{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,20]],"date-time":"2026-03-20T15:35:07Z","timestamp":1774020907930,"version":"3.50.1"},"reference-count":29,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2023,8,8]],"date-time":"2023-08-08T00:00:00Z","timestamp":1691452800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Natural Sciences and Engineering Research Council of Canada (NSERC)"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We investigate the support of a capacity-achieving input to a vector-valued Gaussian noise channel. The input is subjected to a radial even-moment constraint and is either allowed to take any value in Rn or is restricted to a given compact subset of Rn. It is shown that the support of the capacity-achieving distribution is composed of a countable union of submanifolds, each with a dimension of n\u22121 or less. When the input is restricted to a compact subset of Rn, this union is finite. Finally, the support of the capacity-achieving distribution is shown to have Lebesgue measure 0 and to be nowhere dense in Rn.<\/jats:p>","DOI":"10.3390\/e25081180","type":"journal-article","created":{"date-parts":[[2023,8,8]],"date-time":"2023-08-08T12:26:39Z","timestamp":1691497599000},"page":"1180","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Capacity-Achieving Input Distributions of Additive Vector Gaussian Noise Channels: Even-Moment Constraints and Unbounded or Compact Support"],"prefix":"10.3390","volume":"25","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9454-3051","authenticated-orcid":false,"given":"Jonah","family":"Eisen","sequence":"first","affiliation":[{"name":"Department of Electrical & Computer Engineering, University of Waterloo, 200 University Ave W, Waterloo, ON N2L 3G1, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0968-0925","authenticated-orcid":false,"given":"Ravi R.","family":"Mazumdar","sequence":"additional","affiliation":[{"name":"Department of Electrical & Computer Engineering, University of Waterloo, 200 University Ave W, Waterloo, ON N2L 3G1, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6479-1700","authenticated-orcid":false,"given":"Patrick","family":"Mitran","sequence":"additional","affiliation":[{"name":"Department of Electrical & Computer Engineering, University of Waterloo, 200 University Ave W, Waterloo, ON N2L 3G1, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,8,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"700","DOI":"10.1109\/TAP.2011.2173441","article-title":"Compensating for Non-Linear Amplifiers in MIMO Communications Systems","volume":"60","author":"Vaughan","year":"2012","journal-title":"IEEE Trans. Antennas Propag."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1461","DOI":"10.1109\/22.780395","article-title":"Nonlinear amplifier effects in communications systems","volume":"47","author":"Liang","year":"1999","journal-title":"IEEE Trans. Microw. Theory Tech."},{"key":"ref_3","unstructured":"Raich, R., and Zhou, G.T. (2002, January 16). On the modeling of memory nonlinear effects of power amplifiers for communication applications. Proceedings of the 2002 IEEE 10th Digital Signal Processing Workshop, 2002 and the 2nd Signal Processing Education Workshop, Pine Mountain, GA, USA."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"203","DOI":"10.1016\/S0019-9958(71)90346-9","article-title":"The information capacity of amplitude- and variance-constrained sclar Gaussian channels","volume":"18","author":"Smith","year":"1971","journal-title":"Inf. Control."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"6884","DOI":"10.1109\/TIT.2016.2615632","article-title":"On the Capacity of Vector Gaussian Channels With Bounded Inputs","volume":"62","author":"Rassouli","year":"2016","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1060","DOI":"10.1109\/18.391243","article-title":"The capacity of average and peak-power-limited quadrature Gaussian channels","volume":"41","author":"Shamai","year":"1995","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"3907","DOI":"10.1109\/TIT.2018.2890208","article-title":"On the Capacity of the Peak Power Constrained Vector Gaussian Channel: An Estimation Theoretic Perspective","volume":"65","author":"Dytso","year":"2019","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"2006","DOI":"10.1109\/TIT.2019.2948636","article-title":"The Capacity Achieving Distribution for the Amplitude Constrained Additive Gaussian Channel: An Upper Bound on the Number of Mass Points","volume":"66","author":"Dytso","year":"2020","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A mathematical theory of communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Sommerfeld, J., Bjelakovic, I., and Boche, H. (2008, January 6\u201311). On the boundedness of the support of optimal input measures for