Two-Dimensional Constellation Shaping in Fiber-Optic Communications

applied sciences

Article Two-Dimensional Constellation Shaping in Fiber-Optic Communications

Zhen Qu 1,*, Ivan B. Djordjevic 2 and Jon Anderson 1

1 Juniper Networks, 1133 Innovation Way, Sunnyvale, CA 94089, USA; [email protected] 2 Department of Electrical and Computer Engineering, University of Arizona, 1230 E. Speedway Blvd., Tucson, AZ 85721, USA; [email protected] * Correspondence: [email protected]; Tel.: +1-520-442-7197

Received: 5 April 2019; Accepted: 2 May 2019; Published: 8 May 2019

Abstract: Constellation shaping has been widely used in optical communication systems. We review recent advances in two-dimensional constellation shaping technologies for ﬁber-optic communications. The system architectures that are discussed include probabilistic shaping, geometric shaping, and hybrid probabilistic-geometric shaping solutions. The performances of the three shaping schemes are also evaluated for Gaussian-noise-limited channels.

Keywords: constellation shaping; probabilistic shaping; geometric shaping; fiber-optic communications; quadrature amplitude modulation; mutual information; generalized mutual information

1. Introduction The advances in photonics integrated circuit [1,2], software-defined networking [3,4], application-oriented fibers [5,6], and optical amplifiers [7,8], have been greatly pushing forward development and commercialization of optical communications. In the modern optical transport systems, especially in terrestrial and transoceanic fiber-optic communications, advanced modulation formats have been overwhelmingly implemented in optical transceivers [9–14], thanks to the ever-cheaper optical frontend and powerful digital signal processing (DSP) chips with smaller size and lower power consumption. Traditional quadrature amplitude modulation (QAM) formats have been applied extensively to realize high-capacity and long-reach optical communications, and we have witnessed numerous two-dimensional QAM (2D-QAM)-based hero experiments in recent years to explore the highest possible system capacity and longest transmission distance [15–17]. Due to the loss profile of the standard single mode fiber (SSMF) and gain profile of commercial Erbium-doped fiber amplifiers (EDFAs), C-band window (1530–1565 nm) is mostly used for data loading [18]. However, the capacity bottleneck becomes more visible for the traditional QAM-based C-band optical transmissions [19]. In order to meet the increasing bandwidth requirements, in particular the upcoming 5G infrastructure, more advanced solutions are expected to be introduced to optical infrastructures. We can roughly divide the promising solutions into two categories: coded modulation [20–27] and extended multiplexing [17,28–31]. The idea of coded modulation is increasing information bits per channel use, including higher-order 2D modulation formats, like 1024QAM [22], multidimensional QAM formats, like 4D optimized constellations [23], and constellation shaping techniques [24–26], like probabilistic shaping (PS)-64QAM [25]. Alternatively, extended multiplexing is a more straightforward solution, which mainly contains C+L band wavelength multiplexing [17,28] and space-division multiplexing [29–31]. All the solutions come with the trade-off choice between optical complexity and electrical complexity. In order to implement C+L band wavelength multiplexing, C-band EDFAs and L-band EDFAs should be used together to simultaneously amplify the channel loss per span [17]. Raman amplifier may be another option [32], but the telecom industry is not in

Appl. Sci. 2019, 9, 1889; doi:10.3390/app9091889 www.mdpi.com/journal/applsci Appl. Sci. 2019, 9, 1889 2 of 13 favor of it because of its high cost. Space-division multiplexing based on few-mode fiber or multi-core fiber is an efficient way to directly boost the aggregate capacity. However, it seems to have been put at the bottom of the to-do list, because (i) it will be quite challenging to replace and rebuild the current fiber links, (ii) space-division-multiplexing-based optical amplifiers are too expensive for commercial applications, and (iii) complex multi-input multi-output (MIMO) channel equalization may be required to compensate channel crosstalk [33]. The telecom industry always chooses the most cost-efficient way to upgrade their communication systems. Therefore, extended multiplexing solution is barely found in the roadmap of optical communications, especially in the field of long-reach optical communications. On the contrary, coded modulation seems to be a more attracting solution. Higher modulation formats are more common now, but they have more stringent requirements on optical signal-to-noise ratio (OSNR), digital-to-analog converter/analog-to-digital converter (DAC/ADC), and DSP recovery. Multidimensional QAM solution can further enlarge the minimum Euclidean distance between constellation points, but it requires powerful DSP technology to recover signals. What is more, such performance optimization can come at a price of less information bits per channel use [34]. In traditional QAM formats, the constellation points are located on a uniform grid. Such uniform QAM formats are easy for generation and recovery but suffer a 1.53-dB asymptotic loss towards the Shannon limit. In order to close the gap, constellation shaping was introduced to optical communications. Constellation shaping, including PS [25,35–37], geometric shaping (GS) [26,38–41], and hybrid probabilistic-geometric shaping (HPGS) [42–45], is used to mimic a Gaussian distribution with limited constellation points. Although target at approaching Gaussian distribution, PS and GS have completely different generation and detection implementations. GS-QAM is obtained by optimizing some metrics, like mutual information (MI) [40] and generalized MI (GMI) [41], which will relocate the constellation point in geometric space. PS-QAM is applied on a uniform grid, but the constellation points will be transmitted with different probabilities. HPGS can optimize the performance in both geometric space and probabilistic space which should, in principle, provide the optimal performance. In this paper, we review recent advances on 2D constellation shaping in fiber-optic communications. Section2 gives an overview of PS, GS, and HPGS. Section3 discusses the performance of PS, GS, and HPGS in Gaussian-noise-limited channels. Finally, the concluding remarks are given in Section4.

