, phase k k 1 k 2 s (t) mismatch, and centre-of-mass position, as we shall see in the index. Motivated by the results in Sec. II-B, the pulses k are chosen to be next section, their initial values do not affect the marginal amplitude channel law. Under these assumptions, the soliton sk(t) = [cos β0, sin β0] Ak sech [Ak(t − kTs)] , (14) solution at z = 0 is given by [57], [62] where Ts is the symbol period. In principle, it is also possible q(0, t) =[q1(0, t), q2(0, t)]=[cos β0, sin β0] A sech(At), to encode information by changing the polarisation angle (8) β0 from slot to slot. However, in this paper, we fix its value to be the same for all the time slots corresponding <β <π/ where A is the soliton amplitude and 0 0 2isthe to a fixed (generally elliptic) degree of polarisation. Thus, β polarisation angle. The value of 0 can be used to control how the transmitted waveform corresponds to soliton amplitude the signal power is split across the two polarisations. modulation, which is schematically shown in Fig. 2 for the β For any 0, the Manakov soliton solution after propagation scalar (NSE) case. over a distance Z with the initial condition given by (8), At the transmitter, we assume that symbols Xk are mapped is expressed as to soliton amplitudes A via A = X 2. This normalisation is k k k õA2 Z introduced only to simplify the analytical derivations in this q(Z, t) = [cos β , sin β ] A sech(At) exp (9) 0 0 2 paper. To avoid soliton-to-soliton interactions, we also assume that the separation T is large, i.e., exp(−A T ) 1, ∀k. õA2 Z s k s = q(0, t) exp . (10) The receiver in Fig. 1 is assumed to process the received 2 waveform during a window of Ts via the forward NFT [22], The soliton solution for the NSE in (5) can be obtained by [32] and returns the amplitude of the received soliton, which β = 7 = 2 using 0 0 in (8)Ð(10), which gives we denoted by Rk Yk . Before proceeding further, it is important to discuss the role q(0, t) = A sech (At) , (11) of the amplitudes Ak on a potential enhancement of soliton- and soliton interactions. The interaction force prefactor is known

õA2 Z to scale as the amplitude cubed [8, Ch. 9.2], [9, Ch. 5.4]. How- q(Z, t) = A sech (At) exp ever, the interaction also decays exponentially as exp(−Ak Ts). 2 This exponential decay dominates the interaction, and thus, õA2 Z = q(0, t) exp . (12) considering very large amplitudes (or equivalently, very large 2 powers, as we will do later in the paper), is in principle not a problem. At extremely large amplitudes, however, the model 6 Fundamental solitons are “bright” only for the focusing case we consider used in this paper is invalid for different reasons: higher order in this paper, i.e., for anomalous dispersion. 7This corresponds to the case where all the signal power is transmitted in nonlinearities should be taken into account. This includes the first polarisation. stimulated Brilloin scattering (for very large powers) or Raman 2982 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 66, NO. 7, JULY 2018

Fig. 2. Schematic visualisation of the amplitude modulation of soliton sequence (scalar NSE case). scattering (for very short pulses). Studying these effects is, is beyond the scope of this paper. In this context, the channel however, out of the scope of this paper. capacity lower bounds in Sec. IV, should be considered as a We would also like to emphasise that for a fixed pulse first step towards more involved analyses. separation Ts, the channel model we consider in this paper The conditional probability density function (PDF) for the is not applicable for low soliton amplitudes. This is due to received soliton amplitude R given the transmitted amplitude two reasons. The first one is that for low amplitude solitons, A was obtained in [57, eq. (15)] using standard perturbative the perturbation theory used to derive the channel law becomes approach and the Fokker-Planck equation method. The result inapplicable as the signal becomes of the same order as noise. can be expressed as a noncentral chi-squared distribution Secondly, low amplitude solitons are also very broad, and √ 1 r a + r 2 ar thus, nonnegligible soliton interactions are generated. These p | (r|a) = exp − I , (15) R A σ 2 σ 2 2 σ 2 two cases can be overcome if the soliton amplitudes are N a N N always forced to be larger than certain cutoff amplitude aˆ, where which we will now estimate. For the first case (noise-limited), Z the threshold aˆ is proportional to σ 2 . In the second σ 2 = D · (16) noise N N 2 case (interaction-limited), the threshold is proportional to the ˆ ∝ −1 is the normalised variance of accumulated ASE noise, and symbol rate, i.e., ainter Ts . This shows that for fixed system I (·) is the modified Bessel function of the first kind of order parameters, the threshold aˆ = max{ˆanoise, aˆinter} is a constant. 2 The implications of this will be discussed at the end of Sec. IV. two. The expression in (15) is a noncentral chi-squared distri- Having defined the transmitter and receiver, we can now bution with six degrees of freedom (see, e.g., [65, eq. (29.4)]) define a discrete-time channel model, which encompasses providing non-Gaussian statistics for Manakov√ soliton ampli- = the transmitter, the optical fibre, and the receiver, as shown tudes.