{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,14]],"date-time":"2025-10-14T00:35:21Z","timestamp":1760402121733,"version":"build-2065373602"},"reference-count":37,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2020,4,17]],"date-time":"2020-04-17T00:00:00Z","timestamp":1587081600000},"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":["61871303","61771362"],"award-info":[{"award-number":["61871303","61771362"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100013314","name":"111 Project","doi-asserted-by":"publisher","award":["B18039"],"award-info":[{"award-number":["B18039"]}],"id":[{"id":"10.13039\/501100013314","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Remote Sensing"],"abstract":"<jats:p>The problem of target detection in impulsive non-Gaussian sea clutter has attracted a lot of attention in recent years. The positive alpha-stable (P\u03b1S) distribution has been validated as a suitable model for the impulsive non-Gaussian sea clutter. Since the probability density function (PDF) of the P\u03b1S variable cannot be expressed as a closed-form expression, the research into constant false alarm rate (CFAR) detectors in P\u03b1S distributed sea clutter is limited. This paper formulates and evaluates some CFAR detectors, such as Greatest Of-CFAR (GO-CFAR), Smallest Of-CFAR (SO-CFAR), Order Statistic-CFAR (OS-CFAR) and censored mean level (CML) detectors, in P\u03b1S distributed sea clutter. Firstly, the Fox\u2019s H-function is adopted to express the PDF of the P\u03b1S variable, and the cumulative density function based on Fox\u2019s H-function is derived in this paper. Then, by use of the properties of the H-function and P\u03b1S distribution, exact expressions of the probabilities of false alarm and detection for CFAR detectors in the P\u03b1S background are derived. Some CFAR properties of these detectors in the P\u03b1S background are also explored. Numerical results based on derived expressions are given and verified by Monte Carlo simulation. Some analyses of detection performance from a practical perspective are also given.<\/jats:p>","DOI":"10.3390\/rs12081273","type":"journal-article","created":{"date-parts":[[2020,4,21]],"date-time":"2020-04-21T04:49:38Z","timestamp":1587444578000},"page":"1273","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["CFAR Strategy Formulation and Evaluation Based on Fox\u2019s H-function in Positive Alpha-Stable Sea Clutter"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8650-5816","authenticated-orcid":false,"given":"Xu","family":"Liu","sequence":"first","affiliation":[{"name":"National Laboratory of Radar Signal Processing, Xidian University, Xi\u2019an 710071, China"},{"name":"Collaborative Innovation Center of Information Sensing and Understanding at Xidian University, Xi\u2019an 710071, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shuwen","family":"Xu","sequence":"additional","affiliation":[{"name":"National Laboratory of Radar Signal Processing, Xidian University, Xi\u2019an 710071, China"},{"name":"Collaborative Innovation Center of Information Sensing and Understanding at Xidian University, Xi\u2019an 710071, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9213-0085","authenticated-orcid":false,"given":"Shiyang","family":"Tang","sequence":"additional","affiliation":[{"name":"National Laboratory of Radar Signal Processing, Xidian University, Xi\u2019an 710071, China"},{"name":"Collaborative Innovation Center of Information Sensing and Understanding at Xidian University, Xi\u2019an 710071, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,4,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1109\/7.7185","article-title":"Analysis of CFAR processors in nonhomogeneous background","volume":"24","author":"Gandhi","year":"1988","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"492","DOI":"10.1109\/TAES.2015.140967","article-title":"Performance prediction of OS-CFAR for generalized swerling-chi fluctuating targets","volume":"52","author":"Kong","year":"2016","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"233","DOI":"10.1049\/iet-rsn.2014.0053","article-title":"Performance analysis of mean level constant false alarm rate detectors with binary integration in Weibull background","volume":"9","author":"Baadeche","year":"2015","journal-title":"IET Radar Sonar Nav."