2. Typical Constellation Shaping Schemes

2.1. Probabilistic Shaping In a Gaussian-noise-limited channel, PS-QAM yielding a Maxwell–Boltzmann (M–B) distribution is recognized as the optimal format to maximize the channel capacity [46]. In general, the low-amplitude constellation points are sent with a higher probability than the high-amplitude ones. Besides, the constellation points under the same amplitude layer are sent equally likely. Therefore, the average symbol power will be decreased, but at a cost of lower source entropy. The first PS scheme was proposed by Gallager, which is based on many-to-one mapping [47]. Complex forward error correction (FEC) coding is required to be implemented to recover the original bits from the systematic errors after many-to-one demapping. The recently proposed arithmetic coding-based constant composition distribution matcher (CCDM) is an invertible fix-to-fix length distribution matcher, enabling maximum information rate asymptotically in the block-length [48]. Later, there are some methods proposed to further reduce the complexity of CCDM, like multiset-partition distribution matching [49] and streaming distribution matching [50]. The first PS-based coded modulation was probabilistic amplitude shaping (PAS) [51], which could seamlessly combine the binary FEC coding and CCDM in a square M-QAM format. The proposed PAS enables capacity approaching fiber-optic communications, but also brings some issues. Firstly, CCDM and the modified CCDM architectures are hard to be implemented in commercial optical transceivers. Secondly, more bit-to-symbol (B2S) mapping and symbol-to-bit (S2B) Appl. Sci. 2019, 9, 1889 3 of 13 mapping modules are required to be implemented at the transceivers, leading to extra complexity. Thirdly, there will be an entropy loss by shaping on a given M-ary signal constellation format. Fourthly, Appl. Sci. 2019, 9, x FOR PEER REVIEW 3 of 13 intrablock error propagation will loom over the dematching procedure once the FEC coding cannot totallycommercial correct alloptical bit errors. transceivers. Fifthly, Secondly, the applicable more FEC bit-to-symbol code rate is(B2S) limited, mapping which and will symbol-to-bit be lower-bounded (S2B) by mapping[log (M) modules2]/ log are(M required). Sixthly, to DSP be implemented circuit will be at under the transceivers, a higher pressure leading to to recover extra complexity. Gaussian-like 2 − 2 constellationThirdly, there diagrams. will be an Thereafter, entropy loss more by pilot-tonesshaping on area given essential M-ary to signal be used constellation for signal format. recovery. LastFourthly, but not intrablock least, although error propagation could be used will loom for reach over the extension, dematching PAS-MQAM procedure su onceffers the more FEC fromcoding the modulation-dependentcannot totally correct noiseall bit in errors. long-haul Fifthly, fiber-optic the appl communicationsicable FEC code rate [52]. is limited, which will be lower-boundedIn a square MQAM by [log format,(𝑀) −2] the⁄ coordinate log(𝑀). Sixthly, of each DSP constellation circuit will can be be under represented a higher by pressure the Cartesian to productrecover of Gaussian-like two pulse-amplitude constellation modulation diagrams. (PAM) Thereaft coordinates,er, more pilot-tones namely, are essential to be used for signal recovery. Last but not least, although could be used for reach extension, PAS-MQAM suffers n o more from the modulation-dependentX = noise1, in3, long-haul... , √ fiber-opticM 1 communications [52]. (1) ± ± ± − In a square MQAM format, the coordinate of each constellation can be represented by the CartesianThe well-known product of M–B two distributionpulse-amplitude is defined modulation by (PAM) coordinates, namely,

𝑋 ={±1,±3,…,±(2 X√𝑀 −1)}2 (1) v x v x0 PXv (x) = e− | | / e− | | (2) The well-known M–B distribution is defined by x X 0∈ ( ) || where v is a non-negative scaling factor.𝑃 𝑥 If v=𝑒is 0, the/ PAM format 𝑒 follows a uniform distribution.(2) ∈ The capacity of PAS-MQAM format can be deﬁned as [51] where 𝑣 is a non-negative scaling factor. If 𝑣 is 0, the PAM format follows a uniform distribution. The capacity of PAS-MQAM formatC =canH be(p) definedm(1 asR [51]) (3) − − 𝐶=𝐻(𝑝) −𝑚(1−𝑅) (3) where H(p) is the entropy of the PAS-MQAM format, R is the FEC code rate, and m = log2(M). whereFigure 𝐻1(𝑝 shows) is the the entropy conceptual of the diagramPAS-MQAM of PAS-MQAM format, 𝑅 is generation, the FEC code where rate, and 16QAM 𝑚=log is used(𝑀) as. an illustrativeFigure example. 1 shows Inthe such conceptual scheme, diagram the 1D of amplitude PAS-MQAM symbols generation, labeled where by 1-3-1-1-1-316QAM is used... , as will an be probabilisticallyillustrative example. shaped In according such scheme, to the the M–B 1D distribution amplitude (thesymbols probabilities labeled ofby Symbol-11-3-1-1-1-3…, and Symbol-3will be probabilistically shaped according to the M–B distribution (the probabilities of Symbol-1 and are indicated by the blue and black colors, respectively), and the sign bits labeled by 1-0-0-1-0-1 Symbol-3 are indicated by the blue and black colors, respectively), and the sign bits labeled by 1-0-0- ... , will be used to carry the uniformly distributed parity-check bits. The S2B mapping is used to 1-0-1…, will be used to carry the uniformly distributed parity-check bits. The S2B mapping is used to map the Symbol-1 and Symbol-3 to amplitude bits, i.e., Bit-1 and Bit-0, respectively. In the FEC map the Symbol-1 and Symbol-3 to amplitude bits, i.e., Bit-1 and Bit-0, respectively. In the FEC encoder, the party-check bits will be appended to the sign bits, and combine with the amplitude bits. encoder, the party-check bits will be appended to the sign bits, and combine with the amplitude bits. Such encoded bits are remapped to PAM-4 symbols (00 3, 01 1, 11 +1, 10 +3). Therefore, Such encoded bits are remapped to PAM-4 symbols (00→→−−3, 01→−→−1, 11→+1,→ 10→+3).→ Therefore, after afterFEC FEC coding, coding, the theM–B M–B distribution distribution will willnot be not changed, be changed, and Gray-mapping and Gray-mapping rule is rule still is applicable. still applicable.

10 00 01 11 10

+1 +3 Amplitude -3 -1 +1 +3 1000 1001 1011 1010 Binary 1D- S2B bits FEC B2S In-phase +3 data DM 131113... mapping ENC mapping 1100 1101 1111 1110 Sign bits +1 100101... I/Q MOD 010110... -3 -1 +1 +3 Sign bits -1 0100 0101 0111 0110 Binary 1D- 31131... S2B FEC B2S data DM mapping Amplitude ENC mapping -3 Quadrature 0000 0011 0010 bits 0001 10 00 01 11 10 +1 +3 -3 -1 +1 +3

FigureFigure 1. The 1. conceptual The conceptual diagram diagram of probabilistic of probabilistic amplitude amplitude shaping shaping (PAS)-M-aryquadrature (PAS)-M-ary quadrature amplitude modulation (MQAM) generation.amplitude modulation (MQAM) generation.

