Relation between optimal nonlinearity and non-Gaussian noise: Enhancing a weak signal in a nonlinear system

In the study of stochastic resonance, it is often mentioned that nonlinearity can enhance a weak signal embedded in noise. In order to give a systematic proof of the signal enhancement in nonlinear systems, we derive an optimal nonlinearity that maximizes a signal-to-noise ratio (SNR). The obtained optimal nonlinearity yields the maximum unbiased signal estimation performance, which is known in the context of information theory. It is found that a linear system is optimal for a Gaussian noise, but for a non-Gaussian noise, there exist nonlinear systems that can achieve an SNR higher than that obtained from linear systems. This analysis refers to a system subjected to an additive non-Gaussian noise with a small signal input.

Figure 1

SNRs obtained from numerical simulations for the optimal nonlinearity (circles), a linear system (squares), and a simple threshold system (cross marks) in the presence of white mixed Gaussian noise. The solid line is the theoretically predicted SNR for the optimal nonlinearity.

Figure 2

SNRs obtained from numerical simulations for the optimal nonlinearity (solid line) and the input channel (dotted line) in the presence of noise generated from Markovian process. The parameter $\varphi$ corresponds to the noise intensity.