√ By making the change of variables Y R, and using = in Fig. 1. Due to the assumption on solitons well-separated X A, the PDF in (15) can be expressed as in time, we model the channel as memoryless, and thus, 2 y3 x2 + y2 2xy from now on we drop the time index k. This memoryless pY |X (y|x) = exp − I2 , (17) σ 2 x2 σ 2 σ 2 assumption is supported by additional numerical simulations N N N we performed, which are included in Appendix A. Never- which corresponds to the noncentral chi-distribution with six theless, at this point it is important to consider the impli- degrees of freedom. An extra factor 2y before the exponential cations of a potential mismatch between the memoryless function comes from the Jacobian. assumption of the model and the true channel in the context For the NSE, it is possible to show that the channel law of channel capacity lower bounds. In particular, if in some becomes [49], [51], [52] regimes (e.g., low power or large transmission distances) the 2 y2 x2 + y2 2xy memoryless assumption would not hold, considering a mem- p | (y|x) = exp − I , (18) Y X σ 2 x σ 2 1 σ 2 oryless channel model would result in approximated lower N N N bounds on the channel capacity. Provable lower bounds can be which corresponds to a noncentral chi-distribution with four obtained by using mismatched decoding theory [63] (as done degrees of freedom. in [64, Sec. III-A and III-B]) or by considering an average We note that although in this paper we only consider an memoryless channel (as done in [6, Sec. III-F]). Although amplitude modulation Ak (or in the NFT terms the imaginary both approaches can in principle be used in the context of part of each discrete eigenvalue), it is possible to include amplitude-modulated solitons, they both rely on having access other discrete degrees of freedom corresponding to various to samples from the true channel, and not from a (poten- soliton parameters in (14) in order to improve the achievable tially memoryless) model. Such samples can only be obtained information rates. This is, however, beyond the scope of through numerical simulations or an optical experiment, which this paper. Furthermore, the channel models presented in this SHEVCHENKO et al.: CAPACITY LOWER BOUNDS OF THE NONCENTRAL CHI-CHANNEL 2983 section were obtained via a perturbative treatment, and thus, in the context of soliton/eigenvalue communications they are technically valid only at high SNR.8 Despite that, in the current paper we will also study capacity lower bounds of a general noncentral chi-channel with an arbitrary number of degrees of freedom at any range of SNR. While admittedly the low-SNR region is currently only of interest when n = 1 (noncoherent phase channel) we believe its generalisation for n > 1 can still be of interest for the new generation of nonlinear optical regeneration systems.

B. Generalised Discrete-Time Channel Model The results in the previous section show that both scalar and vector soliton channels can be modelled using the same class of the noncentral chi-distribution with an even number of degrees of freedom 2n, with n = 2, 3. The simplest channel of this type corresponds to n = 1, which describes a fibre optical communication channel with zero-dispersion [13] as well as the noncoherent phase channel studied in [66] (see also [67]). Fig. 3. Generalised discrete-time channel model: noncentral chi-channel with 2n degrees of freedom. Motivated by this, here we consider a general communication channel described by the noncentral chi-distribution with an arbitrary (even) degrees of freedom 2n. Although we are As previously explained, the inter-symbol interference due currently not aware of any physically-relevant communication to pulse interaction can be neglected due to the large enough system that can be modelled with n ≥ 4, we present results soliton separation assumed, and thus, the channel can be for arbitrary n to provide an exhaustive treatment for channels treated as a memoryless (see Appendix A for more details). of this type. The channel capacity, in bits per channel use, is then given The channel in question is therefore modelled via the PDF by [68], [69] corresponding to noncentral chi-distribution

n 2 + 2 C(ρ)  max IX,Y (ρ), (21) 2 y x y 2xy ( ): E 2 ≤σ 2 p | (y|x) = exp − I − , pX x [X ] S Y X σ 2 xn−1 σ 2 n 1 σ 2 N N N where (19) pX,Y (X, Y ) with n ∈ N and where N  {1, 2, 3,...}. This channel law I , (ρ)  E X Y log2 ( ) · ( ) (22) corresponds to the following input-output relation pX X pY Y = (ρ) − (ρ), hY hY |X (23) 1 2n X 2 2 = √ + , (ρ)  − E[ ( )] (ρ)  Y Ni (20) and where hY log2 pY Y and hY |X 2 = n − E[ ( | )] i 1 log2 pY |X Y X are the output and conditional differ- where {N }2n is a set of independent and identically dis- ential entropies, respectively. The optimisation in (21) is i i=1 ( ) tributed Gaussian random variables with zero mean and vari- performed over all possible statistical distributions pX x σ 2 that satisfy the power constraint. In our case this constraint ance N . The above input-output relationship is schematically shown in Fig. 3, which particularises to (17) and (18), for corresponds to a fixed second moment of the input symbol n = 3andn = 2, respectively. distribution or, equivalently, to a fixed average signal power in a given symbol period. IV. MAIN RESULTS The exact solution for the power-constrained optimisation problem (21) with the channel law (19) is