2.2.2.2. Geometric Geometric Shaping Shaping GS-QAMGS-QAM is is generatedgenerated by by optimizi optimizingng certain certain criterion criterion under under a give an signal-to-noise given signal-to-noise ratio (SNR). ratio (SNR).Such Suchcriteria criteria can be can maximizing be maximizing MI [40], MI maximizi [40], maximizingng GMI [41], GMI maximizing [41], maximizing constellation constellation figure of merit [53], or minimizing mean-square error of Gaussian distribution [54], etc. Typically, the lower- amplitude constellation points will be more concentrated around the origin. Generalized cross

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figure of merit [53], or minimizing mean-square error of Gaussian distribution [54], etc. Typically, theAppl. lower-amplitude Sci. 2019, 9, x FOR PEER constellation REVIEW points will be more concentrated around the origin. Generalized4 of 13 cross constellations and Voronoi constellations were proposed decades ago [38,39]. GMI-optimized constellations and Voronoi constellations were proposed decades ago [38,39]. GMI-optimized constellations have also been proposed recently to enhance the capacity in a binary FEC coding featured constellations have also been proposed recently to enhance the capacity in a binary FEC coding fiber-optic communication systems. featured fiber-optic communication systems. GS could potentially simplify the process to generate Gaussian-like constellations, but it also brings GS could potentially simplify the process to generate Gaussian-like constellations, but it also severalbrings issues. several Firstly, issues. due Firstly, to the due unavailability to the unavailability of Gray-mapping of Gray-mapping in most cases, in most GMI cases, performance GMI hardlyperformance approach hardly MI performance.approach MI pe Althoughrformance. nonbinary Although nonbinary FEC coding FEC is acoding solution is a solution to close to such close gap, thesuch high gap, complexity the high hinders complexity its commercial hinders its development commercial [55development]. Secondly, [55]. the commonSecondly, asymmetry the common of the constellationsasymmetry mayof the result constellations in more complex may result PAM in constellations more complex in bothPAM in-phaseconstellations and quadrature in both in-phase branches. Therefore,and quadrature the ADC branches./DAC is Therefore, required to the be ADC/DAC implemented is required with higher to be resolutions. implemented Thirdly, with higher the DSP circuitsresolutions. to recover Thirdly, GS-MQAM the DSP circuits formats to arerecover not compatibleGS-MQAM withformats the are ones not tocompatible recover regular-MQAM. with the ones Asto a result,recover new regular-MQAM. DSP algorithms As area result, suggested new toDSP be algorithms developed toare e ffisuggestedciently recover to be developed the GS-MQAM to signals.efficiently Fourthly, recover it isthe hard GS-MQAM to reach signals. standard Fourthly, agreements it is hard between to reach optical standard transceiver agreements manufacturers between dueoptical to the transceiver variety of GS-QAMmanufacturers formats due and to the matchedvariety of DSP GS-QAM algorithms. formats and the matched DSP algorithms.Figure2 shows the conceptual diagram of a GS-QAM-based communication system. If binary FEC codingFigure is 2 used, shows the the suboptimal conceptual B2S diagram mapping of a tableGS-QAM-based has to be found communication to minimize system. the gap If binary between GMIFEC and coding MI, inis orderused, tothe minimize suboptimal the B2S Non-Gray mapping mapping table has penalty. to be found Brutal to forceminimize algorithm the gap may between be used toGMI find suchand MI, mapping in order rule to at minimize a cost of the high Non-Gray computational mapping complexity. penalty. Brutal Alternatively, force algorithm binary data may can be be mappedused to to find the such GS-QAM mapping symbols rule at through a cost of any high B2S computational look-up table, complexity. followed Alternatively, by a reasonable binary complexity data nonbinarycan be mapped FEC encoder to the GS-QAM to sustain symbols reliable through communications any B2S look-up [56]. table, followed by a reasonable complexity nonbinary FEC encoder to sustain reliable communications [56].

FigureFigure 2. 2.The The conceptual conceptual diagram diagram ofof aa geometricgeometric shap shapinging (GS)-QAM-based (GS)-QAM-based communication communication system. system.

FigureFigure3 shows 3 shows 16 /16/32/64QAM32/64QAM formats formats generated generated by by maximizing maximizing constellation constellation ﬁgure figure of of merit merit [53 ], minimizing[53], minimizing mean-square mean-square error of error Gaussian of Gaussian distribution distribution [57], and [57], maximizing and maximizing GMI [41], GMI respectively. [41], Therespectively. GS-16QAM The constellation GS-16QAM shown constellation in Figure shown3a is obtained in Figure by maximizing3a is obtained the by minimum maximizing Euclidean the distanceminimum under Euclidean the same distance average under power the ofsame the av 16-aryerage constellation. power of the 16-ary constellation. Appl. Sci.The 2019 GS-32QAM, 9, x FOR PEER constellation REVIEW shown in Figure 3b can be obtained by the following steps. 5 of 13 1. Choosing 2D Gaussian distribution as the optimal source distribution, and select the uniformly distributed regular-32QAM as the initial constellation. 2. Generating a symbol sequence following the Gaussian distribution. 3. Distributing the symbols into 32 clusters, where the decision is made as per the minimum Euclidean distance from the 32QAM constellation points obtained in previous iteration. 4. Finding the average central positions from the symbols labeled by each cluster. Such 32 points located on the central positions are used as the new MQAM constellation points. 5. Iterating over Steps 2 and 4 until convergence. The GS-64QAM constellation shown in Figure 3b is designed with the constraint of Gray mapping, which can maximize the GMI. Due to the Gray mapping constraint, such scheme cannot fully explore the(a )2D space, but it can also reduce(b capacity) gap towards the Shannon(c) limit. It is an open question about the optimal GS solution, since it depends on lots of implementation scenarios.FigureFigure 3. When3.GS-QAM GS-QAM the formatsamountformats basedbasedof the on constellation ( a)) maximizing points constellation constellation is more figure ﬁgurethan of of64,merit, merit, GS ( bmay) ( bminimizing) minimizing suffer more implementationmean-squaremean-square error penaltieserror of Gaussianof Gaussian than PS. distribution, distribution, and and (c) maximizing(c) maximizing generalized generalized mutual mutual information information (GMI). (GMI).

2.3. Hybrid Probabilistic-Geometric Shaping When more flexible constellation formats are required, especially multi-dimensional QAM formats, HPGS may be an enabling technique,Juniper since Internal each constellation point during the optimization process will not be limited to equal probability or uniform-grid locations. Figure 4 shows HPGS- 5QAM and HPGS-9QAM formats based on Huffman coding [58,59]. The probability of each symbol is predetermined by the corresponding Huffman tree, and the coordinates of the symbols can be obtained by optimizing the MI. As an illustrative example, the generation process of HPGS-9QAM constellation can be found below: 1. Parsing the binary source into nine blocks labeled by unique bit sets {00, 010, 110, 011, 100, 1110, 1111, 1010, 1011}. If the binary sequence is sufficiently long, the resulting blocks should be generated with the probabilities of {P(00) = 1/4, P(010) = 1/8, P(011) = 1/8, P(100) = 1/8, P(1110) = 1/16, P(1111) = 1/16, P(1010) = 1/16, P(1011) = 1/16}. Thereafter, the entropy is 3, i.e., there is no entropy loss. 2. Mapping the 9-block sequence to any 9-QAM symbols with the constraints: (i) The 9-ary constellation points with the same probability are uniformly located in the same power layer, (ii) The constellation points with higher probabilities are located at higher power layers. In other words, such 9-ary constellation should be featured with three power layers and 1, 4, and 4 points are equally spaced in each power layer, respectively. 3. Maximizing the MI by iterating over all amplitude ratios and phase differences of such 9-ary constellation. Huffman coding can be treated as a variable-length and prefix-free PS scheme, which can also be uniquely decodable, but it cannot provide flex rate. There are four major problems for this HPGS scheme. Firstly, it is quite challenging to implement Huffman coding when the amount of symbols increase. Secondly, as any other variable-length coding technology, Huffman coding will also suffer from the overflow or underflow problems. Thirdly, the higher-complexity nonbinary FEC coding is required. Fourthly, error propagation may occur if unexpected symbol errors remained after FEC decoding.

