# Probabilistic Shaping for Finite Blocklengths: Distribution Matching and Sphere Shaping

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Notation and Definitions

#### 2.2. Discrete Constellations and Amplitude Shaping

**Example**

**1**

**.**The BRGC is tabulated for 8-ASK in Figure 2 (bottom). Here, ${B}_{1}$ is symmetric around zero. Furthermore, when X has a distribution which is symmetric around zero, ${B}_{1}$ is uniform and stochastically independent of ${B}_{2}$ and ${B}_{3}$.

#### 2.3. Fundamentals of Amplitude Shaping Schemes

#### 2.4. Shaping Architecture vs. Shaping Algorithm

## 3. Signaling Schemes

#### 3.1. Uniform Signaling

#### 3.2. Probabilistic Amplitude Shaping

**Example**

**2**

**.**Consider the PAS architecture with 8-ASK, a rate ${R}_{\mathrm{c}}=5/6$ FEC code, and a target rate $R=2.25$ bit/1-D. The rate of the extra data that will be carried in the signs of the channel inputs is $\gamma ={R}_{\mathrm{c}}m-(m-1)=0.5$ bit/1-D. Therefore, the rate of the amplitude shaper should be $k/n=R-\gamma =1.75$ bit/1-D. If the length of the FEC code is ${n}_{\mathrm{c}}=648$ bits, the blocklength is $n={n}_{\mathrm{c}}/m=216$ real symbols. Then the output set of the amplitude shaper must consist of at least ${2}^{k}={2}^{216\xb71.75}={2}^{378}$ sequences.

#### 3.3. PAS Receiver

#### 3.4. Selection of Parameters for PAS

**Example**

**3**

**.**In Figure 5, the entropy $\mathbb{H}(X)$ of an input X with $\left|\mathcal{X}\right|/2=4$ MB-distributed amplitude levels (i.e., 8-ASK) vs. $\Delta SNR$ is plotted for $R=2.25$ bit/1-D. On the top horizontal axis, the corresponding FEC code rates (14) are also shown. The rightmost point (indicated by a square) corresponds to uniform signaling where the target rate of 2.25 bit/1-D is obtained by using a FEC code of rate ${R}_{c}=R/m=3/4$. In this trivial case, all 0.75 bits of redundancy are added by the coding operation, and the gap to capacity $\Delta SNR$ is 1.04 dB. The leftmost part of the curve where $\mathbb{H}(X)$ goes to R belongs to the uncoded signaling case, i.e., ${R}_{c}=1$, where R is attained by shaping the constellation such that $\mathbb{H}(X)=R$. Here $\Delta SNR$ is infinite since without coding, reliable communication is only possible over a noiseless channel. The minimum $\Delta SNR$ in Figure 5 is obtained with $\mathbb{H}(X)=2.745$, which corresponds to ${R}_{c}=0.835$ from (14). In IEEE DVB-S2 [77] and 802.11 [78], the code rate that is closest to 0.835 is $5/6\approx 0.833$. Accordingly, the best performance is expected to be provided with rate-$5/6$ FEC code, with an SNR gain over uniform that amounts according to this analysis to 0.83 dB. This will be confirmed by the numerical simulations presented in Section 5.3.

## 4. Distribution Matching and Sphere Shaping Architectures

#### 4.1. Distribution Matching Architectures (Direct Method)

**Remark**

**1**

**Example**

**4**

**.**We consider the target PMF ${P}_{A}=[0.4378,0.3212,0.1728,0.0682]$ over $\mathcal{A}=\{1,3,5,7\}$ with $\mathbb{H}(A)=1.75$ bit. Combined with rate-5/6 FEC coding in PAS framework, an amplitude shaper with this entropy corresponds to a transmission rate of $R=2.25$ bit/1-D from (14), which is a typical target rate with 8-ASK. The composition that is obtained for $n=216$ with the quantization rule proposed in (Algorithm 2 in [93]) is $C=[95,69,37,15]$. The shaping rate (7) of the matcher that produces sequences with composition C is ${R}_{s}=1.6991$ bit/1-D. The input length (8) of this matcher is $k=367$ bits.

**Example**

**5**

**.**We consider the same target PMF and $n=216$ as in Example 4. Pairwise MPDM with tree structure utilizes 945 compositions whose average is again $[95,69,37,15]$. The shaping rate (7) of the matcher that produces sequences with these compositions is ${R}_{s}=1.7315$ bit/1-D. The corresponding input length (8) is $k=374$ which is 7 bits more than that of CCDM which is a 1.9% rate increase.

#### 4.2. Sphere Shaping Architecture (Indirect Method)

**Remark**

**2.**

**Example**

**6**

**.**The sphere shaping set ${\mathcal{A}}^{\u2022}\subset {\mathcal{A}}^{n}$ for the parameters $n=64$, $\mathcal{A}=\{1,3,5,7\}$ and ${E}^{\u2022}=768$, i.e., $L=89$, has the shaping rate ${R}_{s}=1.7538$ bit/1-D. The input length of the corresponding amplitude shaper is $k=112$ bits. The average PMF is ${P}_{A}(a)=[0.42,0.32,0.18,0.08]$ over $\mathcal{A}$, where the average energy per dimension is $E=11.6316$.

**Example**

**7**

**.**If the shaping set ${\mathcal{A}}^{\u2022}$ in Example 6 is constructed with BP using ${n}_{m}=9$ bit mantissas and ${n}_{p}=7$ bit exponents, the resulting rate loss is upper-bounded by 0.0056 bit/1-D. For ESS and SM, the actual rate losses are 0.0021 and 0.0003 bit/1-D, respectively. Since the shaping rate with FP was ${R}_{s}=1.7538$, these rate losses keep ${R}_{s}>1.75$, and consequently, keep $k=112$. Therefore, we claim that when more than a few bytes are used to store mantissas, BP rate loss is smaller than the loss due to the rounding operation in (8). Consequently, the operational rate $k/n$ is not affected. However, the required memory to store an element of the shaping matrix drops from $(k+1)=113$ bits to ${n}_{m}+{n}_{p}=16$ bits.