Rayleigh fading channels. Proceedings of the 2008 IEEE International Symposium on Information Theory, Toronto, ON, Canada.","DOI":"10.1109\/ISIT.2008.4595179"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Dytso, A., Goldenbaum, M., Shamai, S., and Poor, H.V. (2017, January 4\u20138). Upper and Lower Bounds on the Capacity of Amplitude-Constrained MIMO Channels. Proceedings of the GLOBECOM 2017\u20142017 IEEE Global Communications Conference, Singapore.","DOI":"10.1109\/GLOCOM.2017.8254165"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2073","DOI":"10.1109\/TIT.2005.847707","article-title":"Capacity-achieving probability measure for conditionally Gaussian channels with bounded inputs","volume":"51","author":"Chan","year":"2005","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Dytso, A., Goldenbaum, M., Poor, H.V., and Shamai (Shitz), S. (2019). Amplitude Constrained MIMO Channels: Properties of Optimal Input Distributions and Bounds on the Capacity. Entropy, 21.","DOI":"10.3390\/e21020200"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"5302","DOI":"10.1109\/TIT.2012.2197173","article-title":"Using Hermite Bases in Studying Capacity-Achieving Distributions Over AWGN Channels","volume":"58","author":"Fahs","year":"2012","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_15","doi-asserted-by":"crossref","unstructured":"Fahs, J., Tchamkerten, A., and Yousefi, M.I. (2019, January 7\u201312). On the Optimal Input of the Nondispersive Optical Fiber. Proceedings of the 2019 IEEE International Symposium on Information Theory (ISIT), Paris, France.","DOI":"10.1109\/ISIT.2019.8849407"},{"key":"ref_16","unstructured":"Bhaskara Rao, K., and Bhaskara Rao, M. (1983). Theory of Charges, Academic Press. Pure and Applied Mathematics."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Han, T.S. (2003). Information-Spectrum Methods in Information Theory, Springer.","DOI":"10.1007\/978-3-662-12066-8"},{"key":"ref_18","unstructured":"Dudley, R.M. (2002). Real Analysis and Probability, Cambridge University Press. [2nd ed.]. Cambridge Studies in Advanced Mathematics."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1939","DOI":"10.1109\/18.340468","article-title":"A mapping approach to rate-distortion computation and analysis","volume":"40","author":"Rose","year":"1994","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"249","DOI":"10.4169\/amer.math.monthly.121.03.249","article-title":"The Weighted Hermite Polynomials Form a Basis for L2(R)","volume":"121","author":"Johnston","year":"2014","journal-title":"Am. Math. Mon."},{"key":"ref_21","unstructured":"Reed, M., and Simon, B. (1972). Methods of Modern Mathematical Physics, Academic Press."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Alajaji, F., and Chen, P. (2018). An Introduction to Single-User Information Theory, Springer. Springer Undergraduate Texts in Mathematics and Technology.","DOI":"10.1007\/978-981-10-8001-2"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"135","DOI":"10.1007\/BF02880360","article-title":"Uncertainty principles on certain Lie groups","volume":"105","author":"Sitaram","year":"1995","journal-title":"Proc. Indian Acad. Sci. Math. Sci."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Krantz, S., and Parks, H. (2002). A Primer of Real Analytic Functions, Birkh\u00e4user.","DOI":"10.1007\/978-0-8176-8134-0"},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Shiryaev, A., and Chibisov, D. (2016). Probability-1, Springer. Graduate Texts in Mathematics.","DOI":"10.1007\/978-0-387-72206-1"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Dupuis, P., and Ellis, R. (1997). A Weak Convergence Approach to the Theory of Large Deviations, Wiley.","DOI":"10.1002\/9781118165904"},{"key":"ref_27","unstructured":"Billingsley, P. (2013). Convergence of Probability Measures, Wiley."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"1469","DOI":"10.1109\/LCOMM.2017.2689770","article-title":"Existence and Continuity of Differential Entropy for a Class of Distributions","volume":"21","author":"Ghourchian","year":"2017","journal-title":"IEEE Commun. Lett."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Gunning, R., and Rossi, H. (2009). Analytic Functions of Several Complex Variables, Ams Chelsea Publishing, AMS Chelsea Pub.","DOI":"10.1090\/chel\/368"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/8\/1180\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T20:28:08Z","timestamp":1760128088000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/25\/8\/1180"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,8,8]]},"references-count":29,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2023,8]]}},"alternative-id":["e25081180"],"URL":"https:\/\/doi.org\/10.3390\/e25081180","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,8,8]]}}}