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The GS-32QAM constellation shown in Figure3b can be obtained by the following steps.

1. Choosing 2D Gaussian distribution as the optimal source distribution, and select the uniformly distributed regular-32QAM as the initial constellation. 2. Generating a symbol sequence following the Gaussian distribution. 3. Distributing the symbols into 32 clusters, where the decision is made as per the minimum Euclidean distance from the 32QAM constellation points obtained in previous iteration. 4. Finding the average central positions from the symbols labeled by each cluster. Such 32 points located on the central positions are used as the new MQAM constellation points. 5. Iterating over Steps 2 and 4 until convergence.

The GS-64QAM constellation shown in Figure3b is designed with the constraint of Gray mapping, which can maximize the GMI. Due to the Gray mapping constraint, such scheme cannot fully explore the 2D space, but it can also reduce capacity gap towards the Shannon limit. It is an open question about the optimal GS solution, since it depends on lots of implementation scenarios. When the amount of the constellation points is more than 64, GS may suﬀer more implementation penalties than PS.

2.3. Hybrid Probabilistic-Geometric Shaping When more flexible constellation formats are required, especially multi-dimensional QAM formats, HPGS may be an enabling technique, since each constellation point during the optimization process will not be limited to equal probability or uniform-grid locations. Figure4 shows HPGS-5QAM and HPGS-9QAM formats based on Huffman coding [58,59]. The probability of each symbol is predetermined by the corresponding Huffman tree, and the coordinates of the symbols can be obtained by optimizing the MI. Appl. Sci. 2019, 9, x FOR PEER REVIEW 6 of 13

0 1 100 101 0 1 1010 010 1110 1 0 1 0 0 0 1 1 100 110 0 00 1 0 1 1 0 1 0 1 0 0 1011 011 1111 0 1 110 111

(a) (b) (c) (d)

FigureFigure 4. Hu 4.ff Huffmanman coding coding (a) ( anda) and the the constellation constellation format format ( b(b)) forfor 5-QAM;5-QAM; HuHuffmanffman coding coding (c ()c )and and the the constellation format (d) for 9-QAM. constellation format (d) for 9-QAM. A more efficient and practical way to generate HPGS-QAM is based on universal probabilistic As an illustrative example, the generation process of HPGS-9QAM constellation can be found below: shaping scheme and GMI-optimized GS scheme [44,60]. Figure 5 shows the constellation formats for 1. HPGS-32QAM,Parsing the binary as well source as the into regular/PS-32QAM. nine blocks labeled by unique bit sets {00, 010, 110, 011, 100, 1110, 1111,The 1010, constellation 1011}. If diagram the binary of the sequence HPGS-32QAM is suffi cientlyshown long,in Figure the resulting5a is generated blocks by should the be generalized pairwise optimization algorithm. The objective function is maximizing GMI under the generated with the probabilities of {P(00) = 1/4, P(010) = 1/8, P(011) = 1/8, P(100) = 1/8, P(1110) = constraints of zeros mean amplitude and normalized average power. The two constraints can be P P P expressed1/16, (1111) as = 1/16, (1010) = 1/16, (1011) = 1/16}. Thereafter, the entropy is 3, i.e., there is no entropy loss. 𝑝 𝑥 =−𝐴 −𝑝𝑥 (4) 2. Mapping the 9-block sequence to any 9-QAM symbols with the constraints: (i) The 9-ary 𝑝 constellation points with the same𝑝𝑥 probability+𝑏 + 𝑥 are =1−𝐵 uniformly located in the same power(5) layer, 𝑝 (ii) The constellation points with higher probabilities are located at higher power layers. In other wherewords, (𝑥 such,𝑥) is 9-ary one pair constellation of the M-ary should constellation be featured (𝑀=32 with in three this case), power 𝑝 layers is the andprobability 1, 4, and of 4𝑥 points, ∑ ∑ | | 𝐴=are equally,, spaced𝑝𝑥 , inand each 𝐵= power,, layer, respectively.𝑝 𝑥 . The objective of maximizing GMI will be searching the (𝑥,𝑥) pair over a hypersphere with the center and radius determined by the other 𝑀− 2 constellation points. Such generalized pairwise optimization algorithm can converge to the final steady state after iterating all 𝑀(𝑀 − 1)/2 pairs. As a rule of thumb, we suggest start the iteration with the regular-32QAM, and the optimal HPGS-32QAM shown in Figure 5a can be obtained within 1000 iterations. Such HPGS constellation could provide the trade-off between the number of nearest symbols and their Hamming distance. Any constellation format shown in Figure 5 does not belong to the square QAM category. As a result, the well-known PAS scheme cannot be used for the PS purpose.

(a) (b) (c) (d)

Figure 5. Constellation formats for (a) hybrid probabilistic-geometric shaping (HPGS)-32QAM, (b) regular-32QAM, (c) shallowly shaped 32QAM, and (d) deeply shaped 32QAM.

Figure 6 shows the modified probabilistic fold shaping and universal probabilistic shaping schemes [60]. Probabilistic fold shaping can be used for shaping any F-fold rotationally symmetric constellation, like regular-32QAM, as shown in Figure 6a. In such kind of constellations, the bits determining the fold index yield uniform distribution, which can be used to carry the parity-check

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0 1 100 101 0 1 1010 010 1110 1 0 1 0 0 0 1 1 100 110 0 00 1 0 1 1 0 1 0 1 0 0 1011 011 1111 0 1 110 111

(a) (b) (c) (d)

Figure 4. Huffman coding (a) and the constellation format (b) for 5-QAM; Huffman coding (c) and the constellation format (d) for 9-QAM.

