Modeling Noise as a Bernoulli-Gaussian Process

Abstract

Transmission medium is always perturbed by noise with a random nature which can be characterized by taking a sequence of noise samples and, after analyzing the sequence, attributing a probabilistic model to represent the randomness of the noise. If thermal noise (receiver generated) is the only noise impairing the transmission (our only focus is digital transmission) the memoryless stationary discrete-time Gaussian process is the best model to probabilistically represent the noise. The mathematical representation of the transmission medium in such a situation yields the well known Gaussian Channel. As Information Theory points out, for a fixed noise power, the Gaussian channel is the worst channel to send information through. If thermal noise is not the only noise impairing the transmission (as in sonar communication and power line communication) finding the probabilistic model other than the single-parameter Gaussian process, which best match the noise can much improve the communication system design. The Bernoulli-Gaussian process, a three parameters model, is a common considered option. Finding the three parameters of the Bernoulli-Gaussian model (from known noise samples) is a formidable task that can be made simpler by considering the (original) results presented in the current paper. The Bernoulli-Gaussian model can be characterized, analytically, by using the noise power and two additional quantities: the expectation of the absolute value of the noise process plus the expected value of the third power of the absolute value. In practice the parameters would be calculated using estimates of the mentioned expected values. The communication system design can be much improved if a well fit Bernoulli-Gaussian stochastic process is selected to model the noise. This is an alternative to model the communication using power lines which is often modeled as Middleton Class-A. The rate harvested when modeling the medium as a Bernoulli-Gaussian channel, it is shown, is increased when compared to modeling the medium with the easily obtained Gaussian channel.