## 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.

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

In general, information and communications engineering defines the (dimensionless) *rate$R$*of 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), with$R=1/(1+OH)$ and $OH=(1-R)/R$. 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.

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.

#### 2.1 Distribution matcher

In a first step, the DM generates the shaped *positive-amplitude* portion of the constellation symbols $X$of an *M*-PAM constellation following a probability distribution${\mathbb{P}}_{X}(x)$ with corresponding entropy $\mathbb{H}(X)$; for example, the DM creates ${\mathbb{P}}_{X}(x)=\{0.537,0.322,0.116,$ $0.025\}$on $X=\{1,3,5,7\}$, yielding $\mathbb{H}(X)\approx 1.5$. The resulting information rate $\beta $ of this shaped half-PAM constellation is $\beta \le \mathbb{H}(X)-1\le m-1$ b/symbol, with $m={\mathrm{log}}_{2}M$(*M* = 8 in the example in Fig. 2). Each shaped half-PAM symbol is represented by$m-1$ 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)

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, $\mathbb{H}(X)$ is maximized by symbols *X _{MB}* that follow a Maxwell-Boltzmann (MB) distribution, and an ideal DM achieves a maximum information rate of${\beta}^{*}=\mathbb{H}({X}_{MB})-1.$ Any non-ideal DM yields$\beta ={\beta}^{*}-{\Delta}_{\beta}$ with arate loss${\Delta}_{\beta}>0.$ 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 ${\Delta}_{\beta}$asymptotically 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 rate${R}_{c},$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 to${R}_{c}\ge (m-1)/m.$ If the FEC code rate is chosen *higher* than $(m-1)/m,$ such that the FEC overhead does not fully exhaust an entire bit per symbol, a fraction $0\le \gamma \le 1$ of the sign bit may be allocated to carry information, cf. Figure 2; in this case, $\gamma $ and $1-\gamma $ 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 $\gamma $ by

#### 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 $n={\mathrm{log}}_{2}N$b/symbol, different from the constellation used by the data symbols (typically$n\le m$). 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 $(1-{\rho}_{Fr})$ *n*-bit pilot or framing symbols are added per${\rho}_{Fr}$ *m*-bit information symbols, the bit rate of the digital logic after the framer is $\left[m{\rho}_{Fr}+n(1-{\rho}_{Fr})\right]{r}_{s},$with ${r}_{s}$being the symbol rate used on the channel. Defining the framing rate ${R}_{Fr}$as the ratio of the physical bit rate entering the framer to the physical bit rate leaving the framer, we have

#### 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

*effective code rate*𝜌

*relating a serial shaping architecture with shaping rate${R}_{Sh}=(\beta +1)/m\le 1$ and a PAS architecture as${\rho}_{c}\triangleq (\beta +\gamma )/(\beta +1)=1-(1-{R}_{c})/{R}_{Sh},$cf. Figure 3(a). The shaping rate ${R}_{Sh}$ is the ratio of the bit rate before PS to the bit rate after PS for the entire PAM constellation, while ${R}_{DM}$ is defined for the half-PAM constellation. Note that the overall system rate ${R}_{Sys}\ne {R}_{Sh}{R}_{c}{R}_{Fr},$ 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,$$\beta =2.38,$${R}_{c}=(5/6)\times (16/17),$ and${R}_{Fr}=32/33$(assuming that the pilots used the same shaping as the data signals) of [4] in Eq. (5) to obtain${R}_{Sh}=0.676,$ ${\rho}_{c}\approx 0.6809,$ hence ${R}_{Sys}\approx \mathrm{0.4464.}$ Ignoring Eqs. (4) and (5) while invoking the PAS architecture leads to a significant overestimation of system claims, as discussed below.*

_{c}Figure 4(a) shows the effective code rate 𝜌* _{c}* for the PAS architecture with PS 8-PAM (generating PS 64-QAM) with various DM rates ${R}_{DM},$ assuming an ideal DM $({\Delta}_{\beta}=0).$ The code rate constraint is${R}_{c}\ge 2/3,$ cf. Equation (2), leading to the non-permissible code rate regions shown in gray. As ${R}_{DM}$ decreases (i.e., with more shaping; ${R}_{DM}=0$ implies quadrature phase shift keying), and as ${R}_{c}$ decreases (i.e., with a larger coding overhead), the effective code rate ${\rho}_{c}$ becomes much smaller than the true rate ${R}_{c}$ of the underlying FEC code. Figure 4(b) shows the corresponding overall system rate ${R}_{Sys},$ assuming ${R}_{Fr}=1$ (i.e., no pilot symbols). As ${R}_{DM}$ decreases, and as ${R}_{c}$ decreases, the overall system rate is significantly overestimated by the incorrect computation ${R}_{Sh}{R}_{c}$ (dashed) relative to the correct computation${R}_{Sh}{\rho}_{c}$ (solid).

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 as$OH\triangleq 1/R-1$ (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 $O{H}_{c}$but rather have to be obtained using the *effective FEC overhead* $o{h}_{c}\triangleq 1/{\rho}_{c}-1$as (cf. Figure 3(b))

## 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)

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

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 *X _{Fr}* using the same constellation size ($n=m$), the line rate is defined as${R}_{Line}=2vm{r}_{s}.$ 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

*X*and

*X*are uniformly distributed, $\mathbb{H}(X)=m$ and $\mathbb{H}({X}_{Fr})=n,$ and Eq. (10) becomeswhich accurately quantifies the line rate of a multiplexed

_{Fr}*M*

^{2}-QAM payload with an

*N*

^{2}-QAM framing overhead; in particular, when ${R}_{Fr}=1$ (i.e., no pilots), this expression further degenerates to the conventional definition of QAM line rates,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.

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