Spectral Efficient Coding Schemes in Optical Communications

Achieving high spectral efficiency in optical transmissions has recently attracted much attention, aiming to satisfy the ever increasing demand for high data rates in optical fiber communications. Therefore, strong Forward Error Correct ion (FEC) coding in combination with mult ilevel modulat ion schemes is mandatory to approach the channel capacity of the transmission link. In this paper we g ive design rules on the joint optimization of coding and signal constellations under practical considerations. We give trade-offs between spectral efficiency and hardware complexity, by comparing coding schemes using capacity achieving constellations with bit-interleaved coded modulation and iterative decoding (BICM-ID) against applying conventional square quadrature amplitude modulation (QAM) constellations but employing powerful low complexity low-density parity-check (LDPC) codes. Both schemes are suitable for optical single carrier (SC) and optical orthogonal frequency-division multiplexing (OFDM) transmission systems, where we consider the latter one in this paper, due to well-studied equalizat ion techniques in wireless communications. We numerically study the performance of different coded modulation formats in optical OFDM transmission, showing that for a fiber optical transmission link of 960 km reach the net spectral efficiency can be increased by ≈0.4 bit/s/Hz to 8.61 b it/s/Hz at a post FEC BER of <10 by using coded optimized constellations instead of coded 64-QAM.


Introduction
Over the last decade there has been an exponent ial growth of bandwidth-intensive services such as video on demand, cloud storage and social networking which require large volu mes of data to be transmitted over long distances. Based on current trends, this growth is likewise to continue, driving the need to increase transmission capacity. Ongoing commercial develop ments with data rates of 100 Gbit/s in the standardized 50 GHz wavelength-division multip lexing (WDM ) grid consider po larization d ivision mu lt ip lexed quadrature phase shift key ing (PDM-QPSK) as the most s u itab le mo d u latio n fo r mat [ 1]. Th is a llo ws a p er polarization spectral efficiency of up to 2.0 bit/s/Hz. Next generation systems will need to supply data rates beyond 400 Gb it/s. Therefore it is necessary to exp loit the channel capacity of the fiber up to its theoretical limit. The most effective method to increase the spectral efficiency of any co mmun icat ion system is to emp loy larger modu lat ion formats [2]. However, in order to approach channel capacity the constellation itself needs to be modified with respect to informat ion theory. In [3] the design principles of capacity achieving circu lar constellations, based on iterative polar quantization (IPQ), are p roposed. For higher modulation formats the robustness to noise is degraded, claiming the need for powerful FEC coding schemes, relying on the "Turbo Princip le" [4]. The co mbination of iterat ive polar modulation (IPM ) and LPDC codes have already been a matter of investigation in[5], using mu ltilevel coding (M LC) and iterative mult istage decoding [6], but o mitting complexity considerations, investigations on the available channel capacity and error floor evaluations. To cope with the non-linear channel effects and dispersion, we consider OFDM in co mb ination with mu ltip le-input multip le output polarization mu ltip lexed equalization to be most appropriate, as it offers high spectral efficiency and needs only simp le computation operations for equalizat ion [7].
In this paper we propose a high spectral efficient coded modulation scheme for imp lementation in future optical communicat ion systems operating at data rates beyond 400 Gb/s. In detail, we adapt the "Turbo Principle" to BICM -ID [8] and comb ine it with a h igh-rate outer algebraic code to obtain a post-FEC BER <10 -15 , which is a typical demand in optical transponders. Furthermore we give simp le design principles for the design of BICM-ID based on the extrinsic informat ion transfer (EXIT) chart analysis [9]. The optical channel is considered to be weakly-nonlinear. Therefore the proposed techniques are also applicable for single-carrier transmission; however we consider OFDM since it appears to be more appropriate for the high order modulation formats and efficient equalization algorith ms that are well established in wireless communicat ions.
The remainder of the paper is organized as fo llo ws: In Section 2 the design principles of capacity achieving constellations are derived. In Section 3, the investigated optical OFDM system is introduced and the availab le channel capacity for different modulation formats is estimated. The basic coded modulation scheme, namely BICM -ID, is reviewed in Section 4. Design princip les for BICM -ID with IPQ signal constellations, using recursive systematic convolutional (RSC) codes and square signal constellations, using LDPC codes, are given in Sections 5 and 6. We conclude our paper in Sect ion 7.

Design of Capacity Approaching Constellations by Means of Iterative Polar Quantization
In order to explo it the available channel capacity up to its theoretical limit the signal distribution needs to comply with the channel characteristics. In additive wh ite Gaussian noise (AWGN) channels, such as thermal and amplified spontaneous emission (ASE) noise dominated channels, a Gaussian source provides the maximu m shaping gain value.
