{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:02:26Z","timestamp":1760234546840,"version":"build-2065373602"},"reference-count":24,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2021,5,17]],"date-time":"2021-05-17T00:00:00Z","timestamp":1621209600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"University of Trieste","award":["FRA 2020"],"award-info":[{"award-number":["FRA 2020"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Information"],"abstract":"<jats:p>Low Density Parity Check (LDPC) codes are currently being deeply analyzed through algorithms that require the capability of addressing their iterative decoding convergence performance. Since it has been observed that the probability distribution function of the decoder\u2019s log-likelihood ratio messages is roughly Gaussian, a multiplicity of moderate entanglement strategies to this analysis has been suggested. The first of them was proposed in Chung et al.\u2019s 2001 paper, where the recurrent sequence, characterizing the passage of messages between variable and check nodes, concerns the function \u03d5(x), therein specified, and its inverse. In this paper, we review this old approximation to the function \u03d5(x), one variant on it obtained in the same period (proposed in Ha et al.\u2019s 2004 paper), and some new ones, recently published in two 2019 papers by Vatta et al. The objective of this review is to analyze the differences among them and their characteristics in terms of accuracy and computational complexity. In particular, the explicitly invertible, not piecewise defined approximation of the function \u03d5(x), published in the second of the two abovementioned 2019 papers, is shown to have less relative error in any x than most of the other approximations. Moreover, its use conducts to an important complexity reduction, and allows better Gaussian approximated thresholds to be obtained.<\/jats:p>","DOI":"10.3390\/info12050212","type":"journal-article","created":{"date-parts":[[2021,5,17]],"date-time":"2021-05-17T12:19:57Z","timestamp":1621253997000},"page":"212","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A Survey on Old and New Approximations to the Function \u03d5(x) Involved in LDPC Codes Density Evolution Analysis Using a Gaussian Approximation"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4105-8084","authenticated-orcid":false,"given":"Francesca","family":"Vatta","sequence":"first","affiliation":[{"name":"Department of Engineering and Architecture, University of Trieste, 34127 Trieste, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alessandro","family":"Soranzo","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Geosciences, University of Trieste, 34127 Trieste, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7261-1247","authenticated-orcid":false,"given":"Massimiliano","family":"Comisso","sequence":"additional","affiliation":[{"name":"Department of Engineering and Architecture, University of Trieste, 34127 Trieste, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9105-6066","authenticated-orcid":false,"given":"Giulia","family":"Buttazzoni","sequence":"additional","affiliation":[{"name":"Department of Engineering and Architecture, University of Trieste, 34127 Trieste, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0121-5524","authenticated-orcid":false,"given":"Fulvio","family":"Babich","sequence":"additional","affiliation":[{"name":"Department of Engineering and Architecture, University of Trieste, 34127 Trieste, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,5,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1109\/TIT.1981.1056404","article-title":"A recursive approach to low complexity codes","volume":"27","author":"Tanner","year":"1981","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_2","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_3","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_4","doi-asserted-by":"crossref","unstructured":"Vatta, F., Soranzo, A., Comisso, M., Buttazzoni, G., and Babich, F. (2019, January 18\u201320). Performance study of a class of irregular LDPC codes through low complexity bounds on their belief-propagation decoding thresholds. Proceedings of the 2019 AEIT International Annual Conference, AEIT 2019, Florence, Italy.","DOI":"10.23919\/AEIT.2019.8893306"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1038","DOI":"10.1049\/el.2018.0478","article-title":"Low-Complexity bound on irregular LDPC belief-propagation decoding thresholds using a Gaussian approximation","volume":"54","author":"Vatta","year":"2018","journal-title":"Electron. Lett."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2824","DOI":"10.1109\/TIT.2004.836667","article-title":"Rate-compatible puncturing of low-density parity-check codes","volume":"50","author":"Ha","year":"2004","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_7","first-page":"350","article-title":"Low complexity rate compatible puncturing patterns design for LDPC codes","volume":"14","author":"Babich","year":"2018","journal-title":"J. Commun. Softw. Syst. (JCOMSS)"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"90","DOI":"10.1049\/iet-com.2011.0271","article-title":"Bit-error rate analysis of low-density parity- check codes with generalised selection combining over a Rayleigh-fading channel using Gaussian approximation","volume":"6","author":"Tan","year":"2012","journal-title":"IET Commun."