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On line rates, information rates, and spectral efficiencies in probabilistically shaped QAM systems

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Abstract

We carefully revisit the definitions of line rates, information rates, and spectral efficiencies in probabilistically shaped optical transmission systems. Generally accepted definitions for uniform quadrature amplitude modulation (QAM) systems are extended to more generally apply to systems with probabilistically shaped QAM, as well as to systems using pilot symbols of different QAM order than the information symbols. Based on the proper definitions, we correct erroneous claims in a recently reported work.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Probabilistic shaping (PS) of quadrature amplitude modulation (QAM) has recently attracted significant attention and has resulted in several record claims in optical fiber transmission [1–6] in terms of line rates, information rates, and spectral efficiencies. However, properly interpreting seemingly obvious experimental results is not always straightforward for PS-QAM and has even led to erroneous record claims [4]. It is the aim of this paper to properly define for an optical systems experimentalist such basic metrics as line rate, information rate, and spectral efficiency in the context of PS-QAM systems. The notation used in this paper is summarized in Table 1.

Tables Icon

Table 1. List of Symbols and Notation

2. Definition of rates and overheads in a PS-QAM system

In general, information and communications engineering defines the (dimensionless) rateRof a system (or subsystem) as the ratio of the physical bit rate entering the system (or subsystem) to the physical bit rate leaving the system (or subsystem), as shown in Fig. 1(a). Optical communications have historically preferred the equivalent notion of an overhead OH, defined as the ratio of the additional physical bit rate (b/s) added by a system (or subsystem) to the physical bit rate entering the system (or subsystem), as illustrated in Fig. 1(b), withR=1/(1+OH) and OH=(1R)/R. Note that the term “rate” is used in a dimensionless context(R) as well as in a physical context (s−1). The rates and overheads for a PS-QAM system and its constituent subsystems will be discussed in what follows.

 figure: Fig. 1

Fig. 1 Block diagram defining (a) the rateR, and (b) the overheadOH.

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In the context of PS-QAM, we first need to acknowledge that the only practically feasible capacity-approaching PS-QAM architecture known today is based on probabilistic amplitude shaping (PAS) [7]. Hence, a PS M 2-QAM transmitter consists of two orthogonally multiplexed PS M-ary pulse amplitude modulation (PAM) transmitters, as shown in Fig. 2. The binary signal path of each M-PAM transmitter has three components: (i) a distribution matcher (DM) that transforms independent and identically distributed (i.i.d.) input bits to non-uniformly distributed bits that form a PS M-PAM symbol sequence with a desired symbol amplitude distribution after modulation, (ii) a forward error correction (FEC) encoder, and (iii) a framer that adds meta data such as pilot bits (that become pilot symbols after symbol mapping) to aid coherent digital signal processing (DSP) at the receiver.

 figure: Fig. 2

Fig. 2 Data flow within the PAS architecture.

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2.1 Distribution matcher

In a first step, the DM generates the shaped positive-amplitude portion of the constellation symbols Xof an M-PAM constellation following a probability distributionX(x) with corresponding entropy (X); for example, the DM creates X(x)={0.537,0.322,0.116, 0.025}on X={1,3,5,7}, yielding (X)1.5. The resulting information rate β of this shaped half-PAM constellation is β(X)1m1 b/symbol, with m=log2M(M = 8 in the example in Fig. 2). Each shaped half-PAM symbol is represented bym1 b/symbol at the DM output. The DM rate, defined as the ratio of the physical bit rate entering the DM to the physical bit rate leaving the DM, is therefore (see Fig. 2)

RDM=βm11.

A variety of DMs exist, and even time-division hybrid modulation (TDHM) [8] may be thought of as a DM [9], albeit based on a serial concatenation of shaping and FEC as opposed to a PAS architecture. Under an average transmit energy constraint, (X) is maximized by symbols XMB that follow a Maxwell-Boltzmann (MB) distribution, and an ideal DM achieves a maximum information rate ofβ*=(XMB)1. Any non-ideal DM yieldsβ=β*Δβ with arate lossΔβ>0. Finding a practically implementable DM with minimum rate loss is key to the design of PS systems, such as the constant composition DM (CCDM) [10], with vanishing rate loss Δβasymptotically with increasing block length.