A Gaussian distribution at the channel input can be realized by different strategies. One considers shaping, first proposed in [10] by Gallager, where it is shown that optimal coding for the Gaussian channel is utilized by grouping b its fro m a b inary code, where the larger group is assigned to the less probable points in the constellation, and the smaller to the more frequent [11]. In this concept the basis constellation A is square; shaping is performed by addit ional shaping encoders and decoders. A hardware comp lexity considering design of this approach is proposed in [11], but is hardly feasible for high data rates.
Therefore we consider a different strategy to provide a Gaussian source, by applying non-uniform signal constellations, which co mply with the channel characteristics. Design principles of capacity achieving constellations are proposed in [3]. In order to provide Gaussian signaling at the input of the system, we optimize the signal constellation A by minimizing the quantization mean square error (QM SE) of a Gaussian information source by means of IPQ. The constellation points of the obtained constellation are distributed over the circles of radii determined by a Rayleigh distribution [3]. The idea of using a Rayleigh distribution p(r) originates from the fact that the envelope r of a two-dimensional Gaussian distribution is Rayleigh distributed. Let N r denote the number of circles in the constellation and L j denote the number of constellation points per circle of radius m j ; j ∈ {1,2,...,N r }. Then the optimu m nu mber of constellation points at the jth circle fo r a 2 M -ary signal set is determined by min imizing the QM SE (1) The radius of the jth circle calculates to (2) where ∆θ j = 2π/L j . The limits of integration in (1) and (2) are determined by (3) The optimu m signal constellation A for a 2 M -ary signal set is found by applying equations (1)-(3) in an iterative fashion until convergence [3]. Table 1 summarizes the details of the 64-IPM constellation, for other constellation sizes see [5] and [12].  Figure 1 depicts the 64-IPM constellation as specified in Table 1. The distribution of the points is non-uniform and forms a discrete 2-D Gaussian distribution with zero mean. Since the applied mapping is crucial for the performance of BICM-ID, we designed the mapping to be most Gray like, i.e. the b inary decomposition of neighboring symbols differs in a minimu m number of dig its. A pure Gray labeling is not availab le for constellations designed by IPQ, due to a non-uniform symbol spacing and differing numbers of neighboring symbols.
The benefit of applying IPM constellations over QAM constellations is revealed in Figure 2,   We assumed equal a-priori p robability for the input symbols a m ∈ A, p(a)= 1/ |A| and use the conditional probability density function (PDF) of the comp le x valued AWGN channel (5) where the variance σ 2 corresponds to the double-sided noise power spectral density N 0 /2. Obviously, IPM signal constellations approach capacity for low and mediu m signal-to-noise ratios and outperform conventional square-QAM constellations. The objective of this paper is to provide practical design principles for capacity approaching coding schemes. Therefore knowledge of the available channel capacity is inevitable. The optical OFDM system under investigation is depicted in Figure 3.

The Optical OFDM System and Channel Capacity Estimation
A binary information sequence u is encoded and mapped to complex valued 2 M -ary sy mbols a. After mapping to a block of symbols, the symbols a are fed to an OFDM transmitter, electro-optical I-/Q-modulated, polarization mu ltip lexed and transmitted over the fiber-optical channel. At the receiver, the signal is demult iplexed, coherently demodulated sampled and equalized in the frequency domain, providing the complex samples r at the output of the OFDM receiver. For a description of our OFDM system in detail see [13].
In the simu lations of the optical system we modulate 864 subcarriers within a bandwidth of 24.1 GHz in each state of polarization. The cyclic prefix length equals to 1/10 o f the original OFDM symbol duration (not considered in the capacity estimat ion). The lasers line-width is assumed to be 100 kHz. The transmission lin k itself consists of q = 12 identical spans of standard single mode fiber (SSMF) (length: 80 km, chro mat ic dispersion coefficient: 17 ps/nm/km, attenuation: 0.2 d B/km); the whole transmission link covers 960 km. The per-channel optical input power was varied fro m -10 to 0 d Bm. The simulat ion model of the optical channel considers the Kerr effect with the non-linear coefficient 1.33/W/km; polarizat ion dependent loss is neglected. Optical amp lifiers (OA) co mpensate for attenuation; their noise-figure is 4 dB. Further we include five WDM channels at a 25 GHz grid in our simu lations. The receiver contains a 30 GHz optical pass-band filter and a 6 bit analog-to-digital converter (ADC).
