Pink Frequency Profiles

• Lots of things display $1/f^{\alpha}$ frequency spectrums.
• These are linear on a log-log scale (logscale both frequency and amplitude)

This corresponds to fractal structure:

• Picture sinusoid at a low frequency. Now, imagine “zooming out” 2x on this sinusoid in only the x-direction. What you’ll see will look like a sinusoid of the same amplitude, but 2x the frequency. If we keep doing this, we’ll get sinusoids of 1x, 2x, 3x… the frequency, but of equal amplitude. This will give us a flat frequency profile if we add them all together.

= Now, imagine “zooming out” in both the x and the y directions. We’ll see a sinusoid of k times the frequency, and 1/k the amplitude if we zoom out by a factor of k. If we add these sinusoids together, for k from 0 to N, we’ll get an overall signal with a 1/f frequency profile.

• This behavior is closer to self-similarity or a fractal structure, since the summed signal is a sum of sinusoids that are all the same shape, just on different scales.

Last Reviewed: 4/30/25