2.2 Forward error correction

For each DM-generated half-PAM symbol, the negative-amplitude values are generated by one uniformly distributed sign bit ( ± 1) per symbol that multiplies the positive-amplitude half-PAM symbols. The sign bit carries the FEC overhead. This is done to separate shaping and coding operations, which is at the conceptual heart of the PAS architecture. As a consequence, however, at most 1 out of m b/symbol after the FEC encoder may carry FEC overhead, so the FEC code rateRc,defined as usual as the ratio of the aggregate bit rate entering the FEC encoder to the aggregate bit rate leaving the FEC encoder, is constrained toRc(m1)/m. If the FEC code rate is chosen higher than (m1)/m, such that the FEC overhead does not fully exhaust an entire bit per symbol, a fraction 0γ1 of the sign bit may be allocated to carry information, cf. Figure 2; in this case, γ and 1γ denote the fractions of information content and FEC overhead within the sign bit, respectively (The notion of “fractions of bits” assumes a block structure over which shaping and coding is performed, as opposed to a symbol-by-symbol allocation of information and overhead bits). The FEC code rate is then related to γ by

Rc=m1+γmm1m.
Note from Eq. (2) that smaller FEC code rates than (m1)/m, violate the code rate constraint, and hence are not permitted in the PAS architecture. For example, the concatenated FEC code rate of ~0.78 assumed in [4] does not fulfill the necessary condition Rc4/5 for PS 1024-QAM (i.e., two orthogonally multiplexed PS 32-PAMs), invalidating the claimed results based on a practically implementable shaping architecture.

2.3 Framing and pilot insertion

After shaping and FEC encoding, some symbols may be added periodically, e.g., as pilots to aid receive-side DSP. Pilot symbols may use a (typically unshaped) N-PAM constellation with n=log2Nb/symbol, different from the constellation used by the data symbols (typicallynm). The subsequent bit-to-symbol mapper then interprets the periodically inserted pilot bits differently from the coded and shaped information bits. In addition to pilot symbols, the module may also periodically insert symbols for other purposes. We therefore generically call this module “framer” (Note that the term “framer” does not refer to optical transport network (OTN) framing, which is performed prior to the binary bit stream entering Fig. 2 from the left). If we assume that (1ρFr) n-bit pilot or framing symbols are added perρFr m-bit information symbols, the bit rate of the digital logic after the framer is [mρFr+n(1ρFr)]rs,with rsbeing the symbol rate used on the channel. Defining the framing rate RFras the ratio of the physical bit rate entering the framer to the physical bit rate leaving the framer, we have

RFr=mρFrmρFr+n(1ρFr).

2.4 Overall system

With the above three constituent subsystem rates defined, the overall rate of the PS M-PAM system, defined as the ratio of the physical bit rate entering the single-quadrature processing chain to the physical bit rate entering the mapper, is

RSys=β+γmRFr=ρPASRDMRcRFr.
Compared to a (hypothetical as presently non-implementable) serial transmitter architecture including DM, FEC, and framer, Eq. (4) shows that the overall rate of the PAS architecture features a multiplicative correction factorρPAS(γ/RDM+m1)/(γ+m1) for RDM>0. Assuming a serial transmitter architecture implies an as of yet unknown FEC scheme that retains the previously generated non-uniform symbol distribution, and a de-shaping module at the receiver prior to FEC decoding that does not introduce significant errors. Such a serial scheme only exists for very much suboptimum “shaping” solutions like TDHM [8]. Note that ρPASRDM=(β+γ)/(m1+γ)represents the physical bit rate entering the FEC encoder. Alternatively, we have
RSys=RShρcRFr,
with the effective code rate 𝜌c relating a serial shaping architecture with shaping rateRSh=(β+1)/m1 and a PAS architecture asρc(β+γ)/(β+1)=1(1Rc)/RSh,cf. Figure 3(a). The shaping rate RSh is the ratio of the bit rate before PS to the bit rate after PS for the entire PAM constellation, while RDM is defined for the half-PAM constellation. Note that the overall system rate RSysRShRcRFr, as used in [4], since this erroneously assumes a simple serial concatenation of DM, FEC, and framing modules; i.e., we should substitute m=5,β=2.38,Rc=(5/6)×(16/17), andRFr=32/33(assuming that the pilots used the same shaping as the data signals) of [4] in Eq. (5) to obtainRSh=0.676, ρc0.6809, hence RSys0.4464. Ignoring Eqs. (4) and (5) while invoking the PAS architecture leads to a significant overestimation of system claims, as discussed below.

 figure: Fig. 3

Fig. 3 (a) Rates, and (b) overheads of a PS QAM system and its subsystems, shown in a serially concatenated system model.

