{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,7]],"date-time":"2025-11-07T09:21:39Z","timestamp":1762507299950,"version":"build-2065373602"},"reference-count":34,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2018,1,11]],"date-time":"2018-01-11T00:00:00Z","timestamp":1515628800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministerio Economia y Competitividad","award":["TEC2013-4741-C4-R (RACHEL)","TEC2016-75067-C4-R (CARMEN)"],"award-info":[{"award-number":["TEC2013-4741-C4-R (RACHEL)","TEC2016-75067-C4-R (CARMEN)"]}]},{"name":"German Research Foundation","award":["LA 4107\/1-1","SCHR 1384\/6-1"],"award-info":[{"award-number":["LA 4107\/1-1","SCHR 1384\/6-1"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Non-circular or improper Gaussian signaling has proven beneficial in several interference-limited wireless networks. However, all implementable coding schemes are based on finite discrete constellations rather than Gaussian signals. In this paper, we propose a new family of improper constellations generated by widely linear processing of a square M-QAM (quadrature amplitude modulation) signal. This family of discrete constellations is parameterized by    \u03ba   , the circularity coefficient and a phase    \u03d5   . For uncoded communication systems, this phase should be optimized as      \u03d5 *   ( \u03ba )      to maximize the minimum Euclidean distance between points of the improper constellation, therefore minimizing the bit error rate (BER). For the more relevant case of coded communications, where the coded symbols are constrained to be in this family of improper constellations using      \u03d5 *   ( \u03ba )     , it is shown theoretically and further corroborated by simulations that, except for a shaping loss of 1.53 dB encountered at a high signal-to-noise ratio (snr), there is no rate loss with respect to the improper Gaussian capacity. In this sense, the proposed family of constellations can be viewed as the improper counterpart of the standard proper M-QAM constellations widely used in coded communication systems.<\/jats:p>","DOI":"10.3390\/e20010045","type":"journal-article","created":{"date-parts":[[2018,1,11]],"date-time":"2018-01-11T13:36:15Z","timestamp":1515677775000},"page":"45","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["Information-Theoretic Analysis of a Family of Improper Discrete Constellations"],"prefix":"10.3390","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0040-7436","authenticated-orcid":false,"given":"Ignacio","family":"Santamaria","sequence":"first","affiliation":[{"name":"Departamento de Ingenier\u00eda de Comunicaciones, Universidad de Cantabria, 39005 Santander, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pedro","family":"Crespo","sequence":"additional","affiliation":[{"name":"TECNUN, Universidad de Navarra, 20018 San Sebasti\u00e1n, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christian","family":"Lameiro","sequence":"additional","affiliation":[{"name":"Signal & System Theory Group, Paderborn University, 33098 Paderborn, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Peter","family":"Schreier","sequence":"additional","affiliation":[{"name":"Signal & System Theory Group, Paderborn University, 33098 Paderborn, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2018,1,11]]},"reference":[{"key":"ref_1","unstructured":"Cover, T.M., and Thomas, J.A. (2006). Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing), John Wiley & Sons."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"4552","DOI":"10.1109\/TIT.2010.2053895","article-title":"Interference alignment with asymmetric complex signaling\u2014Settling the H\u00f8st-Madsen-Nosratinia conjecture","volume":"56","author":"Cadambe","year":"2010","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"3194","DOI":"10.1109\/TWC.2012.071612.111338","article-title":"Improper Gaussian signaling on the two-user SISO interference channel","volume":"11","author":"Ho","year":"2012","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"2899","DOI":"10.1109\/TSP.2013.2254480","article-title":"Transmit optimization with improper Gaussian signaling for interference channels","volume":"61","author":"Zeng","year":"2013","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"652","DOI":"10.1109\/TSP.2015.2489604","article-title":"Coexisting linear and widely linear transceivers in the MIMO interference channel","volume":"64","author":"Lagen","year":"2016","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1022","DOI":"10.1109\/TCOMM.2016.2641948","article-title":"Rate region boundary of the SISO Z-interference channel with improper signaling","volume":"65","author":"Lameiro","year":"2017","journal-title":"IEEE Trans. Commun."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1109\/LWC.2014.2360179","article-title":"Benefits of improper signaling for underlay cognitive radio","volume":"4","author":"Lameiro","year":"2015","journal-title":"IEEE Wirel. Commun. Lett."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"4875","DOI":"10.1109\/TWC.2016.2547918","article-title":"Underlay cognitive radio systems with improper Gaussian signaling: Outage performance analysis","volume":"15","author":"Amin","year":"2016","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Schreier, P.J., and Scharf, L.L. (2010). Statistical Signal Processing of Complex-Valued Data: The Theory of Improper and Noncircular Signals, Cambridge University Press.","DOI":"10.1017\/CBO9780511815911"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1512","DOI":"10.1109\/TCOMM.2003.816992","article-title":"Receivers with widely linear processing for frequency-selective channels","volume":"51","author":"Gerstacker","year":"2003","journal-title":"IEEE Trans. Commun."