{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,16]],"date-time":"2026-05-16T16:54:51Z","timestamp":1778950491909,"version":"3.51.4"},"reference-count":33,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2019,5,1]],"date-time":"2019-05-01T00:00:00Z","timestamp":1556668800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["51874264, 61571108, 61501158"],"award-info":[{"award-number":["51874264, 61571108, 61501158"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Based on density evolution analysis of the existing belief propagation (BP) algorithm, the Turbo Decoding Message Passing (TDMP) algorithm was analyzed from the perspective of density evolution and Gaussian approximation, and the theoretical analysis process of TDMP algorithm was given. When calculating the prior message of each layer of the TDMP algorithm, the check message of the previous iteration should be subtracted. Therefore, the result will not be convergent, if the TDMP algorithm is directly analyzed based on density evolution and Gaussian approximation. We researched the TDMP algorithm based on the symmetry conditions to obtain the convergent result. When using density evolution (DE) and Gaussian approximation to analyze the decoding convergence of the TDMP algorithm, we can provide a theoretical basis for proving the superiority of the algorithm. Then, based on the DE theory, we calculated the probability density function (PDF) of the check-to-variable information of TDMP and its simplified algorithm, and then gave it a calculation based on the process of the normalization factor. Simulation results show that the decoding convergence speed of the TDMP algorithm was faster and the iterations were smaller compared to the BP algorithm under the same conditions.<\/jats:p>","DOI":"10.3390\/e21050457","type":"journal-article","created":{"date-parts":[[2019,5,2]],"date-time":"2019-05-02T03:15:22Z","timestamp":1556766922000},"page":"457","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Analysis of TDMP Algorithm of LDPC Codes Based on Density Evolution and Gaussian Approximation"],"prefix":"10.3390","volume":"21","author":[{"given":"Xiumin","family":"Wang","sequence":"first","affiliation":[{"name":"College of Information Engineering, China Jiliang University, Hangzhou 310018, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hong","family":"Chang","sequence":"additional","affiliation":[{"name":"College of Information Engineering, China Jiliang University, Hangzhou 310018, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jun","family":"Li","sequence":"additional","affiliation":[{"name":"Binjiang College, Nanjing University of Information Science &amp; Technology, Wuxi 214105, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Weilin","family":"Cao","sequence":"additional","affiliation":[{"name":"College of Information Engineering, China Jiliang University, Hangzhou 310018, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Liang","family":"Shan","sequence":"additional","affiliation":[{"name":"College of Information Engineering, China Jiliang University, Hangzhou 310018, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,5,1]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1109\/TIT.1962.1057683","article-title":"Low-density parity-check codes","volume":"IT-8","author":"Gallager","year":"1962","journal-title":"IRE Trans. Inf. Theory"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1645","DOI":"10.1049\/el:19961141","article-title":"Near Shannon limit performance of low-density parity-check codes","volume":"32","author":"MacKay","year":"1996","journal-title":"IEEE Electron. Lett."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1449","DOI":"10.1109\/26.795809","article-title":"Comparison of constructions of irregular Gallager codes","volume":"47","author":"Mackay","year":"1999","journal-title":"IEEE Trans. Commun."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"619","DOI":"10.1109\/18.910578","article-title":"Design of capacity-approaching irregular low-density parity-check codes","volume":"47","author":"Richardson","year":"2001","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Kulkarni, V., and Sankar, K.J. (2015, January 7\u20138). Design of structured irregular LDPC codes from structured regular LDPC codes. Proceedings of the 2015 Third International Conference on Computer, Communication, Control and Information Technology (C3IT), Hooghly, India.","DOI":"10.1109\/C3IT.2015.7060128"},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Huang, S., and Skoglund, M. (2017). On Linear Coding Over Finite Rings and Applications to Computing. Entropy, 9.","DOI":"10.3390\/e19050233"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"97","DOI":"10.1109\/CC.2016.7563692","article-title":"Construction of regular rate-compatible LDPC convolutional codes","volume":"13","author":"Mu","year":"2016","journal-title":"China Commun."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"399","DOI":"10.1109\/18.748992","article-title":"Good Error-Correcting Codes Based on Very Sparse Matrices","volume":"45","author":"Mackay","year":"1999","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Jayasooriya, S., Johnson, S.J., Ong, L., and Berretta, R. (2014, January 2\u20135). Optimization of graph based on codes for belief propagation decoding. Proceedings of the 2014 IEEE Information Theory Workshop (ITW 2014), Hobart, TAS, Australia.","DOI":"10.1109\/ITW.2014.6970873"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"877","DOI":"10.1109\/TCOMM.2013.011613.120249","article-title":"Check node reliability-based scheduling for BP decoding of non-binary LDPC codes","volume":"61","author":"Han","year":"2013","journal-title":"IEEE Trans. Commun."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"599","DOI":"10.1109\/18.910577","article-title":"The Capacity of Low-density Parity-check Codes Under Message-passing Decoding","volume":"47","author":"Richardson","year":"2001","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_12","unstructured":"Richiardson, T., and Urbanke, R. (,  2002). Multi-edge type LDPC codes. Proceedings of the Workshop Honoring Pof. Bob McEliece 60th Birthday, Pasadena, CA, USA."