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Constellation Shaping for WDM Systems Using 256QAM/1024QAM with Probabilistic Optimization Metodi P

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Constellation Shaping for WDM Systems Using 256QAM/1024QAM with Probabilistic Optimization Metodi P 1 Constellation Shaping for WDM systems using 256QAM/1024QAM with Probabilistic Optimization Metodi P. Yankov, Member, IEEE, Francesco Da Ros Member, IEEE, OSA, Edson P. da Silva Student Member, IEEE, OSA, Søren Forchhammer Member, IEEE, Knud J. Larsen, Leif K. Oxenløwe Member, OSA, Michael Galili Member, IEEE, and Darko Zibar Member, IEEE Abstract—In this paper, probabilistic shaping is numerically a multidimensional ball. This leads to potential shaping gains and experimentally investigated for increasing the transmission exceeding the above-mentioned 1:53dB, suggesting similar reach of wavelength division multiplexed (WDM) optical com- results for discrete constellations of large size, e.g. high order munication systems employing quadrature amplitude modulation (QAM). An optimized probability mass function (PMF) of the quadrature amplitude modulation (QAM). However, memory- QAM symbols is first found from a modified Blahut-Arimoto aware receivers generally require exponential increase in com- algorithm for the optical channel. A turbo coded bit interleaved plexity. If a typical value of 16 symbols is used to express coded modulation system is then applied, which relies on many- the channel memory (see e.g. [3]) in combination with a to-one labeling to achieve the desired PMF, thereby achieving constellation with 16 points, the receiver would generally have shaping gains. Pilot symbols at rate at most 2% are used 4·16 for synchronization and equalization, making it possible to to go through all 2 possible states of the input, something receive input constellations as large as 1024QAM. The system clearly impractical. The independent, identical distribution is evaluated experimentally on a 10 GBaud, 5 channels WDM (i.i.d.) assumption on the input symbols, even though underes- setup. The maximum system reach is increased w.r.t. standard timating the channel capacity, provides a proper estimate of the 1024QAM by 20% at input data rate of 4.65 bits/symbol and up achievable rates by current digital systems, which usually do to 75% at 5.46 bits/symbol. It is shown that rate adaptation does not require changing of the modulation format. The performance not consider the memory. Under this assumption, well known of the proposed 1024QAM shaped system is validated on all 5 digital communication techniques can be used to optimize the channels of the WDM signal for selected distances and rates. transmission strategy, among these probabilistic shaping. Finally, it is shown via EXIT charts and BER analysis that In [4], trellis shaping is used to achieve very high SE, iterative demapping, while generally beneficial to the system, is close to the theoretical lower bound. Probabilistic shaping is not a requirement for achieving the shaping gain. performed in [5] in combination with a low-density parity Index Terms—WDM, probabilistic shaping, experimental check (LDPC) code, where the reach of the link is increased demonstration, 1024QAM, nonlinear transmission. by between two and four spans for a long haul link. In [4] and [5], digital back-propagation is performed, a technique which I. INTRODUCTION will often be impractical, especially in wavelength division High order modulation formats have been a hot topic multiplexed (WDM) network scenarios where the receiver for investigation for the fiber optic channel community in does not have access to the interfering channels. In [6], shell- the last decade due to their capability of transmission with mapping is used to achieve shaping gain, which suffers from high spectral efficiency (SE). In such cases, uniform input a poor complexity scaling with the constellation size. Perfect distribution suffers from shaping loss, which for the linear synchronization and equalization is assumed in [4]–[6]. The AWGN channels and high SNR is found to be 1:53dB [1]. system from [5] is later experimentally demonstrated [7], [8] The fiber optic channel on the other hand is highly non-linear and significant reach increase is shown in the presence of arXiv:1603.07327v2 [cs.IT] 11 Sep 2016 in the high signal-to-noise ratio (SNR) regime, which limits QPSK interfering signals. Rate adaptation with that scheme the maximum achievable SE [2] with standard receivers and is realized by changing the input distribution, which may not makes the ultimate shaping gain analysis difficult. Particularly always be desirable in WDM networks. so due to the memory effect of chromatic dispersion (CD) Methods also exist for geometric shaping of the input PDF and its interaction with the non-linear self and cross phase [9]–[12]. For example in [9], polar modulation is used, which modulation (SPM and XPM, respectively) in