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"2252","DOI":"10.1109\/TSP.2006.873590","article-title":"Widely linear reception strategies for layered space-time wireless communications","volume":"54","author":"Buzzi","year":"2006","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"870","DOI":"10.1109\/TSP.2005.862946","article-title":"New insights into optimal widely linear array receivers for the demodulation of BPSK, MSK, and GMSK signals corrupted by noncircular interferences-application to SAIC","volume":"54","author":"Chevalier","year":"2006","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"746","DOI":"10.1109\/TSP.2003.822359","article-title":"Data-aided and blind stochastic gradient algorithms for widely linear MMSE MAI suppression for DS-CDMA","volume":"52","author":"Schober","year":"2004","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Lameiro, C., and Santamaria, I. (2013, January 9\u201313). Degrees-of-freedom for the 4-user SISO interference channel with improper signaling. Proceedings of the 2013 IEEE International Conference on Communications (ICC), Budapest, Hungary.","DOI":"10.1109\/ICC.2013.6655009"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"857","DOI":"10.1109\/TCOMM.2015.2396511","article-title":"Improper signaling for symbol error rate minimization in K-user interference channel","volume":"63","author":"Nguyen","year":"2015","journal-title":"IEEE Trans. Commun."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"3912","DOI":"10.1109\/TWC.2015.2414913","article-title":"Improper Gaussian signaling scheme for the Z-interference channel","volume":"14","author":"Kurniawan","year":"2015","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Lagen, S., Agustin, A., and Vidal, J. (2014, January 4\u20139). Improper Gaussian signaling for the Z-interference channel. Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), Florence, Italy.","DOI":"10.1109\/ICASSP.2014.6853775"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"3350","DOI":"10.1109\/TCOMM.2016.2584601","article-title":"On the superiority of improper Gaussian signaling in wireless interference MIMO scenarios","volume":"64","author":"Lagen","year":"2016","journal-title":"IEEE Trans. Commun."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Hellings, C., and Utschick, W. (2013, January 26\u201331). Performance gains due to improper signals in MIMO broadcast channels with widely linear receivers. Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), Vancouver, BC, Canada.","DOI":"10.1109\/ICASSP.2013.6638487"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"6303","DOI":"10.1109\/TWC.2013.103013.130439","article-title":"Optimized transmission with improper Gaussian signaling in the K-user MISO interference channel","volume":"12","author":"Zeng","year":"2013","journal-title":"IEEE Trans. Wirel. Commun."},{"key":"ref_21","first-page":"50","article-title":"Overlay spectrum sharing using improper Gaussian signaling","volume":"35","author":"Amin","year":"2017","journal-title":"IEEE J. Sel. Areas Commun."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Gaafar, M., Khafagy, M.G., Amin, O., and Alouini, M.S. (2016, January 22\u201327). Improper Gaussian signaling in full-duplex relay channels with residual self-interference. Proceedings of the IEEE International Conference on Communications (ICC), Kuala Lumpur, Malaysia.","DOI":"10.1109\/ICC.2016.7511009"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"1683","DOI":"10.1109\/LCOMM.2016.2577029","article-title":"On alternate relaying with improper Gaussian signaling","volume":"20","author":"Gaafar","year":"2016","journal-title":"IEEE Commun. Lett."},{"key":"ref_24","unstructured":"Kariminezhad, A., Chaaban, A., and Sezgin, A. (arXiv, 2017). Improper signaling and symbol extensions: How far can we go with Gaussian P2P codebooks in the interfering MAC with TIN?, arXiv."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2384","DOI":"10.1109\/18.720542","article-title":"Modulation and coding for linear Gaussian channels","volume":"44","author":"Forney","year":"1998","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1061","DOI":"10.1109\/TIT.2013.2291865","article-title":"High-SNR asymptotics of mutual information for discrete constellations with applications to BICM","volume":"60","author":"Alvarado","year":"2014","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1293","DOI":"10.1109\/18.243446","article-title":"Proper complex random processes with applications to information theory","volume":"39","author":"Neeser","year":"1993","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"607","DOI":"10.1109\/TIT.1985.1057085","article-title":"On the Gaussian interference channel","volume":"31","author":"Costa","year":"1985","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Tse, D., and Viswanath, P. (2005). Fundamentals of Wireless Communications, Cambridge University Press.","DOI":"10.1017\/CBO9780511807213"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Fisher, R.F.H. (2002). Precoding and Signal Shaping for Digital Transmissions, John Wiley & Sons.","DOI":"10.1002\/0471439002"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1261","DOI":"10.1109\/TIT.2005.844072","article-title":"Mutual information and minimum mean-square error in Gaussian channels","volume":"51","author":"Guo","year":"2005","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1319","DOI":"10.1109\/TIT.2002.1003824","article-title":"Spectral efficiency in the wideband regime","volume":"48","author":"Verdu","year":"2002","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_33","first-page":"101","article-title":"Volume Integrals over n-Dimensional Ellipsoids","volume":"1","author":"Sykora","year":"2005","journal-title":"Stan\u2019s Library"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"2371","DOI":"10.1109\/TIT.2011.2111010","article-title":"Estimation in Gaussian noise: Properties of the minimum mean-square error","volume":"57","author":"Guo","year":"2011","journal-title":"IEEE Trans. Inf. Theory"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/1\/45\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T14:50:53Z","timestamp":1760194253000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/20\/1\/45"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,1,11]]},"references-count":34,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2018,1]]}},"alternative-id":["e20010045"],"URL":"https:\/\/doi.org\/10.3390\/e20010045","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2018,1,11]]}}}