Studies about the noise characteristics show that the whole setup can be treated as weakly non-linear, so the Gaussian PDF of equation (5) is appropriate to describe the channel with respect to the OFDM subcarriers. Using (4) and (5), the informat ion spectral densities of the resulting channel for various per-WDM channel input powers and different modulations format were estimated. The results are shown in Figure 4.  informat ion theory, the signal meets the requirements to approach the channel capacity of the resulting optical channel main ly corrupted by additive Gaussian noise. On the other hand, although we are using an OFDM system, the achievable capacity for IPM on each subcarrier is closer to the Shannon limit than for conventional QAM, for t wo reasons: First, the average symbol power is decreased and second, regarding the OFDM system as part of the channel, the optimu m symbol constellation has to be Gaussian, due to the fact that the symbols r at the receiver side after the FFT are corrupted only by fairly Gaussian distributed noise, no matter what d istortions occurred on the optical channel, if the number of subcarriers is large. As displayed in Figure 4, the optimu m fiber launch power is -7 dBm. For low optical input powers, ASE noise is dominant, the signal-to-noise ratio (SNR) increases by 1 dB per 1 dB input power increment (as expected for a linear system with AWGN). But when the optical power is increased above a certain threshold, the Kerr effect becomes dominant, causing an SNR degradation of 2 dB per 1 dB power increment [13]. The threshold for the Kerr effect to become dominant is at a fiber launch power of -7 dBm, which corresponds to an E s /N 0 of 16.9 d B in an AW GN channel model, supplying a channel capacity of 5.61 bit/s/Hz.
Obviously only the 256-IPM signal constellation is able to provide a capacity close to the channel capacity, leaving a gap of only 0.07 b it/s/Hz. Unfortunately for this constellation a hardware implementation currently seems unfeasible and further we do not expect that switching fro m 64-IPM to 256-IPM is justifiable, since it requires an ADC of more than 6 bit and a h igh code overhead. Therefore we restrict us in the following on the design of coded modulation for 64-IPM and for co mparison 64-QAM, supplying an information spectral density of 5.31 bit/s/Hz and 5.22 bit/s/Hz, respectively, which is still close to the Shannon limit.

Advanced Spectral Efficient Coding in Optical Communications
In classical non-coherent optical co mmunicat ion systems, operating at data rates up to 10 Gb/s, no powerfu l FEC coding is required, since in the systems a pre-FEC BER of <10 -4 is obtained and a post-FEC BER of only <10 -12 is demanded. Hence the algebraic Reed-Solo mon (RS) (255, 239) code, supplying a net coding gain (NCG) of 5.62 dB at a pre-FEC BER of 1.8•10 -4 for a post-FEC BER 10 -12 is the most used code in these systems [14].
But when higher data rates are claimed, mo re powerful coding schemes need to be considered. In today's systems with 40 Gb it/s differential phase-shift keying, enhanced FEC (EFEC) coding, as reco mmend by the ITU in [15], is utilized by the concatenation of the Bose-Chaudhuri-Hocguenghem (BCH) (3860, 3824) code and the BCH (2040, 1930) code, using iterative decoding. The BCH (3860, 3824) + BCH (2040, 1930) concatenated code has a code overhead of 6.69 %. After three iterations this code achieves a NCG of 8.99 dB at a pre-FEC BER of 3.15•10 -3 and a post-FEC BER of 10 -15 . If iterative decoding is to be avoided the ITU reco mmends the RS (2720, 2550) code with a NCG of 8 dB at a pre-FEC BER of 1.1•10 -3 [ 15]. A ll of these codes are binary linear cyclic codes, with decoders processing hard decision values of the received output bit stream. So these schemes are most suitable for low BER, but when coherent modulation formats beyond QPSK are emp loyed and approaching the Shannon limit is desired, soft decision decoding is inevitable.
A straightforward approach to comb ine mu ltilevel signaling and coding is BICM -ID, which can be considered the simplest approach to achieve high spectral efficiency while providing a low decoder comp lexity [2]. However, coding schemes relying on the Turbo Princip le may run into an error floor, wh ich is several decades higher than the desired post-FEC BER of 10 -15 . Therefore, we apply BICM -ID for inner coding and concatenate it with a high-rate outer code to remove residual errors. The proposed coding scheme is depicted in Figure 5. In the following, we consider the binary RS (2720, 2550) code with a code rate of R RS = 0.9375 and a length of 32640 bit for outer coding. The concept inner coded modulation scheme is as follows: The transmission is performed on a block basis: k bits of the outer encoded sequence x are fed to a recursive systematic convolutional (RSC) o r LDPC encoder, which extends the sequence by n-k parity bits, where n is the codeword (CW) length. As explained later, we prefer to use punctured RSC codes for IPM signal constellations and LDPC codes for square QAM signal constellation, see Section 5 and 6.