unknown. For the In this section, we study capacity lower bounds of the noncentral chi-distribution with 2 degrees of freedom (i.e., channel in (19). We will show results as a function of the to the noncoherent additive noise channel), it was shown [66] ρ  σ 2/σ 2 σ 2 effective SNR defined as S N ,where S is the second that the capacity-achieving distribution is discrete with an p σ 2 moment of the input distribution X and N is given by (16). infinite number of mass points. To the best of our knowledge, σ 2 The value of S also corresponds to the average soliton that proof has not been extended to higher number of degrees σ 2 = E 2 = E amplitude, i.e., S X [A]. It can be shown that of freedom, however, we expect that will be the case for (19) for given system parameters, the noise power (in real world too. σ 2 units) is constant and proportional to N , and the signal power In this paper, we do not aim at finding the capacity- σ 2 ρ (in real world units) is proportional to S . The parameter achieving distribution, but instead, we study lower bounds on therefore indeed corresponds to an effective SNR. the capacity. We do this because the capacity problem is in 8More precisely, when the total soliton energy in the time slot is much general very difficult, but also because of the relevance of greater than that of the ASE noise. having nondecreasing lower bounds on the capacity for the 2984 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 66, NO. 7, JULY 2018 optical community. To obtain a lower bound on the capacity, we will simply choose an input distribution pX (x) (as done in, e.g., [5], [49]). Without claiming the generality, we, however, consider four important candidates for the input distribution. First, following [49], we use symbols drawn from a Rayleigh distribution

2x x2 p (x) = exp − , x ∈[0, ∞). (24) X σ 2 σ 2 S S As we will see later, this input distribution is not the one giving the highest lower bound. However, it has one important advantage: it allows some analytical results for the mutual information. The other three distributions are considered later in this section as numerical examples. The next two Lemmas provide an exact closed-form expression for the conditional differential entropy hY |X (ρ) and an asymptotic expression for the output differential entropy hY (ρ). Lemma 1: For the channel in (19) and the input distribu- tion (24) (ρ)   Fig. 4. The MI IX,Y in (23) (numerically calculated) for the chi- n distribution with different degrees of freedom and the channel model (19). (ρ) = ρ + − ψ( ) − hY |X 2 n n log2 e 1 The asymptotic estimate given by Theorem 1 is also shown. Lower and 2 upper bounds for n = 1 are also shown. n −   + 1 ρ + ψ( ) log2 1 log2 e 2 n log e ρ ρ − 2  , 1, n Proof: We expand the function Fn(ρ) in (27) defining 2 ρ + 1 ρ + 1 the conditional entropy in Lemma 1. At fixed large ρ the ( − )/ ρ + 1 n 1 2 integrand asymptotically decays as exp (−ξ/2ρ), i.e., with −ρ−1 F (ρ) e, ρ n log2 (25) small decrement (which can be proven by a standard large argument asymptotes of the Bessel functions). This means ψ( )  ( )/ where x d log x dx is the digamma function and that the main contribution to the integral comes from the (α, , ) 1 n is the special case of the Lerch transcendent func- asymptotic region 1  ξ  ρ in most part of which the tion [70, eq. (9.551)] large argument expansion of both Bessel functions is indeed n−2 + − justified. Using it uniformly we obtain log(1 − α) αk 1 n (α, 1, n)  − − . (26) αn + ρ = k 1 2 1 k 0 Fn(ρ) = 2ρ + log + 1 − log 4π − ψ(1) + O [1] , 2 ρ The function Fn(ρ) is defined as ∞ which used in (25) gives the asymptotic expression    −   F (ρ)  ξ K − ( 1 + ρ 1 ξ) I − (ξ) log I − (ξ) dξ, n n 1 n 1 n 1 (ρ) = 1 π + ρ−1 ,ρ→∞. hY |X log2 e O (30) 0 2 (27) The proof is completed by combining (30) and (28) and Kn(x) is the modified Bessel function of the second kind with (23). of order n. The result in Theorem 1 is a universal and n-independent Proof: See Appendix B. expression. The expression in (29) shows that the capacity Lemma 2: For the channel in (19) and the input distribu- lower bound is asymptotically equivalent to half of logarithm tion (24) of SNR plus a constant which is order-independent. Fig. 4 (ρ) = , , , ψ( )   shows the numerical evaluation of IX,Y for n 1 2 3 12 (ρ) = 1 ρ + − 1 − + ρ−1 , hY log2 1 log2 e 1 O obtained by numerically evaluating all the integrals in the exact 2 2 expressions for the conditional and output entropies in (25) ρ →∞ (28) and (53), as well as the asymptotic expression in Theorem 1. Proof: See Appendix C. Interestingly, we can see that even in the medium-SNR region, The next theorem is one of the main results of this paper. the influence of the number of degrees of freedom on the MI Theorem 1: The MI for the channel in (19) and the input is minimal, and the curves are quite close to each other. In this = distribution (24) admits the following asymptotic expansion figure, we also include the lower and upper bounds for n 1   given by [67, eq. (21)] and [66, eq. (41)], resp. These results 1−ψ(1)   1 e −1 show that the asymptotic results in Theorem 1 correctly follow IX,Y (ρ) = log ρ + O ρ ,ρ→∞. (29) 2 2 4π these two bounds. SHEVCHENKO et al.: CAPACITY LOWER BOUNDS OF THE NONCENTRAL CHI-CHANNEL 2985