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Figure 4(a) shows the effective code rate 𝜌c for the PAS architecture with PS 8-PAM (generating PS 64-QAM) with various DM rates RDM, assuming an ideal DM (Δβ=0). The code rate constraint isRc2/3, cf. Equation (2), leading to the non-permissible code rate regions shown in gray. As RDM decreases (i.e., with more shaping; RDM=0 implies quadrature phase shift keying), and as Rc decreases (i.e., with a larger coding overhead), the effective code rate ρc becomes much smaller than the true rate Rc of the underlying FEC code. Figure 4(b) shows the corresponding overall system rate RSys, assuming RFr=1 (i.e., no pilot symbols). As RDM decreases, and as Rc decreases, the overall system rate is significantly overestimated by the incorrect computation RShRc (dashed) relative to the correct computationRShρc (solid).

 figure: Fig. 4

Fig. 4 Rates of PS 8-PAM: (a) the effective code ratesρc as a function of the true code rates Rcwith various RDM,and (b) the overall system rates RSyscomputed incorrectly byRShRc(dashed) and correctly by RShρc(solid).

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As an alternative to the above defined shaping rate, code rate, and framing rate, we can also define the three system elements in terms of their respective overheads asOH1/R1 (cf. Figures 1(a) and 1(b)). In complete analogy to the rates, the three overheads of a PS PAM transmitter also cannot be simply multiplied using OHcbut rather have to be obtained using the effective FEC overhead ohc1/ρc1as (cf. Figure 3(b))

OHSys=(1+OHSh)(1+ohc)(1+OHFr)1,
or using an additive correction factorαPASγ/(β+γ)OHDM as

OHSys=(1+OHDM+αPAS)(1+OHc)(1+OHFr)1.

3. Line rate, information rate, and spectral efficiency in a PS-QAM system

After modulation to M 2-QAM, the achieved information rate (IR), i.e., the net payload bit rate, is (cf. Table 2)

RInfo=2vmRSysrsb/s,
where v{1,2} accounts for polarization multiplexing, and the underlying FEC is assumed to produce error-free decoding. Error-free decoding can be verified using the normalized generalized mutual information (NGMI), which has been shown to be a reliable FEC threshold for uniform as well as PS-QAM based on empirical studies [11,12]. The operational meaning of NGMI as an achievable binary code rate (ABCR) has been shown in [13], based on rigorous information-theoretic analysis. As a result, the IR reported in [4] is not 680 Gb/s as reported, but 589 Gb/s (v=2,m=5,RSys0.4464,andrs=66×109), disregarding the fact that the code rate condition (2) was violated.

Tables Icon

Table 2. Bit Rates within the PAS Architecture (cf. Figure 2)

Any claim of a spectral efficiency (SE) first needs to correctly assess the IR. Strictly speaking, an SE can only be defined in a wavelength-division multiplexed (WDM) scenario with channel spacing of F Hz as

SE=RInfo/Fb/s/Hz.
If a single channel transmission experiment is only performed using a root-raised cosine (RRC) pulse shaping, the smallest WDM channel spacing that does not introduce excessive inter-channel interference can be calculated asF=rs(1+ρRRC) Hz, whereρRRCdenotes the roll-off factor of the RRC filter, optimistically assuming that this WDM spacing is achieved without any additional guard band (The significant discrepancy between claiming such a “potential” SE versus an actual SE obtained in a WDM experiment is clearly evident when comparing a single-channel experiment [5] and a WDM experiment [6], obtained under the same system conditions). The system reported in [4] uses ρRRC=0.1, leading to a potential SE of 8.11 b/s/Hz withRInfo=589 Gb/s and F=72.6 GHz, but not the SE of 9.35 b/s/Hz as reported, disregarding the fact that the code rate condition (2) was violated and that no WDM experiment was done.