#### 4.3. Geometric Interpretation of the Shaping Architectures

## 5. Performance Comparison

#### 5.1. Rate Loss Analysis

**Example**

**8**

**.**We consider the target distribution ${P}_{A}=[0.438,0.321,0.173,0.068]$ with entropy $\mathbb{H}(A)=1.75$. The n-type distribution that has the minimum informational divergence from ${P}_{A}$ for $n=216$ is ${P}_{\overline{A}}=[0.440,0.320,0.171,0.069]$. The corresponding composition is $C=[95,69,37,15]$. Starting with the same target distribution, i.e., with the same composition, the number of compositions that are employed by MPDM is 945 (Section III in [55]). Since MPDM’s set of compositions consists of pairs whose average is C, the average distribution ${P}_{\overline{A}}$, its entropy $\mathbb{H}(\overline{A})$ and the average symbol energy E are the same as CCDM’s. The smallest ${E}^{\u2022}$ that gives $|{\mathcal{A}}^{\u2022}|\ge {2}^{k}$ is ${E}^{\u2022}=2376$ where k is the input length of MPDM. The corresponding average distribution is ${P}_{\tilde{A}}=[0.439,0.322,0.172,0.067]$. Table 2 shows the input length k, average symbol energy E and rate loss ${R}_{\mathit{loss}}$ of CCDM, MPDM and SpSh for these parameters. We see that MPDM is able to address a larger set of sequences than CCDM, leading to a 7 bit increase in the input length. Since the corresponding average distributions are the same, this is reflected as a decrease in rate loss. Then starting with the same target k, SpSh employs a set of sequences with smaller average energy. This is also translated to a decrease in rate loss as shown in Table 2.

**Remark**

**3**

**.**Example 8 shows that when the entropy of the target distribution is taken to be the target rate $k/n$ (1.75 bit/1-D in Example 8), CCDM and MPDM are not able to obtain ${2}^{k}$ sequences. This is due to the inevitable nonzero rate loss of the DM schemes for finite blocklengths. For such cases, we increase the SNR that the target distribution is optimized for, until we obtain ${2}^{k}$ output sequences for the DM schemes.

#### 5.2. Achievable Information Rates

#### 5.3. End-to-End Decoding Performance

## 6. Approximate Complexity Discussion

#### 6.1. Latency

#### 6.2. Storage Requirements

**Example**

**9**

**.**We consider $\mathcal{A}=\{1,3,5,7\}$, $n=216$ and target rates $k/n=1.5$ and 1.75 bit/1-D. To obtain these rates, MPDM uses 318 and 593 different compositions, respectively. Assuming that numbers in a composition can be stored using at most $\u2308{log}_{2}n\u2309$ bits, at most 10,176 and 18,976 bits of memory are required to store the corresponding LUTs, respectively. Note that these are the parameters that are used for the simulations considered in Figure 11.

**Example**

**10**

**.**To realize ESS or (Algorithm 1 in [23]) for the setup in Example 6, at most $Ln\u2308n{R}_{\mathrm{s}}\u2309=80.46$ kilobytes (kB) of memory is required. On the other hand for SM, at most $L{log}_{2}n\u2308n{R}_{\mathrm{s}}\u2309=7.54$ kB of memory should be allocated.

**Remark**

**4.**

**Example**

**11**

**.**To realize ESS or (Algorithm 1 in [23]) with ${n}_{m}=9$ and ${n}_{p}=7$ for the setup in Example 6, at most $Ln({n}_{m}+{n}_{p})=11.39$ kB of memory is required. On the other hand, when implemented using ${n}_{m}=6$ and ${n}_{p}=7$, SM demands at most $L{log}_{2}n({n}_{m}+{n}_{p})=0.87$ kB of memory. We note that the mantissa lengths ${n}_{m}$ are selected according to the discussion in Remark 4.

#### 6.3. Computational Complexity

**Example**

**12**

**Example**

**13**

**.**When Example 6 is now constructed with ${n}_{m}=9$ and ${n}_{p}=7$, ESS and (Algorithm 1 in [23]) require at most four 9-bit additions per 1-D. Correspondingly, if SM is realized with ${n}_{m}=6$ and ${n}_{p}=7$, at most 89 6-bit multiplications are necessary per 1-D.