Fig. 5. MI estimates (by numerically evaluating (23) via Monte-Carlo Fig. 6. MI estimates (numerically calculated) for equally-spaced M-ASK integration) for different trial continuous input distributions and different constellations with M ={2, 4, 8, 16} constellation points. values of n (different line types). Different distributions are shown with different colours. bits per symbol. The MI (23) in this case can be evaluated as ∞ The main reason for considering a Rayleigh input distrib- (ρ) = 1 ( | ) ution was that it yields a semi-analytical lower bound on the IX,Y pY |X y x M ∈X the capacity. In the following example, we consider three other x 0 p | (y|x) input distributions and numerically calculate the resulting MI. · log  Y X dy, (34) 2 1 ( | ) Example 1: Consider the geometric (exponential), half- M x ∈X pY |X y x Gaussian, and Maxwell-Boltzmann distributions given by where we assumed the symbols are equally likely. √  √  Example 2: Consider ASK constellations X ={0, 1, 2 2 x ...,M − 1} with m = 1, 2, 3, 4 and second moment σ 2, pX (x) = exp − , x ∈[0, ∞), (31) S σS σS which correspond to OOK, 4-ASK, 8-ASK, and 16-ASK, respectively. The MI numerically evaluated for these constel- lations is shown in Fig. 6 for chi-channel with n = 1, 2, 3. √ As a reference, in this figure we also show (black lines) 2 x2 p (x) = √ exp − , x ∈[0, ∞), (32) the MI for the (continuous) half-Gaussian input distribution. X πσ σ 2 S 2 S The results in this figure show that in the low SNR regime, the use of binary modulation is in fact better than the half- and Gaussian distribution. This can, however, be remedied by √ using a geometric distribution, which, as shown in Fig. 5, 3 6 x2 3 x2 p (x) = √ exp − , x ∈[0, ∞), (33) outperforms the half-Gaussian distribution in the low SNR X πσ3 σ 2 S 2 S regime. In the high SNR regime, however, this is not the case. Finally, let us address the impact of the cutoff aˆ we respectively. The MIs for these three distributions for introduced in Sec. III. All our results for continuous input n = 1, 2, 3 are shown in Fig. 5 and show that the lower bound distributions have been obtained for the input distributions given by the geometric input distribution in (31) displays that are not bounded away from zero (see (24), (31)Ð(33)). high MI in the low SNR regime (ρ<10 dB), whereas the Therefore,√ symbols Xk are generated below the threshold half-Gaussian input distribution in (32) is better for medium xˆ = aˆ, where the channel law considered in this paper does and large SNR. On the other hand, the Maxwell-Boltzmann not hold. We shall now only consider here the case of the distribution in (33) gives the lowest MI for all SNR. Numerical Rayleigh input (24) as this distribution was used to obtain the results also indicate that all the presented MIs asymptotically main result of this section. We will prove that in the high- exhibit an equivalent growth irrespective of the number of the power (i.e., high SNR) regime, the effect of the cutoff on the degrees of freedom 2n. achievable data rate tends to zero. To do so, we note that The following example considers the use of discrete con- for fixed fibre parameters and propagation distance, the cutoff ˆ2 =ˆ= {ˆ , ˆ } σ 2 = ρσ2 stellations. In particular, we assume that the soliton amplitudes x a max anoise ainter is also fixed, while S N m take values on a set X  {x1,...,xM },whereM  |X | = 2 grows linearly with SNR. In other words, one can achieve is the cardinality of the constellation, and m is a number of high SNR at the expense of high power solitons for fixed 2986 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 66, NO. 7, JULY 2018 noise variance. One possible way of showing that the effect TABLE I of the cutoff on the achievable rate is zero as SNR tends to SIMULATION SYSTEM PARAMETER infinity is to consider a transmitter which generates a dummy symbol every time Xk ≤ˆx. The value of the threshold xˆ is message-independent and thus, can be assumed to be known to the receiver which will discard sub-threshold symbols. This allows us to keep the main results of the paper at the expense of a data rate loss (since part of the time, dummy symbols are transmitted). The probability of such “outage” event η is given by an the integral of the input distribution from zero to the threshold. For the Raylei gh input PDF (24) this probability η = − −ˆ/σ 2 is given by 1 exp a S (see (64)Ð(67)). Therefore asymptotically η(ρ) ≈ˆa/(ρσ 2 ) → 0whenρ →∞.The N for further investigation. Furthermore, another interesting open average rate loss is then given by 1 − η(ρ), which tends to research problem is the derivation of capacity upper bounds zero as ρ →∞. for amplitude-modulated soliton systems. This is also left for An alternative and more rigorous solution to the problem further investigation. above is to consider directly the difference between the MI asymptote obtained in the current paper (i.e., Theorem 3) APPENDIX A and that obtained by a truncated input Rayleigh distribution MEMORYLESS PROPERTY OF THE DISCRETE-TIME which simply does not generate sub-threshold symbols. This CHANNEL MODEL difference can be shown to tend to zero as ρ →∞. This proof is given in Appendix D. In this section, we present numerical simulations to verify the memoryless assumption for the discrete channel model in Sec. III. To this end, we simulated the propagation V. C ONCLUSIONS of sequences of N = 10 soliton symbols through