A more efficient and practical way to generate HPGS-QAM is based on universal probabilistic shaping scheme and GMI-optimized GS scheme [44,60]. Figure 5 shows the constellation formats for HPGS-32QAM, as well as the regular/PS-32QAM. The constellation diagram of the HPGS-32QAM shown in Figure 5a is generated by the generalized pairwise optimization algorithm. The objective function is maximizing GMI under the constraints of zeros mean amplitude and normalized average power. The two constraints can be Appl. Sci. 2019 9 expressed as, , 1889 6 of 13

𝑝𝑥 =−𝐴 −𝑝𝑥 (4) 3. Maximizing the MI by iterating over all amplitude ratios and phase differences of such 𝑝 9-ary constellation. 𝑝𝑥 +𝑏 + 𝑥 =1−𝐵 (5) 𝑝 Huffman coding can be treated as a variable-length and prefix-free PS scheme, which can also be where (𝑥,𝑥) is one pair of the M-ary constellation (𝑀=32 in this case), 𝑝 is the probability of 𝑥, uniquely decodable, but it cannot provide flex rate. 𝐴=∑,, 𝑝𝑥 , and 𝐵=∑,, 𝑝|𝑥| . The objective of maximizing GMI will be There are four major problems for this HPGS scheme. Firstly, it is quite challenging to implement searching the (𝑥,𝑥) pair over a hypersphere with the center and radius determined by the other 𝑀− 2Hu constellationffman coding points. when theSuch amount generalized of symbols pairwise increase. optimization Secondly, algorithm as any other can variable-length converge to the coding final steadytechnology, state after Huff maniterating coding all will𝑀(𝑀 also − su 1)/2ff er pairs. from As the a rule overflow of thumb, or underflow we suggest problems. start the iteration Thirdly, withthe higher-complexity the regular-32QAM, nonbinary and the optimal FEC coding HPGS-32QAM is required. shown Fourthly, in Figure error propagation5a can be obtained may occur within if 1000unexpected iterations. symbol errors remained after FEC decoding. SuchA more HPGS efficient constellation and practical could way provide to generate the trade-off HPGS-QAM between is basedthe number on universal of nearest probabilistic symbols andshaping their scheme Hamming and GMI-optimizeddistance. Any constellation GS scheme [ 44format,60]. Figureshown5 showsin Figure the 5 constellation does not belong formats to forthe squareHPGS-32QAM, QAM category. as well As as thea result, regular the/PS-32QAM. well-known PAS scheme cannot be used for the PS purpose.

(a) (b) (c) (d)

Figure 5. 5. ConstellationConstellation formats formats for for (a)( ahybrid) hybrid probabilistic-geometric probabilistic-geometric shaping shaping (HPGS)-32QAM, (HPGS)-32QAM, (b) regular-32QAM,(b) regular-32QAM, (c) shallowly (c) shallowly shaped shaped 32QAM, 32QAM, and and (d) (deeplyd) deeply shaped shaped 32QAM. 32QAM.

FigureThe constellation 6 shows the diagram modified of the probabilistic HPGS-32QAM fold shown shaping in Figure and5 auniversal is generated probabilistic by the generalized shaping schemespairwise [60]. optimization Probabilistic algorithm. fold shaping The objective can be functionused for isshaping maximizing any F GMI-fold underrotationally the constraints symmetric of constellation,zeros mean amplitude like regular-32QAM, and normalized as shown average in power. Figure The 6a. twoIn such constraints kind of can constellations, be expressed the as bits determining the fold index yield uniform distribution, which can be used to carry the parity-check p x = A p x (4) i i − − j j

p 2 j 2 pjxj + b + xj = 1 B (5) p 2 − Juniper Internali where (xi, xj) is one pair of the M-ary constellation (M = 32 in this case), pi is the probability of xi, M M P P 2 A = pkxk, and B = pk xk . The objective of maximizing GMI will be searching k=1,k,i,k,j k=1,k,i,k,j | | the (x , x ) pair over a hypersphere with the center and radius determined by the other M 2 i j − constellation points. Such generalized pairwise optimization algorithm can converge to the final steady state after iterating all M(M 1)/2 pairs. As a rule of thumb, we suggest start the iteration − with the regular-32QAM, and the optimal HPGS-32QAM shown in Figure5a can be obtained within 1000 iterations. Such HPGS constellation could provide the trade-off between the number of nearest symbols and their Hamming distance. Any constellation format shown in Figure5 does not belong to the square QAM category. As a result, the well-known PAS scheme cannot be used for the PS purpose. Figure6 shows the modified probabilistic fold shaping and universal probabilistic shaping schemes [60]. Probabilistic fold shaping can be used for shaping any F-fold rotationally symmetric constellation, like regular-32QAM, as shown in Figure6a. In such kind of constellations, the bits determining the fold index yield uniform distribution, which can be used to carry the parity-check bits. There is a major difference between PAS and probabilistic fold shaping. PAS is a 1D shaping scheme, which uses the single bit determining the positive or negative amplitude to carry the parity-check bit Appl. Sci. 2019, 9, x FOR PEER REVIEW 7 of 13

bits. There is a major difference between PAS and probabilistic fold shaping. PAS is a 1D shaping scheme, which uses the single bit determining the positive or negative amplitude to carry the parity- check bit and shapes the 1D-PAM with a 1D M–B distribution. On the contrary, probabilistic fold

shaping is a 2D shaping scheme, which uses the log(𝐹) bits determining the fold indexes to carry the parity-check bits, and shapes the 2D constellation points in one fold with a 2D M–B distribution. As an illustrative example, the 32QAM shown in Figure 6a is featured with four-fold rotational symmetry. The 8-ary constellation points in one fold are firstly shaped with a 2D M–B distribution, where different colors indicate different probabilities. The bit sets {11,01,00,10} determining the fold indexes will carry the parity-check bit after FEC encoding and rotate the 8-ary constellation by 0°, 90°, 180°, 270°, respectively. Therefore, 2D M–B distribution can be applied to the constellation points in Appl. Sci. 2019, 9, 1889 7 of 13 one fold, and the selection of the fold-index can be performed by the parity-check bits. The target distribution will not be changed after the binary FEC coding. and shapesThe universal the 1D-PAM probabilistic with a 1D shaping M–B distribution. scheme show Onn thein Figure contrary, 6b probabilisticcan be applied fold to shaping shape any is a QAM format. The MQAM symbols generated from the CCDM may not yield a M–B distribution. The 2D shaping scheme, which uses the log2(F) bits determining the fold indexes to carry the parity-check bits,binary and bits shapes generated the 2D after constellation the bit labeling points inblock one ar folde used with to a carry 2D M–B the distribution.uniformly distributed As an illustrative parity- example,check bits. the During 32QAM the shown process in Figureof B2S 6mapping,a is featured the withparity-check four-fold bits rotational will be uniformly symmetry. mapped The 8-ary to constellationpartial MQAM points symbols, in one i.e., fold N are-ary firstly QAM shaped (NQAM) with symbols, a 2D M–B where distribution, N is the wherelargest dipowerfferent of colors 2 to indicatecontain dithefferent MQAM probabilities. constellation The points bit sets with {11,01,00,10} the desirable determining probabilities the foldof >(1 indexes − 𝑅)/𝑀 will. Assuming carry the that the desirable probability distribution of the MQAM symbols is 𝑃(𝑥) , the probability parity-check bit after FEC encoding and rotate the 8-ary constellation by 0◦, 90◦, 180◦, 270◦, respectively. 𝑃 (𝑥) Therefore,distribution 2D M–Bof the distribution MQAM symbols can be appliedafter the to CCDM the constellation is , and points probability in one fold, distribution and the selection of the 𝑃 (𝑦) =1/𝑁 𝑃 (𝑥) = ofNQAM the fold-index symbols can is be given performed by by the parity-check. The final bits. relationship The target distribution can be written will not as be changed [𝑅𝑃 (𝑥)⁄log (𝑀) +(1−𝑅)𝑃 (𝑦)⁄]log (𝑁) /[𝑅⁄ log (𝑀) +(1−𝑅)⁄ log (𝑁)] . In such a way, any after the binary FEC coding. desirable distribution of the HPGS-MQAM symbols can be obtained.

