Stochastic resonance in binary composite hypothesis-testing problems in the Neyman–Pearson framework

Abstract

Performance of some suboptimal detectors can be enhanced by adding independent noise to their inputs via the stochastic resonance (SR) effect. In this paper, the effects of SR are studied for binary composite hypothesis-testing problems. A Neyman-Pearson framework is considered, and the maximization of detection performance under a constraint on the maximum probability of false-alarm is studied. The detection performance is quantified in terms of the sum, the minimum, and the maximum of the detection probabilities corresponding to possible parameter values under the alternative hypothesis. Sufficient conditions under which detection performance can or cannot be improved are derived for each case. Also, statistical characterization of optimal additive noise is provided, and the resulting false-alarm probabilities and bounds on detection performance are investigated. In addition, optimization theoretic approaches to obtaining the probability distribution of optimal additive noise are discussed. Finally, a detection example is presented to investigate the theoretical results.

Noise benefits in joint detection and estimation problems

“…The structures of the proposed problems are similar to those in [15,23,25,29]. The same principles can be applied to both of the optimization problems in (12) and (21) and the optimum noise distribution structure can be specified under certain conditions as follows:…”

“…Using the primal-dual concept, [23] reaches PMFs with at most two point masses under certain conditions for binary hypothesis testing problems. In [29] and [25], the proof given in [15] is extended to hypothesis testing problems with ðM À1Þ constraint functions and the optimum noise distribution is found to have M point masses.…”

“…equivalently, V [15,23,25,29]. Then, from Carathéodory's theorem [40], it can be concluded that any point on the boundary of V can be expressed as the convex combination of at most three different points in U.…”

“…They examined the convex structure of the problem and specified the nature of the optimal probability distribution of additive noise as a probability mass function with at most two point masses. This result is generalized for M-ary composite hypothesis testing problems under NP, restricted NP and restricted Bayes criteria [25,29,34]. In estimation problems, additive noise can also be utilized to improve the performance of a given suboptimal estimator [4,19,30].…”

“…Later, it is proved that a suboptimal detector in the Bayesian framework may be improved (i.e., the Bayes risk can be reduced) by adding a constant signal to the observation signal; that is, the optimal probability density function is a single Dirac delta function [14]. This intuition is extended in various directions and it is demonstrated that injection of additive noise to the observation signal at the input of a suboptimal detector can enhance the system performance [15,[18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34]. In this paper, performance improvements through noise benefits are addressed in the context of joint detection and estimation systems by adding an independent noise component to the observation signal at the input of a suboptimal system.…”