the A non-Gaussian channel model for the conditional PDF of scalar waveform channel given by (5). Two launch powers − . . well-separated (in time) soliton amplitudes was used to study ( 1 5and145 dBm) and two propagation distances (500 km lower bounds on the channel capacity. Results for propagation and 2000 km) are considered. The simulations were carried of signals over a nonlinear optical fibre using one and two out via the standard split-step Fourier method. The soliton polarisations were presented. The results in this paper demon- amplitudes were generated as i.i.d. samples from a Rayleigh strated both analytically and numerically that there exist lower input distribution (see (24)) and the variance of X was chosen bounds on the channel capacity that display an unbounded to be 1.25 and 20, so that the resulting soliton waveforms have − . . growth with the effective SNR, similarly to the linear Gaussian powers of 1 5and145 dBm, respectively. The transmitted (τ) channel. All the results in this paper are given in bit per waveform x was created using (13) at a symbol rate of . channel use only, and thus, they should be considered as a first 1 7 GBd. To guarantee an accurate simulation, the time- . step towards analysing the more practically relevant problem domain samples were taken every 4 6 ps and the step size . of channel capacity in bit per second per unit bandwidth. This was 0 1 km. White Gaussian noise was added at each step to is a considerably more challenging problem, which is left for model the ideal DRA process. The simulation parameters are further investigation. similar to those used in [44] and are summarised in Table I. Apart from the ME soliton channel model this paper also Fig. 7 shows the waveforms before and after propagation studied lower bounds on the capacity of an abstract general through the channel given in (5). As expected, the received noncentral chi-channel with an arbitrary number of degrees signal is a noisy version of the transmitted waveform, where of freedom. Similar channel models appear in the study of the noise increases as the propagation distance increases. relatively general systems of noise-driven coupled nonlinear These results show that doubling the transmission distance oscillators [58]. Therefore, we believe that the results for and/or (approximately) doubling the launch power has very large number of degrees of freedom might also some day find little effect in the soliton shapes. applications in nonlinear communication channels. The noisy waveforms shown in Fig. 7 were then used  , ,..., The results obtained in this paper for the general noncentral to obtain soliton amplitudes Y [Y1 Y2 Y10] via the chi-channel are true capacity lower bounds for that channel forward NFT. Each amplitude is obtained by processing the model. For the case of the application considered in this paper corresponding symbol period via the spectral matrix method (amplitude-modulated soliton systems), however, the pre- [28, Sec. IV-B]. To test the memoryless assumption, we per- sented analysis was based on a perturbative-based model form a simple correlation test. In particular, we consider the which holds at high SNR. This model also does not consider normalised output symbol correlation matrix, whose entries potential interaction between solitons, and thus, the results are defined as E ( − E )( − E ) in this paper are limited to solitons well separated in time. [ Yk [Yk] Yk [Yk ] ] ckk  . (35) Another way of interpreting these results is that the obtained E [Yk] E [Yk ] expressions are approximated lower bounds on the capacity of The obtained correlation matrices are shown in Fig. 8, where the true channel. Bounds that consider memory effects are left statistics were gathered by performing 103 Monte-Carlo runs SHEVCHENKO et al.: CAPACITY LOWER BOUNDS OF THE NONCENTRAL CHI-CHANNEL 2987

Fig. 7. Continuous-time input x(τ) and output y(τ) soliton waveforms for 10 solitons and distributed noise due to DRA. Two launch powers are considered: (a) −1.5dBmand(b)1.45 dBm. The solitons are propagated 500 and 2000 km. of the signal propagation. As we can see from Fig. 8, the matri- where (37) follows from (19). In what follows, we will ces are almost diagonal. Since our communication channel compute the 5 expectations in (37). is believed to be non-Gaussian, the absence of correlation The third and fourth terms in (37) can be readily obtained does not of course necessarily imply the memoryless property using (24) (understood here as the statistical independence). However,   1 it does constitute an important quantification of the qualitative E log X = (log ρ + ψ(1)) , (38) criterion exp(−Ak Ts ) 1 as given in Sec. III-A.   2 E X 2 = ρ. (39) APPENDIX B PROOF OF LEMMA 1 To compute the second and fifth terms in (37), we first The MI is invariant under a simultaneous linear re-scaling calculate the output distribution as → /σ → /σ of the variables x x N and y y N . For notation ∞ simplicity, and without loss of generality, throughout this σ 2 = pY (y) = pX,Y (x, y) dx (40) proof we thus assume N 1. Furthermore, we study the ρ = σ 2 conditional entropy as a function of S and all the results 0   − 2 n 2 2 k will be given in nats. 2y − y −α 2 (αy ) = e ρ+1 1 − e y , (41) We express the conditional differential entropy as ραn−2 k! ∞∞ k=0 (ρ) =− ( , ) ( | ) hY |X pX,Y x y log pY |X y x dydx (36) where the joint distribution pX,Y (x, y) can be expressed using 0 0     (19) and (24) as =−log 2 − n E log Y + (n − 1) E log X       4 yn x2 + αy2 +E 2 + E 2 − E ( ) , pX,Y (x, y) = exp − In− (2xy), (42) X Y log In−1 2XY (37) ρ xn−2 α 1 2988 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 66, NO. 7, JULY 2018