001 000 01 00010 00011 10011 10010 11

100 110 010 00110 01110 01010 11010 11110 10110

101 111 011 00111 01111 01011 11011 11111 10111 Bit Bit-sequence Bit-to-symbol R labeling concatenation mapping 10... 10...11... 10...11...01… 00101 01101 01001 11001 11101 10101

00100 01100 01000 11000 11100 10100 Binary 11... Parity-check bits 2D-DM P 1 − R data 01… 0000000 00001 10001 10000 10

(a)

2D- Symbol-sequence 001 000 001/00 000 DM concatenation 010 MQAM 010 110 00 100 100/10 110 011 Parity- 10 011/01 101 101 Binary Bit check bits Bit-to-symbol NQAM 01 P 111 data labeling 11...1 1001…. mapping 11 111/11

(b)

FigureFigure 6. 6.Modiﬁed Modified probabilisticprobabilistic shaping shaping (PS) (PS) schemes schemes based based on ( ona) probabilistic (a) probabilistic fold shaping, fold shaping, (b) (b) universal PS. universal PS.

3. PerformanceThe universal Comparison probabilistic in Gaussian-Noise-Limited shaping scheme shown inChannels Figure6 b can be applied to shape any QAMAs format. a rule The of MQAMthumb, symbolsHPGS-MQAM generated cannot from show the clear CCDM performance may not yield improvement a M–B distribution. over PS- TheMQAM binary and bits GS-MQAM generated when after M the < 32. bit In labeling addition, block if M are ≥ 64, used PS-MQAM to carry could the uniformly closely approach distributed the parity-checkShannon limit, bits. thus During it is not the necessary process to of apply B2S mapping,HPGS to higher the parity-check order QAM bitsformats. will In be this uniformly paper, mappedwe performed to partial Monte MQAM Carlo symbols, simulations i.e., inN MATL-ary QAMAB. The (NQAM) block-length symbols, of the where CCDMN is was the chosen largest power of 2 to contain the MQAM constellation points with the desirable probabilities of >(1 R)/M. more than 5000, so the normalized divergence of the encoder output and the desired distribution− is Assuming that the desirable probability distribution of the MQAM symbols is P (x), the probability M distribution of the MQAM symbols after the CCDM is PD(x), and probability distribution of the NQAM symbols is given by P (y) = 1/N. The ﬁnal relationship can be written as h N Juniper Internali i PM(x) = RPD(x)/ log (M) + (1 R)PN(y)/ log (N) /[R/ log (M) + 1 R)/ log (N) . In such a 2 − 2 2 − 2 way, any desirable distribution of the HPGS-MQAM symbols can be obtained.

3. Performance Comparison in Gaussian-Noise-Limited Channels As a rule of thumb, HPGS-MQAM cannot show clear performance improvement over PS-MQAM and GS-MQAM when M < 32. In addition, if M 64, PS-MQAM could closely approach the Shannon ≥ limit, thus it is not necessary to apply HPGS to higher order QAM formats. In this paper, we performed Monte Carlo simulations in MATLAB. The block-length of the CCDM was chosen more than 5000, Appl. Sci. 2019, 9, 1889 8 of 13 so the normalized divergence of the encoder output and the desired distribution is negligible. The awgn functionAppl. Sci. provided 2019, 9, x FOR by PEER MATLAB REVIEW was used to add white Gaussian noise to the 2D-MQAM8 of signals.13 All thenegligible. results wereThe awgn averaged function over provided 1000 trials. by MATLAB was used to add white Gaussian noise to the In2D-MQAM a numerical signals. simulation, All the results as were shown averaged in Figure over 10007, we trials. compared the MI performances of the minimizingIn a numerical mean-square simulation, error as of sh Gaussian-distribution-basedown in Figure 7, we compared GS-8the MI/16 /performances32QAM, CCDM-based of the PS-8/minimizing16/32QAM, mean-square and regular-8 error/16 /of32QAM. Gaussian-distribution-based Figure7a1,a2,b1,b2,c1,c2 GS-8/16/32QAM, show the constellation CCDM-based diagrams PS- of PS8/16/32QAM,/GS-8/16/32QAM. and regular-8/16/32QAM. Here we chose MI Figure as the 7a1,a2,b1,b2,c1,c2 metric for performance show the constellation comparison, diagrams because of MI can bePS/GS-8/16/32QAM. measured without Here the we considerationchose MI as the ofmetric the for FEC performance coding performance. comparison, because Given MI that can mostbe of the currentmeasured nonbinary without the FEC consideration coding and of binarythe FEC FECcoding coding performance. schemes Given may that vary most in performanceof the current and nonbinary FEC coding and binary FEC coding schemes may vary in performance and implementation penalty, MI instead shows the upper limit of the capacity obtained with the “ideal” implementation penalty, MI instead shows the upper limit of the capacity obtained with the “ideal” FEC coding scheme [61]. In order to easily describe the PS-MQAM with an entropy of A b/s, we adopt FEC coding scheme [61]. In order to easily describe the PS-MQAM with an entropy of A b/s, we adopt the notationthe notation of PS-MQAM of PS-MQAMA. ForA. example,For example, we usewe PS-8QAM2.3use PS-8QAM2.3 to denote to denote the the PS-8QAM PS-8QAM with with an entropyan of 2.3entropy b/s. As of we 2.3 can b/s. see As fromwe can Figure see from7a,b, Figure the best 7a,b, MI the performances best MI performances can be obtainedcan be obtained by GS-8 by /GS-16QAM. GS-8/8/16QAM.16QAM can GS-8/16QAM always outperform can always regular-8 outperform/16QAM. regular-8/16QAM. PS-8QAM PS-8QAM can have comparablecan have comparable performance overperformance GS-8QAM whenover GS-8QAM the SNR when is less the than SNR 6.2 is dB;less thethan performances 6.2 dB; the performances of PS-16QAM of PS-16QAM and GS-16QAM and are quiteGS-16QAM similar are when quite thesimilar SNR when is less the than SNR 11.7is less dB. than In 11.7 the dB. region In the of region high of SNR high region, SNR region, PS-8/ PS-16QAM cannot8/16QAM bring better cannot MI bring performance. better MI performance. In Figure7 Inc, Figure we ﬁnd 7c, that we find the bestthat the performance best performance can be can achieved be by PSachieved/GS-32QAM by PS/GS-32QAM separately. separately. In addition, In GS-32QAMaddition, GS-32QAM is always is always better thanbetter regular-32QAMthan regular-32QAM in terms in terms of MI performance. When the SNR is more than 15.7 dB, GS-32QAM has the best of MI performance. When the SNR is more than 15.7 dB, GS-32QAM has the best performance, performance, while PS-32QAM formats can maximize the MI performance when the SNR is less than while PS-32QAM formats can maximize the MI performance when the SNR is less than 15.7 dB. 15.7 dB.

