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Stochastic Resonance

In signal processing, noise is generally considered a problem to be dealt with as compared to a positive thing to be used. Stochastic Resonance (SR) is a phenomenon that can change this perception. In this, we begin with a nonlinear bi-stable system. This is a differential equation x‘(t) = U‘(x(t)) that has two stable steady-state solutions, separated by a non-stable steady-state solution. A typical example is the nonlinear Langevin equation given by U(x) = ½ax2 – ¼bx4. This has stable steady-state solutions at x = ±(a/b)½and a non-stable steady-state solution at x = 0. We also need a periodic signal y(t) and 0 mean white Gaussian noise ξ(t) with variance σ2. We use these two signals as forces exciting the differential equation, i.e.

x‘(t) = U‘(x(t)) + y(t) + ξ(t) (1)

When properly tuned, i.e., with the appropriate choices of a > 0, b > 0, and σ2 > 0, the signal x(t) found as the solution of (1) will have periodic components at the same frequencies as y(t), but with more energy. What is going on is that x(t) is jumping between the two steady-state domains of attraction. The periodic excitation given by y(t) causes the domains of attraction to increase and lose their ability to attract solutions. And, if y(t) is too powerful as compared to a function of a and b, the bi-stability of the system is lost, and the solution will have a reduction of energy at the frequencies of interest. In the absence of the periodic signal, or if the noise overpowers the periodic signal, x(t) will jump at random times with a given rate depending on a, b, and σ2. When these two phenomena occur together at the appropriate intensities, we achieve the desired resonance. In this case the domains of attraction are fluctuating in sync with y(t), and the noise causes x(t) to jump between them in such a way as to create the increase in power at these
frequencies.

SR shows significant promise for applications in signal processing. It can be used for extracting weak signals from noisy environments. This lends easily to many applications where we are limited in what we
can do to a signal by how noisy the signal is. Signals that previously would have been considered lost could now be extracted and processed without problems.

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