the fiber, which achieves significant shaping gain over uniformly distributed are not considered in standard approaches for estimating and QAM constellations. Superposition coded modulation was maximizing the mutual information (MI) between channel used in [10] as a shaping method. Geometric shaping schemes input and output. generally require high-precision digital-to-analog (DAC) and In [3], the memory is taken into account, and the probability analog-to-digital conversion (ADC) due to the non-uniformity density function (PDF) of a continuous input is optimized to of the signal on the complex plane. Standard QAM constella- The authors are with the Department of Photonics Engineering, Tech- tions are desirable due to their simplicity and the possibility nical University of Denmark, 2800 Kgs. Lyngby, Denmark: e-mail: to directly apply I/Q modulation/demodulation. [email protected] As part of our previous work [13], a near-capacity achieving Copyright (c) 2016 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be coded modulation scheme was designed for an AWGN chan- obtained from the IEEE by sending a request to [email protected] nel, based on turbo-coded bit-interleaved coded modulation 2 (BICM) employing QAM and constellation shaping. In [14], probability solution is x^k = arg maxX qXjY (XjY ) = the QAM probability mass function (PMF) is optimized for arg maxX pX (X)qY jX (Y jX). In either case, some sort of a single channel optical system and up to 1.5 dB gains are auxiliary channel function qY jX is employed for demodulation numerically shown in both the linear and non-linear region of and decoding, the quality of which governs the performance transmission. of the receiver. The lower bound in (2) will thus represent In this paper, the single channel work from [14] is extended the maximum error-free data rate which can be achieved by to WDM scenarios, where both intra- and inter-channel non- the specific receiver, employing a specific auxiliary function. linearities are present. We also perform a distance study, where The lower bounds is therefore also referred to as achievable the shaping gain is evaluated in terms of transmission reach. information rate (AIR). We evaluate the achievable MI rate after the synchronization and equalization, and show that the proposed turbo-coded scheme closely approaches it. A. Probabilistic optimization The main contribution of this paper is the experimental The standard approach to perform the optimization in demonstration of the method in a WDM optical fiber system Eq. (2) for the AWGN channel with constellation constrained with 5 channels. The performance on all channels is validated input is the Blahut-Arimoto algorithm [16]. The algorithm for selected scenarios. is straight-forward in that case due to the simple likelihood function, which is the Gaussian PDF with the variance of II. THEORETICAL BACKGROUND the additive noise. Since the input-output relation of the In information theory, channel capacity is defined as the fiber-optic channel is not available in closed form, and is maximum MI between the channel input and output, where furthermore signal dependent, we model the memoryless aux- i the maximization is performed over all input PDF (PMF in iliary PDF qY jX (Y jX = x ) as a 2D Gaussian distribution T the discrete case). For channels with memory the capacity can N (Σi; µi; [Re [Y ] ; Im [Y ]] ) with covariance matrix Σi and i be found as [2, Eq.(41)] mean µi, where x is the i−th element in X [17]. The param- eters of the Gaussians are estimated from training sequences 1 K K K K C = lim max I(X1 ; Y1 ) fx ; y g by taking the sample mean and covariance matrix K!1 K 1 1 pXK (X1 ) K 1 of the sub-set of points yKi , where Ki is the set of indexes k i 1 K 1 K K for which xk = x . The modified Blahut-Arimoto algorithm = lim max H(X1 ) − H(X1 jY1 ) ; (1) K!1 K for finding an optimized input PMF for the optical channel p K (X1 ) K K X1 at each input power Pin is given in Algorithm 1. The outer- K K where X1 and Y1 are the channel input and output se- most loop over α in Algorithm 1 allows for a basic geometric quences of length K, respectively, I is the MI, and H shaping via scaling of the original constellation [16]. We is the entropy function. The output sequence of the chan- observed that less than 5 iterations are required for the PMF nel can be obtained by solving the non-linear Shrodinger¨ to converge, which with 105 symbols results in a few hours equation (NLSE) by the split-step Fourier method (SSFM). runtime for each parameter set fα; Ping. We note that accurate The input-output relation is governed by some probability MI estimation for constellations larger than 1024QAM would K K function p K K (X jY ), which for the fiber is generally require longer sequences, thereby increasing the runtime of the X1 jY1 1 1 unknown. If the conditional entropy in Eq. (1) is defined algorithm significantly.
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