On the relationship between 1/f and α-stable processes

1/f^β-type spectral behavior has received considerable attention in the past few years because it arises from a wide range of natural phenomena. By expressing a 1/f^β process as a fractional integral of white noise, we show that, if β<1, the process is stationary and follows an α-stable model, while if β>1, the process has stationary α-stable increments. We also provide closed form expressions for the relationship between β and α. The theoretical results are verified via real ultrasound data. Ultrasound breast data, or their increments, which appear to be 1/f^β, are shown to follow reasonably well the α-stable model