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AC | Arithmetic coding |

AIR | Achievable information rate |

ASK | Amplitude-shift keying |

AWGN | Additive white Gaussian noise |

BC | Binomial coefficient |

BL-DM | Bit-level distribution matching |

BMD | Bit-metric decoding |

BP | Bounded-precision |

BRGC | Binary reflected Gray code |

CCDM | Constant composition distribution matching |

CM | Coded modulation |

D&C | Divide and conquer |

DEMUX | Demultiplexer |

DM | Distribution matching |

ESS | Enumerative sphere shaping |

FEC | Forward error correction |

FER | Frame error rate |

FP | Full-precision |

GMI | Generalized mutual information |

GS | Geometric shaping |

LDPC | Low-density parity-check |

LLR | Log-likelihood ratio |

LUT | Lookup table |

MB | Maxwell-Boltzmann |

MC | Multinomial coefficient |

MI | Mutual information |

MLC | Multilevel coding |

MPDM | Multiset-partition distribution matching |

MUX | Multiplexer |

PA | Parallel-amplitude |

PAS | Probabilistic amplitude shaping |

PDM | Product distribution matching |

PMF | Probability mass function |

PS | Probabilistic shaping |

QAM | Quadrature amplitude modulation |

SM | Shell mapping |

SNR | Signal-to-noise ratio |

SpSh | Sphere shaping |

SR | Subset ranking |

## References

- Imai, H.; Hirakawa, S. A new multilevel coding method using error-correcting codes. IEEE Trans. Inf. Theory
**1977**, 23, 371–377. [Google Scholar] [CrossRef] - Wachsmann, U.; Fischer, R.F.H.; Huber, J.B. Multilevel codes: Theoretical concepts and practical design rules. IEEE Trans. Inf. Theory
**1999**, 45, 1361–1391. [Google Scholar] [CrossRef] [Green Version] - Ungerböck, G. Channel coding with multilevel/phase signals. IEEE Trans. Inf. Theory
**1982**, 28, 55–67. [Google Scholar] [CrossRef] - Zehavi, E. 8-PSK trellis codes for a Rayleigh channel. IEEE Trans. Commun.
**1992**, 40, 873–884. [Google Scholar] [CrossRef] - Caire, G.; Taricco, G.; Biglieri, E. Bit-interleaved coded modulation. IEEE Trans. Inf. Theory
**1998**, 44, 927–946. [Google Scholar] [CrossRef] [Green Version] - Guillén i Fàbregas, A.; Martinez, A.; Caire, G. Bit-interleaved coded modulation. Found. Trends Commun. Inf. Theory
**2008**, 5, 1–153. [Google Scholar] [CrossRef] [Green Version] - Martinez, A.; Guillén i Fàbregas, A.; Caire, G.; Willems, F.M.J. Bit-interleaved coded modulation revisited: A mismatched decoding perspective. IEEE Trans. Inf. Theory
**2009**, 55, 2756–2765. [Google Scholar] [CrossRef] [Green Version] - Szczecinski, L.; Alvarado, A. Bit-Interleaved Coded Modulation: Fundamentals, Analysis, and Design; John Wiley & Sons: Chichester, UK, 2015. [Google Scholar]
- Forney, G.; Gallager, R.; Lang, G.; Longstaff, F.; Qureshi, S. Efficient modulation for band-limited channels. IEEE J. Sel. Areas Commun.
**1984**, 2, 632–647. [Google Scholar] [CrossRef] [Green Version] - Fischer, R. Precoding and Signal Shaping for Digital Transmission; John Wiley & Sons: New York, NY, USA, 2002. [Google Scholar]
- Sun, F.W.; van Tilborg, H.C.A. Approaching capacity by equiprobable signaling on the Gaussian channel. IEEE Trans. Inf. Theory
**1993**, 39, 1714–1716. [Google Scholar] - Loghin, N.S.; Zöllner, J.; Mouhouche, B.; Ansorregui, D.; Kim, J.; Park, S. Non-uniform constellations for ATSC 3.0. IEEE Trans. Broadcast.
**2016**, 62, 197–203. [Google Scholar] [CrossRef] - Qu, Z.; Djordjevic, I.B. Geometrically shaped 16QAM outperforming probabilistically shaped 16QAM. In Proceedings of the 2017 European Conference on Optical Communication (ECOC), Gothenburg, Sweden, 17–21 September 2017. [Google Scholar]
- Steiner, F.; Böcherer, G. Comparison of geometric and probabilistic shaping with application to ATSC 3.0. In Proceedings of the 11th International ITG Conference on Systems, Communications and Coding, Hamburg, Germany, 6–9 February 2017. [Google Scholar]
- Boutros, J.J.; Erez, U.; Wonterghem, J.V.; Shamir, G.I.; Zèmorl, G. Geometric shaping: Low-density coding of Gaussian-like constellations. In Proceedings of the 2018 IEEE Information Theory Workshop (ITW), Guangzhou, China, 25–29 November 2018. [Google Scholar]
- Chen, B.; Okonkwo, C.; Hafermann, H.; Alvarado, A. Increasing achievable information rates via geometric shaping. In Proceedings of the 2018 European Conference on Optical Communication (ECOC), Rome, Italy, 23–27 September 2018. [Google Scholar]
- Chen, B.; Okonkwo, C.; Lavery, D.; Alvarado, A. Geometrically-shaped 64-point constellations via achievable information rates. In Proceedings of the 20th International Conference on Transparent Optical Networks (ICTON), Bucharest, Romania, 1–5 July 2018. [Google Scholar]
- Chen, B.; Lei, Y.; Lavery, D.; Okonkwo, C.; Alvarado, A. Rate-adaptive coded modulation with geometrically-shaped constellations. In Proceedings of the Asia Communications and Photonics Conference (ACP), Hangzhou, China, 26–29 October 2018. [Google Scholar]
