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Theory of the stochastic resonance effect in signal detectionPart II: Variable detectors
 IEEE Trans. Signal Process
, 2008
"... Abstract—This paper develops the mathematical framework to analyze the stochastic resonance (SR) effect in binary hypothesis testing problems. The mechanism for SR noise enhanced signal detection is explored. The detection performance of a noise modified detector is derived in terms of the probabil ..."
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Abstract—This paper develops the mathematical framework to analyze the stochastic resonance (SR) effect in binary hypothesis testing problems. The mechanism for SR noise enhanced signal detection is explored. The detection performance of a noise modified detector is derived in terms of the probability of detection D and the probability of false alarm
Adaptive stochastic resonance in noisy neurons based on mutual information
 IEEE Trans. Neural Netw
, 2004
"... Abstract—Noise can improve how memoryless neurons process signals and maximize their throughput information. Such favorable use of noise is the socalled “stochastic resonance ” or SR effect at the level of threshold neurons and continuous neurons. This paper presents theoretical and simulation evid ..."
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Cited by 11 (6 self)
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Abstract—Noise can improve how memoryless neurons process signals and maximize their throughput information. Such favorable use of noise is the socalled “stochastic resonance ” or SR effect at the level of threshold neurons and continuous neurons. This paper presents theoretical and simulation evidence that 1) lone noisy threshold and continuous neurons exhibit the SR effect in terms of the mutual information between random input and output sequences, 2) a new statistically robust learning law can find this entropyoptimal noise level, and 3) the adaptive SR effect is robust against highly impulsive noise with infinite variance. Histograms estimate the relevant probability density functions at each learning iteration. A theorem shows that almost all noise probability density functions produce some SR effect in threshold neurons even if the noise is impulsive and has infinite variance. The optimal noise level in threshold neurons also behaves nonlinearly as the input
Optimal Noise Benefits in Neyman–Pearson and InequalityConstrained Statistical Signal Detection
"... Abstract—We present theorems and an algorithm to find optimal or nearoptimal “stochastic resonance ” (SR) noise benefits for Neyman–Pearson hypothesis testing and for more general inequalityconstrained signal detection problems. The optimal SR noise distribution is just the randomization of two no ..."
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Abstract—We present theorems and an algorithm to find optimal or nearoptimal “stochastic resonance ” (SR) noise benefits for Neyman–Pearson hypothesis testing and for more general inequalityconstrained signal detection problems. The optimal SR noise distribution is just the randomization of two noise realizations when the optimal noise exists for a single inequality constraint on the average cost. The theorems give necessary and sufficient conditions for the existence of such optimal SR noise in inequalityconstrained signal detectors. There exists a sequence of noise variables whose detection performance limit is optimal when such noise does not exist. Another theorem gives sufficient conditions for SR noise benefits in Neyman–Pearson and other signal detection problems with inequality cost constraints. An upper bound limits the number of iterations that the algorithm requires to find nearoptimal noise. The appendix presents the proofs of the main results.
Simulation of circuits demonstrating stochastic resonance
 Microelectronics Journal
, 2000
"... In certain dynamical systems, the addition of noise can assist the detection of a signal and not degrade it as normally expected. This is possible via a phenomenon termed stochastic resonance (SR), where the response of a nonlinear system to a subthreshold periodic input signal is optimal for some n ..."
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Cited by 9 (4 self)
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In certain dynamical systems, the addition of noise can assist the detection of a signal and not degrade it as normally expected. This is possible via a phenomenon termed stochastic resonance (SR), where the response of a nonlinear system to a subthreshold periodic input signal is optimal for some nonzero value of noise intensity. We investigate the SR phenomenon in several circuits and systems. Although SR occurs in many disciplines, the sinusoidal signal by itself is not information bearing. To greatly enhance the practicality of SR, an (aperiodic) broadband signal is preferable. Hence, we employ aperiodic stochastic resonance (ASR) where noise can enhance the response of a nonlinear system to a weak aperiodic signal. We can characterize ASR by the use of crosscorrelationbased measures. Using this measure, the ASR in a simple threshold system and in a FitzHugh–Nagumo neuronal model are compared using numerical simulations. Using both weak periodic
Stochastic resonance in continuous and spiking neuron models with levy noise. Neural Networks
 IEEE Transactions on
, 2008
"... Abstract—Levy noise can help neurons detect faint or subthreshold signals. Levy noise extends standard Brownian noise to many types of impulsive jumpnoise processes found in real and model neurons as well as in models of finance and other random phenomena. Two new theorems and the Itô calculus show ..."