In addition to IRs, optical communications systems frequently quote line rates to indicate the speed of modulation on the channel and compare systems at equal raw speeds, independent of FEC or framing overheads; i.e., the line rate is the gross channel bit rate that includes information and all overheads [14]. For uniform M 2-QAM symbols X and pilot symbols XFr using the same constellation size (n=m), the line rate is defined asRLine=2vmrs. However, this definition cannot be simply applied to PS-QAM, since the entropy rate of PS-QAM varies with the shaping factor of the underlying symbol distribution, leading to vast over-claims if line rates are solely based on the constellation “template”. For example, a wrongly defined line rate could be made arbitrarily large for a PS-QAM system by increasing the constellation template size m for a fixed entropy rate, i.e., through overly strong shaping of a very large QAM constellation that in reality never needs to be physically generated, as its outer symbols are never being used within a limited data record. We therefore propose the use of the constellation entropy rate as a more general definition of the line rate, valid for both uniform and PS-QAM systems. The line rate is then given by

RLine=v[2(X)RFr+2(XFr)(1RFr)]rsb/s,
where the second term denotes the framing overhead. When both X and XFr are uniformly distributed, (X)=m and (XFr)=n, and Eq. (10) becomes
RLine=v[2mRFr+2n(1RFr)]rsb/s,
which accurately quantifies the line rate of a multiplexed M 2-QAM payload with an N 2-QAM framing overhead; in particular, when RFr=1 (i.e., no pilots), this expression further degenerates to the conventional definition of QAM line rates,
RLine=2vmrsb/s,
For PS-QAM, Eq. (10) captures the spirit of the conventional line rate by focusing only on the physical quantities that the transmission medium is truly transporting, no matter what operations are performed in digital circuits (e.g., DM and FEC).

The line rate should not be confused with the digital logic rate, which is often simply called computing, data, or memory bandwidth by digital circuit designers, depending on the context it occurs in. While the line rate quantifies the bit rate on the communication medium, the digital logic rate quantifies the bit rates at various points within the transponder that implement the shaping, coding, and framing signal processing chain in digital logic. The respectively applicable digital logic rates at various points within the PAS architecture are summarized in Table 2.

4. Conclusion

In this paper, we extended the definitions of line rates, information rates, and spectral efficiencies to optical transmission systems with probabilistically shaped constellations. It was shown that, due to the unique structure of the capacity-approaching PAS architecture as the only implementable shaping architecture known today, various subsystem rates cannot simply be multiplied to yield an overall rate, unlike in serially concatenated systems such as a uniform QAM system or a TDHM system. We showed how the line rate can be defined to capture the physical bit rate from the communication channel’s perspective, and pointed out its difference to the digital logic rate. When PS-QAM systems or uniform QAM systems that exploit pilot symbols are used for experimental demonstrations, the resulting rates should be reported according to proper definitions taking full account of the sometimes implicitly assumed underlying processing architecture.

References and links

1. F. Buchali, F. Steiner, G. Böcherer, L. Schmalen, P. Schulte, and W. Idler, “Rate adaptation and reach increase by probabilistically shaped 64-QAM: an experimental demonstration,” J. Lightwave Technol. 34(7), 1599–1609 (2016). [CrossRef]  

2. S. Chandrasekhar, B. Li, J. Cho, X. Chen, E. Burrows, G. Raybon, and P. J. Winzer, “High-spectral-efficiency transmission of PDM 256-QAM with parallel probabilistic shaping at record rate-reach trade-offs,” in Proc. Eur. Conf. Opt. Commun. (2016) paper Th.3.C.1.

3. J. Cho, X. Chen, S. Chandrasekhar, G. Raybon, R. Dar, L. Schmalen, E. Burrows, A. Adamiecki1, S. Corteselli, Y. Pan, D. Correa, B. McKay, S. Zsigmond, P. Winzer, and S. Grubb, “Trans-Atlantic field trial using probabilistically shaped 64-QAM at high spectral efficiencies and single-carrier real-time 250-Gb/s 16-QAM,” in Proc. Opt. Fiber Commun. Conf., 2017, paper Th5B.3.

4. R. Maher, K. Croussore, M. Lauermann, R. Going, X. Xu, and J. Rahn, “Constellation shaped 66 GBd DP-1024QAM transceiver with 400 km transmission over standard SMF,” in Proc. Eur. Conf. Opt. Commun. (2017) paper Th.PDP.B.2.