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1055","DOI":"10.1049\/iet-rsn.2015.0008","article-title":"Distributed fuzzy maximum-censored mean level detector-constant false alarm rate detector based on voting fuzzy fusion rule","volume":"9","author":"Gong","year":"2015","journal-title":"IET Radar Sonar Nav."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Richards, M.A., Scheer, J.A., and Holm, W.A. (2010). Principles of Modern Radar, SciTech Publishing.","DOI":"10.1049\/SBRA021E"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1753","DOI":"10.1016\/j.sigpro.2006.02.036","article-title":"Performance analysis of some CFAR detectors in homogeneous and non-homogeneous Pearson distributed clutter","volume":"86","author":"Meziani","year":"2006","journal-title":"Signal Process."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"557","DOI":"10.1049\/iet-rsn.2013.0115","article-title":"Persymmetric detectors with enhanced rejection capabilities","volume":"8","author":"Hao","year":"2013","journal-title":"IET Radar Sonar Navig."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1830","DOI":"10.1109\/LSP.2016.2618619","article-title":"On the maximal invariant statistic for adaptive radar detection in partially homogeneous disturbance with persymmetric covariance","volume":"23","author":"Ciuonzo","year":"2016","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_9","first-page":"1","article-title":"Advanced radar detection schemes under mismatched signal models","volume":"4","author":"Bandiera","year":"2009","journal-title":"Synth. Lect. Signal Process."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Angelliaume, S., Rosenberg, L., and Ritchie, M. (2019). Modeling the Amplitude Distribution of Radar Sea Clutter. Remote Sens., 11.","DOI":"10.3390\/rs11030319"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Ma, L., Wu, J., Zhang, J., Wu, Z., Jeon, G., Tan, M., and Zhang, Y. (2019). Sea Clutter Amplitude Prediction Using a Long Short-Term Memory Neural Network. Remote Sens., 11.","DOI":"10.3390\/rs11232826"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2024","DOI":"10.1109\/TGRS.2009.2033193","article-title":"Statistical Analysis of a High-Resolution Sea-Clutter Database","volume":"48","year":"2010","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Uchaikin, V.V., and Zolotarev, V.M. (1999). CHANCE and STABILITY, Stable Distributions and Their Applications, Gruyter.","DOI":"10.1515\/9783110935974"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Ward, K., Tough, R., and Watts, S. (2013). Sea Clutter: Scattering, the K Distribution and Radar Performance, The Institution of Engineering and Technology. [2nd ed.].","DOI":"10.1049\/PBRA025E"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1958","DOI":"10.1109\/LGRS.2017.2743686","article-title":"Minimum-Based Sliding Window Detectors in Correlated Pareto Distributed Clutter","volume":"14","author":"Weinberg","year":"2017","journal-title":"IEEE Geosci. Remote Sens. Lett."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"774","DOI":"10.1016\/j.sigpro.2009.07.003","article-title":"Modelling with mixture of symmetric stable distributions using Gibbs sampling","volume":"90","author":"Kuruoglu","year":"2010","journal-title":"Signal Process."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2145","DOI":"10.3390\/rs5052145","article-title":"SAR Images Statistical Modeling and Classification Based on the Mixture of Alpha-Stable Distributions","volume":"5","author":"Peng","year":"2013","journal-title":"Remote Sens."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1016\/j.sigpro.2013.12.024","article-title":"Mixture approximation to the amplitude statistics of isotropic \u03b1-stable clutter","volume":"99","author":"Li","year":"2014","journal-title":"Signal Process."},{"key":"ref_19","unstructured":"Pierce, R.D. (1997, January 21\u201323). Application of the positive alpha-stable distribution. Proceedings of the IEEE Signal Processing Workshop on Higher-Order Statistics, Banff, AB, Canada."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1377","DOI":"10.1109\/7.892685","article-title":"Performance assessment of CFAR processors in Pearson-distributed clutter","volume":"36","author":"Tsakalides","year":"2000","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"021825","DOI":"10.1155\/2007\/21825","article-title":"Performance of distributed CFAR processors in Pearson distributed clutter","volume":"2007","author":"Messali","year":"2007","journal-title":"EURASIP J. Adv. Signal Process."