Discrete Simulation of Colored Noise and Stochastic Processes and 1/fα Power Law Noise Generation

This paper discusses techniques for generating digital sequences of noise which simulate processes with certain known properties or describing equations. Part I of the paper presents a review of stochastic processes and spectral estimation (with some new results) and a tutorial on simulating continuous noise processes with a known autospectral density or autocorrelation function. In defining these techniques for computer generating sequences, it also defines the necessary accuracy criteria. These methods are compared to some of the common techniques for noise generation and the problems, or advantages, of each are discussed. Finally, Part I presents results on simulating stochastic differential equations. A Runge–Kutta (RK) method is presented for numerically solving these equations. Part II of the paper discusses power law, or 1/fα, noises. Such noise processes occur frequently in nature and, in many cases, with nonintegral values for α. A review of 1/f noises in devices and systems is followed by a discussion of the most common continuous 1/f noise models. The paper then presents a new digital model for power law noises. This model allows for very accurate and efficient computer generation of 1/fα noises for any a. Many of the statistical properties of this model are discussed and compared to the previous continuous models. Lastly, a number of approximate techniques for generating power law noises are presented for rapid or real time simulation.