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"5274","DOI":"10.1109\/TIT.2010.2059490","article-title":"Iterative Decoding Threshold Analysis for LDPC Convolutional Codes","volume":"56","author":"Lentmaier","year":"2010","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"15","DOI":"10.1109\/MCOM.2016.7565182","article-title":"The three primary colors of mobile systems","volume":"54","author":"Liu","year":"2016","journal-title":"IEEE Commun. Mag."},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Hao, Y., Xiao, K., Chen, Z., and Xia, B. (2017, January 11\u201313). Density Evolution Analysis of LDPC-coded SCMA Systems. Proceeding of the 2017 9th International Conference on Wireless Communications and Signal Processing, Nanjing, China.","DOI":"10.1109\/WCSP.2017.8171149"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"657","DOI":"10.1109\/18.910580","article-title":"Analysis of Sum-Product Decoding of Low-Density Parity-Check Codes Using a Gaussian Approximation","volume":"47","author":"Chung","year":"2001","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"166","DOI":"10.1109\/TBC.2018.2883918","article-title":"Performance Analysis of LDPC-BICM System Based on Gaussian Approximation","volume":"65","author":"Zhang","year":"2019","journal-title":"IEEE T. Broadcast."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"230","DOI":"10.1109\/LCOMM.2018.2886261","article-title":"More Accurate Analysis of Sum-Product Decoding of LDPC codes Using a Gaussian Approximation","volume":"23","author":"Vatta","year":"2019","journal-title":"IEEE Commun. Lett."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"976","DOI":"10.1109\/TVLSI.2003.817545","article-title":"High-Throughput LDPC Decoder","volume":"11","author":"Mansour","year":"2003","journal-title":"IEEE Trans. Very Large Scale Integr. Syst."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"4376","DOI":"10.1109\/TSP.2006.880240","article-title":"A Turbo-Decoding Message-Passing Algorithm for Sparse Parity-Check Matrix Codes","volume":"54","author":"Mansour","year":"2006","journal-title":"IEEE Tran. Signal. Process."},{"key":"ref_21","first-page":"4044","article-title":"A New Density Evolution Approximation for LDPC and Multi-Edge Type LDPC Codes","volume":"64","author":"Jayasooriya","year":"2016","journal-title":"IEEE Trans. Commun."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Mamani, H., and Saeedi, H. (2011, January 15\u201316). Generalized EXIT chars for irregular LDPC codes. Proceedings of the 2011 International Symposium on Artificial Intelligence and Signal Processing (AISP), Tehran, Iran.","DOI":"10.1109\/AISP.2011.5960973"},{"key":"ref_23","unstructured":"Cheng, S., and Xiong, Z. (2004, January 24\u201329). Channel symmetry in Slepian-Wolf code design based on LDPC codes with application to the quadratic Gaussian Wyner-Ziv problem. Proceedings of the Information Theory Workshop, San Antonio, TX, USA."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1582","DOI":"10.1049\/iet-com.2017.0014","article-title":"Improved min-sum algorithm based on density evolution for low-density parity check codes","volume":"11","author":"Wang","year":"2017","journal-title":"IET Commun."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"406","DOI":"10.1109\/26.990903","article-title":"Near Optimum Universal Belief Propagation Based Decoding of Low-Density Parity Check Codes","volume":"50","author":"Chen","year":"2002","journal-title":"IEEE Trans. Commun."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/s11265-010-0477-6","article-title":"A Flexible LDPC\/Turbo Decoder Architecture","volume":"64","author":"Sun","year":"2011","journal-title":"J. Signal Process. Syst."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1007\/s11265-012-0680-8","article-title":"A Reconfigurable TDMP Decoder for Raptor Codes","volume":"69","author":"Zeineddine","year":"2012","journal-title":"J. Signal Process. Syst."},{"key":"ref_28","first-page":"56","article-title":"Design of a High-Throughput QC-LDPC Decoder with TDMP Scheduling","volume":"62","author":"Zhao","year":"2015","journal-title":"IEEE T. CIRCUITS-II"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Lao, L., Li, L., Zhu, M., and Zhang, H. (2014, January 5\u20138). The Improved Turbo-decoding Message-passing Algorithm and Corresponding Decoder for LDPC Based on LTE. Proceedings of the IEEE International Conference on Signal Processing, Communications and Computing, Guilin, China.","DOI":"10.1109\/ICSPCC.2014.6986326"},{"key":"ref_30","first-page":"76","article-title":"Analysis of TDMP algorithm of LDPC coded based on GA","volume":"28","author":"Wang","year":"2017","journal-title":"J. China Univ. Metrol."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"116","DOI":"10.1049\/el:20092323","article-title":"Simplified sum-product algorithm for decoding LDPC codes with optimal performance","volume":"45","author":"Papaharalabos","year":"2009","journal-title":"Electron. Lett."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1587\/transinf.E95.D.403","article-title":"A Flexible LDPC Decoder Architecture Supporting TPMP and TDMP Decoding Algorithms","volume":"E95D","author":"Huang","year":"2012","journal-title":"IEICE Trans. Inf. Syst."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Li, S., Zhang, Q., Chen, Y., and Zeng, X. (2018, January 19\u201321). A High-Throughput QC-LDPC Decoder for Near-Earth Application. Proceedings of the 2018 IEEE 23rd International Conference on Digital Signal Processing (DSP), Shanghai, China.","DOI":"10.1109\/ICDSP.2018.8631641"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/5\/457\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:48:38Z","timestamp":1760186918000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/5\/457"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,1]]},"references-count":33,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2019,5]]}},"alternative-id":["e21050457"],"URL":"https:\/\/doi.org\/10.3390\/e21050457","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,5,1]]}}}