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Levy noise can help neurons detect faint or subthreshold signals. Levy noise extends standard Brownian noise to many types of impulsive jump-noise processes found in real and model neurons as well as in models of finance and other random phenomena. Two new theorems and the ItO 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 noise-injection 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 infinite-variance stable Levy noise processes even though the theorems themselves apply only to finite-variance Levy noise. The proves the two ItO-theoretic lemmas that underlie the new Levy noise-benefit theorems.

Abstract-Quantizer noise can improve statistical signal detection in array-based nonlinear correlators in Neyman-Pearson and maximum-likelihood (ML) detection. This holds even for infinitevariance symmetric alpha-stable channel noise and for generalized-Gaussian channel noise. Noise-enhanced correlation detection leads to noise-enhanced watermark extraction based on such nonlinear detection at the pixel or bit level. This yields a noise-based 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 false-alarm 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 stochastic-resonance noise benefit if the symmetric channel noise is unimodal. It also shows that the noise-benefit rate improves in the small-quantizer noise limit as the number of array quantizers increases. The second theorem further shows that symmetric uniform quantizer noise gives the optimal rate for an initial noise benefit among all finite-variance symmetric scalefamily noise. Two corollaries give similar results for stochastic-resonance noise benefits in ML detection of a signal sequence with known shape but unknown amplitude. Fig. 1 shows an SR noise benefit in the ML watermark extraction of the "yin-yang" image embedded in the discrete-cosine transform (DCT-2) coefficients of the "Lena" image [32]. The yin-yang image of Fig. 1(a) is the 64 64 binary watermark message embedded in the midfrequency DCT-2 coefficients of the 512 512 gray-scale Lena image using direct-sequence spread spectrum [33]. Fig. 1(b) shows the result when the yin-yang figure watermarks the Lena image. Figs. 1(c)-1(g) shows that small amounts of additive uniform quantizer noise improve the watermark-extraction performance of the noisy quantizer-array ML detector while too much noise degrades the performance. Uniform quantizer noise with standard deviation reduces more than 33% of the pixel-detection errors in the extracted watermark image. Section VI gives the details of such noise-enhanced watermark decoding. IndexThe quantizer-array detector consists of two parts. It consists of a nonlinear preprocessor that precedes a correlator and a likelihood-ratio test of the correlator's output. This nonlinear detector takes samples of a noise-corrupted signal and then sends each sample to the nonlinear preprocessor array of noisy quantizers connected in parallel. Each quantizer in the array adds its independent quantizer noise to the noisy input sample and then quantizes this doubly noisy data sample into a binary value. The quantizer array output for each sample is just the sum of all quantizer outputs. The correlator then correlates these preprocessed samples with the signal. The detector's final stage applies either the NP likelihood-ratio test in Section II or the ML-ratio test in Section V.Sect...

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