Fig. 8. Normalised output symbol correlation matrices for the two launch powers and propagation distances in Fig. 7. with where ρ ∞∞ α  < ,   1 (43) (6) (ρ)  ( , ) ( ) . ρ + 1 hY |X pX,Y x y log In−1 2xy dx dy (47) and where (41) can be obtained using a symbolic integration 0 0 software. Using (41), we obtain (using a symbolic integration (6) (ρ) The last step is to compute the term hY |X ,which software) using (42) can be expressed as   ∞∞ 1 n 2 2 E log Y = (α(α, 1, n) + ψ(n)) , (44) ( ) 4 y x + αy 2 h 6 (ρ) = exp − Y |X ρ xn−2 α ψ( ) (α, , ) where n is the digamma function, 1 n is given 0 0   by (26). The second moment of the output distribution is ·In−1 (2xy) log In−1(2xy) dx dy. (48) obtained directly from the channel input-output relation (20), We then make the change of variables ξ = 2xy, η = y2, with yielding −   the Jacobian ∂(x, y)/∂(ξ, η) = (4y2) 1, yielding E 2 = ρ + . ∞ Y n (45) n−2   (6) 2 h | (ρ) = In−1(ξ) log In−1(ξ) Substituting (38), (39), (44) and (45) into (37), we have Y X ρ 0 − ∞ n n n 1  n−2 hY |X (ρ) =−log 2 − α(α, 1, n) − ψ(n) + η ξ 2 2 2 2 · exp − − η dη dξ. (49) · ( ρ + ψ( )) + ρ + − (6) (ρ), ξ 4ηα log 1 2 n hY |X (46) 0 SHEVCHENKO et al.: CAPACITY LOWER BOUNDS OF THE NONCENTRAL CHI-CHANNEL 2989

η (4)(ρ) The integration over can be performed analytically, yielding Next, one notices that hY is positive and can be upper- ∞ bounded as follows  − η n 2 ξ 2 ∞ − − η η  exp d ( ) 1 ξ 4ηα 4 (ρ) ≤ (− ( ) ( )) hY − f z log f z dz (60) 0 ραn 1 ξ 0 2−n (1−n)/2 = 2 α ξ Kn−1 , (50) C α1/2  . (61) ραn−1 where K (x) is the modified Bessel function of the second n It is therefore only left to prove that the integral converges, kind of order n. Using (50) in (49) gives i.e., that the constant C is finite. This can be done as follows: ∞ (1−n)/2 ∞ (6) α ξ h (ρ) = ξ K − Y |X ρ n 1 α1/2 C = (− f (z) log f (z)) dz 0   0 ·In−1(ξ) log In−1(ξ) dξ (51) ∞ ( − )/ α 1 n 2 ≤ (1 − f (z)) dz = F (ρ). ρ n (52) 0 ∞ − The proof is completed by using (52) in (46), the definition n 2 zk = −z of α in (43), and by returning to logarithm base 2. e ! dz = k 0 k 0 = n − 1 APPENDIX C < ∞, PROOF FOR LEMMA 2 where in the second line we have used an inequality −x ln x ≤ From (41), it follows that the output entropy can then be ( ) 9 ( − ) ∈ ( , ] 4 (ρ) expressed as 1 x , x 0 1 . Therefore, asymptotically hY decays not slower than 1/ρ. ραn−2     1 2 (4) The asymptotic expression for the output entropy can be hY (ρ)=log −E log Y + E Y + h (ρ), 2 ρ + 1 Y written by combining (60), (44), (45) and (53), which yields (53) ψ( )   1 1 −1 hY (ρ) = log ρ + 1 − − log 2 + O ρ . (62) where α is given by (43), 2 2 ∞ ∞ The proof is completed by returning to logarithm base 2. (4)(ρ)  ( ) ( | ) (4)( ) hY pX x pY |X y x gY y dydx (54) APPENDIX D 0 0 ∞ PROOF OF THE ASYMPTOTICALLY VANISHING RATE LOSS = ( ) (4)( ) , pY y gY y dy (55) Here we shall prove that an input distribution bounded (truncated) away from zero gives the same results as 0 Theorem 1 in the limit of large average power σS →∞. where pY (y) is given by (41) and To this end, consider a system where the transmitted ampli- tudes X are drawn from a Rayleigh distribution with PDF (4)( )  − (α 2) gY y log f y (56) given in (24). Let us now introduce a threshold xˆ of amplitudes n−2 k − (z) realisations below which our channel law model is expected f (z)  1 − e z . (57) k! to be inapplicable. Let us now introduce an alternative system k=0 where the symbols X˜ are drawn from a “truncated” Rayleigh Notice that from its definition it follows that the function f (z) distribution with PDF is confined to the interval 0 ≤ f (z) ≤ 1. We shall now ( )   1 4 (ρ) ρ−1 ρ →∞ p ˜ (x) = pX (x) H (x −ˆx), x ∈[ˆx, ∞), (63) prove that hY decays as O or faster when . X 1 − η Indeed, one has where H (x −ˆx) is the Heaviside step function, and η is defined ∞ 2 as (4) 2y − y h (ρ) =− e ρ+1 f (αy2) log f (αy2) dy (58)   Y ραn−2 η  P X < xˆ . (64) 0 ∞ 1 − /ρ This probability can be expressed as =− e z f (z) f (z) dz. n−1 log (59) ρα xˆ 0 η = pX (x) dx (65) 9 Similarly to Appendix B, the results in this proof are in nats. 0 2990 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 66, NO. 7, JULY 2018

xˆ 2 x2 Let us now turn to the first conditional differential entropy = x exp − dx (66) σ 2 σ 2 in (70), for which we have S S 0 ∞∞ 2 xˆ  − ( | ) ( ) ( | ) = 1 − exp − . (67) h ˜ | ˜ pY |X y x p ˜ x log pY |X y x dxdy (81) σ 2 Y X X S 0 0 ∞ As discussed in Sec. III-A and Sec. IV, the threshold xˆ is a η = = p ˜ (x) g(x) dx, (82) constant, and thus, limσS→∞ 0. X To prove that the rate loss tends to zero, we shall prove that 0   − = where lim IX,Y IX˜ ,Y˜ 0 (68) σs →∞ ∞ or equivalently, g(x)  − pY |X (y|x) log pY |X (y|x) dy (83)   0 lim h − h ˜ = 0 (69) σ →∞ Y Y s represents the conditional differential entropy of pY |X (y|x), ( | ) and and pY |X y x is given by the noncentral chi-distribution (19).   