{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,14]],"date-time":"2026-01-14T15:46:22Z","timestamp":1768405582472,"version":"3.49.0"},"reference-count":30,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2025,4,21]],"date-time":"2025-04-21T00:00:00Z","timestamp":1745193600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Zhejiang Provincial Natural Science Foundation of China","award":["LMS25F050004"],"award-info":[{"award-number":["LMS25F050004"]}]},{"name":"Zhejiang Provincial Natural Science Foundation of China","award":["62201507"],"award-info":[{"award-number":["62201507"]}]},{"name":"National Natural Science Foundation of China","award":["LMS25F050004"],"award-info":[{"award-number":["LMS25F050004"]}]},{"name":"National Natural Science Foundation of China","award":["62201507"],"award-info":[{"award-number":["62201507"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The choice of constellations largely affects the performance of both wireless and optical communications. To address increasing capacity requirements, constellation shaping, especially for high-order modulations, is imperative in high-speed coherent communication systems. This paper, thus, proposes novel mutual information neural estimation (MINE)-based geometric, probabilistic, and joint constellation shaping schemes, i.e., the MINE-GCS, MINE-PCS, and MINE-JCS, to maximize mutual information (MI) via emerging deep learning (DL) techniques. Innovatively, we first introduce the MINE module to effectively estimate and maximize MI through backpropagation, without clear knowledge of the channel state information. Then, we train encoder and probability generator networks with different signal-to-noise ratios to optimize the distribution locations and probabilities of the points, respectively. Note that MINE transforms the precise MI calculation problem into a parameter optimization problem. Our MINE-based schemes only optimize the transmitter end, and avoid the computational and structural complexity in traditional shaping. All the designs were verified through simulations as having superior performance for MI, among which the MINE-JCS undoubtedly performed the best for additive white Gaussian noise, compared to the unshaped QAMs and even the end-to-end training and other DL-based joint shaping schemes. For example, the low-order 8-ary MINE-GCS could achieve an MI gain of about 0.1 bits\/symbol compared to the unshaped Star-8QAM. It is worth emphasizing that our proposed schemes achieve a balance between implementation complexity and MI performance, and they are expected to be applied in various practical scenarios with different noise and fading levels in the future.<\/jats:p>","DOI":"10.3390\/e27040451","type":"journal-article","created":{"date-parts":[[2025,4,21]],"date-time":"2025-04-21T20:42:00Z","timestamp":1745268120000},"page":"451","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Mutual Information Neural-Estimation-Driven Constellation Shaping Design and Performance Analysis"],"prefix":"10.3390","volume":"27","author":[{"given":"Xiuli","family":"Ji","sequence":"first","affiliation":[{"name":"Institute of Cyberspace Security, Zhejiang University of Technology, Hangzhou 310023, China"}]},{"given":"Qian","family":"Wang","sequence":"additional","affiliation":[{"name":"Institute of Cyberspace Security, Zhejiang University of Technology, Hangzhou 310023, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6210-2617","authenticated-orcid":false,"given":"Liping","family":"Qian","sequence":"additional","affiliation":[{"name":"Institute of Cyberspace Security, Zhejiang University of Technology, Hangzhou 310023, China"}]},{"given":"Pooi-Yuen","family":"Kam","sequence":"additional","affiliation":[{"name":"The School of Science and Engineering, The Chinese University of Hong Kong (Shenzhen), Shenzhen 518172, China"}]}],"member":"1968","published-online":{"date-parts":[[2025,4,21]]},"reference":[{"key":"ref_1","first-page":"1","article-title":"Capacity-approaching TQC-LDPC convolutional codes enabling power-efficient decoders","volume":"65","author":"Pisek","year":"2017","journal-title":"IEEE Trans. Commun."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1939","DOI":"10.1109\/TCOMM.2015.2424434","article-title":"Trellis-based QC-LDPC convolutional codes enabling low power decoders","volume":"63","author":"Pisek","year":"2015","journal-title":"IEEE Trans. Commun."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1318","DOI":"10.1109\/JLT.2017.2786750","article-title":"High spectral efficiency PM-128QAM comb-based superchannel transmission enabled by a single shared optical pilot tone","volume":"36","author":"Mazur","year":"2018","journal-title":"J. Lightw. Technol."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2154","DOI":"10.1109\/LWC.2020.3015890","article-title":"M-APSK constellation optimization in the presence of phase reference error","volume":"9","author":"Wang","year":"2020","journal-title":"IEEE Wirel. Commun. Lett."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"108","DOI":"10.1109\/MMM.2009.932835","article-title":"High-capacity ethernet backhaul radio systems for advanced mobile data networks","volume":"10","author":"Boch","year":"2009","journal-title":"IEEE Microw. Mag."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"617","DOI":"10.1109\/OJCOMS.2021.3067384","article-title":"A survey on higher-order QAM constellations: Technical challenges, recent advances, and future trends","volume":"2","author":"Singya","year":"2021","journal-title":"IEEE Open J. Commun. Soc."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"846","DOI":"10.1109\/LPT.2023.3284504","article-title":"Triple-convex probabilistic constellation shaping PAM8 in IM\/DD system","volume":"35","author":"Sun","year":"2023","journal-title":"IEEE Photon. Technol. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2119","DOI":"10.1109\/LCOMM.2020.3001253","article-title":"A note on probabilistic and geometric shaping for the AWGN channel","volume":"24","author":"Modonesi","year":"2020","journal-title":"IEEE Commun. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2407","DOI":"10.1109\/LPT.2014.2358274","article-title":"Constellation shaping for fiber-optic channels with QAM and high spectral efficiency","volume":"26","author":"Yankov","year":"2014","journal-title":"IEEE Photon. Technol. Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"501","DOI":"10.1109\/JLT.2017.2752842","article-title":"Comparison of probabilistically shaped 64QAM with lower cardinality uniform constellations in long-haul optical systems","volume":"36","author":"Pilori","year":"2018","journal-title":"J. Lightw. Technol."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1076","DOI":"10.1109\/LPT.2019.2917925","article-title":"Probabilistic amplitude shaping for a 64-QAM OFDM W-Band RoF system","volume":"31","author":"Wu","year":"2019","journal-title":"IEEE Photon. Technol. Lett."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"5494","DOI":"10.1109\/JLT.2021.3087919","article-title":"High spectral efficiency WDM transmission based on hybrid probabilistically and geometrically shaped 256QAM","volume":"39","author":"Ding","year":"2021","journal-title":"J. Lightw. Technol."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Stark, M., Aoudia, F.A., and Hoydis, J. (2019, January 9\u201313). Joint learning of geometric and probabilistic constellation shaping. Proceedings of the 2019 IEEE Globecom Workshops (GC Wkshps), Waikoloa, HI, USA.","DOI":"10.1109\/GCWkshps45667.2019.9024567"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1191","DOI":"10.1162\/089976603321780272","article-title":"Estimation of entropy and mutual information","volume":"15","author":"Paninski","year":"2003","journal-title":"Neural Comput."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"563","DOI":"10.1109\/TCCN.2017.2758370","article-title":"An introduction to deep learning for the physical layer","volume":"3","author":"Hoydis","year":"2017","journal-title":"IEEE Trans. Cogn. Commun. Netw."},{"key":"ref_16","unstructured":"Alemi, A.A., Fischer, I., Dillon, J.V., and Murphy, K. (2016). Deep variational information bottleneck. arXiv."},{"key":"ref_17","unstructured":"Hjelm, R.D., Lavoie-Marchildon, A., Grewal, S., Bachman, K., Trischler, P.A., and Bengio, Y. (2019). Learning deep representations by mutual information estimation and maximization. arXiv."},{"key":"ref_18","first-page":"530","article-title":"Mutual information neural estimation","volume":"80","author":"Belghazi","year":"2018","journal-title":"Proc. Int. Conf. Mach. Learn."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Omidi, A., Zeng, M., Lin, J., and Rusch, L.A. (2021, January 24\u201328). Geometric constellation shaping using initialized autoencoders. Proceedings of the 2021 IEEE International Black Sea Conference on Communications and Networking (BlackSeaCom), Bucharest, Romania.","DOI":"10.1109\/BlackSeaCom52164.2021.9527735"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"3316","DOI":"10.1109\/JLT.2022.3169993","article-title":"End-to-end learning of a constellation shape robust to channel condition uncertainties","volume":"40","author":"Jovanovic","year":"2022","journal-title":"J. Lightw. Technol."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Aref, V., and Chagnon, M. (2022, January 6\u201310). End-to-end learning of joint geometric and probabilistic constellation shaping. Proceedings of the Optical Fiber Communication Conference 2022, San Diego, CA, USA.","DOI":"10.1364\/OFC.2022.W4I.3"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"481","DOI":"10.1109\/JLT.2020.3031017","article-title":"Geometric shaping of 2-D constellations in the presence of laser phase noise","volume":"39","author":"Dzieciol","year":"2021","journal-title":"J. Lightw. Technol."},{"key":"ref_23","unstructured":"Ash, R.B. (2008). Basic Probability Theory, Courier Corporation."},{"key":"ref_24","unstructured":"Kullback, S. (1997). Information Theory and Statistics, Courier Corporation."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1002\/cpa.3160360204","article-title":"Asymptotic evaluation of certain markov process expectations for large time. IV","volume":"36","author":"Donsker","year":"1983","journal-title":"Commun. Pure Appl. Math."},{"key":"ref_26","first-page":"271","article-title":"f-GAN: Training generative neural samplers using variational divergence minimization","volume":"29","author":"Nowozin","year":"2016","journal-title":"Proc. Neural Int. Process. Syst."},{"key":"ref_27","unstructured":"Jang, E., Gu, S., and Poole, B. (2017). Categorical reparameterization with Gumbel-Softmax. arXiv."},{"key":"ref_28","unstructured":"Paszke, A. (2019). PyTorch: An imperative style, high-performance deep learning library. arXiv."},{"key":"ref_29","unstructured":"Kingma, D.P., and Ba, J. (2015). Adam: A method for stochastic optimization. arXiv."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Wang, Q., Ji, X., Qian, L.P., Liu, Z., Du, X., and Kam, P.-Y. (2023, January 10\u201312). MINE-based geometric constellation shaping in AWGN channel. Proceedings of the 2023 IEEE\/CIC International Conference on Communications in China (ICCC Workshops), Dalian, China.","DOI":"10.1109\/ICCCWorkshops57813.2023.10233820"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/4\/451\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T17:18:56Z","timestamp":1760030336000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/27\/4\/451"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,21]]},"references-count":30,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2025,4]]}},"alternative-id":["e27040451"],"URL":"https:\/\/doi.org\/10.3390\/e27040451","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,4,21]]}}}