2.8 Shannon limit GS-8QAM 2.7 R-8QAM PS-8QAM2.875 PS-8QAM2.75 2.6 PS-8QAM2.625 PS-8QAM2.5 2.5

2.4 MI [bits/symbol]

2.3

2.2

6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 SNR [dB] (a)

3.8

3.6

3.4

3.2 Shannon limit GS-16QAM

MI MI [bits/symbol] 3.0 R-16QAM PS-16QAM3.875 PS-16QAM3.75 2.8 PS-16QAM3.625 PS-16QAM3.5 2.6 7 8 9 10 11 12 13 14 SNR [dB] (b)

Figure 7. Cont.

Juniper Internal Appl. Sci. 2019, 9, 1889 9 of 13

Appl. Sci. 2019, 9, x FOR PEER REVIEW 9 of 13

4.8

4.6

4.4

4.2

4.0 Shannon limit GS-32QAM 3.8 R-32QAM MI[bits/symbol] PS-32QAM4.875 3.6 PS-32QAM4.75 PS-32QAM4.625 3.4 PS-32QAM4.5

3.2 10 11 12 13 14 15 16 SNR [dB] (c)

FigureFigure 7. Mutual 7. Mutual information information (MI) (MI) versus versus signal-to-noisesignal-to-noise ratio ratio (SNR) (SNR) performance performance in Gaussian-noise- in Gaussian-noise- limitedlimited channels channels for for (a) (a PS) PS/GS/R-8QAM,/GS/R-8QAM, ((bb)) PSPS/GS/R-16QAM,/GS/R-16QAM, (c) ( cPS/GS/R-32QAM.) PS/GS/R-32QAM. Insets Insets (a1,a2) (a1,a2) The The constellation diagrams of PS-8QAM and GS-8QAM. Insets (b1,b2) The constellation diagrams of PS- constellation diagrams of PS-8QAM and GS-8QAM. Insets (b1,b2) The constellation diagrams of 16QAM and GS-16QAM. Insets (c1,c2) The constellation diagrams of PS-32QAM and GS-32QAM. PS-16QAM and GS-16QAM. Insets (c1,c2) The constellation diagrams of PS-32QAM and GS-32QAM. Although PS-256QAM and PS-1024QAM have been investigated over fiber links [62], we still believeAlthough that PS-256QAMPS-64QAM is andexpected PS-1024QAM to be the most have promising been investigated solution for over high-order fiber links QAM-based [62], we still believefiber-optic that PS-64QAM communications, is expected at least toforbe the the upcoming most promising 400 G/800 G solution Ethernet.for It is high-order mainly because QAM-based that fiber-opticPS-64QAM communications, has near-capacity-approaching at least for the performance, upcoming and 400 its G /implementation800 G Ethernet. penalty It is may mainly also becausebe thatwell PS-64QAM reduced hasto an near-capacity-approaching acceptable level in the near future. performance, While, unfortunately, and its implementation regular-64QAM penaltyand PS- may also be64QAM well reducedhave been to ansuffering acceptable a large level implementa in the neartion future. penalty While, so far. unfortunately, Here we compare regular-64QAM the and PS-64QAMperformances haveof PAS-64QAM been suff eringand GM aI-optimized large implementation HPGS-32QAM penalty (referred so to far. as opti-32QAM Here we compare in this the performancespaper). of PAS-64QAM and GMI-optimized HPGS-32QAM (referred to as opti-32QAM in this paper).In another numerical simulation, Figure 8 shows the post-FEC bit error rate (BER) versus SNR performances of PAS-64QAM, opti-32QAM, PS-32QAM, and regular-32QAM. DVB-S2 irregular low- In another numerical simulation, Figure8 shows the post-FEC bit error rate (BER) versus SNR density parity-check (LDPC) codes were applied for FEC coding. The performance comparisons were performancesexecuted under of PAS-64QAM, the same capacity opti-32QAM, levels, i.e., C PS-32QAM, = 3.33 b/s and and C = 4 regular-32QAM. b/s. Table 1 lists the DVB-S2 parameters irregular low-densityused in the parity-check post-FEC BER (LDPC) analysis codes under were the applied capacity for levels FEC of coding. 3.33 b/s Theand performance4 b/s. In Figure comparisons 8a, the wereperformance executed under of PS-32QAM the same capacityis slightly levels, better i.e.,thanC opti-32QAM,= 3.33 b/s and butC worse= 4 b/ s.than Table PAS-64QAM1 lists the parametersby 0.2 useddB. in Opti-32QAM the post-FEC shows BER a analysis0.8-dB performance under the im capacityprovement levels over ofregular-32QAM 3.33 b/s and in 4 case b/s. of InC = Figure 3.33 8a, the performanceb/s. In Figure 8b, of PS-32QAMthe performance is slightly of opti-32QAM better thanis better opti-32QAM, than PS-32QAM but worseby 0.2 dB, than better PAS-64QAM than by 0.2regular-32QAM dB. Opti-32QAM by 0.8 shows dB when a 0.8-dB the capacity performance is 4 b/s. improvement However, PAS-64QAM over regular-32QAM is also shown in to case of C = 3.33outperform b/s. In Figureopti-32QAM8b, the by performance 0.4 dB. It is reasonable of opti-32QAM to find that is betterPAS-64QAM than PS-32QAM always has the by best 0.2 dB,post- better thanFEC regular-32QAM performance over by 0.8shaped dB when32QAM. the While capacity in a re isalistic 4 b/ s.communication However, PAS-64QAM system, we believe is also that shown the performance of HPGS-32QAM (opti-32QAM) should be similar to that of PAS-64QAM, due to to outperform opti-32QAM by 0.4 dB. It is reasonable to find that PAS-64QAM always has the best the higher implementation penalties that PAS-64QAM may suffer. post-FEC performance over shaped 32QAM. While in a realistic communication system, we believe that the performanceTable 1. The parameters of HPGS-32QAM used in the post-forward (opti-32QAM) error should correction be (FEC) similar bit error to that rate of (BER) PAS-64QAM, analysis due to the higher implementation penalties thatunder PAS-64QAM the same capacity. may su ffer. C [b/s] H/R R-32QAM Opti-32QAM PS-32QAM PAS-64QAM Table 1. The parameters used in the post-forward error correction (FEC) bit error rate (BER) analysis 𝐻(𝑝) 5 5 4.33 4.53 under the3.33 same capacity. 𝑅 2/3 2/3 4/5 4/5 𝐻(𝑝) 5 5 4.55 5.2 C 4[b /s] H/R R-32QAM Opti-32QAM PS-32QAM PAS-64QAM 𝑅 4/5 4/5 8/9 4/5 H(p) 5 5 4.33 4.53 3.33 R 2/3 2/3 4/5 4/5 H(p) 5 5 4.55 5.2 4 R 4/5 4/5 8/9 4/5