- Calderbank, A.R.; Ozarow, L.H. Nonequiprobable signaling on the Gaussian channel. IEEE Trans. Inf. Theory
**1990**, 36, 726–740. [Google Scholar] [CrossRef] - Forney, G.D. Trellis shaping. IEEE Trans. Inf. Theory
**1992**, 38, 281–300. [Google Scholar] [CrossRef] - Kschischang, F.R.; Pasupathy, S. Optimal nonuniform signaling for Gaussian channels. IEEE Trans. Inf. Theory
**1993**, 39, 913–929. [Google Scholar] [CrossRef] [Green Version] - Willems, F.; Wuijts, J. A pragmatic approach to shaped coded modulation. In Proceedings of the IEEE Symposium on Communications and Vehicular Technology in the Benelux, Delft, The Netherlands, 7–8 October 1993. [Google Scholar]
- Laroia, R.; Farvardin, N.; Tretter, S.A. On optimal shaping of multidimensional constellations. IEEE Trans. Inf. Theory
**1994**, 40, 1044–1056. [Google Scholar] [CrossRef] - Böcherer, G.; Mathar, R. Matching dyadic distributions to channels. In Proceedings of the 2011 Data Compression Conference, Snowbird, UT, USA, 29–31 March 2011. [Google Scholar]
- Batshon, H.G.; Mazurczyk, M.V.; Cai, J.X.; Sinkin, O.V.; Paskov, M.; Davidson, C.R.; Wang, D.; Bolshtyansky, M.; Foursa, D. Coded modulation based on 56APSK with hybrid shaping for high spectral efficiency transmission. In Proceedings of the European Conference on Optical Communication (ECOC), Gothenburg, Sweden, 17–21 September 2017. [Google Scholar]
- Cai, J.X.; Batshon, H.G.; Mazurczyk, M.V.; Sinkin, O.V.; Wang, D.; Paskov, M.; Patterson, W.W.; Davidson, C.R.; Corbett, P.C.; Wolter, G.M.; et al. 70.46 Tb/s over 7600 km and 71.65 Tb/s over 6970 km transmission in C+L band using coded modulation with hybrid constellation shaping and nonlinearity compensation. J. Lightwave Technol.
**2018**, 36, 114–121. [Google Scholar] [CrossRef] - Cai, J.; Batshon, H.G.; Mazurczyk, M.V.; Sinkin, O.V.; Wang, D.; Paskov, M.; Davidson, C.R.; Patterson, W.W.; Turukhin, A.; Bolshtyansky, M.A.; et al. 51.5 Tb/s capacity over 17,107 km in C+L bandwidth using single-mode fibers and nonlinearity compensation. J. Lightw. Technol.
**2018**, 36, 2135–2141. [Google Scholar] [CrossRef] - Böcherer, G.; Steiner, F.; Schulte, P. Bandwidth efficient and rate-matched low-density parity-check coded modulation. IEEE Trans. Commun.
**2015**, 63, 4651–4665. [Google Scholar] [CrossRef] [Green Version] - Sommer, D.; Fettweis, G.P. Signal shaping by non-uniform QAM for AWGN channels and applications using turbo coding. In Proceedings of the ITG Conference on Source and Channel Coding, Munich, Germany, 17–19 January 2000. [Google Scholar]
- Le Goff, S.Y. Signal constellations for bit-interleaved coded modulation. IEEE Trans. Inf. Theory
**2003**, 49, 307–313. [Google Scholar] [CrossRef] - Barsoum, M.F.; Jones, C.; Fitz, M. Constellation design via capacity maximization. In Proceedings of the IEEE International Symposium on Information Theory, Nice, France, 24–29 June 2007. [Google Scholar]
- Le Goff, S.Y.; Sharif, B.S.; Jimaa, S.A. A new bit-interleaved coded modulation scheme using shaping coding. In Proceedings of the IEEE Global Telecommunications Conference, 2004. GLOBECOM ’04, Dallas, TX, USA, 29 November–3 December 2004. [Google Scholar]
- Raphaeli, D.; Gurevitz, A. Constellation shaping for pragmatic turbo-coded modulation with high spectral efficiency. IEEE Trans. Commun.
**2004**, 52, 341–345. [Google Scholar] [CrossRef] - Le Goff, S.Y.; Sharif, B.S.; Jimaa, S.A. Bit-interleaved turbo-coded modulation using shaping coding. IEEE Commun. Lett.
**2005**, 9, 246–248. [Google Scholar] [CrossRef] - Valenti, M.C.; Xiang, X. Constellation shaping for bit-interleaved LDPC coded APSK. IEEE Trans. Commun.
**2012**, 60, 2960–2970. [Google Scholar] [CrossRef] [Green Version] - Le Goff, S.Y.; Khoo, B.K.; Tsimenidis, C.C.; Sharif, B.S. Constellation shaping for bandwidth-efficient turbo-coded modulation with iterative receiver. IEEE Trans. Wirel. Commun.
**2007**, 6, 2223–2233. [Google Scholar] [CrossRef] - Guillén i Fàbregas, A.; Martinez, A. Bit-interleaved coded modulation with shaping. In Proceedings of the IEEE Information Theory Workshop, Dublin, Ireland, 30 August–3 September 2010. [Google Scholar]
- Alvarado, A.; Brännström, F.; Agrell, E. High SNR bounds for the BICM capacity. In Proceedings of the IEEE Information Theory Workshop, Paraty, Brazil, 16–20 October 2011. [Google Scholar]
- Böcherer, G.; Altenbach, F.; Alvarado, A.; Corroy, S.; Mathar, R. An efficient algorithm to calculate BICM capacity. In Proceedings of the IEEE International Symposium on Information Theory Proceedings, Cambridge, MA, USA, 1–6 July 2012. [Google Scholar]
- Peng, L.; Guillén i Fàbregas, A.; Martinez, A. Mismatched shaping schemes for bit-interleaved coded modulation. In Proceedings of the IEEE International Symposium on Information Theory Proceedings, Cambridge, MA, USA, 1–6 July 2012. [Google Scholar]