After encoding the CW is fed to a row column interleaver, so the bits in the binary decomposition of a symbol are uncorrelated, which is a pre-requirement for iterative demapping and decoding. After interleaving the grouping operation takes M coded bits t 0,1,..,M-1 ∈ {0,1} to form the complex output symbol a = map (t 0,1,..,M-1 ). This mapping operation map(•) is essential for the iterative demapping and decoding process, since it links up the bits t 0,1,..,M-1 to a symbol and mutual dependencies arise between them [8].
After mapping the comp lex symbols a are transmitted over the channel. In our optical OFDM system we use a high number of subcarriers; therefore the central limit theorem applies when performing the FFT on the received channel symbols, causing that after equalizat ion the symbols r at the input of the decoder are only corrupted by fairly Gaussian distributed noise; r = a + n, where n = n I + jn Q , with n I , n Q being realizations of two independent Gaussian random variables and variance σ 2 =σ I 2 =σ Q 2 . In the soft-demapper the channel symbo ls r are demapped and ungrouped to each M log-likelihood ratio (LLR) values .
In the first iteration these a posteriori LLR values are deinterleaved and soft-in/soft-out decoded in a symbol-by-symbol a posteriori probability (APP) estimator; a BCJR [16]  The decoder performs bitwise soft-input processing, thus the demapper extracts a soft value for each coded bit t 0,1,..,M-1 of a 2 M -ary complex channel symbol r .When a-priori informat ion is available, the soft-demapper calculates the APP LLR of bit k computes to [9] Using the max-log appro ximat ion [8] and assuming that the received symbols are only corrupted by Gaussian, noise we can be rewritten by the soft-demapping algorith m (6) with (5) according to [8] (8) where a k, (1,i) = map (t k =1, t j≠k ≡ b in(i)) and a k,(0,i) = map(t k =0, t j≠k ≡ bin(i)), j ∈ {0..M-1}. This algorith m can even be more simp lified, when only the met rics to those symbols closest to the received sy mbol are co mputed, which are also the most likely ones. In the fo llo wing we use the soft-demapping algorith m of (7), since we expect the soft-demapper implementation in hardware without the ma x-log appro ximat ion unfeasible fo r h igh data rates and the losses are inferior anyway.

Design of BICM-ID for IPM
In this section we discuss the design of BICM-ID, using a 64-IPM signal constellation and RSC codes. We restricted the use of LDPC codes and utilized RSC codes, since they offer a greater degree o f freedo m in design for coded modulation, when the code rate is high. As denoted above, residual errors are removed by an outer RS (2720, 2550) code. In order to design the BICM -ID scheme towards proper convergence and to acquire a post-FEC BER below the threshold BER o f the outer RS code of 1.1•10 -3 , we use the EXIT chart analysis, which is a powerful tool to visualize the flow of ext rinsic information between the soft-demapper and decoder [9]. In the EXIT chart analysis, the mutual information (MI) of the decoder/demapper is plotted versus its a-priori input, i.e. the MI of the demapper/decoder. If an optimu m demapper/decoder is used, the knowledge of the MI contained in the a-priori informat ion is sufficient to derive the MI of the decoder/demapper. Denoting the encoder/demapper input by X a and the corresponding extrinsic decoder/demapper output by X e , the MI I(X e ;X a ) calcu lates to (9) As denoted above the 256-IPM constellation offers a spectral efficiency very close to the Shannon Limit, however designing a mapping fo r a proper convergence of BICM -ID using either RSC or LDPC co mponent codes is still an open problem. Therefo re we restricted us to the design of BICM-ID using the 64-IPM signal constellation only. Figure 6 depicts the EXIT chart of the designed 64-IPM coded modulation scheme at a per-channel input power of -7 dBm after 960 km SSMF transmission.
The mapping was designed to be most Gray like, in order to provide a flat soft-demapper EXIT function, wh ich is equivalent to allow a fast convergence of the iteration loop.  x The RSC code applied has a constraint length of 7 and was punctured to result in a code rate of R RSC = 0.86. The CW length was set to n=20736, to co mply with the number of bits in two OFDM symbols. As indicated by the dashed line, the EXIT functions of soft-demapper and decoder intersect at an MI co rresponding to BER <1.1•10 -3 . The obtained BERs of the designed BICM-ID system for different per-channel input powers and different number of iterations after 960 km SSM F transmission are depicted in Figure 7.  Obviously, 6 iterations are necessary to provide a BER <1.1•10 -3 , wh ich is sufficient for the outer EFEC coding scheme to deliver a post-FEC BER <10 -15 . Considering a 6.69 FEC overhead for outer coding and 10% overhead for OFDM pilots duration and guard interval, this system obtains a spectral efficiency of 8.64 b it/s/Hz in both states of polarizat ion after 960 km SSMF t ransmission.

