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Finite pulse width effects in level crossing detectors and similar systems, such as neurons, can yield an output noise that is, in the low frequency limit, a monotonically increasing function of the frequency. The phenomenon can well be observed at excitations with strong noise when the firing frequency is high. Blue noise effects have been observed in stochastically driven dynamical systems like harmonic oscillators. However it is surprising that it exists in a non-dynamical system like the level crossing detector.

We consider kinematics of a rigid body rotation about a fixed point. An angular velocity is considered to be a sum of deterministic and random vector-functions. We suppose that the components of the latter are statistically independent small stationary processes. The problem is, for example, important for the theory of a rotational Brownian motion or an error analysis of strapped-down inertial navigation systems. We construct a multiplicative perturbation theory so that the desired matrix is represented as a product of deterministic matrix by random perturbation one. The perturbation matrix is approximately found by the first terms of a matrizant. As example, we consider the constant deterministic velocity and the components of the colored random velocity to be white noises (the Brownian motion), the Ornstein-Uhlenbeck processes, or the time derivative of the Ornstein-Uhlenbeck processes ("green" noises). In the second case it is shown that a diffusion of an instanta-neous rotation angle appears only if the deterministic rotation is not zero.

The phenomenon of stochastic resonance in a bacterium growth system that is with two different kinds of time delays and is driven by colored noises is investigated. Based on the extended unified colored noise theory and the method of the probability density approximation, the Fokker–Planck equation and the stationary probability density function are derived. Then via the theory of adiabatic limit, the analytical expression of the signal-to-noise ratio (SNR) is obtained. The different effects of the time delays existed in the nonlinear system and the noise correlation times on the stationary probability density and the signal-to-noise rate are discussed respectively. Finally, numerical simulations are offered and are consistent with approximate analytical results.

Physical layer key generation exploiting inherent channel randomness is an open research area in securing the networks with resource constraint nodes; therefore reduction of numerical computation is desirable to save battery power. However, the correlated components due to colored noise also affect the system performance. In this work, we consider the correlated colored noise components along with the additive white Gaussian noise (AWGN) in the wireless channel and analyze the effect of these correlated components on the system performance. We further propose a hybrid averaging and dimensionality reduction (AvDR), based received signal strength (RSS) preprocessing which is the combination of moving window averaging (Av) and principal component analysis (PCA) as dimensionality reduction technique (DR) to improve the system performance. Further, the system performance was evaluated by numerical simulations, and it is observed that the same improvement in system performance is achieved by generating keys from a fewer number of points selected after PCA as compared to processing all the points. Picking a few of the points in the data sequence instead of all reduces the total number of numerical calculations and saves system power, which is the primary requirement of resource constraint networks like the IoT.

We consider a Richards growth model (modified logistic model) driven by correlated multiplicative and additive colored noises, and investigate the effects of noises on the eventual distribution of population size with the help of steady-state analysis. An approximative Fokker–Planck equation is first derived for the stochastic model. By performing detailed theoretical analysis and numerical simulation for the steady-state solution of the Fokker–Planck equation, i.e., stationary probability distribution (SPD) of the stochastic model, we find that the correlated noises have complex effects on the statistical property of the stochastic model. Specifically, the phenomenological bifurcation may be caused by the noises. The position of extrema of the SPD depends on the model parameter and the characters of noises in different ways.