Generating long streams of $1/f^alpha$ noise

Abstract

We review existing methods for generating long streams of 1/f^alpha noise ($0<α\le 2$) focusing on the digital filtering of white noise. We detail the formalism to conceive an efficient random number generator (white outside some bounds) in order to generate very long streams of noise without an exhaustive computer memory load. For $α=2$ it is shown why the process is equivalent to a random-walk and can be obtained simply by a first order filtering of white noise. As soon as $α<2$ the problem becomes non linear and we show why the exact digital filtering method becomes inefficient. Instead, we work out the formalism of using several 1/f^2 filters spaced logarithmically, to approximate the spectrum at the percent level. Finally, from work on logistic maps, we give hints on how to design generators with $α>2$. The software is available from http://planck.lal.in2p3.fr/article.php3?id\_article=8