François Chapeau-Blondeau

Numerical evaluation of the Lambert W function and application to generation of generalized Gaussian noise with exponent 1/2

We address the problem of synthesizing a generalized Gaussian noise with exponent 1/2 by means of a nonlinear memoryless transformation applied to a uniform noise. We show that this transformation is expressable in terms of a special function known under the name of the Lambert W function. We review the main methods for numerical evaluation of the relevant branch of the (multivalued) Lambert W function with controlled accuracy and complement them with an original rational function approximation. Based on these methods, synthesis of the generalized Gaussian noise can be performed with arbitrary accuracy. We construct a simple and fast evaluation algorithm with prescribed accuracy, which is especially suited for Monte Carlo simulation requiring large numbers of realizations of the generalized Gaussian noise.

A neural network model of the cerebellar cortex performing dynamic associations

The present paper proposes a model which applies formal neural network modeling techniques to construct a theoretical representation of the cerebellar cortex and its performances in motor control. A schema that makes explicit use of propagation delays of neural signals, is introduced to describe the ability to store temporal sequences of patterns in the Golgi-granule cell system. A perceptron association is then performed on these sequences of patterns by the Purkinje cell layer. The model conforms with important biological constraints, such as the known excitatory or inhibitory nature of the various synapses. Also, as suggested by experimental evidence, the synaptic plasticity underlying the learning ability of the model, is confined to the parallel fiber — Purkinje cell synapses, and takes place under the control of the climbing fibers. The result is a neural network model, constructed according to the anatomy of the cerebellar cortex, and capable of learning and retrieval of temporal sequences of patterns. It provides a framework to represent and interpret properties of learning and control of movements by the cerebellum, and to assess the capacity of formal neural network techniques for modeling of real neural systems.

Stochastic Resonance in the Information Capacity of a Nonlinear Dynamic System

We consider a nonlinear bistable dynamic system governed by the quartic potential with twostate quantization at the output | the earliest system to have revealed the phenomenon of periodic stochastic resonance. We devise a scheme in which this system is used to transmit a broadband aperiodic informative signal. With this scheme, we demonstrate that the system can be operated as a memoryless symmetric binary channel, and we develop the characterization of the transmission up to the evaluation of the input{output information capacity of this channel. We show that a regime exists where the information capacity can be increased by means of noise addition, a property we interpret as a form of aperiodic stochastic resonance. In addition, we demonstrate that a positive input{output gain in the ecacy of the signal recovery can be obtained with the stochastic resonator, compared to the recovery that would directly operate on the input signal-plus-noise mixture.

Noise-enhanced performance for an optimal Bayesian estimator

A novel instance of a stochastic resonance effect, under the form of a noise-improved performance, is shown to be possible for an optimal Bayesian estimator. Estimation of the frequency of a periodic signal corrupted by a phase noise is considered. The optimal Bayesian estimator, achieving the minimum of the mean square estimation error, is explicitly derived. Conditions are exhibited where this minimal error is reduced when the noise level is raised, over some ranges, where this occurs essentially with non-Gaussian noise, in the tested configurations. These results contribute a new step in the exploration of stochastic resonance and its potentialities for signal processing.

Input-output gains for signal in noise in stochastic resonance

Abstract A theoretical model for the transmission of a periodic signal added to a noise through a static nonlinearity is considered. Expressions are derived for the gains experienced by the signal, the noise and the signal-to-noise ratio in the input-output nonlinear transmission. The gains are obtained in the presence of a periodic input, a noise distribution and a static nonlinearity, all three being arbitrary. These gains are studied as measures of the phenomenon of stochastic resonance whereby the transmission of the periodic signal can be improved by means of noise addition. As the noise level is raised, resonant evolutions for the signal and signal-to-noise gains or antiresonant evolutions for the noise gain, are demonstrated. At the same time, conditions are exhibited where the signal-to-noise gain is larger than unity, demonstrating several realizations of a signal-to-noise ratio larger at the output than at the input in stochastic resonance.

Noise-assisted signal transmission in a nonlinear electronic comparator: experiment and theory

Abstract A periodic signal superposed to a white noise is input onto a nonlinear two-state threshold comparator. The output signal contains, embedded in random fluctuations originating in the input noise, a part of correlation with the periodic input. We show that a regime exists where this part of correlation can be enhanced by means of an increase of the input noise level. This is the phenomenon of stochastic resonance, whereby the noise becomes beneficial to the transmission of a coherent signal, and where an increase of the noise can result in improved performances. We experimentally demonstrate this property in an electronic circuit. We then develop a complete theoretical analysis of this effect of noise-assisted signal transmission. This simple electronic implementation of a stochastic resonator, together with its theoretical analysis, constitute a unique framework for further investigations on the nonlinear phenomenon of stochastic resonance and its implications for signal processing.

Nonlinear test statistic to improve signal detection in non-Gaussian noise

We compare two simple test statistics that a detector can compute from multiple noisy data in a binary decision problem based on a maximum a posteriori probability (MAP) criterion. One of these statistics is the standard sample mean of the data (linear detector), which allows one to minimize the probability of detection error when the noise is Gaussian. The other statistic is even simpler and consists of a sample mean of a two-state quantized version of the data (nonlinear detector). Although simpler to compute, we show that this nonlinear detector can achieve smaller probability of error compared to the linear detector. This especially occurs for non-Gaussian noises with heavy tails or a leptokurtic character.

Stochastic resonance and improvement by noise in optimal detection strategies

A stochastic resonance effect, under the form of a noise-improved performance, is shown feasible for a whole range of optimal detection strategies, including Bayesian, minimum error-probability, Neyman-Pearson, and minimax detectors. In each case, situations are demonstrated where the performance of the optimal detector can be improved (locally) by raising the level of the noise. This is obtained with a nonlinear signal-noise mixture where a non-Gaussian noise acts on the phase of a periodic signal.

NOISE IMPROVEMENTS IN STOCHASTIC RESONANCE: FROM SIGNAL AMPLIFICATION TO OPTIMAL DETECTION

It is demonstrated that benefits from the noise can be gained at various levels in stochastic resonance. Raising the noise can produce signal amplification as well as signal-to-noise ratio improvement, input–output gain exceeding unity in signal-to-noise ratio, and enhanced performance in optimal processing. This series of benefits is successively exhibited in the processing of a periodic signal coupled to a white noise through essentially static nonlinearities. Especially, it is established that noise benefits in stochastic resonance can extend up to optimal processing, by considering an optimal Bayesian detector whose performance is improvable by raising the level of the noise.

Stochastic resonance and noise-enhanced transmission of spatial signals in optics: the case of scattering

The nonlinear effect of noise-enhanced signal transmission by means of stochastic resonance in optics is studied. We investigate this effect for the novel case of spatial signals or images. With a theoretical model involving a threshold nonlinearity we describe a mechanism whereby the transmission of an image can be improved by the addition of noise. We argue that such a nonlinear mechanism can operate in different types of light scattering. With a stimulated Raman scattering experiment we verify the existence of a stochastic resonance effect in the transmission of a laser image; we show that maximal efficacy is obtained with the assistance of a speckle of sufficient intensity. The results extend the scope of stochastic resonance and can serve as a basis for further development of the effect in optics.