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2530","DOI":"10.1016\/j.sigpro.2011.05.006","article-title":"Decentralized fuzzy CFAR detectors in homogeneous Pearson clutter","volume":"91","author":"Meziani","year":"2011","journal-title":"Signal Process."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1007\/s11760-010-0207-3","article-title":"Optimum second threshold for the CFAR binary integrator in Pearson-distributed clutter","volume":"6","author":"Meziani","year":"2012","journal-title":"Signal Image Video Process."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Aalo, V.A., Peppas, K.P., Efthymoglou, G., Alwakeel, M., and Alwakeel, S. (2013, January 12\u201315). Performance of CA-CFAR receivers in alpha-stable clutter. Proceedings of the IEEE International Symposium on Signal Processing and Information Technology, Athens, Greece.","DOI":"10.1109\/ISSPIT.2013.6781867"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2027","DOI":"10.1109\/TAES.2015.140043","article-title":"Performance of CA-CFAR detectors in nonhomogeneous positive alpha-stable clutter","volume":"51","author":"Aalo","year":"2015","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Sun, Z., and Han, C. (2007, January 15\u201320). Parameter Estimation of Positive Alpha-Stable Distribution Based on Negative-Order Moments. Proceedings of the IEEE International Conference on Acoustics, Speech & Signal Processing, Honolulu, HI, USA.","DOI":"10.1109\/ICASSP.2007.367110"},{"key":"ref_27","unstructured":"Jacoboni, C., and Lugli, P. (2012). The Monte Carlo Method for Semiconductor Device Simulation, Springer Science & Business Media."},{"key":"ref_28","unstructured":"(2020, February 15). The McMaster IPIX Radar Sea Clutter Database. Available online: http:\/\/soma.mcmaster.ca\/ipix.php\/dartmouth\/index.html."},{"key":"ref_29","unstructured":"Samorodnitsky, G., and Taqqu, M.S. (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance, CRC Press."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Kilbas, A.A., and Saigo, M. (2004). H-Transforms Theory and Applications, CRC Press.","DOI":"10.1201\/9780203487372"},{"key":"ref_31","unstructured":"David, H.A. (1981). Order Statistics, Wiley. [3rd ed.]."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"3243","DOI":"10.1364\/OL.37.003243","article-title":"Simple, accurate formula for the average bit error probability of multiple-input multiple-output free-space optical links over negative exponential turbulence channels","volume":"37","author":"Peppas","year":"2012","journal-title":"Opt. Lett."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"340","DOI":"10.1080\/01621459.1976.10480344","article-title":"A method for simulating stable random variables","volume":"71","author":"Chambers","year":"1976","journal-title":"J. Am. Stat. Assoc."},{"key":"ref_34","unstructured":"(2020, February 15). Alpha Stable Distributions for MATLAB. Available online: http:\/\/www.mathworks.com\/matlabcentral\/fileexchange\/37514-stbl-alpha-stable-distributions-for-matlab."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"1357","DOI":"10.1109\/TAES.2013.6494420","article-title":"Performance Analysis of OS-CFAR with Binary Integration for Weibull Background","volume":"49","author":"Meng","year":"2013","journal-title":"IEEE Trans. Aerosp. Electron. Syst."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1109\/TGRS.1983.350506","article-title":"Testing Goodness-of-Fit for the Singly Truncated Normal Distribution Using the Kolmogorov-Smirnov Statistic","volume":"GE-21","year":"1983","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"ref_37","unstructured":"Gradshteyn, I., and Ryzhik, I.M. (2007). Tables of Integrals, Series, and Products, Academic Press. [7th ed.]."}],"container-title":["Remote Sensing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2072-4292\/12\/8\/1273\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,13]],"date-time":"2025-10-13T13:21:24Z","timestamp":1760361684000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2072-4292\/12\/8\/1273"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,4,17]]},"references-count":37,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2020,4]]}},"alternative-id":["rs12081273"],"URL":"https:\/\/doi.org\/10.3390\/rs12081273","relation":{},"ISSN":["2072-4292"],"issn-type":[{"type":"electronic","value":"2072-4292"}],"subject":[],"published":{"date-parts":[[2020,4,17]]}}}