5. S. L. I. Olsson, J. Cho, S. Chandrasekhar, X. Chen, P. J. Winzer, and S. Makovejs, “Probabilistically shaped PDM 4096-QAM transmission over up to 200 km of fiber using standard intradyne detection,” Opt. Express 26(4), 4522–4530 (2018). [CrossRef]   [PubMed]  

6. S. L. I. Olsson, J. Cho, S. Chandrasekhar, X. Chen, E. Burrows, and P. J. Winzer, “Record-high 17.3-bit/s/Hz spectral efficiency transmission over 50 km using probabilistically shaped PDM 4096-QAM,” in Proc. Opt. Fiber Commun. Conf. (2018) paper Th4C.5. [CrossRef]  

7. G. Böcherer, F. Steiner, and P. Schulte, “Bandwidth efficient and rate-matched low-density parity-check coded modulation,” IEEE Trans. Commun. 63(12), 4651–4665 (2015). [CrossRef]  

8. X. Zhou, L. E. Nelson, P. Magill, R. Isaac, B. Zhu, D. W. Peckham, P. I. Borel, and K. Carlson, “High spectral efficiency 400 Gb/s transmission using PDM time-domain hybrid 32-64 QAM and training-assisted carrier recovery,” J. Lightwave Technol. 31(7), 999–1005 (2013). [CrossRef]  

9. J. Cho, S. Chandrasekhar, X. Chen, G. Raybon, and P. J. Winzer, “High spectral efficiency transmission with probabilistic shaping,” Proc. Eur. Conf. Opt. Commun. (2017) paper Th.1.E.1.

10. P. Schulte and G. Böcherer, “Constant composition distribution matching,” IEEE Trans. Inf. Theory 62(1), 430–434 (2016). [CrossRef]  

11. A. Alvarado, E. Agrell, D. Lavery, R. Maher, and P. Bayvel, “Replacing the soft-decision FEC limit paradigm in the design of optical communication systems,” J. Lightwave Technol. 34(2), 707–721 (2016). [CrossRef]  

12. J. Cho, L. Schmalen, and P. Winzer, “Normalized generalized mutual information as a forward error correction threshold for probabilistically shaped QAM,” in Proc. Eur. Conf. Opt. Commun. (2017) paper M.2.D.2.

13. G. Böcherer, “Achievable rates for probabilistic shaping,” https://arxiv.org/abs/1707.01134.

14. P. Winzer, “High-spectral-efficiency optical modulation formats,” J. Lightwave Technol. 30(24), 3824–3835 (2012). [CrossRef]  

References

  • View by:

  1. F. Buchali, F. Steiner, G. Böcherer, L. Schmalen, P. Schulte, and W. Idler, “Rate adaptation and reach increase by probabilistically shaped 64-QAM: an experimental demonstration,” J. Lightwave Technol. 34(7), 1599–1609 (2016).
    [Crossref]
  2. S. Chandrasekhar, B. Li, J. Cho, X. Chen, E. Burrows, G. Raybon, and P. J. Winzer, “High-spectral-efficiency transmission of PDM 256-QAM with parallel probabilistic shaping at record rate-reach trade-offs,” in Proc. Eur. Conf. Opt. Commun. (2016) paper Th.3.C.1.
  3. J. Cho, X. Chen, S. Chandrasekhar, G. Raybon, R. Dar, L. Schmalen, E. Burrows, A. Adamiecki1, S. Corteselli, Y. Pan, D. Correa, B. McKay, S. Zsigmond, P. Winzer, and S. Grubb, “Trans-Atlantic field trial using probabilistically shaped 64-QAM at high spectral efficiencies and single-carrier real-time 250-Gb/s 16-QAM,” in Proc. Opt. Fiber Commun. Conf., 2017, paper Th5B.3.
  4. R. Maher, K. Croussore, M. Lauermann, R. Going, X. Xu, and J. Rahn, “Constellation shaped 66 GBd DP-1024QAM transceiver with 400 km transmission over standard SMF,” in Proc. Eur. Conf. Opt. Commun. (2017) paper Th.PDP.B.2.
  5. S. L. I. Olsson, J. Cho, S. Chandrasekhar, X. Chen, P. J. Winzer, and S. Makovejs, “Probabilistically shaped PDM 4096-QAM transmission over up to 200 km of fiber using standard intradyne detection,” Opt. Express 26(4), 4522–4530 (2018).
    [Crossref] [PubMed]
  6. S. L. I. Olsson, J. Cho, S. Chandrasekhar, X. Chen, E. Burrows, and P. J. Winzer, “Record-high 17.3-bit/s/Hz spectral efficiency transmission over 50 km using probabilistically shaped PDM 4096-QAM,” in Proc. Opt. Fiber Commun. Conf. (2018) paper Th4C.5.
    [Crossref]
  7. G. Böcherer, F. Steiner, and P. Schulte, “Bandwidth efficient and rate-matched low-density parity-check coded modulation,” IEEE Trans. Commun. 63(12), 4651–4665 (2015).
    [Crossref]
  8. X. Zhou, L. E. Nelson, P. Magill, R. Isaac, B. Zhu, D. W. Peckham, P. I. Borel, and K. Carlson, “High spectral efficiency 400 Gb/s transmission using PDM time-domain hybrid 32-64 QAM and training-assisted carrier recovery,” J. Lightwave Technol. 31(7), 999–1005 (2013).
    [Crossref]
  9. J. Cho, S. Chandrasekhar, X. Chen, G. Raybon, and P. J. Winzer, “High spectral efficiency transmission with probabilistic shaping,” Proc. Eur. Conf. Opt. Commun. (2017) paper Th.1.E.1.
  10. P. Schulte and G. Böcherer, “Constant composition distribution matching,” IEEE Trans. Inf. Theory 62(1), 430–434 (2016).
    [Crossref]
  11. A. Alvarado, E. Agrell, D. Lavery, R. Maher, and P. Bayvel, “Replacing the soft-decision FEC limit paradigm in the design of optical communication systems,” J. Lightwave Technol. 34(2), 707–721 (2016).
    [Crossref]
  12. J. Cho, L. Schmalen, and P. Winzer, “Normalized generalized mutual information as a forward error correction threshold for probabilistically shaped QAM,” in Proc. Eur. Conf. Opt. Commun. (2017) paper M.2.D.2.
  13. G. Böcherer, “Achievable rates for probabilistic shaping,” https://arxiv.org/abs/1707.01134 .
  14. P. Winzer, “High-spectral-efficiency optical modulation formats,” J. Lightwave Technol. 30(24), 3824–3835 (2012).
    [Crossref]

