{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:19:48Z","timestamp":1760242788507,"version":"build-2065373602"},"reference-count":28,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2016,6,17]],"date-time":"2016-06-17T00:00:00Z","timestamp":1466121600000},"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":["41404027","61471072","61301224","61103212","61471073","61108086"],"award-info":[{"award-number":["41404027","61471072","61301224","61103212","61471073","61108086"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>In this paper, a new framework for variable detectors is formulated in order to solve different noise enhanced signal detection optimal problems, where six different disjoint sets of detector and discrete vector pairs are defined according to the two inequality-constraints on detection and false-alarm probabilities. Then theorems and algorithms constructed based on the new framework are presented to search the optimal noise enhanced solutions to maximize the relative improvements of the detection and the false-alarm probabilities, respectively. Further, the optimal noise enhanced solution of the maximum overall improvement is obtained based on the new framework and the relationship among the three maximums is presented. In addition, the sufficient conditions for improvability or non-improvability under the two certain constraints are given. Finally, numerous examples are presented to illustrate the theoretical results and the proofs of the main theorems are given in the Appendix.<\/jats:p>","DOI":"10.3390\/e18060213","type":"journal-article","created":{"date-parts":[[2016,6,17]],"date-time":"2016-06-17T10:09:24Z","timestamp":1466158164000},"page":"213","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Optimal Noise Enhanced Signal Detection in a Unified Framework"],"prefix":"10.3390","volume":"18","author":[{"given":"Ting","family":"Yang","sequence":"first","affiliation":[{"name":"College of Communication Engineering, Chongqing University, Chongqing 400044, China"}]},{"given":"Shujun","family":"Liu","sequence":"additional","affiliation":[{"name":"College of Communication Engineering, Chongqing University, Chongqing 400044, China"}]},{"given":"Mingchun","family":"Tang","sequence":"additional","affiliation":[{"name":"College of Communication Engineering, Chongqing University, Chongqing 400044, China"}]},{"given":"Kui","family":"Zhang","sequence":"additional","affiliation":[{"name":"College of Communication Engineering, Chongqing University, Chongqing 400044, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0170-992X","authenticated-orcid":false,"given":"Xinzheng","family":"Zhang","sequence":"additional","affiliation":[{"name":"College of Communication Engineering, Chongqing University, Chongqing 400044, China"}]}],"member":"1968","published-online":{"date-parts":[[2016,6,17]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"453","DOI":"10.1088\/0305-4470\/14\/11\/006","article-title":"The mechanism of stochastic resonance","volume":"14","author":"Benzi","year":"1981","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1103\/RevModPhys.70.223","article-title":"Stochastic resonance","volume":"70","author":"Gammaitoni","year":"1998","journal-title":"Rev. Mod. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"11969","DOI":"10.1088\/0305-4470\/36\/48\/005","article-title":"How to tune the system parameters to realize stochastic resonance","volume":"36","author":"Xu","year":"2003","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"19","DOI":"10.1016\/j.dsp.2004.09.006","article-title":"Stochastic resonance and improvement by noise in optimal detection strategies","volume":"15","author":"Rousseau","year":"2005","journal-title":"Digit. Signal Process."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"488","DOI":"10.1109\/TSP.2010.2091409","article-title":"Noise benefits in quantizer-array correlation detection and watermark decoding","volume":"59","author":"Patel","year":"2011","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1607","DOI":"10.1109\/JPROC.2014.2341554","article-title":"Noise-enhanced information systems","volume":"102","author":"Chen","year":"2014","journal-title":"Proc. IEEE"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1016\/S0375-9601(97)00350-2","article-title":"Input-output gains for signal in noise in stochastic resonance","volume":"232","year":"1997","journal-title":"Phys. Lett. A"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1142\/S0218127499000146","article-title":"Periodic and aperiodic stochastic resonance with output signal-to-noise ratio exceeding that at the input","volume":"9","year":"1999","journal-title":"Int. J. Bifurc. Chaos"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"L181","DOI":"10.1142\/S0219477501000408","article-title":"High signal-to-noise ratio gain by stochastic resonance in a double well","volume":"1","author":"Gingl","year":"2001","journal-title":"Fluctuat. Noise Lett."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"L145","DOI":"10.1142\/S0219477502000750","article-title":"Signal-to-noise ratio gain in non-dynamical and dynamical bistable stochastic resonators","volume":"2","author":"Makra","year":"2002","journal-title":"Fluct. Noise Lett."