On the relationship between 1/f and /spl alpha/-stable processes

1/f/sup /spl beta//-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/sup /spl beta// process as a fractional integral of white noise, we show that, if /spl beta/<1, the process is stationary and follows an /spl alpha/-stable model, while if /spl beta/>1, the process has stationary /spl alpha/-stable increments. We also provide closed form expressions for the relationship between /spl beta/ and /spl alpha/. The theoretical results are verified via real ultrasound data. Ultrasound breast data, or their increments, which appear to be 1/f/sup /spl beta//, are shown to follow reasonably well the /spl alpha/-stable model.

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