- Peng, L. Fundamentals of Bit-Interleaved Coded Modulation and Reliable Source Transmission. Ph.D. Thesis, University of Cambridge, Cambridge, UK, 2012. [Google Scholar]
- Agrell, E.; Alvarado, A. Signal shaping for BICM at low SNR. IEEE Trans. Inf. Theory
**2013**, 59, 2396–2410. [Google Scholar] [CrossRef] [Green Version] - Bouazza, B.S.; Djebari, A. Bit-interleaved coded modulation with iterative decoding using constellation shaping over Rayleigh fading channels. AEÜ Int. J. Electron. Commun.
**2007**, 61, 405–410. [Google Scholar] [CrossRef] - Xiang, X.; Valenti, M.C. Improving DVB-S2 performance through constellation shaping and iterative demapping. In Proceedings of the MILCOM 2011 Military Communications Conference, Baltimore, MD, USA, 7–10 November 2011. [Google Scholar]
- Bliss, W.G. Circuitry for performing error correction calculations on baseband encoded data to eliminate error propagation. IBM Technol. Discl. Bull.
**1981**, 23, 4633–4634. [Google Scholar] - Fan, J.L.; Cioffi, J.M. Constrained coding techniques for soft iterative decoders. In Proceedings of the IEEE Global Telecommunications Conference, Rio de Janeireo, Brazil, 5–9 December 1999. [Google Scholar]
- Prinz, T.; Yuan, P.; Böcherer, G.; Steiner, F.; İşcan, O.; Böhnke, R.; Xu, W. Polar coded probabilistic amplitude shaping for short packets. In Proceedings of the IEEE 18th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), Sapporo, Japan, 3–6 July 2017. [Google Scholar]
- Gültekin, Y.C.; van Houtum, W.J.; Koppelaar, A.; Willems, F.M.J. Enumerative sphere shaping for wireless communications with short packets. IEEE Trans. Wirel. Commun.
**2020**, 19, 1098–1112. [Google Scholar] [CrossRef] [Green Version] - Buchali, F.; Steiner, F.; Böcherer, G.; Schmalen, L.; Schulte, P.; Idler, W. Rate Adaptation and Reach Increase by Probabilistically Shaped 64-QAM: An Experimental Demonstration. J. Lightwave Technol.
**2016**, 34, 1599–1609. [Google Scholar] [CrossRef] - Fehenberger, T.; Lavery, D.; Maher, R.; Alvarado, A.; Bayvel, P.; Hanik, N. Sensitivity Gains by Mismatched Probabilistic Shaping for Optical Communication Systems. IEEE Photonics Technol. Lett.
**2016**, 28, 786–789. [Google Scholar] [CrossRef] [Green Version] - Steiner, F.; Schulte, P.; Böcherer, G. Approaching waterfilling capacity of parallel channels by higher order modulation and probabilistic amplitude shaping. In Proceedings of the 52nd Annual Conference on Information Sciences and Systems (CISS), Princeton, NJ, USA, 21–23 March 2018. [Google Scholar]
- Schulte, P.; Böcherer, G. Constant composition distribution matching. IEEE Trans. Inf. Theory
**2016**, 62, 430–434. [Google Scholar] [CrossRef] [Green Version] - Ramabadran, T.V. A coding scheme for m-out-of-n codes. IEEE Trans. Commun.
**1990**, 38, 1156–1163. [Google Scholar] [CrossRef] - Gültekin, Y.C.; van Houtum, W.J.; Willems, F.M.J. On constellation shaping for short block lengths. In Proceedings of the 2018 Symposium on Information Theory and Signal Processing in the Benelux, Enschede, The Netherlands, 31 May–1 June 2018. [Google Scholar]
- Fehenberger, T.; Millar, D.S.; Koike-Akino, T.; Kojima, K.; Parsons, K. Multiset-partition distribution matching. IEEE Trans. Commun.
**2019**, 67, 1885–1893. [Google Scholar] [CrossRef] [Green Version] - Sayood, K. Lossless Compression Handbook; Academic Press: Cambridge, MA, USA, 2002. [Google Scholar]
- Pikus, M.; Xu, W. Bit-level probabilistically shaped coded modulation. IEEE Commun. Lett.
**2017**, 21, 1929–1932. [Google Scholar] [CrossRef] - Fehenberger, T.; Millar, D.S.; Koike-Akino, T.; Kojima, K.; Parsons, K. Parallel-amplitude architecture and subset ranking for fast distribution Matching. IEEE Trans. Commun.
**2020**. [Google Scholar] [CrossRef] [Green Version] - Lang, G.R.; Longstaff, F.M. A Leech lattice modem. IEEE J. Sel. Areas Commun.
**1989**, 7, 968–973. [Google Scholar] [CrossRef] - Gültekin, Y.C.; van Houtum, W.J.; Şerbetli, S.; Willems, F.M.J. Constellation shaping for IEEE 802.11. In Proceedings of the IEEE 21st International Symposium on Personal, Indoor and Mobile Radio Communications, Montreal, QC, Canada, 8–13 October 2017. [Google Scholar]
- Amari, A.; Goossens, S.; Gültekin, Y.C.; Vassilieva, O.; Kim, I.; Ikeuchi, T.; Okonkwo, C.; Willems, F.M.J.; Alvarado, A. Introducing enumerative sphere shaping for optical communication systems with short blocklengths. J. Lightwave Technol.