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Opti-32QAM 0.01 0.01 PS-32QAM Opti-32QAM R-32QAM PS-32QAM PAS-64QAM R-32QAM PAS-64QAM BER BER

1E-3 1E-3

1E-4 1E-4 10.4 10.6 10.8 11.0 11.2 11.4 12.4 12.6 12.8 13.0 13.2 13.4 13.6 SNR [dB] SNR [dB] (a) (b)

Figure 8. BER versus SNR performances in the scenarios of ( a) C = 3.333.33 b/s, b/s, ( (bb)) CC == 44 b/s. b/s.

4.4. Concluding Concluding Remarks Remarks ConstellationConstellation shaping willwill playplay an an increasingly increasingly important important role role in in fiber-optic fiber-optic communications communications in the in thewake wake of the of boomingthe booming 5G era. 5G Inera. this In paper, this paper, we focused we focused on the on performance the performance of 2D constellation of 2D constellation shaping shapingin Gaussian-noise-limited in Gaussian-noise-limited channels. channels. We have discussedWe have threediscussed key constellation three key constellation shaping schemes, shaping i.e., schemes,PS, GS, and i.e., HPGS, PS, GS, and and analyzed HPGS, theirand analyzed pros and constheir inpros terms and of cons performance in terms andof performance implementation and implementationcomplexity. We alsocomplexity. introduced We twoalso modifiedintroduced CCDM-based two modified shaping CCDM-based schemes, shaping i.e., probabilistic schemes, foldi.e., probabilisticshaping and universalfold shaping PS, whichand universal could enable PS, which applying could PS onenable any 2D applying modulation PS on format. any 2D We modulation found that format.GS-8QAM We and found GS-16QAM that GS-8QAM could outperform and GS-16QAM PS/regular-8QAM could outperform and PS-regular-16QAM, PS/regular-8QAM respectively, and PS- regular-16QAM,in terms of MI performance. respectively, Inin addition,terms of MI the performance. best MI performance In addition, of 32-ary the best QAM MI formatperformance could beof 32-aryreached QAM by PS-32QAM format could and be GS-32QAM reached by separately. PS-32QAM We and compared GS-32QAM the post-FEC separately. BER We performances compared the of post-FECHPGS/PS /BERregular-32QAM performances and of PAS-64QAMHPGS/PS/regular- under32QAM the same and capacity. PAS-64QAM under the same capacity. TheThe performancesperformances ofof HPGS HPGS/PS-32QAM/PS-32QAM were were shown show ton be to similar, be similar, and better and thanbetter regular-32QAM. than regular- 32QAM.What is more,What PAS-64QAMis more, PAS-64QAM could still could have 0.2–0.4still have dB 0.2–0.4 performance dB pe gainsrformance over HPGSgains /overPS-32QAM. HPGS/PS- 32QAM.For a commercial CCDM-based PS implementation, the power consumption is mainly determined by theFor block a commercial length and theCCDM-based entropy of the PS CCDM, implementa as welltion, as FEC the coding power selection. consumption GS-QAM is formatsmainly determinedwill cost more by power the block consumption length and than the regular-QAM entropy of the formats, CCDM, because as well of as that FEC a higher coding complexity selection. DSPGS- QAMcircuit formats is required will tocost be more used power for GS-QAM-based consumption than transceivers regular-QAM to recover formats, the signal because from of that the receiveda higher complexitydata with Gaussian-like DSP circuit is distribution. required to Althoughbe used for the GS-QAM-based best performance transceivers can be theoretically to recover achieved the signal by fromHPGS-QAM, the received the extra data power with consumption Gaussian-like arising distribution. from PS andAlthough GS schemes the be isst nontrivial. performance Considering can be theoreticallythat both PS achieved and GS can by closelyHPGS-QAM, approach the theextra Shannon power consumption limit and only arising limited from margin PS and can GS be schemes reached isby nontrivial. HPGS, HPGS Considering may not bethat in favorboth PS by theand industry GS can dueclosely to the approach low benefit–cost the Shannon ratio. limit and only limited margin can be reached by HPGS, HPGS may not be in favor by the industry due to the low Author Contributions: This research was supervised by I.B.D. and J.A. All works were done by Z.Q. benefit–cost ratio. Conflicts of Interest: The authors declare no conflict of interest. Author Contributions: This research was supervised by I.B.D. and J.A. All works were done by Z.Q. References Conflicts of Interest: The authors declare no conflict of interest. 1. Zhalehpour, S.; Lin, J.; Guo, M.; Sepehrian, H.; Zhang, Z.; Rusch, L.A.; Shi, W. All-silicon IQ modulator Referencesfor 100 GBaud 32QAM transmissions. In Proceedings of the Optical Fiber Communication Conference, San Diego, CA, USA, 3–7 March 2019; p. Th4A.5. 1. Zhalehpour, S.; Lin, J.; Guo, M.; Sepehrian, H.; Zhang, Z.; Rusch, L.A.; Shi, W. All-silicon IQ modulator for 2. 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