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1560","DOI":"10.1049\/iet-com.2010.0677","article-title":"Optimization of LDPC codes with deterministic unequal error protection properties","volume":"5","author":"Chen","year":"2011","journal-title":"IET Commun."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1183","DOI":"10.1049\/el.2019.1349","article-title":"New explicitly invertible approximation of the function involved in LDPC codes density evolution analysis using a Gaussian approximation","volume":"55","author":"Vatta","year":"2019","journal-title":"Electron. Lett."},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Babich, F., Noschese, M., Soranzo, A., and Vatta, F. (2017, January 21\u201323). Low complexity rate compatible puncturing patterns design for LDPC codes. Proceedings of the 2017 International Conference on Software, Telecommunications and Computer Networks, SoftCOM\u201917, Split, Croatia.","DOI":"10.23919\/SOFTCOM.2017.8115558"},{"key":"ref_12","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_13","doi-asserted-by":"crossref","unstructured":"Vatta, F., Babich, F., Ellero, F., Noschese, M., Buttazzoni, G., and Comisso, M. (2019). Role of the product \u03bb\u2032(0)\u03c1\u2032(1) in determining LDPC code performance. Electronics, 8.","DOI":"10.3390\/electronics8121515"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Vatta, F., Babich, F., Ellero, F., Noschese, M., Buttazzoni, G., and Comisso, M. (2019, January 19\u201321). Performance study of a class of irregular LDPC codes based on their weight distribution analysis. Proceedings of the 2019 International Conference on Software, Telecommunications and Computer Networks, SoftCOM\u201919, Split, Croatia.","DOI":"10.23919\/SOFTCOM.2019.8903633"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1949","DOI":"10.1109\/LCOMM.2017.2714684","article-title":"Useful mathematical tools for capacity approaching codes design","volume":"21","author":"Babich","year":"2017","journal-title":"IEEE Commun. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Hale, J., and Ko\u00e7ak, H. (1991). Dynamics and Bifurcations, Springer.","DOI":"10.1007\/978-1-4612-4426-4"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"544","DOI":"10.1109\/WCL.2012.080112.120232","article-title":"Invertible exponential-type approximations for the Gaussian probability integral Q(x) with applications","volume":"1","author":"Olabiyi","year":"2012","journal-title":"IEEE Wirel. Commun. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"639","DOI":"10.1109\/TCOM.1979.1094433","article-title":"Simple approximations of the error function Q(x) for communications applications","volume":"27","author":"Borjesson","year":"1979","journal-title":"IEEE Trans. Commun."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"840","DOI":"10.1109\/TWC.2003.814350","article-title":"New exponential bounds and approximations for the computation of error probability in fading channels","volume":"2","author":"Chiani","year":"2003","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1109\/LCOMM.2011.032111.102440","article-title":"An accurate and efficient approximation to the Gaussian Q-function and its applications in performance analysis in Nakagami-m fading","volume":"15","author":"Shi","year":"2011","journal-title":"IEEE Commun. Lett."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"917","DOI":"10.1109\/TCOMM.2011.012711.100105","article-title":"Versatile, accurate, and analytically tractable approximation for the Gaussian Q-function","volume":"59","author":"Benitez","year":"2011","journal-title":"IEEE Trans. Commun."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Soranzo, A., Vatta, F., Comisso, M., Buttazzoni, G., and Babich, F. (2019, January 19\u201321). New very simply explicitly invertible approximation of the Gaussian Q-function. Proceedings of the 2019 International Conference on Software, Telecommunications and Computer Networks, SoftCOM\u201919, Split, Croatia.","DOI":"10.23919\/SOFTCOM.2019.8903883"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"585","DOI":"10.1109\/18.910576","article-title":"Improved low-density parity-check codes using irregular graphs","volume":"47","author":"Luby","year":"2001","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_24","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"}],"container-title":["Information"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2078-2489\/12\/5\/212\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:02:50Z","timestamp":1760162570000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2078-2489\/12\/5\/212"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,5,17]]},"references-count":24,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2021,5]]}},"alternative-id":["info12050212"],"URL":"https:\/\/doi.org\/10.3390\/info12050212","relation":{},"ISSN":["2078-2489"],"issn-type":[{"type":"electronic","value":"2078-2489"}],"subject":[],"published":{"date-parts":[[2021,5,17]]}}}