**2019**, 37, 5926–5936. [Google Scholar] [CrossRef] [Green Version] - Amari, A.; Goossens, S.; Gültekin, Y.C.; Vassilieva, O.; Kim, I.; Ikeuchi, T.; Okonkwo, C.; Willems, F.M.J.; Alvarado, A. Enumerative sphere shaping for rate adaptation and reach increase in WDM transmission systems. In Proceedings of the European Conference on Communications, Valencia, Spain, 18–21 June 2019. [Google Scholar]
- Goossens, S.; van der Heide, S.; van den Hout, M.; Amari, A.; Gültekin, Y.C.; Vassilieva, O.; Kim, I.; Willems, F.M.J.; Alvarado, A.; Okonkwo, C. First experimental demonstration of probabilistic enumerative sphere shaping in optical fiber communications. In Proceedings of the 2019 24th OptoElectronics and Communications Conference (OECC) and 2019 International Conference on Photonics in Switching and Computing (PSC), Fukuoka, Japan, 7–11 July 2019. [Google Scholar]
- Schulte, P.; Steiner, F. Divergence-optimal fixed-to-fixed length distribution matching with shell mapping. IEEE Wirel. Commun. Lett.
**2019**, 8, 620–623. [Google Scholar] [CrossRef] [Green Version] - Gültekin, Y.C.; Willems, F.M.J.; van Houtum, W.J.; Şerbetli, S. Approximate Enumerative Sphere Shaping. In Proceedings of the IEEE International Symposium on Information Theory (ISIT), Vail, CO, USA, 17–22 June 2018. [Google Scholar]
- Yoshida, T.; Karlsson, M.; Agrell, E. Short-block-length shaping by simple mark ratio controllers for granular and wide-range spectral efficiencies. In Proceedings of the European Conference on Optical Communication (ECOC), Gothenburg, Sweden, 17–21 September 2017. [Google Scholar]
- Yoshida, T.; Karlsson, M.; Agrell, E. Low-complexity variable-length output distribution matching with periodical distribution uniformalization. In Proceedings of the Optical Fiber Communication Conference, San Diego, CA, USA, 11–15 March 2018. [Google Scholar]
- Böcherer, G.; Steiner, F.; Schulte, P. Fast probabilistic shaping implementation for long-haul fiber-optic communication systems. In Proceedings of the European Conference on Optical Communication (ECOC), Gothenburg, Sweden, 17–21 September 2017. [Google Scholar]
- Cho, J.; Winzer, P.J. Multi-rate prefix-free code distribution matching. In Proceedings of the Optical Fiber Communications Conference and Exhibition (OFC), San Diego, CA, USA, 3–7 March 2019. [Google Scholar]
- Cho, J. Prefix-Free Code Distribution Matching for Probabilistic Constellation Shaping. IEEE Trans. Commun.
**2020**, 68, 670–682. [Google Scholar] [CrossRef] - Pikus, M.; Xu, W. Arithmetic coding based multi-composition codes for bit-level distribution matching. In Proceedings of the IEEE Wireless Communications and Networking Conference (WCNC), Marrakesh, Morocco, 15–18 April 2019. [Google Scholar]
- Pikus, M.; Xu, W.; Kramer, G. Finite-precision implementation of arithmetic coding based distribution matchers. arXiv
**2019**, arXiv:1907.12066. [Google Scholar] - Gültekin, Y.C.; van Houtum, W.J.; Koppelaar, A.; Willems, F.M. Partial Enumerative Sphere Shaping. In Proceedings of the IEEE Vehicular Technology Conference (Fall), Honolulu, HI, USA, 22–25 September 2019. [Google Scholar]
- Yoshida, T.; Karlsson, M.; Agrell, E. Hierarchical distribution matching for probabilistically shaped coded modulation. J. Lightwave Technol.
**2019**, 37, 1579–1589. [Google Scholar] [CrossRef] [Green Version] - Civelli, S.; Secondini, M. Hierarchical distribution matching: A versatile tool for probabilistic shaping. In Proceedings of the Optical Fiber Communications Conference and Exhibition (OFC), San Diego, CA, USA, 8–12 March 2020. [Google Scholar]
- Millar, D.S.; Fehenberger, T.; Yoshida, T.; Koike-Akino, T.; Kojima, K.; Suzuki, N.; Parsons, K. Huffman coded sphere shaping with short length and reduced complexity. In Proceedings of the European Conference on Optical Communication, Dublin, Ireland, 22–26 September 2019. [Google Scholar]
- Digital Video Broadcasting (DVB); 2nd Generation Framing Structure, Channel Coding and Modulation Systems for Broadcasting, Interactive Services, News Gathering and Other Broadband Satellite Applications (DVB-S2); (ETSI) Standard EN 302 307, Rev. 1.2.1; European Telecommunications Standards Institute: Sophia Antipolis, France, 2009.
- IEEE Standard for Inform. Technol.-Telecommun. and Inform. Exchange Between Syst. Local and Metropolitan Area Networks-Specific Requirements-Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications; Revision of IEEE Standard 802.11-2012; IEEE Standard 802.11-2016: Piscataway, NJ, USA, 2016.
- Cover, T.M.; Thomas, J.A. Elements of Information Theory; John Wiley & Sons: Hoboken, NJ, USA, 2006. [Google Scholar]
- Böcherer, G. Principles of Coded Modulation. Habilitation Thesis, Department of Electrical and Computer Engineering, Technical University of Munich, Munich, Germany, 2018. [Google Scholar]
- Khandani, A.K.; Kabal, P. Shaping multidimensional signal spaces – Part I: Optimum shaping, shell mapping. IEEE Trans. Inf. Theory
**1993**, 39, 1799–1808. [Google Scholar] [CrossRef] - Tretter, S. Constellation Shaping, Nonlinear Precoding, and Trellis Coding for Voiceband Telephone Channel Modems with Emphasis on ITU-T Recommendation; Springer: New York, NY, USA, 2002. [Google Scholar]
- Vassilieva, O.; Kim, I.; Ikeuchi, T. On the fairness of the performance evaluation of probabilistically shaped QAM. In Proceedings of the European Conference on Communications (ECOC), Athens, Greece, 13–16 May 2019. [Google Scholar]