Using (63), the conditional differential entropy hY˜|X˜ can be − = . lim hY˜|X˜ hY |X 0 (70) expressed as σs →∞ ∞ To prove (69), we have the following: = 1 ( ) ( ) , hY˜ |X˜ pX x g x dx (84) ∞ 1 − η xˆ ( ) = ( | ) ( ) pY˜ y pY |X y x pX˜ x dx (71) xˆ hY |X 1 0 = − g(x) pX (x) dx. (85) ∞ 1 − η 1 − η 1 0 = p | (y|x) p (x) dx − η Y X X (72) 1 The first term on the r.h.s. of (85) tends to the conditional xˆ ∞ entropy of the untruncated distribution. We shall now prove 1 that the last (integral) term in (85) tends to zero when ≤ pY |X (y|x) pX (x) dx (73) 1 − η σS →∞. We note that according to (24) the input distri- 0 bution pX (x) tends to zero uniformly in the interval [0, xˆ] 1 as σ →∞. Then, according to the bounded convergence = pY (y). (74) S 1 − η theorem, in order to prove that integral term in (85) is asymp- The Kullback-Leibler divergence (relative entropy) between totically vanishing, it is sufficient to prove that the function ( ) ( ) g(x) remains bounded within the interval [0, xˆ]. We shall do the distributions pY˜ y and pY y is defined as so by providing separate upper and lower bounds for this   p ˜ (Y ) function. ( )  ( )  E Y D pY˜ y pY y log (75) ( ) pY (Y ) The upper bound for g x can be obtained by considering ∞ a relative entropy between the channel law pY |X (y|x) and p ˜ (y)  ( ) [ , ∞) = ( ) Y an auxiliary distribution pY y supported on 0 .The pY˜ y log dy (76) pY (y) nonnegativeness of the relative entropy immediately provides 0 ∞ an upper bound for the differential entropy (83), namely, 1 ∞ ≤ log p ˜ (y) dy (77)   1 − η Y   ( ) ≤−E ( ) =− | ( | ) ( ) . 0 g x log pY Y pY X y x log pY y dy =−log (1 − η) (78) 0 xˆ2 (86) = . (79) σ 2  S ( ) = Choosing √  a half-Gaussian distribution pY y / π − 2 Using the nonnegativity property of the relative entropy 2 exp y immediately √  gives an upper bound ( ) ≤ E 2 − / π together with (79), we obtain g x Y log 2 . The second moment for the   noncentral chi distribution is readily available, e.g., from lim D p ˜ (y)  p (y) = 0. (80) (20), leading to the following upper bound: σ →∞ Y Y S √ π ( ) ≤ 2 + σ 2 + . Using the fact that the relative entropy is zero if and only g x x n N log (87) ( ) = ( ) 2 if pY˜ y pY y almost everywhere [69, Th. 8.6.1], we conclude that (69) is fulfilled since the integrands in the Note that this upper bound is bounded inside an arbitrary finite differential entropy integrals differ on a set with measure zero. interval [0, xˆ]. SHEVCHENKO et al.: CAPACITY LOWER BOUNDS OF THE NONCENTRAL CHI-CHANNEL 2991

Establishing a lower bound for g(x) is slightly more ACKNOWLEDGMENTS involved. The first step is to transform the noncentral chi distri- The authors would like to thank the anonymous reviewers bution into a noncentral chi-squared distribution by making the = 2/σ 2 for their valuable comments. following change of variable in the integral (83): z 2y N . λ = 2/σ 2 = / Introducing the additional notation 2x N and n k 2, where k is a number of degrees of freedom of noncentral chi- REFERENCES squared distribution, we obtain [1] M. Cvijetic and I. B. Djordjevic, Advanced Optical Communication   (k−2)/4 + λ √ Systems and Networks. Norwood, MA, USA: Artech House, 2013. 1 z z [2] P. J. Winzer, “Scaling optical fiber networks: Challenges and solutions,” pZ|(z|λ) = exp − I(k−2)/2( λz) 2 λ 2 Opt. Photon. News, vol. 26, pp. 28Ð35, Mar. 2015. (88) [3] P. Bayvel et al., “Maximizing the optical network capacity,” Phil. Trans. R. Soc. A: Math., Phys. Eng. 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Nikita A. Shevchenko received the B.Sc. and M.Sc. degrees (summa Alex Alvarado (S’06–M’11–SM’15) was born in Quellón, Chile. He received cum laude) in applied physics from Donetsk National University, Ukraine, the degree (Ingeniero Civil Electrónico) in electronics engineering and the in 2008 and 2009, respectively. He is currently pursuing the Ph.D. degree M.Sc. degree (Magíster en Ciencias de la Ingeniería Electrónica) from with the Optical Networks Group, University College London, London, U.K. Universidad Técnica Federico Santa María, Valparaíso, Chile, in 2003 and In 2009, he enrolled on the Ph.D. programme in condensed matter physics 2005, respectively, and the degree of Licentiate of Engineering (Teknologie with the Donetsk O. O. Galkin Institute for Physics and Engineering, National Licentiatexamen) and the Ph.D. degree from Chalmers University of Technol- Academy of Sciences of Ukraine, with a focus on theoretical investigations in ogy, Gothenburg, Sweden, in 2008 and 2011, respectively. nonlinear optics. During his studies, he spent some time as an Academic Vis- From 2011 to 2012, he was a Newton International Fellow at the University itor with the University of Exeter, where he was involved in a research project of Cambridge, U.K, where he was a Marie Curie Intra-European Fellow from with the Electromagnetic and Acoustic Materials Research Group. In 2013, 2012 to 2014. From 2014 to 2016, he was a Senior Research Associate with he joined the Optical Networks Group, University College London. His main the Optical Networks Group, University College London, U.K. He is currently research interests include nonlinear optics, photonics, optical communications, an Assistant Professor with the Signal Processing Systems Group, Department and information theory. of Electrical Engineering, Eindhoven University of Technology, The Nether- lands. His current research was funded by the Netherlands Organization for Scientific Research through a VIDI