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"228","DOI":"10.1016\/j.physleta.2003.08.042","article-title":"Signal-to-noise ratio gain in stochastic resonators driven by coloured noises","volume":"317","author":"Makra","year":"2003","journal-title":"Phys. Lett. A"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"2310","DOI":"10.1103\/PhysRevLett.84.2310","article-title":"Suprathreshold stochastic resonance in multilevel threshold systems","volume":"84","author":"Stocks","year":"2000","journal-title":"Phys. Rev. Lett."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1526","DOI":"10.1109\/TNN.2004.826218","article-title":"Adaptive stochastic resonance in noisy neurons based on mutual information","volume":"15","author":"Mitaim","year":"2004","journal-title":"IEEE Trans. Neural Netw."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"8","DOI":"10.1109\/97.809511","article-title":"Can detectability be improved by adding noise","volume":"7","author":"Kay","year":"2000","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"3172","DOI":"10.1109\/TSP.2007.893757","article-title":"Theory of the stochastic resonance effect in signal detection: Part I\u2014Fixed detectors","volume":"55","author":"Chen","year":"2007","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"5031","DOI":"10.1109\/TSP.2008.928509","article-title":"Theory of the stochastic resonance effect in signal detection: Part II\u2014Variable detectors","volume":"56","author":"Chen","year":"2007","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"499","DOI":"10.1109\/TIT.2008.2009813","article-title":"Noise enhanced nonparametric detection","volume":"55","author":"Chen","year":"2009","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1655","DOI":"10.1109\/TSP.2009.2012893","article-title":"Optimal noise benefits in Neyman\u2013Pearson and inequality constrained signal detection","volume":"57","author":"Patel","year":"2009","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"391","DOI":"10.1016\/j.dsp.2012.02.003","article-title":"Stochastic resonance in binary composite hypothesis-testing problems in the Neyman\u2013Pearson framework","volume":"22","author":"Bayram","year":"2012","journal-title":"Digit. Signal Process."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1016\/j.dsp.2013.10.014","article-title":"Noise enhanced hypothesis-testing according to restricted Neyman\u2013Pearson criterion","volume":"25","author":"Bayrama","year":"2014","journal-title":"Digit. Signal Process."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"3972","DOI":"10.1109\/TSP.2010.2048107","article-title":"Noise enhanced hypothesis-testing in the restricted bayesian framework","volume":"58","author":"Bayram","year":"2010","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1292","DOI":"10.1109\/TSP.2010.2097257","article-title":"Noise enhanced M-ary composite hypothesis-testing in the presence of partial prior information","volume":"59","author":"Bayram","year":"2011","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"695","DOI":"10.1109\/LSP.2006.879455","article-title":"Reducing probability of decision error using stochastic resonance","volume":"13","author":"Kay","year":"2006","journal-title":"IEEE Signal Process. Lett."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Bayram, S., and Gezici, S. (2009, January 28\u201330). Noise-enhanced M-ary hypothesis-testing in the minimax framework. Proceedings of the 3rd International Conference on Signal Processing and Communication Systems (ICSPCS), Omaha, NE, USA.","DOI":"10.1109\/ICSPCS.2009.5306400"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1016\/j.dsp.2015.03.016","article-title":"Noise enhanced binary hypothesis-testing in a new framework","volume":"41","author":"Liu","year":"2015","journal-title":"Digit. Signal Process."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1140\/epjb\/e2014-50382-y","article-title":"Noise enhanced activity in a complex network","volume":"87","author":"Choudhary","year":"2014","journal-title":"Eur. Phys. J. B"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"957","DOI":"10.1016\/j.physleta.2012.01.039","article-title":"Noise-assisted morphing of memory and logic function","volume":"376","author":"Kohar","year":"2012","journal-title":"Phys. Lett. A"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"70","DOI":"10.1214\/14-BJPS257","article-title":"Adaptative significance levels using optimal decision rules: Balancing by weighting the error probabilities","volume":"30","author":"Pericchi","year":"2016","journal-title":"Braz. J. Probab. Stat."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/18\/6\/213\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T19:25:38Z","timestamp":1760210738000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/18\/6\/213"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,6,17]]},"references-count":28,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2016,6]]}},"alternative-id":["e18060213"],"URL":"https:\/\/doi.org\/10.3390\/e18060213","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2016,6,17]]}}}