2018 (1)

2016 (3)

2015 (1)

G. Böcherer, F. Steiner, and P. Schulte, “Bandwidth efficient and rate-matched low-density parity-check coded modulation,” IEEE Trans. Commun. 63(12), 4651–4665 (2015).
[Crossref]

2013 (1)

2012 (1)

Agrell, E.

Alvarado, A.

Bayvel, P.

Böcherer, G.

P. Schulte and G. Böcherer, “Constant composition distribution matching,” IEEE Trans. Inf. Theory 62(1), 430–434 (2016).
[Crossref]

F. Buchali, F. Steiner, G. Böcherer, L. Schmalen, P. Schulte, and W. Idler, “Rate adaptation and reach increase by probabilistically shaped 64-QAM: an experimental demonstration,” J. Lightwave Technol. 34(7), 1599–1609 (2016).
[Crossref]

G. Böcherer, F. Steiner, and P. Schulte, “Bandwidth efficient and rate-matched low-density parity-check coded modulation,” IEEE Trans. Commun. 63(12), 4651–4665 (2015).
[Crossref]

Borel, P. I.

Buchali, F.

Carlson, K.

Chandrasekhar, S.

Chen, X.

Cho, J.

Idler, W.

Isaac, R.

Lavery, D.

Magill, P.

Maher, R.

Makovejs, S.

Nelson, L. E.

Olsson, S. L. I.

Peckham, D. W.

Schmalen, L.

Schulte, P.

F. Buchali, F. Steiner, G. Böcherer, L. Schmalen, P. Schulte, and W. Idler, “Rate adaptation and reach increase by probabilistically shaped 64-QAM: an experimental demonstration,” J. Lightwave Technol. 34(7), 1599–1609 (2016).
[Crossref]

P. Schulte and G. Böcherer, “Constant composition distribution matching,” IEEE Trans. Inf. Theory 62(1), 430–434 (2016).
[Crossref]

G. Böcherer, F. Steiner, and P. Schulte, “Bandwidth efficient and rate-matched low-density parity-check coded modulation,” IEEE Trans. Commun. 63(12), 4651–4665 (2015).
[Crossref]

Steiner, F.

F. Buchali, F. Steiner, G. Böcherer, L. Schmalen, P. Schulte, and W. Idler, “Rate adaptation and reach increase by probabilistically shaped 64-QAM: an experimental demonstration,” J. Lightwave Technol. 34(7), 1599–1609 (2016).
[Crossref]

G. Böcherer, F. Steiner, and P. Schulte, “Bandwidth efficient and rate-matched low-density parity-check coded modulation,” IEEE Trans. Commun. 63(12), 4651–4665 (2015).
[Crossref]

Winzer, P.