grant, as well as by the European Research Council through an ERC Starting grant. His general research interests include digital communications, coding, and information theory. He was a recipient of the 2009 IEEE Information Theory Workshop Best Poster Award, the 2013 IEEE Communication Theory Workshop Best Poster Award, and the 2015 IEEE TRANSACTION ON COM- MUNICATIONS Exemplary Reviewer Award. He received the 2015 Journal of Lightwave Technology Best Paper Award, honoring the most influential, Stanislav A. Derevyanko received the Ph.D. degree in theoretical physics highest-cited original paper published in the journal in 2015. Since 2018, he from the Institute for Radio Physics and Electronics, Kharkiv, Ukraine, has been in the OFC subcommittee Digital and Electronic Subsystems (S4). He in 2001. From 2002 to 2007, he was twice a Post-Doctoral Research Fellow is an Associate Editor of the IEEE TRANSACTIONS ON COMMUNICATIONS at the Photonics Research Group, Aston University, Birmingham, U.K., and (Optical Coded Modulation and Information Theory). from 2007 to 2012, he was an EPSRC Advanced Fellow at the Nonlinearity and Complexity Research Group at Aston University. From 2013 to 2015, Polina Bayvel received the B.Sc. degree in engineering and the Ph.D. he was a Marie Curie Visiting Fellow at the Weizmann Institute of Science, degree in electronic and electrical engineering from the University of London, Israel, and in 2015, he joined the Department of Electrical and Computer U.K., in 1986 and 1990, respectively. In 1990, she was with the Fiber Engineering, Ben-Gurion University of the Negev, Beersheba, Israel. Optics Laboratory, General Physics Institute, Russian Academy of Sciences, His research interests include nonlinear optics, optical telecommunications, Moscow, under the Royal Society Postdoctoral Exchange Fellowship. She was and information theory. a Principal Systems Engineer with STC Submarine Systems Ltd., London, U.K., and Nortel Networks Harlow, U.K., and Ottawa, ON, Canada, where she was involved in the design and planning of optical fiber transmission networks. From 1994 to 2004, she held the Royal Society University Research Fellowship at University College London (UCL), and in 2002, she became the Chair in optical communications and networks. She is currently the Head of the Optical Networks Group (ONG), UCL, which she also initiated in 1994. She has authored or co-authored over 300 refereed journal and conference papers. Her research interests include wavelength-routed optical networks, high-speed optical transmission, and the study and mitigation of fiber nonlinearities. Dr. Bayvel is a fellow of the Royal Academy of Engineering, the Optical Jaroslaw E. Prilepsky received the M.E. degree in theoretical physic (Hons.) Society of America, the Institute of Physics, U.K., and the Institute of from the National University of Kharkiv, Ukraine, in 1999, and the Ph.D. Engineering and Technology. She was a recipient of the Royal Society degree in theoretical physics from the Verkin Institute for Low Temperature Wolfson Research Merit Award from 2007 to 2012, the 2013 IEEE Photonics Physics and Engineering, Kharkiv, Ukraine, in 2003, with a focus on nonlinear Society Engineering Achievement Award, and the 2014 Royal Society Clifford excitations in low-dimensional systems. From 2003 to 2010, he was a Patterson Prize Lecture and Medal. In 2015, she and five members of the ONG Research Fellow at the Verkin Institute for Low Temperature Physics and received the Royal Academy of Engineering Colin Campbell Mitchell Award Engineering, and in 2004, he was a Visiting Fellow at the Nonlinear Physics for their pioneering contributions to optical communications technology. Center, Research School of Physical Sciences and Engineering, Australian National University, Canberra, Australia. From 2010 to 2012, he was a Sergei K. Turitsyn received the degree from the Department of Physics, Research Associate with the Nonlinearity and Complexity Research Group, Novosibirsk State University, Novosibirsk, Russia, in 1982, and the Ph.D. Aston University, U.K., and since 2012, he has been a Research Associate degree in theoretical and mathematical physics from the Budker Institute with the Aston Institute of Photonics Technologies, Aston University. He of Nuclear Physics, Novosibirsk, in 1986. In 1992, he moved to Germany, has authored over 60 journal papers and conference contributions in the first as a Humboldt Fellow, and then working in the collaborative projects fields of nonlinear physics, solitons, nonlinear signal-noise interaction, optical with Deutsche Telekom. He is currently the Director of the Aston Institute, transmission, and signal processing. His current research interests include (but Photonic Technologies. He is a fellow of the Optical Society of America and are not limited to) optical transmission systems and networks, nonlinearity the Institute of Physics. He received the Royal Society Wolfson Research mitigation methods, nonlinear Fourier-based optical transmission methods, Merit Award in 2005, the European Research Council Advanced Grant soliton usage for telecommunications, information theory, and methods for in 2011, the Lebedev Medal by the Rozhdestvensky Optical Society in 2014, optical signal processing. and the Aston 50th Anniversary Chair Medal in 2016.