PHYSICAL REVIEW E 84, 051107 (2011) Fisher-information condition for enhanced signal detection via stochastic resonance Fabing Duan * Department of Automation Engineering, Qingdao University, Qingdao 266071, PR China Franc ¸ois Chapeau-Blondeau † Laboratoire d’Ing´ enierie des Syst` emes Automatis´ es (LISA), Universit´ e d’Angers, 62 avenue Notre Dame du Lac, 49000 Angers, France Derek Abbott ‡ Centre for Biomedical Engineering (CBME) and School of Electrical and Electronic Engineering, University of Adelaide, SA 5005, Australia (Received 7 September 2011; revised manuscript received 26 October 2011; published 11 November 2011) Various situations where a signal is enhanced by noise through stochastic resonance are now known. This paper contributes to determining general conditions under which improvement by noise can be a priori decided as feasible or not. We focus on the detection of a known signal in additive white noise. Under the assumptions of a weak signal and a sufficiently large sample size, it is proved, with an inequality based on the Fisher information, that improvement by adding noise is never possible, generically, in these conditions. However, under less restrictive conditions, an example of signal detection is shown with favorable action of adding noise. DOI: 10.1103/PhysRevE.84.051107 PACS number(s): 05.40.−a, 02.50.−r I. INTRODUCTION Stochastic resonance (SR) is now a well-established cooper- ative phenomenon wherein the response of a nonlinear system to a weak signal can be optimized at a nonzero noise level [1–11]. Briefly, SR emerged from the field of meteorology [1], and the topic has flourished in physics [2–6] and neuroscience [5–11]. Meanwhile, the promise of applying SR to nonlinear signal processing has been studied over several decades. The improvement of output signal-to-noise ratio of a nonlinear system first attracted much attention [2–5,12–16], and later, noise-enhanced detection was observed in dynamic [17–19] and static nonlinearities [20–29]. An interesting idea explored in Ref. [29] is that, in order to find an optimal processor in the context of SR where injection of more noise into a given signal is an available option, one can continuously update the optimal processor according to the composite noise. Then, as shown by examples in Refs. [27–29], optimal processors acting on the output with added noise can emerge with an improved performance over that of the original optimal processor on the output without added noise. In this context, it is then useful to seek to identify generic conditions under which it is a priori possible to decide whether or not addition of noise can be a favorable option for signal detection. In this paper we focus on the detection of known weak sig- nals in additive white noise in the context of SR. This detection problem can be viewed as a simple binary hypothesis testing. Under assumptions of a weak signal and a sufficiently large number of observation values, the performance of a locally optimum (LO) detector is demonstrated to be asymptotically optimum that its detection probability is maximized for a desired false alarm probability [30–32]. In order to evaluate the performance of the LO detector with respective to the * fabing.duan@gmail.com † chapeau@univ-angers.fr ‡ dabbott@eleceng.adelaide.edu.au Neyman-Pearson detector, the asymptotic relative efficiency of two detectors is introduced [30–32]. With regularity conditions [32], the asymptotic relative efficiency can be computed simply as a ratio of their efficacies [see Eq. (7)] of detection procedures based on the sequence of statistics [30–32]. For a given false alarm probability, the detection probability of the LO detector is a monotonically increasing function of its efficacy, which is simply given by the Fisher information (FI) of the noise probability density function (PDF) [31,32]. When independent noise is added to the signal, we update the exact LO detector for each added-noise condition. Then, it is theoretically proven, by using the FI convolution inequality [33,34], that no improvement in detection can be obtained compared to the initial condition with no added noise. However, beyond these restrictive conditions, the SR method can be an appropriate way of improving the detection performance of a detector [22–29]. Here we present a novel instance of detection of a known weak signal in uniform noise with favorable action of the noise through SR. In this case the FI of a uniform noise PDF is infinite, but the LO detector is physically unrealizable, since the output of the LO detector tends to infinity when the input is larger than unity [31]. It is shown that a realizable LO detector can be constructed by adding a type of noise with a continuous PDF. Furthermore, we observe that the detection performance of a fixed dead-zone limiter (DZL) detector can be infinitely enhanced by adding suitable dichotomous noise in order to better detect the known weak signal in uniform noise. This example shows a potential application of SR in signal detection in the case where a LO detector is physically unrealizable. II. THE OBSERVATION MODEL AND FEASIBILITY OF SR IN SIGNAL DETECTION Consider the observation vector X = (X 1 ,X 2 ,...,X N ) of real-valued components X n by X n = θs n + W n , n = 1, 2, ...,N, (1) 051107-1 1539-3755/2011/84(5)/051107(5) ©2011 American Physical Society