Noise-benefit forbidden-interval theorems for threshold signal detectors based on cross correlations

We show that the main forbidden interval theorems of stochastic resonance hold for a correlation performance measure. Earlier theorems held only for performance measures based on mutual information or the probability of error detection. Forbidden interval theorems ensure that a threshold signal detector benefits from deliberately added noise if the average noise does not lie in an interval that depends on the threshold value. We first show that this result holds for correlation for all finite-variance noise and for all forms of infinite-variance stable noise. A second forbidden-interval theorem gives necessary and sufficient conditions for a local noise benefit in a bipolar signal system when the noise comes from a location-scale family. A third theorem gives a general condition for a local noise benefit for arbitrary signals with finite second moments and for location-scale noise. This result also extends forbidden intervals to forbidden bands of parameters. A fourth theorem gives necessary and sufficient conditions for a local noise benefit when both the independent signal and noise are normal. A final theorem derives necessary and sufficient conditions for forbidden bands when using arrays of threshold detectors for arbitrary signals and location-scale noise.

Figure 1

Forbidden-interval noise benefit in a subthreshold signal system (1) that uses the bipolar yin-yang image as its input. The threshold $\theta$ is $\theta =1$. The bipolar signal has amplitude $A=0.7$. Top panels: The noise is uniform with mean ${\mu }_N=2$. So the mean does not lie in the forbidden interval $(\theta -A,\theta +A)=(0.3,1.7)$. Thus the correlation system benefits from added noise and the correlation curve shows the signature nonmonotonic inverted $\mathsf{U}$ of a global SR noise benefit. The optimal noise is approximately ${\sigma }_N^{*}\approx 0.98$. Bottom panels: The noise is uniform with mean ${\mu }_N=1$. So the mean lies in the forbidden interval $(\theta -A,\theta +A)=(0.3,1.7)$. Thus there is no noise benefit.

Figure 2

Samples of symmetric $\alpha$-stable probability densities and their white-noise realizations. (a) Standard symmetric $\alpha$-stable density functions with zero location $(a=0$) and unit dispersion $\left(\gamma =1\right)$ for $\alpha =$ 2, 1.7, 1.5, and 1. The densities are bell curves that have thicker tails as $\alpha$ decreases and thus that model increasingly impulsive noise as $\alpha$ decreases. The case $\alpha =2$ gives a Gaussian density with variance two (or unit dispersion). The case $\alpha =1$ gives the Cauchy density with infinite variance. (b) White-noise samples of $\alpha$-stable random variables $n_t$ with zero location and unit dispersion. The plots show realizations when $\alpha =$ 2, 1.7, 1.5, and 1. Note the scale differences on the $y$ axes. The $\alpha$-stable variable $N$ becomes more impulsive as the parameter $\alpha$ falls and thus as the bell curves get thicker. The algorithm in Refs. [37, 38] generated these realizations. (c) $\alpha$-stable density functions for $\alpha =1.7$ with dispersions $\gamma =$ 0.5, 1, and 2. (d) Samples of finite and infinite $\alpha$-stable noise $n_t$ for $\alpha =1.7$ with finite dispersions $\gamma =$ 0.5, 1, and 2.

Figure 3

Forbidden-interval noise benefit in an array of threshold detectors (1) that uses the baboon image as its input. (a) Histogram of the gray-scale baboon image. We scale the pixel values to lie in the interval [$-1$,1] and use these values as input signals to the array of threshold detectors. [(b) and (c)] Forbidden intervals (bands) using the Gaussian approximation of Theorem 5 for $\theta =-0.8$ and $\theta =0$. Each interval depends on the signal and noise statistics. Solid lines are intervals from the statistics of the baboon image with uniform noise. Dashed lines are estimated intervals using Gaussian approximation of both signal and noise. [(d)–(g)] Quantized (thresholded) baboon images when the detectors have threshold $\theta =-0.8$. The noise $N$ is uniform with zero mean. Thus ${\mu }_N\notin (\theta -a_2,\theta -a_1)$ and this gives an SR noise benefit. [(h)–(k)] The detectors have threshold $\theta =0$ and the noise $N$ is uniform with zero mean. Thus ${\mu }_N\in (\theta -a_2,\theta -a_1)$ and there is no noise benefit. (l) Cross correlation $C$ when $\theta =-0.8$: This gives an SR noise benefit. (m) Cross correlation $C$ when $\theta =0$: There is no noise benefit.

Figure 4

Cross correlations $C$ in (a) and (b) and derivatives $\frac{\partial C}{\partial \sigma }$ in (c) and (d) of an array of threshold detectors for the baboon image with uniform noise. The forbidden intervals (bands) for noise benefits in an array of threshold detectors (1) in Figure 2 derive from the contours of $\frac{\partial C}{\partial \sigma }=0$.