- Böcherer, G. Probabilistic signal shaping for bit-metric decoding. In Proceedings of the IEEE International Symposium on Information Theory, Honolulu, HI, USA, 29 June–4 July 2014. [Google Scholar]
- Böcherer, G. Probabilistic signal shaping for bit-metric decoding. arXiv
**2014**, arXiv:1401.6190. [Google Scholar] - Kramer, G. Topics in multi-user information theory. Found. Trends Commun. Inf. Theory
**2008**, 4, 265–444. [Google Scholar] [CrossRef] [Green Version] - Böcherer, G. Achievable rates for shaped bit-metric decoding. arXiv
**2016**, arXiv:1410.8075. [Google Scholar] - Merhav, N.; Kaplan, G.; Lapidoth, A.; Shamai (Shitz), S. On information rates for mismatched decoders. IEEE Trans. Inf. Theory
**1994**, 40, 1953–1967. [Google Scholar] [CrossRef] [Green Version] - Böcherer, G. Achievable rates for probabilistic shaping. arXiv
**2018**, arXiv:1707.01134. [Google Scholar] - Amjad, R.A. Information Rates and Error Exponents for Probabilistic Amplitude Shaping. In Proceedings of the IEEE IEEE Information Theory Workshop (ITW), Guangzhou, China, 25–29 November 2018. [Google Scholar]
- Gültekin, Y.C.; Alvarado, A.; Willems, F.M.J. Achievable information rates for probabilistic amplitude shaping: A minimum-randomness approach via random sign-coding arguments. arXiv
**2020**, arXiv:2002.10387. [Google Scholar] - Fischer, R.F.H.; Huber, J.B.; Wachsmann, U. Multilevel coding: Aspects from information theory. In Proceedings of the IEEE Global Telecommunications Conference, London, UK, 18–28 November 1996. [Google Scholar]
- Böcherer, G.; Geiger, B.C. Optimal quantization for distribution synthesis. IEEE Trans. Inf. Theory
**2016**, 62, 6162–6172. [Google Scholar] [CrossRef] [Green Version] - Millar, D.S.; Fehenberger, T.; Koike-Akino, T.; Kojima, K.; Parsons, K. Distribution matching for high spectral efficiency optical communication with multiset partitions. J. Lightwave Technol.
**2019**, 37, 517–523. [Google Scholar] [CrossRef] - Fehenberger, T.; Millar, D.S.; Koike-Akino, T.; Kojima, K.; Parsons, K. Partition-based probabilistic shaping for fiber-optic communication systems. In Proceedings of the Optical Fiber Communication Conference, San Diego, CA, USA, 3–7 March 2019. [Google Scholar]
- Schalkwijk, J. An algorithm for source coding. IEEE Trans. Inf. Theory
**1972**, 18, 395–399. [Google Scholar] [CrossRef] - Cover, T. Enumerative source encoding. IEEE Trans. Inf. Theory
**1973**, 19, 73–77. [Google Scholar] [CrossRef] - Wozencraft, J.M.; Jacobs, I.M. Principles of Communication Engineering; John Wiley & Sons: New York, NY, USA, 1965. [Google Scholar]
- Gültekin, Y.C.; Willems, F.M.J. Building the optimum enumerative shaping trellis. In Proceedings of the Symposium on Information Theory in the Benelux, Bruxelles, Benelux, 28–29 May 2019; p. 34. [Google Scholar]
- Fischer, R.F.H. Calculation of shell frequency distributions obtained with shell-mapping schemes. IEEE Trans. Inf. Theory
**1999**, 45, 1631–1639. [Google Scholar] [CrossRef] [Green Version] - Böcherer, G.; Schulte, P.; Steiner, F. Probabilistic shaping and forward error correction for fiber-optic communication Systems. J. Lightwave Technol.
**2019**, 37, 230–244. [Google Scholar] [CrossRef] - Yoshida, T.; Binkai, M.; Koshikawa, S.; Chikamori, S.; Matsuda, K.; Suzuki, N.; Karlsson, M.; Agrell, E. FPGA implementation of distribution matching and dematching. In Proceedings of the European Conference on Optical Communication, Athens, Greece, 13–16 May 2019. [Google Scholar]
- Li, J.; Zhang, A.; Zhang, C.; Huo, X.; Yang, Q.; Wang, J.; Wang, J.; Qu, W.; Wang, Y.; Zhang, J.; et al. Field trial of probabilistic-shaping-programmable real-time 200-Gb/s coherent transceivers in an intelligent core optical network. In Proceedings of the Asia Communications and Photonics Conference (ACP), Hangzhou, China, 26–29 October 2018. [Google Scholar]
- Zhang, Z.; Wang, J.; Ouyang, S.; Wang, J.; Chen, J.; Liu, X.; Chen, J.; Wang, Y.; Wang, W.; Ding, T.; et al. Real-time measurement of a probabilistic-shaped 200Gb/s DP-16QAM transceiver. Opt. Express
**2019**, 27, 18787–18793. [Google Scholar] [CrossRef] [PubMed] - Yu, Q.; Corteselli, S.; Cho, J. FPGA implementation of prefix-free code distribution matching for probabilistic constellation shaping. In Proceedings of the Optical Fiber Communication Conference, San Diego, CA, USA, 8–12 March 2020. [Google Scholar]
- Rissanen, J.; Langdon, G.G. Arithmetic Coding. IBM J. Res. Dev.
**1979**, 23, 149–162. [Google Scholar] [CrossRef] [Green Version] - Langdon, G.G. An Introduction to Arithmetic Coding. IBM J. Res. Dev.
**1984**, 28, 135–149. [Google Scholar] [CrossRef]