Coded Modulation with LDPC Codes
For optical transponders processing at data rates beyond 400 Gbit/s, another potential candidate for strong soft-decision-based FEC in BICM -ID are LDPC codes, which are linear codes defined by the sparse parity-check matrix invented by Gallager [10]. They allow for an iterative decoding procedure, denoted as belief propagation or sum-product algorithm, where ext rinsic in formation is exchanged between variable nodes, corresponding to informat ion and redundancy bits, and check nodes, representing the parity check equations. The edges between them are given by the parity-check matrix. An LDPC codes is specified by t wo degree d istributions λ(x) and ρ(x), where and . λ i is the fraction of edges that belong to degree-i variable nodes, ρ j is the fraction o f edges that belong to degree-j check nodes, d l is the maximu m variable node degree, and d c is the maximu m check node degree. LDPC codes are expected to exhibit superior error-correcting performance close to the Shannon limit, which imp lies a flat LDPC decoder EXIT function in the EXIT chart analysis. Therefore we intend to use LDPC codes in BICM-ID when square signal constellations with Gray mapping are applied only. This offers the advantage that there is no need for information feedback in BICM -ID, since the soft-demapper EXIT-function of Gray mapped signal constellations is flat too, so there is no gain in computing the extrinsic informat ion when a-priori informat ion is available. Furthermore the interleavers are dispensable, since in an LDPC CW ad jacent bits participate in different checks, so there are no significant statistical dependencies between the bits mapped to symbol. The degree distribution of a proper LDPC code for the system regard ing the capacity of the modulation was found by means of density evolution [17]. Figure 8 shows the EXIT chart of the designed coding scheme at a per-channel input power of Obviously, the EXIT functions of the soft-demapper and decoder intersect at an MI corresponding to a BER <1.1•10 -3 without performing any iterations between the decoder and demapper. The LDPC code applied has a CW length of n=20736, same as the RSC code before. The code rate was R LDPC = 0.8205 and 30 iterations were perfo rmed in the LDPC decoder. The check nodes of the applied LDPC code have degree 30 only, ρ(30)=1; the degree distribution of the variable nodes are denoted in Table 2. The available BERs after performing one and two iterations of the iterative soft-demapping and decoding system versus different per-channel input powers after 960 km SSMF transmission are depicted in Figure 9. As expected by the EXIT chart analysis, there is no significant gain when performing iterat ions between LDPC decoder and soft-demapper. Furthermore, the error floor of the inner coding system is several decades blow the threshold BER of the RS (2720,2500) code, so it is reasonable to consider the simp ler RS (255, 239) code with an NCG of 6.2 d B for an post-FEC BER of 5•10 -15 for outer coding.
Considering the 6.69 % overhead for outer coding and 10% overhead for OFDM pilots duration and guard interval, a spectral efficiency of 8.24 bit/s/Hz was achieved with the system in both states of polarizat ion after 960 km SSMF transmission.

Conclusions
In this paper we presented a simple, but spectral highly efficient coding scheme for optical OFDM-based communicat ions, relying on the concatenation of BICM-ID with a h igh-rate outer code to obtain a post-FEC BER < 10 -15 . Further, we provided design princip les of capacity approaching constellations and showed how to jointly optimize mapping and coding for a proper convergence of the iteration loop.
We demonstrate that non-uniform constellations based on IPQ outperform conventional square signal constellations. In order to co mpare the performance of IPM versus QAM towards BER performance we developed different coding schemes, showing that in both cases an overall post-FEC BER < 10 -15 is achievable, but, when using coded 64-IPM a spectral efficiency of 8.64 b it/s/Hz in both states of polarization after 960 km SSMF transmission could be obtained, which is ≈0.4 bit/s/Hz greater co mpared to coded 64-QAM.
However for a pract ical implementation we suggest coded modulation schemes, based on QAM constellations and LDPC codes, to be more appropriate, since they are implementable more efficiently in hardware to operate at high data rates. Disregarding the loss in spectral efficiency due to OFDM p ilots duration and guard interval, th is coding scheme is able to approach the Shannon limit by 1.85 bit/s/Hz in both states of polarization.