Winzer, P. J.

Zhou, X.

Zhu, B.

IEEE Trans. Commun. (1)

G. Böcherer, F. Steiner, and P. Schulte, “Bandwidth efficient and rate-matched low-density parity-check coded modulation,” IEEE Trans. Commun. 63(12), 4651–4665 (2015).
[Crossref]

IEEE Trans. Inf. Theory (1)

P. Schulte and G. Böcherer, “Constant composition distribution matching,” IEEE Trans. Inf. Theory 62(1), 430–434 (2016).
[Crossref]

J. Lightwave Technol. (4)

Opt. Express (1)

Other (7)

S. L. I. Olsson, J. Cho, S. Chandrasekhar, X. Chen, E. Burrows, and P. J. Winzer, “Record-high 17.3-bit/s/Hz spectral efficiency transmission over 50 km using probabilistically shaped PDM 4096-QAM,” in Proc. Opt. Fiber Commun. Conf. (2018) paper Th4C.5.
[Crossref]

J. Cho, S. Chandrasekhar, X. Chen, G. Raybon, and P. J. Winzer, “High spectral efficiency transmission with probabilistic shaping,” Proc. Eur. Conf. Opt. Commun. (2017) paper Th.1.E.1.

S. Chandrasekhar, B. Li, J. Cho, X. Chen, E. Burrows, G. Raybon, and P. J. Winzer, “High-spectral-efficiency transmission of PDM 256-QAM with parallel probabilistic shaping at record rate-reach trade-offs,” in Proc. Eur. Conf. Opt. Commun. (2016) paper Th.3.C.1.

J. Cho, X. Chen, S. Chandrasekhar, G. Raybon, R. Dar, L. Schmalen, E. Burrows, A. Adamiecki1, S. Corteselli, Y. Pan, D. Correa, B. McKay, S. Zsigmond, P. Winzer, and S. Grubb, “Trans-Atlantic field trial using probabilistically shaped 64-QAM at high spectral efficiencies and single-carrier real-time 250-Gb/s 16-QAM,” in Proc. Opt. Fiber Commun. Conf., 2017, paper Th5B.3.

R. Maher, K. Croussore, M. Lauermann, R. Going, X. Xu, and J. Rahn, “Constellation shaped 66 GBd DP-1024QAM transceiver with 400 km transmission over standard SMF,” in Proc. Eur. Conf. Opt. Commun. (2017) paper Th.PDP.B.2.

J. Cho, L. Schmalen, and P. Winzer, “Normalized generalized mutual information as a forward error correction threshold for probabilistically shaped QAM,” in Proc. Eur. Conf. Opt. Commun. (2017) paper M.2.D.2.

G. Böcherer, “Achievable rates for probabilistic shaping,” https://arxiv.org/abs/1707.01134 .

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Figures (4)

Fig. 1
Fig. 1 Block diagram defining (a) the rateR, and (b) the overhead OH.
Fig. 2
Fig. 2 Data flow within the PAS architecture.
Fig. 3
Fig. 3 (a) Rates, and (b) overheads of a PS QAM system and its subsystems, shown in a serially concatenated system model.
Fig. 4
Fig. 4 Rates of PS 8-PAM: (a) the effective code rates ρ c as a function of the true code rates R c with various R DM ,and (b) the overall system rates R Sys computed incorrectly by R Sh R c (dashed) and correctly by R Sh ρ c (solid).

Tables (2)

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Table 1 List of Symbols and Notation

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Table 2 Bit Rates within the PAS Architecture (cf. Figure 2)

Equations (12)

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R DM = β m1 1.
R c = m1+γ m m1 m .
R Fr = m ρ Fr m ρ Fr +n(1 ρ Fr ) .
R Sys = β+γ m R Fr = ρ PAS R DM R c R Fr .
R Sys = R Sh ρ c R Fr ,
O H Sys =(1+O H Sh )(1+o h c )(1+O H Fr )1,
O H Sys =(1+O H DM + α PAS )(1+O H c )(1+O H Fr )1.
R Info =2vm R Sys r s b/s,
SE= R Info /F b/s/Hz.
R Line =v[ 2(X) R Fr +2( X Fr )(1 R Fr ) ] r s b/s,
R Line =v[ 2m R Fr +2n(1 R Fr ) ] r s b/s,
R Line =2vm r s b/s,

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