**Figure 1.**Taxonomy of shaping in the context of probabilistic amplitude shaping (PAS). We focus on the schemes that are evaluated in this work.

**Figure 2.**(

**Top**) Block diagram of the PAS architecture. Amplitude shaping blocks (green boxes) are examined in the current paper. (

**Bottom**) The binary reflected Gray code (BRGC) for 8-ary amplitude- shift keying (8-ASK). A quadrature amplitude modulation (QAM) symbol is the concatenation of two ASK symbols.

**Figure 3.**Signaling options: (

**top**) uniform signaling with rate $R=k/n$ bit/1-D, (

**middle**) PAS with rate $R=k/n$ bit/1-D (all information is on amplitudes), (

**bottom**) modified PAS with rate $R=k/n+\gamma $ bit/1-D (extra data is carried on signs).

**Figure 5.**Channel input entropy vs. gap-to-capacity for 8-ASK at the target rate of $R=2.25$ bit/1-D. The x-axis above shows the corresponding FEC code rates.

**Figure 6.**The illustration of the employed n-dimensional signal points by CCDM (

**left**), MPDM (

**middle**) and SpSh (

**right**). Each circle represents an n-dimensional shell. Darker portions of the shells indicate the signal points on them which are utilized by the corresponding shaping approach.

**Figure 10.**FER vs. SNR for 64-QAM at a transmission rate of 4.5 bit/2-D. Rate-${R}_{\mathrm{c}}$ DVB-S2 LDPC codes of length ${n}_{\mathrm{c}}$ = 64,800 bits are used. All shaping schemes use a blocklength of $n=180$. At this shaping blocklength, each LDPC codeword consists of 120 shaped blocks.

**Figure 11.**FER vs. SNR for 64-QAM at transmission rates of 4 and 4.5 bit/2-D. Rate-$5/6$ IEEE 802.11 LDPC codes of length ${n}_{\mathrm{c}}=648$ bits are used for shaped signaling. All shaping schemes use a blocklength of $n=216$. At this shaping blocklength, each LDPC codeword consists of 1 shaped block.

**Figure 12.**Approximate illustration of maximum computational complexity vs. maximum required storage for ESS, SM and CCDM. Red- and blue-colored markers indicate FP and BP implementations, respectively. Radii of the markers are proportional to the corresponding blocklength $n\in \{64,216,512\}$. Here we assume that BP AC-CCDM is implemented with finite-precision arithmetic using 16-bit numbers which is comparable to the values selected in [72]. Furthermore, we assume that a k-bit addition is equivalent to k bit operations, while a k-bit multiplication is equivalent to ${k}^{2}$ bit operations as in [23].

**Table 1.**Content of an amplitude sequence as in Figure 4 based on Example 2.

Parameter | Formula (per n-Sequence) | Value per 1-D (Example 2) | Value per 216-D (Example 2) |
---|---|---|---|

Data on amp. | $n\mathbb{H}(A)$ | 1.75 | 378 |

Data on sign | $n\gamma $ | 0.50 | 108 |

Shap. redundancy | $n(m-1-\mathbb{H}(A))$ | 0.25 | 54 |

Cod. redundancy | $n(\mathbb{H}(A)+1-R)$ | 0.50 | 108 |

Redundancy | $n(m-R)$ | 0.75 | 162 |

Data, $nR$ | $n(\mathbb{H}(A)+\gamma )$ | 2.25 | 486 |

Architecture | k | $\mathit{k}/\mathit{n}$ | E | $\mathbb{H}(\overline{\mathit{A}})$ or $\mathbb{H}(\tilde{\mathit{A}})$ | R_{loss} |
---|---|---|---|---|---|

CCDM | 367 | 1.6991 | 11.00 | 1.7504 | 0.0513 |

MPDM | 374 | 1.7315 | 11.00 | 1.7504 | 0.0189 |

SpSh | 374 | 1.7315 | 10.90 | 1.7448 | 0.0133 |

Direct Method (Distribution Matching) | Indirect Method (Energy-Efficient Signal Space) | ||||
---|---|---|---|---|---|

AC-CCDM [52] | SR-DM [58] | ESS [22] and (Algorithm 1 in [23]) | SM [23] | ||

Serialism (no. of loop iter.) | $k+n$ | $min({n}_{1},n-{n}_{1})+1$ | $k+n$ | $k+{log}_{2}n$ | |

Storage Complexity | $\mathcal{O}(logn)$ | $\mathcal{O}(logn)$ | FP: $\mathcal{O}({n}^{3})$ BP [65]: $\mathcal{O}({n}^{2}logn)$ | FP: $\mathcal{O}({n}^{2}logn)$ BP [65]: $\mathcal{O}(n{log}^{2}n)$ | |

Computations (per 1-D) | ${n}_{a}$ divisions, multiplications and comparisons | Sh: $({n}_{a}-1)$ BCs Dsh: $({n}_{a}-1)/2$ BCs | Sh: ${n}_{a}$ comparisons and subtractions Dsh: ${n}_{a}$ additions (and L comparisons/additions per n-D for [23, Algorithm 1]) | Sh: L multiplications, comparisons and subtractions ${}^{\u2020}$ Dsh: L multiplications and additions |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Gültekin, Y.C.; Fehenberger, T.; Alvarado, A.; Willems, F.M.J.
Probabilistic Shaping for Finite Blocklengths: Distribution Matching and Sphere Shaping. *Entropy* **2020**, *22*, 581.
https://doi.org/10.3390/e22050581

**AMA Style**

Gültekin YC, Fehenberger T, Alvarado A, Willems FMJ.
Probabilistic Shaping for Finite Blocklengths: Distribution Matching and Sphere Shaping. *Entropy*. 2020; 22(5):581.
https://doi.org/10.3390/e22050581

**Chicago/Turabian Style**

Gültekin, Yunus Can, Tobias Fehenberger, Alex Alvarado, and Frans M. J. Willems.
2020. "Probabilistic Shaping for Finite Blocklengths: Distribution Matching and Sphere Shaping" *Entropy* 22, no. 5: 581.
https://doi.org/10.3390/e22050581