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Cited by 8 (6 self)
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Abstract—Levy noise can help neurons detect faint or subthreshold signals. Levy noise extends standard Brownian noise to many types of impulsive jumpnoise processes found in real and model neurons as well as in models of finance and other random phenomena. Two new theorems and the Itô calculus show that white Levy noise will benefit subthreshold neuronal signal detection if the noise process’s scaled drift velocity falls inside an interval that depends on the threshold values. These results generalize earlier “forbidden interval ” theorems of neuronal “stochastic resonance ” (SR) or noiseinjection benefits. Global and local Lipschitz conditions imply that additive white Levy noise can increase the mutual information or bit count of several feedback neuron models that obey a general stochastic differential equation (SDE). Simulation results show that the same noise benefits still occur for some infinitevariance stable Levy noise processes even though the theorems themselves apply only to finitevariance Levy noise. The Appendix proves the two Itôtheoretic lemmas that underlie the new Levy noisebenefit theorems. Index Terms—Levy noise, jump diffusion, mutual information, neuron models, signal detection, stochastic resonance (SR). I. STOCHASTIC RESONANCE IN NEURAL SIGNAL DETECTION STOCHASTIC RESONANCE (SR) occurs when noise benefits a system rather than harms it. Small amounts of noise can often enhance some forms of nonlinear signal processing
Nonlinear Estimation from Quantized Signals: Quantizer Optimization and Stochastic Resonance
 in Third International Symposium on Physics in Signal and Image Processing
, 2003
"... We consider a parameter estimation task performed on a signal buried in noise by means of a quantized representation by a twolevel quantizer of the signalplusnoise mixture. By considering the Fisher information, we show that the performance for estimation can be maximized by an optimal choice o ..."
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We consider a parameter estimation task performed on a signal buried in noise by means of a quantized representation by a twolevel quantizer of the signalplusnoise mixture. By considering the Fisher information, we show that the performance for estimation can be maximized by an optimal choice of the quantization threshold. At the optimal threshold, we quantify the loss in performance when the analog input is replaced by its onebit representation for estimation, and demonstrate the existence of conditions where this loss is very small. If constraints require us to work with a fixed threshold, we establish that noise addition at the input prior to quantization (a form of stochastic resonance) offers another means for maximizing the performance. For illustration, we derive the maximum likelihood estimator for estimation of a constant signal from the quantized data. We involve this estimator in adaptive schemes able to bring the quantizer to operate in optimal conditions, either by varying the quantization threshold at fixed noise level, or by increasing the noise level at fixed threshold. Examples are given with noises belonging to the famility of generalized Gaussians, which occur in particular in ocean acoustics and sonar applications. 1.
Noise Benefits in QuantizerArray Correlation Detection and Watermark Decoding
"... Abstract—Quantizer noise can improve statistical signal detection in arraybased nonlinear correlators in NeymanPearson and maximumlikelihood (ML) detection. This holds even for infinitevariance symmetric alphastable channel noise and for generalizedGaussian channel noise. Noiseenhanced correla ..."
