{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,3]],"date-time":"2026-01-03T06:51:26Z","timestamp":1767423086684,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2009,12,2]],"date-time":"2009-12-02T00:00:00Z","timestamp":1259712000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, an analytical expression is developed for the differential entropy of a sinusoid with a Beta-distributed phase angle. This particular signal model is prevalent in optical communications, however an expression for the associated differential entropy does not currently exist. The expression we derive is approximate as it relies on a series expansion for one of the key terms needed in the derivation. However, we are able to show that the approximation is accurate (error \u2264 5%) for a wide variety of Beta parameter choices.<\/jats:p>","DOI":"10.3390\/e11040949","type":"journal-article","created":{"date-parts":[[2009,12,2]],"date-time":"2009-12-02T11:40:11Z","timestamp":1259754011000},"page":"949-958","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Calculation of Entropy for a Sinusoid with Beta-Distributed Phase"],"prefix":"10.3390","volume":"11","author":[{"given":"Joseph  V.","family":"Michalowicz","sequence":"first","affiliation":[{"name":"U.S. Naval Research Laboratory, Optical Science Division, Washington DC 20375, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Colin  C.","family":"Olson","sequence":"additional","affiliation":[{"name":"NRC Postdoctoral Research Associate, Naval Research Laboratory, Optical Science Division, Washington DC 20375, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Frank","family":"Bucholtz","sequence":"additional","affiliation":[{"name":"U.S. Naval Research Laboratory, Optical Science Division, Washington DC 20375, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jonathan  M.","family":"Nichols","sequence":"additional","affiliation":[{"name":"U.S. Naval Research Laboratory, Optical Science Division, Washington DC 20375, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2009,12,2]]},"reference":[{"key":"ref_1","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_2","unstructured":"Cover, T.M., and Thomas, J.A. (2006). Elements of Information Theory, John Wiley and Sons."},{"key":"ref_3","unstructured":"Kullback, S. (1968). Information Theory and Statistics, Dover Publications."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1007\/s10867-007-9047-y","article-title":"Mapping the information landscape: Discerning peaks and valleys for ecological monitoring","volume":"33","author":"Moniz","year":"2007","journal-title":"J. Biol. Phys."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"517","DOI":"10.1107\/S010876738800354X","article-title":"A bayesian statistical theory of the phase problem. I. A multichannel maximum-entropy formalism for constructing generalized joint probability distributions of structure factors","volume":"44","author":"Bricogne","year":"1988","journal-title":"Acta Crystallogr. A"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1103\/PhysRevE.75.061105","article-title":"Optimal stimulus and noise distributions for information transmission via suprathreshold stochastic resonance","volume":"75","author":"McDonnell","year":"2007","journal-title":"Phys. Rev. E"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"177","DOI":"10.1016\/j.optcom.2005.08.006","article-title":"Probability density function of noise statistics for optically pre-amplified DPSK receivers with optical mach-Zehnder interferometer demodulation","volume":"258","author":"Zhang","year":"2006","journal-title":"Opt. Commun."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1403","DOI":"10.1109\/LPT.2004.826054","article-title":"Impact of nonlinear phase noise to DPSK signals: A comparison of different models","volume":"16","author":"Ho","year":"2004","journal-title":"IEEE Photonic. Technol. Lett."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1875","DOI":"10.1364\/JOSAB.20.001875","article-title":"Probability density of nonlinear phase noise","volume":"20","author":"Ho","year":"2003","journal-title":"J. Opt. Soc. Amer. B"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"1936","DOI":"10.1364\/JOSAA.8.001936","article-title":"Probability-density function of noise at the output of a two-beam interferometer","volume":"8","author":"Arie","year":"1991","journal-title":"J. Opt. Soc. Amer. A"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"3592","DOI":"10.1088\/0957-0233\/18\/11\/044","article-title":"Probability density functions for intensity induced phase noise in CW phase demodulation systems","volume":"18","author":"Freitas","year":"2007","journal-title":"Meas. Sci. Technol."},{"key":"ref_12","unstructured":"Damper, R.I. (1995). Introduction to Discrete-Time Signals and Systems, Chapman-Hall."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1016\/S0304-4076(98)00046-3","article-title":"Ordering univariate distributions by entropy and variance","volume":"90","author":"Ebrahimi","year":"1999","journal-title":"J. Econometrics"},{"key":"ref_14","unstructured":"Gradshteyn, I.S., and Ryzhik, I.M. (1965). Table of Integrals, Series and Products, Academic Press. [4th ed.]."},{"key":"ref_15","unstructured":"Hahn, G.J., and Shapiro, S.S. (1968). Statistical Models in Engineering, John Wiley and Sons."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/11\/4\/949\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T22:11:51Z","timestamp":1760220711000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/11\/4\/949"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,12,2]]},"references-count":15,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2009,12]]}},"alternative-id":["e11040949"],"URL":"https:\/\/doi.org\/10.3390\/e11040949","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2009,12,2]]}}}