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Cited by 6 (3 self)
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Abstract—Quantizer noise can improve statistical signal detection in arraybased nonlinear correlators in NeymanPearson and maximumlikelihood (ML) detection. This holds even for infinitevariance symmetric alphastable channel noise and for generalizedGaussian channel noise. Noiseenhanced correlation detection leads to noiseenhanced watermark extraction based on such nonlinear detection at the pixel or bit level. This yields a noisebased algorithm for digital watermark decoding using two new noisebenefit theorems. The first theorem gives a necessary and sufficient condition for quantizer noise to increase the detection probability of a constant signal for a fixed falsealarm probability if the channel noise is symmetric and if the sample size is large. The second theorem shows that the array must contain more than one quantizer for such a stochasticresonance noise benefit if the symmetric channel noise is unimodal. It also shows that the noisebenefit
Reducing probability of decision error using stochastic resonance
 IEEE Signal Processing Lett
, 2006
"... Abstract—The problem of reducing the probability of decision error of an existing binary receiver that is suboptimal using the ideas of stochastic resonance is solved. The optimal probability density function of the random variable that should be added to the input is found to be a Dirac delta funct ..."
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Abstract—The problem of reducing the probability of decision error of an existing binary receiver that is suboptimal using the ideas of stochastic resonance is solved. The optimal probability density function of the random variable that should be added to the input is found to be a Dirac delta function, and hence, the optimal random variable is a constant. The constant to be added depends upon the decision regions and the probability density functions under the two hypotheses and is illustrated with an example. Also, an approximate procedure for the constant determination is derived for the meanshifted binary hypothesis testing problem. Index Terms—Modeling, pattern classification, signal detection. I.
Applications of Forbidden Interval Theorems in Stochastic Resonance
"... Abstract Forbidden interval theorems state whether a stochasticresonance noise benefit occurs based on whether the average noise value falls outside or inside an interval of parameter values. Such theorems act as a type of screening device for mutualinformation noise benefits in the detection of s ..."
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Abstract Forbidden interval theorems state whether a stochasticresonance noise benefit occurs based on whether the average noise value falls outside or inside an interval of parameter values. Such theorems act as a type of screening device for mutualinformation noise benefits in the detection of subthreshold signals. Their proof structure reduces the search for a noise benefit to the often simple task of showing that a zero limit exists. This chapter presents the basic forbidden interval theorem for threshold neurons and four applications of increasing complexity. The first application shows that small amounts of electrical noise can help a carbon nanotube detect faint electrical signals. The second application extends the basic forbidden interval theorem to quantum communication through the judicious use of squeezed light. The third application extends the theorems to noise benefits in standard models of spiking retinas. The fourth application extends the noise benefits in retinal and other neuron models to Levy noise that generalizes Brownian motion and allows for jump and impulsive noise processes. 1 Forbidden Interval Theorems for Stochastic Resonance Stochastic resonance (SR) occurs in a nonlinear system when noise benefits the system [3, 17, 33]. The noise benefit can take the form of an increase in mutual information or a signaltonoise ratio or correlation or a decrease in an error measure. But when will such a noise benefit occur? Forbidden interval theorems answer that SR question for several nonlinear systems. The theorems act as a type of SR screening device because they can give sufficient or necessary conditions for an SR noise benefit. We here restrict noise benefits
Informationtheoretic measures improved by noise in nonlinear systems
 Proc. 14th Int. Conf. on Math. Theory of Networks
, 2000
"... signal, noise. Stochastic resonance is a phenomenon whereby the transmission of a signal by certain nonlinear systems can be improved by addition of noise. We propose a brief overview of this effect, together with an extension based on informationtheoretic concepts. We analyze various conditions of ..."
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signal, noise. Stochastic resonance is a phenomenon whereby the transmission of a signal by certain nonlinear systems can be improved by addition of noise. We propose a brief overview of this effect, together with an extension based on informationtheoretic concepts. We analyze various conditions of nonlinear transmission where the input–output Shannon mutual information, the input–output Kullback divergence, or the input–output Fisher information can receive improvement from noise addition, demonstrating different forms of noiseenhanced transmission. 1 Stochastic resonance phenomenon When a linear system couples linearly a signal and a noise, generally the noise acts as a nuisance spoiling the signal.